On Stability of Sets for Sampled-Data Nonlinear Inclusions via Their Approximate Discrete-Time Models and Summability Criteria

Abstract

On stability of sets for sampled-data nonlinear inclusions via their approximate discrete-time models and summability criteria. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2009Mathematics, , 48 (3), pp.1888Mathematics, -1913 Abstract. This paper consists of two main parts. In the first part, we provide a framework for stabilization of arbitrary (not necessarily compact) closed sets for sampled-data nonlinear differential inclusions via their approximate discrete-time models. We generalize [19, Theorem 1] in several different directions: we consider stabilization of arbitrary closed sets, plants described as sampleddata differential inclusions and arbitrary dynamic controllers in the form of difference inclusions. Our result does not require the knowledge of a Lyapunov function for the approximate model, which is a standing assumption in [21] and [19, Theorem 2]. We present checkable conditions that one can use to conclude semi-global asymptotic (SPA) stability, or global exponential stability (GES), of the sampled-data system via appropriate properties of its approximate discrete-time model.In the second part, we present sufficient conditions for stability of parameterized difference inclusions that involve various summability criteria on trajectories of the system to conclude global asymptotic stability (GAS), or GES, and they represent discrete-time counterparts of results given in [32]. These summability criteria are not Lyapunov based and they are tailored to be used within our above mentioned framework for stabilization of sampled-data differential inclusions via their approximate discrete-time models. We believe that these tools will be a useful addition to the toolbox for controller design for sampled-data nonlinear systems via their approximate discrete-time models.Key words. Sampled-data systems, stability, difference inclusions AMS subject classifications.1. Introduction. Although most controllers are nowadays implemented digitally using sample and hold devices, sampled data nonlinear control has received much less attention than continuous time nonlinear control. The controller design problem for sampled-data systems can be carried out in three essentially different ways: (i) emulation (design continuous time controller and then discretize the controller); (ii) discrete-time design (discretize the plant and design a discrete-time controller directly on the discrete time model); (iii) sampled-data design (use the real model of the sampled-data system that includes the inter-sample behavior to design the controller). For nonlinear systems, some results on emulation can be found in [16], while we are not aware of any results on sampled data design for nonlinear systems (details on the sampled data method for linear systems can be found in [4] and references cited therein).

Uniform stability of sets for difference inclusions under summability criteria

“…A sampled-data counterpart of this results is given in [13]. We note that the notion of cUGES was not used in [15], [13].…”

“…Of broader interest are necessary and sufficient conditions for stability of sets. A sampled-data counterpart of the integrability criteria for differential inclusions from [15] was reported in [13]. In this approach, one needs to establish stability of a family of approximate discrete-time models that are parameterized with the sampling period and the goal is to establish appropriate stability properties of this family that would guarantee that the family of exact discrete-time models will also be stable for sufficiently small sampling periods.…”

“…Results in [13] can be modified and used in the special case when the exact discrete time model of the sampleddata system is known and one does not need to deal with families of discrete-time systems. However, straightforward modifications of results from [13] to this special case would be unnecessarily restrictive and technical.…”

“…However, straightforward modifications of results from [13] to this special case would be unnecessarily restrictive and technical. Hence, in this paper we address the case when the exact discretetime model of the system is known and we prove the results under weaker assumptions than what would be possible by doing the modifications of [13]. In particular, we investigate various stability characterization of uniform global asymptotic (and exponential) stability of arbitrary sets for nonlinear difference inclusions (I).…”

Semiglobal exponential input-to-state stability of sampled-data systems based on approximate discrete-time models

“…They give sufficient conditions for global exponential stability of the closed-loop NCS. [9] provides a framework for stabilization of arbitrary closed sets for sampled-data nonlinear differential inclusions via approximate discrete-time models and present checkable conditions to ensure semiglobal practical stability or global exponential (asymptotic) stability of the sampled-data system. These disturbance-free results are presented for uniform sampling and under the assumption of perfect feedback.…”