User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Hankel Norm of Affine Parameter 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_{\{Q \succeq 0,\gamma^2,\theta\}} \gamma^2 \\ &\quad  s.t.\quad D_{zw}(\theta)=0, \quad  A^\top Q +QA + C_z(\theta)C_z(\theta)\preceq 0,\quad \gamma^2 I-W^{1/2}_c Q W^{1/2}_c \succeq 0, \end{align}$$ where $$ W_c$$ is the controllability Gramian, i.e., $$ W_c \triangleq \int_{0}^{\infty} e^{At}B_wB^\top_w e^{A^\top t}dt $$.

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/0faedcdd9fba92bc27a318d80159c04a0b342f35

Conclusion
The Hanakel norm (i.e., the square root of the maximum eigenvalue) of $$ H_{\theta}$$ is less than $$ {\gamma}$$ if and only if the above LMI holds and the value function returns the maximum provable Hankel norm.

Remark
$$ D_{zw}$$ is assumed to be zero.