## 1. Introduction

Fractional calculus has become very useful over the last years due to its many applications in almost all applied sciences. There are applications in acoustic wave propagation in inhomogeneous porous material, diffusive transport, fluid flow, dynamical processes in self-similar structures, dynamics of earthquakes, optics, geology, viscoelastic materials, biosciences, bioengineering, medicine, economics, probability and statistics, astrophysics, chemical engineering, physics, splines, tomography, fluid mechanics, electromagnetic waves, nonlinear control, signal processing, control of power electronics, converters, chaotic dynamics, polymer science, proteins, polymer physics, electrochemistry, statistical physics, thermodynamics, neural networks, etc. [

1,

2,

3,

4,

5,

6,

7].

Many researchers consider this mathematical tool very useful and provide significant contributions in their field. The work of Podlubny [

8] had a major impact in control engineering. He proposed a generalization of the PID controller, namely the PI

^{λ}D

^{μ} controller, involving an integrator of order

λ and a differentiator of order

μ and of Oustaloup [

9], who introduced the CRONE approach for these systems. They also demonstrated that the response of this type of controller is better, in comparison with the classical PID controller, when used for the control of fractional order systems. There are also numerous different forms of fractional order controllers available, proper in some particular cases. For example, in [

10,

11] is presented a particular fractional-order control scheme, the PDD

^{1/2}, which derives from the classical PD scheme with the introduction of the half-derivative term.

The fractional order controller design techniques are in general based on extensions of the classical PID control theory, with an emphasis on the increased flexibility in the tuning strategy resulting better control performances as compared to classical control tuning methods.

Several works approach the tuning of the fractional order PID controller through frequency domain specifications, firstly described by [

12]. Tuning the fractional order controller implies solving the system of nonlinear equations composed of the design constraints, usually by optimization method [

13] or by approximation methods [

14]. There are also available tuning algorithms using time domain cost functions and optimization routines [

15], constrained integral optimization methods, the fractional extension of the MIGO algorithm designed by Astrom et al. as an improvement to the Ziegler–Nichols rules [

16,

17] and auto-tuning methods [

18]. Several applications use fractional order techniques, as it is presented in the survey [

2].

All of these works use one of the several definitions of fractional order derivative (or integral) described above:

The Riemann–Liouville definition [

12]:

where

$\Gamma \left(n\right)=\underset{0}{\overset{\infty}{\int}}{t}^{n-1}{e}^{-t}dt$ is the Euler’s Gamma function which is a generalization of a factorial and

$n\in {R}^{+}$ is an extension of the fractional integral.

The Caputo definition [

12]:

The Grünwald–Letnikov’s definition of the fractional-order derivative [

12]:

The frequency domain fractional-order controller design methods are generally based on the following design specifications [

12]:

Phase margin φ_{m} and gain crossover frequency ω_{cg}:

Iso-damping property;

High-frequency noise rejection;

Good output disturbance rejection; and

Steady-state error cancellation;

Noting with H_{p}(s) the transfer function of the process and with H_{c}(s) the transfer function of the controller, these design specifications can be mathematically described as:

$\left|{H}_{C}(j{\omega}_{gc})\xb7{H}_{P}\left(j{\omega}_{gc}\right)\right|=0dB$; $\mathrm{arg}\left({H}_{C}(j{\omega}_{gc})\xb7{H}_{P}\left(j{\omega}_{gc}\right)\right)=-\pi +{\varphi}_{m}$;

${\frac{d\mathrm{arg}\left({H}_{C}\left(j\omega \right)\xb7{H}_{P}\left(j\omega \right)\right)}{d\omega}|}_{\omega ={\omega}_{gc}}=0$;

$\left|T\left(j\omega \right)=\frac{{H}_{C}\left(j\omega \right)\xb7{H}_{P}\left(j\omega \right)}{1+{H}_{C}\left(j\omega \right)\xb7{H}_{P}\left(j\omega \right)}\right|\le AdB$, with A the desired noise attenuation for frequencies $\omega \ge {\omega}_{T}$ rad/s;

$\left|S\left(j\omega \right)=\frac{1}{1+{H}_{C}\left(j\omega \right)\xb7{H}_{P}\left(j\omega \right)}\right|\le BdB$, with B the desired value of the sensitivity function for frequencies $\omega \le {\omega}_{S}$ rad/s.

Using the frequency definition of fractional order [

12]:

it is easy to imagine the complexity of the resulted inequality system.

In contrast, the proposed method offers new, simple and tuning-friendly rules for fractional order PID controllers, with guaranteed phase margin and gain crossover frequency, while the fractional order offers an excellent tradeoff between dynamic performances and stability robustness.

The paper is structured as follows. After this first, brief introductory part, the second section describes the proposed controller design method, followed by case studies for different process models, from integer order model to fractional order model. The work ends with concluding remarks.

## 2. The Proposed Controller Design Method

The design method is inspired by the ‘symmetrical optimum’ introduced by Kessler [

19]. The plant to be controlled is assumed to be of the form:

where

T_{pi} correspond to large (compensable) time constants with respect to the sum of the ‘parasitic’ time constants

T_{pj} and time delay

T_{m}, i.e.,:

Therefore, for the frequencies below

$1/{T}_{\Sigma}$ the plant transfer function can be approximated by:

and in the region of the crossover frequency is furthermore approximated by a cascade of pure integrators:

The Kessler’s ‘symmetrical optimum’ can be expressed as follows: ‘The crossover frequency of the compensated system is

ω_{cr} = 1/(2

T_{∑}) and the PI(D) is adjusted such that a region with a slope of −20 dB/s is secured for one octave on the right and

m octaves on the left of the crossover frequency” [

20].

The resulting tuning rules for a PI controller, in the case when

m = 1, are the well-known Kessler’s equations [

20]:

obtained from the open loop:

${H}_{ol}\left(s\right)=\frac{1+4{T}_{\Sigma}s}{8{T}_{\Sigma}^{2}{s}^{2}\left(1+{T}_{\Sigma}s\right)}$, with gain crossover frequency and open loop gain:

In the same research Voda and Landau conclude that in the case of a pure integrator plant (l/

sT_{1}), a damped response with 43% overshoot is obtained–due to the zero (1 + 4

T_{∑}s) in the closed loop–having a rise time of 3.1

T_{∑} and a settling time of 16.3

T_{∑}. The gain margin is GM >2.7 and the phase margin (PM) is 36.8°. These performances, excepting phase margin which cannot be modified, can be corrected by using a reference filter or by using a PI with the proportional part acting only on the output. The above-mentioned performance becomes unacceptable due to a large sensitivity with respect to the modification of the plant gain accompanied by an alleviation of the phase margin. This shortcoming can be much stronger if

T_{∑} corresponds to the sum of parasitic time constants [

18]. A way for control system performance enhancement, including the value of the phase margin, is obtained by the generalization of the tuning rules in terms of [

21]:

with the recommended values for

β constrained to the range [

4,

18]. The exact value of

β is chosen as a result of a compromise between the imposed closed loop performances (overshoot, settling time, etc.) and the desired phase margin. The case of

$\beta =4$ is the solution presented in [

21].

The gain crossover frequency and the gain of the open loop in this case are:

A more generalized form of this method and a new approach, including one more degree of freedom using fractional order derivatives are proposed in the present work.

#### 2.1. The Generalized Optimum Method

For the plant transfer function as in (8), with an integral behavior, the ideal form of the open loop transfer function which can reject a step disturbance is:

The closed loop in this case will ensure perfect disturbance rejection in steady state, but the phase margin is 0°, the system is at stability limit, a highly oscillatory system. To correct this problem, a positive phase element is added to the open loop:

To ensure maximum stability (maximum value of phase margin), the gain crossover frequency is imposed to be at the maximum value of the open loop phase characteristic. Analytically this can be expressed by the equations:

Solving this equation system yields the gain crossover frequency and the open loop gain:

being a generalization of (12), with one more degree of freedom, ensuring better performances than the classical method.

Choosing the time constants:

the particular tuning rules are obtained in terms of Preitl and Precup [

21] and for

$\beta =4$ the Kessler’s optimum method [

20].

#### 2.2. Fractional Order Optimum Method

The above presented method has the disadvantage of compromise between the desired closed loop performances and the desired phase margin. To eliminate this disadvantage, one more degree of freedom can be added using a fractional order correction element in Equation (13):

where

α is the fractional order, the generalization of the classical operation of derivation and integration to orders other than integer [

12]. Theoretically, this parameter can take any real, positive value. However, for the controller to have physical meaning, the interval of the fractional orders of integration and differentiation is usually limited to (0, 2) [

12].

The multiplication term of the time constant is chosen β^{2} instead of β to avoid the square root from the controller’s tuning equations.

Considering this fractional order form of the open loop, the system Equation (14) becomes:

The explicit form of the equations for the gain crossover frequency and phase conditions, using the frequency definition of fractional order, Equation (4), are:

The solution of this system, the fractional order generalization of Equation (14) is:

Using these equations, for any chosen value of the fractional order, at the crossover frequency the phase always reaches its maximum value:

or, expressed in terms of phase margin:

With the particular values $T={T}_{\Sigma}$ and $\alpha =1$ the results from (8), while for $\beta =2$, the Kessler’s form (Equation (9)) is obtained.

The obtained phase margin and gain crossover frequency as function of the fractional order is plotted in

Figure 1 and

Figure 2. For simplicity reasons is considered

T =

T_{∑} = 1, without affecting the conclusions.

It can be observed that both gain crossover frequency and phase margin increases with the fractional order α, meaning increased stability and smaller settling time as higher the α is. For $\alpha =1$ the “classical” Kessler’s phase margin value of 36.8° and gain crossover frequency is obtained, as in Equation (10).

From the step response of the closed control loop with different values of

α the corresponding overshoots can be determined, resulting the plot from

Figure 3. This plot reveals that the overshoot decreases with the increasing fractional order.

The main advantage of the Kessler’s optimum method-the steady state speed error cancellation-is maintained with the fractional order system as well. Moreover, the higher the fractional order, the better the transient response of the closed loop system, as it is presented in

Figure 4.

Having two more degrees of freedom-due to the parameter β and fractional order α–the controller design problem becomes an optimization problem: select the proper β and α to ensure the desired closed loop performance measures.

In

Figure 5 the gain crossover frequency evolution (which is inversely proportional with the settling time) with respect to parameters

β and

α is presented. In a similar manner the maximum phase margin,

Figure 6, or any other desired performance measure of the closed loop system can be represented. With this optimization technique the desired performances of the system can be ensured, no matter how rigorous they are.

The controller parameter tuning algorithm can be described as follows:

Having the process mathematical model of the form of Equation (5), the open loop transfer function form is imposed as in Equation (16) to provide zero steady-state position and velocity error.

Using Equations (18) and (19) the tuning parameters K, α and β for the desired values of gain crossover frequency and phase margin are computed.

Having the open loop in Equation (17) and the process model in Equation (5), the transfer function of the fractional order controller in one of the forms presented in [

22] is obtained.

The controller obtained with the proposed method being a fractional order one, engineers are faced with the problem of implementation. Actually, the fractional-order controller itself is an infinite-dimensional linear filter due to the fractional-order differentiator. A band-limit implementation is important in practice. Finite dimensional approximation of the fractional order controller should be used in a proper range of frequency of practical interest. A possible approximation method, the most widely applicable, is the Oustaloup recursive algorithm [

23].