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 $H_{\infty }$ 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 ${\hat {G}}(s)$.

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

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 ${\hat {D}}\in \mathbb {R} ^{p\times m}$. 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 $H_{\infty }$ norm distance of the two systems. Minimizing $\|G-{\hat {G}}\|_{\infty }$ with respect to ${\hat {G}}$.

The LMI: The Lyapunov Inequality

Objective: $\min \gamma$.

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

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

It can be seen from the above LMI that the second matrix inequality is not linear in ${\hat {A}},{\hat {B}},{\hat {C}},{\hat {D}},P$. 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.