# 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:

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

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

{\displaystyle {\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:

{\displaystyle {\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:

## Related LMIs

[1] - Schur Stabilization