Global Practical Stability Research Articles

Robust Motion Control of an Underactuated Hovercraft

In this paper, we address the tasks of trajectory tracking (TT) and path following (PF) for a hovercraft vehicle in the presence of uncertain parameters and unknown external disturbances. Building on the backstepping technique, the proposed TT (PF) control laws are able to drive a hovercraft toward and stay within a neighborhood of a reference trajectory (path), achieving global practical stability. Moreover, the devised control strategies also guarantee that the actuations remain bounded with respect to the position error. The controller is made robust to constant external disturbances and uncertain model parameters by the introduction of dynamic estimators for the disturbance and friction coefficients. A projection operator is used to ensure the estimates remain within prescribed bounds and are sufficiently smooth to be backstepped. In order to validate the efficacy and performance of the proposed controllers, we present and analyze both simulation and experimental results.

Event-based cooperative global practical output regulation of multi-agent systems with nonlinear leader

This paper presents a distributed event-based high gain feedback design for the cooperative global practical output regulation problem of nonlinear multi-agent systems with unit relative degree subject to the general nonlinear leader system. We first convert global practical output regulation problem into global practical stabilization problem, then prove the input-to-state stability of the closed-loop system by regarding the sampling error and the steady-state of the actuator together as the external input. Based on the high gain feedback technique, the input-to-state gain function can be made sufficiently small, and hence the tracking error can converge to an arbitrary predefined small residual set around the origin. A distinguishing feature of our result is that, our controller is static and purely digital, thus, giving rise to quite a simple structure.

Robust distributed control of quasilinear reaction–diffusion equations via infinite-dimensional sliding modes

This paper investigates the problem of robust tracking control for quasilinear reaction–diffusion partial differential equations subject to external unknown perturbations. The considered class of equations is quite general, and includes classical equations such as the heat equation or the Fisher–KPP equation as special cases. Global practical stabilization of the tracking error system is established under mild conditions on the disturbance term using a regularized infinite-dimensional sliding-mode controller. Extensive simulations support and validate the theoretical results.

Global robust practical output regulation for nonlinear systems in output feedback form by output‐based event‐triggered control

SummaryIn this paper, we study the event‐triggered global robust practical output regulation problem for a class of nonlinear systems in output feedback form with any relative degree. Our approach consists of the following three steps. First, we design an internal model and an observer to form the so‐called extended augmented system. Second, we convert the original problem into the event‐triggered global robust practical stabilization problem of the extended augmented system. Third, we design an output‐based event‐triggered control law and a Zeno‐free output‐based event‐triggered mechanism to solve the stabilization problem, which, in turn, leads to the solvability of the original problem. Finally, we apply our result to the controlled hyperchaotic Lorenz systems.

Event-Based Practical Output Regulation for a Class of Multiagent Nonlinear Systems.

This paper explores a cooperative practical output regulation problem for a class of heterogeneous multiagent nonlinear systems by event-based output feedback. Specifically, we shall restrict our attention to the situation of sampled-data-based local measurements. As usual, due to agents heterogeneity, we first convert the problem into a stabilization one for the so-called augmented system, composed of the agent systems and suitably designed continuous-time internal models. Then, we show that this stabilization can be solved by measurement feedback. It finally allows us to establish a valid event-based protocol, leading to a global practical stability property and meanwhile guaranteeing Zeno-free condition.

Periodic event-triggered robust output feedback control for nonlinear uncertain systems with time-varying disturbance

This paper investigates the periodic event-triggered robust output feedback control problem for a class of nonlinear uncertain systems subject to time-varying disturbance. By means of the feedback domination approach and disturbance compensation technique, a new framework of periodic event-triggered robust control strategy is developed in the form of output feedback, which encompasses a discrete-time event-triggering transmission scheme that is only intermittently monitored at sampling instants and a discrete-time output feedback controller consisting of a set of linear difference equations. The proposed robust method may reduce the communication resource utilization as compared to the non-event triggering schemes while maintaining a desirable closed-loop system performance even in the presence of a general class of time-varying disturbance and nonlinear uncertainties. The closed-loop system under the proposed control scheme is actually modeled as a hybrid system, and it is shown that the global practical stability of the closed-loop hybrid system is guaranteed by selecting a sufficiently large scaling gain and a sufficiently small sampling period. Finally, the experimental results on a DC–DC buck power converter are presented to illustrate the effectiveness of the proposed control approaches.

Razumikhin-type Theorems on Practical Stability of Dynamic Equations on Time Scales

In this work, we investigate some Razumikhin-type criteria for the uniform global practical asymptotic stability on arbitrary time domains, for time-varying dynamic equations. Using Lyapunov-type functions on time scales, we develop appropriate inequalities ensuring that trajectories decay to the neighborhood of the trivial solution asymptotically. Some numerical examples are discussed to illustrate our results.

Stabilization of a certain class of fuzzy control systems with uncertainties

AbstractIn this paper, we investigate the global uniform practical exponential stability for a class of uncertain Takagi-Sugeno fuzzy systems. The uncertainties are supposed uniformly to be bounded by some known integrable functions to obtain an exponential convergence toward a neighborhood of the origin. Therefore, we use common quadratic Lyapunov function (CQLF) and parallel distributed compensation (PDC) controller techniques to show the global uniform practical exponential stability of the closed-loop system. Numeric simulations are given to validate the proposed approach.

Global Practical Mittag Leffler Stabilization by Output Feedback for a Class Of Nonlinear Fractional‐Order Systems

AbstractIn this paper, the problem of a global practical Mittag Leffler feedback stabilization for a class of nonlinear fractional order systems by means of observer is described. The linear matrix inequality approach is used to guarantee the practical stability of the proposed feedback fractional order system. An illustrative example is given to show the applicability of the results.

Quasi‐passivity‐based adaptive stabilization for switched nonlinearly parameterized systems

SummaryThis paper investigates the adaptive quasi‐passification‐based stabilization problem for a class of switched nonlinearly parameterized systems via average dwell time method. First, when all the subsystems have any same relative degree, the global practical stability is achieved by combining the recursive feedback quasi‐passification design technique with a switched adaptive control technique. The states and parameter estimation errors converge to the ball whose sizes can be reduced by choosing appropriate design parameters. Second, when the system states are unavailable for measurements, adaptive output feedback controllers are designed to stabilize the system using quasi‐passivity. The proposed output feedback controllers do not depend on any state observer. Finally, three examples show the effectiveness of the proposed methods.

Adaptive tracking control for a class of stochastic switched systems

ABSTRACTThe problem of adaptive tracking is considered for a class of stochastic switched systems, in this paper. As preliminaries, the criterion of global asymptotical practical stability in probability is first presented by the aid of common Lyapunov function method. Based on the Lyapunov stability criterion, adaptive backstepping controllers are designed to guarantee that the closed-loop system has a unique global solution, which is globally asymptotically practically stable in probability, and the tracking error in the fourth moment converges to an arbitrarily small neighbourhood of zero. Simulation examples are given to demonstrate the efficiency of the proposed schemes.

Model reference adaptive control for nonlinear switched systems under asynchronous switching

SummaryIn this paper, the problem of model reference adaptive control for nonlinear switched systems with parametric uncertainties is investigated. Asynchronous switching between subsystems and adaptive controllers is also considered. Firstly, a state feedback adaptive controller is designed. Then, sufficient conditions ensuring the global practical stability of the error switched system with average dwell time are proposed. The boundedness of all signals in the closed‐loop system is guaranteed by the proposed adaptive controller. Finally, a practical example is given to demonstrate the validity of the main results. Copyright © 2016 John Wiley & Sons, Ltd.

Exponential convergence of nonlinear time-varying differential equations

In this paper we present a converse Lyapunov theorem for global practical uniform exponential stability of nonlinear time-varying systems. The main result shows that the system is practically globally uniformly exponentially stable if and only if it admits a Lyapunov function which satisfies some conditions. An example is also discussed to illustrate the advantage of the proposed result.

On the practical global uniform asymptotic stability of stochastic differential equations

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations (SDEs) by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of SDEs based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.

Event-Based Control and Scheduling Codesign: Stochastic and Robust Approaches

With the advent of networked embedded control systems (NECSs) new opportunities and challenges have arisen. Among others, the challenges result mostly from variable communication delays, access constraints, and resource constraints. An event-based control and scheduling (EBCS) codesign strategy for NECSs involving a set of continuous-time LTI plants is proposed in this paper addressing all aforementioned challenges. A novel representation of the network-induced delay as an uncertain variable belonging to a finite set of different bounded intervals is further proposed. The transition from one bounded interval to another can be arbitrary or according to a stochastic process. Regarding the type of the transition and the resulting discrete-time switched polytopic system of the NECS, two versions of the EBCS problem are introduced: A robust EBCS problem under arbitrary transition and a stochastic EBCS problem under stochastic transition. Global uniform practical stability with guaranteed performance (measured by a quadratic cost function) is guaranteed for both versions after formulating them as LMI optimization problems. The effectiveness of the proposed EBCS strategy is illustrated along with a comparison between its versions for a set of mobile robots. Notably, the EBCS strategy is generally applicable to discrete-time switched polytopic systems.

Observer/Controller With Global Practical Stability for Tracking in Robots Without Velocity Measurement

AbstractIn this work, a global practical stability (GPS) observer/controller is given for a revolute‐jointed multiple degree of freedom robotic plant without velocity measurement and without knowledge of plant parameter values and with a globally bounded disturbance torque. For this considered plant, a global observer/controller that globally drives the combined estimation and position tracking error to a set on which the tracking error can be made arbitrarily small by adjusting parameters/gains of the observer/controller is proposed. For the above‐mentioned plant, the current paper is the first GPS result for which the control torque is not an exponential function of a (filtered) tracking error. This paper is also the first global result for which the convergence to the residual set is exponential in time (Global Exponential Practical Stability). The observer update is discontinuous, however, only at isolated time instants. No sliding modes are used. Also, the torque is continuous.

Adaptive tracking control for uncertain switched systems under asynchronous switching

SummaryThis paper addresses the state‐tracking model reference adaptive control problem for a class of switched systems with parametric uncertainties, where switchings between subsystems and designed adaptive controller are asynchronous. First, we establish a stability criterion for a switched reference model and convert the state‐tracking problem into the stability problem of an error switched system. Then, an adaptive law is designed, and the global practical stability of the error switched system is guaranteed under a class of switching signals characterized by a dwell‐time condition. An electrohydraulic system is given as an example to demonstrate the feasibility and effectiveness of the proposed design method. Copyright © 2014 John Wiley & Sons, Ltd.

A nonlinear inequality and application to global asymptotic stability of perturbed systems

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time‐varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov‐like functions for nonlinear time‐varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.

Robust tracking control for a class of perturbed and uncertain reaction-diffusion equations

Abstract This paper is focused on the design of robust tracking control for a class of reaction-diffusion partial differential equations. Both uncertainties in the model parameters and bounded external perturbations are considered. Global practical stabilization is addressed through a regularized sliding-mode approach; in addition, a scheduled controller is proposed in order to achieve asymptotic stability if some conditions on the reference to be tracked are met. Numerical simulations support theoretical results.

H∞ Adaptive tracking control for switched systems based on an average dwell-time method

This paper investigates the H∞ state tracking model reference adaptive control (MRAC) problem for a class of switched systems using an average dwell-time method. First, a stability criterion is established for a switched reference model. Then, an adaptive controller is designed and the state tracking control problem is converted into the stability analysis. The global practical stability of the error switched system can be guaranteed under a class of switching signals characterised by an average dwell time. Consequently, sufficient conditions for the solvability of the H∞ state tracking MRAC problem are derived. An example of highly manoeuvrable aircraft technology vehicle is given to demonstrate the feasibility and effectiveness of the proposed design method.