204 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANUARY/FEBRUARY 1993

Correspondence

Dynamic System Identification Using Neural Networks

Takayuki Yamada and Tetsuro Yabuta

Abstract-Many studies have heen undertaken in order to apply both the flexibility and learning ability of neural networks to robot controllers. The application of neural networks to servolevel controllers has also been studied. Therefore, we have proposed a neural network direct controller as a servolevel controller and confirmed its characteristics such as stability. However, there is also possibility of applying to the identi6cation of a nonlinear plant. Thus, a practical neural network design method for the identification of both the direct transfer function and inverse transfer function of the object plant is proposed. A nonlinear plant simulator as a practical application of the direct transfer function identilier using a ueural network is also proposed. Simulated and experimental results for a second-order plant show that these identifications can be satisfactorily achieved. They also confirm that neural network identiBen can represent nonlinear plant characteristics very well. The characteristics of a neural network direct controller with a feedback control loop is also proposed and confirmed, which uses the learning results of the inverse transfer tunetion identi6er.

I. INTRODUCTION A neural network is modeled on a biological neural network and

has excellent capabilities such as nonlinear mapping. Recently, many studies such as Kawato’s work have been undertaken in order to apply both the flexibility and learning ability of neural networks to robot controllers [ 11-[6]. However, studies on neural networks as servolevel controllers are still in their early stages. On the other hand, there are many studies on the servolevel controller based on conventional control theories such as adaptive control, learning control and fuzzy control. If it is possible to make an interface between neural networks and conventional control theories, neural networks can not only use the results of conventional theories, but also develop in combination with them. Therefore, we have proposed a neural network direct controller as a servolevel controller in order to compare neural network controllers with conventional control theories. We have also confirmed its characteristics such as stability [7], [8]. However, it is necessary to take into account not only the controller but also the identification of the nonlinear dynamics system through the use of the neural network. However, no such studies have yet been completed and only Jordan has begun to study this problem [9].

This paper proposes practical design methods for the identification of both the direct and inverse transfer functions of a dynamic system through the use of a neural network. Simulated and experimental re- sults confirm that these identifications can be satisfactorily achieved. This paper proposes a nonlinear simulator of a dynamic system as a practical application of the direct transfer function identifier. This paper also proposes a neural network controller with a feedback loop, which uses the learning results of the inverse transfer function identifier. With this control method, this feedback loop is expected to compensate for the remaining control error caused by identification error. Experimental results confirm the control characteristics of this scheme.

Manuscript received March 16, 1990; revised March 6, 1992. The authors are with NTT Telecommunication Field Systems R&D Center,

Tokai-Mura, Naka-Gun, Ibaraki-Ken, 319-1 1, Japan. IEEE Log Number 9202115.

Yd -

Fig. 1. General control system (parallel type).

Fig. 2. General control system (series type).

11. CLASSIFICATION OF CONTROLLERS

Figs. 1 and 2 show a parallel and a series controller, respectively. In this control scheme, both feedforward and feedback controllers are tuned through the use of plant parameters estimated by observers (identifiers). Kawato has applied a neural network to the parallel controller shown in Fig. 1 [4]. His controller uses the neural network as a feedforward block. At the beginning of learning, the feedback block guarantees stability. After learning, the system is mainly controlled by the neural network. With the controller shown in Fig. 2, the feedforward block acts as a dynamic filter of the desired value for control. One typical example is that the feedforward block is used as an inverse kinematics solver of robot manipulators. Although many studies have been concerned with the feedforward feedback control scheme, we can realize a neural network feedforward controller without a feedback block [7], [8]. There have been many attempts to design a controller using a neural network, but the characteristics and stability of neural network observers have not yet been clarified. As PDP type neural networks are trained to minimize error energy

functions, observers (identifiers) using neural networks are classified into two types by definition of their error energy functions. One type is trained to minimize the squared error (energy function) between a neural network output and a plant output. Fig. 3 shows this type of observer that is defined as a direct transfer function identifier in this paper. In this type of identifier, the neural network output converges with the plant output after learning and the direct transfer function of the plant is composed in the neural network. The other type is trained to minimize the squared error (error energy function) between the neural network output and the plant input. Fig. 4 shows this type of observer that is defined as an inverse transfer function identifier. In this type of identifier, the neural network output converges with the plant input after learning and the inverse transfer function of the plant is composed in the neural network.

0018-9472/93$03.00 0 1993 IEEE

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANUARYffEBRUARY 1993 205

Y U Plant 1

Neural Network

L

Fig. 3. Scheme of direct transfer function identifiet

- Y U Plant 1 I *

Neural Network

i Fig. 4. Scheme of inverse transfer function identifier.

Position

1 ,Actual Position

/Desired Trajectory

V I

Time Fig. 5. Concept of adaptive type.

Observers are also classified into two types by learning method [lo]. One type learns at every sampling time. Fig. 5 shows this type of observer that we call the adaptive type. As shown in Fig. 5, the neural network output converges with the teaching signal within one trial. The other type learns at every trial period. Fig. 6 shows this type of observer that we call the learning type. As shown in Fig. 6, the neural network output converges with the teaching signal after several trials. As the converging speed is of no importance in this observer, a cost function type identifier can be realized because a cost function needs much calculation. Fig. 7 shows one example of the learning type, which can operate as a self-tuning controller.

In particular, this paper proposes practical design methods for both direct and inverse transfer function identifiers using neural networks and confirms their characteristics.

111. BASIC CONCEPT A neural network design based on a PDF' model is selected with

a view to control systems application [ll]. There have been many studies on the PDP model with regard to such problems as clustering

Position Actual Position

Trajectory Time

'I 'ocess

Fig. 6. Concept of learning type

Fig. 7. Concept of cost function type.

and logical function. However, no studies on neural networks as servolevel controllers have yet been completed. When we attempt to realize a servolevel controller using neural networks, the following points must be considered.

It is necessary to use a neural network that can deal with analog signals because both plant input and output are analog. It has been proved that a neural network can express any nonlinear logical function if the neural network is assumed to have infinite neurons [12]. However, it is difficult for an actual system to satisfy this assumption. Thus, the required size of the neural network has to be determined. A neural network has a nonlinear sigmoid function in order to achieve a nonlinear mapping ability. This makes it very difficult to analyze the neural network. When a neural network estimates an unknown plant, it needs the input and output of the plant for the teaching signal and the neural network input. However, it is impossible to predetermine the spectra of the input and output of an unknown plant. Thus, it is necessary to establish a practical design method, in which learning results do not depend on the input and output spectra of the plant. to point l), signals in the neural network are assumed to be

analog. Regarding point 2), the required size of a neural network is defined as the minimum size in which the weights between the neurons are such that the neural network output matches the teaching signal. With regard to point 3), the dominant part of the object plant is assumed to be linear so that we can design the neural network for a linear system. In this semi-linear approach, we can analyze the main characteristics of the neural network theoretically. The nonlinear effect is studied on the basis of the linear neural network in addition to the nonlinear sigmoid function by use of both simulation and experiment. Regarding point 4), a teaching signal (which is the plant output for the direct transfer function identifier and the plant input for the inverse case) is divided into the parameter of an unknown plant and a signal vector, based on assumption 3). The unknown plant parameter is learned.

206 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANUARYIFEBRUARY 1993

These are the basic assumptions of the practical design methods. Based on assumption 4), the teaching signal T is divided into both the unknown vector a and the signal vector I by the following:

T = aTI . (1)

This signal vector I is the neural network input vector. If we use the assumption that the neural network is linear, we can divide the neural network output Nout into both weight matrix W and neural network input vector I by the following:

Nout = W T I . (2)

Thus, error E between the neural network and the actual plant is given by

E = (T - Nout) = (aT - W T ) I . (3)

As mentioned previously by using the linear assumption, we can prove the existence of the neural network weights, which can be obtained independent of the input signal I, where the error E is zero. A suitable input vector I is discussed in the following section.

IV. CONTROLLER DESIGN METHOD This section describes general design methods for both the di-

rect and inverse transfer function identifiers. The object plant is a predominantly linear SISO plant system. The leaming is performed every sampling time. Since we assume that the dominant part of the plant is linear, the neural network identifiers are designed for a linear system. The basic neural network structure is a three layer, linear PDP type. As to the nonlinear effect, the sigmoid function is added to the designed linear neural network. The transfer function of the object plant is assumed to be expressed as

A(t-')Y(k) = ~ - ~ G o l ? ( z - ~ ) U ( k )

A(2-l) = 1 + * = 1 m . .

B(z- ' ) = 1 + 1 b , C i i=l

(4)

where a, and b, are unknown parameters and d is dead-time. This is analyzed using the following assumptions

1. The upper limit orders of the object plant n, m are known, and 2. The dead-time of the object plant d is known. The plant output Y ( k ) can be expressed by the following.

n

Y(k) = - xa,Y(k - i) +Go z = 1

m

U ( k - d ) + b ,U(k - i - d ) } (5) i = l

A. Direct Transfer Function Identifier

When the plant input is used as the neural network input, and the neural network output YN(~) converges with the plant output, the direct transfer function of the unknown plant is composed in the neural network. In this identifier, the plant output Y (k) becomes the teaching signal. By using the result of (5), both the neural network input vector I (k) and the unknown parameter vector a are defined by the following equations:

I T @ ) = [U(k-d),Y(k- l),...,Y(k-n), U ( k - d - 1) , U ( k - m - d) ] (6)

aT = [Go,al,...,a,,Gobl,...,Gob,] . (7)

The teaching signal Y(k) (the plant output) is divided into both the unknown vector a of the plant and the signal vector I(k) as (8)

Y(k) = aTI (k ) (8)

where I ( k ) and a are w[= n + m + 11 order vectors. On the other hand, when the neural network is linear, the neural network output y,(k) iS

YN(k) w T ( k ) W ( k ) r ( k ) (9)

where the weights between neurons w ( k ) and W(k) are w x 1 order vector and u x w order matrix, respectively. The error ~ ( k ) = Y(k) - YN(k) is revised as

E ( k ) = Y ( k ) - YN(k) = a T I ( k ) - wT(k)W(k)I(k)

= {a* - W T ( k ) W ( k ) } I ( k ) . (10)

Equation (10) indicates that the error ~ ( k ) becomes zero when the relation aT = w T ( k ) W ( k ) is satisfied. This fact implies the existence of a neural network weight matrix that makes the error ~ ( k ) zero. As shown in (lo), the selection of the weight matrix can be performed independent of both the plant input and output when the signal vector defined by (6) is used as the neural network input. The relation aT = w T ( k ) W ( k ) indicates the existence of infinite weight matrixes w ( k ) and W(k), which satisfy this relation. This means that the weight matrixes w ( k ) and W(k) directly correspond with vector a of the unknown parameter. To cause the weights w T ( k ) W ( k ) to converge with a*, the learning is undertaken every sampling time using the S rule as follows [ll].

W ( k + 1) = W ( k ) + qAW(k) w ( k + 1) = w ( k ) + qAw(k)

(11) (12)

AW(k) = -dE/dW(k) (13) Aw(k) = - d E / d w ( k ) (14)

E = E2(k)

where q is the parameter related to the converging speed. Fig. 8 shows a block diagram of the direct transfer function

identifier, which is applied to an n = 2, m = 1, d = 1 plant. Due to the result of (6), the neural network input is composed of both the plant input and output information. The neural network learning is performed with (11H14) using the error E between the plant output and the neural network output. The broken line in Fig. 8 is a feedback loop to stabilihe the system during learning, which also guarantees the finiteness of the neural network input. If the plant output is finite without the feedback loop, this feedback loop is not necessary. Although the previous design method is applied to a linear system, the neural network can be extended into the nonlinear field when the nonlinear sigmoid function is added to this linear neural network framework. A detailed investigation of the nonlinear sigmoid function is presented in Section V.

A nonlinear simulator that uses the result of the direct transfer function identifier can be obtained when the object plant has nonlinear and parasite terms. Fig. 9 shows a block diagram of a nonlinear simulator applied to an n = 2, m = 1,d = 1 plant. This plant simulator uses the result of the trained identifier shown in Fig. 8 without learning. The broken line of Fig. 9 is necessary when the identifier shown in Fig. 8 uses a feedback loop. As shown here, we can obtain a nonlinear neural network simulator that can not be obtained through the use of a conventional method such as adaptive control.

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1 , JANUARYFEBRUARY 1993 207

w Fig. 8. Block diagram of direct transfer function identifier.

1 - - - - - - - - - - - - - -

I I

Fig. 9. Block diagram of simulator.

B. Inverse Transfer Function Identifier

When the plant output I'(k) is used as the neural network input and the neural network output I" ( k ) converges with the plant input, the inverse transfer function of the unknown plant is obtained in the neural network. Thus, we select the plant input 1 7 ( k ) as the teaching signal. We can obtain the following (15) using (2)

U ( k - d ) = ( l / G o ) ~ ( k ) + C a , l ' ( k - i ) - G o C b , I - k - / - d . n 111

2=1 , = 1

(15)

Both the neural network input vector I ( k ) and the unknown parameter vector cy are defined in the same way as the direct transfer function identifier:

I T ( k ) = [ 1 7 ( k ) , I ' ( k - 1 ) : . ' . I ' (k - n ) .

l r ( k - 1 - d ) . . . . l - ( k - m - d ) ] (16) aT = l / G o [ l , a l ;... a , . G o b l . . . . . Gob,,,]. (17)

The teaching signal L ' ( k ) (the plant input) is divided into both unknown vector a and the signal vector I ( k ) . ( [ - ( k - d ) = n ' l ( k ) ) . On the other hand, when the neural network is linear, its output U N ( ~ ) is obtained by

l - , v ( k ) = d J T ( k ) W ( k ) I ( k ) . (18)

The output error ~ ( k ) = U ( k - d ) - l * v ( k ) is revised by the following equation.

&(IC) = U ( k - d - r v ( k ) = crTI(k) - w T ( k ) W ( k ) l ( k )

= {a' - J ( k ) W ( k ) } 1 ( k ) . (19)

When the relation aT = w T ( k ) W ( k ) is satisfied, the error ~ ( k ) becomes zero. This fact implies the existence of a neural network weight matrix that makes the error ~ ( k ) zero. The weight matrix

r----------- 1 I U I

Fig. 10. Block diagram of inverse transfer function identifier.

selection can be performed independent of both the plant input and output when the signal vector defined by (16) is used as the neural network input. In order to make the weights , ( k ) ' W - ( k ) converge with n T , the learning is undertaken every sampling time using the 5 rule following (1 lF(14). Fig. 10 shows a block diagram of the inverse transfer function identifier applied to an 11 = 2, m = 1, d = 1 plant. As seen here, the neural network input is composed of both the plant input and output information. Neural network learning is performed in order to minimize the error c between the plant input C ( k - d ) and the neural network output 1 - x ( k ) . The broken line in Fig. 10 is a feedback loop to stabilize the system. This feedback loop also guarantees the finiteness of the neural network input. If the plant output is finite without the feedback loop, this loop is unnecessary. As mentioned above, we can realize an inverse transfer function identifier.

The most important application of the identified inverse transfer function result is in the direct controller [SI. Only the local con- vergence of the weight matrix is guaranteed for the direct controller [SI. Thus, when the neural network begins to learn through the use of some initial weight matrixes, the error sometimes does not reach zero even after learning because of the local minimums problem. To avoid this problem, it is necessary to learn from initial weight matrixes in which the output error is nearly zero. If the direct controller uses the learning results of the inverse transfer function identifier as the initial weights, it is expected that the convergence of the direct controller will be guaranteed. This is because the initial weights stay within the local convergence condition. Therefore, the convergence of the neural network direct controller is guaranteed and the output error of the direct controller achieves zero.

V. SIMULATION In this section, both direct and inverse transfer function identifiers

are simulated using a second-order plant in order to confirm their re- alization and to discuss their characteristics. The differential equation of the second-order plant for this simulation is as follows:

where n3 and C,,,,,, are the coefficients of the parasite and nonlinear terms, respectively. In this simulation, we choose a1 = -1 .3 . a2 = 0.3 and b = 0.7.

A. Simulation of Direct Transfer Function Identifier

Fig. 11 shows the learning results of the direct transfer function identifier whose block diagram is shown in Fig. 8. In this simulation, the plant is linear (a = 0, C,,, = 0) and the neural network does not use the nonlinear sigmoid function. As shown in Fig. 11, the neural network output Irv(k) converges with the plant output Y ( k ) as learning progresses. This result indicates that the direct transfer function identifier can be realized. Fig. 12 shows a learning result

208 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANUARYFEBRUARY 1993

c, a - Netwbrk output

z * %

P c,

g 2 0 E (A

q=0.3 I I

0 100 200

4- a P

0

0

c, a

* s c,

t U c 0 c, E Q h

Sampling number

Fig. 11. Learning process of direct transfer function identifier.

q=o.1

- Network output Plant output ----

0 100 200 Sampling number

Fig. 12. Learning process of direct transfer function identifier using different initial value.

using different initial weights under the same simulation conditions. As shown in Fig. 12, the output error decreases as learning progresses even if different initial weights are used. However, a relatively large output error remains after the learning has stopped. It is supposed that the neural network enters the local minimum state and leaming is stopped. The converged weights wT W in the Fig. 12 simulation are very different from the plant parameter a. This result means that the conventional control theories such as adaptive control can not use the leamed neural network weights as the identified plant parameters. This is because the leamed weights are different from the true plant parameters if inadequate initial weights are used. Fig. 13 shows the simulation result for a nonlinear plant where a3 = 0.05 and C,,, = 0.1 in (20). Only the hidden layer of the neural network has a nonlinear input-output function as follows.

Parameter X g changes the shape of the nonlinear function whose input-output relationship is shown in Fig. l.4. As shown in Fig. 14, when X g ---* 00, the nonlinear function (21) is equal to the linear function f(z) = z [7], [8]. The learning result in Fig. 13 shows that the nonlinear neural network output converges with the plant output after learning though the plant includes both the nonlinear and parasite terms.

The follawing discussion is concerned with a nonlinear simulator that uses the result of the direct transfer function identifier. Fig. 15

xg=20 g=o.1 - Network output

Plant output ---- I

0 100 200 Sampling number

' Fig. 13. Learning process of direct transfer function identifier applied to nonlinear plant.

10

h

K v w- +. w z o a

-10 -10 0 10

Input x Fig. 14. Input-output relationship of nonlinear function.

U I J 0 100 200

Sampling number Fig. 15. Nonlinear simulator ushg results of direct transfer function

identifier.

shows the relation between the true plant output and the nonlinear simulator output using the identified result from Fig. 13. The feedback loop is used to stabilize the system with h'p = 0.2 in this simulator. Although the nonlinear simulator output exhibits similar behavior to the plant output, a relatively small error between the nonlinear output and the plant output remains. This is because both the nonlinear and parasite terms prevent perfect identification. Fig. 16 shows the relation between the nonlinear simulator output and the true plant output where the weight matrixes are the same as those in Fig. 15 with a different feedback gain K p = 0.4. Both outputs

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. I , JANUARYFEBRUARY 1993 209

+-

P c 3 0

+- E

a

m n

m

m

iij

- U c

b c - z

- Plant output ---- Simulator (Network) output

I x9=20 I

100 200

Sampling number

Fig. 16. Nonlinear simulator using results of direct transfer function identifier.

- Network output - _ - _ Plant input

q=0.75

I I

0 100 2

Sampling number Fig. 17. Process of inverse transfer function of identifier.

10

are different from those in Fig. 15 because of the difference in the feedback gain lip. However, the nonlinear simulator exhibits behavior similar to that of the true plant though the feedback gain is different. This result indicates the usefulness of the proposed nonlinear simulator.

B. Simulation of Inverse Transfer Function Identifier

Fig. 17 shows the learning result of the inverse transfer function identifier where both the plant and the neural network are linear. As shown in Fig. 17, the neural network output converges with the plant input as learning progresses. This result means that the proposed inverse transfer function identifier can be realized. As mentioned in Section IV, we can use the learned weights of the inverse transfer function identifier as the initial weights of the direct controller. Fig. 18 shows a block diagram of the direct controller without learning. In such a case, a proportional control feedback loop is added to compensate for the identification error. Fig. 19 shows the relation between the plant output and the desired value where the feedback gain l<p = 0. As shown in Fig. 19, the error between the plant output and the desired value remains because no direct controller learning is performed. Fig. 20 shows

I I

Neural

- 1

Fig. 18. Direct controller using results of inverse transfer function identifier.

- Plant output K,=O ---- Desired value

0 200 400

Sampling number Fig. 19. Simulation result of direct controller.

+- a P c. 3 0

+- r m n

U I I 0 200 400

Sampling number Fig. 20. Simulation result of direct controller.

the telation between the plant output and the desired value where the feedback gain l ip = 0.3. The error in Fig. 20 is larger than that in Fig. 19. Fig. 21 shows the relation between the feedback loop output and the neural network output under the same conditions as those of Fig. 20. Since the sign of the feedback loop output is the same as that of the neural network output, the feedback loop increases the error. This result shows that there is a possibility of increasing the output error by adding the direct controller with the feedback loop in order to compensate for the identification error. This phenomenon is caused by using past plant output as the neural network input.

210 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 23, NO. 1, JANUARY/FEBRUARY 1993

0 25 50 Sampling number

Fig. 21. Feedback loop output and neural network output.

3 Amp.

I

Fig. 22. Experimental setup.

Sensor

VI. EXPERIMENT The realization of identifiers in an actual system is confirmed

experimentally for a one freedom force control system. Fig. 22 shows the experimental setup. In this control system, the contact force can not be accurately controlled using the motor current because of the friction of the harmonic drive. Thus, it is necessary to estimate the stiffness of the contact object using adaptive contlol or leaming control. We can approximately express the transfer function G(z) of the one freedom force control system as follows.

where T, is the sampling period, h-,,, is the gain of the system, T, is the time constant of the system and Ke is the stiffness of the contact object. Fig. 23 shows the experimental result for the direct transfer function identifier. As shown in Fig. 23, the neural network output converges with the actual plant output. This result confirms that the proposed direct transfer function identifier is very useful in an actual system. Fig. 24 shows the effect of the lionlinear function of (21). The vertical axis of Fig. 24 is the root mean squared error of 500 sampling errors after learning. As shown in Fig. 24, this is minimum at about Xg = 7. Since the actual control system has a nonlinear effect, the result in Fig. 24 indicates the existence of an optimum

xg=7 tl =0.03

El 1.0 Y v

LL E 0.51 'b- - Plant output

--- Neural network output 0 1000 1100

Sampling nuniber Fig. 23. Expenmentd results.

x 10-3

b

1.5894

1.5892

xg Fig. 24. Effect of nonlinear function.

xg=7

El 1.0 v Y - True plant output

Q) ---- Simulator output LL 2 o 0.51 bp,

0 I 1000 1100

Sampling number Fig. 25. Nonlinear simulator output.

result for the nonlinear simulator using the results from Fig. 24. As shown in Fig. 25, the simulator behaves similarly to an actual plant. This result confirms that the proposed nonlinear simulator is very useful in an actual system.

VII. DISCUSSION As shown in Section V, the direct transfer function identifier has

a local minimum because the 5 rule is used as the leaming laws. This local minimum causes the weight matrix u T w to converge

nonlinear sigmoid function shape. Fig. 25 shows the experimental with a value different form that of the unknown parameter vector

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CY. Therefore, conventional control such as adaptive control can not use the identified results by obtained neural networks as the estimated true plant parameters. This characteristic is a weakness of the identifier using neural networks. This is because adaptive control can identify the true plant parameters if the plant is linear. However, the neural network direct transfer function identifier has other good characteristics. That is, its nonlinear mapping capability can realize a nonlinear simulator that cannot be realized through the use of conventional control theories such as adaptive control.

Although the simulation results also confirm that the inverse transfer function identifier can be realized, it has a local minimum problem because it uses the 5 learning rule. Thus, there is the possibility that output error remains after learning. As shown in the simulation result, the feedback loop in combination with the neural network direct controller increases the output error caused by this remaining identification error. This phenomenon is caused by a phase lag effect between the neural network and feedback loop outputs because of dead-time in the plant. It is necessary to obtain a perfect plant model in order to eliminate this phase lag. However, this is impossible in practice. Therefore, for practical purposes, the learning of the neural network direct controller is useful. That is, the direct controller uses the learned weights of the inverse transfer identifier in order to avoid the initial dangerous condition, and our proposed direct controller learning method reduces the remaining error caused by the identification error [8].

VIII. CONCLUSION This paper proposed practical design methods for the identification

of both the direct and inverse transfer functions of a dynamic system through the use of a neural network. Simulation and experimental results showed that these identifications can be successfully realized. As a practical application, the identification result of the direct transfer function was applied to a nonlinear simulator and that of the inverse transfer function was applied to a direct controller through the matrix information.

ACKNOWLEDGMENT

The authors wish to express their thanks to Dr. Shimada and Dr. Ishihara, N T I Transmission Systems Labs., for their continued encouragement.

REFERENCES

D. Psaltis, A. Siders and A. Yamamura, “Neural controller,” in Proc. I987 IEEE Int. Con$ Neural Networks, vol. IV, San Diego, 1987, pp. 551-558. A. Guez, J. Eilbert and M. Kam, “Neuromophic architecture adaptive control: A preliminary analysis,” in Proc. 1987 IEEE Int. Con/: Neural Networks, vol. IV, San Diego, 1987, pp. 567-572. C. A. Niznik, W. Hoss and C. Watts, “Modeling robot cerebellar neural network pathway information measures,” in Proc. 1987 IEEE Int. Con$ Neural Networks, vol. IV, San Diego, 1987, pp. 583-591. M. Kawato, Y. Uno, M. Isobe and R. Suzuki, “A hierarchical model for voluntary movement and its application to robotics,” Proc. 1987 IEEE Int. Conf: Network, vol. IV, 1987, pp. 573-582. T. Tokita, T. Fukuda, T. Mitsuoka, T. Kurihara, “Force control of robot manipulator by neural network (control of one degree-of-freedom manipulator),” J. Robotics SOC. Japan, vol. 7, no. 1, pp. 47-51, 1989. T. Yabuta, T. Tsujimura, T. Yamada and T. Yasuno, “On the character- istics of the manipulator controller using neural network,” in Proc. Int. Workshop on Machine Intell. and Esion, Apr. 1989, pp. 76-81.

T. Yamada and T. Yabuta, “Direct controller using neural network,” Trans. Inst. Electron., Inform. Commun. Eng. (to be submitted, in Japanese). __ , “Some remarks on characteristics of direct neuro-controller with regard to adaptive control,” Trans. SOC. Instrument Contr. Eng., vol. 27, no. 7, pp. 784-791, 1991 (in Japanese). M. I . Jordan, “Generic constraints on underspecified target trajectories,” in Proc. IJCNN89, Washington, DC, June 1989. T. Yabuta and T. Yamada, “Possibility of neural network controller for robot manipulators,” in Proc. Int. Conf: Robotics Automat., Cincinnati, OH, 1990, pp. 1686-1691. Rumelhart, McClelland and PDP Research Group, Parallel Distributed Processing. K. Funahashi, “On the approximate realization of identity mappings by three-layer neural networks,” The Inst. Electron., Inform. Commum. Eng., vol. J73-A, no. 1, pp. 139-145, 1990.

Cambridge, MA: MIT Press, 1988.

Optimization of Detection Networks: Part 11-Tree Structures

Z.-B. Tang, K. R. Pattipati, and D. L. Kleinman

Abstract-A distributed binary detection problem with multimessage ( 2 1 bit) communications is considered, wherein the nodes (sensors, decision makers (DMs)) of the system are organized in the form of a tree with multiple root nodes. A numerical algorithm is developed for determining the optimal decision rule at each node assuming monotone cost functions imposed only on the root nodes. It is assumed that the observations of each node are conditionally independent of those of the other nodes. It is shown that the problem is equivalent to solving a nonlinear optimal control problem, and the necessary conditions of optimality using Bayes’ risk as the optimization criterion are derived. The optimal control approach provides an interpretation of certain functions of the co-state variables in terms of thresholds, and leads to a computationally efficient min-H algorithm to solve for the optimal decision rule at each node. The numerical algorithm provides a tool to investigate the organizational issues of adaptation, structure, and robustness.

I. INTRODUCTION Information-processing in most large scale systems is effected

by a network of geographically or functionally distributed decision makers (DMs) with limited communications. These problems arise naturally in a wide variety of contexts, including weather forecast- ing, military command and control (C2) , air-traffic control, electric power networks, surveillance systems, oil prospecting, emergency management, product distribution and supply, and medical diagnosis. Each DM (variously referred to as a node, a sensor, a commander, etc. depending on the problem context) has access to only a portion of the information available to the network of DMs. Moreover, even the total information set is generally incomplete and is characterized by a high level of uncertainty due to lax updating, missed detection of events,

Manuscript received March 30, 1991; revised April 10, 1992. This work was supported in part by NSF Grant #IRI-8902755 and in part by ONR Contract #N00014-90-3-1753.

2.-B. Tang was with the Department of Electrical and Systems Engineering, University of Connecticut, Storrs, CT 06269-3157 and is now with the Division of Applied Sciences, Harvard University, Cambridge, MA 02138.

K. R. Pattipati and D. L. Kleinman are with the Department of Electrical and Systems Engineering, University of Connecticut, Storrs, CT 06269-3157.

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