-Gain Control for Positive Switched Systems with Time-Varying Delay via Delta Operator Approach

and Applied Analysis 3 Combining Lemma 2 in [45] and Lemma 1 in [28], one can obtain the remaining proof easily. Remark 4. WhenT → 0, system (2) degenerates to a general continuous-time positive switched system as follows: ?̇? (t) = A σ(t) x (t) + A dσ(t) x (t − d (t)) + D σ(t) w (t) , x (t) = φ (t) , t ∈ [−d, 0] z (t) = C σ(t) x (t) + E σ(t) w (t) , (5) where d(t) denotes the time-varying delay which is everywhere time differentiable and satisfies 0 ≤ d ≤ d(t) ≤ d, and ̇ d(t) ≤ d d < 1 for known constants d, d, and d d . Then, according to [39], system (5) is positive if and only if A p are Metzler matrices, and A dp ⪰ 0, C p ⪰ 0, D p ⪰ 0, and E p ⪰ 0, for all p ∈ m. Definition 5 (see [46]). For any switching signal σ(k) and any k 2 > k 1 ≥ 0, letN σ (k 1 , k 2 ) denote the number of switches of σ(k) over the interval [k 1 , k 2 ). For given τ a > 0 andN 0 ≥ 0, if the inequality N σ (k 1 , k 2 ) ≤ N 0 + k 2 − k 1