User:Margav06/sandbox/Click here to continue/Applications of Linear systems/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{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 $$\hat{D} \in \mathbb{R}^{p\times m}$$. 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{align} \ P &= \begin{bmatrix} \ P11 & P12 \\ \ P21 & P22 \\ \end{bmatrix} \ > 0, \end{align} $$

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

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