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 sense. This methods uses LMI techniques iteratively to obtain the result.

The SystemEdit

Given a state-space representation of a system   and an initial estimate of reduced order model  .


Where   and  . Where   are full order, reduced order, number of inputs and number of outputs respectively.

The DataEdit

The full order state matrices   and the reduced model order  .

The Optimization ProblemEdit

The objective of the optimization is to reduce the   norm distance of the two systems. Minimizing   with respect to  .

The LMI: The Lyapunov InequalityEdit

Objective:  .

Subject to::  


It can be seen from the above LMI that the second matrix inequality is not linear in  . But making   constant it is linear in  . And if   are constant it is linear in  . Hence the following iterative algorithm can be used.

(a) Start with initial estimate   obtained from techniques like Hankel-norm reduction/Balanced truncation.

(b) Fix   and optimize with respect to  .

(c) Fix   and optimize with respect to  .

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


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 LinksEdit

A list of references documenting and validating the LMI.

  • Model order Reduction using LMIs - A conference paper by Helmersson, Anders, Proceedings of the 33rd IEEE Conference on Decision and Control, 1994, p. 3217-3222 vol.4

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