Testing Sets

Abstract

In Chapter 4 several testing methods were presented for checking stability of a family of polynomials

$$P(s,Q)=\lbrace p(s,{\bf\it q})\mid {\bf\it q}\in Q\rbrace$$

((5.0.1))

generated by an uncertain polynomial defined over Q

$$p(s,{\bf\it q})= a_0({\bf\it q})+a_1({\bf\it q})s+\ldots + a_{n-1}({\bf\it q})s^{n-1}+a_n({\bf\it q})s^n, \quad {\bf\it q}\in Q$$

((5.0.2))

As in Chapter 4, real and continuous coefficient functions a _i (q) are assumed with

$$a_n({\bf\it q})>0$$

((5.0.3))

for all q ∈ Q where Q is a parameter box

$$q=\lbrace {\bf\it q}\mid q_i\in\lbrack q_i^-;q_i^+\rbrack,\ i=1,\ldots,\ell\rbrace$$

((5.0.4))

The robustness tests considered so far were all based on the complete set of uncertain parameters, thus, the testing set for the robustness checks of p(s, g) over Q was Q itself. Here, the question arises whether it suffices to check only a subset of Q. Consider for example the polynomial

$$p(s,{\bf\it q})=q_1+q_2s+s^2$$

((5.0.5))

uncertain in two parameters q₁ and q₂ in a two-dimensional box

$$Q=\lbrace {\bf\it q}\mid q_i\in \lbrack q_i^-;q_i^+\rbrack, i=1,2\rbrace$$

((5.0.6))

There is à simple robustness result: P(s, Q) is stable for all q = [q1 q2]^T in Q if and only if q1 < 0 and q2 < 0. This is trivially true if and only if q⁻ > 0 and q₂ : 0. Thus, a simple robustness test for the uncertain second order polynomial is: the polynomial (5.0.5) is robustly stable over the box Q if and only if the vertex

$$q^{--}:=\lbrack q_1^- q_2^-\rbrack^T$$

((5.0.7))

of Q yields a stable polynomial (in short: q^‒‒ is stable). Hence, to check stability, only a single point in the uncertainty box Q has to be checked. The set

$$Q_T:=\lbrace{\bf\it q}^{--}\rbrace\subset Q$$

((5.0.8))

is a testing set for the robustness check of the uncertain polynomial (5.0.5) over Q. A general definition for a testing set will now be given:Definition 5.1. A set Q_T ⊂ Q is called a testing set (of Q) if the stability of the polynomial family P(s, Q_T) implies the stability of P(s, Q).