User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Entropy Bond 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}$$ depend affinely on parameter $$ \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^2,\lambda,\theta\}} \gamma^2 \\ &\quad  s.t.\quad D_{zw}(\theta)=0, \begin{bmatrix} A^\top P +PA & PB_w & C_z(\theta)^\top \\ B^\top_wP &  -\gamma^2 I & 0 \\ C_z(\theta) & 0 & -I \end{bmatrix}\preceq 0, \quad \rm{Tr}(B^\top_w P B_w) \leq \lambda. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/02f31a2d7a22b2464dfe9212eb76409bda9439b1

Conclusion
The value function of the above semi-definite program returns a bound for $$ \gamma $$-entropy of the system, which is defined as

\begin{align} I_{\gamma}(H_{\theta}) \triangleq \begin{cases} \frac{-\gamma^2}{2\pi}\int_{-\infty}^{\infty}\log\det(I-\gamma^2 H_{\theta}(i\omega)H_{\theta}(i\omega)^*)d\omega, \quad \text{if} \ \|H_{\theta}\|_{\infty} < \gamma \\ \infty, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{otherwise.}\end{cases} \end{align} $$

Remark
When it is finite, $$ I_{\gamma}(H_{\theta}) $$ is given by $$ \rm{Tr}(B^\top_wPB_w)$$ where $$ P$$, is asymmetric matrix with the smallest possible maximum singular value among all solutions of a Riccati equation.