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ISSN 1320-0682 | ||||
| Volume 3 | April 1996 | ||||
S.M. Monzurzur Rahman, Xinghuo Yu, Man Zhihong
...Rahman & Yu
Department of Mathematics and Computing, Central Queensland University,
Rockhampton QLD 4702 AUSTRALIA
...Zhihong
Department of Electrical Engineering, University of Tasmania, Hobart, Tasmania
7001 AUSTRALIA
Email:rahmanm@topaz.cqu.edu.au, x.yu@cqu.edu.au, Zhihong.Man@eee.utas.edu.au
An adaptation law for the learning in a feedforward neural network is proposed. This law does not require the boundedness of input signals. The convergence analysis is presented and numerical simulation given for application to two dynamics identification problems.
Feedforward neural networks (FNNs) are composed of layers of neurons, in which the input layer of neurons are connected to the output layer of neurons. The training process of FNN is undertaken by changing the weights such that a desired input-output relationship is realised. A major class of adaptation-learning algorithms are based on the Widrow-Hoff back-propagation algorithm that minimises the error function of output with respect to input by adjusting weights. The learning parameters are adjusted using, basically, the gradient descent method. The back-propagation algorithm is effective, but time-consuming, and the learning parameters may not converge to global optimum.
A substantial departure from the back-propagation procedure is that, instead of employing the gradient descent method, an asymptotically stable error dynamics can be used. This enables better control and speed of convergence by appropriate choice of parameters of the error dynamics. One of the advantages of using the error dynamics is that the convergence to the global optimum is guaranteed. The design principle is based on the sliding mode control (SMC) approach. The research on using this SMC approach for FNN learning has been reported in [3], [2].
One of the problems of the existing results is that there is a stringent requirement of a priori knowledge of the boundedness of input signals and their derivatives. The effectiveness of the SMC approach very much depends on the accuracy of the knowledge of the boundedness. Over- or under-estimation can either cause severe chattering or divergence.
In this paper, we propose an adaptation law for the adaptive learning in the one layer FNN that does not require the boundedness condition to guarantee effective adaptive learning. Conditions for convergence are given. The algorithm is tested for forward and inverse dynamics identification problems. This paper is organised as follows. The next section presents the algorithm for adaptive learning in the FNN and analysis of convergence. The following section discusses application of the algorithm to forward and inverse dynamics identification problems. The conclusion is drawn in the last section.
We consider the one layer neural network model
where y(t), x(t) and w(t) are
defined as follows. The vector
represents a
smooth vector of time-varying inputs. We define
as augmented
inputs, that include a constant input of value
affecting the
threshold weight
in the model.
The vector 
represents the set of time-varying weights. We also defined the augmented
weights by including the bias weight component
For simplicity, we consider a scalar output y(t). The signal
represents the
time-varying desired output of the model.
The learning error e(t) is defined as
. Using the
sliding mode control theory [4],
we define a time-varying sliding manifold
which is the desired condition for the error signal e(t) and
one which guarantees that the output y(t) coincides with the
desired output signal
for all times
where
is the
hitting time of e=0.
For the one layer model (1), if the learning algorithm for the weights is
then the learning error e converges to zero in finite time.
Proof: Consider the Lyapunov function candidate
Differentiating it with respect to t yields
Substituting (3) into (5) yields
Since 
, (6)
becomes
A simple computation of (7) leads to that the time reaching e=0 is
Hence, finite time reachability is realised.
When e=0 is reached, the system state shall stay in it forever. Therefore,
which are ideally satisfied after the sliding mode e=0 is reached. Since
The method of equivalent control [4]
suggests that, from (9),
the equivalent adaptive weight vector
is
and from the first equation in (8)
The question is, when in sliding, how does the system (11) behave?
If the following conditions hold,
and
where 
are
constants and 
, and 
is the
transition matrix of the equation
then, the equivalent adaptation law (10)
yields a bounded trajectory with the weights
.
Proof: The solution of (14) is
and hence,
Therefore, 
is bounded.
The conditions given in Proposition 2 are more relaxed than those presented in [2] which required the weight dynamics to satisfy the stringent exponential stability condition. Here, the boundedness of input signals is not required. Also, the condition (13) allows a wider variety of signals to satisfy - for example, the sinusoidal signals.
Consider the inverted pendulum controlled by a dc motor [1] which is expressed in the following dynamics
We first consider the forward dynamics identification problem; that is, the
output of the neural network tracks the output of the pendulum
and the
following is used as the input to the neural network.
The error function is defined as
, where y(t)
is the output of the neural network. For the simulation, we set the initial
state as 
.
Figure 1 shows the response. The output of the neural net (dotted line) quickly
approaches the output of the pendulum and stays with it.
is the input
to the pendulum.
Figure 1: Output responses. (Solid line-desired output. Dotted
line-neural net output)
For the inverse dynamics identification problem - that is, the output of the neural network tracks the input signal to the pendulum - the input to the neural network is chosen the same as that for the forward dynamics identification, but the error function is defined as e(t)=y(t)-u(t). For the simulation, the initial condition is set to zero. Figure 2 shows the response. The output of the neural net (dotted line) quickly approaches the input signal u(t) and stays with it.
Figure 2: Output responses. (Solid line-given input. Dotted
line-neural net output)
A modification to the existing algorithm has shown that boundedness of input signals is not required. Convergence is analysed that suits a much broader class of systems. Applications to forward and inverse dynamics identification problems have been made and proved to be successful.
The second author wishes to thank the Central Queensland University for a grant.
A Feedforward Neural Network with Adaptive Learning
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