LMIs in Control/Template

This methods uses LMI techniques iteratively to obtain the result.

The System
Given a state-space representation of a system $$

G(s) $$ and an initial estimate of reduced order model $$

\hat{G}(s) $$.



\begin{align} \ G(s) &= C(sI-A)B + D,\\ \ \hat{G}(s) &= \hat{C}(sI-\hat{A})\hat{B} + \hat{D},\\ \end{align}$$

Where $$ A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, C \in \mathbb{R}^{p\times n}, D \in \mathbb{R}^{p\times m}, \hat{A} \in \mathbb{R}^{k\times k}, \hat{B} \in \mathbb{R}^{k\times m}, \hat{C} \in \mathbb{R}^{p\times k}$$ and $$ \hat{D} \in \mathbb{R}^{p\times m}$$.

The Data
The full order state matrices $$ A,B,C,D $$.

The Optimization Problem
The objective of the optimization is to reduce the $$ H_{\infty} $$ norm.

The LMI: The Lyapunov Inequality
Objective: $$ \min \gamma $$.

Subject to:: $$

\begin{align} \ P &= \begin{bmatrix} \ P11 & P12 \\ \ P21 & P22 \\ \end{bmatrix} \ > 0, \end{align} $$

$$ \begin{align} \begin{bmatrix} \ A^{T}P11 + P11A & A^{T}P12 + P12\hat{A} & P11B - P12\hat{B} & C^T \\ \ \hat{A}^{T}P12^T + P12^{T}A & \hat{A}^{T}P22 + P22\hat{A} & P12^{T}B - P22\hat{B} & \hat{C}^T \\ \ B^{T}P11 - \hat{B}^{T}P12^{T} & B^{T}P12 - \hat{B}^{T}P22 & -\gamma{I} & D^{T} -\hat{D}^{T} \\ \ C & \hat{C} & D -\hat{D} & -\gamma{I} \\ \end{bmatrix} \ > 0 \end{align}$$

Conclusion:
The LMI techniques results in model reduction close to the theoretical bounds.