LMIs in Control/Click here to continue/Integral Quadratic Constraints/Quadratic Stability and IQCs

The System

Consider the system of differential equations

${\dot {x}}(t)=(A+B\Delta (t)C)x(t)\quad \Delta (t)\in {\mathfrak {D}}$

where $A,B,C$ are given and $A$ is Hurwitz. ${\mathfrak {D}}$ is the set of all diagonal matrices with the norm not exceeding 1.

The Problem

The system is called quadratically stable if there exists a matrix $P=P^{T}$ such that

$P(A+B\Delta _{i}C)+(A+B\Delta _{i}C)^{T}P<0,\quad \forall {i}$

The stability of the system above is equivalent to the stability of the feedback interconnection:

${\begin{cases}v=Gu+f,\\u=\Delta (v)+r\end{cases}}$

where $G$ is the linear time-invariant operator with transfer function $G(s)=C(sI-A)^{-1}B$, and $\Delta$ is the operator, $\Delta (t)\in {\mathfrak {D}}.$

The Data

Let $\Pi ={\begin{bmatrix}Q&S\\S^{T}&R\end{bmatrix}}$

where $Q=Q^{T},R=R^{T},S$ are real matrices such that

$Q+S\Delta +\Delta ^{T}S^{T}+\Delta ^{T}R\Delta >0,\quad \forall {\Delta }\in {\mathfrak {D}}$

For a fixed matrix $\Pi$ satisfying the inequality above, a sufficient condition of stability is given by

${\begin{bmatrix}G(j\omega )\\I\end{bmatrix}}^{*}\Pi {\begin{bmatrix}G(j\omega )\\I\end{bmatrix}}<0\qquad \forall \omega \in \mathbb {R} \cup \{\infty \}$

The LMI

If there exists a $P=P^{T}$ such that

${\begin{bmatrix}PA+A^{T}P+C^{T}QC&PB+C^{T}S\\B^{T}P+S^{T}C&R\end{bmatrix}}<0$

then the system given by ${\dot {x}}(t)=(A+B\Delta (t)C)x(t)\quad \Delta (t)\in {\mathfrak {D}}$ is quadratically stable.

References

A. Megretski and A. Rantzer, "System analysis via integral quadratic constraints," in IEEE Transactions on Automatic Control, vol. 42, no. 6, pp. 819-830, June 1997, doi: 10.1109/9.587335

P. Seiler, "Stability Analysis With Dissipation Inequalities and Integral Quadratic Constraints," in IEEE Transactions on Automatic Control, vol. 60, no. 6, pp. 1704-1709, June 2015, doi: 10.1109/TAC.2014.2361004