Expert Systems with Applications 37 (2010) 8871–8878
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Fractional-order PID controller optimization via improved electromagnetism-like algorithm Ching-Hung Lee *, Fu-Kai Chang Department of Electrical Engineering, Yuan Ze University, Chungli, Taoyuan, Taiwan, ROC
a r t i c l e
i n f o
Keywords: PID control Fractional-order PID control Electromagnetism-like algorithm Genetic algorithm
a b s t r a c t Based on the electromagnetism-like algorithm, an evolutionary algorithm, improved EM algorithm with genetic algorithm technique (IEMGA), for optimization of fractional-order PID (FOPID) controller is proposed in this article. IEMGA is a population-based meta-heuristic algorithm originated from the electromagnetism theory. It does not require gradient calculations and can automatically converge at a good solution. For FOPID control optimization, IEMGA simulates the ‘‘attraction” and ‘‘repulsion” of charged particles by considering each controller parameters as an electrical charge. The neighborhood randomly local search of EM algorithm is improved by using GA and the competitive concept. IEMGA has the advantages of EM and GA in reducing the computation complexity of EM. Finally, several illustration examples are presented to show the performance and effectiveness. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Evolutionary computation technique has become gradually popular to obtain global optimal solution in many areas. Several algorithms have been proposed by the observation of real-world systems, such as genetic algorithm, evolutionary algorithm, particle swarm optimization, immune algorithm, differential evolution, and so on (ACPA, 2002; Cao & Cao, 2006; Cao, Liang, & Cao, 2005; Fan, Sun, & Zhang, 2007; IEE, 2002; IFAC, 2001; Lee, 2004a, 2004b; Lee & Teng, 2002, 2003; Nataraj & Tharewal, 2007; Podlubny, 1999; Yamato & Hashimoto, 1991). Recently, a novel meta-heuristic algorithm, electromagnetism-like (EM) mechanism, for global optimization was proposed (Biswas, Das, Abraham, & Dasgupta, 2009; Clerc & Kenney, 2002; Farag, Quintana, & Germano, 1998; Goldberg, 1989; Podlubny, 1999b). EM algorithm is originated from the electromagnetism theory in physics, which simulated the electromagnetism theory by considering each particle to be an electrical charge. Subsequently, the movement of attraction and repulsion is introduced by Coulomb’s law, i.e., the force is inversely propositional to the distance between the particles and directly proportional to the product of their charges. Obviously, it has advantages of multiple search, global optimization, and faster convergence procedure and simultaneously evaluates many points in the search space; researchers are more likely to find a better solution (Biswas et al., 2009; Clerc & Kenney, 2002; Farag et al., 1998; Goldberg, 1989; Podlubny, 1999b). However, the local search procedure of EM is stochastic. Hence, the major drawback of * Corresponding author. Tel.: +886 3 4638800; fax: +886 3 46389355. E-mail address:
[email protected] (C.-H. Lee). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.06.009
EM is high computation complexity. To improve the performance of EM, a modified local search phase and the competition concept are adopted. The proportional-integral-derivative (PID) controller is perhaps the most widely used controller in the world; it is easy to design and implement and has been applied well in most control systems. While control theory has been developed significantly, the PID controllers are used in a wide range of process control, motor drives, magnetic and optic memories, automotive control, flight control, instrumentation, and so on. In industrial applications, more than 90% of all control loops are PID type (Gudise & Venayagamoorthy, 2003; Hong & Yuan, 2002; Juang, 2004; Kim, 2002; Price, Storn, & Lampinen, 2005; Srinivas & Patnaik, 1994; Xu, Wei, & Xu, 2000; Yao, 1999). Feedback control systems form one such area, as witnessed in part by recent special issues on the subject (Gudise & Venayagamoorthy, 2003; Hong & Yuan, 2002; Juang, 2004). The concept of fractional-order processes or controllers has been studied considerably (ACPA, 2002; Birbil & Fang, 2003; Birbil, Fang, & Sheu, 2004; Chang, Chen, & Fan, 2009; Tsou & Kao, 2007; Wu, Yang, & Wei, 2004). This is due to the fact that many real-world systems are characterized by fractional-order differential equations (ACPA, 2002; Birbil & Fang, 2003; Birbil et al., 2004; Chang et al., 2009; Tsou & Kao, 2007; Wu et al., 2004). Hence, Podlubny proposed a generalization of the PID controller, namely the fractional-order PID controller (FOPID or PIkDd), where k and d are integers (Tsou & Kao, 2007; Wu et al., 2004). Fractional calculus is usually used to design the FOPID controller. Various studies have demonstrated that the FOPID controller enhances performance and robustness (Birbil et al., 2004; Chang et al., 2009). However, it is difficult and complicated to design the FOPID controller by analytical meth-
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od because of using fraction calculus. In this article, a hybrid optimization method for FOPID controller is presented by the evolutionary algorithm IEMGA. This article proposes an evolutionary algorithm IEMGA for FOPID controller optimization. IEMGA is a population-based metaheuristic algorithm originated from the electromagnetism theory. For FOPID control optimization, IEMGA simulates the ‘‘attraction” and ‘‘repulsion” of charged particles by considering each controller parameters as an electrical charge. The neighborhood random local search is implemented by GA and the competitive concept, which reduces the computation complexity. The IEMGA has the capability of multiple searches, global optimization, and less computation complexity. Furthermore, IEMGA does not need any gradient information for the optimization process. As a result of these advantages, we use IEMGA to solve optimization of FOPID controller design. The article is organized as follows. Section 2 introduces the fractional-order PID controllers. In Section 3, the EM algorithm is introduced. The hybrid algorithm IEMGA for PID controller optimization is introduced in Section 4. Section 5 shows the simulation and comparison results of optimization of FOPID controller. It demonstrates the performance of the proposed IEMGA. The final section offers the conclusion.
Fig. 1. Block diagram of a simple feedback control system.
ð2Þ k
d
and the fractional-order PID controller-PI D is of the form:
GC ðsÞ ¼ K p þ K i sk þ K d sd ;
ð3Þ
where k and d are the orders of the fractional integral and derivative, respectively. Therefore, the PID controller parameters vector is (Kp, Ki, Kd) and the fractional-order PID controller parameters vector is (Kp, Ki, k, Kd, d). Obviously, the PID controller is a special case of FOPID. In addition, the expansion of fractional order of derivative and integral terms could provide much more flexibility in PID controller design. 3. Electromagnetism-like algorithm This section introduces the EM algorithm for optimization problem. The EM algorithm was developed to simulate the electromagnetism theory of physics by each sample point to be a charge (or particle) Podlubny (1999b). The EM for optimization problems with lower and upper bound is in the form of:
Subject to x 2 S;
Proportional-integral-derivative (PID) controllers are widely used to build automation equipment in industries. They are easy to design, implement, and are applied well in most industrial control systems. Even though control theory has been developed significantly, the PID controllers are used for a wide range of process control, motor drives, magnetic and optic memories, automotive control, flight control, instrumentation, etc. In industrial applications, over 90% of all control loops are PID type (Gudise & Venayagamoorthy, 2003; Hong & Yuan, 2002; Juang, 2004; Price et al., 2005; Yao, 1999). As many real-world systems are characterized by fractional-order differential equations, the concept of fractional-order processes or controllers has been the subject of considerable research (ACPA, 2002; Birbil & Fang, 2003; Birbil et al., 2004; Cao & Cao, 2006; Chang et al., 2009; Tsou & Kao, 2007; Wu et al., 2004). In (Tsou & Kao, 2007; Wu et al., 2004), a generalization of the PID controller is introduced, namely the PIkDd controller, where k and d are the indices. Previous results have demonstrated that the PIkDd controller enhances performance and robustness (Birbil et al., 2004; Chang et al., 2009). A block diagram of a simple feedback control system is shown in Fig. 1. The system comprises a process and a controller. The process has one input and one output, denoted u and y, respectively. The desired value yr is called the set point or the reference one. The purpose of the system is to keep the process output y close to the desired one yr in spite of disturbances. Assume that the system is modeled by an nth-order process with time delay L:
bm sm þ bm1 sm1 þ þ b1 s þ b0 Ls e : sn þ an1 sn1 þ þ a1 s þ a0
GC ðsÞ ¼ K p þ K i s1 þ K d s;
Minimize f ðxÞ;
2. Fractional-order PID controllers
Gp ðsÞ ¼
Here, we assume n > m and the system (1) is stable. For the PID controller, the transfer function is
ð1Þ
ð4Þ
where S ¼ fx 2 Rn jlk 6 xk 6 uk ; lk ; uk 2 R; k ¼ 1; . . . ; ng and n, dimension of the problem; uk, corresponding upper bound; lk, corresponding lower bound; and f(x), pointer to the function that is minimized. Herein, each particle x represents a solution with the corresponding fitness function f(x). EM uses the mechanisms of attraction and repulsion to put the sample points toward to the optimum. By the Coulomb’s law, the magnitude of force is proportional to the product of the particles and inversely proportional to the distance between two particles. The principles behind the algorithm are that inferior particles prevent a move in their direction by repelling other particles in the population and that superior particles facilitate moves in their direction. The Pseudo code of EM algorithm is shown in Fig. 2. The following sections discuss each phase. 3.1. Initialization phase For most applications of algorithm, the real-value coding technique is used to represent a solution for a given problem. In realvalue coding implementation, each particle is encoded as a vector of real numbers, of the same lengths as the solution vector. In this article, each particle denotes a weighting vector (Kp, Ki, k, Kd, d), and the EM is used to find the optimum ðK P ; K i ; k; K d ; dÞ. Typically, initial particles are randomly chosen from a feasible solution region. ‘‘Initialization phase” is used to generate m initial particles. At first, the feasible region of solution for tuning param-
P= initial population While the termination criteria is not satisfied Fitness values calculation xbest=best particle of P Local search For each particle, calculate total force F For each particle, Movement end while Fig. 2. Pseudo code of the EM algorithm.
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eters (i.e., uk and lk) should be defined. In addition, the training cycle is chosen to be the termination condition.
Fi ¼
3.2. Evaluation phase
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, u N uX f ðxÞ t e2 ðkÞ N;
ð5Þ
k¼1
f xj < f xi
m j P > qi qj > i > : x x jjxj xi jj2
if
f xj P f x i
j–i
;
8i;
ð7Þ
where f(xj) < f(xi) represents attraction and f(xj) P f(xi) represents repulsion. After comparing the fitness values, (i.e., f(x)), the direction of the forces between the particle and the others is selected. Therefore, xbest plays the role of attraction, i.e., it attracts all points in the population. 3.3.3. Movement After determining the total force vector Fi, particle xi moves in the direction of the total force by a random step length, i.e.,
3.3. Operation phase According to the literature (Cao et al., 2005; Lee, 2004a, 2004b; Lee & Chang, 2008; Lee, Chang, & Chen, 2007; Lee & Teng, 2002, 2003), there are three steps in the EM operation phase. They are ‘‘local search”, ‘‘total force calculation”, and ‘‘movement”, respectively. 3.3.1. Local search of EM algorithm The local search provides the EM with a good balance between the exploration and exploitation. In literature (Podlubny, 1999b), Birbil and Fang propose two approaches – local search applied to all points and local search applied only the current best point. For the first, it is a simple random search algorithm applied coordinate by coordinate. By the results of Lee and Chang (2008), Lee et al. (2007), EM algorithm has the properties of rapid convergence and global optimization. However, it usually time consuming (Lee & Chang, 2008; Lee et al., 2007). According to our experimental results, the local search spends 95.84% of the computation time of the EM algorithm process. It is seen that the EM has a problem of computation complexity in local search. For the second, the local search by using the current best particle, the local minimum may be obtained. Therefore, the genetic algorithm and competitive method are used to develop local search of IEMGA. 3.3.2. Total force calculation In this step, a particle is assigned to each particle of the population like electromagnetic charges. The charge qi of particle xi is determined by
"
8i:
ð6Þ
The particle having the largest charge is called the ‘‘optima particle”. The particle attracts others particles with better fitness values (lower RMSE), and repels other particles with worse fitness values. According to electromagnetic theory, the force is inversely propositional to the distance between two particles and directly proportional to the product of their charges. Hence, the total force vector exerted on xi computed by the superposition principle is
8 < xi þ k
uk xik if x ¼ : xi þ k F i xi lk if k jjF i jj i
where e denotes the error between setting point and plant output (i.e., e = yr y) and N denotes the data number. Subsequently, all particles are ranked and indexed by the corresponding of fitness value. Finally, the particle having the largest fitness value (minimum RMSE) is stored in xbest, i.e., xbest = {x| min RMSE of e}.
# f xi f xbest q ¼ exp n Pm ; k best Þ k¼1 ½f ðx Þ f ðx
if
j–i
This phase calculates the fitness values of all particles. Each particle is evaluated by the given fitness function to decide its survival or extinction in the next generation. The evaluation phase helps us to find superior particles by fitness value. There are three steps in this phase–fitness values evaluation, fitness ranking, and the best particle definition. To evaluate the performance of each particle in optimization, we define the root-mean-square error (RMSE) to be the fitness function:
i
8 m P j i j > > x xi jxjqxq i j2 > < j j
Fi jjF i jj
k ¼ 1; 2; . . . ; n;
F ik > 0 otherwise
i ¼ 1; 2; . . . ; m;
;
i – best;
ð8Þ
where the random step length k = random (0, 1). Note that the optima particle does not move because of having the best fitness value and attracting all other particles. Based on earlier comparisons of the three different learning methods in the literature (Lee et al., 2007), we can see that the use of the EM algorithm method obtains a faster convergence procedure and more multiple searches than the traditional PSO and the traditional GA methods. However, the high computation complexity problem should be solved. In Section 4, based on EM, the hybrid algorithm IEMGA is proposed to enhance the performance. 4. Hybrid algorithm IEMGA GA does not require or use derivative information for optimization, and the most appropriate applications are problems where gradient information is unavailable or difficult to obtain. Because of its global optimization capability, GA has become another useful tool in the automatic adjustment parameters. For this reason, GA is adopted to replace the random local search. 4.1. Improved EM algorithm with GA technique Fig. 3 depicts the IEMGA algorithm for optimization. Obviously, the local search of IEMGA method contains the reproduction (competitive selection), crossover, and mutation. IEMGA is a multi-point search, and it is capable of decreasing the computational complexity and has high-speed convergence. The major modification from EM algorithm is that the random neighborhood local search is replaced by GA, i.e., steps 5–7. After the best particle definition, the competitive selection is adopted to generate the next generation. The 50% front particles are selected to enter the modified local search. The other 50% particles are discarded. Subsequently, the GA optimization procedures, including reproduction (competitive selection), crossover, and mutation, are used to generate the better particles. Finally, the new particles and the remaining 50% particles are combined to be the new population. Fig. 4 shows the flow chart of the modified local search. After the evaluation phase (fitness calculation and ranking), the elites concept is adopted to implement the competitive selection; r is a floating-point number randomly chosen between 0 and 1 for each chromosome. Pc and Pm are crossover rate and mutation rate, respectively. The IEMGA local search consists of three basic operations: reproduction (competition selection), crossover, and mutation. A detailed description follows.
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50%
Start 50%
Step 7 Generate new population
Step 1 Determine the size of particles [m] and choose initial particles
Step 8 Calculation of fitness function
Step 2 Calculation of fitness function
Step 9 Fitness ranking
Step 3 Fitness ranking
Step 10 Total force calculation via (7) and (8)
Step 4 Define the best particle xbest
Step 11 Movement via (9), (10)
Step 5 Competitive selection (better particles)?
No Yes
Step 12 Termination satisfies ?
No
Yes
Discard particles
Step 13 Find the x*
Step 6 New particles are produced by GA
End
Fig. 3. Descriptions of learning algorithm IEMGA.
4.2. Genetic algorithm Population size M is the number of solutions to parallel search in each generation. A large population size takes a longer time for operations. Therefore, for the setting of this parameter, a balance must be achieved between solution quality and operation time. In this article, M is equal to half the total particles according the competitive selection, i.e., M = m/2. 4.2.1. Reproduction by competitive selection The fitness function is used to judge whether the chromosome is good or not. Chromosomes with lower fitness values may be easily eliminated during reproduction. The fitness function is used to test whether the specified is able to survive and reproduce, and its main purpose is to preserve the chromosomes with advantages to the next generation so as to improve the quality of solutions. After fitness function calculation, the population will be decided and whether to keep or get rid of the chromosomes. The species with high fitness is more ascendant than those with low fitness
in the reproduction. The reproduced chromosomes will be placed in the mating pool, waiting for the next operation procedure. During the process of reproduction, the operation is usually divided into two stages, competitive selection and roulette wheel selection (IFAC, 2001; Lee, 2004b; Lee & Teng, 2003). Herein, we choose the first one. Competitive selection is to randomly pick more than two chromosomes to compare their fitness values. The one with a higher fitness value will be reproduced, whereas the one with a lower fitness value will be eliminated; the higher the fitness value is, the larger the area is, and values are produced to decide the falling point at random. Each chromosome is evaluated by the given fitness function to decide its survival or extinction in the next generation. The better the fitness value is, the higher the possibility for a chromosome to survive. 4.2.2. Crossover Based on the pre-assigned crossover rate, a certain amount of pairs of chromosomes are randomly chosen to execute the cross-
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Start of local search The better 50% particles
The better 50% particles
Algorithm
Total time (s)
Local search time (s)
Percentage of local search
EM IEMGA
4.107 0.214
3.936 0.165
95.84 46.45
Competition selection Randomly choose two strings as parents
generation calculation. Competition selection and gradient method also improve the computation efficiency of the EM algorithm. r < Pc ?
No
5. Simulation results
Yes Offspring = parents
A block diagram of simulation with PID and PIkDd controllers using Simulink is shown in Fig. 5. The remaining work is to set up a controller index that can choose the controller model. In this article, we adopt the integrated-square-error (ISE) function to be the corresponding integral cost function:
Uniform crossover Putting offspring into new population
Jt ¼ Satisfying enough new population
r < Pm ?
No
Yes Keeping the same value in the gene
Randomly assign a value to the gene Putting offspring into new population
Produce new population
End of local search Fig. 4. Flow description for IEMGA local search.
over operation. During this operation, one chromosome in a pair will exchange parts of its subsequence with the other. The crossover operation allows the chromosomes to approach the desired solutions. In this article, we select the uniform crossover method (Fan et al., 2007; IFAC, 2001; Lee & Teng, 2003). During the operation of crossover rate Pc, if a high crossover rate is chosen, it will be helpful in producing more chromosomes of new structures. However, this also means that good chromosome structures are also highly likely to be destroyed. Usually, it refers to the rate for chromosomes to be mated in the same generation. 4.2.3. Mutation The use of a higher mutation rate Pm may be somewhat helpful in introducing gene structures that have not been searched repeatedly (Cao & Cao, 2006; IFAC, 2001; Lee, 2004b; Lee & Teng, 2003). Randomly produce a floating-point number r between 0 and 1 for each chromosome; if r < Pm, the chromosome must be mutated. Using the above modifications, simple comparison results of EM and IEMGA are shown in Table 1. Obviously, the proposed IEMGA algorithm obtains a smaller computation time than the traditional EM. IEMGA uses only 0.214 s (about 5.2% of EM algorithm) in a
1
e2 ðtÞdt;
ð9Þ
0
No
Yes
Z
where the error e(t) = yr(t) y(t) is the difference between the reference and the output signal of the closed-loop system. In addition, the initial particles are randomly selected from the feasible region of the search space. Consider the following second-order system:
Gp ðsÞ ¼
400 e0:5 s : s2 þ 50 s
ð10Þ
IEMGA is used to design the parameters of PID and PIkDd controllers so as to minimize the cost function – Jt. The cost function, which quantifies the effectiveness of a given PID and a PIkDd controller, is evaluated at the conclusion of a step-response experiment. The ISE cost function is realized as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, u N uX Jk t e2 ðkÞ N;
ð11Þ
k¼1
where e(k) = yr(k) y(k) is the difference between set point and the output signal of the closed-loop system. 5.1. Optimization of PID controller design To show the effectiveness and efficiency of IEMGA, the method in the literature (Cao et al., 2005) is applied to the same control problem. The initial parameters are chosen randomly in the following range: Kp: [0, 20], Ki: [0, 1], and Kd: [0, 1]. The reference input signal yr(t) is in the form of:
yr ðtÞ ¼
1;
t 6 5ðsecÞ;
1:5; t > 5ðsecÞ:
ð12Þ
The parameters of IEMGA and GA algorithm are chosen as follows: – – – – –
Total time: 10 (sec) Sample time: 0.01 (sec) Parameters number of PID: 3 Population number (or particles): 50 Generation number (or epoch): 30
Simulation results are shown in Figs. 6–8 and Table 2. Fig. 6 shows the dynamic response after optimization (solid line: desired trajectory; dashed line: system actual output). The corresponding control effort is shown in Fig. 7. Comparison results of ISE between IEMGA and GA are shown in Fig. 8 (dashed line: Ref. Cao et al., 2005; solid line: IEMGA). Obviously, the IEMGA algorithm has better performance in ISE and faster convergence than that in the lit-
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KP
sig_out
error
Kp To Workspace 1 s
[T sig_in]
400 e-0.5s 2 s +50s
In1 Out1
yp
Integrator
From Workspace
To Workspace1
KI Transfer Fcn1
du/dt
In1 Out1
u
KD
To Workspace2
Derivative
Fig. 5. Block diagram of simulation with PID and PIkDd controller using Simulink.
erature (Cao et al., 2005) in 30 epochs. Besides, from Table 2, it is seen that the IEMGA algorithm has better performance (ISE: 0.0816) than GA (Cao et al., 2005).
0.12 literature [5] IEMGA
0.115
5.2. Optimization of PIkDd controller design In this example, the IEMGA is also applied to optimize the PIkDd controllers to show the effectiveness and efficiency of IEMGA. The
RMSE
0.11
Kp = 1.3779, Ki = 1.2269, Kd = 0.0586
1.6
0.09 0.085
1.2
0.08
1
Output
0.1 0.095
Desired trajectory System actual output
1.4
0.105
0
5
10
15
20
25
30
Epoch
0.8
Fig. 8. Comparative results of RMSE of system (13) using PID controller (dashed line: literature (Cao et al., 2005); solid line: IEMGA).
0.6 0.4
Table 2 Comparative results of example 1 using PID controller (controller parameters and ISE).
0.2 0
0
1
2
3
4
5
6
7
8
9
10
Time (sec.)
Algorithm
Kp
Ki
Kd
ISE
Reference: Cao et al. (2005) IEMGA
1.5662 1.3779
1.1149 1.2269
0.1196 0.0586
0.0942 0.0816
Fig. 6. Simulation results of system (13) using PID controller (solid line: desired trajectory; dashed line: system actual output).
Kp = 1.0825, Ki = 0.047421, λ = 0.92749, Kd = 0.6317, δ = 0.89606
1.6
5
Desired trajectory System actual output
1.4 4
1.2 1
Output
u
3 2
0.8 0.6
1
0.4 0.2
0
0 -1
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Time (sec.)
Time (sec.) Fig. 7. Simulation results of system (13) using PID controller: control effort.
Fig. 9. Simulation results of system (13) using PIkDd controller (solid line: desired trajectory; dashed line: system actual output).
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1.6
5
Desired trajectory PID Fractional order PID
1.4
4
1.2
Output
u
3 2
1 0.8 0.6
1
0.4 0
0.2
-1 0
1
2
3
4
5
6
7
8
9
0
10
0
1
2
3
4
5
6
7
8
9
10
Time (sec.)
Time (sec.) Fig. 10. Simulation results of system (13) using PIkDd controller: control effort.
Fig. 12. Simulation results of system (13) using IEMGA algorithm (solid line: desired trajectory, dash–dotted line: PID controller, and dashed line: PIkDd controller).
0.17 literature [5] IEMGA
0.16
0.15
0.15
0.14
0.14
0.13
PID Fractional order PID
0.13
RMSE
RMSE
0.16
0.12 0.11
0.12 0.11
0.1 0.1
0.09 0.08
0.09 0
5
10
15
20
25
30
0.08
Epoch Fig. 11. Comparative results of RMSE of system (13) using PIkDd controller (dashed line: reference Cao et al., 2005; solid line: IEMGA).
initial parameters are chosen randomly in the following range: Kp: [0, 20], Ki: [0, 1], k: [0, 1], Kd: [0, 1], and d: [0, 1]. The reference input signal yr(t) is in the form of (12), and the parameters of the IEMGA and GA algorithm are chosen according to the above description. Simulation results are shown in Figs. 9–13 and Table 3. Fig. 9 shows the dynamic response after optimization (solid line: desired trajectory; dashed line: system actual output). The corresponding control effort is shown in Fig. 10. Comparison results of ISE between IEMGA and GA are shown in Fig. 11 (dashed line: reference Cao et al., 2005; solid line: IEMGA). Fig. 12 shows the dynamic response after optimization using the IEMGA algorithm (solid line: desired trajectory, dash–dotted line: PID controller, and dashed line: PIkDd controller). We see that the PIkDd controller enhances the performance. To reach stable effect, the PID controller needs 3 s but using the PIkDd controller needs only 0.5 s. Obviously, the PIkDd controller is better than the PID controller. Comparison results of ISE by the IEMGA algorithm is shown in Fig. 13 (dashed line: PID controller and solid line: PIkDd controller). The PIkDd con-
0
5
10
15
20
25
30
Epoch Fig. 13. Comparative results of RMSE of system (13) using IEMGA algorithm (dash– dotted line: PID controller; dashed line: PIkDd controller).
troller has better performance in ISE and faster convergence than that reported in the literature (Cao et al., 2005) in 30 epochs. Besides, from Table 3, it is seen that the IEMGA algorithm has better performance (ISE: 0.0811) than GA (Cao et al., 2005). 6. Conclusions In this article, an intelligent method, IEMGA, for designing an FOPID controller has been proposed. The IEMGA is an evolutionary method and avoids the use of fraction calculus. It modifies the ‘‘local search” of the EM algorithm by using competitive selection and GA. It is capable of decreasing the computational complexity, i.e., it is highly effective. In addition, the proposed IEMGA method acquires a shorter CPU time. The IEMGA is used to design the PIkDd controller parameters based on the ISE performance index. In addition, the IEMGA optimization method is available when other performance indexes are chosen, e.g., integral absolute error (IAE),
Table 3 Comparative results of example 2 using PIkDd controller (controller parameters and ISE). Algorithm
Kp
Ki
k
Kd
d
ISE
Reference: Cao et al. (2005) IEMGA
1.6230 1.0825
1.1908 0.0474
0.8190 0.9275
0.1135 0.6317
1.5146 0.8961
0.0971 0.0811
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integral square time error (ISTE), etc. From the simulation results, it can be observed that our approach to IEMGA can find the best global solution for FOPID controllers. They show the effectiveness of IEMGA. Acknowledgment This work was partially supported by the National Science Council, Taiwan, ROC, under contracts NSC-97-2221-E-155-033MY3. References ACPA (2002). Special issue on advance in PID control. Asian Journal of Control, 4(4). Birbil, S. I., & Fang, S. C. (2003). An electromagnetism-like mechanism for global optimization. Journal of Global Optimization, 25(3), 263–282. Birbil, S. I., Fang, S. C., & Sheu, R. L. (2004). On the convergence of a population-based global optimization algorithm. Journal of Global Optimization, 30(2), 301–318. Biswas, A., Das, S., Abraham, A., & Dasgupta, S. (2009). Design of fractional-order PIkDl controllers with an improved differential evolution. Engineering Applications of Artificial Intelligence, 22, 343–350. Cao, J. Y., & Cao, B. G. (2006). Design of fractional order controller based on particle swarm optimization. International Journal of Control, Automation, and Systems, 4(6), 775–781. Cao, J. Y., Liang, J., & Cao, B. G. (2005). Optimization of fractional order PID controllers based on genetic algorithms. In Proceedings of the fourth international conference on machine learning and cybernetics, Guangzhou, 18–21 August, 2005 (pp. 5686–5689). Chang, P. C., Chen, S. H., & Fan, C. Y. (2009). A hybrid electromagnetism-like algorithm in single machine scheduling problems with learning effect. Expert Systems with Applications, 36(3), 1259–1267. Clerc, M., & Kenney, J. (2002). The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, 6(1), 58–73. Fan, H., Sun, Y., & Zhang, X. (2007). Research on fractional order controller in servo press control system. In Proceedings of the 2007 IEEE international conference on mechatronics and automation, August 2007 (pp. 2934–2938). Farag, W. A., Quintana, V. H., & Germano, L. T. (1998). A genetic-based neuro-fuzzy approach for modeling and control of dynamical systems. IEEE Transactions on Neural Networks, 9(5), 756–767. Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. Reading: Addison-Wesley. Gudise, V. G., & Venayagamoorthy, G. K. (2003). Comparison of particle swarm optimization and backpropagation as training algorithm for neural networks. In IEEE swarm intelligence symposium, April 2003 (pp. 110–117).
Hong, G., & Yuan, M. Z. (2002). Immune algorithm. In Proceedings of the 4th world congress on intelligent control and automation, Vol. 3 (pp. 1784–1788). IEE (2002). Special issue on PID control. IEE Proceedings D – Control Theory and Applications, 149(1). IFAC (2001). Special issue on PID controller. Control Engineering Practice, 9(11). Juang, C. F. (2004). A hybrid of genetic algorithm and particle swarm optimization for recurrent network design. IEEE Transactions on Systems, Man, and Cybernetics Part B – Cybernetics, 34(2), 997–1006. Kim, D. H. (2002). Parameter tuning of fuzzy neural networks by immune algorithm. In IEEE international conference on fuzzy systems, Vol. 1 (pp. 408–413). Lee, C. H. (2004a). Low order robust controller design for preserving H1 performance: Genetic algorithm approach. ISA Transactions, 43(4), 539–547. Lee, C. H. (2004b). A survey of PID controller design based on gain and phase margins. International Journal of Computational Cognition, 2(3), 63–100. Lee, C. H., & Chang, F. K. (2008). Recurrent fuzzy neural controller design for nonlinear systems using electromagnetism-like algorithm. Far East Journal of Experimental and Theoretical Artificial Intelligence, 1(1), 5–22. Lee, C. H., Chang, F. K., & Chen, C. W. (2007). A modified electromagnetism-like algorithm for training neural fuzzy systems in control applications. In CACS international automatic control conference, November 2007. Taichung, Taiwan. Lee, C. H., & Teng, C. C. (2002). Tuning PID controller of unstable processes: A fuzzy neural network approach. Fuzzy Sets and Systems, 128(1), 95–106. Lee, C. H., & Teng, C. C. (2003). Calculation of PID controller parameters by using fuzzy neural network. ISA Transactions, 42(3), 91–400. Nataraj, P. S. V., & Tharewal, S. (2007). On fractional-order QFT controllers. Journal of Dynamic Systems Measurement and Control – Transactions of the ASME, 129, 212–218. Podlubny, I. (1999a). Fractional differential equations. New York: Academic Press. Podlubny, I. (1999b). Fractional-order systems and PIkDl controllers. IEEE Transactions on Automatic Control, 44(1), 208–214. Price, K., Storn, R., & Lampinen, J. (2005). Differential evolution – A practical approach to global optimization. New York: Springer. Srinivas, M., & Patnaik, L. M. (1994). Adaptive probabilities of crossover and mutation in genetic algorithms. IEEE Transactions on Systems, Man, and Cybernetics, 24(4), 656–667. Tsou, C. S., & Kao, C. H. (2007). Multi-objective inventory control using electromagnetism-like meta-heuristic. International Journal of Production Research, 1, 1–16. Wu, P. T., Yang, W. H., & Wei, N. C. (2004). An electromagnetism algorithm of neural network analysis – An application to textile retail operation. Journal of the Chinese Institute of Industrial Engineers, 21(1), 59–67. Xu, H. G., Wei, X. Y., & Xu, M. S. (2000). Schema analysis of multi-points crossover genetic algorithm. In Proceedings of the 3rd world congress on intelligent control and automation, Vol. 1 (pp. 521–524). Yamato, S., & Hashimoto, I. (1991). Present status and future needs the view from Japanese industry. In Y. Arkun, W. H. Ray, (Eds.), Proceedings of the fourth international conference on chemical process control, TX. In Proceedings of the CPCIV (pp. 1–28). Yao, X. (1999). Evolutionary computation: Theory and applications. World Scientific.
