Journal of the Chinese Institute of Engineers, Vol. 29, No. 4, pp. 747-751 (2006)
747
Short Paper
DYNAMIC RECURRENT WAVELET NETWORK CONTROLLERS FOR NONLINEAR SYSTEM CONTROL
Cheng-Jian Lin*, Chi-Yung Lee, and Cheng-Chung Chin
ABSTRACT To solve nonlinear system control problems, a Recurrent Wavelet Network Controller (RWNC) is proposed in this paper. The proposed RWNC model has four-layer structure. Temporal relations embedded in the network by adding some feedback connections representing the memory units in the second layer. A self-organizing learning algorithm, which consists of structure learning and parameter learning, is proposed and is able to construct the recurrent wavelet network dynamically. The structure learning is based on the input partitions to determine the number of wavelet bases, and the parameter learning is based on the supervised gradient descent method to adjust the shape of wavelet functions, feedback weights, and the connection weights. Computer simulations were conducted to illustrate the performance and applicability of the proposed model. Key Words: recurrent networks, wavelet bases, control, self-organizing learning, degree measure.
I. INTRODUCTION Recently, many studies have proposed wavelet neural networks for identification and control (Ho et al., 2001; Huang and Huang, 2002; Ikonomopoulos and Endou, 1998; Jiao et al., 2001; Lin et al., 2003; Zhang, 1998). Ikonomopoulos and Endou (1998) posited the analytical ability of the discrete wavelet decomposition with the computational power of radial basis function networks. Members of a wavelet family were chosen through a statistical selection criterion that constructs the structure of the network. Ho et al. (2001) used the orthogonal least squares (OLS) algorithm to purify the wavelets from their candidates, which avoided using more wavelets than required. Overuse of wavelets often resulted in an overfitting *Corresponding author. (Fax: 886-4-23742375; Email:
[email protected]) C. J. Lin and C. C. Chin are with the Department of Computer Science and Information Engineering, Chaoyang University of Technology, Taichung, Taiwan 413, R.O.C. C. Y. Lee is with the Department of Computer Science and Information Engineering, Nankai Institute of Technology, Nantou, Taiwan 542, R.O.C.
of the data and a poor situation in (Zhang, 1998). Lin et al. (2003) proposed a wavelet neural network to control the moving table of a linear ultrasonic motor (LUSM) drive system. They chose an initialization for the mother wavelet based on the input domains defined by the examples of the training sequence. Huang and Huang (2002) proposed an evolutionary algorithm for optimally adjusted wavelet networks. However, the selections of wavelet bases were based on practical experimentation or trial-and-error tests. The objective of this paper is to introduce a recurrent wavelet network controller (RWNC) with a self-organizing learning algorithm. The architecture of the RWNC model enables it to preserve past states of the networks. Therefore, the RWNC model has the capability to deal with temporal problems. A selforganizing learning algorithm, which consists of structure learning and parameter learning, is proposed and is able to construct the recurrent wavelet network dynamically. The structure learning is based on the input partitions to determine the number of wavelet bases and memory units, and the parameter learning is based on the supervised gradient descent method to adjust the shape of wavelet functions, feedback
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Journal of the Chinese Institute of Engineers, Vol. 29, No. 4 (2006)
Y1(s)
Yp(s)
Σ
Σ
w1i
1 p wj
wi ψ
ψ
1
θ 11 –1 Z
θ 1j
θ 1m
Z–1
j
w jp
φ dij.tij(h ij(s)) Output layer
ij
Product layer
ψ
m
θ nj
θ nm
Z–1
Z–1
Wavelet layer
Input layer xn(s)
m
weights, and the connection weights. In the initial form, there are no wavelet bases or wavelet functions. They are created and begin to grow as the first training input data arrives. Thus, the user need not be given any a priori knowledge or initial information on the wavelet bases and functions. Finally, the RWNC model is applied in several control problems. The advantages of the proposed RWNC model are summarized as follows: 1) it converges quickly; 2) it is constructed automatically; 3) it requires many fewer hidden nodes and adjustable parameters; and 4) it has much lower rms error. II. STRUCTURE OF THE RWNC MODEL The structure of the RWNC model is shown in Fig. 1. The proposed RWNC model is designed as a four-layer structure, which is comprised of an input layer, wavelet layer, product layer, and output layer. In the wavelet layer, a differentiable Mexican-hat function (Huang and Huang, 2002) is adopted as a mother wavelet. Therefore, the activation function of the jth wavelet node connected with the ith input data is represented as:
φ dij.tij(x i(s)) = 2 d /2(1 – ||2 d x i(s) – t ij|| 2)e –||2 ij
h ij(s) – tij|| 2/2
d ij
x i(s) – tij || 2/2
i = 1, ... , n j = 1, ... , m,
, (1)
where n is the number of input-dimensions and m is the number of wavelets. Temporal relations are embedded in the network by adding some feedback connections representing the memory units in the second layer. In the RWNC model, to store information for later use, dynamic mapping is adopted as follows:
, (2)
where, h ij(s) = x i(s) + θ ij . φ dij.tij(h ij (s – 1)), θ ij is the feedback weight in the memory unit; x i (s) is the ith input variable at time s. It is clear that the input of this layer contains the memory terms hij(s – 1), which store the past information of the network. Then, each wavelet in the product layer is labeled Π . According to the theory of multi-resolution analysis (MRA) (Zhang, 1998), any f ∈ L 2(R) can be regarded as a linear combination of wavelets at different resolution levels. For this reason, the function f is expressed as Y l(s) = Y l(h ij(s)) = f ≈
Fig. 1 The architecture of the RWNC model
ij
d ij
i = 1, ... , n j = 1, ... , m,
wm1
Z–1
x1(s)
ij
wmp
θ n1
Z–1
= 2 d /2(1 – ||2 d h ij(s) – t ij|| 2)e –||2
=
Σ
j=1
m
Σ w ljψ j(h ij(s)) j=1
n
w lj Π φd ij . tij(h ij(s))
(3)
i=1
where Y = [Y 1, ... , Y l, ... , Y p] means the lth output of the RWNC model. III. A SELF-ORGANIZING LEARNING ALGORITHM In this section, the degree measure method and the well-known back propagation (BP) algorithm are used concurrently for constructing and adjusting the RWNC controller. The degree measure method is used to decide the number of wavelet bases in the wavelet layer, the product layer and the memory units. On the other hand, the BP algorithm is used to adjust the parameters of the wavelet bases, feedback weights, and connection weights. The details of the algorithm are presented below. 1. The Structure Learning Scheme Initially, there are no wavelet bases in the RWNC model. The first task is to decide when a new wavelet base is generated. For each incoming pattern x i(s), the firing strength of a wavelet base can be regarded as the degree of the incoming pattern belonging to the corresponding wavelet base. An input datum xi(s) with a higher firing strength means that its spatial location is nearer to the center of the wavelet base tij than those with smaller firing strengths. Based on this concept, the firing strength obtained from Eq. (3) in the product layer can be used as the degree measure Fj = |ψ j| j = 1, ... , m,
(4)
where m is the number of existing wavelet bases and
C. J. Lin et al.: Dynamic Recurrent Wavelet Network Controllers for Nonlinear System Control
| ψ j| is the absolute value of ψ j. According to the degree measure, the criterion of a new wavelet base generated for new incoming data is described as follows: Find the maximum degree F max 1≤ j≤m
– If F max ≤ F, then a new wavelet base is generated, – where F is a pre-specified threshold that should decay during the learning process, limiting the size of the RWNC model (Lin et al., 2004). 2. The Parameter Learning Scheme
⋅ ln2(2 d ijh ij(s) – t ij)[1 +
2 ]} 1 – (2 d ijh ij(s) – t ij) 2
(13) p 2 d ijh ij(s) – t ij ∆θ ij = – η θ ⋅ 1p Σ e l ⋅ w lj ⋅ ψ j{ l=1 1 – (2 d ijh ij(s) – t ij) 2
⋅ φd jt j ⋅ (h ij(s – 1)) ⋅ 2 d ij ⋅ ((2 d ijh ij(s)
– t ij) 2 – 3)} .
After the network structure has been adjusted according to the current training pattern, the network then enters the second learning step to adjust the parameters of the wavelet base, feedback weight and the connection weight (t, d, θ and w) with the same training pattern. The parameter-learning algorithm is based on a set of MIMO pairs {x i(s) – Y ldes(s)}. If the lth des error function el is defined el = (Yl(s) – Yl (s)), where des Y l(s) is the lth model output and Y l (s) is the lth desired output at time s, then the cost function E can be defined as p
E = 1 Σ e l2 2p l = 1
(6)
and can be minimized by all adjustable parameters using an iterative computational scheme. The free parameters adjusted in the RWNC model are as follows: The connection weight of the output layer is updated by wjl(s + 1) = wjl(s) + ∆ wjl
(7)
where
∆w lj = – η w ∂El = – η w ⋅ 1p ⋅ e ⋅ ψ j . ∂w j
(8)
Similarly, the updated laws of tij, dij and θij are shown as follows: t ij(s + 1) = t ij(s) + ∆ t ij
(9)
d ij(s + 1) = d ij(s) + ∆ d ij
(10)
θ ij(s + 1) = θ ij(s) + ∆θ ij
(11)
where p 2 d ijh ij(s) – t ij ∆t ij = – η t ⋅ 1p Σ e l ⋅ w lj ⋅ ψ j{ l=1 1 – (2 d ijh ij(s) – t ij) 2
⋅ [2 + (1 – (2 d ijh ij(s) – t ij) 2)]}
p
∆d ij = – η d ⋅ 1p Σ e l ⋅ w lj ⋅ ψ j{ ln2 – 2 d ijh ij(s) 2 l=1
(5)
Fmax = max F j .
(12)
749
(14)
IV. ILLUSTRATIVE EXAMPLES In this section, two control examples are given to demonstrate the validity of the presented RWNC model. The first example is to control a ball and beam system (Hauser et al., 1992; Lin and Lin, 1997). The second example is to control the temperature of a water bath system (Tanomaru and Omatu, 1992). Example 1: Control of the Ball and Beam System The ball and beam system (Hauser et al., 1992) can be written in state space form as x2 x1 0 x2 B(x1 x42 – G sinx3) 0 u, = + x4 x3 0 1 x4 0
y = x1
(15)
. . where x = (x 1, x2, x 3, x 4) T ≡ (r, r, θ, θ ) T is the state of the system and y = x1 ≡ r is the output of the .. system. The control u is the angular acceleration (θ ) and the parameters B = 0.7143 and G = 9.81 are chosen in this system. We use the input/output-linearization algorithm (Hauser et al., 1992) to generate the input/ output train pair with x obtained from randomly sampling 200 points in the region U = [-5, 5] ×[-3, 3] × [-1, 1] ×[-2, 2]. The learning rate η θ = η d = η t = η w – = 0.05 and the prespecified threshold F = 0.3 are chosen. After the self-organizing structure-parameter learning, there are 9 nodes generated in the wavelet layer. We have tested the trained controller at two initial conditions. Fig. 2 shows the behavior of the four states of the ball and beam system starting at the initial condition x(0) = [-2.4, 0.1, -0.6, -0.1] T and [2.4, -0.1, 0.6, 0.1] T. In this figure, the four states of the system decay to zero gradually. The results show the perfect control capability of the trained RWNC model. We also compare the performance of RWNC with several existing methods (Lin and Chen, 2003;
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Journal of the Chinese Institute of Engineers, Vol. 29, No. 4 (2006)
Table 1 Performance comparison of several existing models in Example 1 RWNC
Neural networks
FALCON (Lin and Lin, 1997)
Neuro-Fuzzy (Nomura et al., 1992)
CNFN controller (Lin and Chen, 2003)
Training steps
200
300
50000
500
250
Number of rules /hidden nodes
9
35
28
81
23
Number of adjustable parameters
117
175
420
243
253
RMS error
0.095
0.1267
0.2
0.1551
0.1212
6
temperature of a water bath using the RWNC. The water bath plan is governed by
4 ×4
State
2
β (1 – e – α Ts) y(t + 1) = e – α Tsy(k) + α 0.5y(k) – 40 u(k) 1+e
×2
0 ×3
+ [1 – e – α Ts]y0 .
-2 ×1 -4 -6
0
50
100
150
200
250
150
200
250
Times (a) 6 4
State
2
×1 ×3
0 ×4
-2
×2
-4 -6
0
50
100 Times (b)
Fig. 2
The responses of the four states of the ball and beam system under the control of the trained RWNC controller. (a) [-2.4, 0.1, -0.6, -0.1] T and (b) [2.4, -0.1, 0.6, 0.1] T.
Lin and Lin, 1997; Nomura et al., 1992). The comparison results are tabulated in Table 1. Simulation results show that the proposed model can obtain smaller rms errors and fewer adjustable parameters than several existing methods.
(16)
The system parameters used in this example are α = 1.0015e –4, β = 8.67973e –3 and Y 0 = 25.0(°C), which were obtained from a real water bath plant in (Tanomaru and Omatu, 1992). The input u(k) is limited to 0 and 5V representing voltage units. The sampling period is Ts = 30. The 120 training patterns are chosen from the input-output characteristics in order to cover the entire reference output. The initial temperature of the water is 25°C, and the temperature rises progressively when random input signals are injected. We choose the same parameters as Example1. After training, there are 12 nodes generated in the wavelet layer. The task is to control the simulated system to follow three set-points.
yref (k) =
35°C , for k ≤ 40 55°C , for 40 < k ≤ 80 75°C , for 80 < k ≤ 120 .
(17)
The regulation performance of the RWNC model is shown in Fig. 3(a). We also test the regulation performance by using CNFN controller (Lin and Chen, 2003). The error curves of the RWNC controller and the CNFN controller between k = 80 and k = 100 are shown in Fig. 3(b). In this figure, the proposed RWNC controller obtains smaller errors than the CNFN controller (Lin and Chen, 2003). V. CONCLUSIONS
Example 2: Control of Water Bath Temperature System The goal of this example is to control the
In this paper, we propose a recurrent wavelet network controller (RWNC) for solving control problems. We adopt the orthogonal function as wavelet neural
C. J. Lin et al.: Dynamic Recurrent Wavelet Network Controllers for Nonlinear System Control
Reference Signal “– –” Actual Signal “–”
Temperature (Degree C)
80 70 60 50 40 30 20 10
Control input
5V
0
Temperature (degree)
0
20
40
60 80 k (a) RWNC: - CNFN: --
20 18 16 14 12 10 8 6 4 2 0 82
Fig. 3
84
86
88
90 92 k (b)
94
100
96
120
98
100
(a) Final regulation performance of the RWNC controller for water bath system. (b) The error curves of the RWNC controller and the CNFN controller (Lin and Chen, 2003) between k = 80 and k = 100.
network bases. A self-organizing learning algorithm is proposed to construct a model and tune parameters automatically. The experimental results strongly demonstrate that the learning scheme is very effective for control of nonlinear systems. ACKNOWLEDGMENTS This research was supported by the National Science Council of R.O.C. under grant NSC 94-2213E-324-004. REFERENCES Ho, D.W.C., Zhang, P. A., and Xu, J., 2001, “Fuzzy Wavelet Networks for Function Learning,” IEEE Transactions on Fuzzy Systems, Vol. 9, No. 1, pp. 200-211.
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Hauser, J., Sastry, S., and Kokotovic, P., 1992, “Nonolinear Control via Approximate Input-Output Linearization: The Ball and Beam Example,” IEEE Transactions on Automatic Control, Vol. 37, pp. 392-398. Huang, Y. C. and Huang, C. M., 2002, “Evolving Wavelet Networks for Power Transformer Condition Monitoring,” IEEE Transactions on Power Delivery, Vol. 17, No. 2, pp. 412-416. Ikonomopoulos, A. and Endou, A., 1998, “Wavelet Decomposition and Radial Basis Function Networks for System Monitoring,” IEEE Transactions on Nuclear Science, Vol. 45, No. 5, pp. 2293-2301. Jiao, L., Pan, J., and Fang, Y., 2001, “Multiwavelet Neural Network and Its Approximation Properties,” IEEE Transactions on Neural Networks, Vol. 12, No. 5, pp. 1060-1066. Lin, C. J. and Chen, C. H., 2003, “Nonlinear System Control Using Compensatory Neuro-Fuzzy Networks,” IEICE Transactions on Fundamentals, Vol. E86-A, No. 9, pp. 2309-2316. Lin, C. J. and Lin, C. T., 1997, “An ART-Based Fuzzy Adaptive Learning Control Network,” IEEE Transactions on Fuzzy Systems, Vol. 5, No. 4, pp. 477-496. Lin, F. J., Wai, R. J., and Chen, M. P., 2003, “Wavelet Neural Network Control for Linear Ultrasonic Motor Drive via Adaptive Sliding-Mode Technique,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 50, No. 6, pp. 686-698. Lin, C. J., Shih, C. C., and Chen, P. Y., 2004, “A Nonlinear Time-Varying Channel Equalizer Using Self-Organizing Wavelet Neural Networks,” International Joint Conference on Neural Networks, Budapest, Hungary, pp. 2089-2094. Nomura, H., Hayashi, I., and Wakami, N., 1992, “A Learning Method of Fuzzy Inference Rules by Descent Method,” IEEE International Conference on Fuzzy Systems, San Diego, CA, USA, pp. 203210. Tanomaru, J. and Omatu, S., 1992, “Process Control by Online Trained Neural Network,” IEEE Transactions on Industrial Electronics, Vol. 39, pp. 511-521. Zhang, Q., 1998, “Using Wavelet Networks in Nonparametric Estimation,” IEEE Transactions on Neural Networks, Vol. 8, pp. 227-236. Manuscript Received: Jun. 04, 2004 Revision Received: Apr. 09, 2005 and Accepted: May 12, 2005
