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A convex optimization procedure to compute ℋ︁ 2 and ℋ︁ ∞ norms for uncertain linear systems in polytopic domains
Author(s) -
Oliveira Ricardo C. L. F.,
Peres Pedro L. D.
Publication year - 2007
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.825
Subject(s) - mathematics , polytope , lyapunov function , linear system , homogeneous polynomial , sequence (biology) , invariant (physics) , linear matrix inequality , degree (music) , matrix (chemical analysis) , convex optimization , matrix norm , polynomial , regular polygon , homogeneous , norm (philosophy) , discrete mathematics , mathematical optimization , combinatorics , matrix polynomial , mathematical analysis , nonlinear system , eigenvalues and eigenvectors , composite material , mathematical physics , biology , genetics , geometry , quantum mechanics , political science , physics , materials science , law , acoustics
In this paper, a convergent numerical procedure to compute ℋ 2 and ℋ ∞ norms of uncertain time‐invariant linear systems in polytopic domains is proposed. The norms are characterized by means of homogeneous polynomially parameter‐dependent Lyapunov functions of arbitrary degree g solving parameter‐dependent linear matrix inequalities. Using an extension of Pólya's Theorem to the case of matrix‐valued polynomials, a sequence of linear matrix inequalities is constructed in terms of an integer d providing a Lyapunov solution for a given degree g and guaranteed ℋ 2 and ℋ ∞ costs whenever such a solution exists. As the degree of the homogeneous polynomial matrices increases, the guaranteed costs tend to the worst‐case norm evaluations in the polytope. Both continuous‐ and discrete‐time uncertain systems are investigated, as illustrated by numerical examples that include comparisons with other techniques from the literature. Copyright © 2007 John Wiley & Sons, Ltd.