Composite Adaptation for Neural Network-Based Controllers

With the motivation of using more information to update the parameter estimates to achieve improved tracking performance, composite adaptation that uses both the system tracking errors and a prediction error containing parametric information to drive the update laws, has become widespread in adaptive control literature. However, despite its obvious benefits, composite adaptation has not been widely implemented in neural network-based control, primarily due to the neural network (NN) reconstruction error that destroys a typical prediction error formulation required for the composite adaptation. This technical note presents a novel approach to design a composite adaptation law for NNs by devising an innovative swapping procedure that uses the recently developed robust integral of the sign of the error (RISE) feedback method. Semi-global asymptotic tracking is proven for a Euler-Lagrange system. Experimental results are provided to illustrate the concept.

Related papers:

title: Passivity-based control of Euler-Lagrange systems : mechanical, electrical and electromechanical applications
abstract: 1 Introduction.- 2 Euler-Lagrange systems.- 3 Set-point regulation.- 4 Trajectory tracking control.- 5 Adaptive disturbance attenuation: Friction compensation.- 6 Modeling of switched DC-to-DC power converters.- 7 Passivity-based control of DC-to-DC power converters.- 8 Nested-loop passivity-based control: An illustrative example.- 9 Generalized AC motor.- 10 Voltage-fed induction motors.- 11 Current-fed induction motors.- 12 Feedback interconnected systems: Robots with AC drives.- 13 Other applications and current research.- A Dissipativity and passivity.- 1 Circuit example.- 3 Passivity and finite-gain stability.- 4 Feedback systems.- 5 Internal stability and passivity.- 6 The Kalman-Yakubovich-Popov lemma.- B Derivation of the Euler-Lagrange equations.- 1 Generalized coordinates and velocities.- 2 Hamilton's principle.- 3 From Hamilton's principle to the EL equations.- 4 EL equations for non-conservative systems.- 5 List of generalized variables.- 6 Hamiltonian formulation.- C Background material.- D Proofs.- 3 The BP transformation.- 3.1 Proof of Proposition 9.20.- 3.2 A Lemma on the BP Transformation.- 4 Proof of Eqs. (10.41) and (10.77).- 4.1 A theorem on positivity of a block matrix.- 4.2 Proof of Eq. (10.77).- 4.3 Proof of Eq. (10.41).- 5 Derivation of Eqs. (10.55) and (10.56).- 5.1 Derivation of Eq. (10.55).- 5.2 Derivation of Eq. (10.56).- 6 Boundedness of all signals for indirect FOC.- 6.1 Proof of Proposition 11.10.

title: Multilayer feedforward networks are universal approximators
abstract: This paper rigorously establishes that standard multilayer feedforward networks with as few as one hidden layer using arbitrary squashing functions are capable of approximating any Borel measurable function from one finite dimensional space to another to any desired degree of accuracy, provided sufficiently many hidden units are available. In this sense, multilayer feedforward networks are a class of universal approximators.

title: Adaptive Neural Network Control of Robotic Manipulators
abstract: There has been considerable research interest in neural network control of robots, and satisfactory results have been obtained in solving some of the special issues associated with the problems of robot control in an "on-and-off" fasion. This text is dedicated to issues on adaptive control of robots based on neural networks. The text has been tailored to give a comprehensive study of robot dynamics, present structured network models for robots, and provide systematic approaches for neural network based adaptive controller design for rigid robots, flexible joint robots, and robots in constraint motion. Rigorous proof of the stability properties of adaptive neural network controllers is provided. Simulation examples are also presented to verify the effectiveness of the controllers, and practical implementation issues associated with the controllers are also discussed.

title: Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities
abstract: From the Publisher: 
About the Author 
Frank L. Lewis is Associate Director for Research and Head of the Advanced Controls, Sensors, and MEMS Group at the Automation and Robotics Research Institute at the University of Texas at Arlington. Javier Campos is Linux Consultant Engineer at Montavista Software, Inc. in Irving, Texas. Rastko Selmic is DSP Systems Engineer at Signalogic, Inc. in Dallas, Texas.

title: Applied Nonlinear Control
abstract: Covers in a progressive fashion a number of analysis tools and design techniques directly applicable to nonlinear control problems in high performance systems (in aerospace, robotics and automotive areas).

title: Composite adaptive control of robot manipulators
abstract: Adaptive control of linear time-invariant single-input single-output systems has been extensively studied, and a number of globally convergent controllers have been derived. While extensions of the results to non-linear or multivariable systems have rarely been achieved, similar global convergence properties can indeed be obtained in the case of robot manipulators, an important and unique class of non-linear multi-input multi-output dynamic systems. Based on the observation that the parameter uncertainty is reflected in both the tracking error in joint motion and the prediction error in the joint torques or the power input, we recently proposed a new class of adaptive robot controllers, the parameter adaptation of which is driven by both tracking error and prediction error. In this paper, following a brief review of our earlier globally convergent direct adaptive controller, we provide a detailed analysis of these “composite” adaptive controllers. Results on global asymptotic and exponential tracking convergence are established and confirmed by simulation.

title: A composite adaptive output feedback tracking controller for robotic manipulators
abstract: This paper provides a solution to the composite adaptive output feedback tracking control problem for robotic manipulators. The proposed controller utilizes an update law that is a composite of a gradient update law driven by the link position tracking error and a least squares update law driven by the prediction error. In order to remove the controller's dependence on link velocity measurements, a linear filter and a new prediction error formulation are designed. The controller provides semi-global asymptotic link position tracking performance. Experimental results illustrate that the proposed controller provides improved link position tracking error transient performance and faster parameter estimate convergence in comparsion to the same controller using a gradient update law.

title: Neural control of nonlinear systems with composite adaptation for improved convergence of Gaussian networks
abstract: The use of composite adaptive laws for control of the affine class of nonlinear systems having unknown dynamics is proposed. These dynamics are approximated by Gaussian radial basis function neural networks whose parameters are updated by a composite law that is driven by both tracking and estimation errors. This is motivated by the need to improve the speed of convergence of the unknown parameters, hence resulting in better system performance. To ensure global stability despite the inevitable network approximation errors, the control law is augmented with a low gain sliding mode component and deadzone adaptation is used for the indirect part of the composite law. The stability of the system is analyzed and the effectiveness of the method is demonstrated by simulation.

title: Functional Adaptive Control: An Intelligent Systems Approach
abstract: I. Introduction.- 1. Introduction.- 1.1 Intelligent Control Systems.- 1.2 Approaches to Intelligent Control.- 1.2.1 Contribution of Adaptive Control.- 1.2.2 Contribution of Artificial Intelligence.- 1.2.3 Confluence of Adaptive Control and AI: Intelligent Control.- 1.3 Enhancing the Performance of Intelligent Control.- 1.3.1 Multiple Model Schemes: Dealing with Complexity.- 1.3.2 Stochastic Adaptive Control: Dealing with Uncertainty.- 1.4 The Objectives and their Rationale.- II. Deterministic Systems.- 2. Adaptive Control of Nonlinear Systems.- 2.1 Introduction.- 2.2 Continuous-time Systems.- 2.2.1 Control by Feedback Linearization.- 2.2.2 Control by Backstepping.- 2.2.3 Adaptive Control.- 2.3 Discrete-time Systems.- 2.3.1 Affine Approximations and Feedback Linearization.- 2.3.2 Adaptive Control.- 2.4 Summary.- 3. Dynamic Strueture Networks for Stahle Adaptive  3.3 Fixed-structure Network Solutions.- 3.4 Dynamic Network Structure.- 3.5 The Control Law and Error Dynamies.- 3.6 The Adaptive System.- 3.7 Stability Analysis.- 3.8 Evaluation of Control Parameters and Implementation.- 3.8.1 The Disturbanee Bound.- 3.8.2 Choice of the Boundary Layer.- 3.8.3 Comments.- 3.8.4 Implementation.- 3.9 Simulation Examples.- 3.9.1 Example 1.- 3.9.2 Example 2.- 3.10 Summary.- 4. Composite Adaptive Control of Continuous-Time Systems.- 4.1 Introduetion.- 4.2 Problem Formulation.- 4.3 The Neural Networks.- 4.4 The Control Law.- 4.5 Composite Adaptation.- 4.5.1 The Identifieation Model.- 4.5.2 The Adaptation Law.- 4.6 Stability Analysis.- 4.7 Determination of the Disturbanee Bounds.- 4.8 Simulation Examples.- 4.8.1 Example 1.- 4.8.2 Example 2.- 4.9 Summary.- 5. Funetional Adaptive Control of Discrete-Time Systems.- 5.1 Introduetion.- 5.2 Problem Formulation.- 5.3 The Neural Network.- 5.4 The Control Law.- 5.5 The Adaptive System.- 5.6 Stability Analysis.- 5.7 Traeking Error Convergenee.- 5.8 Simulation Examples.- 5.8.1 Example 1.- 5.8.2 Example 2.- 5.9 Extension to Adaptive Sliding Mode Control.- 5.9.1 Definitions of a Discrete-time Sliding Mode.- 5.9.2 Adaptive Sliding Mode Control.- 5.9.3 Problem Formulation.- 5.9.4 The Control Law.- 5.9.5 The Adaptive System.- 5.9.6 Stability Analysis.- 5.9.7 Sliding and Tracking Error Convergence.- 5.9.8 Simulation Example.- 5.10 Summary.- III. Stochastic Systems.- 6. Stochastic Control.- 6.1 Introduction.- 6.2 FUndamental Principles.- 6.3 Classes of Stochastic Control Problems.- 6.4 Dual Control.- 6.4.1 Degrees of Interaction.- 6.4.2 Solutions to the Implementation Problem.- 6.5 Conclusions.- 7. Dual Adaptive Control of Nonlinear Systems.- 7.1 Introduction.- 7.2 Problem Formulation.- 7.3 Dual Controller Design.- 7.3.1 GaRBF Dual Controller.- 7.3.2 Sigmoidal MLP Dual Controller.- 7.3.3 Analysis of the Control Laws.- 7.4 Simulation Examples and Performance Evaluation.- 7.4.1 Example 1.- 7.4.2 Example 2.- 7.5 Summary.- 8. Multiple Model Approaches.- 8.1 Introduction.- 8.2 Basic Formulation.- 8.2.1 Multiple Model Adaptive Contro!..- 8.2.2 Jump Systems.- 8.3 Adaptive IO Models.- 8.3.1 Scheduled Mode Transitions.- 8.4 Summary.- 9. Multiple Model Dual Adaptive Control of Jump Nonlinear Systems.- 9.1 Introduction.- 9.2 Problem Formulation.- 9.3 The Estimation Problem.- 9.3.1 Known Mode Case.- 9.3.2 Unknown Mode Case.- 9.4 Self-organized Allocation of Local Models.- 9.5 The Control Law.- 9.5.1 Known Mode Case.- 9.5.2 Unknown Mode Case.- 9.6 Simulation Examples and Performance Evaluation.- 9.6.1 Example 1.- 9.6.2 Example 2.- 9.7 Summary.- 10. Multiple Model Dual Adaptive Control of Spatial Multimodal Systems.- 10.1 Introduction.- 10.2 Problem Formulation.- 10.3 The Modular Network.- 10.4 The Estimation Problem.- 10.4.1 Local Model Parameter Estimation.- 10.4.2 Validity Function Estimation.- 10.5 The Control Law.- 10.5.1 Known System Case.- 10.5.2 Unknown System Case.- 10.6 Simulation Examples and Performance Evaluation.- 10.6.1 Example 1.- 10.6.2 Example 2.- 10.6.3 Performance Evaluation.- 10.7 Summary.- IV.

title: Bounded error adaptive control
abstract: The paper discusses the adaptive control of a linear time-invariant plant when the measurement of the plant output is corrupted by a bounded disturbance. The principal contribution is the demonstration of the boundedness of all the signals of the overall system using a nonlinear adaptive law involving a dead zone. A statement of the prior information needed to determine the adaptive algorithm as well as a discussion of the limitations of the scheme are given. Computer simulations are presented to illustrate the effect of various parameters.

title: Composite adaptive control with locally weighted statistical learning
abstract: This paper introduces a probably stable learning adaptive control framework with statistical learning. The proposed algorithm employs nonlinear function approximation with automatic growth of the learning network according to the nonlinearities and the working domain of the control system. The unknown function in the dynamical system is approximated by piecewise linear models using a nonparametric regression technique. Local models are allocated as necessary and their parameters are optimized on-line. Inspired by composite adaptive control methods, the proposed learning adaptive control algorithm uses both the tracking error and the estimation error to update the parameters. We first discuss statistical learning of nonlinear functions, and motivate our choice of the locally weighted learning framework. Second, we begin with a class of first order SISO systems for theoretical development of our learning adaptive control framework, and present a stability proof including a parameter projection method that is needed to avoid potential singularities during adaptation. Then, we generalize our adaptive controller to higher order SISO systems, and discuss further extension to MIMO problems. Finally, we evaluate our theoretical control framework in numerical simulations to illustrate the effectiveness of the proposed learning adaptive controller for rapid convergence and high accuracy of control.

title: Asymptotic Tracking for Systems With Structured and Unstructured Uncertainties
abstract: The control of systems with uncertain nonlinear dynamics has been a decades-long mainstream area of focus. The general trend for previous control strategies developed for uncertain nonlinear systems is that the more unstructured the system uncertainty, the more control effort (i.e., high gain or high-frequency feedback) is required to cope with the uncertainty, and the resulting stability and performance of the system is diminished (e.g., uniformly ultimately bounded stability). This brief illustrates how the amalgamation of an adaptive model-based feedforward term (for linearly parameterized uncertainty) with a robust integral of the sign of the error (RISE) feedback term (for additive bounded disturbances) can be used to yield an asymptotic tracking result for Euler-Lagrange systems that have mixed unstructured and structured uncertainty. Experimental results are provided that illustrate a reduced root-mean-squared tracking error with reduced control effort.

title: MODEL-BASED LEARNING CONTROLS AND THEIR COMPARISONS USING LYAPUNOV DIRECT METHOD
abstract: Model-based learning control of nonlinear systems is studied. Two types of learning algorithms, described by differential equations and/or difference equations to learn unknown time functions, are designed and compared using the Lyapunov’s direct method. The time functions to be learned are classified into several classes according to their properties such as continuity, periodicity, and value at the origin of the state space. Conditions are found for iterative learning controls to achieve asymptotic stability and asymptotic learning convergence. For a comparative study, learning capability of a control is defined and, using the criterion, other model-based controls with learning capability such as adaptive controls and robust controls are investigated. Through the study, iterative learning control is shown to be the one best suited for learning unknown time functions of known period. Finally, it is shown for the first time that an iterative learning control is directly applicable to systems described by nonlinear partial differential equation.

title: A continuous asymptotic tracking control strategy for uncertain nonlinear systems
abstract: In this note, we present a new continuous control mechanism that compensates for uncertainty in a class of high-order, multiple-input-multiple-output nonlinear systems. The control strategy is based on limited assumptions on the structure of the system nonlinearities. A new Lyapunov-based stability argument is employed to prove semiglobal asymptotic tracking.

title: Asymptotic Tracking for Uncertain Dynamic Systems Via a Multilayer Neural Network Feedforward and RISE Feedback Control Structure
abstract: The use of a neural network (NN) as a feedforward control element to compensate for nonlinear system uncertainties has been investigated for over a decade. Typical NN-based controllers yield uniformly ultimately bounded (UUB) stability results due to residual functional reconstruction inaccuracies and an inability to compensate for some system disturbances. Several researchers have proposed discontinuous feedback controllers (e.g., variable structure or sliding mode controllers) to reject the residual errors and yield asymptotic results. The research in this paper describes how a recently developed continuous robust integral of the sign of the error (RISE) feedback term can be incorporated with a NN-based feedforward term to achieve semi-global asymptotic tracking. To achieve this result, the typical stability analysis for the RISE method is modified to enable the incorporation of the NN-based feedforward terms, and a projection algorithm is developed to guarantee bounded NN weight estimates.

title: Robust adaptive asymptotic tracking of nonlinear systems with additive disturbance
abstract: This note deals with the tracking control of multiple-input-multiple-output (MIMO) nonlinear parametric strict-feedback systems in the presence of additive disturbances and parametric uncertainties. For such systems, C/sup 0/ robust adaptive controllers usually cannot ensure asymptotic tracking or even regulation. In this work, under the assumption the disturbances are C/sup 2/ with bounded time derivatives, we present a C/sup 0/ robust adaptive control construction that guarantees the tracking error is asymptotically driven to zero.

title: NONLINEAR NETWORK STRUCTURES FOR FEEDBACK CONTROL
abstract: A framework is given for controller design using Nonlinear Network Structures, which include both neural networks and fuzzy logic systems. These structures possess a universal approximation property that allows them to be used in feedback control of unknown systems without requirements for linearity in the system parameters or finding a regression matrix. Nonlinear nets can be linear or nonlinear in the tunable weight parameters. In the latter case weight tuning algorithms are not straightforward to obtain. Feedback control topologies and weight tuning algorithms are given here that guarantee closed-loop stability and bounded weights. Extensions are discussed to force control, backstepping control, and output feedback control, where dynamic nonlinear nets are required.

title: Dynamic structure neural networks for stable adaptive control of nonlinear systems
abstract: An adaptive control technique, using dynamic structure Gaussian radial basis function neural networks, that grow in time according to the location of the system's state in space is presented for the affine class of nonlinear systems having unknown or partially known dynamics. The method results in a network that is "economic" in terms of network size, for cases where the state spans only a small subset of state space, by utilizing less basis functions than would have been the case if basis functions were centered on discrete locations covering the whole, relevant region of state space. Additionally, the system is augmented with sliding control so as to ensure global stability if and when the state moves outside the region of state space spanned by the basis functions, and to ensure robustness to disturbances that arise due to the network inherent approximation errors and to the fact that for limiting the network size, a minimal number of basis functions are actually being used. Adaptation laws and sliding control gains that ensure system stability in a Lyapunov sense are presented, together with techniques for determining which basis functions are to form part of the network structure. The effectiveness of the method is demonstrated by experiment simulations.

title: Adaptive control of a class of nonlinear discrete-time systems using neural networks
abstract: Layered neural networks are used in a nonlinear self-tuning adaptive control problem. The plant is an unknown feedback-linearizable discrete-time system, represented by an input-output model. To derive the linearizing-stabilizing feedback control, a (possibly nonminimal) state-space model of the plant is obtained. This model is used to define the zero dynamics, which are assumed to be stable, i.e., the system is assumed to be minimum phase. A linearizing feedback control is derived in terms of some unknown nonlinear functions. A layered neural network is used to model the unknown system and generate the feedback control. Based on the error between the plant output and the model output, the weights of the neural network are updated. A local convergence result is given. The result says that, for any bounded initial conditions of the plant, if the neural network model contains enough number of nonlinear hidden neurons and if the initial guess of the network weights is sufficiently close to the correct weights, then the tracking error between the plant output and the reference command will converge to a bounded ball, whose size is determined by a dead-zone nonlinearity. Computer simulations verify the theoretical result.

title: Adaptive control of linearizable systems
abstract: The authors give some initial results on the adaptive control of minimum-phase nonlinear systems which are exactly input-output linearizable by state feedback. Parameter adaptation is used as a technique to make robust the exact cancellation of nonlinear terms, which is called for in the linearization technique. The application of the adaptive technique to control of robot manipulators is discussed. Only the continuous-time case is considered; extensions to the discrete-time and sampled-data cases are not obvious.

title: Stable adaptive neural control scheme for nonlinear systems
abstract: Based on the Lyapunov synthesis approach, several adaptive neural control schemes have been developed during the last few years. So far, these schemes have been applied only to simple classes of nonlinear systems. This paper develops a design methodology that expands the class of nonlinear systems that adaptive neural control schemes can be applied to and relaxes some of the restrictive assumptions that are usually made. One such assumption is the requirement of a known bound on the network reconstruction error. The overall adaptive scheme is shown to guarantee semiglobal uniform ultimate boundedness. The proposed feedback control law is a smooth function of the state.

title: On the existence and uniqueness of solutions in adaptive control systems
abstract: The issue of existence and uniqueness of solutions, which arises in the analysis of adaptive control systems, is investigated. Sufficient conditions that guarantee existence and uniqueness in positive time of solutions to differential equations describing a wide class of adaptive schemes are given. The concept of Filippov's solution is introduced as a method of dealing with discontinuous adaptive systems.

title: Gaussian Networks for Direct Adaptive Control
abstract: A direct adaptive tracking control architecture is proposed and evaluated for a class of continuous-time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible. The architecture employs a network of gausian radial basis functions to adaptively compensate for the plant nonlinearities. Under mild assumptions about the degree of smoothness exhibited by the nonlinear functions, the algorithm is proven to be stable, with tracking errors converging to a neighborhood of zero. A constructive procedure is detailed, which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy. A stable weight adjustment mechanism is then determined using Lyapunov theory. The network construction and performance of the resulting controller are illustrated through simulations with an example system.