LMIs in Control/pages/Stability of Quadratic Constrained Systems

The System

{\begin{aligned}{\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)\\\int _{0}^{t}&p^{\top }(\tau )p(\tau )\ d\tau \leq \int _{0}^{t}q^{\top }(\tau )q(\tau )\ d\tau .\end{aligned}} 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{aligned}{\text{Find}}\;&\{P\succ 0,\lambda \geq 0\}:\\&s.t.\quad {\begin{bmatrix}A^{\top }P+PA+\lambda C_{q}^{\top }C_{q}&PB_{p}+\lambda C_{q}^{\top }D_{qp}\\(PB_{p}+\lambda C_{q}^{\top }D_{qp})^{\top }&\lambda (I-D_{qp}^{\top }D_{qp})\end{bmatrix}}\prec 0.\end{aligned}} 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.