LMIs in Control/pages/CT-SOFS

LMIs in Control/pages/CT-SOFS

In view of applications, static feedback of the full state is not feasible in general: only a few of the state variables (or a linear combination of them, $y=Cx(t)$, called the output) can be actually measured and re-injected into the system.
So, we are led to the notion of static output feedback

The System

Consider the continuous-time LTI system, with generalized state-space realization $(A,B,C,0)$

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+Bu(t)\\ y(t)& =Cx(t)\\ \end{array}$

The Data

• $A\in {ℝ}^{n\times n},B\in {ℝ}^{n\times m},C\in {ℝ}^{p\times n}$

• $x\in {ℝ}^n,y\in {ℝ}^p,u\in {ℝ}^m$

The Optimization Problem

This system is static output feedback stabilizable (SOFS) if there exists a matrix F such that the closed-loop system
$\dot{x}=(A-BKC)x$
(obtained by replacing $u=-Ky$ which means applying static output feedback)
is asymptotically stable at the origin

The LMI: LMI for Continuous Time - Static Output Feedback Stabilizability

The system is static output feedback stabilizable if and only if it satisfies any of the following conditions:

• There exists a $K\in {ℝ}^{m\times p}$ and $P\in {𝕊}^n$, where $P>0$, such that

$\left[\begin{array}{cc}{A}^{T}P+PA-PB{B}^{T}P& PB+C^T{K}^T\\ KC+B^{T}P& -1\end{array}\right]$$\begin{array}{c}<0\end{array}$

• There exists a $K\in {ℝ}^{m\times p}$ and $Q\in {𝕊}^n$, where $Q>0$, such that

$\left[\begin{array}{cc}Q{A}^T+AQ-Q{C}^{T}CQ& BK+Q{C}^T\\ C{Q}^T+K^T{B}^T& -1\end{array}\right]$$\begin{array}{c}<0\end{array}$

• There exists a $K\in {ℝ}^{m\times p}$ and $Q\in {𝕊}^n$, where $Q>0$, such that

$\left[\begin{array}{cc}Q{A}^T+AQ-B{B}^T& B+Q{C}^T{K}^T\\ B^T+KC{Q}^T& -1\end{array}\right]$$\begin{array}{c}<0\end{array}$

• There exists a $K\in {ℝ}^{m\times p}$ and $P\in {𝕊}^n$, where $P>0$, such that

$\left[\begin{array}{cc}{A}^{T}P+PA-C^{T}C& PBK+C^T\\ K^T{B}^{T}P& -1\end{array}\right]$$\begin{array}{c}<0\end{array}$

Conclusion

On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 2 output matrices one of which is the Symmeteric matrix $P$ (or $Q$) and $K$

Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:

https://github.com/yashgvd/LMI_wikibooks

Related LMIs

Discrete time Static Output Feedback Stabilizability
Static Feedback Stabilizability

External Links

• [1] - LMI in Control Systems Analysis, Design and Applications
• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.