Robust local stability of multilayer recurrent neural networks J.A.K. Suykens, B. De Moor, J. Vandewalle Katholieke Universiteit Leuven Department of Electrical Engineering, ESAT-SISTA Kardinaal Mercierlaan 94, B-3001 Leuven (Heverlee), Belgium Tel: 32/16/32 18 02 Fax: 32/16/32 19 70 E-mail:
[email protected] (Corresponding author: Johan Suykens) TNN 3342 Rev.
This research work was carried out at the ESAT laboratory and the Interdisciplinary Center of Neural Networks ICNN of the Katholieke Universiteit Leuven, in the framework of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Oce for Science, Technology and Culture (IUAP P4-02 & P4-24) and in the framework of a Concerted Action Project MIPS (Modelbased Information Processing Systems) of the Flemish Community. Johan Suykens is a postdoctoral researcher with the National Fund for Scienti c Research - Flanders. Bart De Moor is Associate Professor and Senior Research Associate with the National Fund for Scienti c Research - Flanders. We thank the reviewers for constructive comments.
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Abstract In this paper we derive a condition for robust local stability of multilayer recurrent neural networks with two hidden layers. The stability condition follows from linking theory about linearization, robustness analysis of linear systems under nonlinear perturbation and matrix inequalities. A characterization of the basin of attraction of the origin is given in terms of the level set of a quadratic Lyapunov function. In a similar way like for NLq theory, local stability is imposed around the origin and the apparent basin of attraction is made large by applying the criterion, while the proven basin of attraction is relatively small due to conservatism of the criterion. Modifying dynamic backpropagation by the new stability condition is discussed and illustrated by simulation examples.
Keywords. Multilayer recurrent neural networks, NLq theory, linearization, matrix in-
equalities, dynamic backpropagation.
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1 Introduction Because of their universal approximation ability [1, 9], multilayer perceptrons are powerful tools in order to parametrize general nonlinear models and controllers. A well-known method for training the resulting recurrent neural network models and controllers is dynamic backpropagation [15]. When employing this method for training a controller to track speci c reference inputs, a cost function is de ned for minimizing the di erence between the reference input and the actual output of the control scheme. Dynamic backpropagation is basically a gradient based method for minimizing the cost function, where the gradient of the cost function is computed by means of a sensitivity model [15]. However by applying this method, internal stability of the neural control scheme is not guaranteed. While a good performance might be obtained on the training data, the extrapolation in time can be bad or even drive the system towards instability. This has been reported e.g. for the application of dynamic backpropagation to the control of a real-life ball and beam system [28]. Recently, NLq theory has been introduced as a stability theory for discrete time multilayer recurrent neural networks [23, 25]. Conditions for global asymptotic stability, input/output stability with nite L2 -gain and robust stability of multilayer recurrent neural networks have been given, which are expressed in terms of diagonal scaling, diagonal dominance or condition numbers. Links between NLq theory and robust control theory have been demonstrated by considering perturbed NLq representations, with parametric uncertainties upon the interconnection matrices e.g. due to modelling errors. The NLq stability criteria have been used in order to modify the dynamic backpropagation procedure. The cost function of the tracking problem is constrained then with an additional stability condition for the recurrent neural network, which is represented in NLq form. It has been demonstrated how models with a unique or multiple equilibria, periodic, quasi-periodic behaviour and chaos can be controlled based upon these stability criteria [23, 26, 28]. Results for the continuous-time case with conditions for absolute stability, dissipativity and links with Lur'e forms have been given in [27]. NLq theory can be employed in various ways towards applications. The conditions of input/output stability with nite L2-gain can be regarded as nonlinear H1 control for multilayer recurrent neural network representations related to the closed-loop scheme of the 3
model and controller. They can be used also in combination with dynamic backpropogation, by imposing stability conditions as constraints to the backpropagation procedure. The latter has been successfully applied to control of a real-life ball and beam system in [28]. Furthermore, the NLq stability criteria can also be used to check and impose stability of recurrent neural networks (with global and/or local feedback) for nonlinear modelling [26]. In the application of NLq theory to control problems one assumes that a recurrent model is identi ed from input/output data measured on the plant, that the controller is designed based upon the identi ed model and then applied to the plant, assuming that the certainty equivalence principle holds. This is similar to the classical emulator approach in [16]. The NLq framework is basically non-adaptive and rather related to the direction of absolute stability and H1 control theory than adaptive control theory. We also want to mention that within neural adaptive control theory stability criteria are available [5, 11, 19, 20] and [10]. Local stability constraints have been imposed in neural optimal control problems for swinging up an inverted and double inverted pendulum in [23, 22]. Although the NLq stability criteria related to diagonal dominance or condition numbers can be conservative, it has been shown that they still can be employed for neural control design. This is done by imposing local stability at the origin and in uencing the basin of attraction by application of these stability criteria. The apparent basin of attraction can be made very large in this way. The aim of this paper is to investigate these issues in more detail by combining theory about local linearization [29], robust stability [21] and matrix inequalities [3]. In this way we derive a stability criterion which makes the link between the local linearization around the origin, its robustness and the basin of attraction of the point more explicit. Simulation results clearly indicate an apparent basin which is very large (probably globally asymptotically stable), while the proven basin of attraction is relatively small, due to conservatism. In this paper we focus on the NL2 case. The q = 1 case is directly related to discrete time Lur'e forms for which absolute stability criteria are available in literature [3, 4, 12, 14, 29]. This paper is organized as follows. In Section 2 we describe the two-hidden layer recurrent neural networks. In Section 3 the linearization of the recurrent neural networks around the origin and its robustness is discussed. In Section 4 related matrix inequalities and the condition for stability are derived. In Section 5 constrained dynamic backpropagation is discussed. In Section 6 simulation examples are presented. 4
2 Two-hidden layer recurrent neural networks Let us consider the following nonlinear models and controllers that are parametrized by multilayer perceptrons with recurrent neural network model M, controller C1 and reference model R:
R: M: C1 :
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