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SICE Annual Conference 2007 Sept. 17-20, 2007, Kagawa University, Japan

Simple Adaptive Control for SISO Nonlinear Systems Using Multiple Neural Networks Muhammad Yasser, Agus Trisanto, Ayman Haggag, Takashi Yahagi, Hiroo Sekiya, and Jianming Lu Graduate School of Science and Technology Chiba University Chiba-shi, 263-8522 Japan Email: [email protected] Abstract—This paper presents a method of continuous-time simple adaptive control (SAC) using multiple neural networks for a single-input single-output (SISO) nonlinear systems with unknown parameters and dynamics, bounded-input boundedoutput, and bounded nonlinearities. The control input is given by the sum of the output of the simple adaptive controller and the sum of the outputs of the parallel small-scale neural networks. The parallel small-scale neural networks are used to compensate the nonlinearity of plant dynamics that is not taken into consideration in the usual SAC. The role of the parallel smallscale neural networks is to construct a linearized model by minimizing the output error caused by nonlinearities in the control systems. Finally, the stability analysis of the proposed method is carried out, and the effectiveness of this method is confirmed through computer simulations.

I.

INTRODUCTION

Adaptive control methods were developed as an attempt to overcome difficulties connected with the ignorance of systems structure and critical parameter values as well as changing control regimes [1]－[3]. Most self-tuning and adaptive control algorithms usually use reference models, controllers, or identifiers of almost the same order as the controlled plant. Since the dimension of the plants in the real world may be very large or unknown, implementation of adaptive control procedures may be very difficult or impossible. To overcome this problem, simple adaptive control (SAC) method was developed by Sobel et al. [4] as an attempt to simplify the adaptive controllers, since no observers or identifiers are needed in the feedback loop [5]. Furthermore, the reference model is allowed to be of very low order compared with the controlled plant. For linear plants with unknown structures, SAC is an important class of adaptive control scheme [5], [6]. However, for nonlinear plants with unknown structures, it may not be possible to ensure perfect plant output that follows the output of a reference model by using SAC [7]. For nonlinear plants, many methods for the control using neural network are proposed. It has been proved that these control methods show excellent performance for nonlinearity [8], [9], [10]. The combination of SAC and neural network for a single-input single-output (SISO) nonlinear plant has been proposed and proven to give a perfect result [7], [11].

The method of the combination of SAC and neural network for nonlinear continuous-time systems and its stability proof has been proposed in [11]. The control input is given by the sum of the output of a simple adaptive controller and the output of neural network. The role of neural network is to compensate for constructing a linearized model so as to minimize the output error caused by nonlinearities in the control system by using backpropagation learning algorithm. The role of simple adaptive controller is to perform the model matching for the linear system with unknown structures to a given linear reference model. In other control methods using neural network, when the size of neural network increased, the calculation time for each of iteration of the learning process will also increase. When the size of neural network is very large then its calculation process will be very time consuming. This will cause the control system to require large sampling time. If it is applied to an actual plant, it is necessary to drastically reduce the calculation time of the learning process of neural network. Thus the sampling-time of the controller can also be reduced to control the plant well. The proposed method of this paper is an improvement of the methods in [11] and [12]. In this paper, we propose a design method using backpropagation learning algorithm for multiple neural networks, which consists of several small-scale neural networks with identical structures connected in parallel, in order to design SAC required for real-time processing. This proposed method is designed for a class of SISO nonlinear plants with unknown parameters and dynamics, bounded-input bounded-output (BIBO), and bounded nonlinearities. The number of time required per iteration for the calculation in the learning process of neural network to update its weights can be decreased by using the method of multiple neural networks [12]. Moreover, by parallel training of several small-scale neural networks, the learning efficiency is improved. Finally, the stability analysis of the proposed method is carried out, and the effectiveness of this method is confirmed through computer simulations. II. SIMPLE ADAPTIVE CONTROL In this section, we briefly describe SAC for a linear SISO plant. Simple adaptive controller is designed to realize a plant

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output which converges to reference model output. Let us consider the following controllable and observable but with unknown parameter SISO linear plant of order n p

x p (t ) = Ap x p (t ) + B p u (t ) y p (t ) = C p x p (t )

(1) (2)

where x p (t ) is an n p th-order plant state vector, u (t ) is the

The control objective is to achieve the following relation lim e y (t ) = 0 . (9) t →∞

Since the plant is unknown, the actual control of the plant will be generated by the following adaptive algorithm using the values that can be measured, which are ey (t ) , xm (t ) and um (t ) , to get the low-order adaptive controller u p (t ) = K (t )r (t ) = K e (t )ey (t ) + K x (t )x m (t ) + K u (t )um (t ) (10)

control input, y p (t ) is the plant output, Ap , B p , and C p are

matrices with appropriate dimensions. It is necessary for us in the realization of linear SAC to control the plant in (1), (2) to make the following assumption. Assumption 1 (a) Plant (1), (2) is ASPR (almost strictly positive real). That is, there exists a constant gain ke* such that the transfer function G p ( s ) = C p ( sI − Ac ) −1 B p is SPR (strictly positive real), where G p ( s) is the plant transfer function, and Ac = Ap + ke* B p C p .

⎡ Ap B p ⎤ det ⎢ ⎥≠0 . ⎣C p 0 ⎦ Furthermore, let us consider that the plant (1), (2) is required to follow the input-output behavior of a reference model of the form x m (t ) = Am x m (t ) + Bm um (t ) (3)

(b)

ym (t ) = Cm x m (t )

(4)

where x m (t ) is an nm th-order reference model state vector,

um (t ) is the reference model input, ym (t ) is the reference model output, Am , Cm are matrices with appropriate dimensions, and Bm is a scalar value. The reference model can be independent of the controlled plant, and it is permissible to assume nm n p . It is necessary to add the supplementary values of the augmented plant which are defined as (5) ya (t ) = y p (t ) + ys (t )

where

K (t ) = [ K e (t ) K x (t ) K u (t ) ]

(11)

rT (t ) = ⎡⎣ey (t ) xTm (t ) um (t ) ⎤⎦

(12)

and the adaptive gains are obtained as a combination of 'proportional' and 'integral' terms as follows K (t ) = K p (t ) + K i (t ) (13) K p (t ) = ⎡⎢ e y (t )ey (t )Tp e y ⎣ = e y (t )r T (t )Tp

ey (t ) xTm (t )Tp x

m

e y (t )um (t )Tp u ⎤⎥ m ⎦ (14)

K i (t ) = ey (t )rT (t )Ti − σ K i (t )

(T

p

(15)

= T > 0, Ti = Ti > 0 ) . T p

T

Then we apply SAC control input in (10) to control the SISO linear plant (1), (2), so that the control input u (t ) for the plant will be u (t ) = u p (t ) . (16) III. SIMPLE ADAPTIVE CONTROL USING MULTIPLE NEURAL NETWORK FOR NONLINEAR SYSTEM When the input-output characteristic of the controlled plant is nonlinear, it is not possible to express as in (1) and (2). First let us consider an SISO nonlinear plant with BIBO to be expressed by a system that consists of a linear part and a nonlinear part as (17) x p (t ) = Ap x p (t ) + B p u (t ) + f x ( x p (t ), u (t ))

y p (t ) = C p x p (t ) + f y (x p (t ))

(18)

ys ( s ) = D p ( s )u ( s)

(6)

where x p (t ) is an n p th-order plant state vector, u (t ) is the

ey (t ) = ym (t ) − ya (t )

(7)

control input, y p (t ) is the plant output, Ap , B p , and C p are

where D p ( s) is a simple parallel feedforward compensator

matrices with appropriate dimensions, f x (⋅) is a nonlinear

(PFC)

function vector ∈ R p , and f y (⋅) is a nonlinear function. We

n

Dp (8) 1+ ρs across the controlled plant to fulfill the condition in assumption 1(a) to guarantee its robust stability [5], [6], [13], where ρ is a positive constant. The augmented plant we use here must satisfy the following conditions: (i) plant (13) is ASPR, (ii) ya (t ) = y p (t ) + ys (t ) ≅ y p (t ) , which can be fulfilled by setting Dp ( s) =

the value of D p to be very small [5], [6], and (iii) D p ( s) is physically realizable.

further assume that the system (17), (18) is controllable and observable. Then, it is necessary to make the following assumption [11]. Assumption 2 (a) The linear part and the nonlinear part of the plant in (17), (18) are unknown. (b) For the plant in (17), (18), there exists an augmented plant where its linear part satisfies assumption 1(a). This augmented plant, as in (21), is formed by incorporating the plant in (17), (18) with the supplementary values in (5)—(8)

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[15]. (c) The nonlinear part of the plant in (17), (18), which is represented by f x (⋅) and f y (⋅) , is bounded. However, in this case, when the input in (16) is used to control the nonlinear system in (17) and (18), the problem of output error will arise [7], [11]. To keep the plant output y p (t ) converge to the reference model output ym (t ) , the control input can be expressed as u (t ) = h( ymT (t ), yaT (t ), yTp (t ), xTp (t ))

(19)

according to (7), (17), and (18), where h(.) is an unknown nonlinear function vector. In this paper, we synthesize the control input u (t ) by the following equation (20) u (t ) = u p (t ) + u p (t ) where u p (t ) is the output of SAC, as mentioned in (10). And u p (t ) is the total control input of neural network given as u p (t ) = α uˆ p (t ) = α f ZOH ( uˆ p (k ) ) nv

uˆ p ( k ) = ∑ uˆ p v ( k )

(21) (22)

v =1

where α is a positive constant, uˆ p (t ) is the total continuous-

For the v -th parallel small-scale neural network ( v = 1, 2,..., nv ), let xiv (k ) be the input to the i -th neuron in the input layer ( i = 1, 2,..., ni ), hq v ( k ) be the input to the q -th neuron in the hidden layer ( q = 1, 2,..., nq ), ov (k ) be the input to the neuron in the output layer. Furthermore, let miq v ( k ) be the weight between the input layer and the hidden layer, mqj v ( k ) be the weight between the hidden layer and the output layer. The control input is given by the sum of the output of simple adaptive controller and the output of neural network. The neural network is used to compensate the nonlinearity of the plant dynamics that is not taken into consideration in the usual SAC. The role of the neural network is to construct a linearized model by minimizing the output error caused by nonlinearities in the control systems. Refering to (23), the input iv (k ) of the neural network is given as (26) iv (k ) = ⎡⎣ ym ( k − 1), y p (k − 1),..., y p ( k − n) ⎤⎦ . Therefore, the nonlinear function of the system can be approximated by neural network. Furthermore, values n should be chosen appropriately according to practical nonlinear systems.

time output of neural network, uˆ p (k ) is the total discrete-time output of neural network, f ZOH ( ⋅) is a zero-order hold function [11], uˆ p v (k ) is the output of the v -th small-scale neural network, and nv is the number of the parallel small-scale neural networks. As in [11], a sampler is implemented in front of the neural network with appropriate sampling period to obtain discretetime multi-input of the neural network, and a zero-order hold is implemented to change the discrete-time output u p ( k ) of the neural network back to continuous-time output u p (t ) as shown in.(21). Consequently, we can assume the discrete-time output uˆ p ( k ) as follows

uˆ p (k ) = hˆ ( ym (k − 1), y p ( k − 1),..., y p (k − n) )

(23) ˆ where h ( ⋅) is an unknown nonlinear function and is the number of the past data of output of the plant. Using the above approach, the parallel small-scale neural networks will be trained until the output error e(t ) given as (24) e(t ) = ym (t ) − y p (t ) satisfies the following relation lim e(t ) = lim ym (t ) − y p (t ) ≤ ε t →∞

t →∞

(25)

where ε is a small positive value.

V. LEARNING OF NEURAL NETWORK The dynamics of the v -th parallel small-scale neural network are given as hq v (k ) = ∑ xi v (k )miq v (k ) (27) i

ov (k ) = ∑ S1 (hq v (k ))mqj v (k )

(28)

uˆ p v (k ) = S 2 ( ov ( k ) )

(29)

q

where S1 (.) is a tangent sigmoid function, S 2 (.) is a pure linear function. The tangent sigmoid function is chosen as 2 −1 (30) S1 ( X ) = 1 + exp(− µ X ) where µ > 0 , and the pure linear function is chosen as S2 ( X ) = X .

(31)

Consider the case when S1 ( X ) = a . Then the derivative of the tangent sigmoid function S1 (.) and the pure linear function S 2 (.) are as follows S1′( X ) =

µ

(32) (1 − a 2 ) 2 S 2′ ( X ) = 1 . (33) The objective of training is to minimize the error function E (k ) by taking the error gradient with respect to the parameters or weight vector mv ( k ) , that is to be adapted. The error function is defined as

IV. COMPOSITION OF NEURAL NETWORK Each of the parallel small-scale neural networks consists of three layers: an input layer, an output layer and a hidden layer.

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2 1 ⎡⎣ ym (k ) − y p ( k ) ⎤⎦ . 2 The weights are then adapted by using ∂E ( k ) ∆mqj v (k ) = −c ⋅ ∂mqj v ( k )

E (k ) =

∆miq v ( k ) = −c ⋅

C = ⎡⎣C p

(34)

D p ⎤⎦

(47)

and δˆ i (x(t ), u (t ), u p (t )) and δˆ o (x(t )) represent the nonlinear (35)

∂E (k ) ∂miq v (k )

part of the augmented plant described as follows ⎡f ( x (t ), u (t )) ⎤ δˆ i (x(t ), u p (t ), uc (t )) = ⎢ x p ⎥ + Bu p (t ) 0 ⎣ ⎦

(36)

(48)

= δi (x(t ), u (t )) + Bu p (t )

δˆo (x(t )) = f y (x p (t )) . (49) where c > 0 is the learning parameter. For the learning process, To prove the stability of our method, we start by defining its (35) and (36) will be expanded as follows Lyapunov function as follows ∂E (k ) ∂y p ( k ) ∂S2 ( ov (k ) ) ∂ov (k ) ∆mqj v (k ) = −c ⋅ ⋅ ⋅ ⋅ VSACNN (t ) = VSAC (t ) + VNN (t ) (50) ∂y p (k ) ∂uˆ p v (k ) ov (k ) ∂mqj v (k ) (37) where VSAC (t ) is the Lyapunov function of SAC of our method = c ⋅ [ ym (k ) − y p (k )] ⋅ J plant v ⋅ S2′ ( ov (k ) ) ⋅ S1 (hq v ( k )) which is a modification from the Lyapunov function of SAC ∂uˆ p v (k ) ∂S 2 ( ov (k ) ) ∂E (k ) ∂y p ( k ) presented in [11], [14], and VNN (t ) is the total of the Lyapunov ∆miq v ( k ) = −c ⋅ ⋅ ⋅ ⋅ ∂y p (k ) ∂uˆ p v ( k ) ∂S 2 ( ov (k ) ) ov (k ) function of NN of our method. The Lyapunov function of SAC of our method is described ∂ov (k ) ∂S1 (hq (k )) ∂hq (k ) ⋅ ⋅ ⋅ (38) as follows ∂S1 ( hq (k )) ∂hq (k ) ∂miq (k ) T VSAC (t ) = eTx (t ) Pe x (t ) + tr ⎡⎣ K i (t ) − K ⎤⎦ Ti −1 ⎡⎣ K i (t ) − K ⎤⎦ (51) = c ⋅ [ ym (k ) − y p ( k )] ⋅ J plant ⋅ S2′ (ov (k )) ⋅ mqj ( k ) v

{

v

}

where P is a real symmetric positive definite matrix, tr ( ⋅) is a

S1′(hq (k )) ⋅ iv (k )

trace function, and e x (t ) is a state error vector given as

where ⎛ ∂y p ( k ) ⎞ J plant v = SGN ⎜ (39) ⎟ ⎜ ∂uˆ p (k ) ⎟ v ⎝ ⎠ which is derived from the one mentioned in reference [9], [10]. VI. CONVERGENCE AND STABILITY For the stability analysis of our method, we use and modify the stability analysis presented in [11] and [14]. As mentioned in assumption 2(b), the PFC in (6), (8) is incorporated with the nonlinear system in (1), (2) to form the augmented plant, as in (5), where its linear part is ASPR. For convenience, first, it is necessary for the PFC in (6), (8) to be expressed in a statespace form as follows x s (t ) = As x s (t ) + Bs u (t ) (40)

( n −1) e x (t ) = xˆ (t ) − x(t ) = ⎡⎣ ex (t ), ex (t )," , ex p (t ) ⎤⎦

where the Lyapunov function of SAC of our method VSAC (t ) in (51) is developed by replacing the disturbances used in the Lyapunov function of SAC in [14] with the nonlinear part represented by δˆ i (x(t ), u (t ), u p (t )) and δˆo ( x(t )) , then its derivation, V (t ) , can be described as follows

(43)

where

SAC

⎡B ⎤ B = ⎢ p⎥ ⎣ Bs ⎦

(44)

(45) (46)

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{

T VSAC (t ) = −eTx (t )Qe x (t ) − 2σ tr ⎡⎣ K i (t ) − K ⎤⎦ Ti −1 ⎡⎣ K i (t ) − K ⎤⎦

−2e (t )e y (t )e (t )Tp e ey (t ) T y

T y

y

−2e (t )e y (t ) ⎡ xmT (t )Tp x xm (t ) + umT (t )Tp u um (t ) ⎤ m m ⎣ ⎦ T T − 1 −2σ tr ⎡⎣ ⎡⎣ K i (t ) − K ⎤⎦ Ti K ⎤⎦ − 2ex (t ) P F(t ) −2δˆ (x(t )) K (t ) − K r (t ) T y

o

⎡ x (t ) ⎤ x (t ) = ⎢ p ⎥ ⎣ x s (t ) ⎦ 0⎤ ⎡A A=⎢ p ⎥ 0 A s⎦ ⎣

(52)

where xˆ (t ) is the ideal target state vector of the system, and K = ⎡⎣ K e K x K u ⎤⎦ is the unknown ideal gain of SAC. The derivation of Lyapunov function in (50) will be VSACNN (t ) = VSAC (t ) + VNN (t ) . (53)

ys (t ) = Cs x s (t ) (41) then, by applying (40), (41) to (6), (17), and (18), the augmented plant can be described as follows x (t ) = Ax(t ) + Bueq (t ) + δˆ i (x(t ), u (t ), u p (t )) (42) ya (t ) = Cx(t ) + δˆo ( x(t ))

T

(

i

)

−2δˆo (x(t ))ey (t )eTy (t )Tp e ey (t ) y

−2δˆo (x(t ))ey (t ) ⎡ xmT (t )Tp x xm (t ) + umT (t )Tp u um (t ) ⎤ m m ⎣ ⎦ −1 T T −2σ tr ⎡⎣ ⎡⎣ K i (t ) − K ⎤⎦ Ti K ⎤⎦ − 2ex (t ) P F(t ) −2δˆ (x(t )) K (t ) − K r (t ) o

(

i

)

}

−2δˆo (x(t ))ey (t )eTy (t )Tp e ey (t )

nlower and an upper limit at nupper , and the parameters

y

−2δˆo (x(t ))ey (t ) ⎡ xmT (t )Tp x xm (t ) + umT (t )Tp u um (t ) ⎤ m m ⎣ ⎦ where Q is a real matrix, and F(t ) is given as F(t ) = E (t ) − BK δˆ (x(t )) + δˆ (x(t ), u (t ), u (t )) Bias

e o

i

p

(54)

in

(8),

ρ =1

Tp = diag (5 ×10 ,5 ×10 ,5 ×10 ) 3

(55)

where E Bias (t ) is a bias term as explained in [16]. For V (t ) , we assume that it can be approximated as NN

nv

D p = 0.001

VNN (t ) = ∑ VNN v (t )

(56)

VNN v (t ) ≅ ∆VNN v (k ) / ∆T

(56)

v =1

where ∆VNN v (k ) is the derivation of a discrete-time Lyapunov function, and ∆T is a sampling time. According to [11], ∆VNN v (k ) can be guaranteed to be negative definite if the

3

3

in

in

(8),

(14),

Ti = diag (5 ×104 ,5 ×104 ,5 ×104 ) in (15), σ = 1 in (15), µ = 2 in

(30), c = 0.001 in (35)－(38), and ∆T = 0.01 in (56) are fixed. Furthermore, we assume a first-order reference model (3), (4) with parameters Am = −10 , Bm = 10 , and Cm = 1 . Fig. 1 shows the output of the reference model ym (t ) and the plant output y p (t ) using only SAC. The result in Fig. 1 shows that the error between y p (t ) and ym (t ) is large. Fig. 2 shows the output of reference model ym (t ) and the plant output y p (t ) using our proposed method of SAC using

(57)

multiple neural networks with nv = 1 . It can be seen that the error of the system has been reduced, and the plant output y p (t ) can follow closer the output of the reference model ym (t )

for the weights between the hidden layer and the output layer, mqj v ( k ) , and

compared to using only SAC. Fig. 3 shows the output of reference model ym (t ) and the

learning parameter c fulfills the following conditions 0 < c < 2 / nq

0

Lihat lebih banyak...
Simple Adaptive Control for SISO Nonlinear Systems Using Multiple Neural Networks Muhammad Yasser, Agus Trisanto, Ayman Haggag, Takashi Yahagi, Hiroo Sekiya, and Jianming Lu Graduate School of Science and Technology Chiba University Chiba-shi, 263-8522 Japan Email: [email protected] Abstract—This paper presents a method of continuous-time simple adaptive control (SAC) using multiple neural networks for a single-input single-output (SISO) nonlinear systems with unknown parameters and dynamics, bounded-input boundedoutput, and bounded nonlinearities. The control input is given by the sum of the output of the simple adaptive controller and the sum of the outputs of the parallel small-scale neural networks. The parallel small-scale neural networks are used to compensate the nonlinearity of plant dynamics that is not taken into consideration in the usual SAC. The role of the parallel smallscale neural networks is to construct a linearized model by minimizing the output error caused by nonlinearities in the control systems. Finally, the stability analysis of the proposed method is carried out, and the effectiveness of this method is confirmed through computer simulations.

I.

INTRODUCTION

Adaptive control methods were developed as an attempt to overcome difficulties connected with the ignorance of systems structure and critical parameter values as well as changing control regimes [1]－[3]. Most self-tuning and adaptive control algorithms usually use reference models, controllers, or identifiers of almost the same order as the controlled plant. Since the dimension of the plants in the real world may be very large or unknown, implementation of adaptive control procedures may be very difficult or impossible. To overcome this problem, simple adaptive control (SAC) method was developed by Sobel et al. [4] as an attempt to simplify the adaptive controllers, since no observers or identifiers are needed in the feedback loop [5]. Furthermore, the reference model is allowed to be of very low order compared with the controlled plant. For linear plants with unknown structures, SAC is an important class of adaptive control scheme [5], [6]. However, for nonlinear plants with unknown structures, it may not be possible to ensure perfect plant output that follows the output of a reference model by using SAC [7]. For nonlinear plants, many methods for the control using neural network are proposed. It has been proved that these control methods show excellent performance for nonlinearity [8], [9], [10]. The combination of SAC and neural network for a single-input single-output (SISO) nonlinear plant has been proposed and proven to give a perfect result [7], [11].

The method of the combination of SAC and neural network for nonlinear continuous-time systems and its stability proof has been proposed in [11]. The control input is given by the sum of the output of a simple adaptive controller and the output of neural network. The role of neural network is to compensate for constructing a linearized model so as to minimize the output error caused by nonlinearities in the control system by using backpropagation learning algorithm. The role of simple adaptive controller is to perform the model matching for the linear system with unknown structures to a given linear reference model. In other control methods using neural network, when the size of neural network increased, the calculation time for each of iteration of the learning process will also increase. When the size of neural network is very large then its calculation process will be very time consuming. This will cause the control system to require large sampling time. If it is applied to an actual plant, it is necessary to drastically reduce the calculation time of the learning process of neural network. Thus the sampling-time of the controller can also be reduced to control the plant well. The proposed method of this paper is an improvement of the methods in [11] and [12]. In this paper, we propose a design method using backpropagation learning algorithm for multiple neural networks, which consists of several small-scale neural networks with identical structures connected in parallel, in order to design SAC required for real-time processing. This proposed method is designed for a class of SISO nonlinear plants with unknown parameters and dynamics, bounded-input bounded-output (BIBO), and bounded nonlinearities. The number of time required per iteration for the calculation in the learning process of neural network to update its weights can be decreased by using the method of multiple neural networks [12]. Moreover, by parallel training of several small-scale neural networks, the learning efficiency is improved. Finally, the stability analysis of the proposed method is carried out, and the effectiveness of this method is confirmed through computer simulations. II. SIMPLE ADAPTIVE CONTROL In this section, we briefly describe SAC for a linear SISO plant. Simple adaptive controller is designed to realize a plant

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output which converges to reference model output. Let us consider the following controllable and observable but with unknown parameter SISO linear plant of order n p

x p (t ) = Ap x p (t ) + B p u (t ) y p (t ) = C p x p (t )

(1) (2)

where x p (t ) is an n p th-order plant state vector, u (t ) is the

The control objective is to achieve the following relation lim e y (t ) = 0 . (9) t →∞

Since the plant is unknown, the actual control of the plant will be generated by the following adaptive algorithm using the values that can be measured, which are ey (t ) , xm (t ) and um (t ) , to get the low-order adaptive controller u p (t ) = K (t )r (t ) = K e (t )ey (t ) + K x (t )x m (t ) + K u (t )um (t ) (10)

control input, y p (t ) is the plant output, Ap , B p , and C p are

matrices with appropriate dimensions. It is necessary for us in the realization of linear SAC to control the plant in (1), (2) to make the following assumption. Assumption 1 (a) Plant (1), (2) is ASPR (almost strictly positive real). That is, there exists a constant gain ke* such that the transfer function G p ( s ) = C p ( sI − Ac ) −1 B p is SPR (strictly positive real), where G p ( s) is the plant transfer function, and Ac = Ap + ke* B p C p .

⎡ Ap B p ⎤ det ⎢ ⎥≠0 . ⎣C p 0 ⎦ Furthermore, let us consider that the plant (1), (2) is required to follow the input-output behavior of a reference model of the form x m (t ) = Am x m (t ) + Bm um (t ) (3)

(b)

ym (t ) = Cm x m (t )

(4)

where x m (t ) is an nm th-order reference model state vector,

um (t ) is the reference model input, ym (t ) is the reference model output, Am , Cm are matrices with appropriate dimensions, and Bm is a scalar value. The reference model can be independent of the controlled plant, and it is permissible to assume nm n p . It is necessary to add the supplementary values of the augmented plant which are defined as (5) ya (t ) = y p (t ) + ys (t )

where

K (t ) = [ K e (t ) K x (t ) K u (t ) ]

(11)

rT (t ) = ⎡⎣ey (t ) xTm (t ) um (t ) ⎤⎦

(12)

and the adaptive gains are obtained as a combination of 'proportional' and 'integral' terms as follows K (t ) = K p (t ) + K i (t ) (13) K p (t ) = ⎡⎢ e y (t )ey (t )Tp e y ⎣ = e y (t )r T (t )Tp

ey (t ) xTm (t )Tp x

m

e y (t )um (t )Tp u ⎤⎥ m ⎦ (14)

K i (t ) = ey (t )rT (t )Ti − σ K i (t )

(T

p

(15)

= T > 0, Ti = Ti > 0 ) . T p

T

Then we apply SAC control input in (10) to control the SISO linear plant (1), (2), so that the control input u (t ) for the plant will be u (t ) = u p (t ) . (16) III. SIMPLE ADAPTIVE CONTROL USING MULTIPLE NEURAL NETWORK FOR NONLINEAR SYSTEM When the input-output characteristic of the controlled plant is nonlinear, it is not possible to express as in (1) and (2). First let us consider an SISO nonlinear plant with BIBO to be expressed by a system that consists of a linear part and a nonlinear part as (17) x p (t ) = Ap x p (t ) + B p u (t ) + f x ( x p (t ), u (t ))

y p (t ) = C p x p (t ) + f y (x p (t ))

(18)

ys ( s ) = D p ( s )u ( s)

(6)

where x p (t ) is an n p th-order plant state vector, u (t ) is the

ey (t ) = ym (t ) − ya (t )

(7)

control input, y p (t ) is the plant output, Ap , B p , and C p are

where D p ( s) is a simple parallel feedforward compensator

matrices with appropriate dimensions, f x (⋅) is a nonlinear

(PFC)

function vector ∈ R p , and f y (⋅) is a nonlinear function. We

n

Dp (8) 1+ ρs across the controlled plant to fulfill the condition in assumption 1(a) to guarantee its robust stability [5], [6], [13], where ρ is a positive constant. The augmented plant we use here must satisfy the following conditions: (i) plant (13) is ASPR, (ii) ya (t ) = y p (t ) + ys (t ) ≅ y p (t ) , which can be fulfilled by setting Dp ( s) =

the value of D p to be very small [5], [6], and (iii) D p ( s) is physically realizable.

further assume that the system (17), (18) is controllable and observable. Then, it is necessary to make the following assumption [11]. Assumption 2 (a) The linear part and the nonlinear part of the plant in (17), (18) are unknown. (b) For the plant in (17), (18), there exists an augmented plant where its linear part satisfies assumption 1(a). This augmented plant, as in (21), is formed by incorporating the plant in (17), (18) with the supplementary values in (5)—(8)

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[15]. (c) The nonlinear part of the plant in (17), (18), which is represented by f x (⋅) and f y (⋅) , is bounded. However, in this case, when the input in (16) is used to control the nonlinear system in (17) and (18), the problem of output error will arise [7], [11]. To keep the plant output y p (t ) converge to the reference model output ym (t ) , the control input can be expressed as u (t ) = h( ymT (t ), yaT (t ), yTp (t ), xTp (t ))

(19)

according to (7), (17), and (18), where h(.) is an unknown nonlinear function vector. In this paper, we synthesize the control input u (t ) by the following equation (20) u (t ) = u p (t ) + u p (t ) where u p (t ) is the output of SAC, as mentioned in (10). And u p (t ) is the total control input of neural network given as u p (t ) = α uˆ p (t ) = α f ZOH ( uˆ p (k ) ) nv

uˆ p ( k ) = ∑ uˆ p v ( k )

(21) (22)

v =1

where α is a positive constant, uˆ p (t ) is the total continuous-

For the v -th parallel small-scale neural network ( v = 1, 2,..., nv ), let xiv (k ) be the input to the i -th neuron in the input layer ( i = 1, 2,..., ni ), hq v ( k ) be the input to the q -th neuron in the hidden layer ( q = 1, 2,..., nq ), ov (k ) be the input to the neuron in the output layer. Furthermore, let miq v ( k ) be the weight between the input layer and the hidden layer, mqj v ( k ) be the weight between the hidden layer and the output layer. The control input is given by the sum of the output of simple adaptive controller and the output of neural network. The neural network is used to compensate the nonlinearity of the plant dynamics that is not taken into consideration in the usual SAC. The role of the neural network is to construct a linearized model by minimizing the output error caused by nonlinearities in the control systems. Refering to (23), the input iv (k ) of the neural network is given as (26) iv (k ) = ⎡⎣ ym ( k − 1), y p (k − 1),..., y p ( k − n) ⎤⎦ . Therefore, the nonlinear function of the system can be approximated by neural network. Furthermore, values n should be chosen appropriately according to practical nonlinear systems.

time output of neural network, uˆ p (k ) is the total discrete-time output of neural network, f ZOH ( ⋅) is a zero-order hold function [11], uˆ p v (k ) is the output of the v -th small-scale neural network, and nv is the number of the parallel small-scale neural networks. As in [11], a sampler is implemented in front of the neural network with appropriate sampling period to obtain discretetime multi-input of the neural network, and a zero-order hold is implemented to change the discrete-time output u p ( k ) of the neural network back to continuous-time output u p (t ) as shown in.(21). Consequently, we can assume the discrete-time output uˆ p ( k ) as follows

uˆ p (k ) = hˆ ( ym (k − 1), y p ( k − 1),..., y p (k − n) )

(23) ˆ where h ( ⋅) is an unknown nonlinear function and is the number of the past data of output of the plant. Using the above approach, the parallel small-scale neural networks will be trained until the output error e(t ) given as (24) e(t ) = ym (t ) − y p (t ) satisfies the following relation lim e(t ) = lim ym (t ) − y p (t ) ≤ ε t →∞

t →∞

(25)

where ε is a small positive value.

V. LEARNING OF NEURAL NETWORK The dynamics of the v -th parallel small-scale neural network are given as hq v (k ) = ∑ xi v (k )miq v (k ) (27) i

ov (k ) = ∑ S1 (hq v (k ))mqj v (k )

(28)

uˆ p v (k ) = S 2 ( ov ( k ) )

(29)

q

where S1 (.) is a tangent sigmoid function, S 2 (.) is a pure linear function. The tangent sigmoid function is chosen as 2 −1 (30) S1 ( X ) = 1 + exp(− µ X ) where µ > 0 , and the pure linear function is chosen as S2 ( X ) = X .

(31)

Consider the case when S1 ( X ) = a . Then the derivative of the tangent sigmoid function S1 (.) and the pure linear function S 2 (.) are as follows S1′( X ) =

µ

(32) (1 − a 2 ) 2 S 2′ ( X ) = 1 . (33) The objective of training is to minimize the error function E (k ) by taking the error gradient with respect to the parameters or weight vector mv ( k ) , that is to be adapted. The error function is defined as

IV. COMPOSITION OF NEURAL NETWORK Each of the parallel small-scale neural networks consists of three layers: an input layer, an output layer and a hidden layer.

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2 1 ⎡⎣ ym (k ) − y p ( k ) ⎤⎦ . 2 The weights are then adapted by using ∂E ( k ) ∆mqj v (k ) = −c ⋅ ∂mqj v ( k )

E (k ) =

∆miq v ( k ) = −c ⋅

C = ⎡⎣C p

(34)

D p ⎤⎦

(47)

and δˆ i (x(t ), u (t ), u p (t )) and δˆ o (x(t )) represent the nonlinear (35)

∂E (k ) ∂miq v (k )

part of the augmented plant described as follows ⎡f ( x (t ), u (t )) ⎤ δˆ i (x(t ), u p (t ), uc (t )) = ⎢ x p ⎥ + Bu p (t ) 0 ⎣ ⎦

(36)

(48)

= δi (x(t ), u (t )) + Bu p (t )

δˆo (x(t )) = f y (x p (t )) . (49) where c > 0 is the learning parameter. For the learning process, To prove the stability of our method, we start by defining its (35) and (36) will be expanded as follows Lyapunov function as follows ∂E (k ) ∂y p ( k ) ∂S2 ( ov (k ) ) ∂ov (k ) ∆mqj v (k ) = −c ⋅ ⋅ ⋅ ⋅ VSACNN (t ) = VSAC (t ) + VNN (t ) (50) ∂y p (k ) ∂uˆ p v (k ) ov (k ) ∂mqj v (k ) (37) where VSAC (t ) is the Lyapunov function of SAC of our method = c ⋅ [ ym (k ) − y p (k )] ⋅ J plant v ⋅ S2′ ( ov (k ) ) ⋅ S1 (hq v ( k )) which is a modification from the Lyapunov function of SAC ∂uˆ p v (k ) ∂S 2 ( ov (k ) ) ∂E (k ) ∂y p ( k ) presented in [11], [14], and VNN (t ) is the total of the Lyapunov ∆miq v ( k ) = −c ⋅ ⋅ ⋅ ⋅ ∂y p (k ) ∂uˆ p v ( k ) ∂S 2 ( ov (k ) ) ov (k ) function of NN of our method. The Lyapunov function of SAC of our method is described ∂ov (k ) ∂S1 (hq (k )) ∂hq (k ) ⋅ ⋅ ⋅ (38) as follows ∂S1 ( hq (k )) ∂hq (k ) ∂miq (k ) T VSAC (t ) = eTx (t ) Pe x (t ) + tr ⎡⎣ K i (t ) − K ⎤⎦ Ti −1 ⎡⎣ K i (t ) − K ⎤⎦ (51) = c ⋅ [ ym (k ) − y p ( k )] ⋅ J plant ⋅ S2′ (ov (k )) ⋅ mqj ( k ) v

{

v

}

where P is a real symmetric positive definite matrix, tr ( ⋅) is a

S1′(hq (k )) ⋅ iv (k )

trace function, and e x (t ) is a state error vector given as

where ⎛ ∂y p ( k ) ⎞ J plant v = SGN ⎜ (39) ⎟ ⎜ ∂uˆ p (k ) ⎟ v ⎝ ⎠ which is derived from the one mentioned in reference [9], [10]. VI. CONVERGENCE AND STABILITY For the stability analysis of our method, we use and modify the stability analysis presented in [11] and [14]. As mentioned in assumption 2(b), the PFC in (6), (8) is incorporated with the nonlinear system in (1), (2) to form the augmented plant, as in (5), where its linear part is ASPR. For convenience, first, it is necessary for the PFC in (6), (8) to be expressed in a statespace form as follows x s (t ) = As x s (t ) + Bs u (t ) (40)

( n −1) e x (t ) = xˆ (t ) − x(t ) = ⎡⎣ ex (t ), ex (t )," , ex p (t ) ⎤⎦

where the Lyapunov function of SAC of our method VSAC (t ) in (51) is developed by replacing the disturbances used in the Lyapunov function of SAC in [14] with the nonlinear part represented by δˆ i (x(t ), u (t ), u p (t )) and δˆo ( x(t )) , then its derivation, V (t ) , can be described as follows

(43)

where

SAC

⎡B ⎤ B = ⎢ p⎥ ⎣ Bs ⎦

(44)

(45) (46)

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{

T VSAC (t ) = −eTx (t )Qe x (t ) − 2σ tr ⎡⎣ K i (t ) − K ⎤⎦ Ti −1 ⎡⎣ K i (t ) − K ⎤⎦

−2e (t )e y (t )e (t )Tp e ey (t ) T y

T y

y

−2e (t )e y (t ) ⎡ xmT (t )Tp x xm (t ) + umT (t )Tp u um (t ) ⎤ m m ⎣ ⎦ T T − 1 −2σ tr ⎡⎣ ⎡⎣ K i (t ) − K ⎤⎦ Ti K ⎤⎦ − 2ex (t ) P F(t ) −2δˆ (x(t )) K (t ) − K r (t ) T y

o

⎡ x (t ) ⎤ x (t ) = ⎢ p ⎥ ⎣ x s (t ) ⎦ 0⎤ ⎡A A=⎢ p ⎥ 0 A s⎦ ⎣

(52)

where xˆ (t ) is the ideal target state vector of the system, and K = ⎡⎣ K e K x K u ⎤⎦ is the unknown ideal gain of SAC. The derivation of Lyapunov function in (50) will be VSACNN (t ) = VSAC (t ) + VNN (t ) . (53)

ys (t ) = Cs x s (t ) (41) then, by applying (40), (41) to (6), (17), and (18), the augmented plant can be described as follows x (t ) = Ax(t ) + Bueq (t ) + δˆ i (x(t ), u (t ), u p (t )) (42) ya (t ) = Cx(t ) + δˆo ( x(t ))

T

(

i

)

−2δˆo (x(t ))ey (t )eTy (t )Tp e ey (t ) y

−2δˆo (x(t ))ey (t ) ⎡ xmT (t )Tp x xm (t ) + umT (t )Tp u um (t ) ⎤ m m ⎣ ⎦ −1 T T −2σ tr ⎡⎣ ⎡⎣ K i (t ) − K ⎤⎦ Ti K ⎤⎦ − 2ex (t ) P F(t ) −2δˆ (x(t )) K (t ) − K r (t ) o

(

i

)

}

−2δˆo (x(t ))ey (t )eTy (t )Tp e ey (t )

nlower and an upper limit at nupper , and the parameters

y

−2δˆo (x(t ))ey (t ) ⎡ xmT (t )Tp x xm (t ) + umT (t )Tp u um (t ) ⎤ m m ⎣ ⎦ where Q is a real matrix, and F(t ) is given as F(t ) = E (t ) − BK δˆ (x(t )) + δˆ (x(t ), u (t ), u (t )) Bias

e o

i

p

(54)

in

(8),

ρ =1

Tp = diag (5 ×10 ,5 ×10 ,5 ×10 ) 3

(55)

where E Bias (t ) is a bias term as explained in [16]. For V (t ) , we assume that it can be approximated as NN

nv

D p = 0.001

VNN (t ) = ∑ VNN v (t )

(56)

VNN v (t ) ≅ ∆VNN v (k ) / ∆T

(56)

v =1

where ∆VNN v (k ) is the derivation of a discrete-time Lyapunov function, and ∆T is a sampling time. According to [11], ∆VNN v (k ) can be guaranteed to be negative definite if the

3

3

in

in

(8),

(14),

Ti = diag (5 ×104 ,5 ×104 ,5 ×104 ) in (15), σ = 1 in (15), µ = 2 in

(30), c = 0.001 in (35)－(38), and ∆T = 0.01 in (56) are fixed. Furthermore, we assume a first-order reference model (3), (4) with parameters Am = −10 , Bm = 10 , and Cm = 1 . Fig. 1 shows the output of the reference model ym (t ) and the plant output y p (t ) using only SAC. The result in Fig. 1 shows that the error between y p (t ) and ym (t ) is large. Fig. 2 shows the output of reference model ym (t ) and the plant output y p (t ) using our proposed method of SAC using

(57)

multiple neural networks with nv = 1 . It can be seen that the error of the system has been reduced, and the plant output y p (t ) can follow closer the output of the reference model ym (t )

for the weights between the hidden layer and the output layer, mqj v ( k ) , and

compared to using only SAC. Fig. 3 shows the output of reference model ym (t ) and the

learning parameter c fulfills the following conditions 0 < c < 2 / nq

0

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