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:

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

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:

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

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

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

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
- Schur Stabilization

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