An ISS characterization for discontinuous discrete‐time systems

Input‐to‐state stability (ISS) is an important concept to study robustness properties of nonlinear control systems. The ISS property can be, in particular, characterized by ISS Lyapunov functions. For discrete‐time nonlinear control systems two different forms of ISS Lyapunov functions (implication‐form and dissipation‐form) are known to be equivalent if the dynamics are continuous. However, for discontinuous dynamics the equivalence is no longer satisfied. In this work, we discuss this phenomenon and, eventually, give a complete characterization of ISS in terms of ISS Lyapunov functions.

We consider nonlinear discrete-time systems of the form with state x ∈ R n , input u ∈ R m and G : R n × R m → R n saisfying G(0, 0) = 0. Solutions at time k ∈ N with initial state x(0) = ξ and input u(·) are denoted by x(k, ξ, u(·)). For discrete time systems of the form (1) with unconstrained state and input spaces, solutions are uniquely defined. Thus, no additional properties on G have to be required. In the following, P denotes the class of positive definite function φ : R + → R + and K, K ∞ , KL are the standard comparison functions, see eg. [1]. Definition 1 We call system (1) input-to-state stable (ISS) if there exist functions β ∈ KL and γ ∈ K ∞ such that for all initial values ξ ∈ R n , all input functions u(·) ∈ ∞ and all times k ∈ N we have There are two standard forms of ISS Lyapunov functions to characterize the ISS property of system (1). Definition 2 Let V : R n → R + satisfy α 1 ( ξ ) ≤ V (ξ) ≤ α 2 ( ξ ) for some α 1 , α 2 ∈ K ∞ . Then V is said to be a • dissipation-form ISS Lyapunov function if there exist functions α ∈ K ∞ and σ ∈ K such that for all ξ ∈ R n and µ ∈ R m we have • implication-form ISS Lyapunov function if there exist functions χ ∈ K and θ ∈ P such that for all ξ ∈ R n and µ ∈ R m we have Note that if G in (1) is continuous then also the ISS LF can be assumed to be continuous. This follows from [2, Theorem 1], where the authors prove the equivalence between ISS of system (1) and the existence of a continuous (even smooth) dissipation-form ISS Lyapunov function. Moreover, by [2, Remark 3.3] both dissipation-form and implication-form ISS Lyapunov functions are equivalent for continuous dynamics G.
For discontinuous systems, it is shown in [3, Example 3.3] that the concept of an implication-form ISS Lyapunov functions is not strong enough to conclude ISS. Here, we give a slightly modified version of this example to show that not only continuity in the origin is necessary but also boundedness on bounded sets.
Motivated by Example 3, we see that for ISS of a system not only continuity in the origin is necessary but also boundedness on bounded sets. Therefore, we follow a different procedure than that in [3]. In particular, we develop a new Lyapunov characterization of ISS, which does not require an additional property on the implication-form ISS Lyapunov function, but on the system dynamics itself. The essential condition is the following, which was originally introduced in [4]. (1) is globally K-bounded if there exist functions η 1 , η 2 ∈ K such that for all ξ ∈ R n and µ ∈ R m , we have Note that ISS of (1) implies global K-boundedness, see [5,Remark 3.3]. Therefore, the global K-boundedness is a necessary condition for ISS. Moreover, from [6, Lemma 5] we know that G is K-bounded if G satisfies G(0, 0) = 0, is continuous in (0, 0) and it is bounded on bounded sets. This exactly reflects the behaviour of the discontinuous system given in Example 3. In particular, any continuous G satisfying G(0, 0) = 0 is globally K-bounded.
Eventually, we are able to prove that the existence of an implication-form ISS Lyapunov function indeed implies ISS of the discontinuous system (1), provided that G is globally K-bounded (which is necessary for ISS). This main conclusion is summarized in the following result. Theorem 5 Consider system (1). Then the following are equivalent.

2.
There exists a dissipation-form ISS Lyapunov function.

3.
There exists an implication-form ISS Lyapunov function and G is globally K-bounded.
The proof of the theorem can be found in [6, Theorem 12], where we particularly present an extension of this observation for a more general ISS property with respect to closed sets. In addition to the dissipation and implication forms of ISS Lyapunov functions, an ISS Lyapunov function can be given in a so-called max-form, which is particularly useful in the context of large-scale systems, see [7] for more details.