Deepyaman Maiti and Amit Konar Department of Electronics and Telecommunication Engineering, Jadavpur University, Kolkata - 700 032 E-mail: deepyamanmaiti@gmail.com, konaramit@yahoo.co.in

Abstract System identification is a necessity in control theory. Classical control theory usually considers processes with integer order transfer functions. Real processes are usually of fractional order as opposed to the ideal integral order models. A simple and elegant scheme is presented for approximation of such a real world fractional order process by an ideal integral order model. A population of integral order process models is generated and updated by PSO technique, the fitness function being the sum of squared deviations from the set of observations obtained from the actual fractional order process. Results show that the proposed scheme offers a high degree of accuracy.

1. INTRODUCTION Proper estimation of the parameters of a real process, fractional or otherwise, is a challenge to be encountered in the context of system identification [1], [2]. Accurate knowledge of the transfer function of a system is often the first step in designing controllers. Computation of transfer characteristics of the fractional order dynamic systems has been the subject of several publications, e.g. by numerical methods [3], as well as by analytical methods [4]. Many classical statistical and geometric methods such as least square and regression models are widely used for real-time system identification. The problem of system identification becomes more difficult for a fractional order system compared to an integral order one. The real world objects or processes that we want to estimate are generally of fractional order [5] (for example, the voltage-current relation of a semi-infinite lossy RC line or diffusion of heat into a semi-infinite solid). The usual practice when dealing with such a fractional order process is to use an integer order approximation. In general, this approximation can cause significant differences between a real system and its mathematical model. However, the concept of approximation of a real fractional order process by an integral order one is not without its merits, provided the approximation is sufficiently accurate. Classical control theory deals with integral order processes. A good approximation would enable us to analyze and control fractional order processes with the conventional theory. In this paper, we propose a general method for the estimation of parameters of a fractional order system using PSO technique. PSO, a stochastic optimization strategy from the family of evolutionary computation, is a biologically-inspired technique originally proposed by Kennedy and Eberhart. PSO offers optimal or sub-optimal solution to multi-dimensional rough objective functions. We use this technique to find the integral order process model whose outputs match the set of observations from the actual fractional order system most closely. This method enables us to achieve a very good approximation. It is necessary to understand the theory of fractional calculus in order to realize the significance of fractional order integration and derivation.

2. THEORY OF FRACTIONAL CALCULUS The fractional calculus is a generalization of integration and derivation to non-integer order operators. At first, we generalize the differential and integral operators into one fundamental operator α t a D where: ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ < α ℜ τ = α ℜ

α ℜ

∫ α − α α t a α t a 0 ) ( , ) d ( 0 ) ( ,1 0 ) ( , dt d D (1)
The two definitions used for fractional differintegral are the Riemann-Liouville definition [6] and the Grunwald-Letnikov definition. The Riemann-Liouville definition is given as τ τ − τ α − Γ

∫ + − α α d ) t( ) ( f dt d ) n ( 1 )t( f D t a 1 n n n t a (2)
for ) n 1 n ( < α < − and Γ(x) is Euler’s gamma function. The Grunwald-Letnikov definition is ) jh t( f j )1 ( h 1 lim )t( f D h a t 0 j j 0 h t a − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛α −

∑ ⎥⎦ ⎤ ⎢⎣ ⎡−

α → α (3)
IEEE Sponsored Conference on Computational Intelligence, Control And Computer Vision In Robotics & Automation © IEEE CICCRA 2008 150 where [ ] x means the integer part of x. Derived from the Grunwald-Letnikov definition, the numerical calculation formula of fractional derivative can be achieved as: [ ] ) jh t( x b h )t( x D T L 0 j j t L t − ≈ ∑

α − α − (4)
where L is the length of memory

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