Design Templates for Some Fractional Order Control Systems

Time domain characteristics of first and second order systems are well known. But the same simplicity and explicitness do not exist for low order fractional order systems (FOSs). Considering the step response, the templates are developed for designing the behavior of simple FOSs with a 2-term denominator polynomial (one is unity and the other involves fractional power). Although the explicit relations between design parameters and the performance parameters such as time constant, rise time, overshoot, settling time for fractional order control systems (FOCSs) do not exist and can't be obtainable as in the ordinary integer order control systems, the obtained templates in this paper can be used for designing low order FOCSs. Hence, the drawback of non-existence of similar explicit formulas for FOCSs is eliminated by using these templates.


I. INTRODUCTION
FOSs have founded many applications in the last two decades and a great deal of literature has appeared for analyzing and designing these systems [1][2][3][4][5][6][7]. Especially fractional order proportional integral derivative (FOPID) controllers have appeared very frequently in control system design [8][9][10]. Focusing on some very recent literature, for example [11] proposes an adaptive FOPI control method based on enhanced virtual reference feedback tuning to meet high precision and speed requirements for controlling flexible swing arm system in the light-emitting diode (LED) packaging industry. E. Cokmez et. all have obtained and visualized stability regions based on specified gain and phase margins for a FOPI controller to control integrating processes with time delay [12]. J. R. Nayak and B. Shaw have shown how to enhance the performance of the automatic generation control by adopting cascade proportional derivative (PD) -FOPID controller in a two-area mutually connected thermal power plant with generation rate constraint; group hunting search algorithm is adopted to explore the gain parameters of the controllers [13]. In [14], PI controller design is performed by using optimization for FOSs; first, controller parameters for a stable control are calculated by using the stability boundary locus method and then optimization is used to provide the best control. In [15], a new robust FOPID controller to stabilize a perturbed nonlinear chaotic system on one of its unstable fixed points is proposed based on the PID actions using the bifurcation diagram. In [16], fractional-order discrete synchronization of a new fourth-order memristor chaotic oscillator and the dynamic properties of the fractional-order discrete system are investigated; a new method for synchronizing is proposed and validated. In spite of the existence of a great deal of publications about FOSs some of which have just been mentioned above, most of the present analysis and design techniques deal with sophisticated and rather special applications [17][18][19][20][21][22][23][24]. Although the step response characteristics such as rise time, settling time, delay time, overshoot and some others are well known by explicit formulas for simple integer order systems [25], such formulas are not available for FOSs. And a compact publication yielding the relations between the design parameters and the step response characteristics of even simple FOSs are not yet present. The purpose of this paper is to fulfill this vacancy and to supply some design tools for simple order FOSs. The paper is organized as follows; in Section 2, basic definitions of time domain characteristics of first and second integer order systems are given. Section 3 introduces the FOSs that is studied and the investigation of its step responses depending on the fractional power. Section 4 gives and discusses the templates that can be used for the design of low order FOSs. Finally, Section 5 finishes with conclusions . (1) It is assumed that 1 ≥ 0 for stability. Since the study is confined to step response characteristics, it is easily obtained by applying the unit step input ( ) = 0 for < 0, ( ) = 1 for ≥ 0, and compute the step response y(t) as The variation of the step response is shown in Fig. 1. It is an increasing exponential starting from 0 at = 0, and rising to the steady-state value of 1 as → ∞. The following time domain characteristics are defined for a response of the type shown in Fig. 1. ( 3b ) Settling time TS: It is the time required for the response to stay around its final value with an error less than 2 %. = 50 = 3.912023 ≅ 4 . (3c) Note that all the time characteristics depend on only the coefficient = 1 in Eq. (1). So, 1 is chosen according to satisfy all the specifications on , , . Note also that for 1 = 0, the system is a unity gain system which yields ( ) = ( ); that is all the characteristic times , , are zero and no delay occurs at the response. In summary, the following properties of time domain characteristics are valid: i) The response increases exponentially to its steady state value without any oscillations; ii) Rise time and settling time are some multiples of time constant = 1 . It is well known that a time domain normalization with respect to corresponds to replacing by / which also corresponds to frequency domain normalization by writing instead of . In Eq. (1), if we replace by / 1 where = 1 is the time constant, then we have which has normalized time constant = 1.
As the reference transfer function for a second order system is considered [25]; obviously dc gain is equal to 1; is undamped natural frequency and ≥ 0 is the damping ratio. For > 1 which corresponds to overdamped case, Eq. (5) yields the step response which starts from 0 and rises to 1 monotonically as shown in Fig. 2 (zeta=2.0). In this expression, there are two time constants 1 and 2 so that where s1 and s2 are the poles of transfer function. It can be shown by using (6b) that is the period of sustained oscillations of the undamped ( = 0) system. For ξ = 1, the system in Eq. (5) is said to be critically damped; in this case there is only one time constant which is = 1/ = /2 . The step response is found as which increases monotonically from 0 to the steady-state value 1 as shown in Fig. 2 (zeta=1.0). For 0 < < 1, the system is said to be underdamped; for = 0.2 the step response is shown in Fig. 2 (zeta=0.2). It is seen that the response is stable and approaches to the reference value 1 in a damped oscillatory manner. It is a routine process to show that the response is given by Finally, for = 0, the system is undamped and the step response is which represents sustained oscillations as shown in Fig. 2 (zeta=0) with undamped natural frequency = 1 and undamped oscillation period = 2 .
For the step responses corresponding to underdamped case two new time characteristics are defined. The oscillation period from Eq. (8) is The settling time is approximately obtained from Eq. Another time which is important is when the first peak occurs in the response. From Eq. (8) and ( ) = are found to be The overshoot ℎ and percent overshoot ( For the succeeding peaks, Eqs. (12a), (12b), (12c) are modified by replacing by ( 2 − 1 ) where represents the peak numbers; for the first peak = 1. The reduction ratio ( ) is defined as the ratio of successive overshoots, and from Eq. (11c) (11d) For the undamped system ( = 0), Eq. (9) yields the following time domain characteristics:  Step response of FOS in Eq. (14b) for different values of . 1) It is obvious that dc gain of this system is 1, which results from Eq. (13b) by inserting = 0 and assuming ≠ 0. Therefore, all the step responses for stable cases will tend to 1 as → ∞; ( = 0.5, 1.2, 1.7).
2) For = 0, the transfer function will be equal to a constant gain of 1/2, hence the step response is 0.5 ( = 0). 3) For = 1, the transfer function is equal to a first order integer type transfer function, hence, the step response is an increasing exponential with a time constant = 1 ( = 1, see als o Fig. 1). 4) For = 2, the fractional system in Eq. (13b) is equivalent to the second order integer type system in (5) with = 1, = 0; hence, the step response is sustained oscillation ( = 2, see also Fig. 2).

5) For
> 2, the system is not stable, and the step response increases exponentially (and oscillatory)-like manner ( = 2.018). 6) For 1 < < 2, the system is stable, and it has step responses ( = 1.2, = 1.7). The first of these responses ( = 1.2) is a decaying curve after an overshoot; and the second ( = 1.7) is an oscillatorylike motion with exponentially-like decaying. 7) For 0 < < 1, step response is a stable exponentially-like increasing behavior ( = 0.5). We note that those responses for = 2.018, 1.7, 0.5, resemble to those of a second order (for = 2.018, 1.7) and of a first order ( = 0.5) integer order systems. But the explicit formulas as in Eqs. (3,10,11,12) between the system parameter and the step response characteristics do not exist for the considered FOSs. So, in the following section, instead of using explicit formulas, some templates are obtained to be used for designing FOSs.      with the damping ratio of a second order system. This plot is useful for finding the fractional order and damping ratio     V. CONCLUSIONS Time domain characteristics of the FOS with a 2-term denominator polynomial involving a single fractional power is investigated in this presentation. Dependence of important step response characteristics, namely rise time, settling time, delay time, overshoot, and oscillation period on the fractional order are derived, and the results are presented in graphical forms that can be used as templates for design purposes. The study is conducted comparatively by considering integer order systems of 1 st and 2 nd order types. It is shown that the same simplicity and explicitness present for second order systems do not exist between the transfer function parameters and the step response characteristics for low order fractional systems. The results bring light for designing s imp le FOCSs , thus a vacancy has been fulfilled by this work.