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.
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 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
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.
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