Fractional Order Control and Its Application of PIaD Controller for Robust Two-inertia Speed Control

This paper deals with the speed control of two-inertia system by fractional order PID controller which means the order of I controller will not only be integer but also can be any real number. The significance of fractional order control is that it is a generalization and “interpolation” of the classical integer order control theory, which can achieve more adequate modeling and clear-cut design of robust control system. However, most of fractional order control researches were originated and concentrated on the control of chemical processes, while in motion control the research is still in a primitive stage. In this paper, a frequency-band fractional order PID controller is proposed to speed control of the two-inertia system, which is a basic control problem in motion control. A frequency-band broken-line approximation method is introduced to realize the designed fractional order PIDcontroller that has a satisfactory accuracy in frequency domain. The better robustness performances of the PID control system against saturation nonlinearity and load inertia variation are shown by the comparison of fractional order PID control’s experimental time responses with integer order PID control’s. The superior robustness and clear-cut control design highlight the promising aspects of applying fractional order control in motion control. keywords fractional order control; two-inertia system, speed control; robustness I. I NTRODUCTION A. History Review Fractional Order Control (FOC) means controlled systems and/or controllers described by fractional order differential equations. Expanding calculus to fractional orders is by no means new and actually had a firm and long standing theoretical foundation. Leibniz mentioned it in a letter to L’Hospital over three hundred years ago (1695). The earliest more or less systematic studies seem to have been made in the beginning and middle of the 19th century by Liouville (1832), Holmgren (1864) and Riemann (1953), although Eular, Lagrange, and others made contribution even earlier [1] [2]. As one of fractional order calculus’s applications in control engineering, FOC was introduced by Tustin for the position control of massive objects half a century ago, where actuator saturation requires sufficient phase margin around and below the critical point [3]. For such kind of 1/s system, taking fractional order α (1 < α < 2) will give a desirable tradeoff between control system’s stability and robustness against saturation non-linearity. Some other pioneering works were also carried out around 60’s by Manabe [4]. However FOC was not widely incorporated into control engineering mainly due to the unfamiliar idea of taking fractional order, so few physical applications and limited computational power available at that time [5]. B. Present Situation In last few decades, researchers found that fractional order differential equations could model various materials more adequately than integer order ones and provide an excellent tool for describing dynamic processes [1] [2] [6]. The fractional order models need fractional order controllers for more effective control of dynamic systems [7]. This necessity motivated renewed interest in various applications of FOC [8] [9] [10]. And with the rapid development of computer performances, modeling and realization of FOC systems also became possible and much easier than before. The researches on FOC are mainly centered in European universities at present. The CRONE (Non-integer order robust control in France) team in France is leaded by Alain Oustaloup and Patrick Lanusse from Bordeaux University, France. Their practices include applying FOC into car suspension control, flexible transmission, hydraulic actuator etc. Denis Matignon, a researcher from ENST, Signal Dept.& CNRS, URA, France, contributed to the theoretical aspects of FOC concept, such as stability, controllability, and observability of the fractional order systems. Slovak researchers, Ivo Petras and Igor Podlubny from the Technical University of Kosice, are well-known for their efforts in modeling, realization and implementation of fractional order systems. J. A. Tenreiro Machado and Yangquan Chen, from Polytechnic Institute of Porto, Portugal, and Utah State University, Logan, are playing important roles in developing the implementation methods for fractional order controllers and applying FOC i robotics control, disturbance observer, etc. Fractional differentiation’s applications in automatic control is now an important issue for the international scientific community. The First Symposium on Fractional Derivatives and Their Applications (FDTA) of the 19th Biennial Conference on Mechanical Vibration and Noise was held from September 2 to September 6, 2003 in Chicago, IL, USA [11]. This conference was part of the ASME 2003 Design Technical Conferences. 29 papers concerning FDTA in Automatic Control, Automatic Control and System, Robotics and Dynamic Systems, Analysis Tools and Numerical Methods, Modeling, Visco-elasticity and Thermal Systems were presented in the symposium. A sub-committee called “Fractional Dynamic System” under ASME “Multi-body Systems and Nonlinear Dynamics” committee was formed during the symposium. And first IFAC Workshop on Fractional Differentiation and its Applications will be held in this year’s summer, July 19-21, in Bordeaux, France [12]. The following areas will be covered by the workshop: Representation tools, analysis tools, synthesis tools, simulation tools, modeling, identification, observation, control, vibration insulation, filtering, pattern recognition, edge detection. Besides the presentation of theoretical works and applications, this workshop can also give rise to benchmark, testing bench and software presentations. The article is organized as follows: in section II, mathematical aspects of fractional order control are mentioned; in section III, a integer order PID controller is designed for the speed control; in section IV, a frequency band PID controller is proposed and its broken-line realization method is also introduced; in Section V, Experimental results are presented to show the robustness of proposed fractional orderPID controllers. Finally, in section V, conclusions are drawn. II. M ATHEMATICAL ASPECTS A. Mathematical Definitions The mathematical definition of fractional derivatives and integrals has been the subject of several different approaches[1][2]. The most frequently encountered definition is called Riemann-Liouville definition, in which the fractional order integrals are defined as t0D −α t = 1 Γ(α) ∫ t t0 (t− ξ)α−1f(ξ)d(ξ) (1) while the definition of fractional order derivatives is t0D α t = d dtn [ t0D −(n−α) t ] (2)