# 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 System**
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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 Data**
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The full order state matrices and the reduced model order .

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

**The LMI:** The Lyapunov Inequality
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Objective: .

Subject to::

It can be seen from the above LMI that the second matrix inequality is not linear in . By 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.

**Conclusion:**
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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 edit

A list of references documenting and validating the LMI.