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Using the modified PSO method to identify a Scott-Russell mechanism actuated by a piezoelectric element
ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 1652–1661
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Using the modified PSO method to identify a Scott-Russell mechanism actuated by a piezoelectric element Chih-Cheng Kao a, Rong-Fong Fung b,c, a
Department of Electrical Engineering, Kao-Yuan University, No. 1821, Jhongshan Road, Lujhu Township, Kaohsiung County 821, Taiwan, ROC Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung County 824, Taiwan, ROC c Graduate Institute of Electro-Optical Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung County 824, Taiwan, ROC b
a r t i c l e i n f o
abstract
Article history: Received 7 May 2008 Received in revised form 21 November 2008 Accepted 14 December 2008 Available online 24 December 2008
This paper mainly proposes an efficient modified particle swarm optimization (MPSO) method, to identify a Scott-Russell (SR) magnification mechanism driven by a piezoelectric actuator (PA), in which Bouc-Wen model is employed to describe the hysteresis phenomenon. In system identification, we adopt the MPSO to find parameters of the SR mechanism and the PA. This new algorithm is added with ‘‘distance’’ term in the traditional PSO’s fitness function to avoid converging to a local optimum. It is found that the MPSO method can obtain optimal high-quality solution, high calculation efficiency, and its feasibility and effectiveness are demonstrated for the modified IEEE 30-bus system. Finally, the comparisons of numerical simulations and experimental results prove that the MPSO identification method for the SR magnification mechanism is feasible. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Scott-Russell mechanism Piezoelectric actuator System identification Modified particle swarm optimization IEEE 30-bus system
1. Introduction In recent years, the piezoelectric actuator (PA) is more and more widely used in modern precision and driving systems because it has the advantages: large-force generation, fast-frequency response, high-positioning precision form 10 pm to 100 mm, high electrical–mechanical coupling efficiency, miniature and small thermal expansion during actuation. The Scott-Russell (SR) amplifying mechanism [1] is designed for a cutting tool to amplify the PA displacement and carries out high precision for the motion control system. Although the PA has many advantages, its main drawback is the hysteresis phenomena, which arises from materials polarization and molecule friction [2]. Therefore, the maximum error caused by the hysteresis phenomena may be as much as 10–15% of the full motion ranges [3,4]. To analyze the hysteresis phenomenon of the PA, there are many researches [5–8] concerning about modeling and identifying the hysteresis behavior for the purpose of solving nonlinear problems. Therefore, mathematical modeling of the physical system with hysteresis phenomena is an important tool [9]. In this paper, we consider Bouc-Wen hysteresis model, which is established by the piezoelectric coefficients and differential equations [4,5] to describe the hysteresis phenomenon. Particle swarm optimization (PSO) has been shown to be an efficient, roust and simple optimization algorithm. Most studies of the PSO are empirical with only a few theoretical analyses that concentrate on understanding particle
Corresponding author at: Graduate Institute of Electro-Optical Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung County 824, Taiwan, ROC. Fax: +886 7 6011066. E-mail address: [email protected] (R.-F. Fung).
0888-3270/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.12.003
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trajectories. The PSO is a stochastic population based optimization approach, and was first published by Kennedy and Eberhart in 1995 [10,11]. Since its first publication, a large body of research has been done to study and improve the performance of PSO. From these studies, much effort has been invested to obtain a better understanding of its convergence properties. These studies concentrated mostly on a better understanding of the basic PSO control parameters, namely the acceleration coefficients, inertia weight, velocity clamping and swarm size [12–15]. The main difference of the modified particle swarm optimization (MPSO) from the PSO is its fitness function considering the distance to avoid converging to a local optimum. From these empirical studies, it can be concluded that the MPSO is sensitive to control parameter choices, specifically the inertia weight, acceleration coefficients and velocity clamping. However, wrong initialization of these parameters may lead to divergent or cyclic behavior. Zhang et al. [16] have presented the genetic algorithm (GA) to identify the hysteresis parameters of Bouc-Wen model and demonstrated the identification method to be feasible [17–21]. In this paper, we utilize Bouc-Wen hysteresis model to describe the hysteresis phenomena of a PA [22,23], which actuates a SR amplifying mechanism. It is found that Bouc-Wen hysteresis model presents the best matching with the experimental results of the SR mechanism driven by the PA. This study successfully demonstrates that the dynamic formulation can give a wonderful interpretation of a SR mechanism in comparison with the experimental results. Furthermore, a new identified method using the MPSO is proposed, and it is confirmed that the method can perfectly search the parameters of the SR mechanism driven by the PA through the numerical simulations and experimental results. The organization of this paper is as follows. In Section 2, Bouc-Wen hysteresis model is introduced. The SR mechanism and experimental setup are explained in Section 3. The parameter identifications of the PSO and MPSO are described in Section 4. In Section 5, comparisons between numerical simulations and experimental results are shown. Finally, some conclusions are drawn.
2. Bouc-Wen hysteresis models Bouc-Wen model [5] is a typical hysteretic nonlinear model, and was originally proposed by Bouc in 1967 and later generalized by Wen in 1976. This model is described by a first-order nonlinear differential equation as follows: ðn1Þ _ _ n h_ ¼ aV_ bjVjhjhj gVjhj
(1)
where h represents the hysteretic state variable and V is the input voltage. The parameter a controls the hysteretic amplitude, b and g control the shapes of hysteresis loop and n controls the smoothness of transition from elastic to plastic responses. Up to now, this model has been successfully used to model piezoelectric elements, magnetorheological dampers, wood joints and base isolation devices for buildings [24]. The identified model has not only been used to forecast the behavior of hysteretic elements but also design controllers for nonlinear systems. However, the hysteresis behavior of this model is limited by the input type. The schematic diagram of the SR amplifying mechanism actuated by a PA is shown in Fig. 1, and its dynamic equation is established as follows [4]: mx€ þ cx_ þ kx ¼ kðde V hÞ
(2)
Fig. 1. The schematic diagram of the SR amplifying mechanism.
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where m is the equivalent mass, c is the equivalent damping, k is the equivalent stiffness, x is the output displacement at point D of the SR amplifying mechanism, and de is the piezoelectric coefficient. 3. The Scott-Russell mechanism and experimental setup 3.1. The Scott-Russell mechanism The structural application of flexure hinges in the SR amplifying mechanism is especially important to practically play an important role in the positioning stage and precision cutting [1]. The SR amplifying mechanism is a simple structure with main features of displacement amplification and straight-line output motion [25]. The rigid-body diagram of the SR amplifying mechanism is shown in Fig. 2, and it is composed of two links AC and BD, where a pin at point A connects these two links, point C is a fixed point, and lengths of AB, AC and AD are the same. The piezoelectric actuator is excited by an applied voltage and acts at point B against the Y-axis. Then, through the SR amplifying mechanism, the motion is transferred to point D in the X-axis. The PA will elongate or shorten due to the applied voltage being positive or negative value and the link AC rotates and generates linear motion at point D. The displacement analysis of the SR amplifying mechanism is also shown in Fig. 2, where y is the initial angle, Dy is the angular displacement due to an input incremental displacement DyB at point B, and DxD is the output incremental displacement at point D. 3.2. Experimental setup The configured experimental setup is shown in Fig. 3(a), while the photograph of the whole experimental setup is shown in Fig. 3(b). The PA is produced by Tokin Company of Japan and has the dimensions of 5 5 20 mm3. The input and output displacements of the SR amplifying mechanism can be measured by the capacitor-type gap sensor (Micro Sense: 3401-RA2), which owns the measuring range 725/250 mm and the corresponding precision is 10/100 nm. The waveform of the applied input voltage is generated by the LABVIEW, which is employed to communicate with the digital and analog signals during experimental process. The DA/AD converter (PCI-6052E) with a resolution of 12 bits is used to transform the waveform to the power amplifier (Chichibu Onoda), which has the voltage output range 7200 V. 4. Identifications of the system 4.1. Particle swarm optimization Birds (particles) flocking optimizes a certain objective function in a PSO system. Each agent knows its best value (pbest) so far and its position. This information has analogy to personal experiences of each agent. Moreover, each agent knows the
Fig. 2. The rigid-body diagram of the SR amplifying mechanism. The input Y-axis motion of point B is transferred to the output X-axis motion at point D.
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Gap Sensor
SR Mechanism
A/D Converter
Piezoelectric Actuator Power Amplifier
D/A Converter
Gap Sensor
Scott-Russel Mechanism
Piezoelectric Actuator
Fig. 3. Experimental setup (a) configured experimental setup and (b) photograph.
best value so far in the group (gbest) among pbests. The PSO concept consists of changing the velocity of each particle towards its pbest and gbest locations. In the PSO, each particle moves to a new position according to new velocity and the previous positions of the particle. This is compared with the best position generated by previous particles in the fitness function, and the best one is kept; so each particle accelerates in the direction of not only the local best solution but also the global best position. If a particle discovers a new probable solution, other particles will move closer to it to explore the region in the process more completely. In general, there are three attributes, current position xj, current velocity vj and past best position pbestj, for particles in the search space to present their features. Each particle in the swarm is iteratively updated according to the aforementioned attributes. For example [5–9], the jth particle is represented as xj ¼ (xj,1, xj,2, y, xj,g) in the g-dimensional space. The best previous position of the jth particle is recorded and represented as pbestj ¼ (pbestj,1, pbestj,2, y, pbestj,g). The index of best particle among all particles in the group is represented by the gbestg. The rate of the position change (velocity) for particle j is represented as vj ¼ (vj,1, vj,2, y, vj,g). The modified velocity and position of each particle can be calculated using the current velocity and distance from pbestj,g to gbestj,g as shown in the following formulas [9]:
¼ w vðtÞ þ c1 RandðÞ ðpbestðtÞ xðtÞ Þ þ c2 Rand ðÞ ðgbestðtÞ xðtÞ Þ, vðtþ1Þ j;g j;g j;g j;g j;g j;g xðtþ1Þ ¼ xðtÞ þ vðtþ1Þ . j;g j;g j;g j ¼ 1; 2; . . . ; n;
g ¼ 1; 2; . . . ; m
(3)
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where n is the number of particles in a group; m is the number of members in a particle; t is the pointer of iterations (generations); vj,g(t) is the velocity of the particle j at iteration t, Vgminpvj,g(t)pVgmax; w is the inertia weighting factor; c1, c2 are the acceleration constants; Rand(), Rand*() are the random numbers between 0 and 1; xj,g(t) is the current position of the particle j at iteration t; pbestj is the pbest of particle j; gbestg is the gbest of the group g. In the above procedures, the parameter Vgmax determines the resolution or fitness, with which regions are searched between the present position and the target position. If Vgmax is too high, particles might fly past good solutions. If Vgmax is too low, particles may not explore sufficiently beyond local solutions. The constants c1 and c2 represent the weighting of the stochastic acceleration terms that pull each particle toward pbest and gbest positions. Low values allow particles to roam far from the target regions before being tugged back. On the other hand, high values result in abrupt movement towards or past target regions. Suitable selection of inertia weighting factor w provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution. As originally developed, w often decreases linearly from about 0.9 to 0.4 during a run. In general, the inertia weighting factor w is set according to the following equation [10–14]: w ¼ wmax
wmax wmin iter iter max
(4)
where itermax is the maximum number of iterations (generations), and iter is the current number of iterations. 4.2. Modified particle swarm optimization The main point of the MPSO, differs from the PSO, is to consider the ‘‘distance’’ in its fitness function to avoid converging to a local optimum. Assign a rank (i.e., the number place 1, 2, 3, y, etc.) REk to the calculated error of each new individual, vk, k ¼ 1, y, PS, PS is the population size. A combined population with 2 PS individuals is formed. Unlike previously developed statistic methods, the concept of ‘‘distance’’ is added to the fitness function to prevent from being trapped in a local minimum. The fitness score of the kth individual is modified by [15] F k ¼ REk þ r RDk ;
k ¼ 1; ; 2 PS.
(5)
where r is an adaptive decay scale, rmax is set as 0.7 and rmin is set as 0.005 in this paper. RDk is the rank of Dk for the kth individual, and Dk is the distance from the individual to the current best solution, and is given by Dk ¼ jjvk vbest jj
(6)
where vk is the vector of the kth individual in the combined population, and vbest is the current best solution vector. An adaptive scheme is defined as
rD ¼ R ðrmax rmin Þ=g max (
rðg þ 1Þ ¼
rðgÞ rD ; rðgÞ;
F min ðgÞ ¼ F min ðg 1Þ F min ðgÞoF min ðg 1Þ
(7)
(8)
and
rðg þ 1Þ ¼ rmin ;
if rðgÞ rD ormin
(9)
where rD is the step size; Fmin is the minimum value of fitness functions; R is the regulating scale and is set as 1.25 in this paper, and gmax is the maximum allowable number of iterations. Individuals will be ranked in ascending according to their fitness scores by a sorting algorithm. The PS individuals are transcribed along with their fitness for the next generation. If the new population does not include the current best solution, the best solution must be replaced with the last individual in the new population. In addition, a gradually decreased decay scale can satisfy a successive statistic searching process by first using the diversification (bigger r) to explore more regions, and then the intensification (smaller r) to exploit the neighborhood of an elite solution. The current best solution (point A) for a minimum fitness problem as shown in Fig. 4 may not reach the global optimum, and there are three electable solutions existing. Generally, solutions with slightly better fitness (point C or B) prevailed, so the solution trapped into the valley prematurely. The more attractive solution (point G) is relatively far away from point A, but it nears the global optimal. To prevent prematurity, point G with slightly worse fitness than C, it needs a higher rank to be selected. That is, a higher RDk is awarded to a longer Dk. Stopping Criteria: Stopping criteria is given in the following order: (1) Maximum allowable number of iterations reached. (2) Number of iterations reached without improving the current best solution. Fig. 5 shows the flowchart of the proposed algorithm.
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farther distance
Fitness value
shorter distance
A
G
B C
global optimum local optimum Search Space Fig. 4. The concept of distances.
Fig. 5. The flowchart of the MPSO.
4.3. Comparison with PSO To show that the proposed MPSO method is superior to the PSO method [10–14] for solving the optimal generation scheduling of units, we compare performances between these two methods in the modified IEEE 30-bus system with 10 units by using the same evaluation function and individual definition. Total 100 trials are run using these two methods to observe their convergence tendencies, to examine the variation in their evaluation values and total generation costs, and to compare their qualities of solutions. The results have shown that the proposed MPSO method is superior to the PSO method. Fig. 6 shows the convergence properties of these two methods. Table 1 shows the solution qualities that are obtained from these two methods. The IEEE 30-bus system is used to demonstrate the data preparation and the use of the power flow programs, and is made available to the electric utility industry as a standard test case for evaluating various analytical methods and other optimal algorithms for the solution of power system problems. A unit commitment (UC) of the electrical power system is a minimum cost problem of determining the optimal set of generators that are in service during a scheduling horizon, the total power generation must satisfy the load consumption and system loss with considering the transmission line power
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8236.5 PSO MPSO
Fitness (Cost (S))
8236 8235.5 8235 8234.5 8234 8233.5
0
10
20
30
40 50 60 Iteration
70
80
90 100
Fig. 6. The comparisons in the convergence tendency of the IEEE 30-bus system between the PSO and MPSO methods.
Table 1 Comparison of solution quality in the modified IEEE 30-bus system (100 iterations). Method
MPSO
PSO
Worst Average Best The number of trial reaching optimum Average number of generations to converge Average execution time (s)
8234.2046 8234.1267 8233.6607 35 21 0.1971
8250.2047 8241.4033 8234.2507 21 47 0.2473
flow limitations. A modified IEEE 30-bus system was used to show the superior of MPSO. The test system contains ten thermal units, 30 buses and 41 transmission lines. The fuel cost function of all generation units are taking into account the valve-point effect [26], which makes UC become a multiple minima problem. The upper and lower limits of the PV-bus and load-bus voltages are 0.90 and 1.1 pu, respectively. From the results of contingency selection, the critical fault is proven when lines 12–15 faulted, then lines 7–8 are overloaded. The detailed characteristics of the ten thermal units are given in Ref. [27]. The unit commitment problem consists of 10 units and a 24 h scheduling horizon. In this simulation, the dimensions of an individual and a population are 10 24 and 10 24 100, respectively.
4.4. Parameter identifications How to define the fitness function is the key point of the MPSO, since the fitness function is a figure of merit, and could be computed by using any domain knowledge. In this paper, we adopt the fitness function as follows [16]: FðparametersÞ ¼
n X
E2i
(10)
i¼1
Ei ¼ xðiÞ xnðiÞ
(11) (i)
where n is the total number of samples and Ei is the calculated error of the ith sampling time, x* is a solution by using the fourth-order Runge-Kutta method to solve the dynamic Eqs. (1) and (2) of the SR amplifying mechanism with the parameters identified from these two methods, and x(i) is the displacement measured experimentally at the ith sampling time. For the parameter identification of Bouc-Wen hysteresis model, we consider the SR amplifying mechanism subject to an applied voltage V ¼ 24+24 sin 16pt. Here, the input voltage is a positive waveform so that the displacement x is positive. Here, we take D ¼ 1 107, the population size is 10,000, the inertia weighting factor w ¼ 0.65; the acceleration constants are c1 ¼ 0.7 and c2 ¼ 2.05. The physical model of the SR amplifying mechanism driven by the PA is described by Eqs. (1) and (2). In the parameter identification, the structure material can be assumed as elasticity [4,8] and set n ¼ 1 in Eq. (1) to combine with the dynamic Eq. (2) of the SR amplifying mechanism. We utilize the MPSO and PSO methods to identify the seven parameters
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Table 2 The identified values of Bouc-Wen hysteresis model. Parameters
m (kg)
c (Ns/m)
k (N/m)
de (m/V)
a
b
g
n¼1 MPSO PSO
0.1025 0.1
57.665 60
4.1805E3 4.497E3
2.1672E-7 2.17481E-7
0.2161 0.2112
0.00504 0.005
0.0239 0.0299
na1 MPSO PSO
0.1008 0.1
55.905 58.074
4.0935E3 4.499E3
2.1603E-7 2.168E-7
0.1815 0.198
0.005 0.005
0.0237 0.0251
x 10-9 4
Average Error (m)
3 2 1 0 -1 -2
PSO MPSO
-3 0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time (Sec) 5
x 10-9
4 Average Error (m)
3 2 1 0 -1 -2 MPSO PSO
-3 -4 0.15
0.2
0.25
0.3 0.35 Time (Sec)
0.4
0.45
0.5
Fig. 7. The comparisons in displacement errors between the PSO and MPSO methods. (a) n ¼ 1 and (b) na1.
m, c, k, de, a, b and g simultaneously [8], and the fitness function is described as Eq. (10). Alternatively, we also could consider the plasticity and set na1 in Eq. (1) to identify the eight parameters m, c, k, de, a, b, g and n simultaneously.
5. Comparisons between numerical simulations and experimental results The identified value for Bouc-Wen hysteresis model by both the PSO and MPSO methods are shown in Table 2, where the results clearly show that Bouc-Wen hysteresis model presents the best matching with the experimental results and
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7.4
x 10-5 PSO MPSO
7.2 7 Error
6.8 6.6 6.4 6.2 6 5.8
7.2
0
10
20
30
40 50 60 Generation
70
80
90 100
x 10-5 PSO MPSO
7 6.8 Error
6.6 6.4 6.2 6 5.8 5.6
0
10
20
30
40 50 60 Generation
70
80
90 100
Fig. 8. Comparison of convergence characteristics in the PSO and MPSO methods for Bouc-Wen hysteresis models. (a) n ¼ 1 and (b) na1.
converges to one optimum value. Besides, we can find that by using Bouc-Wen hysteresis model, the identified dynamic responses are almost the same as the experimental results for both n ¼ 1 and na1. Fig. 7 shows the displacement errors of the SR amplifying mechanism, where Bouc-Wen hysteresis model is identified by both the PSO and MPSO methods. It is seen that both the displacement errors by using Bouc-Wen hysteresis model (n ¼ 1) and (na1) are almost identical. Furthermore, their displacement errors of Bouc-Wen hysteresis model are all about 70.2 mm. Therefore, it is demonstrated that Bouc-Wen hysteresis model has the best matching with the experimental results for the hysteretic phenomenon. Fig. 8 shows the convergence characteristics in the PSO and MPSO methods for Bouc-Wen hysteresis models. It is seen from Fig. 8 that the proposed MPSO method is superior to the PSO method.
6. Conclusions The main objective of this study is to utilize Bouc-Wen hysteresis models to describe the hysteretic phenomenon of the PA, which actuates a Scott-Russell mechanism. According to the comparisons between identified results and displacement errors among the hysteresis models, it is found that Bouc-Wen model has the best matching with the experimental results for hysteretic phenomenon. It is concluded that the implementations of MPSO are different from the PSO in five aspects. Firstly, it’s fitness function considers the distance to avoid converging to a local optimum. Secondly, for the MPSO, vectors with good enough fitness scores would be used as candidates to create new solutions. Thirdly, it has the advantage of the MPSO to conquer various constraints without using the fitness function with penalties, and can perform better. Fourthly, the solution is coded with a decimal representation and saves computer memory. Finally, the gradually decaying parameters can satisfy a successive statistic searching process by first using the diversification (bigger parameters) to
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Using the modified PSO method to identify a Scott-Russell mechanism actuated by a piezoelectric element Chih-Cheng Kao a, Rong-Fong Fung b,c, a
Department of Electrical Engineering, Kao-Yuan University, No. 1821, Jhongshan Road, Lujhu Township, Kaohsiung County 821, Taiwan, ROC Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung County 824, Taiwan, ROC c Graduate Institute of Electro-Optical Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung County 824, Taiwan, ROC b
a r t i c l e i n f o
abstract
Article history: Received 7 May 2008 Received in revised form 21 November 2008 Accepted 14 December 2008 Available online 24 December 2008
This paper mainly proposes an efficient modified particle swarm optimization (MPSO) method, to identify a Scott-Russell (SR) magnification mechanism driven by a piezoelectric actuator (PA), in which Bouc-Wen model is employed to describe the hysteresis phenomenon. In system identification, we adopt the MPSO to find parameters of the SR mechanism and the PA. This new algorithm is added with ‘‘distance’’ term in the traditional PSO’s fitness function to avoid converging to a local optimum. It is found that the MPSO method can obtain optimal high-quality solution, high calculation efficiency, and its feasibility and effectiveness are demonstrated for the modified IEEE 30-bus system. Finally, the comparisons of numerical simulations and experimental results prove that the MPSO identification method for the SR magnification mechanism is feasible. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Scott-Russell mechanism Piezoelectric actuator System identification Modified particle swarm optimization IEEE 30-bus system
1. Introduction In recent years, the piezoelectric actuator (PA) is more and more widely used in modern precision and driving systems because it has the advantages: large-force generation, fast-frequency response, high-positioning precision form 10 pm to 100 mm, high electrical–mechanical coupling efficiency, miniature and small thermal expansion during actuation. The Scott-Russell (SR) amplifying mechanism [1] is designed for a cutting tool to amplify the PA displacement and carries out high precision for the motion control system. Although the PA has many advantages, its main drawback is the hysteresis phenomena, which arises from materials polarization and molecule friction [2]. Therefore, the maximum error caused by the hysteresis phenomena may be as much as 10–15% of the full motion ranges [3,4]. To analyze the hysteresis phenomenon of the PA, there are many researches [5–8] concerning about modeling and identifying the hysteresis behavior for the purpose of solving nonlinear problems. Therefore, mathematical modeling of the physical system with hysteresis phenomena is an important tool [9]. In this paper, we consider Bouc-Wen hysteresis model, which is established by the piezoelectric coefficients and differential equations [4,5] to describe the hysteresis phenomenon. Particle swarm optimization (PSO) has been shown to be an efficient, roust and simple optimization algorithm. Most studies of the PSO are empirical with only a few theoretical analyses that concentrate on understanding particle
Corresponding author at: Graduate Institute of Electro-Optical Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung County 824, Taiwan, ROC. Fax: +886 7 6011066. E-mail address: [email protected] (R.-F. Fung).
0888-3270/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.12.003
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trajectories. The PSO is a stochastic population based optimization approach, and was first published by Kennedy and Eberhart in 1995 [10,11]. Since its first publication, a large body of research has been done to study and improve the performance of PSO. From these studies, much effort has been invested to obtain a better understanding of its convergence properties. These studies concentrated mostly on a better understanding of the basic PSO control parameters, namely the acceleration coefficients, inertia weight, velocity clamping and swarm size [12–15]. The main difference of the modified particle swarm optimization (MPSO) from the PSO is its fitness function considering the distance to avoid converging to a local optimum. From these empirical studies, it can be concluded that the MPSO is sensitive to control parameter choices, specifically the inertia weight, acceleration coefficients and velocity clamping. However, wrong initialization of these parameters may lead to divergent or cyclic behavior. Zhang et al. [16] have presented the genetic algorithm (GA) to identify the hysteresis parameters of Bouc-Wen model and demonstrated the identification method to be feasible [17–21]. In this paper, we utilize Bouc-Wen hysteresis model to describe the hysteresis phenomena of a PA [22,23], which actuates a SR amplifying mechanism. It is found that Bouc-Wen hysteresis model presents the best matching with the experimental results of the SR mechanism driven by the PA. This study successfully demonstrates that the dynamic formulation can give a wonderful interpretation of a SR mechanism in comparison with the experimental results. Furthermore, a new identified method using the MPSO is proposed, and it is confirmed that the method can perfectly search the parameters of the SR mechanism driven by the PA through the numerical simulations and experimental results. The organization of this paper is as follows. In Section 2, Bouc-Wen hysteresis model is introduced. The SR mechanism and experimental setup are explained in Section 3. The parameter identifications of the PSO and MPSO are described in Section 4. In Section 5, comparisons between numerical simulations and experimental results are shown. Finally, some conclusions are drawn.
2. Bouc-Wen hysteresis models Bouc-Wen model [5] is a typical hysteretic nonlinear model, and was originally proposed by Bouc in 1967 and later generalized by Wen in 1976. This model is described by a first-order nonlinear differential equation as follows: ðn1Þ _ _ n h_ ¼ aV_ bjVjhjhj gVjhj
(1)
where h represents the hysteretic state variable and V is the input voltage. The parameter a controls the hysteretic amplitude, b and g control the shapes of hysteresis loop and n controls the smoothness of transition from elastic to plastic responses. Up to now, this model has been successfully used to model piezoelectric elements, magnetorheological dampers, wood joints and base isolation devices for buildings [24]. The identified model has not only been used to forecast the behavior of hysteretic elements but also design controllers for nonlinear systems. However, the hysteresis behavior of this model is limited by the input type. The schematic diagram of the SR amplifying mechanism actuated by a PA is shown in Fig. 1, and its dynamic equation is established as follows [4]: mx€ þ cx_ þ kx ¼ kðde V hÞ
(2)
Fig. 1. The schematic diagram of the SR amplifying mechanism.
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where m is the equivalent mass, c is the equivalent damping, k is the equivalent stiffness, x is the output displacement at point D of the SR amplifying mechanism, and de is the piezoelectric coefficient. 3. The Scott-Russell mechanism and experimental setup 3.1. The Scott-Russell mechanism The structural application of flexure hinges in the SR amplifying mechanism is especially important to practically play an important role in the positioning stage and precision cutting [1]. The SR amplifying mechanism is a simple structure with main features of displacement amplification and straight-line output motion [25]. The rigid-body diagram of the SR amplifying mechanism is shown in Fig. 2, and it is composed of two links AC and BD, where a pin at point A connects these two links, point C is a fixed point, and lengths of AB, AC and AD are the same. The piezoelectric actuator is excited by an applied voltage and acts at point B against the Y-axis. Then, through the SR amplifying mechanism, the motion is transferred to point D in the X-axis. The PA will elongate or shorten due to the applied voltage being positive or negative value and the link AC rotates and generates linear motion at point D. The displacement analysis of the SR amplifying mechanism is also shown in Fig. 2, where y is the initial angle, Dy is the angular displacement due to an input incremental displacement DyB at point B, and DxD is the output incremental displacement at point D. 3.2. Experimental setup The configured experimental setup is shown in Fig. 3(a), while the photograph of the whole experimental setup is shown in Fig. 3(b). The PA is produced by Tokin Company of Japan and has the dimensions of 5 5 20 mm3. The input and output displacements of the SR amplifying mechanism can be measured by the capacitor-type gap sensor (Micro Sense: 3401-RA2), which owns the measuring range 725/250 mm and the corresponding precision is 10/100 nm. The waveform of the applied input voltage is generated by the LABVIEW, which is employed to communicate with the digital and analog signals during experimental process. The DA/AD converter (PCI-6052E) with a resolution of 12 bits is used to transform the waveform to the power amplifier (Chichibu Onoda), which has the voltage output range 7200 V. 4. Identifications of the system 4.1. Particle swarm optimization Birds (particles) flocking optimizes a certain objective function in a PSO system. Each agent knows its best value (pbest) so far and its position. This information has analogy to personal experiences of each agent. Moreover, each agent knows the
Fig. 2. The rigid-body diagram of the SR amplifying mechanism. The input Y-axis motion of point B is transferred to the output X-axis motion at point D.
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Gap Sensor
SR Mechanism
A/D Converter
Piezoelectric Actuator Power Amplifier
D/A Converter
Gap Sensor
Scott-Russel Mechanism
Piezoelectric Actuator
Fig. 3. Experimental setup (a) configured experimental setup and (b) photograph.
best value so far in the group (gbest) among pbests. The PSO concept consists of changing the velocity of each particle towards its pbest and gbest locations. In the PSO, each particle moves to a new position according to new velocity and the previous positions of the particle. This is compared with the best position generated by previous particles in the fitness function, and the best one is kept; so each particle accelerates in the direction of not only the local best solution but also the global best position. If a particle discovers a new probable solution, other particles will move closer to it to explore the region in the process more completely. In general, there are three attributes, current position xj, current velocity vj and past best position pbestj, for particles in the search space to present their features. Each particle in the swarm is iteratively updated according to the aforementioned attributes. For example [5–9], the jth particle is represented as xj ¼ (xj,1, xj,2, y, xj,g) in the g-dimensional space. The best previous position of the jth particle is recorded and represented as pbestj ¼ (pbestj,1, pbestj,2, y, pbestj,g). The index of best particle among all particles in the group is represented by the gbestg. The rate of the position change (velocity) for particle j is represented as vj ¼ (vj,1, vj,2, y, vj,g). The modified velocity and position of each particle can be calculated using the current velocity and distance from pbestj,g to gbestj,g as shown in the following formulas [9]:
¼ w vðtÞ þ c1 RandðÞ ðpbestðtÞ xðtÞ Þ þ c2 Rand ðÞ ðgbestðtÞ xðtÞ Þ, vðtþ1Þ j;g j;g j;g j;g j;g j;g xðtþ1Þ ¼ xðtÞ þ vðtþ1Þ . j;g j;g j;g j ¼ 1; 2; . . . ; n;
g ¼ 1; 2; . . . ; m
(3)
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where n is the number of particles in a group; m is the number of members in a particle; t is the pointer of iterations (generations); vj,g(t) is the velocity of the particle j at iteration t, Vgminpvj,g(t)pVgmax; w is the inertia weighting factor; c1, c2 are the acceleration constants; Rand(), Rand*() are the random numbers between 0 and 1; xj,g(t) is the current position of the particle j at iteration t; pbestj is the pbest of particle j; gbestg is the gbest of the group g. In the above procedures, the parameter Vgmax determines the resolution or fitness, with which regions are searched between the present position and the target position. If Vgmax is too high, particles might fly past good solutions. If Vgmax is too low, particles may not explore sufficiently beyond local solutions. The constants c1 and c2 represent the weighting of the stochastic acceleration terms that pull each particle toward pbest and gbest positions. Low values allow particles to roam far from the target regions before being tugged back. On the other hand, high values result in abrupt movement towards or past target regions. Suitable selection of inertia weighting factor w provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution. As originally developed, w often decreases linearly from about 0.9 to 0.4 during a run. In general, the inertia weighting factor w is set according to the following equation [10–14]: w ¼ wmax
wmax wmin iter iter max
(4)
where itermax is the maximum number of iterations (generations), and iter is the current number of iterations. 4.2. Modified particle swarm optimization The main point of the MPSO, differs from the PSO, is to consider the ‘‘distance’’ in its fitness function to avoid converging to a local optimum. Assign a rank (i.e., the number place 1, 2, 3, y, etc.) REk to the calculated error of each new individual, vk, k ¼ 1, y, PS, PS is the population size. A combined population with 2 PS individuals is formed. Unlike previously developed statistic methods, the concept of ‘‘distance’’ is added to the fitness function to prevent from being trapped in a local minimum. The fitness score of the kth individual is modified by [15] F k ¼ REk þ r RDk ;
k ¼ 1; ; 2 PS.
(5)
where r is an adaptive decay scale, rmax is set as 0.7 and rmin is set as 0.005 in this paper. RDk is the rank of Dk for the kth individual, and Dk is the distance from the individual to the current best solution, and is given by Dk ¼ jjvk vbest jj
(6)
where vk is the vector of the kth individual in the combined population, and vbest is the current best solution vector. An adaptive scheme is defined as
rD ¼ R ðrmax rmin Þ=g max (
rðg þ 1Þ ¼
rðgÞ rD ; rðgÞ;
F min ðgÞ ¼ F min ðg 1Þ F min ðgÞoF min ðg 1Þ
(7)
(8)
and
rðg þ 1Þ ¼ rmin ;
if rðgÞ rD ormin
(9)
where rD is the step size; Fmin is the minimum value of fitness functions; R is the regulating scale and is set as 1.25 in this paper, and gmax is the maximum allowable number of iterations. Individuals will be ranked in ascending according to their fitness scores by a sorting algorithm. The PS individuals are transcribed along with their fitness for the next generation. If the new population does not include the current best solution, the best solution must be replaced with the last individual in the new population. In addition, a gradually decreased decay scale can satisfy a successive statistic searching process by first using the diversification (bigger r) to explore more regions, and then the intensification (smaller r) to exploit the neighborhood of an elite solution. The current best solution (point A) for a minimum fitness problem as shown in Fig. 4 may not reach the global optimum, and there are three electable solutions existing. Generally, solutions with slightly better fitness (point C or B) prevailed, so the solution trapped into the valley prematurely. The more attractive solution (point G) is relatively far away from point A, but it nears the global optimal. To prevent prematurity, point G with slightly worse fitness than C, it needs a higher rank to be selected. That is, a higher RDk is awarded to a longer Dk. Stopping Criteria: Stopping criteria is given in the following order: (1) Maximum allowable number of iterations reached. (2) Number of iterations reached without improving the current best solution. Fig. 5 shows the flowchart of the proposed algorithm.
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farther distance
Fitness value
shorter distance
A
G
B C
global optimum local optimum Search Space Fig. 4. The concept of distances.
Fig. 5. The flowchart of the MPSO.
4.3. Comparison with PSO To show that the proposed MPSO method is superior to the PSO method [10–14] for solving the optimal generation scheduling of units, we compare performances between these two methods in the modified IEEE 30-bus system with 10 units by using the same evaluation function and individual definition. Total 100 trials are run using these two methods to observe their convergence tendencies, to examine the variation in their evaluation values and total generation costs, and to compare their qualities of solutions. The results have shown that the proposed MPSO method is superior to the PSO method. Fig. 6 shows the convergence properties of these two methods. Table 1 shows the solution qualities that are obtained from these two methods. The IEEE 30-bus system is used to demonstrate the data preparation and the use of the power flow programs, and is made available to the electric utility industry as a standard test case for evaluating various analytical methods and other optimal algorithms for the solution of power system problems. A unit commitment (UC) of the electrical power system is a minimum cost problem of determining the optimal set of generators that are in service during a scheduling horizon, the total power generation must satisfy the load consumption and system loss with considering the transmission line power
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8236.5 PSO MPSO
Fitness (Cost (S))
8236 8235.5 8235 8234.5 8234 8233.5
0
10
20
30
40 50 60 Iteration
70
80
90 100
Fig. 6. The comparisons in the convergence tendency of the IEEE 30-bus system between the PSO and MPSO methods.
Table 1 Comparison of solution quality in the modified IEEE 30-bus system (100 iterations). Method
MPSO
PSO
Worst Average Best The number of trial reaching optimum Average number of generations to converge Average execution time (s)
8234.2046 8234.1267 8233.6607 35 21 0.1971
8250.2047 8241.4033 8234.2507 21 47 0.2473
flow limitations. A modified IEEE 30-bus system was used to show the superior of MPSO. The test system contains ten thermal units, 30 buses and 41 transmission lines. The fuel cost function of all generation units are taking into account the valve-point effect [26], which makes UC become a multiple minima problem. The upper and lower limits of the PV-bus and load-bus voltages are 0.90 and 1.1 pu, respectively. From the results of contingency selection, the critical fault is proven when lines 12–15 faulted, then lines 7–8 are overloaded. The detailed characteristics of the ten thermal units are given in Ref. [27]. The unit commitment problem consists of 10 units and a 24 h scheduling horizon. In this simulation, the dimensions of an individual and a population are 10 24 and 10 24 100, respectively.
4.4. Parameter identifications How to define the fitness function is the key point of the MPSO, since the fitness function is a figure of merit, and could be computed by using any domain knowledge. In this paper, we adopt the fitness function as follows [16]: FðparametersÞ ¼
n X
E2i
(10)
i¼1
Ei ¼ xðiÞ xnðiÞ
(11) (i)
where n is the total number of samples and Ei is the calculated error of the ith sampling time, x* is a solution by using the fourth-order Runge-Kutta method to solve the dynamic Eqs. (1) and (2) of the SR amplifying mechanism with the parameters identified from these two methods, and x(i) is the displacement measured experimentally at the ith sampling time. For the parameter identification of Bouc-Wen hysteresis model, we consider the SR amplifying mechanism subject to an applied voltage V ¼ 24+24 sin 16pt. Here, the input voltage is a positive waveform so that the displacement x is positive. Here, we take D ¼ 1 107, the population size is 10,000, the inertia weighting factor w ¼ 0.65; the acceleration constants are c1 ¼ 0.7 and c2 ¼ 2.05. The physical model of the SR amplifying mechanism driven by the PA is described by Eqs. (1) and (2). In the parameter identification, the structure material can be assumed as elasticity [4,8] and set n ¼ 1 in Eq. (1) to combine with the dynamic Eq. (2) of the SR amplifying mechanism. We utilize the MPSO and PSO methods to identify the seven parameters
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Table 2 The identified values of Bouc-Wen hysteresis model. Parameters
m (kg)
c (Ns/m)
k (N/m)
de (m/V)
a
b
g
n¼1 MPSO PSO
0.1025 0.1
57.665 60
4.1805E3 4.497E3
2.1672E-7 2.17481E-7
0.2161 0.2112
0.00504 0.005
0.0239 0.0299
na1 MPSO PSO
0.1008 0.1
55.905 58.074
4.0935E3 4.499E3
2.1603E-7 2.168E-7
0.1815 0.198
0.005 0.005
0.0237 0.0251
x 10-9 4
Average Error (m)
3 2 1 0 -1 -2
PSO MPSO
-3 0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time (Sec) 5
x 10-9
4 Average Error (m)
3 2 1 0 -1 -2 MPSO PSO
-3 -4 0.15
0.2
0.25
0.3 0.35 Time (Sec)
0.4
0.45
0.5
Fig. 7. The comparisons in displacement errors between the PSO and MPSO methods. (a) n ¼ 1 and (b) na1.
m, c, k, de, a, b and g simultaneously [8], and the fitness function is described as Eq. (10). Alternatively, we also could consider the plasticity and set na1 in Eq. (1) to identify the eight parameters m, c, k, de, a, b, g and n simultaneously.
5. Comparisons between numerical simulations and experimental results The identified value for Bouc-Wen hysteresis model by both the PSO and MPSO methods are shown in Table 2, where the results clearly show that Bouc-Wen hysteresis model presents the best matching with the experimental results and
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7.4
x 10-5 PSO MPSO
7.2 7 Error
6.8 6.6 6.4 6.2 6 5.8
7.2
0
10
20
30
40 50 60 Generation
70
80
90 100
x 10-5 PSO MPSO
7 6.8 Error
6.6 6.4 6.2 6 5.8 5.6
0
10
20
30
40 50 60 Generation
70
80
90 100
Fig. 8. Comparison of convergence characteristics in the PSO and MPSO methods for Bouc-Wen hysteresis models. (a) n ¼ 1 and (b) na1.
converges to one optimum value. Besides, we can find that by using Bouc-Wen hysteresis model, the identified dynamic responses are almost the same as the experimental results for both n ¼ 1 and na1. Fig. 7 shows the displacement errors of the SR amplifying mechanism, where Bouc-Wen hysteresis model is identified by both the PSO and MPSO methods. It is seen that both the displacement errors by using Bouc-Wen hysteresis model (n ¼ 1) and (na1) are almost identical. Furthermore, their displacement errors of Bouc-Wen hysteresis model are all about 70.2 mm. Therefore, it is demonstrated that Bouc-Wen hysteresis model has the best matching with the experimental results for the hysteretic phenomenon. Fig. 8 shows the convergence characteristics in the PSO and MPSO methods for Bouc-Wen hysteresis models. It is seen from Fig. 8 that the proposed MPSO method is superior to the PSO method.
6. Conclusions The main objective of this study is to utilize Bouc-Wen hysteresis models to describe the hysteretic phenomenon of the PA, which actuates a Scott-Russell mechanism. According to the comparisons between identified results and displacement errors among the hysteresis models, it is found that Bouc-Wen model has the best matching with the experimental results for hysteretic phenomenon. It is concluded that the implementations of MPSO are different from the PSO in five aspects. Firstly, it’s fitness function considers the distance to avoid converging to a local optimum. Secondly, for the MPSO, vectors with good enough fitness scores would be used as candidates to create new solutions. Thirdly, it has the advantage of the MPSO to conquer various constraints without using the fitness function with penalties, and can perform better. Fourthly, the solution is coded with a decimal representation and saves computer memory. Finally, the gradually decaying parameters can satisfy a successive statistic searching process by first using the diversification (bigger parameters) to
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