Input-to-state stability for discrete-time nonlinear systems

Design of observer-based controllers for LPV systems with unknown parameters

“…We assume that ω k ∈ Ω, k ∈ N for some set Ω ⊂ R dω . Next, we define the notions of global asymptotic stability (GAS) and global exponential stability (GES) for (1) and input-tostate stability (ISS) [10], [11] for (2). Definition 1: [10], [11] The system (1) with uncertainty set Ω is called globally asymptotically stable (GAS), if there exists a KL-function β such that, for each x 0 ∈ R n and all {ω k } k∈N with ω k ∈ Ω, k ∈ N, it holds that the corresponding state trajectory satisfies…”

“…Next we state sufficient conditions for ISS using so-called ISS Lyapunov functions. The proofs are omitted for shortness, but can be based on [11], [15] by including the uncertainty parameter and adopting parameter-dependent Lyapunov functions.…”

“…Interestingly, the conditions in (15) also guarantee that the hypotheses of Theorem 4 are satisfied (as will be shown in Theorem 7 below). This shows the non-conservatism of the LMIs (11) as the existence of a nominal observer for the exact LPV system, with a parameter-dependent quadratic Lyapunov function proving GES of the error dynamics, is sufficient for (11) to hold. This also shows that any GES observer for the exact LPV system has some degree of robustness in the sense that it has some steady state relative error as in (13) for some σ ev > 0.…”

“…Convexity is only obtained in case of stability analysis [9]. In [9] it is also shown that an observer that is asymptotically recovering the state when the parameters are exactly measured, is inputto-state stable (ISS) [10], [11] with respect to mismatch between the true and the available parameters. However, [9] does not study the observer synthesis nor the outputbased stabilization problem.…”

Numerical ISS controller design via a dynamic game approach

“…In order to avoid this problem, in this paper we propose an approach which consists of converting the ISS controller design problem into a uniform stabilization problem for a perturbed system which can be accomplished without increasing the dimension. The equivalence between ISS and robust stability was already exploited in a theoretical context in [17] (for a continuous time version of this result see [23]) and thus our approach can be seen as a constructive numerical interpretation of the results in [17]. In order to solve the auxiliary stabilization problem under perturbations we use the game theoretic algorithmic approach from [8] which in turn relies on [9], [19].…”