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Brief Papers_______________________________________________________________________________ Development of Command-Based Iterative Learning Control Algorithm With Consideration of Friction, Disturbance, and Noise Effects Meng-Shiun Tsai, Ming-Tzong Lin, and Hong-Tzong Yau
Abstract—In this brief, a command-based iterative learning control (ILC) architecture is proposed to compensate for friction effect and to reduce tracking error caused by servo lag. In contrast to a feedback–feedforward control structure, the proposed methodology utilizes the learning algorithm that updates the input commands based on the tracking errors from the previous machining process. The effects of noise accumulations from each learning process of the ILC are analyzed by formulating the equivalent error dynamic and updated command equations, and the P-type ILC with a zero-phase filter is applied to alleviate noise and disturbance effects. It is shown that, for tracking a circle, the quadrant protrusions caused by friction can be reduced substantially by the updated command containing a concave shape located at the crossing of the zero velocity. Finally, analytical simulation and experimental results demonstrate that the command-based ILC algorithm can enhance the tracking performance significantly. Index Terms—Command-based iterative learning control (ILC), friction compensation, noise/disturbance accumulation, zero-phase filtering.
I. INTRODUCTION
I
N SERVO control systems, friction is an important factor that affects the quality of machining process. Its nonlinear phenomena, such as the Stribeck effect, hysteresis, springlike behavior and varying breakaway force [1], [2] might degrade machining accuracy. Various friction compensation methods have been proposed such as the observer-based approach [3]–[5], the adaptive method [6], [7], the feedforward-repetitive compensation [8], and the iterative learning control technique [9], [10]. Canudas de Wit et al. [3] first proposed the LuGre dynamic model and adopted the model-based control with the friction observer to reduce tracking errors. Canudas [11] utilized LaSalle’s invariant principle to prove stability of the closed-loop system with consideration of friction effect. The disadvantage of the model-based approach is that an accurate friction model is not easy to obtain in real machining. Tang et al. [8] adopted the repetitive control scheme such that the control signal is adjusted iteratively to compensate for the friction effect using a lookup table. Cho and Ha [9] analyzed the convergence of the iterative learning control (ILC) algorithm in which the friction
Manuscript received July 4, 2004. Manuscript received in final form April 8, 2005. Recommended by Associate Editor K. Kozlowski. The authors are with the Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi 621, Taiwan, R.O.C. (e-mail:
[email protected];
[email protected];
[email protected]). Digital Object Identifier 10.1109/TCST.2005.860521
effect is included in the linear model. The paper was mainly focused on mathematical derivation for the convergence analysis, and no physical interpretation was given. The limit cycle behaviors and convergence analysis for the stick–slip friction were analyzed in [12] and [13]. No simulation and experimental results were provided to verify the concepts. Many applications of servo control systems often perform the same task repeatedly. Typical examples such as robotic manipulators and Computer Numerical Control (CNC) machine tools are used extensively in industry. Arimoto et al. [14] first proposed the concept of ILC for repetitive processes. Since then, the ILC has been widely studied by many researchers in the past two decades [15], [16]. One of the main issues for the ILC algorithm is its convergence condition, which has been studied both in time domain [15] and frequency domain [17]. Technical considerations on transients, disturbances, measurement noises, and system uncertainty were discussed in [18]–[23]. A zero-phase filter is normally applied to the learning process [18]–[22]. However, the tradeoff between the convergence and noise/disturbance accumulation was not analyzed. In this brief, we propose a command-based iterative learning algorithm to compensate for the friction effect and to reduce tracking errors due to servo lag. One of the advantages for the command-based ILC is that it can improve system performance without changing the existing control architecture. The equivalent error dynamic and updated command equations are derived to analyze the noise and disturbance accumulation problems during the iterating process. The tradeoff between convergence and noise/disturbance accumulation is discussed. The ILC algorithm shows that a concave shape embedded in the updated command can compensate for the quadrant protrusion caused by friction effectively. Simulations and experiments are performed to verify the proposed algorithm. II. MODELING OF SERVO CONTROL SYSTEM WITH FRICTION A. General System Dynamic Model A general model of an ac servo control system is shown in Fig. 1, which includes the linear dynamic model of the servo system, the friction model, the velocity, and position loops with . The function is a velocity feedforward controller designed as , where is the feedforward gain. The paand in the velocity loop are designed by rameters specifying the damping ratio and bandwidth of the closed-loop in transfer function of the velocity loop. The position gain
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Fig. 1. Servo control system dynamics with friction model.
Fig. 2. Command-based iterative learning control architecture.
the position loop is determined by the designed bandwidth of is designed by the position loop [24]. The feedforward gain using an optimization approach, which minimizes the sum of the magnitudes of the error transfer function at various frequencies [10]. The general linear system description in the continuous-time domain has the following Laplace transforms: (1) , , , and denote the output, input, where disturbance, and sensor noise, respectively. For simplicity, the friction model is not included but will be considered later. , , and As shown in Fig. 1, the transfer functions are described as follows:
where is the relative angular velocity between two contact is the stiffness of the bristle, is the viscous cosurfaces, efficient, is the Coulomb friction torque, is the stiction is the Stribeck angular velocity. torque, and III. COMMAND-BASED ITERATIVE LEARNING CONTROL ALGORITHM In the previous section, the general model of the servo control system, which includes the friction model, is developed. The command-based ILC algorithm is introduced in this section. The convergence condition is analyzed together with the accumulation of measurement noise and disturbances during the iterating process. A. Iterative Learning Control Algorithm
(2)
and are the position and velocity feedforward where is equal to . is the closed-loop controllers and transfer function of the velocity loop and is given as (3)
Since the learning process is implemented in discrete-time domain, (1) is converted to the discrete-time model using the zero-order hold technique and is given as (4)
B. Friction Dynamic Model
The methodology of the command-based ILC algorithm is instead of upto iteratively update the reference command based on the tracking errors of dating the control input the previous machining cycle. The ILC architecture shown in Fig. 2 consists of an internal closed-loop servo system and an external ILC algorithm. The analysis will be first focused on the linear part of the systems, and thus the friction model is not included. However, it will be demonstrated by simulation and experiments that the friction effects on the tracking accuracy can be reduced significantly using the proposed algorithm. Although the proof of the convergence condition for the model with friction can be obtained by following the procedures in [12] and [13], geometric interpretation will be provided in this paper. The command-based ILC algorithm with the given closed-loop servo system is given as follows: (6) (7) (8)
(9)
The LuGre model [3] is adopted to describe the friction effect in this paper. The steady-state friction force related to angular velocity is given as
(5)
where is the iteration number, and denote the iteration and the iteration, input command at the , , , , and denote the respectively. desired trajectory, the system output signal, the tracking error, iteration, the disturbance, and the measurement noise at the and denote the measured output signal respectively.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 13, MAY 2006
and the tracking error with measurement noise, respectively. The initial conditions are reset to zero at every learning cycle. is a discrete-time filter, and is a noncausal zero-phase filter. Since the updated command is calculated offline according to the command-based ILC algorithm, the noncausal can be applied, and the zero-phase property ensures filter that no extra phase delay will be introduced in the updated command. To implement the ILC algorithm successfully to the CNC machining, one should not only consider the convergence condition but also investigate the disturbance and measurement noise effects. The error dynamic equation for the command-based ILC is formulated as the following lemma. Lemma III.1: Given the command-based ILC algorithm (6) and the servo system dynamics (4), (7)–(9), the error signal is updated according to
(10) Proof: Please see the Appendix. B. Convergence Condition Without Consideration of Disturbance and Noise It is shown that the error equation is driven by three types of inputs: the tracking error, the measurement noise, and the disturbance. By following the same terminology in [22], transient response is used to describe the process when the disturbance and noise are assumed to be zero. Under such a case, the error dynamic (10) is reduced to the following: (11) The sufficient condition for the convergence in frequency domain is given as (12) is the sampling time of the controller. The tracking where error will tend to zero when approaches infinity. Since the main purpose of this paper is to demonstrate the command behavior after each iteration, the updated command equation can be derived as follows: (13) As approaches to infinite, the command converges to . The proof of (13) is given in Lemma III.2 in the Appendix. It is clear that the updated command will converge to is obtained to cancel the the signal such that the inverse of closed-loop dynamic function under the condition of no disturbance and noise. C. Effects of Disturbance and Measurement Noise Although the convergence condition of the transient response is given, it is not adequate to analyze the behaviors of noise and disturbance for the proposed algorithm. To systematically evaluate the noise/disturbance accumulation effect, the equivalent
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error dynamic and the updated command equations are formulated and given in the following lemma. Lemma III.2: Given (4) and (6)–(10), the equivalent error dynamic and the updated command equations are derived and given as follows:
(14)
(15) . where is the sensitivity function which is defined as Proof: Please see the Appendix. In contrast to the conventional error dynamic (10), the equivalent error dynamic (14) can be represented by the summation form if one substitutes (15) into (14). The first two terms at the right-hand side of (15) correspond to the convergence of the transient response, which is equal to those in (13). The third and fourth terms illustrate how the noise and disturbance sigto enter the current updated nals at each iteration from . Obviously, the noise/disturbance accumulation in signal the command will deteriorate the learning accuracy because the error signal is a function of . Equation (15) indicates , , and play an importhat the functions tant role in filtering the measurement noise and the disturbance. With the designed control parameters of the closed-loop system , and ), the noise and disturbance accumulafixed ( , tions might cause the system diverge without proper design of and . Interpretation of (14) and (15) is best illustrated by an example which is given in Section V. IV. SIMULATION ANALYSIS In this section, analytical simulations are performed to demonstrate the effectiveness of the proposed algorithm. Although the previous convergence analysis does not include the nonlinear friction model, the simulation results will demonstrate that the friction effects can be reduced, and some interesting phenomenon of the command-based ILC will be observed. To compare the simulation results with the experimental results in the next section, open-loop system identification algorithm using the frequency-domain approach [25] is utilized is to identify the plant parameters. The nominal plant measured with a linear swept-frequency signal between the magnitudes from 1 to 1 V and the frequencies from 0.1 to 500 Hz. The plant is identified by using the empirical transfer function estimate (ETFE) method. The friction model is included, and the parameters of the LuGre model are determined experimentally by using the technique in [5]. The damping ratio and the bandwidth of the velocity closed-loop transfer function are chosen as 1.0 and 502.65 rad/s, respectively. The bandwidth of the position closed-loop transfer function is chosen as
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TABLE I PARAMETERS OF THE ac SERVO SYSTEM AND CONTROLLERS
Fig. 4.
X axis tracking errors at different learning cycles.
k
Fig. 5. Comparison of contouring errors between = 0 and (solid: the desired output ( ), dashed–dotted: the output errors amplified 2000 times).
y t
k = 20 learning y(t), contouring
utilized to ensure that the convergence condition for transient response is satisfied. It is shown that the Nyquist will cross the unit circle if both curve of and are selected to be 1. For the case of the being equal to 0.3 and being equal to 1, gain the convergence condition still violates at the high-frequency region. Finally, a sixth-order Butterworth zero-phase filter is de, and the convergence condition is satisfied as signed for shown in Fig. 3. The cutoff frequency of the filter is selected to 288.77 rad/s. The design criterion is based on the error spectrum obtained from the experiments, and more discussions will be given in Section V. B. Compensation of Friction and Servo Lag
Fig. 3. Nyquist plot of the closed-loop transfer functions learning gains and filters.
Q
Q8G
for different
251.33 rad/s and the velocity feedforward gain is selected to 0.95. All parameters of the closed-loop systems are listed in Table I unless stated otherwise. A. Convergence Condition of Analytical Simulation shown in Fig. 3 The Nyquist plot of and the filter is used to select the proper gain to satisfy the convergence condition. The Nyquist plot is
To validate the command-based ILC algorithm, simulations are performed for tracking a circle of 20-mm radius. Since there is no information about the noise and disturbance, they are not included in the simulation. The input commands to axis and axis are and mm, respectively. The linear acceleration/deceleration time is 20 ms. The LuGre friction model is included in the simulation. As shown in Fig. 4, the tracking errors of axis decrease as increases from 0 to 20. The sinusoidal shape in the error signal due to servo lag is , and the protrusion caused reduced almost to zero at by the friction is also attenuated significantly. The contouring and are shown in errors for tracking the circle at Fig. 5. It is clear that the command-based ILC can reduce both
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Fig. 6. Zoom in the friction effect of
Fig. 7.
X axis at k = 0 learning in simulation.
Fig. 8. Divergence of
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X axis tracking errors at different learning cycles.
Zoom in the desired output, position output, and updated command at
k = 20 learning in simulation.
the tracking error due to servo lag and the quadrant protrusions caused by the friction effect. To gain future insight into the learning properties of the ILC axis, , and the output algorithm, the commands of the localized at the crossing of the zero velocity are shown in Fig. 6, and the updated command , the output , and are shown in Fig. 7. Fig. 6 shows that the desired output the stick–slip friction effect causes the protrusion at the output as the axis is decelerated from a positive to a negative velocity (crossing the zero velocity). With the learning, the upcontains a concave shape to compensate dated command for the protrusion. This interesting phenomenon indicates that the tracking error due to the nonlinear friction can be reduced by modifying the original command to the desired command using the ILC algorithm without changing the existing control architecture. Note that the working principle is similar to the function of the feedforward controller such as the zero-phase error tracking control (ZPETC) [26]. However, the ZPETC algorithm was proposed to solve the servo-lag problem, and only linear dynamic systems are considered. V. EXPERIMENTAL VALIDATION Since the noise and disturbance spectrum strongly depend on experimental conditions, we will use the experimental results to demonstrate how the noise/disturbance accumulations affect
Fig. 9. Error spectrum of
X axis tracking errors at k = 0 and k = 9 learning.
the learning algorithm. The design criterion of the filter is discussed by using the experimental results. An – table using a PC-based controller is utilized for performing the closed-loop control and applying the command-based ILC algorithm. The control parameters used in experiments are listed in Table I. A. Disturbance Rejection and Noise Attenuation via
Filter
Following the simulation results, the experiment for tracking a circle of 20-mm radius at the speed of 1 rad/s is performed. is chosen to be 0.3, and the Initially, the learning gain filter is selected to be 1. Although this set of parameters might not guarantee the transient convergence, we use this example to illustrate the phenomenon of the noise/disturbance accumula0, 5, 9 are shown in Fig. 8. As the tion. The tracking errors at to , the error due to the learning process starts from servo lag is reduced. However, the magnitude of the high-frequency components increases as the learning process starts from to . The system will resonate if more iteration is performed. This situation could be improved if the Butteris applied. Since the tracking error worth zero-phase filter is mainly caused by servo lag, friction, and noise/disturbance, the power spectrum of the signal is very important in designing and are shown the filter. The error spectrum at , it contains in Fig. 9. Observing the error spectrum at the main frequency component at 1 rad/s, which is caused by
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Fig. 10.
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Convergence of X axis tracking errors at different learning cycles. TABLE II rms AND MAXIMUM TRACKING ERRORS (UNIT: m)
the servo lag. The frequency components around 60–130 rad/s (about 10–20 Hz) are possibly caused by the friction effect. The clearly indicates that both the error error spectrum at frequency components due to the servo lag and friction effect are reduced. However, the frequency components centered at 376.99 rad/s (60 Hz) become dominant. This signal comes from the power electronics of the driving and sensing circuits that are powered from the 60-Hz mains supply. The discussion that follows is based on a 60-Hz supply and would need to be modified for a main power supply at a different frequency. To solve this problem, a sixth-order Butterworth low-pass filter with the cutoff frequency 288.77 rad/s (36.41 Hz) is applied to attenuate the unwanted signal. The passband and stopband frequency of the filter is 188.50 rad/s (30 Hz) and 376.99 rad/s (60 Hz), respectively. The design criterion for this zero-pass filter is that must keep the useful error inforthe updated command mation due to the servo lag and friction after applying the filtering. However, the unwanted signal due to noise/disturbance accumulation should be removed. The sixth-order zero-phase low-pass filter satisfies the criterion. The performances of applying the ILC with the zero-phase filter are shown in Fig. 10, and the noise/disturbance accumulation has been reduced significantly as compared with the case in Fig. 8. The statistics data for the maximum and root-mean-square (rms) errors at and are summarized in Table II. To quantitatively evaluate different designs of the filter, the and filtering functions for noise and disturbance in (15) are analyzed. The frequency and as responses for equal to 1 are shown in Figs. 11 and 12. It is clear that the design
Fig. 11. Bode plot of the transfer functions Q8S (1 Q filters.
Fig. 12. Bode plot of the transfer functions Q8G Q filters.
0 Q8G ) for different
(1 0 Q8G ) for different
being the low-pass filter can provide higher noise/diswith turbance attenuation as compared to that with being equal to 1. It is also noticed that (15) can be applied to investigate the tradeoff between the performance of the transient and stochastic process when one designs different values of the . With the fixed design of the filter , learning gain and the frequency response functions of become smaller as the learning gain decreases. Smaller learning gain is beneficial in filtering the unwanted signal as shown in Fig. 13. However, a smaller results in a slower convergence rate for the transient response. To achieve a satisfactory transient response, more iteration is needed. The number of iteration increases might cause more accumulations as indicated by (15). This observation shows that a smaller can achieve better filtering, but more iteration is needed due to the slower convergence rate which might worsen the noise/disturbance accumulation. The above discussions indicate that in developing the ILC algorithm, the transient response, and the iteration times for stopping the learning process due to noise/disturbance accumulation should be considered simultaneously. Development for an optimal
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Fig. 13.
Q8G
Bode plot of the transfer functions ) for different learning gains 8.
Fig. 14.
Comparison of
Q8S (10Q8G ) and Q8G (10
X axis tracking errors for different Q filters.
, which investigates the tradeoff between these two effects, will be presented in the near future. Fig. 14 compares the tracking performance for the cases of being equal to 1 and being equal to the low-pass filter during the iteration process. Both the maximum and rms errors can be reduced significantly when the low-pass filter is applied. To further validate the proposed concepts, experiments using different feed rates are performed, and the results are shown in Table II. It clearly demonstrates the effectiveness of the command-based ILC algorithm. B. Experimental Validation of Friction Compensation As indicated in the previous section, the friction effect can be compensated by the updated command containing a concave shape. Fig. 15 shows the experimental results for the updated , which agree quite well with the simulation command at results shown in Fig. 7. The comparison of Figs. 7 and 15 shows in the exthat a saw-toothed signal is found at the output perimental results. It could be caused by the resolution limit of the optical sensor. To verify this presumption, the 1- m resolution of the optical encoder is included in the simulation. Fig. 16 shows that the saw-toothed-type behavior is also found in the simulation which is close to that of the experimental results.
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Fig. 15. Zoom in the desired output, position output, and updated command at = 20 learning in experiment.
k
Fig. 16. Zoom in the quantization effect of learning in simulation.
X -axis tracking errors at k = 20
VI. CONCLUSION In this brief, we propose a command-based iterative learning control (ILC) architecture for the high-precision machining process. In contrast to a feedback–feedforward control structure, the proposed methodology utilizes the learning algorithm, which updates the input commands based on the tracking errors from the previous machining process. One advantage of the proposed algorithm is that it can systematically generate the desired commands while keeping the feedback control architecture intact. It is shown that the proposed algorithm not only can compensate the tracking error due to servo lag, but also can reduce the quadrature protrusions significantly by the updated command containing a concave shape. The effects of the noise/disturbance accumulation are analyzed by using the equivalent error dynamic and updated command equations. To the best of the authors’ knowledge, this brief is the first attempt to derive these two equations in summation forms. The two equations provide valuable information in designing the zero-phase low-pass filter. The tradeoff between the transient and stochastic processes in selecting different values of learning gain is also discussed. Simulation and experimental results validate that high-precision machining can be achieved with the application of the command-based ILC algorithm.
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APPENDIX
REFERENCES
PROOF OF LEMMA III.1 into
Proof: By replacing by , it gives
in (8), and substituting it (16)
Substituting (6) and (9) into (16) gives (17) PROOF OF LEMMA III.2 Proof: By substituting (8) into (10), it gives (18) Substituting (7) into (18) and applying the sensitivity function gives (19) By substituting (7) and (9) into (6) and applying the sensitivity , one can obtain the following: function (20) Replacing
by , the above equation becomes (21)
By substituting (21) into (20), it becomes
(22) to , the updated By repeating the recursive process from command equation in the summation form is given as
(23) Given the convergence condition , and assuming condition is given as updated command at
, the initial for all , the
(24)
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