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@ -8,7 +8,7 @@ Tags
: [System Identification]({{< relref "system_identification" >}}), [Reference Books]({{< relref "reference_books" >}}), [Modal Analysis]({{< relref "modal_analysis" >}})
Reference
: ([Ewins 2000](#org8088c4f))
: ([Ewins 2000](#org3f74bcc))
Author(s)
: Ewins, D.
@ -16,6 +16,9 @@ Author(s)
Year
: 2000
PDF version
: [link](/ox-hugo/ewins00_modal.pdf)
## Overview {#overview}
@ -141,7 +144,7 @@ The main measurement technique studied are those which will permit to make **dir
The type of test best suited to FRF measurement is shown in figure [fig:modal_analysis_schematic](#fig:modal_analysis_schematic).
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{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
@ -215,7 +218,7 @@ This assumption allows us to use the circular nature of a modulus/phase polar pl
This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
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{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
@ -253,7 +256,7 @@ Theoretical foundations of modal testing are of paramount importance to its succ
The three phases through a typical theoretical vibration analysis progresses are shown on figure [fig:vibration_analysis_procedure](#fig:vibration_analysis_procedure).
Generally, we start with a description of the structure's physical characteristics (mass, stiffness and damping properties), this is referred to as the **Spatial model**.
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{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
@ -298,7 +301,7 @@ Three classes of system model will be described:
The basic model for the SDOF system is shown in figure [fig:sdof_model](#fig:sdof_model) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
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{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
@ -374,7 +377,7 @@ which is a single mode of vibration with a complex natural frequency having two
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [fig:sdof_response](#fig:sdof_response)
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{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
@ -418,7 +421,7 @@ The damping effect of such a component can conveniently be defined by the ratio
| ![](/ox-hugo/ewins00_material_histeresis.png) | ![](/ox-hugo/ewins00_dry_friction.png) | ![](/ox-hugo/ewins00_viscous_damper.png) |
|-----------------------------------------------|----------------------------------------|------------------------------------------|
| <a id="orgbcdc049"></a> Material hysteresis | <a id="org18191a0"></a> Dry friction | <a id="org945cf1f"></a> Viscous damper |
| <a id="org9b2bd66"></a> Material hysteresis | <a id="orga9fe99b"></a> Dry friction | <a id="org8634a73"></a> Viscous damper |
| height=2cm | height=2cm | height=2cm |
Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
@ -537,7 +540,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [fig
| ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
|-------------------------------------------|-----------------------------------------|--------------------------------------------|
| <a id="org2eaa0c3"></a> Receptance FRF | <a id="org7e1d881"></a> Mobility FRF | <a id="org4e9fb38"></a> Accelerance FRF |
| <a id="org944c0ec"></a> Receptance FRF | <a id="orgfe4fe4a"></a> Mobility FRF | <a id="org286e856"></a> Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@ -560,7 +563,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
| ![](/ox-hugo/ewins00_plot_receptance_real.png) | ![](/ox-hugo/ewins00_plot_receptance_imag.png) |
|------------------------------------------------|------------------------------------------------|
| <a id="orgd1238ba"></a> Real part | <a id="org82cd2e0"></a> Imaginary part |
| <a id="org7615541"></a> Real part | <a id="org77a3c15"></a> Imaginary part |
| width=\linewidth | width=\linewidth |
@ -578,7 +581,7 @@ Figure [fig:inverse_frf_mixed](#fig:inverse_frf_mixed) shows an example of a plo
| ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
|---------------------------------------------|-----------------------------------------------|
| <a id="orgc522f0d"></a> Mixed | <a id="org6c88c76"></a> Viscous |
| <a id="orge8fcd8a"></a> Mixed | <a id="org7403b02"></a> Viscous |
| width=\linewidth | width=\linewidth |
@ -595,7 +598,7 @@ The missing information (in this case, the frequency) must be added by identifyi
| ![](/ox-hugo/ewins00_nyquist_receptance_viscous.png) | ![](/ox-hugo/ewins00_nyquist_receptance_structural.png) |
|------------------------------------------------------|---------------------------------------------------------|
| <a id="org0c0c036"></a> Viscous damping | <a id="org2f9ce1d"></a> Structural damping |
| <a id="org1f9bdad"></a> Viscous damping | <a id="org9c802e3"></a> Structural damping |
| width=\linewidth | width=\linewidth |
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@ -1103,7 +1106,7 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [fig:real_complex_modes](#fig:real_complex_modes)).
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{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
@ -1118,7 +1121,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
| ![](/ox-hugo/ewins00_argand_diagram_a.png) | ![](/ox-hugo/ewins00_argand_diagram_b.png) | ![](/ox-hugo/ewins00_argand_diagram_c.png) |
|--------------------------------------------|--------------------------------------------|-----------------------------------------------|
| <a id="orgddfe5f4"></a> Almost-real mode | <a id="org2799378"></a> Complex Mode | <a id="org35c2595"></a> Measure of complexity |
| <a id="org8589a1f"></a> Almost-real mode | <a id="orgbf11e51"></a> Complex Mode | <a id="org48d76d6"></a> Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -1235,7 +1238,7 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
| ![](/ox-hugo/ewins00_mobility_frf_mdof_point.png) | ![](/ox-hugo/ewins00_mobility_frf_mdof_transfer.png) |
|---------------------------------------------------|------------------------------------------------------|
| <a id="org006a892"></a> Point FRF | <a id="org9c58289"></a> Transfer FRF |
| <a id="org75cf239"></a> Point FRF | <a id="orgd1132b4"></a> Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [fig:mobility_frf_mdof_transfer](#fig:mobility_frf_mdof_transfer), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
@ -1260,7 +1263,7 @@ Most mobility plots have this general form as long as the modes are relatively w
This condition is satisfied unless the separation between adjacent natural frequencies is of the same order as, or less than, the modal damping factors, in which case it becomes difficult to distinguish the individual modes.
<a id="orgfc61536"></a>
<a id="org802e394"></a>
{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
@ -1281,7 +1284,7 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
| ![](/ox-hugo/ewins00_nyquist_point.png) | ![](/ox-hugo/ewins00_nyquist_transfer.png) |
|------------------------------------------|---------------------------------------------|
| <a id="orgecd04e1"></a> Point receptance | <a id="org8471776"></a> Transfer receptance |
| <a id="org47146fd"></a> Point receptance | <a id="orga5e6c6f"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
In the two figures [fig:nyquist_nonpropdamp_point](#fig:nyquist_nonpropdamp_point) and [fig:nyquist_nonpropdamp_transfer](#fig:nyquist_nonpropdamp_transfer), we show corresponding data for **non-proportional** damping.
@ -1296,7 +1299,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
| ![](/ox-hugo/ewins00_nyquist_nonpropdamp_point.png) | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_transfer.png) |
|-----------------------------------------------------|--------------------------------------------------------|
| <a id="orgad9090b"></a> Point receptance | <a id="org1385d00"></a> Transfer receptance |
| <a id="orged7cd3d"></a> Point receptance | <a id="orgf48b2a8"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
@ -1450,7 +1453,7 @@ Examples of random signals, autocorrelation function and power spectral density
| ![](/ox-hugo/ewins00_random_time.png) | ![](/ox-hugo/ewins00_random_autocorrelation.png) | ![](/ox-hugo/ewins00_random_psd.png) |
|---------------------------------------|--------------------------------------------------|------------------------------------------------|
| <a id="org8bf5606"></a> Time history | <a id="org2675b47"></a> Autocorrelation Function | <a id="orgd639dba"></a> Power Spectral Density |
| <a id="orgb0bc05b"></a> Time history | <a id="org0cee034"></a> Autocorrelation Function | <a id="org1896dbe"></a> Power Spectral Density |
| width=\linewidth | width=\linewidth | width=\linewidth |
A similar concept can be applied to a pair of functions such as \\(f(t)\\) and \\(x(t)\\) to produce **cross correlation** and **cross spectral density** functions.
@ -1547,7 +1550,7 @@ Then in [fig:frf_feedback_model](#fig:frf_feedback_model) is given a more detail
| ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png) |
|------------------------------------------|--------------------------------------------------|
| <a id="org02275d0"></a> Basic SISO model | <a id="org30e7018"></a> SISO model with feedback |
| <a id="orgabcc011"></a> Basic SISO model | <a id="org146b5c9"></a> SISO model with feedback |
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
@ -1580,7 +1583,7 @@ We obtain two alternative formulas:
In practical application of both of these formulae, care must be taken to ensure the non-singularity of the spectral density matrix which is to be inverted, and it is in this respect that the former version may be found to be more reliable.
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<a id="org14f8997"></a>
{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
@ -1852,7 +1855,7 @@ The experimental setup used for mobility measurement contains three major items:
A typical layout for the measurement system is shown on figure [fig:general_frf_measurement_setup](#fig:general_frf_measurement_setup).
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{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
@ -1909,7 +1912,7 @@ This can modify the response of the system in those directions.
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [fig:shaker_rod](#fig:shaker_rod).
Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}\\) in diameter, if the rod is longer, it may introduce the effect of its own resonances.
<a id="orgc17da13"></a>
<a id="org544daaf"></a>
{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
@ -1930,7 +1933,7 @@ Figure [fig:shaker_mount_3](#fig:shaker_mount_3) shows an unsatisfactory setup.
| ![](/ox-hugo/ewins00_shaker_mount_1.png) | ![](/ox-hugo/ewins00_shaker_mount_2.png) | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
| <a id="orgd87a497"></a> Ideal Configuration | <a id="orgad477d2"></a> Suspended Configuration | <a id="org826f9f9"></a> Unsatisfactory |
| <a id="orgf336e75"></a> Ideal Configuration | <a id="orgc6a61e4"></a> Suspended Configuration | <a id="org2b9e9bf"></a> Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -1948,7 +1951,7 @@ The frequency range which is effectively excited is controlled by the stiffness
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [fig:hammer_impulse](#fig:hammer_impulse).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [fig:hammer_impulse](#fig:hammer_impulse).
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<a id="orgfd37987"></a>
{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
@ -1979,7 +1982,7 @@ By suitable design, such a material may be incorporated into a device which **in
The force transducer is the simplest type of piezoelectric transducer.
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [fig:piezo_force_transducer](#fig:piezo_force_transducer)).
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{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
@ -1992,7 +1995,7 @@ In an accelerometer, transduction is indirect and is achieved using a seismic ma
In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
Thus, so long as the body and the seismic mass move together, the output of the transducer will be proportional to the acceleration of its body \\(x\\).
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{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
@ -2040,7 +2043,7 @@ Shown on figure [fig:transducer_mounting_response](#fig:transducer_mounting_resp
| ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png) |
|-----------------------------------------------------|------------------------------------------------------------|
| <a id="orgeaa12fc"></a> Attachment methods | <a id="org069b94e"></a> Frequency response characteristics |
| <a id="org4b6e4a8"></a> Attachment methods | <a id="orgd39cc35"></a> Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@ -2127,7 +2130,7 @@ Aliasing originates from the discretisation of the originally continuous time hi
With this discretisation process, the **existence of very high frequencies in the original signal may well be misinterpreted if the sampling rate is too slow**.
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [fig:aliasing](#fig:aliasing).
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{{< figure src="/ox-hugo/ewins00_aliasing.png" caption="Figure 14: The phenomenon of aliasing. On top: Low-frequency signal, On the bottom: High frequency signal" >}}
@ -2144,7 +2147,7 @@ This is illustrated on figure [fig:effect_aliasing](#fig:effect_aliasing).
| ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png) |
|--------------------------------------------------|-----------------------------------------------------|
| <a id="org0b90972"></a> True spectrum of signal | <a id="orgc8fbf25"></a> Indicated spectrum from DFT |
| <a id="org0213de5"></a> True spectrum of signal | <a id="org15546cb"></a> Indicated spectrum from DFT |
| width=\linewidth | width=\linewidth |
The solution of the problem is to use an **anti-aliasing filter** which subjects the original time signal to a low-pass, sharp cut-off filter.
@ -2165,7 +2168,7 @@ Leakage is a problem which is a direct **consequence of the need to take only a
| ![](/ox-hugo/ewins00_leakage_ok.png) | ![](/ox-hugo/ewins00_leakage_nok.png) |
|--------------------------------------|----------------------------------------|
| <a id="org33f3403"></a> Ideal signal | <a id="org6e8c780"></a> Awkward signal |
| <a id="orgbd19d54"></a> Ideal signal | <a id="org6111739"></a> Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [fig:leakage](#fig:leakage).
@ -2190,7 +2193,7 @@ Windowing involves the imposition of a prescribed profile on the time signal pri
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [fig:windowing_examples](#fig:windowing_examples).
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{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
@ -2211,7 +2214,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
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##### Increasing transform size {#increasing-transform-size}
@ -2247,10 +2250,10 @@ If we apply a band-pass filter to the signal, as shown on figure [fig:zoom_bandp
| ![](/ox-hugo/ewins00_zoom_range.png) | ![](/ox-hugo/ewins00_zoom_bandpass.png) |
|------------------------------------------------|------------------------------------------|
| <a id="org9e7d260"></a> Spectrum of the signal | <a id="orgdd4792e"></a> Band-pass filter |
| <a id="org11c8a57"></a> Spectrum of the signal | <a id="orgfa4dd48"></a> Band-pass filter |
| width=\linewidth | width=\linewidth |
<a id="org7f4ded4"></a>
<a id="org29bca60"></a>
{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
@ -2322,7 +2325,7 @@ This is the traditional method of FRF measurement and involves the use of a swee
It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [fig:sweep_distortions](#fig:sweep_distortions).
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<a id="org1599296"></a>
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
@ -2440,7 +2443,7 @@ It is known that a low coherence can arise in a measurement where the frequency
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [fig:coherence_resonance](#fig:coherence_resonance).
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{{< figure src="/ox-hugo/ewins00_coherence_resonance.png" caption="Figure 18: Coherence \\(\gamma^2\\) and FRF estimate \\(H\_1(\omega)\\) for a lightly damped structure" >}}
@ -2483,7 +2486,7 @@ For the chirp and impulse excitations, each individual sample is collected and p
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [fig:burst_excitation](#fig:burst_excitation)).
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{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
@ -2502,7 +2505,7 @@ The chirp consist of a short duration signal which has the form shown in figure
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
<a id="org4023c80"></a>
<a id="orgc4efbaa"></a>
{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
@ -2513,7 +2516,7 @@ The hammer blow produces an input and response as shown in the figure [fig:impul
This and the chirp excitation are very similar in the analysis point of view, the main difference is that the chirp offers the possibility of greater control of both amplitude and frequency content of the input and also permits the input of a greater amount of vibration energy.
<a id="orgb5fef72"></a>
<a id="org9fac12d"></a>
{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
@ -2523,7 +2526,7 @@ However, it should be recorded that in the region below the first cut-off freque
On some structures, the movement of the structure in response to the hammer blow can be such that it returns and **rebounds** on the hammer tip before the user has had time to move that out of the way.
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [fig:double_hits](#fig:double_hits)).
<a id="org9f9eb63"></a>
<a id="org4123ad7"></a>
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
@ -2598,7 +2601,7 @@ Suppose the response parameter is acceleration, then the FRF obtained is inertan
Figure [fig:calibration_setup](#fig:calibration_setup) shows a typical calibration setup.
<a id="org701a673"></a>
<a id="orgbe28ade"></a>
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
@ -2613,7 +2616,7 @@ This is because near resonance, the actual applied force becomes very small and
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [fig:mass_cancellation](#fig:mass_cancellation).
<a id="org5a93559"></a>
<a id="org9733cd1"></a>
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
@ -2670,7 +2673,7 @@ There are two problems to be tackled:
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [fig:rotational_measurement](#fig:rotational_measurement).
<a id="org91fd0a9"></a>
<a id="orga7e45cb"></a>
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
@ -2691,7 +2694,7 @@ First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneou
Then, the same excitation force is applied at the second position that gives a force \\(F\_0 = F\_2\\) and moment \\(M\_0 = F\_2 l\_2\\).
By adding and subtracting the responses produced by these two separate excitations conditions, we can deduce the translational and rotational responses to the translational force and the rotational moment separately, thus enabling the measurement of all four types of FRF: \\(X/F\\), \\(\Theta/F\\), \\(X/M\\) and \\(\Theta/M\\).
<a id="orgdeaeca9"></a>
<a id="orgc1246ae"></a>
{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
@ -3031,7 +3034,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_numerical_FRF.png) | ![](/ox-hugo/ewins00_PRF_numerical_svd.png) | ![](/ox-hugo/ewins00_PRF_numerical_PRF.png) |
|---------------------------------------------|---------------------------------------------|---------------------------------------------|
| <a id="org5e4d562"></a> FRF | <a id="org9ceb30f"></a> Singular Values | <a id="orga08a8a7"></a> PRF |
| <a id="orgfa46a8b"></a> FRF | <a id="org72e5201"></a> Singular Values | <a id="org6644f6d"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
<a id="table--fig:PRF-measured"></a>
@ -3042,7 +3045,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_measured_FRF.png) | ![](/ox-hugo/ewins00_PRF_measured_svd.png) | ![](/ox-hugo/ewins00_PRF_measured_PRF.png) |
|--------------------------------------------|--------------------------------------------|--------------------------------------------|
| <a id="org8035f6d"></a> FRF | <a id="org31cc333"></a> Singular Values | <a id="orgdaca43e"></a> PRF |
| <a id="org022ec27"></a> FRF | <a id="org1f277c1"></a> Singular Values | <a id="org4075cf4"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -3084,7 +3087,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
- the left singular vector \\(\\{U(\omega\_r)\\}\_1\\) which approximates the **mode shape** of that mode
- the right singular vector \\(\\{V(\omega\_r)\\}\_1\\) which represents the approximate **force pattern necessary to generate a response on that mode only**
<a id="orgc3aeceb"></a>
<a id="org8755428"></a>
{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
@ -3179,7 +3182,7 @@ The peak-picking method is applied as follows (illustrated on figure [fig:peak_a
It must be noted that the estimates of both damping and modal constant depend heavily on the accuracy of the maximum FRF level \\(|\hat{H}|\\) which is difficult to measure with great accuracy, especially for lightly damped systems.
Only real modal constants and thus real modes can be deduced by this method.
<a id="orgeef94b9"></a>
<a id="org05af8a6"></a>
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
@ -3214,7 +3217,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [fig:modal_circle
| ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png) |
|----------------------------------------|--------------------------------------------------------------------|
| <a id="orgfdf30cd"></a> Properties | <a id="org5d77a2c"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| <a id="org05973f0"></a> Properties | <a id="orgf4ab26e"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@ -3328,7 +3331,7 @@ The sequence is:
5. **Determine modal constant modulus and argument**.
The magnitude and argument of the modal constant is determined from the diameter of the circle and from its orientation relative to the Real and Imaginary axis.
<a id="org8bcb59c"></a>
<a id="orgfe9a7d1"></a>
{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
@ -3453,7 +3456,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
| ![](/ox-hugo/ewins00_residual_without.png) | ![](/ox-hugo/ewins00_residual_with.png) |
|--------------------------------------------|-----------------------------------------|
| <a id="org8959f1c"></a> without residual | <a id="org11cae37"></a> with residuals |
| <a id="org1ab142d"></a> without residual | <a id="org99525fc"></a> with residuals |
| width=\linewidth | width=\linewidth |
If we regenerate an FRF curve from the modal parameters we have extracted from the measured data, we shall use a formula of the type
@ -3484,7 +3487,7 @@ The three terms corresponds to:
These three terms are illustrated on figure [fig:low_medium_high_modes](#fig:low_medium_high_modes).
<a id="orgc4073ee"></a>
<a id="orgc97325b"></a>
{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
@ -3785,7 +3788,7 @@ As an example, a set of mobilities measured are shown individually in figure [fi
| ![](/ox-hugo/ewins00_composite_raw.png) | ![](/ox-hugo/ewins00_composite_sum.png) |
|-------------------------------------------|-----------------------------------------|
| <a id="org29cb641"></a> Individual curves | <a id="org2cc32c6"></a> Composite curve |
| <a id="orgdc23da1"></a> Individual curves | <a id="org4f39bca"></a> Composite curve |
| width=\linewidth | width=\linewidth |
The global analysis methods have the disadvantages first, that the computation power required is high and second that there may be valid reasons why the various FRF curves exhibit slight differences in their characteristics and it may not always be appropriate to average them.
@ -4332,7 +4335,7 @@ Measured coordinates of the test structure are first linked as shown on figure [
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [fig:static_display](#fig:static_display) (b).
The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
<a id="org101a589"></a>
<a id="org40e9e07"></a>
{{< figure src="/ox-hugo/ewins00_static_display.png" caption="Figure 31: Static display of modes shapes. (a) basic grid (b) single-frame deflection pattern (c) multiple-frame deflection pattern (d) complex mode (e) Argand diagram - quasi-real mode (f) Argand diagram - complex mode" >}}
@ -4377,7 +4380,7 @@ If we consider the first six modes of the beam, whose mode shapes are sketched i
All the higher modes will be indistinguishable from these first few.
This is a well known problem of **spatial aliasing**.
<a id="org3a0ba51"></a>
<a id="org6c7b3d5"></a>
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
@ -4440,7 +4443,7 @@ The inclusion of these two additional terms (obtained here only after measuring
| ![](/ox-hugo/ewins00_H22_without_residual.png) | ![](/ox-hugo/ewins00_H22_with_residual.png) |
|--------------------------------------------------------|-----------------------------------------------------------|
| <a id="org7065b88"></a> Using measured modal data only | <a id="orgaf204de"></a> After inclusion of residual terms |
| <a id="org403cf7b"></a> Using measured modal data only | <a id="orgec20458"></a> After inclusion of residual terms |
| width=\linewidth | width=\linewidth |
The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
@ -4495,7 +4498,7 @@ If the transmissibility is measured during a modal test which has a single excit
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [fig:transmissibility_plots](#fig:transmissibility_plots).
This may explain why transmissibilities are not widely used in modal analysis.
<a id="orgc659f30"></a>
<a id="orgaf8efc0"></a>
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
@ -4516,7 +4519,7 @@ The fact that the excitation force is not measured is responsible for the lack o
| ![](/ox-hugo/ewins00_conventional_modal_test_setup.png) | ![](/ox-hugo/ewins00_base_excitation_modal_setup.png) |
|---------------------------------------------------------|-------------------------------------------------------|
| <a id="org1352af3"></a> Conventional modal test setup | <a id="orga3e4265"></a> Base excitation setup |
| <a id="orga37d2ba"></a> Conventional modal test setup | <a id="orgddfcb08"></a> Base excitation setup |
| height=4cm | height=4cm |
@ -4559,4 +4562,4 @@ Because the rank of each pseudo matrix is less than its order, it cannot be inve
## Bibliography {#bibliography}
<a id="org8088c4f"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
<a id="org3f74bcc"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.

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content/book/skogestad07_multiv_feedb_contr.md
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@ -8,7 +8,7 @@ Tags
: [Reference Books]({{< relref "reference_books" >}}), [Multivariable Control]({{< relref "multivariable_control" >}})
Reference
: ([Skogestad and Postlethwaite 2007](#org57bef6b))
: ([Skogestad and Postlethwaite 2007](#org11783d5))
Author(s)
: Skogestad, S., & Postlethwaite, I.
@ -16,10 +16,13 @@ Author(s)
Year
: 2007
PDF version
: [link](/ox-hugo/skogestad07_multiv_feedb_contr.pdf)
## Introduction {#introduction}
<a id="orga0078c7"></a>
<a id="org00c11b7"></a>
### The Process of Control System Design {#the-process-of-control-system-design}
@ -190,7 +193,7 @@ Notations used throughout this note are summarized in tables&nbsp;[table:notatio
## Classical Feedback Control {#classical-feedback-control}
<a id="org7271725"></a>
<a id="orga4a5e87"></a>
### Frequency Response {#frequency-response}
@ -239,7 +242,7 @@ 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\\).
<a id="org77fbf8e"></a>
<a id="org5e365ec"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_alt.png" caption="Figure 1: Configuration for one degree-of-freedom control" >}}
@ -551,7 +554,7 @@ 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.&nbsp;[fig:classical_feedback_2dof_alt](#fig:classical_feedback_2dof_alt)), rather than operating on their difference \\(r - y\_m\\).
<a id="org81824cd"></a>
<a id="org129af4f"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_alt.png" caption="Figure 2: 2 degrees-of-freedom control architecture" >}}
@ -560,7 +563,7 @@ The controller can be slit into two separate blocks (Fig.&nbsp;[fig:classical_fe
- 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
<a id="org787203e"></a>
<a id="org4122018"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_sep.png" caption="Figure 3: 2 degrees-of-freedom control architecture with two separate blocs" >}}
@ -629,7 +632,7 @@ With (see Fig.&nbsp;[fig:performance_weigth](#fig:performance_weigth)):
</div>
<a id="orgca017f5"></a>
<a id="org4c3b768"></a>
{{< figure src="/ox-hugo/skogestad07_weight_first_order.png" caption="Figure 4: Inverse of performance weight" >}}
@ -653,7 +656,7 @@ After selecting the form of \\(N\\) and the weights, the \\(\hinf\\) optimal con
## Introduction to Multivariable Control {#introduction-to-multivariable-control}
<a id="orgbf0f66e"></a>
<a id="org2f9ff86"></a>
### Introduction {#introduction}
@ -696,7 +699,7 @@ For negative feedback system (Fig.&nbsp;[fig:classical_feedback_bis](#fig:classi
- \\(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\\)
<a id="org8fc2a9c"></a>
<a id="org70ec6ea"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_bis.png" caption="Figure 5: Conventional negative feedback control system" >}}
@ -1011,7 +1014,7 @@ The **structured singular value** \\(\mu\\) is a tool for analyzing the effects
The general control problem formulation is represented in Fig.&nbsp;[fig:general_control_names](#fig:general_control_names).
<a id="org0f9e5cf"></a>
<a id="org7ec4fea"></a>
{{< figure src="/ox-hugo/skogestad07_general_control_names.png" caption="Figure 6: General control configuration" >}}
@ -1041,7 +1044,7 @@ 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\\))
<a id="org5d57dcb"></a>
<a id="org822c0eb"></a>
{{< figure src="/ox-hugo/skogestad07_general_plant_weights.png" caption="Figure 7: General Weighted Plant" >}}
@ -1084,7 +1087,7 @@ where \\(F\_l(P, K)\\) denotes a **lower linear fractional transformation** (LFT
The general control configuration may be extended to include model uncertainty as shown in Fig.&nbsp;[fig:general_config_model_uncertainty](#fig:general_config_model_uncertainty).
<a id="orgc0d2312"></a>
<a id="orga4222dc"></a>
{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta.png" caption="Figure 8: General control configuration for the case with model uncertainty" >}}
@ -1112,7 +1115,7 @@ MIMO systems are often **more sensitive to uncertainty** than SISO systems.
## Elements of Linear System Theory {#elements-of-linear-system-theory}
<a id="org9517705"></a>
<a id="orga54f9d1"></a>
### System Descriptions {#system-descriptions}
@ -1398,7 +1401,7 @@ RHP-zeros therefore imply high gain instability.
### Internal Stability of Feedback Systems {#internal-stability-of-feedback-systems}
<a id="orgbd7faac"></a>
<a id="orgbcfb6c1"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_stability.png" caption="Figure 9: Block diagram used to check internal stability" >}}
@ -1545,7 +1548,7 @@ 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}
<a id="org92b7ead"></a>
<a id="orgc858ea4"></a>
### Input-Output Controllability {#input-output-controllability}
@ -1937,7 +1940,7 @@ Uncertainty in the crossover frequency region can result in poor performance and
### Summary: Controllability Analysis with Feedback Control {#summary-controllability-analysis-with-feedback-control}
<a id="orgcf527a3"></a>
<a id="orgf5efc91"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_meas.png" caption="Figure 10: Feedback control system" >}}
@ -1966,7 +1969,7 @@ In summary:
Sometimes, the disturbances are so large that we hit input saturation or the required bandwidth is not achievable. To avoid the latter problem, we must at least require that the effect of the disturbance is less than \\(1\\) at frequencies beyond the bandwidth:
\\[ \abs{G\_d(j\w)} < 1 \quad \forall \w \geq \w\_c \\]
<a id="org6de05c1"></a>
<a id="orge21a990"></a>
{{< figure src="/ox-hugo/skogestad07_margin_requirements.png" caption="Figure 11: Illustration of controllability requirements" >}}
@ -1988,7 +1991,7 @@ 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}
<a id="org2a52a06"></a>
<a id="org4dfb00d"></a>
### Introduction {#introduction}
@ -2299,7 +2302,7 @@ We here focus on input and output uncertainty.
In multiplicative form, the input and output uncertainties are given by (see Fig.&nbsp;[fig:input_output_uncertainty](#fig:input_output_uncertainty)):
\\[ G^\prime = (I + E\_O) G (I + E\_I) \\]
<a id="orge254987"></a>
<a id="org3fbf73d"></a>
{{< figure src="/ox-hugo/skogestad07_input_output_uncertainty.png" caption="Figure 12: Plant with multiplicative input and output uncertainty" >}}
@ -2435,7 +2438,7 @@ However, the situation is usually the opposite with model uncertainty because fo
## Uncertainty and Robustness for SISO Systems {#uncertainty-and-robustness-for-siso-systems}
<a id="org7590b78"></a>
<a id="org5704899"></a>
### Introduction to Robustness {#introduction-to-robustness}
@ -2509,7 +2512,7 @@ which may be represented by the diagram in Fig.&nbsp;[fig:input_uncertainty_set]
</div>
<a id="org59d99b4"></a>
<a id="orgd35c110"></a>
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set.png" caption="Figure 13: Plant with multiplicative uncertainty" >}}
@ -2563,7 +2566,7 @@ 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**
<a id="org9aee3fc"></a>
<a id="org4a26dbb"></a>
{{< figure src="/ox-hugo/skogestad07_uncertainty_region.png" caption="Figure 14: Uncertainty regions of the Nyquist plot at given frequencies" >}}
@ -2586,7 +2589,7 @@ At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped regi
</div>
<a id="org25a3a51"></a>
<a id="orgcfd9931"></a>
{{< figure src="/ox-hugo/skogestad07_uncertainty_disc_generated.png" caption="Figure 15: Disc-shaped uncertainty regions generated by complex additive uncertainty" >}}
@ -2643,7 +2646,7 @@ To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_
</div>
<a id="org15a3cec"></a>
<a id="org998234c"></a>
{{< 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}\\)" >}}
@ -2682,7 +2685,7 @@ The magnitude of the relative uncertainty caused by neglecting the dynamics in \
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.&nbsp;[fig:neglected_time_delay](#fig:neglected_time_delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
\\[ 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} \\]
<a id="org45ae2b1"></a>
<a id="orgb55c79b"></a>
{{< figure src="/ox-hugo/skogestad07_neglected_time_delay.png" caption="Figure 17: Neglected time delay" >}}
@ -2692,7 +2695,7 @@ Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{m
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.&nbsp;[fig:neglected_first_order_lag](#fig:neglected_first_order_lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
\\[ w\_I(s) = \frac{\tau\_{\text{max}} s}{\tau\_{\text{max}} s + 1} \\]
<a id="orga754d5c"></a>
<a id="orgf44e8cb"></a>
{{< figure src="/ox-hugo/skogestad07_neglected_first_order_lag.png" caption="Figure 18: Neglected first-order lag uncertainty" >}}
@ -2709,7 +2712,7 @@ However, as shown in Fig.&nbsp;[fig:lag_delay_uncertainty](#fig:lag_delay_uncert
It is suggested to start with the simple weight and then if needed, to try the higher order weight.
<a id="org51e6318"></a>
<a id="org93262ff"></a>
{{< figure src="/ox-hugo/skogestad07_lag_delay_uncertainty.png" caption="Figure 19: Multiplicative weight for gain and delay uncertainty" >}}
@ -2749,7 +2752,7 @@ 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\*}
<a id="org8a7056b"></a>
<a id="orgbad333b"></a>
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback.png" caption="Figure 20: Feedback system with multiplicative uncertainty" >}}
@ -2765,7 +2768,7 @@ Encirclements are avoided if none of the discs cover \\(-1\\), and we get:
&\Leftrightarrow \quad \abs{w\_I T} < 1, \ \forall\w \\\\\\
\end{align\*}
<a id="org8bc35ca"></a>
<a id="orge9928b5"></a>
{{< figure src="/ox-hugo/skogestad07_nyquist_uncertainty.png" caption="Figure 21: Nyquist plot of \\(L\_p\\) for robust stability" >}}
@ -2803,7 +2806,7 @@ And we obtain the same condition as before.
We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty (Fig.&nbsp;[fig:inverse_uncertainty_set](#fig:inverse_uncertainty_set)) in which
\\[ G\_p = G(1 + w\_{iI}(s) \Delta\_{iI})^{-1} \\]
<a id="org3e7ba07"></a>
<a id="org13fd454"></a>
{{< figure src="/ox-hugo/skogestad07_inverse_uncertainty_set.png" caption="Figure 22: Feedback system with inverse multiplicative uncertainty" >}}
@ -2855,7 +2858,7 @@ The condition for nominal performance when considering performance in terms of t
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.&nbsp;[fig:nyquist_performance_condition](#fig:nyquist_performance_condition).
<a id="org22ac31b"></a>
<a id="orgd24a4f6"></a>
{{< 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}\\)" >}}
@ -2880,7 +2883,7 @@ Let's consider the case of multiplicative uncertainty as shown on Fig.&nbsp;[fig
The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \Delta\_I\\) and the set of possible loop transfer functions is
\\[ L\_p = G\_p K = L (1 + w\_I \Delta\_I) = L + w\_I L \Delta\_I \\]
<a id="orga419cf8"></a>
<a id="org069ce98"></a>
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight_bis.png" caption="Figure 24: Diagram for robust performance with multiplicative uncertainty" >}}
@ -3046,7 +3049,7 @@ with \\(\Phi(s) \triangleq (sI - A)^{-1}\\).
This is illustrated in the block diagram of Fig.&nbsp;[fig:uncertainty_state_a_matrix](#fig:uncertainty_state_a_matrix), which is in the form of an inverse additive perturbation.
<a id="org7cd4d84"></a>
<a id="org9a3d0a4"></a>
{{< figure src="/ox-hugo/skogestad07_uncertainty_state_a_matrix.png" caption="Figure 25: Uncertainty in state space A-matrix" >}}
@ -3064,7 +3067,7 @@ We also derived a condition for robust performance with multiplicative uncertain
## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis}
<a id="org2be789b"></a>
<a id="org4f20e35"></a>
### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty}
@ -3075,13 +3078,13 @@ where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g.
If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig.&nbsp;[fig:general_control_delta](#fig:general_control_delta). This form is useful for controller synthesis.
<a id="orgc602523"></a>
<a id="orgb0aeaeb"></a>
{{< 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.&nbsp;[fig:general_control_Ndelta](#fig:general_control_Ndelta).
<a id="orgb849575"></a>
<a id="orga6aaa95"></a>
{{< figure src="/ox-hugo/skogestad07_general_control_Ndelta.png" caption="Figure 27: \\(N\Delta\text{-structure}\\) for robust performance analysis" >}}
@ -3101,7 +3104,7 @@ Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\),
To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig.&nbsp;[fig:general_control_Mdelta_bis](#fig:general_control_Mdelta_bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
<a id="org8eb2223"></a>
<a id="org54767ec"></a>
{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta_bis.png" caption="Figure 28: \\(M\Delta\text{-structure}\\) for robust stability analysis" >}}
@ -3153,7 +3156,7 @@ 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) |
|----------------------------------------------------|----------------------------------------------------------|-----------------------------------------------------------|
| <a id="org27b2961"></a> Additive uncertainty | <a id="org269d6fa"></a> Multiplicative input uncertainty | <a id="org51f791f"></a> Multiplicative output uncertainty |
| <a id="orge5a4bc6"></a> Additive uncertainty | <a id="orgaad9985"></a> Multiplicative input uncertainty | <a id="orgb80428f"></a> Multiplicative output uncertainty |
In Fig.&nbsp;[fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
@ -3176,7 +3179,7 @@ In Fig.&nbsp;[fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feed
| ![](/ox-hugo/skogestad07_inv_additive_uncertainty.png) | ![](/ox-hugo/skogestad07_inv_input_uncertainty.png) | ![](/ox-hugo/skogestad07_inv_output_uncertainty.png) |
|--------------------------------------------------------|------------------------------------------------------------------|-------------------------------------------------------------------|
| <a id="orgc0a9c0b"></a> Inverse additive uncertainty | <a id="org90a3fb2"></a> Inverse multiplicative input uncertainty | <a id="orgb1747a9"></a> Inverse multiplicative output uncertainty |
| <a id="org661c497"></a> Inverse additive uncertainty | <a id="org63413ec"></a> Inverse multiplicative input uncertainty | <a id="orgd2892d0"></a> Inverse multiplicative output uncertainty |
##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation}
@ -3246,7 +3249,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig.&nbsp;[fig:input_uncertainty_set_feedback_weight](#fig:input_uncertainty_set_feedback_weight).
\\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
<a id="org31ea15f"></a>
<a id="orgad900ed"></a>
{{< 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" >}}
@ -3406,7 +3409,7 @@ 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.
<a id="org3996f6b"></a>
<a id="orga1e768f"></a>
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="Figure 30: Coprime Uncertainty" >}}
@ -3438,7 +3441,7 @@ where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same di
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.&nbsp;[fig:block_diagonal_scalings](#fig:block_diagonal_scalings).
This clearly has no effect on stability.
<a id="org5377248"></a>
<a id="orgca7a936"></a>
{{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="Figure 31: Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
@ -3754,7 +3757,7 @@ with the decoupling controller we have:
\\[ \bar{\sigma}(N\_{22}) = \bar{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right| \\]
and we see from Fig.&nbsp;[fig:mu_plots_distillation](#fig:mu_plots_distillation) that the NP-condition is satisfied.
<a id="org6561d8a"></a>
<a id="org6367e11"></a>
{{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="Figure 32: \\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
@ -3877,7 +3880,7 @@ The latter is an attempt to "flatten out" \\(\mu\\).
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.&nbsp;[fig:weights_distillation](#fig:weights_distillation).
<a id="org4a75c93"></a>
<a id="org217e6d7"></a>
{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 33: Uncertainty and performance weights" >}}
@ -3900,11 +3903,11 @@ 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.
<a id="org316e326"></a>
<a id="orgb5d1c3d"></a>
{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 34: Change in \\(\mu\\) during DK-iteration" >}}
<a id="org585c918"></a>
<a id="org5c1f5e2"></a>
{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 35: Change in D-scale \\(d\_1\\) during DK-iteration" >}}
@ -3912,13 +3915,13 @@ The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\
The objectives of RS and NP are easily satisfied.
The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\bar{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
<a id="orgc63d84a"></a>
<a id="org5e38f39"></a>
{{< 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.&nbsp;[fig:perturb_s_k3](#fig:perturb_s_k3).
<a id="orgfc73254"></a>
<a id="org4119d9f"></a>
{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 37: Perturbed sensitivity functions \\(\bar{\sigma}(S^\prime)\\) using \\(\mu\\) \"optimal\" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
@ -3973,7 +3976,7 @@ If resulting control performance is not satisfactory, one may switch to the seco
## Controller Design {#controller-design}
<a id="org81cd286"></a>
<a id="org501dbc0"></a>
### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design}
@ -3993,7 +3996,7 @@ We have the following important relationships:
\end{align}
\end{subequations}
<a id="orgfc101bb"></a>
<a id="org2b5ff7f"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 38: One degree-of-freedom feedback configuration" >}}
@ -4035,7 +4038,7 @@ 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.&nbsp;[fig:design_trade_off_mimo_gk](#fig:design_trade_off_mimo_gk).
<a id="orgb8b3048"></a>
<a id="org8eb6fc6"></a>
{{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="Figure 39: Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
@ -4092,7 +4095,7 @@ The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}
We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig.&nbsp;[fig:lqg_separation](#fig:lqg_separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
<a id="org0ee8748"></a>
<a id="org905be3a"></a>
{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 40: The separation theorem" >}}
@ -4142,7 +4145,7 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
</div>
<a id="org9328dd8"></a>
<a id="org835e4e9"></a>
{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="Figure 41: The LQG controller and noisy plant" >}}
@ -4163,7 +4166,7 @@ It has the same degree (number of poles) as the plant.<br />
For the LQG-controller, as shown on Fig.&nbsp;[fig:lqg_kalman_filter](#fig:lqg_kalman_filter), 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.&nbsp;[fig:lqg_integral](#fig:lqg_integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
<a id="orgb96cd46"></a>
<a id="org54ba593"></a>
{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 42: LQG controller with integral action and reference input" >}}
@ -4176,18 +4179,18 @@ Their main limitation is that they can only be applied to minimum phase plants.
### \\(\htwo\\) and \\(\hinf\\) Control {#htwo--and--hinf--control}
<a id="orga525cc0"></a>
<a id="org61b4f13"></a>
#### General Control Problem Formulation {#general-control-problem-formulation}
<a id="orgfc02b74"></a>
<a id="org4a10a34"></a>
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.&nbsp;[fig:general_control](#fig:general_control).
<a id="orgd8daf7e"></a>
<a id="orgbc33bb3"></a>
{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 43: General control configuration" >}}
@ -4438,7 +4441,7 @@ The elements of the generalized plant are
\end{array}
\end{equation\*}
<a id="orgf4bf125"></a>
<a id="orgee004ae"></a>
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 44: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
@ -4447,7 +4450,7 @@ 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.&nbsp;[fig:mixed_sensitivity_dist_rejection](#fig:mixed_sensitivity_dist_rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
<a id="orge63037d"></a>
<a id="orgfd3e068"></a>
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 45: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
@ -4471,7 +4474,7 @@ The elements of the generalized plant are
\end{array}
\end{equation\*}
<a id="orgcae1c61"></a>
<a id="org8c0d917"></a>
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 46: \\(S/T\\) mixed-sensitivity optimization in standard form" >}}
@ -4499,7 +4502,7 @@ 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.&nbsp;[fig:input_uncertainty_hinf](#fig:input_uncertainty_hinf), 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\\).
<a id="orgcbbbe4d"></a>
<a id="org4894d3a"></a>
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="Figure 47: Multiplicative dynamic uncertainty model" >}}
@ -4521,7 +4524,7 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
w = \begin{bmatrix}d\\r\\n\end{bmatrix},\ z = \begin{bmatrix}z\_1\\z\_2\end{bmatrix}, \ v = \begin{bmatrix}r\_s\\y\_m\end{bmatrix},\ u = u
\end{equation\*}
<a id="orgbf6e2ac"></a>
<a id="org8550dcd"></a>
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 48: A signal-based \\(\hinf\\) control problem" >}}
@ -4532,7 +4535,7 @@ This problem is a non-standard \\(\hinf\\) optimization.
It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) procedure can be applied where we require the structured singular value:
\\[ \mu(M(j\omega)) < 1, \quad \forall\omega \\]
<a id="org8211ec2"></a>
<a id="org1100ab1"></a>
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 49: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
@ -4590,7 +4593,7 @@ For the perturbed feedback system of Fig.&nbsp;[fig:coprime_uncertainty_bis](#fi
Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bmatrix}u\\y\end{bmatrix}\\) and \\((I-GK)^{-1}\\) is the sensitivity function for this positive feedback arrangement.
<a id="org6ec03ef"></a>
<a id="orga4f86fd"></a>
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 50: \\(\hinf\\) robust stabilization problem" >}}
@ -4637,7 +4640,7 @@ 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}
<a id="org9136674"></a>
<a id="org6c9fad4"></a>
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.
@ -4650,7 +4653,7 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
as shown in Fig.&nbsp;[fig:shaped_plant](#fig:shaped_plant).
<a id="orgef11ed5"></a>
<a id="org5cc80bb"></a>
{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 51: The shaped plant and controller" >}}
@ -4687,7 +4690,7 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
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\\)
<a id="orgbfd1976"></a>
<a id="orgcca7b5c"></a>
{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 52: A practical implementation of the loop-shaping controller" >}}
@ -4713,7 +4716,7 @@ But in cases where stringent time-domain specifications are set on the output re
A general two degrees-of-freedom feedback control scheme is depicted in Fig.&nbsp;[fig:classical_feedback_2dof_simple](#fig:classical_feedback_2dof_simple).
The commands and feedbacks enter the controller separately and are independently processed.
<a id="org02d3783"></a>
<a id="orgd29e8b7"></a>
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 53: General two degrees-of-freedom feedback control scheme" >}}
@ -4724,7 +4727,7 @@ The design problem is illustrated in Fig.&nbsp;[fig:coprime_uncertainty_hinf](#f
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**.
<a id="org79631f7"></a>
<a id="orge2361a6"></a>
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 54: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
@ -4749,7 +4752,7 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\
The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig.&nbsp;[fig:hinf_synthesis_2dof](#fig:hinf_synthesis_2dof).
<a id="org44f4511"></a>
<a id="org4f84c7d"></a>
{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 55: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
@ -4821,7 +4824,7 @@ where \\(u\_a\\) is the **actual plant input**, that is the measurement at the *
The situation is illustrated in Fig.&nbsp;[fig:weight_anti_windup](#fig:weight_anti_windup), where the actuators are each modeled by a unit gain and a saturation.
<a id="org6787ef9"></a>
<a id="org33e91bc"></a>
{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 56: Self-conditioned weight \\(W\_1\\)" >}}
@ -4869,14 +4872,14 @@ Moreover, one should be careful about combining controller synthesis and analysi
## Controller Structure Design {#controller-structure-design}
<a id="orgb7b170f"></a>
<a id="orgfdae959"></a>
### Introduction {#introduction}
In previous sections, we considered the general problem formulation in Fig.&nbsp;[fig:general_control_names_bis](#fig:general_control_names_bis) 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\\).
<a id="orgfc83c01"></a>
<a id="orgc01d2f1"></a>
{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 57: General Control Configuration" >}}
@ -4911,7 +4914,7 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
Additional layers are possible, as is illustrated in Fig.&nbsp;[fig:control_system_hierarchy](#fig:control_system_hierarchy) which shows a typical control hierarchy for a chemical plant.
<a id="orgb38fb33"></a>
<a id="orgb5b26e9"></a>
{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 58: Typical control system hierarchy in a chemical plant" >}}
@ -4933,7 +4936,7 @@ 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) |
|--------------------------------------------------|--------------------------------------------------------------------------------|-------------------------------------------------------------|
| <a id="org46243fe"></a> Open loop optimization | <a id="org9054ed1"></a> Closed-loop implementation with separate control layer | <a id="org7c9ab5b"></a> Integrated optimization and control |
| <a id="org5ccb7d9"></a> Open loop optimization | <a id="orgf6234ce"></a> Closed-loop implementation with separate control layer | <a id="org9e71e3f"></a> Integrated optimization and control |
### Selection of Controlled Outputs {#selection-of-controlled-outputs}
@ -5140,7 +5143,7 @@ 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) |
|-------------------------------------------------------|---------------------------------------------------|
| <a id="orgb5eec16"></a> Extra measurements \\(y\_2\\) | <a id="org17e03b3"></a> Extra inputs \\(u\_2\\) |
| <a id="org125610c"></a> Extra measurements \\(y\_2\\) | <a id="org98bcc42"></a> Extra inputs \\(u\_2\\) |
#### Cascade Control: Extra Measurements {#cascade-control-extra-measurements}
@ -5189,7 +5192,7 @@ With reference to the special (but common) case of cascade control shown in Fig.
</div>
<a id="orga224135"></a>
<a id="orgcb932fe"></a>
{{< 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\\)" >}}
@ -5239,7 +5242,7 @@ We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\)
</div>
<a id="org96aa47c"></a>
<a id="org4db6270"></a>
{{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="Figure 60: Control configuration with two layers of cascade control" >}}
@ -5354,7 +5357,7 @@ We get:
\end{aligned}
\end{equation}
<a id="orgf93ad55"></a>
<a id="org5d7da0f"></a>
{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 61: Partial Control" >}}
@ -5474,7 +5477,7 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig.&nbsp;[fig:decentralized_diagonal_control](#fig:decentralized_diagonal_control)).
<a id="org6e9e0ea"></a>
<a id="orgeffcd1b"></a>
{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 62: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
@ -5861,7 +5864,7 @@ The conditions are also useful in an **input-output controllability analysis** f
## Model Reduction {#model-reduction}
<a id="org01b0041"></a>
<a id="org9778e55"></a>
### Introduction {#introduction}
@ -6268,4 +6271,4 @@ In such a case, using truncation or optimal Hankel norm approximation with appro
## Bibliography {#bibliography}
<a id="org57bef6b"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
<a id="org11783d5"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.

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content/book/taghirad13_paral.md
View File

@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Reference Books]({{< relref "reference_books" >}})
Reference
: ([Taghirad 2013](#orge267b8c))
: ([Taghirad 2013](#orgf5fcdad))
Author(s)
: Taghirad, H.
@ -16,10 +16,13 @@ Author(s)
Year
: 2013
PDF version
: [link](/ox-hugo/taghirad13_paral.pdf)
## Introduction {#introduction}
<a id="org157e93f"></a>
<a id="org1585c2f"></a>
This book is intended to give some analysis and design tools for the increase number of engineers and researchers who are interested in the design and implementation of parallel robots.
A systematic approach is presented to analyze the kinematics, dynamics and control of parallel robots.
@ -44,7 +47,7 @@ The control of parallel robots is elaborated in the last two chapters, in which
## Motion Representation {#motion-representation}
<a id="org363db2c"></a>
<a id="org57436ee"></a>
### Spatial Motion Representation {#spatial-motion-representation}
@ -59,7 +62,7 @@ Let us define:
The absolute position of point \\(P\\) of the rigid body can be constructed from the relative position of that point with respect to the moving frame \\(\\{\bm{B}\\}\\), and the **position and orientation** of the moving frame \\(\\{\bm{B}\\}\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
<a id="org5533e51"></a>
<a id="org237be68"></a>
{{< figure src="/ox-hugo/taghirad13_rigid_body_motion.png" caption="Figure 1: Representation of a rigid body spatial motion" >}}
@ -84,7 +87,7 @@ It can be **represented in several different ways**: the rotation matrix, the sc
##### Rotation Matrix {#rotation-matrix}
We consider a rigid body that has been exposed to a pure rotation.
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#org6f4a801)).
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#org62850f5)).
A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
@ -106,7 +109,7 @@ in which \\({}^A\hat{\bm{x}}\_B, {}^A\hat{\bm{y}}\_B\\) and \\({}^A\hat{\bm{z}}\
The nine elements of the rotation matrix can be simply represented as the projections of the Cartesian unit vectors of frame \\(\\{\bm{B}\\}\\) on the unit vectors of frame \\(\\{\bm{A}\\}\\).
<a id="org6f4a801"></a>
<a id="org62850f5"></a>
{{< figure src="/ox-hugo/taghirad13_rotation_matrix.png" caption="Figure 2: Pure rotation of a rigid body" >}}
@ -132,7 +135,7 @@ The term screw axis for this axis of rotation has the benefit that a general mot
The screw axis representation has the benefit of **using only four parameters** to describe a pure rotation.
These parameters are the angle of rotation \\(\theta\\) and the axis of rotation which is a unit vector \\({}^A\hat{\bm{s}} = [s\_x, s\_y, s\_z]^T\\).
<a id="orgee29ec0"></a>
<a id="org7be28ae"></a>
{{< figure src="/ox-hugo/taghirad13_screw_axis_representation.png" caption="Figure 3: Pure rotation about a screw axis" >}}
@ -158,7 +161,7 @@ Three other types of Euler angles are consider with respect to a moving frame: t
The pitch, roll and yaw angles are defined for a moving object in space as the rotations along the lateral, longitudinal and vertical axes attached to the moving object.
<a id="orgb162cd6"></a>
<a id="orgda1c23e"></a>
{{< figure src="/ox-hugo/taghirad13_pitch-roll-yaw.png" caption="Figure 4: Definition of pitch, roll and yaw angles on an air plain" >}}
@ -363,7 +366,7 @@ There exist transformations to from screw displacement notation to the transform
Let us consider the motion of a rigid body described at three locations (Figure [fig:consecutive_transformations](#fig:consecutive_transformations)).
Frame \\(\\{\bm{A}\\}\\) represents the initial location, frame \\(\\{\bm{B}\\}\\) is an intermediate location, and frame \\(\\{\bm{C}\\}\\) represents the rigid body at its final location.
<a id="orgc1a7048"></a>
<a id="orgd7ad6e9"></a>
{{< figure src="/ox-hugo/taghirad13_consecutive_transformations.png" caption="Figure 5: Motion of a rigid body represented at three locations by frame \\(\\{\bm{A}\\}\\), \\(\\{\bm{B}\\}\\) and \\(\\{\bm{C}\\}\\)" >}}
@ -426,7 +429,7 @@ Hence, the **inverse of the transformation matrix** can be obtain by
## Kinematics {#kinematics}
<a id="orgdc3f0a7"></a>
<a id="orgb134fb4"></a>
### Introduction {#introduction}
@ -533,7 +536,7 @@ The position of the point \\(O\_B\\) of the moving platform is described by the
\end{bmatrix}
\end{equation}
<a id="org02bb8ee"></a>
<a id="orga8573da"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_schematic.png" caption="Figure 6: Geometry of a Stewart-Gough platform" >}}
@ -586,7 +589,7 @@ The complexity of the problem depends widely on the manipulator architecture and
## Jacobian: Velocities and Static Forces {#jacobian-velocities-and-static-forces}
<a id="org7462b3d"></a>
<a id="org7012945"></a>
### Introduction {#introduction}
@ -683,7 +686,7 @@ The matrix \\(\bm{\Omega}^\times\\) denotes a **skew-symmetric matrix** defined
Now consider the general motion of a rigid body shown in Figure [fig:general_motion](#fig:general_motion), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
<a id="org38d1c64"></a>
<a id="orgb6a6b15"></a>
{{< figure src="/ox-hugo/taghirad13_general_motion.png" caption="Figure 7: Instantaneous velocity of a point \\(P\\) with respect to a moving frame \\(\\{\bm{B}\\}\\)" >}}
@ -942,7 +945,7 @@ We obtain that the **Jacobian matrix** constructs the **transformation needed to
As shown in Figure [fig:stewart_static_forces](#fig:stewart_static_forces), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
<a id="org1e5d0ac"></a>
<a id="orgaf66685"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_static_forces.png" caption="Figure 8: Free-body diagram of forces and moments action on the moving platform and each limb of the Stewart-Gough platform" >}}
@ -1099,7 +1102,7 @@ in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the small
In this section, we restrict our analysis to a 3-6 structure (Figure [fig:stewart36](#fig:stewart36)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
<a id="org5661020"></a>
<a id="org4537864"></a>
{{< figure src="/ox-hugo/taghirad13_stewart36.png" caption="Figure 9: Schematic of a 3-6 Stewart-Gough platform" >}}
@ -1129,7 +1132,7 @@ The largest axis of the stiffness transformation hyper-ellipsoid is given by thi
## Dynamics {#dynamics}
<a id="org7575df8"></a>
<a id="orgf66f22c"></a>
### Introduction {#introduction}
@ -1260,7 +1263,7 @@ For the case where \\(P\\) is a point embedded in the rigid body, \\({}^B\bm{v}\
In this section, the properties of mass, namely **center of mass**, **moments of inertia** and its characteristics and the required transformations are described.
<a id="org1e725b6"></a>
<a id="orgdbdd46c"></a>
{{< figure src="/ox-hugo/taghirad13_mass_property_rigid_body.png" caption="Figure 10: Mass properties of a rigid body" >}}
@ -1374,7 +1377,7 @@ in which \\({}^A\bm{v}\_C\\) denotes the velocity of the center of mass with res
This result implies that the **total linear momentum** of differential masses is equal to the linear momentum of a **point mass** \\(m\\) located at the **center of mass**.
This highlights the important of the center of mass in dynamic formulation of rigid bodies.
<a id="org5ee285a"></a>
<a id="org0fc91f4"></a>
{{< figure src="/ox-hugo/taghirad13_angular_momentum_rigid_body.png" caption="Figure 11: The components of the angular momentum of a rigid body about \\(A\\)" >}}
@ -1519,7 +1522,7 @@ The position vector of these two center of masses can be determined by the follo
\bm{p}\_{i\_2} &= \bm{a}\_{i} + ( l\_i - c\_{i\_2}) \hat{\bm{s}}\_{i}
\end{align}
<a id="org503dfcf"></a>
<a id="org733ec88"></a>
{{< figure src="/ox-hugo/taghirad13_free_body_diagram_stewart.png" caption="Figure 12: Free-body diagram of the limbs and the moving platform of a general Stewart-Gough manipulator" >}}
@ -1733,7 +1736,7 @@ in which
As shown in Figure [fig:stewart_forward_dynamics](#fig:stewart_forward_dynamics), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
<a id="org3d13427"></a>
<a id="org8a9c98e"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_forward_dynamics.png" caption="Figure 13: Flowchart of forward dynamics implementation sequence" >}}
@ -1766,7 +1769,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
\bm{\tau} = \bm{J}^{-T} \left( \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) - \bm{\mathcal{F}}\_d \right)
\end{equation}
<a id="org7d05c5f"></a>
<a id="org03df75b"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_inverse_dynamics.png" caption="Figure 14: Flowchart of inverse dynamics implementation sequence" >}}
@ -1791,7 +1794,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
## Motion Control {#motion-control}
<a id="org9d0691b"></a>
<a id="orgfd7dc2a"></a>
### Introduction {#introduction}
@ -1812,7 +1815,7 @@ However, using advanced techniques in nonlinear and MIMO control permits to over
### Controller Topology {#controller-topology}
<a id="orgd508d64"></a>
<a id="orgc91286c"></a>
<div class="cbox">
<div></div>
@ -1861,7 +1864,7 @@ Figure [fig:general_topology_motion_feedback](#fig:general_topology_motion_feedb
In such a structure, the measured position and orientation of the manipulator is compared to its desired value to generate the **motion error vector** \\(\bm{e}\_\mathcal{X}\\).
The controller uses this error information to generate suitable commands for the actuators to minimize the tracking error.<br />
<a id="org46eaf39"></a>
<a id="orgeae8325"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback.png" caption="Figure 15: The general topology of motion feedback control: motion variable \\(\bm{\mathcal{X}}\\) is measured" >}}
@ -1871,7 +1874,7 @@ The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) an
It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in Figure [fig:general_topology_motion_feedback_bis](#fig:general_topology_motion_feedback_bis) to implement such a controller.
<a id="org1c41f76"></a>
<a id="org1ab4f0e"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_bis.png" caption="Figure 16: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured" >}}
@ -1885,7 +1888,7 @@ To overcome the implementation problem of the control topology in Figure [fig:ge
In this topology, depicted in Figure [fig:general_topology_motion_feedback_ter](#fig:general_topology_motion_feedback_ter), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_q\\).
<a id="org3605dfc"></a>
<a id="org7760bf4"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_ter.png" caption="Figure 17: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured, and the inverse kinematic analysis is used" >}}
@ -1899,7 +1902,7 @@ For the topology in Figure [fig:general_topology_motion_feedback_ter](#fig:gener
To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [fig:general_topology_motion_feedback_quater](#fig:general_topology_motion_feedback_quater).
A force distribution block is added which maps the generated wrench in the task space \\(\bm{\mathcal{F}}\\), to its corresponding actuator forces/torque \\(\bm{\tau}\\).
<a id="org1ee8224"></a>
<a id="org9063bb5"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_quater.png" caption="Figure 18: The general topology of motion feedback control in task space: the motion variable \\(\bm{\mathcal{X}}\\) is measured, and the controller output generates wrench in task space" >}}
@ -1909,7 +1912,7 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
### Motion Control in Task Space {#motion-control-in-task-space}
<a id="orgfa59a60"></a>
<a id="orge26d8ba"></a>
#### Decentralized PD Control {#decentralized-pd-control}
@ -1918,7 +1921,7 @@ In the control structure in Figure [fig:decentralized_pd_control_task_space](#fi
The decentralized controller consists of **six disjoint linear controllers** acting on each error component \\(\bm{e}\_x = [e\_x,\ e\_y,\ e\_z,\ e\_{\theta\_x},\ e\_{\theta\_y},\ e\_{\theta\_z}]\\).
The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), in which \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(6 \times 6\\) **diagonal matrices** denoting the derivative and proportional controller gains for each error term.
<a id="org1010d4c"></a>
<a id="org0b1f26b"></a>
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_task_space.png" caption="Figure 19: Decentralized PD controller implemented in task space" >}}
@ -1941,7 +1944,7 @@ A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the
This term is generated from the dynamic model of the manipulator in the task space, represented in a closed form by the following equation:
\\[ \bm{\mathcal{F}}\_{ff} = \bm{\hat{M}}(\bm{\mathcal{X}}\_d)\ddot{\bm{\mathcal{X}}}\_d + \bm{\hat{C}}(\bm{\mathcal{X}}\_d, \dot{\bm{\mathcal{X}}}\_d)\dot{\bm{\mathcal{X}}}\_d + \bm{\hat{G}}(\bm{\mathcal{X}}\_d) \\]
<a id="org93f8916"></a>
<a id="org6217e2e"></a>
{{< figure src="/ox-hugo/taghirad13_feedforward_control_task_space.png" caption="Figure 20: Feed forward wrench added to the decentralized PD controller in task space" >}}
@ -2004,7 +2007,7 @@ Furthermore, mass matrix is added in the forward path in addition to the desired
As for the feedforward control, the **dynamics and kinematic parameters of the robot are needed**, and in practice estimates of these matrices are used.<br />
<a id="org0b8b7da"></a>
<a id="orge65e32a"></a>
{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_task_space.png" caption="Figure 21: General configuration of inverse dynamics control implemented in task space" >}}
@ -2126,14 +2129,14 @@ in which
\\[ \bm{\eta} = \bm{M}^{-1} \left( \tilde{\bm{M}} \bm{a}\_r + \tilde{\bm{C}} \dot{\bm{\mathcal{X}}} + \tilde{\bm{G}} \right) \\]
is a measure of modeling uncertainty.
<a id="org6507df1"></a>
<a id="orgc52d977"></a>
{{< figure src="/ox-hugo/taghirad13_robust_inverse_dynamics_task_space.png" caption="Figure 22: General configuration of robust inverse dynamics control implemented in the task space" >}}
#### Adaptive Inverse Dynamics Control {#adaptive-inverse-dynamics-control}
<a id="orga4ee1ee"></a>
<a id="orgdd9d529"></a>
{{< figure src="/ox-hugo/taghirad13_adaptative_inverse_control_task_space.png" caption="Figure 23: General configuration of adaptative inverse dynamics control implemented in task space" >}}
@ -2218,7 +2221,7 @@ In this control structure, depicted in Figure [fig:decentralized_pd_control_join
The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), where \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(n \times n\\) **diagonal** matrices denoting the derivative and proportional controller gains, respectively.<br />
<a id="org33c84d2"></a>
<a id="orgfb556a4"></a>
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_joint_space.png" caption="Figure 24: Decentralized PD controller implemented in joint space" >}}
@ -2240,7 +2243,7 @@ To remedy these shortcomings, some modifications have been proposed to this stru
The tracking performance of the simple PD controller implemented in the joint space is usually not sufficient at different configurations.
To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in Figure [fig:feedforward_pd_control_joint_space](#fig:feedforward_pd_control_joint_space).
<a id="org8250688"></a>
<a id="org23beb34"></a>
{{< figure src="/ox-hugo/taghirad13_feedforward_pd_control_joint_space.png" caption="Figure 25: Feed forward actuator force added to the decentralized PD controller in joint space" >}}
@ -2288,7 +2291,7 @@ Furthermore, the mass matrix is acting in the **forward path**, in addition to t
Note that to generate this term, the **dynamic formulation** of the robot, and its **kinematic and dynamic parameters are needed**.
In practice, exact knowledge of dynamic matrices are not available, and there estimates are used.<br />
<a id="org3e13d23"></a>
<a id="orgca195f2"></a>
{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_joint_space.png" caption="Figure 26: General configuration of inverse dynamics control implemented in joint space" >}}
@ -2564,7 +2567,7 @@ Hence, it is recommended to design and implement controllers in the task space,
## Force Control {#force-control}
<a id="org0954a98"></a>
<a id="orgf34f8a2"></a>
### Introduction {#introduction}
@ -2620,7 +2623,7 @@ The output control loop is called the **primary loop**, while the inner loop is
</div>
<a id="orgfa4a25b"></a>
<a id="org55aeea5"></a>
{{< figure src="/ox-hugo/taghirad13_cascade_control.png" caption="Figure 27: Block diagram of a closed-loop system with cascade control" >}}
@ -2654,7 +2657,7 @@ As seen in Figure [fig:taghira13_cascade_force_outer_loop](#fig:taghira13_cascad
The output of motion controller is also designed in the task space, and to convert it to implementable actuator force \\(\bm{\tau}\\), the force distribution block is considered in this topology.<br />
<a id="orgcb034a8"></a>
<a id="orgf8b5788"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="Figure 28: Cascade topology of force feedback control: position in inner loop and force in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
@ -2662,7 +2665,7 @@ Other alternatives for force control topology may be suggested based on the vari
If the force is measured in the joint space, the topology suggested in Figure [fig:taghira13_cascade_force_outer_loop_tau](#fig:taghira13_cascade_force_outer_loop_tau) can be used.
In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped into its corresponding wrench in the task space by the Jacobian transpose mapping \\(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\\).<br />
<a id="org7a885fa"></a>
<a id="org6a52697"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="Figure 29: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
@ -2673,7 +2676,7 @@ However, as the inner loop is constructed in the joint space, the desired motion
Therefore, the structure and characteristics of the position controller in this topology is totally different from that given in the first two topologies.<br />
<a id="orgddea488"></a>
<a id="org20969a6"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau_q.png" caption="Figure 30: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
@ -2691,7 +2694,7 @@ By this means, when the manipulator is not in contact with a stiff environment,
However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
This configuration may be seen as if the **outer loop generates a desired force trajectory for the inner loop**.<br />
<a id="org13367ca"></a>
<a id="org92b76a8"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="Figure 31: Cascade topology of force feedback control: force in inner loop and position in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
@ -2699,7 +2702,7 @@ Other alternatives for control topology may be suggested based on the variations
If the force is measured in the joint space, control topology shown in Figure [fig:taghira13_cascade_force_inner_loop_tau](#fig:taghira13_cascade_force_inner_loop_tau) can be used.
In such case, the Jacobian transpose is used to map the actuator force to its corresponding wrench in the task space.<br />
<a id="orgdc8007d"></a>
<a id="org9f8b940"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="Figure 32: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
@ -2708,7 +2711,7 @@ The inner loop is based on the measured actuator force vector in the joint space
In this topology, the desired motion in the task space is mapped into the joint space using **inverse kinematic** solution, and **both the position and force feedback controllers are designed in the joint space**.
Thus, independent controllers for each joint may be suitable for this topology.
<a id="org1575595"></a>
<a id="orga5155ab"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau_q.png" caption="Figure 33: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
@ -2727,7 +2730,7 @@ Thus, independent controllers for each joint may be suitable for this topology.
### Direct Force Control {#direct-force-control}
<a id="org53ade2e"></a>
<a id="orgf40bcf9"></a>
{{< figure src="/ox-hugo/taghira13_direct_force_control.png" caption="Figure 34: Direct force control scheme, force feedback in the outer loop and motion feedback in the inner loop" >}}
@ -2818,7 +2821,7 @@ The impedance of the system may be found from the Laplace transform of the above
</div>
<a id="orgfca539d"></a>
<a id="org251429f"></a>
{{< figure src="/ox-hugo/taghirad13_impedance_control_rlc.png" caption="Figure 35: Analogy of electrical impedance in (a) an electrical RLC circuit to (b) a mechanical mass-spring-damper system" >}}
@ -2877,7 +2880,7 @@ Moreover, direct force-tracking objective is not assigned in this control scheme
However, an auxiliary force trajectory \\(\bm{\mathcal{F}}\_a\\) is generated from the motion control law and is used as the reference for the force tracking.
By this means, no prescribed force trajectory is tracked, while the **motion control scheme would advise a force trajectory for the robot to ensure the desired impedance regulation**.<br />
<a id="orgf8596ce"></a>
<a id="org79bfe67"></a>
{{< figure src="/ox-hugo/taghira13_impedance_control.png" caption="Figure 36: Impedance control scheme; motion feedback in the outer loop and force feedback in the inner loop" >}}
@ -2915,4 +2918,4 @@ However, note that for a good performance, an accurate model of the system is re
## Bibliography {#bibliography}
<a id="orge267b8c"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.
<a id="orgf5fcdad"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.

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