Input-to-State Stability of Discrete-Time Lur'e Systems

Abstract

An input-to-state stability theory, which subsumes results of circle criterion type, is developed in the context of discrete-time Lur'e systems. The approach developed is inspired by the complexified Aizerman conjecture.

Input-to-State Stability of Periodic Orbits of Systems With Impulse Effects via Poincaré Analysis

“…where the first inequality follows from Proposition 1; the second from (9) with the solution starting from ∆(x 0 , v 0 ); the third from (46); the fourth follows by [42,Lemma 14] on β(L ∆ x 0 − x * + v ∞ , t k − t 0 ) followed by absorbing the v ∞ term in α 2 ; and the last inequality follows from the fact that β is monotonically decreasing in time and t k+1 − t k = T I (x k , u k , v k ) ≥ T for all k ∈ Z + . Noting that (1/λ)β ∈ KL, (1/λ)α 1 ∈ K and (1/λ)α 2 ∈ K, and comparing the last inequality with (10), we get that the solution of (7) satisfies Definition 3, thus completing the proof.…”

Semi-global persistence and stability for a class of forced discrete-time population models

“…Note that (2.4) implies that the zero equilibrium of the unforced system x ∇ = Ax + bh(c T x) is globally asymptotically stable (GAS). Simple counterexamples [54] show that there exist systems of the form (2.1) such that the zero equilibrium of the unforced system is GAS, but the system is not ISS and, moreover, that there exist initial conditions and bounded inputs which generate unbounded state trajectories.…”

“…Under the assumption that the nonlinearity h satisfies h(0, u) = 0, it follows that 0 is an equilibrium of (1.1). A so-called absolute stability criterion for (1.1) is a sufficient condition for the stability of this equilibrium, usually formulated in terms of the linear components (A, b, c) and related sector or boundedness conditions for h: stability is guaranteed for every nonlinearity h satisfying these sector or boundedness properties, thereby ensuring robustness of stability with respect to uncertainty in h. There is a large and contemporary body of work on absolute stability theory; see, for example, [26,34,35,38,42,53,54,67,68].…”

Input-to-state stability of Lur’e systems

“…Traditionally, Lyapunov approaches to the stability theory of systems of the form (1.1) consider unforced Lur'e systems (i.e., v = 0 in (1.1)), whilst Lur'e systems with forcing (usually acting through B, that is, B e = B) have been studied using the input-output framework initiated by Sandberg and Zames in the 1960s, see, for example [27]. More recently, forced Lur'e systems have been analysed in the context of input-to-state stability (ISS) theory, see [1,2,12,13] (and [22] for discrete-time systems). In [1], an ISS result is obtained for Lur'e systems (1.1) under the assumptions that B e = B, the underlying linear system has the positive real property and the nonlinearity (which may have superlinear growth) satisfies a suitable cone condition.…”