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Stability analysis for the linear system with structured uncertainties using pseudo-remainder sequences. (English) Zbl 1026.93044

The authors are interested in analyzing the stability of a polynomial affected by parametric uncertainty, a problem typically arising in the robust control of linear systems. It is assumed that the coefficients of the polynomial are multivariate polynomial functions of the uncertain parameters belonging to given intervals. At this level of generality, the problem is known to be NP-hard, and thus it is very unlikely that there exists a computational algorithm to solve the problem with a running time which is a polynomial function of the problem dimension (degree of the polynomial and number of uncertain parameters). It is well known that robust stability of such a polynomial is equivalent to positivity of the last-but-one Hurwitz determinant of the polynomial (itself a multivariate polynomial) as soon as there exists at least one stable polynomial for some admissible uncertain parameters. Then the authors focus on two distinct problems: (1) computing the last-but-one Hurwitz determinant; (2) ensuring positivity of this determinant over the whole set of interval uncertain parameters, i.e. over a given hyper-rectangle. Problem (1) is tackled via symbolic computation, by computing iteratively all the Hurwitz determinants with the Routh-Hurwitz stability criterion. Several algorithms are compared, and the authors’ preference goes to an optimal fraction free algorithm avoiding intermediate expression swell. For problem (2) monotonicity properties are invoked, which are sufficient conditions under which positivity of a multivariate polynomial over an hyper-rectangle is equivalent to positivity at the hyper-rectangle vertices only.

MSC:

93D09 Robust stability
93B25 Algebraic methods
93B51 Design techniques (robust design, computer-aided design, etc.)
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