User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Output Energy Bound for 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(t)&=C_qx(t),\\ 0 &\leq \sigma \phi_i(\sigma) \leq \sigma^2 \ \forall \sigma \in \mathbb{R} \end{align}$$

The Data
The matrices $$ A,B_p,B_w,C_q,C_z,x(0) $$.

The Optimization Problem:
The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.



\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} x^\top(0)(P+C^\top_q \Lambda C_q)x(0)\\ & \quad \quad \quad \quad \quad \quad \quad \begin{bmatrix} A^\top P+PA & PB_p+A^\top C^\top_q \Lambda +C^\top_q T \\ B^\top_p P + \Lambda C_qA+TC_q & \Lambda C_q B_p+B^\top_p C^\top_q \Lambda-2T \end{bmatrix} \preceq 0\\ \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/blob/master/LMIs%20for%20Output%20Energy%20Bounds%20of%20Lure's%20Systems

Conclusion
The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., $$ J=\int_{0}^{\infty} z^\top z \ dt $$ with initial conditions $$ x(0) $$.

Remark
The key step in the proof is to satisfy $$ \frac{d}{dt}V(x)+z^\top z \leq 0 $$, where $$ V(.)$$ is Lyapunov function in a special form.