User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/H infinity Norm for Affine Parametric Varying Systems

The System


\begin{align} \dot x(t)&=Ax(t)+B_w w(t),\\ z(t)&=C_z(\theta)x(t)+D_{zw}(\theta)w(t), \end{align}$$ where $$ C_z $$ and $$ D_{zw}$$ depend affinity on parameter $$ \theta \in \mathbb{R}^p $$.

The Data
The matrices $$ A,B_w,C_z(.),D_{zw}(.) $$.

The Optimization Problem:
Solve the following semi-definite program

\begin{align} &\min_{\{P \succ 0,\gamma \geq 0\}} \gamma \\ &\quad  s.t. \begin{bmatrix} A^\top P +PA & PB_w \\ B^\top_wP &  -\gamma^2 I \end{bmatrix}+ \begin{bmatrix} C^\top_z(\theta) \\ D^\top_{zw}(\theta) \end{bmatrix}\begin{bmatrix} C_z(\theta) & D_{zw}(\theta) \end{bmatrix}\preceq 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/5462bc1dc441bc298d50a2c35075e9466eba8355

Conclusion
The value function of the above semi-definite program returns the $$ \mathcal{H}_{\infty}$$ norm of the system.

Remark
It is assumed that $$ A $$ is stable and $$ (A,B_w) $$ is controllable and the semi-infinite convex constraint $$ \|H_{\theta}(j\omega)\| < \gamma $$ for all $$ \omega \in \mathbb{R}$$, is converted into a finite-dimensional convex LMI.