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Indirect adaptive control of nonlinear dynamic systems using self recurrent wavelet neural networks via adaptive learning rates
Information Sciences 177 (2007) 3074–3098 www.elsevier.com/locate/ins
Indirect adaptive control of nonlinear dynamic systems using self recurrent wavelet neural networks via adaptive learning rates Sung Jin Yoo a
a,*
, Jin Bae Park a, Yoon Ho Choi
b
Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Republic of Korea b School of Electronic Engineering, Kyonggi University, Suwon, Kyonggi-Do 443-760, Republic of Korea Received 26 December 2005; received in revised form 31 January 2007; accepted 11 February 2007
Abstract This paper proposes an indirect adaptive control method using self recurrent wavelet neural networks (SRWNNs) for dynamic systems. The architecture of the SRWNN is a modified model of the wavelet neural network (WNN). However, unlike the WNN, since a mother wavelet layer of the SRWNN is composed of self-feedback neurons, the SRWNN can store the past information of wavelets. In the proposed control architecture, two SRWNNs are used as both an identifier and a controller. The SRWNN identifier approximates dynamic systems and provides the SRWNN controller with information about the system sensitivity. The gradient-descent method using adaptive learning rates (ALRs) is applied to train all weights of the SRWNN. The ALRs are derived from discrete Lyapunov stability theorem, which are applied to guarantee the convergence of the proposed control system. Finally, we perform some simulations to verify the effectiveness of the proposed control scheme. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Indirect adaptive control; Self recurrent wavelet neural network; Adaptive learning rate; Intelligent control; Dynamic system
1. Introduction During the past few years, neural networks (NNs) have received much attention as a popular universal approximator [4–6] for the identification and control of nonlinear dynamic systems [2,10,20,21]. Many researchers used multi-layer perceptron (MLP) based on the back propagation (BP) or gradient descent (GD) training algorithm using static learning rates (SLRs) to solve the dynamical problems [26]. However, this method has two disadvantages. First, since the MLP is a static mapping, without the aid of tapped delays, it cannot represent a dynamic mapping. Second, since it is difficult to find optimal learning rates in the BP algorithm using SLRs, guaranteeing the fast convergence of the NN controller is hard. To solve the first problem, many works apply recurrent neural networks (RNNs) with merits such as attractor dynamics and information storage for later use to the identification and control of nonlinear dynamic systems [12,15–17,25,33]. *
Corresponding author. Tel.: +82 2 2123 2773; fax: +82 2 362 4539. E-mail addresses: [email protected] (S.J. Yoo), [email protected] (J.B. Park), [email protected] (Y.H. Choi).
0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.02.009
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However, since the RNNs have complex network structures [8], the weight training of them is more difficult than that of MLPs and the heavy computation efforts are also required. To solve the second problem, the global optimization techniques such as the genetic algorithm and the evolutionary program were proposed [14,18,30]. However, the computational burden of these algorithms degrades the performance of real-time control systems. Recently, wavelet neural networks (WNNs), which absorbs the advantages such as the multi-resolution of wavelets and the learning of NN, were proposed to guarantee the good convergence and were used to identify and control nonlinear systems [3,7,24,34,35]. The WNN is suitable for the approximation of unknown nonlinear functions with local nonlinearities and fast variations because of its intrinsic properties of finite support and self-similarity. Despite these advantages, the WNN has a disadvantage that it can be only used for static problems due to its feedforward network structure. That is, the WNN is inefficient in solving temporal problems. Hence, self recurrent wavelet neural networks (SRWNNs), which combines the properties such as attractor dynamics of RNN and the fast convergence of WNN, were proposed to identify and control nonlinear systems [31,32]. Since the SRWNN has a mother wavelet layer with self-feedback neurons, it can capture the past information of the network and adapt rapidly to sudden changes of the control environment. Owing to these properties, the structure of SRWNN can be simpler than that of WNN [31]. In this paper, we propose the indirect adaptive control method using SRWNN for nonlinear dynamic systems. In the proposed control structure, we use two SRWNNs as an identifier and a controller. The SRWNN identifier approximates dynamic systems and provides the SRWNN controller with information about the system sensitivity. In addition, unlike the conventional GD algorithm, the GD algorithm using adaptive learning rates (ALRs) [12] is employed for the on-line training of all weights of SRWNN. The ALRs for training the SRWNN are derived from discrete Lyapunov stability theorem, which are used to guarantee the convergence of the SRWNN identifier and controller in the proposed control system. Finally, we simulate the chaotic system, the dynamic system, and the water bath system to demonstrate the on-line adapting and disturbance rejection ability of the proposed control method. This paper is organized as follows. In Section 2, we describe the architecture and training algorithm of SRWNN. The identification mechanism and the SRWNN-based indirect adaptive control method are presented in Section 3. Section 4 presents the stability analysis via ALRs for the convergence of the SRWNN identifier and controller. Simulation results are discussed in Section 5. Section 6 gives the conclusion of this paper. 2. Preliminaries 2.1. Description of the SRWNN structure A schematic diagram of the SRWNN structure shown in Fig. 1 has Ni inputs, one output, and N i N w mother wavelets. The SRWNN structure consists of four layers [31]. The layer 1 is an input layer. This layer accepts the input variables and transmits the accepted inputs to the next layer directly. The layer 2 is a mother wavelet layer. Each node of this layer has a mother wavelet and a self-feedback loop. In this paper, we select the first derivative of a gaussian function, /ðxÞ ¼ x expð12x2 Þ as a mother wavelet function. A wavelet /jk of each node is derived from its mother wavelet / as follows: ujk mjk ujk mjk /jk ðzjk Þ ¼ / ; ð1Þ ; with zjk ¼ d jk d jk where mjk and djk are the translation factor and the dilation factor of the wavelets, respectively. The subscript jk indicates the kth input term of the jth wavelet. In addition, the inputs of this layer for discrete time n can be denoted by ujk ðnÞ ¼ xk ðnÞ þ /jk ðn 1Þ hjk ;
ð2Þ
where hjk denotes the weight of the self-feedback loop. The input of this layer contains the memory term /jk ðn 1Þ, which can store the past information of networks. That is, the current dynamics of the system
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Fig. 1. The SRWNN structure.
is conserved for the next sample step. Thus, even if the SRWNN has less mother wavelets than the WNN, the SRWNN can attract well the system with complex dynamics. Here, hjk is a factor to represent the rate of information storage. These aspects are the apparently dissimilar point between the WNN and the SRWNN. And the SRWNN is a generalization system of the WNN because the structure of SRWNN is the same as that of WNN when hjk ¼ 0. The layer 3 is a product layer. The nodes in this layer are given by the product of the mother wavelets as follows: " 2 !# Ni Ni Y Y ujk mjk 1 ujk mjk Uj ðxÞ ¼ /ðzjk Þ ¼ exp : ð3Þ 2 d jk d jk k¼1 k¼1 The layer 4 is an output layer. The node output is a linear combination of consequences obtained from the output of the layer 3. In addition, the output node accepts directly input values from the input layer. Therefore, the output of SRWNN is composed of each self recurrent wavelet and parameters as follows: yðnÞ ¼
Nw X
wj Uj ðxÞ þ
j¼1
Ni X
ð4Þ
ak x k ;
k¼1
where wj is the connection weight between product nodes and output nodes, and aj is the connection weight between the input nodes and the output nodes. By using the direct term, the SRWNN has advantages of a direct linear feedthrough network such as the initialization of network parameters based on process knowledge and enhanced extrapolation outside of examples of the learning data sets [9]. W is the weighting vector of SRWNN represented by: W ¼ ½ ak
mjk
d jk
hjk
T
wj ;
ð5Þ
where the initial values of tuning parameters ak, mjk, and wj are given randomly in the range of [1 1], and the initial values of hjk and djk are given by 0 and in the range of [0 1], respectively. That is, there are no feedback units initially.
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2.2. Training of the SRWNN Our goal is to minimize the following quadratic cost function: 1 1 2 ð6Þ J ðnÞ ¼ ½y d ðnÞ yðnÞ ¼ e2 ðnÞ; 2 2 where yd(n) is the desired output and yðnÞ ¼ ^y d ðnÞ is the current output of SRWNN for the discrete time n. By using the GD method, the weight values of SRWNN are adjusted so that the error is minimized after a given number of training cycles. The GD method may be defined as oJ ðnÞ W ðn þ 1Þ ¼ W ðnÞ þ DW ðnÞ ¼ W ðnÞ þ g ; ð7Þ oW ðnÞ where g ¼ diag½ga ; gm ; gd ; gh ; gw represents the learning rate matrix for the weights of SRWNN and W is weighting vector, which is defined in Section 2.1. Here, diag[Æ] denotes the diagonal matrix. The partial derivative of the cost function with respect to W(n) is oJ ðnÞ oyðnÞ ¼ eðnÞ : oW ðnÞ oW ðnÞ
ð8Þ
By applying the chain rule recursively, the error term for each layer is first calculated, and then the parameters in the corresponding layers are adjusted. The components of the weighting vector are as follows: oyðnÞ ¼ xk ; ð9Þ oak ðnÞ oyðnÞ wj oUj ðxÞ ¼ ; omjk ðnÞ d jk ozjk
ð10Þ
oyðnÞ wj oUj ðxÞ ¼ zjk ; od jk ðnÞ ozjk d jk
ð11Þ
oyðnÞ wj oUj ðxÞ ¼ / ðn 1Þ ; ohjk ðnÞ d jk jk ozjk
ð12Þ
oyðnÞ ¼ Uj ðxÞ; owj ðnÞ
ð13Þ
_ jk Þ /ðzjN Þ, /ðz _ jk Þ ¼ o/ =ozjk ¼ ðz2 1Þ expðð1=2Þz2 Þ. where oUj =ozjk ¼ /ðzj1 Þ/ðzj2 Þ /ðz j jk jk i 3. Indirect adaptive control using SRWNN 3.1. Identification architecture In this paper, the series-parallel method [7,21,23] is used for the identification of dynamic systems. The identification architecture is shown in Fig. 2. The identification model of the dynamic system is composed
y(n) Dynamic Systems +
z –1
Σ
eI (n)
-
uI (n)
SRWNN Identifier
yI (n) GD
Fig. 2. Identification architecture using the SRWNN.
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of the SRWNN and the tapped delay line. The inputs of the SRWNN identifier are the current input and the most recent output of the dynamic system, and the error eI(n) between the actual system output and the SRWNN output is used to train all weights of the SRWNN identifier. The output of SRWNN will attract the output trajectories of the dynamic system as the weights of the SRWNN identifier are tuned by the GD method. The current output of SRWNN represents as follows: y I ðnÞ ¼ f ðyðn 1Þ; uðnÞÞ;
ð14Þ
where y(n) and u(n) are the dynamic system output and the identification input, respectively. In (14), only yðn 1Þ and u(n) are fed into the identification model. Since this reduces the number of the wavelets of SRWNN identifier, the network structure can be simplified. 3.2. Indirect adaptive control architecture To control the dynamic system, we use the indirect adaptive control technique using the SRWNN. The indirect adaptive control architecture usually requires an identified system model, and the parameters of the controller are updated by adaptation laws. Accordingly, the SRWNN identifier is firstly trained off-line by the identification process as described in Section 3.1. Then, the trained SRWNN identifier is applied to control the dynamic systems on-line. The inputs of the SRWNN controller are the reference signal and the previous plant output, and the output of the SRWNN controller is the control signal. That is, only two inputs are used. Note that this simplifies the structure of the SRWNN controller. The overall controller architecture based on the indirect adaptive control scheme is shown in Fig. 3. In this architecture, the identifier approximates dynamic systems and provides a controller with information about the system sensitivity. The controller generates a control signal so that the error between the actual system output and the reference model output is minimized. In the structure, the identification error eI(n) defined as an error between the outputs of the identifier and the dynamic system is used for tuning the parameters of the SRWNN identifier using the ALR-based GD method. The parameters of the SRWNN controller are also adjusted via the ALR-based GD method where the difference between the reference model output and the dynamic system output is used as a control error eC(n). To select optimal control signal uC(n), let us define a cost function as 1 1 J C ðnÞ ¼ ½y r ðnÞ yðnÞ2 ¼ e2C ðnÞ; 2 2
ð15Þ
where yr(n) and y(n) are the output of the nth reference model and the dynamic system, respectively. Next, using (15), the gradient of cost function JC(n) with respect to a weighing vector WC of the controller is oJ C ðnÞ oyðnÞ oyðnÞ ouC ðnÞ ¼ eC ðnÞ ¼ eC ðnÞ ; oW C ðnÞ oW C ðnÞ ouC ðnÞ oW C ðnÞ
ð16Þ
where WC(n) and uC(n) indicate the weighting vector of the SRWNN controller and the control signal, respectively. oyðnÞ=ouC ðnÞ is the system sensitivity, but cannot be directly calculated by the output of the dynamic system. Thus, the identifier output yI(n) must be used to compute the system sensitivity. It can be computed by the chain rule as follows: " # Nw X oyðnÞ oy I ðnÞ oy I ðnÞ ox wj oUj ðxÞ ¼ ¼ ¼ þ ak ; ð17Þ ouC ðnÞ ouC ðnÞ ox ouC ðnÞ d jk ozjk j¼1 k¼N s þ1
where ox=ouC ðnÞ is the column vector as follows: h iT ox ouC ðnÞ ¼ oyðn1Þ ¼ ½ 0 1 T ; ouC ðnÞ ouC ðnÞ ouC ðnÞ where x is the inputs of the SRWNN identifier and ouC ðnÞ=oW C ðnÞ can be calculated by using (9)–(13).
ð18Þ
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4. Convergence analysis via ALRs We now analyze the convergence of the proposed identifier and controller. The convergence of the SRWNN is related to select the appropriate learning rate. Since the learning rate is an essential factor for determining the performance of the neuroidentifier and the neurocontroller trained via the GD method, it is important to find the optimal learning rate [19]. However, in the conventional GD method, it is difficult to choose an appropriate learning rate because the appropriate learning rate is usually selected as a timeinvariant constant by trial and error. Accordingly, the ALRs, which can adapt rapidly the change of the plant, have been researched for the various neural networks [12,13,32]. Based on this progress about the ALRs, we devise some theorems for the convergence of the proposed indirect adaptive control system. 4.1. Convergence analysis for identification Let us define a discrete Lyapunov function as 1 V I ðnÞ ¼ J I ðnÞ ¼ e2I ðnÞ; 2
ð19Þ
where eI(n) is the identification error. The change in the Lyapunov function is obtained by 1 DV I ðnÞ ¼ V I ðn þ 1Þ V I ðnÞ ¼ ½e2I ðn þ 1Þ e2I ðnÞ: 2 The error difference can be represented by [29] T oeI ðnÞ DW iI ðnÞ; DeI ðnÞ ¼ eI ðn þ 1Þ eI ðnÞ oW iI ðnÞ
ð20Þ
ð21Þ
where W iI ðnÞ is an arbitrary component of the weighting vector WI(n) of the SRWNN identifier. And the corresponding change of W iI ðnÞ is denoted by DW iI ðnÞ. Using (15) and (8), DWI is obtained by DW iI ¼ giI eI ðnÞ
oy I ðnÞ ; oW iI ðnÞ
ð22Þ
where giI is an arbitrary diagonal element of the learning rate matrix gI corresponding to the weight component W iI ðnÞ. d h w Proposition 1. Let gI ¼ diag½g1I ; g2I ; g3I ; g4I ; g5I ¼ diag½gaI ; gm I ; gI ; gI ; gI be the learning rates for the weights of the SRWNN identifier and define CI;max as T
CI; max ¼ ½C 1I; max C 2I; max C 3I; max C 4I; max C 5I; max oy I ðnÞ max oy I ðnÞ ¼ max n n oaI omI
oy I ðnÞ max n od I
oy I ðnÞ max n ohI
oy I ðnÞ T ; max n owI
where j k represents the Euclidean norm. Then, the convergence is guaranteed if giI are chosen to satisfy 2 0 < giI < 2=ðC iI; max Þ ; i ¼ 1; . . . ; 5. Proof. See Appendix A.
h
Corollary 1. From conditions of Proposition 1, the learning rate which guarantees the maximum convergence is gi;M ¼ 1=ðC iI;max Þ2 . I Proof. See Appendix B. h Proposition 2. Let gaI be the learning rate of the input direct weight for the SRWNN identifier. The convergence is 2 guaranteed if the learning rate gaI satisfies 0 < gaI < 2=ðN i jxI;max j Þ, where Ni denotes the input number of the SRWNN identifier and jxI;max j is the maximum of the absolute value of each controller’s input.
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Proof. See Appendix C.
h
In order to prove Proposition 3, the following lemmas are used [13]. Lemma 1. Let f ðtÞ ¼ t expðt2 Þ: Then, jf ðtÞj < 1 8t 2 R. Lemma 2. Let gðtÞ ¼ t2 expðt2 Þ: Then, jgðtÞj < 1 8t 2 R. Proposition 3. Let gmI , gdI and ghI be the learning rates of the translation, dilation and self-feedback weights for the SRWNN identifier, respectively. The convergence is guaranteed if the learning rates satisfy 2 2 jd I; min j ; 0 < gmI ; ghI < N w N i 2 expð0:5ÞjwI; max j 2 2 jd I; min j d 0 < gI < : N w N i 2 expð0:5ÞjwI; max j Proof. See Appendix D. h Proposition 4. Let gwI be the learning rate for weights wI of the SRWNN identifier. Then, the convergence is guaranteed if the learning rate satisfies 0 < gwI < 2=N w , where Nw is the number of nodes in the product layer of the SRWNN identifier. Proof. See Appendix E.
h
Remark 1. From Corollary 1, the learning rates of the SRWNN identifier for the maximum convergence are as follows: ga;M ¼ I gm;M I gd;M I
1 N i jxI; max j2
;
2 1 jd I; min j ¼ ¼ ; N w N i 2 expð0:5ÞjwI; max j 2 1 jd I; min j ¼ ; N w N i 2 expð0:5ÞjwI; max j gh;M I
gw;M ¼ 1=N w : I Remark 2. The GD method is widely used for training multilayer networks by means of error propagation via variational calculus. But the global optimization in the GD method depends upon various factors such as the quality of the training data, the network complexity and so on, thus, many techniques are developed to solve this problem [1,28,22]. These methods can be used for the proposed SRWNN. Since this paper focuses on the dynamical mapping and the control ability of SRWNN, we do not discuss the application of these solutions. If these techniques are applied for the SRWNN successfully, the following convergence analysis for the SRWNN can be performed. Using (21) and (22), we obtain eI ðn þ 1Þ ¼ eI ðnÞ ðC iI ðnÞÞT giI eI ðnÞðC iI ðnÞÞ:
ð23Þ
Then, keI ðn þ 1Þk ¼ keI ðnÞ ðC iI ðnÞÞT giI eI ðnÞðC iI ðnÞÞk 6 keI ðnÞkk1 giI ðC iI ðnÞÞT ðC iI ðnÞÞk:
ð24Þ
If giI are chosen as the learning rates for the maximum convergence of SRWNN identifier in Remark 1, then T the term k1 giI ðC iI ðnÞÞ ðC iI ðnÞÞk in (24) is less than 1. Thus, from (19) and (20), V I > 0 and DV I < 0 are guaranteed. The tracking error will converge to zero as t ! 1.
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4.2. Convergence analysis for indirect adaptive control Similar to Section 4.1, let us define a discrete Lyapunov function and the error difference, respectively, as follows: 1 V C ðnÞ ¼ e2C ðnÞ; 2 T T oeC ðnÞ oeC ðnÞ oyðnÞ ouC ðnÞ i DeC ðnÞ DW C ¼ gC eC ðnÞ ; i i ouC ðnÞ oW iC ðnÞ oW C oW C
ð25Þ ð26Þ
where W iC ðnÞ is an arbitrary component of the weighting vector WC(n) of SRWNN controller and the corresponding change of W iC ðnÞ is denoted by DW iC ðnÞ. oyðnÞ=ouC ðnÞ is defined in (17). d h w Proposition 5. Let gC ¼ diag½g1C ; g2C ; g3C ; g4C ; g5C ¼ diag½gaC ; gm C ; gC ; gC ; gC be the learning rates for weights of the SRWNN controller and define CC; max as T
CC; max ¼ ½C 1C; max C 2C; max C 3C; max C 4C; max C 5C; max ouC ðnÞ ouC ðnÞ ouC ðnÞ max max ¼ max n n n oaC omC od C
ouC ðnÞ max n ohC
ouC ðnÞ T : max n owC 2
Then, the convergence is guaranteed if giC are chosen to satisfy 0 < giC < 2=ðy s C iC; max Þ ; i ¼ 1; . . . ; 5 where ys is the system sensitivity. Proof. See Appendix F. h Corollary 2. From conditions of Proposition 5, the learning rate which guarantees the maximum convergence is 2 i gM C ¼ 1=ðy s ðnÞC C;max Þ . Proof. This proof is straightforward from Corollary 1.
h
From now on, we use Ni and Nw as the number of inputs and the nodes in the product layer of SRWNN controller, respectively. Proposition 6. Let gaC be the learning rate for the input direct weight of SRWNN controller. The convergence is guaranteed if the learning rate gaC satisfies 0 < gaC < 2=ðy 2s ðnÞN i jxC;max j2 Þ. Proof. See Appendix G.
h
Proposition 7. Let gmC , gdC and ghC be the learning rates of the translation, dilation and self-feedback weight for the SRWNN controller, respectively. The convergence is guaranteed if the learning rates satisfy 2 2 jd C; min j ; 0 < gmC ; ghC < 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j 2 2 jd C; min j d : 0 < gC < 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j Proof. See Appendix H.
h
Proposition 8. Let gwC be the learning rate for weights WC of the SRWNN controller. Then, the convergence is guaranteed if the learning rate satisfies 0 < gwC < 2=ðy 2s ðnÞN w Þ, where Nw is the number of nodes in the product layer for the SRWNN controller. Proof. See Appendix I.
h
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Remark 3. From Corollary 2, the learning rates of the SRWNN controller for the maximum convergence are as follows: ga;M ¼ 1=ðy 2s ðnÞN i jxC; max j2 Þ; C 2 1 jd C; min j m;M h;M ; gC ¼ gC ¼ 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j 2 1 jd C; min j gd;M ¼ ; C y 2s ðnÞN w N i 2 expð0:5ÞjwC; max j gw;M ¼ 1=ðy 2s ðnÞN w Þ: C Remark 4. The system sensitivity ys(n) is provided by the SRWNN identifier. Therefore, the system sensitivity ys(n) must be replaced by y s; max ðnÞ as follows: X Nw Ni oy I ðnÞ X oUj ox ¼ wI;j aI;k y s ðnÞ ¼ þ ouC ouC ouC j¼1 k¼1 9 8 N Qi > > > > /ðz Þ jk N N < w i X X o/ðzjk Þ ozjk oxk = k¼1 þ aI;N i þ1 ¼ wI;j > ozjk oxk ouC > /ðzjk Þ > > j¼1 ; : k¼1 o/ðzjk Þ ozjk < wI;j max þ aI;N i þ1 ozjk oxk j¼1 Nw X 1 < wI;j max 2 expð0:5Þ þ aI;N i þ1 d I j¼1
pffiffiffiffiffiffiffi
2 expð0:5Þ
þ aI;N þ1 ; < N w jwI; max j
i d I; min
Nw X
where ox=ouC is calculated by (16). Thus pffiffiffiffiffiffiffi 2 expð0:5Þ þ aI;N i þ1 : y s; max ðnÞ ¼ N w jwI; max j jd I; min j
ð27Þ
Remark 5. Similar to Remark 2, we can obtain keC ðn þ 1Þk ¼ keC ðnÞ ðC iC ðnÞÞT giC eC ðnÞy s ðnÞðC iC ðnÞÞk 6 keC ðnÞkk1 giC y s ðnÞðC iC ðnÞÞT ðC iC ðnÞÞk:
ð28Þ
If giC are chosen as the learning rates for the maximum convergence of the SRWNN identifier in Remark 3, the T term k1 giC y s ðnÞðC iI ðnÞÞ ðC iC ðnÞÞk in (28) is less than 1. Thus, V C > 0 and DV C < 0 are guaranteed. The tracking error will converge to zero as t ! 1. 5. Simulation results To demonstrate the on-line adapting and recovering ability from disturbance of the SRWNN controller, we simulate three examples such as a chaotic system, a dynamic system, and a water bath temperature system. In order to evaluate the performance of the proposed control method, we compare the performance of SRWNN controllers via the SLR and the ALR, respectively, with that of the WNN controller. The number of the training parameters for the SRWNN and the WNN are chosen as ðN w N i 3Þ þ ðN w þ N i Þ and ðN w N i 2Þ þ ðN w þ N i Þ, respectively.
S.J. Yoo et al. / Information Sciences 177 (2007) 3074–3098 ∂ y I ( n) ∂uC (n)
GD
yI (n)
Identifier (SRWNN)
-
Σ
z –1
+
Controller (SRWNN)
uC (n)
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Dynamic system
eI (n)
y(n) -
GD
r (n)
Reference model
eC (n)
Σ +
yr (n)
Fig. 3. Indirect adaptive control architecture using the SRWNNs.
Table 1 ID performance comparison of the SRWNN and the WNN Identification
No. of product nodes No. of inputs No. of training parameters Sampling rate Learning rate Iteration ID MSEx1 ID MSEx2
SRWNN (adaptive)
SRWNN (static)
WNN
2 2 16 0.02 Adaptive 1500 0.0001 0.0007
2 2 16 0.02 0.02 1500 0.005 0.004
5 3 38 0.02 0.02 1500 0.008 0.009
IDerror (x1)
1
0.5
0
–0.5 0
5
10
15 time
20
25
30
0
5
10
15 time
20
25
30
IDerror (x2)
1
0.5
0
–0.5
Fig. 4. The identification errors of SRWNN via ALRs (solid line), SRWNN via SLRs (dash-dotted line), and WNN (dotted line).
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5.1. Example 1: chaotic system This subsection considers Duffing system, which is the representative continuous-time chaotic system. The state equation of Duffing system is [11] x2 þ vd1 x_1 ¼ ; ð29Þ a1 x1 x31 a2 x2 þ b cosðwtÞ þ u þ vd2 x_2 where a1 ¼ 1:1; a2 ¼ 0:4, b ¼ 2:1 and w ¼ 1:8. (x1, x2), u, and (vd1 ; vd2 ) denote the states, the control input, and the disturbances, respectively.
3
2
x
1
1
0
–1
–2
–3 0
5
10
15
20
25
30
20
25
30
time 5
4
3
2
x
2
1
0
–1
–2
–3
–4
–5
0
5
10
15
time
Fig. 5. The control results for Duffing system (solid line: reference signal, dotted line: SRWNN with ALRs, dash-dotted line: SRWNN with SLRs, dashed line: WNN) (a) state x1 (b) state x2.
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Control error(x1)
2 1 0 –1 –2 0
5
10
15 time
20
25
30
5
10
15 time
20
25
30
Control error(x2)
2 1 0 –1 –2 0
Fig. 6. The control errors of SRWNN with ALRs (solid line), SRWNN with SLRs (dotted line), and WNN (dash-dotted line).
–3
700
6
600
5 4 ηC
m
400
ηa
C
500
x 10
3
300 2
200
1
100 0
0
10
20
0
30
0
10
Time(sec) –3
1
20
30
20
30
Time(sec) –3
x 10
6
x 10
5
0.8
4 α
ηC
ηdC
0.6 3
0.4 2 0.2 0
1 0
10
20 Time(sec)
30
0
0
10 Time(sec)
Fig. 7. The ALRs of SRWNN controller for Duffing system.
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5.1.1. The on-line adapting ability of SRWNN controller We examine the on-line adapting ability of the proposed control system for the chaotic system without disturbances. Since the indirect adaptive control method needs an identification model for controlling Duffing system, we first identify Duffing system off-line. Then, the identified model is used as the identifier of the control architecture, as shown in Fig. 3. Here, the learning rates of the SRWNN identifier are chosen based on Remark 1. In addition, we choose initial system states as (1, 0). The identification environments and results are compared in Table 1 and Fig. 4, respectively. From these results, note that the SRWNN via ALRs guarantees faster initial convergence and better identification performance than other cases. The tracking control objective for Duffing system is to follow the unstable periodic solution of Duffing system. As the value of b varies, Duffing system may have either a chaotic or a periodic solution. In this simulation, the reference signal is defined as one periodic solution in the case of b ¼ 2:3, and the learning rates of the SRWNN controller are chosen based on Remark 3. The initial states of the reference signal and the plant are ð1; 0Þ and ð3; 1Þ, respectively. Figs. 5 and 6 compare the tracking results and errors of each network structure for Duffing system, respectively. These figures reveal that the SRWNN controller has smaller errors and faster convergence than the WNN controller. The time evolution of the ALRs for the SRWNN controller is shown in Fig. 7. In Fig. 7, note that the optimal learning rates are found rapidly by the ALRs algorithm as the reference signal is changed. The control results after 1 s are compared in Table 2 where the mean-squared error (MSE) is used as the performance index. As shown in Table 2, we can observe that the SRWNN controller using ALRs has smaller MSE than the WNN controller under the same training steps although the SRWNN consists of smaller training parameters than the WNN. 5.1.2. The recovering ability of SRWNN controller from disturbances To examine the recovering ability of the proposed control system from disturbances, two disturbances vd1 ðnÞ ¼ vd2 ðnÞ ¼ 2 and vd1 ðnÞ ¼ vd2 ðnÞ ¼ 5 are applied to the output of Duffing system at 6 and 20 s, respectively. The SRWNN controller via ALRs can recover quickly from the disturbance after about 3 s, as shown in Fig. 8. At this time, the ALRs are optimized rapidly to follow the reference signal, as shown in Fig. 9. 5.2. Example 2: dynamic system 5.2.1. The on-line adapting ability of SRWNN controller We consider the dynamic system represented by [21] yðn þ 1Þ ¼ f ½yðnÞ; yðn 1Þ þ uðnÞ þ vd ðnÞ; where u(n) and vd(n) denote the control input and a disturbance, respectively. The function f ½yðnÞ; yðn 1Þ ¼
yðnÞyðn 1ÞðyðnÞ þ 2:5Þ 1 þ y 2 ðnÞ þ y 2 ðn 1Þ
is assumed to be unknown. A reference model is described as Table 2 Control performance comparison of the SRWNN and the WNN Control
No. of product nodes No. of inputs No. of training parameters Sampling rate Learning rate Iteration Control MSEx1 (after 1 s) Control MSEx2 (after 1 s)
SRWNN (adaptive)
SRWNN (static)
WNN
1 2 9 0.02 Adaptive 1500 0.0002 0.0007
1 2 9 0.02 0.00005 1500 0.0145 0.0207
3 3 24 0.02 0.002 1500 0.0157 0.0318
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y m ðn þ 1Þ ¼ 0:6y m ðnÞ þ 0:2y m ðn 1Þ þ rðnÞ;
ð30Þ
where r(n) is the bounded reference input. We identify the dynamic system off-line using random identification inputs, and the identified model is used as the identifier of the control architecture. The control input is as follows: uðnÞ ¼ f^ ðyðnÞ; yðn 1ÞÞ þ 0:6yðnÞ þ 0:2yðn 1Þ þ rðnÞ; ð31Þ ^ where rðnÞ ¼ sinð2pn=25Þ and f is the on-line estimated function of unknown function f ½. Finally, both identification and control are implemented simultaneously. Fig. 10 represents the control architecture. The simulation environments and MSEs of the SRWNN and the WNN are listed in Table 3. Figs. 11 and 12 compare
5
4
3
x
1
2
1
0
–1
–2
–3
0
5
10
15
20
25
30
20
25
30
time 6
4
x
2
2
0
–2
–4
–6
0
5
10
15
time
Fig. 8. The control results for Duffing system with output disturbances (solid line: SRWNN with ALRs, dotted line: reference signal) (a) state x1 (b) state x2.
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3
60
2.5 2 ηm C
40
a
ηC
50
1.5
30 1
20
0.5
10 0
0
10
20
0
30
0
10
Time(sec) 0.35
3
0.3
2.5
0.25
30
20
30
2 ηα C
0.2
ηd
C
20 Time(sec)
1.5
0.15 1
0.1
0.5
0.05 0
0
10
20
30
0
0
Time(sec)
10 Time(sec)
Fig. 9. The ALRs of SRWNN controller for Duffing system with output disturbances.
Fig. 10. Control structure for Example 2.
Table 3 Comparison of the simulation environments and results
No. of product nodes No. of inputs No. of training parameters Sampling rate Learning rate Iteration On-line ID MSE Control MSE
SRWNN
WNN
1 2 9 0.01 Adaptive 10,000 0.0003 0.0005
5 2 27 0.01 0.01 10,000 0.0167 0.2364
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the control results of the SRWNN and the WNN, respectively. The time evolution of the ALRs for the SRWNN controller is shown in Fig. 13. Here, the SRWNN identifier and controller are trained by the GD method using the ALRs. These figures represent that the WNN controller cannot tract the dynamic system. On the other hand, the SRWNN identifier and controller can successfully approximate the continuous function f ½ and attract dynamics of the dynamic system. Thus, we can draw the conclusion that the SRWNN control system has better identification and control performance for the nonlinear dynamic system than the WNN control system.
6
4
y and y
m
2
0
–2
–4
–6 0
10
20
30
40
50 time
60
70
80
90
100
Fig. 11. The control result of SRWNN controller for the dynamic system (solid line: reference output dotted-line: plant output).
6
4
yp and ym
2
0
–2
–4
–6
0
10
20
30
40
50 time
60
70
80
90
100
Fig. 12. The control result of WNN controller for the dynamic system (solid line: reference output dotted-line: plant output).
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2000
3.1345
x 10
3.134 1500 m1
1000
3.133
η
η
a1
3.1335
3.1325 500 3.132 0
0
20
40 60 Time(sec)
80
3.1315
100
–6
4.242
0
20
40 60 Time(sec)
80
100
20
40 60 Time(sec)
80
100
–5
x 10
3.1345
x 10
3.134 4.241 α1
4.24
3.133
η
ηd1
3.1335
3.1325 4.239 3.132 4.238
0
20
40 60 Time(sec)
80
100
3.1315
0
Fig. 13. The ALRs of SRWNN controller for the dynamic system.
10
8
6
y and ym
4
2
0
–2
–4
–6
–8
0
10
20
30
40
50 time
60
70
80
90
100
Fig. 14. The control result of SRWNN controller for the dynamic system with output disturbances (solid line: reference output dottedline: plant output).
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5.2.2. The recovering ability of SRWNN controller from disturbances We verify the tolerance ability of the proposed control system to disturbances under the same simulation environment of Example 2. The sinusoidal and the constant disturbance, vd ðnÞ ¼ 3 sinð2pn=50Þ and vd ðnÞ ¼ 5 are applied to the plant output during 20–23 s and at 60 s, respectively. The SRWNN controller has the fast
–5
3500
3.135
3000
x 10
3.1345 3.134
2000
ηm1
ηa1
2500
1500
3.1335
1000 3.133
500 0
0
20
40 60 Time(sec)
80
3.1325
100
–6
4.2425
0
20
40 60 Time(sec)
80
100
20
40 60 Time(sec)
80
100
–5
x 10
3.135
4.242
x 10
3.1345
4.2415 ηα1
η
d1
3.134 4.241
3.1335 4.2405 3.133
4.24 4.2395
0
20
40 60 Time(sec)
80
3.1325
100
0
Fig. 15. The ALRs of SRWNN controller for dynamic system with output disturbances.
90
80
70
Temperature ºC
60
50
40
30
20
10
Control input
0 0
20
40
60
80
100
120
Time
Fig. 16. The control results of the water bath system (solid line: system output, dash-dotted line: reference output, dash-dashed line: control input).
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recovering capability from the disturbances, as shown in Fig. 14. Then, the optimal learning rates are found quickly using the ALRs algorithm to follow the reference signal, as shown in Fig. 15.
–3
x 10 2
0.06
1.5 m
ηC
η
a C
0.04 1
0.02 0.5
0
0
50
100
0
150
0
50
Time(sec)
100
150
Time(sec)
–3
x 10 8
0.06
6 α
ηC
η
d C
0.04 4
0.02 2 0
0 0
50
100
150
0
50
Time(sec)
100
150
Time(sec)
Fig. 17. The ALRs of SRWNN controller for the water bath system.
90
80
70
Temperature ºC
60
50
40
30
20
10
Control input
0 0
20
40
60
80
100
120
Time
Fig. 18. The control result of SRWNN controller for the water bath system with output disturbances (solid line: system output, dashdotted line: reference output, dash-dashed line: control input).
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5.3. Example 3: water bath temperature system 5.3.1. The on-line adapting ability of SRWNN controller In this simulation, the proposed controller is applied to control the temperature of the water bath system as follows [27]: dyðtÞ uðtÞ y 0 yðtÞ ¼ þ ð32Þ dt C RC here y(t) and u(t) denote the system output temperature in °C and the heat flowing inward the system, respectively. Here, y0 is the room temperature, C is the equivalent system thermal capacity, and R is the equivalent thermal resistance between the system borders and surroundings. Assuming that R and C are essentially constant, the system (32) can be expressed as the discrete-time form described by b ð1 a
eaTs Þ uðnÞ þ ½1 eaTs y 0 þ vd ðnÞ ð33Þ 1 þ e0:5yðnÞc where a and b are some constant values depending on R and C, respectively. vd(n) denotes the output disturbance. In this simulation, we consider the water bath system with vd ðnÞ ¼ 0. The control input u(n) is limited between 0 V and 5 V. The sampling period Ts is chosen as 30 s. The parameters used in this simulation are defined as a ¼ 1:00151e4 ; b ¼ 8:67973e3 ; c ¼ 40, and y 0 ¼ 25°C obtained from a real water bath plant [27]. We first employ the SRWNN for off-line identification of the water bath system using random inputs. Then, we design the SRWNN controller to control the system output y(n) to track the reference trajectory given by 8 < 35 C; n 6 40; rðnÞ ¼ 55 C; 40 < n 6 80; ð34Þ : 75 C; 80 < n 6 120: yðn þ 1Þ ¼ eaTs yðnÞ þ
–3
x 10 2
0.06
1.5 m
ηC
ηa
C
0.04 1
0.02 0.5
0
0 0
50
100
150
0
50
Time(sec)
100
150
Time(sec)
–3
x 10 8
0.06
6 α
ηC
η
d C
0.04 4
0.02 2
0
0 0
50
100
Time(sec)
150
0
50
100
150
Time(sec)
Fig. 19. The ALRs of SRWNN controller for the water bath system with output disturbances.
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Both the SRWNN identifier and controller consist of simple structures: two inputs, two mother wavelet nodes, and one output. In addition, we use the ALRs algorithm to train all weights of the SRWNN identifier and controller. Fig. 16 shows the tracking results of SRWNN controller. The time evolution of the ALRs is shown in Fig. 17. From the simulated results, note that the proposed control system has the good regulation performance and fast initial convergence. 5.3.2. The recovering ability of SRWNN controller from disturbances We carry out the additional simulation to investigate the noise-rejection ability of the proposed control system. Assume that the output disturbances given by vd ðnÞ ¼ 5 and vd ðnÞ ¼ 5 influence the water bath system at 60 and 100 s, respectively. Fig. 18 shows that the proposed control system has the fast rejection ability under the output disturbance. Besides, according to the change of the output disturbance, the learning rates adapt quickly via the ALRs algorithm (see Fig. 19). 6. Conclusion In this paper, we have proposed the SRWNN-based indirect adaptive control method for the dynamic nonlinear systems. The SRWNN, which is a generalized network of the WNN, consists of four layers including a mother wavelet layer with self-feedback neurons. Since the self-feedback units act as memory elements, the SRWNN has the capability of temporarily storing information. Thus, the SRWNN identifier has been used to obtain an accurate model for an indirect adaptive control of the dynamic nonlinear system. Using the Lyapunov approach, the convergence theorems for SRWNN have been proven and, from this process, the optimal ALRs have been established. Finally, the proposed control system was applied to Duffing system, a dynamic system, and the water bath temperature system, respectively. Simulation results have shown that the SRWNN has the following advantages. First, the SRWNN identifier successfully can approximate a dynamic system as accurately as desired although the SRWNN has the simpler network structure than the WNN. Second, the SRWNN controller using ALRs has an on-line adapting ability for controlling dynamic systems. Third, the SRWNN controller using ALRs has a fast recovering ability from various disturbances such as a constant and a sinusoidal function form. Acknowledgments This work was supported by the Brain Korea 21 Project in 2006. The authors faithfully appreciate the Associate Editor and the anonymous reviewers for their valuable comments and kind suggestions to improve the presentation quality of this paper. Appendix A. The proof of Proposition 1 From (19), V I ðnÞ > 0. Thus, from (21 and 22), the change of the Lyapunov function is as follows: 1 DV I ðnÞ ¼ ½e2I ðn þ 1Þ e2I ðnÞ 2 1 ¼ DeI ðnÞ½eI ðnÞ þ DeI ðnÞ 2 ( ) T T oeI ðnÞ oy ðnÞ 1 oe ðnÞ oy ðnÞ I ¼ giI eI ðnÞ I i eI ðnÞ þ giI eI ðnÞ I i 2 oW iI oW iI oW I oW I ( ) T T oy I ðnÞ oy I ðnÞ 1 oy I ðnÞ oy I ðnÞ i i ¼ gI eI ðnÞ eI ðnÞ gI eI ðnÞ 2 oW iI oW iI oW iI oW iI " 2 2 !# 1 i oy I ðnÞ 2 i oy I ðnÞ ¼ eI ðnÞ gI 1 gI ¼ cI e2I ðnÞ; 2 oW iI oW iI
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where
2 2 ! 2 oy ðnÞ 1 oy ðnÞ 1 i i 2 i I i I i oy I ðnÞ c I ¼ gI 1 gI 1 gI ðC I; max Þ P 0: P gI 2 2 oW iI oW iI oW iI
ðA:1Þ
2
From (A.1), if giI ð1 12giI ðC iI; max Þ Þ > 0 is satisfied, then the convergence of the SRWNN identifier is guaran2 teed. Here, we obtain 0 < giI < 2=ðC iI; max Þ ; i ¼ 1; . . . ; 5. This completes the proof of the theorem. h Appendix B. The proof of Corollary 1 cI can represent as follows: 2 1 i i 2 i oy I ðnÞ cI P gI 1 gI ðC I; max Þ 2 oW iI !2 2 i oy I ðnÞ2 1 oy ðnÞ 1 1 2 I i ðC Þ g þ ¼ I i : 2 2 2 oW iI I; max ðC iI; max Þ 2ðC iI; max Þ oW I
ðB:1Þ 2
From (B.1), the maximum learning rate which guarantees convergence is gi;M ¼ 1=ðC iI; max Þ . This completes I the proof. h Appendix C. The proof of Proposition 2 From (4), we obtain C 1I ðnÞ ¼ oy I ðnÞ=oaI ¼ X , p where, X ¼ ½xI;1 xI;2 xI;N i T is the input vector of the ffiffiffiffiffi 1 SRWNN identifier. Then we have kC I ðnÞk 6 N i jxI; max j. Therefore, from Theorem 1, we find that 2 2 0 < gaI < 2=ðC 1I; max Þ ¼ 2=ðN i jxI; max j Þ. h Appendix D. The proof of Proposition 3 (1) The learning rate gmI of the translation weight mI: Nw oy ðnÞ X oUj C 2I ðnÞ ¼ I ¼ wI;j omI omI j¼1 ( ) Nw Ni X X o/ðzjk Þ ozjk < wI;j max ozjk omI j¼1 k¼1 ( ) Nw Ni X X 1 wI;j max 2 expð0:5Þ : < dI j¼1 k¼1
ðD:1Þ ðD:2Þ
According to Lemma 2,
1 2
z 1 exp 1 z2 1 < 1: jk
2 jk 2 2 2
Thus, (D.1) is obviously smaller than (D.2). Then we have
Nw X pffiffiffiffiffi 2 expð0:5Þ pffiffiffiffiffiffiffipffiffiffiffiffi
2 expð0:5Þ
2
: wI;j N i kC I ðnÞk ¼ < N w N i jwI; max j
d I; min d I; min
j¼1 Accordingly, from Theorem 1, we find 2 2 2 jd I; min j ¼ : 0 < gmI < 2 ðC I; max Þ2 N w N i 2 expð0:5ÞjwI; max j
ðD:3Þ
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(2) The learning rate gdI of the dilation weight dI: Nw oy ðnÞ X oUj C 3I ðnÞ ¼ I ¼ wI;j od I od I j¼1 ( ) Nw Ni X X o/ðzjk Þ ozjk < wI;j max ozjk od I j¼1 k¼1 ( ) Nw Ni X X 1 wI;j max 2 expð0:5Þ : < d I j¼1 k¼1
ðD:4Þ ðD:5Þ
According to Lemmas 1 and 2, jzjk expðz2jk Þj < 1; and
1 1 2
1 1 2
z
< 1: z exp jk
2 2 jk
2 2 Thus, (D.4) is obviously smaller than (D.5). Then we have kC 3I ðnÞk
¼
Nw X j¼1
pffiffiffiffiffi 2 expð0:5Þ pffiffiffiffiffiffiffipffiffiffiffiffi
2 expð0:5Þ
: wI;j N i < N w N i jwI; max j
d I; min d I; min
ðD:6Þ
Accordingly, from Theorem 1, we find 0 < gdI <
2 ðC 3I; max Þ
2
¼
2 2 jd I; min j : N w N i 2 expð0:5ÞjwI; max j
(3) The learning rate ghI of the self-feedback weight hI : Nw oy I ðnÞ X oUj 4 C I ðnÞ ¼ ¼ wI;j ohI ohI j¼1 ( ) Nw Ni X X o/ðzjk Þ ozjk < wI;j max ozjk ohI j¼1 k¼1 ( ) Nw Ni X X /jk ðn 1Þ wI;j max 2 expð0:5Þ : < dI j¼1 k¼1
ðD:7Þ
ðD:8Þ
According to Lemma 2,
1 2
z 1 exp 1 z2 1 < 1: jk
2 jk 2 2 2
Thus, (D.7) is obviously smaller than (D.8). Then we have
Nw X pffiffiffiffiffi 2 expð0:5Þ pffiffiffiffiffiffiffipffiffiffiffiffi
2 expð0:5Þ
: kC 4I ðnÞk ¼ wI;j N i < N w N i jwI; max j
d I; min d I; min
j¼1 Accordingly, from Theorem 1, we find 2 2 2 jd I; min j h 0 < gI < 4 ¼ : 2 N w N i 2 expð0:5ÞjwI; max j ðC I; max Þ This completes the proof.
h
ðD:9Þ
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3097
Appendix E. The proof of Proposition 4 T
C 5I ðnÞ ¼ oy I ðnÞ=owI ¼ U, where U ¼ ½U1 U2 UN w is the outputpvector of the product layer of the ffiffiffiffiffiffiffi SRWNN identifier. Then, since we have Uj 6 1 for all j, kC 5I ðnÞk 6 N w . Accordingly, from Theorem 1, we find that 0 < gwI < 2=N w . h Appendix F. The proof of Proposition 5 Similar to the proof of Theorem 1, from (25), V C ðnÞ > 0. The change of the Lyapunov function is 1 DV C ðnÞ ¼ ½e2C ðn þ 1Þ e2C ðnÞ 2 1 ¼ DeC ðnÞ½eC ðnÞ þ DeC ðnÞ " 2 2 2 ! # ou ðnÞ 1 ou ðnÞ C C i 2 ¼ e2C ðnÞy 2s ðnÞ giC ðF:1Þ ¼ cC e2C ðnÞ: oW i 1 2 gC y s ðnÞ oW i C C If giC ð1 12giC ðy s ðnÞC iC; max Þ2 Þ > 0 is satisfied, then the convergence of the SRWNN controller is guaranteed. Here, we obtain 0 < giC < 2=ðy s ðnÞC iC; max Þ2 ; i ¼ 1; . . . ; 5. This completes the proof of the theorem. h Appendix G. The proof of Proposition 6 Similar to the proof of Theorem 2, pffiffiffiffiffi kC 1C ðnÞk < N i jxC; max j: Therefore, from Theorem 5, we find that 0 < gaC < 2=ðy s ðnÞC 1C; max Þ2 ¼ 2=ðy 2s ðnÞN i jxC; max j2 Þ. h Appendix H. The proof of Proposition 7 Similar to the proof of Theorem 3, pffiffiffiffiffiffiffipffiffiffiffiffi N w N i jwC; max jj2 expð0:5Þ j. From Theorem 5, we find d C; min 2 2 2 jd C; min j m ¼ 2 : 0 < gC < 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j ðy s ðnÞC 2C; max Þ pffiffiffiffiffiffiffipffiffiffiffiffi (2) d C : kC 3C ðnÞk < N w N i jwC; max jj2 expð0:5Þ j. From Theorem 5, we find d C; min 2 2 2 jd C; min j d 0 < gC < ¼ 2 : 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j ðy s ðnÞC 3C; max Þ pffiffiffiffiffiffiffipffiffiffiffiffi (3) hC : kC 4C ðnÞk < N w N i jwC; max jj2 expð0:5Þ j. From Theorem 5, we find d C; min 2 2 2 jd C; min j h ¼ 2 : 0 < gC < 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j ðy s ðnÞC 4C; max Þ (1) mC : kC 2C ðnÞk <
Appendix I. The proof of Proposition 8 Similar to the proof of Theorem 4, kC 5C ðnÞk 6 2 0 < gwC < 2=ðy s ðnÞC 5C; max Þ ¼ 2=ðy 2s ðnÞN w Þ. h
pffiffiffiffiffiffiffi N w . Accordingly, from Theorem 5, we find that
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Indirect adaptive control of nonlinear dynamic systems using self recurrent wavelet neural networks via adaptive learning rates Sung Jin Yoo a
a,*
, Jin Bae Park a, Yoon Ho Choi
b
Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Republic of Korea b School of Electronic Engineering, Kyonggi University, Suwon, Kyonggi-Do 443-760, Republic of Korea Received 26 December 2005; received in revised form 31 January 2007; accepted 11 February 2007
Abstract This paper proposes an indirect adaptive control method using self recurrent wavelet neural networks (SRWNNs) for dynamic systems. The architecture of the SRWNN is a modified model of the wavelet neural network (WNN). However, unlike the WNN, since a mother wavelet layer of the SRWNN is composed of self-feedback neurons, the SRWNN can store the past information of wavelets. In the proposed control architecture, two SRWNNs are used as both an identifier and a controller. The SRWNN identifier approximates dynamic systems and provides the SRWNN controller with information about the system sensitivity. The gradient-descent method using adaptive learning rates (ALRs) is applied to train all weights of the SRWNN. The ALRs are derived from discrete Lyapunov stability theorem, which are applied to guarantee the convergence of the proposed control system. Finally, we perform some simulations to verify the effectiveness of the proposed control scheme. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Indirect adaptive control; Self recurrent wavelet neural network; Adaptive learning rate; Intelligent control; Dynamic system
1. Introduction During the past few years, neural networks (NNs) have received much attention as a popular universal approximator [4–6] for the identification and control of nonlinear dynamic systems [2,10,20,21]. Many researchers used multi-layer perceptron (MLP) based on the back propagation (BP) or gradient descent (GD) training algorithm using static learning rates (SLRs) to solve the dynamical problems [26]. However, this method has two disadvantages. First, since the MLP is a static mapping, without the aid of tapped delays, it cannot represent a dynamic mapping. Second, since it is difficult to find optimal learning rates in the BP algorithm using SLRs, guaranteeing the fast convergence of the NN controller is hard. To solve the first problem, many works apply recurrent neural networks (RNNs) with merits such as attractor dynamics and information storage for later use to the identification and control of nonlinear dynamic systems [12,15–17,25,33]. *
Corresponding author. Tel.: +82 2 2123 2773; fax: +82 2 362 4539. E-mail addresses: [email protected] (S.J. Yoo), [email protected] (J.B. Park), [email protected] (Y.H. Choi).
0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.02.009
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However, since the RNNs have complex network structures [8], the weight training of them is more difficult than that of MLPs and the heavy computation efforts are also required. To solve the second problem, the global optimization techniques such as the genetic algorithm and the evolutionary program were proposed [14,18,30]. However, the computational burden of these algorithms degrades the performance of real-time control systems. Recently, wavelet neural networks (WNNs), which absorbs the advantages such as the multi-resolution of wavelets and the learning of NN, were proposed to guarantee the good convergence and were used to identify and control nonlinear systems [3,7,24,34,35]. The WNN is suitable for the approximation of unknown nonlinear functions with local nonlinearities and fast variations because of its intrinsic properties of finite support and self-similarity. Despite these advantages, the WNN has a disadvantage that it can be only used for static problems due to its feedforward network structure. That is, the WNN is inefficient in solving temporal problems. Hence, self recurrent wavelet neural networks (SRWNNs), which combines the properties such as attractor dynamics of RNN and the fast convergence of WNN, were proposed to identify and control nonlinear systems [31,32]. Since the SRWNN has a mother wavelet layer with self-feedback neurons, it can capture the past information of the network and adapt rapidly to sudden changes of the control environment. Owing to these properties, the structure of SRWNN can be simpler than that of WNN [31]. In this paper, we propose the indirect adaptive control method using SRWNN for nonlinear dynamic systems. In the proposed control structure, we use two SRWNNs as an identifier and a controller. The SRWNN identifier approximates dynamic systems and provides the SRWNN controller with information about the system sensitivity. In addition, unlike the conventional GD algorithm, the GD algorithm using adaptive learning rates (ALRs) [12] is employed for the on-line training of all weights of SRWNN. The ALRs for training the SRWNN are derived from discrete Lyapunov stability theorem, which are used to guarantee the convergence of the SRWNN identifier and controller in the proposed control system. Finally, we simulate the chaotic system, the dynamic system, and the water bath system to demonstrate the on-line adapting and disturbance rejection ability of the proposed control method. This paper is organized as follows. In Section 2, we describe the architecture and training algorithm of SRWNN. The identification mechanism and the SRWNN-based indirect adaptive control method are presented in Section 3. Section 4 presents the stability analysis via ALRs for the convergence of the SRWNN identifier and controller. Simulation results are discussed in Section 5. Section 6 gives the conclusion of this paper. 2. Preliminaries 2.1. Description of the SRWNN structure A schematic diagram of the SRWNN structure shown in Fig. 1 has Ni inputs, one output, and N i N w mother wavelets. The SRWNN structure consists of four layers [31]. The layer 1 is an input layer. This layer accepts the input variables and transmits the accepted inputs to the next layer directly. The layer 2 is a mother wavelet layer. Each node of this layer has a mother wavelet and a self-feedback loop. In this paper, we select the first derivative of a gaussian function, /ðxÞ ¼ x expð12x2 Þ as a mother wavelet function. A wavelet /jk of each node is derived from its mother wavelet / as follows: ujk mjk ujk mjk /jk ðzjk Þ ¼ / ; ð1Þ ; with zjk ¼ d jk d jk where mjk and djk are the translation factor and the dilation factor of the wavelets, respectively. The subscript jk indicates the kth input term of the jth wavelet. In addition, the inputs of this layer for discrete time n can be denoted by ujk ðnÞ ¼ xk ðnÞ þ /jk ðn 1Þ hjk ;
ð2Þ
where hjk denotes the weight of the self-feedback loop. The input of this layer contains the memory term /jk ðn 1Þ, which can store the past information of networks. That is, the current dynamics of the system
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Fig. 1. The SRWNN structure.
is conserved for the next sample step. Thus, even if the SRWNN has less mother wavelets than the WNN, the SRWNN can attract well the system with complex dynamics. Here, hjk is a factor to represent the rate of information storage. These aspects are the apparently dissimilar point between the WNN and the SRWNN. And the SRWNN is a generalization system of the WNN because the structure of SRWNN is the same as that of WNN when hjk ¼ 0. The layer 3 is a product layer. The nodes in this layer are given by the product of the mother wavelets as follows: " 2 !# Ni Ni Y Y ujk mjk 1 ujk mjk Uj ðxÞ ¼ /ðzjk Þ ¼ exp : ð3Þ 2 d jk d jk k¼1 k¼1 The layer 4 is an output layer. The node output is a linear combination of consequences obtained from the output of the layer 3. In addition, the output node accepts directly input values from the input layer. Therefore, the output of SRWNN is composed of each self recurrent wavelet and parameters as follows: yðnÞ ¼
Nw X
wj Uj ðxÞ þ
j¼1
Ni X
ð4Þ
ak x k ;
k¼1
where wj is the connection weight between product nodes and output nodes, and aj is the connection weight between the input nodes and the output nodes. By using the direct term, the SRWNN has advantages of a direct linear feedthrough network such as the initialization of network parameters based on process knowledge and enhanced extrapolation outside of examples of the learning data sets [9]. W is the weighting vector of SRWNN represented by: W ¼ ½ ak
mjk
d jk
hjk
T
wj ;
ð5Þ
where the initial values of tuning parameters ak, mjk, and wj are given randomly in the range of [1 1], and the initial values of hjk and djk are given by 0 and in the range of [0 1], respectively. That is, there are no feedback units initially.
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2.2. Training of the SRWNN Our goal is to minimize the following quadratic cost function: 1 1 2 ð6Þ J ðnÞ ¼ ½y d ðnÞ yðnÞ ¼ e2 ðnÞ; 2 2 where yd(n) is the desired output and yðnÞ ¼ ^y d ðnÞ is the current output of SRWNN for the discrete time n. By using the GD method, the weight values of SRWNN are adjusted so that the error is minimized after a given number of training cycles. The GD method may be defined as oJ ðnÞ W ðn þ 1Þ ¼ W ðnÞ þ DW ðnÞ ¼ W ðnÞ þ g ; ð7Þ oW ðnÞ where g ¼ diag½ga ; gm ; gd ; gh ; gw represents the learning rate matrix for the weights of SRWNN and W is weighting vector, which is defined in Section 2.1. Here, diag[Æ] denotes the diagonal matrix. The partial derivative of the cost function with respect to W(n) is oJ ðnÞ oyðnÞ ¼ eðnÞ : oW ðnÞ oW ðnÞ
ð8Þ
By applying the chain rule recursively, the error term for each layer is first calculated, and then the parameters in the corresponding layers are adjusted. The components of the weighting vector are as follows: oyðnÞ ¼ xk ; ð9Þ oak ðnÞ oyðnÞ wj oUj ðxÞ ¼ ; omjk ðnÞ d jk ozjk
ð10Þ
oyðnÞ wj oUj ðxÞ ¼ zjk ; od jk ðnÞ ozjk d jk
ð11Þ
oyðnÞ wj oUj ðxÞ ¼ / ðn 1Þ ; ohjk ðnÞ d jk jk ozjk
ð12Þ
oyðnÞ ¼ Uj ðxÞ; owj ðnÞ
ð13Þ
_ jk Þ /ðzjN Þ, /ðz _ jk Þ ¼ o/ =ozjk ¼ ðz2 1Þ expðð1=2Þz2 Þ. where oUj =ozjk ¼ /ðzj1 Þ/ðzj2 Þ /ðz j jk jk i 3. Indirect adaptive control using SRWNN 3.1. Identification architecture In this paper, the series-parallel method [7,21,23] is used for the identification of dynamic systems. The identification architecture is shown in Fig. 2. The identification model of the dynamic system is composed
y(n) Dynamic Systems +
z –1
Σ
eI (n)
-
uI (n)
SRWNN Identifier
yI (n) GD
Fig. 2. Identification architecture using the SRWNN.
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of the SRWNN and the tapped delay line. The inputs of the SRWNN identifier are the current input and the most recent output of the dynamic system, and the error eI(n) between the actual system output and the SRWNN output is used to train all weights of the SRWNN identifier. The output of SRWNN will attract the output trajectories of the dynamic system as the weights of the SRWNN identifier are tuned by the GD method. The current output of SRWNN represents as follows: y I ðnÞ ¼ f ðyðn 1Þ; uðnÞÞ;
ð14Þ
where y(n) and u(n) are the dynamic system output and the identification input, respectively. In (14), only yðn 1Þ and u(n) are fed into the identification model. Since this reduces the number of the wavelets of SRWNN identifier, the network structure can be simplified. 3.2. Indirect adaptive control architecture To control the dynamic system, we use the indirect adaptive control technique using the SRWNN. The indirect adaptive control architecture usually requires an identified system model, and the parameters of the controller are updated by adaptation laws. Accordingly, the SRWNN identifier is firstly trained off-line by the identification process as described in Section 3.1. Then, the trained SRWNN identifier is applied to control the dynamic systems on-line. The inputs of the SRWNN controller are the reference signal and the previous plant output, and the output of the SRWNN controller is the control signal. That is, only two inputs are used. Note that this simplifies the structure of the SRWNN controller. The overall controller architecture based on the indirect adaptive control scheme is shown in Fig. 3. In this architecture, the identifier approximates dynamic systems and provides a controller with information about the system sensitivity. The controller generates a control signal so that the error between the actual system output and the reference model output is minimized. In the structure, the identification error eI(n) defined as an error between the outputs of the identifier and the dynamic system is used for tuning the parameters of the SRWNN identifier using the ALR-based GD method. The parameters of the SRWNN controller are also adjusted via the ALR-based GD method where the difference between the reference model output and the dynamic system output is used as a control error eC(n). To select optimal control signal uC(n), let us define a cost function as 1 1 J C ðnÞ ¼ ½y r ðnÞ yðnÞ2 ¼ e2C ðnÞ; 2 2
ð15Þ
where yr(n) and y(n) are the output of the nth reference model and the dynamic system, respectively. Next, using (15), the gradient of cost function JC(n) with respect to a weighing vector WC of the controller is oJ C ðnÞ oyðnÞ oyðnÞ ouC ðnÞ ¼ eC ðnÞ ¼ eC ðnÞ ; oW C ðnÞ oW C ðnÞ ouC ðnÞ oW C ðnÞ
ð16Þ
where WC(n) and uC(n) indicate the weighting vector of the SRWNN controller and the control signal, respectively. oyðnÞ=ouC ðnÞ is the system sensitivity, but cannot be directly calculated by the output of the dynamic system. Thus, the identifier output yI(n) must be used to compute the system sensitivity. It can be computed by the chain rule as follows: " # Nw X oyðnÞ oy I ðnÞ oy I ðnÞ ox wj oUj ðxÞ ¼ ¼ ¼ þ ak ; ð17Þ ouC ðnÞ ouC ðnÞ ox ouC ðnÞ d jk ozjk j¼1 k¼N s þ1
where ox=ouC ðnÞ is the column vector as follows: h iT ox ouC ðnÞ ¼ oyðn1Þ ¼ ½ 0 1 T ; ouC ðnÞ ouC ðnÞ ouC ðnÞ where x is the inputs of the SRWNN identifier and ouC ðnÞ=oW C ðnÞ can be calculated by using (9)–(13).
ð18Þ
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4. Convergence analysis via ALRs We now analyze the convergence of the proposed identifier and controller. The convergence of the SRWNN is related to select the appropriate learning rate. Since the learning rate is an essential factor for determining the performance of the neuroidentifier and the neurocontroller trained via the GD method, it is important to find the optimal learning rate [19]. However, in the conventional GD method, it is difficult to choose an appropriate learning rate because the appropriate learning rate is usually selected as a timeinvariant constant by trial and error. Accordingly, the ALRs, which can adapt rapidly the change of the plant, have been researched for the various neural networks [12,13,32]. Based on this progress about the ALRs, we devise some theorems for the convergence of the proposed indirect adaptive control system. 4.1. Convergence analysis for identification Let us define a discrete Lyapunov function as 1 V I ðnÞ ¼ J I ðnÞ ¼ e2I ðnÞ; 2
ð19Þ
where eI(n) is the identification error. The change in the Lyapunov function is obtained by 1 DV I ðnÞ ¼ V I ðn þ 1Þ V I ðnÞ ¼ ½e2I ðn þ 1Þ e2I ðnÞ: 2 The error difference can be represented by [29] T oeI ðnÞ DW iI ðnÞ; DeI ðnÞ ¼ eI ðn þ 1Þ eI ðnÞ oW iI ðnÞ
ð20Þ
ð21Þ
where W iI ðnÞ is an arbitrary component of the weighting vector WI(n) of the SRWNN identifier. And the corresponding change of W iI ðnÞ is denoted by DW iI ðnÞ. Using (15) and (8), DWI is obtained by DW iI ¼ giI eI ðnÞ
oy I ðnÞ ; oW iI ðnÞ
ð22Þ
where giI is an arbitrary diagonal element of the learning rate matrix gI corresponding to the weight component W iI ðnÞ. d h w Proposition 1. Let gI ¼ diag½g1I ; g2I ; g3I ; g4I ; g5I ¼ diag½gaI ; gm I ; gI ; gI ; gI be the learning rates for the weights of the SRWNN identifier and define CI;max as T
CI; max ¼ ½C 1I; max C 2I; max C 3I; max C 4I; max C 5I; max oy I ðnÞ max oy I ðnÞ ¼ max n n oaI omI
oy I ðnÞ max n od I
oy I ðnÞ max n ohI
oy I ðnÞ T ; max n owI
where j k represents the Euclidean norm. Then, the convergence is guaranteed if giI are chosen to satisfy 2 0 < giI < 2=ðC iI; max Þ ; i ¼ 1; . . . ; 5. Proof. See Appendix A.
h
Corollary 1. From conditions of Proposition 1, the learning rate which guarantees the maximum convergence is gi;M ¼ 1=ðC iI;max Þ2 . I Proof. See Appendix B. h Proposition 2. Let gaI be the learning rate of the input direct weight for the SRWNN identifier. The convergence is 2 guaranteed if the learning rate gaI satisfies 0 < gaI < 2=ðN i jxI;max j Þ, where Ni denotes the input number of the SRWNN identifier and jxI;max j is the maximum of the absolute value of each controller’s input.
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Proof. See Appendix C.
h
In order to prove Proposition 3, the following lemmas are used [13]. Lemma 1. Let f ðtÞ ¼ t expðt2 Þ: Then, jf ðtÞj < 1 8t 2 R. Lemma 2. Let gðtÞ ¼ t2 expðt2 Þ: Then, jgðtÞj < 1 8t 2 R. Proposition 3. Let gmI , gdI and ghI be the learning rates of the translation, dilation and self-feedback weights for the SRWNN identifier, respectively. The convergence is guaranteed if the learning rates satisfy 2 2 jd I; min j ; 0 < gmI ; ghI < N w N i 2 expð0:5ÞjwI; max j 2 2 jd I; min j d 0 < gI < : N w N i 2 expð0:5ÞjwI; max j Proof. See Appendix D. h Proposition 4. Let gwI be the learning rate for weights wI of the SRWNN identifier. Then, the convergence is guaranteed if the learning rate satisfies 0 < gwI < 2=N w , where Nw is the number of nodes in the product layer of the SRWNN identifier. Proof. See Appendix E.
h
Remark 1. From Corollary 1, the learning rates of the SRWNN identifier for the maximum convergence are as follows: ga;M ¼ I gm;M I gd;M I
1 N i jxI; max j2
;
2 1 jd I; min j ¼ ¼ ; N w N i 2 expð0:5ÞjwI; max j 2 1 jd I; min j ¼ ; N w N i 2 expð0:5ÞjwI; max j gh;M I
gw;M ¼ 1=N w : I Remark 2. The GD method is widely used for training multilayer networks by means of error propagation via variational calculus. But the global optimization in the GD method depends upon various factors such as the quality of the training data, the network complexity and so on, thus, many techniques are developed to solve this problem [1,28,22]. These methods can be used for the proposed SRWNN. Since this paper focuses on the dynamical mapping and the control ability of SRWNN, we do not discuss the application of these solutions. If these techniques are applied for the SRWNN successfully, the following convergence analysis for the SRWNN can be performed. Using (21) and (22), we obtain eI ðn þ 1Þ ¼ eI ðnÞ ðC iI ðnÞÞT giI eI ðnÞðC iI ðnÞÞ:
ð23Þ
Then, keI ðn þ 1Þk ¼ keI ðnÞ ðC iI ðnÞÞT giI eI ðnÞðC iI ðnÞÞk 6 keI ðnÞkk1 giI ðC iI ðnÞÞT ðC iI ðnÞÞk:
ð24Þ
If giI are chosen as the learning rates for the maximum convergence of SRWNN identifier in Remark 1, then T the term k1 giI ðC iI ðnÞÞ ðC iI ðnÞÞk in (24) is less than 1. Thus, from (19) and (20), V I > 0 and DV I < 0 are guaranteed. The tracking error will converge to zero as t ! 1.
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4.2. Convergence analysis for indirect adaptive control Similar to Section 4.1, let us define a discrete Lyapunov function and the error difference, respectively, as follows: 1 V C ðnÞ ¼ e2C ðnÞ; 2 T T oeC ðnÞ oeC ðnÞ oyðnÞ ouC ðnÞ i DeC ðnÞ DW C ¼ gC eC ðnÞ ; i i ouC ðnÞ oW iC ðnÞ oW C oW C
ð25Þ ð26Þ
where W iC ðnÞ is an arbitrary component of the weighting vector WC(n) of SRWNN controller and the corresponding change of W iC ðnÞ is denoted by DW iC ðnÞ. oyðnÞ=ouC ðnÞ is defined in (17). d h w Proposition 5. Let gC ¼ diag½g1C ; g2C ; g3C ; g4C ; g5C ¼ diag½gaC ; gm C ; gC ; gC ; gC be the learning rates for weights of the SRWNN controller and define CC; max as T
CC; max ¼ ½C 1C; max C 2C; max C 3C; max C 4C; max C 5C; max ouC ðnÞ ouC ðnÞ ouC ðnÞ max max ¼ max n n n oaC omC od C
ouC ðnÞ max n ohC
ouC ðnÞ T : max n owC 2
Then, the convergence is guaranteed if giC are chosen to satisfy 0 < giC < 2=ðy s C iC; max Þ ; i ¼ 1; . . . ; 5 where ys is the system sensitivity. Proof. See Appendix F. h Corollary 2. From conditions of Proposition 5, the learning rate which guarantees the maximum convergence is 2 i gM C ¼ 1=ðy s ðnÞC C;max Þ . Proof. This proof is straightforward from Corollary 1.
h
From now on, we use Ni and Nw as the number of inputs and the nodes in the product layer of SRWNN controller, respectively. Proposition 6. Let gaC be the learning rate for the input direct weight of SRWNN controller. The convergence is guaranteed if the learning rate gaC satisfies 0 < gaC < 2=ðy 2s ðnÞN i jxC;max j2 Þ. Proof. See Appendix G.
h
Proposition 7. Let gmC , gdC and ghC be the learning rates of the translation, dilation and self-feedback weight for the SRWNN controller, respectively. The convergence is guaranteed if the learning rates satisfy 2 2 jd C; min j ; 0 < gmC ; ghC < 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j 2 2 jd C; min j d : 0 < gC < 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j Proof. See Appendix H.
h
Proposition 8. Let gwC be the learning rate for weights WC of the SRWNN controller. Then, the convergence is guaranteed if the learning rate satisfies 0 < gwC < 2=ðy 2s ðnÞN w Þ, where Nw is the number of nodes in the product layer for the SRWNN controller. Proof. See Appendix I.
h
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Remark 3. From Corollary 2, the learning rates of the SRWNN controller for the maximum convergence are as follows: ga;M ¼ 1=ðy 2s ðnÞN i jxC; max j2 Þ; C 2 1 jd C; min j m;M h;M ; gC ¼ gC ¼ 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j 2 1 jd C; min j gd;M ¼ ; C y 2s ðnÞN w N i 2 expð0:5ÞjwC; max j gw;M ¼ 1=ðy 2s ðnÞN w Þ: C Remark 4. The system sensitivity ys(n) is provided by the SRWNN identifier. Therefore, the system sensitivity ys(n) must be replaced by y s; max ðnÞ as follows: X Nw Ni oy I ðnÞ X oUj ox ¼ wI;j aI;k y s ðnÞ ¼ þ ouC ouC ouC j¼1 k¼1 9 8 N Qi > > > > /ðz Þ jk N N < w i X X o/ðzjk Þ ozjk oxk = k¼1 þ aI;N i þ1 ¼ wI;j > ozjk oxk ouC > /ðzjk Þ > > j¼1 ; : k¼1 o/ðzjk Þ ozjk < wI;j max þ aI;N i þ1 ozjk oxk j¼1 Nw X 1 < wI;j max 2 expð0:5Þ þ aI;N i þ1 d I j¼1
pffiffiffiffiffiffiffi
2 expð0:5Þ
þ aI;N þ1 ; < N w jwI; max j
i d I; min
Nw X
where ox=ouC is calculated by (16). Thus pffiffiffiffiffiffiffi 2 expð0:5Þ þ aI;N i þ1 : y s; max ðnÞ ¼ N w jwI; max j jd I; min j
ð27Þ
Remark 5. Similar to Remark 2, we can obtain keC ðn þ 1Þk ¼ keC ðnÞ ðC iC ðnÞÞT giC eC ðnÞy s ðnÞðC iC ðnÞÞk 6 keC ðnÞkk1 giC y s ðnÞðC iC ðnÞÞT ðC iC ðnÞÞk:
ð28Þ
If giC are chosen as the learning rates for the maximum convergence of the SRWNN identifier in Remark 3, the T term k1 giC y s ðnÞðC iI ðnÞÞ ðC iC ðnÞÞk in (28) is less than 1. Thus, V C > 0 and DV C < 0 are guaranteed. The tracking error will converge to zero as t ! 1. 5. Simulation results To demonstrate the on-line adapting and recovering ability from disturbance of the SRWNN controller, we simulate three examples such as a chaotic system, a dynamic system, and a water bath temperature system. In order to evaluate the performance of the proposed control method, we compare the performance of SRWNN controllers via the SLR and the ALR, respectively, with that of the WNN controller. The number of the training parameters for the SRWNN and the WNN are chosen as ðN w N i 3Þ þ ðN w þ N i Þ and ðN w N i 2Þ þ ðN w þ N i Þ, respectively.
S.J. Yoo et al. / Information Sciences 177 (2007) 3074–3098 ∂ y I ( n) ∂uC (n)
GD
yI (n)
Identifier (SRWNN)
-
Σ
z –1
+
Controller (SRWNN)
uC (n)
3083
Dynamic system
eI (n)
y(n) -
GD
r (n)
Reference model
eC (n)
Σ +
yr (n)
Fig. 3. Indirect adaptive control architecture using the SRWNNs.
Table 1 ID performance comparison of the SRWNN and the WNN Identification
No. of product nodes No. of inputs No. of training parameters Sampling rate Learning rate Iteration ID MSEx1 ID MSEx2
SRWNN (adaptive)
SRWNN (static)
WNN
2 2 16 0.02 Adaptive 1500 0.0001 0.0007
2 2 16 0.02 0.02 1500 0.005 0.004
5 3 38 0.02 0.02 1500 0.008 0.009
IDerror (x1)
1
0.5
0
–0.5 0
5
10
15 time
20
25
30
0
5
10
15 time
20
25
30
IDerror (x2)
1
0.5
0
–0.5
Fig. 4. The identification errors of SRWNN via ALRs (solid line), SRWNN via SLRs (dash-dotted line), and WNN (dotted line).
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5.1. Example 1: chaotic system This subsection considers Duffing system, which is the representative continuous-time chaotic system. The state equation of Duffing system is [11] x2 þ vd1 x_1 ¼ ; ð29Þ a1 x1 x31 a2 x2 þ b cosðwtÞ þ u þ vd2 x_2 where a1 ¼ 1:1; a2 ¼ 0:4, b ¼ 2:1 and w ¼ 1:8. (x1, x2), u, and (vd1 ; vd2 ) denote the states, the control input, and the disturbances, respectively.
3
2
x
1
1
0
–1
–2
–3 0
5
10
15
20
25
30
20
25
30
time 5
4
3
2
x
2
1
0
–1
–2
–3
–4
–5
0
5
10
15
time
Fig. 5. The control results for Duffing system (solid line: reference signal, dotted line: SRWNN with ALRs, dash-dotted line: SRWNN with SLRs, dashed line: WNN) (a) state x1 (b) state x2.
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Control error(x1)
2 1 0 –1 –2 0
5
10
15 time
20
25
30
5
10
15 time
20
25
30
Control error(x2)
2 1 0 –1 –2 0
Fig. 6. The control errors of SRWNN with ALRs (solid line), SRWNN with SLRs (dotted line), and WNN (dash-dotted line).
–3
700
6
600
5 4 ηC
m
400
ηa
C
500
x 10
3
300 2
200
1
100 0
0
10
20
0
30
0
10
Time(sec) –3
1
20
30
20
30
Time(sec) –3
x 10
6
x 10
5
0.8
4 α
ηC
ηdC
0.6 3
0.4 2 0.2 0
1 0
10
20 Time(sec)
30
0
0
10 Time(sec)
Fig. 7. The ALRs of SRWNN controller for Duffing system.
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5.1.1. The on-line adapting ability of SRWNN controller We examine the on-line adapting ability of the proposed control system for the chaotic system without disturbances. Since the indirect adaptive control method needs an identification model for controlling Duffing system, we first identify Duffing system off-line. Then, the identified model is used as the identifier of the control architecture, as shown in Fig. 3. Here, the learning rates of the SRWNN identifier are chosen based on Remark 1. In addition, we choose initial system states as (1, 0). The identification environments and results are compared in Table 1 and Fig. 4, respectively. From these results, note that the SRWNN via ALRs guarantees faster initial convergence and better identification performance than other cases. The tracking control objective for Duffing system is to follow the unstable periodic solution of Duffing system. As the value of b varies, Duffing system may have either a chaotic or a periodic solution. In this simulation, the reference signal is defined as one periodic solution in the case of b ¼ 2:3, and the learning rates of the SRWNN controller are chosen based on Remark 3. The initial states of the reference signal and the plant are ð1; 0Þ and ð3; 1Þ, respectively. Figs. 5 and 6 compare the tracking results and errors of each network structure for Duffing system, respectively. These figures reveal that the SRWNN controller has smaller errors and faster convergence than the WNN controller. The time evolution of the ALRs for the SRWNN controller is shown in Fig. 7. In Fig. 7, note that the optimal learning rates are found rapidly by the ALRs algorithm as the reference signal is changed. The control results after 1 s are compared in Table 2 where the mean-squared error (MSE) is used as the performance index. As shown in Table 2, we can observe that the SRWNN controller using ALRs has smaller MSE than the WNN controller under the same training steps although the SRWNN consists of smaller training parameters than the WNN. 5.1.2. The recovering ability of SRWNN controller from disturbances To examine the recovering ability of the proposed control system from disturbances, two disturbances vd1 ðnÞ ¼ vd2 ðnÞ ¼ 2 and vd1 ðnÞ ¼ vd2 ðnÞ ¼ 5 are applied to the output of Duffing system at 6 and 20 s, respectively. The SRWNN controller via ALRs can recover quickly from the disturbance after about 3 s, as shown in Fig. 8. At this time, the ALRs are optimized rapidly to follow the reference signal, as shown in Fig. 9. 5.2. Example 2: dynamic system 5.2.1. The on-line adapting ability of SRWNN controller We consider the dynamic system represented by [21] yðn þ 1Þ ¼ f ½yðnÞ; yðn 1Þ þ uðnÞ þ vd ðnÞ; where u(n) and vd(n) denote the control input and a disturbance, respectively. The function f ½yðnÞ; yðn 1Þ ¼
yðnÞyðn 1ÞðyðnÞ þ 2:5Þ 1 þ y 2 ðnÞ þ y 2 ðn 1Þ
is assumed to be unknown. A reference model is described as Table 2 Control performance comparison of the SRWNN and the WNN Control
No. of product nodes No. of inputs No. of training parameters Sampling rate Learning rate Iteration Control MSEx1 (after 1 s) Control MSEx2 (after 1 s)
SRWNN (adaptive)
SRWNN (static)
WNN
1 2 9 0.02 Adaptive 1500 0.0002 0.0007
1 2 9 0.02 0.00005 1500 0.0145 0.0207
3 3 24 0.02 0.002 1500 0.0157 0.0318
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y m ðn þ 1Þ ¼ 0:6y m ðnÞ þ 0:2y m ðn 1Þ þ rðnÞ;
ð30Þ
where r(n) is the bounded reference input. We identify the dynamic system off-line using random identification inputs, and the identified model is used as the identifier of the control architecture. The control input is as follows: uðnÞ ¼ f^ ðyðnÞ; yðn 1ÞÞ þ 0:6yðnÞ þ 0:2yðn 1Þ þ rðnÞ; ð31Þ ^ where rðnÞ ¼ sinð2pn=25Þ and f is the on-line estimated function of unknown function f ½. Finally, both identification and control are implemented simultaneously. Fig. 10 represents the control architecture. The simulation environments and MSEs of the SRWNN and the WNN are listed in Table 3. Figs. 11 and 12 compare
5
4
3
x
1
2
1
0
–1
–2
–3
0
5
10
15
20
25
30
20
25
30
time 6
4
x
2
2
0
–2
–4
–6
0
5
10
15
time
Fig. 8. The control results for Duffing system with output disturbances (solid line: SRWNN with ALRs, dotted line: reference signal) (a) state x1 (b) state x2.
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3
60
2.5 2 ηm C
40
a
ηC
50
1.5
30 1
20
0.5
10 0
0
10
20
0
30
0
10
Time(sec) 0.35
3
0.3
2.5
0.25
30
20
30
2 ηα C
0.2
ηd
C
20 Time(sec)
1.5
0.15 1
0.1
0.5
0.05 0
0
10
20
30
0
0
Time(sec)
10 Time(sec)
Fig. 9. The ALRs of SRWNN controller for Duffing system with output disturbances.
Fig. 10. Control structure for Example 2.
Table 3 Comparison of the simulation environments and results
No. of product nodes No. of inputs No. of training parameters Sampling rate Learning rate Iteration On-line ID MSE Control MSE
SRWNN
WNN
1 2 9 0.01 Adaptive 10,000 0.0003 0.0005
5 2 27 0.01 0.01 10,000 0.0167 0.2364
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the control results of the SRWNN and the WNN, respectively. The time evolution of the ALRs for the SRWNN controller is shown in Fig. 13. Here, the SRWNN identifier and controller are trained by the GD method using the ALRs. These figures represent that the WNN controller cannot tract the dynamic system. On the other hand, the SRWNN identifier and controller can successfully approximate the continuous function f ½ and attract dynamics of the dynamic system. Thus, we can draw the conclusion that the SRWNN control system has better identification and control performance for the nonlinear dynamic system than the WNN control system.
6
4
y and y
m
2
0
–2
–4
–6 0
10
20
30
40
50 time
60
70
80
90
100
Fig. 11. The control result of SRWNN controller for the dynamic system (solid line: reference output dotted-line: plant output).
6
4
yp and ym
2
0
–2
–4
–6
0
10
20
30
40
50 time
60
70
80
90
100
Fig. 12. The control result of WNN controller for the dynamic system (solid line: reference output dotted-line: plant output).
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2000
3.1345
x 10
3.134 1500 m1
1000
3.133
η
η
a1
3.1335
3.1325 500 3.132 0
0
20
40 60 Time(sec)
80
3.1315
100
–6
4.242
0
20
40 60 Time(sec)
80
100
20
40 60 Time(sec)
80
100
–5
x 10
3.1345
x 10
3.134 4.241 α1
4.24
3.133
η
ηd1
3.1335
3.1325 4.239 3.132 4.238
0
20
40 60 Time(sec)
80
100
3.1315
0
Fig. 13. The ALRs of SRWNN controller for the dynamic system.
10
8
6
y and ym
4
2
0
–2
–4
–6
–8
0
10
20
30
40
50 time
60
70
80
90
100
Fig. 14. The control result of SRWNN controller for the dynamic system with output disturbances (solid line: reference output dottedline: plant output).
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5.2.2. The recovering ability of SRWNN controller from disturbances We verify the tolerance ability of the proposed control system to disturbances under the same simulation environment of Example 2. The sinusoidal and the constant disturbance, vd ðnÞ ¼ 3 sinð2pn=50Þ and vd ðnÞ ¼ 5 are applied to the plant output during 20–23 s and at 60 s, respectively. The SRWNN controller has the fast
–5
3500
3.135
3000
x 10
3.1345 3.134
2000
ηm1
ηa1
2500
1500
3.1335
1000 3.133
500 0
0
20
40 60 Time(sec)
80
3.1325
100
–6
4.2425
0
20
40 60 Time(sec)
80
100
20
40 60 Time(sec)
80
100
–5
x 10
3.135
4.242
x 10
3.1345
4.2415 ηα1
η
d1
3.134 4.241
3.1335 4.2405 3.133
4.24 4.2395
0
20
40 60 Time(sec)
80
3.1325
100
0
Fig. 15. The ALRs of SRWNN controller for dynamic system with output disturbances.
90
80
70
Temperature ºC
60
50
40
30
20
10
Control input
0 0
20
40
60
80
100
120
Time
Fig. 16. The control results of the water bath system (solid line: system output, dash-dotted line: reference output, dash-dashed line: control input).
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recovering capability from the disturbances, as shown in Fig. 14. Then, the optimal learning rates are found quickly using the ALRs algorithm to follow the reference signal, as shown in Fig. 15.
–3
x 10 2
0.06
1.5 m
ηC
η
a C
0.04 1
0.02 0.5
0
0
50
100
0
150
0
50
Time(sec)
100
150
Time(sec)
–3
x 10 8
0.06
6 α
ηC
η
d C
0.04 4
0.02 2 0
0 0
50
100
150
0
50
Time(sec)
100
150
Time(sec)
Fig. 17. The ALRs of SRWNN controller for the water bath system.
90
80
70
Temperature ºC
60
50
40
30
20
10
Control input
0 0
20
40
60
80
100
120
Time
Fig. 18. The control result of SRWNN controller for the water bath system with output disturbances (solid line: system output, dashdotted line: reference output, dash-dashed line: control input).
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5.3. Example 3: water bath temperature system 5.3.1. The on-line adapting ability of SRWNN controller In this simulation, the proposed controller is applied to control the temperature of the water bath system as follows [27]: dyðtÞ uðtÞ y 0 yðtÞ ¼ þ ð32Þ dt C RC here y(t) and u(t) denote the system output temperature in °C and the heat flowing inward the system, respectively. Here, y0 is the room temperature, C is the equivalent system thermal capacity, and R is the equivalent thermal resistance between the system borders and surroundings. Assuming that R and C are essentially constant, the system (32) can be expressed as the discrete-time form described by b ð1 a
eaTs Þ uðnÞ þ ½1 eaTs y 0 þ vd ðnÞ ð33Þ 1 þ e0:5yðnÞc where a and b are some constant values depending on R and C, respectively. vd(n) denotes the output disturbance. In this simulation, we consider the water bath system with vd ðnÞ ¼ 0. The control input u(n) is limited between 0 V and 5 V. The sampling period Ts is chosen as 30 s. The parameters used in this simulation are defined as a ¼ 1:00151e4 ; b ¼ 8:67973e3 ; c ¼ 40, and y 0 ¼ 25°C obtained from a real water bath plant [27]. We first employ the SRWNN for off-line identification of the water bath system using random inputs. Then, we design the SRWNN controller to control the system output y(n) to track the reference trajectory given by 8 < 35 C; n 6 40; rðnÞ ¼ 55 C; 40 < n 6 80; ð34Þ : 75 C; 80 < n 6 120: yðn þ 1Þ ¼ eaTs yðnÞ þ
–3
x 10 2
0.06
1.5 m
ηC
ηa
C
0.04 1
0.02 0.5
0
0 0
50
100
150
0
50
Time(sec)
100
150
Time(sec)
–3
x 10 8
0.06
6 α
ηC
η
d C
0.04 4
0.02 2
0
0 0
50
100
Time(sec)
150
0
50
100
150
Time(sec)
Fig. 19. The ALRs of SRWNN controller for the water bath system with output disturbances.
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Both the SRWNN identifier and controller consist of simple structures: two inputs, two mother wavelet nodes, and one output. In addition, we use the ALRs algorithm to train all weights of the SRWNN identifier and controller. Fig. 16 shows the tracking results of SRWNN controller. The time evolution of the ALRs is shown in Fig. 17. From the simulated results, note that the proposed control system has the good regulation performance and fast initial convergence. 5.3.2. The recovering ability of SRWNN controller from disturbances We carry out the additional simulation to investigate the noise-rejection ability of the proposed control system. Assume that the output disturbances given by vd ðnÞ ¼ 5 and vd ðnÞ ¼ 5 influence the water bath system at 60 and 100 s, respectively. Fig. 18 shows that the proposed control system has the fast rejection ability under the output disturbance. Besides, according to the change of the output disturbance, the learning rates adapt quickly via the ALRs algorithm (see Fig. 19). 6. Conclusion In this paper, we have proposed the SRWNN-based indirect adaptive control method for the dynamic nonlinear systems. The SRWNN, which is a generalized network of the WNN, consists of four layers including a mother wavelet layer with self-feedback neurons. Since the self-feedback units act as memory elements, the SRWNN has the capability of temporarily storing information. Thus, the SRWNN identifier has been used to obtain an accurate model for an indirect adaptive control of the dynamic nonlinear system. Using the Lyapunov approach, the convergence theorems for SRWNN have been proven and, from this process, the optimal ALRs have been established. Finally, the proposed control system was applied to Duffing system, a dynamic system, and the water bath temperature system, respectively. Simulation results have shown that the SRWNN has the following advantages. First, the SRWNN identifier successfully can approximate a dynamic system as accurately as desired although the SRWNN has the simpler network structure than the WNN. Second, the SRWNN controller using ALRs has an on-line adapting ability for controlling dynamic systems. Third, the SRWNN controller using ALRs has a fast recovering ability from various disturbances such as a constant and a sinusoidal function form. Acknowledgments This work was supported by the Brain Korea 21 Project in 2006. The authors faithfully appreciate the Associate Editor and the anonymous reviewers for their valuable comments and kind suggestions to improve the presentation quality of this paper. Appendix A. The proof of Proposition 1 From (19), V I ðnÞ > 0. Thus, from (21 and 22), the change of the Lyapunov function is as follows: 1 DV I ðnÞ ¼ ½e2I ðn þ 1Þ e2I ðnÞ 2 1 ¼ DeI ðnÞ½eI ðnÞ þ DeI ðnÞ 2 ( ) T T oeI ðnÞ oy ðnÞ 1 oe ðnÞ oy ðnÞ I ¼ giI eI ðnÞ I i eI ðnÞ þ giI eI ðnÞ I i 2 oW iI oW iI oW I oW I ( ) T T oy I ðnÞ oy I ðnÞ 1 oy I ðnÞ oy I ðnÞ i i ¼ gI eI ðnÞ eI ðnÞ gI eI ðnÞ 2 oW iI oW iI oW iI oW iI " 2 2 !# 1 i oy I ðnÞ 2 i oy I ðnÞ ¼ eI ðnÞ gI 1 gI ¼ cI e2I ðnÞ; 2 oW iI oW iI
S.J. Yoo et al. / Information Sciences 177 (2007) 3074–3098
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where
2 2 ! 2 oy ðnÞ 1 oy ðnÞ 1 i i 2 i I i I i oy I ðnÞ c I ¼ gI 1 gI 1 gI ðC I; max Þ P 0: P gI 2 2 oW iI oW iI oW iI
ðA:1Þ
2
From (A.1), if giI ð1 12giI ðC iI; max Þ Þ > 0 is satisfied, then the convergence of the SRWNN identifier is guaran2 teed. Here, we obtain 0 < giI < 2=ðC iI; max Þ ; i ¼ 1; . . . ; 5. This completes the proof of the theorem. h Appendix B. The proof of Corollary 1 cI can represent as follows: 2 1 i i 2 i oy I ðnÞ cI P gI 1 gI ðC I; max Þ 2 oW iI !2 2 i oy I ðnÞ2 1 oy ðnÞ 1 1 2 I i ðC Þ g þ ¼ I i : 2 2 2 oW iI I; max ðC iI; max Þ 2ðC iI; max Þ oW I
ðB:1Þ 2
From (B.1), the maximum learning rate which guarantees convergence is gi;M ¼ 1=ðC iI; max Þ . This completes I the proof. h Appendix C. The proof of Proposition 2 From (4), we obtain C 1I ðnÞ ¼ oy I ðnÞ=oaI ¼ X , p where, X ¼ ½xI;1 xI;2 xI;N i T is the input vector of the ffiffiffiffiffi 1 SRWNN identifier. Then we have kC I ðnÞk 6 N i jxI; max j. Therefore, from Theorem 1, we find that 2 2 0 < gaI < 2=ðC 1I; max Þ ¼ 2=ðN i jxI; max j Þ. h Appendix D. The proof of Proposition 3 (1) The learning rate gmI of the translation weight mI: Nw oy ðnÞ X oUj C 2I ðnÞ ¼ I ¼ wI;j omI omI j¼1 ( ) Nw Ni X X o/ðzjk Þ ozjk < wI;j max ozjk omI j¼1 k¼1 ( ) Nw Ni X X 1 wI;j max 2 expð0:5Þ : < dI j¼1 k¼1
ðD:1Þ ðD:2Þ
According to Lemma 2,
1 2
z 1 exp 1 z2 1 < 1: jk
2 jk 2 2 2
Thus, (D.1) is obviously smaller than (D.2). Then we have
Nw X pffiffiffiffiffi 2 expð0:5Þ pffiffiffiffiffiffiffipffiffiffiffiffi
2 expð0:5Þ
2
: wI;j N i kC I ðnÞk ¼ < N w N i jwI; max j
d I; min d I; min
j¼1 Accordingly, from Theorem 1, we find 2 2 2 jd I; min j ¼ : 0 < gmI < 2 ðC I; max Þ2 N w N i 2 expð0:5ÞjwI; max j
ðD:3Þ
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(2) The learning rate gdI of the dilation weight dI: Nw oy ðnÞ X oUj C 3I ðnÞ ¼ I ¼ wI;j od I od I j¼1 ( ) Nw Ni X X o/ðzjk Þ ozjk < wI;j max ozjk od I j¼1 k¼1 ( ) Nw Ni X X 1 wI;j max 2 expð0:5Þ : < d I j¼1 k¼1
ðD:4Þ ðD:5Þ
According to Lemmas 1 and 2, jzjk expðz2jk Þj < 1; and
1 1 2
1 1 2
z
< 1: z exp jk
2 2 jk
2 2 Thus, (D.4) is obviously smaller than (D.5). Then we have kC 3I ðnÞk
¼
Nw X j¼1
pffiffiffiffiffi 2 expð0:5Þ pffiffiffiffiffiffiffipffiffiffiffiffi
2 expð0:5Þ
: wI;j N i < N w N i jwI; max j
d I; min d I; min
ðD:6Þ
Accordingly, from Theorem 1, we find 0 < gdI <
2 ðC 3I; max Þ
2
¼
2 2 jd I; min j : N w N i 2 expð0:5ÞjwI; max j
(3) The learning rate ghI of the self-feedback weight hI : Nw oy I ðnÞ X oUj 4 C I ðnÞ ¼ ¼ wI;j ohI ohI j¼1 ( ) Nw Ni X X o/ðzjk Þ ozjk < wI;j max ozjk ohI j¼1 k¼1 ( ) Nw Ni X X /jk ðn 1Þ wI;j max 2 expð0:5Þ : < dI j¼1 k¼1
ðD:7Þ
ðD:8Þ
According to Lemma 2,
1 2
z 1 exp 1 z2 1 < 1: jk
2 jk 2 2 2
Thus, (D.7) is obviously smaller than (D.8). Then we have
Nw X pffiffiffiffiffi 2 expð0:5Þ pffiffiffiffiffiffiffipffiffiffiffiffi
2 expð0:5Þ
: kC 4I ðnÞk ¼ wI;j N i < N w N i jwI; max j
d I; min d I; min
j¼1 Accordingly, from Theorem 1, we find 2 2 2 jd I; min j h 0 < gI < 4 ¼ : 2 N w N i 2 expð0:5ÞjwI; max j ðC I; max Þ This completes the proof.
h
ðD:9Þ
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Appendix E. The proof of Proposition 4 T
C 5I ðnÞ ¼ oy I ðnÞ=owI ¼ U, where U ¼ ½U1 U2 UN w is the outputpvector of the product layer of the ffiffiffiffiffiffiffi SRWNN identifier. Then, since we have Uj 6 1 for all j, kC 5I ðnÞk 6 N w . Accordingly, from Theorem 1, we find that 0 < gwI < 2=N w . h Appendix F. The proof of Proposition 5 Similar to the proof of Theorem 1, from (25), V C ðnÞ > 0. The change of the Lyapunov function is 1 DV C ðnÞ ¼ ½e2C ðn þ 1Þ e2C ðnÞ 2 1 ¼ DeC ðnÞ½eC ðnÞ þ DeC ðnÞ " 2 2 2 ! # ou ðnÞ 1 ou ðnÞ C C i 2 ¼ e2C ðnÞy 2s ðnÞ giC ðF:1Þ ¼ cC e2C ðnÞ: oW i 1 2 gC y s ðnÞ oW i C C If giC ð1 12giC ðy s ðnÞC iC; max Þ2 Þ > 0 is satisfied, then the convergence of the SRWNN controller is guaranteed. Here, we obtain 0 < giC < 2=ðy s ðnÞC iC; max Þ2 ; i ¼ 1; . . . ; 5. This completes the proof of the theorem. h Appendix G. The proof of Proposition 6 Similar to the proof of Theorem 2, pffiffiffiffiffi kC 1C ðnÞk < N i jxC; max j: Therefore, from Theorem 5, we find that 0 < gaC < 2=ðy s ðnÞC 1C; max Þ2 ¼ 2=ðy 2s ðnÞN i jxC; max j2 Þ. h Appendix H. The proof of Proposition 7 Similar to the proof of Theorem 3, pffiffiffiffiffiffiffipffiffiffiffiffi N w N i jwC; max jj2 expð0:5Þ j. From Theorem 5, we find d C; min 2 2 2 jd C; min j m ¼ 2 : 0 < gC < 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j ðy s ðnÞC 2C; max Þ pffiffiffiffiffiffiffipffiffiffiffiffi (2) d C : kC 3C ðnÞk < N w N i jwC; max jj2 expð0:5Þ j. From Theorem 5, we find d C; min 2 2 2 jd C; min j d 0 < gC < ¼ 2 : 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j ðy s ðnÞC 3C; max Þ pffiffiffiffiffiffiffipffiffiffiffiffi (3) hC : kC 4C ðnÞk < N w N i jwC; max jj2 expð0:5Þ j. From Theorem 5, we find d C; min 2 2 2 jd C; min j h ¼ 2 : 0 < gC < 2 y s ðnÞN w N i 2 expð0:5ÞjwC; max j ðy s ðnÞC 4C; max Þ (1) mC : kC 2C ðnÞk <
Appendix I. The proof of Proposition 8 Similar to the proof of Theorem 4, kC 5C ðnÞk 6 2 0 < gwC < 2=ðy s ðnÞC 5C; max Þ ¼ 2=ðy 2s ðnÞN w Þ. h
pffiffiffiffiffiffiffi N w . Accordingly, from Theorem 5, we find that
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