State observer and filter design for delayed neural networks

Abstract

The study of recurrent neural networks has been an active topic during the past few years because of their advantages in terms of parallel computation, learning capability, function approximation, and fault tolerance. Neural networks have been enjoying wide and successful applications in signal processing, automatic control, classi¯cation, knowledge acquisition, pattern recognition, combinatorial optimization, machine learning and other ¯elds. When a neural network is designed to handle complex nonlinear problems, a great number of neurons with tremendous connections are often required. On the other hand, in analog VLSI (very large scale integration) implementations of neural networks, networks parameters are most likely subject to some variations due to the tolerances of the utilized electronic elements, and noises may also be introduced. It is thus very di±cult or expensive (even may be impossible) to obtain the complete information of the neuron states in relatively large-scale neural networks. At the same time, in many practical applications of neural networks such as system mod- eling and state feedback control, the information on the states of neurons is needed and utilized to achieve certain objectives. Therefore, it is of great importance and practical signi¯cance to study the issues on estimating the neuron states via available output measurements. Time delay inevitably occurs in the electronic implementations of neural networks because of the ¯nite switching speeds of ampli¯ers. It has been well recognized that the existence of time delay can change the dynamic behaviors and/or deteriorate the performance of the underlying neural networks. On the other hand, it can be more e±cient to solve some engineering problems (for example, the speed detection of moving objects and processing of moving images) when time delay is introduced in neural networks. This thesis is thus focused on investigating the state observer and ¯lter design problems for delayed neural networks. First of all, the state observer design problem is studied for a class of recurrent neural networks with time-varying delay. The restrictions in some existing results that the time-varying delay was di®erentiable and its time-derivative was smaller than a constant scalar are removed. Instead, the time-varying delay is only required to be continuous and bounded. An improved approach to estimating the neuron states is developed based on a delay-dependent condition, under which the resulting error system is globally asymptotically stable. The design of the gain matrix of the state observer can be achieved by solving a linear matrix inequality, which is facilitated readily by resorting to standard numerical algorithms. Then, a novel delay partition approach is proposed to further address the state observer design problem for neural networks with time-varying delay. The basic idea of this approach lies in that the time-varying delay and its upper bound are respectively divided into di®erent slices. By de¯ning a new Lyapunov-Krasovskii functional, a delay-dependent design criterion is provided and formulated by means of a linear matrix inequality. With the increase of the segments in the two partitions, less conservative conditions are achieved. In practice, some fundamental coe±cients, such as the ¯ring rates of neurons and the interconnection weights between neurons are generally acquired and processed by the statistical method. They may su®er from some variations. As a result, pa- rameter uncertainties should be taken into account when modeling neural networks. The robust state observer design problem is then studied for neural networks with parameter uncertainties and time-varying delay. Based on a newly-established in- tegral inequality, delay-dependent criteria are obtained to ensure the existence of a desired state observer for such neural networks. It is shown that the robust state observer can be designed by means of the feasibility of a linear matrix inequality. It should be pointed out that slack variables are introduced by the integral inequality to e®ectively reduce the conservatism of the developed results. Furthermore, the state observer design problem for static neural networks, which is one of the two typical models of neural networks with the other being local ¯eld neu- ral networks classi¯ed by the modeling approaches, is addressed. The time-derivative of the time-varying delay is no longer required to be smaller than one. In particular, a delay partition approach is developed to deal with this issue. A delay-dependent condition in terms of a linear matrix inequality is presented for the design of a proper state observer, under which the resulting error system is globally asymptotically sta- ble. Finally, the robust ¯ltering design problems are addressed for delayed neural networks disturbed by noises. Two types of ¯lters are considered: H1 ¯lter and gen- eralized H2 ¯lter. Both delay-independent and delay-dependent criteria are developed such that the resulting ¯ltering error system is globally stable with guaranteed H1 or generalized H2 performance. The design of the appropriate ¯lters and the optimal performance indexes can be obtained by solving some convex optimization problems subject to linear matrix inequalities.

Date of Award	2 Oct 2009
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Gang Gary FENG (Supervisor)