Discrete-Time Adaptive Backstepping Nonlinear Control via High-Order Neural Networks

This paper deals with adaptive tracking for discrete-time multiple-input-multiple-output (MIMO) nonlinear systems in presence of bounded disturbances. In this paper, a high-order neural network (HONN) structure is used to approximate a control law designed by the backstepping technique, applied to a block strict feedback form (BSFF). This paper also includes the respective stability analysis, on the basis of the Lyapunov approach, for the whole controlled system, including the extended Kalman filter (EKF)-based NN learning algorithm. Applicability of the scheme is illustrated via simulation for a discrete-time nonlinear model of an electric induction motor.     

Robust adaptive NN control for a class of uncertain discrete-time nonlinear MIMO systems

A robust adaptive NN output feedback control is proposed to control a class of uncertain discrete-time nonlinear multi-input–multi-output (MIMO) systems. The high-order neural networks are utilized to approximate the unknown nonlinear functions in the systems. Compared with the previous research for discrete-time MIMO systems, robustness of the proposed adaptive algorithm is obvious improved. Using Lyapunov stability theorem, the results show all the signals in the closed-loop system are semi-globally uniformly ultimately bounded, and the tracking errors converge to a small neighborhood of zero by choosing the design parameters appropriately.     

H∞-control of discrete-time nonlinear systems

This paper presents an explicit solution to the problem of disturbance attenuation with internal stability via full information feedback, state feedback, and dynamic output feedback, respectively, for discrete-time nonlinear systems. The H_∞-control theory is first developed for affine systems and then extended to general nonlinear systems based on the concepts of dissipation inequality, differential game, and LaSalle's invariance principle in discrete time. A substantial difficulty that V(A(x)+B(x)u+E(x)w) [respectively, V(f(x,u,w))] is no longer quadratic in [_w^u] arising in the case of discrete-time nonlinear systems has been surmounted in the paper. In the case of a linear system, we show how the results reduce to the well-known ones recently proposed in the literature     

On robust stability of discrete-time adaptive nonlinear control

We consider adaptive control of discrete-time nonlinear systems with a single unknown parameter in this paper. We demonstrate that the necessary and sufficient condition for the existence of a feedback stabilizer that is robust to bounded noise is that the nonlinear growth rate of the system dynamics is less than 4. This result further confirms the conclusion of Guo [On critical stability of discrete-time adaptive nonlinear control, IEEE Trans. Automat. Control 42 (1997) 1488–1499] which addresses unbounded noise in a stochastic setting. Also in our worst-case approach, we find that much simpler adaptive stabilizers can be constructed when the nonlinear growth rate is less than 4.     

Output feedback stabilisation of high-order nonlinear feedforward time-delay systems

This paper studies the output feedback control problem for high-order nonlinear feedforward time-delay systems. Systems become more general due to both low-order and high-order in nonlinearities taking any value in certain intervals. By constructing the new Lyapunov–Krasovskii functional and reduced-order observer, based on the homogeneous domination theory, an output feedback controller is developed to guarantee high-order nonlinear feedforward time-delay systems globally asymptotically stable. A simulation example demonstrates the theoretical result.     

单参数离散时间非线性系统的可镇定性定理

本文对基本的离散时间非线性单参数随机系统建立了可镇定性定理. 该定理推进了文献[1]的结果, 进一步完 善了关于离散时间自适应控制的反馈能力刻画. 离散时间单参数系统可镇定的一个重要非线性临界常数是4, 用以刻画 关于幂函数类系统的反馈能力. 而作为本文定理的应用, 本文对一类典型的单参数离散时间非线性随机系统发现了新的 可镇定临界常数2.     

Global stability of an age-structured population model

We consider a nonlinear discrete-time population model for the dynamics of an age-structured species. This model has the form of a Lure feedback system (well-known in control theory) and is a particular case of the system studied by Townley et al. in Townley et al. (2012). The main objective is to show that, in this case, the range of nonlinearities for which the existence of globally asymptotically stable non-zero equilibrium can be guaranteed is considerably larger than that in the main result in Townley et al. (2012). We illustrate our results with several biologically meaningful examples.     

On the “Uniform” Observability of Discrete-Time Nonlinear Systems

In constructing an observer for a discrete-time nonlinear system, the system is commonly required to satisfy a certain kind of "uniform" observability condition, that is, the state should always be reconstructible from observation windows of a specific length, irrespective of the values of the state and inputs. In this technical note, it is proved that this "uniform" requirement is unnecessary in the sense that if the initial state and inputs are on a compact set, then the "uniform" observability is derived from its non-uniform counterpart.     

Several algorithms for finite-model adaptive control

Finite-model adaptive control problem is studied for a class of discrete-time nonlinear uncertain systems. This problem was motivated by recent efforts on the capability and limitation of feedback mechanism and has the characteristics of “essentially” finite internal uncertainties. To solve this type of problem, based on different ideas, we introduce several approaches, controller falsification, controller combination, and pseudo-parameter estimation, to design the feedback control law and rigorously establish the stability of closed-loop system for several typical algorithms in these approaches. Our results show that, under reasonably weak conditions, capable feedback control laws exist dealing with the finite internal uncertainties of the system. These results together with related results in companion papers provide partial answers to the finite-model adaptive control problem and may lead to deeper understanding on the capability of the whole feedback mechanism.     

Robust predictor feedback for discrete-time systems with input delays

This work studies the design problem of feedback stabilisers for discrete-time systems with input delays. A backstepping procedure is proposed for disturbance-free discrete-time systems. The feedback law designed by using backstepping coincides with the predictor-based feedback law used in continuous-time systems with input delays. However, simple examples demonstrate that the sensitivity of the closed-loop system with respect to modelling errors increases as the value of the delay increases. The paper proposes a Lyapunov redesign procedure that can minimise the effect of the uncertainty. Specific results are provided for linear single-input discrete-time systems with multiplicative uncertainty. The feedback law that guarantees robust global exponential stability is a nonlinear, homogeneous of degree 1 feedback law.     

Output feedback control of a class of discrete MIMO nonlinear systems with triangular form inputs

In this paper, adaptive neural network (NN) control is investigated for a class of discrete-time multi-input-multi-output (MIMO) nonlinear systems with triangular form inputs. Each subsystem of the MIMO system is in strict feedback form. First, through two phases of coordinate transformation, the MIMO system is transformed into input-output representation with the triangular form input structure unchanged. By using high-order neural networks (HONNs) as the emulators of the desired controls, effective output feedback adaptive control is developed using backstepping. The closed-loop system is proved to be semiglobally uniformly ultimate bounded (SGUUB) by using Lyapunov method. The output tracking errors are guaranteed to converge into a compact set whose size is adjustable, and all the other signals in the closed-loop system are proved to be bounded. Simulation results show the effectiveness of the proposed control scheme.     

一类非线性多变量系统的多模型自适应控制

针对一类不确定的非线性多变量离散时间动态系统,提出了一种基于切换的多模型自适应控制方法.该控制方法的特点在于以下两个方面:首先,引入一个高阶差分算子使得非线性系统的非线性项的限制条件不再要求全局有界;其次,提出的控制方法由线性自适应控制器、神经网络非线性自适应控制器以及切换机构组成:线性控制器用来保证闭环系统的输入输出信号有界,神经网络非线性控制器用来改善闭环系统的性能,基于性能指标的切换机构在每一时刻选择性能指标较好的控制器对系统进行控制.理论分析和仿真实验说明了提出的多模型自适应控制方法的有效性.     

关于自适应非线性镇定的某些结果

不确定非线性系统的反馈控制一直是控制科学的中心问题之一,迄今已经取得很大进展。然而,目前现有大部分工作所研究的反馈控制规律,或是连续时间形式的,或是采样反馈形式但需要采样频率充分快,或是离散时间反馈形式,但需要被控离散时间系统的非线性函数增长速度不超过线性。要消除或减弱这些约束条件,一般来讲是相当困难的。这就促使我们探究反馈机制的最大能力和根本局限。尽管近年来在这个方向有许多重要进展,但仍有许多非平凡的重要问题有待研究。例如,在反馈通道中有时滞情形,或者系统状态是高维的情形。在本文中,我们将探索两类比较特殊的离散时间不确定非线性动力系统的控制问题,给出关于全局自适应反馈镇定的某些初步结果。     

Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators

This paper deals with the stabilization of a class of nonlinear discrete-time systems under control saturations including time-varying parameter dependency. The control law studied consists of the gain-scheduled feedback of the measured output and of the nonlinearity present in the dynamics of the controlled system. Furthermore, the saturations are taken into account by modeling the nonlinear saturated system through a dead-zone nonlinearity satisfying a modified sector condition. Thus, as for precisely known systems, linear matrix inequality (LMI) stabilization conditions are proposed for such a generic system. These conditions can be cast into convex programming problems for design purposes. An illustrative example stresses the efficiency of the main result.     

Improving transient performance in tracking control for linear multivariable discrete-time systems with input saturation

In this paper, we present a composite nonlinear feedback (CNF) control technique for linear discrete-time multivariable systems with actuator saturation. The CNF control law serves to improve the transient performance of the closed-loop system by adding an additional nonlinear feedback. The linear feedback can be designed to yield a quick response at the initial stage, then the nonlinear feedback is introduced to smooth out overshoots when the system output approaches the target reference. As such, the resulting closed-loop system typically has very fast transient response and small overshoots. The goal of this work is to complete the theory for general discrete-time systems. The technique is applied to a magnetic-tape-drive servo system design and yields a huge improvement in settling time compared to that of a purely linear controller.     

Stability analysis of hybrid stochastic delay differential equations with asynchronous switching and discrete observations

This paper studies the hybrid stochastic delay differential equations (SDDEs) with asynchronous switching and discrete observations. For SDDEs based on discrete observations, there are two methods: The discrete-time approach and the input time delay method. For linear solvable equations, the discrete-time approach is feasible but for unsolvable nonlinear hybrid SDSs, the results of the discrete-time approach have not been discussed. So, it is natural to ask: Is the discrete-time approach still workable for nonlinear hybrid SDSs? This paper focuses on this problem. By using tools of stochastic analysis, constructing Lyapunov functional and using the discrete-time approach, the stability of hybrid SDSs by discrete-time feedback control is obtained. Finally, a numerical example is presented to verify the theoretical result.     

Actuator fault detection and compensation under feedback control

The problem of unknown input estimation and compensation is studied for actuated nonlinear systems with noisy measurements. The proposed solution is based on high-order sliding-mode differentiation and discrete-time optimization technique. Accuracy of the proposed hybrid estimation scheme is evaluated and stability conditions of the compensating mechanism are established. It is shown that the fault detection delay as well as the smallest detectable fault magnitude can be estimated. Efficiency of the proposed approach is demonstrated through oscillatory failure detection and compensation in aircraft surface servo loops.     

A functional equations approach to nonlinear discrete-time feedback stabilization through pole-placement

The present work proposes a new approach to the nonlinear discrete-time feedback stabilization problem with pole-placement. The problem's formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived. Using tools from functional equations theory, one can prove that the solution to the above system of nonlinear functional equations is locally analytic, and an easily programmable series solution method can be developed. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete-time state feedback control law that are both computed through the solution of the system of nonlinear functional equations, the feedback stabilization with pole-placement design objective can be attained under rather general conditions. The key idea of the proposed single-step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach. However, since the proposed method does not involve an external reference input, it cannot meet other control objectives such as trajectory tracking and model matching.     

Stabilization of nonlinear discrete-time systems using statedetection

Some recent results concerning stabilization of continuous-time systems using state detection are extended to the case of discrete-time nonlinear systems. It is shown that if a discrete-time locally detectable system can be stabilized by a state feedback law, then it can also be stabilized by a feedback law that depends on the output of a weak detector     

Identification of nonlinear discrete-time systems using raised-cosine radial basis function networks

An effective technique for identifying nonlinear discrete-time systems using raised-cosine radial basis function (RBF) networks is presented. Raised-cosine RBF networks are bounded-input bounded-output stable systems, and the network output is a continuously differentiable function of the past input and the past output. The evaluation speed of an n-dimensional raised-cosine RBF network is high because, at each discrete time, at most 2ⁿ RBF terms are nonzero and contribute to the output. As a consequence, raised-cosine RBF networks can be used to identify relatively high-order nonlinear discrete-time systems. Unlike the most commonly used RBFs, the raised-cosine RBF satisfies a constant interpolation property. This makes raised-cosine RBF highly suitable for identifying nonlinear systems that undergo saturation effects. In addition, for the important special case of a linear discrete-time system, a first-order raised-cosine RBF network is exact on the domain over which it is defined, and it is minimal in terms of the number of distinct parameters that must be stored. Several examples, including both physical systems and benchmark systems, are used to illustrate that raised-cosine RBF networks are highly effective in identifying nonlinear discrete-time systems.