Fourier-Neural-Network-Based Learning Control for a Class of ...

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 1, JANUARY 2009

139

Fourier-Neural-Network-Based Learning Control for a Class of Nonlinear Systems With Flexible Components Wei Zuo, Yang Zhu, and Lilong Cai, Member, IEEE

Abstract—This paper considers an output feedback learning control for a class of uncertain nonlinear systems with flexible components. The distinct time delay caused by system flexibility leads to the phase lag phenomenon and low system bandwidth. Therefore, the tracking problem of such systems is very difficult and challenging. To improve the tracking performance of such systems, an iterative learning control scheme using the Fourier neural network (FNN) is presented in this paper. This scheme uses only local output information for feedback. FNN employs orthogonal complex Fourier exponentials as its activation functions and the physical meaning of its hidden-layer neurons is clear. The FNN-based learning controller introduced here relies on the frequency-domain method, which converts the tracking problem in the time domain into a number of regulation problems in the frequency domain. A novel phase compensation method is introduced to deal with the phase lag phenomenon, so that the bandwidth of the closed-loop system is increased. Experiments on a belt-driven positioning table are conducted to show the effectiveness of the proposed controller. Index Terms—Fourier neural network (FNN), iterative learning control, orthogonal activation function, output feedback, phase compensation.

I. INTRODUCTION N industrial motion control applications, flexible components have become attractive over the past few years due to potential advantages, such as faster operation, lighter weight, and less costly structures. To achieve precise position tracking control, the flexibility between the actuator and the tip sensor cannot be neglected, which inevitably introduces high nonlinear dynamics and increases the complexity of system modeling. In addition, system flexibility is a significant factor that causes low natural frequency, narrow bandwidth, and small phase margin. Therefore, regardless of the flexibility, rigid control schemes fail to provide satisfactory control performance. Especially in a system with a small damping ratio, elastic oscillation can lead to strongly resonant behavior and even instability.

I

Manuscript received October 02, 2007; revised March 28, 2008 and July 17, 2008; accepted August 05, 2008. First published December 22, 2008; current version published January 05, 2009. This work was supported by the Research Grants Council of Hong Kong, China under Project HKUST6114/03E. W. Zuo and Y. Zhu are with HyFun Technology Limited, Kowloon Bay, Hong Kong (e-mail: [email protected]; [email protected]). L. Cai is with the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNN.2008.2006496

Much research has been done in recent years on different approaches to attenuate the effect of flexibility. Spong proposed one of the first adaptive controllers for flexible-joint robots based on the singular perturbation formulation of robot dynamics [1]. The control law includes three parts, i.e., the rigid control, the fast control, and the corrective control [2]. The rigid control is derived from the rigid models of manipulators by neglecting joint flexibility. The fast control is used to compensate for the deviations in the system states from the integral manifold and damp out oscillation due to joint flexibility. By setting the perturbation parameter to zero, the corrective control can make the system behavior approximate that of a rigid system. However, without accurate modeling of the system’s dynamics, the convergence of the parameters in the adaptive controller cannot be guaranteed and the control performance is limited. Recently, robust control of flexible-joint manipulators with unmodeled parameters and unknown disturbances has been reported [3], [4]. Dawson et al. employed a robust tracking controller for a robot manipulator with bounded disturbances and parameter uncertainty and achieved global uniform ultimate bounded tracking performance [5]. Chen and Fukuda investigated the robust tip position control problem in flexible arms by using the sliding-mode method [6]. Although under certain conditions sliding-mode control is robust with respect to system uncertainties and external disturbances, this control strategy may incur large control chattering and excite unstable system dynamics [7]. The application of the concepts of fuzzy set theory to vibration control has attracted increasing interest in the past years. Fuzzy controllers afford a simple and robust framework for specific nonlinear control laws that accommodate uncertainty and imprecision [8]. Wang et al. proposed a fuzzy-neural controller for the direct adaptive control of uncertain nonlinear systems [9]. Lin et al. used a hierarchical fuzzy approach to supervisory control of robot manipulators with oscillatory bases [10]. Such control scheme can be applied to civil, mechanical, transport, and aeronautical engineering. In recent years, much has been written about neural networks (NNs), which are regarded as a powerful control technique due to their ability to learn, adapt, and approximate nonlinear functions to desired degrees of accuracy [11]–[14]. Compared with other control techniques, NN-based control does not require knowledge about the system model. Gutierrez and Lewis used an NN tracking controller on a single flexible link, which was capable of learning unmodeled nonlinear dynamics [15]. Tian et al. proposed a recurrent NN to estimate the underlying dynamics of the manipulator and applied it to the motion control of constraint flexible manipulators, where both the contact force

1045-9227/$25.00 © 2008 IEEE

140

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 1, JANUARY 2009

exerted by the manipulator and the position of the end-effect contacting with a surface were controlled [16]. NN-based control has been a good candidate in the nonlinear control field due to its advantages, including fault tolerance, learning, and parallelism. However, when using traditional NN in real applications, one faces difficulties, such as the selection of the network structure, the low convergent speed, the local minimum, and the stability analysis of the closed-loop system. Iterative learning control is a method designed to exploit repetitiveness in various control situations [17]–[21]. It was first proposed by Arimoto et al. in 1984 [17]. Huang et al. applied a decentralized learning control scheme for the tracking control of a 3-degrees-of-freedom (3-DOF) flexible-joint robot [18]. Gunnarsson et al. presented an iterative learning algorithm for a laboratory flexible robot arm, which improved the control performance by using the motor angle and the arm’s angular acceleration [19]. Currently, most of the learning controllers only employ the input/output information in the time domain. In investigating the frequency response of a system with flexible elements, the phase lags in different frequency components between the system input and output are quite obvious. We know that output errors may include high-frequency and resonant-frequency components. If an output feedback is directly added to the controller, due to the phase lags, these frequency components will be stimulated and cause the system to be unstable. The learning controllers designed in the time domain may lack the capability of dealing with a system’s phase lags, which cannot be ignored when flexible elements exist. In this paper, a new output feedback control method, a Fourier neural network (FNN)-based iterative learning controller, is proposed to improve the tracking performance of a class of uncertain nonlinear systems with flexible components. The underlying idea in the design of the FNN is to extend the Fourier learning control method [22]–[24] to NN-based control theory. The proposed FNN uses orthogonal complex Fourier exponentials as its basis functions. Therefore, it has a clear physical meaning, i.e., each neuron can be regarded as the frequency filter of the respective frequency component, so that the network structure is limited by the system bandwidth. Compared with traditional nonlinear controllers, a salient feature of the proposed control methodology is that the control algorithm is essentially carried out in the frequency domain. By employing a phase compensation technique in the frequency domain, the phase of each frequency component in the control signal is adjusted and the influence of the phase lags can be effectively eliminated. In addition, all the nonlinearities and uncertainties of the dynamical system are lumped together and iteratively compensated for by FNN so that a priori knowledge of the system model is not required. However, the computational cost of the FNN is still much heavier than traditional proportional–integral–derivative (PID), robust, and fuzzy controllers. Fortunately, because the structure of the FNN is limited by system bandwidth and the development of hardware technique is fast, this problem can be well solved in real implementations. This paper is organized as follows. In Section II, a system description and a problem formulation are provided. Section III introduces the FNN-based learning controller. The stability of the

system is analyzed in Section IV. Section V presents the experimental results. Finally, conclusions are offered in Section VI. II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION A. The System’s Dynamical Equation Consider a class of nonlinear systems that can be described by the following Lagrange equation: (1) is the system’s input, is the vector of the generalized coordinates, the system’s Lagrangian function satisfies , and and are the system kinetic energy and the potential energy, respectively. Generally, for rigid systems, the dimensions of the system’s generalized coordinates are identical with those of the system’s input. However, for systems with flexible components, these two dimensions are not the same because flexible components increase the number of the generalized coordinates. In this paper, we deal with the control problem of a class of nonlinear systems with flexible components that can also be expressed by (1). The system input vector is rewritten as with , , and , and the vector of the with generalized coordinates is . In most real industrial applications, the above system can be divided into a number of subsystems that are described as where

(2) .. .

In the presence of flexibility, the inputs cannot be directly applied to the controlled objects. Therefore, internal dynamics exist and we have and . In general, the flexibility of the th subsystem can be evaluated by the value of . A larger implies that the subsystem’s internal dynamics is more complicated. as the output of the subsystem, we have When choosing to take the internal dynamics into account to obtain a satisfactory description of the system’s dynamical behavior. In order to reveal the input/output relationship explicitly, all the dynamical equations in (2) are lumped together. Note that the total potential can be divided into two parts: and , energy where the former is all the potential energy stored in the flexible components and the latter is the remainder of the energy. Supposing that all the flexible components in (2) are connected in series, we have (3)

ZUO et al.: FNN-BASED LEARNING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

The physical meaning of the above equation is that all the generalized internal forces in the flexible components can be canceled out. Therefore, dynamical equations in (2) are combined as follows: (4) In practical applications, a number of systems can be represented by (4), such as a single-link robot with joint flexibility [16]–[25], a belt-driven system [26], and a traditional spring damping system. It is important to realize that the dynamic equation (4) we have derived does not encompass all the effects acting on the system. In mechanical systems, the most important source of forces that are not included is friction, which contains viscous friction, Coulomb friction, etc. A fairly complex friction [27]–[29]. There are model would take the form of other effects that are also neglected in this model, e.g., bending effects (which give rise to resonances). With all the discussed effects considered, a more general expression of (4) is obtained (5) The system’s dynamical equation briefly described in (5) has already appeared in [22], [24], and [30]. It applies to almost all mechanical, electrical, and hydraulic systems. Both rigid and flexible systems can be described by (5). In a flexible component, e.g., a spring, its elastic coefficient is relatively small, which means that only a slight external force will cause a notable deformation or displacement. Therefore, vibration often occurs and introduces high nonlinearities and uncertainties. From a frequency-domain point of view, a flexible component has a low natural frequency due to its small elastic coefficient . As we know, the bandwidth is an excellent measurement of the range of fidelity of the response of a system. When flexible components are utilized, the whole system’s bandwidth is narrowed so that the system’s transient response will be slow. Moreover, due to the flexibility, an obvious time delay between the system’s input and output can be observed, which incurs a remarkable phase lag phenomenon in the frequency domain. contains not Notice that the nonlinear function only the internal dynamics but also couplings from other subsystems. We can adopt the time domain’s decentralized control scheme to handle each subsystem [18]. The couplings in are treated as deterministic disturbances. If they are bounded, a decentralized controller can be designed to stabilize each subsystem by using only the local output information. When is perfectly compensated for, each subsystem would be totally decoupled and decentralized control would work as well as centralized control. For simplicity, in the following discussion, we focus only on the design of a single-input control system and the results can be easily extended to multiple-input systems. B. The Tracking Control Problem Formulation Based on (5), we consider a class of single-input–singleoutput (SISO) nonlinear systems described by

(6)

141

represent the generalized where coordinates of the system, is the system output, and is the system input. For the tracking problem of system (6), we should find a conto make follow the given desired trajectory trol action , which is defined on the interval . When using an iterative learning controller to perform the control task, the obsuch that jective is to design a learning algorithm the input function generated by the iteration (7) . converges to a fixed optimal system input function To solve this problem, we need to find a nonlinear feedback , which, when substituted into (6), results in a control input linear closed-loop system. Therefore, we design the controller as (8) where is a conventional propordenotes tional–derivative (PD) controller, and are positive constant PD the output position error, gains, and is the feedforward control action to be designed. With this control law (8), the closed-loop system dynamics becomes (9) If one chooses an optimal feedforward to compensate for , then the closed-loop system becomes a linear system (10) and the tracking error converges to zero as time passes. However, in a nonlinear system with flexible components, such a control action is quite difficult to find. The feedforward input cannot be straightforwardly calculated since the structure and parameters of the system are not precisely known. In particular, when only using the local output information as feedback, the control performance may be unacceptable due to the complex internal dynamics. It is a challenging problem to guarantee the system’s stability and control performance in the presence of flexibility. Due to the low bandwidth and the phase lags caused by system flexibility, the phase margin of the system is small. As a result, the controller’s design as well as the gain selection should be done with care to avoid pushing the system over the edge of stability [31]. In the design of output feedback controllers, an important key to dealing with flexibility is to compensate for the phase lags between the system’s input and output. If the phases of the input are properly shifted, then after transmission in the flexible components, the phases of the output may be desirable. A suitable controller should meet the requirements of compensating for the nonlinearities and uncertainties in the time domain and reducing the effects of phase lags in the frequency domain. In this paper, an FNN-based learning controller is proposed to generate the feedforward input function of the system. This control scheme is essentially a frequency-domain method that

142

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 1, JANUARY 2009

Fig. 1. Block diagram of the closed-loop system with the proposed controller.

deals with the harmonic components of the feedforward function. By properly tuning the magnitude and phase of each frequency component, the output of the FNN can efficiently eliminate oscillation in the system and improve the control performance. However, the FNN can only be applied to open-loop stable systems or to systems that have been stabilized. In addition, it is not able to response to random disturbances immediately during the control process because its weights are updated at the end of each trial. Thus, the time-domain PD controller has to be used to handle disturbances and decrease system variability. Fig. 1 presents a block diagram of the overall controller. III. THE FNN-BASED LEARNING CONTROLLER DESIGN A. Fourier Neural Networks The ability of NNs to approximate continuous functions has been widely studied [11], [32], [33]. A nonlinear function can be expressed by the NN in a compact matrix form (11) is the network weight, , denotes the is called the NN functional reactivation function, and construction error. Typical examples of are the sigmoid , the hyperbolic tangent function function , and the Gaussian function . By means of NN, a complicated nonlinear function is represented by the combination of a number of simple functions. If we can properly choose the structure, activation functions, and learning algorithm for an NN, the estimation error can be reduced to any desired accuracy. Fourier analysis is typically thought of as decomposing a signal into its component frequencies with different amplitudes and phases. For a given function , defined on , the Fourier harmonic basis functions are determined by , where represents the base fre, and quency, is the imaginary unit that satisfies is in accordance with Euler’s formula. The orthogonality of the complex Fourier functions can be demonstrated by where

Fig. 2. Block diagram of the FNN learning process in the frequency domain.

The FNN is proposed in light of the Fourier series and neural network theory, which employs complex Fourier harmonic basis functions as its activation functions. The feedforward function , defined on , can be generated by FNN in the following form: (13) where is the family of the orthogonal activation functions, represents the net. work weights, and the node number of the network is Constructed on the basis of orthogonal Fourier activation functions, FNN can be looked at as a particular case of NN. The update of the network weights is based upon the spectrum of the PD output in the previous trial, which indicates the obtained performance and serves as a “sensor” of the FNN learning controller. Fig. 2 shows the block diagram of the FNN learning process, which deals with the historical spectral information in the frequency domain. If a learning algorithm is to the spectrum of the optimal found to drive the value of feedforward function, then the tracking problem is solved. Due to the clear physical meaning of FNN, the selection of the relies on the system bandwidth, and the link term number between the network coefficients and the Fourier transform is explicit. Moreover, because of the orthogonality of the basis functions, FNN may perform excellently in nonlinear function modeling and decomposition and achieve a faster convergent speed, i.e., fewer trials are required to achieve satisfactory control performance. B. A Learning Algorithm Design in Fourier Space According to (8) and (9), the closed-loop system dynamics in the th trial is given by (14) Projecting each term in (14) to FNN orthogonal activation functions, we get (15)

(12)

where coefficients of the system’s dynamical function

are Fourier ,

ZUO et al.: FNN-BASED LEARNING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

143

are network weights of the FNN controller, and are Fourier coefficients of the PD controller , which are represented by (16)

, (16) is Due to the orthogonality of the basis functions equations in Fourier space decentralized into (17) represent the spectrum of the optimal feedforward Let . According to Fourier analysis, should be a function is unknown, we cannot directly make constant vector. Since approximate to . Thus, the control strategy is to make tend to , i.e., make tend to zero. When this objective , which means that approaches is achieved, we have . Therefore, the tracking problem in the constant spectrum the time domain is now transformed into a series of regulation problems in the frequency domain, which are easy to cope with. tend to zero, we intuitively update the network To make weight with the following learning algorithm [23], [30]: (18) where is the learning gain. Learning algorithm (18) exploits historical information in the frequency domain to update the network weights. The tracking is simply integrated in by multiplying output spectrum the learning gain . The update law (18) has proved to be efficient for the tracking control of rigid systems in [18], [22], and [30]. However, it is not suitable for the control of nonlinear systems with flexible components. Due to flexibility, the phases of output harmonics components are quite different from those of input harmonic components. Distinct phase lags can be found, especially in high-frequency and resonant-frequency components. Lacking in the ability of compensating for the phase lags, the update law (18) may introduce positive output feedback and cause oscillation in the system. Therefore, an advanced learning algorithm with phase compensation in different frequency components is required. In the following, for the sake of convenience, we use and to represent the magnitude and the phase of a vector, and , respectively. Let . Then, we define a vector as and obtain (19) and the phase is where the magnitude is . Similarly, we also define , , , , and their magnitudes and phases. The phase lag between and in the th trial can be expressed as (20)

Fig. 3. Illustration of the phase lag and phase compensation method.

The control objective is to make tend to zero and the network weights converge to the optimal coefficients. From . learning algorithm (18), we know that This implies that both the magnitude and the phase of need to be considered to make converge. However, with (18), we cannot guarantee the convergence of in the presence of the phase lag . As illustrated in Fig. 3, if the phase of is not appropriate, the magnitude of becomes even larger. The only solution is to modify the feedforward input of the system by adjusting the FNN weights, i.e., changing the , so that a proper output phase can be input phase obtained. Hence, a phase compensation method is introduced to with improve the learning algorithm by means of shifting a compensative phase . should be Obviously, the optimal phase of , i.e., the same direction as the vector in Fig. 4. Therefore, the selection of the compensative phase should be based on the value of . We should increase when , and decrease when (we assume that the value of is adjusted to fall in the range ). However, in the beginning of the th trial, we do not know the exact phase th trial lag in this trial, so we will use the phase lag in the instead, which can be calculated from and . It is worth mentioning that due to the existence of the phase estimation error, optimal phase compensation may be not achievable. Based on the above discussion, the new learning algorithm with is designed as follows: the compensative phase (21a) (21b) and in one harIn linear systems, the variations of monic component are independent of other harmonics. However, in nonlinear systems, the variation in one harmonic component will alter other harmonic components, i.e., the harmonic components are cross related. Therefore, with the control law

144

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 1, JANUARY 2009

Fig. 4. Block diagram of the FNN architecture.

[(21a) and (21b)], the condition may not be sufficient converges to zero. In the next section, the to ensure that system stability and the tracking performance of the proposed controller are analyzed. A sufficient condition is given to ensure that the tracking error converges as the number of trial increases.

From the control law (21a), we know that (23) Substituting (23) into (22) yields (24)

IV. STABILITY ANALYSIS OF THE CLOSED-LOOP SYSTEM Before the stability analysis, we assume that the controlled system satisfies the following conditions. of the system Assumption 1: The desired trajectory lasts for a finite time. It complies with the actuator’s capacity and is continuous. Assumption 2: The system’s internal dynamics as well as external disturbances are bounded. Assumption 3: The system can be stabilizable by output feedback control, although the control performance may be limited. th trial Assumption 4: From the th trial to the , the variation of the phase lag in the th harmonic is bounded by , such that . Remark 1: The last assumption is about the characteristics of the system’s phase lags. In classical control theory, it is proved that the phase lags of a linear system are invariant. However, those of nonlinear systems are affected by both the structure of the plant and the system input. In different iterations, due to the change of the system input, the phase lags are quite different. The meaning of Assumption 4 is that although the phase lag of a certain frequency component is not a constant value, its variation lies in a certain range with a width of . If Assumptions 1–4 are all satisfied, the sufficient condition for system stability and convergence of the tracking error is presented by the following theorem. Theorem 1: With the controller (8) and the update law (21a) and (21b), the sufficient condition for the convergence of the , and the tracking error is the learning gain closed-loop system input/output relation in the Fourier space for all and . satisfies Proof: Based on (17), we have (22)

Using the triangle inequality in the vector space, we have (25) The learning gain is chosen as , so that holds for any . Then, after substituting into (25), we obtain (26) With the phase compensation method (21b), our aim is to ad. just the phase of the input to make However, due to the phase lag variation in different trials, we cannot accomplish accurate phase compensation. In the pressatisfies the ence of the estimation error , the phase of following: (27) Then, the magnitude of

is calculated by

(28) In the last step of the above deduction, the Euler formula is used. Considering , we get (26) and

(29)

ZUO et al.: FNN-BASED LEARNING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

Thus, the relation between (28) as follows:

and

145

is obtained from

(30) Since the initial value is bounded, as the trial number increases, each Fourier coefficient of the PD controller output will diminish asymptotically. We will finally obtain the dynamic will equation (10) in the time domain. As a result, the error tend to zero along the sliding surface constructed by (10) as time goes on. Remark 2: The parameters of FNN’s activation functions can be physically determined. No extra computational cost is needed to tune any parameter of the basis functions as required in Gaussian or sigmoid networks. According to [34] and [35], if we use a traditional NN to represent a given function , there are many possible solutions that have different layer numbers, node numbers, and network weights. In contrast, due to the orthogonality of the basis functions, the solution of FNN is unique. A wavelet neural network employs scaling functions as its basis functions, which are also orthogonal [7]. However, due to the lack of clear physical meaning, the network structure is still not very easy to determine. For the proposed FNN, each of its hidden-layer neurons can be regarded as the frequency filter of the respective frequency component, and the complexity of the network structure is limited by the system bandwidth. and vary from trial to trial. Remark 3: Both only gives a The condition conservative robust bound. In linear systems, and are independent of other harmonic components, so that the actual bound of is smaller and the learning gain can be selected to be larger for fast convergence rates. In nonlinear and are affected by other harmonic systems, both is larger. A components, so the actual bound of smaller learning gain should be selected to ensure the stability. Remark 4: In using the FNN-based learning controller for a nonlinear system with flexible components, the phase lags are carefully compensated for. The estimation of the phase lags plays a very important role in the improvement of the tracking performance. If the estimation is more precise, i.e., is smaller, a larger learning gain can be selected in the range , so that a faster convergence rate of the tracking error is achieved. On the other hand, if we do not adopt the phase compensation technique or the phase lag estimation is will be larger and lead to a smaller learning less accurate, gain and a slower convergence speed. In other words, due to the phase compensation, it is possible to apply high gains in FNN without exciting the high-frequency components. In real applications, the implementation of the FNN learning algorithm is in sine/cosine form. The detail steps are presented here. Step 1) Search PD controller gains and choose the learning , let gain . Set the learning counter to , and set . , where we Step 2) Calculate have and .

Fig. 5. Experimental setup.

and for all and obtain the Step 3) Calculate compensative angle , based on the learning algorithm (21b). and with the following algorithm, Step 4) Update which is the sin/cosine form of (21a): (31a) (31b) Step 5) Calculate the output of FNN, which is . Step 6) If the error is acceptable, then the learning process ends. Otherwise, increase by 1 and go to Step 2. To obtain a fast convergent speed of the error as increases, we may repeat the above steps with different PD gains and learning gains. In this case, we reduce the learning gain first and then modify PD gains as required. Fig. 4 shows the block diagram of the structure of FNN. V. EXPERIMENTAL RESULTS In this section, experimental results are provided to show the effectiveness of the proposed controller. Comparisons between the controllers without phase compensation and with phase compensation are made. In addition, the spectrum of the controller output is analyzed. A. Experimental Setup We selected the Daedal 500000PD series belt-driven positioning table as our experimental setup. A linear encoder with 1- m resolution measures the position of the moving plate. The velocity of the plate is obtained by numerically differentiating the position with respect to time. The permanent magnet direct current (dc) servomotors were made by Galil Motion Control Inc. (Rocklin, CA) and driven by a 25A8 pulse width modulation (PWM) amplifiers. A computer equipped with an analog-todigital (A/D), digital-to-analog (D/A) card and a decoder card was used to process the control algorithm. The experimental setup and its schematic diagram are given in Figs. 5 and 6, respectively. The definitions of notation of the system parameters are listed in Table I.

146

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 1, JANUARY 2009

TABLE I DEFINITIONS OF NOTATIONS OF THE BELT-DRIVEN POSITIONING TABLE

When the input/output mapping is considered, (32) can be rewritten as

(33) This system is actually a sixth-order system with strong flexibilities. The internal dynamics of the positioning table is complex and the vibration of the belts introduces extra nonlinearities and uncertainties. However, although the dynamical model of the system is complicated, when using the proposed FNN-based learning controller, we need not model the system and measure the parameters. B. Experiments on the Belt-Driven Positioning Table The desired output position trajectory is given by (34)

Fig. 6. Schematic diagram of a belt-driven positioning table.

As shown in Fig. 6, the belt can be divided into three parts and there are three generalized coordinates , , and . The corresponding tensions can be written as , , and . Neglecting the inductance of the motor, the relation between the and the input voltage is . motor torque Thus, the dynamics of the system can be briefly modeled as follows:

(32)

First, we used the learning algorithm (18) without phase comto perform the control. The controller strucpensation ture and parameters are presented in Table II. Two FNNs, which have different hidden-layer neurons, were employed to do the same task. Fig. 7 shows the position error of an FNN controller with 13 hidden-layer neurons. Fig. 8 shows the tracking error of an FNN controller with 41 hidden-layer neurons. Note that the hidden-layer neuron numbers of the two FNNs correspond to the frequency bandwidths of 15 and 50 Hz, respectively. In the first FNN with fewer neurons, only a narrow frequency band was covered. The convergence of the tracking error almost stopped after seven trials. Fig. 9 shows the spectrum of the tracking error in the tenth trial. The spectrum shows that the error was dominated by the high-frequency components. This means we

ZUO et al.: FNN-BASED LEARNING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

147

TABLE II AFNN CONTROLLER DESIGN AND PARAMETER SELECTION

Fig. 9. Error spectrum (tenth trial) of a 13-hidden-layer-neuron FNN (without phase compensation).

Fig. 10. Error spectrum (tenth trial) of a 41-hidden-layer-neuron FNN (without phase compensation).

Fig. 7. Tracking error of a 13-hidden-layer-neuron FNN (without phase compensation).

Fig. 11. Tracking error of a 41-hidden-layer-neuron FNN (with phase compensation).

Fig. 8. Tracking error of a 41-hidden-layer-neuron FNN (without phase compensation).

cannot handle frequencies larger than 15 Hz. Therefore, a reasonable solution is to increase the number of hidden-layer neurons as in the second FNN. However, although we employed more frequency components, the control performance in Fig. 8 was worse than that in Fig. 7. The tacking error of the closedloop system even became divergent. In fact, an obvious vibration of the belt can be observed during the experiment, which is the major resource of the tracking error. Fig. 10 is the spectrum of the error in the tenth trial when using the second FNN controller. The amplitudes of the high-frequency components are almost ten times larger than those in Fig. 9. This phenomenon was caused by the flexibility of the system, which led to notable phase lags in the high-frequency components. From Figs. 9 and 10, we can see that the resonant frequency of the belt is

around 20 Hz (the eighth harmonic term). We face a dilemma on whether to employ the resonant-frequency components. On one hand, without using these terms, the control performance is not good due to the high-frequency errors. On the other hand, if we only employ the control law (18) and do not compensate for the phase lags, the resonant-frequency components will be stimulated and cause instability of the closed-loop system. Then, we used the updated learning algorithm (31a) and (31b) with the second FNN to perform the same experiment. In this experiment, the vibration of the belt is not comparable with the vibration in the previous experiment. The tracking error is shown in Fig. 11. In the tenth trial, the position error was very small 10 m). For the spectrum of the error in the tenth (within trial shown in Fig. 12, the amplitudes of the high-frequency components are greatly reduced. This fact proved that the phase compensation method is efficient in dealing with the phase lags and capable of highly depressing the belt vibration. It is well known that most controllers have a bandwidth limitation due to the phase margin. The phase shifting in the proposed controller makes it possible to use some frequency components outside the

148

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 1, JANUARY 2009

Fig. 12. Error spectrum (tenth trial) of a 41-hidden-layer-neuron FNN (with phase compensation).

Fig. 13. Tracking error of a PID controller.

bandwidth to improve the control performance without destabilizing the closed-loop system. Therefore, in some sense, the bandwidth of the closed-loop system is widened. We revisit the learning algorithm of FNN in (31a) and (31b). There are two cross terms in the control law, i.e., in the expression of , and in the expression of . These two terms not only change the phase of each frequency component but also suggest a recurrent structure of the FNN. NNs are closely modeled on biological processes for information processing. We know that biological systems always have feedback in their operations, such as the cerebellum and its associated nerves. This fact shows that an NN may have to be recurrent to improve its learning capability. In FNN, the link and is not arbitrarily selected but between each pair of mathematically determined, so that it has the potential to offer a new recurrent structure. Future research will investigate the recurrent network structure of FNN and compare it with common feedforward structures.

Fig. 14. Tracking error of the NN controller.

C. Analysis of the Experimental Results Positioning tables are widely used in many industrial applications and various control methods have proved to be efficient to control such systems. For comparison purposes, we also employed a PID controller and an NN controller [35] for the same experimental system. The parameters of the PID controller were carefully tuned so that good tracking performances were ob, , and ). tained ( For the design of the NN-based controller, there is no need for the dynamical system model. A three-layer neural network was employed to construct the controller, whose activation function of hidden-layer neurons was selected as the standard sigmoid function. We changed the hidden-layer neuron number in the experiments and the best control performance was obtained with the layer number of 35. The controller parameters were and , and a novel adapselected as tive weight tuning algorithm based on the system dynamics was employed to update the network weights, which can be found in [35]. Figs. 13 and 14 show the tracking errors of the PID controller and the NN controller, respectively. Fig. 15 shows the position error (tenth trial) of the FNN controller with phase compensation. Compared with the PID controller and the NN controller, the FNN controller provided a much better control performance. The outputs of the three controllers are also illustrated in Figs. 16, 17, and 18, respectively. It can be seen that

Fig. 15. Tracking error of the FNN controller (tenth trial).

Fig. 16. Output voltage of the PID controller.

the output voltage of the FNN controller was smoother than that of the PID controller. The reason is that FNN only employs finite-frequency components and high-frequency noises are filtered. In order to gain a deeper understanding of the proposed FNN control scheme, the spectra of the outputs of the PID and FNN controllers were analyzed. Because the magnitudes of the highfrequency components in the spectra were small, we focused on the behavior of the low-frequency components. In Figs. 19 and

ZUO et al.: FNN-BASED LEARNING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

149

Fig. 21. Phase lags of sixth and 11th harmonic components in different trials. Fig. 17. Output voltage of the NN controller.

Fig. 22. Phase lags of different harmonic components (20th trial). Fig. 18. Output voltage of the FNN controller (tenth trial).

Fig. 19. Amplitude spectrum of the output voltage of the PID controller and that of the FNN controller.

Fig. 20. Phase spectrum of the output voltage of the PID controller and that of the FNN controller.

20, only the first ten harmonic components are presented. Fig. 19 shows the amplitude spectra of the output voltages and Fig. 20 presents the phase spectra of the output voltages. The dc components of the input voltages are not zero in the figures. This means that the nonlinearities in the closed-loop system are asymmetric. The other phenomenon is that the amplitude of each frequency

term decreases as the order of the harmonic increases. The dominance of the low-frequency components shows that it is possible for FNN to use finite neurons to achieve satisfactory performance. As indicated in Fig. 19, the amplitudes of the FNN controller’s output are a little bit smaller than those of the PID controller’s output. Meanwhile, remarkable differences between the corresponding phases are clearly depicted in Fig. 20, especially in the high-frequency components. This implies that although the energy that was input into the systems was nearly the same, due to the distinct phases, the tracking performances of the two controllers were quite different. In fact, when the FNN controller was used, the energy consumption was lower. The phase compensation in the FNN controller plays a very important role in improving the control performance. By using the phase compensation method, the phase lag of each frequency component can be well handled. For a certain frequency component, we found that its phase lag varies in different trials. In Fig. 21, we can see the change of the phase lags of the sixth and 11th harmonic components from trial to trial. It is quite clear that the phase lag is input dependent. As a result, its variation in the first ten trials was large because the system input changed a lot. However, as the trial number increased, the tracking error became smaller and the input did not change much, so that the phase lag tended to a constant value. In Fig. 22, we investigated the phase lags of different harmonic components in the 20th trial. We found that they were not in a monotonic increasing or decreasing sequence. It is not feasible to treat the phase lags simply as a constant time delay. Therefore, phase compensation is difficult to achieve with common controllers in the time domain. VI. CONCLUSION Flexibility in a system causes extreme difficulty in system modeling. It is a potential source of uncertainty that can degrade

150

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 1, JANUARY 2009

the control performance and even destabilize the system in some cases. In this research, an FNN-based output feedback learning control scheme is proposed to improve the tracking control performance of a class of uncertain nonlinear systems with flexible components. Constructed on the family of orthogonal Fourier functions, FNN has a clear physical meaning that makes the determination of the network topology convenient. This model-free method can iteratively compensate for nonlinearities and uncertainties of the system, which are lumped together with no restrictions on their relationship. By using the proposed FNN controller, both the magnitude and phase of each frequency component of the system input can be well handled. In particular, the phase compensation technique is capable of dealing with phase lags, which is very common in systems with flexible components. The novel learning algorithm is actually carried out in the Fourier space, and it can guarantee the stability of the closed-loop system and the convergence of the tracking error. The experiments conducted on a belt-driven positioning table verified the effectiveness of the proposed learning controller.

REFERENCES [1] M. W. Spong, “Adaptive control of flexible-joint manipulator,” Syst. Control Lett., vol. 13, pp. 15–21, 1989. [2] W. Zhu, Z. Bien, and J. D. Schutter, “Adaptive motion/force control of multiple manipulators with joint flexibility based on virtual decomposition,” IEEE Trans. Autom. Control, vol. 43, no. 1, pp. 46–60, Jan. 1998. [3] P. Tomei, “Tracking control of flexible joint robots with uncertain parameters and disturbances,” IEEE Trans. Autom. Control, vol. 39, no. 5, pp. 1067–1072, May 1994. [4] M. M. Bridges, D. M. Dawson, Z. Qu, and S. C. Martindale, “Robust control of rigid-link flexible-joint robots with redundant joint actuators,” IEEE Trans. Syst. Man Cybern., vol. 24, no. 7, pp. 961–970, Jul. 1994. [5] D. M. Dawson, Z. Qu, M. M. Bridges, and J. J. Carroll, “Robust tracking of rigid-link flexible-joint electrically-driven robots,” presented at the 30th IEEE Conf. Decision Control, Brighton, U.K., 1991, unpublished. [6] X. Chen and T. Fukuda, “Robust sliding-mode tip position control for flexible arms,” IEEE Trans. Ind. Electron., vol. 48, no. 6, pp. 1048–1056, Dec. 2001. [7] R. J. Wai and R. Y. Duan, “Wavelet neural network control for induction motor drive using sliding-mode design technique,” IEEE Trans. Ind. Electron., vol. 50, no. 4, pp. 733–748, Aug. 2003. [8] J. Lin, “Multi-timescale fuzzy controller for a continuum with a moving oscillator,” Inst. Electr. Eng. Proc. Control Theory Appl., vol. 151, pp. 310–318, 2004. [9] Y. C. Wang, C. J. Chien, and C. C. Teng, “Direct adaptive iterative learning control of nonlinear systems using an output-recurrent fuzzy neural network,” IEEE Trans. Syst. Man Cybern. B. Cybern., vol. 34, no. 3, pp. 1348–1359, Jun. 2004. [10] J. Lin and Z. Z. Huang, “A hierarchical fuzzy approach to supervisory control of robot manipulators with oscillatory bases,” IFAC Mechatronics, vol. 17, pp. 589–600, 2007. [11] F. L. Lewis, A. Yegildirek, and K. Liu, “Multilayer neural-net robot controller with guaranteed tracking performance,” IEEE Trans. Neural Netw., vol. 7, no. 2, pp. 388–399, Mar. 1996. [12] W. Wang, C. Cheng, and Y. Leu, “An online GA-based output-feedback direct adaptive fuzzy-neural controller for uncertain nonlinear systems,” IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 34, no. 1, pp. 334–345, Feb. 2004. [13] C. F. Hsu, “Self-organizing adaptive fuzzy neural control for a class of nonlinear systems,” IEEE Trans. Neural Netw., vol. 18, no. 4, pp. 1232–1241, Jul. 2007. [14] R. J. Wai and J. D. Lee, “Adaptive fuzzy-neural-network control for maglev transportation systems,” IEEE Trans. Neural Netw., vol. 19, no. 1, pp. 54–70, Jan. 2008.

[15] L. B. Gutierrez, F. L. Lewis, and J. A. Lowe, “Implementation of a neural network tracking controller for a single flexible link: Comparison with PD and PID controllers,” IEEE Trans. Ind. Electron., vol. 45, no. 2, pp. 307–318, Apr. 1998. [16] L. Tian, J. Wang, and Z. Mao, “Constrained motion control of flexible robot manipulators based on recurrent neural networks,” IEEE Trans. Syst. Man Cybern. B. Cybern., vol. 34, no. 3, pp. 1541–1552, Jun. 2004. [17] S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operation of robots by learning,” J. Robot. Syst., vol. 1, pp. 706–712, 1984. [18] W. Huang, L. Cai, and X. Tang, “Decentralized learning control for robot manipulators,” in Proc. IEEE Int. Conf. Robot. Autom., Detroit, MI, May 1999, vol. 2, pp. 843–848. [19] S. Gunnarsson, M. Norrlof, E. Rahic, and M. Oabek, “Iterative learning control of a flexible robot arm using accelerometers,” in Proc. IEEE Int. Conf. Control Appl., Taipei, Taiwan, Sep. 2004, vol. 2, pp. 1012–1016. [20] X. Ruan, “Decentralized iterative learning control to large-scale industrial processes for nonrepetitive trajectory tracking,” IEEE Trans. Syst. Man Cybern. A, Syst. Humans, vol. 38, no. 1, pp. 238–252, Jan. 2008. [21] H. S. Ahn, Y. Q. Chen, and K. L. Moore, “Iterative learning control: Brief survey and categorization,” IEEE Trans. Syst. Man Cybern. C, Appl. Rev., vol. 37, no. 6, pp. 1099–1121, Nov. 2007. [22] L. Cai and W. Huang, “Fourier series based learning control and application to positioning table,” Robot. Autonom. Syst., vol. 32, pp. 89–100, 1999. [23] X. Tang, L. Cai, and W. Q. Huang, “A learning controller for robot manipulators using Fourier series,” IEEE Trans. Robot. Autom., vol. 16, no. 1, pp. 36–45, Feb. 2000. [24] W. Zuo and L. Cai, “Tracking control of nonlinear systems using Fourier neural network,” in Proc. IEEE/ASME Int. Conf. Adv. Intell. Mechatron., Monterey, CA, Jul. 2005, pp. 670–675. [25] M. W. Spong and M. Vidyasagar, Robot Dynamics and Control. New York: Wiley, 1989. [26] Z. Yang and L. Cai, “Tracking control of a belt-driven position table using Fourier series based learning control scheme,” in Proc. IEEE Int. Conf. Robot. Intell. Syst. Signal Process., Changsha, China, Oct. 2003, vol. 1, pp. 196–201. [27] S. Yang and M. Tomizuka, “Adaptive pulse width control for precise positioning under the influence of stiction and Coulomb friction,” J. Dyn. Syst. Meas. Control, vol. 110, pp. 221–227, 1988. [28] B. Armstrong-Helouvry, P. Dupont, and C. C. d. Wit, “A survey of models, analysis tools, and compensation methods for control of machines with friction,” Automatica, vol. 30, pp. 1083–1138, 1994. [29] C. C. d. Wit, H. Olsson, K. J. Astrom, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Trans. Autom. Control, vol. 40, no. 3, pp. 419–425, Mar. 1995. [30] W. Huang and L. Cai, “New hybrid controller for systems with deterministic uncertainties,” IEEE/ASME Trans. Mechatron., vol. 5, no. 4, pp. 342–348, Dec. 2000. [31] R. C. Dorf and R. H. Bishop, Modern Control Systems, 10th ed. Englewood Cliffs, NJ: Prentice-Hall, 2005. [32] S. Lin and A. A. Goldenberg, “Neural network control of mobile manipulators,” IEEE Trans. Neural Netw., vol. 12, no. 5, pp. 1121–1133, Sep. 2001. [33] J. Huang and F. L. Lewis, “Neural-network predictive control for nonlinear dynamic systems with time-delay,” IEEE Trans. Neural Netw., vol. 14, no. 2, pp. 377–389, Mar. 2003. [34] W. Duch and J. N. , “Survey of neural transfer functions,” Neural Comput. Surv., vol. 2, pp. 163–212, 1999. [35] F. Lewis and S. Jagannathan, Neural Network Control of Robotic Manipulators and Nonlinear Systems. London, U.K.: Taylor & Francis, 1999. Wei Zuo received the B.S. degree in machine design and manufacture and the M.S. degree in mechatronic engineering from the Department of Precision Instrument and Mechanology, Tsinghua University, Beijing, China, in 1998 and 2001, respectively, and the Ph.D. degree from the Department of Mechanical Engineering, Hong Kong University of Science and Technology (HKUST), Hong Kong, China, in 2008. After graduation from HKUST, he joined HyFun Technology Limited, Kowloon Bay, Hong Kong, a company aiming at offering complete and integrated solutions to design and development of control and automation systems. He is now the Project Manager of the Division of Intelligent and Process Control. His current interests include intelligent systems, robotics, process control, sensor fusion, etc.

ZUO et al.: FNN-BASED LEARNING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

Yang Zhu received the B.S. degree in mechanical engineering from Tsinghua University, Beijing, China, in 2002, and the Mphil. degree in mechanical engineering from the Hong Kong University of Science and Technology (HKUST), Hong Kong, China, in 2005. During 2005–2007, he was with the Design and Manufacturing Services Department, HKUST, where he was responsible for development of motion control systems for various industrial automation projects. After that, he joined HyFun Technology Limited, Kowloon Bay, Hong Kong, as Senior R&D Engineer, leading the R&D team for development of innovative automation technologies with application to industrial automation machines and embedded systems. His current interests include real-time and embedded systems, motor control, motion control, digital signal processing/field-programmable gate array (DSP/FPGA), advanced RISC microprocessor/microcontroller unit (ARM/MCU) technologies, etc.

151

Lilong Cai (S’88–M’90) received the B.S. degree in precision instrumentation engineering from Tianjin University, Tianjin, China, in 1982 and the Ph.D. degree in robotics from the University of Toronto, Toronto, ON, Canada, in 1990. From 1990 to 1993, he was an Assistant Professor with the Department of Mechanical Engineering, Columbia University, New York City, NY. He is currently a Professor at the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Hong Kong. His research interests lie in the control of nonlinear systems, robotics, optics, and mechatronics. He has published 42 referred international journal papers and 58 papers at international conferences proceedings. He holds three U.S. patents.