User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Stability of Quadratic Constrained Systems

The System


\begin{align} \dot x(t)&=Ax(t)+B_pp(t)+B_u u(t)+B_ww(t),\\ q(t)&=C_qx(t)+D_{qp}p(t)+D_{qu}u(t)+D_{qw}w(t),\\ z(t)&=C_zx(t)+D_{zp}p(t)+D_{zu}u(t)+D_{zw}w(t)\\ \int_0^t &p^\top(\tau)p(\tau) \ d\tau \leq \int_{0}^t q^\top(\tau)q(\tau) \ d\tau. \end{align}

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The Data
The matrices $$ A,B_p,B_w,C_q,C_z,D_{qp},D_{zw} $$.

The LMI:
The following feasibility problem should be solved.

\begin{align} \text{Find} \; &\{P\succ0,\lambda \geq 0\}:\\ &  s.t. \quad \begin{bmatrix} A^\top P+PA+\lambda C^\top_q C_q & PB_p+ \lambda C^\top_q D_{qp}   \\ (PB_p+ \lambda C^\top_q D_{qp})^\top &  \lambda (I-D^\top_{qp}D_{qp})  \end{bmatrix} \prec 0. \end{align}$$

Implementation
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/38f3b55ca7060a1260384a96e9dc31142af07a9a

Conclusion
The integral quadratic constrained system is stable if the provided LMI is feasible

Remark
The key point of the proof is to satisfy $$ \dot{V}<0 $$ whenever $$ p^\top p \leq q^\top q $$, using $$ \mathcal{S} $$-procedure.