LMIs in Control/pages/Stability of Quadratic Constrained Systems

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+B_{p}p(t)+B_{u}u(t)+B_{w}w(t),\\ q(t)& =C_{q}x(t)+D_{qp}p(t)+D_{qu}u(t)+D_{qw}w(t),\\ z(t)& =C_{z}x(t)+D_{zp}p(t)+D_{zu}u(t)+D_{zw}w(t)\\ \underset{0}{\overset{t}{\int }}& p^{\mathrm{\top }}(\tau )p(\tau )d\tau \le \underset{0}{\overset{t}{\int }}{q}^{\mathrm{\top }}(\tau )q(\tau )d\tau .\end{array}$

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.

$\begin{array}{cc}\text{Find}& \{P\succ 0,\lambda \ge 0\}:\\ & s.t.\left[\begin{array}{cc}{A}^{\mathrm{\top }}P+PA+\lambda C_q^{\mathrm{\top }}{C}_q& P{B}_p+\lambda C_q^{\mathrm{\top }}{D}_{qp}\\ (P{B}_p+\lambda C_q^{\mathrm{\top }}{D}_{qp}{)}^{\mathrm{\top }}& \lambda (I-D_{qp}^{\mathrm{\top }}{D}_{qp})\end{array}\right]\prec 0.\end{array}$

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 $p^{\mathrm{\top }}p\le q^{\mathrm{\top }}q$, using $𝒮$-procedure.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.