LMIs in Control/Applications/Hinf optimal Model Reduction

Given a full order model and an initial estimate of a reduced order model it is possible to obtain a reduced order model optimal in $H_{\mathrm{\infty }}$ sense. 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 $\hat{G}(s)$.

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

Where $A\in {ℝ}^{n\times n},B\in {ℝ}^{n\times m},C\in {ℝ}^{p\times n},D\in {ℝ}^{p\times m},\hat{A}\in {ℝ}^{k\times k},\hat{B}\in {ℝ}^{k\times m},\hat{C}\in {ℝ}^{p\times k}$ and $\hat{D}\in {ℝ}^{p\times m}$. Where $n,k,m,p$ are full order, reduced order, number of inputs and number of outputs respectively.

The Data

The full order state matrices $A,B,C,D$ and the reduced model order $k$.

The Optimization Problem

The objective of the optimization is to reduce the $H_{\mathrm{\infty }}$ norm distance of the two systems. Minimizing $\Vert G-\hat{G}{\Vert }_{\mathrm{\infty }}$ with respect to $\hat{G}$.

The LMI: The Lyapunov Inequality

Objective: $\min \gamma$.

Subject to:: $\begin{array}{cc}P& =\left[\begin{array}{cc}P11& P12\\ P21& P22\end{array}\right]>0,\end{array}$

$\begin{array}{c}\left[\begin{array}{cccc}{A}^{T}P11+P11A& A^{T}P12+P12\hat{A}& P11B-P12\hat{B}& C^T\\ {\hat{A}}^{T}P12^T+P12^{T}A& {\hat{A}}^{T}P22+P22\hat{A}& P12^{T}B-P22\hat{B}& {\hat{C}}^T\\ B^{T}P11-{\hat{B}}^{T}P12^T& B^{T}P12-{\hat{B}}^{T}P22& -\gamma I& D^T-{\hat{D}}^T\\ C& \hat{C}& D-\hat{D}& -\gamma I\end{array}\right]>0\end{array}$

It can be seen from the above LMI that the second matrix inequality is not linear in $\hat{A},\hat{B},\hat{C},\hat{D},P$. By making $\hat{A},\hat{B}$ constant it is linear in $\hat{C},\hat{D},P$. And if $P12,P22$ are constant it is linear in $\hat{A},\hat{B},\hat{C},\hat{D},P11$. Hence the following iterative algorithm can be used.

(a) Start with initial estimate $\hat{G}$ obtained from techniques like Hankel-norm reduction/Balanced truncation.

(b) Fix $\hat{A},\hat{B}$ and optimize with respect to $\hat{C},\hat{D},P$.

(c) Fix $P12,P22$ and optimize with respect to $\hat{A},\hat{B},\hat{C},\hat{D},P11$.

(d) Repeat steps (b) and (c) until the solution converges.

Conclusion:

The LMI techniques results in model reduction close to the theoretical limits set by the largest removed hankel singular value. The improvements are often not significant to that of Hankel-norm reduction. Due to high computational load it is recommended to only use this algorithm if optimal performance becomes a necessity.

External Links

A list of references documenting and validating the LMI.