Robust adaptive control for a class of MIMO nonlinear systems with guaranteed error bounds

The design of stabilizing controllers for multiple-input-multiple-output (MIMO) nonlinear plants with unknown nonlinearities is a challenging problem. The high dimensionality coupled with the inability to identify the nonlinearities online or offline accurately motivates the design of stabilizing controllers based on approximations or on approximate estimates of the plant nonlinearities that are simple enough to be generated in real time. The price paid in such case, could be lack of theoretical guarantees for global stability, and nonzero tracking or regulation error at steady state. In this paper, a nonlinear robust adaptive control algorithm is designed and analyzed for a class of MIMO nonlinear systems with unknown nonlinearities. The proposed control scheme provides a general approach to bypass the stabilizability problem where the estimated plant becomes uncontrollable without any restrictive assumptions. The controller is continuous and guarantees closed-loop semi-global stability and convergence of the tracking error to a small residual set. The size of the tracking error at steady state can be specified a priori and guaranteed by choosing certain design parameters. A procedure for choosing these parameters is presented. The properties of the proposed control algorithm are demonstrated using simulations.     

Adaptive tracking control for stochastic uncertain non-linear systems satisfying short- and long-term cost criteria

The tracking control problem for a class of stochastic and uncertain non-linear systems is addressed. The proposed controller uses suitable radial basis function neural network designs for the approximation of the unknown non-linearities while it is arbitrarily regulated in order to effectively penalize the tracking error. This regulation is implemented through a risk-sensitivity parameter. A stability analysis based on Lyapunov functions obtained by the backstepping technique, proves that all the error variables are bounded in probability; simultaneously, for any given risk-sensitivity parameter the system performance is regulated with respect to both a desired small average tracking error and low long-term average cost in accordance to a risk-sensitive cost criterion. Moreover, the larger this parameter is, the mean square tracking error becomes semiglobally uniformly ultimately bounded in a smaller area while a lower level of a long-term average cost is achieved. The effectiveness of the design approach is illustrated by simulation results wherein it becomes clear how one can achieve a tradeoff between good response and control effort.     

Distributed adaptive output consensus control of second-order systems containing unknown non-linear control gains

In this paper, we address the output consensus problem of tracking a desired trajectory for a group of second-order agents on a directed graph with a fixed topology. Each agent is modelled by a second-order non-linear system with unknown non-linear dynamics and unknown non-linear control gains. Only a subset of the agents is given access to the desired trajectory information directly. A distributed adaptive consensus protocol driving all agents to track the desired trajectory is presented using the backstepping technique and approximation technique of Fourier series (FSs). The FS structure is taken not only for tracking the non-linear dynamics but also the unknown portion in the controller design procedure, which can avoid virtual controllers containing the uncertain terms. Stability analysis and parameter convergence of the proposed algorithm are conducted based on the Lyapunov theory and the algebraic graph theory. It is also demonstrated that arbitrary small tracking errors can be achieved by appropriately choosing design parameters. Though the proposed work is applicable for second-order non-linear systems containing unknown non-linear control gains, the proposed controller design can be easily extended to higher-order non-linear systems containing unknown non-linear control gains. Simulation results show the effectiveness of the proposed schemes.     

Decentralized Adaptive Control of Interconnected Non-Linear Systems Using High Gain Observer

This paper investigates a decentralized adaptive control strategy for a class of interconnected unknown non-linear systems. The idea of the strategy is based on the feedback linearizing control and perturbation estimation. A high gain observer is designed in association with each sub-system to estimate the states and a fictitious state which is defined to represent the system perturbation including the combined effect of system non-linearities, unknown system dynamics, disturbances and interactions between sub-systems. Subject to the availability of sub-system states, two local controllers — decentralized non-linear adaptive state-feedback controller (DNASFC) and decentralized non-linear adaptive output-feedback controller (DNAOFC) — are developed using a high gain perturbation observer (HGPO) or a high gain state and perturbation observer (HGSPO) respectively. The stability and error analysis of the high gain observers and the closed-loop control systems are addressed in detail. The two proposed controllers are evaluated on an interconnected non-linear system involving two inverted pendulums on carts without velocity measurements.     

Robust tracking and disturbance rejection of bounded rate signals for uncertain/non-linear systems

We consider the problem of designing controllers for non-linear/uncertain systems to achieve tracking of a reference output signal in the presence of a disturbance input signal. When the exogenous signal (combined reference and disturbance) is constant, we require that the tracking error eventually go to zero. When the exogenous signal has a bounded rate, we require that the tracking error be eventually bounded with a bound which only depends on the bound on the magnitude of the rate of the exogenous signal. We also require that the state and control input are bounded when the exogenous signal is bounded. We propose controllers which have a classical PI (proportional integral) structure using state feedback. For linear systems and specific classes of non-linear/uncertain systems we present conditions whose satisfaction guarantees the existence of controllers which achieve the desired behaviour. Satisfaction of these conditions also yields the controller gain matrices.     

Adaptive tracking of switched nonlinear systems with prescribed performance using a reference-dependent reparametrisation approach

In this paper, adaptive tracking control of switched nonlinear systems in the parametric strict-feedback form is investigated. After defining a reparametrisation lemma in the presence of a non-zero reference signal, we propose a new adaptive backstepping design of the virtual controllers that can handle the extra terms arising from the reparametrisation (and that the state-of-the-art backstepping designs cannot dominate). The proposed adaptive design guarantees, under arbitrarily fast switching, an a priori bound for the steady-state performance of the tracking error and a tunable bound for the transient error. Finally, the proposed method, by overcoming the need for subsystems with common sign of the input vector field, enlarges the class of uncertain switched nonlinear systems for which the adaptive tracking problem can be solved. A numerical example is provided to illustrate the proposed control scheme.     

A linear differential operator approach to flatness based tracking for linear and non-linear systems

This contribution presents a flatness based solution to the tracking for linear systems in differential operator representation. Since the differential operator representation is a flat system representation, tracking controllers can easily be designed using dynamic output feedback. Then, the differential operator approach for flatness based tracking of linear systems is extended to non-linear systems. The design of the resulting linear time varying dynamic output feedback controller is based on a linearization about the trajectory, which directly yields the differential operator representation. Different from the non-linear flatness based controller design the new approach uses linear methods, both in stabilizing the tracking and in computing the output feedback controller. The proposed design procedure assures exact tracking in the steady state when no disturbances are present. A simple example demonstrates the design of a dynamic output feedback controller for the tracking of a non-linear system.     

Robust adaptive control of uncertain non-linear systems using neural networks

This paper presents a robust adaptive output feedback control design method for uncertain non-affine non-linear systems, which does not rely on state estimation. The approach is applicable to systems with unknown but bounded dimensions and with known relative degree. A neural network is employed to approximate the unknown modelling error. In fact, a neural network is considered to approximate and adaptively make ineffective unknown plant non-linearities. An adaptive law for the weights in the hidden layer and the output layer of the neural network are also established so that the entire closed-loop system is stable in the sense of Lyapunov. Moreover, the robustness of the system against the approximation error of neural network is achieved with the aid of an additional adaptive robustifying control term. In addition, the tracking error is guaranteed to be uniformly and asymptotically stable, rather than uniformly ultimately bounded, by using this additional control term. The proposed control algorithm is relatively straightforward and no restrictive conditions on the design parameters for achieving the systems stability are required. The effectiveness of the proposed scheme is shown through simulations of a non-affine non-linear system with unmodelled dynamics, and is compared with a second-sliding mode controller.     

ROBUST TRACKING FOR NONLINEAR SYSTEMS VIA HYBRID CONTROL

In this paper, the problem of output tracking for single-input/single-output nonlinear systems in the presence of mismatched uncertainty is studied. In our problem, the so-called matching condition in the literature is further relaxed, and a more general condition on the uncertainty is given. To attenuate the effects of uncertainties on the tracking error, a design method which is referred to as the Stable Combined Variable Perturbation Method (SCVPM) is presented. Based on this design method, a new robust tracking controller is derived using hyb rid control strategy. This controller, taken as a root-controller, is then used to generate two other controllers. All these controllers guarantee robustness of the closed-loop system, only with different tracking accuracies. The design method as well as the robust controllers is characterized by a small robust design parameter, ϵ. The tracking error converges to an ϵ-neighbourhood of the origin, and, by letting ϵ go to zero, the accuracy of tracking can be improved to any desired degree. Finally, an example is given and the simulation results confirm the theoretical analyses, thus show the effectiveness of the new design method and controllers.     

Robust analysis of flatness based control using interval methods

In this paper a new approach to the robustness analysis of flatness based tracking controllers using interval methods is proposed. This methodology allows us to explicitely determine admissible intervals for the uncertain parameters such that specified error bounds for the state space trajectories are not violated. In contrast to earlier approaches no additional controllers have to be considered to take robustness properties into account. The presented approach also poses no a priori restrictions on the velocity of the reference trajectory. The application of the robustness analysis is demonstrated for a feedforward as well as a feedback tracking controller for the Van de Vusse type continuous stirred tank reactor (CSTR).     

Adaptive neuro-genetic control of chaos applied to the attitude control problem

Conventional adaptive control techniques have, for the most part, been based on methods for linear or weakly non-linear systems. More recently, neural network and genetic algorithm controllers have started to be applied to complex, non-linear dynamic systems. The control of chaotic dynamic systems poses a series of especially challenging problems. In this paper, an adaptive control architecture using neural networks and genetic algorithms is applied to a complex, highly nonlinear, chaotic dynamic system: the adaptive attitude control problem (for a satellite), in the presence of large, external forces (which left to themselves led the system into a chaotic motion). In contrast to the OGY method, which uses small control adjustments to stabilize a chaotic system in an otherwise unstable but natural periodic orbit of the system, the neuro-genetic controller may use large control adjustments and proves capable of effectively attaining any specified system state, with no a prioriknowledge of the dynamics, even in the presence of significant noise.This work was partly supported by SERC grant 90800355.     

Adaptive Regulation Against Unknown Narrow Band Disturbances Applied to the Flying Height Control in Data Storage Systems

This paper presents an adaptive regulation approach in linear systems against exogenous narrow band inputs such as disturbances or reference signals consisting of a linear combination of biased sinusoids with unknown amplitudes, frequencies, and phases. The design of the regulator is based on considering a Q‐parameterized set of stabilizing controllers for the linear system, where an adaptive FIR filter with fixed IIR filtering is adopted as the Q parameter. The goal of the adaptation is to search within the set of stabilizing controllers for a controller, or equivalently a Q parameter, that yields regulation in the closed loop system. The proposed adaptive regulation algorithm is applied to an active suspension beam system, which is motivated by the flying height control problem in data storage systems. The experimental result of the closed loop system shows the effectiveness of the proposed adaptive regulator in achieving the desired tracking performance under unknown exogenous disturbances.     

Integrated design of a linear/neuro-adaptive controller in the presence of norm-bounded uncertainties

Adaptive neural controllers are often criticised for the lack of clear and easy design methodologies that relate adaptive neural network (NN) design parameters to performance requirements. This study proposes a methodology for the design of an integrated linear-adaptive model reference controller that guarantees component-wise boundedness of the tracking error within an a priori specified compact domain. The approach is based on the design of a robust invariant ellipsoidal set where both the NN reconstruction error and the neuro-adaptive control are considered as bounded persistent uncertainties. We show that all the performance and control requirements for the closed-loop system can be expressed as linear matrix inequality constraints. This brings the advantage that feasibility and optimal design parameters can be effectively computed while solving a linear optimisation problem. An advantage of the method is that it allows a systematic and quantitative evaluation of the interplay between the design parameters and their impact on the requirements. This produces an integrated linear/neuro-adaptive performance-oriented design methodology. A numerical example is used to illustrate the approach.     

不确定非线性系统的多模反演滑模控制

对一般形式的仿射非匹配不确定非线性系统,研究了一种具有任意小跟踪误差的稳定控制器的新方法,结合反演（backstepping）设计和变结构控制,提出了反演变结构控制策略,对存在非匹配的不确定性和未知干扰的系统,设计的反演结构控制器实现了鲁棒输出跟踪,闭环系统在有限时间进入滑动模态,仿真算例证实了理论结果。     

Control of nonlinear systems with output tracking error constraints and its application to magnetic bearings

A constructive method is presented to design controllers that force the output of nonlinear systems in a strict feedback form to track a bounded and sufficiently smooth reference trajectory asymptotically. Under a suitable condition on the initial output tracking error, the proposed controllers guarantee the output tracking error within a symmetric or an asymmetric pre-specified limit range, and boundedness of all signals of the closed-loop system. A transformation is introduced to take care of the output tracking error constraint. Smooth and/or p-times differentiable step functions are proposed and incorporated in the output tracking error transformation to overcome difficulties due to the asymmetric limit range on the output tracking error. As a result, there are no switchings in the proposed controllers despite the asymmetric limit range. The proposed control design is then applied to design a tracking controller for active magnetic bearings as an illustrating application.     

Transient performance analysis in robust nonlinear adaptive control

This paper deals with the problem of controlling unknown linear systems in the presence of strictly proper unmodelled dynamics and bounded disturbances. Adaptive controllers that ensure the closed-loop global (uniform) stability and asymptotic performances can be designed following either the backstepping approach or the certainty-equivalence method. The main shortcoming of the involved controllers is that they do not allow quantification of the closed-loop transient behaviour. In this paper, the transient issue is addressed for backstepping adaptive controllers. A L_∞ bound on the tracking error is explicitly given as a function of the design parameters. This shows that the error can be made arbitrarily small by sufficiently increasing the design gains.     

Robust trajectory tracking for a scale model autonomous helicopter

This paper considers the question of obtaining an a priori bound on the tracking performance, for an arbitrary trajectory, of closed‐loop control of an idealized model of a scale model autonomous helicopter. The problem is difficult due to the presence of small body forces that cannot be directly incorporated into the control design. A control Lyapunov function is derived for an approximate model (in which the small body forces are neglected) using backstepping techniques. The Lyapunov function derived is used to analyse the closed‐loop performance of the full system. A theorem is proved that provides a priori bounds on initial error and the trajectory parameters (linear acceleration and its derivatives) that guarantees acceptable tracking performance of the system. The analysis is expected to be of use in verification of trajectory planning procedures. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive recurrent neural network control using a structure adaptation algorithm

This paper proposes an adaptive recurrent neural network control (ARNNC) system with structure adaptation algorithm for the uncertain nonlinear systems. The developed ARNNC system is composed of a neural controller and a robust controller. The neural controller which uses a self-structuring recurrent neural network (SRNN) is the principal controller, and the robust controller is designed to achieve L ² tracking performance with desired attenuation level. The SRNN approximator is used to online estimate an ideal tracking controller with the online structuring and parameter learning algorithms. The structure learning possesses the ability of both adding and pruning hidden neurons, and the parameter learning adjusts the interconnection weights of neural network to achieve favorable approximation performance. And, by the L ² control design technique, the worst effect of approximation error on the tracking error can be attenuated to be less or equal to a specified level. Finally, the proposed ARNNC system with structure adaptation algorithm is applied to control two nonlinear dynamic systems. Simulation results prove that the proposed ARNNC system with structure adaptation algorithm can achieve favorable tracking performance even unknown the control system dynamics function.     

Stable direct adaptive fuzzy control for a class of MIMO non-linear systems

Two new schemes of direct adaptive fuzzy controller for a class of multi-input multi-output non-linear systems with unknown constant gains or function gains are proposed in this paper. The design is based on a modified Lyapunov function and the approximation capability of the first type fuzzy system. The approach is able to avoid the requirement of the upper bound of the first-time derivative of the control gain, which is assumed to know a priori in some of the existing adaptive fuzzy/neural network control schemes. In addition, it is also able to avoid the controller singularity problem. By theoretical analysis, the closed-loop fuzzy control system is proven to be globally stable in the sense that all signals involved are bounded, with tracking errors converging to zero. The simulation results verify the effectiveness the proposed controllers and the theoretical discussion.     

A flat-zone modification for robust adaptive control of nonlinearoutput feedback systems with unknown high-frequency gains

This note deals with adaptive control of perturbed nonlinear output feedback systems with unknown high-frequency gains. The disturbances in the systems are assumed to be bounded, but the bounds are unknown. A flat-zone modification is proposed to incorporate both the bound estimation and Nussbaum gain design in the nonlinear adaptive control. To ensure the differentiability of stabilizing functions needed for backstepping design, high order terms are introduced in the Lyapunov function candidate with a flat zone around the neighborhood of the origin. The output tracking error converges to an arbitrarily small interval around zero