# 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 ${\displaystyle H_{\infty }}$ sense. This methods uses LMI techniques iteratively to obtain the result.

## The System

Given a state-space representation of a system ${\displaystyle G(s)}$  and an initial estimate of reduced order model ${\displaystyle {\hat {G}}(s)}$ .

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

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

## The Data

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

## The Optimization Problem

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

## The LMI: The Lyapunov Inequality

Objective: ${\displaystyle \min \gamma }$ .

Subject to:: {\displaystyle {\begin{aligned}\ P&={\begin{bmatrix}\ P11&P12\\\ P21&P22\\\end{bmatrix}}\ >0,\end{aligned}}}

{\displaystyle {\begin{aligned}{\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{aligned}}}

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

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

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

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

(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.