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.

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