Fractional Calculus: From Simple Control Solutions to Complex Implementation Issues
Fractional calculus is currently gaining more and more popularity in the control engineering world. Several tuning algorithms for fractional order controllers have been proposed so far. This chapter describes a simple tuning rule for fractional order PI c
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Fractional Calculus: From Simple Control Solutions to Complex Implementation Issues Cristina I. Muresan
Abstract Fractional calculus is currently gaining more and more popularity in the control engineering world. Several tuning algorithms for fractional order controllers have been proposed so far. This chapter describes a simple tuning rule for fractional order PI controllers for single-input–single-output processes and an extension of this method to the multivariable case. The implementation of a fractional order PI on an FPGA target for controlling the DC motor speed, as well as the implementation of a multivariable fractional order PI controller for a time delay system is presented. Experimental results are given to show the efficiency and robustness of the tuning algorithm. Keywords Fractional calculus • Control algorithm • Multivariable processes DC motor speed control • Multivariable fractional order controller • Decoupling FPGA implementation • Micro-controller implementation • Time delays Experimental results • Robustness
7.1 Introduction Fractional calculus represents the generalization of the integration and differentiation to an arbitrary order. The beginning of fractional calculus dates back to the early days of classical differential calculus, although its inherent complexity postponed its use and application to the engineering world [1]. Nowadays, its use in control engineering has been gaining more and more popularity in both modeling and identification, as well as in the controller tuning. The approach of fractional calculus to modeling is based on the concepts of viscoelasticity, diffusion, and C.I. Muresan () Department of Automatic Control, Technical University of Cluj-Napoca, Baritiu str. 26-28, Cluj-Napoca, Romania e-mail: [email protected] J.A.T. Machado et al. (eds.), Discontinuity and Complexity in Nonlinear Physical Systems, Nonlinear Systems and Complexity 6, DOI 10.1007/978-3-319-01411-1__7, © Springer International Publishing Switzerland 2014
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fractal structures that several processes may exhibit, which are more easily and accurately described using fractional order models [2–6]. In terms of controller tuning, the fractional order PI Dœ controller is in fact a generalization of the classical integer order PID controller. It is generally accepted that the fractional order PI Dœ controller, due to the two supplementary tuning variables, and œ, is able to meet more performance criteria and behave more robustly than the traditional PID controller [7–11]. Several approaches to tuning fractional order PI Dœ exist, with some notable works that use the theory of fractional calculus in controlling both integer order and fractional order dynamical systems [12–15]. Usually, the design of the fractional order controllers is done by imposing various performance criteria that restrict the open loop system to a certain gain crossover frequency, a given phase margin, a boundary on open loop amplification at certain frequencies, or a robustness to open loop gain
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