Input-to-state stability and sliding mode control of the nonlinear singularly perturbed systems via trajectory-based small-gain theorem

This paper presents the trajectory-based input-to-state stability (ISS) and input-to-output stability (IOS) small-gain theorem, and the finite-time ISS (FTISS) and finite-time IOS (FTIOS) of nonlinear singularly perturbed systems. The contribution of this paper is threefold. Firstly, a novel idea is proposed to analyze the stability of the nonlinear singularly perturbed system, which is regarded as an interconnected system by using two-time-scale decomposition. Secondly, the trajectory-based approach is applied to establish ISS and IOS small-gain theorem for singularly perturbed systems and the FTISS and FTIOS properties are proposed. Thirdly, a novel sliding mode controller is developed for a class of nonlinear singularly perturbed systems. Finally, the effectiveness of proposed method is illustrated by using a numerical example, a DC motor simulation and a multi-agent singularly perturbed system.     

Iterative learning control for discrete-time switched singular systems

This paper deals with the problem of iterative learning control for a class of discrete-time switched singular systems with arbitrary switching rules. According to the characteristics of the systems, two types of iterative learning algorithms are proposed and the corresponding convergence conditions of the algorithms are established. Under some given assumptions, the algorithms can ensure the system state converges to the desired state trajectory on a finite time interval. Finally, two numerical examples are constructed to support the theoretical analysis.     

Terminal sliding mode control without singular for a class of uncertain system

An improved nonsingular terminal sliding mode method is proposed for a class of nonlinear systems with unmodeled dynamics. The proposed method can effectively avoid the singularity problem. The stability of the proposed procedure which could guarantee the robustness against uncertain unmodeled dynamic and external disturbances is proven by using the Lyapunov theory in finite time. An example is given to show the proposed improved terminal sliding mode control law without singular effectively. © 2016 Wiley Periodicals, Inc. Complexity 21: 566–572, 2016     

线性定常广义系统的D型迭代学习控制收敛性

给出了线性定常广义系统在D型学习律作用下的迭代控制收敛性结果,这一结果是全新的,其对时变系统也成立,而且,本文结果和方法大部分可以移植到线性定常广义离散时间系统.     

Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty

A novel type of control strategy combining the fractional calculus with terminal sliding mode control called fractional terminal sliding mode control is introduced for a class of dynamical systems subject to uncertainties. A fractional-order switching manifold is proposed and the corresponding control law is formulated based on the Lyapunov stability theory to guarantee the sliding condition. The proposed fractional-order terminal sliding mode controller ensures the finite time stability of the closed-loop system. Finally, numerical simulation results are presented and compared to illustrate the effectiveness of the proposed method.     

Nonlinear control of systems with multiple equilibria and unknown sinusoidal disturbance

In this paper, nonlinear systems having multiple equilibrium points and low order dynamics are investigated. Roll motions of ships are studied by means of modern nonlinear techniques to exemplify the behavior of such nonlinear systems in the case when they are under the influence of external sinusoidal disturbances with unknown amplitudes. The main objective is to analyze the performance of this system at different operating conditions, including those giving rise to chaos, and to design a controller with an overparameterized structure to stabilize the system at the origin. A nonlinear recursive backstepping controller is proposed and the transient performance is investigated. Lyapunov-based techniques are used to force systematic following of a reference model while introducing a nonlinear parameter estimator to guarantee adaptivity. Robustness problems as well as ways to tune the controller parameters are examined. Simulation results are submitted for the uncontrolled and controlled cases, verifying the effectiveness of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.     

Iterative learning control design of nonlinear multiple time-delay systems

Most available results on iterative learning control address trajectory tracking problem for systems without multiple time-delay. This paper is concerned with iterative learning control design for nonlinear systems with multiple delays, in which external disturbances and output measurement noises are involved. We obtain some new and interesting criteria to guarantee the convergence of the tracking error in the sense of the λ − norm. It will be shown that the convergence of the system outputs to the desired trajectory is ensured in the absence of disturbances and output measurement noises. In the presence of disturbance and measurement noises, we estimate the upper bound of the tracking error. In order to confirm the validity of the present results, numerical simulation is also presented.     

一类非线性中立型系统的渐近稳定性

基于李雅普诺夫泛函方法,本文讨论了一类非线性中立型系统渐近稳定性的充分条件,推广了中立型系统稳定性结果。最后,通过实例及其仿真数值模拟验证了上述理论结果。     

Comments on “Fuzzy fractional order sliding mode controller for nonlinear systems” [Commun Nonlinear Sci Numer Simulat 15 (2010) 963-978]

The aim of this note is to point out some comments to the article [Delavari H, Ghaderi R, Ranjbar A, Momani S. Fuzzy fractional order sliding mode controller for nonlinear systems, Commun Nonlinear Sci Numer Simulat 15 (2010) 963-978].     

Iterative identification algorithm for Wiener nonlinear systems using the Newton method

The Wiener system is an important class of output nonlinear systems. This paper presents a Newton iterative parameter estimation algorithm for Wiener nonlinear systems. The simulation results show that the proposed algorithm is effective. The proposed algorithm can be combined with other iterative methods to identify other linear or nonlinear systems with colored noises.     

Non-coercive Lyapunov functions for infinite-dimensional systems

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give some examples showing the necessity of our assumptions.     

线性滞后广义系统的二阶滑模控制方法

讨论了时滞广义系统在不同条件下的变结构控制,根据终端滑模控制的特点,提出了一种由线性滑模与终端滑模构成的二阶终端滑模及相应控制策略.研究结果表明,该方法能够有效地清除系统的高频抖振,同时保证闭环系统的渐近稳定,实现滑模运动.举例说明了设计的合理性和有效性.     

A new control method and its robustness for a class of nonlinear systems

For a class of time-varying nonlinear systems described by the equation , the precalculating control is not available if the input matrixg(x,t) is not invertible. With Lyapunov's second method, a stabilizing controller which makes the system practically stable is constructed in this paper. It is shown that the implementation of this scheme depends on some so-called posi-invertibility conditions forg(x,t). In case the system is partly stable, the method, named part-calculating control, can simplify the on-line computations. Without the assumption that the nominal system is asymptotically stable, the method is applied to the problems of control for the corresponding uncertain system that satisfies the matching condition. When the matching condition is not satisfied, the mismatching control problem is also studied with Lyapunov's second method.This work was supported by the Science Fund of the Chinese Academy of Science.     

A new method for control of nonlinear state-dependent switched systems

In this paper, a new method for the control of input-affine nonlinear switched systems is introduced. The system switching conditions are assumed to be state-dependent, rather than the simpler input-dependent case. The main contribution of this research is that the effects of switched dynamics are interpreted as a model uncertainty bounded within a polynomial of states norms, with unknown coefficients. In order to prevent extra conservativeness, coefficients are tuned adaptively, so that a minimal state-varying bound could be achieved. This is unlike the conventional sliding mode control (SMC) scheme, where the existence of a constant and usually large upper bound must be presumed. To address the challenge of coping with such a new concept of uncertainty, an extended form of the original adaptive fuzzy sliding mode control scheme is proposed. Adaptation laws are used to tune a fuzzy controller and also real-time estimation of the instantaneous bound of uncertainties. Closed-loop stability is guaranteed by proposing a group of multiple Lyapunov functions (MLF) with tunable parameters. Except for the mild condition that the largest difference between the magnitudes of the sub-manifolds of the switched system is bounded by a polynomial of states with uncertain coefficients, the proposed method has the distinct advantage that no information about the dynamic equations or switching conditions is required in the control design stage. The proposed method is applied to the two challenging case studies, depicting the outstanding effectiveness of the method.     

A generic design methodology for sliding mode control of switched systems

The aim of this paper is to present a generic methodology to design sliding mode controllers for multivariable switched systems affine in control such as dc–dc power converters. An original formulation of the so-called reachability condition, suitable for this class of systems, is established. Based on the choice of a Lyapunov-like function and parameterized by a single weighting matrix, it allows several kinds of control strategies to be derived, namely conventional piecewise continuous strategies as well as discrete (Boolean) strategies. Its application to the important subclass of linear time invariant systems is investigated more specifically. In the Boolean case, the present approach is also compared to another hybrid one called the stabilizing approach. Eventually, its efficiency as a design methodology, as well as the performance of the resulting control, are shown by simulating it on non-trivial examples of power converters.     

Pattern-forming systems for control of large arrays of actuators

Summary. We consider an approach for coordinating the activity of a large array of microactuators via diffusive (i.e., nearest-neighbor) coupling combined with reactive growth and decay, implemented via interconnection templates which have been artificially engineered into the system (for example, in collocated microelectronic circuitry, or through the formulation of active material layers). Such coupled systems can support interesting spatiotemporal patterns, which in turn determine the actuation patterns. Generating such spatiotemporal patterns typically involves stressing the interconnections by raising or lowering a parameter resulting in the crossing of stability thresholds. The possibility of making such parametric adjustments via feedback on a slower timescale offers a solution to the problem of communicating effectively within a large array: The communication is achieved through the interconnection template. The mathematics behind this idea leads us into the rich domain of nonlinear partial differential equations (PDEs) with spatiotemporal pattern solutions. We present a global nonlinear stability analysis that applies to certain model pattern-forming systems. The nonlinear stability analysis could serve as a starting point for control system design for systems containing large microactuator arrays. Received August 1, 2000; accepted August 16, 2001 Online publication November 5, 2001     

A new proof of a theorem on optimal control of switched systems

Recently Sorin C. Bengea and Raymond A. DeCarlo [Sorin C. Bengea, Raymond A. DeCarlo, Optimal control of switching systems, Automatica J. IFAC 41 (2005) 11-27] have offered a key result that the set of trajectories of the two-switched system is dense in the set of trajectories of the embedded system. This result was proven by means of relaxed controls and the Chattering Lemma. In this paper we use the Lyapunov theorem to give a new simple proof.     

非线性系统的高阶迭代学习控制算法

本文对一般非线性系统,提出了三种高阶迭代学习控制算法:(Ⅰ)u_(h 1)=sum from j=1 to r(P_ju_(k-j 1) Г_je_(h-j 1));(Ⅱ)u_(k 1)=sum from j=1 to r(P_ju_(k-j 1) F_je_(k-j 1));(Ⅲ)u_(k 1)=sum from j=1 to r{P_ju_(k-j 1) (Г_j F_jd/(dt))e_(k-j 1)},其中u_(k 1)=u_(k 1)(t)表示系统第k 1次运行时的输入;e_k=y_k-y_d;y_d是系统所期望的输出;y_k是系统第k次运行时的输出;P_j,Г_j,F_j(j=1,…,r)是常数阵;进而给出了比较弱的收敛性条件。     

Stabilization of nonlinear switched systems using control Lyapunov functions

In this paper, we study the stabilization of general nonlinear switched systems by using control Lyapunov functions. The concept of control Lyapunov function for nonlinear control systems is generalized to switched control systems. The first part of our contribution provides a necessary and sufficient condition of stabilization. The main idea is to use a common control Lyapunov function; this is achieved with the converse Lyapunov theorem dedicated to switched systems. In the second part, an explicit construction of a common control Lyapunov function is addressed with respect to a finite family of switched systems. The approach uses a family of control Lyapunov functions attached to the subsystems.