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content/book/skogestad07_multiv_feedb_contr.md
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@@ -63,9 +63,7 @@ PDF version
</div>
## Introduction {#introduction}
<span class="org-target" id="org-target--sec-introduction"></span>
## Introduction {#sec:introduction}
### The Process of Control System Design {#the-process-of-control-system-design}
@@ -87,10 +85,10 @@ The process of designing a control system is a step by step design procedure as
13. Choose hardware and software and implement the controller
14. Test and validate the control system, and tune the controller on-line, if necessary
Input-output controllability analysis is studied in section for SISO systems and in section for MIMO systems.
The steps 4, 5, 6 and 7 are corresponding to the **control structure design**. This is treated in section .
The design of the controller is described in section .
The analysis of performance and robustness of a controlled system is studied in sections and .
Input-output controllability analysis is studied in section [Limitations on Performance in SISO Systems](#sec:perf_limit_siso) for SISO systems and in section [Limitations on Performance in MIMO Systems](#sec:perf_limit_mimo) for MIMO systems.
The steps 4, 5, 6 and 7 are corresponding to the **control structure design**. This is treated in section [Controller Structure Design](#sec:controller_structure_design).
The design of the controller is described in section [Controller Design](#sec:controller_design).
The analysis of performance and robustness of a controlled system is studied in sections [Uncertainty and Robustness for SISO Systems](#sec:uncertainty_robustness_siso) and [Robust Stability and Performance Analysis](#sec:robust_perf_mimo).
### The Control Problem {#the-control-problem}
@@ -238,9 +236,7 @@ Notations used throughout this note are summarized in [Table 1](#table--tab:nota
| \\(T\\) | Complementary sensitivity function: \\(T = (I + L)\*(I + L)^{-1}\\) |
## Classical Feedback Control {#classical-feedback-control}
<span class="org-target" id="org-target--sec-classical-feedback"></span>
## Classical Feedback Control {#sec:classical_feedback}
### Frequency Response {#frequency-response}
@@ -721,9 +717,7 @@ To combined these **mixed sensitivity specifications**, a **stacking approach**
After selecting the form of \\(N\\) and the weights, the \\(\hinf\\) optimal controller is obtained by solving the problem \\(\min\_K\hnorm{N(K)}\\).
## Introduction to Multivariable Control {#introduction-to-multivariable-control}
<span class="org-target" id="org-target--sec-multivariable-control"></span>
## Introduction to Multivariable Control {#sec:multivariable_control}
### Introduction {#introduction}
@@ -1235,9 +1229,7 @@ MIMO systems are often **more sensitive to uncertainty** than SISO systems.
</div>
## Elements of Linear System Theory {#elements-of-linear-system-theory}
<span class="org-target" id="org-target--sec-linear-sys-theory"></span>
## Elements of Linear System Theory {#sec:linear_sys_theory}
### System Descriptions {#system-descriptions}
@@ -1768,9 +1760,7 @@ The Hankel norm is a kind of induced norm from past inputs to future outputs.
It may be shown that the Hankel norm is equal to \\(\left\\|G(s)\right\\|\_H = \sqrt{\rho(PQ)}\\) where \\(\rho\\) is the spectral radius, \\(P\\) is the controllability Gramian and \\(Q\\) the observability Gramian.
## Limitations on Performance in SISO Systems {#limitations-on-performance-in-siso-systems}
<span class="org-target" id="org-target--sec-perf-limit-siso"></span>
## Limitations on Performance in SISO Systems {#sec:perf_limit_siso}
### Input-Output Controllability {#input-output-controllability}
@@ -2292,9 +2282,7 @@ They are not sufficient since among other things they only consider one effect a
The rules may be used to **determine whether or not a given plant is controllable**.
## Limitations on Performance in MIMO Systems {#limitations-on-performance-in-mimo-systems}
<span class="org-target" id="org-target--sec-perf-limit-mimo"></span>
## Limitations on Performance in MIMO Systems {#sec:perf_limit_mimo}
### Introduction {#introduction}
@@ -2808,9 +2796,7 @@ For RHP-zeros, RHP-poles and disturbances, the issue of directions usually makes
However, the situation is usually the opposite with model uncertainty because for MIMO systems, there is also uncertainty associated with plant directionality.
## Uncertainty and Robustness for SISO Systems {#uncertainty-and-robustness-for-siso-systems}
<span class="org-target" id="org-target--sec-uncertainty-robustness-siso"></span>
## Uncertainty and Robustness for SISO Systems {#sec:uncertainty_robustness_siso}
### Introduction to Robustness {#introduction-to-robustness}
@@ -3551,9 +3537,7 @@ Similarly, the inverse multiplicative uncertainty imposes an upper bound on the
We also derived a condition for robust performance with multiplicative uncertainty, \\(\abs{w\_P S} + \abs{w\_I T} < 1, \ \forall\w\\).
## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis}
<span class="org-target" id="org-target--sec-robust-perf-mimo"></span>
## Robust Stability and Performance Analysis {#sec:robust_perf_mimo}
### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty}
@@ -4474,7 +4458,7 @@ The peak value is close to 6, meaning that even with 6 times less uncertainty, t
We here consider the relationship between \\(\mu\\) for RP and the condition number of the plant or of the controller.
We consider unstructured multiplicative uncertainty (i.e. \\(\Delta\_I\\) is a full matrix) and performance is measured in terms of the weighted sensitivity.
With \\(N\\) given by \ref{eq:n\_delta\_structure\_clasic}, we have:
With \\(N\\) given by \eqref{eq:n\_delta\_structure\_clasic}, we have:
\begin{equation\*}
\overbrace{\mu\_{\tilde{\Delta}}(N)}^{\text{RP}} \le [ \overbrace{\overline{\sigma}(w\_I T\_I)}^{\text{RS}} + \overbrace{\overline{\sigma}(w\_P S)}^{\text{NP}} ] (1 + \sqrt{k})
@@ -4711,9 +4695,7 @@ If resulting control performance is not satisfactory, one may switch to the seco
4. \\(\mu\\) is most commonly used for analysis. If \\(\mu\\) is used for synthesis, then we recommend that you keep the uncertainty fixed and adjust the parameters in the performance weight until \\(\mu\\) is close to 1
## Controller Design {#controller-design}
<span class="org-target" id="org-target--sec-controller-design"></span>
## Controller Design {#sec:controller_design}
### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design}
@@ -4921,14 +4903,11 @@ These procedure are somehow difficult to use in practice.
Their main limitation is that they can only be applied to minimum phase plants.
### \\(\htwo\\) and \\(\hinf\\) Control {#htwo-and-hinf-control}
<span class="org-target" id="org-target--sec-htwo-and-hinf"></span>
### \\(\htwo\\) and \\(\hinf\\) Control {#sec:htwo_and_hinf}
#### General Control Problem Formulation {#general-control-problem-formulation}
#### General Control Problem Formulation {#sec:htwo_inf_assumptions}
<span class="org-target" id="org-target--sec-htwo-inf-assumptions"></span>
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.
@@ -5466,13 +5445,12 @@ for a specified \\(\gamma > \gamma\_\text{min}\\), is given by
L &= (1-\gamma^2) I + XZ
\end{align}
The Matlab function `coprimeunc` can be used to generate the controller in \ref{eq:control\_coprime\_factor}.
It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from \ref{eq:gamma\_min\_coprime} we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
The Matlab function `coprimeunc` can be used to generate the controller in \eqref{eq:control\_coprime\_factor}.
It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from \eqref{eq:gamma\_min\_coprime} we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
#### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic-hinf-loop-shaping-design-procedure}
#### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#sec:hinf_loop_shaping_procedure}
<span class="org-target" id="org-target--sec-hinf-loop-shaping-procedure"></span>
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.
@@ -5552,7 +5530,7 @@ The commands and feedbacks enter the controller separately and are independently
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="<span class='figure-number'>Figure 55: </span>General two degrees-of-freedom feedback control scheme" >}}
The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy.
The presented \\(\mathcal{H}\_\infty\\) loop-shaping design procedure in section [A Systematic \\(\hinf\\) Loop-Shaping Design Procedure](#sec:hinf_loop_shaping_procedure) is a one-degree-of-freedom design, although a **constant** pre-filter can be easily implemented for steady-state accuracy.
However, this may not be sufficient and a dynamic two degrees-of-freedom design is required.<br />
The design problem is illustrated in [Figure 56](#figure--fig:coprime-uncertainty-hinf).
@@ -5586,7 +5564,7 @@ The purpose of the prefilter is to ensure that:
The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller are:
1. Design a one degree-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller (section ) but without a post-compensator \\(W\_2\\)
1. Design a one degree-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller (section [A Systematic \\(\hinf\\) Loop-Shaping Design Procedure](#sec:hinf_loop_shaping_procedure)) but without a post-compensator \\(W\_2\\)
2. Select a desired closed-loop transfer function \\(T\_{\text{ref}}\\) between the commands and controller outputs
3. Set the scalar parameter \\(\rho\\) to a small value greater than \\(1\\); something in the range \\(1\\) to \\(3\\) will usually suffice
4. For the shaped \\(G\_s = G W\_1\\), the desired response \\(T\_{\text{ref}}\\), and the scalar parameter \\(\rho\\), solve the standard \\(\mathcal{H}\_\infty\\) optimization problem to a specified tolerance to get \\(K = [K\_1,\ K\_2]\\)
@@ -5733,9 +5711,7 @@ Sometimes, one might consider synthesizing a \\(\mu\text{-optimal}\\) controller
Moreover, one should be careful about combining controller synthesis and analysis into a single step.
## Controller Structure Design {#controller-structure-design}
<span class="org-target" id="org-target--sec-controller-structure-design"></span>
## Controller Structure Design {#sec:controller_structure_design}
### Introduction {#introduction}
@@ -6214,7 +6190,7 @@ Four applications of partial control are:
The outputs \\(y\_1\\) have an associated control objective but are not measured.
Instead, we aim at indirectly controlling \\(y\_1\\) by controlling the secondary measured variables \\(y\_2\\).
The table [Table 4](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
The [Table 4](#table--tab:partial-control) shows clearly the differences between the four applications of partial control.
In all cases, there is a control objective associated with \\(y\_1\\) and a feedback involving measurement and control of \\(y\_2\\) and we want:
- The effect of disturbances on \\(y\_1\\) to be small (when \\(y\_2\\) is controlled)
@@ -6314,7 +6290,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
Consider the conventional cascade control system in [Figure 62a](#org-target--fig-cascade-extra-meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
From \ref{eq:partial\_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
From \eqref{eq:partial\_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
These arguments particularly apply at high frequencies.
More precisely, we want the input-output controllability of \\([P\_u\ P\_r]\\) with disturbance model \\(P\_d\\) to be better that of the plant \\([G\_{11}\ G\_{12}]\\) with disturbance model \\(G\_{d1}\\).
@@ -6330,7 +6306,7 @@ A set of outputs \\(y\_1\\) may be left uncontrolled only if the effects of all
</div>
To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in \ref{eq:partial\_control\_partitioning} and \ref{eq:partial\_control}, and compute \\(P\_d\\) using \ref{eq:tight\_control\_y2}.
To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in \eqref{eq:partial\_control\_partitioning} and \eqref{eq:partial\_control}, and compute \\(P\_d\\) using \eqref{eq:tight\_control\_y2}.
#### Measurement Selection for Indirect Control {#measurement-selection-for-indirect-control}
@@ -6500,7 +6476,7 @@ We then derive **necessary conditions for stability** which may be used to elimi
For decentralized diagonal control, it is desirable that the system can be tuned and operated one loop at a time.
Assume therefore that \\(G\\) is stable and each individual loop is stable by itself (\\(\tilde{S}\\) and \\(\tilde{T}\\) are stable).
Using the **spectral radius condition** on the factorized \\(S\\) in \ref{eq:S\_factorization}, we have that the overall system is stable (\\(S\\) is stable) if
Using the **spectral radius condition** on the factorized \\(S\\) in \eqref{eq:S\_factorization}, we have that the overall system is stable (\\(S\\) is stable) if
\begin{equation}
\rho(E\tilde{T}(j\omega)) < 1, \forall\omega
@@ -6725,7 +6701,7 @@ When a reasonable choice of pairings have been made, one should rearrange \\(G\\
|1 + L\_i| > \max\_{k,j}\\{|\tilde{g}\_{dik}|, |\gamma\_{ij}|\\}
\end{equation}
To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by \ref{eq:decent\_contr\_one\_loop} is achievable.
To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by \eqref{eq:decent\_contr\_one\_loop} is achievable.
Note that RHP-zeros in the diagonal elements may limit achievable decentralized control, whereas they may not pose any problems for a multivariable controller.
Since with decentralized control, we usually want to use simple controllers, the achievable bandwidth in each loop will be limited by the frequency where \\(\angle g\_{ii}\\) is \\(\SI{-180}{\degree}\\)
6. Check for constraints by considering the elements of \\(G^{-1} G\_d\\) and make sure that they do not exceed one in magnitude within the frequency range where control is needed.
@@ -6741,7 +6717,7 @@ When a reasonable choice of pairings have been made, one should rearrange \\(G\\
If the plant is not controllable, then one may consider another choice of pairing and go back to Step 4.
If one still cannot find any pairing which are controllable, then one should consider multivariable control.
7. If the chosen pairing is controllable, then \ref{eq:decent\_contr\_one\_loop} tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
7. If the chosen pairing is controllable, then \eqref{eq:decent\_contr\_one\_loop} tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
This can be used as a basis for designing the controller \\(k\_i(s)\\) for loop \\(i\\)
@@ -6761,16 +6737,14 @@ Thus sequential design may involve many iterations.
#### Conclusion on Decentralized Control {#conclusion-on-decentralized-control}
A number of **conditions for the stability**, e.g. \ref{eq:decent\_contr\_cond\_stability} and \ref{eq:decent\_contr\_necessary\_cond\_stability}, and **performance**, e.g. \ref{eq:decent\_contr\_cond\_perf\_dist} and \ref{eq:decent\_contr\_cond\_perf\_ref}, of decentralized control systems have been derived.
A number of **conditions for the stability**, e.g. \eqref{eq:decent\_contr\_cond\_stability} and \eqref{eq:decent\_contr\_necessary\_cond\_stability}, and **performance**, e.g. \eqref{eq:decent\_contr\_cond\_perf\_dist} and \eqref{eq:decent\_contr\_cond\_perf\_ref}, of decentralized control systems have been derived.
The conditions may be useful in **determining appropriate pairings of inputs and outputs** and the **sequence in which the decentralized controllers should be designed**.
The conditions are also useful in an **input-output controllability analysis** for determining the viability of decentralized control.
## Model Reduction {#model-reduction}
<span class="org-target" id="org-target--sec-model-reduction"></span>
## Model Reduction {#sec:model_reduction}
### Introduction {#introduction}
@@ -7193,3 +7167,8 @@ In such a case, using truncation or optimal Hankel norm approximation with appro
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Doyle, John C. 1983. “Synthesis of Robust Controllers and Filters.” In <i>The 22nd IEEE Conference on Decision and Control</i>, 109–14. IEEE.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Skogestad, S., and I. Postlethwaite. 2007. <i>Multivariable Feedback Control: Analysis and Design - Second Edition</i>. John Wiley.</div>
</div>