LMIs in Control/Matrix and LMI Properties and Tools/Schur Stabilizability

LMI for Schur Stabilizability

Schur Stabilization is one method of ensuring that a controller can be made to stabilize a system. The following LMI is one that determines whether or not a system is indeed Schur Stabilizable, or having the property of being able to be Schur Stabilized.

The System

We consider the following system:

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

or the matrix pair (A,B). In both cases, the matrices $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times r}$, $x\in \mathbb {R} ^{n}$, and $u\in \mathbb {R} ^{r}$ are the state matrix, input matrix, state vector, and the input vector, respectively.

The Data

The data required is both the matrices A and B as seen in the form above.

The Optimization Problem

The goal of the optimization is to find a valid symmetric P such that the following LMI is satisfied.

The LMI: LMI for Schur stabilizability

The LMI problem is to find a symmetric matrix P and a matrix W satisfying:

${\begin{aligned}{\begin{bmatrix}-P&AP+BW\\(AP+BW)^{T}&-P\end{bmatrix}}<0\\\end{aligned}}$

Another LMI with the same result of finding Schur Stabilizability is to find a symmetric matrix P such that:

${\begin{aligned}{\begin{bmatrix}-P&PA^{T}\\AP&-P-\gamma BB^{T}\end{bmatrix}}<0,\gamma \leq 1\\\end{aligned}}$

Conclusion:

If the one of the above LMIs is found to be feasible, then the system is Schur Stabilizable and the Schur Stabilization LMI will always give a feasible result as well, in addition to a controller K that will Schur Stabilize the system.

Implementation

A link to Matlab codes for this problem in the Github repository:

https://github.com/maxwellpeterson99/MAE509Code

Related LMIs

[1] - Schur Stabilization

External Links

[2] - LMI in Control Systems Analysis, Design and Applications

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[3] -Matrix and LMI Properties and Tools