User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/L2 Gain of Lure's Systems

The System


\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_ww(t),\\ z(t)&=C_zx(t)\\ p_i(t)&=\phi_i(q_i(t)), i=1,\dots, n_p\\ q&=C_qx,\\ 0 &\leq \sigma \phi_i(\sigma) \leq \sigma^2 \ \forall \sigma \in \mathbb{R} \end{align}$$

The Data
The matrices $$ A,B_p,B_w,C_q,C_z $$.

The Optimization Problem:
The following semi-definite problem should be solved.

\begin{align} &\min_{\{P \succ 0, \Lambda=diag(\lambda_1,\dots,\lambda_{n_p})\succeq 0,T=diag(\tau_1,\dots,\tau_{n_p}) \succeq 0\}} \gamma^2 \; \\ &\quad \quad \quad \quad \quad \quad \quad \quad  s.t. \quad \begin{bmatrix} A^\top P+PA+C^\top_zC_z & PB_p+A^\top C^\top_q \Lambda +C^\top_q T & PB_w \\ B^\top_p P + \Lambda C_qA+TC_q & \Lambda C_q B_p+B^\top_p C^\top_q \Lambda-2T & \Lambda C_q B_w \\ B^\top_wP & B^\top_w C^\top_q\Lambda & -\gamma^2I \end{bmatrix} \preceq 0\\ \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/12a7039f9e3d966e24b43fd58a3cce3725282ed2

Conclusion
The value function returns the square of the smallest provable upper bound on the $$ \mathcal{L}_2 $$ gain of the Lure's system.

Remark
The Lyapunov function which is used to proof is similar to the one for the systems with unknown parameters.