Optimal Design of LQR Weighting Matrices based on ...

Optimal Design of LQR Weighting Matrices based on Intelligent Optimization Methods S.Amir Ghoreishi, Mohammad Ali Nekoui, S. Omid Basiri International Journal of Intelligent Information Processing, Volume 2, Number 1, March 2011

Optimal Design of LQR Weighting Matrices based on Intelligent Optimization Methods S.Amir Ghoreishi, Mohammad Ali Nekoui and S. Omid Basiri South Tehran Branch, Islamic Azad University, Tehran, Iran. [email protected], [email protected] doi:10.4156/ijiip.vol2. issue1.7

Abstract In this paper, considering some important indices such as closed-loop pole locations, speed of response, and maximum level of control effort, and combining them into an objective function, an optimization problem is defined to find the optimal weighting matrices in LQR controller. To solve this optimization problem four intelligent optimization methods are utilized: Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and Imperialist Competitive Algorithm (ICA). The proposed method is applied to a nonlinear flexible robot manipulator model, and obtained results from the algorithms are compared.

Keywords: Optimal Control, Genetic Algorithm (GA), Differential Evolution (DE) 1. Introduction Optimal control uses some methods and mathematical tools to design controllers for dynamical systems such that a criterion to be optimized. Usually the criterion considers performance, energy consumption, response time and final state situations. For example designing a controller to transfer the state of a dynamical system to a desired state in minimum time can be categorized in an optimal control problem. The Linear Quadratic Regulator (LQR) is a method used abundantly to design linear controllers for linear systems. The LQR controller possesses suitable robustness with minimum gain margin -6db and maximum gain margin to infinity and at the same time it can reach to 60 degree of phase margin. The design parameters for LQR are the weighting matrices in the objective function and should be selected by the designer. Since these matrices directly affect the optimal control performance many discussions have been done to select these matrices called eigen-structure assignment [1-3]. Besides the classical methods, the algorithms based on intelligent optimization and soft computing have been used gradually for selecting these matrices. For example, Genetic Algorithm (GA) [4-10], combination of GA and Simulated Annealing (SA) [11] and Ant Colony Optimization (ACO) [12] have been used for solving this problem. In this paper, by considering some important criteria like speed of response, the closed-loop pole locations, and maximum level of control effort, and combining them into an objective function, give us an optimization problem, solution of which leads us to the best weighting matrices in LQR controller. Since we have a nonlinear, complicated optimization problem GA [13], Particle Swarm Optimization (PSO) [14,15,16], Differential Evolution (DE) [17,18], and Imperialist Competitive Algorithm (ICA) [19,20] are used to solve this problem. The proposed method is applied to flexible robot manipulator. This paper is organized as follows: Section 2 briefly describes the LQR controller. The proposed method and its related criterion are described in Section 3. Section 4, briefly explains the intelligent optimization tools, GA, PSO, DE, and ICA, which are used to solve the optimization problem. In Section 5 the flexible robot manipulator model is described, which in Section 6 the proposed method is applied to this model. Finally, Section 7 concludes the paper.


2. LQR Controller One of the state space based optimal control method is Linear Quadratic Regulator (LQR). In this section we briefly describe this method. Consider the following linear, continuous-time and controllable system: (1) x = Ax + Bu The following objective function is defined:

J =

1 ¥ T [x Qx + uT Ru ]dt ò 0 2

(2)

where, Q and R are weighting matrices and should be positive-semi-definite and positivedefinite, respectively. Since system (1) is controllable, the method which is able to minimize (2) is called LQR. Considering the functional (2) in LQR the following Riccati equation should be solved:

PA + AT P - PBR-1BT P + Q = 0

(3)

By solving the above Riccati equation the positive-definite matrix P is obtained, thus the optimal gain and controller are calculated as: (4) -1 T

K =R B P

u = -Kx

(5)

Therefore, the closed-loop poles are the eigenvalues of A - BK . In this paper the eigenvalues of A - BK are shown as l =

{l1, l2, , ln } .

Q and R have profound effect on controller performance. On the other hand, finding the best Q and R needs many computer simulation In LQR problem, the weighting matrices

and trial and errors, which are very time-consuming. Thus using intelligent optimization methods for finding Q and R is more effective. In this paper, we use Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and Imperialist Competitive Algorithm (ICA) as optimization tools. Firstly, in the next section we are going to describe the optimization problem for finding Q and R .

3. Proposed Approach An optimization problem has at least two distinct solutions. The process of optimization is to search for the best solution of the optimization problem. In this paper, we are searching for best weighting matrices of LQR approach. So it is necessary to define terms better and best. A good LQR controller, must make the closed-loop system (i) as fast as possible and (ii) as stable as possible, using (iii) as lowest as possible control effort. We define three indices for the mentioned requirements of a good LQR. Definitions of the indices are listed below: Stability Index. This index is related to real parts of closed-loop poles and defined as follows:

SI = -

1 max Re{li } i

(6)


Obviously the better Q and R give smaller stability index. Settling-time Index. For a time response of a system this index is the minimum time that the response reaches to absolute error of 0.05. In mathematical form:

ì ï ST = min ï ít ï ï î where

y(t ) - yd (t ) ïü £ 0.05 ïý ïþï yd (t )

(7)

y(t ) is the time response and yd (t ) is desired output trajectory. If the system has more than

one output, the largest settling time among outputs, is the settling time of system. If desired value of the output is zero, then setting time is defined as follows, conventionally.

ì ï ï ST = min ïí t ï ï ï î

y(t ) £

max y(t ) ü ï ï ïý t 100 ï ï ï þ

(8)

Maximum Control Effort Index. Because of the limitations of actuators, the objectives should be satisfied with minimum control effort, usually. Using the actuators with higher saturation level needs more expenditure. Maximum control effort to the system can be defined as:

umax = max u(t )

(9)

t

Finding the best Q and R matrices, using above indices, is a multi-objective optimization problem. Thus to avoid complication we define the following objective function:

JTotal = w1 SI + w 2 ST + w 3 umax

(10)

where, w 1 , w 2 and w 3 should be selected by the designer. Remark 1: The most important point in Q and

R matrices is the constraints on them. As mentioned

before, the Q and R matrices should be positive-semi-definite and positive-definite, respectively. However, these constraints cannot be easily satisfied. It is hard to define a simple relation between matrix elements and its positive definiteness. Thus, in this paper, a method is used to make positiveT

semi-definite matrices. It is known that for any real matrix A , B = A A is nonnegative, i.e., it can be said that matrix B is positive-semi-definite. Usually matrix A is known as square root of the matrix B . The above method is used in this paper to make initial responses and to code the responses of the problem. In other words, instead of using Q and R matrices as unknown variables the matrices

W and V that satisfy Q = W TW and R = V TV are used as unknown variables. Thus optimization algorithms first find W and V , then Q and R matrices are calculated by above equations.

4. Optimization Tools In this section, the utilized intelligent optimization methods are briefly reviewed.


4.1. Genetic Algorithm Genetic Algorithm (GA) [13] is one of the evolutionary algorithms inspired by natural evolution. In GA, the proposed responses for an optimization problem are considered as living creatures and GA provides a virtual environment for growth and activity of these proposed responses. In this virtual environment the better responses are increased easier and more than others. In this virtual environment, similar to any phenomenon in nature some mechanisms are considered. The most important operators and mechanisms used in GA with their equivalencies in nature are:  Selection operator which is equivalent to natural selection phenomenon  Crossover operator which is equivalent to reproduction phenomenon  Mutation operator which is equivalent to genetic mutation phenomenon By using these operators on current population, new population emerges which the average of them are not worse than the current population, nevertheless the average quite often is better. Thus by time passing the GA gives better response for optimization problem.

4.2. Particle Swarm Optimization The PSO method was first introduced by James Kennedy and Rassel Eberhart [14] in 1995. They were essentially aimed at producing computational intelligence by exploiting simple analogues of social interaction, rather than purely individual cognitive abilities. Their work developed into a powerful optimization method called Particle Swarm Optimization (PSO) [14,15]. In PSO, a number of simple entities, namely the particles, are placed in the search space of some problem or function, and each evaluates the objective function at its current location. Each particle then determines its movement through the search space by combining some aspect of the history of its own current and best (best-fitness) locations with those of one or more members of the swarm, with some random perturbations. The algorithm, searches a space by adjusting the trajectories of particles as they are conceptualized as moving points in multidimensional space. The individual particles are drawn stochastically toward the positions of their own previous best performance and the best previous performance of their neighbors. If the search space is considered as n-dimensional space, the j position and velocity of a particle can be shown by n-dimensional vectors. Consider x i [t ] and

vij [t ] to be the j -th element of position and velocity of the i -th particle in t -th iteration, respectively. The position and velocity of i -th particle in (t + 1) -th iteration are defined as [14,15]:

xij [t + 1] = xij [t ] + vij [t ] j vij [t + 1] = wvij [t ] + r1c1 ( x ij,best [t ] - x ij [t ] ) + r2c2 ( x gbest [t ] - x ij [t ] )

(11) (12)

where, w is inertia coefficient and can be constant, variable or random. This coefficient guarantees that the particles which give the best response are not halted and continue their pervious trajectories [14,15]. The constants c1 and c2 are learning coefficients and they are selected in the interval [0, 4]

c1 + c2 = 4 [15]. r1 and r2 are random numbers with uniform distribution in the interval [0,1] . x i,best [t ] is the best response that is found by the i -th particle until t -th iteration and

and usually

x gbest [t ] is the best response of total population until t -th iteration.


According to [16], it is possible to define constriction coefficients, which guarantee the stability and good performance for PSO. For two scalars f1 and f2 , where f1 + f2 > 4 , the constriction coefficients are defined by:

2

w =

(f1 + f2 ) - 2 + (f1 + f2 )2 - 4(f1 + f2 ) c1 = f1w c2 = f2w

(13) (14) (15)

A good choice is f1 = f2 = 2.05 , which implies that w » 0.7298 and c1 = c2 » 1.4962 [15,16].

4.3. Differential Evolution Price and Storn proposed Differential Evolution (DE) in mid 1990s [17,18], to deal with optimization problems, defined in continuous domains. DE has similarities with both GA and PSO. DE uses information of all individuals, and differences between them, to create new solutions for optimization problem. The new solutions are created using difference and trial vectors. To create a new solution y , an old solution a is perturbed using the following rule:

y = a + F Ä (b - c)

(16)

c are two individuals, randomly selected from population, and a ¹ b ¹ c . Vector F is the scaling factor, and its elements are uniformly distributed random numbers in [Fmin , Fmax ] . Operator Ä is the element-wise multiplication operator. To create final solution z , crossover operator is applied to y and another randomly selected individual x . There are various methods of crossover. Simplest case is formulated as follows: where b and

ìïy , r £ CR or i = i0 zi = ïí i ïï xi , otherwise î where x i indicates i -th element of vector

(17)

x , scalar r is a uniformly distributed random number in

[0,1] , CR is the Crossover Rate parameter, and i0 is a random integer index in the set {1, 2, 3,..., n} . It is assumed that number of search space dimensions (also number of elements of solution vectors) are equal to n . In this way, DE uses information of current population, to create individuals of the next iteration (generation). This process is carried out, until termination conditions are satisfied.

4.4. Imperialist Competitive Algorithm Imperialist Competitive Algorithm (ICA) is proposed by Atashpaz-Gargari et al. [19,20], and it is inspired by imperialist competition. Similar to GA, which simulates the natural evolution process to solve an optimization problem, ICA simulates the socio-political evolution to deal with optimization problems. In ICA, individual solutions are referred as (virtual) countries. Some of good countries in the initialization phase, which are named imperialists, form their own imperial. They capture their colonies


from other non-imperialist (normal) countries. In every iteration (decade) of ICA, the following operations are carried out: Assimilation of Colonies. Colonies of each imperialist are assimilated to their respective imperialist. Assimilation is formulated as following: new old old xcol = xcol + b r Ä (ximp - xcol )

where

(18)

b is assimilation factor, and r is a vector, and its elements are uniformly distributed random

numbers in

old new [0,1] . x imp , xcol , and x col are position of imperialist, old position of colony, new

position of colony, respectively. In [19,20], the new position of colony is angularly deviated. For more information about deviation, refer to [19] and [20]. Revolution of Colonies. Similar to mutation operator in GA, selected colonies of every imperialist are changed randomly, or revolved. Revolution is applied to a colony, with a probability of

pr .

Exchange with Best Colony. If after assimilation and revolution steps, there are colonies which are better than their respective imperialists, the imperialist is exchanged with its best colony. In other words, imperialist will be colony, and the best colony will be the new imperialist. Imperialist Competition. Weakest imperialist among others, loses its weakest colony. One of other imperialists will capture the lost colony, randomly. The better the imperial, the more probable it will possess the colony. An imperialist without colony will collapse. It will become a colony, and captured by other imperialists. The mentioned steps are carried out, while stop conditions are not satisfied.

5. Flexible Robot Manipulator A single-link flexible robot manipulator is used in this paper, to implement and check the proposed method. Nonlinear differential equations which are governing the single-link flexible robot manipulator system are listed below:

Iq1 + MgL sin q1 + k (q1 - q 2 ) = 0 Jq2 - k (q1 - q 2 ) = u

(19)

where q1 and q 2 are angular positions, I and J are moment inertia, M is the mass of link, L is the length of link, k is the stiffness, g is the gravitational acceleration, and u is the control input. Values of parameters of the system are given in Table 1. Table 1. Value of Parameters of System (4) Parameter Value Unit Kg.m2 I 0.031 Kg.m2 J 0.004 N.m/rad k 31 m L 0.255 Kg M 0.32 Defining state vector as T T x(t ) = éê x1(t ) x 2(t ) x 3(t ) x 4 (t ) ùú = éê q1(t ) q1(t ) q 2(t ) q2(t ) ùú ë û ë û

(20)


state space model of the flexible robot manipulator is as follows:

x1 = x 2 x 2 = -

MgL k sin x1 - (x1 - x 3 ) I I

x 3 = x 4 1 k x 4 = (x1 - x 3 ) + u J J

(21)

If the model (21) is linearized around the origin (the equilibrium point), the obtained linear model is: é 0 ê ê MgL + k êê I x (t ) = ê 0 ê ê k ê ê ë J

1

0 k 0 I 0 0 k 0 J

0ù ú ú 0ú ú ú x(t ) + 1ú ú ú 0ú û

é0ù ê ú ê0ú ê ú ê 0 ú u(t ) ê ú ê1ú ê ú êë J úû

(22)

Outputs of system are defined by:

é1 0 0 0ù ú x(t ) = y(t ) = êê ú 0 0 1 0 úû ëê

é q1(t ) ù ê ú ê q (t ) ú ëê 2 ûú

(23)

Desired set point of both outputs is zero. In all of simulations, initial state of system is defined as T x 0 = éê 0.3 0 0.1 0 ùú ë û

(24)

6. Simulation Results In this section the proposed method is applied to rotational inverted pendulum introduced in Section 5. The simulation time is considered 5 seconds, with step size of 0.01 seconds. Weighting factors in cost function

JTotal , defined by Eq. (10), are set to w1 = 10 , w 2 = 5

and w 3 = 1 . First of all, GA is used to solve the optimization problem. GA results are shown in Figs. 1 and 2. In Fig. 1, total cost functional, Stability Index, Settling time Index and Maximum control effort are plotted versus iteration. After 100 iterations (generations), GA reaches to 5.9678 for cost functional. Fig. 2 shows the time response of y1(t ) = q1(t ) ,

y 2 (t ) = q 2 (t ) and the control signal u(t ) for different iterations. It can be seen that the performance of the controller is modified by increasing the number of iterations.

Optimal Design of LQR Weighting Matrices based on Intelligent Optimization Methods S.Amir Ghoreishi, Mohammad Ali Nekoui, S. Omid Basiri International Journal of Intelligent Information Processing, Volume 2, Number 1, March 2011 20 Total Cost

Best Total Cost 15 10 5

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Fig 1. GA: Cost functional JTotal , Stability Index, Settling Time Index and Maximum control effort 0.3 Iteration Iteration Iteration Iteration Iteration

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Fig 2. GA: Outputs y1 and y2 , and the control signal The above optimization problem is solved by PSO, as second approach. The simulation results are depicted in Figs. 3 and 4. In Fig. 3, the total cost functional, Stability Index, Settling time Index and Maximum control effort are shown, versus iteration count. PSO reaches to 4.6057 for the cost functional after 100 iterations. At the same time Fig. 4 shows the time response of y1(t ) = q1(t ) , y 2 (t ) = q 2 (t ) and the control signal u(t ) for different iterations. 15 Total Cost


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Fig 3. PSO: Cost functional JTotal , Stability Index, Settling time Index and Maximum control effort

Optimal Design of LQR Weighting Matrices based on Intelligent Optimization Methods S.Amir Ghoreishi, Mohammad Ali Nekoui, S. Omid Basiri International Journal of Intelligent Information Processing, Volume 2, Number 1, March 2011 0.3 Iteration Iteration Iteration Iteration Iteration

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Fig 4. PSO: Outputs y1 and y2 , and the control signal As third approach, DE is utilized to solve the optimization problem and find the optimal values of weighting matrices. The simulation results are sketched in Figs. 5 and 6. In Fig. 5, the total cost functional, Stability Index, Settling time Index and Maximum control effort are shown, versus iteration count. DE finally reaches to 4.6967 for the cost functional after 100 iterations. Time response of

y1(t ) = q1(t ) , y 2 (t ) = q 2 (t ) and the control signal u(t ) for different iterations, are shown in Fig. 6. 12 Best Total Cost

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Fig 5. DE: Cost functional JTotal , Stability Index, Settling time Index and Maximum control effort 0.3 Iteration Iteration Iteration Iteration Iteration

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Fig 6. DE: Outputs y1 and y2 , and the control signal

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Finally ICA is used to solve the optimization problem, as fourth method. The simulation results are shown in Figs. 7 and 8. Total cost functional, Stability Index, Settling time Index and Maximum control effort are shown, versus iteration count, in Fig. 7. ICA reaches to 4.5430 for the cost functional after 100 iterations. Time response of y1(t ) = q1(t ) , y 2 (t ) = q 2 (t ) and the control signal

u(t ) for different iterations, are shown in Fig. 8. 15 Total Cost


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Fig 7. ICA: Cost functional JTotal , Stability Index, Settling time Index and Maximum control effort 0.3 Iteration Iteration Iteration Iteration Iteration

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Fig 8. ICA: Outputs y1 and y2 , and the control signal For all of algorithms, population size is set to 50 and maximum number of iterations is set to 100. Results obtained from used algorithms are summarized in Table 2. The rank of each algorithm is presented in parenthesis, for each of indices. Considering Stability Index, DE has best performance among others. Stability Index for DE is 0.8040, which implies that greatest real part of the closed-loop poles is -1.2438. ICA with Settling Time of 0.37 sec, results the fastest closed-loop system among others. PSO needs least control effort among others and uses Maximum Control Effort of 1.7738 to stabilize and control the system. Considering total cost value, ICA has best performance. PSO, DE, and GA has second, third, and fourth rank, respectively.


Table 2. Optimization Results Algorithm

Stability Index

Settling Time

Maximum Control Effort

Total Cost

GA

0.1263 (4)

0.5200 (4)

2.1046 (4)

5.9678 (4)

PSO

0.0882 (3)

0.3900 (3)

4.6057 (2)

DE

0.0804 (1)

0.3800 (2)

1.7738 (1) 1.9923 (3)

4.6967 (3)

ICA

0.0866 (2)

0.3700 (1)

1.8270 (2)

4.5430 (1)

7. Conclusions In this paper, the problem of finding weighting matrices for an LQR controller has been formulated as an optimization problem. By considering some important indices in designing controllers such as; closed-loop pole locations, speed of response and maximum control effort, and combing them into an objective functional, the problem of finding the weighting matrices for an LQR controller has been converted to an optimization one. To solve this optimization problem four intelligent optimization methods are utilized: Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and Imperialist Competitive Algorithm (ICA). The proposed method has been applied to a single-link flexible robot manipulator system, and the obtained results from these four algorithms have been compared. Considering stability index, settling time, and maximum control effort, DE, ICA, and PSO have better performance among others, respectively. Considering all of indices, ICA has the best performance among all.

8. References [1] F. L. Lewis and V. L. Syrmos, Optimal Control, 2nd Edition, John Wiley & Sons, Inc., 1995. [2] G. P. Liu and R. J. Patton, Eigenstructure Assignment for Control System Design, John Wiley & Sons, Inc., 1998. [3] J. W. Choi and Y. B. Seo, “Control design methodology using EALQR,” in Proceedings of the IEEE 37th SICE Annual Conference, 1998. [4] J. V. da Fonseca Neto and C. P. Bottura, “Parallel Genetic Algorithm Fitness Function Team for Eigenstructure Assignmentvia LQR Designs,” in Proceedings of IEEE Congress on Evolutionary Computation, 1999. [5] C. P. Bottura and J. V. da Fonseca Neto, “Parallel Eigenstructure Assignment via LQR Design and Genetic Algorithms,” in Proceedings of the American Control Conference, San Diego, California, 1999. [6] J. V. da Fonseca Neto, et al., “Modelos e Convergência de um AlgoritmoGenéticoparaAlocação de Auto-estrutura via RLQ,” in IEEE Latin America Transactions, vol. 6, no. 1, pp. 1-9, March 2008. [7] R. Davis and T. Clarke, “A parallel implementation of the genetic algorithm applied to the flight control problem,” in IEE Colloquium on High Performance Computing for Advanced Control, pp. 6/1 - 6/3, 1994. [8] C. Wongsathan and C. Sirima, “Application of GA to Design LQR Controller for an Inverted Pendulum System,” in Proceedings of the IEEE International Conference on Robotics and Biomimetics, 2009. [9] A. H. Zaeri, M. BayatiPoodeh, and S. Eshtehardiha, “Improvement of Cûk Converter Performance with Optimum LQR Controller Based on Genetic Algorithm,” in Proceedings of International Conference on Intelligent and Advanced Systems, 2007. [10] M. BayatiPoodeh, et al., “Optimizing LQR and Pole placement to Control Buck Converter by Genetic Algorithm,” in Proceedings of International Conference on Control, Automation and Systems, 2007.


[11] Y. J. Lee and K, H. Cho, “Determination of the Weighting Parameters of the LQR System for Nuclear Reactor Power Control using the Stochastic Searching Methods,” in Journal of the Korean Nuclear Society, vol. 29, no. 1, pp. 68-77, Feb 1997. [12] D. Ali, L. Hend, and M. Hassani, “Optimized Eigenstructure Assignment by Ant System and LQR Approaches,” in International Journal of Computer Science and Applications, vol. 5, no. 4, pp. 45-56, 2008. [13] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, 2nd Edition, John Wiley & Sons, Inc., 2004. [14] J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” in Proceedings of IEEE International Conference on Neural Networks, 1995. [15] R. Poli, J. Kennedy, and T. Blackwell, “Particle Swarm Optimization: An Overview,” in Swarm Intelligence, no. 1, pp. 33-57, 2007. [16] M. Clerc and J. Kennedy, “The Particle Swarm – Explosion, Stability, and Convergence in a Multi-Dimensional Complex Space,” IEEE Transactions on Evolutionary Computation, vol. 6, pp. 58-73, 2002. [17] R. Storn and K. Price, “Differential Evolution: A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces,” Technical Report TR-95-012, International Computer Science Institute, Berkeley, California, March 1995. [18] R. Storn and K. Price, “Differential Evolution – A Fast and Efficient Heuristic for Global Optimization over Continuous Spaces,” Journal of Global Optimization, vol. 11, pp. 341–359, 1997. [19] E. Atashpaz-Gargari and C. Lucas, “Imperialist Competitive Algorithm: An algorithm for optimization inspired by imperialistic competition,” Proceedings of IEEE Congress on Evolutionary Computation 2007 (CEC2007), Singapore, 2007. [20] E. Atashpaz-Gargari, F. Hashemzadeh, R. Rajabioun, and C. Lucas “Colonial Competitive Algorithm: a novel approach for PID controller design in MIMO distillation column process,” Int. J. Intell. Comput, Cybernet., vol. 1, no. 3, pp. 337-355, 2008.