Adaptive finite-time neural network control for nonlinear stochastic systems with state constraints

This paper focuses on the adaptive finite-time neural network control problem for nonlinear stochastic systems with full state constraints. Adaptive controller and adaptive law are designed by backstepping design with log-type barrier Lyapunov function. Radial basis function neural networks are employed to approximate unknown system parameters. It is proved that the tracking error can achieve finite-time convergence to a small region of the origin in probability and the state constraints are confirmed in probability. Different from deterministic nonlinear systems, here the stochastic system is affected by two random terms including continuous Brownian motion and discontinuous Poisson jump process. Therefore, it will bring difficulties to the controller design and the estimations of unknown parameters. A simulation example is given to illustrate the effectiveness of the designed control method.     

Adaptive Neural Network-Based Control for a Class of Nonlinear Pure-Feedback Systems With Time-Varying Full State Constraints

In this paper, an adaptive neural network (NN) control approach is proposed for nonlinear pure-feedback systems with time-varying full state constraints. The pure-feedback systems of this paper are assumed to possess nonlinear function uncertainties. By using the mean value theorem, pure-feedback systems can be transformed into strict feedback forms. For the newly generated systems, NNs are employed to approximate unknown items. Based on the adaptive control scheme and backstepping algorithm, an intelligent controller is designed. At the same time, time-varying Barrier Lyapunov functions (BLFs) with error variables are adopted to avoid violating full state constraints in every step of the backstepping design. All closedloop signals are uniformly ultimately bounded and the output tracking error converges to the neighborhood of zero, which can be verified by using the Lyapunov stability theorem. Two simulation examples reveal the performance of the adaptive NN control approach.      

Adaptive control Lyapunov function based model predictive control for continuous nonlinear systems

A design of adaptive model predictive control (MPC) based on adaptive control Lyapunov function (aCLF) is proposed in this article for nonlinear continuous systems with part of its dynamics being unknown at the starting time. Specifically, to guarantee the convergence of the closed-loop system with online predictive model updating, a stability constraint is designed. It limits the aCLF of the system under the MPC to be less than that under an online updated auxiliary adaptive control. The auxiliary adaptive control which implements in a sampling-hold fashion can guarantee the convergence of the controlled system. The sufficient conditions that guarantee the states to be steered to a small region near the equilibrium by the proposed MPC are provided. The calculation of the proposed algorithm does not depend on the model mismatch at the starting time. And it does not require the Lyapunov function of the state of the real system always to be reduced at each time. These provide the potential to improve the performance of the closed-loop system. The effectiveness of the proposed method is illustrated through a chemical process example.     

A sum of squares approach to backstepping controller synthesis for piecewise affine and polynomial systems

This paper develops a backstepping controller synthesis methodology for piecewise polynomial (PWP) systems in strict form. The main contribution of the paper is to formulate sufficient conditions for controller design for PWP systems in strict form as a sum of squares feasibility problem under the assumption that an initial control Lyapunov function exists to start the iterative backstepping procedure. This problem can then be translated into a convex SDP problem and solved by available software packages. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares polynomial control Lyapunov function for PWP systems with discontinuous vector fields. The second case addresses the construction of a PWP control Lyapunov function for PWP systems with continuous vector fields. One major advantage of the proposed method is the fact that it can handle systems with discontinuous vector fields and sliding modes. The new synthesis method is applied to several numerical examples. One of these examples offers the first convex optimization solution to piecewise affine (PWA) control of a benchmark circuit system addressed before in the literature using non‐convex PWA control solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust adaptive tracking control for a class of switched time-delay systems with application of vehicle roll dynamic control

The problems of stability and robust tracking control for a class of switched nonlinear systems with uncertain input and state delays are investigated in this study. In the presence of unknown functions and external disturbances, an adaptive fuzzy technique based on nonlinear disturbance observer is used to counteract the effects of external disturbances and approximate unknown functions. For this purpose, it is first assumed that a common Lyapunov function exists for the switched system, and then a new common Lyapunov–Krasovskii functional is built using the terms of the hypothetical common Lyapunov function. By employing this new function, the delay-dependent input-to-state stability (ISS) under an arbitrary switching signal is provided. The robust tracking control problem for this system is investigated next. Finally, to assure ISS and robust tracking performance, an adaptive control law is developed that ensures conditions of the aforementioned hypothetical common Lyapunov function. In addition, to demonstrate the efficiency of the proposed strategy, the aforesaid method is applied to a vehicle roll dynamic as well as a mass-spring system.     

Asymptotic switched composite adaptive control with application to robotic interaction tasks

This article proposes a new switched adaptive control design for uncertain switched systems with composite (time-driven and state-dependent) switching and shows its applicability in switched impedance control. A composite switched adaptive control design, consisting of the direct switched adaptive control and the indirect switched adaptive control counterpart, is developed to improve the control performance. Specifically, a new stability condition for composite switching is proposed by making use of differential matrix equations and Sylvester matrix equations, which are a generalization of Lyapunov matrix equations. The design results in a time-varying multiple Lyapunov function that is decreasing at the switching instants. From the theoretical point of view, the relevance of this work is the construction of the adaptive laws that guarantee asymptotic tracking error and asymptotic estimation for the direct and indirect switched adaptive control loops, respectively. From the practical point of view, the relevance of this work is validated in a new switched impedance control for the robot interaction with uncertain and discontinuous environments.     

具有参数不确定性的轮式移动机器人自适应backstepping控制

针对参数不确定的轮式移动机器人的轨迹跟踪问题,设计自适应跟踪控制器.基于移动机器人的动力学模型,采用backstepping积分方法,通过逐步递推选择适当的Lyapunov函数,设计基于状态反馈的自适应控制器,并进行了相应的稳定性分析.与传统PID控制进行仿真对比,结果表明提出的自适应控制策略能较好地补偿系统参数摄动的影响,提高了移动机器人的轨迹跟踪性能和鲁棒性.     

Observer based adaptive neuro-sliding mode control for MIMO nonlinear systems

In this paper, a stable adaptive neural sliding mode controller is developed for a class of multivariable uncertain nonlinear systems. For these systems not all state variables are available for measurements. By designing a state observer, adaptive neural systems, which are used to model unknown functions, can be constructed using the state estimations. Based on Lyapunov stability theorem, the proposed adaptive neural control system can guarantee the stability of the whole closed loop system and obtain good tracking performances. Adaptive laws are proposed to adjust the free parameters of the neural models. Simulation results illustrate the design procedure and demonstrate the tracking performances of the proposed controller.     

基于分段线性化Hammerstein模型的自校正控制器设计

许多实际系统可以表示为不连续非线性块状结构模型,其不连续非线性部分常采用符号函数参数化,该处理方法适用于递推参数辨识,但自适应控制器的设计较为困难。鉴于此,针对一类含有不连续非线性环节的Hammerstein模型,采用一系列线性分段函数参数化不连续非线性环节,提出自校正控制方法。根据线性分段函数的逆函数特性,求解自适应控制律。理论分析证明了闭环系统的稳定性,仿真结果验证了所提出方法的有效性。     

Stable adaptive control for a class of nonlinear systems using amodified Lyapunov function

Investigates the adaptive control design for a class of nonlinear systems using Lyapunov's stability theory. The proposed method is developed based on a novel Lyapunov function, which removes the possible controller singularity problem in some of the existing adaptive control schemes using feedback linearization techniques. The resulting closed-loop system is proven to be globally stable, and the output tracking error converges to an adjustable neighborhood of zero     

GA-based decoupled adaptive FSMC for nonlinear systems by a singular perturbation scheme

Generally, the difficulty with multivariable system control is how to overcome the coupling effects for each degree of freedom. The computational burden and dynamic uncertainty of multivariable systems makes the model-based decoupling approach hard to implement in a real-time control system. In this study, an intelligent adaptive controller is proposed to handle these behaviors. The structure of these model-free new controllers is based on fuzzy systems for which the initial parameter vector values are found based on the genetic algorithm. One modified adaptive law is derived based on Lyapunov stability theory to control the system for tracking a user-defined reference model. The requirement of the Kalman–Yacubovich lemma is fulfilled. In addition, a non-square multivariable system can be decoupled into several isolated reduced-order square multivariable subsystems by using the singular perturbation scheme for different time-scale stability analysis. The adjustable parameters for the intelligent system can be initialized using a genetic algorithm. Novel online parameter tuning algorithms are developed based on the Lyapunov stability theory. A boundary-layer function is introduced into these updating laws to cover parameter and modeling errors and to guarantee that the state errors converge into a specified error bound. Finally, a numerical simulation is carried out to demonstrate the control methodology that can rapidly and efficiently control nonlinear multivariable systems.     

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.     

Asymptotic tracking control for constrained nonstrict‐feedback MIMO nonlinear systems via parameter compensations

This paper studies the problem of adaptive fuzzy asymptotic tracking control for multiple input multiple output nonlinear systems in nonstrict‐feedback form. Full state constraints, input quantization, and unknown control direction are simultaneously considered in the systems. By using the fuzzy logic systems, the unknown nonlinear functions are identified. A modified partition of variables is introduced to handle the difficulty caused by nonstrict‐feedback structure. In each step of the backstepping design, the symmetric barrier Lyapunov functions are designed to avoid the breach of the state constraints, and the issues of overparametrization and unknown control direction are settled via introducing two compensation functions and the property of Nussbaum function, respectively. Furthermore, an adaptive fuzzy asymptotic tracking control strategy is raised. Based on Lyapunov stability analysis, the developed control strategy can effectually ensure that all the system variables are bounded, and the tracking errors asymptotically converge to zero. Eventually, simulation results are supplied to verify the feasibility of the proposed scheme.     

Fuzzy adaptive output feedback control for a class of switched non-triangular structure nonlinear systems with time-varying delays

This paper investigates an adaptive fuzzy output feedback control design problem for switched nonlinear system in non-triangular structure form. The discussed system contains unknown nonlinear dynamics, unmeasured states and unknown time-varying delays under a batch of switching signals. Fuzzy logic systems are utilised to learn unknown nonlinear dynamics and construct a fuzzy switched nonlinear observer. By combining the property of fuzzy basis function with Lyapunov–Krasovskii functional and the command filter, a novel observer-based fuzzy adaptive backstepping schematic design algorithm is presented. Furthermore, the stability of the closed-loop control system is proved via Lyapunov stability theory and average dwell time method. The simulation results are presented to verify the validity of the proposed control scheme.     

Adaptive dynamic surface-based differential games for a class of pure-feedback nonlinear systems with output constraints

ABSTRACT

This paper investigates the zero-sum differential game problem for a class of uncertain nonlinear pure-feedback systems with output constraints and unknown external disturbances. A barrier Lyapunov function is introduced to tackle the output constraints. By constructing an affine variable at each dynamic surface control design step rather than utilising the mean-value theorem, the tracking control problem for pure-feedback systems can be transformed into an equivalent zero-sum differential game problem for affine systems. Then, the solution of associated Hamilton–Jacobi–Isaacs equation can be obtained online by using the adaptive dynamic programming technique. Finally, the whole control scheme that is composed of a feedforward dynamic surface controller and a feedback differential game control strategy guarantees the stability of the closed-loop system, and the tracking error is remained in a bounded compact set. The simulation results demonstrate the effectiveness of the proposed control scheme.     

PI tracking control for nonlinear time-varying delay Markov jump systems

This paper considers robust stochastic stability and PI tracking control problem for Markov jump systems with both input delay and an unknown nonlinear function. Based on the traditional PI control strategy, a new controller design scheme is proposed for nonlinear time-delay Markov jump systems which can realize multiple control objectives including robust stochastic stability and tracking performance. By using the Lyapunov stability theory and LMI algorithms, a sufficient condition for the solution to robust stochastic stability and tracking control problem is obtained. Then, the desired controller with PI structure is designed, which ensures the resulting closed-loop system is robust stochastically stable and the system state has favorable tracking performance. Finally, a numerical example is provided to illustrate the effectiveness of the proposed results.     

Adaptive fuzzy dynamic surface control for uncertain nonlinear systems in pure-feedback form with input and state constraints using noisy measurements

This paper is concerned with the design of an adaptive fuzzy dynamic surface control for uncertain nonlinear pure-feedback systems with input and state constraints using a set of noisy measurements. The design approach is described as follows. The nonlinear uncertainties are approximated by using the fuzzy logic systems at the first stage, secondly the adaptive fuzzy dynamic surface control is introduced to remove the problem of the explosion of complexity for the derivation of the adaptive fuzzy backstepping control, thirdly a new saturation function for state constraints is proposed to design the controllers based on the Lyapunov function, fourthly the number of the adjustable parameters is reduced by using the simplified extended single input rule modules, and finally the weighted least squares estimator to take the estimates for the un-measurable states and the adjustable parameters is in a simplified structure designed. The proposed approach provides effective system performance in the simulation experiment.     

Robust state‐feedback control of stochastic state‐multiplicative discrete‐time linear switched systems with dwell time

Linear discrete‐time switched stochastic systems are considered, where the problems of mean square stability, stochastic l₂‐gain and state‐feedback control design are treated and solved. Solutions are obtained for both nominal and polytopic‐type uncertain systems. In all these problems, the switching obeys a dwell time constraint. In our solution, to each subsystem of the switched system, a Lyapunov function is assigned that is nonincreasing at the switching instants. The latter function is allowed to vary piecewise linearly, starting at the end of the previous switch instant, and it becomes time invariant after the dwell. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain Linear Matrix Inequalities conditions. Based on the solution of the stochastic l₂‐gain problem, we derive a solution to the state‐feedback control design, where we treat a variety of special cases. Being affine in the system matrices, all the aforementioned solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a practical example taken from the field of flight control. Copyright © 2015 John Wiley & Sons, Ltd.     

Lyapunov-based adaptive state estimation for a class of nonlinear stochastic systems

This paper is concerned with an adaptive state estimation problem for a class of nonlinear stochastic systems with unknown constant parameters. These nonlinear systems have a linear-in-parameter structure, and the nonlinearity is assumed to be bounded in a Lipschitz-like manner. Using stochastic counterparts of Lyapunov stability theory, we present adaptive state and parameter estimators with ultimately exponentially bounded estimator errors in the sense of mean square for both continuous-time and discrete-time nonlinear stochastic systems. Sufficient conditions are given in terms of the solvability of LMIs. Moreover, we also introduce a suboptimal design approach to optimizing the upper bound of the mean-square error of parameter estimation. This suboptimal design procedure is also realized by LMI computations. By a martingale method, we also show that the related Lyapunov function has a non-negative Lyapunov exponent.