Proceedings of 2015 RAECS UIET Panjab University Chandigarh 21-22nd December 2015
FOPID Controller Optimization Employing PSO and TRSBF Function Rohit Gupta
Sanjay Gairola
Dept. of Electrical Instrumentation and Control Engineering Thapar University, Patiala, Punjab, India e.mail:
[email protected]
Dept. of Electrical and Electronics Engineering NIET, Gr. Noida, U.P., India e.mail:
[email protected]
Abstract—This paper presents a technique of PID controller tuning for fractional order industrial process and proposes a Time-Response Specification Based Fitness (TRSBF) function required by Particle Swarm Optimization (PSO) for the controller tuning. The results are compared with the conventional Integral Time Absolute Error (ITAE) based and analytic controllers. The performance and robustness is analyzed in time-domain and frequency-domain to establish the suitability of proposed fitness function. MATLAB/Simulink has been employed for PSO based tuning algorithm validation and comparison of parameters like peak overshoot, settling time, disturbance rejection time, stability in frequency domain, etc. Keywords— Fitness Function; Fractional order controller; Heating Furnace; Particle Swarm Optimization
I. INTRODUCTION There is an increasing interest in dynamic systems which are represented by differential equations of non-integer orders, called fractional order (FO) systems. Such systems are employed in extending the order of derivatives and integrals from integer to non-integer order due to its advantages of robustness and improved controllability [1]. Leibnitz introduced the concept of fractional order system over three hundred years ago and the systematic studies were made by Liouville, Riemann and Holmgren in the 19th century. In 1974, K. B. Oldham and J. Spanier presented theoretical and practical aspects of computing methods for mathematical modeling of non-linear systems by considering number of computing techniques. These methods mainly involved operator approximation, operator interpolation techniques including a non-Lagrange interpolation. Fractional order control has been also tried in the last decade for controlling various types of processes. Some design approaches have been proposed to improve the closed loop performance of systems and these have been tested by researches using different simulation conditions [2]. The advantages of fractional order controllers in terms of robustness and controllability have motivated renewed interest in several applications of fractional order control [3]. In industries, the process control techniques have made some advancement and strategies like adaptive control, fuzzy control, neural control and neuro-fuzzy control are becoming popular [4-7]. However in most of these cases, the control strategies mainly used is feedback with Proportional-Integral-Derivative (PID) controller in some way. Although PID controller alone
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can achieve adequate results, advanced control techniques capable of improving their benefits are very much required in the automotive field. In the past few years, fractional-order PID (FO-PID) controllers which are the generalization of traditional PID controllers, are recognized to guarantee better closed-loop performance and robustness with respect to the classical controllers [8]. The tuning of these FOPID controllers is also crucial and some methods have been proposed which are analytic [9] while some of them are based on numerical optimization algorithms. The neural network based method is also available to tune the controller but have disadvantage like unavailability of training data. To overcome such problem nature inspired optimization techniques can be used and Particle Swarm Optimization (PSO) is one of them. The PSO technique can generate a high-quality solution within shorter calculation time and has stable convergence characteristic compared to other stochastic methods, by selecting proper fitness function [10-11]. In this paper, electric furnace is used as an industrial application. In industries, the electric furnaces are used for brazing, annealing, carburizing, forging, galvanizing, melting, hardening, enameling, sintering, and tempering metals, copper, steel and iron and alloys of magnesium. It is the central process of any mini mills, which produce steel mainly from scrap. A typical electric arc furnace generally operates at high power levels from 10MW to 100MW. The power level is directly related to production throughput, so it is very important to control the furnace efficiently [12]. An example of a heating furnace is described by Podlubny et al. [13] in year 1997 that had been referred by many researchers. The same furnace model has been used in this paper for analysis, for that Merrikh et al. [9] proposed an analytic method where some parameters are chosen by trial and error, the controller proposed by Merrikh et al. [9] is given by equation,
Gc ( s ) = K ∗ s
0.31
+ K ∗a + K ∗b∗ s
−0.01 X
In above transfer function of controller value of X will be either 50 or 100, and all the other parameters are user dependent, which is a major drawback of this method. This paper develops a PSO based FOPID controller to search the optimal FOPID parameters. A Time-Response Specification Based Fitness (TRSBF) function is proposed
Proceedings of 2015 RAECS UIET Panjab University Chandigarh 21-22nd December 2015 which is employed in PSO algorithm for tuning the FOPID controller. The PSO based tuning algorithm has been developed in MATLAB/Simulink for validation and comparison of performance indices like peak overshoot, settling time, disturbance rejection time, stability in frequency domain, etc. II. FRACTIONAL ORDER CONTROLLER The classical PID controllers have three parameters Kp, Ki and Kd however, fractional-order PID (FO-PID) controller introduces two additional adjustable parameters λ and μ (λ and μ are non-integer orders of derivative and integral terms). The differential equation of a fractional-order PID controller is given as: −λ
μ
(1)
y (t ) = K p e (t ) + K i Dt e(t ) + K i Dt e(t )
where y(t) is an output of the controller, e(t) is error which is used as input of the controller Kp, Ki, Kd are the proportional, integral and differential gain respectively, λ and μ are real number (λ, μ>0) which defines the order of fractional PID −λ
μ
controller and Dt , Dt are differintegral operator, defined as (2) [14].
⎧ dα ⎪ α ⎪ dt ⎪ α 1 a Dt = ⎨ ⎪t ⎪ ( dτ )−α ⎪⎩ ∫a
α >0 α =0
(2)
α
