23-09-2013, 04:31 PM
Fast algorithms for solving H∞-norm minimization problems
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Abstract
In this paper we propose an efficient computational ap-
proach to minimize the H∞ -norm of a transfer-function
matrix depending affinely on a set of free parame-
ters. The minimization problem, formulated as a semi-
infinite convex programming problem, is solved via a
relaxation approach over a finite set of frequency val-
ues. In this way, a significant speed up is achieved by
avoiding the solution of high order LMIs resulting by
equivalently formulating the minimization problem as
a high dimensional semidefinite programming problem.
Numerical results illustrate the superiority of proposed
approach over LMIs based techniques in solving zero
order H∞ -norm approximation problems.
Introduction
Let Gθ (s) be a p × m stable transfer-function matrix
(TFM) of a linear, continuous time-invariant system
with a state space realization (A, B, Cθ , Dθ ) satisfying
Conclusions
We have shown that several affine approximation prob-
lems encountered in model reduction can be solved us-
ing a fast algorithm which avoids the conversion to an
equivalent high order SDP formulation. Since the so-
lution of SDP problems via LMIs is not feasible even
for moderately large dimensions, the new method of-
fers a viable alternative to solve this class of approxi-
mation problems. Important speedup can be achieved
by fully exploiting the structure of the underlying op-
timization problem. Preliminary experimental results
indicate a good potential of the proposed approach to
address other similar problems in the control theory.
An open aspect related to the SIP algorithm is the
development of a rigorous theory for its convergence
rate, independently of or in conjunction with adaptive
precision computations to solve subproblems.