Stability Analysis of Linear Uncertain Systems via Checking Positivity of Forms on Simplices

Given a continuous linear time-invariant system in the state space model, its Hurwitz stability is determined by the distribution of eigenvalues of the system matrix. When entries of the system matrix are uncertain, e.g., they are varying in intervals, the robust stability of such a system have been studied in a large amount of literatures. First, attempts were made to find a Kharitonov-like criterion [1] of the stability of an interval matrix which only checks some extreme matrices [2], but the criterion was found to be false [3]. Later, necessary and sufficient criterions were proposed for interval matrices with special properties (e.g., real symmetric interval matrices [7] or Hermitian interval matrices [8]). At the same time, various sufficient criterions were found to check the stability of interval matrices [4,5,6,7].

In this paper, we study a more general kind of uncertainty of system matrices, i.e., entries of system matrices are rational functions of uncertain parameters which are bounded by intervals. we will present a complete method which can check the robust stability of such systems in finite steps.

Denote by R the field of real numbers, the system matrix A ∈ R n×n in ẋ(t) = Ax(t) is called Hurwitz stable if all its eigenvalues lie in the open left half complex plane. When A is continuously varying in R n×n , i.e., A is in a connected set A ⊂ R n×n , we say A is robustly Hurwitz stable if each A ∈ A is Hurwitz stable. [14] showed that the system matrix with polytopic uncertainty is robustly Hurwitz stable if and only if a Hurwitz stable matrix exists and two forms (i.e., homogenous polynomials) are positive on the standard simplex. In fact, we could come to a similar conclusion for A. Denote the characteristic polynomial of A by

and the Hurwitz matrix of f A (s) by ∆ A , which is an n × n matrix defined as

Proof. The proof is exactly the same as that of Theorem 1 in [14].

In this paper, we are interested in a type of matrix uncertainty in which case the entries of the matrix are rational functions of parameters varying in intervals, i.e.,

where q = (q 1 , . . . , q m ) T , a ij (q) are rational functions of q, and q k ∈ [q k , q k ], k = 1, . . . , m. We have Theorem 2. The robust Hurwitz stability of the matrix set A can be checked in finite steps.

The proof of the above theorem will be given in Section 5.

In our method of checking robust Hurwitz stability of A, we need transform this problem to a problem of checking positivity of forms on simplices. Since the uncertain parameters are varying in hypercubic, we first introduces the procedure [10] of subdividing the unit hypercubic [0, 1] m into nonoverlapping simplices in this section.

Denote by Θ m the set of all m! permutations of {1, 2, . . . , m}.

spanning a simplex can be formed using following equations.

Denote by S θ the simplex spanned by {a 0 , . . . , a m }, i.e.,

it could be readily shown that such constructed S θ has the following equivalent definition

According to [10], these simplices have no common interior points with each other, and

Denote by N the set of all nonnegative integers, let α = (α 1 , α 2 , . . . , α m ) ∈ N m , and

it is immediate that f is strict positive on the standard (m -1)-simplex Sm if all c α are positive, where Sm = {(t 1 , . . . , t m ) :

In fact this condition is not only sufficient, but also necessary in the following sense.

Theorem 3 (Pólya’s Theorem, [15]). If a form f (x 1 , . . . , x m ) is strict positive on Sm , then for sufficiently large integer N, all coefficients of

are positive.

[16] gave an explicit bound for N, that is

where

and λ is the minimum of f on Sm . A newly proposed method, i.e., the WDS (i.e., weighted difference substitution) method [12], can also be used to check positivity of forms efficiently, we will introduce this method below.

Suppose θ = (k 1 k 2 . . . k m ) ∈ Θ m , let P θ = (p ij ) m×m be the permutation matrix corresponding θ, that is

Given T m ∈ R m×m , where

let

and call it the WDS matrix determined by the permutation θ. The variable substitution x ← A θ x corresponding θ is called a WDS. In fact, each variable substitution corresponds an assumption of sizes of x 1 , x 2 , . . . , x m in Sm . If for each θ ∈ Θ m , all coefficients of f (A θ x) are positive, then f (x) is positive on Sm . More generally, if there exists k ∈ N, such that all forms in WDS (k) (f ) have no nonnegative coefficients, then f (x) is positive on Sm , where

is the kth WDS set of f (x). In fact, the reverse is also true.

). If f (x 1 , . . . , x m ) is a form of degree d, the magnitudes of its coefficients are bounded by M, then f is positive on Sm , if and only if there exists k ≤ C p (M, m, d), such that each form in WDS (k) (f ) has no nonnegative coefficients, where

Remark: The C p (M, m, d) in ( 8) provides a theoretical upper bound of the number of steps of substitutions required to check positivity of an integral form. In practice, numbers of steps used are gen

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