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

The System

We will consider the following feedback interconnection $S(G,\mathrm{\Delta })$:

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

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

The Problem

Let $\mathrm{\Pi }:jℝ\to ℂ$ be a measurable Hermitian-valued function, $G\in ℝ{ℍ}_{\mathrm{\infty }}$ and $\mathrm{\Delta }$ be a bounded casual operator. $\mathrm{\exists }ϵ>0$ such that

${\left[\begin{array}{c}G(j\omega )\\ I\end{array}\right]}^{*}\mathrm{\Pi }(j\omega )\left[\begin{array}{c}G(j\omega )\\ I\end{array}\right]\le -ϵI\mathrm{\forall }\omega \in ℝ$

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

The Data

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

$\{\begin{array}{c}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$

where $x$ is the state.

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

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

The LMI

If there exists a matrix $P=P^T$ such that

$\left[\begin{array}{cc}{\hat{A}}^{T}P+P\hat{A}& P\hat{B}\\ {\hat{B}}^{T}P& 0\end{array}\right]+\left[\begin{array}{c}{\hat{C}}^T\\ {\hat{D}}^T\end{array}\right]M\left[\begin{array}{cc}\hat{C}& \hat{D}\end{array}\right]<0$

then the feedback interconnection $S(G,\mathrm{\Delta })$ is 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