Particle Sw Optimizer with Integral ControllerJianchao Zeng Zbihua Cui

Division of System Simulaton and Computer ApplicationTaiyuan Universiy of Science and Technology,Shani, P.R.China, 030024

E-mail: zengianchao@263.netcui_zhi_hua_76450sohu.com

Abstract-TRhe evoluionary equations of PSO are second-order discree ime hear stochastic system. It can be regrdedas a control plant, and Introduci a controller to regulatethe dynamic evolutionary behaviors of Pso. In the paper, z-tA_formi n used to evolutiona equations, and the systemstructure diarm Is obtained. Then, an IneFgal controler IsIntrodued 0tP80 to give an Integrl-PSO algorithm Meanwhlde,the Intora-PSO is analyzed through the support set of samplespace of partles and Ihear system theory for the convergencecondtioua and diversity of particles., Finally, the smulationresult for the test functions are given to show that the integral-PSO is more efficiency than the original PSO.

I. INTRODUCTIONSince the particle swarm optimion was developed by

JKney and R.CEberiart in 1995,[l] its theories and ap-plications have been studied by many schola in the world.Particularly, to improve the global convergence ofPSO, severalmodified algorithms have been presented and tested by thetypical test functions to show the efficiency, for examples,Hybrid Particle Swarm Optimizer with breeding and sub-populations developed by M.Lovbjerg and T.K.Rasmussen[2];a diversity-guided PSO with "Attractive" and "Repulsive"opcrators;[3] the guaranteed convergence particle swarm op-timizer(GCPSO)[4] and the stochastic PSO with no memoryfor velocity [5] etc..

Solis and Wets have studied the convergence of stochasticsch algorims, providing criteria under which algorithmscan be cosideed to be global convergence, or only localconvergence[6]. Solis and Wets's definition are used by Fransvan den Bergh in the study of the convergence characteisticsof PSO, and concluded tha the original PSO is not globalconvergence or local convergence, and the GCPSO is localconvergence algoitn [7]. But the stochastic PSO can beconvergence to global optimum with probability one [5].

In order to improve the global convergence of PSO, it ispivotal how to e ce the global exploration ability of al-gorithm. Generally, the dynamic adaptive parameters adjustedmethiod, o the hybrid algorith which is formed by integrationofPSO and other evolutionary algorithm are adapted to resolvethe global convergence. In other words, the paramets orsructure (integation with other optimization algorithm) areregulated or changed to guatee the global optimality ofconvergence through balancing the global exploration and thelocal exploitation. In this paper, the update equations of PSOare regarded as a discrete time linear system plants, andan integal controller is introduced to control the dynamic

evolutionary behaviour of PSO, so as to enhance the globalconvergence.

II. ANALYSIS OF PARTICLE SWARM OPTIMIZER

Consider the update equation of the original particle swamoptimiz (PSO):

vi(t+ 1) = wvi(t) +clrl(pi-xi (t)) +c2r2 (pg - xi(t)) (1)

xi(t + 1) = xi(t) + vi(t + 1) (2)

Where, piis the personal best position of particle i, p9is thebest position discovered by any of the particles so far, wis an inertia weight constant, and cl,c2are two acceleradoncoefficients, and rl U(0, 1), r2 U(O, 1) are two indepen-dent uniform distrbution random sequences which are used toeffect the stochastic nature of the PSO algorithm

Let P1 = c1rl, W2 = C2r2, and( = l + W2. Substuingequation (1) into equation (2) and eliminating the vi(t), resultsin

Xi(t+l) = (W+1)Xi(t)-WXi(t)+pi(Pi-Xi(t))+P2(Pg-Xi(t))(3)

z-transformation is made and reanranged, the following equa-tion is deduced.

(4)

+c2(Pg(z) - Xi(z))]

The system diagram expressed by equation (4) can be ob-tained, as shown in Fig.l.The following conclusion can be seen from Fig.l. (1)Pi(pi -i (t)) + P2(pg -xi (t))in the velocity update equation

coreponds to the parallel connection of two integral units.(2) The dynamic evolutionary behaviour of PSO corspondsto a second-order linear stochastic system, its operationalbehaviour can be analyzed by the second-order linear systemtheory, and its convergence is very easy to be guarateed.(3) PSO can be simulated using the Malab software. (4)A controller may be introduced to change the parameteand structure of the PSO algorihm, so as to regulate theoperational behaviour of PSO.

0-7803-9422-4/05/$20.00 02005 IEEE1840

Xi (z) = z [wi (Pi (z) - XiW)Z2 (W + J)Z + W

III. PSO WITH INTEGRAL CONTROLLER

As above-mentioned, a controller may be introduced to theoriginal PSO algorthm to regulate the evolutionary behaviourofPSO, so that the global optimality ofPSO convergence canbe guaranteed with larger probability. The system diagrm ofPSO algorithm with controllers is shown as Fig.2 Where, cl(z)and c2(z)are the z-transfer fumctions of controllers.The following equation can be deduced from Fig.2

zXi(z) = - (qu+l)Z+W[C1(z)(Pi(z)-Xi(z)) (5)

+p2C2(z)(Pg(z) - Xi(z))]

From Fig.2 and equation (5), it can be seen that the differentdynamic evolutionary behaviour of PSO with controllers canappears if the controllers cl(z) and c2(z) have different struc-tures. In the paper,Cq(z) = c2(Z) = , that is, an integralcontroller is only considerd, and the other type of controllerswould be considered in our future research works.

For cl(z) = c2(z) =zz 1 , equation (5) results in follows.

x2

z -(2 + W)z2 + (2w+ ()z-6w6)(Pi(z)-Xi(z))

+W2(P9(Z) - Xi(Z))]

The inverse z-transformation is made, we have

Xi(t+1) = (2+w)xi(t)-(2w+1)xi(t-1)+wxi(t-2) (7)

+Wl(Pi -Xi(t)) + P2(Pg - xi(t))

Definevi(t + 1) = xi(t + 1) -xi(t) (8)

ai(t+l) = v,(t+1)-v,(t) = x5(t+1)-2xj(t)+xj(t-1) (9)

Then, the update equations ofPSO algorithm with the integralcontrollers are as follows

ai(t + 1) =WtVi(t) + (i(Pi -Xi(t)) + 2(Pg -Xi(t)) (10)

vi(t + 1) = Vi(t) + ai(t + 1)

-xi(t + 1) = xi(t) + vi(t + 1)

It can be seen hat the PSO algoritim with integral con-

trollers is equalent to ading an acceleration term in theupdate equations, and the evolutionary behaviour of PSOhas the property of the hird-order linear systeuL For thesame parameters, convergence speed becomes slower, and thediversity of particles produced is enhanced, so that the globalexploration ability is increased.

IV. ANALYSIS OF CONVERGENCE BEHAVIOUR OFINTEGRAL-PSO

In the following, ihe convergence of the integal-PSO algo-rithm is discussed based on linear control theory. The integral-PSO can be regarded as a two-input and one-output linearsystem for one particle evolution, and the open z-transferfunction GK1(Z) (for input Pi) is as follows.

(PiZ2Gkl(Z) = [Z2 + (W + 1)z+ W][Z - 1]

Its eigenequation is 1+GIl(z)= O, is

3- (2+w+pi)Z2+(2w+1)z-w=0

(11)

(12)

For the stability conditions derived, above equation is trans-formed by z = , and the stability judged equation isobtained as follows.

y3+ y2+ (6-p1)y+ (4 + 4w-p1) =0 (13)For convenience to discuss, equation (15) can be rewritten inthe ordinary form:

bo 3+ bly2 +b2y +b3 = 0 (14)

According to Routh's stability criterion, the system specifiedby equation (16) is stable, that is, the integal-PSO is conver-gent, if and only if:

bo > 0, bi > 0, b2 >0, b3 > 0, b1b2-bob.3 > 0 (15)

For equation (15), bo > 0, bi > 0 are evidently satisfied.b2 > 0 implies W, < 6, and b3 > 0 implies w > 4 -1 > 0

4meanwhile, in order to satisfy b1b2 - bob3 > 0, w must be

less than 0.5. From above-discussed, the stability conditionsof output for input pi

(16)4< pi <6Wi -1 < w < 14 2

The same method is used for input pg, and the stabilityconditions of system output for input pg can be obtained asfollows.

4<wp1 <642 14 -< <2

(18)

(19)

Note that the above analysis assumed that pl,ilrLeainedconstant, which is not the fact in the PSO. Ihe vau ofcl and c2 can be considered an upper bound for (plandml.The average behaviour of PSO algorithm can be observed bythe expected value of cplandkol, becse of r1andr2 beinguniform distribution, we have

E[1pl] = C1 ITdx= (20)

and

E[p2] = C2j Zdx= X (21)

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(17)

Combining equations (18) to (23), the convergence condi-tions of the integral PSO algorithm are gained as followingform:

8< cl,c2<12Ci C2 1

A"max--i- --1<w< -8 '.8 2

(22)

(23)

The above convergence conditions make the choice ofparameters of the integmal-PSO become very easy. And thesimulation results in the next section should prove the conclu-sion.

integral-PSO are set as cl=c2=10 and w decreasing form 0.5to 0.4 linearly.

aik (t + 1) = Amaxt if(a,k (t + 1) > Amax)

ajk(t + 1) = -Amazi if(ajk (t + 1) < -Amax)

ajk(t + 1) = ajk(t + 1), otherwise

Vjk (t + 1) = Vma,if(Vjk (t + 1) > Vmax)

VJk (t + 1) = -Vmaz,if(Vjk (t + 1) < -Vmaz)

Vjk(t + 1) = vjk(t + 1), otherwise

where Amafis the maximum acceleration, denotes the max-imum speed.The experiment results is showed in Tab.1.

TABLE ICOMPARATION RESULTS

Fig. 1. The System Diagram of PSO

Fig. 2. The System Digram of PSO with Controllers

V. EXAMPLES

(l)Rastrigin Functionn

f1(X) = (xj2 - A cos(2irxj) + A),-5.12 < xj < 5.12j=l

This function's optimum is xj = 0,j = 1,2,...,n, andthere are almost lOn local minimums in the space S =xi E (-5.12, 5.12),j = 1, 2, ...,n.

(2)Griewank FunctionX2 n

xf2(X) 4000- Cos 4 +1

This fimction's global minimum is 0 when xi = 0,j =1,2 ...,n, and the local minimums are get when x = kirj/j,j = 1,2,...,n,k = 0,1,...,n .

In these experiments, every algorithm plements 50 timeswith its maximum generations 5000. In PSO, w decreasingform 0.9 to 0.4 linearly and cl=c2=2.05. The parame in

Fun Dim Err AlgFl 10 0.0001 PSOFl 10 0.0001 IPSOF2 10 0.0001 PSOF2 10 0.0001 IPSO

IF-Feval Fper1487.45 22310.97 ' 741525.20 40341.36 66

Notes:Feval means The means of generations to convergence.Fper means The means of convergence rate.

ACKNOWLEDGMENTThis work is supported by Educational Department Science

and Technology Project (No.204018)REFERENCES

[1] J.Kennedy, and R.C.Eberhart "Particle Swarm Optimization," Proceed-ings of the IEEE International Conference on Neural Networby,1995,pp. 1942-1948.

[2] M.Lovbjerg, TK.Rasmussen, and T.Krink, "Hybrid Particle Swarm Op-timiser With Breeding and Subpopulations," Proceedings of the Geneticand Evolutionay Computation Conference,2001, pp. 156-162.

[3] J.Riget, and J.S.Vesterstrom, "A Diversity-Guided Particle SwarmOpimizr-the ARPSO," Proceedngs ofthe the Congress on EvolutionayComputation,2001, pp. 1178-1182.

[4] F Van den Bergh, and A.Engelbrecht, "A New Locally ConvergentParticle Swarm Optimizer," Proceedngs of the the IEEE Congress onSystem, Man And Cybernetics,2002, pp. 1942-1948.

[5] Zeng Jianchao, and Cui Zhihua, "A Guaranteed Global Conver-gence PSO Algorithm," Computer Research and Development,2004,voL 41,no. 8,pp. 1333-1338.(in Chinese)

[6] F Van den Bergh, "An Analysis of Particle Swarm Optimizers," PhDissrtation, Pretoria: University of Pretoria, 2001.

[7] F.Solis, and R.Wet, 'Minimiztion By Random Search Techniques,"Mathematics of Operations Reearch, 1981, vol. 6,no. 1,pp. 19-30.

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