User:Margav06/sandbox/Click here to continue/Controller synthesis/Quadratic Schur Satbilization

LMI for Quadratic Schur Stabilization A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.

The System
Consider discrete time system

\begin{align} x_{k+1}=Ax_k+Bu_k,\\ \end{align}$$ where $$x_k\in \R^n$$, $$u_k\in \R^m$$, at any $$t\in \R$$. The system consist of uncertainties of the following form

\begin{align} \Delta_{A(t)} = A_1 \delta_1 (t)+....+A_k \delta_k (t)\\ \Delta_{B(t)} = B_1 \delta_1 (t)+....+B_k \delta_k (t)\\ \end{align}$$

where $$x\in\mathbb{R}^{m}$$,$$u\in\mathbb{R}^{n}$$,$$A\in\mathbb{R}^{mxm}$$ and $$B\in\mathbb{R}^{mxn}$$

The Data
The matrices necessary for this LMI are $$A$$,$$\Delta_{A(t)} \, ie \, A_i $$ ,$$B$$ and $$\Delta_{B(t)} \, ie \, B_i $$

The LMI:
There exists some X > 0 and Z such that

\begin{align} \begin{bmatrix} X && AX+BZ \\ (AX+BZ)^T && X \end{bmatrix} +

\begin{bmatrix} 0 && A_iX + B_iZ \\(A_iX + B_iZ)^T&&0 \end{bmatrix} > 0 \quad i = 1,......,k

\end{align}$$

The Optimization Problem
The optimization problem is to find a matrix $$\begin{align} K \in \mathbb{R}^{r\times n} \end{align}$$ such that:

$$\begin{align} ||A + BK||_{2} < \gamma \end{align}$$

According to the definition of the spectral norms of matrices, this condition becomes equivalent to:

$$\begin{align} (A + BK)^{T}(A + BK) < \gamma^{2}I \end{align}$$

Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:

$$\begin{align} \begin{bmatrix} -\gamma I & (A + BK) \\ (A + BK)^{T} & -\gamma I\end{bmatrix} < 0\\ \end{align}$$

Conclusion:
The Controller gain matrix is extracted as $$ F = ZX^{-1} $$ $$ u_k = Fx_k $$

\begin{align} x_{k+1}=Ax_k+Bu_k,\\ \quad\quad\quad = Ax_k + BFx_k\\ \quad\quad = (A+BF)x_k \end{align}$$

It follows that the trajectories of the closed-loop system (A+BK) are stable for any $$ \, \Delta \, \in \, C_0(\Delta_1,...,\Delta_k) $$

Implementation
https://github.com/JalpeshBhadra/LMI/blob/master/quadratic_schur_stabilization.m

Related LMIs
Schur Complement Schur Stabilization