Neural Adaptive Control Technology

The following sections are included:

• PREFACE

• CONTENTS

A unified framework for discrete-time control of continuous-time nonlinear systems based on a neural plant model is presented. The exploitation of a-priori structural information for setting up neural model structures is combined with a special type of basis function for modelling and a two-degrees of freedom nonlinear internal model control strategy.

The continuous-time version of the Local Model Network (LMN) approach to modelling non-linear systems of Johansen and Foss is developed and shown to provide a useful alternative to the ubiquitous NARMAX based neural network approach. It is shown how local-state LMN representations of non-linear systems can be used to design LMN-based controllers via state observer/state feedback methods. An input-output emulator-based interpretation of these LMNs is also given; it is suggested that this approach provides an alternative foundation for non-linear self-tuning control.

This Chapter describes a novel approach to modelling of non-linear control systems (given by NARMA models) with feedforward neural networks. The method is based on Fourier analysis and nonuniform multi-dimensional (N-D) sampling. In order to obtain an approximation of system’s dynamics with a neural network the right-hand side (RHS) of the NARMA model is reconstructed from irregularly spaced N-D samples. The algorithm uses an approximation to the Fourier transform and the distortions involved are discussed in detail. Real-world constraints on the system’s inputs and outputs make the function defining the RHS have a bounded support and thus its Fourier transform (which is a Paley-Wiener function) can be approximately reconstructed and inverted. Then an RBF neural network may be applied as an implementation of a multidimensional interpolating filter. The neural model obtained this way is smooth and suitable for the purpose of non-linear geometric control. A tutorial survey of the theory of Paley-Wiener functions, with emphasis on the neural modelling aspects, completes the presentation.

This Chapter is a survey of some geometric methods used in nonlinear control theory. After recalling main tools from Differential Geometry the following basic control theoretic concepts are discussed in the nonlinear context: controllability, observability, equivalence of systems, linearization, and decoupling. The presented results are illustrated by some engineering examples.

New models of Fornasini-Marchesini type and Roesser type of 2-D bilinear systems are introduced and relationships between them are established. It is shown that the models of 2-D bilinear systems are equivalent to models of 2-D linear systems with variable coefficients. The general response formula for the new models are derived. Necessary and sufficient conditions are established for the local reachability, local controllability for given boundary conditions and observability of a class of 2-D bilinear systems. It is shown that the local reachability for given boundary conditions of a 2-D bilinear system implies always its local controllability for given boundary conditions.

The properties of an adaptive control system based on a general class of non-linear models are analyzed. The class of models contains NARX models represented by feedforward neural networks with sigmoidal non-linearity, some radial basis-function expansions, local model networks, and some fuzzy systems. We apply a simple adaptive feedback linearizing controller. The analysis takes into account modeling error and slowly time-varying nominal model parameters. We discuss stability of the closed loop, as well as robustness, and derive performance bounds on the tracking error. Due to the general setup, the results are mainly of a qualitative nature. Finally, we discuss the relevance of the results, and outline some practical modifications and possible extensions.

In this manuscript, we introduce two different weight adaptation laws, based on Lyapunov stability theory, for the adaptive control of a certain class of MIMO continuous-time processes. The adaptation strategy is indirect in that it uses a sensitivity model of the plant (in our case, the sensitivity derivatives are estimated with a neural network that identifies the plant) in order to implement the adaptation law. The resulting adjustment rules are surprisingly simple in comparison with the gradient-based algorithm (also similar to the dynamic back-propagation, Williams & Zipser 1989), introduced by Narendra & Parthasarathy (1990, 1992). However, the stability statements, based on a first-order expansion, are only valid if the initial weight values are not too far from their optimal values that allow perfect model matching (local stability). We therefore propose to initialize the weights with values that solve the linear problem. After the treatment of the ideal case, we derive a robust adjustment law that takes noise and modelling errors into account. Indeed, due to noise, disturbances and modelling errors, the perfect model matching condition cannot be verified, and it is not possible to ensure that the error converges to zero, but only that it remains bounded. Moreover, it has been shown that in certain cases, the simple adaptive scheme designed for the ideal case can be driven unstable in the presence of bounded disturbances and unmodelled dynamics. Finally, in practical situations, the sensitivity model needed for the adaptation strategy is always imperfect. We therefore use a modified adjustment law known as the “e1 -modification” scheme, originally introduced by Narendra & Annaswamy (1987) for the adaptive control of linear systems, that does not require information about the bounds of the perturbation effects. This adaptive control scheme is proved to be robust in that all the signals in the adaptive loop are bounded for bounded disturbances. Finally, we show the effectiveness of the robust adaptation law by applying it to the control of a simulated two inputs - two outputs flowmixer.

Neural network black-box modeling is usually performed using nonlinear inputoutput models. The goal of this paper is to show that there are advantages in using nonlinear state-space models, which constitute a larger class of nonlinear dynamical models, and their corresponding state-space neural predictors. We recall the fundamentals of both input-output and state-space black-box modeling, and show the state-space neural networks to be potentially more efficient and more parsimonious than their conventional input-output counterparts. This is examplified on simulated processes as well as on a real one, the hydraulic actuator of a robot arm.

We have developed an approach to modeling nonlinear dynamical systems (with inputs) by an identifier consisting of tapped delay lines used as the input layer to a feedforward neural network with a bottleneck architecture. After training, the identifier is used in externally recurrent mode to approximate the dynamic behavior of the given system. The adequacy of approximation is evaluated by commanding both the given system and the identifier with the same control signal and comparing their responses. We demonstrate the effectiveness of this approach on three challenging dynamic systems: a Saturn engine model (with model retrained and verified on an actual Saturn engine plant) a van der Pol oscillator, and a lead-acid storage battery. Further, we have developed an approach to neural adaptive control of the engine plant based upon dynamic programming approaches of optimal control theory. The goal of the auto engine controller is to optimize multiple engine parameters consisting of drivability, emissions performance, and fuel economy.

CMAC is one of the first neural networks successfully applied to real world control problems. Its ability to locally “generalise” an input/output behaviour based on a non-linear input point processing and a linear algorithm for modifying internal states guarantees fast convergence to an implicit model. Thus it is a good candidate for learning control. In this Chapter some criteria for learning elements in control are first presented, and various approaches are judged against these criteria. Then the features of CMAC (in a sense a winner of this comparison) are described together with some improvements, one of them probably being a new result. Finally, guidelines for a CMAC hardware realisation are discussed, as they were used for the implementation of an ASIC, which now is available in a first version.

The ASMOD (Adaptive Spline Modelling of Observation Data) algorithm automatically identifies non-linear multivariate models from empirical data, using B-splines as the internal representation for the identified dependencies. The model structure is automatically adapted to the problem by an iterative search. The capability of ASMOD to find model structures which are both parsimonious and relevant for the problem have earlier been demonstrated for many real and simulated problems.

We here show that models of this type are equivalent to a certain type of fuzzy logic models. The B-spline basis functions in an ASMOD model correspond to the fuzzy membership functions of the input set in the equivalent fuzzy system, and each basis function corresponds to one rule in the fuzzy model. An automatically identified ASMOD model can thus be directly mapped into a set of fuzzy rules for manual interpretation and modification, or a fuzzy system set up from expert knowledge can be mapped into an ASMOD model structure for further refinement and adaptation to training data.

We discuss how this relation can be used to combine the strengths of both paradigms, and the potentials of this approach is demonstrated on a simulated problem where the task is to identify a state space model for a highly non-linear batch fermentation process.

The following sections are included:

• INDEX