System Parameter Estimation using PSO Algorithm

System Parameter Estimation using PSO Algorithm

Arun M K[1], Biju U[2], Neeraj Nair Rajagopal[3], Prof. Bagyaveereswaran.V[4], SELECT,VIT UNIVERSITY, Vellore

AbstractThe paper proposes a new method of identifying a system by sample data. The identification technique involves the process of parameter estimation and application of PSO algorithm. Particle swarm optimization (PSO) aims to realize both structure and parameter identification of system in real time. The paper explains how the PSO is a better option for system identification process than other conventional algorithms.

Index TermsPSO, combinatorial optimization, System Identification.

I.INTRODUCTION

System identification[1] is a challenging task which determines a model from a set of several models based on the input and output data. It is an advanced process which defines a system mathematically through its physical working. Sample data based system identification is the most used method in these processes. System Identification can be done through analytical approach or through experimental approach. New identification methods have also been developed using genetic algorithm[2], neural network,fuzzy logic, wavelet transform, et al. These methods have limitations that they fail when the structure and parameters of the systems are completely unknown. Optimization algorithms have helped in solving many problems like estimation bias, controller design, et al. Hence, we consider Particle Swarm Optimization(PSO) algorithm for identification problem.

Particle Swarm Optimization algorithm[3]is an evolutionary technique used for optimization put forwarded by Kennedy and Eberhart in 1995. The algorithm is based on the predation of birds and swarm intelligence. The main attraction of this algorithm includes simple principle, fast convergence and easy implementation. The rapid convergence property and simple computation makes it applicable to various fields of studies. PSO is useful for parameter estimation which is a key step towards identification of systems[7].

A system model is comprised of several mathematical models through inter combination. A fitness function is defined for each sub-model which is evaluated in the main program to find a local optimum and consequently a global optimum for the parameters of the system. Thus by using PSO algorithm, the structure and parameters of the model are identified. The effectiveness of

the method is verified by using simulation experiment of the model.

1. PARTICLE SWARM OPTIMISATION

   This comes under the category of global random optimization[4]. Here every solution is considered as a particle or bird in a search space or region. With an optimized function we determine fitness function of all particles given. The fastness of determining distance and the direction differs for every particle. For every iteration, iteration of particle depends on two extrema i.e. local extremum and global extremum. If both extrema are known, particle updates its velocities and position according to the formula,

   ( + ) = ( )+ ( – ( )+ (

   () (1)

   ( + ) = ( + )+ () (2)

   Where,

   inertial weight

   () component of dimension no.j of velocity vector of particle i at timet,

   xij (t) – component of dimension no.j of theposition vector of particlei at time t,

   – component of dimension no.j of the historicallyoptimal position vector of particle i at time t,

   – component of dimension no.j of thehistorically optimal position vector of particle swarmat time t,

   ,- two random numbers between 0 and 1

   ,- group cognitive functions(correction factors).

2. DESIGN OF FITNESS FUNCTION

   The aim of the system identification process is to fit a sample model with the original system in an optimal way.It means that the calculated system output ymkshould match the actual system outputyk as much as possible. So a criterion functionshould be taken for the validation of each output withthe actual one[5]. The cost function used here is,

   Where

   n

   2

   y ( yk

   k 1

   • ymk )

   (3)

   For experimental proof,we consider a first order continuous stirred tank reactor(CSTR) which is isothermal and has a constant volume as shown in figure(1). Here

   ymk – the model output result of the sample data of ith set of input-output

   yk – the actual output of known model for ith set of input- output.

3. PROCEDURE FOR SYSTEM IDENTIFICATION

   Fi,Fo- Inlet,outlet flow

   i,o- Densities of the inlet and outlet flow

   -Density of the mixture

   Ti , To-Temperature of inlet and outlet flow

   T-Temperature of the mixture

   C C – Specific heat of inlet and outlet fluid

   pi , po

   System identification using PSO requires a lot of pre-processing stages before the actual identification[7]. Below given are the main steps undertaken to achieve the

   Cp-Specific heat of the mixture

   From first principle the balance equation for the CSTRprocess is given as

   same:

   Step 1: This step consists of initialization of PSO

   A

   = FAO

   – FA

   – kV A(4)

   parameters like number of birds,maximum number of

   bird steps,dimension of the problem(no. of parameters),

   By rearranging we get the following equation,

   correction factorsand inertia.

   Step 2: Initialising random variables r1 and r2 and the

   C

   A

   + ( + )C

   A

   = C

   AO

   (5)

   current fitness function.

   Step 3:Initialize the swarm velocities and its positions. Current position is defined and the local best position is calculated. Velocity of the particle is also initialised.

   Step 4: Evaluate the initial population using the initialized parameters and the current position and proceed to next step.

   Hence the time constant of the system can be deduced as

   1

   V

   = F + k

   Writing in terms of residence time, we get

   =

   Step 5: Now we evaluate for a new swarm where a current fitness function is defined through the initialized parameters and input-output data set. Assign it as the local best fitness and compare with the global best. If the new one is a global best then proceed to Step 6 or go back to Step 2.

   That is

   = 1

   1 +

   R

   =

   1+

   (6)

   Step 6: Velocity update- Velocity of each particle is updated using the equation (1) when the global best is

   found.

   The rearranged model of the system is then

   + 1 C = 1 C (7)

   Step 7: Swarm update- current position is updated using

   A A R AO

   the velocity obtained from the previous step. The whole procedure is repeated for n iterations and an optimized

   Converting eqn. (4) in transfer function form,

   function is obtained.

   = 1

   1+ (S+1)

   =

   S +1

   (8)

4. PROBLEMDESCRIPTION AND ANALYSIS

   Let =k=1, = 0.5 and = 1 we can deduce that,

   R

   1+

   == 0.5

   0.5S+1

   (9)

   Figure(3) shows the response of the system given in equation (9) which is the reference curve.

   From MATLABSystem Identification Toolbox,we get

   1

   = 0.511

   4.6904 s+1

   (10)

   Figure(1). Experimental setup of CSTR

   That is =0.511and= 4.6904

   Figure(3) gives the response for the above transfer function.Equation (9) an (10) infers that the estimated model from the system identification toolbox is an inaccurate estimation.

   While applying PSO algorithm to same set of input- output datathrough MATLAB,we get

   A=1.8111and B=3.6894 (11)

   Where A/B=K and 1/B=

   Hence the estimated transfer function would be of the form

   Size of the swarm = 56,

   Maximum number of bird steps = 56, Dimension(parameters) of the problem = 2 Correction factor 1= 1.2,

   Correction factor 2 =0.12 and Momentum or inertia = 0.9.

   Parameter s\Method	Actual Model	SI Toolbox Estimatio n	LQR Estimatio n	PSO Estimatio n
   K	0.5	0.5112	0.5110	0.492
   	0.5	4.6904	4.6901	0.274

   Table 1 showsestimated parameters given for different estimation techniques :

   2 =

   0.4962

   0.274+1

   (12)

   Table (1) Comparison of Estimated Parameters

   Figure(3) depicts the unit step response for the estimated model. Equation (9) and (12) implies that PSO algorithm has produced a better approximation than the other methods.

5. SIMULATION RESULTS

6. CONCLUSION

   There is large gap between the estimated values computed by various methods. The modelobtained from PSO is the most accurate one compared to that obtained from other algorithms.So, system identification is found to be more accurate when we use PSO algorithm. Also, this algorithm can be used with more advanced algorithms to produce hybrid ones which may produce a more accurate result.

7. REFERENCES

Figure(2) Block diagram of simulation of estimated models.

Figure(3) Unit step response ofactual model and estimated models.

For the given problem of CSTR, the following specifications are considered to compute PSO algorithm[6].

1. Y.J. Li, Theory and application of system Identification, National Defence Industry press,Beijing 2003.

2. S.A Liu,F.Tang, Study on system Identification method based on Genetic Algorithms,System Engineering Theory and practise,2007.

3. J.Kennedy, R.Eberhart, Particle Swarm Optimization, Proceedings of IEEE International Conference on Neural Networks, Perth 1995.

4. Y. Shi, R. Eberhart, A modified particle swarm optimizer, Proceedings of IEEE International Conference on Evolutionary Computation, Piscataway, USA:IEEE 1998.

5. X.Q Deng, Application of particle Swarm Optimization in point- pattern Matching, Computing technology and Automation Conference, Beijing 2008.

6. G.M Chen, J.Y. Jia, Study on the strategy of decreasing inertial weight in particle swarm optimization algorithm, Journal of Xian Jiaotong University, 2006.

7. Qibing Jin, A Novel Hybrid PSO Identification Method, International Journal of Applied Mathematics and Information Sciences, Beijing 2013.