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A BMI approach for ℋ︁ ∞ gain scheduling of discrete time‐varying systems
Author(s) -
Borges Renato A.,
Oliveira Ricardo C. L. F.,
Abdallah Chaouki T.,
Peres Pedro L. D.
Publication year - 2010
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.1507
Subject(s) - control theory (sociology) , mathematics , convex optimization , discrete time and continuous time , gain scheduling , lyapunov function , parametric statistics , mathematical optimization , upper and lower bounds , computer science , nonlinear system , regular polygon , control (management) , artificial intelligence , mathematical analysis , statistics , physics , geometry , quantum mechanics
The problem of gain‐scheduled state feedback control for discrete‐time linear systems with time‐varying parameters is considered in this paper. The time‐varying parameters are assumed to belong to the unit simplex and to have bounded rates of variation, which depend on the values of the parameters and can vary from slow to arbitrarily fast. An augmented state vector is defined to take into account possible time‐delayed inputs, allowing a simplified closed‐loop analysis by means of parameter‐dependent Lyapunov functions. A gain‐scheduled state feedback controller that minimizes an upper bound to the ℋ ∞ performance of the closed‐loop system is proposed. No grids in the parametric space are used. The design conditions are expressed in terms of bilinear matrix inequalities (BMIs) due to the use of extra variables introduced by the Finsler's lemma. By fixing some of the extra variables, the BMIs reduce to a convex optimization problem, providing an alternate semi‐definite programming algorithm to solve the problem. Robust controllers for time‐invariant uncertain parameters, as well as gain‐scheduled controllers for arbitrarily time‐varying parameters, can be obtained as particular cases of the proposed conditions. As illustrated by numerical examples, the extra variables in the BMIs can provide better results in terms of the closed‐loop ℋ ∞ performance. Copyright © 2009 John Wiley & Sons, Ltd.