the design of a genetic algorithm-based fuzzy pulse

Journal of the Chinese Institute of Engineers, Vol. 30, No. 1, pp. 91-102 (2007)

91

THE DESIGN OF A GENETIC ALGORITHM-BASED FUZZY PULSE PUMP CONTROLLER FOR A FREQUENCY-LOCKED SERVO SYSTEM

Liang-Rui Chen*, Guan-Chyun Hsieh, and Hahn- Ming Lee

ABSTRACT In this paper, a Genetic Algorithm-based Fuzzy Pulse Pump Controller (GAFPPC) is proposed to realize a Frequency-Locked Servo system (FLS). In order to get a better convergence and a higher diversity in the genetic algorithm, a Mixed Elitist and Fuzzy Clustering (MEFC) selection strategy is proposed. A prototype of the GFCFLS is tested to assess the system performance. In comparison with the FPC and the SVDPC, the acquisition time of the GA-FPPC is improved over than 18%. In particular, there is no overshoot in GA-FPPC for any servo distances. In addition, the GA-FPPC has the best tracking performance of these three controllers. These demonstrate that GA-FPPC can actually provide an appropriate pump voltage to fast locking response without overshoot, meeting the theoretical prediction. Key Words: genetic algorithm-based fuzzy pulse pump controller, mixed elitist and fuzzy clustering selection strategy, frequency-locked servo system.

I. INTRODUCTION The control strategy in a conventional servo control system is described by three key parameters, including Proportional, Integral, and Derivative constants (PID) (Nagrath and Gopal, 1982). It is difficult to produce an adaptive controller to acquire a satisfactory performance, such as nearly zero steady-state error, fast and stable response, and low overshoot in the servo process (Geiger, 1981; Nagrath and Gopal, 1982). To produce better performance in the servo control system, the phase locked loop technique was proposed for motor servo systems in the 1970’s (Moore, 1973). Presented *Corresponding author. (Tel: 886-4-7232105 ext. 7716; Fax: 886-4-7211094; Email: [email protected]) L. R. Chen is with the Department of Aeronautical Engineering, National Formosa University Huwei, Yunlin, Taiwan 632, R.O.C., and is currently at the Department of Electrical Engineering, National Changhua University of Education, Changhua, Taiwan 500, R.O.C. G. C. Hsieh is with the Department of Electrical Engineering, TungNan Institute of Technology, ShenKeng, Taipei, Taiwan 222, R.O.C H. M. Lee is with the Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan 106, R.O.C

Phase-Locked Servo systems (PLS) can provide a stable speed with high regulation of up to 0.002%. In 1985, a voltage pump controller was proposed to achieve nearly zero steady-state error in the PLS (Margaris and Petridis, 1985). Later, an Adaptive Digital Pump Controller (ADPC) and Frequency Pump Controller (FPC) were proposed to build the PLS and FrequencyLocked Servo system (FLS), respectively (Hsieh et al.,1987; Hsieh, 1989; Li and Hsieh, 1992). In the ADPC-based PLS and FPC-based FLS, better servo performance, such as nearly zero steady-state error, stable response, and low overshoot, were achieved. The application of phase/frequency locked technique in motor control creates many advantages, for example auto-tracking, auto-lock and high accuracy (Emura et al., 2000; Hsieh et al., 1987; Hsieh and Hung, 1996; Li and Hsieh, 1992; Moore, 1973). However, in these approaches there still exist some problems, such as slow locking process and overshoot. In order to speed up the locking process of the FLS, a Slope-Varied Digital Pump Controller (SVDPC) was proposed for the FLS (Hsieh et al., 1992). The SVDPC can effectively reduce the acquisition time compared to the conventional pump controller. In 1995, a Variable Slope Pulse Pump Controller (VSPPC) was also proposed

92

Journal of the Chinese Institute of Engineers, Vol. 30, No. 1 (2007)

for the stepping position servo system (Lai et al., 1995). The VSPPC can also reduce the acquisition time. Lately, fuzzy logic has been applied to get a proper slope to be used in SVDPC and VSPPC (Lai et al., 1996). This fuzzy controlled PLS, SVDPC and VSPPC can only improve the motion behavior in the out-of-lock range, but the motion profile in the in-lock range does not change. As a result, the acquisition time for the short servo distance (i.e., within the in-lock range) is the same as the conventional pump controller. In addition, many properties, such as overshoot and stability are not improved, since they depend on the motion behavior in the in-lock range (Feng et al., 2000, Roland, 1993; Tal, 1977). It is well-known that the most effective method to improve the performance of a PLS/ FLS is to modify the motion behavior in the in-lock range. A novel PLL built by digital techniques was presented to achieve a very fast locking process (Kobayashi et al., 1996). Unfortunately, the overshoot of this very fast PLL is too large to fit the servo requirement. In this paper, we use the fuzzy control approach to build a Fuzzy Pulse Pump Controller (FPPC) for programming the motion profile in the in-locked range. In the proposed FPPC-based FLS, the motor position is periodically sampled. To effectively improve the performance of the FPPC, a Genetic Algorithm (GA) is used to optimize the FPPC. This new kind of pump controller is called the Genetic Algorithm-based Fuzzy Pulse Pump Controller (GA-FPPC). To optimize the FPPC with GA creates two challenging problems. One is how to simultaneously reduce the convergence time and get the near-optimal solutions, the other is to find an appropriate objective function to correctly indicate the servo performance. Thus, we propose a Fuzzy Clustering-based Genetic Algorithm (FC-GA) for converging at a better value and reducing the convergence time. Additionally, a Weighted Integral Absolute Error (WIAE) (Chen, 2005) is used as an objective function for estimating the performance index of the GA-FPPC-based FrequencyLocked Servo system (GFC-FLS). A prototype of the GFC-FLS is tested to assess the system performance. Experiments and simulations show the proposed GAFPPC actually provides an appropriate pump voltage to fast locking response without overshoot. This meets the theoretical prediction. II. SYSTEM DESCRIPTION The functional scheme of the GFC-FLS is shown in Fig. 1. This system is composed of two Position/ Frequency Converters (PFCs), a Discrete Frequency/ Pulse Comparator (DFPC), a GA-FPPC, a DC motor with a motor driver, an optical encoder, an up/down counter, and a lead screw. The PFC i and PFC o are, respectively, used to

pi

fi PFCi

pe DFPC Delay

−

+

∆pe

GAFPPC

Vp

Motor & driver

Lead D screw

fo PFCo

po

Up/down counter

Optical encorder

Fig. 1 The block diagram of the GFC-FLS

Fuzzy clustering-based genetic algorithm, FC-GA

Rule library ∆pe pe

Fuzzifier

Inference engine

Nvp Defuzzifier

Digital pump controller

Vp

Fuzzy-logic controller, FLC Fuzzy pulse pump controller, FPPC

Fig. 2 The block diagram of the GA-FPPC

convert the input pulse number pi (i.e., input position) and feedback pulse number po (i.e., feedback position) into the discrete frequencies f i and fo. The DFPC then compares the discrete frequencies f i and f o, and converts the period error 1/f i - 1/f o into a pulse error p e (Hsieh et al., 1998). After that, the GA-FPPC infers a suitable pump voltage V p according to the instant pulse error p e and its time change ∆ p e to drive the motor. The lead screw is mounted on this motor to produce linear displacement D from the motor shaft rotation θ . An optical encoder and an up/down counter are also mounted on the shaft of the DC motor to sense and measure the feedback position represented as pulse number po . The configuration of the GA-FPPC is shown in Fig. 2. The GA-FPPC consists of an FC-GA, an FLC, and a digital-pump controller. The FC-GA is a new kind of GA. Details of this will be given in Section III. Firstly, the FLC infers the input fuzzy variables with the rule library to produce a digital-coded value NVp which indicates the pump voltage V p. This digital-coded value N Vp is then sent to the digital pump controller to generate an analog pump voltage V p as motor drives. Since the control rules and membership functions in the FLC are tuned by the FC-GA, the near-optimal pump voltage Vp for FLS can be obtained. In Fig. 2, GA-FPPC has two inputs, pe and ∆pe, defined as pe = DFPC(f i, f o) and

(1)

∆p e = pe(n) – p e(n – 1),

(2)

93

L. R. Chen et al.: The Design of a GA-FPPC for a FLS

where DFPC is a comparator and converter, which compares the input frequency f i with the output one fo and converts the difference to a position error pulse number pe (Hsieh et al, 1998). In this paper, p e has 2m + 1 linguistic terms p pe named MF –m , ... , MF 0 e , ... , MF mpe, ∆ p e has 2n + 1 ∆ p e ... linguistic terms named MF –n , , MF 0∆p e, ... , MF n∆pe , and V p has 2j + 1 linguistic terms named MF –jVp, ... , MF 0Vp, ... , MF jVp. In which, m, n and j are positive pe ... , MF mpe– 1 , integers. The linguistic terms MF –m + 1, V ∆p e ∆ p MF –n + 1 , ... , MF –n e– 1 , and MF –j p+ 1 , ... , MF jVp– 1 are defined as the form of Gaussian function written as:

fGaussian(x, σ g, m g) = e

–

,

(3)

1 , 1 + e – σ s ⋅ (x – m s)

IF p e = MF0pe AND ∆ pe = MF0∆pe THEN Vp = MF 0Vp pe When |p e| is large enough (i.e., p e = MF –m or p e pe = MF m ), the FLS requires large pump voltage (i.e., Vp V p = MF –j or V p = MF Vj p) to reduce the acquisition time. Thus, we can set these fuzzy rules as follows: Vp pe ∆p e IF pe = MF –m AND ∆ p e = MF –n THEN V p = MF –j

IF pe = MF–m AND ∆ pe = MF –n + 1 THEN Vp = MF –j .. . ∆pe

Vp

IF pe = MF –m AND ∆ p e = MF 0 THEN V p = MF –j .. . pe

∆p e

p

p

MF1 e

p

p

p

e e MFm-2 MFm-1 MFm e

0.5 0

Pe p

∆p

1

p

p∆

e MF-n e MF-n+1 MF-n+2e

∆p

∆pe

MF-1 e MF0

∆pe

MF1

∆p

∆p

∆pe

MFn-2 e MFn-1e MFn

0.5 0

Pe

1

Vp

Vp

MF-1 MF-j+1 MF-j+2

V

V

MF-1p MF0 p

V

MF1 p

V

V

MFj-2p MFj-1p

Vp

MFj

0

Fig. 3

Vp

The membership functions used for the GA-FPPC, (Vertical axis is the degree of the membership functions)

∆p e THEN V p = MF jVp IF p e = MF mpe AND ∆p e = MF –n ∆p e Vp IF p e = MF mpe AND ∆p e = MF –n + 1 THEN V p = MF j .. .

IF p e = MF mpe AND ∆p e = MF0∆pe THEN Vp = MF jVp .. . IF p e = MF mpe AND ∆p e = MF n∆p–e 1 THEN Vp = MF jVp IF p e = MF mpe AND ∆p e = MFn∆pe THEN Vp = MF jVp As stated above, in total 2(2n + 1) + 1 rules are pe set. Remarkably, if pe is not for MF –m or MF mpe, suitable rules are not easy to determine. Since the number of the complete rule set is (2m + 1) × (2n + 1), there are still Nrule rules needed to be set. The Nrule can be obtained as: Nrule = (2m + 1) × (2n + 1) – [2(2n + 1) + 1] = (2m – 1) × (2n + 1) –1

(5)

On the other hand, there are N MF membership functions used. The NMF is: NMF = (2m + 1) + (2n + 1) + (2j + 1)

(6)

Vp

Vp pe IF pe = MF–m AND ∆ pe = MF n∆p–e 1 THEN V p = MF –j Vp pe IF pe = MF –m AND ∆ p e = MF n∆pe THEN V p = MF –j

and

p

MF-1e MF0 e

(4)

where σ s and m s are respectively the decreasing rate with direction and the center position of the linguistic terms. Fig. 3 shows the membership functions used for the GA-FPPC. All membership functions can be determined by properly setting σg, σs, mg and ms values. A fuzzy rule set of FPPC built by intuitive experiment for the FLS with the criteria to make pe → 0 and ∆ p e → 0 is depicted in (Chen et al., 2004; Guillemin, 1996; Lai et al., 1996). When the system settles at the target position, where pe = 0 (i.e., p e = MF 0∆p e ) and ∆ p e = 0 (i.e., ∆ p e = MF 0∆p e), zero pump voltage (i.e., V p = MF 0Vp) is required to keep the FLS in the locked state. Therefore, we can determine the corresponding fuzzy rule as:

pe

p

0.5

where σg and mg are the variance and the center position of the linguistic terms, respectively. On the other pe ∆p e , MFmpe, MF–n , MFn∆pe, MF–jVp, and MFVj p are hand, MF–m defined as the form of Sigmoidal function written as: fSigmoidal(x, σ s, m s) =

p

e e MF-meMF-m+1 MF-m+2

Vp

(x – m g) 2 2σ g2

p

1

Since each membership function is characterized by two parameters σ g and m g (or σ s and m s), 2 × NMF variables are required. Therefore, a total of 2 × NMF + Nrule variables are needed in GA-FPPC. We assume that the system acts symmetrically and the number of variables can then be reduced to NMF + N rule . 2

94

Journal of the Chinese Institute of Engineers, Vol. 30, No. 1 (2007)

Fig. 4 The structure of FC-GA for FLS

III. FUZZY CLUSTERING-BASED GENETIC ALGORITHM The structure of FC-GA is shown in Fig. 4. First, a feasible set of solutions of the GA-FPPC’s variables is represented by a set of parameters. These parameters are regarded as genes, and a set of genes is regarded as a chromosome. Many kinds of chromosome representation have

been presented (Gen and Cheng, 1997; Man et al., 1996). The real-value chromosome representation is adopted in the FC-GA since it can deal well with real parameter problems. Since the number of genes in each chromosome is equal to the number of variables in the controller, each chromosome has N MF + N rule 2 genes in the FC-GA. For convenience in description, the chromosome x of the FC-GA is shown in Fig. 5.

L. R. Chen et al.: The Design of a GA-FPPC for a FLS

95

Fig. 5 The chromosome x of the FC-GA for the FLS

x p e In which, σ MF m is the variance of the membership function MF mp e, m xMF mpe is the center position of the membership function MF mpe , and so on. In addition, the r x(m – 1) × n is the consequence of the inference rule whose antecedent is “pe is MF mpe – 1 and ∆p e is MF n∆pe”, and so on. These encoded chromosomes are used as parents to evolve the next generation (i.e., offspring) by crossover and mutation processes. The crossover process mixes the genes of the parents to generate offspring for retaining some of the parental characteristics. The convex crossover (Gen and Cheng, 1997) is the most commonly-used and is adopted in FC-GA. For simplicity in description, the ith gene in the new generated chromosome is denoted as gi , and the ith genes in the parent chromosomes x and y are respectively denoted as g xi and giy. How to select the parent chromosomes x and y will be given in Section IV. The convex crossover can be represented as:

g i = wg xi + (1 – w) giy, w = (0,1),

(7)

where w is the weight and (0,1) are limits. The mutation process is to produce random change in genes to avoid converging to a local optimal region. Uniform mutation is adopted in FC-GA since it fits real-value chromosome representation and can be easily implemented (Gen and Cheng, 1997). Uniform mutation replaces a gene g xi with the sum of the gxi and a random real value within a specified range [-R,R], which can be represented as: g i = g xi + R · Random [–1,1],

(8)

where the Random[-1,1] is a procedure which generates a random real value in the range [-1,1], and R is a real number. Finally, the new chromosomes are selected using a selection strategy. The widely used selection strategies of GAs are the elitist selection strategy and the roulette wheel selection strategy (Gen and Cheng, 1997; Man et al., 1996). Since an ideal GA must have a high diversity (Gen and Cheng, 1997), we propose

a Mixed Elitist and Fuzzy Clustering (MEFC) selection strategy to maintain high diversity. This new selection strategy will be discussed and compared with other selection strategies in Section IV. In order to correctly indicate the servo performance, a WIAE is used as an object function of GA (Chen, 2005), and defined as: ∞

J WIAE =

≅

0

WSS(t) ⋅ e(t) dt

∞

Σ

n=0

WSS(nT) ⋅ e(nT) ⋅ T ,

(9)

where T is the sampling period, and W SS () is a StairShape Weighting Function (SSWF) defined as: WSS(t) =

1 , t ≤ ta , wo , t > t a

(10)

where w o is the overshoot weight, and t a is the arriving time that maps the first zero-error point (i.e., moving distance just equal to the input reference distance) in a step response. IV. SELECTION STRATEGIES OF GAS The distribution of chromosome selection by elitist selection, roulette wheel selection, and MEFC selection strategies are shown in Fig. 6. In Fig. 6, “
” is the selected chromosome, and gray level indicates the fitness value. The elitist selection strategy always selects the elite chromosomes to generate the next population. In the roulette wheel selection strategy, each chromosome is given a fitness value, which is a real number with respect to the selected probability. In particular, the diversity of the roulette wheel selection strategy is higher than that of the elitist selection strategy. This means that the elitist selection strategy searches in a smaller area and quickly converges to the local optimal solution. Fuzzy clustering (Lin and Lee, 1996) is a process which can assign each data element to different fuzzy sets (i.e., cluster) according to the variance of

96

Journal of the Chinese Institute of Engineers, Vol. 30, No. 1 (2007)

Solution space

Table 1 The initial fuzzy inference rules

Solution space

pe PB PS ZO NS NB

∆pe

(a) Solution space

PB

PS

ZO

NS

NB

PB PB PB PB PB

rPS × PB rPS × PS r PS × ZO r PS × NS r PS × NB

rZO × PB rZO × PS ZO r ZO × NS r ZO × NB

rNS × PB r NS × PS r NS × ZO rNS × NS r NS × NB

NB NB NB NB NB

(b) GA-FPPC

Solution space pi

PFCi

DFPC Te Te(nT)

1/fi

T

FC-GA pe

Tb

T

Vp

z.o.h.

D

1 s

FLC

To lead screw

1/fo PFCo

p0

Kc

(a) pi

cluster (c) Fig. 6

each data element. In this paper, the cluster centers are selected to work as parents to generate offspring. Since cluster centers have the typical characteristics for each cluster and all clusters cover all characteristics of the parents, the offspring almost always maintain all the characteristics of the parents. Thus, fuzzy clustering can be used as a selection strategy to maintain high diversity at each generation. The fuzzy Cmean clustering algorithm (FCM) (Lin and Lee, 1996) is adopted as the fuzzy clustering process in this paper. In FC-GA, a fuzzy C-mean clustering selection and an elitist selection are combined. The Mixed Elitist and Fuzzy Clustering (MEFC) selection strategy not only selects the cluster centers to maintain high diversity, but also selects the elite to speed up the convergence. V. A DESIGN EXAMPLE Figure 7(a) shows the complete mathematical model of the GFC-FLS, while Fig. 7(b) shows its simplified one. The relevant parameters are defined as follows: T Tb τm Km

p0

(d)

Distribution of chromosome selection. The gray level indicates fitness value, and “
” is the selected chromosome by (a) elitist selection strategy, (b) roulette wheel selection strategy, (c) fuzzy clustering selection strategy, and (d) MEFC selection strategy.

the sampling period, the time base period, the mechanical time constant of the motor, the DC gain including the motor and driver (rad/sec/v),

pe

z.o.h. T

Vp GA-FPPC

1 s

Ks

D To lead screw

Kc

(b) Fig. 7

Mathematical model of the GFC-FLS, (a) complete model, and (b) simplified model.

Ks Kc

the conversion ratio of the angular displacement to the length of lead screw (cm/rad), the conversion ratio of the angular displacement to the number of pulses (1/rad).

In this design example, the mechanical time constant τm of the DC motor is 115 msec, the DC gain Km of the motor with driver is 6.7 cm/sec/V, the lead screw has a pitch of 1 cm, and the optical encoder generates 1000 pulses per round. Therefore, the minimum position resolution equals ±10 µm. The in-lock range is specified within ±1 cm and the full speed in out-oflock range is set at 6.7 cm/sec. According to the specified in-lock range, pe, max =1000 (i.e., 1 cm) is specified. In the GA-FPPC, the number of linguistic terms for each fuzzy variable (i.e., pe, ∆pe and Vp) is 5 (i.e., m = n = j = 2), as proposed in the motor control systems (Chen et al., 2004; Guillemin, 1996; Lai et al, 1996). These linguistic terms are named PB, PS, ZO, NS, and NB. In which, PB means positive big, PS indicates positive small, ZO is zero, NS means negative small, and NB is negative big. The inference rules are given in Table 1. In which, r PS × PB is the consequence of the inference rule whose antecedent is “p e is PS and ∆ pe is PB”, and so on. According to Eqs. (5) and (6), we know that there are N MF + N rule = 22 genes in a 2 chromosome. The parameters used for FC-GA are listed in Table. 2. They are selected empirically, except the number of genes.

97

L. R. Chen et al.: The Design of a GA-FPPC for a FLS

Table 2 Parameters of FC-GA The number of genes Crossover rate Mutation rate Number of elitist Number of cluster Max. number of iterations Min. amount of improvement overshoot weight wo of SSWF

Table 3 The fuzzy inference rule table after tuning by FC-GA

22 0.8 0.2 4 3 100 0.000001 30

pe

∆pe

PB PS ZO NS NB

PB

PS

ZO

NS

NB

PB PB PB PB PB

PB PB PS PS ZO

PS PS ZO NS NS

ZO NS NS NB NB

NB NB NB NB NB

(a) 1 NB

PS

ZO

NS

PB

0.5 0 -1000

-800

-600

-400

-200

200

0

400

600

800

1000

Pulse error, Pe 1

NS

NB

PS

ZO

PB

0.5 0 -1000

-800

-600

-400

-200

0

200

400

600

800

1000

Change of pulse error, ∆Pe 1 NB

NS

ZO

PS

PB

0.5 0 -1

-0.8

-0.6

-0.4

0.2 -0.2 0 Pump voltage, Vp (b)

0.4

0.6

0.8

1

Fig. 8 (a) The chromosome and (b) the membership functions of GA-FPPC after tuned by FC-GA.

The simplified mathematical model of GFCFLS is realized by MATLAB with SIMULINK, and the FC-GA as described in Section III is also coded by MATLAB. After that, the near-optimal chromosome, membership functions, and inference rules can be obtained by computer simulation and are respectively shown in Figs. 8(a) and 8(b) and Table 3. In order to assess the performance of the proposed GFC-FLS, the behaviors after and before tuning are compared. Remarkably, the before tuned fuzzy control rules for the GA-FPPC are same as the FPPC (Chen, 2004), and the initial membership functions are normal distribution as shown in Fig. 3. Comparing before and after tuned membership functions, the

membership function ZO in the input fuzzy variable pe, the membership functions PS and NS in the input fuzzy variable ∆ p e, and the membership function ZO in the output fuzzy variable Vp are strongly changed. Furthermore, comparing before and after tuned fuzzy control rule tables, both fuzzy control rule tables are almost the same besides r NS × NS, r NS × PS, r PS × NS and r NS × PS . In order to clearly see the effects produced by the above-mentioned changes in GA-FPPC, the before and after tuned 3-D mesh surface descriptions of output pump voltage Vp are respectively shown in Figs. 9(a) and 9(b). It is clear that the after tuned output pump voltage |Vp| is larger than before tuned when the pulse error |pe| is large, and both are almost the same when

98

1

1

0.5

0.5

0

Vp

Vp

Journal of the Chinese Institute of Engineers, Vol. 30, No. 1 (2007)

-0.5

-0.5

-1 1000 ∆pe

0

1000 0

0 -1000 -1000

pe

(a)

-1 1000 ∆pe

1000 0

0 -1000 -1000

pe

(b)

Fig. 9 3-D mesh surface description of the output of GA-FPPC (a) before and (b) after tuned.

Fig. 10 (a) The block diagram and (b) the circuit diagram for realizing the GFC-FLS.

the pulse error |p e| is close to zero. Note that, a typical Fuzzy Pulse Pump Controller (FPPC) with optimal parameters obtained by the

proposed FC-GA is used instead of the proposed GAFPPC in real-time control as shown in Figs. 10(a) and 10(b). Figs. 10(a) and 10(b) are the realization block

99

L. R. Chen et al.: The Design of a GA-FPPC for a FLS

3500

3500

3000

3000

2500

2500

Distance (10–3 cm)

Distance

GA-FPPC

2000 1500 1000

2000 SVDPC 1500 FPC 1000 500

500 0

0

0

0.5

1 Time

1.5

0

2

Fig. 11 Computer simulations of 3 cm distance servos before (.....) and after (––––) tuning

0.5

1

1.5 Time(s) (a)

2

2.5

3

12000 GA-FPPC

VI. EXPERIMENTS The results of the GFC-FLS from before and after tuning for 3cm distance are shown in Fig. 11. From Fig. 11, we can see that the proposed FC-GA can effectively obtain near-optimal parameters to reduce the overshoot. The conventional FPC (Hsieh, 1989) and SVDPC (Hsieh et al., 1992) are also included for comparison. Fig. 12 shows the position responses for the three controllers mentioned above. The SVDPC effectively reduces the acquisition time for the middle and long distance servos. However, SVDPC cannot reduce the acquisition time for the short distance servos. On the other hand, the GAFPPC not only reduces the acquisition time in the middle and long distance servos, but also reduces the acquisition time in the short distance servos. In addition, there is no overshoot in the GFC-FLS for any servo distances. Remarkably, there is little overshoot in the FPC- and SVDPC-based FLS. In comparison with the FPC, the acquisition time of the GFCFLS for the short, middle and long distance servos is improved by 57%, 62% and 67%, respectively. For SVDPC, the acquisition time of the GFC-FLS for the short, middle and long distance servos is improved by 57%, 41% and 18%, respectively. Figure 13 shows the position tracking responses for the GA-FPPC, the VSDPC, and the FPC. The GAFPPC and the SVDPC effectively track the sine wave input reference. However, the tracking performance of the GA-FPPC is better than that of the SVDPC. Figures. 14(a), 14(b) and 14(c) show the experiment results of GFC-FLS for 3cm, 10cm and 30cm distances.

8000 6000 FPC 4000 SVDPC 2000 0 0

3.5

1

2

× 104

3 Time(s) (b)

4

5

6

GA-FPPC 3 Distance (10–3 cm)

diagram and circuit diagram of the GFC-FLS, respectively.

Distance (10–3 cm)

1000

2.5 SVDPC 2 1.5 FPC 1 0.5 0

0

5

10 Time(s) (c)

15

20

Fig. 12 Simulation of GA-FPPC, SVDPC and FPC for short-(3 cm), middle-(10 cm), and long-(30 cm) distance servos

From Figs. 14(a), 14(b) and 14(c), we can see that the computer simulation and the realized system results are very close. This indicates that the GA-FPPC can actually provide near-optimum motion profiles as theoretically predicted. Since an ideal GA always must have high

100

Journal of the Chinese Institute of Engineers, Vol. 30, No. 1 (2007)

15000

3500

GA-FPPC Reference FPC SVDPC

10000

3000 2500 Diversity

Distance

5000 0 -5000

1500 100

-10000 -15000

2000

500 0 0

5

10

15 Time

20

25

30

0

0.2

0.4

0.6

0.8

1

Time (a)

Fig. 13 Simulation of GA-FPPC, SVDPC and FPC for sine wave tracking response.

12000 10000 8000 Distance

diversity (Gen and Cheng, 1997), the diversity can be used as an index to measure the performance of selection strategies. For convenience in describing the degree of diversity, a diversity_index is defined as:

600 400

N

diversity_index =

Σ

i=1

std i

N

,

200

(11)

0

3.5

Σ

l=1

1

1.5

2

2.5

3

Time (b)

× 104

M

[g il – ( 1 Σ g il)] , Ml=1

(12)

where M is the number of selected chromosomes for generating the next population, and g il is the ith gene of the selected chromosome l. The diversity_index and fitness values of the MEFC selection strategy, the elitist selection strategy, and the roulette wheel selection strategy are respectively shown in Figs. 15 and 16. From these figures, we can see that: 1). The diversity of the elitist selection strategy is the lowest in these three selection strategies. It means that its search area is the smallest and it often converges at local minimal value. The largest fitness value from the elitist selection strategy is reasonable. 2). The diversity of the roulette wheel selection strategy is higher than that of the elitist selection strategy. Therefore, the roulette wheel selection strategy can converge at a better value than the elitist selection strategy. However, the convergence time of the roulette wheel selection strategy is longer than that of the elitist selection strategy. 3). The MEFC selection strategy has the highest

3 2.5 Distance

M

std i =

0.5

0

where N is the number of genes in each chromosome, and stdi is the standard deviation at the same position in each chromosome. std i is defined as:

2 1.5 1 0.5 0

0

2

4

6

8

Time (c) Fig. 14 Experimental results of (a) short-(3 cm), (b) middle-(10 cm), and (c) long-(30 cm) distance servos; computer simulation (––––) and realized system results (ooo) after tuning.

diversity. This means that this selection strategy can search a large area and converge at near global optimum value. It is interesting that the MEFC selection strategy not only converges at near optimum

101

L. R. Chen et al.: The Design of a GA-FPPC for a FLS

800

0.3

700

Mixed elitist and fuzzy clustering selection strategy Roulette wheel selection strategy 0.1

500 400 300

Elitist selection strategy

0

Fitness

Diversity index

600 0.2

200 100

0

50

100

150

Generation Fig. 15 The diversities of the selection strategies: MEFC selection, elitist selection and roulette wheel selection

value, but also converges quickly. The convergence speed of the MEFC selection strategy is slower than that of the elitist selection strategy. However, the convergence speed of the mixed elitist and fuzzy clustering selection strategy is often faster than that of the roulette wheel selection strategy. VII. CONCLUSIONS In this paper, we propose a Genetic Algorithmbased Fuzzy Pulse Pump Controller (GA-FPPC) for use in the Frequency-Locked Servo system (FLS). A set of fuzzy inference rules and three sets of membership functions are determined by Fuzzy Clustering-based Genetic Algorithm (FC-GA). This new algorithm can effectively obtain a set of near-optimum values. Therefore, the GA-FPPC can get a proper motion profile and can overcome the traditional FLS’s inherent drawbacks (e.g., slow locking process and overshoot) to get a fast response without overshoot. A prototype GFC-FLS is realized, and the results of a realized system and computer simulation show that the GA-FPPC can provide near-optimum motion profiles for the GFC-FLS to achieve a fast locking response without overshoot. This meets the theoretical prediction. Simulations of GA-FPPC compared with Frequency Pump Controller (FPC) and Slope-Varied Digital Pump Controller (SVDPC) are also conducted. In comparison with the FPC-based FLS, the acquisition time of the GFC-FLS for the short, middle and long distance servos is improved by 57%, 62% and 67%, respectively. For SVDPCbased FLS, the acquisition time of the GFC-FLS for the short, middle and long distance servos is improved by 57%, 41% and 18%, respectively. In particular, there is no overshoot in the GFC-PLS for any servo distances.

0

50

100

150

Generation Fig. 16 The fitness values of the selection strategies: MEFC selection (––––), elitist selection ( ..... ), and roulette wheel selection (- - - -)

REFERENCES Chen, L. R., Hsieh, G. C., and Lee, H. M., 2004, “A Design of Fast-Locking without Overshoot Frequency-Locked Servo Control System by Fuzzy Control Technology,” Journal of the Chinese Institute of Engineers, Vol. 27, No. 1, pp. 147-151. Chen, L. R., 2005, “Improvement of Transient Response using GA with a Weighed Integral Absolute Error Objective Function,” Transactions of the Aeronautical and Astronautical Society of the Republic of China, Vol. 37, No. 3, pp. 139-246. Emura, T., Wang, L., Yamanaka, M., and Nakamura, H., 2000, “A High-Precision Positioning Servo Controller Based on Phase/Frequency Detecting Technique of Two-Phase-Type PLL,” IEEE Transactions on Industrial Electronics, Vol. 47 No. 6, pp. 1298-1306. Feng, X., Chan, C. C., Jie, W., Shuying, G., and Shouzhi, Q., 2000, “Modeling and Bifurcation of Phase-Locked Loop Speed Control of Induction Motor Drive,” Proceedings of the 26 th Annual Conference of the IEEE (IECON), Nagoya, Japan, Nagoya, pp. 2327-2333. Geiger, D. F., 1981, Phaselock Loops for DC Motor Speed Control, Wiley, USA. Gen, M., and Cheng, R., 1997, Genetic Algorithms & Engineering Design, Wiley, USA. Guillemin, P., 1996, “Fuzzy logic applied to motor control,” IEEE Transactions on Industrial Applications, Vol. 32, No. 1, pp. 51-56. Hsieh, G. C., Wu, P. P., Lee, C. H., and Liu, C. H., 1987, “An Adaptive Digital Pump Controller for Phase-Locked Servo Systems,” IEEE Transactions on Industrial Electronics., Vol. IE-34, pp. 379-386. Hsieh, G. C., 1989, “A Study on Position Servo Control System by Frequency-Locked Technique,”

102

Journal of the Chinese Institute of Engineers, Vol. 30, No. 1 (2007)

IEEE Transactions on Industrial Electronics, Vol. 36, No. 3, pp. 365-373. Hsieh, G. C., Lin, Y. S., and Jou, R. N., 1992, “A Microprocessor-Based Phase-Locked Servo System Using the Slope-varied Digital-Pump Technique,” Journal of Chinese Institute of Engineers, Vol. 15, No. 4, pp. 405-414. Hsieh, G. C., and Hung, J. C., 1996, “Phase-Locked Loop Techniques-A Survey,” IEEE Transactions on Industrial Electronics, Vol. 43, No. 6, pp. 609615. Hsieh G. C., Chen L. R., Hsieh K. L., Wu H. C., and Hsu J. H., 1998, “Adaptive Pulse-Pump Controller for X-Y Mode Frequency-Locked Stepping Servomechanism,” Journal of Chinese Institute of Engineers, Vol. 21, No. 1, pp. 47-58. Kobayashi, F., Haratsu, M., Yabumoto, M., and Nakano, M., 1996, “Efficient Digital Techniques for Implementing a Class of Fast Phase-Locked Loops (PLL’s),” IEEE Transactions on Industrial Electronics, Vol. 43, No. 6, pp. 616-620. Lai, M. F., Hsieh, G. C., and Wu, Y. P., 1995, “Variable Slope Pulse Pump Controller for Stepping Position Servo Control System Using FrequencyLocked Technique,” IEEE Transactions on Industrial Electronics, Vol. 42, No. 3, pp. 290-299. Lai, M. F., Nakano, M., and Hsieh, G. C., 1996, “Application of Fuzzy Logic in the Phase-Locked Loop Speed Control of Induction Motor Drive,” IEEE Transactions on Industrial Electronics, Vol.

43, No. 6, pp. 630-639. Li, J. C., and Hsieh G. C., 1992, “A Phase/FrequencyLocked Controller for Stepping Servo Control Systems,” IEEE Transactions on Industrial Electronics, Vol. 39, No. 2, pp. 112-119. Lin, C. T., and Lee, C. S. G., 1996, Neural Fuzzy Systems, Prentice-Hall, USA. Margaris, M., and Petridis, V., 1985, “Voltage Pump Phase-Locked Loop,” IEEE Transactions on Industrial Electronics, Vol. IE-3, pp. 41-49. Man, K. F., Tang, K. S., and Kwong, S., 1996, “Genetic Algorithms: Concepts and Applications,” IEEE Transactions on Industrial Electronics, vol. 43, no. 5, pp. 519-534. Moore, A. W., 1973, “Phase Locked Loops for Motor-Speed Control,” IEEE Spectrum, pp. 61-67. Nagrath, I. J., and Gopal, M., 1982, Control Systems Engineering, Wiley, USA. Roland, E. B., 1993, Phase-Locked Loops Theory, Design, and Applications, 2 nd ed., McGraw-Hill, USA. Tal, J., 1977, “Speed Control by Phase-Locked Servo System-New Possibility and Limitation,” Transactions on Industrial Electronics and Control Instrumentation, Vol. IECI-24, pp. 118-125. Manuscript Received: Jul. 08, 2005 Revision Received: Mar. 26, 2006 and Accepted: Apr. 28, 2006