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

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

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.