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content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md
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@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}) : [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference Reference
: ([Zuo 2004](#orgdb2a627)) : ([Zuo 2004](#orgb4186fb))
Author(s) Author(s)
: Zuo, L. : Zuo, L.
@ -16,6 +16,20 @@ Author(s)
Year Year
: 2004 : 2004
<div style="display: none;">
\(
\newcommand{\eatLabel}[2]{}
\newenvironment{subequations}{\eatLabel}{}
\)
</div>
\begin{equation}
\begin{align}
\left[ H\_{xf}(\omega) \right]\_{n \times n} &= \left[ S\_{x^\prime v}(\omega) \right]\_{n \times n} \left[ S\_{f^\prime v}(\omega) \right]\_{n \times n}^{-1} \\\\\\
\left[ H\_{xf}(\omega) \right]\_{n \times n} &= \left[ S\_{f^\prime f^\prime}(\omega) \right]\_{n \times n}^{-1} \left[ S\_{x^\prime f^\prime}(\omega) \right]\_{n \times n}
\end{align}
\end{equation}
> Vibration isolation systems can have various system architectures. > Vibration isolation systems can have various system architectures.
> When we configure an active isolation system, we can use compliant actuators (such as voice coils) or stiff actuators (such as PZT stacks). > When we configure an active isolation system, we can use compliant actuators (such as voice coils) or stiff actuators (such as PZT stacks).
> We also need to consider how to **combine the active actuation with passive elements**: we can place the actuator in parallel or in series with the passive elements. > We also need to consider how to **combine the active actuation with passive elements**: we can place the actuator in parallel or in series with the passive elements.
@ -26,23 +40,23 @@ Year
> They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback. > 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. > Note that reaction force actuators can also work with soft mounts or hard mounts.
<a id="org8018206"></a> <a id="orgd66c057"></a>
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}} {{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
<a id="org8874676"></a> <a id="org1008b43"></a>
{{< 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_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
<a id="orga7046e2"></a> <a id="orgab03e30"></a>
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}} {{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
<a id="org735f298"></a> <a id="orgc03f8c8"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil 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} ## Bibliography {#bibliography}
<a id="orgdb2a627"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology. <a id="orgb4186fb"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.

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content/book/du10_model_contr_vibrat_mechan_system.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference Reference
: ([Du and Xie 2010](#orgad87753)) : ([Du and Xie 2010](#orge0a6379))
Author(s) Author(s)
: Du, C., & Xie, L. : Du, C., & Xie, L.
@ -21,4 +21,4 @@ Read Chapter 1 and 3.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgad87753"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>. <a id="orge0a6379"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>.

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content/book/du19_multi_actuat_system_contr.md
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Reference Reference
: ([Du and Pang 2019](#org86410b0)) : ([Du and Pang 2019](#orgfa82dec))
Author(s) Author(s)
: Du, C., & Pang, C. K. : Du, C., & Pang, C. K.
@ -17,6 +17,19 @@ Author(s)
Year Year
: 2019 : 2019
<div style="display: none;">
\(
\newcommand{\SI}[2]{#1\,#2}
% Simulate SIunitx
\newcommand{\ang}[1]{#1^{\circ}}
\newcommand{\degree}{^{\circ}}
\newcommand{\radian}{\text{rad}}
\newcommand{\percent}{\%}
\newcommand{\decibel}{\text{dB}}
\newcommand{\per}{/}
\)
</div>
## Mechanical Actuation Systems {#mechanical-actuation-systems} ## Mechanical Actuation Systems {#mechanical-actuation-systems}
@ -68,9 +81,9 @@ and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following fo
#### Secondary Actuators {#secondary-actuators} #### Secondary Actuators {#secondary-actuators}
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#orgf5ea358)) and the microactuator. We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org2501c9c)) and the microactuator.
<a id="orgf5ea358"></a> <a id="org2501c9c"></a>
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}} {{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
@ -92,9 +105,9 @@ There characteristics are shown on table [1](#table--tab:microactuator).
### Single-Stage Actuation Systems {#single-stage-actuation-systems} ### Single-Stage Actuation Systems {#single-stage-actuation-systems}
A typical closed-loop control system is shown on figure [2](#orga949a40), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller. A typical closed-loop control system is shown on figure [2](#org3f9b6d4), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
<a id="orga949a40"></a> <a id="org3f9b6d4"></a>
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}} {{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
@ -104,7 +117,7 @@ A typical closed-loop control system is shown on figure [2](#orga949a40), where
Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head. Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit. The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
<a id="orgc544eee"></a> <a id="org59cb446"></a>
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}} {{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
@ -130,7 +143,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
### Control Schemes {#control-schemes} ### Control Schemes {#control-schemes}
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#orgb533d71). A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org371fae9).
- \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\). - \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
- \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\). - \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@ -138,7 +151,7 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
- \\(n\\) is the measurement noise - \\(n\\) is the measurement noise
- \\(d\_u\\) stands for external vibration - \\(d\_u\\) stands for external vibration
<a id="orgb533d71"></a> <a id="org371fae9"></a>
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}} {{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
@ -160,14 +173,14 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\). And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
Thus, the dual-stage system control design can be decoupled into two independent controller designs. Thus, the dual-stage system control design can be decoupled into two independent controller designs.
Another type of control scheme is the **parallel structure** as shown in figure [5](#org3c23d1d). Another type of control scheme is the **parallel structure** as shown in figure [5](#org3d4cd09).
The open-loop transfer function from \\(pes\\) to \\(y\\) is The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\] \\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\] \\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
<a id="org3c23d1d"></a> <a id="org3d4cd09"></a>
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}} {{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
@ -177,7 +190,7 @@ Because of the limited displacement range of the secondary actuator, the control
### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain} ### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators. \\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org2b9887b) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function. The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org6dcd465) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
@ -187,11 +200,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha
is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\). is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
<a id="org2b9887b"></a> <a id="org6dcd465"></a>
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}} {{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
Equation [1](#orgcd45840) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\). Equation [1](#orgd1210d7) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
One form of \\(W(s)\\) is taken as One form of \\(W(s)\\) is taken as
\begin{equation} \begin{equation}
@ -204,16 +217,16 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
The primary and secondary actuator control loops are designed separately for the dual-stage control systems. The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority. But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
As seen in figure [7](#orgec2571e), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range. As seen in figure [7](#orgbe7f7d1), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
<a id="orgec2571e"></a> <a id="orgbe7f7d1"></a>
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}} {{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
The sensitivity functions are shown in figure [8](#orgc3be866), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible. The sensitivity functions are shown in figure [8](#orgfed486e), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop. This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
<a id="orgc3be866"></a> <a id="orgfed486e"></a>
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}} {{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
@ -246,13 +259,13 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
### Control Strategy and Controller Design {#control-strategy-and-controller-design} ### Control Strategy and Controller Design {#control-strategy-and-controller-design}
Figure [9](#org0a6e6a1) shows the control structure for the three-stage actuation system. Figure [9](#orga4074a3) shows the control structure for the three-stage actuation system.
The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system. The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop. The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately. The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
<a id="org0a6e6a1"></a> <a id="orga4074a3"></a>
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}} {{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
@ -281,15 +294,15 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible. The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes. The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
The open-loop frequency responses of the three stages are shown on figure [10](#org596d540). The open-loop frequency responses of the three stages are shown on figure [10](#org11a6581).
<a id="org596d540"></a> <a id="org11a6581"></a>
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}} {{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}
The obtained sensitivity function is shown on figure [11](#orgb011ee0). The obtained sensitivity function is shown on figure [11](#org58e9561).
<a id="orgb011ee0"></a> <a id="org58e9561"></a>
{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}} {{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
@ -304,7 +317,7 @@ Otherwise, saturation will occur in the control loop and the control system perf
Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance. Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations. Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
For the three-stage control architecture as shown on figure [9](#org0a6e6a1), the position error is For the three-stage control architecture as shown on figure [9](#orga4074a3), the position error is
\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\] \\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
The control signals and positions of the actuators are given by The control signals and positions of the actuators are given by
@ -320,11 +333,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
### Different Configurations of the Control System {#different-configurations-of-the-control-system} ### Different Configurations of the Control System {#different-configurations-of-the-control-system}
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org13d72d1)). A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org066c259)).
The overall sensitivity function is The overall sensitivity function is
\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\] \\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org8cccb61) and with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org7c7e2b1) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\] \\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\) Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@ -333,7 +346,7 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
The open-loop transfer function of the overall system is The open-loop transfer function of the overall system is
\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\] \\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
<a id="org13d72d1"></a> <a id="org066c259"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}} {{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
@ -345,9 +358,9 @@ The control signals and the positions of the three actuators are
u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
\end{align\*} \end{align\*}
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org96a1b82)). The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org8c9fc90)).
<a id="org96a1b82"></a> <a id="org8c9fc90"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}} {{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
@ -658,4 +671,4 @@ As a more advanced concept, PZT elements being used as actuator and sensor simul
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org86410b0"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press. <a id="orgfa82dec"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.

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@ -1,7 +1,7 @@
+++ +++
title = "Design, modeling and control of nanopositioning systems" title = "Design, modeling and control of nanopositioning systems"
author = ["Thomas Dehaeze"] author = ["Thomas Dehaeze"]
draft = false draft = true
+++ +++
Tags Tags
@ -9,7 +9,7 @@ Tags
Reference Reference
: ([Fleming and Leang 2014](#org378bdb9)) : ([Fleming and Leang 2014](#org53722bc))
Author(s) Author(s)
: Fleming, A. J., & Leang, K. K. : Fleming, A. J., & Leang, K. K.
@ -821,11 +821,11 @@ Year
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models} ### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
<a id="org80070ee"></a> <a id="orgaaa53eb"></a>
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}} {{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
Consider the electrical circuit shown in Figure [1](#org80070ee) where a voltage source is connected to a piezoelectric actuator. Consider the electrical circuit shown in Figure [1](#orgaaa53eb) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\). The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively. The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
@ -948,4 +948,4 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org378bdb9"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>. <a id="org53722bc"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.

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@ -8,7 +8,7 @@ Tags
: [Finite Element Model]({{< relref "finite_element_model" >}}) : [Finite Element Model]({{< relref "finite_element_model" >}})
Reference Reference
: ([Hatch 2000](#org50dffcf)) : ([Hatch 2000](#orgebf8ccb))
Author(s) Author(s)
: Hatch, M. R. : Hatch, M. R.
@ -21,14 +21,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction} ## Introduction {#introduction}
<a id="orgda412b2"></a> <a id="org4115a7e"></a>
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. 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} ### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#org4d6ba0d) shows the methodology for analyzing a lightly damped structure using normal modes. The diagram in Figure [1](#org242701e) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important"> <div class="important">
<div></div> <div></div>
@ -46,7 +46,7 @@ The steps are:
</div> </div>
<a id="org4d6ba0d"></a> <a id="org242701e"></a>
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}} {{< 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
<div class="important"> <div class="important">
<div></div> <div></div>
Figure [2](#org1081f0b) shows such process, the steps are: Figure [2](#org398c443) shows such process, the steps are:
- start with the finite element model - start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model) - compute the eigenvalues and eigenvectors (as many as dof in the model)
@ -71,14 +71,14 @@ Figure [2](#org1081f0b) shows such process, the steps are:
</div> </div>
<a id="org1081f0b"></a> <a id="org398c443"></a>
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}} {{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
### Notations {#notations} ### Notations {#notations}
Tables [3](#orgb6964ec), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document. Tables [3](#org02d84e8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a> <a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption"> <div class="table-caption">
@ -127,22 +127,22 @@ Tables [3](#orgb6964ec), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems} ## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<a id="org8996806"></a> <a id="orga20292c"></a>
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. 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**. 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](#orgb6964ec) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs. Figure [3](#org02d84e8) 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\\). The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="orgb6964ec"></a> <a id="org02d84e8"></a>
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}} {{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
<div class="important"> <div class="important">
<div></div> <div></div>
([Miu 1993](#org03acd9e)) 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](#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.
The resonances of the "overhanging appendages" of the constrained system create the zeros. 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} ## State Space Analysis {#state-space-analysis}
<a id="org8166c96"></a> <a id="org24eb004"></a>
## Modal Analysis {#modal-analysis} ## Modal Analysis {#modal-analysis}
<a id="org331466a"></a> <a id="orgc6df38d"></a>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section. 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} #### Equation of Motion {#equation-of-motion}
Let's consider the model shown in Figure [4](#org627cff8) 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](#org0c2921d) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="org627cff8"></a> <a id="org0c2921d"></a>
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}} {{< 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{bmatrix}
\end{equation} \end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#org0396b30), [6](#orgd3bc915) and [7](#orgc82dccd). Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc90fe3a), [6](#orgfd8222c) and [7](#orgaf9cc36).
<a id="org0396b30"></a> <a id="orgc90fe3a"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}} {{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
<a id="orgd3bc915"></a> <a id="orgfd8222c"></a>
{{< 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_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
<a id="orgc82dccd"></a> <a id="orgaf9cc36"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/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-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. 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](#org2a145bc). This procedure is schematically shown in Figure [8](#orgf9a2963).
<a id="org2a145bc"></a> <a id="orgf9a2963"></a>
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}} {{< 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} ## Frequency Response: Modal Form {#frequency-response-modal-form}
<a id="orgcf74144"></a> <a id="orgfd97109"></a>
The procedure to obtain the frequency response from a modal form is as follow: 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 - 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 - 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](#org5228de8). This will be applied to the model shown in Figure [9](#org48b68a4).
<a id="org5228de8"></a> <a id="org48b68a4"></a>
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}} {{< 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
</div> </div>
Figure [10](#org36b2696) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\). Figure [10](#org87763b9) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="org36b2696"></a> <a id="org87763b9"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_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} ## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
<a id="org6520d55"></a> <a id="org031e9ac"></a>
### Introduction {#introduction} ### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org332d1e7). In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orga66d597).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output. 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. The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
<a id="org332d1e7"></a> <a id="orga66d597"></a>
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}} {{< 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} ## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<a id="orgd3512da"></a> <a id="org9a76f4b"></a>
### Model Description {#model-description} ### 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} ## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<a id="org17e706f"></a> <a id="orga8b2a2f"></a>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org6d55a33)). 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)).
### Actuator Description {#actuator-description} ### Actuator Description {#actuator-description}
<a id="org6d55a33"></a> <a id="org94e126d"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}} {{< 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](#org482c35b)). 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)).
<a id="org482c35b"></a> <a id="org4a20950"></a>
{{< 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" >}} {{< 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. For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#org482c35b) shows the nodes used for the reduced Matlab model. Figure [13](#org4a20950) 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 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. The arrows at the nodes indicate the direction of forces.
@ -1045,6 +1045,9 @@ A small section of the exported `.eig` file from ANSYS is shown bellow..
<div class="exampl"> <div class="exampl">
<div></div> <div></div>
<div class="monoblock">
<div></div>
LOAD STEP= 1 SUBSTEP= 1 LOAD STEP= 1 SUBSTEP= 1
FREQ= 8.1532 LOAD CASE= 0 FREQ= 8.1532 LOAD CASE= 0
@ -1059,6 +1062,8 @@ NODE UX UY UZ ROTX ROTY ROTZ
</div> </div>
</div>
Important information are: Important information are:
- `SUBSTEP`: mode number - `SUBSTEP`: mode number
@ -1082,7 +1087,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction} ## Balanced Reduction {#balanced-reduction}
<a id="org2c8e979"></a> <a id="org56fcc2f"></a>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods. In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@ -1197,14 +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} ## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<a id="org0b45098"></a> <a id="orga7cf69e"></a>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#orgdc24ed7)). In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org1453e17)).
### Actuator Description {#actuator-description} ### Actuator Description {#actuator-description}
<a id="orgdc24ed7"></a> <a id="org1453e17"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
@ -1226,9 +1231,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description} ### Ansys Model Description {#ansys-model-description}
In Figure [15](#org40d5587) are shown the principal nodes used for the model. In Figure [15](#orge94bde1) are shown the principal nodes used for the model.
<a id="org40d5587"></a> <a id="orge94bde1"></a>
{{< 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" >}} {{< 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" >}}
@ -1347,11 +1352,11 @@ And we note:
G = zn * Gp; G = zn * Gp;
``` ```
<a id="org12f3141"></a> <a id="org22a3db4"></a>
{{< 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_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
<a id="orgd9eb688"></a> <a id="org33c49a2"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}} {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -1449,13 +1454,13 @@ G_f = ss(A, B, C, D);
### Simple mode truncation {#simple-mode-truncation} ### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#org152bcb2)). Let's plot the frequency of the modes (Figure [18](#orga04e866)).
<a id="org152bcb2"></a> <a id="orga04e866"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
<a id="orge00504f"></a> <a id="org0c4b8bc"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
@ -1524,7 +1529,7 @@ dc_gain = abs(xn(i_input, :).*xn(i_output, :))./(2*pi*f0).^2;
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend'); [dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
``` ```
<a id="orga1ddc35"></a> <a id="orga62ba4f"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
@ -1868,7 +1873,7 @@ wo = gram(G_m, 'o');
And we plot the diagonal terms And we plot the diagonal terms
<a id="org27ebe1f"></a> <a id="orgab7d0ba"></a>
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}} {{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@ -1886,7 +1891,7 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m); [G_b, G, T, Ti] = balreal(G_m);
``` ```
<a id="org801e76e"></a> <a id="orgb25e36e"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@ -2131,6 +2136,6 @@ pos_frames = pos([1, i_input, i_output], :);
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org50dffcf"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press. <a id="orgebf8ccb"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="org03acd9e"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York. <a id="org39eead7"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.

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@ -8,7 +8,7 @@ Tags
: [Metrology]({{< relref "metrology" >}}) : [Metrology]({{< relref "metrology" >}})
Reference Reference
: ([Leach 2014](#orgc3e03e3)) : ([Leach 2014](#orgc132434))
Author(s) Author(s)
: Leach, R. : Leach, R.
@ -89,4 +89,4 @@ This type of angular interferometer is used to measure small angles (less than \
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgc3e03e3"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>. <a id="orgc132434"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.

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@ -8,7 +8,7 @@ Tags
: [Precision Engineering]({{< relref "precision_engineering" >}}) : [Precision Engineering]({{< relref "precision_engineering" >}})
Reference Reference
: ([Leach and Smith 2018](#org545df46)) : ([Leach and Smith 2018](#org50ae2e1))
Author(s) Author(s)
: Leach, R., & Smith, S. T. : Leach, R., & Smith, S. T.
@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org545df46"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press. <a id="org50ae2e1"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.

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