An MDL-based Hammerstein recurrent neural network for control applications

Neurocomputing 74 (2010) 315–327

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

An MDL-based Hammerstein recurrent neural network for control applications Jeen-Shing Wang , Yu-Liang Hsu Department of Electrical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC

a r t i c l e in fo

abstract

Article history: Received 14 August 2009 Received in revised form 8 January 2010 Accepted 15 March 2010 Communicated by W. Yu Available online 7 April 2010

This paper presents an efficient control scheme using a Hammerstein recurrent neural network (HRNN) based on the minimum description length (MDL) principle for controlling nonlinear dynamic systems. In the proposed control approach, an unknown system is first identified by the MDL-based HRNN, which consists of a static nonlinear model cascaded by a dynamic linear model and can be expressed in a state-space representation. For high-accuracy system modeling, we have developed a selfconstruction algorithm that integrates the MDL principle and recursive recurrent learning algorithm for constructing a parsimonious HRNN in an efficient manner. To ease the control of the system, we have established a nonlinearity eliminator that functions as the inverse of the static nonlinear model to remove the global nonlinear behavior of the unknown system. If the system modeling and the inverse of the nonlinear model are accurate, the compound model, the unknown system cascaded with the nonlinearity eliminator, will behave like the linear dynamic model. This assumption turns the task of complex nonlinear control problems into a simple feedback linear controller design. Hence, welldeveloped linear controller design theories can be applied directly to achieve satisfactory control performance. Computer simulations on unknown nonlinear system control problems have successfully validated the effectiveness of the proposed MDL-based HRNN and its control scheme as well as its superiority in control performance. & 2010 Elsevier B.V. All rights reserved.

Keywords: Hammerstein model Recurrent neural networks Minimum description length Nonlinearity eliminator

1. Introduction In most cases, the mathematical model of a plant and its linear approximation are required for the design of controllers [24,33]. However, due to uncertainties in parameters, unexpected disturbances and noises, practical systems are nonlinear and the acquisition of their mathematical expressions may become difficult if not impossible. In the past decades, many research efforts have been directed to addressing these issues in the areas of adaptive control [22,26], nonlinear mathematics [17,29], and robust control [2,5]. In response to the increasingly nonlinear and complex dynamic systems, many researchers have turned away from conventional control approaches to intelligent-based approaches [3,32]. Among these intelligent-based approaches, neural-network (NN) based controllers have been developed to compensate for the effects of nonlinearities and system uncertainties so that the stability, convergence, and robustness of the control system can be improved. Thorough reviews on neuralnetwork-based control systems can be found in [9,13,35,36]. Among the diverse neural network structures, recurrent neural networks (RNNs) have been recognized as one of the most effective

 Corresponding author.

E-mail address: [email protected] (J.-S. Wang). 0925-2312/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2010.03.011

tools for modeling and controlling complex dynamic systems due to their learning capability and flexibility in incorporating ‘‘dynamics’’ into the structures. Among the successful applications [11,16,31,34], to name a few, Ku and Lee [16] proposed a partially connected diagonal recurrent neural network (DRNN) that has better learning capability than fully connected RNNs do. To control an unknown plant, they adopted a model reference adaptive control structure by using the DRNN as the plant identifier to provide a channel for propagating the error signal to fine-tune a DRNN-based controller. Huang and Lewis [11] developed an RNNbased predictive feedback control scheme for uncertain nonlinear time-delayed systems. In their approach, an RNN was used to estimate the dynamics of the delay-free nonlinear system, and a nonlinear compensator was extracted from the RNN to remove the system’s nonlinear effects. This results in a so-called linearized local subsystem that facilitates the design of the remote predictive controller for handling time-delay problems. Recently, Zhu and Guo [34] designed a linear generalized minimum variance (GMV) controller based on the linearization of a nonlinear system around operating points. To reinforce the control performance, a nonlinear compensator implemented by an RNN was devised to provide compensation for the system nonlinearities. From the above studies, a general agreement can be reached that a linearized model derived from an unknown nonlinear system can often capture the significant dynamics of the system

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around operating points and thus can provide a good basis for controller design. In this study, we have developed an efficient control approach that integrates system modeling and controller design into a unified framework. That is, we first design a Hammerstein recurrent neural network (HRNN) consisting of a static nonlinear model and a dynamic linear model for modeling an unknown system into a state-space representation. Then, we develop a self-construction algorithm, integrating the MDL principle and recursive recurrent learning algorithm, to automate the construction of the proposed recurrent network. Based on the identified Hammerstein network, which models the dynamics of the unknown system, we generate a nonlinearity eliminator (NLE) to remove the nonlinear effects caused by the static nonlinear model of the Hammerstein network. This results in an unknown system with a trained NLE that behaves like the dynamic linear model of the Hammerstein network. With the linear model, we can design a linear controller by using well-developed linear control theories directly and effortlessly. The major contribution of this paper is as follows. In most controller design problems, identification and control are usually treated as two separable tasks. However, we regard these two tasks as integral and take full advantage of the procedures conducted for system modeling to design an effective but simple linear controller. The advantages of our approach include: (1) the proposed HRNN with the proposed identification algorithm can describe the nonlinear dynamics of a given unknown system as a state-space equation; (2) the proposed MDL algorithm can automatically determine a compact network structure for a system identification problem; and (3) the network controllability and observability can be analyzed by the state-space equation. However, the proposed MDL-based structure learning algorithm cannot be performed online. This is one of the limitations for using the MDL principle in the structure learning phase of recurrent network-based modeling problems. The rest of this paper is organized as follows. In Section 2, we introduce the network architecture of the HRNN and the selfconstruction algorithm for establishing a parsimonious HRNN. The concept of the efficient control scheme, the design procedure of the nonlinear eliminator for removing the nonlinearity of the HRNN, and the relative control strategy are presented in Section 3. Section 4 provides the computer simulations to validate the effectiveness of the control approach, while Section 5 is devoted to conclusions.

2. Structure of MDL-based HRNN and its identification algorithm According to [23], adaptive control can be classified into two major categories: direct and indirect approaches. In the direct approach, the controller is learned directly so as to minimize the difference between the desired outputs and plant outputs. In the indirect approach, the design of the controller requires two procedures: modeling of the unknown plant and training of the controller based on the identified model. Here, we propose a recurrent-network based indirect control scheme that unifies an MDL-based HRNN with a self-construction algorithm to systematize the designs of all necessary control components without a trial-and-error approach or any user manipulation. To explain our philosophy in developing such a control scheme, we first introduce the structure of our MDL-based HRNN and its relative identification algorithm. The overall control scheme and the methodology of controller design are provided in Section 3.

from the model we can get a physical understanding of the system and develop a simulation from which a control law is designed [19]. To obtain good model identification for the system requires the following considerations: (1) the selection of models for describing the system, (2) the construction of quality models to best fit the system, and (3) the accuracy of models in representing the system. These issues are integral parts for the development of effective system identification tools. Recent research studies have shown that the Hammerstein model, consisting of a nonlinear static subsystem cascaded with a dynamic linear subsystem, is one of the effective models for dealing with nonlinear problem [4,12,15]. In this paper, we present a HRNN that is capable of precisely capturing the dynamics of the true system with a transparent and concise network representation. The conventional Hammerstein model is shown in Fig. 1a. The novelty of this network is to incorporate dynamic elements in such a way that each state corresponds to a neuron that linearly combines its own output history and the outputs from other neurons. Such a deployment of dynamic elements enables the proposed structure to be mapped into a state-space equation from its internal structure. Fig. 2a shows the structure of the proposed HRNN. The structure consists of four layers, which can be expressed by the block diagram shown in Fig. 2b. The whole HRNN can be classified into two major components: a static nonlinear model and a dynamic linear model. The static nonlinear model maps the input space into a state space via a nonlinear transformation and then the state space is transformed into the output space through a linear dynamic mapping. The state space equations of the proposed network are expressed as xðk þ 1Þ ¼ AxðkÞ þ BNðuðkÞÞ, yðkÞ ¼ CxðkÞ,

ð1Þ

where A A RJJ , B A RJJ , C A RmJ , N A RJ , u¼[u1, y, up]T is the input vector, y¼[y1, y, ym]T is the output vector, and p and m are the dimensions of the input and output layers, respectively. The elements of matrix A stand for the degree of inter-correlation among the states. B is a diagonal matrix with diagonal elements [b11, y, bJJ]T, representing the weights of the inputs of the dynamic layer. The elements of matrix C are the weights of the states. N¼[n1, y, nJ]T is the nonlinear function vector. x¼[x1, y, xJ]T is the state vector, where J is the total number of state variables and is equal to the number of neurons in the hidden layer and the dynamic layer. The current ith output yi(k) and the state variables x(k) are obtained by calculating the activities of all nodes on each layer, and the corresponding functions are

Static Nonlinear

Dynamic Linear

Static Nonlinear

Dynamic Linear

2.1. Structure of HRNN The model identification of unknown systems (plants) is an important and integral part of control design methodology, since

Fig. 1. The block diagram of Hammerstein models. (a) A general single-inputsingle-output Hammerstein model. (b) The proposed single-input-single-output recurrent neural network with J system order.

J.-S. Wang, Y.-L. Hsu / Neurocomputing 74 (2010) 315–327

Static nonlinear model

Dynamic linear model

d1 w11

f

a11

z-1

b1

w1p d

u1(k)

2

a1J

w21

a21

x1(k)

c12

y1(k)

a22

c1J

z-1

b2

c11

a12

f w2p

317

x2(k)

a2J cm1

up(k)

aJ2

wJ1 dJ wJp Input Layer

u(k)

f

aJ1 bJ

Hidden Layer

N(u(k))

cm2

z-1

Dynamic Layer

B

ym(k)

aJJ cmJ

xJ(k) State Variables

z-1

Output Layer

x(k)

C

y(k)

A Fig. 2. (a) The topology of the proposed Hammerstein recurrent neural network. (b) The network block diagram.

summarized as follows: yi ðkÞ ¼ ci xðkÞ ¼

J X

cij xj ðkÞ,

ð2Þ

ðaji xi ðk1ÞÞþ bjj nj ðk1Þ,

ð3Þ

j¼1

xj ðkÞ ¼

J X

structure can be completely established once the number of the system order is identified. After the structure learning phase, we continuously fine-tune the parameters of the network to capture the dynamics of the unknown system via a recursive recurrent learning algorithm. We now introduce the two identification procedures in detail as follows.

i¼1

expðpj ðkÞÞexpðpj ðkÞÞ , nj ðkÞ ¼ f ðpj ðkÞÞ ¼ expðpj ðkÞÞ þ expðpj ðkÞÞ pj ðkÞ ¼

p X

ðwji ui ðkÞÞ þ dj ,

ð4Þ

ð5Þ

i¼1

where wji is the weight between the ith input neuron and the jth hidden neuron; dj is the bias of the jth hidden neuron. The hyperbolic tangent sigmoid function in (4) is selected as the nonlinear activation function of the hidden layer because it can provide a dual polarity signal to the output. In the HRNN, the number of outputs in the static nonlinear subsystem is the same as the number of the system order, J, as shown in Fig. 1b. Therefore, the subsystem is always a multivariable system if the system order of the HRNN is greater than one. 2.2. Self-construction algorithm In this paper, we developed a self-construction algorithm to cope with practical applications. The objectives of the selfconstruction algorithm include: (1) to identify a minimal network size with optimal parameters and (2) to closely capture the dynamics of the unknown system within the minimal network. The self-construction algorithm is composed of the procedures of structure learning and parameter learning. The procedures of the self-construction algorithm are implemented as follows. In the structure learning phase, we devised an order determination algorithm that utilizes the MDL principle to detect the intrinsic system order (or minimal system order) from the input-output measurements of the unknown system to be identified. The HRNN

2.2.1. Minimum description length (MDL) principle The system order estimation for nonlinear systems typically requires considerations of the spatial and/or temporal characteristics of the training data used for system modeling. In general, the order of input–output models can be cast as the memory of the system or the region of the past states that affects the evolution of the system dynamics. For the order determination of the network, we adopted the MDL principle to obtain the intrinsic system order (or minimal system order) from the input–output measurements. The MDL principle was first proposed by Rissanen [25] and based on the concept of data compression. The idea is to precisely describe the data using fewer symbols than the number of symbols in the data. The MDL principle can be used to not only construct the structure of the model but also estimate the parameters against overfitting. The MDL principle has been successfully used to determine a minimum number of neurons of feedforward neural networks. For example, Lappalainen [20] used the MDL principle as the cost function for selecting a feedforward neural network. Small and Tse [28] applied the MDL principle as a stop criterion to construct a feedforward neural network which contains a minimum number of neurons to mimic a nonlinear system well. In this paper, we future extended the applicability to estimate the minimum number of neurons of RNNs. The MDL principle can be used as a criterion to determine the model dimensions/orders of the nonlinear systems because of its good performance in nonlinear systems. The MDL principle can be separated into two parts that include L(M) and L(Noff|M), where L(M) is the length of describing the candidate hypothesis (model), and L(Noff|M) represents the length of describing the data when the data is encoded by the hypothesis (model), where Noff is the

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J.-S. Wang, Y.-L. Hsu / Neurocomputing 74 (2010) 315–327

number of observed data. The best hypothesis (model) is the one that minimizes the values of description length (DL) which is the sum of L(M)+ L(Noff|M). From [8], we can turn L(M) and L(Noff|M) into (2log(J)+ 1) and (–logP(Noff|M(J))) for regression problems, respectively, where M(J) is the candidate model with Jth-degree polynomial and P(  ) is a probability function. Hence, we can express

where e(k) ¼yp(k)–y(k); yp(k) and y(k) are the desired output and the current output, respectively. Let w be the adjustable parameters of a node and the correction Dw applied to w is defined by the ordered derivative with a momentum term  þ  @ E þ aDwðk1Þ, DwðkÞ ¼ x ð9Þ @w

0 Noff  2 1 P yðiÞwðuðiÞÞ C  Noff B   1 C B P Noff jM ðJÞ ¼ Pðyju, s,wÞ ¼ pffiffiffiffiffiffi expB i ¼ 1 C, A @ 2s2 2ps

wðkÞ ¼ wðk1Þ þ DwðkÞ: ð6Þ

where w is the candidate Jth-degree polynomial model, y(k) and u(k) are the input–output pairs of the unknown system, and s is the standard deviation of errors. The DL can be rewritten as 9 8 0 Noff   2 1> > P > > > yðiÞwðuðiÞÞ C> > > Noff = < B 1 B i¼1 C pffiffiffiffiffiffi DL ¼ log expB C þ2logðJÞ þ 1: 2 > @ A> 2s 2ps > > > > > > ; : ð7Þ For RNNs, we can summarize the procedure of finding a minimum number of neurons as follows: Step 1: Randomly generate a set of candidate neurons. Noff

Step 2: Calculate the value of errg ¼

P

ei hig for each candidate

i¼1

neuron, where g ¼1, y, R. R is the number of candidate neurons, Noff the number of data, ei the error of the current network, and hig the output of the candidate neuron. J P Step 3: Let Hg ¼ j eT hm jþ jerrg j, where e is the error of the

ð10Þ +

where x is a learning rate and q E/qw is the ordered derivative that considers the direct (current state) and indirect (previous states) effects of changing a structure parameter. The term (aDw) is the momentum where aA[0, 1] is a learning rate. The general update rule is expressed as (10). The adjustable parameters w of the proposed network consist of the output link weights ðC A RmJ Þ, the elements of the state matrix (AA RJJ ), the elements of the input diagonal matrix (B A RJJ ), and the link weights (wji) and bias (dj) of the hidden layer. The following equations summarize the parameter update rules cj ðkÞ ¼ cj ðk1Þ þðzac eðkÞxj ðkÞ þ aDcj ðk1ÞÞ  @ þ xj ðk1Þ þ aDaji ðk1ÞÞ; aji ðkÞ ¼ aji ðk1Þ þ ðzac eðkÞcj ðk1Þ xi ðk1Þ þ ajj ðk1Þ @aji

ð12Þ 

bjj ðkÞ ¼ bjj ðk1Þ þ ðzeðkÞcj ðk1Þ nj ðk1Þ þajj ðk1Þ

2.2.2. Recursive recurrent learning algorithm In the parameter learning phase, the parameters of the whole network are recursively adjusted to memorize the desired dynamic trajectories of the unknown system. Here, an online recursive learning algorithm based on the ordered derivative [30] combined with momentum terms has been developed for the parameter learning. The inclusion of momentum terms with a suitable learning rate usually yields the convergence to a local minimum with a lower number of learning epochs [10]. To ease our discussion, the optimization target is characterized to minimize the following error function with respect to the adjustable parameters (w) of the network in the case of a MISO system Eðw,kÞ ¼ 1=2ðyp ðkÞyðkÞÞ2 ¼ 1=2eðkÞ2 ,

ð8Þ



@ þ xj ðk1Þ þ aDbjj ðk1ÞÞ @bjj

ð13Þ

wji ðkÞ ¼ wji ðk1Þ þ ðzeðkÞcj ðk1Þ " # @ þ xj ðk1Þ 4 þ aDwji ðk1ÞÞ þ a ðk1Þ  ui ðkÞbjj ðk1Þ  p jj 2 @wji ðe j þ epj Þ

ð14Þ

m¼1

current network, hm the hidden neuron output of the current network, and J the number of hidden neurons (model size) of the current network. Find out which candidate neuron occurs at the maximum value of Hg and add it to the current network. Step 4: Calculate the value of Wi ¼ jeT hi j, where i¼1, y, J. Find out which hidden neuron causes the minimum value and drops it out of the current network. If the neuron is added in this iteration, then keep it in the network. Step 5: Calculate the value of DL. Step 6: If the criterion is reached (usually we say the value of DL is minimum if it is smaller than the following ten DL values), then stop. Otherwise, go to Step 1. After utilizing the MDL principle to decide the system order of the HRNN, we can obtain not only the minimum number of neurons but also the initial values of the HRNN parameters. Subsequently, we can use the following parameter learning algorithm to update the parameters of the HRNN.

ð11Þ



dj ðkÞ ¼ dj ðk1Þ þ ðzeðkÞcj ðk1Þ " # @ þ xj ðk1Þ 4  bjj ðk1Þ  p þa ðk1Þ þ adj ðk1ÞÞ; jj @dj ðe j þ epj Þ2 ð15Þ where zac is the learning rate for adjusting aji and cj, and z is the learning rate for adjusting bjj, wji, and dj. Moreover, we have developed a stable learning algorithm to guarantee the HRNN to be stable during the parameter learning phase. Please refer to Appendix for the stable learning algorithm. In addition, we have shown that state space equations in (1) can be extracted from the proposed network topology directly. Theoretically, one of the difficult problems encountered in nonlinear system control is the stability of the overall system. The state-space equations will facilitate not only the stability analysis of the HRNN by using the well-developed linear systems theory but also the design of the control scheme, including the construction of an accurate inverse model as well. Next, we will introduce the control scheme based on this network.

3. An efficient control approach for HRNN To control an unknown nonlinear dynamic system, we propose an efficient control approach that takes advantage of the proposed HRNN, describing the unknown system by a static nonlinear model cascaded with a dynamic linear model. The philosophy of our control approach is to apply a divide-and-conquer strategy that transfers a complex nonlinear control problem to a simple linear one by eliminating the effect of the static nonlinear model. The conventional control strategy of Hammerstein models is

J.-S. Wang, Y.-L. Hsu / Neurocomputing 74 (2010) 315–327

shown in Fig. 3. The procedures of our approach include: (1) the model identification of the unknown system using the HRNN, (2) the construction of a nonlinearity eliminator (NLE) for removing the unknown system nonlinearity, and (3) the design of the linear controller based on the linearized dynamic linear model obtained from the HRNN. To begin with, we model the unknown system using the HRNN as shown in Fig. 4 and express the identification error over the region of a compact domain D by the following equation: :HRNNðuðkÞÞf ðuðkÞÞ: ¼ :eðuðkÞÞ:o e8u A D,

ð16Þ

where f(  ) represents the model of the unknown system that is identified by the HRNN, denoted as HRNN(  ), and e is arbitrarily a small value when the HRNN consists of a sufficiently large number of hidden nodes. In the second procedure, we first construct a dynamic linear system in a standard state-space form as a reference model to remove the nonlinearity of the identified HRNN. The matrices A and C of the linear reference model are

r(k)

+

e(k) _

Linear Controller

NL-1

319

directly copied from the identified HRNN and the matrix B^ is selected to optimize the steady-state response of the linear reference model for step inputs. Using the dynamic linear model, we then train the NLE network via the block diagram as shown in Fig. 5. Note that the NLE network is a three-layered neural network whose output number is the same as the input number of the unknown system, and the numbers of input and hidden neurons are selected as the output number of the static nonlinear model. The reason for such a setup is that the NLE can be treated as the inverse of the static nonlinear model. Intuitively, if the NLE training is perfect, the output of the HRNN will match the output of the reference model, that is HRNNðNLEðB^ vðkÞÞÞ ¼ CðAxðkÞ þ B^ vðkÞÞ:

ð17Þ

From (16) and (17), we obtain f ðuðkÞÞ þeðuðkÞÞ ¼ HRNNðuðkÞÞ ¼ CðAxðkÞ þ B^ vðkÞÞ,

ð18Þ

where uðkÞ ¼ NLEðB^ vðkÞÞ and :eðÞ:o e: From (18), we can

Hammerstein model Nonlinear Linear Static Dynamic Element Element

Fig. 3. The control strategy for Hammerstein model.

Fig. 4. The diagram of model identification.

Fig. 5. The block diagram of the construction of the nonlinear eliminator.

yp (k)

320

J.-S. Wang, Y.-L. Hsu / Neurocomputing 74 (2010) 315–327

Fig. 6. The overall control scheme of our proposed approach.

conclude that the compound system, which is the NLE cascaded with the unknown system, behaves approximately like a linear system. Consequently, the task of controlling the compound system can be fulfilled by using the well-developed linear control theory such as PID [6], pole placement [6], or optimal control [18]. Fig. 6 illustrates the proposed efficient control scheme. The linear controller design is based on the linear reference model in (17). For a tracking problem, the error along the desired trajectory is sent to the linear controller to generate the required control command. During the control phase, the NLE continuously adapts its parameters using the error signals backpropagated through the HRNN to minimize the difference between the actual output of the unknown system and the desired trajectory. That is, the controlling performance of the proposed approach can be improved continuously and the global nonlinearity of the unknown system can be eliminated effectively via the online parameter tuning of the NLE. The proposed control approach possesses several desirable features if we compare with model reference adaptive control (MRAC) [21] or internal model control (IMC) [7]. In the MRAC, the reference model is provided by the user; that is, the user must be sufficiently familiar with the controlled system so that s/he can specify the desired behavior by a reference model. Contrarily, the linear reference model in the proposed control scheme can be obtained directly from the identified HRNN without prior knowledge of the unknown system. In the IMC, the modeling procedures for the internal model, the inverse model, and the filter design are performed separately without informative guidelines. Our approach, however, utilizes the structure of the HRNN to exempt trial-and-error efforts from the design procedures. Moreover, the stability analysis is transparent by examining the algebraic conditions of the state-space equations obtained from the HRNN. That is, if all eigenvalues of the system matrix A in (1) lie within a unit circle, the HRNN is stable.

linear feedback control theory can be directly applied for such a task. From Fig. 1b, the HRNN is a nonlinear dynamic model consisting of a static nonlinear model cascaded with a dynamic linear model. In establishing a dynamic linear model as a reference model, we can utilize the information of the dynamic linear model of the HRNN. If we compare (1) with the following general linear state-space model ^ þ 1Þ ¼ A xðkÞ ^ þ B^ vðkÞ, xðk ^yðkÞ ¼ C xðkÞ, ^

the difference between them is only the input transformation. Obviously, the transformation in (1) is nonlinear, while in (19) it is linear. To remove the nonlinear component of the trained HRNN, we train the NLE by the scheme shown in Fig. 5. Due to the operation region of activation functions in neural networks and the performance requirement of steady-state step response, the matrix B^ is selected randomly but satisfies the pre-specified region of steady-state value for the linear reference model. The NLE is a three-layered neural network whose numbers of the neurons in the input layer and the hidden layer are the same as the number of the estimated system order, and the dimension of output neurons is the same as those of the inputs of the unknown system. The activation functions are selected as the hyperbolic tangent sigmoid transfer functions. The functions of the NLE are summarized as follows: fsð2Þ ¼

With the identified HRNN, we develop the learning algorithm of the nonlinearity eliminator to remove the nonlinearity of the unknown system. Finally, based on the linear reference model, we present the design of a simple feedback linear controller for controlling the compound system, which is the NLE cascaded with the unknown system, to follow desired input signals. In our control scheme, the NLE plays a role in removing the nonlinear behavior of the unknown system so that the compound model, which is the system with the trained NLE, behaves like a dynamic linear model obtained from the HRNN. Based on the dynamic linear model, the design task of a feedback controller for the compound model becomes simple since the well-developed

J X

ð2Þ wð2Þ sr vr þ ds ,

ð20Þ

r¼1

    exp fsð2Þ exp fsð2Þ    , oð2Þ s ¼ exp fsð2Þ þ exp fsð2Þ

fqð3Þ ¼ 3.1. Learning algorithm of nonlinearity eliminator

ð19Þ

J X

ð2Þ ð3Þ ð3Þ wð3Þ qs os þdq ¼ oq ,

ð21Þ

ð22Þ

s¼1

where wsr(2) is the weight between the rth input neuron and the sth hidden neuron, and ds(2) the bias of the sth hidden neuron; wqs(3) the weight between the sth hidden neuron and the qth output neuron, and dq(3) the bias of the qth output neuron. Based on the errors between the linear reference model and the trained HRNN with the NLE, we can define the error function with respect to the adjustable parameters (w) as 2 ^ El ðw,kÞ ¼ 1=2ðyðkÞyðkÞÞ ¼ 1=2e2l ðkÞ,

ð23Þ

^ ^ and y(k) and y(k) are the actual output of where el ðkÞ ¼ yðkÞyðkÞ the trained HRNN and the desired linear reference output, respectively. The error signal in (23) is not directly applicable to tuning the NLE parameters. It needs to be propagated from the

J.-S. Wang, Y.-L. Hsu / Neurocomputing 74 (2010) 315–327

00

wð3Þ qs ðkÞ ¼

J X @@ wð3Þ cj ðk1Þel ðkÞ qs ðk1Þ þ zl j¼1

0

321

1 )1 @xjþ ðk1Þ ð2Þ A A  os , bjj ðk1Þ wjq þ ajj ðk1Þ ðexpðpj Þ þ expðpj ÞÞ2 @oð3Þ q ðkÞ

(

4

ð24Þ

)1 þ @x ðk1Þ j ð3Þ @ A, dð3Þ cj ðk1Þel ðkÞ bjj ðk1Þ wjq þ ajj ðk1Þ q ðkÞ ¼ dq ðk1Þ þ zl ðexpðpj Þ þexpðpj ÞÞ2 @oð3Þ q ðkÞ j¼1 0

wð2Þ sr ðkÞ

J X

00

(

p J X X @ @@ ¼ wð2Þ cj ðk1Þel ðkÞ sr ðk1Þ þ zl q¼1 j¼1

0

dð2Þ s ðkÞ ¼

00

4

(

p J X X @ @@ dð2Þ cj ðk1Þel ðkÞ s ðk1Þ þ zl q¼1 j¼1

bjj ðk1Þ

4 ðexpðpj Þ þ expðpj ÞÞ2

( bjj ðk1Þ

wjq þ ajj ðk1Þ

4 ðexpðpj Þ þ expðpj ÞÞ2

@xjþ ðk1Þ @oð3Þ q ðkÞ

wjq þ ajj ðk1Þ

ð25Þ

1 )1 A  wð3Þ A 

@xjþ ðk1Þ @oð3Þ q ðkÞ

qs

ðexpðfsð2Þ Þ þ expðfsð2Þ ÞÞ2

1 )1 A  wð3Þ A  qs

1

4

vr A,

4 ðexpðfsð2Þ Þ þexpðfsð2Þ ÞÞ2

ð26Þ 1 A

ð27Þ

system output through the HRNN to the corresponding parameters of the NLE. The recursive update rules for the parameters of the NLE are summarized as follows: where zl is the learning rate for adjusting all parameters of the NLE. If the NLE learning is perfect, the unknown system cascaded with the NLE will behave approximately like the linear reference model. Consequently, a conventional linear feedback controller can be designed effortlessly to control the unknown system. 3.2. Linear controller design We use a pole placement approach as an example to design a feedback linear controller based on the linear reference model obtained from the identified HRNN. Other linear controller design approaches can be applied to such a design task without much complexity. Assuming that (19) is a completely controllable system, we can thus choose the desired pole positions of the close-loop system to achieve satisfactory performance. According to the pole placement method proposed in [6], the plant input can ^ be set as uðkÞ ¼ NrðkÞKxðkÞ: First, we find the parameters of K so ^ ¼ 0 are in the desired locations. Then, that the roots of jzIAþ BKj to track the desired trajectory, the system is assumed to operate at a steady state, and the command matrix N ¼ Nu þ KNx can be determined by the following equation: " #" # " # 0Jm Nx B^ AIJJ ¼ , ð28Þ Nu Imm C 0mp where Nx A RJm , and Nu A Rpm . More detailed information can be found in [6]. During the control stage (as shown in Fig. 7), the pole placement method requires the feedback of actual state variables. However, the state variables are unobtainable for most practical systems. To solve this problem, we adopt the state variables of the HRNN as an approximation of the actual state variables since the HRNN state variables have been trained to depict the unknown system dynamics.

4. Simulation results To validate the effectiveness as well as performance of the proposed control approach, we have conducted extensive computer simulations for benchmark examples of unknown system control. These examples are used to illustrate our research advantages including the effectiveness of the self-construction algorithm and the superiority of our control scheme through comparisons of some existing methodologies. Example 1. Single-input-single-output (SISO) dynamic plant control. The controlled plant is the same as in [14,23,27] and is

given by " # yp ðkÞyp ðk1Þðyp ðkÞ þ 2:5Þ yp ðkþ 1Þ ¼ 0:35 þ uðkÞ : 1 þ y2p ðkÞ þ y2p ðk1Þ

ð29Þ

Note that in our simulations, the plant model was only used for the generation of input–output data. None of the plant information was used in the modeling or control procedures. We first performed the system modeling of the above dynamic plant using the HRNN with the input–output data. The modeling procedure consists of two phases: the structure learning phase and the parameter learning phase. The structure learning phase, the MDL principle is used to obtain a parsimonious but efficient structure. Subsequently, the recursive learning algorithm is utilized to optimize the performance of the HRNN in the parameter learning phase. The same procedure used in [14] was adopted to train the HRNN for comparisons of some existing approaches. An i.i.d. uniform sequence within [–1, 1] was generated as the input signal u(k) for half of the training time, and a sinusoid signal, sin(pk/45), was subsequently given for the remaining training time. A total of 4000 time steps were used for training the HRNN. We set the learning rates zac ¼0.006 and z ¼0.06. The testing input signal (as shown in (30)) was adopted from [14] to verify the identification performance of the HRNN 8 sinðpk=25Þ, 0 r k o250 > > > > > 1:0, 250 rk o500 > < 1:0, 500 r ko 750 ð30Þ uðkÞ ¼ > > > p k=25Þ þ 0:1sinð p k=32Þ 0:3sinð > > > : þ0:6sinðpk=10Þ: 750 rk o1000 In the beginning, the first 500 time steps were employed to decide the system order and initial values of the HRNN parameters by the MDL principle. The system order of the HRNN was identified as 2. Fig. 8 shows the curve of DLs for different system orders from 1 to 10. From Fig. 8, the value of DL is minimum when the estimated system order was 2. Subsequently, the online recursive learning algorithm was performed to optimize the parameters of the HRNN for the remaining training time steps. After the parameter learning process, the state-space equations extracted from the trained HRNN are as follows:     0:3504 0:2640 0:3543 0 xðk þ 1Þ ¼ xðkÞ þ NðuðkÞÞ; 0:0495 0:1239 0 0:3488

yðkÞ ¼ 0:1697 3:2404 xðkÞ, ð31Þ where x(k)¼ [x1(k), x2(k)] are the state variables and the total number of parameters of the trained HRNN is 12. The eigenvalues of the matrix A are 0.3980 and 0.0762. Fig. 9 illustrates the

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+

NLE Learning Algorithm

e

r

… …

B

+

yp

Unknown Dynamic System

…

^

…

_ N

N(u(k))

z

B

1

x^

A

K Fig. 7. The proposed control scheme using the state feedback control technique.

−100 −200 −300

DL

−400 −500 −600 −700 −800 −900 1

2

3

4

5 6 7 No. of system orders

8

9

10

Fig. 8. The DLs of the HRNN with different system order from 1 to 10 for Example 1.

Output

0.5

0

−0.5 100

200

300

400

500 Time Steps

600

700

800

900

1000

Fig. 9. The outputs of the SISO plant (solid curve) and the HRNN (dotted curve) for Example 1.

outputs of the unknown plant and the trained HRNN for the test input signal in (30). The trained HRNN closely emulates the given plant and achieves the accuracy of a mean squared error (MSE)¼7.9904  10  4.

In the NLE constructing phase, we first build a linear reference system with the matrices A and C directly copied from the trained HRNN, and matrix B^ is selected as [  0.5262, 1.4470]T. The above linear system is controllable since its controllability matrix is of

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where r(k) is a bounded reference input and we used the following reference input adopted from [27]. 8 sinðpk=25Þ, ko 500 > > > > > 1:0, 500 rk o 1000 > < 1000r k o1500 ð33Þ rðkÞ ¼ 1:0, > > > p k=25Þ þ 0:4sinð p k=32Þ 0:3sinð > > > : þ 0:3sinðpk=40Þ: kZ 1500

full rank. A total of 1000 time steps of an i.i.d. uniform sequence within the limits [–2, 2] are generated to train the NLE with a learning rate equal to 0.5. The training structure is shown in Fig. 5 and the MSE between the outputs of the linear reference model and the compound system, the HRNN cascaded with the NLE, is 8.9203  10  3. The NLE contains two hidden neurons and one output neuron, and the total number of parameters is 9. After training the NLE, a state feedback controller based on the pole placement approach and the linear reference system is designed to control the unknown system. By setting the system roots equal to {0.15, –0.25}, we obtained the state feedback gain K¼[–0.0098, 0.0437] and the state command matrix N ¼ 1.3628. To test the controller’s performance, we used a tracking model which is a second-order linear system given by ym ðk þ1Þ ¼ 0:6ym ðkÞ þ 0:2ym ðk1Þ þ 0:1rðkÞ,

323

The objective of this control problem is to determine a bounded control force u(k) such that limk-N ec(k) ¼ym(k)– yp(k)¼0. Figs. 10 a and b show the outputs of the trained NLE block and the controlled output trajectory of the unknown plant using our control approach (dotted curve), respectively. The MSE between the desired and controlled trajectories is 5.2721  10  5. This result is better than those of the RSONFIN [14] and the memory neural network [27]. We also performed the same

ð32Þ

Output

1 0

−1 −2 200

400

600

800

1000 1200 Time Steps

1400

1600

1800

2000

Desired trajectory Actual output (without using the propsed control scheme) Actual output (without the NLE) Actual output (with the proposed control scheme)

Output

0.5 0

−0.5 200

400

600

800

1000 1200 Time Steps

1400

1600

1800

2000

Fig. 10. (a) The output of the trained NLE. (b) The simulation results of the SISO plant under different situations.

Desired trajectory Actual output (without online tuning) Actual output (with the proposed control scheme)

Output

0.5 0 −0.5 200

400

600

800

1000

1200

1400

1600

1800

Time Steps Fig. 11. The simulation results of the proposed control scheme with measurement noise.

2000

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experiment with different situations: (1) without using the proposed control scheme and (2) without using the trained NLE. The corresponding output trajectories of the unknown plant are shown in Fig. 10b. The MSEs are 6.6360  10  2 and 5.0302  10  2 for the above two different situations, respectively. To demonstrate the robustness of the proposed control scheme, we added measurement noise to the SISO plant. The measurement noise was generated by 0.1 sin(k/45)+ 0.03 cos(k/ 25). First, we applied the proposed control scheme to control the SISO plant with the measurement noise but without online tuning the parameters of the NLE. The MSE of the controlled trajectory compared to the desired one is 3.1531  10  2. Next, we applied the whole control scheme to the same problem and the MSE is 7.9597  10  5. Fig. 11 shows the reference and controlled trajectories with the measurement noise. Table 1 summarizes the control performance of the aforementioned cases and validates the effectiveness of the proposed control scheme.

Table 1 Control performance comparisons for Example 1. Case

MSE

Without using our controller Our control scheme without the NLE Our control scheme Our control scheme without online tuning (measurement noise) Our control scheme (measurement noise)

6.6360  10  2 5.0302  10  2 5.2721  10  5 3.1531  10  2 7.9597  10  5

700 650

yp ðkþ 1Þ ¼ f ½yp ðkÞ,yp ðk1Þ,yp ðk2Þ,uðkÞ,uðk1Þ

ð34Þ

where f ½x1 ,x2 ,x3 ,x4 ,x5  ¼

x1 x2 x3 x5 ðx3 1Þ þ x4 1þ x23 þ x22

ð35Þ

The HRNN training procedure is similar to the previous example. A total of 9000 time steps, including 5000 time steps of an i.i.d. uniform sequence generated within [  2, 2] and a single sinusoid signal given by 1.05  sin(pk/45) for the remaining training time are generated to model the nonlinear plant. In the structure learning phase, we first applied the MDL principle to decide the system order and initial values of the HRNN parameters. The order determination algorithm was stopped after 500 time steps. The final system order was identified as 3. Fig. 12 shows the curve of DLs for different system orders from 1 to 10. The value of DL is minimum when the estimated system order was 3. The online recursive learning algorithm was subsequently performed for the remaining training time steps to optimize the network parameters with the learning rates xac ¼0.0006 and x ¼0.006. The total parameter number of the HRNN is 21 and its state-space equation extracted from the trained network is written as 2 3 0:3215 0:1556 0:0593 6 7 xðk þ 1Þ ¼ 4 0:1662 0:3384 0:1501 5xðkÞ 0:0315 0:0141 0:6293 2 3 0:8723 0 0 6 7 0:5617 0 þ4 0 5NðuðkÞÞ, 0 0 0:9324

yðkÞ ¼ 0:7398 1:4284 0:5831 xðkÞ: ð36Þ The eigenvalues of the matrix A are 0.2755,  0.2850, and  0.6368. Fig. 13 depicts the outputs of the nonlinear plant and the trained HRNN for the test input signal in (30). Table 2 shows the identification performance comparisons of the HRNN with some existing recurrent networks. These results illustrate the parsimony and outstanding performance of the HRNN for dynamic plant identification.

600 DL

Example 2. Dynamic plant control. The nonlinear plant with multiple time-delays [23] is given as

550 500

Table 2 Identification performance comparisons of the HRNN with some recurrent networks for Example 2.

450 400 1

2

3

4

5

6

7

8

9

10

No. of system orders Fig. 12. The DLs of the HRNN with different system order from 1 to 10 for Example 2.

Network type

No. of parameters

Training time (time steps)

MSE

HRNN RSONFIN [14] MNN [27]

21 38 81

9000 11,000 620,000

3.67  10  3 4.41  10  2 7.52  10  2

1

Output

0.5 0 −0.5 −1 100

200

300

400

500

600

700

800

900

1000

Time Steps Fig. 13. The actual output of the nonlinear plant (solid curve) and the output of the HRNN (dotted curve) for Example 2.

J.-S. Wang, Y.-L. Hsu / Neurocomputing 74 (2010) 315–327

325

Desired trajectory Actual output (without using the propsed control scheme) Actual output (without the NLE) Actual output (with the proposed control scheme)

Output

0.5 0 −0.5 200

400

600

800

1000 1200 Time Steps

1400

1600

1800

2000

Fig. 14. The actual outputs of the nonlinear plant under different situations.

The construction procedure of the NLE is the same as that in Example 1. The linear reference model is obtained and expressed as 2 3 2 3 0:3215 0:1556 0:0593 0:8710 6 7^ 6 7 ^ xðkþ 1Þ ¼ 4 0:1662 0:3384 0:1501 5xðkÞ þ 4 0:0513 5uðkÞ, ^ yðkÞ ¼ 0:7398

0:0141 1:4284

0:6293

^ 0:5831 xðkÞ:

75

70

0:8543 ð37Þ

DL

0:0315



80

65 The above linear system is controllable since its controllability matrix is of full rank. The system roots were selected to be {  0.35, –0.6, 0.4}, the learning rate was set as 0.5, and the state feedback gain K ¼[ 0.1082, 0.0650, 0.0062]. The state command matrix N ¼ 0:7315. The NLE is composed of three hidden neurons and one output neuron, and the total number of parameters of the NLE is 16. We used the same tracking model (32). Fig. 14 shows the output trajectories of the nonlinear system controlled by the proposed control approach with the above feedback controller (dotted curve) and the desired trajectory (solid curve). The MSE between the desired and controlled trajectories is 1.5982  10  4. Furthermore, in Fig. 14, we display the output trajectory of the nonlinear system without using the controller and the trajectory controlled by the proposed control scheme without using the NLE. The MSEs for the above two cases are 6.2059  10  3 and 1.8753  10  2, respectively. Example 3. Hammerstein dynamic plant control. The Hammerstein plant modified from [23] is given as yp ðk þ1Þ ¼ 0:7f0:3yp ðkÞ þ 0:6yp ðk1Þ þf ½uðkÞg,

ð38Þ

where f ½u ¼ 0:6sinðpuÞ þ 0:3sinð3puÞ þ 0:1sinð5puÞ:

ð39Þ

The HRNN training procedure is similar to the previous examples. A total of 10,000 time steps, including 5000 time steps of an i.i.d. uniform sequence generated within [  0.5, 0.5] and a single sinusoid signal given by 0.5  sin(pk/45) for the remaining training time are generated to model the nonlinear plant. And the magnitude of the test input signal is half of Example 1. The system order was identified as 2 via the MDL principle. Fig. 15 shows the curve of DLs for different system orders from 1 to 10. The value of DL is minimum when the system order was estimated at 2. The online recursive learning algorithm was subsequently performed for the remaining training time steps to optimize the network parameters with the learning rates xac ¼0.0006 and x ¼0.006. The total parameter number of the HRNN is 12 and its state-space equation extracted from the

60

55 1

2

3

4

5

6

7

8

9

10

No. of system orders Fig. 15. The DLs of the HRNN with different system order from 1 to 10 for Example 3.

trained network is written as    0:3726 1811 1:0197 xðk þ 1Þ ¼ xðkÞ þ 0:6767 0:6971 0

yðkÞ ¼ 0:4285 0:3980 xðkÞ:

 0 NðuðkÞÞ, 0:7870 ð40Þ

The eigenvalues of the matrix A are  0.4770 and 0.8014. Fig. 16 depicts the outputs of the nonlinear plant and the trained HRNN with MSE¼7.8315  10  3. The construction procedure of the NLE is the same as that in Example 1. The linear reference model is obtained and expressed as     0:3726 1811 0:0914 ^ þ 1Þ ¼ ^ þ xðk xðkÞ uðkÞ, 0:6767 0:6971 0:3930

^ ^ yðkÞ ¼ 0:4285 0:3980 xðkÞ: ð41Þ The above linear system is controllable since its controllability matrix is of full rank. The system roots were selected to be {0.15, 0.15}, the learning rate was set as 0.5, and the state feedback gain K¼[4.5852,1.8919]. The state command matrix N ¼ 3:4705. The NLE is composed of three hidden neurons and one output neuron, and the total number of parameters of the NLE is 9. In this example, we used the same tracking model (32) but the magnitude of the reference input r(k) is the double of (33). Fig. 17 shows the output trajectories of the nonlinear system controlled by the proposed control approach with the above

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Output

1 0.5 0 −0.5 −1 100

200

300

400

500 600 Time Steps

700

800

900

1000

Fig. 16. The actual output of the nonlinear plant (solid curve) and the output of the HRNN (dotted curve) for Example 3.

Output

1 0.5 0 −0.5 −1 200

400

600

800

1000

1200

1400

1600

1800

2000

Time Steps Fig. 17. The desired trajectory (solid curve) and the controlled output of the nonlinear plant (dotted curve).

feedback controller (dotted curve) and the desired trajectory (solid curve). The MSE between the desired and controlled trajectories is 2.7214  10  3.

5. Conclusions Identification and control are usually treated as two separate tasks in most controller design problems. Here, we regard these two tasks as an integral whole and take full advantage of the procedures conducted for system modeling to design an effective but simple linear controller. The design of the proposed control approach is based on the structure of the HRNN, which models an unknown system with a static nonlinear model and a dynamic linear model in a state-space representation. A self-construction algorithm is employed to construct a parsimonious HRNN with accurate identification capability for unknown system modeling. Based on the identified MDL-based HRNN, the design of the NLE and the linear feedback controllers can be done systematically and effortlessly. Unlike a traditional linearization method that linearizes unknown systems with respect to operating points; our linear reference model is obtained by removing the global nonlinear behavior of the unknown system using the NLE. If the system modeling and the NLE are perfect, the compound model, the unknown system cascaded with the inverse model, will behave like a pure linear model. Hence, the feedback linear controller designed based on the linear model can perform well in a wide operating range. Several remarkable results on unknown system control problems have confirmed the effectiveness and superior performance of the proposed control approach.

training is very sensitive to the system matrix A, and that a small change in matrix A can cause a dramatic variation in the evolution of the system dynamics. Hence, if all the eigenvalues of matrix A are located inside the unit circle, the overall stability of the HRNN can be ensured. Here, a multiplier method is used to develop a stable learning algorithm by introducing an additional cost function. The additional cost function is used to monitor the stability of matrix A and acts as a penalty function. Our cost function contains an error function and a penalty function for instability caused by unsuitable parameter adjustments for matrix A. The penalty function includes the stable constraints of matrix A, where the stable constraints can avoid the instability of the HRNN during the parameter learning procedure. We modified the original optimization problem in (8) to form the following objective function: minimize EðwÞ þ gPðaji Þ,

ðA1Þ

Subject to w A Rn , aji A R:

where g A R is a positive constant and is called the penalty parameter, P the penalty function, and n the total number of adjustable parameters. We applied the Schur–Cohn criterion [1] to derive the stability constraints of the HRNN as the penalty function. The Schur–Cohn criterion provides us a way to determine whether the roots of the characteristic polynomial lie inside the unit circle just by examining the coefficients of the polynomial, rather than directly solving for the roots. Here, we take the HRNN whose system order is 2 as an illustrative example. The penalty function can be written as Pðaji Þ ¼

2  X

giþ ðaji Þ

2

,

ðA2Þ

i¼1

Appendix where We have developed a stable learning algorithm to guarantee the stability of the network during/after the parameter learning phase. From the HRNN topology, we know that the parameter

( giþ ðaji Þ ¼ maxð0,gi ðaji ÞÞ ¼

0,ifgi ðaji Þ r 0, gi ðaji Þ, ifgi ðaji Þ 4 0,

ðA3Þ

J.-S. Wang, Y.-L. Hsu / Neurocomputing 74 (2010) 315–327

and based on the Schur–Cohn criterion, we obtain the following two constraints: g1 ðaji Þ ¼ ja11 þ a22 jða11 a22 a12 a21 Þ1o 0, g2 ðaji Þ ¼ ja11 a22 a12 a21 j1 o 0:

ðA4Þ

Subsequently, the stable update term @ þ Pðaji Þ=@aji can be obtained by  2  2  @ g1þ ðaji Þ þ g2þ ðaji Þ @Pðaji Þ ¼ , ðA5Þ @aji @aji where    @Pðaji Þ ¼ 2 g1þ ðaji Þ  signða11 þ a22 Þa22 þ g2þ ðaji Þ @a11    signða11 a22 a12 a21 Þa22 ,

ðA6Þ

 @Pðaji Þ ¼ 2 g1þ ðaji Þ  a21 g2þ ðaji Þ @a12    signða11 a22 a12 a21 Þa21

ðA7Þ

 @Pðaji Þ ¼ 2 g1þ ðaji Þ  a12 g2þ ðaji Þ @a21    signða11 a22 a12 a21 Þa12 ,

ðA8Þ

   @Pðaji Þ ¼ 2 g1þ ðaji Þ  signða11 þ a22 Þa11 þ g2þ ðaji Þ @a11    signða11 a22 a12 a21 Þa11 ,

ðA9Þ

Note that the update term @ þ Pðaji Þ=@aji only contains the derivatives of the unsatisfied constraints since giþ ðaji Þ ¼ 0, if giþ ðaji Þ r 0. This stable learning algorithm can ensure the network stability during and after the online parameter learning phase.

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Jeen-Shing Wang received his B.S. and M.S. degrees in electrical engineering from the University of Missouri, Columbia, MO, in 1996 and 1997, respectively, and Ph.D. degree from Purdue University, West Lafayette, IN, in 2001. He is currently an Associate Professor with the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan. His research interests include computational intelligence, intelligent control, clustering analysis, and optimization.

Yu-Liang Hsu received his B.S. degree in automatic control engineering from the Feng Chia University in 2004, and the M.S. degree in electrical engineering from National Cheng Kung University, Taiwan, R.O.C., in 2007. He is currently working toward the Ph.D. degree in electrical engineering in the Department of Electrical Engineering, National Cheng Kung University. His research interests include intelligent control, nonlinear system identification, and inertial sensing.