A New Analytic Method to Tune a Fractional Order PID Controller

T his paper proposes a new method to tune a fractional order PID controller. This method utilizes both the analytic and numeric approach to determine the controller parameters. The control design specifications that must be achieved by the control system are gain crossover frequency, phase margin, and peak magnitude at the resonant frequency, where the latter is a new design specification suggested by this paper. These specifications results in three equations in five unknown variables. Assuming that certain relations exist between two variables and discretizing one of them, a performance index can be evaluated and the optimal controller parameters that minimize this performance index are selected. As a case study, a third order linear time invariant system is taken as a process to be controlled and the proposed method is applied to design the controller. The resultant control system exactly fulfills the control design specification, a feature that is laked in numerical design methods. Through matlab simulation, the step response of the closed loop system with the proposed controller and a conventional PID controller demonstrate the performance of the system in terms of time domain transient response specifications (rise time, overshoot, and settling time)


INTRODUCTION
The fractional order PID controller (also called ) was proposed by Podlubny, 1994 and Podliubny, 1999 and has a transfer function where , , and and are the parameters of the controller that must be tuned.Parameters and increase the degree of freedom in tuning the controller, which makes the design of the control system more flexible.Fractional order PID controllers have less sensitivity to parameter variation due to these two additional parameters, Zhao, et al  The remaining of this paper is organized as follows: In section 2, the proposed tuning method is presented, in section 3, a design and simulation example is presented to demonstrate the application of this method, and in section 4, the conclusions are drawn from the simulation results.

THE PROPOSE TUNING METHOD
Consider the unity feedback control system shown in Fig. 1.The transfer functions of the controller and the plant are ( ) and ( ), respectively.The sinusoidal transfer function of the controller is Substituting Eq. ( 2) in Eq. ( 3) and equating the real part and imaginary part of both sides yields The steady state error should be zero; therefore, (Final value theorem).The two relations and are assumed to exist between and ; the reason for choosing these relations is to get the sine and cosine functions for the same angle, namely, .i) First, assume that (6) There is one degree of freedom in choosing and (choose one and evaluate the other) as shown in Fig. 2. Substituting Eq. (6) in Eq. ( 4) and Eq. ( 5) yields ( ) Now, take discrete values of , from 0 to 1, say 0.01, 0.02, …, 1 (100 values).For each value of , solve Eq. ( 7) and Eq. ( 8) for and in terms of and .
At the resonant frequency of the plant , the magnitude of the open loop transfer function is Squaring both sides of Eq. ( 11) yields Substituting Eq. ( 9) and Eq.(10) in Eq. ( 12), the value of is obtained as follows where , ( ( ) ), ( ( ) ) The performance index that will be used to select the optimal value of ( ) ∫ ( ) ii) Second, assume that (15) Substituting Eq. ( 15) in Eq. ( 4) and Eq. ( 5) yields For the same discrete values of , solve Eq. ( 16) and Eq. ( 17) for and in terms of and .

DESIGN AND SIMULATION
Consider applying the proposed design procedure to the plant given by the transfer function The resultant controller is ) are all achieved by this controller.The step response of the close loop system with this controller is shown in Fig. 5.For the purpose of comparison, a conventional PID controller is designed using matlab pidtune command, which designes a PID controller for a given transfer function.The transfer function of this controller is The step response of the close loop system with the PID controller is shown in Fig. 6.Table 1 shows the transient response specifications of the two systems.The system with the controller has better percentage overshoot, delay time, and rise time than that with the PID controller, while the settling time is much greater.This is because the specifications that are fulfilled by the controller are the gain crossover frequency which enhances the rise time and delay time, and the phase margin which enhances the percentage overshoot.

CONCLUSIONS
A conclusion can be drawn from the design procedure of this paper that if the domain of some of the design variables is restricted by a certain mathematical relation (restriction) between these variables ( and in this case), an analytic, rather than a numerical, solution can be obtained.Unlike the optimization problems that may be subjected to certain constraints (such as the parameters should be positive), the analytic solution gives exact solution of the design specification equations; this solution may be positive, negative, or even a complex number.This is evident in the value of , which is negative.While a controller that is tuned by optimization techniques can fulfill the design specifications with some sufficiently small error, the analytically tuned controller fulfill the design specifications exactly, since it is the solution of a set of simultaneous equations; this is evident in enhancing the percentage overshoot and rise time of the closed loop system.= set of positive real numbers.= fractional order of integration of the controller, dimensionless.= optimal value of , dimensionless.= fractional order of differentiation of the controller dimensionless.= optimal value of , dimensionless.= gain crossover frequency of the open loop transfer function, radian/s.= resonance frequency of the plant, radian/s.= real part of a complex number.= imaginary part of a complex number.
Badri, and Tavazoei, 2013 solved the equations graphically by finding the intersection point of two curves.Sadati, 2007 used a performance index in time domain to determine the controller parameters, while Tepljakov, et al., 2015 used a performance index in frequency domain to determine these parameters.Lazarevic, 2013 tuned the controller parameters using genetic algoritm to minimize a performance index.The objective functions in such problems have complex surfaces such that the analytic methods of optimization often fail Dorcak, et al, 2006.Another approach to tune a fractional order PID controller was given in Xue, et al., 2006, Khalil, et.Al., 2009, and Joshi and Talange, 2013, by taking cerain values for the fractional order of integration and differentiation and finding the optimal values for the remaining gain parameters.
The bode plot of the open loop transfer function is shown in Fig. 4. The control design specifications ( , , and


Xue, D., Zhao, C., and Chenm, Y., 2006, Fractional  Order PID Control of a DC-Motor with Elastic Shaft: A Case Study, Proceeding of the 2006 Ammerican Control Conference, Minneapolis, Minnesota, USA, June 14-16, pp.3182-3187. Zhao, C, Xue, D., and Chen, Y, 2005, A FOPID Tuning Algorithm for a Class of Fractional Order Plants, IEEE International Conference on Mechatronics and Automation, vol.1, pp.216-221.NOMENCLATURE e= error between the desired and actual output.j= imaginary unit (= √ ).= derivative gain of the PID or controller, dimensionless.= integral gain of the PID or controller, dimensionless.= proportional gain of the PID or controller, dimensionless.=magnitude of the open loop transfer function at the resonant frequency, dimensionless.( ), ( )= performance indices.R= set of real numbers.

, and Costa, 2010. In Zhao, et. al, 2005 and Caponetto, et al., 2004, the
controller parameters were derived analytically to achieve gain (phase) margin and phase (gain) crossover frequency specifications.

In Monje, et al., 2008, one
of the equations was taken as an objective function and the rest of the specifications were taken as constraints.