LMIs in Control/Stability Analysis/Stability 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 LMI: The Lure's System's Stability
The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.



\begin{align} \text{Find} \; &P>0,\Lambda=diag(\lambda_1,\dots,\lambda_{n_p})\succeq 0, T=diag(\tau_1,\dots,\tau_{n_p}) \succeq 0:\\ &\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} \prec 0\\ \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/9030b507b39f658fb34fef8563ce9130d5dc2baa https://codeocean.com/capsule/0232754/tree

Conclusion
If the feasibility problem with LMI constraints has solution, then the Lure's system is stable.

Remark
The LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system. It is also necessary when there is only one nonlinearity, i.e., when $$ n_p = 1 $$.