Stabilization Of General Nonlinear Systems Research Articles

Event-triggered distributed delayed impulsive control for nonlinear systems with applications to complex networks

This paper studies the Lyapunov stability of general nonlinear systems and the synchronization of complex networks via event-triggered impulsive control (ETIC), where the distributed delayed impulses are fully considered. Some sufficient conditions are given to deal with the problem of Lyapunov stability for nonlinear systems and avoid Zeno behavior based on distributed delayed ETIC. Moreover, some relationships between the distributed delays in impulses, the triggering parameters, and the impulsive parameters are constructed. Then, as an application, we apply the above theoretical results to complex networks and design a kind of distributed delayed ETIC strategy to ensure synchronization of complex networks under distributed delayed ETIC. In the end, the validity of our proposed results is proved by two numerical examples.

Stabilization of nonlinear time-delay systems: Distributed-delay dependent impulsive control

This technical note studies impulsive stabilization of general nonlinear systems with time-delay. Distributed time-delay is considered in the proposed nonlinear impulsive controller. Using Lyapunov–Razumikhin method, an exponential stability criterion is constructed, which is then applied to investigate stabilization of a linear time-delay system under linear distributed-delay dependent impulsive control. Sufficient conditions on the system parameters, impulsive control gains, impulsive instants and distributed delays are obtained in the form of an inequality for global exponential stability. In these results, it is shown that an unstable time-delay system can be successfully stabilized by distributed-delay dependent impulses. It is worth noting that the proposed impulsive controllers are independent of the system states at each impulsive instant, and the states with distributed delays play the key role in the stabilization process. A numerical example is provided to demonstrate the efficiency of the main results.

Compensation of actuator dynamics governed by quasilinear hyperbolic PDEs

We present a methodology for stabilization of general nonlinear systems with actuator dynamics governed by a certain class of, quasilinear, first-order hyperbolic PDEs. Since for such PDE–ODE cascades the speed of propagation depends on the PDE state itself (which implies that the prediction horizon cannot be a priori known analytically), the key design challenge is the determination of the predictor state. We resolve this challenge and introduce a PDE predictor-feedback control law that compensates the transport actuator dynamics. Due to the potential formation of shock waves in the solutions of quasilinear, first-order hyperbolic PDEs (which is related to the fundamental restriction for systems with time-varying delays that the delay rate is bounded by unity), we limit ourselves to a certain feasibility region around the origin and we show that the PDE predictor-feedback law achieves asymptotic stability of the closed-loop system, providing an estimate of its region of attraction. Our analysis combines Lyapunov-like arguments and ISS estimates. Since it may be intriguing as to what is the exact relation of the cascade to a system with input delay, we highlight the fact that the considered PDE–ODE cascade gives rise to a system with input delay, with a delay that depends on past input values (defined implicitly via a nonlinear equation). The developed control design methodology is applied to the control of vehicular traffic flow at distant bottlenecks.

Finite-Time Stabilization of a Class of T–S Fuzzy Systems

This paper considers the finite-time stabilization problem for a class of nonlinear systems that can be described by Takagi–Sugeno (T–S) fuzzy models. We propose a novel finite-time switching fuzzy control scheme for T–S fuzzy models, and the scheme is based on the Lyapunov stability theory and the control Lyapunov function technique. It is shown that the finite-time fuzzy controller and the quadratic control Lyapunov function can be obtained at the same time by solving a set of linear matrix inequalities, which can be easily facilitated by available software packages. It is also shown that the potential control law singularity can be avoided with the proposed control scheme. Unlike many existing approaches to finite-time stabilization of general nonlinear systems, the proposed approach does not require the restrictive assumption on the existence of a control Lyapunov function before the corresponding control law is constructed. Furthermore, a finite upper bound on the settling time is estimated, which indicates that within the settling time, the system trajectory would arrive and stay at the origin thereafter. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Stabilization of Nonlinear Systems via Potential-Based Realization

This technical note considers the problem of representing a sufficiently smooth control affine system as a structured potential-driven system and to exploit the obtained representation for stability analysis and state feedback controller design. These problems have been studied in recent years for particular classes of potential-driven systems. To recover the advantages of those representations for the stabilization of general nonlinear systems, the present note proposes a geometric decomposition technique, based on the Hodge decomposition theorem, to re-express a given vector field into a potential-driven form. Using the proposed decomposition technique, stability conditions are developed based on the convexity of a computed potential. Finally, stabilization is studied in the context of the proposed decomposition by reshaping the Hessian matrix of the obtained potential using damping feedback.

Global asymptotic stabilization of periodic nonlinear systems with stable free dynamics

We consider a nonlinear control system with periodic coefficients. We study the problem of asymptotic stabilization of the equilibrium x=0 of the closed-loop system by state feedback. We assume that the free dynamic system possesses a periodic Lyapunov function ensuring Lyapunov stability of the equilibrium x=0. We have developed a method for constructing a damping control for affine systems with periodic coefficients based on a generalization of weak Jurdjevic–Quinn conditions, using an extension of the notion of the commutator to non-stationary vector fields. Using this approach, we obtain sufficient conditions for uniform local and global asymptotic stabilization of general nonlinear systems, in particular, affine control systems, with periodic coefficients. Stabilization results generalize known results for time-invariant systems to time-varying periodic systems.

A Sufficient Condition for the Stability of Discrete-Time Systems With State-Dependent Coefficient Matrices

There is no systematic method of checking the stability of general nonlinear systems. In this study, we focus on discrete-time autonomous systems with state-dependent coefficient matrices. Sufficient conditions for local and global asymptotic stabilities are given for upper-triangularizable systems. The transformation matrix for triangularization is not required explicitly for checking such conditions. Moreover, the system may have nondifferentiability at the origin, which is an advantage of our approach over Lyapunov's indirect method.

Stabilization of Nonlinear Systems via Potential-based Realization

This paper considers the problem of representing a sufficiently smooth nonlinear system as a structured potential-driven system and to exploit the obtained structure for the design of nonlinear state feedback stabilizing controllers. The problem has been studied in recent years for systems modeled as structured potential-driven systems, for example gradient systems, generalized Hamiltonian systems and systems given in Brayton-Moser form. To recover the advantages of those representations for the stabilization of general nonlinear systems, the present note proposes a geometric decomposition technique to re-express a given vector field into a desired potential-driven form. The decomposition method is based on the Hodge decomposition theorem, where a one-form associated to the given vector field is decomposed into its exact, co-exact, and harmonic parts.

Technical stability of nonlinear time-varying systems with small parameters

Technical stability allowing quantitative estimation of trajectory behavior of a dynamical system over a given time interval was considered. Based on a differential comparison principle and a basic monotonicity condition, technical stability relative to certain prescribed state constraint sets of a class of nonlinear time-varying systems with small parameters was analyzed by means of vector Liapunov function method. Explicit criteria of technical stability are established in terms of coefficients of the system under consideration. Conditions under which the technical stability of the system can be derived from its reduced linear time-varying (LTV) system were further examined, as well as a condition for linearization approach to technical stability of general nonlinear systems. Also, a simple algebraic condition of exponential asymptotic stability of LTV systems is presented. Two illustrative examples are given to demonstrate the availability of the presently proposed method.

Stabilization of Nonlinear Systems Using Vector Lyapunov Functions

This paper deals with the stabilization problem of nonlinear systems using vector Lyapunov methods. The stabilization of general nonlinear systems is posed in terms of the existence of some vector Lyapunov function which together with comparison principles allows us to compute the stabilizing controller. The design procedure consists of giving a comparison system satisfying the desired stability requirements and proposing some vector Lyapunov function then, the stabilizing control law is computed from certain differential inequality. For nonlinear systems affine in the control input the computation of a closedform expression for the control law is indeed guaranteed by the existence of some control Lyapunov function.

Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback

This paper develops a systematic method for global asymptotic stabilization (GAS) of general nonlinear control systems with stable free dynamics. The approach is based on the development of non-affine passive systems theory, the idea of feedback equivalence and the technique of bounded state feedback. Using the approach presented here, we show that, as in the case of affine systems, a nonlinear system whose free dynamics are stable is GAS if it satisfies appropriate controllability-like rank conditions characterized by the Lie brackets of affine-part vector fields. We then show how a globally stabilizing smooth state feedback control law can be constructed explicitly. As an application, a dynamic state (respectively output) feedback compensator is presented for a class of input-output non-affine systems.

Receding horizon control and discontinuous state feedback stabilization

This paper addresses three aspects of receding horizon control in discrete-time: (1) feedback stabilization of general nonlinear systems with receding horizon control; (2) the generation of stabili...