ISS inequalities for vector versions of Halanay’s inequality and of the trajectory-based approach

Building coercive Lyapunov–Krasovskii functionals based on Razumikhin and Halanay approaches

“…A second possible line of research is the use of vector versions of Halanay conditions. It was shown in References 33–35 that matrix‐based conditions can be derived to conclude stability of systems involving several Lyapunov functions, which turns out particularly useful for networks of interconnected dynamical systems. While the techniques presented here allow for the construction of an LKF in that setup, the conditions we obtained so far appear too demanding as compared to the flexibility offered the original vector extension of Halanay's inequality.…”

Event-Triggered Control Under Unknown Input and Unknown Measurement Delays Using Interval Observers

“…In the second part, we use the bound on |x(t) − x(t i )| from the first part to find a vector valued upper bound on ṡ(t) where s(t) = x(t) − x(t) is the difference between upper and lower vectors in an interval observer (x, x) for x. In the final part, we use the bound on ṡ(t) from the second part and a variant of the ISS vector Halanay's inequality from [6] to obtain our ISS estimate (8).…”

“…We apply [6, Theorem 1] to the function w(t) = s(t) and τ = 2(T + ∆ ♯ ), after extending s to the domain [−τ, +∞) by defining s to be constant on [−τ, 0], so we use a piecewise C 1 function w instead of assuming that w is C 1 as was assumed in [6]; this is valid because the proof of [6, Theorem 1] remains valid if the C 1 requirement on w in [6] is relaxed to only requiring it to be piecewise C 1 . This ensures that s satisfies an ISS property when…”

“…Since |x(t)| ≤ s(t) holds for all t ≥ 0, and since the values x(t) do not depend on δ(ℓ) values for times ℓ ≥ t, we can substitute (33) into (30) and then use the fact that β 0 (a + b, t) ≤ β 0 (2a, t) + β 0 (2b, 0) holds for all a ≥ 0, b ≥ 0 and t ≥ 0, to satisfy the ISS requirement with β(r, t) = β 0 (2(3 + 2h)c 3 r, t) and γ(s) = γ 0 (s) + β 0 (2c 3 s, 0). Remark 2: In terms of diagonal matrices D whose main diagonal entries are all positive, matrices P > 0, constants h > 0 and τ > 0, and a constant c 0 ∈ (0, 1) and U 0 ∈ [1, +∞) n such that M 0 U 0 = c 0 U 0 where M 0 = e −Dh + 0 −h e Dℓ dℓP , the β 0 from [6, Theorem 1] and [6, Remark 1] is β 0 (r, t) = n||U 0 ||e (t−h) ln(c 0 )/(τ +h) r. The existence of c 0 and U 0 follows because M 0 > 0 is Schur stable [6]. This and the last part of the proof of Theorem 1 ensure the exponential stability condition from Remark 1.…”

“…We allow time-varying delays, and unknown disturbances. Although neither this work nor our prior eventtriggered controlled works assume that our systems to be controlled are positive, our analysis uses new synergies of the theory of positive systems, which has been shown in our prior works to lead to less triggering than traditional approaches, and the vector Halanay's inequality approach that was presented in [6] in the absence of event triggering.…”