Optimal control of steel structures by improved particle swarm

International Journal of Steel Structures June 2014, Vol 14, No 2, 223-230 DOI 10.1007/s13296-014-2003-3

www.springer.com/journal/13296

Optimal Control of Steel Structures by Improved Particle Swarm Saeid Aghajanian1, Hadi Baghi2, Fereidoun Amini3,*, and Masoud Zabihi Samani1 1

Ph.D. Candidate, School of Civil Engineering, Iran University of Science and Technology, P.O.Box 16765-163, Narmak, Tehran, Iran 2 Ph.D. Candidate, Department of Civil Engineering, University of Minho, Azurem, Portugal 3 Professor, School of Civil Engineering, Iran University of Science and Technology, P.O.Box 16765-163, Narmak, Tehran, Iran

Abstract Active control is one of the modern approaches in seismic design of steel structures. Recently, induced by economic considerations, especially high expenses of control systems, optimality has become an important issue. In this paper an active system is used to control a steel structure’s displacements by a simplified pole assignment method. To optimize the number, the locations, and the total driving force of the required actuators, an improved particle swarm algorithm is presented focusing on the parameters of the velocity equation. A Geographical neighborhood topology and an adaptive inertia weight are used to improve the standard PSO algorithm. In addition to the local and global best solutions, the positions of the best particles in the geographical neighborhood are mathematically represented in an additional term. The performance of the proposed algorithm is compared with the traditional Genetic Algorithm (GA) and the standard particle swarm considering the optimal control of a 12-story steel structure as a numerical example. High capabilities of the proposed method in terms of the control target, convergence rate, and accuracy are simultaneously clarified by the results. Keywords: optimization, improved particle swarm, active control, genetic algorithm

1. Introduction Over the past two decades, modern control approaches have become an integral part of the structural seismic design and retrofitting system, especially in steel structures. Judicious supervision of active control systems, as a main class of control outlines, mitigates a structures’ response by applying suitable control forces. However, a major problem with this kind of practice is its high power requirements and maintenance costs. Therefore, the design doctrine in these systems is to soothe the structural response to an acceptable level by utilizing limited applied forces. This approach is constrained by the number of actuators and their required driving energy. In recent years, there has been an increasing interest in optimal control studies and the associated optimization strategies. In one of the primary research works, genetic algorithm has been applied in conjunction with gradientbased optimization techniques for simultaneous placement and design of an effective structural control system. The Note.-Discussion open until November 1, 2014. This manuscript for this paper was submitted for review and possible publication on January 21, 2013; approved on December 2, 2013. © KSSC and Springer 2014 *Corresponding author Tel: +98-21-77240332; Fax: +98-21-77240398 E-mail: [email protected]

proposed method of simultaneously placing sensors/ actuators has been compared to a commonly used method of placement (Abdullah et al., 2001). In the extant literature, the effect of semi-active controllers’ locations on the control forces and control performance of the structures has been discussed. Two optimization methods have been used for semi-active control. In the first method, all cases have been studied to find the optimal case and several evaluation criteria have been used to do so. The second method was based on a performance index related to each story (Amini and Karagah, 2006). In another case, a multi-start meta-heuristic algorithm called Multiple Start Guided Neighborhood Search (MSGNS) algorithm was presented, which made use of the best features of guided local searches techniques like Simulated Annealing (SA) and Tabu Search (TS). Four distinct design criteria influencing the active control design were considered to study the optimal placement of actuators. The sensitivities of these four optimization criteria were explored with respect to different earthquake records. Furthermore, the usual practice of using shear building (or simple lumped mass) models was implemented in active control research for finding optimal actuator locations (Mohan Rao and Sivasubramanian, 2008a). Another related study optimized the location of piezoelectric actuators and sensors for active vibration control. Genetic algorithm was used to find the optimal

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configurations considering two variables for each piezoelectric device: the location of its center and its orientation (Bruant et al., 2010). In a more recent study, active control of a vibrating beam has been considered by piezoelectric patches. The optimal scheme of this control theory has been formulated with the objective function specified as a weighted quadratic function of the dynamic responses. This function has been minimized at a specified terminal time. The numerical results clarified the effectiveness and the capabilities of piezo actuation to damp out the vibrations (Kucuk et al., 2011). Along with the abovementioned techniques, one of the optimization methods that can be used to deal with control issues is a new population-based technique, named Particle Swarm Optimization (PSO), which is recently proposed as an alternative to the more traditional methods such as Genetic Algorithm (GA). PSO is a relatively recent evolutionary method that is inspired by the behavior of biological populations such as birds flocking. In PSO, despite the traditional operators of GA, each particle is assigned with a velocity. Particles adjust movements based on some deterministic and probabilistic rules using their own experience and those of their companions (Kennedy and Eberhart, 1995). Nowadays, researchers have shown much interest in this evolutionary computation branch due to the simple nature of its scheme and its capability to effectively handle a wide range of optimization problems. One of the first studies in this field introduced an inertia weight parameter into PSO and illustrated its significant impact on the particle swarm optimizer. The algorithm was evaluated by the values of inertia weight tested in several ranges in order to obtain better performance. It is believed that this parameter results in a bigger chance to find the global optimum within a reasonable number of iterations (Shi and Eberhart, 1998). In another case, two evolutionary computation paradigms were compared genetic algorithm and particle swarm optimization. The operators of each paradigm were analyzed by focusing on the way each of them affects the search behavior in the problem space (Eberhart and Shi, 1998). A novel PSO approach was introduced in 2007 for multi-objective optimization, called Time Variant MultiObjective PSO. TV-MOPSO is made adaptive in nature by allowing its vital parameters to change with iterations. The method was compared with some developed multiobjective PSO techniques and the results showed that TVMOPSO was good not only in approximating the Paretooptimal front, but also in terms of diversity of the solutions on the front (Tripathy et al., 2007). In another research study, a multi-objective fuzzy logic controller (PSO-FLC) was presented for active vibration control of seismically excited buildings. Accordingly, a new self-configurable multi-objective particle swarm optimization algorithm was combined with a fuzzy logic controller

(Mohan Rao and Sivasubramanian, 2008b). An optimal fuzzy logic control algorithm was introduced for vibration mitigation of buildings using MagnetoRheological (MR) dampers in another study. The voltage monitoring of the MR dampers was accomplished using an evolutionary fuzzy system, where the fuzzy system was optimized using Evolutionary Algorithms (EAs). A micro-Genetic Algorithm (µ-GA) and a Particle Swarm Optimization (PSO) were used to optimize the FLC parameters. Finally, the study evaluated the performance of the fuzzy controller, optimized off-line, on a threestory building model under seismic excitations (Faruque and Ramaswamy, 2008). In 2010, a new approach was introduced for scheduling reactive power control variables for voltage stability enhancement using particle swarm optimization. The aim of the optimization problem was the maximization of the system’s reactive reserves. The developed algorithm was implemented in 6-bus, 7-line and 25-bus, 35-line standard test systems. The results were finally compared with those obtained using Devidon-Fletcher-Powell’s (Arya et al., 2010). In another research, a particle swarm optimization was subsequently proposed for calculation of the free parameters in active control systems. The fuzzy control was especially considered, which is a suitable tool for the systematic development of active control strategies and can be finetuned if no experience exists or if more complicated control schemes are to be designed. The usage of PSO with a combination of continuous and discrete variables in the optimal design of the controller was proposed. Furthermore, numerical applications on smart piezoelastic beams were presented (Marinaki et al., 2010). As summarized above, all of the mentioned researches works employed some complicated methods in their control schemes. In the present paper however, an active control procedure is employed to decrease the story drift of a steel structure based on a simplified pole assignment method that can result in lower computational cost while maintaining acceptable accuracy. In order to satisfy both the structural demands and the economic considerations, an optimal control system is conducted by incorporation of an optimization algorithm. In addition to the optimal number and locations of the actuators, their minimum motivated forces are sought. Unlike the prior research studies, where traditional methods were used, Particle swarm optimization method is utilized here in order to provide better optimization capabilities. An innovated improved adaptive PSO, which makes use of a geographical neighborhood, is employed. The learning coefficients of the modified PSO are selected in order that the algorithm’s convergence velocity is improved. Furthermore, the same optimization procedure is also performed using GA and the standard PSO in order to make a comparison between the performances and convergence velocities of the algorithms. Finally, structural responses of the

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Optimal Control of Steel Structures by Improved Particle Swarm

uncontrolled and the controlled systems are evaluated by both the common pole assignment method and the simplified one.

2. Simplified Pole Assignment Method Generally, for a Multi Degree Of Freedom (MDOF) structural system subjected to an earthquake excitation and control forces, the equation of motion is found to be as:

·· · ·· [ m ] {x } + [ c ] {x } + k { x } = – [ m ] {xg } – {Uc }

(1)

where [m], [c], and [k] are the mass, damping, and ·· · stiffness matrices, respectively; {x } , {x } and {x} are the acceleration, velocity, and displacement vectors of the structure, respectively; and {Uc} is the control force vector. In a state space description, Eq. 1 can be written as follows:

· ·· {q} = [ A]{q} + {Be}{xg} + [ Bu]{Uc}

(2)

The state vector and the system matrices are given by the following equations:

{q } = A=

x · x

(3)

The gain matrix [F] of order (n×2n) is partitioned as

I

–1

–1

–m k –m c

[ Bu] =

natural frequencies and damping ratios. There are several techniques that can be adopted to shift the poles’ locations to the proper areas in order to improve the structural response under external excitations. The Pole Assignment Method is one of the most widely used approaches and plays a key role in the modern control concepts. In a closed-loop system with the state feedback, the control force vector {Uc} may be defined as the gain matrix [F] multiplied by the state vector {q}.

{Uc } = [ F ] { q }

0

{Be} =

Figure 1. Relationship between system’s poles, frequencies and damping ratios.

0 –I

(4)

where in order to obtain a smooth analytical solution, it is assumed that the parts of the stiffness and the damping are:

0 –m

[ F ] = [ Fk Fc ]

–1

where 0 and I are N-by-N zero and identity matrices for a N-degree of freedom system. The eigenvalues of the system matrix (A) are named as the uncontrolled system poles, which can be plotted in a complex coordinate system as shown in the Fig. 1. Furthermore, natural frequency and damping ratio of each mode can be defined using the real and imaginary components of the poles. The natural frequency is equal to the magnitude of each complex conjugate pole. The related damping ratio can also be calculated as the ratio of the real part of the pole to the natural frequency, which is equal to the sine of the angle (è) between the pole and the positive direction of the imaginary axis. In terms of control theory, a structural system is classified as a stable system when its poles are on the left half area of the complex coordinate system as presented in Fig. 1. Vibration control strategies, either active schemes or passive ones, are generally based on altering the poles’ locations by modifying the structural parameters including

[ Fk] = [ β ][ K'] and [ Fc] = [ α][ C']

(5)

where

β1 … 0 0 [ β] =

[ Κ'] =

α1 … 0 0

β2 0 , [ α ] =

α2 0

0 0 0 βn

0 0 0 αn

K1 … 0 0

C1 … 0 0

K2

0 , [ C'] =

0 0 0 Kn

C2 0 0 0 0

Cn

As is evident from associated equations, [ Κ'] and [ C'] are the simplified diagonal stiffness and damping matrices, where [β] and [α] are the diagonal coefficient matrices. The modal damping coefficient of the ith mode is given by:

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Ci =2ξωi Mi

(6)

where ξ, ωi , and Mi are the damping coefficient, the natural frequency, and the diagonal mass matrices corresponding to the ith mode of vibration, respectively. Based on Eq. 3, the control force {Uc} is given so that the state space formulation and the system matrix of the controlled structure (Acon) take the form

· ·· {q} = [ Acon]{q} + {Be}∗xg , [ Acon] = [ A] + [ Bu][ F] (7) The eigenvalues of the above matrix lead to the new poles of the closed-loop system. The presented simplified method is suggested as an alternative to the conventional pole placement procedure. Generally, for an n-DOF structural system, the direct calculations with 2×n variables (αi, βi) are offered as a substitute for the complex inverse calculations of the gain matrix using 2×n poles of the closed-loop system. Obviously, the new procedure is more feasible than the other one, especially on account of the computational cost required for the optimization algorithms’ iterations.

3. Multi-Objective Optimization Multi-Objective or Multi-Criteria Optimization is the process of simultaneous optimization of a series of conflicting objective functions. The general M-objective minimization problem can be mathematically posed as follows: Given x = [ x1, x2, …, xn] as the optimization variables, optimize objective functions

f(x) = [ fi(x), for i = 1, 2, …, M] subjected to the following inequality and equality constraints:

gj(x) ≤ 0 , for j = 1, 2, …, J and hk(x) = 0 , for k = 1, 2, …, K . A traditional method to satisfy multiple objectives is the so-called weighted-sum optimization. This technique is based on weighting the goals according to their importance and leads to the construction of an overall objective function. The standard optimization methods can be used to find a solution by reformulating the functions in a single objective format. However, the weights are imprecise and the optimization provides no information about semi-optimal solutions. In other words, the most important issue is how the trade-off between the conflicting objectives should be done. Alternatively, Pareto is a distinct method based on finding the multi-criteria solutions that could be partially ordered without making any preference choices a priori. An engineer/economist, Vilfredo Pareto initially presented his relatively simple idea

of optimality, which can be verbally described as follows (Lampinen, 2000): A solution is Pareto optimal if it is dominated by no other feasible solution, which means that there exists no other solution that is superior at least in case of one objective function value and equal or superior with respect to the other objective function values. Unlike single objective optimization, solutions to multiobjective problems are sets of Pareto points instead of a unique point. These points define the Pareto optimal curve i.e. the points with the best possible trade-off between the objectives to be optimized. Accordingly, in this paper a suitable trade-off is tried to be conducted between the objective functions so that its solution would be placed on the optimal curve.

4. Particle Swarm Optimization As an alternative to the traditional evolutionary optimization methods, Eberhart and Kennedy developed a swarm intelligence technique called Particle Swarm Optimization (PSO). It was inspired by social behavior of animals such as flocks of birds or schools of fish. Each particle’s treatment in an n-dimension search space is modeled with its position and velocity. Therefore, a PSO k k k k swarm characteristics are S = {x1, x2, x3, …, xn} and k k k k V = {v1, v2, v3, …, vn} as positions and velocities, respectively. In the above equations k, xi, vi, and n are denoted as iteration number, ith particle’s position, ith particle’s velocity, and the number of particles, respectively. In each iteration, the individuals update their positions based on two parameters representing local and global optimality: k k Pbesti (Pi ) and Gbest k(Gk); The personal and global best solutions which are the best position found, until the last terminated (kth) iteration, respectively by the ith particle and by its group. The position and the velocity of each particle are determined by the following equations: k+1

= ω∗vi + r1∗c1∗(Pi – xi ) + r2∗c2∗(G – xi )

k+1

= xi + v i

vi xi

k

k

k+1

k

k

k

k

(8) (9)

where r1 and r2 are two uniformly distributed random numbers generated independently in the range of [0,1]; c1 and c2 are individual and group learning parameters called acceleration coefficients. The inertia weight (ω) controls the effect of the previous velocity on the new one and can be changed between 0 and 1. As it is shown in Fig. 2, each particle tracks its path toward the new position based on the aforementioned parameters’ values. k k+1 It moves from xi toward xi by (k+1)th velocity vector, which itself contains three main parts. The first one is the k previous velocity ( vi ), multiplied by an adaptive inertia weight (ω). The last two terms are the differences between

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Optimal Control of Steel Structures by Improved Particle Swarm

Figure 2. Particles’ position and velocity update.

the personal and global best solutions and the previous position of the particle. These terms are scaled by the corresponding random and learning parameters.

5. Improved PSO One of the most important issues in evolutionary optimization and convergence is the proper control of the procedure, which depends on the best trade-off between global exploration and local exploitation. On the other hand, it is clear that the best performance of PSO depends on its parameters and the inertia weight, specifically. As noted above, the inertia weight controls the impact of the previous velocity, and therefore, its value manipulates the balance between exploitation and exploration in the optimization process. In order to achieve a proper trade-off to find the optimum solution efficiently, the inertia weight could be adaptively adjusted according to the objective values of the particles. In particular, an adaptive inertia weight factor (AIWF) has been introduced (Liu et al., 2005). In another similar research work, a dynamic inertia weight was discussed, and its performance was evaluated in comparison with the standard PSO. The results showed the effectiveness of the proposed approach (Jiao et al., 2008). In the present study, a novel approach of inertia weight treatment is implemented based on the two aforementioned ideas as follows:

ω = ω' × U

–k

(10)

where

⎧ (ω max – ωmin)(f – fmin) ⎪ωmin + ------------------------------------------------, favg – fmin ω' = ⎨ ⎪ , ⎩ωmax ⎧ (Umax – Umin)(f – fmin) ⎪Umin + -------------------------------------------------, favg – fmin U=⎨ ⎪ , ⎩Umax

f ≤ favg f ≥ favg f ≤ favg f ≥ favg

In the above formulations, U=[1.0001,1.005] and ω'= [0.4,0.9]; f, favg, and fmin are the values for the target function, its average value and its minimum value among all particles at each iteration, respectively; K denotes the current iteration number. Based on the above equations, the inertia weight varies in its range depending on objective function values. This means that for a particle with a low objective value ω decreases and the particle tends to perform exploitation accordingly. On the other hand, high objective values result in increased inertia weight values and more exploration tendency. As noted above, PSO algorithm is based on social interaction and the obtained information circulates among the swarm particles. This procedure in the neighborhood form (NPSO), leads to the development of the individual memories of each particle, the collective memory of the entire swarm, and the collective memories of the neighbors, respectively denoted as Pbest, Gbest, and Rbest. In other words, in NPSO, the performance of each particle depends on the best position in its predefined neighborhood as well as the local and the global best positions. Different types of neighborhood topologies such as Star, Ring, Geographical, and Wheel topology have been developed by different researchers. These neighborhoods affect convergence velocity and accuracy. Accordingly, regardless of the user-defined neighborhood structure, the particles’ velocities are manipulated by the following equation: k+1

vi

k

k

k

= ω × vi + r1 × c1 × (Pbesti – xi ) + r2 × c2 k

k

k

k

(11)

× (Gbesti – xi ) + r3 × c3 × (Rbesti – xi ) where c3 is the neighbor learning coefficient and r3 is a uniform random number in the range of [0,1]. Therefore, the particle moves toward the best position among its neighbors in addition to the global best. This leads to increased convergence velocity and computational efficiency (Tang et al., 2006). In the present paper, the geographical neighborhood is used based on the above equation. In this type of neighborhood, each particle communicates with a limited number of individuals that are located within a certain distance from it. This proposed neighborhood in combination with the presented adaptive inertia weight leads to a novel improved PSO with a convergence velocity suitable for obtaining optimal solutions.

6. Numerical Study In order to evaluate the performance of the proposed algorithm, an optimization procedure is conducted to obtain the optimal values and positions of the active control forces. The design objective is to find the best controller scheme with state feedback, which reduces the maximum structural displacement below a certain value in addition

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Z = 0.55Z1 + 0.4Z2 + 0.05Z3 In this equation, Z1, Z2, and Z3 are the individual parts representing the maximum displacement of the roof, the maximum number of actuators, and the total required driving force, respectively, which are multiplied by suitable scaling coefficients based on Pareto method. In order to consider the effect of the aforementioned parameter values as well as involving probable constraints, the following equations were utilized:

Z1 = Z11(1 + α1(*max(Z11 – 0.16), 0.00)) Z2 = Z22(1 + α2(*max(Z22 – 10), 0.00)) Figure 3. Shear frame model of the 12-story shear frame steel structure.

Figure 4. El Centro earthquake time history.

to optimizing the total control force and the number of controllers. As a simple practical example, a 12-story shear frame steel structure (Fig. 3) is modeled and subjected to the El Centro earthquake ground motion (Fig. 4). Table 1 shows the steel frame’s properties, including the stories’ mass and stiffness. The modal damping ratios of the vibration modes are taken to be equal to 2% of the corresponding critical value. Controlling the structure using active actuators prevents the story displacements from exceeding the maximum allowable value. In order to obtain an optimal control scheme, an objective function is defined incorporating the number and the total force of the used actuators as well as the inadmissible displacement values. In this paper, the cost function for the optimization is defined as follows:

Z3 = Z33(1 + α3(*max(Z33 – 30), 0.00)) The parameters Z11, Z22, and Z33 are the maximum displacement of the roof in meters, the number of actuators, and the total value of the maximum forces of actuator, respectively. α1, α2, and α3 are the corresponding penalty coefficients, which are considered to be equal to 100,000, 6, and 10, respectively, based on the regarded design circumstances. It is clear that higher values of the first coefficient lead to the limitation of the maximum displacement to 0.16m. At the same time, relatively lower values for the other two coefficients might result in their constraints less severely considered i.e. the number of actuators and the maximum total force can exceed their limit values 100 and 300 (KN), respectively. As a first evaluation of the proposed adaptive inertia weight (IPSO), its performance on obtaining the optimal control solution of the sample structure was investigated. The adaptive inertia weight method proposed by Liu (APSO1) and the dynamic inertia weight presented by Jiao (DPSO2) are used to compare with IPSO. Figure 5 shows the objective function values obtained by these three methods versus the iteration number. As illustrated in the convergence procedure of Fig. 5, the proposed method converges to the desirable optimal solution faster than the other ones, especially the dynamic parameter method. It clearly indicates that the algorithm’s performance is effectively improved by combining the two previously proposed approaches. Subsequently, the proposed algorithm, standard PSO, and GA are used to attain the optimal values and positions of minimum required active forces under the same circumstances. The population size is considered to be equal to 350. The parameters c1 and c2 are set as 2 in the PSO while in IPSO the parameter values c1=c2=1.5 and

Table 1. Frame properties Stories

1

2

3

4

5

6

7

8

9

10

11

12

Mass (kN) Stiffness (kN/m)

80 7497

80 7009

80 6347

80 5864

80 5088

80 4654

80 3818

80 3294

80 2478

80 1878

80 1288

80 612.7

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Optimal Control of Steel Structures by Improved Particle Swarm

Figure 5. Convergence procedure of the adaptive algorithms.

Figure 6. Structural displacement.

c3=1 are used. The gain matrix’s coefficients (α, β) are generated in the range of [0, 10]. The optimal solution is obtained based on the simplified pole assignment method. Table 2 shows the aforementioned coefficients for each structure’s stories. Based on the above mentioned cost function and control method, the optimal solution requires the use of six actuators in the 1st, 4th, 5th, 6th, 8th, and 9th story levels for active control of the structure. The optimal value for the total required force of the actuators is found to be 34% of the structure’s total weight. Figure 6 shows the time variation of the maximum structural displacement of the ordinary and the controlled structure under the El Centro strong motion. As illustrated in the figure, the optimal control design can significantly decrease the displacement of structure. The maximum roof displacement is decreased up to 50%, from about 37 centimeters to 16 centimeters, which is defined as the maximum allowable value. This clearly illustrates the efficiency of the control outline implemented for the structure under consideration. It is worth to be noted that the maximum displacement is determined based on the allowable drift in “Iranian Seismic Code”, equal to 1/300:

Figure 7. Roof displacement of the optimally controlled structure.

simplified pole assignment method and the standard one. In addition, a second comparison has been made between the controlled structural displacements of the two methods. As shown in Fig. 7, the results found by the simplified method are in good agreement with those of the standard method, especially in the useful peak values. Finally, in order to analyze the performance of the proposed algorithm in comparison with the standard PSO and GA, the convergence velocities are observed. Fig. 8 shows the objective function values versus the iteration number for all three algorithms. It is illustrated that GA’s convergence velocity is lower than the other ones'. It is also observable that the modified method reaches the optimal solution with less number of iterations in comparison with the standard PSO. The importance of this issue becomes more clear paying attention to the high computational cost of the required iterations.

Dmax =n* hi* (1/300)=12*400/300=16 cm where n is the number of stories and hi is story height. It clearly illustrates the efficiency of the control outline implemented in the mentioned structure. In order to validate the simplified proposed method a comparison has been made between the coefficients (α, β) calculated by the standard and the simplified pole assignment methods. As summarized in Table 2, the results apparently show that the coefficients are the same. Also, the same total force value is obtained by the

Table 2. Alpha and beta coefficients Story

1

2

3

4

5

6

7

8

9

10

11

12

α β

7.212 10

0 0

0 0

0 10

0 10

0 10

0 0

0 10

0 10

0 0

0 0

0 0

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Figure 8. Convergence procedure of the proposed and conventional algorithms.

7. Conclusion This paper presents an Improved Particle Swarm Optimization (IPSO) algorithm to optimize the active control design of steel structures by focusing on their effective parameters. In order to improve the procedure’s accuracy and computational performance, an adaptive inertia weight is introduced. A simplified pole assignment method is used to facilitate the poles’ placement of state feedback control. The optimization procedure is conducted by three methods, namely GA, PSO, and IPSO. The results show that similar gain coefficients and total control force are obtained in the simplified method and the standard pole assignment. Although slight differences can be found among the controlled displacements, the simplified method provides reasonable information, especially in terms of peak values. It is understood from the convergence procedure that the proposed algorithm provides the same optimal solution as PSO. However, the optimal solution obtained by GA is significantly different in terms of objective function and control force values. Moreover, the modified method is more efficient compared to PSO and reaches a desirable solution with less iteration, which results in a significant reduction of total computational cost.

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