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, ${\displaystyle y=Cx(t)}$, 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 ${\displaystyle (A,B,C,0)}$

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

The Data

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

• ${\displaystyle x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{p},u\in \mathbb {R} ^{m}}$

The Optimization Problem

This system is static output feedback stabilizable (SOFS) if there exists a matrix F such that the closed-loop system
${\displaystyle {\dot {x}}=(A-BKC)x}$
(obtained by replacing ${\displaystyle u=-Ky}$ 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 ${\displaystyle K\in \mathbb {R} ^{m\times p}}$ and ${\displaystyle P\in \mathbb {S} ^{n}}$, where ${\displaystyle P>0}$, such that

${\displaystyle {\begin{bmatrix}A^{T}P+PA-PBB^{T}P&PB+C^{T}K^{T}\\KC+B^{T}P&-1\end{bmatrix}}}$${\displaystyle {\begin{aligned}<0\end{aligned}}}$

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

${\displaystyle {\begin{bmatrix}QA^{T}+AQ-QC^{T}CQ&BK+QC^{T}\\CQ^{T}+K^{T}B^{T}&-1\end{bmatrix}}}$${\displaystyle {\begin{aligned}<0\end{aligned}}}$

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

${\displaystyle {\begin{bmatrix}QA^{T}+AQ-BB^{T}&B+QC^{T}K^{T}\\B^{T}+KCQ^{T}&-1\end{bmatrix}}}$${\displaystyle {\begin{aligned}<0\end{aligned}}}$

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

${\displaystyle {\begin{bmatrix}A^{T}P+PA-C^{T}C&PBK+C^{T}\\K^{T}B^{T}P&-1\end{bmatrix}}}$${\displaystyle {\begin{aligned}<0\end{aligned}}}$

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 ${\displaystyle P}$ (or ${\displaystyle Q}$) and ${\displaystyle K}$

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

• - 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.

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