Stable receding horizon control based on recurrent networks zyxwv C.Kambhampati A. Delgado J.D. Mason K. Warwick zyxwvutsrqp Indexing terms: Neural networks, Recurvent, Receding hosizon, Feedback, Control, Stability, Optimality zyxwvu Abstract: zyxwvutsrqp The last decade has seen the re- emergence of artificial neural networks as an alternative to traditional modelling techniques for the control of nonlinear systems. Numerous control schemes have been proposed and have been shown to work in simulations. However, very few analyses have been made of the working of these networks. The authors show that a receding horizon control strategy based on a class of recurrent networks can stabilise nonlinear systems. 1 Introduction With the emergence of artificial neural networks (ANNs) process control engineers have a tool which is capable of reflecting most, if not all, of the complexities of nonlinear systems. The use of ANNs for identifying the input-output mappingsirelationships of nonlinear chemical processes is, as a result, well documented [l-41. The key feature of these networks which has been exploited successfully is their ability to mimic or model arbitrary nonlinear mappings/functions [5-71. As a result these networks have become ideal candidates around which to develop control strategies. The main class of neural networks investigated has been feedforward networks. These networks are basi- cally static representations of a specific family of parameterised maps. Consequently, the utility of this type of network for the identification and control of nonlinear systems has depended upon regarding the feedforward network as a nonlinear input-output dis- crete-time model. A popular alternative to the feedforward network is the recurrent neural network which was introduced by Hopfield [8, 91. Recurrent neural networks are feedfor- ward ANNs with feedback connections. The introduc- tion of feedback enables the description of temporal behaviour and hence the capacity to directly account for the dynamics of nonlinear systems. This temporal behaviour then depends upon both the network inputs and the current state of the neurons in the hidden layer zyxwvu 0 IEE, 1997 zyxwvutsrqponml ZEE ProceedtngJ online 110. 19970950 Paper first received 26th Julie 1996 The authors are with the Department of Cybernetics, The University of Reading, Whiteknights, Reading, RG6 2AB, UK of the networks. Narendra and Parthasarathy [4] have proposed a type of recurrent network configuration for identifying and controlling nonlinear systems. A feed- forward ANN is combined with dynamical elements in the form of stable filters to construct the recurrent net- work configuration. The feasibility of applying this type of architecture to various control problems was demonstrated via simulations. Polycarpou and Ioannou [lo] have shown that this recurrent network configura- tion is capable of approximating a large class of dynamical systems. More recently Funahashi and Nakamura [6] have demonstrated that any finite time trajectory of a given autonomous dynamical system can be approximately realised by the internal state of the output units of a continuous time recurrent network. Although existence proofs of neural networks are available, results regarding the stabilising and regulatory properties of control strategies are rare. Of course there have been reports of the availability of networks for inverse systems [3]. These in the main dealt with the ability of networks to represent inverse systems, and not with the properties of the control strategies themselves. In the literature reports are available regarding the stabilising properties of receding horizon control of nonlinear systems [Ill. In this paper, these basic results are utilised to establish the properties of receding horizon control based on recurrent networks. The manner in which this is established is to show that recurrent networks satisfy each and every assumption made by them. Detailed proofs regarding the continuous differentiability of the Lyapunov functions and the optimality of the receding horizon strategy are omitted, as these follow from the fact that the assumptions and conditions laid down by Mayne and Michalska [l 11 are satisfied. 2 Receding horizon control problem In receding horizon control, the current control at state zy x, and time I, is obtained by determining online the open-loop optimal control zyxw ii over the interval [t, t + T] and setting the current control equal to zi(t). Repeated calculation yields a feedback control. The control problem P(x, t) is posed as minimising a cost function, over the interval [t, t + zyx T), subject to the terminal constraint x(t + T, = 0. The problem is stated as follows; min V(x, t; U) Master problem s.t. x = f(z, U); x(T) = 0 249 IEE Proc.-Control Theory Appl., Vol. 144, No. 3, Muy 1997