Positive trigonometric polynomials for strong stability of difference equations

Abstract

We follow a polynomial approach to analyse strong stability of continuous-time linear difference equations with several delays Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5.

On strong stability and robust strong stability of linear difference equations with two delays

“…whose left hand side is a linear function of (A, B) . Thus, the most important feature of Theorem 1, when compared with the results in [2] (see Lemma 3), the result in [6] (see Lemma 2) and the method in [13], is that the coefficient (A, B) appears as a linear function. Such a property is helpful for solving the robust stability analysis problem, as made clear below.…”

“…The strong stability concept is important since in practical applications the delays are generally subject to small errors [10]. The test of strong stability is however rather complex [13]. Indeed, condition (2) is not tractable in general since the spectral radius should be tested for all θ i ∈ [0, 2π] , i = 1, 2, .…”

“…, N. Strong stability of (1) was tested via deciding positive definiteness of a multivariate trigonometric polynomial matrix, which is then solved as a converging hierarchy of LMIs [13]. The condition in [13] needs to compute the characteristic equation of (1), which is not explicitly expressed as functions of the coefficients, and thus seems difficult to be used for robust stability analysis. For a single delay, strong stability can be checked by computing the generalized eigenvalues of a pair of matrices [16,17] as well as the matrix pencil based approach [19].…”

“…The method of cluster treatment of characteristic roots was used in [20] to derive the stability maps of (1) with three delays. For more related work, see [10,12,13,20] and the references therein.…”

Minimizing trigonometric matrix polynomials over semi-algebraic sets

“…For the case of multiple variable and scalar trigonometric polynomials, a method based on sums of squares (SOS) of trigonometric polynomials has been proposed in [4]. This method has been extended to the case of trigonometric matrix polynomials in [5] where its application to strong stability analysis is described. See also [6] which proposes a simplified method for trigonometric polynomials in two variables and describes its application to FIR filter design.…”