From f975b4bd900b2a65f9a0a5fe01209a0243d8cb76 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Sun, 2 May 2021 20:37:00 +0200 Subject: [PATCH] Update Content - 2021-05-02 --- .../butler11_posit_contr_lithog_equip.md | 5 +- ...0_ident_decoup_contr_flexur_joint_hexap.md | 13 +- .../article/dasgupta00_stewar_platf_manip.md | 9 +- ...eming13_review_nanom_resol_posit_sensor.md | 17 +- .../furqan17_studies_stewar_platf_manip.md | 5 +- ...tani04_nanom_cuttin_machin_using_stewar.md | 5 +- .../gao15_measur_techn_precis_posit.md | 5 +- .../garg07_implem_chall_multiv_contr.md | 5 +- content/article/hanieh03_activ_stewar.md | 5 +- .../holler12_instr_x_ray_nano_imagin.md | 7 +- ..._activ_dampin_based_decoup_colloc_contr.md | 5 +- .../jiao18_dynam_model_exper_analy_stewar.md | 5 +- .../legnani12_new_isotr_decoup_paral_manip.md | 9 +- .../li01_simul_fault_vibrat_isolat_point.md | 59 +- ...cinroy00_desig_contr_flexur_joint_hexap.md | 5 +- content/article/mcinroy99_dynam.md | 25 +- ...omen18_advan_motion_contr_precis_mechat.md | 7 +- .../preumont07_six_axis_singl_stage_activ.md | 17 +- .../saxena12_advan_inter_model_contr_techn.md | 5 +- .../sayed01_survey_spect_factor_method.md | 5 +- ...hroeck01_compen_desig_linear_time_invar.md | 5 +- .../sebastian12_nanop_with_multip_sensor.md | 5 +- ...eille18_concep_activ_mount_space_applic.md | 23 +- .../tang18_decen_vibrat_contr_voice_coil.md | 5 +- .../wang12_autom_marker_full_field_hard.md | 5 +- ...wang16_inves_activ_vibrat_isolat_stewar.md | 11 +- .../yun20_inves_two_stage_vibrat_suppr.md | 5 +- ...system_desig_activ_passiv_vibrat_isolat.md | 13 +- .../book/albertos04_multiv_contr_system.md | 114 +++- .../du10_model_contr_vibrat_mechan_system.md | 521 +++++++++++++++++- content/book/hatch00_vibrat_matlab_ansys.md | 105 ++-- .../book/leach14_fundam_princ_engin_nanom.md | 5 +- .../leach18_basic_precis_engin_edition.md | 5 +- ..._vibrat_contr_activ_struc_fourt_edition.md | 89 +-- .../book/skogestad07_multiv_feedb_contr.md | 324 +++++------ .../monkhorst04_dynam_error_budget.md | 17 +- content/posts/homepages.md | 18 - ...dynamical_systems_with_machine_learning.md | 2 - content/zettels/acquisition_systems.md | 2 - content/zettels/actuator_fusion.md | 13 +- content/zettels/actuators.md | 16 +- .../zettels/analog_to_digital_converters.md | 9 +- content/zettels/capacitive_sensors.md | 2 - content/zettels/charge_amplifiers.md | 13 +- content/zettels/complementary_filters.md | 5 +- content/zettels/cubic_architecture.md | 11 +- .../zettels/digital_to_analog_converters.md | 2 - content/zettels/direct_velocity_feedback.md | 2 - content/zettels/encoders.md | 2 - content/zettels/flexible_joints.md | 37 +- content/zettels/granite.md | 2 - content/zettels/h_infinity_control.md | 6 - content/zettels/inertial_sensors.md | 15 +- content/zettels/instrumented_hammer.md | 2 - content/zettels/interferometers.md | 19 +- content/zettels/irr_and_fir_filters.md | 5 +- content/zettels/mass_spring_damper_systems.md | 6 +- content/zettels/matlab.md | 39 +- content/zettels/metrology.md | 6 - content/zettels/modal_analysis.md | 9 - 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the payload mass/inertia matrix is diagonal - the geometry of the platform and attachment of the payload must be carefully chosen @@ -43,9 +43,9 @@ The algorithm derived herein removes these constraints, thus greatly expanding t ## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods} -The derivation of the dynamic model is done in ([McInroy 1999](#org0e9e807)) ([Notes]({{< relref "mcinroy99_dynam" >}})). +The derivation of the dynamic model is done in ([McInroy 1999](#orgaddeeaf)) ([Notes]({{< relref "mcinroy99_dynam" >}})). - + {{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="Figure 1: A flexured joint Hexapod. {P} is a cartesian coordiante frame located at (and rigidly connected to) the payload's center of mass. {B} is a frame attached to the (possibly moving) base, and {U} is a universal inertial frame of reference" >}} @@ -100,8 +100,9 @@ where ## Experimental Results {#experimental-results} + ## Bibliography {#bibliography} -Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . +Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . -McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. . +McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. . diff --git a/content/article/dasgupta00_stewar_platf_manip.md b/content/article/dasgupta00_stewar_platf_manip.md index 0a04eb7..57f72be 100644 --- a/content/article/dasgupta00_stewar_platf_manip.md +++ b/content/article/dasgupta00_stewar_platf_manip.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}) Reference -: ([Dasgupta and Mruthyunjaya 2000](#orgab59d3a)) +: ([Dasgupta and Mruthyunjaya 2000](#orgb54a8e7)) Author(s) : Dasgupta, B., & Mruthyunjaya, T. @@ -24,16 +24,17 @@ Year | | **Advantages** | **Disadvantages** | |--------------|---------------------------|-----------------------| -| **Serial** | Manoeuverability | Poor precision | +| **Serial** | Maneuverability | Poor precision | | | Large workspace | Bends under high load | | | | Vibrate at high speed | | **Parallel** | High stiffness | Small workspace | | | Good dynamic performances | | | | Precise positioning | | -The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platoform) connected through six extensible legs, each with sherical joints at both ends. +The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platform) connected through six extensible legs, each with spherical joints at both ends. + ## Bibliography {#bibliography} -Dasgupta, Bhaskar, and T.S. Mruthyunjaya. 2000. “The Stewart Platform Manipulator: A Review.” _Mechanism and Machine Theory_ 35 (1):15–40. 00006-3. +Dasgupta, Bhaskar, and T.S. Mruthyunjaya. 2000. “The Stewart Platform Manipulator: A Review.” _Mechanism and Machine Theory_ 35 (1):15–40. 00006-3. diff --git a/content/article/fleming13_review_nanom_resol_posit_sensor.md b/content/article/fleming13_review_nanom_resol_posit_sensor.md index b1f2ddc..b32adae 100644 --- a/content/article/fleming13_review_nanom_resol_posit_sensor.md +++ b/content/article/fleming13_review_nanom_resol_posit_sensor.md @@ -4,15 +4,11 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Position Sensors]({{< relref "position_sensors" >}}) - Tags : [Position Sensors]({{< relref "position_sensors" >}}) Reference -: ([Fleming 2013](#org35f9cea)) +: ([Fleming 2013](#org336a947)) Author(s) : Fleming, A. J. @@ -37,7 +33,7 @@ Usually quoted as a percentage of the fill-scale range (FSR): With \\(e\_m(v)\\) is the mapping error. - + {{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="Figure 1: The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}} @@ -46,7 +42,7 @@ With \\(e\_m(v)\\) is the mapping error. If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error. - + {{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="Figure 2: The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}} @@ -151,9 +147,9 @@ The empirical rule states that there is a \\(99.7\%\\) probability that a sample This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\). Another important parameter that must be specified when quoting resolution is the sensor bandwidth. -There is usually a trade-off between bandwidth and resolution (figure [3](#org2ee752d)). +There is usually a trade-off between bandwidth and resolution (figure [3](#orge95682f)). - + {{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}} @@ -186,6 +182,7 @@ A convenient method for reporting this ratio is in parts-per-million (ppm): | Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR | + ## Bibliography {#bibliography} -Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. . +Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. . diff --git a/content/article/furqan17_studies_stewar_platf_manip.md b/content/article/furqan17_studies_stewar_platf_manip.md index 609ae16..88cf1f9 100644 --- a/content/article/furqan17_studies_stewar_platf_manip.md +++ b/content/article/furqan17_studies_stewar_platf_manip.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}) Reference -: ([Furqan, Suhaib, and Ahmad 2017](#org7520991)) +: ([Furqan, Suhaib, and Ahmad 2017](#orgd310f71)) Author(s) : Furqan, M., Suhaib, M., & Ahmad, N. @@ -19,6 +19,7 @@ Year Lots of references. + ## Bibliography {#bibliography} -Furqan, Mohd, Mohd Suhaib, and Nazeer Ahmad. 2017. “Studies on Stewart Platform Manipulator: A Review.” _Journal of Mechanical Science and Technology_ 31 (9):4459–70. . +Furqan, Mohd, Mohd Suhaib, and Nazeer Ahmad. 2017. “Studies on Stewart Platform Manipulator: A Review.” _Journal of Mechanical Science and Technology_ 31 (9):4459–70. . diff --git a/content/article/furutani04_nanom_cuttin_machin_using_stewar.md b/content/article/furutani04_nanom_cuttin_machin_using_stewar.md index 0c2818b..e63b3f3 100644 --- a/content/article/furutani04_nanom_cuttin_machin_using_stewar.md +++ b/content/article/furutani04_nanom_cuttin_machin_using_stewar.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) Reference -: ([Furutani, Suzuki, and Kudoh 2004](#orgaf9f119)) +: ([Furutani, Suzuki, and Kudoh 2004](#org37254be)) Author(s) : Furutani, K., Suzuki, M., & Kudoh, R. @@ -35,6 +35,7 @@ To minimize the errors, a calibration is done between the required leg length an Then, it is fitted with 4th order polynomial and included in the control architecture. + ## Bibliography {#bibliography} -Furutani, Katsushi, Michio Suzuki, and Ryusei Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” _Measurement Science and Technology_ 15 (2):467–74. . +Furutani, Katsushi, Michio Suzuki, and Ryusei Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” _Measurement Science and Technology_ 15 (2):467–74. . diff --git a/content/article/gao15_measur_techn_precis_posit.md b/content/article/gao15_measur_techn_precis_posit.md index 459bbcd..7e68bd6 100644 --- a/content/article/gao15_measur_techn_precis_posit.md +++ b/content/article/gao15_measur_techn_precis_posit.md @@ -8,7 +8,7 @@ Tags : [Position Sensors]({{< relref "position_sensors" >}}) Reference -: ([Gao et al. 2015](#org7f49efc)) +: ([Gao et al. 2015](#orgb71276a)) Author(s) : Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., … @@ -17,6 +17,7 @@ Year : 2015 + ## Bibliography {#bibliography} -Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” _CIRP Annals_ 64 (2):773–96. . +Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” _CIRP Annals_ 64 (2):773–96. . diff --git a/content/article/garg07_implem_chall_multiv_contr.md b/content/article/garg07_implem_chall_multiv_contr.md index 738ef92..2593bf4 100644 --- a/content/article/garg07_implem_chall_multiv_contr.md +++ b/content/article/garg07_implem_chall_multiv_contr.md @@ -8,7 +8,7 @@ Tags : [Multivariable Control]({{< relref "multivariable_control" >}}) Reference -: ([Garg 2007](#orga5ede8d)) +: ([Garg 2007](#orgae24e63)) Author(s) : Garg, S. @@ -35,6 +35,7 @@ The control rate should be weighted appropriately in order to not saturate the s - importance of scaling the plant prior to synthesis and also replacing pure integrators with slow poles + ## Bibliography {#bibliography} -Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In _AIAA Guidance, Navigation and Control Conference and Exhibit_, nil. . +Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In _AIAA Guidance, Navigation and Control Conference and Exhibit_, nil. . diff --git a/content/article/hanieh03_activ_stewar.md b/content/article/hanieh03_activ_stewar.md index 7e1a4be..f50f307 100644 --- a/content/article/hanieh03_activ_stewar.md +++ b/content/article/hanieh03_activ_stewar.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Active Damping]({{< relref "active_damping" >}}) Reference -: ([Hanieh 2003](#org5da54db)) +: ([Hanieh 2003](#orgd0b61f4)) Author(s) : Hanieh, A. A. @@ -17,6 +17,7 @@ Year : 2003 + ## Bibliography {#bibliography} -Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium. +Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium. diff --git a/content/article/holler12_instr_x_ray_nano_imagin.md b/content/article/holler12_instr_x_ray_nano_imagin.md index 499ad67..da46c3d 100644 --- a/content/article/holler12_instr_x_ray_nano_imagin.md +++ b/content/article/holler12_instr_x_ray_nano_imagin.md @@ -8,7 +8,7 @@ Tags : [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}}) Reference -: ([Holler et al. 2012](#org289d119)) +: ([Holler et al. 2012](#org76dfce6)) Author(s) : Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O. @@ -19,7 +19,7 @@ Year Instrument similar to the NASS. Obtain position stability of 10nm (standard deviation). - + {{< figure src="/ox-hugo/holler12_station.png" caption="Figure 1: Schematic of the tomography setup" >}} @@ -39,6 +39,7 @@ Obtain position stability of 10nm (standard deviation). - **Feedback Loop**: Using the signals from the 2 interferometers, the loop is closed to compensate low frequency vibrations and thermal drifts. + ## Bibliography {#bibliography} -Holler, M., J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk. 2012. “An Instrument for 3d X-Ray Nano-Imaging.” _Review of Scientific Instruments_ 83 (7):073703. . +Holler, M., J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk. 2012. “An Instrument for 3d X-Ray Nano-Imaging.” _Review of Scientific Instruments_ 83 (7):073703. . diff --git a/content/article/holterman05_activ_dampin_based_decoup_colloc_contr.md b/content/article/holterman05_activ_dampin_based_decoup_colloc_contr.md index e0baaa5..4b62f35 100644 --- a/content/article/holterman05_activ_dampin_based_decoup_colloc_contr.md +++ b/content/article/holterman05_activ_dampin_based_decoup_colloc_contr.md @@ -8,7 +8,7 @@ Tags : [Active Damping]({{< relref "active_damping" >}}) Reference -: ([Holterman and deVries 2005](#orgfb04a8c)) +: ([Holterman and deVries 2005](#org69e08df)) Author(s) : Holterman, J., & deVries, T. @@ -17,6 +17,7 @@ Year : 2005 + ## Bibliography {#bibliography} -Holterman, J., and T.J.A. deVries. 2005. “Active Damping Based on Decoupled Collocated Control.” _IEEE/ASME Transactions on Mechatronics_ 10 (2):135–45. . +Holterman, J., and T.J.A. deVries. 2005. “Active Damping Based on Decoupled Collocated Control.” _IEEE/ASME Transactions on Mechatronics_ 10 (2):135–45. . diff --git a/content/article/jiao18_dynam_model_exper_analy_stewar.md b/content/article/jiao18_dynam_model_exper_analy_stewar.md index 9e24c5f..bd8afa8 100644 --- a/content/article/jiao18_dynam_model_exper_analy_stewar.md +++ b/content/article/jiao18_dynam_model_exper_analy_stewar.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) Reference -: ([Jiao et al. 2018](#orga067c93)) +: ([Jiao et al. 2018](#org72b03e6)) Author(s) : Jiao, J., Wu, Y., Yu, K., & Zhao, R. @@ -17,6 +17,7 @@ Year : 2018 + ## Bibliography {#bibliography} -Jiao, Jian, Ying Wu, Kaiping Yu, and Rui Zhao. 2018. “Dynamic Modeling and Experimental Analyses of Stewart Platform with Flexible Hinges.” _Journal of Vibration and Control_ 25 (1):151–71. . +Jiao, Jian, Ying Wu, Kaiping Yu, and Rui Zhao. 2018. “Dynamic Modeling and Experimental Analyses of Stewart Platform with Flexible Hinges.” _Journal of Vibration and Control_ 25 (1):151–71. . diff --git a/content/article/legnani12_new_isotr_decoup_paral_manip.md b/content/article/legnani12_new_isotr_decoup_paral_manip.md index 717c3c6..6209693 100644 --- a/content/article/legnani12_new_isotr_decoup_paral_manip.md +++ b/content/article/legnani12_new_isotr_decoup_paral_manip.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}) Reference -: ([Legnani et al. 2012](#org4d21607)) +: ([Legnani et al. 2012](#orga1dea1c)) Author(s) : Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D. @@ -22,15 +22,16 @@ Year Example of generated isotropic manipulator (not decoupled). - + {{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}} - + {{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}} + ## Bibliography {#bibliography} -Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” _Mechanism and Machine Theory_ 58 (nil):64–81. . +Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” _Mechanism and Machine Theory_ 58 (nil):64–81. . diff --git a/content/article/li01_simul_fault_vibrat_isolat_point.md b/content/article/li01_simul_fault_vibrat_isolat_point.md index 4c31f7a..99b2982 100644 --- a/content/article/li01_simul_fault_vibrat_isolat_point.md +++ b/content/article/li01_simul_fault_vibrat_isolat_point.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Multivariable Control]({{< relref "multivariable_control" >}}) Reference -: ([Li 2001](#org9c3830b)) +: ([Li 2001](#org7799941)) Author(s) : Li, X. @@ -24,7 +24,7 @@ Year - Cubic (mutually orthogonal) - Flexure Joints => eliminate friction and backlash but add complexity to the dynamics - + {{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 1: Flexure jointed Stewart platform used for analysis and control" >}} @@ -38,18 +38,18 @@ Year The origin of \\(\\{P\\}\\) is taken as the center of mass of the payload. **Decoupling**: -If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#orgdda87ef) or in Figure [3](#orga45a9ca), we obtain a decoupled plant provided that: +If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#org7d1b81f) or in Figure [3](#orgc8972f9), we obtain a decoupled plant provided that: 1. the payload mass/inertia matrix must be diagonal (the CoM is coincident with the origin of frame \\(\\{P\\}\\)) 2. the geometry of the hexapod and the attachment of the payload to the hexapod must be carefully chosen > For instance, if the hexapod has a mutually orthogonal geometry (cubic configuration), the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts. - + {{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 2: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}} - + {{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 3: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}} @@ -75,15 +75,15 @@ The control bandwidth is divided as follows: ### Vibration Isolation {#vibration-isolation} -The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#org2bd4cf2). +The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#orgabd1d61). - + {{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 4: Figure caption" >}} -One of the subsystem plant transfer function is shown in Figure [4](#org2bd4cf2) +One of the subsystem plant transfer function is shown in Figure [4](#orgabd1d61) - + {{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 5: Plant transfer function of one of the SISO subsystem for Vibration Control" >}} @@ -97,9 +97,9 @@ The unity control bandwidth of the isolation loop is designed to be from **5Hz t ### Pointing Control {#pointing-control} -A block diagram of the pointing control system is shown in Figure [6](#orge508ebf). +A block diagram of the pointing control system is shown in Figure [6](#orga9c8d31). - + {{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 6: Figure caption" >}} @@ -108,9 +108,9 @@ The compensators are design with inverse-dynamics methods. The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**. -A feedforward control is added as shown in Figure [7](#org68e9af0). +A feedforward control is added as shown in Figure [7](#org78d2b0e). - + {{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 7: Feedforward control" >}} @@ -122,17 +122,17 @@ The simultaneous vibration isolation and pointing control is approached in two w 1. design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators 2. the reverse design order -Figure [8](#orgdb53c99) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length. +Figure [8](#org94ed578) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length. - + {{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 8: A parallel scheme" >}} The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled. The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation. -The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#org3ec32bc). +The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#orgae8c117). - + {{< figure src="/ox-hugo/li01_effect_isolation_loop_closed.png" caption="Figure 9: \\(\theta\_x/\theta\_{x\_d}\\) transfer function with the isolation loop closed (simulation)" >}} @@ -143,19 +143,19 @@ The effect of pointing control on the isolation plant has not much effect. The dynamic interaction effect: - only happens in the unity bandwidth of the loop transmission of the first closed loop. -- affect the closed loop transmission of the loop first closed (see Figures [10](#org4a1beff) and [11](#org945793e)) +- affect the closed loop transmission of the loop first closed (see Figures [10](#org00362f7) and [11](#org77c322d)) -As shown in Figure [10](#org4a1beff), the peak resonance of the pointing loop increase after the isolation loop is closed. +As shown in Figure [10](#org00362f7), the peak resonance of the pointing loop increase after the isolation loop is closed. The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness. - + {{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 10: Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}} -The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#org945793e)). +The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#org77c322d)). The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop. - + {{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 11: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}} @@ -165,18 +165,18 @@ The first peak resonance of the vibration isolation loop at 15Hz is increased wh ### Experimental results {#experimental-results} -Two hexapods are stacked (Figure [12](#orgbdc7900)): +Two hexapods are stacked (Figure [12](#org5977cb3)): - the bottom hexapod is used to generate disturbances matching candidate applications - the top hexapod provide simultaneous vibration isolation and pointing control - + {{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 12: Stacked Hexapods" >}} -Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#org7a6d899). +Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#org4707691). - + {{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 13: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}} @@ -185,9 +185,9 @@ The simultaneous control is of dual use: - it provide simultaneous pointing and isolation control - it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking -The results of simultaneous control is shown in Figure [14](#org15d93e5) where the bandwidth of the isolation control is expanded to very low frequency. +The results of simultaneous control is shown in Figure [14](#orge4b1f73) where the bandwidth of the isolation control is expanded to very low frequency. - + {{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 14: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}} @@ -216,6 +216,7 @@ Proposed future research areas include: - **Geophones** to provide payload and base velocity information + ## Bibliography {#bibliography} -Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming. +Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming. diff --git a/content/article/mcinroy00_desig_contr_flexur_joint_hexap.md b/content/article/mcinroy00_desig_contr_flexur_joint_hexap.md index 185c7aa..185bce1 100644 --- a/content/article/mcinroy00_desig_contr_flexur_joint_hexap.md +++ b/content/article/mcinroy00_desig_contr_flexur_joint_hexap.md @@ -9,7 +9,7 @@ Tags Reference -: ([McInroy and Hamann 2000](#orgb4fc604)) +: ([McInroy and Hamann 2000](#org6ce0f37)) Author(s) : McInroy, J., & Hamann, J. @@ -18,6 +18,7 @@ Year : 2000 + ## Bibliography {#bibliography} -McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. . +McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. . diff --git a/content/article/mcinroy99_dynam.md b/content/article/mcinroy99_dynam.md index f06f9eb..4caa8a9 100644 --- a/content/article/mcinroy99_dynam.md +++ b/content/article/mcinroy99_dynam.md @@ -4,15 +4,11 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}}) - Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) Reference -: ([McInroy 1999](#orgfc7fa52)) +: ([McInroy 1999](#org788f3dd)) Author(s) : McInroy, J. @@ -20,7 +16,7 @@ Author(s) Year : 1999 -This conference paper has been further published in a journal as a short note ([McInroy 2002](#org7752c60)). +This conference paper has been further published in a journal as a short note ([McInroy 2002](#org6bd1808)). ## Abstract {#abstract} @@ -42,22 +38,22 @@ The actuators for FJHs can be divided into two categories: 1. soft (voice coil), which employs a spring flexure mount 2. hard (piezoceramic or magnetostrictive), which employs a compressive load spring. - + {{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="Figure 1: A general Stewart Platform" >}} -Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org26f1840). +Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#orgc6987ef). In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs. - + {{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="Figure 2: The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}}
Table 1: - Definition of quantities on Figure 2 + Definition of quantities on Figure 2
| **Symbol** | **Meaning** | @@ -74,11 +70,11 @@ In order to provide low frequency passive vibration isolation, the hard actuator | \\(v\_i = p\_i - q\_i\\) | vector pointing from the bottom to the top | | \\(\hat{u}\_i = v\_i/l\_i\\) | unit direction of the strut | -It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#org7752c60))). +It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#org6bd1808))). This is a good approximation unless the spherical joints and extremely stiff or massive, of high inertia struts are used. This allows to reduce considerably the complexity of the model. -From Figure [2](#org26f1840) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term): +From Figure [2](#orgc6987ef) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term): \begin{equation} m\_i \hat{u}\_i^T \ddot{p}\_i = f\_{m\_i} - f\_{p\_i} - m\_i \hat{u}\_i^Tg - k\_i(l\_i - l\_{r\_i}) - b\_i \dot{l}\_i @@ -166,8 +162,9 @@ In the next section, a connection between the two will be found to complete the ## Control Example {#control-example} + ## Bibliography {#bibliography} -McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. . +McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. . -———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. . +———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. . diff --git a/content/article/oomen18_advan_motion_contr_precis_mechat.md b/content/article/oomen18_advan_motion_contr_precis_mechat.md index 507063b..859ebeb 100644 --- a/content/article/oomen18_advan_motion_contr_precis_mechat.md +++ b/content/article/oomen18_advan_motion_contr_precis_mechat.md @@ -8,7 +8,7 @@ Tags : [Motion Control]({{< relref "motion_control" >}}) Reference -: ([Oomen 2018](#org18923fa)) +: ([Oomen 2018](#org93151b9)) Author(s) : Oomen, T. @@ -16,11 +16,12 @@ Author(s) Year : 2018 - + {{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="Figure 1: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}} + ## Bibliography {#bibliography} -Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” _IEEJ Journal of Industry Applications_ 7 (2):127–40. . +Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” _IEEJ Journal of Industry Applications_ 7 (2):127–40. . diff --git a/content/article/preumont07_six_axis_singl_stage_activ.md b/content/article/preumont07_six_axis_singl_stage_activ.md index 56b1af3..90b49b9 100644 --- a/content/article/preumont07_six_axis_singl_stage_activ.md +++ b/content/article/preumont07_six_axis_singl_stage_activ.md @@ -8,7 +8,7 @@ Tags : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) Reference -: ([Preumont et al. 2007](#org3b370bf)) +: ([Preumont et al. 2007](#org66383b9)) Author(s) : Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., … @@ -18,34 +18,35 @@ Year Summary: -- **Cubic** Stewart platform (Figure [3](#org0274a43)) +- **Cubic** Stewart platform (Figure [3](#orga6bbb17)) - Provides uniform control capability - Uniform stiffness in all directions - minimizes the cross-coupling among actuators and sensors of different legs -- Flexible joints (Figure [2](#orge9ceb91)) +- Flexible joints (Figure [2](#orga37017c)) - Piezoelectric force sensors - Voice coil actuators - Decentralized feedback control approach for vibration isolation -- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#orgb3c0578)) +- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#orgec64e08)) - The Stewart platform has 6 suspension modes at different frequencies. Thus the gain of the IFF controller cannot be optimal for all the modes. It is better if all the modes of the platform are near to each other. - Discusses the design of the legs in order to maximize the natural frequency of the local modes. - To estimate the isolation performance of the Stewart platform, a scalar indicator is defined as the Frobenius norm of the transmissibility matrix - + {{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="Figure 1: Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}} - + {{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}} - + {{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}} + ## Bibliography {#bibliography} -Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” _Journal of Sound and Vibration_ 300 (3-5):644–61. . +Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” _Journal of Sound and Vibration_ 300 (3-5):644–61. . diff --git a/content/article/saxena12_advan_inter_model_contr_techn.md b/content/article/saxena12_advan_inter_model_contr_techn.md index c8dd637..7bb7551 100644 --- a/content/article/saxena12_advan_inter_model_contr_techn.md +++ b/content/article/saxena12_advan_inter_model_contr_techn.md @@ -8,7 +8,7 @@ Tags : [Complementary Filters]({{< relref "complementary_filters" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion" >}}) Reference -: ([Saxena and Hote 2012](#org0284b71)) +: ([Saxena and Hote 2012](#org6d87fa3)) Author(s) : Saxena, S., & Hote, Y. @@ -85,6 +85,7 @@ Issues: The interesting feature regarding IMC is that the design scheme is identical to the open-loop control design procedure and the implementation of IMC results in a feedback system, thereby copying the disturbances and parameter uncertainties, while open-loop control is not. + ## Bibliography {#bibliography} -Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” _IETE Technical Review_ 29 (6):461. . +Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” _IETE Technical Review_ 29 (6):461. . diff --git a/content/article/sayed01_survey_spect_factor_method.md b/content/article/sayed01_survey_spect_factor_method.md index 086432f..bd0b73b 100644 --- a/content/article/sayed01_survey_spect_factor_method.md +++ b/content/article/sayed01_survey_spect_factor_method.md @@ -9,7 +9,7 @@ Tags Reference -: ([Sayed and Kailath 2001](#orgaf03ac4)) +: ([Sayed and Kailath 2001](#org9b5be86)) Author(s) : Sayed, A. H., & Kailath, T. @@ -18,6 +18,7 @@ Year : 2001 + ## Bibliography {#bibliography} -Sayed, A. H., and T. Kailath. 2001. “A Survey of Spectral Factorization Methods.” _Numerical Linear Algebra with Applications_ 8 (6-7):467–96. . +Sayed, A. H., and T. Kailath. 2001. “A Survey of Spectral Factorization Methods.” _Numerical Linear Algebra with Applications_ 8 (6-7):467–96. . diff --git a/content/article/schroeck01_compen_desig_linear_time_invar.md b/content/article/schroeck01_compen_desig_linear_time_invar.md index b6a4f4c..796be57 100644 --- a/content/article/schroeck01_compen_desig_linear_time_invar.md +++ b/content/article/schroeck01_compen_desig_linear_time_invar.md @@ -9,7 +9,7 @@ Tags Reference -: ([Schroeck, Messner, and McNab 2001](#orgf8182bc)) +: ([Schroeck, Messner, and McNab 2001](#org5e2e067)) Author(s) : Schroeck, S., Messner, W., & McNab, R. @@ -18,6 +18,7 @@ Year : 2001 + ## Bibliography {#bibliography} -Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” _IEEE/ASME Transactions on Mechatronics_ 6 (1):50–57. . +Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” _IEEE/ASME Transactions on Mechatronics_ 6 (1):50–57. . diff --git a/content/article/sebastian12_nanop_with_multip_sensor.md b/content/article/sebastian12_nanop_with_multip_sensor.md index f4ae452..85357e4 100644 --- a/content/article/sebastian12_nanop_with_multip_sensor.md +++ b/content/article/sebastian12_nanop_with_multip_sensor.md @@ -8,7 +8,7 @@ Tags : [Sensor Fusion]({{< relref "sensor_fusion" >}}) Reference -: ([Sebastian and Pantazi 2012](#org03bb39f)) +: ([Sebastian and Pantazi 2012](#orgacf93b4)) Author(s) : Sebastian, A., & Pantazi, A. @@ -17,6 +17,7 @@ Year : 2012 + ## Bibliography {#bibliography} -Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” _IEEE Transactions on Control Systems Technology_ 20 (2):382–94. . +Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” _IEEE Transactions on Control Systems Technology_ 20 (2):382–94. . diff --git a/content/article/souleille18_concep_activ_mount_space_applic.md b/content/article/souleille18_concep_activ_mount_space_applic.md index b77c40a..abed57d 100644 --- a/content/article/souleille18_concep_activ_mount_space_applic.md +++ b/content/article/souleille18_concep_activ_mount_space_applic.md @@ -8,7 +8,7 @@ Tags : [Active Damping]({{< relref "active_damping" >}}) Reference -: ([Souleille et al. 2018](#org5546d0c)) +: ([Souleille et al. 2018](#org5548942)) Author(s) : Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, & Collette, C. @@ -23,10 +23,10 @@ This article discusses the use of Integral Force Feedback with amplified piezoel ## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator} -Figure [1](#org8634178) shows a picture of the amplified piezoelectric stack. +Figure [1](#org0952ca2) shows a picture of the amplified piezoelectric stack. The piezoelectric actuator is divided into two parts: one is used as an actuator, and the other one is used as a force sensor. - + {{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="Figure 1: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}} @@ -61,38 +61,39 @@ and the control force is given by: f = F\_s G(s) = F\_s \frac{g}{s} \end{equation} -The effect of the controller are shown in Figure [2](#orgcb733df): +The effect of the controller are shown in Figure [2](#orgb36ba37): - the resonance peak is almost critically damped - the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies - the degradation of the compliance \\(\frac{x\_1}{F}\\) induced by feedback is limited at \\(\frac{1}{k\_1}\\) - the fraction of the force transmitted to the payload that is measured by the force sensor is reduced at low frequencies - + {{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="Figure 2: Matrix of transfer functions from input (w, f, F) to output (Fs, x1) in open loop (blue curves) and closed loop (dashed red curves)" >}} - + {{< figure src="/ox-hugo/souleille18_root_locus.png" caption="Figure 3: Single DoF system. Comparison between the theoretical (solid curve) and the experimental (crosses) root-locus" >}} ## Flexible payload mounted on three isolators {#flexible-payload-mounted-on-three-isolators} -A heavy payload is mounted on a set of three isolators (Figure [4](#org09ac00a)). +A heavy payload is mounted on a set of three isolators (Figure [4](#orgbe1030b)). The payload consists of two masses, connected through flexible blades such that the flexible resonance of the payload in the vertical direction is around 65Hz. - + {{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="Figure 4: Right: picture of the experimental setup. It consists of a flexible payload mounted on a set of three isolators. Left: simplified sketch of the setup, showing only the vertical direction" >}} -As shown in Figure [5](#org2dcbc51), both the suspension modes and the flexible modes of the payload can be critically damped. +As shown in Figure [5](#orgbb573b8), both the suspension modes and the flexible modes of the payload can be critically damped. - + {{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}} + ## Bibliography {#bibliography} -Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65. +Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65. diff --git a/content/article/tang18_decen_vibrat_contr_voice_coil.md b/content/article/tang18_decen_vibrat_contr_voice_coil.md index 57e1a89..7432538 100644 --- a/content/article/tang18_decen_vibrat_contr_voice_coil.md +++ b/content/article/tang18_decen_vibrat_contr_voice_coil.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}) Reference -: ([Tang, Cao, and Yu 2018](#org45ebb6f)) +: ([Tang, Cao, and Yu 2018](#org44aa05e)) Author(s) : Tang, J., Cao, D., & Yu, T. @@ -17,6 +17,7 @@ Year : 2018 + ## Bibliography {#bibliography} -Tang, Jie, Dengqing Cao, and Tianhu Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ 233 (1):132–45. . +Tang, Jie, Dengqing Cao, and Tianhu Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ 233 (1):132–45. . diff --git a/content/article/wang12_autom_marker_full_field_hard.md b/content/article/wang12_autom_marker_full_field_hard.md index e822ae8..e12525b 100644 --- a/content/article/wang12_autom_marker_full_field_hard.md +++ b/content/article/wang12_autom_marker_full_field_hard.md @@ -8,7 +8,7 @@ Tags : [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}) Reference -: ([Wang et al. 2012](#org187cf70)) +: ([Wang et al. 2012](#org172c2d6)) Author(s) : Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M. @@ -26,6 +26,7 @@ There is a need for markerless nano-tomography It uses calibrated metrology disc and capacitive sensors + ## Bibliography {#bibliography} -Wang, Jun, Yu-chen Karen Chen, Qingxi Yuan, Andrei Tkachuk, Can Erdonmez, Benjamin Hornberger, and Michael Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” _Applied Physics Letters_ 100 (14):143107. . +Wang, Jun, Yu-chen Karen Chen, Qingxi Yuan, Andrei Tkachuk, Can Erdonmez, Benjamin Hornberger, and Michael Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” _Applied Physics Letters_ 100 (14):143107. . diff --git a/content/article/wang16_inves_activ_vibrat_isolat_stewar.md b/content/article/wang16_inves_activ_vibrat_isolat_stewar.md index 946944f..9165318 100644 --- a/content/article/wang16_inves_activ_vibrat_isolat_stewar.md +++ b/content/article/wang16_inves_activ_vibrat_isolat_stewar.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) Reference -: ([Wang et al. 2016](#orgda82aa7)) +: ([Wang et al. 2016](#org6a7c8f9)) Author(s) : Wang, C., Xie, X., Chen, Y., & Zhang, Z. @@ -25,7 +25,7 @@ Year The model is compared with a Finite Element model and is shown to give the same results. The proposed model is thus effective. - + {{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="Figure 1: Stewart Platform" >}} @@ -35,11 +35,11 @@ Combines: - the FxLMS-based adaptive inverse control => suppress transmission of periodic vibrations - direct feedback of integrated forces => dampen vibration of inherent modes and thus reduce random vibrations -Force Feedback (Figure [2](#orgaf56f94)). +Force Feedback (Figure [2](#org25780ff)). - the force sensor is mounted **between the base and the strut** - + {{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}} @@ -54,6 +54,7 @@ Sorts of HAC-LAC control: - Effectiveness of control methods are shown + ## Bibliography {#bibliography} -Wang, Chaoxin, Xiling Xie, Yanhao Chen, and Zhiyi Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” _Journal of Sound and Vibration_ 383 (November). Elsevier BV:1–19. . +Wang, Chaoxin, Xiling Xie, Yanhao Chen, and Zhiyi Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” _Journal of Sound and Vibration_ 383 (November). Elsevier BV:1–19. . diff --git a/content/article/yun20_inves_two_stage_vibrat_suppr.md b/content/article/yun20_inves_two_stage_vibrat_suppr.md index bf909e6..84e972d 100644 --- a/content/article/yun20_inves_two_stage_vibrat_suppr.md +++ b/content/article/yun20_inves_two_stage_vibrat_suppr.md @@ -9,7 +9,7 @@ Tags Reference -: ([Yun et al. 2020](#orgd3a0930)) +: ([Yun et al. 2020](#org2a2f02a)) Author(s) : Yun, H., Liu, L., Li, Q., & Yang, H. @@ -18,6 +18,7 @@ Year : 2020 + ## Bibliography {#bibliography} -Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” _Aerospace Science and Technology_ 96 (nil):105543. . +Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” _Aerospace Science and Technology_ 96 (nil):105543. . diff --git a/content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md b/content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md index 0e89e59..b44cec2 100644 --- a/content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md +++ b/content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md @@ -8,7 +8,7 @@ Tags : [Vibration Isolation]({{< relref "vibration_isolation" >}}) Reference -: ([Zuo 2004](#orgb4186fb)) +: ([Zuo 2004](#orgd9c4f73)) Author(s) : Zuo, L. @@ -40,23 +40,24 @@ Year > They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback. > Note that reaction force actuators can also work with soft mounts or hard mounts. - + {{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}} - + {{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}} - + {{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}} - + {{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}} + ## Bibliography {#bibliography} -Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology. +Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology. diff --git a/content/book/albertos04_multiv_contr_system.md b/content/book/albertos04_multiv_contr_system.md index 1851186..27fc7f4 100644 --- a/content/book/albertos04_multiv_contr_system.md +++ b/content/book/albertos04_multiv_contr_system.md @@ -8,7 +8,7 @@ Tags : [Multivariable Control]({{< relref "multivariable_control" >}}) Reference -: ([Albertos and Antonio 2004](#org911b8ba)) +: ([Albertos and Antonio 2004](#orgb06343d)) Author(s) : Albertos, P., & Antonio, S. @@ -17,6 +17,116 @@ Year : 2004 +## Introduction to Multivariable Control {#introduction-to-multivariable-control} + + +## Linear System Representation: Models and Equivalence {#linear-system-representation-models-and-equivalence} + + +## Linear Systems Analysis {#linear-systems-analysis} + + +## Solutions to the Control Problem {#solutions-to-the-control-problem} + + +## Decentralised and Decoupled Control {#decentralised-and-decoupled-control} + + +### Decoupling {#decoupling} + +In cases when multi-loop control is not effective in reaching the desired specifications, a possible strategy for tackling the MIMO control could be to transform the transfer function matrix into a diagonal dominant one. +This strategy is called **decoupling**. + +[Decoupled Control]({{< relref "decoupled_control" >}}) can be achieved in two ways: + +- feedforward cancellation of the cross-coupling terms +- based on state measurements, via a feedback law + + +#### Feedforward Decoupling {#feedforward-decoupling} + +A pre-compensator can be added to transform the open-loop characteristics into a new one as chosen by the designer. +This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros. + +**Approximate decoupling**: +To design low-bandwidth loops, insertion of the inverse DC-gain before the loop ensures decoupling at least at steady-state. +If further bandwidth extension is desired, an approximation of \\(G^{-1}\\) valid in low frequencies can be used. + + +#### Feedback Decoupling {#feedback-decoupling} + +Although at first glance, decoupling seems an appealing idea, there are some drawbacks: + +- as decoupling is achieved via the coordination of sensors and actuators to achieve an "apparent" diagonal behavior, the failure of one the actuators may heavily affects all loops. +- a decoupling design (inverse-based controller) may not be desirable for all disturbance-rejection tasks. +- many MIMO non-minimum phase systems, when feedforward decoupled, increase the RHP-zero multiplicity so performance limitations due to its presence are exacerbated. +- decoupling may be very sensitive to modeling errors, specially for ill-conditionned plants +- feedback decoupling needs full state measurements + + +#### SVD Decoupling {#svd-decoupling} + +A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition]({{< relref "singular_value_decomposition" >}}) as: + +\begin{equation} +M = U \Sigma V^T +\end{equation} + +where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal. + +The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators. +In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants. + +If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [1](#org3d5b40c), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used. +Usually, the sensor and actuator transformations are obtained using the DC gain, or a real approximation of \\(G(j\omega)\\), where \\(\omega\\) is around the desired closed-loop bandwidth. + + + +{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}} + +The transformed sensor-actuator pair corresponding to the maximum singular value is the direction with biggest "gain" on the plant, that is, the combination of variables being "easiest to control". + +In ill-conditioned plants, the ratio between the biggest and lower singular value is large (for reference, greater than 20). +They are very sensitive to input uncertainty as some "input directions" have much bigger gain than other ones. + +SVD decoupling produces the most suitable combinations for independent "multi-loop" control in the transformed variables, so its performance may be better than RGA-based design (at the expense of losing physical interpretability). +If some of the vectors in \\(V\\) (input directions) have a significant component on a particular input, and the corresponding output direction is also significantly pointing to a particular output, that combination is a good candidate for an independent multi-loop control. + + +## Fundamentals of Centralised Closed-loop Control {#fundamentals-of-centralised-closed-loop-control} + + +## Optimisation-based Control {#optimisation-based-control} + + +## Designing for Robustness {#designing-for-robustness} + + +## Implementation and Other Issues {#implementation-and-other-issues} + + +## Appendices {#appendices} + + +### Summary of SISO System Analysis {#summary-of-siso-system-analysis} + + +### Matrices {#matrices} + + +### Signal and System Norms {#signal-and-system-norms} + + +### Optimisation {#optimisation} + + +### Multivariable Statistics {#multivariable-statistics} + + +### Robust Control Analysis and Synthesis {#robust-control-analysis-and-synthesis} + + + ## Bibliography {#bibliography} -Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. . +Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. . diff --git a/content/book/du10_model_contr_vibrat_mechan_system.md b/content/book/du10_model_contr_vibrat_mechan_system.md index 0728860..f26c7c7 100644 --- a/content/book/du10_model_contr_vibrat_mechan_system.md +++ b/content/book/du10_model_contr_vibrat_mechan_system.md @@ -8,7 +8,7 @@ Tags : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}) Reference -: ([Du and Xie 2010](#orge0a6379)) +: ([Du and Xie 2010](#org5093366)) Author(s) : Du, C., & Xie, L. @@ -16,9 +16,524 @@ Author(s) Year : 2010 -Read Chapter 1 and 3. + +## 1. Mechanical Systems and Vibration {#1-dot-mechanical-systems-and-vibration} + + +### 1.1 Magnetic recording system {#1-dot-1-magnetic-recording-system} + + +### 1.2 Stewart platform {#1-dot-2-stewart-platform} + + +### 1.3 Vibration sources and descriptions {#1-dot-3-vibration-sources-and-descriptions} + + +### 1.4 Types of vibration {#1-dot-4-types-of-vibration} + + +#### 1.4.1 Free and forced vibration {#1-dot-4-dot-1-free-and-forced-vibration} + + +#### 1.4.2 Damped and undamped vibration {#1-dot-4-dot-2-damped-and-undamped-vibration} + + +#### 1.4.3 Linear and nonlinear vibration {#1-dot-4-dot-3-linear-and-nonlinear-vibration} + + +#### 1.4.4 Deterministic and random vibration {#1-dot-4-dot-4-deterministic-and-random-vibration} + + +#### 1.4.5 Periodic and nonperiodic vibration {#1-dot-4-dot-5-periodic-and-nonperiodic-vibration} + + +#### 1.4.6 Broad-band and narrow-band vibration {#1-dot-4-dot-6-broad-band-and-narrow-band-vibration} + + +### 1.5 Random vibration {#1-dot-5-random-vibration} + + +#### 1.5.1 Random process {#1-dot-5-dot-1-random-process} + + +#### 1.5.2 Stationary random process {#1-dot-5-dot-2-stationary-random-process} + + +#### 1.5.3 Gaussian random process {#1-dot-5-dot-3-gaussian-random-process} + + +### 1.6 Vibration analysis {#1-dot-6-vibration-analysis} + + +#### 1.6.1 Fourier transform and spectrum analysis {#1-dot-6-dot-1-fourier-transform-and-spectrum-analysis} + + +#### 1.6.2 Relationship between the Fourier and Laplace transforms {#1-dot-6-dot-2-relationship-between-the-fourier-and-laplace-transforms} + + +#### 1.6.3 Spectral analysis {#1-dot-6-dot-3-spectral-analysis} + + +## 2. Modeling of Disk Drive System and Its Vibration {#2-dot-modeling-of-disk-drive-system-and-its-vibration} + + +### 2.1 Introduction {#2-dot-1-introduction} + + +### 2.2 System description {#2-dot-2-system-description} + + +### 2.3 System modeling {#2-dot-3-system-modeling} + + +#### 2.3.1 Modeling of a VCM actuator {#2-dot-3-dot-1-modeling-of-a-vcm-actuator} + + +#### 2.3.2 Modeling of friction {#2-dot-3-dot-2-modeling-of-friction} + + +#### 2.3.3 Modeling of a PZT microactuator {#2-dot-3-dot-3-modeling-of-a-pzt-microactuator} + + +#### 2.3.4 An example {#2-dot-3-dot-4-an-example} + + +### 2.4 Vibration modeling {#2-dot-4-vibration-modeling} + + +#### 2.4.1 Spectrum-based vibration modeling {#2-dot-4-dot-1-spectrum-based-vibration-modeling} + + +#### 2.4.2 Adaptive modeling of disturbance {#2-dot-4-dot-2-adaptive-modeling-of-disturbance} + + +### 2.5 Conclusion {#2-dot-5-conclusion} + + +## 3. Modeling of [Stewart Platforms]({{< relref "stewart_platforms" >}}) {#3-dot-modeling-of-stewart-platforms--stewart-platforms-dot-md} + + +### 3.1 Introduction {#3-dot-1-introduction} + + +### 3.2 System description and governing equations {#3-dot-2-system-description-and-governing-equations} + + +### 3.3 Modeling using adaptive filtering approach {#3-dot-3-modeling-using-adaptive-filtering-approach} + + +#### 3.3.1 Adaptive filtering theory {#3-dot-3-dot-1-adaptive-filtering-theory} + + +#### 3.3.2 Modeling of a Stewart platform {#3-dot-3-dot-2-modeling-of-a-stewart-platform} + + +### 3.4 Conclusion {#3-dot-4-conclusion} + + +## 4. Classical Vibration Control {#4-dot-classical-vibration-control} + + +### 4.1 Introduction {#4-dot-1-introduction} + + +### 4.2 Passive control {#4-dot-2-passive-control} + + +#### 4.2.1 Isolators {#4-dot-2-dot-1-isolators} + + +#### 4.2.2 Absorbers {#4-dot-2-dot-2-absorbers} + + +#### 4.2.3 Resonators {#4-dot-2-dot-3-resonators} + + +#### 4.2.4 Suspension {#4-dot-2-dot-4-suspension} + + +#### 4.2.5 An application example – Disk vibration reduction via stacked disks {#4-dot-2-dot-5-an-application-example-and-8211-disk-vibration-reduction-via-stacked-disks} + + +### 4.3 Self-adapting systems {#4-dot-3-self-adapting-systems} + + +### 4.4 Active vibration control {#4-dot-4-active-vibration-control} + + +#### 4.4.1 Actuators {#4-dot-4-dot-1-actuators} + + +#### 4.4.2 Active systems {#4-dot-4-dot-2-active-systems} + + +#### 4.4.3 Control strategy {#4-dot-4-dot-3-control-strategy} + + +### 4.5 Conclusion {#4-dot-5-conclusion} + + +## 5. Introduction to Optimal and Robust Control {#5-dot-introduction-to-optimal-and-robust-control} + + +### 5.1 Introduction {#5-dot-1-introduction} + + +### 5.2 H2 and H∞ norms {#5-dot-2-h2-and-h-and-8734-norms} + + +#### 5.2.1 H2 norm {#5-dot-2-dot-1-h2-norm} + + +#### 5.2.2 H∞ norm {#5-dot-2-dot-2-h-and-8734-norm} + + +### 5.3 H2 optimal control {#5-dot-3-h2-optimal-control} + + +#### 5.3.1 Continuous-time case {#5-dot-3-dot-1-continuous-time-case} + + +#### 5.3.2 Discrete-time case {#5-dot-3-dot-2-discrete-time-case} + + +### 5.4 H∞ control {#5-dot-4-h-and-8734-control} + + +#### 5.4.1 Continuous-time case {#5-dot-4-dot-1-continuous-time-case} + + +#### 5.4.2 Discrete-time case {#5-dot-4-dot-2-discrete-time-case} + + +### 5.5 Robust control {#5-dot-5-robust-control} + + +### 5.6 Controller parametrization {#5-dot-6-controller-parametrization} + + +### 5.7 Performance limitation {#5-dot-7-performance-limitation} + + +#### 5.7.1 Bode integral constraint {#5-dot-7-dot-1-bode-integral-constraint} + + +#### 5.7.2 Relationship between system gain and phase {#5-dot-7-dot-2-relationship-between-system-gain-and-phase} + + +#### 5.7.3 Sampling {#5-dot-7-dot-3-sampling} + + +### 5.8 Conclusion {#5-dot-8-conclusion} + + +## 6. Mixed H2/H∞ Control Design for Vibration Rejection {#6-dot-mixed-h2-h-and-8734-control-design-for-vibration-rejection} + + +### 6.1 Introduction {#6-dot-1-introduction} + + +### 6.2 Mixed H2/H∞ control problem {#6-dot-2-mixed-h2-h-and-8734-control-problem} + + +### 6.3 Method 1: slack variable approach {#6-dot-3-method-1-slack-variable-approach} + + +### 6.4 Method 2: an improved slack variable approach {#6-dot-4-method-2-an-improved-slack-variable-approach} + + +### 6.5 Application in servo loop design for hard disk drives {#6-dot-5-application-in-servo-loop-design-for-hard-disk-drives} + + +#### 6.5.1 Problem formulation {#6-dot-5-dot-1-problem-formulation} + + +#### 6.5.2 Design results {#6-dot-5-dot-2-design-results} + + +### 6.6 Conclusion {#6-dot-6-conclusion} + + +## 7. Low-Hump Sensitivity Control Design for Hard Disk Drive Systems {#7-dot-low-hump-sensitivity-control-design-for-hard-disk-drive-systems} + + +### 7.1 Introduction {#7-dot-1-introduction} + + +### 7.2 Problem statement {#7-dot-2-problem-statement} + + +### 7.3 Design in continuous-time domain {#7-dot-3-design-in-continuous-time-domain} + + +#### 7.3.1 H∞ loop shaping for low-hump sensitivity functions {#7-dot-3-dot-1-h-and-8734-loop-shaping-for-low-hump-sensitivity-functions} + + +#### 7.3.2 Application examples {#7-dot-3-dot-2-application-examples} + + +#### 7.3.3 Implementation on a hard disk drive {#7-dot-3-dot-3-implementation-on-a-hard-disk-drive} + + +### 7.4 Design in discrete-time domain {#7-dot-4-design-in-discrete-time-domain} + + +#### 7.4.1 Synthesis method for low-hump sensitivity function {#7-dot-4-dot-1-synthesis-method-for-low-hump-sensitivity-function} + + +#### 7.4.2 An application example {#7-dot-4-dot-2-an-application-example} + + +#### 7.4.3 Implementation on a hard disk drive {#7-dot-4-dot-3-implementation-on-a-hard-disk-drive} + + +### 7.5 Conclusion {#7-dot-5-conclusion} + + +## 8. Generalized KYP Lemma-Based Loop Shaping Control Design {#8-dot-generalized-kyp-lemma-based-loop-shaping-control-design} + + +### 8.1 Introduction {#8-dot-1-introduction} + + +### 8.2 Problem description {#8-dot-2-problem-description} + + +### 8.3 Generalized KYP lemma-based control design method {#8-dot-3-generalized-kyp-lemma-based-control-design-method} + + +### 8.4 Peak filter {#8-dot-4-peak-filter} + + +#### 8.4.1 Conventional peak filter {#8-dot-4-dot-1-conventional-peak-filter} + + +#### 8.4.2 Phase lead peak filter {#8-dot-4-dot-2-phase-lead-peak-filter} + + +#### 8.4.3 Group peak filter {#8-dot-4-dot-3-group-peak-filter} + + +### 8.5 Application in high frequency vibration rejection {#8-dot-5-application-in-high-frequency-vibration-rejection} + + +### 8.6 Application in mid-frequency vibration rejection {#8-dot-6-application-in-mid-frequency-vibration-rejection} + + +### 8.7 Conclusion {#8-dot-7-conclusion} + + +## 9. Combined H2 and KYP Lemma-Based Control Design {#9-dot-combined-h2-and-kyp-lemma-based-control-design} + + +### 9.1 Introduction {#9-dot-1-introduction} + + +### 9.2 Problem formulation {#9-dot-2-problem-formulation} + + +### 9.3 Controller design for specific disturbance rejection and overall error minimization {#9-dot-3-controller-design-for-specific-disturbance-rejection-and-overall-error-minimization} + + +#### 9.3.1 Q parametrization to meet specific specifications {#9-dot-3-dot-1-q-parametrization-to-meet-specific-specifications} + + +#### 9.3.2 Q parametrization to minimize H2 performance {#9-dot-3-dot-2-q-parametrization-to-minimize-h2-performance} + + +#### 9.3.3 Design steps {#9-dot-3-dot-3-design-steps} + + +### 9.4 Simulation and implementation results {#9-dot-4-simulation-and-implementation-results} + + +#### 9.4.1 System models {#9-dot-4-dot-1-system-models} + + +#### 9.4.2 Rejection of specific disturbance and H2 performance minimization {#9-dot-4-dot-2-rejection-of-specific-disturbance-and-h2-performance-minimization} + + +#### 9.4.3 Rejection of two disturbances with H[sub(2)] performance minimization {#9-dot-4-dot-3-rejection-of-two-disturbances-with-h-sub--2--performance-minimization} + + +### 9.5 Conclusion {#9-dot-5-conclusion} + + +## 10. Blending Control for Multi-Frequency Disturbance Rejection {#10-dot-blending-control-for-multi-frequency-disturbance-rejection} + + +### 10.1 Introduction {#10-dot-1-introduction} + + +### 10.2 Control blending {#10-dot-2-control-blending} + + +#### 10.2.1 State feedback control blending {#10-dot-2-dot-1-state-feedback-control-blending} + + +#### 10.2.2 Output feedback control blending {#10-dot-2-dot-2-output-feedback-control-blending} + + +### 10.3 Control blending application in multi-frequency disturbance rejection {#10-dot-3-control-blending-application-in-multi-frequency-disturbance-rejection} + + +#### 10.3.1 Problem formulation {#10-dot-3-dot-1-problem-formulation} + + +#### 10.3.2 Controller design via the control blending technique {#10-dot-3-dot-2-controller-design-via-the-control-blending-technique} + + +### 10.4 Simulation and experimental results {#10-dot-4-simulation-and-experimental-results} + + +#### 10.4.1 Rejecting high-frequency disturbances {#10-dot-4-dot-1-rejecting-high-frequency-disturbances} + + +#### 10.4.2 Rejecting a combined mid and high frequency disturbance {#10-dot-4-dot-2-rejecting-a-combined-mid-and-high-frequency-disturbance} + + +### 10.5 Conclusion {#10-dot-5-conclusion} + + +## 11. H∞-Based Design for Disturbance Observer {#11-dot-h-and-8734-based-design-for-disturbance-observer} + + +### 11.1 Introduction {#11-dot-1-introduction} + + +### 11.2 Conventional disturbance observer {#11-dot-2-conventional-disturbance-observer} + + +### 11.3 A general form of disturbance observer {#11-dot-3-a-general-form-of-disturbance-observer} + + +### 11.4 Application results {#11-dot-4-application-results} + + +### 11.5 Conclusion {#11-dot-5-conclusion} + + +## 12. Two-Dimensional H2 Control for Error Minimization {#12-dot-two-dimensional-h2-control-for-error-minimization} + + +### 12.1 Introduction {#12-dot-1-introduction} + + +### 12.2 2-D stabilization control {#12-dot-2-2-d-stabilization-control} + + +### 12.3 2-D H2 control {#12-dot-3-2-d-h2-control} + + +### 12.4 SSTW process and modeling {#12-dot-4-sstw-process-and-modeling} + + +#### 12.4.1 SSTW servo loop {#12-dot-4-dot-1-sstw-servo-loop} + + +#### 12.4.2 Two-dimensional model {#12-dot-4-dot-2-two-dimensional-model} + + +### 12.5 Feedforward compensation method {#12-dot-5-feedforward-compensation-method} + + +### 12.6 2-D control formulation for SSTW {#12-dot-6-2-d-control-formulation-for-sstw} + + +### 12.7 2-D stabilization control for error propagation containment {#12-dot-7-2-d-stabilization-control-for-error-propagation-containment} + + +#### 12.7.1 Simulation results {#12-dot-7-dot-1-simulation-results} + + +### 12.8 2-D H2 control for error minimization {#12-dot-8-2-d-h2-control-for-error-minimization} + + +#### 12.8.1 Simulation results {#12-dot-8-dot-1-simulation-results} + + +#### 12.8.2 Experimental results {#12-dot-8-dot-2-experimental-results} + + +### 12.9 Conclusion {#12-dot-9-conclusion} + + +## 13. Nonlinearity Compensation and Nonlinear Control {#13-dot-nonlinearity-compensation-and-nonlinear-control} + + +### 13.1 Introduction {#13-dot-1-introduction} + + +### 13.2 Nonlinearity compensation {#13-dot-2-nonlinearity-compensation} + + +### 13.3 Nonlinear control {#13-dot-3-nonlinear-control} + + +#### 13.3.1 Design of a composite control law {#13-dot-3-dot-1-design-of-a-composite-control-law} + + +#### 13.3.2 Experimental results in hard disk drives {#13-dot-3-dot-2-experimental-results-in-hard-disk-drives} + + +### 13.4 Conclusion {#13-dot-4-conclusion} + + +## 14. Quantization Effect on Vibration Rejection and Its Compensation {#14-dot-quantization-effect-on-vibration-rejection-and-its-compensation} + + +### 14.1 Introduction {#14-dot-1-introduction} + + +### 14.2 Description of control system with quantizer {#14-dot-2-description-of-control-system-with-quantizer} + + +### 14.3 Quantization effect on error rejection {#14-dot-3-quantization-effect-on-error-rejection} + + +#### 14.3.1 Quantizer frequency response measurement {#14-dot-3-dot-1-quantizer-frequency-response-measurement} + + +#### 14.3.2 Quantization effect on error rejection {#14-dot-3-dot-2-quantization-effect-on-error-rejection} + + +### 14.4 Compensation of quantization effect on error rejection {#14-dot-4-compensation-of-quantization-effect-on-error-rejection} + + +### 14.5 Conclusion {#14-dot-5-conclusion} + + +## 15. Adaptive Filtering Algorithms for Active Vibration Control {#15-dot-adaptive-filtering-algorithms-for-active-vibration-control} + + +### 15.1 Introduction {#15-dot-1-introduction} + + +### 15.2 Adaptive feedforward algorithm {#15-dot-2-adaptive-feedforward-algorithm} + + +### 15.3 Adaptive feedback algorithm {#15-dot-3-adaptive-feedback-algorithm} + + +### 15.4 Comparison between feedforward and feedback controls {#15-dot-4-comparison-between-feedforward-and-feedback-controls} + + +### 15.5 Application in Stewart platform {#15-dot-5-application-in-stewart-platform} + + +#### 15.5.1 Multi-channel adaptive feedback AVC system {#15-dot-5-dot-1-multi-channel-adaptive-feedback-avc-system} + + +#### 15.5.2 Multi-channel adaptive feedback algorithm for hexapod platform {#15-dot-5-dot-2-multi-channel-adaptive-feedback-algorithm-for-hexapod-platform} + + +#### 15.5.3 Simulation and implementation {#15-dot-5-dot-3-simulation-and-implementation} + + +### 15.6 Conclusion {#15-dot-6-conclusion} + ## Bibliography {#bibliography} -Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. . +Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. . diff --git a/content/book/hatch00_vibrat_matlab_ansys.md b/content/book/hatch00_vibrat_matlab_ansys.md index 200a0f7..574e0cf 100644 --- a/content/book/hatch00_vibrat_matlab_ansys.md +++ b/content/book/hatch00_vibrat_matlab_ansys.md @@ -8,7 +8,7 @@ Tags : [Finite Element Model]({{< relref "finite_element_model" >}}) Reference -: ([Hatch 2000](#orgebf8ccb)) +: ([Hatch 2000](#orgf661cf4)) Author(s) : Hatch, M. R. @@ -21,14 +21,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen ## Introduction {#introduction} - + The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models. ### Modal Analysis {#modal-analysis} -The diagram in Figure [1](#org242701e) shows the methodology for analyzing a lightly damped structure using normal modes. +The diagram in Figure [1](#org3a8e4cc) shows the methodology for analyzing a lightly damped structure using normal modes.
@@ -46,7 +46,7 @@ The steps are:
- + {{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}} @@ -58,7 +58,7 @@ Because finite element models usually have a very large number of states, an imp
-Figure [2](#org398c443) shows such process, the steps are: +Figure [2](#org0a7e8db) shows such process, the steps are: - start with the finite element model - compute the eigenvalues and eigenvectors (as many as dof in the model) @@ -71,14 +71,14 @@ Figure [2](#org398c443) shows such process, the steps are:
- + {{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}} ### Notations {#notations} -Tables [3](#org02d84e8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document. +Tables [3](#orgc82b5d8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
@@ -127,22 +127,22 @@ Tables [3](#org02d84e8), [2](#table--tab:notations-eigen-vectors-values) and [3] ## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems} - + The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency. We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**. -Figure [3](#org02d84e8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs. +Figure [3](#orgc82b5d8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs. The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\). - + {{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
-([Miu 1993](#org39eead7)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function. +([Miu 1993](#org849cfe4)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function. The resonances of the "overhanging appendages" of the constrained system create the zeros. @@ -151,12 +151,12 @@ The resonances of the "overhanging appendages" of the constrained system create ## State Space Analysis {#state-space-analysis} - + ## Modal Analysis {#modal-analysis} - + Lightly damped structures are typically analyzed with the "normal mode" method described in this section. @@ -196,9 +196,9 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and #### Equation of Motion {#equation-of-motion} -Let's consider the model shown in Figure [4](#org0c2921d) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\). +Let's consider the model shown in Figure [4](#orgebf4457) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\). - + {{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}} @@ -297,17 +297,17 @@ One then find: \end{bmatrix} \end{equation} -Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc90fe3a), [6](#orgfd8222c) and [7](#orgaf9cc36). +Virtual interpretation of the eigenvectors are shown in Figures [5](#org520a99d), [6](#org722a9ff) and [7](#org9e25b28). - + {{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}} - + {{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}} - + {{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}} @@ -346,9 +346,9 @@ There are many options for change of basis, but we will show that **when eigenve The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems. The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate. -This procedure is schematically shown in Figure [8](#orgf9a2963). +This procedure is schematically shown in Figure [8](#orgfbabf08). - + {{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}} @@ -696,7 +696,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of ## Frequency Response: Modal Form {#frequency-response-modal-form} - + The procedure to obtain the frequency response from a modal form is as follow: @@ -704,9 +704,9 @@ The procedure to obtain the frequency response from a modal form is as follow: - use Laplace transform to obtain the transfer functions in principal coordinates - back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident -This will be applied to the model shown in Figure [9](#org48b68a4). +This will be applied to the model shown in Figure [9](#orge102983). - + {{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}} @@ -888,9 +888,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha
-Figure [10](#org87763b9) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\). +Figure [10](#org3024448) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\). - + {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}} @@ -899,16 +899,16 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f ## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model} - + ### Introduction {#introduction} -In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orga66d597). +In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org2292476). A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output. The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics. - + {{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}} @@ -987,7 +987,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the ## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model} - + ### Model Description {#model-description} @@ -1001,25 +1001,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the ## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model} - + -In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org94e126d)). +In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org143e4e8)). ### Actuator Description {#actuator-description} - + {{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}} -The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#org4a20950)). +The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orgc294fc5)). - + {{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}} For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required. -Figure [13](#org4a20950) shows the nodes used for the reduced Matlab model. +Figure [13](#orgc294fc5) shows the nodes used for the reduced Matlab model. The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force. The arrows at the nodes indicate the direction of forces. @@ -1087,7 +1087,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\ ## Balanced Reduction {#balanced-reduction} - + In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods. @@ -1202,14 +1202,14 @@ The **states to be kept are the states with the largest diagonal terms**. ## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model} - + -In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org1453e17)). +In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org7003388)). ### Actuator Description {#actuator-description} - + {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}} @@ -1231,9 +1231,9 @@ Since the same forces are being applied to both piezo elements, they represent t ### Ansys Model Description {#ansys-model-description} -In Figure [15](#orge94bde1) are shown the principal nodes used for the model. +In Figure [15](#org472d510) are shown the principal nodes used for the model. - + {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}} @@ -1352,11 +1352,11 @@ And we note: G = zn * Gp; ``` - + {{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}} - + {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}} @@ -1454,13 +1454,13 @@ State Space Model ### Simple mode truncation {#simple-mode-truncation} -Let's plot the frequency of the modes (Figure [18](#orga04e866)). +Let's plot the frequency of the modes (Figure [18](#org6e52a4a)). - + {{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}} - + {{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}} @@ -1529,7 +1529,7 @@ Let's sort the modes by their DC gains and plot their sorted DC gains. [dc_gain_sort, index_sort] = sort(dc_gain, 'descend'); ``` - + {{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}} @@ -1873,7 +1873,7 @@ Then, we compute the controllability and observability gramians. And we plot the diagonal terms - + {{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}} @@ -1891,7 +1891,7 @@ We use `balreal` to rank oscillatory states. [G_b, G, T, Ti] = balreal(G_m); ``` - + {{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}} @@ -2134,8 +2134,9 @@ Reduced Mass and Stiffness matrices in the physical coordinates: ``` + ## Bibliography {#bibliography} -Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press. +Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press. -Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York. +Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York. diff --git a/content/book/leach14_fundam_princ_engin_nanom.md b/content/book/leach14_fundam_princ_engin_nanom.md index 2af7db2..2c9e6d7 100644 --- a/content/book/leach14_fundam_princ_engin_nanom.md +++ b/content/book/leach14_fundam_princ_engin_nanom.md @@ -8,7 +8,7 @@ Tags : [Metrology]({{< relref "metrology" >}}) Reference -: ([Leach 2014](#orgc132434)) +: ([Leach 2014](#org023e404)) Author(s) : Leach, R. @@ -87,6 +87,7 @@ The measurement of angles is then relative. This type of angular interferometer is used to measure small angles (less than \\(10deg\\)). + ## Bibliography {#bibliography} -Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. . +Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. . diff --git a/content/book/leach18_basic_precis_engin_edition.md b/content/book/leach18_basic_precis_engin_edition.md index 397cb91..356733f 100644 --- a/content/book/leach18_basic_precis_engin_edition.md +++ b/content/book/leach18_basic_precis_engin_edition.md @@ -8,7 +8,7 @@ Tags : [Precision Engineering]({{< relref "precision_engineering" >}}) Reference -: ([Leach and Smith 2018](#org50ae2e1)) +: ([Leach and Smith 2018](#orgdc805b5)) Author(s) : Leach, R., & Smith, S. T. @@ -17,6 +17,7 @@ Year : 2018 + ## Bibliography {#bibliography} -Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press. +Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press. diff --git a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md index 5c663aa..83b8960 100644 --- a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md +++ b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md @@ -8,7 +8,7 @@ Tags : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}}) Reference -: ([Preumont 2018](#orgd83c544)) +: ([Preumont 2018](#org29acb4a)) Author(s) : Preumont, A. @@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback #### Feedback {#feedback} - + {{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}} -The principle of feedback is represented on figure [1](#orgda21dda). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance. +The principle of feedback is represented on figure [1](#orga09f785). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance. The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner. In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**. @@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi #### Feedforward {#feedforward} - + {{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}} The method relies on the availability of a **reference signal correlated to the primary disturbance**. -The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#orgf75c047). +The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org57ee378). The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized. @@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of ### The Various Steps of the Design {#the-various-steps-of-the-design} - + {{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}} -The various steps of the design of a controlled structure are shown in figure [3](#org1939c0d). +The various steps of the design of a controlled structure are shown in figure [3](#org8ea735d). The **starting point** is: @@ -154,14 +154,14 @@ If the dynamics of the sensors and actuators may significantly affect the behavi ### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget} -From the block diagram of the control system (figure [4](#orgaf01f6c)): +From the block diagram of the control system (figure [4](#orga135390)): \begin{align\*} y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\ z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w \end{align\*} - + {{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}} @@ -186,12 +186,12 @@ Even more interesting for the design is the **Cumulative Mean Square** response It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\). \\(\sigma\_z(0)\\) is then the global RMS response. -A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org7ddcf2a). +A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orge835b98). It is useful to **identify the critical modes** in a design, at which the effort should be targeted. The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration. - + {{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}} @@ -398,11 +398,11 @@ With: D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i} \end{equation} - + {{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}} -If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#orga618336)). +If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#orga21e5bb)). And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**: \\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\] @@ -441,9 +441,9 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real: \\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\] -\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgecdb253)). +\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org4ad84e0)). - + {{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}} @@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
-If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org2e6ee6b). +If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org0d5b542). - + {{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}} @@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi -By looking at figure [7](#orgecdb253), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right. +By looking at figure [7](#org4ad84e0), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right. - + {{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}} @@ -486,7 +486,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)} \end{equation} -The corresponding Bode plot is represented in figure [9](#org8e5acfb). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\). +The corresponding Bode plot is represented in figure [9](#org6f76f34). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\). In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property. @@ -508,12 +508,12 @@ Two broad categories of actuators can be distinguish: A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa. -The system consists of (see figure [10](#org5b9842b)): +The system consists of (see figure [10](#orgc872907)): - A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap - A coil which is free to move axially - + {{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}} @@ -551,9 +551,9 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed #### Proof-Mass Actuator {#proof-mass-actuator} -A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#org608f53f)). +A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#org4783db3)). - + {{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}} @@ -583,9 +583,9 @@ with: -Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org21ce10b)). +Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org1a21332)). - + {{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}} @@ -610,7 +610,7 @@ By using the two equations, we obtain: Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\). - + {{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}} @@ -619,9 +619,9 @@ Designing geophones with very low corner frequency is in general difficult. Acti ### General Electromechanical Transducer {#general-electromechanical-transducer} -The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org98492c9). +The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#orgf2af0aa). - + {{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}} @@ -646,7 +646,7 @@ With: Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals. Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained. -To do so, the bridge circuit as shown on figure [15](#org3b85763) can be used. +To do so, the bridge circuit as shown on figure [15](#orgeb5cb84) can be used. We can show that @@ -656,7 +656,7 @@ We can show that which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals. - + {{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}} @@ -664,9 +664,9 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term ### Smart Materials {#smart-materials} Smart materials have the ability to respond significantly to stimuli of different physical nature. -Figure [16](#org6279c77) lists various effects that are observed in materials in response to various inputs. +Figure [16](#org1e5bcfc) lists various effects that are observed in materials in response to various inputs. - + {{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}} @@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect -If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org8006b4a)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer: +If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#orgffdc1af)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer: \begin{equation} \begin{bmatrix}Q\\\Delta\end{bmatrix} @@ -782,7 +782,7 @@ where - \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\)) - \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\)) - + {{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}} @@ -802,7 +802,7 @@ Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain #### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer} -Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org7c30411). +Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org890e9f3). The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is @@ -810,7 +810,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\ dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta \end{equation} - + {{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}} @@ -844,10 +844,10 @@ The ratio between the remaining stored energy and the initial stored energy is #### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer} -Consider the system of figure [19](#org5060008), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\). +Consider the system of figure [19](#org87aa6cd), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\). The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\). - + {{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}} @@ -866,9 +866,9 @@ And one can see that \frac{z^2 - p^2}{z^2} = k^2 \end{equation} -Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org7f3b3bf)). +Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org12f8fb9)). - + {{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}} @@ -1566,7 +1566,7 @@ Their design requires a model of the structure, and there is usually a trade-off When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC. -The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org62e1395). +The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgca40454). The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages: @@ -1574,7 +1574,7 @@ This approach has the following advantages: - The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin) - The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin) - + {{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}} @@ -1816,6 +1816,7 @@ This approach has the following advantages: ### Problems {#problems} + ## Bibliography {#bibliography} -Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. . +Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. . diff --git a/content/book/skogestad07_multiv_feedb_contr.md b/content/book/skogestad07_multiv_feedb_contr.md index 0fcd715..5f89b55 100644 --- a/content/book/skogestad07_multiv_feedb_contr.md +++ b/content/book/skogestad07_multiv_feedb_contr.md @@ -54,7 +54,7 @@ draft = false ## Introduction {#introduction} - + ### The Process of Control System Design {#the-process-of-control-system-design} @@ -231,7 +231,7 @@ Notations used throughout this note are summarized in tables [1](#table--tab:not ## Classical Feedback Control {#classical-feedback-control} - + ### Frequency Response {#frequency-response} @@ -273,14 +273,14 @@ We note \\(N(\w\_0) = \left( \frac{d\ln{|G(j\w)|}}{d\ln{\w}} \right)\_{\w=\w\_0} #### One Degree-of-Freedom Controller {#one-degree-of-freedom-controller} -The simple one degree-of-freedom controller negative feedback structure is represented in Fig. [1](#orgc915299). +The simple one degree-of-freedom controller negative feedback structure is represented in Fig. [1](#orgf7f1eea). The input to the controller \\(K(s)\\) is \\(r-y\_m\\) where \\(y\_m = y+n\\) is the measured output and \\(n\\) is the measurement noise. Thus, the input to the plant is \\(u = K(s) (r-y-n)\\). The objective of control is to manipulate \\(u\\) (design \\(K\\)) such that the control error \\(e\\) remains small in spite of disturbances \\(d\\). The control error is defined as \\(e = y-r\\). - + {{< figure src="/ox-hugo/skogestad07_classical_feedback_alt.png" caption="Figure 1: Configuration for one degree-of-freedom control" >}} @@ -599,18 +599,18 @@ For reference tracking, we typically want the controller to look like \\(\frac{1 We cannot achieve both of these simultaneously with a single feedback controller. -The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller (Fig. [2](#orgcad0d3b)), rather than operating on their difference \\(r - y\_m\\). +The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller (Fig. [2](#org8614c12)), rather than operating on their difference \\(r - y\_m\\). - + {{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_alt.png" caption="Figure 2: 2 degrees-of-freedom control architecture" >}} -The controller can be slit into two separate blocks (Fig. [3](#org75fd726)): +The controller can be slit into two separate blocks (Fig. [3](#org49e93e8)): - the **feedback controller** \\(K\_y\\) that is used to **reduce the effect of uncertainty** (disturbances and model errors) - the **prefilter** \\(K\_r\\) that **shapes the commands** \\(r\\) to improve tracking performance - + {{< figure src="/ox-hugo/skogestad07_classical_feedback_sep.png" caption="Figure 3: 2 degrees-of-freedom control architecture with two separate blocs" >}} @@ -681,7 +681,7 @@ Which can be expressed as an \\(\mathcal{H}\_\infty\\): W\_P(s) = \frac{s/M + \w\_B^\*}{s + \w\_B^\* A} \end{equation\*} -With (see Fig. [4](#orge64d111)): +With (see Fig. [4](#org3e8c784)): - \\(M\\): maximum magnitude of \\(\abs{S}\\) - \\(\w\_B\\): crossover frequency @@ -689,7 +689,7 @@ With (see Fig. [4](#orge64d111)): - + {{< figure src="/ox-hugo/skogestad07_weight_first_order.png" caption="Figure 4: Inverse of performance weight" >}} @@ -723,7 +723,7 @@ After selecting the form of \\(N\\) and the weights, the \\(\hinf\\) optimal con ## Introduction to Multivariable Control {#introduction-to-multivariable-control} - + ### Introduction {#introduction} @@ -760,13 +760,13 @@ The main rule for evaluating transfer functions is the **MIMO Rule**: Start from #### Negative Feedback Control Systems {#negative-feedback-control-systems} -For negative feedback system (Fig. [5](#orgd083aa9)), we define \\(L\\) to be the loop transfer function as seen when breaking the loop at the **output** of the plant: +For negative feedback system (Fig. [5](#orgd170e54)), we define \\(L\\) to be the loop transfer function as seen when breaking the loop at the **output** of the plant: - \\(L = G K\\) - \\(S \triangleq (I + L)^{-1}\\) is the transfer function from \\(d\_1\\) to \\(y\\) - \\(T \triangleq L(I + L)^{-1}\\) is the transfer function from \\(r\\) to \\(y\\) - + {{< figure src="/ox-hugo/skogestad07_classical_feedback_bis.png" caption="Figure 5: Conventional negative feedback control system" >}} @@ -1124,9 +1124,9 @@ The **structured singular value** \\(\mu\\) is a tool for analyzing the effects ### General Control Problem Formulation {#general-control-problem-formulation} -The general control problem formulation is represented in Fig. [6](#orgdd7d1c0) (introduced in ([Doyle 1983](#orgf1c2fab))). +The general control problem formulation is represented in Fig. [6](#orgf26fedd) (introduced in ([Doyle 1983](#org4d36284))). - + {{< figure src="/ox-hugo/skogestad07_general_control_names.png" caption="Figure 6: General control configuration" >}} @@ -1157,13 +1157,13 @@ Then we have to break all the "loops" entering and exiting the controller \\(K\\ #### Controller Design: Including Weights in \\(P\\) {#controller-design-including-weights-in--p} -In order to get a meaningful controller synthesis problem, for example in terms of the \\(\hinf\\) norms, we generally have to include the weights \\(W\_z\\) and \\(W\_w\\) in the generalized plant \\(P\\) (Fig. [7](#org155d538)). +In order to get a meaningful controller synthesis problem, for example in terms of the \\(\hinf\\) norms, we generally have to include the weights \\(W\_z\\) and \\(W\_w\\) in the generalized plant \\(P\\) (Fig. [7](#orge0e26e7)). We consider: - The weighted or normalized exogenous inputs \\(w\\) (where \\(\tilde{w} = W\_w w\\) consists of the "physical" signals entering the system) - The weighted or normalized controlled outputs \\(z = W\_z \tilde{z}\\) (where \\(\tilde{z}\\) often consists of the control error \\(y-r\\) and the manipulated input \\(u\\)) - + {{< figure src="/ox-hugo/skogestad07_general_plant_weights.png" caption="Figure 7: General Weighted Plant" >}} @@ -1216,9 +1216,9 @@ where \\(F\_l(P, K)\\) denotes a **lower linear fractional transformation** (LFT #### A General Control Configuration Including Model Uncertainty {#a-general-control-configuration-including-model-uncertainty} -The general control configuration may be extended to include model uncertainty as shown in Fig. [8](#orgc9c09f7). +The general control configuration may be extended to include model uncertainty as shown in Fig. [8](#orgda7871a). - + {{< figure src="/ox-hugo/skogestad07_general_control_Mdelta.png" caption="Figure 8: General control configuration for the case with model uncertainty" >}} @@ -1246,7 +1246,7 @@ MIMO systems are often **more sensitive to uncertainty** than SISO systems. ## Elements of Linear System Theory {#elements-of-linear-system-theory} - + ### System Descriptions {#system-descriptions} @@ -1619,18 +1619,18 @@ RHP-zeros therefore imply high gain instability. ### Internal Stability of Feedback Systems {#internal-stability-of-feedback-systems} - + {{< figure src="/ox-hugo/skogestad07_classical_feedback_stability.png" caption="Figure 9: Block diagram used to check internal stability" >}} -Assume that the components \\(G\\) and \\(K\\) contain no unstable hidden modes. Then the feedback system in Fig. [9](#org59e6d83) is **internally stable** if and only if all four closed-loop transfer matrices are stable. +Assume that the components \\(G\\) and \\(K\\) contain no unstable hidden modes. Then the feedback system in Fig. [9](#orgc94e8c5) is **internally stable** if and only if all four closed-loop transfer matrices are stable. \begin{align\*} &(I+KG)^{-1} & -K&(I+GK)^{-1} \\\\\\ G&(I+KG)^{-1} & &(I+GK)^{-1} \end{align\*} -Assume there are no RHP pole-zero cancellations between \\(G(s)\\) and \\(K(s)\\), the feedback system in Fig. [9](#org59e6d83) is internally stable if and only if **one** of the four closed-loop transfer function matrices is stable. +Assume there are no RHP pole-zero cancellations between \\(G(s)\\) and \\(K(s)\\), the feedback system in Fig. [9](#orgc94e8c5) is internally stable if and only if **one** of the four closed-loop transfer function matrices is stable. ### Stabilizing Controllers {#stabilizing-controllers} @@ -1797,7 +1797,7 @@ It may be shown that the Hankel norm is equal to \\(\left\\|G(s)\right\\|\_H = \ ## Limitations on Performance in SISO Systems {#limitations-on-performance-in-siso-systems} - + ### Input-Output Controllability {#input-output-controllability} @@ -2283,11 +2283,11 @@ Uncertainty in the crossover frequency region can result in poor performance and ### Summary: Controllability Analysis with Feedback Control {#summary-controllability-analysis-with-feedback-control} - + {{< figure src="/ox-hugo/skogestad07_classical_feedback_meas.png" caption="Figure 10: Feedback control system" >}} -Consider the control system in Fig. [10](#orgd57ead2). +Consider the control system in Fig. [10](#org506ba18). Here \\(G\_m(s)\\) denotes the measurement transfer function and we assume \\(G\_m(0) = 1\\) (perfect steady-state measurement).
@@ -2317,7 +2317,7 @@ Sometimes, the disturbances are so large that we hit input saturation or the req \abs{G\_d(j\w)} < 1 \quad \forall \w \geq \w\_c \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_margin_requirements.png" caption="Figure 11: Illustration of controllability requirements" >}} @@ -2339,7 +2339,7 @@ The rules may be used to **determine whether or not a given plant is controllabl ## Limitations on Performance in MIMO Systems {#limitations-on-performance-in-mimo-systems} - + ### Introduction {#introduction} @@ -2719,13 +2719,13 @@ The issues are the same for SISO and MIMO systems, however, with MIMO systems th In practice, the difference between the true perturbed plant \\(G^\prime\\) and the plant model \\(G\\) is caused by a number of different sources. We here focus on input and output uncertainty. -In multiplicative form, the input and output uncertainties are given by (see Fig. [12](#orge6b79ad)): +In multiplicative form, the input and output uncertainties are given by (see Fig. [12](#org2e2a888)): \begin{equation\*} G^\prime = (I + E\_O) G (I + E\_I) \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_input_output_uncertainty.png" caption="Figure 12: Plant with multiplicative input and output uncertainty" >}} @@ -2869,7 +2869,7 @@ However, the situation is usually the opposite with model uncertainty because fo ## Uncertainty and Robustness for SISO Systems {#uncertainty-and-robustness-for-siso-systems} - + ### Introduction to Robustness {#introduction-to-robustness} @@ -2943,11 +2943,11 @@ In most cases, we prefer to lump the uncertainty into a **multiplicative uncerta G\_p(s) = G(s)(1 + w\_I(s)\Delta\_I(s)); \quad \abs{\Delta\_I(j\w)} \le 1 \, \forall\w \end{equation\*} -which may be represented by the diagram in Fig. [13](#org42ae66c). +which may be represented by the diagram in Fig. [13](#orgf804642).
- + {{< figure src="/ox-hugo/skogestad07_input_uncertainty_set.png" caption="Figure 13: Plant with multiplicative uncertainty" >}} @@ -3012,7 +3012,7 @@ This is of course conservative as it introduces possible plants that are not pre #### Uncertain Regions {#uncertain-regions} -To illustrate how parametric uncertainty translate into frequency domain uncertainty, consider in Fig. [14](#org71afec9) the Nyquist plots generated by the following set of plants +To illustrate how parametric uncertainty translate into frequency domain uncertainty, consider in Fig. [14](#org56df3ef) the Nyquist plots generated by the following set of plants \begin{equation\*} G\_p(s) = \frac{k}{\tau s + 1} e^{-\theta s}, \quad 2 \le k, \theta, \tau \le 3 @@ -3022,7 +3022,7 @@ To illustrate how parametric uncertainty translate into frequency domain uncerta In general, these uncertain regions have complicated shapes and complex mathematical descriptions - **Step 2**. We therefore approximate such complex regions as discs, resulting in a **complex additive uncertainty description** - + {{< figure src="/ox-hugo/skogestad07_uncertainty_region.png" caption="Figure 14: Uncertainty regions of the Nyquist plot at given frequencies" >}} @@ -3041,11 +3041,11 @@ The disc-shaped regions may be generated by **additive** complex norm-bounded pe \end{aligned} \end{equation} -At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in Fig. [15](#org80856e5). +At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped region with radius 1 centered at 0, so \\(G(j\w) + w\_A(j\w)\Delta\_A(j\w)\\) generates at each frequency a disc-shapes region of radius \\(\abs{w\_A(j\w)}\\) centered at \\(G(j\w)\\) as shown in Fig. [15](#org53103ab). - + {{< figure src="/ox-hugo/skogestad07_uncertainty_disc_generated.png" caption="Figure 15: Disc-shaped uncertainty regions generated by complex additive uncertainty" >}} @@ -3119,12 +3119,12 @@ To simplify subsequent controller design, we select a delay-free nominal model \end{equation\*} To obtain \\(l\_I(\w)\\), we consider three values (2, 2.5 and 3) for each of the three parameters (\\(k, \theta, \tau\\)). -The corresponding relative errors \\(\abs{\frac{G\_p-G}{G}}\\) are shown as functions of frequency for the \\(3^3 = 27\\) resulting \\(G\_p\\) (Fig. [16](#org472e7e8)). +The corresponding relative errors \\(\abs{\frac{G\_p-G}{G}}\\) are shown as functions of frequency for the \\(3^3 = 27\\) resulting \\(G\_p\\) (Fig. [16](#org7f7b0b3)). To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_I(j\w)}\\) lies above all the dotted lines. - + {{< figure src="/ox-hugo/skogestad07_uncertainty_weight.png" caption="Figure 16: Relative error for 27 combinations of \\(k,\ \tau\\) and \\(\theta\\). Solid and dashed lines: two weights \\(\abs{w\_I}\\)" >}} @@ -3167,26 +3167,26 @@ The magnitude of the relative uncertainty caused by neglecting the dynamics in \ ##### Neglected delay {#neglected-delay} -Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{max}}\\). We want to represent \\(G\_p(s) = G\_0(s)e^{-\theta\_p s}\\) by a delay-free plant \\(G\_0(s)\\) and multiplicative uncertainty. Let first consider the maximum delay, for which the relative error \\(\abs{1 - e^{-j \w \theta\_{\text{max}}}}\\) is shown as a function of frequency (Fig. [17](#org03548f2)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then: +Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{max}}\\). We want to represent \\(G\_p(s) = G\_0(s)e^{-\theta\_p s}\\) by a delay-free plant \\(G\_0(s)\\) and multiplicative uncertainty. Let first consider the maximum delay, for which the relative error \\(\abs{1 - e^{-j \w \theta\_{\text{max}}}}\\) is shown as a function of frequency (Fig. [17](#org64b7e6c)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then: \begin{equation\*} l\_I(\w) = \begin{cases} \abs{1 - e^{-j\w\theta\_{\text{max}}}} & \w < \pi/\theta\_{\text{max}} \\ 2 & \w \ge \pi/\theta\_{\text{max}} \end{cases} \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_neglected_time_delay.png" caption="Figure 17: Neglected time delay" >}} ##### Neglected lag {#neglected-lag} -Let \\(f(s) = 1/(\tau\_p s + 1)\\), where \\(0 \le \tau\_p \le \tau\_{\text{max}}\\). In this case the resulting \\(l\_I(\w)\\) (Fig. [18](#org7cfcb10)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where +Let \\(f(s) = 1/(\tau\_p s + 1)\\), where \\(0 \le \tau\_p \le \tau\_{\text{max}}\\). In this case the resulting \\(l\_I(\w)\\) (Fig. [18](#org490106f)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where \begin{equation\*} w\_I(s) = \frac{\tau\_{\text{max}} s}{\tau\_{\text{max}} s + 1} \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_neglected_first_order_lag.png" caption="Figure 18: Neglected first-order lag uncertainty" >}} @@ -3206,7 +3206,7 @@ There is an exact expression, its first order approximation is w\_I(s) = \frac{(1+\frac{r\_k}{2})\theta\_{\text{max}} s + r\_k}{\frac{\theta\_{\text{max}}}{2} s + 1} \end{equation\*} -However, as shown in Fig. [19](#orgf42f7c2), the weight \\(w\_I\\) is optimistic, especially around frequencies \\(1/\theta\_{\text{max}}\\). To make sure that \\(\abs{w\_I(j\w)} \le l\_I(\w)\\), we can apply a correction factor: +However, as shown in Fig. [19](#org3160a7e), the weight \\(w\_I\\) is optimistic, especially around frequencies \\(1/\theta\_{\text{max}}\\). To make sure that \\(\abs{w\_I(j\w)} \le l\_I(\w)\\), we can apply a correction factor: \begin{equation\*} w\_I^\prime(s) = w\_I \cdot \frac{(\frac{\theta\_{\text{max}}}{2.363})^2 s^2 + 2\cdot 0.838 \cdot \frac{\theta\_{\text{max}}}{2.363} s + 1}{(\frac{\theta\_{\text{max}}}{2.363})^2 s^2 + 2\cdot 0.685 \cdot \frac{\theta\_{\text{max}}}{2.363} s + 1} @@ -3214,7 +3214,7 @@ However, as shown in Fig. [19](#orgf42f7c2), the weight \\(w\_I\\) is optim It is suggested to start with the simple weight and then if needed, to try the higher order weight. - + {{< figure src="/ox-hugo/skogestad07_lag_delay_uncertainty.png" caption="Figure 19: Multiplicative weight for gain and delay uncertainty" >}} @@ -3243,7 +3243,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th #### RS with Multiplicative Uncertainty {#rs-with-multiplicative-uncertainty} -We want to determine the stability of the uncertain feedback system in Fig. [20](#org1e3fbdd) where there is multiplicative uncertainty of magnitude \\(\abs{w\_I(j\w)}\\). +We want to determine the stability of the uncertain feedback system in Fig. [20](#orgf2f99e4) where there is multiplicative uncertainty of magnitude \\(\abs{w\_I(j\w)}\\). The loop transfer function becomes \begin{equation\*} @@ -3258,14 +3258,14 @@ We use the Nyquist stability condition to test for robust stability of the close &\Longleftrightarrow \quad L\_p \ \text{should not encircle -1}, \ \forall L\_p \end{align\*} - + {{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback.png" caption="Figure 20: Feedback system with multiplicative uncertainty" >}} ##### Graphical derivation of RS-condition {#graphical-derivation-of-rs-condition} -Consider the Nyquist plot of \\(L\_p\\) as shown in Fig. [21](#orgc973f2e). \\(\abs{1+L}\\) is the distance from the point \\(-1\\) to the center of the disc representing \\(L\_p\\) and \\(\abs{w\_I L}\\) is the radius of the disc. +Consider the Nyquist plot of \\(L\_p\\) as shown in Fig. [21](#orgf665b41). \\(\abs{1+L}\\) is the distance from the point \\(-1\\) to the center of the disc representing \\(L\_p\\) and \\(\abs{w\_I L}\\) is the radius of the disc. Encirclements are avoided if none of the discs cover \\(-1\\), and we get: \begin{align\*} @@ -3274,7 +3274,7 @@ Encirclements are avoided if none of the discs cover \\(-1\\), and we get: &\Leftrightarrow \quad \abs{w\_I T} < 1, \ \forall\w \\\\\\ \end{align\*} - + {{< figure src="/ox-hugo/skogestad07_nyquist_uncertainty.png" caption="Figure 21: Nyquist plot of \\(L\_p\\) for robust stability" >}} @@ -3313,13 +3313,13 @@ And we obtain the same condition as before. #### RS with Inverse Multiplicative Uncertainty {#rs-with-inverse-multiplicative-uncertainty} -We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty (Fig. [22](#org8e1e6e1)) in which +We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty (Fig. [22](#orgd01a6c1)) in which \begin{equation\*} G\_p = G(1 + w\_{iI}(s) \Delta\_{iI})^{-1} \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_inverse_uncertainty_set.png" caption="Figure 22: Feedback system with inverse multiplicative uncertainty" >}} @@ -3369,9 +3369,9 @@ The condition for **nominal performance** when considering performance in terms Now \\(\abs{1 + L}\\) represents at each frequency the distance of \\(L(j\omega)\\) from the point \\(-1\\) in the Nyquist plot, so \\(L(j\omega)\\) must be at least a distance of \\(\abs{w\_P(j\omega)}\\) from \\(-1\\). -This is illustrated graphically in Fig. [23](#org7634047). +This is illustrated graphically in Fig. [23](#org4671df1). - + {{< figure src="/ox-hugo/skogestad07_nyquist_performance_condition.png" caption="Figure 23: Nyquist plot illustration of the nominal performance condition \\(\abs{w\_P} < \abs{1 + L}\\)" >}} @@ -3392,21 +3392,21 @@ For robust performance, we require the performance condition to be satisfied for -Let's consider the case of multiplicative uncertainty as shown on Fig. [24](#orgf1efa0c). +Let's consider the case of multiplicative uncertainty as shown on Fig. [24](#orgf6bf053). The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \Delta\_I\\) and the set of possible loop transfer functions is \begin{equation\*} L\_p = G\_p K = L (1 + w\_I \Delta\_I) = L + w\_I L \Delta\_I \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight_bis.png" caption="Figure 24: Diagram for robust performance with multiplicative uncertainty" >}} ##### Graphical derivation of RP-condition {#graphical-derivation-of-rp-condition} -As illustrated on Fig. [23](#org7634047), we must required that all possible \\(L\_p(j\omega)\\) stay outside a disk of radius \\(\abs{w\_P(j\omega)}\\) centered on \\(-1\\). +As illustrated on Fig. [23](#org4671df1), we must required that all possible \\(L\_p(j\omega)\\) stay outside a disk of radius \\(\abs{w\_P(j\omega)}\\) centered on \\(-1\\). Since \\(L\_p\\) at each frequency stays within a disk of radius \\(|w\_I(j\omega) L(j\omega)|\\) centered on \\(L(j\omega)\\), the condition for RP becomes: \begin{align\*} @@ -3606,9 +3606,9 @@ In the transfer function form: with \\(\Phi(s) \triangleq (sI - A)^{-1}\\). -This is illustrated in the block diagram of Fig. [25](#org11db4af), which is in the form of an inverse additive perturbation. +This is illustrated in the block diagram of Fig. [25](#orgfa5a97d), which is in the form of an inverse additive perturbation. - + {{< figure src="/ox-hugo/skogestad07_uncertainty_state_a_matrix.png" caption="Figure 25: Uncertainty in state space A-matrix" >}} @@ -3626,7 +3626,7 @@ We also derived a condition for robust performance with multiplicative uncertain ## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis} - + ### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty} @@ -3644,15 +3644,15 @@ The starting point for our robustness analysis is a system representation in whi where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g. input uncertainty \\(\Delta\_I\\) or parametric uncertainty \\(\delta\_i\\). -If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig. [26](#orgee753cc). This form is useful for controller synthesis. +If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig. [26](#orgfea2ba4). This form is useful for controller synthesis. - + {{< figure src="/ox-hugo/skogestad07_general_control_delta.png" caption="Figure 26: General control configuration used for controller synthesis" >}} -If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in Fig. [27](#orgb58d779). +If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in Fig. [27](#org5fa9870). - + {{< figure src="/ox-hugo/skogestad07_general_control_Ndelta.png" caption="Figure 27: \\(N\Delta\text{-structure}\\) for robust performance analysis" >}} @@ -3670,9 +3670,9 @@ Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\), &\triangleq N\_{22} + N\_{21} \Delta (I - N\_{11} \Delta)^{-1} N\_{12} \end{align\*} -To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig. [28](#org19c7321) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations. +To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig. [28](#orgf7c6f1a) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations. - + {{< figure src="/ox-hugo/skogestad07_general_control_Mdelta_bis.png" caption="Figure 28: \\(M\Delta\text{-structure}\\) for robust stability analysis" >}} @@ -3732,7 +3732,7 @@ Three common forms of **feedforward unstructured uncertainty** are shown Fig.&nb | ![](/ox-hugo/skogestad07_additive_uncertainty.png) | ![](/ox-hugo/skogestad07_input_uncertainty.png) | ![](/ox-hugo/skogestad07_output_uncertainty.png) | |----------------------------------------------------|----------------------------------------------------------|-----------------------------------------------------------| -| Additive uncertainty | Multiplicative input uncertainty | Multiplicative output uncertainty | +| Additive uncertainty | Multiplicative input uncertainty | Multiplicative output uncertainty | In Fig. [5](#table--fig:feedback-uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty. @@ -3757,7 +3757,7 @@ In Fig. [5](#table--fig:feedback-uncertainty), three **feedback or inverse | ![](/ox-hugo/skogestad07_inv_additive_uncertainty.png) | ![](/ox-hugo/skogestad07_inv_input_uncertainty.png) | ![](/ox-hugo/skogestad07_inv_output_uncertainty.png) | |--------------------------------------------------------|------------------------------------------------------------------|-------------------------------------------------------------------| -| Inverse additive uncertainty | Inverse multiplicative input uncertainty | Inverse multiplicative output uncertainty | +| Inverse additive uncertainty | Inverse multiplicative input uncertainty | Inverse multiplicative output uncertainty | ##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation} @@ -3853,10 +3853,10 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th ### Obtaining \\(P\\), \\(N\\) and \\(M\\) {#obtaining--p----n--and--m} -Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig. [29](#org8b37adb). +Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig. [29](#org3660233). \\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight. - + {{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="Figure 29: System with multiplicative input uncertainty and performance measured at the output" >}} @@ -4040,7 +4040,7 @@ In order to get tighter condition we must use a tighter uncertainty description Robust stability bound in terms of the \\(\hinf\\) norm (\\(\text{RS}\Leftrightarrow\hnorm{M}<1\\)) are in general only tight when there is a single full perturbation block. An "exception" to this is when the uncertainty blocks enter or exit from the same location in the block diagram, because they can then be stacked on top of each other or side-by-side, in an overall \\(\Delta\\) which is then full matrix. -One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in Fig. [30](#orgd5c7823), for which the set of plants is +One important uncertainty description that falls into this category is the **coprime uncertainty description** shown in Fig. [30](#org46554f9), for which the set of plants is \begin{equation\*} G\_p = (M\_l + \Delta\_M)^{-1}(Nl + \Delta\_N), \quad \hnorm{[\Delta\_N, \ \Delta\_N]} \le \epsilon @@ -4050,7 +4050,7 @@ Where \\(G = M\_l^{-1} N\_l\\) is a left coprime factorization of the nominal pl This uncertainty description is surprisingly **general**, it allows both zeros and poles to cross into the right-half plane, and has proven to be very useful in applications. - + {{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="Figure 30: Coprime Uncertainty" >}} @@ -4096,10 +4096,10 @@ To this effect, introduce the block-diagonal scaling matrix where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same dimension as the \\(i\\)'th perturbation block \\(\Delta\_i\\). -Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in Fig. [31](#org73da59c). +Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in Fig. [31](#org9d2ee79). This clearly has no effect on stability. - + {{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="Figure 31: Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}} @@ -4397,7 +4397,7 @@ Note that \\(\mu\\) underestimate how bad or good the actual worst case performa ### Application: RP with Input Uncertainty {#application-rp-with-input-uncertainty} -We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity (Fig. [29](#org8b37adb)). +We will now consider in some detail the case of multiplicative input uncertainty with performance defined in terms of weighted sensitivity (Fig. [29](#org3660233)). The performance requirement is then @@ -4511,9 +4511,9 @@ with the decoupling controller we have: \overline{\sigma}(N\_{22}) = \overline{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right| \end{equation\*} -and we see from Fig. [32](#org2a07dc9) that the NP-condition is satisfied. +and we see from Fig. [32](#org854aa7f) that the NP-condition is satisfied. - + {{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="Figure 32: \\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}} @@ -4526,7 +4526,7 @@ In this case \\(w\_I T\_I = w\_I T\\) is a scalar times the identity matrix: \mu\_{\Delta\_I}(w\_I T\_I) = |w\_I t| = \left|0.2 \frac{5s + 1}{(0.5s + 1)(1.43s + 1)}\right| \end{equation\*} -and we see from Fig. [32](#org2a07dc9) that RS is satisfied. +and we see from Fig. [32](#org854aa7f) that RS is satisfied. The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may increase the uncertainty by a factor of \\(1/0.53 = 1.89\\) before the worst case uncertainty yields instability. @@ -4534,7 +4534,7 @@ The peak value of \\(\mu\_{\Delta\_I}(M)\\) is \\(0.53\\) meaning that we may in ##### RP {#rp} Although the system has good robustness margins and excellent nominal performance, the robust performance is poor. -This is shown in Fig. [32](#org2a07dc9) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\). +This is shown in Fig. [32](#org854aa7f) where the \\(\mu\text{-curve}\\) for RP was computed numerically using \\(\mu\_{\hat{\Delta}}(N)\\), with \\(\hat{\Delta} = \text{diag}\\{\Delta\_I, \Delta\_P\\}\\) and \\(\Delta\_I = \text{diag}\\{\delta\_1, \delta\_2\\}\\). The peak value is close to 6, meaning that even with 6 times less uncertainty, the weighted sensitivity will be about 6 times larger than what we require. @@ -4672,9 +4672,9 @@ The latter is an attempt to "flatten out" \\(\mu\\). #### Example: \\(\mu\text{-synthesis}\\) with DK-iteration {#example--mu-text-synthesis--with-dk-iteration} For simplicity, we will consider again the case of multiplicative uncertainty and performance defined in terms of weighted sensitivity. -The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in Fig. [33](#orgaea3d40). +The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in Fig. [33](#org909e969). - + {{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 33: Uncertainty and performance weights" >}} @@ -4688,8 +4688,8 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex - Iteration No. 1. Step 1: with the initial scalings, the \\(\mathcal{H}\_\infty\\) synthesis produced a 6 state controller (2 states from the plant model and 2 from each of the weights). - Step 2: the upper \\(\mu\text{-bound}\\) is shown in Fig. [34](#org09f3acc). - Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in Fig. [35](#org6506673) + Step 2: the upper \\(\mu\text{-bound}\\) is shown in Fig. [34](#orge9b0734). + Step 3: the frequency dependent \\(d\_1(\omega)\\) and \\(d\_2(\omega)\\) from step 2 are fitted using a 4th order transfer function shown in Fig. [35](#orgfe7e6eb) - Iteration No. 2. Step 1: with the 8 state scalings \\(D^1(s)\\), the \\(\mathcal{H}\_\infty\\) synthesis gives a 22 state controller. Step 2: This controller gives a peak value of \\(\mu\\) of \\(1.02\\). @@ -4697,25 +4697,25 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex - Iteration No. 3. Step 1: The \\(\mathcal{H}\_\infty\\) norm is only slightly reduced. We thus decide the stop the iterations. - + {{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 34: Change in \\(\mu\\) during DK-iteration" >}} - + {{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 35: Change in D-scale \\(d\_1\\) during DK-iteration" >}} -The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in Fig. [36](#org958a788). +The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\\) are shown in Fig. [36](#org1ee37e3). The objectives of RS and NP are easily satisfied. The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\overline{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants. - + {{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="Figure 36: \\(mu\text{-plots}\\) with \\(\mu\\) \"optimal\" controller \\(K\_3\\)" >}} -To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in Fig. [37](#org11d1180). +To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in Fig. [37](#org973f563). - + {{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 37: Perturbed sensitivity functions \\(\overline{\sigma}(S^\prime)\\) using \\(\mu\\) \"optimal\" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}} @@ -4782,7 +4782,7 @@ If resulting control performance is not satisfactory, one may switch to the seco ## Controller Design {#controller-design} - + ### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design} @@ -4792,7 +4792,7 @@ By multivariable transfer function shaping, therefore, we mean the shaping of th The classical loop-shaping ideas can be further generalized to MIMO systems by considering the singular values. -Consider the one degree-of-freedom system as shown in Fig. [38](#org659c83b). +Consider the one degree-of-freedom system as shown in Fig. [38](#orgeae3b60). We have the following important relationships: \begin{align} @@ -4800,7 +4800,7 @@ We have the following important relationships: u(s) &= K(s) S(s) \big(r(s) - n(s) - d(s) \big) \end{align} - + {{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 38: One degree-of-freedom feedback configuration" >}} @@ -4848,9 +4848,9 @@ Thus, over specified frequency ranges, it is relatively easy to approximate the -Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in Fig. [39](#org3f1452a). +Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in Fig. [39](#orgba303b8). - + {{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="Figure 39: Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}} @@ -4909,9 +4909,9 @@ The optimal state estimate is given by a **Kalman filter**. The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}\\) to give \\(u(t) = -K\_r \hat{x}\\). -We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig. [40](#orgfd91a23): the optimal state feedback and the optimal state estimator (the Kalman filter). +We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig. [40](#orgf5fc4a4): the optimal state feedback and the optimal state estimator (the Kalman filter). - + {{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 40: The separation theorem" >}} @@ -4943,7 +4943,7 @@ and \\(X\\) is the unique positive-semi definite solution of the algebraic Ricca
-The **Kalman filter** has the structure of an ordinary state-estimator, as shown on Fig. [41](#org0d33287), with: +The **Kalman filter** has the structure of an ordinary state-estimator, as shown on Fig. [41](#org070ef62), with: \begin{equation} \label{eq:kalman\_filter\_structure} \dot{\hat{x}} = A\hat{x} + Bu + K\_f(y-C\hat{x}) @@ -4963,11 +4963,11 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
- + {{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="Figure 41: The LQG controller and noisy plant" >}} -The structure of the LQG controller is illustrated in Fig. [41](#org0d33287), its transfer function from \\(y\\) to \\(u\\) is given by +The structure of the LQG controller is illustrated in Fig. [41](#org070ef62), its transfer function from \\(y\\) to \\(u\\) is given by \begin{align\*} L\_{\text{LQG}}(s) &= \left[ \begin{array}{c|c} @@ -4982,9 +4982,9 @@ The structure of the LQG controller is illustrated in Fig. [41](#org0d33287 It has the same degree (number of poles) as the plant.
-For the LQG-controller, as shown on Fig. [41](#org0d33287), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in Fig. [42](#orgeb465c8). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states. +For the LQG-controller, as shown on Fig. [41](#org070ef62), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in Fig. [42](#orgddbf177). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states. - + {{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 42: LQG controller with integral action and reference input" >}} @@ -4997,18 +4997,18 @@ Their main limitation is that they can only be applied to minimum phase plants. ### \\(\htwo\\) and \\(\hinf\\) Control {#htwo--and--hinf--control} - + #### General Control Problem Formulation {#general-control-problem-formulation} - + There are many ways in which feedback design problems can be cast as \\(\htwo\\) and \\(\hinf\\) optimization problems. It is very useful therefore to have a **standard problem formulation** into which any particular problem may be manipulated. -Such a general formulation is afforded by the general configuration shown in Fig. [43](#orgdd7aac9). +Such a general formulation is afforded by the general configuration shown in Fig. [43](#org0aa2c5b). - + {{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 43: General control configuration" >}} @@ -5188,7 +5188,7 @@ Then the LQG cost function is #### \\(\hinf\\) Optimal Control {#hinf--optimal-control} -With reference to the general control configuration on Fig. [43](#orgdd7aac9), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize +With reference to the general control configuration on Fig. [43](#org0aa2c5b), the standard \\(\hinf\\) optimal control problem is to find all stabilizing controllers \\(K\\) which minimize \begin{equation\*} \hnorm{F\_l(P, K)} = \max\_{\omega} \maxsv\big(F\_l(P, K)(j\omega)\big) @@ -5300,7 +5300,7 @@ In general, the scalar weighting functions \\(w\_1(s)\\) and \\(w\_2(s)\\) can b This can be useful for **systems with channels of quite different bandwidths**. In that case, **diagonal weights are recommended** as anything more complicated is usually not worth the effort.
-To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in Fig. [44](#org09986bf). +To see how this mixed sensitivity problem can be formulated in the general setting, we can imagine the disturbance \\(d\\) as a single exogenous input and define and error signal \\(z = [z\_1^T\ z\_2^T]^T\\), where \\(z\_1 = W\_1 y\\) and \\(z\_2 = -W\_2 u\\) as illustrated in Fig. [44](#orgc7fb577). We can then see that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\) as required. The elements of the generalized plant are @@ -5317,16 +5317,16 @@ The elements of the generalized plant are \end{array} \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 44: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}} -Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of Fig. [45](#org33510ad). +Another interpretation can be put on the \\(S/KS\\) mixed-sensitivity optimization as shown in the standard control configuration of Fig. [45](#org2126ed4). Here we consider a tracking problem. The exogenous input is a reference command \\(r\\), and the error signals are \\(z\_1 = -W\_1 e = W\_1 (r-y)\\) and \\(z\_2 = W\_2 u\\). -As the regulation problem of Fig. [44](#org09986bf), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\). +As the regulation problem of Fig. [44](#orgc7fb577), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\). - + {{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 45: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}} @@ -5339,7 +5339,7 @@ Another useful mixed sensitivity optimization problem, is to find a stabilizing The ability to shape \\(T\\) is desirable for tracking problems and noise attenuation. It is also important for robust stability with respect to multiplicative perturbations at the plant output. -The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in Fig. [46](#org88af2c4). +The \\(S/T\\) mixed-sensitivity minimization problem can be put into the standard control configuration as shown in Fig. [46](#org9fe7aaa). The elements of the generalized plant are @@ -5356,7 +5356,7 @@ The elements of the generalized plant are \end{array} \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 46: \\(S/T\\) mixed-sensitivity optimization in standard form" >}} @@ -5382,9 +5382,9 @@ The focus of attention has moved to the size of signals and away from the size a Weights are used to describe the expected or known frequency content of exogenous signals and the desired frequency content of error signals. -Weights are also used if a perturbation is used to model uncertainty, as in Fig. [47](#orgdd7d2be), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\). +Weights are also used if a perturbation is used to model uncertainty, as in Fig. [47](#orgf05c9aa), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\). - + {{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="Figure 47: Multiplicative dynamic uncertainty model" >}} @@ -5393,9 +5393,9 @@ As we have seen, the weights \\(Q\\) and \\(R\\) are constant, but LQG can be ge When we consider a system's response to persistent sinusoidal signals of varying frequency, or when we consider the induced 2-norm between the exogenous input signals and the error signals, we are required to minimize the \\(\hinf\\) norm. In the absence of model uncertainty, there does not appear to be an overwhelming case for using the \\(\hinf\\) norm rather than the more traditional \\(\htwo\\) norm. -However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in Fig. [47](#orgdd7d2be).
+However, when uncertainty is addressed, as it always should be, \\(\hinf\\) is clearly the more **natural approach** using component uncertainty models as in Fig. [47](#orgf05c9aa).
-A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of Fig. [48](#org64a744b). +A typical problem using the signal-based approach to \\(\hinf\\) control is illustrated in the interconnection diagram of Fig. [48](#org3a823ee). \\(G\\) and \\(G\_d\\) are nominal models of the plant and disturbance dynamics, and \\(K\\) is the controller to be designed. The weights \\(W\_d\\), \\(W\_r\\), and \\(W\_n\\) may be constant or dynamic and describe the relative importance and/or the frequency content of the disturbance, set points and noise signals. The weight \\(W\_\text{ref}\\) is a desired closed-loop transfer function between the weighted set point \\(r\_s\\) and the actual output \\(y\\). @@ -5416,11 +5416,11 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co \end{bmatrix},\ u = u \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 48: A signal-based \\(\hinf\\) control problem" >}} -Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in Fig. [49](#orgd303863). +Suppose we now introduce a multiplicative dynamic uncertainty model at the input to the plant as shown in Fig. [49](#org7c422a7). The problem we now want to solve is: find a stabilizing controller \\(K\\) such that the \\(\hinf\\) norm of the transfer function between \\(w\\) and \\(z\\) is less that 1 for all \\(\Delta\\) where \\(\hnorm{\Delta} < 1\\). We have assumed in this statement that the **signal weights have normalized the 2-norm of the exogenous input signals to unity**. This problem is a non-standard \\(\hinf\\) optimization. @@ -5430,7 +5430,7 @@ It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) proc \mu(M(j\omega)) < 1, \quad \forall\omega \end{equation\*} - + {{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 49: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}} @@ -5483,7 +5483,7 @@ The objective of robust stabilization is to stabilize not only the nominal model where \\(\epsilon > 0\\) is then the **stability margin**.
-For the perturbed feedback system of Fig. [50](#orga89c9a7), the stability property is robust if and only if the nominal feedback system is stable and +For the perturbed feedback system of Fig. [50](#org4d5c7c2), the stability property is robust if and only if the nominal feedback system is stable and \begin{equation\*} \gamma \triangleq \hnorm{\begin{bmatrix} @@ -5494,7 +5494,7 @@ For the perturbed feedback system of Fig. [50](#orga89c9a7), the stability Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bmatrix}u \cr y\end{bmatrix}\\) and \\((I-GK)^{-1}\\) is the sensitivity function for this positive feedback arrangement. - + {{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 50: \\(\hinf\\) robust stabilization problem" >}} @@ -5550,7 +5550,7 @@ It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) #### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic--hinf--loop-shaping-design-procedure} - + Robust stabilization alone is not much used in practice because the designer is not able to specify any performance requirements. To do so, **pre and post compensators** are used to **shape the open-loop singular values** prior to robust stabilization of the "shaped" plant. @@ -5561,9 +5561,9 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the G\_s = W\_2 G W\_1 \end{equation} -as shown in Fig. [51](#org877f64b). +as shown in Fig. [51](#org26a1619). - + {{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 51: The shaped plant and controller" >}} @@ -5596,11 +5596,11 @@ Systematic procedure for \\(\hinf\\) loop-shaping design: - A small value of \\(\epsilon\_{\text{max}}\\) indicates that the chosen singular value loop-shapes are incompatible with robust stability requirements 7. **Analyze the design** and if not all the specification are met, make further modifications to the weights 8. **Implement the controller**. - The configuration shown in Fig. [52](#org100612a) has been found useful when compared with the conventional setup in Fig. [38](#org659c83b). + The configuration shown in Fig. [52](#org5d5ea56) has been found useful when compared with the conventional setup in Fig. [38](#orgeae3b60). This is because the references do not directly excite the dynamics of \\(K\_s\\), which can result in large amounts of overshoot. The constant prefilter ensure a steady-state gain of \\(1\\) between \\(r\\) and \\(y\\), assuming integral action in \\(W\_1\\) or \\(G\\) - + {{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 52: A practical implementation of the loop-shaping controller" >}} @@ -5623,25 +5623,25 @@ Many control design problems possess two degrees-of-freedom: Sometimes, one degree-of-freedom is left out of the design, and the controller is driven by an error signal i.e. the difference between a command and the output. But in cases where stringent time-domain specifications are set on the output response, a one degree-of-freedom structure may not be sufficient.
-A general two degrees-of-freedom feedback control scheme is depicted in Fig. [53](#org93a0165). +A general two degrees-of-freedom feedback control scheme is depicted in Fig. [53](#org1a7c974). The commands and feedbacks enter the controller separately and are independently processed. - + {{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 53: General two degrees-of-freedom feedback control scheme" >}} The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy. However, this may not be sufficient and a dynamic two degrees-of-freedom design is required.
-The design problem is illustrated in Fig. [54](#orgf69c18e). +The design problem is illustrated in Fig. [54](#org6fb264c). The feedback part of the controller \\(K\_2\\) is designed to meet robust stability and disturbance rejection requirements. A prefilter is introduced to force the response of the closed-loop system to follow that of a specified model \\(T\_{\text{ref}}\\), often called the **reference model**. - + {{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 54: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}} -The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in Fig. [54](#orgf69c18e). +The design problem is to find the stabilizing controller \\(K = [K\_1,\ K\_2]\\) for the shaped plant \\(G\_s = G W\_1\\), with a normalized coprime factorization \\(G\_s = M\_s^{-1} N\_s\\), which minimizes the \\(\mathcal{H}\_\infty\\) norm of the transfer function between the signals \\([r^T\ \phi^T]^T\\) and \\([u\_s^T\ y^T\ e^T]^T\\) as defined in Fig. [54](#org6fb264c). This problem is easily cast into the general configuration. The control signal to the shaped plant \\(u\_s\\) is given by: @@ -5671,9 +5671,9 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\ 5. Replace the prefilter \\(K\_1\\) by \\(K\_1 W\_i\\) to give exact model-matching at steady-state. 6. Analyze and, if required, redesign making adjustments to \\(\rho\\) and possibly \\(W\_1\\) and \\(T\_{\text{ref}}\\) -The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig. [55](#orgbff52aa). +The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig. [55](#org1b57f23). - + {{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 55: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}} @@ -5755,9 +5755,9 @@ When implemented in Hanus form, the expression for \\(u\\) becomes where \\(u\_a\\) is the **actual plant input**, that is the measurement at the **output of the actuators** which therefore contains information about possible actuator saturation. -The situation is illustrated in Fig. [56](#org4e19257), where the actuators are each modeled by a unit gain and a saturation. +The situation is illustrated in Fig. [56](#org602e9ea), where the actuators are each modeled by a unit gain and a saturation. - + {{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 56: Self-conditioned weight \\(W\_1\\)" >}} @@ -5813,14 +5813,14 @@ Moreover, one should be careful about combining controller synthesis and analysi ## Controller Structure Design {#controller-structure-design} - + ### Introduction {#introduction} -In previous sections, we considered the general problem formulation in Fig. [57](#org66e5235) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\). +In previous sections, we considered the general problem formulation in Fig. [57](#orgf3d33fe) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\). - + {{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 57: General Control Configuration" >}} @@ -5853,19 +5853,19 @@ The reference value \\(r\\) is usually set at some higher layer in the control h - **Optimization layer**: computes the desired reference commands \\(r\\) - **Control layer**: implements these commands to achieve \\(y \approx r\\) -Additional layers are possible, as is illustrated in Fig. [58](#orga2d88eb) which shows a typical control hierarchy for a chemical plant. +Additional layers are possible, as is illustrated in Fig. [58](#org6aa007e) which shows a typical control hierarchy for a chemical plant. - + {{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 58: Typical control system hierarchy in a chemical plant" >}} -In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see Fig. [6](#org62ab823)). +In general, the information flow in such a control hierarchy is based on the higher layer sending reference values (setpoints) to the layer below reporting back any problems achieving this (see Fig. [6](#org252ea5b)). There is usually a time scale separation between the layers which means that the **setpoints**, as viewed from a given layer, are **updated only periodically**.
The optimization tends to be performed open-loop with limited use of feedback. On the other hand, the control layer is mainly based on feedback information. The **optimization is often based on nonlinear steady-state models**, whereas we often use **linear dynamic models in the control layer**.
-From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see Fig. [6](#orgff3e8fb)). +From a theoretical point of view, the optimal performance is obtained with a **centralized optimizing controller**, which combines the two layers of optimizing and control (see Fig. [6](#org9109832)). All control actions in such an ideal control system would be perfectly coordinated and the control system would use on-line dynamic optimization based on nonlinear dynamic model of the complete plant. However, this solution is normally not used for a number a reasons, included the cost of modeling, the difficulty of controller design, maintenance, robustness problems and the lack of computing power. @@ -5877,7 +5877,7 @@ However, this solution is normally not used for a number a reasons, included the | ![](/ox-hugo/skogestad07_optimize_control_a.png) | ![](/ox-hugo/skogestad07_optimize_control_b.png) | ![](/ox-hugo/skogestad07_optimize_control_c.png) | |--------------------------------------------------|--------------------------------------------------------------------------------|-------------------------------------------------------------| -| Open loop optimization | Closed-loop implementation with separate control layer | Integrated optimization and control | +| Open loop optimization | Closed-loop implementation with separate control layer | Integrated optimization and control | ### Selection of Controlled Outputs {#selection-of-controlled-outputs} @@ -5993,7 +5993,7 @@ Thus, the selection of controlled and measured outputs are two separate issues. ### Selection of Manipulations and Measurements {#selection-of-manipulations-and-measurements} -We are here concerned with the variable sets \\(u\\) and \\(v\\) in Fig. [57](#org66e5235). +We are here concerned with the variable sets \\(u\\) and \\(v\\) in Fig. [57](#orgf3d33fe). Note that **the measurements** \\(v\\) used by the controller **are in general different from the controlled variables** \\(z\\) because we may not be able to measure all the controlled variables and we may want to measure and control additional variables in order to: - Stabilize the plant, or more generally change its dynamics @@ -6086,9 +6086,9 @@ Then when a SISO control loop is closed, we lose the input \\(u\_i\\) as a degre A cascade control structure results when either of the following two situations arise: - The reference \\(r\_i\\) is an output from another controller. - This is the **conventional cascade control** (Fig. [7](#orgcbe29d0)) + This is the **conventional cascade control** (Fig. [7](#org1d28d6d)) - The "measurement" \\(y\_i\\) is an output from another controller. - This is referred to as **input resetting** (Fig. [7](#org3ca11c7)) + This is referred to as **input resetting** (Fig. [7](#org6c3853f))
@@ -6098,7 +6098,7 @@ A cascade control structure results when either of the following two situations | ![](/ox-hugo/skogestad07_cascade_extra_meas.png) | ![](/ox-hugo/skogestad07_cascade_extra_input.png) | |-------------------------------------------------------|---------------------------------------------------| -| Extra measurements \\(y\_2\\) | Extra inputs \\(u\_2\\) | +| Extra measurements \\(y\_2\\) | Extra inputs \\(u\_2\\) | #### Cascade Control: Extra Measurements {#cascade-control-extra-measurements} @@ -6122,7 +6122,7 @@ where in most cases \\(r\_2 = 0\\) since we do not have a degree-of-freedom to c ##### Cascade implementation {#cascade-implementation} -To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in Fig. [7](#orgcbe29d0): +To obtain an implementation with two SISO controllers, we may cascade the controllers as illustrated in Fig. [7](#org1d28d6d): \begin{align\*} r\_2 &= K\_1(s)(r\_1 - y\_1) \\\\\\ @@ -6132,13 +6132,13 @@ To obtain an implementation with two SISO controllers, we may cascade the contro Note that the output \\(r\_2\\) from the slower primary controller \\(K\_1\\) is not a manipulated plant input, but rather the reference input to the faster secondary controller \\(K\_2\\). Cascades based on measuring the actual manipulated variable (\\(y\_2 = u\_m\\)) are commonly used to **reduce uncertainty and non-linearity at the plant input**. -In the general case (Fig. [7](#orgcbe29d0)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_. -However, it is common to encounter the situation in Fig. [59](#org8de87b1) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of Fig. [7](#orgcbe29d0). +In the general case (Fig. [7](#org1d28d6d)) \\(y\_1\\) and \\(y\_2\\) are not directly related to each other, and this is sometimes referred to as _parallel cascade control_. +However, it is common to encounter the situation in Fig. [59](#org0106b04) where the primary output \\(y\_1\\) depends directly on \\(y\_2\\) which is a special case of Fig. [7](#org1d28d6d).
-With reference to the special (but common) case of cascade control shown in Fig. [59](#org8de87b1), the use of **extra measurements** is useful under the following circumstances: +With reference to the special (but common) case of cascade control shown in Fig. [59](#org0106b04), the use of **extra measurements** is useful under the following circumstances: - The disturbance \\(d\_2\\) is significant and \\(G\_1\\) is non-minimum phase. If \\(G\_1\\) is minimum phase, the input-output controllability of \\(G\_2\\) and \\(G\_1 G\_2\\) are the same and there is no fundamental advantage in measuring \\(y\_2\\) @@ -6147,7 +6147,7 @@ With reference to the special (but common) case of cascade control shown in Fig.
- + {{< figure src="/ox-hugo/skogestad07_cascade_control.png" caption="Figure 59: Common case of cascade control where the primary output \\(y\_1\\) depends directly on the extra measurement \\(y\_2\\)" >}} @@ -6175,7 +6175,7 @@ Then \\(u\_2(t)\\) will only be used for **transient control** and will return t ##### Cascade implementation {#cascade-implementation} -To obtain an implementation with two SISO controllers we may cascade the controllers as shown in Fig. [7](#org3ca11c7). +To obtain an implementation with two SISO controllers we may cascade the controllers as shown in Fig. [7](#org6c3853f). We again let input \\(u\_2\\) take care of the **fast control** and \\(u\_1\\) of the **long-term control**. The fast control loop is then @@ -6197,7 +6197,7 @@ It also shows more clearly that \\(r\_{u\_2}\\), the reference for \\(u\_2\\), m
-Consider the system in Fig. [60](#orgebbe1c4) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)). +Consider the system in Fig. [60](#org9e7696a) with two manipulated inputs (\\(u\_2\\) and \\(u\_3\\)), one controlled output (\\(y\_1\\) which should be close to \\(r\_1\\)) and two measured variables (\\(y\_1\\) and \\(y\_2\\)). Input \\(u\_2\\) has a more direct effect on \\(y\_1\\) than does input \\(u\_3\\) (there is a large delay in \\(G\_3(s)\\)). Input \\(u\_2\\) should only be used for transient control as it is desirable that it remains close to \\(r\_3 = r\_{u\_2}\\). The extra measurement \\(y\_2\\) is closer than \\(y\_1\\) to the input \\(u\_2\\) and may be useful for detecting disturbances affecting \\(G\_1\\). @@ -6209,7 +6209,7 @@ We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\)
- + {{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="Figure 60: Control configuration with two layers of cascade control" >}} @@ -6313,7 +6313,7 @@ By partitioning the inputs and outputs, the overall model \\(y = G u\\) can be w \end{aligned} \end{equation} -Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) (Fig. [61](#org72a0667)). +Assume now that feedback control \\(u\_2 = K\_2(r\_2 - y\_2 - n\_2)\\) is used for the "secondary" subsystem involving \\(u\_2\\) and \\(y\_2\\) (Fig. [61](#org06b827b)). We get: \begin{equation} \label{eq:partial\_control} @@ -6324,7 +6324,7 @@ We get: \end{aligned} \end{equation} - + {{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 61: Partial Control" >}} @@ -6383,7 +6383,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy ##### Sequential design of cascade control systems {#sequential-design-of-cascade-control-systems} -Consider the conventional cascade control system in Fig. [7](#orgcbe29d0) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\). +Consider the conventional cascade control system in Fig. [7](#org1d28d6d) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\). The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\). From \eqref{eq:partial_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\). @@ -6453,9 +6453,9 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1 ### Decentralized Feedback Control {#decentralized-feedback-control} -In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig. [62](#orgc4d3361)). +In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig. [62](#org4a88902)). - + {{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 62: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}} @@ -6855,7 +6855,7 @@ The conditions are also useful in an **input-output controllability analysis** f ## Model Reduction {#model-reduction} - + ### Introduction {#introduction} @@ -7285,4 +7285,4 @@ In such a case, using truncation or optimal Hankel norm approximation with appro ## Bibliography {#bibliography} -Doyle, John C. 1983. “Synthesis of Robust Controllers and Filters.” In _The 22nd IEEE Conference on Decision and Control_, 109–14. IEEE. +Doyle, John C. 1983. “Synthesis of Robust Controllers and Filters.” In _The 22nd IEEE Conference on Decision and Control_, 109–14. IEEE. diff --git a/content/phdthesis/monkhorst04_dynam_error_budget.md b/content/phdthesis/monkhorst04_dynam_error_budget.md index f1fa888..5c734a5 100644 --- a/content/phdthesis/monkhorst04_dynam_error_budget.md +++ b/content/phdthesis/monkhorst04_dynam_error_budget.md @@ -8,7 +8,7 @@ Tags : [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}) Reference -: ([Monkhorst 2004](#orgf3d75f1)) +: ([Monkhorst 2004](#org5371372)) Author(s) : Monkhorst, W. @@ -95,9 +95,9 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_ In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs. -This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#org43cbab3)). +This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#orgbc6c1df)). - + {{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}} @@ -108,23 +108,23 @@ The PSD \\(S\_w(f)\\) of the weighted signal is: Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_. However, this can be avoided if the modelling of the disturbances is directly done in terms of weighting filters. -Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#orge8654a7)). +Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org5bbc30c)). - + {{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}} #### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve} -In this research, the outputs of the closed loop system (Figure [3](#org55e6862)) are: +In this research, the outputs of the closed loop system (Figure [3](#orgff3035c)) are: - the performance (error) signal \\(e\\) - the controller output \\(u\\) In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output. - + {{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}} @@ -148,6 +148,7 @@ When an \\(\mathcal{H}\_2\\) controller is synthesized for a particular system, Drawbacks however are, that no robustness guarantees can be given and that the order of the \\(\mathcal{H}\_2\\) controller will generally be too high for implementation. + ## Bibliography {#bibliography} -Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University. +Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University. diff --git a/content/posts/homepages.md b/content/posts/homepages.md deleted file mode 100644 index 18c3275..0000000 --- a/content/posts/homepages.md +++ /dev/null @@ -1,18 +0,0 @@ -+++ -author = ["Thomas Dehaeze"] -draft = false -+++ - -## Homepage for Papers {#main} - -Here is the list of papers I took note about. - - -## Homepage for Books {#main} - -Here is the list of books I took note about. - - -## Homepage for Zettels {#main} - -Here is the list of subjects I took note about. diff --git a/content/websites/data_driven_dynamical_systems_with_machine_learning.md b/content/websites/data_driven_dynamical_systems_with_machine_learning.md index 617e125..9000861 100644 --- a/content/websites/data_driven_dynamical_systems_with_machine_learning.md +++ b/content/websites/data_driven_dynamical_systems_with_machine_learning.md @@ -179,5 +179,3 @@ Use powerful optimization techniques from machine learning to learn what are goo ### Applications {#applications} - -<./biblio/references.bib> diff --git a/content/zettels/acquisition_systems.md b/content/zettels/acquisition_systems.md index 5a8cbd7..b8e5f2c 100644 --- a/content/zettels/acquisition_systems.md +++ b/content/zettels/acquisition_systems.md @@ -15,5 +15,3 @@ Tags | [Dewesoft](https://dewesoft.com/) | Slovenia | | [Oros](https://www.oros.com/) | France | | [National Instruments](https://www.ni.com/fr-fr/shop/pc-based-measurement-and-control-system.html) | USA | - -<./biblio/references.bib> diff --git a/content/zettels/actuator_fusion.md b/content/zettels/actuator_fusion.md index 4ad6fca..483a345 100644 --- a/content/zettels/actuator_fusion.md +++ b/content/zettels/actuator_fusion.md @@ -4,20 +4,17 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Sensor Fusion]({{< relref "sensor_fusion" >}}) - Tags : [Complementary Filters]({{< relref "complementary_filters" >}}) -([Beijen et al. 2019](#orgaff80f9)) +([Beijen et al. 2019](#orgc359149)) + +([Beijen 2018](#org585205d)) (section 6.3.1) -([Beijen 2018](#org35f402d)) (section 6.3.1) ## Bibliography {#bibliography} -Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven. +Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven. -Beijen, Michiel A., Marcel F. Heertjes, Hans Butler, and Maarten Steinbuch. 2019. “Mixed Feedback and Feedforward Control Design for Multi-Axis Vibration Isolation Systems.” _Mechatronics_ 61:106–16. . +Beijen, Michiel A., Marcel F. Heertjes, Hans Butler, and Maarten Steinbuch. 2019. “Mixed Feedback and Feedforward Control Design for Multi-Axis Vibration Isolation Systems.” _Mechatronics_ 61:106–16. . diff --git a/content/zettels/actuators.md b/content/zettels/actuators.md index 345d752..5dd9256 100644 --- a/content/zettels/actuators.md +++ b/content/zettels/actuators.md @@ -4,13 +4,6 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Collocated Control]({{< relref "collocated_control" >}}) -- [Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation]({{< relref "ito16_compar_class_high_precis_actuat" >}}) -- [Voice Coil Actuators]({{< relref "voice_coil_actuators" >}}) -- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}) - Tags : @@ -24,18 +17,19 @@ Links to specific actuators: For vibration isolation: -- In ([Ito and Schitter 2016](#org7d0bcd5)), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat" >}})) +- In ([Ito and Schitter 2016](#org4bbf168)), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat" >}})) ## Brush-less DC Motor {#brush-less-dc-motor} -- ([Yedamale 2003](#org87858d4)) +- ([Yedamale 2003](#org1638958)) + ## Bibliography {#bibliography} -Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” _IEEE/ASME Transactions on Mechatronics_ 21 (2):1169–78. . +Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” _IEEE/ASME Transactions on Mechatronics_ 21 (2):1169–78. . -Yedamale, Padmaraja. 2003. “Brushless Dc (BLDC) Motor Fundamentals.” _Microchip Technology Inc_ 20:3–15. +Yedamale, Padmaraja. 2003. “Brushless Dc (BLDC) Motor Fundamentals.” _Microchip Technology Inc_ 20:3–15. diff --git a/content/zettels/analog_to_digital_converters.md b/content/zettels/analog_to_digital_converters.md index 8c54b6c..f8d3aa1 100644 --- a/content/zettels/analog_to_digital_converters.md +++ b/content/zettels/analog_to_digital_converters.md @@ -12,7 +12,7 @@ Tags -- Delta Sigma ([Baker 2011](#org9db2758)) +- Delta Sigma ([Baker 2011](#orgf10fad8)) - Successive Approximation @@ -31,9 +31,9 @@ Let's suppose that the ADC is ideal and the only noise comes from the quantizati Interestingly, the noise amplitude is uniformly distributed. The quantization noise can take a value between \\(\pm q/2\\), and the probability density function is constant in this range (i.e., it’s a uniform distribution). -Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#org79dc805)). +Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#org0a7db3b)). - + {{< figure src="/ox-hugo/probability_density_function_adc.png" caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}} @@ -85,6 +85,7 @@ The quantization is: {{< youtube b9lxtOJj3yU >}} + ## Bibliography {#bibliography} -Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” _Analog Applications_ 7. +Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” _Analog Applications_ 7. diff --git a/content/zettels/capacitive_sensors.md b/content/zettels/capacitive_sensors.md index 9707411..d1d785a 100644 --- a/content/zettels/capacitive_sensors.md +++ b/content/zettels/capacitive_sensors.md @@ -28,5 +28,3 @@ Tags | [Capacitec](https://www.capacitec.com/Displacement-Sensing-Systems) | USA | | [MTIinstruments](https://www.mtiinstruments.com/products/non-contact-measurement/capacitance-sensors/) | USA | | [Althen](https://www.althensensors.com/sensors/linear-position-sensors/capacitive-position-sensors/) | Netherlands | - -<./biblio/references.bib> diff --git a/content/zettels/charge_amplifiers.md b/content/zettels/charge_amplifiers.md index 351cf32..b09bfd4 100644 --- a/content/zettels/charge_amplifiers.md +++ b/content/zettels/charge_amplifiers.md @@ -17,19 +17,19 @@ This can be typically used to interface with piezoelectric sensors. ## Basic Circuit {#basic-circuit} -Two basic circuits of charge amplifiers are shown in Figure [1](#org4fccf5a) (taken from ([Fleming 2010](#org17ae69b))) and Figure [2](#orgad97f51) (taken from ([Schmidt, Schitter, and Rankers 2014](#orge90efed))) +Two basic circuits of charge amplifiers are shown in Figure [1](#org45de288) (taken from ([Fleming 2010](#org2341229))) and Figure [2](#org8955723) (taken from ([Schmidt, Schitter, and Rankers 2014](#orgf9a1421))) - + {{< figure src="/ox-hugo/charge_amplifier_circuit.png" caption="Figure 1: Electrical model of a piezoelectric force sensor is shown in gray. The op-amp charge amplifier is shown on the right. The output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}} - + {{< figure src="/ox-hugo/charge_amplifier_circuit_bis.png" caption="Figure 2: A piezoelectric accelerometer with a charge amplifier as signal conditioning element" >}} The input impedance of the charge amplifier is very small (unlike when using a voltage amplifier). -The gain of the charge amplified (Figure [1](#org4fccf5a)) is equal to: +The gain of the charge amplified (Figure [1](#org45de288)) is equal to: \\[ \frac{V\_s}{q} = \frac{-1}{C\_s} \\] @@ -47,8 +47,9 @@ The gain of the charge amplified (Figure [1](#org4fccf5a)) is equal to: | [L-Card](https://en.lcard.ru/products/accesories/le-41) | Rusia | + ## Bibliography {#bibliography} -Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. . +Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. . -Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press. +Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press. diff --git a/content/zettels/complementary_filters.md b/content/zettels/complementary_filters.md index 84b7c9d..9d37ad4 100644 --- a/content/zettels/complementary_filters.md +++ b/content/zettels/complementary_filters.md @@ -10,9 +10,10 @@ Tags ## Complementary Filters Synthesis {#complementary-filters-synthesis} -The shaping of complementary filters can be done using the \\(\mathcal{H}\_\infty\\) synthesis ([Dehaeze, Vermat, and Christophe 2019](#orgc79060a)). +The shaping of complementary filters can be done using the \\(\mathcal{H}\_\infty\\) synthesis ([Dehaeze, Vermat, and Christophe 2019](#org0c35169)). + ## Bibliography {#bibliography} -Dehaeze, Thomas, Mohit Vermat, and Collette Christophe. 2019. “Complementary Filters Shaping Using \\(mathcalH\_Infty\\) Synthesis.” In _7th International Conference on Control, Mechatronics and Automation (ICCMA)_, 459–64. . +Dehaeze, Thomas, Mohit Vermat, and Collette Christophe. 2019. “Complementary Filters Shaping Using \\(mathcalH\_Infty\\) Synthesis.” In _7th International Conference on Control, Mechatronics and Automation (ICCMA)_, 459–64. . diff --git a/content/zettels/cubic_architecture.md b/content/zettels/cubic_architecture.md index e9fed17..ac08c82 100644 --- a/content/zettels/cubic_architecture.md +++ b/content/zettels/cubic_architecture.md @@ -4,12 +4,6 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}}) -- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}}) -- [Sensors and control of a space-based six-axis vibration isolation system]({{< relref "hauge04_sensor_contr_space_based_six" >}}) - Tags : @@ -19,12 +13,13 @@ Tags ## Special Properties {#special-properties} -Cubic Stewart Platforms can be decoupled provided that (from ([Chen and McInroy 2000](#org4014064))) +Cubic Stewart Platforms can be decoupled provided that (from ([Chen and McInroy 2000](#org0969434))) > 1. The payload mass-inertia matrix is diagonal > 2. If a mutually orthogonal geometry has been selected, the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts. + ## Bibliography {#bibliography} -Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . +Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . diff --git a/content/zettels/digital_to_analog_converters.md b/content/zettels/digital_to_analog_converters.md index 661f72c..fc3b78d 100644 --- a/content/zettels/digital_to_analog_converters.md +++ b/content/zettels/digital_to_analog_converters.md @@ -6,5 +6,3 @@ draft = false Tags : [Electronics]({{< relref "electronics" >}}) - -<./biblio/references.bib> diff --git a/content/zettels/direct_velocity_feedback.md b/content/zettels/direct_velocity_feedback.md index 70bc9bc..61a118d 100644 --- a/content/zettels/direct_velocity_feedback.md +++ b/content/zettels/direct_velocity_feedback.md @@ -6,5 +6,3 @@ draft = false Tags : [Active Damping]({{< relref "active_damping" >}}) - -<./biblio/references.bib> diff --git a/content/zettels/encoders.md b/content/zettels/encoders.md index baf4e7a..fbc0c29 100644 --- a/content/zettels/encoders.md +++ b/content/zettels/encoders.md @@ -18,5 +18,3 @@ There are two main types of encoders: optical encoders, and magnetic encoders. | [MicroE Systems](https://www.celeramotion.com/microe/products/linear-encoders/) | USA | | [Renishaw](https://www.renishaw.com/en/browse-encoder-range--6440) | UK | | [Celera Motion](https://www.celeramotion.com/microe/) | USA | - -<./biblio/references.bib> diff --git a/content/zettels/flexible_joints.md b/content/zettels/flexible_joints.md index 5d3282c..c084534 100644 --- a/content/zettels/flexible_joints.md +++ b/content/zettels/flexible_joints.md @@ -4,18 +4,6 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}}) -- [Dynamic modeling and experimental analyses of stewart platform with flexible hinges]({{< relref "jiao18_dynam_model_exper_analy_stewar" >}}) -- [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}}) -- [Investigation on active vibration isolation of a stewart platform with piezoelectric actuators]({{< relref "wang16_inves_activ_vibrat_isolat_stewar" >}}) -- [Nanometre-cutting machine using a stewart-platform parallel mechanism]({{< relref "furutani04_nanom_cuttin_machin_using_stewar" >}}) -- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}}) -- [Dynamic modeling of flexure jointed hexapods for control purposes]({{< relref "mcinroy99_dynam" >}}) -- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}}) -- [Flexures]({{< relref "flexures" >}}) - Tags : @@ -24,32 +12,33 @@ Tags Books: -- ([Lobontiu 2002](#org74b9989)) -- ([Henein 2003](#org1491e2e)) -- ([Smith 2005](#orgcdbef5f)) -- ([Soemers 2011](#org9626592)) -- ([Cosandier 2017](#org9b28dc9)) +- ([Lobontiu 2002](#orgf96bd1c)) +- ([Henein 2003](#org77c1a30)) +- ([Smith 2005](#orgdf03b02)) +- ([Soemers 2011](#orgc441221)) +- ([Cosandier 2017](#orgc637f07)) ## Flexure Joints for Stewart Platforms: {#flexure-joints-for-stewart-platforms} -From ([Chen and McInroy 2000](#org4bbdddf)): +From ([Chen and McInroy 2000](#org26c43a0)): > To avoid the extremely non-linear micro-dynamics of joint friction and backlash, these hexapods employ flexure joints. > A flexure joint bends material to achieve motion, rather than sliding of rolling across two surfaces. > This does eliminate friction and backlash, but adds spring dynamics and limits the workspace. + ## Bibliography {#bibliography} -Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . +Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . -Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press. +Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press. -Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes. +Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes. -Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press. +Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press. -Smith, Stuart T. 2005. _Foundations of Ultra-Precision Mechanism Design_. Vol. 2. CRC Press. +Smith, Stuart T. 2005. _Foundations of Ultra-Precision Mechanism Design_. Vol. 2. CRC Press. -Soemers, Herman. 2011. _Design Principles for Precision Mechanisms_. T-Pointprint. +Soemers, Herman. 2011. _Design Principles for Precision Mechanisms_. T-Pointprint. diff --git a/content/zettels/granite.md b/content/zettels/granite.md index e59aa9a..62dc55d 100644 --- a/content/zettels/granite.md +++ b/content/zettels/granite.md @@ -14,5 +14,3 @@ Tags |--------------------------------------------------|---------| | [Microplan](https://www.microplan-group.com/fr/) | France | | [Zali](http://zali-precision.it/en/products/) | Italy | - -<./biblio/references.bib> diff --git a/content/zettels/h_infinity_control.md b/content/zettels/h_infinity_control.md index c4dc1d7..b14811a 100644 --- a/content/zettels/h_infinity_control.md +++ b/content/zettels/h_infinity_control.md @@ -4,10 +4,6 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Guidelines for the selection of weighting functions for h-infinity control]({{< relref "bibel92_guidel_h" >}}) - Tags : @@ -19,5 +15,3 @@ From _Rosenbrock, H. H. (1974). Computer-Aided Control System Design, Academic P > Solutions are constrained by so many requirements that it is virtually impossible to list them all. > The designer finds himself threading a maze of such requirements, attempting to reconcile conflicting demands of cost, performance, easy maintenance, and so on. > A good design usually has strong aesthetic appeal to those who are competent in the subject. - -<./biblio/references.bib> diff --git a/content/zettels/inertial_sensors.md b/content/zettels/inertial_sensors.md index 7fed4f4..3454d93 100644 --- a/content/zettels/inertial_sensors.md +++ b/content/zettels/inertial_sensors.md @@ -10,10 +10,10 @@ Tags ## Review of Absolute (inertial) Position Sensors {#review-of-absolute--inertial--position-sensors} -- Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement ([Collette, Janssens, Fernandez-Carmona, et al. 2012](#orgb31e055)) -- Collette, C. et al., Comparison of new absolute displacement sensors ([Collette, Janssens, Mokrani, et al. 2012](#orgcd873cb)) +- Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement ([Collette, Janssens, Fernandez-Carmona, et al. 2012](#orge266e77)) +- Collette, C. et al., Comparison of new absolute displacement sensors ([Collette, Janssens, Mokrani, et al. 2012](#orga0b31ea)) - + {{< figure src="/ox-hugo/collette12_absolute_disp_sensors.png" caption="Figure 1: Dynamic range of several types of inertial sensors; Price versus resolution for several types of inertial sensors" >}} @@ -35,7 +35,7 @@ Wireless Accelerometers - - + {{< figure src="/ox-hugo/inertial_sensors_characteristics_accelerometers.png" caption="Figure 2: Characteristics of commercially available accelerometers collette11_review" >}} @@ -52,13 +52,14 @@ Wireless Accelerometers | [Guralp](https://www.guralp.com/products/surface) | UK | | [Nanometric](https://www.nanometrics.ca/products/seismometers) | Canada | - + {{< figure src="/ox-hugo/inertial_sensors_characteristics_geophone.png" caption="Figure 3: Characteristics of commercially available geophones collette11_review" >}} + ## Bibliography {#bibliography} -Collette, C., S. Janssens, P. Fernandez-Carmona, K. Artoos, M. Guinchard, C. Hauviller, and A. Preumont. 2012. “Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement.” _Bulletin of the Seismological Society of America_ 102 (4):1289–1300. . +Collette, C., S. Janssens, P. Fernandez-Carmona, K. Artoos, M. Guinchard, C. Hauviller, and A. Preumont. 2012. “Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement.” _Bulletin of the Seismological Society of America_ 102 (4):1289–1300. . -Collette, C, S Janssens, B Mokrani, L Fueyo-Roza, K Artoos, M Esposito, P Fernandez-Carmona, M Guinchard, and R Leuxe. 2012. “Comparison of New Absolute Displacement Sensors.” In _International Conference on Noise and Vibration Engineering (ISMA)_. +Collette, C, S Janssens, B Mokrani, L Fueyo-Roza, K Artoos, M Esposito, P Fernandez-Carmona, M Guinchard, and R Leuxe. 2012. “Comparison of New Absolute Displacement Sensors.” In _International Conference on Noise and Vibration Engineering (ISMA)_. diff --git a/content/zettels/instrumented_hammer.md b/content/zettels/instrumented_hammer.md index 9184f10..7363d29 100644 --- a/content/zettels/instrumented_hammer.md +++ b/content/zettels/instrumented_hammer.md @@ -17,5 +17,3 @@ And instrumented hammer consist of a regular hammer with a force sensor fixed at | [PCB](https://www.pcb.com/sensors-for-test-measurement/impact-hammers-electrodynamic-shakers/impact-hammers) | USA | | [DJB](https://www.djbinstruments.com/products/instrumentation/impact-hammers) | UK | | [Dewesoft](https://dewesoft.com/fr/products/interfaces-and-sensors/accelerometers-and-modal-hammers) | Slovenia | - -<./biblio/references.bib> diff --git a/content/zettels/interferometers.md b/content/zettels/interferometers.md index 694ee9b..be3a139 100644 --- a/content/zettels/interferometers.md +++ b/content/zettels/interferometers.md @@ -24,7 +24,7 @@ Tags ## Effect of Refractive Index - Environmental Units {#effect-of-refractive-index-environmental-units} -The measured distance is proportional to the refractive index of the air that depends on several quantities as shown in Table [1](#table--tab:index-air) (Taken from ([Thurner et al. 2015](#org68c8bbb))). +The measured distance is proportional to the refractive index of the air that depends on several quantities as shown in Table [1](#table--tab:index-air) (Taken from ([Thurner et al. 2015](#org1b86993))).
@@ -59,16 +59,16 @@ Typical characteristics of commercial environmental units are shown in Table [2] ## Interferometer Precision {#interferometer-precision} -Figure [1](#org960bbd9) shows the expected precision as a function of the measured distance due to change of refractive index of the air (taken from ([Jang and Kim 2017](#orgd724d07))). +Figure [1](#org3490ef0) shows the expected precision as a function of the measured distance due to change of refractive index of the air (taken from ([Jang and Kim 2017](#org3b0a481))). - + {{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 1: Expected precision of interferometer as a function of measured distance" >}} ## Sources of uncertainty {#sources-of-uncertainty} -Sources of error in laser interferometry are well described in ([Ducourtieux 2018](#orgeacbea1)). +Sources of error in laser interferometry are well described in ([Ducourtieux 2018](#org588696d)). It includes: @@ -78,18 +78,19 @@ It includes: - Pressure: \\(K\_P \approx 0.27 ppm hPa^{-1}\\) - Humidity: \\(K\_{HR} \approx 0.01 ppm \% RH^{-1}\\) - These errors can partially be compensated using an environmental unit. -- Air turbulence (Figure [2](#orgd403994)) +- Air turbulence (Figure [2](#orgceb0667)) - Non linearity - + {{< figure src="/ox-hugo/interferometers_air_turbulence.png" caption="Figure 2: Effect of air turbulences on measurement stability" >}} + ## Bibliography {#bibliography} -Ducourtieux, Sebastien. 2018. “Toward High Precision Position Control Using Laser Interferometry: Main Sources of Error.” . +Ducourtieux, Sebastien. 2018. “Toward High Precision Position Control Using Laser Interferometry: Main Sources of Error.” . -Jang, Yoon-Soo, and Seung-Woo Kim. 2017. “Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: A Review.” _International Journal of Precision Engineering and Manufacturing_ 18 (12):1881–90. . +Jang, Yoon-Soo, and Seung-Woo Kim. 2017. “Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: A Review.” _International Journal of Precision Engineering and Manufacturing_ 18 (12):1881–90. . -Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” _Applied Optics_ 54 (10). Optical Society of America:3051–63. +Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” _Applied Optics_ 54 (10). Optical Society of America:3051–63. diff --git a/content/zettels/irr_and_fir_filters.md b/content/zettels/irr_and_fir_filters.md index fa2042f..d6e1061 100644 --- a/content/zettels/irr_and_fir_filters.md +++ b/content/zettels/irr_and_fir_filters.md @@ -32,7 +32,7 @@ Tags > > The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filter. -From ([Shaw and Srinivasan 1990](#org9a58282)) +From ([Shaw and Srinivasan 1990](#orge62ce0f)) > The FIR are capable of realizing filters with linear phase shift characteristics and furthermore are less susceptible to signal input and filter coefficient quantization effects. > However, their computational demands are excessively large because of the large number of multiplications and additions to be performed at each sampling interval. @@ -49,6 +49,7 @@ From > - Feed-forward control. FIR filters are useful for producing filters that approximate arbitrary frequency responses, hence they can be used to shape a reference signal. A typical example is to use an FIR filter with the inverse frequency response of the plant -- trying to counteract the dynamics of the plant in order to get a desired output. Phase/time-delay is not interfering with the stability or performance since the computation can be done offline. FIR filters can often produce higher performance than IIR filters, especially where there are non-minimum phase zeros. + ## Bibliography {#bibliography} -Shaw, F.R., and K. Srinivasan. 1990. “Bandwidth Enhancement of Position Measurements Using Measured Acceleration.” _Mechanical Systems and Signal Processing_ 4 (1):23–38. 90038-m. +Shaw, F.R., and K. Srinivasan. 1990. “Bandwidth Enhancement of Position Measurements Using Measured Acceleration.” _Mechanical Systems and Signal Processing_ 4 (1):23–38. 90038-m. diff --git a/content/zettels/mass_spring_damper_systems.md b/content/zettels/mass_spring_damper_systems.md index d189389..59745d3 100644 --- a/content/zettels/mass_spring_damper_systems.md +++ b/content/zettels/mass_spring_damper_systems.md @@ -10,7 +10,7 @@ Tags ## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system} -Let's consider Figure [1](#orgeec8f0f) where: +Let's consider Figure [1](#orga358a0b) where: - \\(m\\) is the mass in [kg] - \\(ḱ\\) is the spring stiffness in [N/m] @@ -20,7 +20,7 @@ Let's consider Figure [1](#orgeec8f0f) where: - \\(w\\) is ground motion - \\(x\\) is the absolute mass motion - + {{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="Figure 1: Mass Spring Damper System" >}} @@ -54,5 +54,3 @@ with: \begin{equation} \frac{x}{F\_d}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1} \end{equation} - -<./biblio/references.bib> diff --git a/content/zettels/matlab.md b/content/zettels/matlab.md index eb5e5e9..3d6cc8b 100644 --- a/content/zettels/matlab.md +++ b/content/zettels/matlab.md @@ -4,10 +4,6 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Simulink]({{< relref "simulink" >}}) - Tags : [Simulink]({{< relref "simulink" >}}) @@ -16,11 +12,11 @@ Tags Books: -- ([Higham 2017](#org9e841d2)) -- ([Attaway 2018](#org1d7c8f3)) -- ([OverFlow 2018](#orgdf04055)) -- ([Johnson 2010](#org329fa4a)) -- ([Hahn and Valentine 2016](#org7d00587)) +- ([Higham 2017](#org68f863c)) +- ([Attaway 2018](#org3441bfb)) +- ([OverFlow 2018](#org8e0ff2b)) +- ([Johnson 2010](#org019531d)) +- ([Hahn and Valentine 2016](#orgbeacac3)) ## Useful Commands {#useful-commands} @@ -56,12 +52,12 @@ Books: ### Do not show legend for one plot {#do-not-show-legend-for-one-plot} ```matlab -figure; -hold on; -plot(x, y1, 'DisplayName, 'lengendname'); -plot(x, y2, 'HandleVisibility', 'off'); -hold off; -legend('Location', 'northeast'); + figure; + hold on; + plot(x, y1, 'DisplayName, 'lengendname'); + plot(x, y2, 'HandleVisibility', 'off'); + hold off; + legend('Location', 'northeast'); ``` @@ -70,7 +66,7 @@ legend('Location', 'northeast'); If a single user is using the Matlab installation on the machine: ```bash -sudo chown -R $LOGNAME: /usr/local/MATLAB/R2017b + sudo chown -R $LOGNAME: /usr/local/MATLAB/R2017b ``` Then, Toolboxes can be installed by the user without any problem. @@ -109,14 +105,15 @@ Nice functions: | `echo` | Display statements during function execution | + ## Bibliography {#bibliography} -Attaway, Stormy. 2018. _MATLAB : a Practical Introduction to Programming and Problem Solving_. Amsterdam: Butterworth-Heinemann. +Attaway, Stormy. 2018. _MATLAB : a Practical Introduction to Programming and Problem Solving_. Amsterdam: Butterworth-Heinemann. -Hahn, Brian, and Daniel T Valentine. 2016. _Essential MATLAB for Engineers and Scientists_. Academic Press. +Hahn, Brian, and Daniel T Valentine. 2016. _Essential MATLAB for Engineers and Scientists_. Academic Press. -Higham, Desmond. 2017. _MATLAB Guide_. Philadelphia: Society for Industrial and Applied Mathematics. +Higham, Desmond. 2017. _MATLAB Guide_. Philadelphia: Society for Industrial and Applied Mathematics. -Johnson, Richard K. 2010. _The Elements of MATLAB Style_. Cambridge University Press. +Johnson, Richard K. 2010. _The Elements of MATLAB Style_. Cambridge University Press. -OverFlow, Stack. 2018. _MATLAB Notes for Professionals_. GoalKicker.com. +OverFlow, Stack. 2018. _MATLAB Notes for Professionals_. GoalKicker.com. diff --git a/content/zettels/metrology.md b/content/zettels/metrology.md index fa766db..8f5a81b 100644 --- a/content/zettels/metrology.md +++ b/content/zettels/metrology.md @@ -4,11 +4,5 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Fundamental principles of engineering nanometrology]({{< relref "leach14_fundam_princ_engin_nanom" >}}) - Tags : - -<./biblio/references.bib> diff --git a/content/zettels/modal_analysis.md b/content/zettels/modal_analysis.md index 32efb1f..27d4fb7 100644 --- a/content/zettels/modal_analysis.md +++ b/content/zettels/modal_analysis.md @@ -4,14 +4,5 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Modal testing: theory, practice and application]({{< relref "ewins00_modal" >}}) -- [Force Sensors]({{< relref "force_sensors" >}}) -- [Instrumented Hammer]({{< relref "instrumented_hammer" >}}) -- [System Identification]({{< relref "system_identification" >}}) - Tags : [Inertial Sensors]({{< relref "inertial_sensors" >}}), [Shaker]({{< relref "shaker" >}}) - -<./biblio/references.bib> diff --git a/content/zettels/model_predictive_control.md b/content/zettels/model_predictive_control.md index 9c17cb9..ae8def6 100644 --- a/content/zettels/model_predictive_control.md +++ b/content/zettels/model_predictive_control.md @@ -6,5 +6,3 @@ draft = false Tags : - -<./biblio/references.bib> diff --git a/content/zettels/multivariable_control.md b/content/zettels/multivariable_control.md index d0ed4de..b756e10 100644 --- a/content/zettels/multivariable_control.md +++ b/content/zettels/multivariable_control.md @@ -7,9 +7,10 @@ draft = false Tags : [Norms]({{< relref "norms" >}}) -A very nice book about Multivariable Control is ([Skogestad and Postlethwaite 2007](#org8b835f5)) +A very nice book about Multivariable Control is ([Skogestad and Postlethwaite 2007](#org2f5ed44)) + ## Bibliography {#bibliography} -Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley. +Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley. diff --git a/content/zettels/norms.md b/content/zettels/norms.md index c517adf..dcf7167 100644 --- a/content/zettels/norms.md +++ b/content/zettels/norms.md @@ -11,9 +11,9 @@ Tags Resources: -- ([Skogestad and Postlethwaite 2007](#orgf64cea0)) -- ([Toivonen 2002](#org3ddabae)) -- ([Zhang 2011](#orge43be7a)) +- ([Skogestad and Postlethwaite 2007](#org4c8b20e)) +- ([Toivonen 2002](#org81db503)) +- ([Zhang 2011](#orgc4d2be1)) ## Definition {#definition} @@ -176,17 +176,18 @@ In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as fo The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}). -As explained in ([Monkhorst 2004](#org45aca82)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation: +As explained in ([Monkhorst 2004](#orgc401feb)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation: > The squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input. + ## Bibliography {#bibliography} -Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University. +Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University. -Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley. +Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley. -Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University. +Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University. -Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press. +Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press. diff --git a/content/zettels/operational_amplifiers.md b/content/zettels/operational_amplifiers.md index 3f76fdb..745a7fc 100644 --- a/content/zettels/operational_amplifiers.md +++ b/content/zettels/operational_amplifiers.md @@ -11,5 +11,3 @@ Tags ## Defaults of Operational Amplifiers {#defaults-of-operational-amplifiers} {{< youtube nF104EvI0HM >}} - -<./biblio/references.bib> diff --git a/content/zettels/positioning_stations.md b/content/zettels/positioning_stations.md index 8e21e7e..85ac4f7 100644 --- a/content/zettels/positioning_stations.md +++ b/content/zettels/positioning_stations.md @@ -16,5 +16,3 @@ Tags | [PI](https://www.physikinstrumente.com/en/) | USA | | [Attocube](https://www.attocube.com/en/products/nanopositioners) | Germany | | [Newport](https://www.newport.com/c/manual-positioning) | | - -<./biblio/references.bib> diff --git a/content/zettels/power_spectral_density.md b/content/zettels/power_spectral_density.md index 3e32a1a..2c3cb0f 100644 --- a/content/zettels/power_spectral_density.md +++ b/content/zettels/power_spectral_density.md @@ -9,9 +9,10 @@ Tags Tutorial about Power Spectral Density is accessible [here](https://research.tdehaeze.xyz/spectral-analysis/). -A good article about how to use the `pwelch` function with Matlab ([Schmid 2012](#org23bcfe9)). +A good article about how to use the `pwelch` function with Matlab ([Schmid 2012](#org6fb2cbe)). + ## Bibliography {#bibliography} -Schmid, Hanspeter. 2012. “How to Use the FFT and Matlab’s Pwelch Function for Signal and Noise Simulations and Measurements.” _Institute of Microelectronics_. +Schmid, Hanspeter. 2012. “How to Use the FFT and Matlab’s Pwelch Function for Signal and Noise Simulations and Measurements.” _Institute of Microelectronics_. diff --git a/content/zettels/precision_engineering.md b/content/zettels/precision_engineering.md index 67b27c2..182481d 100644 --- a/content/zettels/precision_engineering.md +++ b/content/zettels/precision_engineering.md @@ -4,12 +4,5 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Basics of precision engineering - 1st edition]({{< relref "leach18_basic_precis_engin_edition" >}}) -- [Design for precision: current status and trends]({{< relref "schellekens98_desig_precis" >}}) - Tags : - -<./biblio/references.bib> diff --git a/content/zettels/sensor_noise_estimation.md b/content/zettels/sensor_noise_estimation.md index fedb54a..858fb6f 100644 --- a/content/zettels/sensor_noise_estimation.md +++ b/content/zettels/sensor_noise_estimation.md @@ -12,13 +12,13 @@ Tags Measuring the noise level of inertial sensors is not easy as the seismic motion is usually much larger than the sensor's noise level. -A technique to estimate the sensor noise in such case is proposed in ([Barzilai, VanZandt, and Kenny 1998](#org7fe766e)) and well explained in ([Poel 2010](#org964c18e)) (Section 6.1.3). +A technique to estimate the sensor noise in such case is proposed in ([Barzilai, VanZandt, and Kenny 1998](#org4702c9a)) and well explained in ([Poel 2010](#orgeaef46f)) (Section 6.1.3). The idea is to mount two inertial sensors closely together such that they should measure the same quantity. -This is represented in Figure [1](#org53e9426) where two identical sensors are measuring the same motion \\(x(t)\\). +This is represented in Figure [1](#org030f5c0) where two identical sensors are measuring the same motion \\(x(t)\\). - + {{< figure src="/ox-hugo/huddle_test_setup.png" caption="Figure 1: Schematic representation of the setup for measuring the noise of inertial sensors." >}} @@ -49,23 +49,23 @@ where: The Matlab function `mscohere` can be used to compute the coherence: ```matlab -%% Parameters -Fs = 1e4; % Sampling Frequency [Hz] -win = hanning(ceil(10*Fs)); % 10 seconds Hanning Windows + %% Parameters + Fs = 1e4; % Sampling Frequency [Hz] + win = hanning(ceil(10*Fs)); % 10 seconds Hanning Windows -%% Coherence between x and y -[pxy, f] = mscohere(x, y, win, [], [], Fs); % Coherence, frequency vector in [Hz] + %% Coherence between x and y + [pxy, f] = mscohere(x, y, win, [], [], Fs); % Coherence, frequency vector in [Hz] ``` Alternatively, it can be manually computed using the `cpsd` and `pwelch` commands: ```matlab -%% Manual Computation of the Coherence -[pxy, f] = cpsd(x, y, win, [], [], Fs); % Cross Spectral Density between x and y -[pxx, ~] = pwelch(x, win, [], [], Fs); % Power Spectral Density of x -[pyy, ~] = pwelch(y, win, [], [], Fs); % Power Spectral Density of y + %% Manual Computation of the Coherence + [pxy, f] = cpsd(x, y, win, [], [], Fs); % Cross Spectral Density between x and y + [pxx, ~] = pwelch(x, win, [], [], Fs); % Power Spectral Density of x + [pyy, ~] = pwelch(y, win, [], [], Fs); % Power Spectral Density of y -pxy_manual = abs(pxy).^2./abs(pxx)./abs(pyy); + pxy_manual = abs(pxy).^2./abs(pxx)./abs(pyy); ```
@@ -76,7 +76,7 @@ Now suppose that: - sensor noises are modelled as input noises \\(n\_1(t)\\) and \\(n\_2(s)\\) - sensor noises are uncorrelated and each are uncorrelated with \\(x(t)\\) -Then, the system can be represented by the block diagram in Figure [2](#org0e1cf4a), and we can write: +Then, the system can be represented by the block diagram in Figure [2](#orgec7c79b), and we can write: \begin{align} P\_{y\_1y\_1}(\omega) &= |H\_1(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_1}(\omega) ) \\\\\\ @@ -90,7 +90,7 @@ And the CSD between \\(y\_1(t)\\) and \\(y\_2(t)\\) is: \gamma^2\_{y\_1y\_2}(\omega) = \frac{|C\_{y\_1y\_2}(j\omega)|^2}{P\_{y\_1}(\omega) P\_{y\_2}(\omega)} \end{equation} - + {{< figure src="/ox-hugo/huddle_test_block_diagram.png" caption="Figure 2: Huddle test block diagram" >}} @@ -113,8 +113,9 @@ If we assume the two sensor dynamics to be the same \\(H\_1(s) \approx H\_2(s)\\
+ ## Bibliography {#bibliography} -Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” _Review of Scientific Instruments_ 69 (7):2767–72. . +Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” _Review of Scientific Instruments_ 69 (7):2767–72. . -Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. . +Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. . diff --git a/content/zettels/sensors.md b/content/zettels/sensors.md index 873d610..9d20cd7 100644 --- a/content/zettels/sensors.md +++ b/content/zettels/sensors.md @@ -12,5 +12,3 @@ Notes about sensors: - [Force Sensors]({{< relref "force_sensors" >}}) - [Position Sensors]({{< relref "position_sensors" >}}) - [Inertial Sensors]({{< relref "inertial_sensors" >}}) - -<./biblio/references.bib> diff --git a/content/zettels/signal_conditioner.md b/content/zettels/signal_conditioner.md index 2813b0a..319a9b5 100644 --- a/content/zettels/signal_conditioner.md +++ b/content/zettels/signal_conditioner.md @@ -4,11 +4,6 @@ author = ["Thomas Dehaeze"] draft = false +++ -Backlinks: - -- [Position Sensors]({{< relref "position_sensors" >}}) -- [Force Sensors]({{< relref "force_sensors" >}}) - Tags : [Sensors]({{< relref "sensors" >}}), [Electronics]({{< relref "electronics" >}}) @@ -33,5 +28,3 @@ Depending on the electrical quantity that is meaningful for the measurement, dif - Current to Voltage ([Transimpedance Amplifiers]({{< relref "transimpedance_amplifiers" >}})) - Charge to Voltage ([Charge Amplifiers]({{< relref "charge_amplifiers" >}})) - Voltage to Voltage ([Voltage Amplifier]({{< relref "voltage_amplifier" >}})) - -<./biblio/references.bib> diff --git a/content/zettels/simulink.md b/content/zettels/simulink.md index df53b5c..e65b0ce 100644 --- a/content/zettels/simulink.md +++ b/content/zettels/simulink.md @@ -36,5 +36,3 @@ Tips: ## Linearize portion of Simulink file {#linearize-portion-of-simulink-file} - -<./biblio/references.bib> diff --git a/content/zettels/simulink_real_time_target_machines.md b/content/zettels/simulink_real_time_target_machines.md index ba4b987..155b2bc 100644 --- a/content/zettels/simulink_real_time_target_machines.md +++ b/content/zettels/simulink_real_time_target_machines.md @@ -34,5 +34,3 @@ Comparison | Encoder (quadrature) | 2 | 4 | 4 | 2 | | Sampling Frequency | ? | ? | 1kHz (USB), 15kHz (Serial) | 2kHz | | Price | waiting for quote | 1000 | 900 | 1300 | - -<./biblio/references.bib> diff --git a/content/zettels/stewart_platforms.md b/content/zettels/stewart_platforms.md index 8cdb794..f61db00 100644 --- a/content/zettels/stewart_platforms.md +++ b/content/zettels/stewart_platforms.md @@ -36,36 +36,37 @@ Tags Papers by J.E. McInroy: -- ([O’Brien et al. 1998](#org301ae65)) -- ([McInroy, O’Brien, and Neat 1999](#org43a0fe2)) -- ([McInroy 1999](#org41ba097)) -- ([McInroy and Hamann 2000](#org73060fc)) -- ([Chen and McInroy 2000](#org2b98584)) -- ([McInroy 2002](#org2d6222b)) -- ([Li, Hamann, and McInroy 2001](#org6598adc)) -- ([Lin and McInroy 2003](#orgfc1736f)) -- ([Jafari and McInroy 2003](#org72de1d8)) -- ([Chen and McInroy 2004](#org6bdfb26)) +- ([O’Brien et al. 1998](#org413fc20)) +- ([McInroy, O’Brien, and Neat 1999](#orgc5005e3)) +- ([McInroy 1999](#orgb4c311e)) +- ([McInroy and Hamann 2000](#org8285ab1)) +- ([Chen and McInroy 2000](#org709b3d5)) +- ([McInroy 2002](#org349aaf8)) +- ([Li, Hamann, and McInroy 2001](#orgaa83268)) +- ([Lin and McInroy 2003](#org055e9ff)) +- ([Jafari and McInroy 2003](#org26e42d2)) +- ([Chen and McInroy 2004](#orgd92590e)) + ## Bibliography {#bibliography} -Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. . +Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. . -Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . +Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. . -Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. . +Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. . -Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. . +Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. . -Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. . +Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. . -McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. . +McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. . -———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. . +———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. . -McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. . +McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. . -McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. . +McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. . -O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. . +O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. . diff --git a/content/zettels/test.md b/content/zettels/test.md index 4c8f3a8..491b608 100644 --- a/content/zettels/test.md +++ b/content/zettels/test.md @@ -78,13 +78,13 @@ Sed aliquam Here is a list of links to: -- Figure [3](#org54379fd) +- Figure [3](#orgcbf9e46) - Table [3](#table--tab:table-with-equations) - Listing [1](#code-snippet--lst:matlab-figure) - Specific line of code - Equation \eqref{eq:numbered} - Section -- Bibliographic Reference ([Stanisic and Legrand 2014](#orgfe85fe6)), and ([Schulte and Davison 2011](#orgdf0380b); [Dominik 2010](#orgb0733db); [Stanisic and Legrand 2014](#orgfe85fe6)) +- Bibliographic Reference ([Stanisic and Legrand 2014](#org0ed95e1)), and ([Schulte and Davison 2011](#org7b9fb79); [Dominik 2010](#org4f5b6d0); [Stanisic and Legrand 2014](#org0ed95e1)) ### Maths {#maths} @@ -157,7 +157,7 @@ Some text. ## Headlines {#headlines} - + ### Second level Headline with tags {#second-level-headline-with-tags} @@ -304,7 +304,7 @@ Cras non mauris ex. Morbi ut eros eu tellus egestas dapibus et et est. Aenean so xlabel('Time [s]'); ylabel('Voltage [V]'); ``` - + {{< figure src="figs/matlab_fig_example.png" caption="Figure 1: Matlab Figure" >}} @@ -375,7 +375,7 @@ Moreover, we can link to specific bode blocks (Listing [1](#code-snippet--lst:ma Code to produce a nice contour plot - + {{< figure src="figs/matlab_logo.png" caption="Figure 2: Obtained Contour Plot" >}} @@ -450,7 +450,7 @@ Numbering can be continued by using `+n` option as shown below. ### Normal Image {#normal-image} -Figure [3](#org54379fd) shows the results of the Tikz code of listing [4](#code-snippet--lst:tikz-test). +Figure [3](#orgcbf9e46) shows the results of the Tikz code of listing [4](#code-snippet--lst:tikz-test). ```latex @@ -477,10 +477,10 @@ Figure [3](#org54379fd) shows the results of the Tikz code of listing [4](#code-
Code Snippet 4: - Tikz code that is used to generate Figure 3 + Tikz code that is used to generate Figure 3
- + {{< figure src="figs/general_control_names.png" caption="Figure 3: General Control Configuration" >}} @@ -493,7 +493,7 @@ Figure [3](#org54379fd) shows the results of the Tikz code of listing [4](#code- ### Wrap Image {#wrap-image} - + {{< figure src="figs/general_control_names.png" caption="Figure 4: General Control Configuration" >}} @@ -509,7 +509,7 @@ Fusce blandit mauris dui, sed lobortis sapien tincidunt ac. Maecenas vitae moles [[file:figs/general_control_names.png]] ``` - + {{< figure src="figs/general_control_names.png" caption="Figure 5: General Control Configuration" >}} @@ -518,7 +518,7 @@ Fusce blandit mauris dui, sed lobortis sapien tincidunt ac. Maecenas vitae moles ### Sub Images {#sub-images} -Link to subfigure [2](#orga5ea12b). +Link to subfigure [2](#org0dc182a). ```md #+name: fig:subfigure @@ -536,7 +536,7 @@ Link to subfigure [2](#orga5ea12b). | ![](figs/general_control_names.png) | ![](figs/general_control_names.png) | |--------------------------------------------|--------------------------------------------| -| sub figure caption | sub figure caption | +| sub figure caption | sub figure caption | ## Tables {#tables} @@ -647,11 +647,11 @@ It is approximately **12,742 km** ## Bibliography {#bibliography} -Dominik, Carsten. 2010. _The Org Mode 7 Reference Manual-Organize Your Life with GNU Emacs_. Network Theory Ltd. +Dominik, Carsten. 2010. _The Org Mode 7 Reference Manual-Organize Your Life with GNU Emacs_. Network Theory Ltd. -Schulte, Eric, and Dan Davison. 2011. “Active Documents with Org-Mode.” _Computing in Science & Engineering_ 13 (3). IEEE Computer Society:66–73. +Schulte, Eric, and Dan Davison. 2011. “Active Documents with Org-Mode.” _Computing in Science & Engineering_ 13 (3). IEEE Computer Society:66–73. -Stanisic, Luka, and Arnaud Legrand. 2014. “Effective Reproducible Research with Org-Mode and Git.” In _European Conference on Parallel Processing_, 475–86. Springer. +Stanisic, Luka, and Arnaud Legrand. 2014. “Effective Reproducible Research with Org-Mode and Git.” In _European Conference on Parallel Processing_, 475–86. Springer. -[^fn:1]: A long foot note. Lorem ipsum dolor sit amet, consectetur adipiscing elit. With a reference to Figure [3](#org54379fd). +[^fn:1]: A long foot note. Lorem ipsum dolor sit amet, consectetur adipiscing elit. With a reference to Figure [3](#orgcbf9e46). [^fn:2]: An other footnote. diff --git a/content/zettels/trainings.md b/content/zettels/trainings.md index ceecdbb..3df287c 100644 --- a/content/zettels/trainings.md +++ b/content/zettels/trainings.md @@ -19,5 +19,3 @@ Mechatronics: Matlab: - [Mathworks](https://www.mathworks.com/training-schedule/) - -<./biblio/references.bib> diff --git a/content/zettels/transconductance_amplifiers.md b/content/zettels/transconductance_amplifiers.md index 7f7e2d0..94721d9 100644 --- a/content/zettels/transconductance_amplifiers.md +++ b/content/zettels/transconductance_amplifiers.md @@ -13,5 +13,3 @@ Tags A Transconductance Amplifier converts the control voltage into current with a current source characteristic. 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