A New Iterative Learning Controller Using Variable ... - IEEE Xplore

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 40, NO. 2, APRIL 2010

A New Iterative Learning Controller Using Variable Structure Fourier Neural Network Wei Zuo and Lilong Cai, Member, IEEE

Abstract—A new iterative learning control approach based on Fourier neural network (FNN) is presented for the tracking control of a class of nonlinear systems with deterministic uncertainties. The proposed controller consists of two loops. The inner loop is a feedback control action that decreases system variability and reduces the influence of random disturbances. The outer loop is an FNN-based learning controller that generates the system input to suppress the error caused by system nonlinearities and deterministic uncertainties. The FNN employs orthogonal complex Fourier exponentials as its activation functions. Therefore, it is essentially a frequency-domain method that converts the tracking problem in the time domain into a number of regulation problems in the frequency domain. Through a novel phase compensation technique, this model-free method makes it possible to use higher-frequency components in the FNN to improve the tracking performance. In addition, the structure of the FNN can be reconfigured according to the system output information to make the learning more efficient and increase the convergent speed of the tracking error. Experiments on both a commercial gear box and a belt-driven positioning table are conducted to show the effectiveness of the proposed controller. Index Terms—Fourier neural network (FNN), iterative learning control (ILC), orthogonal activation function, phase compensation.

I. I NTRODUCTION

I

N MODERN control systems, plants are becoming more complex on one hand, and the specifications of control quality are getting sharper on the other hand. Meanwhile, to meet the requirement of high productivity, the operation speed is increasing. Therefore, the nonlinear dynamical effects can no longer be overlooked, which include viscous and static friction forces, vibration of flexible links, inertia effect of machine bodies, saturation of servo amplifiers, etc. They deteriorate the position and velocity accuracy and degrade the overall control performance of the system. In addition, both structured and unstructured uncertainties exist when modeling a complex nonlinear system, which make the control problem more difficult. In the past years, much effort has been devoted to apply the iterative learning control (ILC) method to nonlinear dyManuscript received October 5, 2007; revised February 17, 2008, December 7, 2008, and March 14, 2009. First published September 11, 2009; current version published March 17, 2010. This work was supported by the Research Grants Council of Hong Kong, China, under Project HKUST6114/03E. This paper was recommended by Associate Editor J. Wang. W. Zuo is with HyFun Technology Ltd., Kowloon Bay, Hong Kong (e-mail: [email protected]). L. Cai is with the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong (e-mail: melcai@ ust.hk). 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/TSMCB.2009.2026729

namical systems. The main idea of this method is to make use of the tracking errors and the system inputs recorded from previous executions to improve the tracking performance for the next execution. ILC has a well-established research history, as shown in [1]–[8]. The most widely used ILC method is the proportional-integral-differential (PID)-type approach because it essentially forms a PID-like system [4], [6]. This kind of controller was developed for nonlinear plants with nonlinearities satisfying global Lipschitz continuous condition [9], [10]. Employing the concept in the typical adaptive control, Park et al. developed the so-called adaptive ILC (AILC), which can be applied to non-Lipschitz nonlinear plants [11], [12]. The control parameters, instead of the control input itself, are updated between successive iterations in the design of AILC. Recently, 2-D system theory has been introduced to the ILC approach [13], [14]. Kurek and Zaremba proposed an ILC algorithm for a linear discrete-time multivariable system and stated the necessary and sufficient conditions for convergence of the algorithm based on 2-D system theory [15]. Recently, learning control methods using neural networks (NNs) were found in the ILC literature. NN is regarded as a powerful control technique due to its ability to learn, adapt, and approximate nonlinear functions to desired degrees of accuracy [16], [17]. It has been well known and extensively studied in the control of nonlinear systems for many years [18], [19]. Compared with other modern control techniques, NN-based control does not require knowledge about the system model. Chow et al. presented a real-time ILC approach for a nonlinear continuous-time system using recurrent NNs with time-varying weights [8]. Chien and Fu employed a feedforward NN with sigmoid hidden units to design an NN-based ILC for nonlinear systems with state-dependent input gains [20]. Wang et al. proposed a direct AILC based on output-recurrent fuzzy NN for a class of repeatable nonlinear systems with unknown nonlinearities and variable initial resetting errors [12]. In this paper, a new Fourier NN (FNN)-based ILC is proposed for the tracking control of a class of nonlinear systems. The underlying idea in the design of the FNN is to extend the Fourier learning control method [3], [21], [22] to NN-based control theory. The main feature of this learning controller and its contributions relative to previous work are summarized as follows: 1) Different from traditional NNs, the proposed FNN employs orthogonal complex Fourier exponentials as its basis functions. Therefore, it has a clear physical meaning and makes the determination of the network structure convenient. Without a priori knowledge of the system model, all the nonlinearities and uncertainties of the dynamical system are lumped together and iteratively compensated for by the FNN. 2) Most of the

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ZUO AND CAI: NEW ILC USING VARIABLE STRUCTURE FOURIER NEURAL NETWORK

learning controllers designed in the time domain lack the capability of dealing with a system’s phase lags. In particular, for systems with flexible components, the remarkable phase lags caused by system flexibility cannot well be handled, and the control performance is not satisfied. We know that output errors may include high-frequency and resonant-frequency components of a system. If an output feedback is directly added to the controller, then due to the phase lags, such frequency components may be stimulated and lead to instability. To avoid this phenomenon, a bandwidth limitation has to be imposed on many control algorithms. The proposed FNN is very closely related to the frequency response method. Compared with the commonly used frequency-domain approaches in ILC [23]– [25], the FNN can deal with not only linear time-invariant systems but also linear time-variant and nonlinear systems. Employing a phase compensation technique, the FNN can effectively eliminate the influence of phase lags by means of adjusting the phase of each frequency component in the control signal. The phase compensation method has been studied in [26]–[28]. It generally requires the transfer function of the system and treats the complex phase lag as a constant time delay. In contrast, the novel phase compensation method proposed in this paper iteratively updates the input phase based on the output information so that better compensative results can be achieved. 3) In traditional NNs, the network structure is predetermined and does not change during learning. To perform the mapping task in a more efficient way in terms of accuracy and network complexity, the structure of the FNN can be reconfigured. After analyzing the spectrum information of the system output, the FNN adjusts its network structure so that it can handle different input patterns and exhibit better performance. The remainder of this paper is organized as follows: In Section II, a description of a class of nonlinear singleinput–single-output (SISO) systems is given, and the tracking control problem is formulated. Then, we introduce the FNN and propose the FNN-based ILC in Section III. The sufficient condition for the stability of the closed-loop system and the convergence of the network is given in Section IV. The rules to construct a variable structure FNN are provided in Section V. Experimental setup and results are presented in Section VI. Finally, conclusions are offered in Section VII.

II. S YSTEM D ESCRIPTION AND P ROBLEM F ORMULATION In this section, we consider a class of nonlinear SISO systems, which are represented by the following state equation:

x(n) = f (x) + b(x)u + d y=x

(1)

where u ∈ R is the system input, y ∈ R is the system output, x = [x, x, ˙ . . . , x(n−1) ]T ∈ Rn is the state vector of the system, f (x) is an unknown but bounded nonlinear continuous function, d denotes the deterministic external disturbances, and b(x) represents the nonzero input coefficient. For convenience, we assume that the controlled system satisfies the following conditions.

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Assumption 1: The desired trajectory yd (t) of the system lasts for finite time. It complies with the actuator’s capacity and is continuous such that yd ≤ Qx , where Qx is a known scalar bound. Assumption 2: The system is state feedback stabilizable. Assumption 3: The system and the actuators have a finite bandwidth. That is, the overall system can only respond to certain frequency inputs. Only the system performance within the bandwidth can be ensured since the signal outside the bandwidth cannot correctly be detected. The objective of this paper is to design a controller for the tracking control of nonlinear systems [see (1)] under plant uncertainties and external disturbances, whereas all the variables of the closed-loop system and the output tracking error of a reference yd (t) defined on [0, T ] are guaranteed to be bounded. ˙ . . . , e(n−1) ]T , the erTaking e(t) = yd (t) − y(t) and e = [e, e, ror dynamics of the system can be expressed as (n) e˙ = Ae + B yd − f (x) − b(x)u − d (2) where ⎡

0 ⎢0 ⎢. . A=⎢ ⎢. ⎣0 0

1 0 .. . 0 0

0 ··· 1 ··· .. . . . . 0 ··· 0 ···

⎤ 0 0⎥ .. ⎥ .⎥ ⎥ 1⎦ 0

n×n

⎡ ⎤ 0 ⎢0⎥ ⎢.⎥ .⎥ B=⎢ ⎢.⎥ ⎣0⎦ 1

.

n×1

Choose a coefficient vector K = [ k0 k1 · · · kn−1 ]1×n such that A = A¯ − BK is Hurwitz, and rewrite (2) as (n) e˙ = Ae + B Ke + yd − f (x) − b(x)u − d . (3) According to [29], if b(x) = 1, then the foregoing system can be feedback linearized by introducing a suitable control law, (n) e.g., u = Ke + yd − f (x) + ua for some auxiliary control input ua , so that satisfactory tracking performance can be achieved with the resulting closed-loop system. The necessary condition for the design of this controller is that the nonlinear function f (x) should be available for feedback. However, in most of the practical applications, due to the existence of uncertainties, b(x) is time variant, and f (x) may not be available for direct control design. From (3), we can deduce (n) ˙ + Ke + yd − f − d = ue + ψ u = b−1 B T (Ae − e) (4) where ˙ ue = B T (Ae − e) = − k0 e + k1 e˙ + · · · + kn−1 e(n−1) + e(n) (n) ψ = b−1 yd − Ke − f (x) − d ˙ + (b−1 − 1)B T (Ae − e).

(5)

(6)

In this paper, the effects of deterministic uncertainties and disturbances will simultaneously be considered. The term ψ,

460


which represents the behavior of the unknown dynamics, will iteratively be learned and compensated for. Now, let us design the controller as follows: u(t) = ur (t) + uF (t)

(7)

where ur (t) = (1 + kr )ue is a feedback controller with a positive constant gain kr , and uF (t) is the learning controller to be designed. With the control law [see (7)], the closed-loop system dynamics becomes kr ue + uF (t) = ψ.

(8)

If one can find an optimal feedforward input uF (t) to compensate for all the uncertainties and nonlinearities in ψ, then we have ue = 0, and the closed-loop system becomes a stable linear system e˙ = Ae. Therefore, the tracking error exponentially converges to zero as time increases, so that even when the given task is defined on a finite time interval, the error will be small at the end of each operation. However, the optimal feedforward function is nonlinear and hard to obtain using linear control techniques. ILC controllers have the advantage to generate such a feedforward input function in the presence of uncertainties. It refers to a class of self-tuning controllers where the system performance of a specified task is gradually improved or perfected based on the previous performance of identical tasks. With a learning algorithm F(u, y, yd , t), the controller is given by the iteration uk (t) = F [uk−1 (t), yk−1 (t), yd (t), t]

(9)

where k is the iteration number. The aim is to make uk (t) converge to a fixed optimal function u∗ (t) so that the L2 norm lim yd (t) − y (uk (t), t) = yd (t) − y (u∗ (t), t)

k→∞

(10)

is minimized on the interval [0, T ]. In this paper, a new ILC controller based on the FNN is proposed to generate uF (t). The FNN employs orthogonal Fourier functions as its activation functions. Therefore, it is essentially a frequency-domain method that deals with the harmonic components of the feedforward function. Since the FNN can only be applied to open-loop stable systems or the systems have been stabilized, and the response of the FNN is slow because its weights are updated once a trial, the feedback controller ur (t) is incorporated with uF (t) to handle the random disturbance and decrease system variability. Fig. 1 presents the block diagram of the overall controller. III. FNN-B ASED ILC D ESIGN A. Network Architecture of FNN According to [16], [30], and [31], a continuous function defined at [0, T ] can be approximated by an NN as follows: f (x) = W T σ ˜ (x)

Fig. 1. Block diagram of the closed-loop system with FNN-based learning controller.

(11)

Fig. 2.

Structure of a SISO FNN.

where x ∈ R and f ∈ R are the input and output, respectively, σ ˜ (x) = [ σ1 (x) · · · σl (x) ]T ∈ Rl , σi (·), i = 1, 2, . . . , l, denotes the activation function, and W ∈ Rl is the network weight vector. To design an FNN, instead of the commonly used sigmoid function or Gaussian function, σi (·) is selected as the family of orthogonal Fourier harmonic functions, i.e., σm (t) = ejωm t , where ωm = 2mπ/T with m = 0, ±1, ±2, . . ., j is the imaginary unit with j 2 = −1, and ejωm t = cos ωm t + j sin ωm t is in accordance with Euler’s formula. The block diagram of a SISO FNN is illustrated in Fig. 2, and the output of the FNN is given by

uF (t) =

M

cm ejωm t

(12)

m=−M

where cm represents the network weight, and M is a positive integer based upon the system bandwidth. In real implementation, (12) is often expressed as the following sine/cosine form: uF (t) = PF Ψ(t)

(13)

where Ψ(t) = [ 1 cos ω1 t sin ω1 t · · · cos ωM t sin ωM t ]T is the family of activation functions, and PF = [ a0 a1 b1 · · · aM bM ] is the vector of network weights. As shown in Fig. 2, in the kth iteration, the coefficients akr,m , bkr,m , and akm , bkm are given as

T akr,m = T2 0 kr uke (t) cos ωm t, m = 0, 1, . . . , M

T bkr,m = T2 0 kr uke (t) sin ωm t, m = 1, 2, . . . , M

(14)


⎧ k−1 j ⎪ k j ⎪ a Δar,m cos θm = γ ⎪ F,m ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ m = 0, 1, . . . , M ; ⎪ ⎪ ⎨ j , j − Δbjr,m sin θm Δbr,0 = 0 ⎪ ⎪ ⎪ ⎪ k−1 j ⎪ ⎪ j ⎪ Δbr,m cos θm bkF,m = γ ⎪ ⎪ ⎪ j=1 ⎪ ⎩ j , m = 1, . . . , M . + Δajr,m sin θm (15) In the following discussion, the derivation of the foregoing equations and the meaning of the parameters will be explained in detail. The FNN is a particular case of NN, which was constructed on the basis of orthogonal Fourier activation functions. The input of the network is ur (t), which is an indication of the tracking performance obtained in the previous trial. The network weights [ a0 a1 b1 . . . aM bM ] represent the amplitudes of the spectra of uF (t). With this clear physical meaning, M can be determined by the system bandwidth, and the link between the network coefficients and the Fourier transform is explicit. In addition, the orthogonality of the basis functions may excellently perform in function modeling and decomposition and achieve a faster convergent speed. B. Design of the FNN-Based Learning Controller In this section, the learning algorithm of the FNN is considered, which is a dynamical process in terms of training iteration k. At the end of each iteration, historical information needs to be recorded, including the current and past neural weights and output, the error between the system output and the desired response, the output of the feedback controller, etc. Then, during the training process, the network weights are updated based on these records to generate a suitable feedforward function for the next iteration. Recall the closed-loop system dynamical equation [see (8)] and project its components to the frequency domain. In the kth iteration, we can obtain the following equation: Prk Ψ(t)

+

PFk Ψ(t)

=

Pfk Ψ(t)

461

suitable FNN weights to approximate P ∗ so that the output of the FNN in the time domain converges to u∗ (t). Therefore, the tracking problem in the time domain is converted to a number of regulation problems in the frequency domain, which are easy to cope with. Since P ∗ is unknown, the control strategy is to make PFk tend to Pfk , i.e., to achieve Prk → 0. Hence, the learning algorithm should be designed to drive both akr,m and bkr,m to zero. The learning process of updating the network weights is given as follows:

k k ak+1 F,m = aF,m + ΔaF,m , m = 0, 1, . . . , M k+1 bF,m = bkF,m + ΔbkF,m , m = 1, . . . , M

(18)

where ΔakF,m and ΔbkF,m are the learning rules in the kth iteration. On the basis of (17a) and (17b), the learning rules should be derived to guarantee the convergence of the network weights. Since akr,m and bkr,m are related to not only the amplitude but also the phase of a certain frequency component, the phase information should be considered. For the sake of convenience, some vectors are defined and employed in the following discussion to investigate kr,m = akr,m + the phase information. We define a vector A k jbkr,m = Akr,m ejφr,m , where Akr,m = (akr,m )2 + (bkr,m )2 and φkr,m = arctan(bkr,m /akr,m ) represent the magnitude and the kr,m = A k+1 k phase, respectively. Next, we define ΔA r,m − Ar,m = k jΔφ r,m with the magnitude and the phase given by ΔAkr,m e k+1 k k 2 k 2 ΔAkr,m = (ak+1 r,m − ar,m ) + (br,m − br,m ) and Δφr,m = k k+1 k arctan[(bk+1 r,m − br,m )/(ar,m − ar,m )]. Similarly, we also dek k k ,A k fine A F,m , ΔAf,m , and ΔAF,m . f,m It is well known that the phase lag phenomenon is very common due to system flexibility, time delay, etc. In the presence kr,m . of phase lags, it is difficult to ensure the convergence of A Therefore, in designing the FNN learning algorithm, the effect of phase lags should be suppressed. According to (17a) and (17b), we have

kr,m + ΔA kF,m . kf,m = ΔA ΔA

(19)

(16)

where Pfk = [ akf,0 akf,1 bkf,1 · · · akf,M bkf,M ] are Fourier coefficients of the system dynamical function ψ k (t), PFk = [ akF,0 akF,1 bkF,1 · · · akF,M bkF,M ] are network weights of the FNN, and Prk = [ akr,0 akr,1 bkr,1 · · · akr,M bkr,M ] are Fourier coefficients of kr uke (t) and represented by (14). Due to the orthogonality of the basis functions Ψ(t), (16) is decentralized into 2M + 1 equations in Fourier space as akf,m = akr,m + akF,m ,

m = 0, 1, . . . , M

(17a)

bkf,m = bkr,m + bkF,m ,

n = 1, 2, . . . , M.

(17b)

Obviously, the spectra of the optimal feedforward function u∗ (t) should be a constant vector P ∗ . Our aim is to find

Based on the dynamical equation [see (19)], in the following k discussion, the input–output phase lag between ΔA F,m and k k ΔAr,m , i.e., ϕm , will be investigated. In general, for the system in (8), the input–output phase lag refers to the phase k , i.e., αk . We choose ϕk to k and A difference between A r,m m m F,m k investigate because of the following reasons. First, ϕkm and αm are mathematically related so that they can be calculated from each other. Second, considering the structure of the control law [see (18)], it is convenient to derive the update law for ΔakF,m and ΔbkF,m with dynamical equation [see (19)] and phase k k lag ϕkm . Third, the magnitudes of ΔA F,m and ΔAr,m are k and A k , so when usually much smaller than those of A r,m F,m k we use ϕm , the variation of the phase lag in different iterations can clearly be observed and well compensated for.

462


In the kth iteration, consider the phase lag between ϕkm , which can be expressed as ϕkm = ΔφkF,m − Δφkr,m .

(20)

k + ΔA k , to make A k+1 tend to zero, the k+1 = A Since A r,m r,m r,m r,m kr,m is Δφkr,m = φkr,m − π. This implies optimal phase of ΔA k should be shifted to Δφk = φk + that the phase of ΔA r,m F,m F,m k k − π with a compensative phase θm = ϕkm . However, ϕkm is θm unknown in the beginning of the kth trial, so the estimation of ϕkm is required. An intuitive solution is to employ the historical phase lags to approximate ϕkm based on the principle of k is Taylor’s expansion. Therefore, the compensative phase θm designed as k k−1 θm = ϕk−1 m + Δϕm

(21)

k−1 k−2 where Δϕk−1 m = ϕm − ϕm . We know that, for a vector p = a + jb, when its phase is shifted with a constant angle θ, the following equation holds: pejθ = (a cos θ − b sin θ) + j(b cos θ + a sin θ). Therefore, we design the learning algorithm for the FNN as follows: ⎧ ΔakF,0 = γΔakr,0 ⎪ ⎪ k ⎪ ⎪ k k ⎪ ⎨ ΔaF,m = γ Δar,m cos θm k − Δb sin θk , m = 1, . . . , M (22) k r,m k m ⎪ k ⎪ ⎪ Δb = γ Δbr,m cos θm ⎪ ⎪ F,m ⎩ k , m = 1, . . . , M + Δakr,m sin θm

where γ > 0 is the learning gain. k can In the preceding control law, the magnitude of ΔA F,m be modified by changing the value of γ, and the influence of the phase lags will be suppressed by the phase compensation method. For linear systems, the variation of akr,m (bkr,m ) and akf,m (bkf,m ) in one harmonic component is independent of other harmonics. However, for nonlinear systems, the input–output relationship is quite complicated. The variation of magnitude and phase in one harmonic component will alter other harmonic components, i.e., the harmonic components are cross related. Hence, with the control law [see (22)], the condition γ > 0 may not be sufficient to ensure akr,m and bkr,m converge to zero. In the next section, the system stability and the tracking performance of the FNN-based learning controller are analyzed. A sufficient condition is given to ensure that the tracking error converges as the number of trials increases. IV. S TABILITY A NALYSIS OF THE C LOSED -L OOP S YSTEM The ability of phase compensation is essential to the FNN-based learning controller, so before we analyze the stability of the closed-loop system, the characteristics of the phase lags in nonlinear systems are discussed. In classical control theory, it has proved that the phase lags of a linear system are invariant. However, for nonlinear systems, the phase lag in each harmonic component is quite different and complicated, which may be 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. Assumption 4: From the kth trial to the (k + 1)th trial (k = 1, 2, . . .), the variation of the phase lag in the mth harmonic k k k = ϕk+1 δm m − ϕm is bounded such that |δm | < ζp , where 0 ≤ ζp < π/2 is known. The physical 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 ζp . If Assumptions 1–4 are all satisfied, the sufficient condition for the system stability and convergence of the tracking error is given by the following theorem. Theorem 1: With the controller in (7) and the update law in (21) and (22), the sufficient condition for the convergence of the tracking error is the learning gain 0 < γ < 2 cos ζp , and the closed-loop system input–output relation in the Fourier space satisfies (ΔAkf,m /ΔAkr,m ) < 1 − (γ/2 cos ζp ) for all m and k. Proof: According to (17a) and (17b), we have kr,m + ΔA kF,m . kf,m = ΔA ΔA

(23)

From the control law in (22), we know k k = γA k ejθm ΔA . F,m r,m

(24)

Substituting (24) into (23) yields k k + γA k ejθm k = ΔA . ΔA f,m r,m r,m

(25)

Using the triangle inequality in vector space, we have ΔAkr,m ≤ ΔAkf,m + γAkr,m .

(26)

We choose γ as 0 < γ < 2 cos ζp , so 1 − γ/(2 cos ζp ) > 0. When substituting ΔAkf,m /ΔAkr,m < 1 − γ/(2 cos ζp ) into (26), we obtain ΔAkr,m − 2 cos ζp < 0. Akr,m

(27)

Using the phase compensation method, our aim is to adjust the phase of the input to make Δφkr,m − φkr,m + π = 0 hold. k in different trials, However, due to the phase lag variation δm we cannot accomplish accurate phase compensation. In the k kr,m , the phase of ΔA presence of the estimation error δm satisfies the following equation: k Δφkr,m − φkr,m + π = δm .

k+1 is calculated by Then, the magnitude of ΔA r,m k k Ak+1 r,m = ΔAr,m + Ar,m k k = ΔAkr,m ejΔφr,m + Akr,m ejφr,m k k k jφkr,m ΔAr,m j (δm −π ) = Ar,m e + 1 k e Ar,m ΔAkr,m ΔAkr,m k k = Ar,m 1 + − 2 cos δm . Akr,m Akr,m

(28)

(29)


k Considering (27) and cos δm > cos ζp , we get

ΔAkr,m k k − 2 cos δm < 2 cos ζp − 2 cos δm < 0. Akr,m

(30)

k Thus, the relation between Ak+1 r,m and Ar,m is

Ak+1 r,m

=

Akr,m

ΔAkr,m 1+ Akr,m

ΔAkr,m k − 2 cos δm Akr,m

< Akr,m . (31)

A1r,m

is bounded, as the trial number k Since the initial value increases, each Fourier coefficient of ur (t) will asymptotically diminish. We will finally obtain the error dynamical equation ue = −(k0 e + k1 e˙ + · · · + kn−1 e(n−1) + e(n) ) = 0 in the time domain. As a result, the error e(t) will tend to zero as time lasts. Remark 1: To improve the tracking performance, particularly for systems containing flexible components, phase lags should carefully be treated. The proposed FNN-based learning controller is capable of reducing the effect of phase lags by introducing the phase compensation method. The accuracy of phase lag estimation plays a very important role in increasing the efficiency of the learning controller. The convergent speed of the network weights depends on not only the learning k . If the phase estigain γ but also the compensative phase θm mation is more precise, i.e., ζp is smaller, then a larger learning gain γ can be selected in the range (0, 2 cos ζp ), so that a faster convergent rate of the tracking error is achieved. In contrast, if we do not adopt the phase compensation technique or the phase lag estimation is less accurate, then ζp will be larger and lead to a smaller learning gain and a slower convergent speed. Remark 2: Both ΔAkf,m and ΔAkr,m vary from trial to trial. The condition (ΔAkf,m /ΔAkr,m ) < 1 − (γ/2 cos ζp ) only gives a conservative robust bound. In linear systems, ΔAkf,m and ΔAkr,m are independent of other harmonic components, so the actual bound of ΔAkf,m /ΔAkr,m is smaller, and the learning gain can be selected larger for fast convergence rate. In nonlinear systems, both ΔAkf,m and ΔAkr,m are affected by other harmonic components, so the actual bound of ΔAkf,m /ΔAkr,m is larger. A smaller learning gain should be selected to ensure stability. V. V ARIABLE S TRUCTURE FNN M ODEL In the previous section, it has been proved that an FNN with the structure shown in Fig. 2 is capable of ensuring the stability of the closed-loop system and the convergence of the tracking error. However, this does not mean that it provides an optimal structure for FNN. In this section, we consider a variable structure model for FNN, which is based on the idea that one can change the parameters of basis functions and network neuron number in different learning trials. This way, the network structure is gradually allocated. The objective of using a dynamical structure FNN is to reduce the node number of FNN and increase the convergence speed of the network weights.

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The FNN structure described in Fig. 2 and the parameters of activation functions are predetermined. The base frequency ω is simply determined by the duration of the given task T with ω = 2π/T . Suppose the highest frequency component contained in the optimal feedforward input function is ωh . Then, the neuron number should be chosen as 2M + 1, where M ≥ [ωh T /2π]. In the very beginning of the learning, since there is no priori knowledge of the system, the foregoing method to construct an FNN is feasible. However, in some cases, the foregoing structure of the FNN may not be efficient for learning purposes. For example, in a mechanical system, backlash causes obvious tracking error, which only covers for a short period and only contains highfrequency components. Therefore, the profile of the optimal input can be divided into two categories: “low-frequency” segments and “high-frequency” segments. If we design an FNN that contains a number of sub-FNNs to deal with different segments, then the total neuron number may be decreased, and the convergence speed of the tracking error may be increased. Meanwhile, when using the variable structure FNN, the closedloop system stability can still be guaranteed, because the given task can be looked as a number of subtasks, and each subtask is achieved by a sub-FNN. Assume that the input function is partitioned into n segments as si , i = 1, . . . , n. Each si is defined on [Ti−1 , Ti ), where T0 = 0, and Tn = T . All si ’s are classified as “low-frequency” segments Sl and “high-frequency” segments Sh . Next, assume that the highest frequency components contained in Sl and Sh are ωl and ωh , respectively. Now, we construct an FNN with n sub-FNNs. For the ith sub-FNN, the base frequency is selected as ωi = 2π/(Ti − Ti−1 ), and the neuron number is 2Mi + 1, where Mi = [ωl (Ti − Ti−1 )/2π] for the low-frequency segment, and Mi = [ωh (Ti − Ti−1 )/2π] for the high-frequency segment. Therefore, the total neuron number satisfies n n ωh T ωh MT = Mi < (Ti − Ti−1 ) = = M. 2π i=1 2π i=1 (32) Equation (32) shows that, when we increase the base frequencies in sub-FNNs, fewer neuron numbers are required to cover a limit bandwidth. Therefore, the whole FNN is less redundant, and the convergence speed of both the network weights and tracking error will be increased. To design such sub-FNNs, one needs to know how to partition the input function and how to obtain ωl and ωh . The following procedure provides a possible solution to solve these problems, which is based on the analysis of output tracking errors. Step 1) Use the Proportional Derivative (PD) controller to perform the tracking for the first trial. Step 2) After the first trial is finished, analyze the tracking error and find all the points when the error curve crosses zero. Partition the error into segments according to these points. Step 3) Calculate the mean square error (MSE) of the tracking error on [0, T ], which is denoted as Ew . Then, calculate the MSE of each segment of the tracking error, which is denoted as Ei .

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Fig. 4. Tracking error of the gear box with FNN controller (without phase compensation).

Fig. 3. Experimental setup: a dc motor-driven gear box.

Step 4) If two adjacent segments satisfy Ei−1 < 2Ew and Ei < 2Ew or Ei−1 > 2Ew and Ei > 2Ew , then combine Ei−1 and Ei as one new segment. Continue to do the combination until no two adjacent segments satisfy the foregoing condition. Step 5) Analyze the spectrum of each new segment and find proper bandwidths for them, i.e., determine ωl and ωh . The bandwidth should be selected so that the FNN approximation error is acceptable. Step 6) Reconfigure the FNN with a number of new subFNNs and then continue the learning process until the control performance is satisfied. The foregoing procedure offers a practical method to design a variable structure FNN. Notice that in Steps 4 and 5 some thresholds are required to do the judgment, which are generally chosen by experiences. Future research will focus on how to use intelligent methods to optimize these steps. Fuzzy theory, knowledge-based method, and expert system are all good candidates to achieve this goal. VI. E XPERIMENTAL R ESULTS A. Experiments on a Commercial Gear Box Experiments have been conducted on a dc motor-driven commercial gear box shown in Fig. 3. The gear box has a double-reduction gear train, and the reduction ratio is 9.31 : 1. An optical rotary encoder with a resolution of 5000 lines/rev is mounted on the output shaft. A computer equipped with an Analog/Digital, a Digital/Analog card, and a decoder card is applied to carry out the control algorithm. The angular position of the output shaft used for feedback is measured by the optical rotary encoder. The gear box in Fig. 3 is a SISO system, and the dynamical model can be represented as follows [32]: ¨ + C θ(t) ˙ + Kθ(t) + fd (t) = Km v(t) M θ(t)

(33)

where θ(t) is the output angle, M denotes the assembled system inertia, C and K are the damping and stiffness coefficients, fd (t) represents uncertain disturbance, Km stands for the equivalent constant gain of the amplifier and motor, and v(t) is the input voltage of the actuator. The desired output trajectory is θd (t) = 30 × (1 − cos 4πt)(deg .),

t ∈ [0, 0.5]s.

(34)

Fig. 5. Tracking error of the gear box with FNN controller (with phase compensation).

Using the proposed FNN learning controller, there is no need to model the gear box. The gains in the PD controller were selected as Kp = 0.04 and Kd = 0.0012, and the FNN learning gain is empirically selected as γn = 0.4. The hidden layer neuron number was chosen as 21, which corresponds to a frequency bandwidth of 20 Hz. First, we did the experiment without the phase compensation k = 0. Fig. 4 shows the angular tracking error, method, i.e., θm which is convergent trial by trial. After six trials, the error is within the range of ±0.4◦ , and only some high-frequency spikes are left. Because the phase compensation method was not employed, high-frequency components are not handled well. Then, we performed the same task using the learning algorithm [see (22)] and the phase compensation method [see (21)]. Fig. 5 presents the experimental result. In the sixth trial, even the high-frequency spikes are eliminated so that the tracking error is very small (about ±0.15◦ ). Since the gear box is a rigid system, the system’s phase lag is not serious. Although we do not compensate for it, the closed-loop system remained stable, and the performance in Fig. 4 was still acceptable. B. Experiments on a Belt-Driven Positioning Table Compared with the gear box, a belt-driven system is more complicated. Due to the flexibility of the belt, such a system contains high nonlinearities and remarkable phase lag phenomenon. We selected the Daedal 500000PD series belt-driven positioning table as our experimental setup. A linear encoder with 1-μm 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 same computer is used to run the control algorithm. The experimental setup and its


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Fig. 8. Tracking error of a 13-hidden-layer-neuron FNN (without phase compensation).

Fig. 6.

Experimental setup: a belt-driven positioning table.

Fig. 9. Tracking error of a 41-hidden-layer-neuron FNN (without phase compensation).

Fig. 7.

Schematic of the experimental setup.

schematic diagram are given in Figs. 6 and 7, respectively. The system dynamical equation can be simplified as [33] ˙ × r + Ji θ¨i (t) + Ti θ˙i (t) [Me y¨(t) + Be y(t)] + d y(t), ˙ y(t), θ˙i (t), θi (t) = Km v(t) (35) where y(t) represents the position of the plate, θi (t), i = 1, 2, are used to indicate the angular positions of the wheels, Me denotes the equivalent mass of the moving parts, Be denotes the coefficient of viscous friction, r is the radius of the driving wheel, Ji represents the equivalent inertias of the wheels, Ti denotes the coefficients of viscous friction of the wheels, d represents the uncertain torque caused by vibration of the belt, Km is the equivalent gain constant of the amplifier and the motor, and v is the input voltage to the amplifier of the dc motor driver. The desired output position trajectory is given by y d (t) = 10 000(1 − cos 5πt) (in micrometers), t ∈ [0, 0.4]s.

(36)

First, we used the learning algorithm in (22) without phase compensation to perform the control. We selected Kp = 0.002 and Kd = 0.000008 to construct the PD controller and chose γn = 0.3 as the learning gain. Note that, due to the high nonlinearities in this flexible belt-driven system, the learning gain is smaller than that of the learning controller for the gear box. Two FNNs, which have different hidden-layer neurons,

Fig. 10. Error spectrum (tenth trial) of a 13-hidden-layer-neuron FNN (without phase compensation).

were employed to do the same task. Fig. 8 shows the position error of an FNN controller with 13 hidden layer neurons. Fig. 9 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 bandwidth of 15 and 50 Hz, respectively. For the first FNN with fewer neurons, only a narrow frequency band is covered. The convergence of the tracking error almost stops after seven trials. Fig. 10 shows the spectrum of the tracking error in the tenth trial. Analyzing the spectrum, we found that the error was dominated by high-frequency components. This means that we cannot handle frequencies larger than 15 Hz. Therefore, a reasonable solution is to increase the neuron number as in the second FNN. However, although we have employed more frequency components, the control performance in Fig. 9 is worse than that in Fig. 8. The tacking error of the closed-loop system even becomes divergent. Fig. 11 shows the spectrum of the error in the tenth trial when using the second FNN controller. The amplitudes of the high-frequency components were almost ten times larger than those in Fig. 10. This phenomenon was caused by system flexibility, which incurs a notable phase lag in

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Fig. 11. Error spectrum (tenth trial) of a 41-hidden-layer-neuron FNN (without phase compensation).

Fig. 14.

Tracking error of a 41-hidden-layer-neuron FNN (fixed structure).

Fig. 12. Tracking error of a 41-hidden-layer-neuron FNN (with phase compensation).

Fig. 15.

Tracking error in the first trial with PD controller.

Fig. 13. Error spectrum (tenth trial) of a 41-hidden-layer-neuron FNN (with phase compensation).

Fig. 16.

Tracking error of the variable structure FNN.

high-frequency components. If we do not compensate for the phase lags, then the high-frequency components will be stimulated and cause instability of the closed-loop system. Then, we used the phase compensation method with the second FNN to perform the same experiment. The tracking error is shown in Fig. 12. In the tenth trial, the position error is very small (within ±10 μm). For the spectrum of the error in the tenth trial shown in Fig. 13, the amplitudes of the highfrequency components are greatly reduced. This fact proved that the phase compensation method is efficient to deal with the phase lags in a flexible system. C. Experiments With Variable Structure FNN To investigate the effectiveness of the variable structure FNN, experiments were carried out on the belt-driven positioning table with a more complicated task given as ⎧ 3000 1 − cos 20 πt ⎪ ⎪ 3 ⎨ (in micrometers), t ∈ [0, 0.3)s . (37) y d (t) = ⎪ 3000(1 − cos 20πt) ⎪ ⎩ (in micrometers), t ∈ [0.3, 0.4]s

The given task contains two continuous trajectories, which are in different frequencies. We still used the 41-hidden-layerneuron FNN with phase compensation method to perform the tracking. Fig. 14 shows the converging process of the tracking error. As indicated in Fig. 14, the position error peaks are reduced as the trial number increases. After 12 trials, the error spikes are in the range of ±60 μm. Then, we used the variable structure FNN to perform the same task. In the first trial, only the PD controller was employed. The tracking performance is illustrated in Fig. 15. After investigating the property of the tracking error, we constructed a new FNN with two sub-FNNs, which were defined on t ∈ [0, 0.307) and t ∈ [0.307, 0.4], respectively. The partition was based on the rules provided in the previous section. Based on the analysis of the spectra of the two segments, the neuron numbers were selected as 19 and 11, which correspond to the frequency bandwidth at about 30 and 50 Hz, respectively. Then, the total neuron number of the variable structure FNN is 30. The tracking performance of the FNN was presented in Fig. 16. Compared with Fig. 14, the convergent speed of the tracking error was improved. It only took seven trials to reduce the error


467

Fig. 17. Tracking error of a PID controller.

Fig. 20. Output voltage of the FNN controller (tenth trial).

Fig. 18. Tracking error of the FNN controller (tenth trial).

Fig. 21. Amplitude spectrum of the output voltage of the PID controller and that of the FNN controller.

Fig. 19. Output voltage of the PID controller.

within ±60 μm. After 12 trials, the tracking error was in the range of ±20 μm. D. 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 for the same experimental system to perform the trajectory [see (36)]. The parameters of the PID controller were carefully tuned so that good tracking performances were obtained. Fig. 17 shows the tracking errors of the PID controller, and Fig. 18 shows the position error (tenth trial) of the 41-hidden-layer-neuron FNN controller with phase compensation method. The control performance of the FNN controller is much better than that of the PID controller. The outputs of the two controllers are also illustrated in Figs. 19 and 20, respectively. It can be seen that the output voltage of the FNN controller is smoother than that of the PID controller. To gain deeper understanding of the proposed FNN control scheme, the spectra of the outputs of the two controllers were analyzed. Because the magnitudes of the high-frequency components in the spectra were very small, we focused on the

Fig. 22. Phase spectrum of the output voltage of the PID controller and that of the FNN controller.

behavior of the low-frequency components. In the following figures, only the first ten harmonic components are presented. Fig. 21 shows the amplitude spectra of the output voltages, and Fig. 22 presents the phase spectra of the output voltages. As indicated in the figures, there is not much difference in the amplitudes. However, remarkable differences between the corresponding phases are clearly depicted. This implies that, although the energy inputted into the systems was nearly the same, due to the distinct phases, the tracking performances of the two controllers were quite different. The phase compensation in the FNN controller is very significant to improve the control performance. VII. C ONCLUSION This paper has introduced a variable structure FNN to deal with the tracking control of nonlinear dynamical systems. It should be noticed that this approach is different from traditional ILCs because its learning algorithm actually handles the spectrum of the system input. The major advantage of the FNN learning controller is that the phase information of each frequency component can well be utilized. The FNN learning controller converts the tracking problem in the time

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domain into a number of regulation problems in the frequency domain. Then, a novel phase compensation method is proposed to handle the phase lag phenomenon. Experimental results on both the gear box and the belt-driven positioning table prove that the FNN learning controller with the phase compensation method is capable of improving the tracking performance by employing high-frequency components. In particular, for systems containing flexible components, even frequencies outside the bandwidth can be used. Meanwhile, by properly adjusting the structure of the FNN, the neuron number can be reduced, and a fast learning process is achieved. Future research will focus on the improvement of the robustness for the learning controller and the selection of a suitable learning gain so that a fast learning process can be achieved without driving the system unstable. R EFERENCES [1] 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, 2004, pp. 1012–1016. [2] W. Huang and L. Cai, “New hybrid controller for systems with deterministic uncertainties,” IEEE/ASME Trans. Mechatronics, vol. 5, no. 4, pp. 342–348, Dec. 2000. [3] L. Cai and W. Huang, “Fourier series based learning control and application to positioning table,” Robot. Auton. Syst., vol. 32, no. 2/3, pp. 89–100, Aug. 2000. [4] S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operation of robots by learning,” J. Robot Syst., vol. 1, no. 2, pp. 706–712, 1984. [5] H. S. Ahn, K. L. Moore, and Y. Chen, “Stability analysis of discrete-time iterative learning control systems with interval uncertainty,” Automatica, vol. 43, no. 5, pp. 892–902, May 2007. [6] Z. Bien, Iterative Learning Control: Analysis, Design, Integration and Applications. Boston, MA: Kluwer, 1998. [7] S. Arimoto, “Learning control theory for robotic motion,” Int. J. Adapt. Control Signal Process., vol. 4, no. 6, pp. 543–564, 1990. [8] T. W. S. Chow, X. D. Li, and Y. Fang, “A real-time learning control approach for nonlinear continuous-time system using recurrent neural networks,” IEEE Trans. Ind. Electron., vol. 47, no. 2, pp. 478–486, Apr. 2000. [9] J. X. Xu and Y. Tan, Linear and Nonlinear Iterative Learning Control. Berlin, Germany: Springer-Verlag, 2003. [10] Y. Chen, C. Wen, J. X. Xu, and M. Sun, “An initial state learning method for iterative learning control of uncertain time-varying systems,” in Proc. 35th IEEE Decision Control, 1996, pp. 3996–4001. [11] B. H. Park, T. Y. Kuc, and J. S. Lee, “Adaptive learning of uncertain robotic systems,” Int. J. Control, vol. 65, no. 5, pp. 725–744, 1996. [12] 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. [13] X. D. Li, T. W. S. Chow, and J. K. L. Ho, “2-D system theory based iterative learning control for linear continuous systems with time delays,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 7, pp. 1421–1430, Jul. 2005. [14] T. W. S. Chow and Y. Fang, “An iterative learning control method for continuous-time systems based on 2-D system theory,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 45, no. 6, pp. 683–689, Jun. 1998. [15] J. E. Kurek and M. B. Zaremba, “Iterative learning control synthesis based on 2-D system theory,” IEEE Trans. Autom. Control, vol. 38, no. 1, pp. 121–125, Jan. 1993. [16] 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. [17] 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. [18] 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.

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Wei Zuo received the B.S. degree in machine design and manufacture and the M.S. degree in mechatronic engineering from Tsinghua University, Beijing, China, in 1998 and 2001, respectively, and the Ph.D. degree from the Hong Kong University of Science and Technology, Kowloon, Hong Kong, in 2008. Since 2008, he has been with HyFun Technology Ltd., Kowloon Bay, Hong Kong, which is a company aiming at offering complete and integrated solutions to the design and development of control and automation systems, where he is currently the Project Manager of the Intelligent and Process Control Division. His current interests include intelligent systems, robotics, process control, sensor fusion, etc.

Lilong Cai (S’88–M’91) 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. He is currently a Professor with the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong. He has published 42 referred international journal papers and 58 papers at international conference proceedings. He is the holder of three U.S. patents. His research interests lie in the control of nonlinear systems, robotics, optics, and mechatronics.