Robust generation of almost‐periodic oscillations in a class of nonlinear systems

This paper deals with the problem of generating, by state feedback, stable oscillations in high order nonlinear systems. The desired oscillations are robust in the presence of disturbances, such as unmodelled dynamics and bounded noise signals, which result in bounded deviations from the nominal target orbit. The method consists of two steps. First, a globally attractive oscillation is generated in a nominal second‐order subsystem. Based on a partition of the state space and solving the Lyapunov equation on each part, a strict Lyapunov function is obtained that ensures exponential convergence, even in the presence of disturbances, to a ring‐shaped region containing the target limit cycle. Then, the oscillation stabilizing controller and the strict Lyapunov function are extended to arbitrary order systems, via backstepping. Notwithstanding backstepping is intended for cascade systems, the acquired ability to deal with unmodelled dynamics permits the analytical treatment of non‐triangular structures, as is illustrated with the Ball and Beam example. Copyright © 2006 John Wiley & Sons, Ltd.     

The oscillations killer: a mechanism to eliminate undesired limit cycles in a class of nonlinear systems

In this paper, we present a mechanism, called oscillation killer, intended to eliminate limit cycles that may occur as a consequence of unexpected external disturbances, interactions with the environment, changes in systems set‐points, physical parameters, and so on. Examples range from controlled systems with friction to electrical networks with varying loads and impedances. The proposed mechanisms is shown to be particulary appropriate for nonlinear systems displaying a locally asymptotically equilibrium point surrounded by a locally asymptotically stable limit set outside this local domain. Copyright © 2012 John Wiley & Sons, Ltd.     

Output transient trajectory shaping control for a class of nonlinear systems

This paper presents a transient trajectory shaping (TTS) control method for the SISO strict feedback nonlinear systems. The TTS control refers to explicitly constraining the system output tracking error transient trajectories within predesigned time‐varying boundaries while they are converging to equilibrium. By this method, the boundaries of system output transient trajectories can be designed a priori according to the system transient control performance requirements in both symmetric and asymmetric ways. With a class of time‐varying boundary functions, control laws can be devised by utilizing the enhanced differential unbounded function techniques. Such control laws can ascertain that the system output tracking errors travel within their respectively predesigned time‐varying boundaries while converging to origin. To handle the control input exaggeration issue in TTS, input constraint control techniques are proposed to effectively reduce the required control input magnitude for second‐order systems. A numerical example is utilized to show the effectiveness of the proposed TTS control methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Generation of stable oscillations in uncertain nonlinear systems with matched and unmatched uncertainties

ABSTRACT

In this paper, a robust limit cycle control technique is proposed for generation of stable oscillations in a class of uncertain nonlinear systems with both matched and unmatched uncertainties. For this purpose, first, the modified Lyapunov function is introduced which is appropriate for stability analysis of invariant sets (instead of equilibrium points). The structure of the proposed Lyapunov function is related to the shape of the desirable limit cycle. Next, in order to design the robust limit cycle control input, the backstepping and Lyapunov redesign methods are employed, simultaneously. The The classical Lyapunov redesign controller is discontinuous and robust with respect to matched uncertainties. To overcome unmatched uncertainties, a modified version of the Lyapunov redesign controller is suggested in each step of backstepping which results in a continuous robust control law. Furthermore, the convergence of the phase trajectories of the uncertain closed-loop system to the target limit cycle is proved using the extended Lyapunov stability theorem. Finally, computer simulations are performed to show the applicability of the given approach. In this regard, two uncertain nonlinear practical systems are considered and robust stable oscillations are generated in these systems via the proposed controller. Simulation results confirm the effectiveness of the proposed technique.     

Stable adaptive fuzzy control for MIMO nonlinear systems

In this paper, an indirect adaptive fuzzy control scheme is presented for a class of multi-input and multi-output (MIMO) nonlinear systems whose dynamics are poorly understood. Within this scheme, fuzzy systems are employed to approximate the plant’s unknown dynamics. In order to overcome the controller singularity problem, the estimated gain matrix is decomposed into the product of one diagonal matrix and two orthogonal matrices, a robustifying control term is used to compensate for the lumped errors, and all parameter adaptive laws and robustifying control term are derived based on Lyapunov stability analysis. The proposed scheme guarantees that all the signals in the resulting closed-loop system are uniformly ultimately bounded (UUB). Moreover, the tracking errors can be made small enough if the designed parameter is chosen to be sufficiently large. A simulation example is used to demonstrate the effectiveness of the proposed control scheme.     

Nonlinear output regulation for invertible nonlinear MIMO systems

In this paper, we address the problem of output regulation for a broad class of multi‐input multi‐output (MIMO) nonlinear systems. Specifically, we consider input–affine systems, which are invertible and input–output linearizable. This class includes, as a trivial special case, the class of MIMO systems which possess a well‐defined vector relative degree. It is shown that if a system in this class is strongly minimum phase, in a sense specified in the paper, the problem of output regulation can be solved via partial‐state feedback or via (dynamic) output feedback. The result substantially broadens the class of nonlinear MIMO systems for which the problem in question is known to be possible. Copyright © 2015 John Wiley & Sons, Ltd.     

Event-triggered adaptive fuzzy tracking control of nonlinear MIMO systems

This paper addresses the problem of an adaptive fuzzy event-triggered control (ETC) for uncertain multi-input and multi-output nonlinear systems. To reduce the communication burden of the network control systems, a novel state-dependent event-triggering condition is designed to decide when to update the controllers. By combining the backstepping and event-trigged techniques, the adaptive fuzzy ETC strategies are developed and the resulting closed-loop system is semi-global bounded. Finally, the analytical results are substantiated using simulation studies.     

On semi-global stabilizability of MIMO nonlinear systems by output feedback

For multi-input multi-output (MIMO) nonlinear systems, we prove that global stabilizability via smooth state feedback and uniform observability imply semi-global stabilizability by dynamic output feedback. This result incorporates and generalizes Teel-Praly's theorem established previously for single-input single-output (SISO) nonlinear systems. The generalization is made possible by employing, in contrast to the complicated proof in Teel and Praly [(1994). Global stabilizability and observability imply semi-global stabilizability by output feedback. Systems and Control Letters, 22, 313-325], a simple and intuitive argument that involves no intricate Lyapunov functions.     

Analysis and control of spatial limit sets and spatial chaos appearance in mimo cascade connected nonlinear systems

The oscillatory and chaotic dynamics of MIMO cascade connected nonlinear systems are analyzed in this paper. For that purpose the known theorems of limit sets existence in appropriate nonlinear discrete systems are used. Control of spatial limit sets and spatial chaos appearance in MIMO cascade connected nonlinear systems using modified Pyragas method is also analyzed. The results are illustrated by examples and confirmed with simulations.     

Robust adaptive neural control for a class of uncertain MIMO nonlinear systems

In this paper, a novel robust adaptive neural control scheme is proposed for a class of uncertain multi-input multi-output nonlinear systems. The proposed scheme has the following main features: (1) a kind of Hurwitz condition is introduced to handle the state-dependent control gain matrix and some assumptions in existing schemes are relaxed; (2) by introducing a novel matrix normalisation technique, it is shown that all bound restrictions imposed on the control gain matrix in existing schemes can be removed; (3) the singularity problem is avoided without any extra effort, which makes the control law quite simple. Besides, with the aid of the minimal learning parameter technique, only one parameter needs to be updated online regardless of the system input–output dimension and the number of neural network nodes. Simulation results are presented to illustrate the effectiveness of the proposed scheme.     

Adaptive dynamic surface control for MIMO nonlinear time-varying systems with prescribed tracking performance

In this paper, an adaptive dynamic surface control scheme is proposed for a class of multi-input multi-output (MIMO) nonlinear time-varying systems. By fusing a bound estimation approach, a smooth function and a time-varying matrix factorisation, the obstacle caused by unknown time-varying parameters is circumvented. The proposed scheme is free of the problem of explosion of complexity and needs only one updated parameter at each design step. Moreover, all tracking errors can converge to predefined arbitrarily small residual sets with a prescribed convergence rate and maximum overshoot. Such features result in a simple adaptive controller which can be easily implemented in applications with less computational burden and satisfactory tracking performance. Simulation results are presented to illustrate the effectiveness of the proposed scheme.     

Synchronisation of high-order MIMO nonlinear systems using distributed neuro-adaptive control

This paper addresses synchronisation problem of high-order multi-input/multi-output (MIMO) multi-agent systems. Each agent has unknown nonlinear dynamics and is subject to uncertain external disturbances. The agents must follow a reference trajectory. An adaptive distributed controller based on relative information of neighbours of each agent is designed to solve the problem for any undirected connected communication topology. A radial basis function neural network is used to represent the controller's unknown structure. Lyapunov stability analysis is employed to guarantee stability of the overall system. By the theoretical analysis, the closed-loop control system is shown to be uniformly ultimately bounded. Finally, simulations are provided to show effectiveness of the proposed control method against uncertainty and disturbances.     

Output feedback stabilization of nonlinear MIMO systems having uncertain high-frequency gain matrix

The purpose of this paper is to provide a method for (semi-global) asymptotic stabilization of a nonlinear minimum-phase MIMO system, under a mild hypothesis of the so-called “high-frequency gain” matrix. This result is based on a non-trivial extension, to the MIMO setting, of the approach based on the use of extended observers. As a byproduct, a dynamic output feedback control is obtained, that asymptotically stabilizes the equilibrium of the closed-loop system, in spite of uncertainties in the high-frequency gain matrix.     

Synthesis of MIMO controllers for extending the stability range of periodic solutions in forced nonlinear systems

This paper deals with a central issue in bifurcations and chaos control applications, i.e., the stabilization of periodic motions in sinusoidally forced nonlinear systems. Specifically, the problem of designing multi-input-multi-output (MIMO) finite-dimensional linear time-invariant controllers maximizing the amplitude of the sinusoidal input for which the corresponding periodic solutions are guaranteed to be stable, is considered. By exploiting linearization techniques and the multi-variable circle criterion, a synthesis algorithm is developed to determine the controller which maximizes a suitable lower bound of the amplitude of the input. The algorithm requires the solution of a sequence of linear matrix inequalities (LMIs) of increasing size. The Brusselator oscillator is employed as a case study to show that the synthesized controllers, though optimizing a lower bound, provide quite satisfactory control performance.     

Direct adaptive fuzzy control for a class of MIMO nonlinear systems

This article presents a direct adaptive fuzzy control scheme for a class of uncertain continuous-time multi-input multi-output nonlinear (MIMO) dynamic systems. Within this scheme, fuzzy systems are employed to approximate an unknown ideal controller that can achieve control objectives. The adjustable parameters of the used fuzzy systems are updated using a gradient descent algorithm that is designed to minimize the error between the unknown ideal controller and the fuzzy controller. The stability analysis of the closed-loop system is performed using a Lyapunov approach. In particular, it is shown that the tracking errors are bounded and converge to a neighborhood of the origin. Simulations performed on a two-link robot manipulator illustrate the approach and exhibit its performance.     

Direct adaptive fuzzy control of a class of MIMO non-affine nonlinear systems

An adaptive fuzzy control approach is proposed for a class of multiple-input–multiple-output (MIMO) nonlinear systems with completely unknown non-affine functions. The global implicit function theorem is first used to prove the existence of an unknown ideal implicit controller that can achieve the control objectives. Within this scheme, fuzzy systems are employed the approximate the unknown ideal implicit controller, and robustifying control terms are used to compensate the approximation errors and external disturbances. The adjustable parameters of the used fuzzy systems are deduced from the stability analysis of the closed-loop system in the sense of Lyapunov. To show the efficiency of the proposed controllers, two simulation examples are presented.     

基于神经网络MIMO非线性系统自适应输出反馈控制

针对一类具有对象不确定和外部干扰的MIMO(多输入多输出)非线性系统提出了自适应鲁棒输出跟踪控制方案.使用了高斯径向基神经网络自适应补偿对象非线性,高增益观测器被用来估计不能直接测量的输出导数.此方法所设计的控制器不仅保证闭环系统稳定,而且所有状态有界以及跟踪误差一致终值有界.仿真结果充分表明了该方案的有效性和可行性.     

Observer-based fuzzy adaptive fault control for a class of MIMO nonlinear systems

In this paper, the fault-tolerant control (FTC) problem is investigated for a class of multi-input multiple output nonlinear systems with time-varying delays, and an active FTC method is proposed. The controlled system contains unknown nonlinear functions, unknown control gain functions and actuator faults, which integrates time-varying bias and gain faults. Then, fuzzy logic systems are used to approximate the unknown nonlinear functions and unknown control gain functions, fuzzy adaptive observers are used for fault detection and isolation. Further, based on the obtained information, an accommodation method is proposed for compensating the actuator faults. It is shown that all the variables of the closed-loop system are semi-globally uniformly bounded, the tracking error converges to an arbitrary small neighbourhood of the origin. A simulation is given to demonstrate the effectiveness of the proposed approach.     

多变量非线性系统的间接模糊输出反馈自适应控制

针对一类多输入多输出非线性不确定系统，提出一种基于观测器的模糊间接自适应控制方法，并基于李亚普诺夫函数方法，导出了输出反馈控制律以及参数的自适应律，证明了整个控制方案不但能保证闭环系统稳定，而且取得了良好的跟踪控制性能。     

Extended state observer for MIMO nonlinear systems with stochastic uncertainties

ABSTRACT

In this paper, both linear extended state observer (ESO) and nonlinear ESO with homogeneous weighted functions are proposed for a class of multi-input multi-output (MIMO) nonlinear systems composed of coupled subsystems with large stochastic uncertainties. The stochastic uncertainties in each subsystem including internal coupled unmodelled dynamics and external stochastic disturbance without known statistical characteristics are lumped together as the stochastic total disturbance (extended state) of each subsystem. The linear ESO and nonlinear ESO are designed separately for real-time estimation of not only the unmeasured state but also the stochastic total disturbance of each subsystem. The practical mean square convergence of these two classes of ESOs are developed. Some numerical simulations are presented to demonstrate the effectiveness of the ESOs with the advantages of smaller peaking values and more accurate estimation by the nonlinear ESO.