Artificial Neural Networks for Modelling and Control of Non-Linear Systems

In this Introduction we give first a short explanation about neural information processing systems in Section 1.1, including basic architectures, learning modes and some brief history. In Section 1.2 we motivate the use of artificial neural networks for modelling and control. In Section 1.3 we sketch the broad picture of this book, together with a Chapter by Chapter overview. In Section 1.4 own contributions are listed.

In this Chapter we discuss two basic types of artificial neural network architectures that are used in the sequel for modelling and control purposes: the multilayer perceptron and the radial basis function network. This Chapter is organized as follows. In Section 2.1 we give a description of the architectures. In Section 2.2 we present an overview of universal approximation theorems, together with a brief historical context. In Section 2.3 classical learning paradigms for feedforward and recurrent neural networks and RBF networks are reviewed.

In this Chapter we treat the problem of nonlinear system identification using neural networks. Model structures and their parametrization by multilayer perceptrons are discussed, together with learning algorithms, practical aspects and examples. The Chapter is organized as follows. In Section 3.1 we review model structures such as NARX, NARMAX and nonlinear state space models. In Section 3.2 parametrizations of these models by multilayer neural nets are made. Neural state space models are introduced. In Section 3.3 classical as well as advanced on- and off-line learning algorithms are presented and their relation with nonlinear optimization theory is explained in Section 3.4. Section 3.5 concerns practical aspects of model validation, model complexity and aspects of pruning and regularization. In Section 3.6 neural network models are interpreted as uncertain linear systems. Finally in Section 3.7 simulated and real life examples are presented on nonlinear system identification using feedforward as well as recurrent type of neural networks. New contributions are made in Sections 3.2.2, 3.2.3, 3.3.2, 3.6 and 3.7.

In this Chapter we provide the reader with some background material on neural control strategies and we discuss neural optimal control in more detail. The Chapter is organized as follows. In Section 4.1 the basic principles of existing methods in neural control are presented, including direct and indirect adaptive control, reinforcement learning, neural optimal control, internal model control and model predictive control. In Section 4.2 neural optimal control is discussed with respect to classical theory of nonlinear optimal control. The emphasis in this Section is on the formulation of control problems as parametric optimization problems, where static and dynamic nonlinear controllers are parametrized by multilayer perceptrons. Furthermore an efficient way for including a priori results from linear control theory into the neural controller is highlighted. The latter is illustrated on the examples of swinging up an inverted and double inverted pendulum system. New contributions are stated in Section 4.2.6.

In this Chapter we develop a model based neural control framework which consists of neural state space models and neural state space controllers. Like in modern (robust) control theory standard plant forms are considered. In order to analyse and synthesize neural controllers within this framework, the so-called NL _q system form is introduced. NL _q s represent a large class of nonlinear dynamical systems in state space form and contain a number of q layers of an alternating sequence of linear and static nonlinear operators that satisfy a sector condition. All system descriptions are transformed into this NL _q form and sufficient conditions for global asymptotic stability, input/output stability with finite L₂-gain and robust performance are derived. It turns out that NL _q s have a unifying nature, in the sense that many problems arising in neural networks, systems and control can be considered as special cases. Moreover, certain results in H_∞ and μ control theory can be interpreted as special cases of NL _q theory. Examples show that following principles from NL _q theory, stabilization and control of several types of nonlinear systems are possible, including mastering chaos.

In this book we discussed the use of artificial neural networks for modelling and control of nonlinear systems in a systemtheoretical context. After a short introduction on neural information processing systems in Chapter 1, we have reviewed basic neural network architectures and their learning rules in Chapter 2, for feedforward as well as recurrent networks. In Chapter 3 we have treated the problem of nonlinear system identification using neural networks. Existing models such as NARX and NARMAX were discussed and neural state space models are introduced. Off- and on-line learning algorithms are presented. An interpretation of neural network models as uncertain linear systems, representable as linear fractional transformations, has been given. Examples on nonlinear system identification of a simulated nonlinear system with hysteresis, a glass furnace with real data and chaotic systems, show the effectiveness of neural state space models. In Chapter 4 a short overview of neural control strategies is given. Neural optimal control has been discussed in more detail. The stabilization problem and tracking problem are discussed. Furthermore it is shown how results from linear control theory can be used as constraint on the neural control design, in order to achieve local stability at a target point. The latter method has been successfully applied to the problems of swinging up an inverted and double inverted pendulum. In Chapter 5 we introduced a neural state space model and control framework with stability criteria. Closed loop systems were transformed into NL_q system forms. Sufficient conditions for global asymptotic stability and I/O stability with finite L₂-gain are derived. Links with H_∞ control and μ theory are revealed. The criteria are formulated as linear matrix inequalities. NL _q theory has been applied to the control of several types of nonlinear behaviour, including chaos and to the real life example of controlling nonlinear distortion in electrodynamic loudspeakers. Furthermore, several types of recurrent neural networks are represented as NL _q s, such as generalized cellular neural networks, multilayer Hopfield networks and locally recurrent globally feedforward networks.