The Kalman Particle Swarm OptimizationAlgorithm and Its Application in Soft-sensor of

Acrylonitrile YieldGUO Wei, Chen Guo-chu, YU Jin-shou

(Research Institute of Automation, East China University of Science and Technology, Shahnghai 200237, ChianCorresponding author: Guo Wei, E - mail: vadgw@l63.com or vadgw@hotnail.com)

Abstract- This paper proposes the Kalaman ParticleSwarm Optimization Algorithm (KPSO), which combines theKalman Filter and PSO. KPSO assumes that particle movesaccording to the Kalman Filer. The comparison ofoptimization performance between KPSO and PSO to threewidely used test functions shows that the optimizationperformance of KPSO is much better than that of PSO. Thecombination of KPSO and ANN is also introduced (KPSONN).Then, KPSONN is applied to construct a practical soft-sensorof acrylonitrile yield. After comparing with practical industrialdata, the obtained result shows that the KPSONN is feasibleand effective in soft-sensing of acrylonitrile yield.

Keywords: KPSO, PSO, acrylonitrile yield; soft-sensor

I. Introduction

Particle swarm optimization (PSO) is a stochasticoptimization technique developed by Dr. Eberhart and Dr.Kennedy in 1995[1], inspired by social behavior of birdflocking or fish schooling. The advantages of PSOincludes: PSO is easy to implement and there are fewparameters to adjust. PSO has been successfully appliedin many areas: function optimization, artificial neuralnetwork training, fuzzy system control, and other areaswhere GA can be applied.

Even so, it is not without problems. PSO suffersfrom premature convergence, tending to get stuck in localminima [2, 3]. We have also found that it suffers from anineffective exploration strategy, especially around localminima, and thus does not find good solutions as quicklyas it could. Moreover, adjusting the tunable parameters ofPSO to obtain good performance can be a difficult task.

Research addressing the shortcomings of PSO isongoing and includes such changes as increased particlediversity [4], evolutionary selection mechanisms [5], andof course tunable parameters in the velocity updateequations [6]. Some work has been done that alters basicparticle motion with some success, but the possibility forimprovement in this area is still open.

This paper presents an approach to particle motionthat significantly speeds the search for optima while

simultaneously improving on the premature convergenceproblems that often plague PSO. The algorithm presentedhere, Kalman Particle Swarm Optimization (KPSO),bases its particle motion on Kalman filtering andprediction.

II. The Kalman Particle Swarn Optimization

A. The Basic PSO AlgorithmPSO is an optimization strategy generally employed

to find a global minimum. The basic PSO algorithmbegins by scattering a number of "particles" in thefunction domain space. Each particle is essential a datastructure that keeps track of its current position x andits current velocity v . Additionally, each particleremembers the "best" (best valued) position it hasobtained in the past, denoted p . The best of these valuesamong all particles (the global best remembered position)is denoted g. Each step, a particle updates its positionand velocity by the following equations [1, 7]:vt+l =oj v, +c *ran4 @-(pxt) +C2 * Rani0-(g-xt ) ( 1)Xt+1 = xt + Vt+1 (2)

d) is inertia weight, Cl and c2 are accelerationconstants, rand() and Rand() are randomnumbers between [0,1]. Particles' velocities on eachdimension are clamped to a maximum velocity and aminimum velocity. Particle's positions on each dimensionare damped to a maximum position and a minimumposition.B. Particle Motion Issues

Given that PSO is closely related to an approach assimple as randomized hill climbing, it is no surprise thatattempts to improve the velocity update equation withvarious scaling terms have met with marginal success.Instead, more fundamental changes such as increasedswarm diversity, selection, and collision avoidingparticles have shown the greatest promise [4, 5].

Unfortunately these methods are not without

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problems either, as they generally fail to reduce theiterations required to reach suitable minima. They focusprimarily on eliminating stagnation, eventually findingbetter answers than the basic PSO without finding themany faster.

It has been pointed out that nonlinear programmingis subject to a fundamental tradeoff between convergencespeed and final fitness [4], suggesting that it is notgenerally possible to improve one without hurting theother. Fortunately, this tradeoff point has not yet beenreached in the context of particle swarm optimization, asit is still possible to find good solutions more quicklywithout damaging fmal solution fitness.

For example, the development of a PSOvisualization tool served to expose a particularlyinteresting inefficiency in the basic PSO algorithm. Asthe particles close in on g they tend to lose their lateralmomentum very quickly, each settling into a simpleperiodic linear motion as they repeatedly overshoot (andundershoot) the target. This exploration strategy aroundlocal minima is very inefficient, suggesting that a changeto particle motion may speed the search by improvingexploration. Such a change should ideally preserve theexisting desirable characteristics of the algorithm. PSO isessentially a social algorithm, which gives it usefulemergent behavior. Additionally, PSO motion isstochastic, allowing for randomized exploration. Particlesalso have momentum, adding direction to the randomsearch. The constriction coefficient indicates a need forstability. Alterations to particle motion shouldpresumably maintain these properties, making theKalman Filter a suitable choice [9].C. The Kalman Filter

Kalman filters involve taking noisy observationsover time and using model information to estimate thetrue state of the environment [8, 9]. Kalman filtering isgenerlly applied to motion tracking problems. It mayalso be used for prediction by applying the systemtransition model to the filtered estimate.

The Kalman Filter is limited to normal noisedistributions and linear transition and sensor functionsand is therefore completely described by several constantmatrices and vectors. Specifically, given an observationcolumn vector Zt+1 the Kalman Filter is used togenerate a normal distribution over a belief about the truestate. The parameters mt+1 and Vt+j of thismultivariate distribution are determined by the followingequations:mt+ F + Kt+1 (Zt+1 -HFm) (3)

Vt+1 (I-Kt+1)(FVtF + X) (4)IK~ =(Ff[FT +Vx)HT(H(F[FT +V )H +J') (5)

In these equations, F and V. describe the

system transition model while H and V, describe thesensor model. The equations require a starting point forthe filtered belief, represented by a normal distributionwith parameters mo and V0 , which must be provided.

The filtered or "true" state is then represented by adistribution:xt Normal(mt,JV-) (6)

This distribution may be used in more than one way.In some applications, the mean m, is sampled once toobtain the value. In others, the distribution is assumed tobe the true value. In this paper, the first is done.

After describing how to construct Kalman filtering,yielding m, from an observation Zt, a simple form ofprediction involves applying the transition model toobtain a belief about the next state mt+,:m= Fm1 (7)

There are other forms of prediction, but this simpleapproach is sufficient for the introduction of thealgorithm in the next section, and for its use in particleswarms.D. The Kalman Particle Swarm Optimization (KPSO)

KPSO defines particle's motion entirely according toKalman prediction. Each particle keeps track of its mt ,

Vt, and K1. The particle then generates an observationfor the Kalman filter with the following formulae:zv= S(g - x) (8)

zp =x+Zvz (ZTT Z-T)

(9)

x p 7'V ' (10)Scalar fb is drawn uniformly from [0, 2.05). The

observation is then used to generate M1 and VJ using(3), (4), and (5). Once the filtered value is obtained, aprediction m1, is generated using (7), then,

xt+lNormal(m'+,V+ ll

The new state of the particle is obtained by samplingonce from this distribution. The position of the particlemay be obtained from the first half of xt,l.

KPSO algorithm can be summarized in the followingsteps:

Stepl: Initialize size of the swarm, maximum numberof iterations, the dimensionality of KPSO; initialize thevector W which contains the size of domain in eachdimension; initialize vector moi, V0, H, Vz, Fand Vx; stochastically initialize the position of eachparticle;

Step2: Evaluate the fitness of each particle.

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Step3: Update the parameters of system according toequation (3)-(10).

Step4: Update the new position of each particleaccording to equation (1 1).

StepS: Update the fitness of each particle.Step6: If necessary, update and store the global best

position and global best fitness of whole swarm.Step7: If the stopping condition is not satisfied, go to

step 3. Otherwise, stop and obtain the result from theglobal best position and the global best fitness of thewhole swarm.

III. Experiment

KPSO was compared to PSO to three common testfunctions: Sphere, DejongF4, and Griewank. The firsttwo are unimodal functions while the last one ismultimodal function. The three functions are describedas:

d

Sphere(x) = Lx2i=l

-50< xi <50 (12)This function has only one global minima: (0, .......

0), with minima value is: 0.d

DeJongF4(x) = Zixil

-20.x.<20 (13)This function has only one global minima: (0, ......

0), its value is: 0.1 d d x.

Griewank(x) = 0 EX - Hcos(-') +14000,4

- 600 <xi < 600 (14)This function has only one global minima: (0, ......

0), its value is: 0. If the function's dimensionality is 2, ithas local minima:(3.1400, 4.4384), with value 0.0074.

In all experiment, swarm size is set to 20, thedimensionality is set to 30. Each experiment was run 100times for 2000 iterations, and the results were averaged toaccount for stochastic differences. Table I shows theresults reached by each algorithm.

Table I. PSO vs. KPSO Final ValuesFunction Algorithm Ave_ . B

finess Probability Ave

______ -______ fitness fi83%ss stepSphere KPSO 0.0000 0.0000 100% 863.7ere____ _PSO 1.8572 0.0000 83% 1812.4

DeJongF4 KPSO 0.0000 0.0000 100% 507.8DeJongF4_ - PSO 5.3389 0.0000 63% 1744.6

Griewank KPSO 0.1220 0.0000 94% 713.1l_______w PSO 26.1992 0.0000 53% 1800.1

In the table I, "Probability" means the probability toget the best fitness. It is clear from the table that theKPSO obtains values that are better than the original PSO

algorithm. These results represent a clear and substantialimprovement over the basic PSO.

IV. KPSO's Application in Soft-sensor of Selectivity to AcrylonitrileAcrylonitrile is the main material of polypropylene

fibre. In process of acrylonitrile, the selectivity toacrylonitrile is one important guideline. It is hard andexpensive to detect selectivity to acrylonitrile online. Andthe online way is too time-lapse to reach the producingrequirement. So it has important economic meaning to gett selectivity to acrylonitrile by soft-sensor.

KPSO is a potential NN training algorithm. TheKPSO is applied to train artificial neural networks (NN)to construct a neural networks which we called KPSONN.The text uses KPSONN to construct a practicalsoft-sensor of selectivity to acrylonitrile. Duringanalyzing of technical mechanism, the text chooses 6input signal that includes pressure of reacting,temperature in middle unit, quantity of pure acrylonitrie,air percent, ammonia percent, and velocity of reacting.The output signal of this neural network is selectivity toacrylonitrile.

The text trained the soft-sensor model by 267 groupsof training samples which is processed by error-detection,smoothing, normalization. After training and accounting,average variance is 0.4822%; average value of error is0.3117%. The experiment data describes that the fitprecision between the model's prediction and real data isgood. Therefore, the training process contents theindustrial requirement. Then the text tested thesoft-sensor model by 50 groups of test samples which isnot take part in training and processed by error-detection,smoothing, normalization in same way. The comparisonbetween test result and real data is showed in figure 1.

73.5 -

- real wlues--- predictiw wlue

-i

700 5 10 15 20 25 30 35 40 45 50

sample number

Figure 1. The comparison between test result and real data by KPSONNIn 50 groups of test samples, average variance is

0.2800%; average value of error is 0.2256%. Theexperiment data and figure 1 describe that the

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I-

I

extrapolation precision between the model's predictionand real data is good.

Then PSONN is used to train the same samples, andthe results is not good as KPSONN's. The comparison isshowed in table II.Table II. The comparison between KPSONN and PSONN

modelTraining Triaging Test Testaverage averageaverage value of average value ofvariance erof variance erof

e rror error

KPSONN 0.4822% 0.3117% 0.2800% 0. 2256%model 5.7281% 283% .9% 301PSONN 5.7281% 2.8337% 5.5988% 3.0814%

odel

The table II shows the soft-sensor model based onKPSONN has better precision and performance thanPSONN soft-sensor model. So it has good potential forapplication.

V. Conclusion

KPSO algorithm is a new improved PSO algorithm.The KPSO can fimd the global best solution effectively, andit has better performance and more advantages in manyaspects than PSO. The text uses KPSO to train artificialneural network (NN) to construct a soft-sensor ofselectivity to acrylonitrile. This soft-sensor model has goodprecision and performance. And it has good potential forapplication.

References[1] Kennedy J, Eberhart R. Particle Swarm Optimization[A]. Proc

IEEE Int. Conf on Neural Networks[C]. Perth , 1995:1942-1948.

[2] Eberhart R C, and Shi Y. Particle swarm optimization:developments, applications and resources[A]. Proc. 2001Congress on Evolutionary Computation[Cj. Soul, South Korea,2001:81-86.

[3] K. E. Parsopoulos and M. N. Vrahatis. Recent approaches to globaloptimization problems through Particle Swarm Optimization[J].Natural Computing: Netherlands, 2002:235-306.

[4] Riget, J. Vesterstroem, J.S. A diversity-guided particle swarmoptimizer-the ARPSO. Technical Report 2002-02, Department ofComputer Science, University ofAarhus. 2002.

[5] Angeline, P.J. Using selection to improve particle swarmoptimization. Proceedings of the IEEE Congress on EvolutionaryComputation (CEC 1998), Anchorage, Alaska. 1998.

[6] Clerc, M., Kennedy, J. The particle swarm. Explosion, stability, andconvergence in a multidimensional complex space. IEEETransactions on Evolutionary Computation 6,2002: 58-73.

[7] Shi Yuhui Eberhart R . A modified particle swarmoptimizer[A]. Proc IEEE Int. Conference on EvolutionaryComputation[C]. Anchorage, 1998: 69 73.

[8] Russel, S., Norvig, P. Artificial Intelligence: A Modern Approach.Second edn. Prentice Hall, Englewood Cliffs, New Jersey, 2003

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