Iterative Learning Control with Extended State Observer for Telescope System

Abstract

An Iterative Learning Control (ILC) method with Extended State Observer (ESO) is proposed to enhance the tracking precision of telescope. Telescope systems usually suffer some uncertain nonlinear disturbances, such as nonlinear friction and unknown disturbances. Thereby, to ensure the tracking precision, the ESO which can estimate system states (including parts of uncertain nonlinear disturbances) is introduced. The nonlinear system is converted to an approximate linear system by making use of the ESO. Besides, to make further improvement on the tracking precision, we make use of the ILC method which can find an ideal control signal by the process of iterative learning. Furthermore, this control method theoretically guarantees a prescribed tracking performance and final tracking accuracy. Finally, a few comparative experimental results show that the proposed control method has excellent performance for reducing the tracking error of telescope system.

Huaxiang Cai 1, 2, 3 and Yongmei Huang 1, 2 and Junfeng Du 1, 2 and Tao Tang 1, 2 and Dan Zuo 1, 2, 3 and Jinying Li 1, 2

Academic Editor:Xinggang Yan

1, Chinese Academy of Sciences Institute of Optics and Electronics, P.O. Box 350, Shuangliu, Chengdu 610209, China
2, Key Laboratory of Optical Engineering, Chinese Academy of Sciences, Chengdu, China
3, University of Chinese Academy of Sciences, Beijing 100039, China

Received 4 March 2015; Revised 24 June 2015; Accepted 1 July 2015; 12 October 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A high performance telescope system has attracted lots of attention, since it is widely used in space applications, such as space observer [1] and satellite surveillance [2]. However, it is inevitable that there are always some nonlinear characteristics in the telescope tracking control systems, such as deadzone [3], friction [4], and some uncertainly disturbances [5]. Without doubt, all the nonlinear features will deteriorate the tracking performance of systems. Simultaneously, the conventional linear control method (proportional-integral) has not guaranteed a higher tracking precision. To solve the above nonlinear problems, there are many control methods being proposed, such as adaptive control [6], slide control [7], compensating control based on DOB [8], active disturbance rejection control (ADRC) [9], and iterative learning control (ILC) [10].

In recent years, since the iterative learning control (ILC) needs too little knowledge of system dynamics, it has received a great deal of attention [11]. ILC is proposed by Arimoto et al. in 1984 [12]. It is essentially a feedforward control approach that fully utilizes the previous control information [13]. Over the past three decades, it has been successfully used in extensive research fields such as industrial robotics [14], manufacturing process [15], stochastic process control system [16], and hysteretic system [17]. Nevertheless, the ILC still has some of its inherent shortages. It can only eliminate some disturbances which emerge repeatedly. If there are some nonlinear nonperiod disturbances involved in the systems, the sole ILC method will face difficulty to get a higher tracking precision. The ADRC is different from the ILC, which can estimate unknown nonlinear disturbances involved in the systems. As the core technology of ADRC, the ESO also needs to know little knowledge of system dynamics, which only needs the information of system's input and output. Thereby, it has been widely used in many fields such as missile control system [18], optical system [19], and other high precision occasions [20]. Even though the ESO has excellent estimation ability for system states, its estimation ability is still restricted to the bandwidth of ESO.

In this paper, since there are some uncertain nonlinear disturbances in the telescope system, which reduce severely the tracking precision of system, it is needed to eliminate the nonlinear disturbances and find the best input signal for the system. Under the circumstances, an ILC technique with ESO is proposed, which takes use of the advantages of the two control methods (ILC and ESO). The function of ESO is to eliminate parts of uncertain nonlinear disturbances and to transform the nonlinear system into an approximate linear system. And the function of ILC is to get the best input signal for the approximate linear system. In the ILC controller, a PD-type learning algorithm with the forgetting factor is chosen. The forgetting factor can balance the learning precision and the robustness of system. To illustrate the excellent performance of the proposed control method, three comparative experiments will be given in this paper.

This remainder of this paper is organized as follows. Section 2 gets the dynamic models. Section 3 introduces the controller design and the theory analysis. The experiment results are presented in Section 4. And some conclusions can be found in Section 5.

2. Dynamic Models

The tracking control system is made of brushless DC, load, high precision encoder, servo actuator, and host computer. The goal is to make the system track the given reference motion trajectory as far as possible. The control structure is shown in Figure 1.

Figure 1: Architecture of tracking system.

[figure omitted; refer to PDF]

In the telescope system, since the electrical response of the actuator is very fast, the current dynamics can be neglected. And the dynamics of the tracking system is [figure omitted; refer to PDF]

In (1), [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the motor inertia and the load inertia, respectively. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent for the load angle position and the load angle velocity, respectively. [figure omitted; refer to PDF] represents the torque imposed on the motor. [figure omitted; refer to PDF] is the control input current of the motor. [figure omitted; refer to PDF] represents nonlinear disturbances (such as the friction disturbance, external disturbances, and some unmodeled dynamics). [figure omitted; refer to PDF] is the viscous coefficient.

Rewrite the dynamic model (1) as [figure omitted; refer to PDF]

[figure omitted; refer to PDF] , which includes the nonlinear disturbances [figure omitted; refer to PDF] and the inner disturbances [figure omitted; refer to PDF] . Here, [figure omitted; refer to PDF] is a parameter that can be adjusted in the controller.

3. Controller Design

The aim of designing controller is eliminating the nonlinear disturbance [figure omitted; refer to PDF] and enhancing the tracking precision of system. The ESO makes the nonlinear system transform into an approximate linear system. And the ILC with ESO gets the ideal input control signal.

In most of telescope systems, the multiple-loops control structure is usually applied to guarantee the tracking precision of systems, which includes position loop, velocity loop, and current loop. In this paper, since the current dynamic has higher frequency range than other loops, it has been ignored. We mainly analyze position loop and speed loop.

3.1. Design PI Controller with ESO

3.1.1. Design the ESO

Before designing the ESO, we assume that the disturbance [figure omitted; refer to PDF] is continuous and differential.

Assumption 1.

The disturbance [figure omitted; refer to PDF] is continuous and differential, and [figure omitted; refer to PDF] . Besides, [figure omitted; refer to PDF] is bounded; namely, there is a positive constant [figure omitted; refer to PDF] to meet: [figure omitted; refer to PDF]

Consequently, we extend the [figure omitted; refer to PDF] as a separate state [figure omitted; refer to PDF] of system. Then, the original plant of system (2) can be described as [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

By (4), we can know that it is observable. Hence, an ESO can be constructed as (5) [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the parameters that will be designed in the ESO.

The aim of ESO is to make [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , denote the estimation error; subtracting (4) from (5), then we can obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Therefore, if the matrix [figure omitted; refer to PDF] is Hurwitz, the ESO will be Bounded-Input Bounded-Output (BIBO) stable. In the matrix [figure omitted; refer to PDF] , the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] can be designed as [21] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the bandwidth of ESO. Equation (9) can promise the matrix [figure omitted; refer to PDF] being Hurwitz.

Lemma 2.

If the [figure omitted; refer to PDF] is bounded, there will be a positive constant [figure omitted; refer to PDF] and a finite time [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

The proof can be found in the literature [22].

Remark 3.

Equation (9) makes the ESO (5) a BIBO stable observer. The result of Lemma 2 shows that the ESO has an excellent estimation ability for disturbances. Its estimation ability depends on the bandwidth [figure omitted; refer to PDF] of ESO. When the bandwidth [figure omitted; refer to PDF] is big enough, it can estimate accurately [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

According to the analysis in Lemma 2, when designing an appropriate parameter [figure omitted; refer to PDF] , it can make [figure omitted; refer to PDF] . And there will be [figure omitted; refer to PDF] ; namely, the [figure omitted; refer to PDF] . So we can transform the input [figure omitted; refer to PDF] into [figure omitted; refer to PDF] ; namely, [figure omitted; refer to PDF] .

Substituting the [figure omitted; refer to PDF] to (2), we can get [figure omitted; refer to PDF]

3.1.2. Design Velocity Loop PI Controller

The nonlinear system (2) is converted to an approximate linear integral system (11). So we can design a PI controller in the velocity loop: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the input signal of velocity loop and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are the proportion gain and the integral gain in the velocity loop controller, respectively.

3.1.3. Design Position Loop PI Controller

Similarly, we can design a PI controller in the position loop: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the input signal of position loop and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are the proportion gain and the integral gain in the position loop controller, respectively.

3.2. Iterative Learning Controller with ESO Design

From (11), we can see that the system is only an approximate linear system. Thereby, to make further improvement on the tracking precision of system, we will find an ideal input [figure omitted; refer to PDF] by the method of iterative learning. And it can make the output [figure omitted; refer to PDF] of system close to the input [figure omitted; refer to PDF] as far as possible. In this section, we will design an ILC to substitute the PI controller in the velocity loop.

According to (11), the state space equation of system can be obtained as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF]

To make the system track the desired reference trajectory, we assume that there is an ideal input signal meeting the condition.

Assumption 4.

The ideal input signal [figure omitted; refer to PDF] can make the system track the desired reference trajectory [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] represent the ideal states of system, respectively.

Since system (14) will be tracked a period trajectory, there will be once input in every period. Then the state space equation of system (14) can be rewritten: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the times. Equation (16) means the [figure omitted; refer to PDF] th states of system under the [figure omitted; refer to PDF] th input [figure omitted; refer to PDF] .

For system (16), a PD-type learning algorithm with the forgetting factor [figure omitted; refer to PDF] is selected. The factor [figure omitted; refer to PDF] is introduced to make a tradeoff between perfect learning and robustness, which can increase the robustness of ILC against noise, initialization error and fluctuation of system dynamics [23]. The selected iterative learning scheme is found in (17), whose schematic diagram is depicted in Figure 2: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the proportion gain matrix and deviation gain matrix, respectively. [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are bounded. [figure omitted; refer to PDF] is the times. [figure omitted; refer to PDF] is the iterative error in the [figure omitted; refer to PDF] th times.

Figure 2: Schematic diagram of the proposed iterative learning control with ESO.

[figure omitted; refer to PDF]

3.3. Iterative Learning Controller Convergence Analysis

Theorem 5.

If the system described by (16) satisfies assumptions and uses the update law (18). Given a desired trajectory [figure omitted; refer to PDF] and an initial state [figure omitted; refer to PDF] which are achievable, if [figure omitted; refer to PDF] then, as [figure omitted; refer to PDF] , the iterative output [figure omitted; refer to PDF] of system will converge to the desired reference trajectory [figure omitted; refer to PDF] .

Proof.

Consider [figure omitted; refer to PDF]

In (19), by the state space equation (15) and (16), the [figure omitted; refer to PDF] can be transformed into [figure omitted; refer to PDF]

Substituting (20) to (19), we can have [figure omitted; refer to PDF]

Taking the norm [figure omitted; refer to PDF] on both sides of (21), we obtain [figure omitted; refer to PDF]

Multiplying by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and we have [figure omitted; refer to PDF]

Definition 6.

Define [figure omitted; refer to PDF] norm for a function [figure omitted; refer to PDF] [24]; then [figure omitted; refer to PDF]

Then [figure omitted; refer to PDF]

Since the [figure omitted; refer to PDF] , we can find a [figure omitted; refer to PDF] which makes [figure omitted; refer to PDF]

So we can get [figure omitted; refer to PDF]

Remark 7.

It shows that [figure omitted; refer to PDF] converges to the [figure omitted; refer to PDF] of radius [figure omitted; refer to PDF] with respect to the [figure omitted; refer to PDF] norm. Besides, we can know that the convergent radius is bounded. And the boundary depends on the initial input states of system.

Consequently, the tracking error of system will be [figure omitted; refer to PDF]

Similarly, taking the norm [figure omitted; refer to PDF] on both sides of (28), we can get [figure omitted; refer to PDF]

Specially, the system usually can meet the initial condition; that is, [figure omitted; refer to PDF]

That is said as [figure omitted; refer to PDF]

To sum up, when the PD-type learning algorithm is applied to the ILC with ESO, the input [figure omitted; refer to PDF] of system can converge to the ideal input [figure omitted; refer to PDF] , and the output of system can converge to the ideal output [figure omitted; refer to PDF] .

4. Experiment Setup and Result

The verification platform is the rotary table of telescope system, which consists of a DC motor, a tracking load, an electrical driver, a control system, and a high precision encoder whose accuracy is about [figure omitted; refer to PDF] (arc-second). The control scheme includes two loops: position loop and velocity loop.

The following three controllers are compared.

(1) PI-PI: this is the traditional Proportional-Integral (PI) controller. In the velocity loop and position loop, we use two PI controllers. The velocity signal can be obtained by differentiating the position signal.

(2) PI-ESO: in both of the position loop and the velocity loop, the PI controllers are used, but the velocity signal will be gotten by the ESO. The bandwidth [figure omitted; refer to PDF] of ESO is [figure omitted; refer to PDF] . The parameters of position loop's PI controller remain the same.

(3) PI-ILC-ESO: the PI controller is used in the position loop, and the ILC controller is used in the speed loop. The parameters of position loop's PI controller still remain the same. The velocity signal is also the signal which is gotten by the ESO. To satisfy the condition (18), The factor [figure omitted; refer to PDF] and the deviation gain matrix [figure omitted; refer to PDF] are chosen as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . By the state space equation of system (14), it will have [figure omitted; refer to PDF]

The three controllers are tested for a sinusoidal signal, whose velocity and acceleration are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The period of [figure omitted; refer to PDF] sinusoidal signal is [figure omitted; refer to PDF] .

The desired reference trajectory is shown in Figure 3. And the corresponding tracking performance under the three controllers is shown in Figures 4-6. As seen, the PI-PI controller has a larger tracking error whose maximum value is about [figure omitted; refer to PDF] . Comparatively speaking, the PI controller with ESO has a better tracking performance, which illustrates that ESO has an excellent estimation ability for disturbances involved in the system. Furthermore, the proposed PI-ILC-ESO controller has the least tracking error than other control methods, whose maximum value of error is about [figure omitted; refer to PDF] . This controller combines the advantages of ESO with the advantages of ILC. It makes use of ESO to eliminate most of the nonlinear disturbances. Simultaneously, the ILC gets the best control input by learning previous control information. Some detailed results of the three controllers are shown in Table 1.

Table 1: The experimental results of the three controllers.

	PI-PI	PI-ESO	PI-ILC-ESO
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
Root mean square error	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]

Figure 3: The desired reference signal.

[figure omitted; refer to PDF]

Figure 4: The tracking error of PI-PI.

[figure omitted; refer to PDF]

Figure 5: The tracking error of PI-ESO.

[figure omitted; refer to PDF]

Figure 6: The tracking error of PI-ILC-ESO.

[figure omitted; refer to PDF]

From Table 1, it can be seen that the PI-ILC-ESO has better performance than other control methods. However, comparing the PI-ESO controller with the PI-ILC-ESO controller, it can be seen that the effectiveness of using ILC is not obvious. The maximum error reduces only to [figure omitted; refer to PDF] and the root mean square error reduces to [figure omitted; refer to PDF] . In fact, Comparing with Figures 5 and 6, it can be found that the errors of many position points exceed [figure omitted; refer to PDF] and the errors are close to [figure omitted; refer to PDF] , when the PI-ESO controller is used. But when the PI-ILC-ESO controller is used, almost all the errors of position points are near [figure omitted; refer to PDF] . So the conclusion that the comprehensive performance of PI-ILC-ESO is better than other control methods can be gotten. The ILC method helps the system get the best input signal. Besides, in Figures 5 and 6, it can be found that the errors are evenly distributed. Comparing with the Figure 4 (PI-PI controller), both of the two controllers have reduced almost all the turning errors of system. The remaining errors are mostly caused by the signal noise. So it is difficult to reduce the remaining error and further improve the precision of system, unless the problem of signal noise is solved.

5. Conclusion

Uncertain nonlinear disturbances have been the major factors which restrict the performance of tracking control systems. This is because the systems will have a low tracking precision when some uncertain nonlinear disturbances are induced in the systems. Therefore, to reduce the influence introduced by the uncertain nonlinear disturbances, an ILC method with ESO is proposed in this paper. The ILC can get an excellent input signal by learning previous control information. It owns a better ability for eliminating some period disturbances. Meanwhile, an ESO is designed for estimating some uncertain nonlinear disturbances. It compensates the shortage of the ILC's disablement for nonperiod disturbances. In addition, the ESO can estimate an accurate velocity signal and supply the velocity as the feedback input signal of iterative learning controller. Furthermore the convergence analysis of the proposed control method guarantees the robustness of system. Finally, the experiment results show that the proposed control method has excellent performance for reducing the tracking error of telescope system.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] G. G. Valyavin, V. D. Bychkov, M. V. Yushkin, G. A. Galazutdinov, S. V. Drabek, V. S. Shergin, A. N. Sarkisyan, E. A. Semenko, T. E. Burlakova, V. M. Kravchenko, D. O. Kudryavtsev, A. M. Pritychenko, P. G. Kryukov, S. L. Semjonov, F. A. Musaev, S. N. Fabrika, "High-resolution fiber-fed echelle spectrograph for the 6-m telescope. I. Optical scheme, arrangement, and control system," Astrophysical Bulletin , vol. 69, no. 2, pp. 224-239, 2014.

[2] K. Lu, Y. Xia, "Adaptive attitude tracking control for rigid spacecraft with finite-time convergence," Automatica , vol. 49, no. 12, pp. 3591-3599, 2013.

[3] Y.-J. Sun, "Composite tracking control for generalized practical synchronization of Duffing-Holmes systems with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities," Abstract and Applied Analysis , vol. 2012, 2012.

[4] Y. Fu, T. Chai, "Self-tuning control with a filter and a neural compensator for a class of nonlinear systems," IEEE Transactions on Neural Networks and Learning Systems , vol. 24, no. 5, pp. 837-843, 2013.

[5] M. M. Bridges, D. M. Dawson, J. Hu, "Adaptive control for a class of direct drive robot manipulators," International Journal of Adaptive Control and Signal Processing , vol. 10, no. 4-5, pp. 417-441, 1996.

[6] A. Tadayoni, W.-F. Xie, B. W. Gordon, "Adaptive control of harmonic drive with parameter varying friction using structurally dynamic wavelet network," International Journal of Control, Automation and Systems , vol. 9, no. 1, pp. 50-59, 2011.

[7] Y. Alipouri, J. Poshtan, "Designing a robust minimum variance controller using discrete slide mode controller approach," ISA Transactions , vol. 52, no. 2, pp. 291-299, 2013.

[8] B.-Z. Guo, F.-F. Jin, "Sliding mode and active disturbance rejection control to stabilization of one-dimensional anti-stable wave equations subject to disturbance in boundary input," IEEE Transactions on Automatic Control , vol. 58, no. 5, pp. 1269-1274, 2013.

[9] Y.-S. Lu, S.-M. Lin, "Disturbance-observer-based adaptive feedforward cancellation of torque ripples in harmonic drive systems," Electrical Engineering , vol. 90, no. 2, pp. 95-106, 2007.

[10] T. Kavli, "Frequency domain synthesis of trajectory learning controllers for robot manipulators," Journal of Robotic Systems , vol. 9, no. 5, pp. 663-680, 1992.

[11] Y.-C. Wang, C.-J. Chien, "An observer-based adaptive iterative learning control using filtered-FNN design for robotic systems," Advances in Mechanical Engineering , vol. 6, 2014.

[12] S. Arimoto, S. Kawamura, F. Miyazaki, "Bettering operation of Robots by learning," Journal of Robotic Systems , vol. 1, no. 2, pp. 123-140, 1984.

[13] J.-X. Xu, D. Huang, V. Venkataramanan, The Cat Tuong Huynh, "Extreme precise motion tracking of piezoelectric positioning stage using sampled-data iterative learning control," IEEE Transactions on Control Systems Technology , vol. 21, no. 4, pp. 1432-1439, 2013.

[14] M. Norrlöf, "An adaptive iterative learning control algorithm with experiments on an industrial robot," IEEE Transactions on Robotics and Automation , vol. 18, no. 2, pp. 245-251, 2002.

[15] J.-X. Xu, Y. Chen, T. H. Lee, S. Yamamoto, "Terminal iterative learning control with an application to RTPCVD thickness control," Automatica , vol. 35, no. 9, pp. 1535-1542, 1999.

[16] S. S. Saab, "A discrete-time stochastic learning control algorithm," IEEE Transactions on Automatic Control , vol. 46, no. 6, pp. 877-887, 2001.

[17] K. K. Leang, S. Devasia, "Design of hysteresis-compensating iterative learning control for piezo-positioners: application to atomic force microscopes," Mechatronics , vol. 16, no. 3-4, pp. 141-158, 2006.

[18] Z. Zhu, D. Xu, J. Liu, Y. Xia, "Missile guidance law based on extended state observer," IEEE Transactions on Industrial Electronics , vol. 60, no. 12, pp. 5882-5891, 2013.

[19] M. Pizzocaro, D. Calonico, C. Calosso, C. Clivati, G. A. Costanzo, F. Levi, A. Mura, "Active disturbance rejection control of temperature for ultrastable optical cavities," IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control , vol. 60, no. 2, pp. 273-280, 2013.

[20] J. Yao, Z. Jiao, D. Ma, "Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping," IEEE Transactions on Industrial Electronics , vol. 61, no. 11, pp. 6285-6293, 2014.

[21] J. Yao, Z. Jiao, D. Ma, "Adaptive robust control of dc motors with extended state observer," IEEE Transactions on Industrial Electronics , vol. 61, no. 7, pp. 3630-3637, 2014.

[22] Q. Zheng, L. Q. Gao, Z. Gao, "On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknown dynamics," in Proceedings of the 46th IEEE Conference on Decision and Control (CDC '07), pp. 3501-3506, New Orleans, La, USA, December 2007.

[23] W. Qian, S. K. Panda, J.-X. Xu, "Torque ripple minimization in PM synchronous motors using iterative learning control," IEEE Transactions on Power Electronics , vol. 19, no. 2, pp. 272-279, 2004.

[24] G. Heinzinger, D. Fenwick, B. Paden, F. Miyazaki, "Stability of learning control with disturbances and uncertain initial conditions," IEEE Transactions on Automatic Control , vol. 37, no. 1, pp. 110-114, 1992.