The Application of Stochastic Optimization Algorithms to the Design of a Fractional-order PID Controller

978-1-4244-2806-9/08/$25.00© 2008 IEEE

1 The Application of Stochastic Optimization Algorithms to the Design of a Fractional-order PID Controller Mithun Chakraborty, Deepyaman Maiti, and Amit Konar
Department of Electronics and Telecommunication Engineering Jadavpur University Kolkata, India mithun.chakra108@gmail.com, deepyamanmaiti@gmail.com, konaramit@yahoo.co.in

Abstract—The Proportional-Integral-Derivative Controller is widely used in industries for process control applications. Fractional-order PID controllers are known to outperform their integer-order counterparts. In this paper, we propose a new technique of fractional-order PID controller synthesis based on peak overshoot and rise-time specifications. Our approach is to construct an objective function, the optimization of which yields a possible solution to the design problem. This objective function is optimized using two popular bio-inspired stochastic search algorithms, namely Particle Swarm Optimization and Differential Evolution. With the help of a suitable example, the superiority of the designed fractional-order PID controller to an integer-order PID controller is affirmed and a comparative study of the efficacy of the two above algorithms in solving the optimization problem is also presented. Keywords-Differential evolution; dominant poles; integer-order and fractional-order PID controllers; particle Swarm Optimization I. INTRODUCTION
The merit of using a Proportional-Integral-Derivative (PID) controller lies in its simplicity of design and good performance, including low percentage overshoot and small settling time (which is essential for slow industrial processes). PID controllers belong to the class of dominating industrial controllers and, therefore, continuous efforts are being made to improve their quality and robustness. An elegant way of enhancing the performance of PID controllers is to use fractional-order controllers where the I- and D-actions have, in general, non-integer orders. In order to grasp the significance of fractional-order PID controllers, an understanding of the theory of fractional calculus is necessary. Fractional calculus is that branch of mathematical analysis [13], which generalizes the order of the derivative or integral of a function to a real number (not necessarily an integer). If D denotes first-order differentiation, then, we know D2 denotes two iterations of differentiation. Likewise, D1/2 may be interpreted as some operator which, when applied twice to a function successively, will have the same effect as a single differentiation [13]. Similar explanations hold for fractional integration too. Just as first- order differentiation (or integration) of a function in time- domain maps to multiplication by s1 (or s-1) of the Laplace Transform of the function in s-domain, sα indicates time- domain derivation to the order α if α > 0 or time-domain integration to the order |α| if α < 0. The name given to this generalized differential/integral operation is differintegration. Of the several definitions of fractional differintegrals, the Grünwald-Letnikov and Riemann-Liouville definitions [14] are the most used. These definitions are required for the realization of discrete control algorithms.
In a fractional PID controller, besides the proportional, integral and derivative constants, denoted by Kp, Ti and Td respectively, we have two more adjustable parameters: the powers of s in integral and derivative actions, -λ and δ respectively. As such, this type of controller has a wider scope of design, while retaining the advantages of classical PID controllers. Finding the appropriate settings of the values of the five parameters p i d {K ,T ,T ,λ,δ} to achieve optimal system performance thus calls for optimization on the five-dimensional space. Classical optimization techniques are not applicable here because of the roughness of the multidimensional objective function surface. We, therefore, use derivative-free optimization techniques: the first one –– Particle Swarm Optimization (PSO) –– draws inspiration from the intelligent, collective behavior of a swarm of social insects (particularly bees) foraging for food together and the other –– Differential Evolution (DE) –– is an evolutionary algorithm that is guided by the principles of Darwinian Evolution and Natural Genetics [12]. Traces of work on fractional-order PID controllers are available in the current literature [1]-[9] on control engineering. A frequency domain approach based on the expected crossover frequency and phase margin is mentioned in [2]. A method based on pole distribution of the characteristic equation in the complex plane was proposed in [5]. A state-space design method based on feedback poles placement can be viewed in [6]

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