A.R. Teel

Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations

Given a parameterized (by sampling period T) family of approximate discrete-time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all T sufficiently small, we present conditions which guarantee that the same family of controllers semi-globally practically stabilizes the exact discrete-time model of the plant for sufficiently small sampling periods. When the family of controllers is locally bounded, uniformly in the sampling period, the inter-sample behavior can also be uniformly bounded so that the (time-varying) sampled-data model of the plant is uniformly semi-globally practically stabilized. The result justifies controller design for sampled-data nonlinear systems based on the approximate discrete-time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied. Our analysis is applicable to a wide range of time-discretization schemes and general plant models.

Periodic Event-Triggered Control for Linear Systems

Event-triggered control (ETC) is a control strategy that is especially suited for applications where communication resources are scarce. By updating and communicating sensor and actuator data only when needed for stability or performance purposes, ETC is capable of reducing the amount of communications, while still retaining a satisfactory closed-loop performance. In this paper, an ETC strategy is proposed by striking a balance between conventional periodic sampled-data control and ETC, leading to so-called periodic event-triggered control (PETC). In PETC, the event-triggering condition is verified periodically and at every sampling time it is decided whether or not to compute and to transmit new measurements and new control signals. The periodic character of the triggering conditions leads to various implementation benefits, including a minimum inter-event time of (at least) the sampling interval of the event-triggering condition. The PETC strategies developed in this paper apply to both static state-feedback and dynamical output-based controllers, as well as to both centralized and decentralized (periodic) event-triggering conditions. To analyze the stability and the L2-gain properties of the resulting PETC systems, three different approaches will be presented based on 1) impulsive systems, 2) piecewise linear systems, and 3) perturbed linear systems. Moreover, the advantages and disadvantages of each of the three approaches will be discussed and the developed theory will be illustrated using a numerical example.

Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem

A Razumikhin-type theorem that guarantees input-to-state stability for functional differential equations with disturbances is established using the nonlinear small-gain theorem. The result is used to show that input-to-state stabilizability for nonlinear finite-dimensional control systems is robust, in an appropriate sense, to small time delays at the input. Also, relaxed Razumikhin-type conditions guaranteeing global asymptotic stability for differential difference equations are given.

Changing supply functions in input/state stable systems

We consider the problem of characterizing possible supply functions for a given dissipative nonlinear system and provide a result which allows some freedom in the modification of such functions. >

Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems

Abstract We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time system and a K function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a closed-loop system, implies uniform ISS of the sampled-data nonlinear system. Our results serve to quantify these facts by means of comparison functions. Our results can be used as an alternative to prove and extend results in [1] or extend some results in [4] to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate.

A note on input-to-state stability of sampled-data nonlinear systems

It is shown for nonlinear systems that sampling sufficiently fast an input-to-state stabilizing (ISS) continuous time control law results in an ISS sampled-data control law. Two main features of our approach are: we show how the nonlinear sampled-data system can be modeled by a functional differential equation (FDE); and we exploit a Razumikhin type theorem for ISS of FDE that was proved by Teel (1998) to analyze the sampled-data system.

Linear matrix inequalities for full and reduced order anti-windup synthesis

This work considers the design of fixed-order antiwindup compensators guaranteeing stability and a given level of L2 performance. The main results show how the design of such a compensator can be cast as a nonconvex optimization problem. It is demonstrated how, under given conditions, this optimization problem can be reduced to a standard LMI (linear matrix inequality) feasibility problem, and situations in which these compensators coincide with those in the literature are described. Moreover, given this formulation, an algorithm for the construction of anti-windup compensators which meet an optimal C2 performance bound is proposed. Furthermore, a lower bound on the C2 performance achievable is shown to be precisely the greater of that predicted by the bounded real lemmas for the linear open-loop plant and for the linear unsaturated closed-loop system.

Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

In this note we consider stability analysis of discrete-time discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discrete-time discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several input-to-state stability tests that explicitly employ an available discontinuous Lyapunov function.

STABILITY PROPERTIES OF RESET SYSTEMS

Abstract Stability properties for a class of reset systems, such as systems containing a Clegg integrator, are investigated. We present Lyapunov based results for verifying L 2 and exponential stability of reset systems. Our results generalize the available results in the literature and can be easily modified to cover L p stability for arbitrary p ∈ [1, ∞]. Several examples illustrate that introducing resets in a linear system may reduce the L 2 gain if the reset controller parameters are carefully tuned.

Periodic event-triggered control based on state feedback

In this paper, a novel event-triggered control (ETC) strategy is proposed by striking a balance between periodic sampled-data control and ETC. This leads to so-called periodic event-triggered control (PETC), in which the advantage of reduced resource utilisation is preserved on the one hand, while, on the other hand, the conditions that trigger the events still have a periodic character. The latter aspect has the advantage that the event-triggering condition has to be verified only at the periodic sampling times, instead of continuously, as in conventional ETC. To analyse the stability and the L2-gain properties of the resulting PETC systems, two different approaches will be presented based on (i) piecewise linear systems, and (ii) impulsive systems, respectively. Moreover, the advantages and disadvantages of each of the methods will be highlighted. The developed theory will be illustrated using a numerical example.