Articles - Mathematics and its Applications

Robust stability of discrete-time linear stochastic systems 161into account that ˆM(i) ≤ 0, we conclude that ˆX(i) < 0, 1 ≤ i ≤ N. TakingX(i) = − ˆX(i) one sees that X = (X(1), X(2), ..., X(N)) solves (48) andX(i) > 0, 1 ≤ i ≤ N.This completes the proof of the implication (iii) → (ii).The equivalence (ii) ↔ (iv) follows immediately by a Schur complementtechnique. This completes the proof of the theorem.If the system (4) is either in the case N = 1 or N ≥ 2, with A k (i) = 0,B k (i) = 0, 1 ≤ k ≤ r, i ∈ D, the result proved in Theorem 3.6 recover as specialcases the stochastic version of the Bounded Real Lemma for discrete-timelinear stochastic systems perturbed by independent random perturbationsand the discrete-time linear stochastic systems with Markovian switching,respectively.Let us remark that if the zero state equilibrium of (6) is ESMS fromTheorem 3.6, it follows that‖T ‖ = inf{γ > 0, for which it exists X ∈ S N n , X > 0,such that (48) holds} = inf{γ > 0,DTSARE (38) has a positive semidefinite solution verifying (39)}4 The small gain theorem and robust stabilityOne of the important consequence of the Bounded Real Lemma is the socalled Small Gain Theorem. It is known that this result is a powerful toolin the derivation of some estimates of the stability radius with respect toseveral classes of parametric uncertainties. We start with an auxiliary resultwhich is interesting in itself:Theorem 4.1 Regarding the system (4) we assume that the followingassumptions are fulfilled:a) the number of inputs equals the number of outputs (i.e. m v = n z = m);b) the zero state equilibrium of the corresponding linear system (6) isESMS;c) the input-output operator T associated to the system (4) satisfies‖T ‖