Robust pole placement for plants with semialgebraic parametric uncertainty

Abstract

Abstract-In this paper we address the problem of robust pole placement for linear-time-invariant systems whose uncertain parameters are assumed to belong to a semialgebraic region. A dynamic controller is designed in order to constrain the coefficients of the closed-loop characteristic polynomial within prescribed intervals. Two main topics arising from the problem of robust pole placement are tackled by means of polynomial optimization. First, necessary conditions on the plant parameters for the existence of a robust controller are given. Then, the set of all admissible robust controllers is sought. Convex relaxation techniques based on sum-of-square decomposition of positive polynomials are used to efficiently solve the formulated optimization problems through semidefinite programming techniques.

Computation of the Structured Singular Value via Moment LMI Relaxations

“…The following theorem (originally presented by the author and coworkers in [19]), provides necessary and sufficient conditions to test robust non-singularity of I − M ∆ against the set of structured uncertainties B r ∆ .…”

Embedding of Nonlinear Systems in a Linear Parameter-Varying Representation

“…, d n−r dt n−r u and the ranges of the latter variables for which ∇Φ| x0 is nonsingular should be specified. This can be tackled by using the approach proposed by [Cerone et al, 2012] to evaluate non-singularity of matrices with uncertain entries by formulating this problem as a non-convex polynomial optimization problem, whose approximate solution is efficiently computed by means of sum-of-square convex-relaxation techniques. Therefore, one can check that ∇Φ| x0 is nonsingular and obtain the corresponding ranges of u, d…”