Robustness analysis of linear parameter varying systems using integral quadratic constraints

Abstract

SUMMARYA general approach is presented to analyze the worst case input/output gain for an interconnection of a linear parameter varying (LPV) system and an uncertain or nonlinear element. The LPV system is described by state matrices that have an arbitrary, that is not necessarily rational, dependence on the parameters. The input/output behavior of the nonlinear/uncertain block is described by an integral quadratic constraint (IQC). A dissipation inequality is proposed to compute an upper bound for this gain. This worst-case gain condition can be formulated as a semidefinite program and efficiently solved using available optimization software. Moreover, it is shown that this new condition is a generalization of the well-known bounded real lemma type result for LPV systems. The results contained in this paper complement known results that apply IQCs for analysis of LPV systems whose state matrices have a rational dependence on the parameters. The effectiveness of the proposed method is demonstrated on simple numerical examples.

Robustness analysis of uncertain discrete‐time systems with dissipation inequalities and integral quadratic constraints

“…The J -spectral factorization result (Lemma 6) is an important technical result on its own and has other potential applications, for example, formulating topological separation theorems [12].This paper complements several existing results in the literature. First, this paper provides a discrete-time counterpart to the continuous-time results in [13][14][15][16]. Moreover, Section 3 contains intermediate results regarding discrete-time IQC factorizations and a related open-loop linear quadratic (LQ) difference game.…”

“…The current paper will present discrete-time derivations assuming the nominal system is LTI. However, the extension to uncertain LPV and uncertain nonlinear systems follows along the lines of the continuous-time results in [13,16,21]. The standard IQC homotopy theory developed for both continuous and discrete-time systems [1,[22][23][24][25] can also be generalized for systems that do not have frequency domain interpretations [26].…”

On computational issues for stability analysis of LPV systems using parameter‐dependent Lyapunov functions and LMIs

“…with e T = [1, 1, … , 1] ∈ ℝ r . Therefore, the time derivatives of the parameters are confined into a manifold with dimension r − 1, given by the intersection of the hyperrectangle in (15) and the hyperplane in (21) .…”

“…6,[9][10][11] Recently, some results appear using high-order time derivatives of the parameters to combine the LFs, leading to improvements (see the works of Mozelli and Palhares 12,13 and references therein). In the work of Trofino and Dezuo, 14 general rational dependence on the parameters is considered, and in the work of Pfifer and Seiler, 15 a griding approach is used to include arbitrary dependence on the parameters.…”

Conic sector analysis using integral quadratic constraints

“…Generally, w 22 is not positive, so slightly different transformations will be required to convert the IQC (55) to an IQC of the form (13), depending on the sign of w 22 . In the case that w 22 > 0, we have that the IQC (55) is equivalent to…”