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, ${\displaystyle y=Cx(t)}$, 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:

Related LMIs

Discrete time Static Output Feedback Stabilizability
Static Feedback Stabilizability