An Identification and Control Algorithm for Nonlinear Systems

Copyright © IFAC System Identification, Kitakyushu, Fukuoka, Japan, 1997

AN IDENTIFICATION AND CONTROL ALGORITHM FOR NONLINEAR SYSTEMS

Wang Jin, Wang Full and Gao Wenzbong

Dept. ofAutomatic Control. Northeastern University. Shenyang 110006. Liaoning. P.RChina e-mail: [email protected]

Abstract: A new adaptive control algorithm for unknown nonlinear plants is presented. The paper first describes a modified neural network(MNN) as well as the associated learning algorithm. The learning algorithm converges considerably faster because of the introduction of recursive least squares(RLS) algorithm. And then designs adaptive pole placement controller based on the modified neural network. Simulation results show that the proposed control algorithm can effectively control a class of highly nonlinear plants. Keywords: adaptive algorithm

control, nonlinear plants, neural networks, recursive least squares

1. INTRODUCTION

The main drawback is the slow rate at which the BP algorithm converges. The BP algorithm is an iterative gradient descent algorithm designed to minimize the mean-squared error between the desired outputs and the actual outputs for a set of particular inputs to the net. Many iterations are required to train small networks for even the simplest problems. Hence, its on-line learning ability and adaptability are not satisfactory with real time applications. Another drawback is the complexity of controller design based on the MFNN model. A MFNN model represents a complex nonlinear mapping, thus many well developed controller design approaches can not be directly applied and, in contrast to the linear case, the control law can not be given an explicit form without some approximation. Although this problem can be avoided by constructing a neural network which serves as a direct controller, the performance of the resulting control system can not be guaranteed because of the lack of appropriate teaching signal at each time step of a transient process and slowly learning rate of the net. Therefore it would be of considerable value to develop a modified neural network and controller design approach that can overcome the problems described above.

The possibility of estimating linear models, offered by the well developed theory of black-box recursive identification, has greatly contributed to the popularity of these control methods. However, the performance of the controlled system is strongly dependent on the dynamics of the specific plant and, in particular, on the type of nonlinearity. The spread of neural networks has contributed considerably to the application of nonlinear control method. There has been considerable interest in the past decade in exploring the applications of artificial neural networks for identification and adaptive control of nonlinear dynamic systems, and various neural network models and learning strategies have been reported (M.S.Ahmed,LA.Tasadduq, 1994; K.S., Narenda, and K,Partbasarathy,1990). Among these network models and learning strategies, multilayered feedforward neural networks (MFNN) and their associated back propagation (BP) learning algorithm are most widely used because of the theoretically proven approximating ability of MFNN to nonlinear functions and the computational simplicity of BP algorithm. Although they are successfully used in many cases, the BP algorithm and MFNN are suffering from a number of drawbacks .

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the local linearization dynamics of the controlled plant, and the other is DRNN (K.chao-Chee and Y.L.Kwang,1994; L.Jin, P.N.Nikiforuk and M.M.Gupa,1994), which approximates the nonlinear dynamics not being modeled by the linear model. In the network learning strategy, the LNNs parameters are identified by RLS algorithm, while the weights and thresholds of the DRNN are learned by BP algorithm. The structure of the MNN shown in Fig. 1 has three advantages: first, the LNN part can be trained at a high convergent speed; secondly, the system order needs not be known prior; thirdly, the MNN becomes relatively simple and its learning will cost less time than that of the feedforward neural networks and DRNN.

Many researchers have developed the control scheme (Y,-M.Park, M .-S.Choi, and K.Y.Lee,1996; H.AIDuwaish, M.N.Karim,1996) that is based on plant linearisation at each operating point. Since control design for linear plants has been well developed, it is natural to make use of it in nonlinear plants. One practical use of this idea is the gain-scheduling control, where the control is based on a finite number of linearised models calculated at a preselected range of operating points. This approach is also called multiple-model adaptive control. In contrast, the approach in (M.S.Ahmed, I.A.Tasadduq,1994) assumes continuous change of the linearised model with the operating point. Although in gain scheduling such continuous parameter values of the controller may be generated using interpolation, it would add extra computational load, making the strategy unattracti ve. For this purpose, the paper presents a MNN, and designs an adaptive pole placement controller for nonlinear dynamic systems based on this MNN. The proposed network is composed of two parts: one is linear neural network(LNN), which models the local linearisation dynamics of the controlled plant, and the other is multilayered diagonal "Tecurrent neural network (DRNN) which approximates the nonlinear dynamics not being modeled by the linear model. In the network learning strategy, the linear neural network parameters are identified by recursive least square algorithm, while the weights and thresholds of the DRNN are learned by BP algorithm. For the controller design, a pole placement control law is derived based on the linear network model; on the other hand, the output of the DRNN is considered as a measurable disturbance, which represents the unmodeled dynamics of the controlled plant by the linear model, and is eliminated through feedforward control .

u(k·2)

y(k.I)

9 '" y(k·2)

Fig. 1. Modified recurrent neural network Consider the following nonlinear controlled plant:

y(k) = f(y(k -l),y(k -2), ... y(k -n), u(k-1),u(k-2),oo.u(k-m)

(1)

Note that, . n and m are not system order here, but selective by experience for the use of identifying the linear part model; usually with n,m increase, the LNN modeling gets more accurate, but the calculation burden will be increased. However, the inaccuracy of LNN due to the selection of m,n and other factors will be compensated by DRNN. MNN is described by the following network model:

The proposed nonlinear adaptive pole placement control algorithm has several advantages. First, the learning algorithm is considerably faster than the back propagation algorithm due to the introduction of recursive least squares(RLS) algorithm. Secondly, the calculation of the control variable is very simple, and the controller parameters can be easily adjusted online since the feedback control law is determined based on the linear network model. Finally, the control algorithm has stronger robustness to nonlinear dynamics with no loss of control accuracy because of the DRNN's capability of approximating nonlinearity .

y(k) =.h (k)+ YN(k)+E(k)

=T x(k -1) + N(u(k -l),y(k -1)) + c(k) (2)

where,

x(k -1)

= [y(k -l) ,y(k -

2), ... y(k - n);

u(k -l),u(k - 2), ... u(k - m)f , = [a p a 2 , ••• a n, bp b 2 , .• • bm]T ,

it

is

the

parameter vector of LNN; N(u(k -I),y(k -I)) is 2. THE MODIFIED NEURAL NETWORK(IDENTIFICATION)

a DRNN with (u(k -I),y(k -I)) as the input layer, 1\

one hidden layer and one single output;

Fig. 1 shows the structure of the MNN that is composed of two parts: one is LNN, which models

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h (k) is

/\

neuron; and

the output of the LNN, y N (k) is the output of the

i .1

i

i

output and diagonal weights respectively.

DRNN, E(k) is the modeling error. The activation fimction

Wl , Wo, and WD represent input,

of the neurons at the hidden layers is

The learning algorithm of the DRNN is described as follows (Wang Jin, Gao Wenzhong,). First, let us define:

p(x) = 1/1 + e - x , the neurons at the input and output layers are linear. The parameter vector et> of the LNN is identified by the RLS algorithm, while the weights and thresholds of the DRNN are learned by the BP algorithm. The detailed procedure is performed as follows :

eN

(k) = (y(k) -

~ L (k) - ; N (k», E = e~ (k) / 2 .

_Wj O (k + 1) = ~o (k) - 17.. oE / ~o

(i). Initialize the weights and thresholds of the . DRNN randomly and the parameter vector et> properly.

= ~o(k)+ 17.. eN(k+l)Hj(k)

(ii). Use the RLS algorithm to identify the parameter vector et> . An important concept in the online linear identification is its ability to track varying process dynamics. To achieve this , the algorithm must incorporate a method of discounting old data, effectively giving more weight to recent data. This is done by introducing a forgetting factor y as follows

= ~D (k) + 17.. eN(k + I)G j (k)~o

(8)

-wt (k + 1) = wt (k) - 17.. oE / ~D G j (k)

= OHj (k) /

(9)

owt

=p'(Sj (k»(Hj (k -1) + ~D (k)G j (k -1», (10)

(Fang Chong Zhe, Xiao De Yun,1987):

cI>(k) =cI>(k - 1) + K(k)[y(k) - x T (k -l)cI>(k -1)] (3) (11)

T

K(k) =P(k -l)x(k -l)[x (k-1)P(k -1)x(k-1) +yr1 (4)

P(k) = [I - K(k)x T (k -l)]P(k -1)/y , P(O) =01 (0) 0)

J Q .. (k) = 8Hi (k)/OWiJ. i~

(5)

where y(k + 1) is the latest process output value.

= p'(Sj (k»(lj (k) + ~D (k)Qj,i (k -1»,

Qj,i (0) = 0

ois a lager constant(e.g.,IO,OOO). I is a unit matrix.

where,

(iii). Then fix the parameter vector et> and use the BP algorithm to learn the weights and thresholds of the DRNN. The mathematical model for the DRNN shown in Fig.1 is given by the following equations (K.chao-Chee and Y.L.Kwang,1994):

llw

(12)

is

learning

rate

of

DRNN;

p'(.) = dp(.)/d •. Note that, we start with the off-line training phase using the algorithm as described above, eqn.(3-5) are run through all the training data, thus obtaining the estimation of the parameter vector et>. Then go to Step 3 using eqn.(8-12) to learn the weights until modeling error E(k) is less than the given small

where,

positive value. While in the on-line learning phase, first run Step 2 one time at each sampling time, and then run Step 3 for some iterations. The simulation results show that the learning speed of the learning algorithm using the MNN model structure as described above is much faster than that of multilayered feed-forward networks and the pure DRNN.

B . is threshold of the jth recurrent neuron. i

For each discrete time k, I j (k) is the ith input to the DRNN, the inputs consist of u(k-1) and y(k-1);

S j (k ) is the sum of inputs to the jth recurrent 3. ADAPTIVE POLE PLACEMENT CONTROL ALGORITHM

neuron; H j (k) is the output of the jth recurrent

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