LMIs in Control/Robustness/Continuous Time/DKIteration

This methods uses LMI techniques iteratively to obtain the result.

The System

Given a state-space representation of a system $G(s)$ and an initial estimate of reduced order model $\hat{G}(s)$.

$\begin{array}{cc}G(s)& =C(sI-A)B+D,\\ \hat{G}(s)& =\hat{C}(sI-\hat{A})\hat{B}+\hat{D},\\ \end{array}$

Where $A\in {ℝ}^{n\times n},B\in {ℝ}^{n\times m},C\in {ℝ}^{p\times n},D\in {ℝ}^{p\times m},\hat{A}\in {ℝ}^{k\times k},\hat{B}\in {ℝ}^{k\times m},\hat{C}\in {ℝ}^{p\times k}$ and $\hat{D}\in {ℝ}^{p\times m}$.

The Data

The full order state matrices $A,B_1,B_2,C_1,C_2,D_{11},D_{12},D_{21}$.

The Optimization Problem

Alternate fixing $K,\mathrm{\Theta }$ in an iterative process to estimate a solution for Dynamic Output Feedback Synthesis with Structured Norm-Bounded Uncertainty.

The LMI: D-K Iteration

Objective: $\min \gamma$.

Subject to:: Initialize: $\mathrm{\Theta }=I$\\ Define:\\

$\begin{array}{cc}{\hat{G}}_{\mathrm{\Theta }}(s)& =\left[\begin{array}{cccc}A& |& B_1{\mathrm{\Theta }}^{-1/2}& B_2\\ {\mathrm{\Theta }}^{1/2}{C}_1& |& {\mathrm{\Theta }}^{1/2}{D}_{11}{\mathrm{\Theta }}^{-1/2}& {\mathrm{\Theta }}^{1/2}{D}_{12}\\ C_2& |& D_{12}{\mathrm{\Theta }}^{-1/2}& 0\end{array}\right]\end{array}$

Step 1:

Fix: $\mathrm{\Theta }$ and solve:

$\underset{k}{\inf }\Vert \underset{\_}{S}(G_{\mathrm{\Theta }},K){\Vert }_{H_{\mathrm{\infty }}}$

Step 2:

Fix K and minimize $\gamma$ such that there exists $\mathrm{\Theta }\in 𝐏\mathrm{\Theta }$ and $X>0$ such that

$\begin{array}{c}\left[\begin{array}{cc}{A}_{cl}^{T}X+X{A}_{xl}& X{B}_{cl}\\ B_{cl}^{T}X& -\mathrm{\Theta }\end{array}\right]+{\gamma }^{-2}\left[\begin{array}{c}{C}_{cl}^T\\ D_{cl}^T\end{array}\right]\mathrm{\Theta }\left[\begin{array}{cc}{C}_{cl}& D_{cl}\end{array}\right]>0\end{array}$ where $A_{cl},B_{cl},C_{cl},D_{cl}$ define $\underset{k}{\inf }\Vert \underset{\_}{S}(G_{\mathrm{\Theta }},K){\Vert }_{H_{\mathrm{\infty }}}$ (Bisection)

Step 3: Go to Step 1

Conclusion:

This is less of an LMI and more of a heuristic that allows us to solve for time-invariant scalings $\mathrm{\Theta }$ and controller K. However, there are no guarantees that this process will return an globally optimized result.

Implementation

• https://github.com/mosmith3asu/WikibooksLMIs/blob/main/DKiteration.m - Example Code

Related LMIs

• https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Discrete_Time_H%E2%88%9E_Optimal_Dynamic_Output_Feedback_Control - $H_{\mathrm{\infty }}$- Dynamic Output Feedback Controller Synthesis

External Links

A list of references documenting and validating the LMI.

• http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control- A course on LMIs in Control by Matthew Peet.