User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Dissipativity of 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,\theta\}} \gamma \\ &\quad  s.t.\quad \begin{bmatrix} A^\top P +PA & PB_w- C_z(\theta)^\top \\ B^\top_wP-C_z(\theta) &  2\gamma I-D_{zw}(\theta)-D_{zw}(\theta)^\top \end{bmatrix}\preceq 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/b6cd6b81f75be4a2052ba3fa76cad1a2f9c49caa

Conclusion
The dissipativity of $$ H_{\theta}$$ (see [Boyd,eq:6.59]) exceeds $$ {\gamma}$$ if and only if the above LMI holds and the value function returns the minimum provable dissipativity.

Remark
It is worth noticing that passivity corresponds to zero dissipativity.