LMIs in Control/Click here to continue/Integral Quadratic Constraints/Frequency Domain

The System
We will consider the following feedback interconnection $$S(G, \Delta)$$:



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

where $$r$$ and $$f$$ are exogeneous inputs. $$G$$ and $$\Delta$$ are two casual operators.

The Problem
Let $$\Pi:j\mathbb{R}\rightarrow \mathbb{C}$$ be a measurable Hermitian-valued function, $$G\in \mathbb{R}\mathbb{H}_{\infty} $$ and $$\Delta$$ be a bounded casual operator. $$\exists \epsilon>0 $$ such that

$$ \begin{bmatrix}G(j\omega) \\ I \end{bmatrix}^{*} \Pi(j\omega) \begin{bmatrix}G(j\omega) \\ I \end{bmatrix} \leq - \epsilon I \qquad \forall \omega \in \mathbb{R} $$

Then the feedback interconnection of $$G$$ and $$\Delta$$ is stable.

The Data
$$G$$ is a linear time-invariant system with the state space realization:

\begin{cases} \dot{x}=Ax+Bu \\ y=Cx+Du \end{cases} $$ where $$x$$ is the state.

Any $$\Pi \in \mathbb{R}\mathbb{H}_{\infty} $$ can be factorized as $$\Pi=\Psi^{*}M\Psi$$ where $$M=M^{T}$$ and $$\Psi \in \mathbb{R}\mathbb{H}_{\infty}$$. Denote the state space realization of $$\Psi$$ by $$(A_\psi,[B_{\psi 1}, B_{\psi 2}], C_\psi, [D_{\psi 1}, D_{\psi 2}])$$.

A state space realization for the system $$\Psi \begin{bmatrix}G \\ I \end{bmatrix}$$ is $$ (\hat{A}, \hat{B}, \hat{C}, \hat{D}):=\left( \begin{bmatrix} A & 0 \\ B_{\psi 1}C & A_{\psi} \end{bmatrix}, \begin{bmatrix} B \\ B_{\psi 2} + B_{\psi 1} D \end{bmatrix}, \begin{bmatrix} D_{\psi 1} C & C_{\psi} \end{bmatrix}, D_{\psi 2} + D_{\psi 1}D \right) $$

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



\begin{bmatrix} \hat{A}^{T}P+P\hat{A} & P\hat{B} \\ \hat{B}^{T}P & 0 \end{bmatrix} + \begin{bmatrix} \hat{C}^{T} \\ \hat{D}^{T} \end{bmatrix}M \begin{bmatrix}\hat{C} & \hat{D}\end{bmatrix}< 0 $$ then the feedback interconnection $$S(G, \Delta)$$ is stable.