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Fuzzy control optimized by PSO for vibration suppression of beams
ARTICLE IN PRESS Control Engineering Practice 18 (2010) 618–629
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Fuzzy control optimized by PSO for vibration suppression of beams Magdalene Marinaki , Yannis Marinakis, Georgios E. Stavroulakis Technical University of Crete, Department of Production Engineering and Management, University Campus, 73100 Chania, Crete, Greece
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 July 2008 Accepted 8 March 2010 Available online 23 March 2010
Smart structures include elements of active, passive or hybrid control. In this paper, particle swarm optimization (PSO) for the calculation of the free parameters in active control systems is proposed and tested. Especially, the fuzzy control is considered, which is a suitable tool for the systematic development of active control strategies and can be fine tuned if no experience exists or if one designs more complicated control schemes. The usage of PSO with a combination of continuous and discrete variables for the optimal design of the controller is proposed. Numerical applications on smart piezoelastic beams are presented. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Particle swarm optimization Active control of structures Fuzzy control Smart structures Nature inspired methods
1. Introduction Nature-inspired approaches represent successful animal and micro-organism team behavior. These methods are used when the task is optimization within complex domains of data or information. One of the most popular nature inspired approaches is particle swarm optimization. Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm that was originally proposed by Kennedy and Eberhart (1995) as a simulation of the social behavior of social organisms such as bird flocking and fish schooling. PSO uses the physical movements of the individuals in the swarm and has a flexible and well-balanced mechanism to enhance and adapt to the global and local exploration abilities. Most applications of PSO have concentrated on the optimization in continuous space while recently, some work has been done for the solution of discrete optimization problems (Kennedy & Eberhart, 1997; Shi & Eberhart, 1998). The wide use of PSO, mainly during the last years, is due to the number of advantages that this method has, compared to other optimization methods. Some of the key advantages are that this optimization method does not need the calculation of derivatives, that the knowledge of good solutions is retained by all particles and that particles in the swarm share information between them. Furthermore, PSO is less sensitive to the nature of the objective function, can be used for stochastic objective functions and can easily escape from local minima. Concerning its implementation, PSO can easily be programmed, has few parameters to regulate and the assessment of the optimum is independent of the initial solution.
In this paper, the particle swarm optimization is used in order to optimize the parameters of a fuzzy control system that is used for the vibration control problem of a flexible structure (smart beam). The results obtained are compared with the results of the fuzzy control system when its parameters are not optimized with PSO. Beams are fundamental elements in many mechanisms and structures (Foutsitzi, Marinova, Hadjigeorgiou, & Stavroulakis, 2003; Kermani, Moallem, & Patel, 2004; Trindade, Benjeddou, & Ohayon, 2001). Therefore, the modeling of the dynamic behavior as well as of the control of beams is a fundamental problem. A smart structure with bonded sensors and actuators as well as an associated control system, which enable the structure to respond to external excitations in such a way that it suppresses undesired effects is considered. The choice of the control technique is important for the design of controllers which ensure the performance of the flexible structure under required conditions and at the same time can be easily applied. It should be mentioned here that the proposed method is quite general and can be used for the design of other smart structures like plates, shells, etc. Results in this direction will be reported in the future. The rest of the paper is organized as follows: In Section 2, the modeling of the smart beams is outlined. In Section 3, an analytical description of the proposed fuzzy control system optimized by PSO is given while in Section 4 the numerical results are presented. In the last section, the conclusions and the proposals for further research are given.
2. Models of smart beams and structures Corresponding author. Tel.: + 30 28210 37282; fax: + 30 28210 69410.
E-mail addresses: [email protected] (M. Marinaki), [email protected] (Y. Marinakis), [email protected] (G.E. Stavroulakis). 0967-0661/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2010.03.001
A smart laminated composite beam with rectangular crosssection having length L, width b, and thickness h is considered
ARTICLE IN PRESS M. Marinaki et al. / Control Engineering Practice 18 (2010) 618–629
(Fig. 1) where the control actuators (thickness hA) and the sensors (thickness hS) are piezoelectric patches symmetrically bonded on the top and the bottom surfaces of the host beam. Both piezoelectric layers are positioned with identical poling directions and can be used as sensors or actuators. The mathematical formulation of the model is based on the shear formulation beam theory (Timosenko theory) and the linear theory of piezoelectricity. Furthermore, quasi-static motion is assumed, which means that the mechanical and electrical forces are balanced at any given instant. The numerical solution of the model is based on the development of superconvergent finite elements using the form of exact solution of the Timoshenko beam theory and Hamilon’s principle (see Aldraihem, Wetherhold, & Sigh, 1997; Foutsitzi et al., 2003; Hadjigeorgiou, Stavroulakis, & Massalas, 2006; Stavroulakis,
619
Foutsitzi, Hadjigeorgiou, Marinova, & Baniotopoulos, 2005; Tairidis, Stavroulakis, Marinova, & Zacharenakis, 2007). The linear constitutive equations of the two coupled fields read: fsg ¼ ½Q ðfeg½dT fEgÞ
ð1Þ
fDg ¼ ½d½Q feg þ ½xfEg
ð2Þ
where fsg61 is the stress vector, feg61 is the strain vector, fDg31 is the electric displacement, fEg31 is the strength of applied electric field acting on the surface of the piezoelectric layer, ½Q 66 is the elastic stiffness matrix, ½d36 is the piezoelectric matrix and ½x33 is the permittivity matrix. Eq. (1) describes the inverse piezoelectric effect (which is exploited for the design of the actuator). Eq. (2) describes the direct piezoelectric effect (which is used for the sensor). Additional assumptions are used for the construction of the simplified model: (a) sensor and actuator (S/A) layers are thin compared with the beam thickness; (b) the polarization direction of the S/A is the thickness direction (z axis); (c) the electric field loading of the S/A is uniform uni-axial in the x-direction; and (d) piezoelectric material is homogeneous, transverse isotropic and elastic. Therefore, the set of equations (1) and (2) is reduced as follows: ( ) " # ( ) ! ex Q11 0 sx d31 ¼ ð3Þ Ez gxz 0 Q55 txz 0 Dz ¼ Q11 d31 ex þ x33 Ez
Fig. 1. Laminated beam with piezoelectric sensors, actuators and the schematic control system.
The electric field intensity Ez can be expressed as Ez ¼
Table 1 Fuzzy inference system rules, e.g. if displacement is far up and velocity is up then control force is max.
FarUp
CloseUp
Equilibrium
CloseDn
FarDn
Max Med + High +
Med+ Low+ Null
Low Null Low+
Null Low Med
High Med Min
V hA
ð5Þ
where V is the applied voltage across the thickness direction of the actuator and hA is the thickness of the actuator layer. Since only strains produced by the host beam act on the sensor layer and no electric field is applied to it, the output charge from the sensor can be calculated using Eq. (4). The charge measured through the electrodes of the sensor is given by 9 8 ! ! Z Z = 1< ð6Þ Dz dS þ Dz dS qðtÞ ¼ ; 2 : Sef Sef z ¼ h=2
z ¼ h=2 þ hS
Input 1 MF Degree of membership
Up Null Down
Displacement
CloseL Equilibrium CloseR
FarL
1
FarR
0.8 0.6 0.4 0.2 0 −4
Degree of membership
Velocity
ð4Þ
1
−3
−2
−1
Left
0 1 Displacement Input 2 MF
2
Null
3
4 x 10−4
Right
0.8 0.6 0.4 0.2 0 −8
−6
−4
−2
0 Velocity
2
4
Fig. 2. Membership functions for the inputs of the classic fuzzy controller.
6
8 x 10−3
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where Sef is the effective surface of the electrode placed on the sensor layer. The current on the surface of the sensor is given by iðtÞ ¼
dqðtÞ dt
based on the Timoshenko beam theory which, in turn, means that the axial displacement is proportional to z and to the rotation cðx,tÞ of the beam cross section about the positive y-axis and that the transverse displacement is equal to the transverse displacement w(x,t) of the point of the centroidal axis (y¼z ¼0). The strain–displacement relationships read:
ð7Þ
The current is converted into open-circuit sensor voltage output by S
V ¼ GS iðtÞ
ex ¼ z
ð8Þ
Degree of membership
exz ¼ c þ
@w @x
ð9Þ
The kinetic energy of the beam with the layers can be expressed as
where GS is the gain of the current amplifier. Furthermore, it is supposed that the bending–torsion coupling and the axial vibration of the beam centerline are negligible and that the components of the displacement field fug of the beam are
T¼
1 2
Z
_ T fug _ dV ¼ rfug
V
b 2
Z
Input 1 MF CloseL Equilibrium CloseR
FarL
1
@c , @x
0
L
Z
h=2 þ hS
r½ðzc_ Þ2 þ w_ 2 dz dx
h=2hA
FarR
0.8 0.6 0.4 0.2 0 −6
−4
−2
0
2
4
6 x 10−4
Displacement Degree of membership
Input 2 MF Left
1
Null
Right
0.8 0.6 0.4 0.2 0 −3
−2
−1
0 Velocity
1
2
3 x 10−4
Fig. 3. Membership functions for the inputs of the fuzzy controller optimized by PSO.
Degree of membership
Input 1 MF FarL
1
CloseL
Equilibrium CloseR
FarR
0.8 0.6 0.4 0.2 0 −2
−1
0
1
2 x 10−4
Displacement Degree of membership
Input 2 MF Left
1
Null
Right
0.8 0.6 0.4 0.2 0 −3
−2
−1
0
1
2
Velocity Fig. 4. Membership functions for the inputs of the fuzzy controller optimized by GA.
3 x 10−4
ð10Þ
ARTICLE IN PRESS M. Marinaki et al. / Control Engineering Practice 18 (2010) 618–629
on the assumption that the host beam and the piezoelectric patches have identical densities. The strain (potential) energy is given by h=2 þ hS
h=2hA
"
# 2 @c @w 2 Q11 z þ Q55 c þ dz dx @x @x
ð11Þ If the only loading consists of moments induced by the piezoelectric actuators and since the structure has no bending–twisting couple then the first variation of the work has the form: Z L @c dx ð12Þ dW ¼ b MA d @x 0 where d is the first variation operator and MA is the moment per unit length induced by the actuator layer and is given by Z h=2 Z h=2 V ð13Þ MA ¼ zsAx dz ¼ zQ 11 d31 EAz dz EAz ¼ A hA h=2hA h=2hA
MX€ þ LX_ þ KX ¼ Fm þ Fe
ð14Þ
where M and K are the generalized mass and stiffness matrices, Fe is the generalized control force vector produced by electromechanical coupling effects, L is the viscous damping matrix and Fm is the external loading vector. It should be mentioned here that technical bending theories for plates can be constructed analogously.
0
ax
−0.2
M
−L ow
1
H ig h− M ed
M
in
Output MF
Degree of membership
0.8
0.6
0.4
0.2
0 −1
−0.8
−0.6
−0.4
0.2
0.4
0.6
0.8
1
Control Force Fig. 5. Membership functions for the output of the classic fuzzy controller.
+
M ax
ed
− Lo w − N ul l Lo w +
1
M
H
M
in
ig h−
Output MF h+
Z
ig
0
L
H
Z
−N ul l Lo w + M ed + H ig h+
b feg fsg dV ¼ 2 V T
ed
Z
Using Hamilton’s principle the equations of motion of the beam are derived. For the finite element discretization beam, finite elements are used with two degrees of freedom (d.o.fs) at each node: the transversal deflection wi and the rotation ci . These are gathered to form the degrees of freedom vector Xi ¼ ½wi ci . After assembling the mass and stiffness matrices for all elements, the equation of motion is obtained in the form:
M
1 U¼ 2
621
Degree of membership
0.8
0.6
0.4
0.2
0 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Control Force Fig. 6. Membership functions for the output of the fuzzy controller optimized by PSO.
0.5
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The main objective is to design control laws for the smart beam bonded with piezoelectric S/A subjected to external induced vibrations. For this purpose, a fuzzy control system optimized by
PSO is applied in this paper. A two-input, single-output fuzzy inference controller is tested. This configuration is suitable for a feedback control force based on the displacement and velocity at a
M ax
H ig h
M in
1
Degree of membership
− M ed Lo − w N − u Lo ll w M + ed + H ig h+
Output MF
0.8 0.6 0.4 0.2 0 −0.6
−0.4
−0.2
0
0.2
0.4
0.6
Control Force Fig. 7. Membership functions for the output of the fuzzy controller optimized by GA.
1
1.5
0.5
1
0
0.5 Force
Force
Excitation (blue−−), Control force (red−.) and Control force rate (black−)
Excitation (blue−−), Control force (red−.) and Control force rate (black−)
−0.5
0
−1
−0.5
−1.5
−1 −1.5
−2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0.2
0.4
Time (sec)
0.6
0.8
1
Time (sec) Excitation (blue−−), Control force (red−.) and Control force rate (black−)
1.5 1
Force
0.5 0 −0.5 −1 −1.5 0
0.2
0.4
0.6
0.8
1
Time (sec) Fig. 8. Control force at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
ARTICLE IN PRESS M. Marinaki et al. / Control Engineering Practice 18 (2010) 618–629
x 10
Displacement before (blue) and after (red) fuzzy 2.5
2
2
1.5
1.5
Displacement (m)
Displacement (m)
2.5
−4
1 0.5 0 −0.5
Displacement before (blue) and after (red) fuzzy
1 0.5 0 −0.5
−1
−1
−1.5
−1.5
−2
x 10
−4
623
−2 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Time (sec)
Time (sec) Displacement before (blue) and after (red) fuzzy
−4
2.5
x 10
2 Displacement (m)
1.5 1 0.5 0 −0.5 −1 −1.5 −2 0
0.2
0.4 0.6 Time (sec)
0.8
1
Fig. 9. Displacement at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
given point (collocated configuration of the local controller). If needed, several independent and decentralized (local) fuzzy controllers, in general with different characteristics, can be installed at the various points of a larger structure.
3. Fuzzy control system optimized by particle swarm optimization 3.1. Fuzzy control Fuzzy systems (Driankov, Hellendoorn, & Reinfrak, 1996) can be applied in many fields and can solve different kinds of problems in various application domains. Fuzzy inference rules systematize existing experience and can be used for the mathematical formulation of nonlinear controllers. The feedback is based on fuzzy inference and may be nonlinear and complicated. Knowledge or experience on the controlled system is required for the application of this technique. In the area of smart systems the application of neuro-fuzzy control has been adopted by many authors. In particular, it seems to be suitable for the control of structures with complicated or nonlinear characteristics, like the tuned mass damper systems for aseismic design
(Pourzeynali, Lavasani, & Modarayi, 2007; Wang & Lin, 2007) or the semi-active control using active friction devices or electrorheological fluid dampers (Reigles & Symans, 2006). In order to design a fuzzy control system (construct the control rules, state the membership functions, tune the parameters, etc.) various approaches have been proposed. In the last years, there is an increasing interest to optimize the fuzzy control systems with heuristic, metaheuristic, and nature inspired approaches. Mainly, genetic algorithms have been used for these purposes while nature inspire approaches, like PSO, have not been used so extensively. For example in Belarbi and Titel (2000), a method for designing fuzzy logic controllers with symmetric partitioning of the universe of discourse is presented where both the rule base and membership functions of the input and output variables are designed optimally using a genetic algorithm. In Chou (2006), a genetic algorithm based optimal fuzzy controller design is proposed where the design procedure is accomplished by establishing an index function as the consequent part of the fuzzy control rule. An efficient genetic reinforcement learning algorithm for designing fuzzy controllers is proposed in Juang, Lin, and Lin (2000). In Poursamad and Montazeri (2008) a geneticfuzzy control strategy for parallel hybrid electric vehicles (HEVs) is presented in which the genetic-fuzzy control strategy is a fuzzy
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Displacement Velocity before (blue) and after (red) fuzzy
x 10−3
5
5
4
4 Displacement Velocity (m/sec)
Displacement Velocity (m/sec)
x 10−3
3 2 1 0 −1 −2 −3 −4 −5
Displacement Velocity before (blue) and after (red) fuzzy
3 2 1 0 −1 −2 −3 −4 −5
0
0.2
0.4
0.6
0.8
1
0
0.2
Time (sec)
5
0.4
0.6
0.8
1
Time (sec)
x
10−3
Displacement Velocity before (blue) and after (red) fuzzy
Displacement Velocity (m/sec)
4 3 2 1 0 −1 −2 −3 −4 −5 0
0.2
0.4
0.6
0.8
1
Time (sec) Fig. 10. Displacement velocity at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
logic controller that is tuned by a genetic algorithm. In Lin (2004) the proposed algorithm combines the genetic algorithm (GA) and the least-squares estimate (LSE) method to construct the genetic algorithm-based neural fuzzy system for temperature control. An extensive analysis of the genetic fuzzy systems can be found in Cordo´n, Gomide, Herrera, Hoffmann, and Magdalena (2004). Welch and Venayagamoorthy (2007) presents a particle swarm optimization method for optimizing a fuzzy logic controller for a photovoltaic grid independent system consisting of a PV collector array, a storage battery, and loads (critical and non-critical loads). A learning approach based on particle swarm optimization for the determination of the consequent parameters and premise parameters of a Takagi and Sugeno type fuzzy model is formulated and explained in Niu, Zhu, He, and Shen (2008). Youssef, Yousef, Sebakhy, and Wahba (2009) introduces the use of the adaptive particle swarm optimization (APSO) for adapting the weights of fuzzy neural networks. In Juang and Lo (2008) proposes a zeroorder Takagi–Sugeno–Kang (TSK)-type fuzzy system learning using a two-phase swarm intelligence algorithm (TPSIA) where the first phase of TPSIA learns fuzzy system structure and parameters by on-line clustering-aided ant colony optimization and phase two aims to further optimize all of the free parameters
in the fuzzy system using particle swarm optimization. In Zamani, Karimi-Ghartemani, Sadati, and Parniani (2009), particle swarm optimization algorithm is used to carry out the design procedure of fractional order PID (FOPID) controller.
3.2. Fuzzy control system for vibration control of smart structures In order to reduce the displacement field of the cantilever beam system, a nonlinear fuzzy controller (Tairidis et al., 2007) was constructed by using the Fuzzy Toolbox of Matlab. More specifically, a Mamdani-type Fuzzy Inference System, consisted of two inputs and one output, was developed. The system receives as _ while gives as inputs the displacement (u) and the velocity ðuÞ, output the increment of the control force (z). Triangular and trapezoidal shape membership functions were chosen both for inputs and output. In order to describe the present system-controller, 15 rules were used. All rules have weights equal to 1 and use the AND-type logical operator. These rules are presented in the Table 1. The implication method was set to minimum (min), while the aggregation method was set to maximum (max). The defuzzified
ARTICLE IN PRESS M. Marinaki et al. / Control Engineering Practice 18 (2010) 618–629
x
Rotation without (blue−−) and with (red−) fuzzy control 4
3
3
2
2 Rotation (rad)
Rotation (rad)
4
10−4
1 0 −1
Rotation without (blue−−) and with (red−) fuzzy control
1 0 −1
−2
−2
−3
−3
−4
x
10−4
625
−4 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
Time (sec)
0.6
0.8
1
Time (sec)
4
x
10−4
Rotation without (blue−−) and with (red−) fuzzy control
3
Rotation (rad)
2 1 0 −1 −2 −3 −4 0
0.2
0.4 0.6 Time (sec)
0.8
1
Fig. 11. Rotation at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
output value has been created by using the MOM (mean of maximum) defuzzification method.
_ Both the control force and the (displacement u and velocity u). external loads provide the next step’s ðt þ DtÞ values of displacement and velocity.
3.3. Structural dynamics and fuzzy control The Houmbolt numerical integration algorithm was chosen for the solution of the equations of motion (Eq. (14)) (Tairidis et al., 2007). According to this method, and on the assumption that the acceleration is constant within each time integration step, the Houmbolt factors are set equal to
b ¼ 0:25, g ¼ 0:5
ð15Þ
The total integration time for the simulation was chosen equal to 1 s while the time step ðDtÞ was chosen equal to 0.001 s. The integration constants are c1 ¼ c4 ¼
1
bðDtÞ2
g bDt
,
,
1
c2 ¼
c5 ¼
g b
1 c3 ¼ , 2b g c6 ¼ D t 1 2b
bDt ,
,
ð16Þ
In each step (t) of the numerical integration, the fuzzy controller provides a control force (z), according to the given input values
3.4. Fuzzy control system optimized by particle swarm optimization for the vibration control of smart structures Some parameters of the fuzzy control system described in Section 3.2 were optimized by particle swarm optimization. More precisely, the parameters (the break points) of the triangular and trapezoidal membership functions, the weights of the rules, and the logical operator are considered as variables. First, a number (swarm) of particles is randomly initialized, where a particle is a solution to the problem. In each particle, the first values correspond to the parameters of the membership functions and can take continuous values, the next 15 values correspond to the weights of the rules and can take continuous values in the range [0,1] while the last 15 values correspond to the AND/OR-type logical operator and can take discrete values equal to 1 for AND and 2 for OR. It should be noted that in PSO implementations usually only continuous or discrete values exist. In this paper, there is a combination of continuous and discrete values in each particle and in order to solve the problem a combination of the
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Rotation Velocity without (blue−−) and with (red−) fuzzy control
x 10−3
8
8
6
6
Rotation Velocity (rad/sec)
Rotation Velocity (rad/sec)
x 10−3
4 2 0 −2 −4 −6
Rotation Velocity without (blue−−) and with (red−) fuzzy control
4 2 0 −2 −4 −6
−8 −10
−8 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.2
0.4
0.6
0.8
1
Time (sec)
Time (sec)
x 10−3
Rotation Velocity without (blue−−) and with (red−) fuzzy control
0
0.2
8
Rotation Velocity (rad/sec)
6 4 2 0 −2 −4 −6 −8 0.4 0.6 Time (sec)
0.8
1
Fig. 12. Rotation velocity at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
equations used for finding the new position of the particle for continuous and discrete problems is used. More precisely, the position of each particle is represented by a d-dimensional vector in problem space si ¼ (si1, si2,y, sid), i ¼ 1,2,y, N (N is the population size, d is set equal to the values’ parameters plus the values of weights plus the values of the logical type operator). The fitness function that has to be minimized is the following error function:
previous best position and the previous best position attained by any particle of the swarm, namely pid and pgd. The velocities and positions of particles are updated using the following formulas. The formula used for the calculation of velocities is the same for continuous and discrete values while different formulas are used for the calculation of the positions of the particles for continuous and discrete values:
err ¼ JXX J
(for the values of the particle that are continuous) and
ð17Þ
where X is the L2 norm in Rm (m is the total number of the assumed d.o.fs), X ¼ [X1,y, Xm] is the nodal displacements and rotations array and X ¼ ½X1 , . . . , Xm is the wished value of the nodal displacements and rotations array (it is set equal to zero). Thus, each particle is randomly placed in the d-dimensional space as a candidate solution. The velocity of the i-th particle vi ¼ (vi1, vi2,y, vid) is defined as the change of its position. The flying direction of each particle is the dynamical interaction between the individual and the social flying experience. The algorithm completes the optimization through following the personal best solution of each particle and the global best value of the whole swarm. Each particle adjusts its trajectory towards its own
vid ðt þ1Þ ¼ vid ðtÞ þe1 rand1ðpid sid ðtÞÞ þ e2 rand2ðpgd sid ðtÞÞ
ð18Þ
sid ðt þ 1Þ ¼ sid ðtÞ þ vid ðt þ 1Þ
ð19Þ
sigðvid Þ ¼
1 1þ expðvid Þ (
sid ðt þ 1Þ ¼
1
if rand3 osigðvid Þ
0
if rand3 Zsigðvid Þ
ð20Þ
ð21Þ
(for the values of the particle that are discrete) where t is the iteration counter; e1 and e2 are the acceleration coefficients; rand1, rand2, rand3 are three random numbers in [0, 1]. The acceleration coefficients e1 and e2 control how far a particle will move in a single iteration. Low values allow particles to roam far from target regions before being tugged back, while high values result in abrupt movement towards, or past, target regions. Typically, these are both
ARTICLE IN PRESS M. Marinaki et al. / Control Engineering Practice 18 (2010) 618–629
x 10−4
627
Displacement for different loadings
2.5 2
Displacement (m)
1.5 1 0.5 0 −0.5 −1 −1.5 −2 0
0.1
0.2
0.3
0.4
without fuzzy control fuzzy control − loading: sin (20t) fuzzy control − loading: sin (10t) fuzzy control − loading: sin (5t)
0.5
0.6
0.7
0.8
0.9
1
0.8
0.9
1
Time (sec)
Rotation Velocity for different loadings
x 10−3 8
Rotation Velocity (rad/sec)
6 4 2 0 −2 −4 −6 −8 0
0.1
0.2
0.3
0.4
without fuzzy control fuzzy control − loading: sin (20t) fuzzy control − loading: sin (10t) fuzzy control − loading: sin (5t)
0.5
0.6
0.7
Time (sec)
Fig. 13. Displacement (a) and rotation velocity (b) at the free end for fuzzy control optimized by PSO for different loadings.
set equal to a value of 2.0, although assigning different values to e1 and e2 sometimes leads to improved performance. It should be noted that in the discrete space, a particle moves in a state space restricted to zero and one on each dimension where each vi represents the probability of bit si taking the value 1. Thus, the particles’ trajectories are defined as the changes in the probability and vid is a measure of individual’s current probability of taking 1. If the velocity is higher it is more likely to choose 1, and lower values favor choosing 0. The sigmoid function (Eq. (20)) is applied to transform the velocity from real number space to the probability space.
4. Numerical results The problem of a cantilever beam is studied in the present paper. The beam has a total length equal to 0.8 m and the structure has been discretized with four finite elements resulting in a model
with eight degrees of freedom. A dynamic loading is used as disturbance that simulates a strong wind and is a periodic sinusoidal loading pressure (sin(20t)) that influences the displacement of the second and the fourth (free end) finite elements. The purpose of the fuzzy controller is to reduce the oscillation. The controller is collocated, takes as input the displacement and the velocity of the free end and gives back the control force to be applied at the same point. The results obtained with the fuzzy controller optimized by PSO are compared with those of the classic fuzzy controller (without any optimization procedure). The results of the classic fuzzy controller have been compared with the results that arose from classical control (LQR) in Tairidis et al. (2007) giving, with suitably chosen parameters for the fuzzy controller, comparable reduction of the vibrations with the LQR control by using fewer inputs (two instead of sixteen measurements in this example) and less effective reduction of the velocities. The parameters of the PSO algorithm are selected after thorough
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testing. These parameters are: the number of swarms is set equal to 1, the number of particles is equal to 15, the number of generations is equal to 20 and the parameters e1 and e2 are both set equal to 2. The whole algorithmic approach was implemented in Matlab on a Centrino Mobile Intel Pentium M 750 at 1.86 GHz. In order to see the effectiveness of the proposed PSO based fuzzy controller, it is compared with another stochastic population-based algorithm, a fuzzy controller optimized by a genetic algorithm. Genetic algorithms (GAs) (Holland, 1975) are search procedures based on the mechanics of natural selection and natural genetics. A GA is a stochastic iterative procedure that maintains the population size constant in each iteration, called a generation. Their basic operation is the mating of two solutions in order to form a new solution. To form a new population, a binary operator called crossover, and a unary operator, called mutation are used (Engelbrecht, 2007; Goldberg, 1989). In genetic algorithm implementation, the parameters were chosen in such a way that in all algorithms the same number of function evaluations are needed. Thus, the parameters for the GA are: the number of individuals is equal to 15, the number of generations is equal to 20, the mutation rate is set equal to 0.5 and, finally, 1-point crossover was selected with crossover rate equal to 0.8. The load, the displacement, the displacement velocity, the rotation and the rotation velocity, for the three kinds of control (classic fuzzy control, fuzzy control optimized by PSO and fuzzy control optimized by GA), are presented in the graphs below. The values without control are shown with a blue dashed line. The membership functions obtained by the fuzzy controller without any optimization procedure (Figs. 2 and 5), when optimized by PSO (Figs. 3 and 6) and when optimized by GA (Figs. 4 and 7) are presented. Fig. 8 presents the control force at the free end for classic fuzzy controller, for the fuzzy controller optimized by PSO and for the fuzzy controller optimized by GA. As it can be seen, the fuzzy controllers optimized by PSO and by GA give superior control results compared to the classical fuzzy controller. More precisely, the displacements (Fig. 9) and the rotations (Fig. 11) are almost zero in the case of the fuzzy controller optimized by PSO and in the case of the fuzzy controller optimized by GA. The displacement (Fig. 10) and rotation (Fig. 12) velocities have significantly smaller values for the fuzzy controller optimized by PSO and for fuzzy controller optimized by GA compared with the ones obtained by the classical fuzzy controller. It should be noted that the fuzzy controller optimized by PSO gives better control results than the fuzzy controller optimized by GA, as the displacement velocities (Fig. 10(b)) and the rotation velocities (Fig. 12(b)) obtained by the fuzzy controller optimized by PSO have smaller values compared with the ones of the fuzzy controller optimized by GA. Of course, as the differences in the results are small, it can be said that both methods, PSO and GA, can be applied with very good results in order to optimize the fuzzy controller used for the vibration control of beams. Furthermore, in order to investigate the robustness of the fuzzy control optimized by PSO and its ability to work for different loading, the controller gain calculated by the PSO based on one specific sinusoidal loading is used for different sinusoidal loading pressures. More specifically, the controller gain is calculated from the PSO algorithm when the loading is equal to sin(20t) and this controller gain is used when the loadings are equal to sin(10t) and sin(5t). As it can be seen from Fig. 13 the results obtained are very efficient and the controller gain obtained by using one loading can be applied successfully when other loadings are used.
5. Conclusion In this paper, a fuzzy control system optimized by particle swarm optimization (PSO) was used for the vibration control of
beams with piezoelectric sensors and actuators. The parameters of the fuzzy controller where selected optimally by using the PSO algorithm. The results obtained for a sinusoidal loading pressure using the fuzzy controller system optimized by PSO are very efficient. A comparison of the proposed method with a classic fuzzy controller for this specific problem revealed the high performance of the proposed method. Future research is intended to be focused in the application of the proposed algorithm for the design of other smart structures like plates, shells and in the modification of the method in order to be used for optimally design other parts of the fuzzy system like the definition of the control rules. References Aldraihem, O. J., Wetherhold, R. C., & Sigh, T. (1997). Distributed control of laminated beams: Timoshenko theory vs. Euler–Bernoulli theory. Journal of Intelligent Mathematical Systems and Structures, 8, 149–157. Belarbi, K., & Titel, F. (2000). Genetic algorithm for the design of a class of fuzzy controllers: An alternative approach. IEEE Transactions on Fuzzy Systems, 8(4), 398–405. Chou, C. H. (2006). Genetic algorithm-based optimal fuzzy controller design in the linguistic space. IEEE Transactions on Fuzzy Systems, 14(3), 372–385. Cordo´n, O., Gomide, F., Herrera, F., Hoffmann, F., & Magdalena, L. (2004). Ten years of genetic fuzzy systems: Current framework and new trends. Fuzzy Sets and Systems, 141(1), 5–31. Driankov, D., Hellendoorn, H., & Reinfrak, M. (1996). An introduction to fuzzy control (2nd ed.). New York: Springer Verlag. Engelbrecht, A. P. (2007). Computational intelligence: An introduction. England: John Wiley and Sons. Foutsitzi, G., Marinova, D. G., Hadjigeorgiou, E., & Stavroulakis, G. E. (2003). Robust H2 vibration control of beams with piezoelectric sensors and actuators. International Conference Physics and Control, 1, 157–162. Goldberg, D. E. (1989). Genetic algorithms in search, optimization, and machine learning. Massachusetts: Addison-Wesley Publishing Company, INC. Hadjigeorgiou, E. P., Stavroulakis, G. E., & Massalas, C. V. (2006). Shape control and damage identification of beams using piezoelectric actuation and genetic optimization. International Journal of Engineering Science, 44, 409–421. Holland, J. H. (1975). Adaptation in natural and artificial systems. Ann Arbor, MI: University of Michigan Press. Juang, C. F., Lin, J. Y., & Lin, C. T. (2000). Genetic reinforcement learning through symbiotic evolution for fuzzy controller design. IEEE Transactions on Systems Man and Cybernetics, Part B: Cybernetics, 30(2), 290–302. Juang, C. F., & Lo, C. (2008). Zero-order TSK-type fuzzy system learning using a two-phase swarm intelligence algorithm. Fuzzy Sets and Systems, 159(21), 2910–2926. Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. Proceedings of 1995 IEEE International Conference on Neural Networks, 4, 1942–1948. Kennedy, J., & Eberhart, R. (1997). A discrete binary version of the particle swarm algorithm. In Proceedings of 1997 IEEE international conference on systems, man, and cybernetics (Vol. 5, pp. 4104–4108). Kermani, M. R., Moallem, M., & Patel, R. V. (2004). Parameter selection and control design for vibration suppression using piezoelectric transducers. Control Engineering Practice, 12(8), 1005–1015. Lin, C. J. (2004). A GA-based neural fuzzy system for temperature control. Fuzzy Sets and Systems, 134, 311–333. Niu, B., Zhu, Y., He, X., & Shen, H. (2008). A multi-swarm optimizer based fuzzy modeling approach for dynamic systems processing. Neurocomputing, 71, 1436–1448. Poursamad, A., & Montazeri, M. (2008). Design of genetic-fuzzy control strategy for parallel hybrid electric vehicles. Control Engineering Practice, 16, 861–873. Pourzeynali, S., Lavasani, H. H., & Modarayi, A. H. (2007). Active control of high rise building structures using fuzzy logic and genetic algorithms. Engineering Structures, 29, 346–357. Reigles, D. G., & Symans, M. D. (2006). Supervisory fuzzy control of a base isolated benchmark building utilizing a neuro-fuzzy model of controllable fluid viscous dampers. Journal of Structural Control and Health Monitoring, 13, 724–747. Shi, Y., & Eberhart, R. (1998). A modified particle swarm optimizer. In Proceedings of 1998 IEEE world congress on computational intelligence (pp. 69–73). Stavroulakis, G. E., Foutsitzi, G., Hadjigeorgiou, E., Marinova, D. G., & Baniotopoulos, C. C. (2005). Design and robust optimal control of smart beams with application on vibrations suppression. Advances in Engineering Software, 36, 806–813. Tairidis, G. K., Stavroulakis, G. E., Marinova, D. G., & Zacharenakis, E. C. (2007). Classical and soft robust active control of smart beams. In M. Papadrakakis, D. C. Charmpis, N. D. Lagaros, & Y. Tsompanakis (Eds.), ECCOMAS thematic conference on computational methods in structural dynamics and earthquake engineering, Rethymno, Crete, Greece. Trindade, M. A., Benjeddou, A., & Ohayon, R. (2001). Piezoelectric active vibration control of damped sandwich beams. Journal of Sound and Vibration, 246(4), 653–677.
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Wang, A.-P., & Lin, Y.-H. (2007). Vibration control of a tall building subjected to earthquake excitation. Journal of Sound and Vibration, 299, 757–773. Welch, R., & Venayagamoorthy, G. K. (2007). A fuzzy-PSO based controller for a grid independent photovoltaic system. In Proceedings of the 2007 IEEE swarm intelligence symposium (SIS 2007).
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Contents lists available at ScienceDirect
Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Fuzzy control optimized by PSO for vibration suppression of beams Magdalene Marinaki , Yannis Marinakis, Georgios E. Stavroulakis Technical University of Crete, Department of Production Engineering and Management, University Campus, 73100 Chania, Crete, Greece
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 July 2008 Accepted 8 March 2010 Available online 23 March 2010
Smart structures include elements of active, passive or hybrid control. In this paper, particle swarm optimization (PSO) for the calculation of the free parameters in active control systems is proposed and tested. Especially, the fuzzy control is considered, which is a suitable tool for the systematic development of active control strategies and can be fine tuned if no experience exists or if one designs more complicated control schemes. The usage of PSO with a combination of continuous and discrete variables for the optimal design of the controller is proposed. Numerical applications on smart piezoelastic beams are presented. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Particle swarm optimization Active control of structures Fuzzy control Smart structures Nature inspired methods
1. Introduction Nature-inspired approaches represent successful animal and micro-organism team behavior. These methods are used when the task is optimization within complex domains of data or information. One of the most popular nature inspired approaches is particle swarm optimization. Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm that was originally proposed by Kennedy and Eberhart (1995) as a simulation of the social behavior of social organisms such as bird flocking and fish schooling. PSO uses the physical movements of the individuals in the swarm and has a flexible and well-balanced mechanism to enhance and adapt to the global and local exploration abilities. Most applications of PSO have concentrated on the optimization in continuous space while recently, some work has been done for the solution of discrete optimization problems (Kennedy & Eberhart, 1997; Shi & Eberhart, 1998). The wide use of PSO, mainly during the last years, is due to the number of advantages that this method has, compared to other optimization methods. Some of the key advantages are that this optimization method does not need the calculation of derivatives, that the knowledge of good solutions is retained by all particles and that particles in the swarm share information between them. Furthermore, PSO is less sensitive to the nature of the objective function, can be used for stochastic objective functions and can easily escape from local minima. Concerning its implementation, PSO can easily be programmed, has few parameters to regulate and the assessment of the optimum is independent of the initial solution.
In this paper, the particle swarm optimization is used in order to optimize the parameters of a fuzzy control system that is used for the vibration control problem of a flexible structure (smart beam). The results obtained are compared with the results of the fuzzy control system when its parameters are not optimized with PSO. Beams are fundamental elements in many mechanisms and structures (Foutsitzi, Marinova, Hadjigeorgiou, & Stavroulakis, 2003; Kermani, Moallem, & Patel, 2004; Trindade, Benjeddou, & Ohayon, 2001). Therefore, the modeling of the dynamic behavior as well as of the control of beams is a fundamental problem. A smart structure with bonded sensors and actuators as well as an associated control system, which enable the structure to respond to external excitations in such a way that it suppresses undesired effects is considered. The choice of the control technique is important for the design of controllers which ensure the performance of the flexible structure under required conditions and at the same time can be easily applied. It should be mentioned here that the proposed method is quite general and can be used for the design of other smart structures like plates, shells, etc. Results in this direction will be reported in the future. The rest of the paper is organized as follows: In Section 2, the modeling of the smart beams is outlined. In Section 3, an analytical description of the proposed fuzzy control system optimized by PSO is given while in Section 4 the numerical results are presented. In the last section, the conclusions and the proposals for further research are given.
2. Models of smart beams and structures Corresponding author. Tel.: + 30 28210 37282; fax: + 30 28210 69410.
E-mail addresses: [email protected] (M. Marinaki), [email protected] (Y. Marinakis), [email protected] (G.E. Stavroulakis). 0967-0661/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2010.03.001
A smart laminated composite beam with rectangular crosssection having length L, width b, and thickness h is considered
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(Fig. 1) where the control actuators (thickness hA) and the sensors (thickness hS) are piezoelectric patches symmetrically bonded on the top and the bottom surfaces of the host beam. Both piezoelectric layers are positioned with identical poling directions and can be used as sensors or actuators. The mathematical formulation of the model is based on the shear formulation beam theory (Timosenko theory) and the linear theory of piezoelectricity. Furthermore, quasi-static motion is assumed, which means that the mechanical and electrical forces are balanced at any given instant. The numerical solution of the model is based on the development of superconvergent finite elements using the form of exact solution of the Timoshenko beam theory and Hamilon’s principle (see Aldraihem, Wetherhold, & Sigh, 1997; Foutsitzi et al., 2003; Hadjigeorgiou, Stavroulakis, & Massalas, 2006; Stavroulakis,
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Foutsitzi, Hadjigeorgiou, Marinova, & Baniotopoulos, 2005; Tairidis, Stavroulakis, Marinova, & Zacharenakis, 2007). The linear constitutive equations of the two coupled fields read: fsg ¼ ½Q ðfeg½dT fEgÞ
ð1Þ
fDg ¼ ½d½Q feg þ ½xfEg
ð2Þ
where fsg61 is the stress vector, feg61 is the strain vector, fDg31 is the electric displacement, fEg31 is the strength of applied electric field acting on the surface of the piezoelectric layer, ½Q 66 is the elastic stiffness matrix, ½d36 is the piezoelectric matrix and ½x33 is the permittivity matrix. Eq. (1) describes the inverse piezoelectric effect (which is exploited for the design of the actuator). Eq. (2) describes the direct piezoelectric effect (which is used for the sensor). Additional assumptions are used for the construction of the simplified model: (a) sensor and actuator (S/A) layers are thin compared with the beam thickness; (b) the polarization direction of the S/A is the thickness direction (z axis); (c) the electric field loading of the S/A is uniform uni-axial in the x-direction; and (d) piezoelectric material is homogeneous, transverse isotropic and elastic. Therefore, the set of equations (1) and (2) is reduced as follows: ( ) " # ( ) ! ex Q11 0 sx d31 ¼ ð3Þ Ez gxz 0 Q55 txz 0 Dz ¼ Q11 d31 ex þ x33 Ez
Fig. 1. Laminated beam with piezoelectric sensors, actuators and the schematic control system.
The electric field intensity Ez can be expressed as Ez ¼
Table 1 Fuzzy inference system rules, e.g. if displacement is far up and velocity is up then control force is max.
FarUp
CloseUp
Equilibrium
CloseDn
FarDn
Max Med + High +
Med+ Low+ Null
Low Null Low+
Null Low Med
High Med Min
V hA
ð5Þ
where V is the applied voltage across the thickness direction of the actuator and hA is the thickness of the actuator layer. Since only strains produced by the host beam act on the sensor layer and no electric field is applied to it, the output charge from the sensor can be calculated using Eq. (4). The charge measured through the electrodes of the sensor is given by 9 8 ! ! Z Z = 1< ð6Þ Dz dS þ Dz dS qðtÞ ¼ ; 2 : Sef Sef z ¼ h=2
z ¼ h=2 þ hS
Input 1 MF Degree of membership
Up Null Down
Displacement
CloseL Equilibrium CloseR
FarL
1
FarR
0.8 0.6 0.4 0.2 0 −4
Degree of membership
Velocity
ð4Þ
1
−3
−2
−1
Left
0 1 Displacement Input 2 MF
2
Null
3
4 x 10−4
Right
0.8 0.6 0.4 0.2 0 −8
−6
−4
−2
0 Velocity
2
4
Fig. 2. Membership functions for the inputs of the classic fuzzy controller.
6
8 x 10−3
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where Sef is the effective surface of the electrode placed on the sensor layer. The current on the surface of the sensor is given by iðtÞ ¼
dqðtÞ dt
based on the Timoshenko beam theory which, in turn, means that the axial displacement is proportional to z and to the rotation cðx,tÞ of the beam cross section about the positive y-axis and that the transverse displacement is equal to the transverse displacement w(x,t) of the point of the centroidal axis (y¼z ¼0). The strain–displacement relationships read:
ð7Þ
The current is converted into open-circuit sensor voltage output by S
V ¼ GS iðtÞ
ex ¼ z
ð8Þ
Degree of membership
exz ¼ c þ
@w @x
ð9Þ
The kinetic energy of the beam with the layers can be expressed as
where GS is the gain of the current amplifier. Furthermore, it is supposed that the bending–torsion coupling and the axial vibration of the beam centerline are negligible and that the components of the displacement field fug of the beam are
T¼
1 2
Z
_ T fug _ dV ¼ rfug
V
b 2
Z
Input 1 MF CloseL Equilibrium CloseR
FarL
1
@c , @x
0
L
Z
h=2 þ hS
r½ðzc_ Þ2 þ w_ 2 dz dx
h=2hA
FarR
0.8 0.6 0.4 0.2 0 −6
−4
−2
0
2
4
6 x 10−4
Displacement Degree of membership
Input 2 MF Left
1
Null
Right
0.8 0.6 0.4 0.2 0 −3
−2
−1
0 Velocity
1
2
3 x 10−4
Fig. 3. Membership functions for the inputs of the fuzzy controller optimized by PSO.
Degree of membership
Input 1 MF FarL
1
CloseL
Equilibrium CloseR
FarR
0.8 0.6 0.4 0.2 0 −2
−1
0
1
2 x 10−4
Displacement Degree of membership
Input 2 MF Left
1
Null
Right
0.8 0.6 0.4 0.2 0 −3
−2
−1
0
1
2
Velocity Fig. 4. Membership functions for the inputs of the fuzzy controller optimized by GA.
3 x 10−4
ð10Þ
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on the assumption that the host beam and the piezoelectric patches have identical densities. The strain (potential) energy is given by h=2 þ hS
h=2hA
"
# 2 @c @w 2 Q11 z þ Q55 c þ dz dx @x @x
ð11Þ If the only loading consists of moments induced by the piezoelectric actuators and since the structure has no bending–twisting couple then the first variation of the work has the form: Z L @c dx ð12Þ dW ¼ b MA d @x 0 where d is the first variation operator and MA is the moment per unit length induced by the actuator layer and is given by Z h=2 Z h=2 V ð13Þ MA ¼ zsAx dz ¼ zQ 11 d31 EAz dz EAz ¼ A hA h=2hA h=2hA
MX€ þ LX_ þ KX ¼ Fm þ Fe
ð14Þ
where M and K are the generalized mass and stiffness matrices, Fe is the generalized control force vector produced by electromechanical coupling effects, L is the viscous damping matrix and Fm is the external loading vector. It should be mentioned here that technical bending theories for plates can be constructed analogously.
0
ax
−0.2
M
−L ow
1
H ig h− M ed
M
in
Output MF
Degree of membership
0.8
0.6
0.4
0.2
0 −1
−0.8
−0.6
−0.4
0.2
0.4
0.6
0.8
1
Control Force Fig. 5. Membership functions for the output of the classic fuzzy controller.
+
M ax
ed
− Lo w − N ul l Lo w +
1
M
H
M
in
ig h−
Output MF h+
Z
ig
0
L
H
Z
−N ul l Lo w + M ed + H ig h+
b feg fsg dV ¼ 2 V T
ed
Z
Using Hamilton’s principle the equations of motion of the beam are derived. For the finite element discretization beam, finite elements are used with two degrees of freedom (d.o.fs) at each node: the transversal deflection wi and the rotation ci . These are gathered to form the degrees of freedom vector Xi ¼ ½wi ci . After assembling the mass and stiffness matrices for all elements, the equation of motion is obtained in the form:
M
1 U¼ 2
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Degree of membership
0.8
0.6
0.4
0.2
0 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Control Force Fig. 6. Membership functions for the output of the fuzzy controller optimized by PSO.
0.5
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The main objective is to design control laws for the smart beam bonded with piezoelectric S/A subjected to external induced vibrations. For this purpose, a fuzzy control system optimized by
PSO is applied in this paper. A two-input, single-output fuzzy inference controller is tested. This configuration is suitable for a feedback control force based on the displacement and velocity at a
M ax
H ig h
M in
1
Degree of membership
− M ed Lo − w N − u Lo ll w M + ed + H ig h+
Output MF
0.8 0.6 0.4 0.2 0 −0.6
−0.4
−0.2
0
0.2
0.4
0.6
Control Force Fig. 7. Membership functions for the output of the fuzzy controller optimized by GA.
1
1.5
0.5
1
0
0.5 Force
Force
Excitation (blue−−), Control force (red−.) and Control force rate (black−)
Excitation (blue−−), Control force (red−.) and Control force rate (black−)
−0.5
0
−1
−0.5
−1.5
−1 −1.5
−2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0.2
0.4
Time (sec)
0.6
0.8
1
Time (sec) Excitation (blue−−), Control force (red−.) and Control force rate (black−)
1.5 1
Force
0.5 0 −0.5 −1 −1.5 0
0.2
0.4
0.6
0.8
1
Time (sec) Fig. 8. Control force at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
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x 10
Displacement before (blue) and after (red) fuzzy 2.5
2
2
1.5
1.5
Displacement (m)
Displacement (m)
2.5
−4
1 0.5 0 −0.5
Displacement before (blue) and after (red) fuzzy
1 0.5 0 −0.5
−1
−1
−1.5
−1.5
−2
x 10
−4
623
−2 0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Time (sec)
Time (sec) Displacement before (blue) and after (red) fuzzy
−4
2.5
x 10
2 Displacement (m)
1.5 1 0.5 0 −0.5 −1 −1.5 −2 0
0.2
0.4 0.6 Time (sec)
0.8
1
Fig. 9. Displacement at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
given point (collocated configuration of the local controller). If needed, several independent and decentralized (local) fuzzy controllers, in general with different characteristics, can be installed at the various points of a larger structure.
3. Fuzzy control system optimized by particle swarm optimization 3.1. Fuzzy control Fuzzy systems (Driankov, Hellendoorn, & Reinfrak, 1996) can be applied in many fields and can solve different kinds of problems in various application domains. Fuzzy inference rules systematize existing experience and can be used for the mathematical formulation of nonlinear controllers. The feedback is based on fuzzy inference and may be nonlinear and complicated. Knowledge or experience on the controlled system is required for the application of this technique. In the area of smart systems the application of neuro-fuzzy control has been adopted by many authors. In particular, it seems to be suitable for the control of structures with complicated or nonlinear characteristics, like the tuned mass damper systems for aseismic design
(Pourzeynali, Lavasani, & Modarayi, 2007; Wang & Lin, 2007) or the semi-active control using active friction devices or electrorheological fluid dampers (Reigles & Symans, 2006). In order to design a fuzzy control system (construct the control rules, state the membership functions, tune the parameters, etc.) various approaches have been proposed. In the last years, there is an increasing interest to optimize the fuzzy control systems with heuristic, metaheuristic, and nature inspired approaches. Mainly, genetic algorithms have been used for these purposes while nature inspire approaches, like PSO, have not been used so extensively. For example in Belarbi and Titel (2000), a method for designing fuzzy logic controllers with symmetric partitioning of the universe of discourse is presented where both the rule base and membership functions of the input and output variables are designed optimally using a genetic algorithm. In Chou (2006), a genetic algorithm based optimal fuzzy controller design is proposed where the design procedure is accomplished by establishing an index function as the consequent part of the fuzzy control rule. An efficient genetic reinforcement learning algorithm for designing fuzzy controllers is proposed in Juang, Lin, and Lin (2000). In Poursamad and Montazeri (2008) a geneticfuzzy control strategy for parallel hybrid electric vehicles (HEVs) is presented in which the genetic-fuzzy control strategy is a fuzzy
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Displacement Velocity before (blue) and after (red) fuzzy
x 10−3
5
5
4
4 Displacement Velocity (m/sec)
Displacement Velocity (m/sec)
x 10−3
3 2 1 0 −1 −2 −3 −4 −5
Displacement Velocity before (blue) and after (red) fuzzy
3 2 1 0 −1 −2 −3 −4 −5
0
0.2
0.4
0.6
0.8
1
0
0.2
Time (sec)
5
0.4
0.6
0.8
1
Time (sec)
x
10−3
Displacement Velocity before (blue) and after (red) fuzzy
Displacement Velocity (m/sec)
4 3 2 1 0 −1 −2 −3 −4 −5 0
0.2
0.4
0.6
0.8
1
Time (sec) Fig. 10. Displacement velocity at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
logic controller that is tuned by a genetic algorithm. In Lin (2004) the proposed algorithm combines the genetic algorithm (GA) and the least-squares estimate (LSE) method to construct the genetic algorithm-based neural fuzzy system for temperature control. An extensive analysis of the genetic fuzzy systems can be found in Cordo´n, Gomide, Herrera, Hoffmann, and Magdalena (2004). Welch and Venayagamoorthy (2007) presents a particle swarm optimization method for optimizing a fuzzy logic controller for a photovoltaic grid independent system consisting of a PV collector array, a storage battery, and loads (critical and non-critical loads). A learning approach based on particle swarm optimization for the determination of the consequent parameters and premise parameters of a Takagi and Sugeno type fuzzy model is formulated and explained in Niu, Zhu, He, and Shen (2008). Youssef, Yousef, Sebakhy, and Wahba (2009) introduces the use of the adaptive particle swarm optimization (APSO) for adapting the weights of fuzzy neural networks. In Juang and Lo (2008) proposes a zeroorder Takagi–Sugeno–Kang (TSK)-type fuzzy system learning using a two-phase swarm intelligence algorithm (TPSIA) where the first phase of TPSIA learns fuzzy system structure and parameters by on-line clustering-aided ant colony optimization and phase two aims to further optimize all of the free parameters
in the fuzzy system using particle swarm optimization. In Zamani, Karimi-Ghartemani, Sadati, and Parniani (2009), particle swarm optimization algorithm is used to carry out the design procedure of fractional order PID (FOPID) controller.
3.2. Fuzzy control system for vibration control of smart structures In order to reduce the displacement field of the cantilever beam system, a nonlinear fuzzy controller (Tairidis et al., 2007) was constructed by using the Fuzzy Toolbox of Matlab. More specifically, a Mamdani-type Fuzzy Inference System, consisted of two inputs and one output, was developed. The system receives as _ while gives as inputs the displacement (u) and the velocity ðuÞ, output the increment of the control force (z). Triangular and trapezoidal shape membership functions were chosen both for inputs and output. In order to describe the present system-controller, 15 rules were used. All rules have weights equal to 1 and use the AND-type logical operator. These rules are presented in the Table 1. The implication method was set to minimum (min), while the aggregation method was set to maximum (max). The defuzzified
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Rotation without (blue−−) and with (red−) fuzzy control 4
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4
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Rotation without (blue−−) and with (red−) fuzzy control
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−2
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x
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Rotation without (blue−−) and with (red−) fuzzy control
3
Rotation (rad)
2 1 0 −1 −2 −3 −4 0
0.2
0.4 0.6 Time (sec)
0.8
1
Fig. 11. Rotation at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
output value has been created by using the MOM (mean of maximum) defuzzification method.
_ Both the control force and the (displacement u and velocity u). external loads provide the next step’s ðt þ DtÞ values of displacement and velocity.
3.3. Structural dynamics and fuzzy control The Houmbolt numerical integration algorithm was chosen for the solution of the equations of motion (Eq. (14)) (Tairidis et al., 2007). According to this method, and on the assumption that the acceleration is constant within each time integration step, the Houmbolt factors are set equal to
b ¼ 0:25, g ¼ 0:5
ð15Þ
The total integration time for the simulation was chosen equal to 1 s while the time step ðDtÞ was chosen equal to 0.001 s. The integration constants are c1 ¼ c4 ¼
1
bðDtÞ2
g bDt
,
,
1
c2 ¼
c5 ¼
g b
1 c3 ¼ , 2b g c6 ¼ D t 1 2b
bDt ,
,
ð16Þ
In each step (t) of the numerical integration, the fuzzy controller provides a control force (z), according to the given input values
3.4. Fuzzy control system optimized by particle swarm optimization for the vibration control of smart structures Some parameters of the fuzzy control system described in Section 3.2 were optimized by particle swarm optimization. More precisely, the parameters (the break points) of the triangular and trapezoidal membership functions, the weights of the rules, and the logical operator are considered as variables. First, a number (swarm) of particles is randomly initialized, where a particle is a solution to the problem. In each particle, the first values correspond to the parameters of the membership functions and can take continuous values, the next 15 values correspond to the weights of the rules and can take continuous values in the range [0,1] while the last 15 values correspond to the AND/OR-type logical operator and can take discrete values equal to 1 for AND and 2 for OR. It should be noted that in PSO implementations usually only continuous or discrete values exist. In this paper, there is a combination of continuous and discrete values in each particle and in order to solve the problem a combination of the
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Rotation Velocity without (blue−−) and with (red−) fuzzy control
x 10−3
8
8
6
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Rotation Velocity (rad/sec)
Rotation Velocity (rad/sec)
x 10−3
4 2 0 −2 −4 −6
Rotation Velocity without (blue−−) and with (red−) fuzzy control
4 2 0 −2 −4 −6
−8 −10
−8 0
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Time (sec)
x 10−3
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0
0.2
8
Rotation Velocity (rad/sec)
6 4 2 0 −2 −4 −6 −8 0.4 0.6 Time (sec)
0.8
1
Fig. 12. Rotation velocity at the free end for classical fuzzy controller (a), for the fuzzy controller optimized by PSO (b) and for the fuzzy controller optimized by GA (c).
equations used for finding the new position of the particle for continuous and discrete problems is used. More precisely, the position of each particle is represented by a d-dimensional vector in problem space si ¼ (si1, si2,y, sid), i ¼ 1,2,y, N (N is the population size, d is set equal to the values’ parameters plus the values of weights plus the values of the logical type operator). The fitness function that has to be minimized is the following error function:
previous best position and the previous best position attained by any particle of the swarm, namely pid and pgd. The velocities and positions of particles are updated using the following formulas. The formula used for the calculation of velocities is the same for continuous and discrete values while different formulas are used for the calculation of the positions of the particles for continuous and discrete values:
err ¼ JXX J
(for the values of the particle that are continuous) and
ð17Þ
where X is the L2 norm in Rm (m is the total number of the assumed d.o.fs), X ¼ [X1,y, Xm] is the nodal displacements and rotations array and X ¼ ½X1 , . . . , Xm is the wished value of the nodal displacements and rotations array (it is set equal to zero). Thus, each particle is randomly placed in the d-dimensional space as a candidate solution. The velocity of the i-th particle vi ¼ (vi1, vi2,y, vid) is defined as the change of its position. The flying direction of each particle is the dynamical interaction between the individual and the social flying experience. The algorithm completes the optimization through following the personal best solution of each particle and the global best value of the whole swarm. Each particle adjusts its trajectory towards its own
vid ðt þ1Þ ¼ vid ðtÞ þe1 rand1ðpid sid ðtÞÞ þ e2 rand2ðpgd sid ðtÞÞ
ð18Þ
sid ðt þ 1Þ ¼ sid ðtÞ þ vid ðt þ 1Þ
ð19Þ
sigðvid Þ ¼
1 1þ expðvid Þ (
sid ðt þ 1Þ ¼
1
if rand3 osigðvid Þ
0
if rand3 Zsigðvid Þ
ð20Þ
ð21Þ
(for the values of the particle that are discrete) where t is the iteration counter; e1 and e2 are the acceleration coefficients; rand1, rand2, rand3 are three random numbers in [0, 1]. The acceleration coefficients e1 and e2 control how far a particle will move in a single iteration. Low values allow particles to roam far from target regions before being tugged back, while high values result in abrupt movement towards, or past, target regions. Typically, these are both
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Displacement for different loadings
2.5 2
Displacement (m)
1.5 1 0.5 0 −0.5 −1 −1.5 −2 0
0.1
0.2
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without fuzzy control fuzzy control − loading: sin (20t) fuzzy control − loading: sin (10t) fuzzy control − loading: sin (5t)
0.5
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0.7
0.8
0.9
1
0.8
0.9
1
Time (sec)
Rotation Velocity for different loadings
x 10−3 8
Rotation Velocity (rad/sec)
6 4 2 0 −2 −4 −6 −8 0
0.1
0.2
0.3
0.4
without fuzzy control fuzzy control − loading: sin (20t) fuzzy control − loading: sin (10t) fuzzy control − loading: sin (5t)
0.5
0.6
0.7
Time (sec)
Fig. 13. Displacement (a) and rotation velocity (b) at the free end for fuzzy control optimized by PSO for different loadings.
set equal to a value of 2.0, although assigning different values to e1 and e2 sometimes leads to improved performance. It should be noted that in the discrete space, a particle moves in a state space restricted to zero and one on each dimension where each vi represents the probability of bit si taking the value 1. Thus, the particles’ trajectories are defined as the changes in the probability and vid is a measure of individual’s current probability of taking 1. If the velocity is higher it is more likely to choose 1, and lower values favor choosing 0. The sigmoid function (Eq. (20)) is applied to transform the velocity from real number space to the probability space.
4. Numerical results The problem of a cantilever beam is studied in the present paper. The beam has a total length equal to 0.8 m and the structure has been discretized with four finite elements resulting in a model
with eight degrees of freedom. A dynamic loading is used as disturbance that simulates a strong wind and is a periodic sinusoidal loading pressure (sin(20t)) that influences the displacement of the second and the fourth (free end) finite elements. The purpose of the fuzzy controller is to reduce the oscillation. The controller is collocated, takes as input the displacement and the velocity of the free end and gives back the control force to be applied at the same point. The results obtained with the fuzzy controller optimized by PSO are compared with those of the classic fuzzy controller (without any optimization procedure). The results of the classic fuzzy controller have been compared with the results that arose from classical control (LQR) in Tairidis et al. (2007) giving, with suitably chosen parameters for the fuzzy controller, comparable reduction of the vibrations with the LQR control by using fewer inputs (two instead of sixteen measurements in this example) and less effective reduction of the velocities. The parameters of the PSO algorithm are selected after thorough
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testing. These parameters are: the number of swarms is set equal to 1, the number of particles is equal to 15, the number of generations is equal to 20 and the parameters e1 and e2 are both set equal to 2. The whole algorithmic approach was implemented in Matlab on a Centrino Mobile Intel Pentium M 750 at 1.86 GHz. In order to see the effectiveness of the proposed PSO based fuzzy controller, it is compared with another stochastic population-based algorithm, a fuzzy controller optimized by a genetic algorithm. Genetic algorithms (GAs) (Holland, 1975) are search procedures based on the mechanics of natural selection and natural genetics. A GA is a stochastic iterative procedure that maintains the population size constant in each iteration, called a generation. Their basic operation is the mating of two solutions in order to form a new solution. To form a new population, a binary operator called crossover, and a unary operator, called mutation are used (Engelbrecht, 2007; Goldberg, 1989). In genetic algorithm implementation, the parameters were chosen in such a way that in all algorithms the same number of function evaluations are needed. Thus, the parameters for the GA are: the number of individuals is equal to 15, the number of generations is equal to 20, the mutation rate is set equal to 0.5 and, finally, 1-point crossover was selected with crossover rate equal to 0.8. The load, the displacement, the displacement velocity, the rotation and the rotation velocity, for the three kinds of control (classic fuzzy control, fuzzy control optimized by PSO and fuzzy control optimized by GA), are presented in the graphs below. The values without control are shown with a blue dashed line. The membership functions obtained by the fuzzy controller without any optimization procedure (Figs. 2 and 5), when optimized by PSO (Figs. 3 and 6) and when optimized by GA (Figs. 4 and 7) are presented. Fig. 8 presents the control force at the free end for classic fuzzy controller, for the fuzzy controller optimized by PSO and for the fuzzy controller optimized by GA. As it can be seen, the fuzzy controllers optimized by PSO and by GA give superior control results compared to the classical fuzzy controller. More precisely, the displacements (Fig. 9) and the rotations (Fig. 11) are almost zero in the case of the fuzzy controller optimized by PSO and in the case of the fuzzy controller optimized by GA. The displacement (Fig. 10) and rotation (Fig. 12) velocities have significantly smaller values for the fuzzy controller optimized by PSO and for fuzzy controller optimized by GA compared with the ones obtained by the classical fuzzy controller. It should be noted that the fuzzy controller optimized by PSO gives better control results than the fuzzy controller optimized by GA, as the displacement velocities (Fig. 10(b)) and the rotation velocities (Fig. 12(b)) obtained by the fuzzy controller optimized by PSO have smaller values compared with the ones of the fuzzy controller optimized by GA. Of course, as the differences in the results are small, it can be said that both methods, PSO and GA, can be applied with very good results in order to optimize the fuzzy controller used for the vibration control of beams. Furthermore, in order to investigate the robustness of the fuzzy control optimized by PSO and its ability to work for different loading, the controller gain calculated by the PSO based on one specific sinusoidal loading is used for different sinusoidal loading pressures. More specifically, the controller gain is calculated from the PSO algorithm when the loading is equal to sin(20t) and this controller gain is used when the loadings are equal to sin(10t) and sin(5t). As it can be seen from Fig. 13 the results obtained are very efficient and the controller gain obtained by using one loading can be applied successfully when other loadings are used.
5. Conclusion In this paper, a fuzzy control system optimized by particle swarm optimization (PSO) was used for the vibration control of
beams with piezoelectric sensors and actuators. The parameters of the fuzzy controller where selected optimally by using the PSO algorithm. The results obtained for a sinusoidal loading pressure using the fuzzy controller system optimized by PSO are very efficient. A comparison of the proposed method with a classic fuzzy controller for this specific problem revealed the high performance of the proposed method. Future research is intended to be focused in the application of the proposed algorithm for the design of other smart structures like plates, shells and in the modification of the method in order to be used for optimally design other parts of the fuzzy system like the definition of the control rules. References Aldraihem, O. J., Wetherhold, R. C., & Sigh, T. (1997). Distributed control of laminated beams: Timoshenko theory vs. Euler–Bernoulli theory. Journal of Intelligent Mathematical Systems and Structures, 8, 149–157. Belarbi, K., & Titel, F. (2000). 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