- Email: [email protected]
A simple hybrid fuzzy PD controller
Mechatronics 14 (2004) 877–890
A simple hybrid fuzzy PD controller Y.X. Su
a,c,* ,
Simon X. Yang b, Dong Sun a, B.Y. Duan
c
a
c
Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong b School of Engineering, University of Guelph, Guelph, Ontario N1G 2W1, Canada School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, PR China Accepted 12 May 2004
Abstract In many practical applications, the performance of a controlled system highly depends on whether a high quality differential signal can be obtained based on the non-continuous measured position signal only. In this paper, a simple hybrid fuzzy PD (proportional and derivative) controller is developed, by combining two nonlinear tracking differentiators to a conventional fuzzy PD controller. The main improvement of the proposed simple hybrid fuzzy PD controller is that it has high robustness against noise and easy for engineering implementation. Simulation and experimental comparison studies on motion control of a permanent magnet synchronous AC motor validate the effectiveness and efficiency of the proposed controller. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Fuzzy control; Nonlinear tracking differentiator; Measurement noise; Motion control
1. Introduction Since the first successful application of the fuzzy sets [1] to the control of a dynamic plant by Mamdani and Assilian [2], fuzzy logic control has been a practical alternative for a variety of challenging control applications, as it provides a convenient method for constructing nonlinear controllers via the use of heuristic information and the development of fuzzy controllers has progressed by leaps and bounds over the past two decades [3–9].
*
Corresponding author. Fax: +852-2788-8423. E-mail address: [email protected] (Y.X. Su).
0957-4158/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2004.05.002
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Y.X. Su et al. / Mechatronics 14 (2004) 877–890
Recently, fuzzy logic and conventional control techniques have been combined to design various hybrid fuzzy controllers. A hybrid controller can provide better system performance over a simple fuzzy controller alone [5–9]. For example, Ying et al. [6] was one of the pioneers in the formulation of a hybrid controller and the analysis of its control performance. Hsu et al. [7] successfully combined a simple fuzzy control and a variable structure control strategy to implement a high accuracy tracking control of a manipulator. Li et al. [8] proposed an enhanced hybrid fuzzy PID (proportional, integral and derivative) controller to improve the control performance in both transient and steady-state periods for mechanical manipulators under uncertainties. Er et al. [9] developed a new approach toward optimal design of a hybrid PID controller for linear as well as nonlinear systems using genetic algorithms. In many practical applications of PD control systems, the optical encoder is still the most popular accurate position sensor used in industry because of its simple detection circuit, high resolution, high accuracy, and relatively easy adaptation [10,11]. Conversely, the velocity measurement obtained by tachometers is often contaminated by noise (discontinuity in the magnetic field of the tachometer stator at low velocities and high-frequency phenomena contribute to diminish the quality of the measured velocities) [11,12]. This circumstance may reduce the dynamic performances of the control system. Furthermore, it is desirable that controllers be designed to require fewer measurements. Therefore it is advantageous to numerically reconstruct the velocity from the position measurement with noise, instead of direct velocity measurement [10–14]. In this paper, a simple hybrid fuzzy PD controller (HFPDC) is proposed to improve the system performance. A nonlinear tracking differentiator (TD) is first developed to obtain a high quality differential signal based on the rotor position measurement only. Based on this high quality differential signal, a simple hybrid fuzzy PD controller is designed to achieve an enhanced performance. This approach keeps the simple structure of a fuzzy PD controller so that it is not necessary to modify any hardware parts of the fuzzy PD control system for practical implementation. Comparative simulations and experiments on motion control of a permanent magnet synchronous AC (PMAC) motor system are presented to demonstrate the effectiveness and efficiency of the proposed controller. The paper is organized as follows. In Section 2, a simple hybrid fuzzy PD controller is developed, including the design of the TD in greater details. The comparative simulation and experimental results are presented in Section 3. Finally, some concluding remarks are summarized in Section 4.
2. Hybrid fuzzy PD controller The schematic diagram of the proposed simple hybrid fuzzy PD controller is shown in Fig. 1. It consists of two nonlinear tracking differentiators (TDs) and a simple fuzzy PD controller. The TD in the feedforward path is motivated by [15,16], which aims to arrange the transition and avoid the oscillation caused by the phase
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879
Fig. 1. Schematic diagram of the proposed simple hybrid fuzzy PD controller.
shift between the desired and actual velocities. Signal u is the plant command signal, and r1 and r2 are the two outputs of TD (I) to track the reference signal r and its differential, respectively. Signals z1 and z2 are the two outputs of TD (II) to track the measured signal and its differential, respectively. Signals e and c are the tracking error and its differential, respectively. 2.1. Nonlinear tracking differentiator design The so-called nonlinear tracking differentiator (TD) [17,18] is referred to as the following system: given a reference signal rðtÞ, the system provides two signals x1 ðtÞ and x2 ðtÞ, such that x1 ðtÞ ¼ rðtÞ and x2 ðtÞ ¼ r_ ðtÞ, respectively. Lemma 1. Suppose zðtÞ is a continuous function defined in ½0; 1Þ and satisfies limt!1 zðtÞ ¼ 0, if rðtÞ ¼ zðRtÞ; R > 0, then for an arbitrarily given T > 0, the following expression holds Z T lim jrðtÞj dt ¼ 0 ð1Þ R!1
0
Proof. Lemma 1 can be easily proved by using the integral transform. Based on Lemma 1 and the following transform 8 < s ¼ t=R x ðsÞ ¼ z1 ðtÞ þ c : 1 x2 ðsÞ ¼ Rz2 ðtÞ
ð2Þ
where c is an arbitrarily constant, Lemma 2 is obtained as follows. h Lemma 2. If the solutions to the system ( z_ 1 ¼ z2 z_ 2 ¼ f ðz1 ; z2 Þ
ð3Þ
hold that z1 ðtÞ ! 0 and z2 ðtÞ ! 0 as t ! 1, then for an arbitrarily constant c and T > 0, the solution x1 ðtÞ to the system
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(
x_ 1 ¼ x2 x2 x_ 2 ¼ R2 f x1 c; R
ð4Þ
satisfies Z lim
R!1
T
jx1 ðtÞ cj dt ¼ 0
ð5Þ
0
Theorem 1. If the arbitrarily solutions to system in (4) satisfy z1 ðtÞ ! 0 and z2 ðtÞ ! 0 as t ! 1, then for any arbitrarily bounded integrable function rðtÞ and a given constant T > 0, the solution x1 ðtÞ to the system ( x_ 1 ¼ x2 x2 ð6Þ x_ 2 ¼ R2 f x1 r; R should be satisfied Z T lim jx1 ðtÞ rðtÞj dt ¼ 0 R!1
ð7Þ
0
Proof. The proof can be completed by dividing the problem into the following two cases. Case 1: If rðtÞ is a constant function, then Theorem 1 is hold from Lemma 2. Case 2: If rðtÞðt 2 ½0; T Þ is a bounded integrable function, then it is an element of L1 ½0; T , where L1 ½0; T denotes the set of all first-integrable function in ½0; T . For an arbitrarily given e > 0, there exists a simple series un ðtÞ; ðn ¼ 1; 2; . . .Þ that uniformly converges to a continuous function wðtÞ 2 C½0; T such that [17] Z
T
jrðtÞ uðtÞj dt <
0
e 2
ð8Þ
Therefore, there exists an integer N0 such that juðtÞ uM ðtÞj < 4Te for all M > N0 . Consequently, the following inequality holds Z T Z T Z T e ð9Þ jrðtÞ uM ðtÞj dt 6 jrðtÞ wðtÞj dt þ jwðtÞ uM ðtÞj dt < 2 0 0 0 Because wðtÞ is a continuous function, the simple function series uM ðtÞ partitions the range ½0; T into some bounded intervals denoted by li ; i ¼ 1; 2; . . . ; m. Selecting uM ðtÞ to be a deterministic constant in each bounded interval, and based on Lemma 2, there exists R0 > 0 such that Z e ; i ¼ 1; 2; . . . ; m ð10Þ jx1 ðtÞ uM ðtÞj dt < 2m i for all R > R0 . So that,
Y.X. Su et al. / Mechatronics 14 (2004) 877–890
Z
T
jx1 ðtÞ uM ðtÞj dt <
0
e 2
ð11Þ
Thereby, the following inequality holds Z T Z T Z jx1 ðtÞ rðtÞj dt < jx1 ðtÞ uM ðtÞj dt þ 0
0
for all R > R0 . The proof is completed.
881
T
juM ðtÞ rðtÞj dt < e
ð12Þ
0
h
Theorem 1 shows that x1 ðtÞ averagely converges to rðtÞ. If the bounded integrable function rðtÞ is viewed as a generalized function, then x2 ðtÞ weakly converges to the generalized differential of rðtÞ [17]. Therefore, system in (6) can be used as a nonlinear tracking differentiator to smooth approach to an original generalized function and its generalized differential in the sense of average convergence and weak convergence, respectively. One feasible second-order TD for the reference signal can be expressed as [17] 8 < r_ 1 ¼ r2 r2 jr2 j ð13Þ ;d : r_ 2 ¼ Rsat r1 r þ 2R where satðA; dÞ is a nonlinear saturation function, which can be described as ( sgnðAÞ; jAj > d satðA; dÞ ¼ A ; jAj 6 d; d > 0 d
ð14Þ
in which sgnðÞ stands for a standard sign function. It is noted that the developed TD has high robustness to the variation of the design parameters R and d. For the reference signal disturbed by a white noise component with maximum amplitude of 0.01, R can be chosen, from the range of ½2:5; 50. For the reference signal with large noise magnitude than 0.01, R will be increased to preserve a better tracking performance at the sacrifice of the differential signal obtained. Therefore, the compromise method is to use the smallest R for achieving the better tracking performance. For all cases, the filtering factor can be determined as d ¼ 0:00005R. Replacing r by y in (13) and using (13) and (14), z1 and z2 of TD (II) can be easily obtained, in which z1 ðtÞ ¼ yðtÞ, z2 ðtÞ ¼ y_ ðtÞ, and y denotes the output of the control system. 2.2. Hybrid fuzzy PD controller The rotor position error and the change in error are used as the inputs to a conventional fuzzy PD controller (FPDC), so the normalized inputs can be defined by using the high quality differential signal obtained by the developed TDs eðtÞ ¼ Ke ðr1 ðtÞ z1 ðtÞÞ
ð15Þ
cðtÞ ¼ Kc ðr2 ðtÞ z2 ðtÞÞ
ð16Þ
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Table 1 Rule base uij;k Ej
Ci )1 )0.8 )0.6 )0.4 )0.2 0 0.2 0.4 0.6 0.8 1
)1
)0.8
)0.6
)0.4
)0.2
0
0.2
0.4
0.6
0.8
1
1 1 1 1 1 1 0.8 0.6 0.4 0.2 0
1 1 1 1 1 0.8 0.6 0.4 0.2 0 )0.2
1 1 1 1 0.8 0.6 0.4 0.2 0 )0.2 )0.4
1 1 1 0.8 0.6 0.4 0.2 )0 )0.2 )0.4 )0.6
1 1 0.8 0.6 0.4 0.2 0 )0.2 )0.4 )0.6 )0.8
1 0.8 0.6 0.4 0.2 0 )0.2 )0.4 )0.6 )0.8 )1
0.8 0.6 0.4 0.2 0 )0.2 )0.4 )0.6 )0.8 )1 )1
0.6 0.4 0.2 0 )0.2 )0.4 )0.6 )0.8 )1 )1 )1
0.4 0.2 0 )0.2 )0.4 )0.6 )0.8 )1 )1 )1 )1
0.2 0 )0.2 )0.4 )0.6 )0.8 )1 )1 )1 )1 )1
0 )0.2 )0.4 )0.6 )0.8 )1 )1 )1 )1 )1 )1
where Ke and Kc are the scaling gains. The control voltage u to the servo drive is the output of the controller. The FPDC has 11 fuzzy sets with membership functions uniformly distributed on each normalized input universe of discourse. All membership functions used in this FPDC are triangular type with a base-width of 0.4. Another 11 fuzzy sets with membership functions uniformly distributed on normalized output universe of discourse are used for the output of the FPDC. Zadeh’s compositional rule of inference and the standard center of gravity defuzzification technique are used to obtain the crisp output [4]. The normalized rulebase that we use for the PMAC motor is shown in Table 1. Since the convergence of the developed TD has been justified in the previous subsection, it is straightforward that the stability of the proposed HFPD controller (HFPDC) by incorporating two TDs into a conventional FPD controller, can be guaranteed according to the stability proofs of the fuzzy controller given in [4,6]. 3. Simulation and experimental results To demonstrate the effectiveness and efficiency of the proposed simple hybrid fuzzy PD controller, simulation and experimental comparison studies on the motion control of a permanent magnet synchronous AC (PMAC) motor system are conducted in this section. First the comparison study of the proposed TD and a conventional backward differentiator is presented. Then the performance using the proposed hybrid fuzzy PD controller is compared to that using a conventional fuzzy PD controller. 3.1. Comparative simulations of the TD and a backward differentiator The prevailing method to obtain the differential signal used in the available FPDC is the conventional backward differentiator [2–9], which can be expressed as
Y.X. Su et al. / Mechatronics 14 (2004) 877–890
cðtÞ ¼
eðtÞ eðt T Þ T
883
ð17Þ
where T is the sampling period. The superior performance of TD is shown in Fig. 2. The reference input is rðtÞ ¼ sinðtÞ rad and it is perturbed by an additive white noise component with maximum amplitude of 0.01 rad. For comparison, the differential signal obtained by the conventional backward differentiator is also shown in Fig. 3. The simulations were programmed in Matlab with a fourth-order Runge–Kutta and run on a PC Celeron-433. The simulation step was chosen as h ¼ 0:01, and the initial values of r1 and r2 were r1 ð0Þ ¼ 0 and r2 ð0Þ ¼ 0. The parameters of the TD were selected as R ¼ 100 and d ¼ 0:005. The first panel in Fig. 2 shows the rotor position, where the solid line represents the desired position while the dashed line represents the estimated position using the proposed TD. The corresponding estimation error is shown in the second panel of Fig. 2. The third panel of Fig. 2 shows the desired and the estimated velocities by solid and dashed lines, respectively. The corresponding estimation error of velocity is shown in the last panel of Fig. 2. For comparison, the differential signal obtained by the conventional backward differentiator is shown in Fig. 3, where the upper panel shows the desired (solid line) and estimated (dashed line) velocities, and lower panel shows the corresponding estimation error of velocity. It shows that the differential tracking performance of the proposed TD is much better over that of the conventional backward difference method. 3.2. Comparative simulations of the HFPDC and a FPDC The proposed hybrid fuzzy PD controller (HFPDC) is applied to the motion control of a PMAC motor, and its performance is compared to that using a conventional fuzzy PD controller (FPDC). When rotor reference coordinates (d–q axes) are chosen as the reference coordinates, the dynamic model for a PMAC motor can be described as [19] 8 dh > > ¼x > > dt > > > dx KN B > > > < dt ¼ J iq J x did R 1 > > > ¼ id þ Niq x þ td > > L L dt > > > > > : diq ¼ R i KN x Ni x þ 1 t q d q L L L dt
ð18Þ
in which, N denotes the number of pole pairs, R is the resistance of the stator, L is the inductance of the stator, K is a torque constant, J is the moment inertia of the rotor, x is the angular velocity, id ; iq are the currents on d–q reference frame, and td , tq are the voltages on the same frame, respectively.
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Fig. 2. Performance of a TD with noise.
Y.X. Su et al. / Mechatronics 14 (2004) 877–890
885
Fig. 3. Performance of a backward differentiator with noise.
In the motor velocity mode for most industrial used PMAC servo drives, the desired tracking error converges in a manner as Z ð19Þ e_ x þ KP1 ex þ KI1 ex dt ¼ 0 then the desired current component in the q coordinate iq is given as Z J TL iq ¼ þ x þ KP1 ex þ KI1 ex dt KN J
ð20Þ
where x is the target velocity, ex ¼ x x is the tracking error, KP1 is the velocity loop proportional gain, KI1 is the velocity loop integral gain, and TL is the external load. The desired current component in the q coordinate iq can be reached quickly, when voltage control laws md and mq are adopted as Z R _ td ¼ L id Niq x þ id KP2 ðid id Þ KI2 ðid id Þ dt ð21Þ L
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Z R KN x þ Nid x þ i_q KP2 ðiq iq Þ KI2 ðiq iq Þ dt tq ¼ L i q þ L L
ð22Þ
where KP2 and KI2 are adjustable parameters denoting the current loop proportional and integral gains, respectively. The PMAC motor system used in this research is a MSMA042A1G AC servomotor and a matched MSDA043A1A servo drive made by Panasonic Inc. Its parameters are: rated power-0.4 KW, rated voltage-200 V, rated current-2.5 A, rated speed-3000 r/min, N ¼ 4, R ¼ 0:20 X, L ¼ 0:0012 H, K ¼ 0:18 Nm/A, and J ¼ 0:36 104 kg m2 . The servo drive was set in the velocity mode, and the gains were selected as KP1 ¼ 100, KI1 ¼ 5, KP2 ¼ 150, and KI2 ¼ 80. The sampling period was selected as T ¼ 10 ms. After several tests, the parameters for the HFPDC and the FPDC were determined as Ke ¼ 10, Kc ¼ 0:5, and Ku ¼ 3. The initial values of r1 , r2 , z1 and z2 were selected as r1 ð0Þ ¼ 0, r2 ð0Þ ¼ 0, z1 ð0Þ ¼ 0 and z2 ð0Þ ¼ 0. The comparative simulation results for the reference input rðtÞ ¼ sinðtÞ rad with the perturbation of an additive white noise component with maximum amplitude of 0.01 rad using the proposed HFPDC and the FPDC are shown in Fig. 4, where the solid and dashed lines represents the position errors using the HFPDC and the FPDC, respectively. The simulation results without any noise are shown in Fig. 5. It can be seen from Fig. 4 that the position tracking performance using the proposed HFPDC is better than that using the conventional FPDC except the initial transition. The rotor position error using HFPDC holds in the range of ±0.02 rad, and the position error using FPDC holds in the range of ±0.03 rad. The large tracking error by HFPDC at the initial stage is resulted from the bad guess of the initial value of r2 . In fact, its initial value is r2 ð0Þ ¼ 1 instead of r2 ð0Þ ¼ 0. Fig. 5 also shows the position tracking performance using the proposed HFPDC is much better than that using the conventional FPDC in the case without any noise.
Fig. 4. Comparative simulation of HFPDC and FPDC with noise.
Y.X. Su et al. / Mechatronics 14 (2004) 877–890
887
Fig. 5. Comparative simulation of HFPDC and FPDC without noise.
The rotor position error by the HFPDC holds in the range of ±0.006 rad, and the position error by the FPDC is in the range of ±0.016 rad. Note that the proposed HFPDC uses two estimated states of the original reference input and the measured output. In the absence of noise, the results obtained using the conventional FPDC should be better than that using the HFPDC. But it can be seen from Fig. 5, a reverse result is obtained. Therefore, it seems that the proposed simple HFPDC is capable of improving the gains of the control system. The better performance results from the feedforward TD, which has the function to arrange the transition of the system. To further verify this observation, additional comparative simulation study on the tracking of a unit step command is conducted. The simulation results with noise are shown in Fig. 6. The results without noise are shown in Fig. 7. It can be seen that the above observation is obviously correct. The response using the proposed HFPDC to a unit step command is faster than that using the conventional FPDC. Furthermore, the result is obtained using the same gains. 3.3. Comparative experiments of the HFPDC and a FPDC After the simulation validation of the performance enhancement of the developed HFPDC over the conventional FPDC, experiments were further performed to verify the practical effectiveness. The experimental setup consists of the aforementioned MSMA042A1G permanent magnet synchronous motor and the matched MSDA043A1A servo drive, a six-channel D/A output card PCL-726, and a three-axis quadrature encoder card PCL-833. An incremental optical encoder with the resolution of 4000 pulse/rev was used to sense the feedback position signal. The first experiment is to test the position tracking performance for a unit periodical sinusoidal command sinðtÞ rad. Experimental results are shown in Fig. 8. Obviously, the better tracking performance is also obtained using the proposed
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Y.X. Su et al. / Mechatronics 14 (2004) 877–890
Fig. 6. Comparative simulation of HFPDC and FPDC to a unit step with noise.
Fig. 7. Comparative simulation of HFPDC and FPDC to a unit step without any noise.
HFPDC in comparison than that of the FPDC. The steady-state tracking error of the HFPDC is within ±0.022 rad, whereas the steady-state tracking error of the FPDC reaches ±0.038 rad. Furthermore, the result obtained by the HFPDC is less noisy than that of the FPDC. The second experiment is to test the position tracking performance for a unit step. Experimental results are shown in Fig. 9. It can be seen that the HFPDC has fast response than the FPDC. The better transient performance is mainly resulted from the high quality differential signal provided by the TD with simple calculation, since the other gains are the same. Therefore, we can conclude that the developed HFPDC is better than the conventional FPDC. The better performance is mainly owed to the high quality differential signal selected by the proposed TD.
Y.X. Su et al. / Mechatronics 14 (2004) 877–890
889
Fig. 8. Comparative experiment of HFPDC and FPDC to a unit sinusoidal position profile.
Fig. 9. Comparative experiment of HFPDC and FPDC to a unit step position profile.
4. Conclusions A simple hybrid fuzzy PD controller is proposed in this paper, by combining two nonlinear tracking differentiators to a conventional fuzzy PD controller. A nonlinear tracking differentiator is first developed to obtain a high quality differential signal with measurement noise, which builds a solid basis for a better performance of the proposed hybrid fuzzy PD controller. The important improvement of the developed controller is that it has high robustness against noise and is easy to implement. The effectiveness and efficiency of the proposed simple hybrid fuzzy PD controller is verified by simulation and experimental comparison studies on motion control of a permanent magnet synchronous AC motor system.
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Acknowledgements The authors are grateful to the anonymous reviewers for the valuable comments and suggestions which have lead to significant improvements of this paper. The authors would like to acknowledge the support from the Natural Science Foundation of Shaanxi Province under Grant 2000C22, and Robotics Laboratory in Shenyang Institute of Automation, Chinese Academy of Sciences under Grant RL200104.
References [1] Zadeh LA. Fuzzy sets. Inform Contr 1965;8:338–53. [2] Mamdani EH, Assilian S. An experiment in linguistic synthesis with a fuzzy logic controller. IntJ Man-Mach Stud 1974;7(1):1–13. [3] Kim E. A fuzzy disturbance observer and its application to control. IEEE Trans Fuzzy Syst 2002;10(1):77–84. [4] Passino KM, Yurkovich S. Fuzzy control. CA: Addison Wesley Longman; 1998. [5] Chen GR, Pham TT. Introduction to fuzzy sets, fuzzy logic, and fuzzy control systems. Boca Raton, FL: CRC Press; 2000. [6] Ying H, Siler W, Buckley JJ. Fuzzy control theory: a nonlinear case. Automatica 1990;26(3):513–20. [7] Hsu YC, Chen GR, Li HX. A fuzzy adaptive variable structure controller with applications to robot manipulators. IEEE Trans Syst Man, Cybern Part B 2001;31(3):331–40. [8] Li W, Chang XG, Farrell J, Wahl FM. Design of an enhanced hybrid fuzzy P+ID controller for a mechanical manipulator. IEEE Trans Syst Man, Cybern Part B 2001;31(6):938–45. [9] Er MJ, Sun YL. Hybrid fuzzy proportional plus conventional derivative control of linear and nonlinear systems. IEEE Trans Ind Electron 2001;48(6):1109–17. [10] Martinez-Guerra R, Poznyak A, Gortcheva E, Diaz de Leon V. Robot angular link velocity estimation in the presence of high-level mixed uncertainties. IEE Proc––Control Theory Appl 2000;147(5):515–22. [11] Feemster M, Aquino P, Dawson DM, Behal A. Sensorless rotor velocity tracking control for induction motors. IEEE Trans Contr Syst Technol 2001;9(4):645–53. [12] Loria A, Melhem K. Position feedback global tracking control of EL systems: a state transformation approach. IEEE Trans Automat Contr 2002;47(5):841–7. [13] Yao B, Xu L. Adaptive robust motion control of linear motors for precision manufacturing. Mechatronics 2002;12(4):595–616. [14] Moore ML, Musacchio JT, Passino KM. Genetic adaptive control for an inverted wedge: experiments and comparative analysis. Eng Appl Artif Intell 2001;14(1):1–14. [15] Jaritz A, Spong MW. An experimental comparison of robust control algorithms on a direct drive manipulator. IEEE Trans Contr Syst Technol 1996;4(6):627–40. [16] Kempf CJ, Kobayashi S. Disturbance observer and feedforward design for a high-speed direct-drive positioning table. IEEE Trans Contr Syst Technol 1999;7(5):513–26. [17] Han JQ, Wang W. Nonlinear tracking-differentiator. J Syst Sci Math Scis 1994;14(2):177–83 (in Chinese). [18] Su YX, Duan BY, Zhang YF. Robust precision motion control for AC servo systems. In: The 4th World Congress on Intelligent Control and Automation. vol. 4. 2002. p. 3319–23. [19] Wang GJ, Fong CT, Chang KJ. Neural-network-based self-tuning PI controller for precise motion control of PMAC motors. IEEE Trans Ind Electron 2001;48(2):408–15.
A simple hybrid fuzzy PD controller Y.X. Su
a,c,* ,
Simon X. Yang b, Dong Sun a, B.Y. Duan
c
a
c
Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong b School of Engineering, University of Guelph, Guelph, Ontario N1G 2W1, Canada School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, PR China Accepted 12 May 2004
Abstract In many practical applications, the performance of a controlled system highly depends on whether a high quality differential signal can be obtained based on the non-continuous measured position signal only. In this paper, a simple hybrid fuzzy PD (proportional and derivative) controller is developed, by combining two nonlinear tracking differentiators to a conventional fuzzy PD controller. The main improvement of the proposed simple hybrid fuzzy PD controller is that it has high robustness against noise and easy for engineering implementation. Simulation and experimental comparison studies on motion control of a permanent magnet synchronous AC motor validate the effectiveness and efficiency of the proposed controller. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Fuzzy control; Nonlinear tracking differentiator; Measurement noise; Motion control
1. Introduction Since the first successful application of the fuzzy sets [1] to the control of a dynamic plant by Mamdani and Assilian [2], fuzzy logic control has been a practical alternative for a variety of challenging control applications, as it provides a convenient method for constructing nonlinear controllers via the use of heuristic information and the development of fuzzy controllers has progressed by leaps and bounds over the past two decades [3–9].
*
Corresponding author. Fax: +852-2788-8423. E-mail address: [email protected] (Y.X. Su).
0957-4158/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2004.05.002
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Y.X. Su et al. / Mechatronics 14 (2004) 877–890
Recently, fuzzy logic and conventional control techniques have been combined to design various hybrid fuzzy controllers. A hybrid controller can provide better system performance over a simple fuzzy controller alone [5–9]. For example, Ying et al. [6] was one of the pioneers in the formulation of a hybrid controller and the analysis of its control performance. Hsu et al. [7] successfully combined a simple fuzzy control and a variable structure control strategy to implement a high accuracy tracking control of a manipulator. Li et al. [8] proposed an enhanced hybrid fuzzy PID (proportional, integral and derivative) controller to improve the control performance in both transient and steady-state periods for mechanical manipulators under uncertainties. Er et al. [9] developed a new approach toward optimal design of a hybrid PID controller for linear as well as nonlinear systems using genetic algorithms. In many practical applications of PD control systems, the optical encoder is still the most popular accurate position sensor used in industry because of its simple detection circuit, high resolution, high accuracy, and relatively easy adaptation [10,11]. Conversely, the velocity measurement obtained by tachometers is often contaminated by noise (discontinuity in the magnetic field of the tachometer stator at low velocities and high-frequency phenomena contribute to diminish the quality of the measured velocities) [11,12]. This circumstance may reduce the dynamic performances of the control system. Furthermore, it is desirable that controllers be designed to require fewer measurements. Therefore it is advantageous to numerically reconstruct the velocity from the position measurement with noise, instead of direct velocity measurement [10–14]. In this paper, a simple hybrid fuzzy PD controller (HFPDC) is proposed to improve the system performance. A nonlinear tracking differentiator (TD) is first developed to obtain a high quality differential signal based on the rotor position measurement only. Based on this high quality differential signal, a simple hybrid fuzzy PD controller is designed to achieve an enhanced performance. This approach keeps the simple structure of a fuzzy PD controller so that it is not necessary to modify any hardware parts of the fuzzy PD control system for practical implementation. Comparative simulations and experiments on motion control of a permanent magnet synchronous AC (PMAC) motor system are presented to demonstrate the effectiveness and efficiency of the proposed controller. The paper is organized as follows. In Section 2, a simple hybrid fuzzy PD controller is developed, including the design of the TD in greater details. The comparative simulation and experimental results are presented in Section 3. Finally, some concluding remarks are summarized in Section 4.
2. Hybrid fuzzy PD controller The schematic diagram of the proposed simple hybrid fuzzy PD controller is shown in Fig. 1. It consists of two nonlinear tracking differentiators (TDs) and a simple fuzzy PD controller. The TD in the feedforward path is motivated by [15,16], which aims to arrange the transition and avoid the oscillation caused by the phase
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Fig. 1. Schematic diagram of the proposed simple hybrid fuzzy PD controller.
shift between the desired and actual velocities. Signal u is the plant command signal, and r1 and r2 are the two outputs of TD (I) to track the reference signal r and its differential, respectively. Signals z1 and z2 are the two outputs of TD (II) to track the measured signal and its differential, respectively. Signals e and c are the tracking error and its differential, respectively. 2.1. Nonlinear tracking differentiator design The so-called nonlinear tracking differentiator (TD) [17,18] is referred to as the following system: given a reference signal rðtÞ, the system provides two signals x1 ðtÞ and x2 ðtÞ, such that x1 ðtÞ ¼ rðtÞ and x2 ðtÞ ¼ r_ ðtÞ, respectively. Lemma 1. Suppose zðtÞ is a continuous function defined in ½0; 1Þ and satisfies limt!1 zðtÞ ¼ 0, if rðtÞ ¼ zðRtÞ; R > 0, then for an arbitrarily given T > 0, the following expression holds Z T lim jrðtÞj dt ¼ 0 ð1Þ R!1
0
Proof. Lemma 1 can be easily proved by using the integral transform. Based on Lemma 1 and the following transform 8 < s ¼ t=R x ðsÞ ¼ z1 ðtÞ þ c : 1 x2 ðsÞ ¼ Rz2 ðtÞ
ð2Þ
where c is an arbitrarily constant, Lemma 2 is obtained as follows. h Lemma 2. If the solutions to the system ( z_ 1 ¼ z2 z_ 2 ¼ f ðz1 ; z2 Þ
ð3Þ
hold that z1 ðtÞ ! 0 and z2 ðtÞ ! 0 as t ! 1, then for an arbitrarily constant c and T > 0, the solution x1 ðtÞ to the system
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(
x_ 1 ¼ x2 x2 x_ 2 ¼ R2 f x1 c; R
ð4Þ
satisfies Z lim
R!1
T
jx1 ðtÞ cj dt ¼ 0
ð5Þ
0
Theorem 1. If the arbitrarily solutions to system in (4) satisfy z1 ðtÞ ! 0 and z2 ðtÞ ! 0 as t ! 1, then for any arbitrarily bounded integrable function rðtÞ and a given constant T > 0, the solution x1 ðtÞ to the system ( x_ 1 ¼ x2 x2 ð6Þ x_ 2 ¼ R2 f x1 r; R should be satisfied Z T lim jx1 ðtÞ rðtÞj dt ¼ 0 R!1
ð7Þ
0
Proof. The proof can be completed by dividing the problem into the following two cases. Case 1: If rðtÞ is a constant function, then Theorem 1 is hold from Lemma 2. Case 2: If rðtÞðt 2 ½0; T Þ is a bounded integrable function, then it is an element of L1 ½0; T , where L1 ½0; T denotes the set of all first-integrable function in ½0; T . For an arbitrarily given e > 0, there exists a simple series un ðtÞ; ðn ¼ 1; 2; . . .Þ that uniformly converges to a continuous function wðtÞ 2 C½0; T such that [17] Z
T
jrðtÞ uðtÞj dt <
0
e 2
ð8Þ
Therefore, there exists an integer N0 such that juðtÞ uM ðtÞj < 4Te for all M > N0 . Consequently, the following inequality holds Z T Z T Z T e ð9Þ jrðtÞ uM ðtÞj dt 6 jrðtÞ wðtÞj dt þ jwðtÞ uM ðtÞj dt < 2 0 0 0 Because wðtÞ is a continuous function, the simple function series uM ðtÞ partitions the range ½0; T into some bounded intervals denoted by li ; i ¼ 1; 2; . . . ; m. Selecting uM ðtÞ to be a deterministic constant in each bounded interval, and based on Lemma 2, there exists R0 > 0 such that Z e ; i ¼ 1; 2; . . . ; m ð10Þ jx1 ðtÞ uM ðtÞj dt < 2m i for all R > R0 . So that,
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Z
T
jx1 ðtÞ uM ðtÞj dt <
0
e 2
ð11Þ
Thereby, the following inequality holds Z T Z T Z jx1 ðtÞ rðtÞj dt < jx1 ðtÞ uM ðtÞj dt þ 0
0
for all R > R0 . The proof is completed.
881
T
juM ðtÞ rðtÞj dt < e
ð12Þ
0
h
Theorem 1 shows that x1 ðtÞ averagely converges to rðtÞ. If the bounded integrable function rðtÞ is viewed as a generalized function, then x2 ðtÞ weakly converges to the generalized differential of rðtÞ [17]. Therefore, system in (6) can be used as a nonlinear tracking differentiator to smooth approach to an original generalized function and its generalized differential in the sense of average convergence and weak convergence, respectively. One feasible second-order TD for the reference signal can be expressed as [17] 8 < r_ 1 ¼ r2 r2 jr2 j ð13Þ ;d : r_ 2 ¼ Rsat r1 r þ 2R where satðA; dÞ is a nonlinear saturation function, which can be described as ( sgnðAÞ; jAj > d satðA; dÞ ¼ A ; jAj 6 d; d > 0 d
ð14Þ
in which sgnðÞ stands for a standard sign function. It is noted that the developed TD has high robustness to the variation of the design parameters R and d. For the reference signal disturbed by a white noise component with maximum amplitude of 0.01, R can be chosen, from the range of ½2:5; 50. For the reference signal with large noise magnitude than 0.01, R will be increased to preserve a better tracking performance at the sacrifice of the differential signal obtained. Therefore, the compromise method is to use the smallest R for achieving the better tracking performance. For all cases, the filtering factor can be determined as d ¼ 0:00005R. Replacing r by y in (13) and using (13) and (14), z1 and z2 of TD (II) can be easily obtained, in which z1 ðtÞ ¼ yðtÞ, z2 ðtÞ ¼ y_ ðtÞ, and y denotes the output of the control system. 2.2. Hybrid fuzzy PD controller The rotor position error and the change in error are used as the inputs to a conventional fuzzy PD controller (FPDC), so the normalized inputs can be defined by using the high quality differential signal obtained by the developed TDs eðtÞ ¼ Ke ðr1 ðtÞ z1 ðtÞÞ
ð15Þ
cðtÞ ¼ Kc ðr2 ðtÞ z2 ðtÞÞ
ð16Þ
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Table 1 Rule base uij;k Ej
Ci )1 )0.8 )0.6 )0.4 )0.2 0 0.2 0.4 0.6 0.8 1
)1
)0.8
)0.6
)0.4
)0.2
0
0.2
0.4
0.6
0.8
1
1 1 1 1 1 1 0.8 0.6 0.4 0.2 0
1 1 1 1 1 0.8 0.6 0.4 0.2 0 )0.2
1 1 1 1 0.8 0.6 0.4 0.2 0 )0.2 )0.4
1 1 1 0.8 0.6 0.4 0.2 )0 )0.2 )0.4 )0.6
1 1 0.8 0.6 0.4 0.2 0 )0.2 )0.4 )0.6 )0.8
1 0.8 0.6 0.4 0.2 0 )0.2 )0.4 )0.6 )0.8 )1
0.8 0.6 0.4 0.2 0 )0.2 )0.4 )0.6 )0.8 )1 )1
0.6 0.4 0.2 0 )0.2 )0.4 )0.6 )0.8 )1 )1 )1
0.4 0.2 0 )0.2 )0.4 )0.6 )0.8 )1 )1 )1 )1
0.2 0 )0.2 )0.4 )0.6 )0.8 )1 )1 )1 )1 )1
0 )0.2 )0.4 )0.6 )0.8 )1 )1 )1 )1 )1 )1
where Ke and Kc are the scaling gains. The control voltage u to the servo drive is the output of the controller. The FPDC has 11 fuzzy sets with membership functions uniformly distributed on each normalized input universe of discourse. All membership functions used in this FPDC are triangular type with a base-width of 0.4. Another 11 fuzzy sets with membership functions uniformly distributed on normalized output universe of discourse are used for the output of the FPDC. Zadeh’s compositional rule of inference and the standard center of gravity defuzzification technique are used to obtain the crisp output [4]. The normalized rulebase that we use for the PMAC motor is shown in Table 1. Since the convergence of the developed TD has been justified in the previous subsection, it is straightforward that the stability of the proposed HFPD controller (HFPDC) by incorporating two TDs into a conventional FPD controller, can be guaranteed according to the stability proofs of the fuzzy controller given in [4,6]. 3. Simulation and experimental results To demonstrate the effectiveness and efficiency of the proposed simple hybrid fuzzy PD controller, simulation and experimental comparison studies on the motion control of a permanent magnet synchronous AC (PMAC) motor system are conducted in this section. First the comparison study of the proposed TD and a conventional backward differentiator is presented. Then the performance using the proposed hybrid fuzzy PD controller is compared to that using a conventional fuzzy PD controller. 3.1. Comparative simulations of the TD and a backward differentiator The prevailing method to obtain the differential signal used in the available FPDC is the conventional backward differentiator [2–9], which can be expressed as
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cðtÞ ¼
eðtÞ eðt T Þ T
883
ð17Þ
where T is the sampling period. The superior performance of TD is shown in Fig. 2. The reference input is rðtÞ ¼ sinðtÞ rad and it is perturbed by an additive white noise component with maximum amplitude of 0.01 rad. For comparison, the differential signal obtained by the conventional backward differentiator is also shown in Fig. 3. The simulations were programmed in Matlab with a fourth-order Runge–Kutta and run on a PC Celeron-433. The simulation step was chosen as h ¼ 0:01, and the initial values of r1 and r2 were r1 ð0Þ ¼ 0 and r2 ð0Þ ¼ 0. The parameters of the TD were selected as R ¼ 100 and d ¼ 0:005. The first panel in Fig. 2 shows the rotor position, where the solid line represents the desired position while the dashed line represents the estimated position using the proposed TD. The corresponding estimation error is shown in the second panel of Fig. 2. The third panel of Fig. 2 shows the desired and the estimated velocities by solid and dashed lines, respectively. The corresponding estimation error of velocity is shown in the last panel of Fig. 2. For comparison, the differential signal obtained by the conventional backward differentiator is shown in Fig. 3, where the upper panel shows the desired (solid line) and estimated (dashed line) velocities, and lower panel shows the corresponding estimation error of velocity. It shows that the differential tracking performance of the proposed TD is much better over that of the conventional backward difference method. 3.2. Comparative simulations of the HFPDC and a FPDC The proposed hybrid fuzzy PD controller (HFPDC) is applied to the motion control of a PMAC motor, and its performance is compared to that using a conventional fuzzy PD controller (FPDC). When rotor reference coordinates (d–q axes) are chosen as the reference coordinates, the dynamic model for a PMAC motor can be described as [19] 8 dh > > ¼x > > dt > > > dx KN B > > > < dt ¼ J iq J x did R 1 > > > ¼ id þ Niq x þ td > > L L dt > > > > > : diq ¼ R i KN x Ni x þ 1 t q d q L L L dt
ð18Þ
in which, N denotes the number of pole pairs, R is the resistance of the stator, L is the inductance of the stator, K is a torque constant, J is the moment inertia of the rotor, x is the angular velocity, id ; iq are the currents on d–q reference frame, and td , tq are the voltages on the same frame, respectively.
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Fig. 2. Performance of a TD with noise.
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Fig. 3. Performance of a backward differentiator with noise.
In the motor velocity mode for most industrial used PMAC servo drives, the desired tracking error converges in a manner as Z ð19Þ e_ x þ KP1 ex þ KI1 ex dt ¼ 0 then the desired current component in the q coordinate iq is given as Z J TL iq ¼ þ x þ KP1 ex þ KI1 ex dt KN J
ð20Þ
where x is the target velocity, ex ¼ x x is the tracking error, KP1 is the velocity loop proportional gain, KI1 is the velocity loop integral gain, and TL is the external load. The desired current component in the q coordinate iq can be reached quickly, when voltage control laws md and mq are adopted as Z R _ td ¼ L id Niq x þ id KP2 ðid id Þ KI2 ðid id Þ dt ð21Þ L
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Z R KN x þ Nid x þ i_q KP2 ðiq iq Þ KI2 ðiq iq Þ dt tq ¼ L i q þ L L
ð22Þ
where KP2 and KI2 are adjustable parameters denoting the current loop proportional and integral gains, respectively. The PMAC motor system used in this research is a MSMA042A1G AC servomotor and a matched MSDA043A1A servo drive made by Panasonic Inc. Its parameters are: rated power-0.4 KW, rated voltage-200 V, rated current-2.5 A, rated speed-3000 r/min, N ¼ 4, R ¼ 0:20 X, L ¼ 0:0012 H, K ¼ 0:18 Nm/A, and J ¼ 0:36 104 kg m2 . The servo drive was set in the velocity mode, and the gains were selected as KP1 ¼ 100, KI1 ¼ 5, KP2 ¼ 150, and KI2 ¼ 80. The sampling period was selected as T ¼ 10 ms. After several tests, the parameters for the HFPDC and the FPDC were determined as Ke ¼ 10, Kc ¼ 0:5, and Ku ¼ 3. The initial values of r1 , r2 , z1 and z2 were selected as r1 ð0Þ ¼ 0, r2 ð0Þ ¼ 0, z1 ð0Þ ¼ 0 and z2 ð0Þ ¼ 0. The comparative simulation results for the reference input rðtÞ ¼ sinðtÞ rad with the perturbation of an additive white noise component with maximum amplitude of 0.01 rad using the proposed HFPDC and the FPDC are shown in Fig. 4, where the solid and dashed lines represents the position errors using the HFPDC and the FPDC, respectively. The simulation results without any noise are shown in Fig. 5. It can be seen from Fig. 4 that the position tracking performance using the proposed HFPDC is better than that using the conventional FPDC except the initial transition. The rotor position error using HFPDC holds in the range of ±0.02 rad, and the position error using FPDC holds in the range of ±0.03 rad. The large tracking error by HFPDC at the initial stage is resulted from the bad guess of the initial value of r2 . In fact, its initial value is r2 ð0Þ ¼ 1 instead of r2 ð0Þ ¼ 0. Fig. 5 also shows the position tracking performance using the proposed HFPDC is much better than that using the conventional FPDC in the case without any noise.
Fig. 4. Comparative simulation of HFPDC and FPDC with noise.
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Fig. 5. Comparative simulation of HFPDC and FPDC without noise.
The rotor position error by the HFPDC holds in the range of ±0.006 rad, and the position error by the FPDC is in the range of ±0.016 rad. Note that the proposed HFPDC uses two estimated states of the original reference input and the measured output. In the absence of noise, the results obtained using the conventional FPDC should be better than that using the HFPDC. But it can be seen from Fig. 5, a reverse result is obtained. Therefore, it seems that the proposed simple HFPDC is capable of improving the gains of the control system. The better performance results from the feedforward TD, which has the function to arrange the transition of the system. To further verify this observation, additional comparative simulation study on the tracking of a unit step command is conducted. The simulation results with noise are shown in Fig. 6. The results without noise are shown in Fig. 7. It can be seen that the above observation is obviously correct. The response using the proposed HFPDC to a unit step command is faster than that using the conventional FPDC. Furthermore, the result is obtained using the same gains. 3.3. Comparative experiments of the HFPDC and a FPDC After the simulation validation of the performance enhancement of the developed HFPDC over the conventional FPDC, experiments were further performed to verify the practical effectiveness. The experimental setup consists of the aforementioned MSMA042A1G permanent magnet synchronous motor and the matched MSDA043A1A servo drive, a six-channel D/A output card PCL-726, and a three-axis quadrature encoder card PCL-833. An incremental optical encoder with the resolution of 4000 pulse/rev was used to sense the feedback position signal. The first experiment is to test the position tracking performance for a unit periodical sinusoidal command sinðtÞ rad. Experimental results are shown in Fig. 8. Obviously, the better tracking performance is also obtained using the proposed
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Fig. 6. Comparative simulation of HFPDC and FPDC to a unit step with noise.
Fig. 7. Comparative simulation of HFPDC and FPDC to a unit step without any noise.
HFPDC in comparison than that of the FPDC. The steady-state tracking error of the HFPDC is within ±0.022 rad, whereas the steady-state tracking error of the FPDC reaches ±0.038 rad. Furthermore, the result obtained by the HFPDC is less noisy than that of the FPDC. The second experiment is to test the position tracking performance for a unit step. Experimental results are shown in Fig. 9. It can be seen that the HFPDC has fast response than the FPDC. The better transient performance is mainly resulted from the high quality differential signal provided by the TD with simple calculation, since the other gains are the same. Therefore, we can conclude that the developed HFPDC is better than the conventional FPDC. The better performance is mainly owed to the high quality differential signal selected by the proposed TD.
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Fig. 8. Comparative experiment of HFPDC and FPDC to a unit sinusoidal position profile.
Fig. 9. Comparative experiment of HFPDC and FPDC to a unit step position profile.
4. Conclusions A simple hybrid fuzzy PD controller is proposed in this paper, by combining two nonlinear tracking differentiators to a conventional fuzzy PD controller. A nonlinear tracking differentiator is first developed to obtain a high quality differential signal with measurement noise, which builds a solid basis for a better performance of the proposed hybrid fuzzy PD controller. The important improvement of the developed controller is that it has high robustness against noise and is easy to implement. The effectiveness and efficiency of the proposed simple hybrid fuzzy PD controller is verified by simulation and experimental comparison studies on motion control of a permanent magnet synchronous AC motor system.
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Acknowledgements The authors are grateful to the anonymous reviewers for the valuable comments and suggestions which have lead to significant improvements of this paper. The authors would like to acknowledge the support from the Natural Science Foundation of Shaanxi Province under Grant 2000C22, and Robotics Laboratory in Shenyang Institute of Automation, Chinese Academy of Sciences under Grant RL200104.
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