Fractional-Order PID Controller Design for Time-Delay Systems Based on Modified Bode’s Ideal Transfer Function

This paper proposes a design method for a robust fractional-order PID (FOPID) controller for time-delay systems. A new Bode’s ideal transfer function is introduced to tolerance the time-delay in loop. Robust stability is analyzed for the Bode’s model in terms of gain and phase margins to help the parameter tuning. To simplify FOPID design from solving nonlinear equations, five unknown parameters are reduced to one in the Bode shaping by data fitting between a parametric model and the real plant. Then, the problem is simply solved by one-dimensional searching. Furthermore, the proposed FOPID controller design is extended to multi-input and multi-output (MIMO) systems by using disturbance observer (DOB). Finally, simulation results are presented to show the effectiveness of the proposed method.


I. INTRODUCTION
In the past decades, an increasing number of studies are focusing on the application of fractional calculus in many areas of industrial engineering. Fractional system provides a better understanding of system characteristics for many phenomena such as heat transfer [1] and wave propagation [2]. Due to the reliable system description, fractional-order (FO) model have been paid more and more attention in the academics. A recent survey on fractional system is presented in [3], [4] and its applications are introduced in [5], [6]. In the control community, Oustaloup [7] introduced the FO algorithm for the control system and demonstrated superior performance.
In the industry and engineering, proportional-integralderivative (PID) controller still serves as the most widely used controller because of its simple structure and easily tuning [8], [9]. To improve robustness and performance, The associate editor coordinating the review of this manuscript and approving it for publication was Nasim Ullah .
Podlubny [10] proposed a generalized PID controller, called FOPID, including a fractional integrator and differentiator. This new form, FOPID, has success among research because of its flexibility and robustness. Some literature has shown the better response of this controller, in comparison with the classical integer PID controller [11], [12]. Different from the classical linear integer-order PID (IOPID) controller, FOPID controller is nonlinear naturally regarding the fractional order, which brings the main difficulties in system design and analysis.
Nowadays, various design methods have been proposed for FOPID controller tuning [13]- [19]. The well-known Ziegler-Nichols tuning rule has been extended to FOPID controller with S-shaped step response, but it only works well on some lag-dominant process [18]. For a typical first-order-plusdead-time model, some tuning rules are developed to minimize integrated absolute error (IAE) subject to a constraint on the maximum sensitivity [19]. To obtain good control performance, dominant pole placement for FOPID controller is developed based on D-decomposition method [15]. In the last years, the internal-model-control (IMC) tuning method is also introduced in the design of FOPID controller [16], [17].
Bode shaping is an effective method for control system design in frequency domain [20]- [26] As a good open-loop model, Bode's ideal transfer function is discussed in [22], which shows its stronger robustness against loop gain variation. To realize auto-tuning of PID controller, online optimization technologies are employed to optimize the Bode's model [20] and controller parameters [21]. Due to the robustness to the gain variation, Bode shaping in the frequency domain is also used in the FOPID controller design [22]- [24] by imposing the open-loop phase to be flat in a frequency band. In [24], a model-based analytical method is developed for FOPID controller design via internal model control (IMC) principle and Bode's model. In [25], a robust FOPID controller tuning strategy is developed based on the flat phase property and optimal Bode's model. This popular scheme can also be found in [13], [14], which is based on the robustness specifications, including the desired phase margin, gain crossover frequency and the flatness of the phase Bode plot. However, for time-delay process, Bode shaping become complicated since the selection of Bode model is constrained by the time-delay item, and often results in complex analytical design and numerical optimization [26], which motivate us to investigate the Bodel's transfer function design and its parameter selection for time delay system in FOPID design.
Time-delay is often encountered in industrial engineering, such as the chemical process and networked control system [27], and often brings in great difficulty in control system design. For example, networked control systems are typical time-delay systems and the time-delay introduced by the network inevitably makes the system performance degraded and may even damage the system stability [28], [29]. Although many modern control strategies have been developed for time-delay systems, PID control is considered as a simple yet effective manner to possess great robustness against system time-delay [30]- [32]. In recent years, an increasing number of studies can be found related to FOPID control for time-delay systems. The basic idea to cope with time-delay is to design a robust controller that can tolerate the time-delay in the feedback control loop. In [33], an analytical approach for the design of FOPID controller is proposed on the basis of the IMC scheme and the maximum sensitivity for FO systems with time-delay. As time-delay systems have infinite-dimensional property naturally, the design of the FOPID controller is more complex than the delay-free case. Thus, finding the set of stabilizing FOPID controller parameters for time-delay systems has been paid great attention. Several methods [12], [34], [35] have been proposed to draw the stability boundary graphically in parameter space based on the D-decomposition method with stability margin specifications [36].
In this paper, we aim to propose a robust FOPID controller design for time-delay systems by Bode shaping. To realize this goal, two problems are solved in this paper: A new Bode's ideal transfer function is design to tolerance the time-delay in loop. We select a time delay Bode's model for FOPID design. It is well known that, the bandwidth design for time-delay system is a critical problem in control system design. However, this problem is seldom discussed in the current Bode shaping methods [37], [38]. We solve this problem by investigating the gain and phase margins for the proposed time-delay Bode's model.
Bode shaping for FOPID design is solved by onedimensional searching, rather than five parameter optimizations. To simplify the design of FOPID from solving nonlinear equations, data fitting at steady state and the crossover frequency are derived, such that five unknown parameters are reduced to one. Then, the problem is easily solved by one-dimensional searching.
The rest of this paper is organized as follows. Section 2 shows the preliminaries. Section 3 describes the proposed FOPID design method. Section 4 extends the proposed method to multivariable systems. Section 5 illustrates comparative numerical results. Finial, Section 6 concludes this study with final remarks.

A. FRACTIONAL CALCULUS
Fractional calculus is a generalization of the integration and differentiation to the non-integer order operator. The differ-integral operator, denoted by a D α t , is defined by where α ∈ R is the order of the operator and a and t are called lower and upper terminal, respectively. There are several definitions for fractional derivatives [23], [39], such as Grünwald-Letnikov (G-L) and Riemann-Liouville (R-L) definitions. As one of the most commonly used definitions, G-L fractional differential/integral definition has the form [39]: where h is the step size, n = (t − a)/h, representing the number of sampling points in a time interval [a, t], and ω (α) j = (−1) j α j is a polynomial coefficient of the sampling signal Note, that, fractional operator is a nonlocal operation over an interval [a, t], i.e. the operator symbol a D α t is explicitly expressed over the upper and lower terminals. VOLUME 8, 2020 While the R-L definition is written as where (·) is Euler's gamma function. The Laplace transform of the R-L fractional derivative/integral (3) under zero initial conditions for the order α(0 < α < 1) is presented as One point should be emphasized is that, for a wide class of functions, the G-L and R-L definitions are equivalent [39].

B. FOPID CONTROLLER
The generalized transfer function of the FOPID controller is first introduced by Podlubny [10], and is defined as follows where k p is the proportional gain, k i is the integration gain, and k d is the derivative gain; λ and µ are the integral and derivative orders, respectively, satisfying 0 < λ, µ < 2. It can be easily found that, by selecting λ = µ = 1, a standard IOPID controller is obtained. As discussed in [39], the FOPID owns an important feature that it allows for a continuous slope compensation of the controller's Bode plot both at low and high frequencies, depending, respectively, on λ and µ. Thus, this feature can be utilized for a more effective loop shaping and better control performance [22], [23]. However, the problem is also raised by these two additional order parameters, which results in complex controller tuning rules in practice because the FOPID transfer function is nonlinear with respect to the coefficients λ and µ.

C. BODE'S IDEAL TRANSFER FUNCTION
An ideal open-loop transfer function is proposed in [22], that is where ω c is the gain crossover frequency of L (s), and 0 < α < 2 is a real. The parameter α determines both the slope of the magnitude curve on Bode plot and the phase margin of the system. In the Bode diagrams, the amplitude of L (s) is a straight line of constant slope −20αdB/dec, and its phase curve is a horizontal line at −απ/2 rad, which indicates the Bode's ideal transfer function L (s) possesses strong robustness against gain variation. It means that the variation of the process gain only changes the crossover frequency ω c but maintains the phase margin constant π (1 − α/2)rad.
The robustness against gain variation is commonly used in FOPID design as an additional specification [13], [14], [24], [35], which demands that the phase Bode plot of the designed with infinite gain margin and constant phase margin.

III. THE PROPOSED METHOD A. BASIC IDEA
The block diagram of the FOPID control system is shown in Fig.1, where G p (s) is the controlled plant. The following time-delay system is considered in this paper, where T is the time-delay, and G 0 (s) is the normal model delay-free. With FOPID, the closed-loop system is given by Note that the time-delay item always appears in the closedloop system (9). Combined with the Bode's ideal transfer function and the time-delay item, a desired closed-loop model for time-delay systems is chosen as such that the time-delay in H (s) equals to the real one. Aimed at the desired closed-loop model H (s), suppose there is an process modelG p (s), satisfying We havẽ Note that,G p (s) in (12) can be viewed as a parametric model of G p (s), since all the five unknown parameters of FOPID and as well as two parameters of the desired model are appearing explicitly in (12). The basical idea for the FOPID controller design is to solve all the unknown parameters to make the parametric modelG p (s) closed to the real plant G p (s) by data fitting. Some optimal searching algorithms, such as particle swarm optimization [40] and neural network [41], can be used to solve the problem. The main difficulties will come from three aspects: 1) The parametric modelG p (s) in (12) is an infinitedimensional system because of the time-delay involved in (s); 2) The nonlinear optimization with multi-parameter often falls into the local optimality. 3) No stabilization controller can be found if the desired model inappropriate. To develop a simple yet effective design method, some prior knowledge of the process is assumed and employed in the design. Two assumptions are made on the process: 1) the plant G p (s) is stable, and 2) the plant has a nonzero steady value, G p (j0) = 0. These two assumptions are satisfied in most of industry processes.

B. APPROXIMATE ROBUST STABILITY ANALYSIS
It is well known that the specified control performance can be achieved by feedback control for delay-free systems theoretically. Different from this case, the stability and control performance of time-delay systems is physically limited by the time-delay in the closed-loop. Intuitively, bandwidth, say ω c , cannot be arbitrary large under the limitation of time-delay or sampling time.
Denote the frequency response of a transfer function W (s) as W (jω). With assumptions discussed previously, data fitting at the steady-state requiresG p (j0) = G p (j0) = 0, yielding λ − α = 0 and Observing the parametric modelG p (s) in (13), it is obvious that limiting | (s)| in a small value can reduce the difference betweenG p (s) and G p (s) to benefit the data fitting. In this manner, given a small constant ε, satisfying 0 < ε < 1, and specified (s) ∞ < ε, the possible choice of ω c should satisfy ω α c ≤ ( Assuming that data fitting will be carried on with the constraint in (14), to have G p (s) ≈G p (s). In this way, the openloop transfer function satisfies Recalling that (s) ∞ < ε and ε is a small positive constant, we have As discussed previously, a system with the Bode's ideal transfer function owns a finite gain margin and constant phase margin. However, this merit is changed by the time-delay item of the process. Based on the approximation in (16), the gain and phase margins of G p (s)G c (s) are expressed by respectively. To guarantee the closed-loop stability, it requires A m > 1 and γ > 0, to have when 0 < α < 2. The approximate stability analysis presented above is meaningful because it provides some guidelines to select two important parameters in (10). Furthermore, (17) and (18) give the estimation on the stability margins. We can design them (A m and γ ) large enough to guaranteed the closed-loop stability. Several remarks are given to show the details.
Remark 1: The developed conditions in (14) and (19) can be viewed as stability constraints in the proposed FOPID design. The selection of α and ω c based on (14) and (19) guarantees the existing of stabilized FOPID as well as the closed-loop performance.
Remark 2: The closed-loop bandwidth has a major impact on the system overshoot, response speed, and disturbance rejection performance. (14) and (19) provides an analytical result on the performance limitation by the time-delay of a control system. A suitable closed-loop bandwidth selection benefits a successful controller with closed-loop stability and performance guaranteed.
Remark 3: the analytical results on the stability in (17) and (18) reveal a fact that a large closed-loop bandwidth will reduce the gain and phase margins of time-delay systems. One can also specify the robustness stability based on (17) and (18) to have A m > A * m and γ > γ * , where A * m and γ * are the specified gain and phase margins, respectively.

C. CONTROLLER DESIGN
This paper presents a simple yet effective FOPID design in the frequency domain with one-dimensional searching. Based on the discussion in [22], α ≈ 1 is recommended for the tradeoff between fast response and small overshoot. Therefore, the remaining bandwidth parameter ω c is selected under the stability constraints in (14) and (19).
Note that, the real plant G p (s) may have high-order dynamic or uncertainties. The parametric modelG p (s) can be viewed as its reduced-order nominal model which dominants the dynamic characteristic of G p (s). We make data fitting in a certain frequency range [0, ω x ], where ω x is selected at the phase crossover frequency of G p (s), such that G p (jω x ) = −π . Procedures of the proposed FOPID design are given as following.
Step 1: Set α ≈ 1 and select ω c under the stability constraints (14) and (19) for good robustness in terms of the gain and phase margins in (17) and (18), respectively; Step 2: Data fitting at the steady state. LetG p (j0) = G p (j0) = 0.With the integral order λ = α, the integration gain k i is derived Step 3: Data fitting at the phase crossover frequency ω = ω where Using Euler's formula, j α and j t are expressed as where a = cos π 2 t , b = sin π 2 t , c = cos π 2 α and d = sin π 2 α . Then, equation (16) yields Till now, two parameters λ and k i are determined. k d and k p are well formulated with the variable µ. Further calculation for the derivative order µ is by one-dimensional searching to realize model matching.
Step 4: Data fitting in the frequency range (0, ω x ). The fitting error in the frequency range (0, ω x ) can be numerically expressed by We can minimize J to determine the value of µ in the range of 0 < µ < 2 In (22), k d and k p can be uniquely determined if µ is iterated. In this way, the design of FOPID is solved in (24) by one-dimensional searching technology.

IV. EXTENSION TO MIMO SYSTEM
Consider the FOPID design for n × n MIMO systems G(s). Suppose the diagonal elements of G(s) are G p1 (s), G p2 (s), · · · , G pn (s) . The system outputs can be expressed as Y = [y 1 , y 2 , · · · , y n ] T with where u g i is the input signal for i-th loop, d i is the external disturbance and w i is the equivalent coupling disturbance caused by the other loops. Obviously, the coupling disturbance w i will affect the control performance greatly, which is the major difficulty in MIMO control system design.
Recently, the decoupling control by disturbance observer (DOB) has been well discussed and demonstrated in [44]. The diagonal element G pi is considered as a single loop system in DOB design such that the total disturbance d i + w i can be estimated and compensated in real time. A diagonal element G pi is firstly decomposed into a minimum phase transfer function, g q (s), and a non-reversible function g p (s), containing input/output time-delay or NMP elements with unity gain. Suppose thatg q (s) is the nominal model of g q (s). The low-pass filter Q i (s) with unity gain is used to ensure the physical realization of Q(s)g −1 q (s). The transfer function from u to y can be simply calculated by (25) Note that, if 1 − Q i (s)g p (s) is shaped zero in a wide frequency range, G pi (s) can be recovered as its nominal model g p (s)g q (s), that is G pi (s) ≈ g p (s)g q (s). Thus, the proposed FOPID can be applied to G pi (s) directly. Based on the DOB decoupling, the proposed FOPID for MIMO system in each loop is shown in Fig.2. The design procedures will be discussed in the simulation study in details.

V. SIMULATIONS
In this section, some typical plants, including first-order-plustime-delay (FOPTD) system, second-order-plus-time-delay (SOPTD) system and uncertain system, are used to illustrate the effectiveness of the proposed method.  Comparisons are made to some existing methods. Some metrics are used to evaluate controller performance, including, gain margin A m , phase margin γ , gain crossover frequency ω γ , overshoot σ , rise time t r , settling time t s and ISE index A. EXAMPLE 1; FOPTD SYSTEM Consider a FOPTD system in literature [35] We set ε = 0.36. With the stability constraints in (14) and (19), the parameters ω c = 4.85 and α = 1.01 are selected. The gain and phase margins are estimated by (17) and (18) respectively, that is A m ≈ 3.2 and γ ≈ 61 o .
Then, the integration gain k i is derived to have k i = 4.9272. As ω x = 18.6 determined in Fig.3, the optimization is carried on for (24) and the differential order is determined µ = 0.68. We calculate the remaining parameters by (22), and the finial FOPID controller is obtained For this process, an integer order (IO) PI controller is optimized following the recognized method by Astrom and Hagglund [42] as Step responses using controllers in (28)-(30).

FIGURE 5. Bode diagrams with the controllers (28)-(30).
and an FO proportional-integral (FOPI) controller is designed to fulfill a flat phase constraint by Luo [35] as To illustrate the set-point tracking and disturbance rejection performance, step response and load disturbance response are presented in Fig.4. Clearly, the proposed FOPID in (28) provides better control performance than the controllers in (29) and (30), which is also demonstrated by their frequency response of open-loop transfer functions in Fig.5.
All the compared indexes are shown in Table 1 for the setpoint response. The effectiveness of the proposed method can also be observed that the achieved gain and phase margins in Table 1 are closed to the estimated ones in (17) and (18).
To compare the robustness of three controllers, the step responses with ±20% loop gain variations are presented in Fig.6. The performance shows the system robustness to the gain uncertainties using the proposed FOPID controller. Compared with the step responses using the controllers (29) and (30), the overshoots of the proposed FOPID are smaller and with shorter settling time.
For further investigation, the proposed FOPID design is tested for different time-delay cases, T = 0.3 and T = 0.5. Two FOPID controllers are obtained as follows VOLUME 8, 2020
The step responses are shown in Fig.7 for the control systems with (28), (31) and (32). It is obvious that, with proper parameter selection for ω c and α, the proposed method is applicable to large time-delay case. Step responses using controllers in (34)-(36).

B. EXAMPLE 2: SOPTD SYSTEM
Consider the SOPTD system in literature [43] G 2 (s) = 0.5 ε = 0.36 is set to limit the value of (s) in (16). We select ω c = 2.5 and α = 0.98 for the robust stability according to (14), (17)- (19). The gain and phase margins are estimated by (17) and (18) Table 2. The results demonstrated that, compared to five parameters searching schemes, one-dimensional searching used in this paper also provides a satisfactory control performance but with a simple calculation. The Bode plots are shown in Fig. 9, which shows a larger phase margin of the proposed FOPID (34) and the flatness of the phase plot.
For further investigation on the robustness of three controllers, Fig.10 shows the step responses when ±40% gain variation occurring in the open-loop plant. The simulation results indicate the proposed FOPID owns better robustness on the gain variation than the controllers (34) and (35). This result can be well understood due to the flatness on the phase plot providing more robustness to gain variation.

C. EXAMPLE 3: UNCERTAIN SYSTEM
In this example, the real system is:  Step responses with open-loop gain variations ±40%. As discussed in [37], this plant can be approximated by the following time-delay low-order model Step responses and load disturbance responses of the control system using controllers in (39)-(41). We use (38) respectively. Simulation comparisons are carried on for the real plant (37). Fig.11 shows the step responses and load disturbance responses. The corresponding control performances are given in Table 3. It can easily be seen that both the step response and disturbance rejection performance are better than two compared FOPID controllers. These results are also supported by their frequency response in Bode plots, as shown in Fig. 12. The open-loop system using the FOPID controller (39) provides a larger phase crossover frequency to generate a fast system response and a flatness of phase plot around ω = ω c .
As a robustness analysis, the step responses of the three FOPID controllers in Fig. 13 are obtained when the open-loop system has ±50% gain variations. It can be seen that even large gain variation occurring, the control performance still maintains quite well. VOLUME 8, 2020

D. EXAMPLE 4: MIMO SYSTEM
A 24-tray tower separating a mixture of methanol and water, examined by Luyben [45] has the following transfer function matrix According to section IV, MIMO processes can be compensated by DOB in the inner loops for performance recovery as shown in Fig.2. According to the design method in [44], we select , Q 2 (s) = 0.35s + 1 0.5s + 1   for loop 1 and loop 2, respectively. Then, we design FOPID controller just like a single loop for the diagonal processes g 1 (s) = −2.2e −s 7s + 1 , g 2 (s) = 4.3e −0.35s 9.2s + 1 directly. Consequently, ε 1 = 0.36 and ε 2 = 0.5 are set to limit the value of (s) in (16). We select ω c1 = 1.5, ω c2 = 0.14 and α 1 = 1.06, α 2 = 0.95 for the robust stability according to (14), (17)- (19). The crossover frequency ω x1 = 5.28 and ω x2 = 5.65 are selected according to the Nyquist plot of g 1 (s) and g 2 (s). The proposed FOPID controllers are calculated by data fitting between the frequency range [0, ω x1 ) and [0, ω x2 ), to have For this process, we make comparisons to the FOPID design methods in [46] and [47]. The comparison results are depicted in Fig.14, Fig.15 and Table 4. In the simulation study, the step inputs for two loops are set r 1 (t) = 1(t) and r 2 (t) = 1(t − 50), respectively. From Figs.14 and 15, we can observe that, the coupling effects between two loops are very strong in the cases of [46] and [47], generate large overshoot and oscillations in two loops response. While in our case, the coupling effects are well overcome by the proposed control scheme, provide good control performance. The results are also confirmed by ISE index in Table 4.

VI. CONCLUSION
This paper presents a simple yet effective FOPID controller tuning method for time-delay systems. A time-delay closedloop model with Bode's ideal transfer function is introduced for control system design. Robust stability is analyzed in terms of gain and phase margins. Bandwidth selection is a critical problem in time-delay system design but is seldom discussed in the current FOPID works [22], [23], [37], [38]. We give an analytical method to solve this problem based on the proposed stability conditions. To simplify the design of FOPID from solving nonlinear equations, five unknown parameters are reduced to one by data fitting. Then, onedimensional searching is used to find the solution. Furthermore, the proposed FOPID controller design is extended to MIMO systems. Finally, the effectiveness of the proposed method is illustrated by numerical simulations and comparisons. The further work is to investigate the discretization for fractional operator such that the proposed FOPID controller can be applied in a real control system.