Design of a Complex fractional Order PID controller for a First Order Plus Time Delay system

ISA Transactions xxx (xxxx) xxx

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Design of a Complex fractional Order PID controller for a First Order Plus Time Delay system Omar Waleed Abdulwahhab College of Engineering, University of Baghdad, Iraq

article

info

Article history: Received 8 August 2019 Received in revised form 22 October 2019 Accepted 26 October 2019 Available online xxxx Keywords: PID controller Fractional Order PID controller Complex Order PID controller First Order Plus Time Delay system Gain crossover frequency Phase margin

a b s t r a c t Complex fractional Order PID (COPID) controller is an extension to the Real fractional Order PID (ROPID) controller by extending the orders of differentiation and integration to include complex numbers, i.e., two extra parameters (the imaginary parts of the orders of the differentiator and the integrator) are introduced into the formula of the controller. The purpose is to overcome the limitation stemmed from restricting the parameters of the ROPID controller to belong to certain intervals, where this limitation results in a control system that does not satisfy the required design specification accurately. In this paper, analysis and design of COPID controller is presented, and for comparison purposes, both ROPID and COPID controllers are designed for a low pressure flowing water circuit, which is a First Order Plus Time Delay (FOPTD) system. The design specifications are given in frequency domain, which are gain crossover frequency, phase margin, and robustness against gain variation. The design specifications are taken as two cases, simple an rigorous, where the latter is considered to demonstrate the superiority of the COPID controller over the ROPID controller to achieve hard specifications. Although the design of the COPID controller is more complex than that of the ROPID controller, the first achieves the required design specification more accurately. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Fractional order calculus extends the order of the integration and the differentiation to include fractional numbers. It is a novel mathematical tool to model and to control dynamic systems. It results in more accurate models and more flexible controllers [1]. Fractional calculus extends modeling and control such that they can be represented by noninteger order differential equations. This introduces dynamic models and controllers that have fractional orders [2]. Although powerful control techniques have been developed, the conventional PID controller is still popular, since it is simple, applicable to several industrial control problems, and capable to provide cost/benefit ratio [3,4]. Many methods were developed to tune the PID controller, therefore, the control designer can design this controller easily and achieve the required performance for various systems [4]. Thus, the PID controller is still the dominant controller and the object of many researches [5], and during the last decades, several methods were developed for tuning the parameters of P, PI, and PID controllers [6]. As a generalization to the conventional PID controller, Podlubny suggested to extend the orders of differentiation and integration to include fractional numbers. The generalized controller E-mail address: [email protected].

is referred to as Fractional Order PID (FOPID) or PIλ Dµ controller [7,8]. He investigated the response of this controller and demonstrated that it outperforms the conventional PID controller when it is used to control fractional order systems [9]. Recently, FOPID controllers have become a research topic in control engineering. An FOPID controller has been proposed to stabilize Linear Time Invariant (LTI) systems in [10–16], where in [11,12] the plants were modeled as fractional order systems. For nonlinear systems, [17] designed an optimal FOPID controller for ElectroHydraulic servo system, [18] proposed an FOPID controller for hydroturbine governing system, [19] proposed an FOPID controller for satellite attitude system, and [20] designed a PID and an FOPID controllers for a crane system. Also, FOPID controllers were designed to stabilize unstable nonlinear systems in [1,21] and to control FOPTD systems in [2,3]. For Multi Input Multi Output (MIMO) systems, [22] designed an FOPID controller to control a robotic manipulator with two-links and [23,24] designed FOPID controllers for a Twin Rotor Aerodynamic System (TRAS), where in [24] the TRAS was modeled as a fractional order MIMO system. All the previous researches have demonstrated that the FOPID controller outperforms the PID controller in many respects concerning performance and robustness. To design a control systems, it was stated – in literature – that the number of design specifications that can be satisfied by the system equals the number of design parameters of the controller.

https://doi.org/10.1016/j.isatra.2019.10.010 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: O.W. Abdulwahhab, Design of a Complex fractional Order PID controller for a First Order Plus Time Delay system. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.10.010.

2

O.W. Abdulwahhab / ISA Transactions xxx (xxxx) xxx

From a mathematical point of view, this is true, since each design specification yields an equation, and these equations can be solved simultaneously for a number of parameters that equals the number of design specifications. Practically, this is not always achievable, since some (or all) of the parameters of the controller may be constrained to belong to certain intervals (for an FOPID controller, the three gain parameters must be positive and the orders of the integration and the differentiation must belong to the interval [0, 2]). In such cases, the simultaneous solution of the design specification equations may result in parameters that violate the constraints. In such cases, the objective becomes to find parameters that satisfy the constraints and best fit the required design specifications. This represents a limitation in the design solution, since the resultant parameters may not satisfy the required design specifications accurately. This paper proposes a solution to this problem by extending the differentiator and the integrator orders of the FOPID controller to include complex numbers, i.e., two additional parameters (the imaginary parts of the orders of the differentiator and the integrator) are introduced into the control system, which gives a sufficient flexibility to the control system to satisfy the design specifications more accurately. In literature, the FOPID controller is investigated against the PID controller; thus, the PID controller – in this context – is called Integer Order PID (IOPID) controller; therefore, the letter F in the acronym FOPID is essential to distinguish between fractional order and integer order controllers. However, this paper extends the orders of the integration and the differentiation of the FOPID controller to include complex numbers, where the resultant controller is called COPID controller. Thus, the distinction between the orders of the FOPID and COPID controllers is in being complex or real, not in being fractional or integer, therefore, the letter F is replaced by the letter R in the acronym FOPID to become ROPID. The organization of this paper is: Section 2 presents the mathematical model of the controlled plant, which is a FOPTD system; Section 3 presents the derivation of the sinusoidal transfer function of the COPID controller and its magnitude phase curve, as well as the design of both an ROPID and a COPID control systems; Section 4 discusses the results; and finally, the conclusion is exhibited in Section 5. 2. Mathematical model of the plant The proposed controller will be designed to control an FOPTD system with transfer function P (s) =

k

τs + 1

e

−Ls

where C jωgc and P jωgc are the sinusoidal transfer functions of the controller and the plant, respectively. In rectangular form

(

(

d dω

)

)

(2)

̸

C jωgc P jωgc

( (

) (

))

=0

(3)

Dividing (2) by P jωgc yields:

(

C jωgc =

(

)

)

P jωgc

(

(

ej(φm −π )

ej(φm −π )

e

) =⏐ ( = )⏐ ⏐P jωgc ⏐ ej̸ P (jωgc )

cos(φm − π − ̸ P jωgc )

(

=

)

⏐ ( )⏐ ⏐P jωgc ⏐

( ) ⏐ ( )⏐ ⏐P jωgc ⏐

j φm −π −̸ P jωgc

sin(φm − π − ̸ P jωgc )

(

+j

)

)

(4)

⏐ ( )⏐ ⏐P jωgc ⏐

Eq. (3) can be rewritten as: d

̸

C jωgc +

(

dω From (1)

)

d dω

⏐ ( )⏐ ⏐P jωgc ⏐ = √

d dω

P jωgc = 0

(

̸

)

(5)

3.13

(6)

2 +1 1.877 × 105 ωgc

̸

P jωgc = − tan−1 433.33ωgc − 50ωgc

̸

P jωgc = −

(

)

(

)

433.33 (433.33ωgc )2 + 1

(7)

− 50

(8)

3.1. ROPID controller The ROPID controller proposed by I. Podlubny provides a control action u(t) given by u(t) = KP e(t) + KI D−λ e(t) + KD Dµ e(t)

(9)

The corresponding transfer function is C (s) =

U (s) E(s)

1

= KP + KI

sλ

+ KD s µ

(10)

where Kp , KI , KD ∈ R+ and λ, µ ∈ [0, 2]. The latter five variables are the controller’s parameters. The sinusoidal transfer function is C (jω) = KP + KI cos

+ j(−KI sin |C (jω)| √ =

(KP + KI cos

λπ 2

λπ

ω−λ + KD cos

2

λπ 2

ω−λ + KD sin

ω−λ + KD cos

µπ 2

µπ 2

µπ 2

ωµ

ωµ )

ωµ )2 + (−KI sin

(11)

λπ 2

ω−λ + KD sin

µπ 2

ω µ )2

(12)

̸

) (

) (

(1)

To investigate the performance of the proposed COPID controller and for comparison purposes, both ROPID and COPID control systems are designed. The design requirements that must be fulfilled by the control system are achieving certain values of gain crossover frequency ωgc and phase margin φm and being robust against gain variation. To achieve the first two requirements, the complex value of the open-loop sinusoidal transfer function at ωgc must have magnitude and phase equal to 1 and φm − π , respectively, i.e.

)

To achieve the third requirement, the phase curve of C (jω) P (jω) at ωgc must be flat, i.e.

3. Control system design

(

(

C jωgc P jωgc = cos (φm − π) + j sin(φm − π )

Without loss of generality and for the purpose of computations, a real system with specific parameters is taken as a case study. This system is a circuit of a flowing water with low pressure, organized against a vertical panel. After identifying the system, its parameters is found to be: time delay L = 50 s, gain k = 3.13 and time constant τ = 433.33 s [25].

C jωgc P jωgc = 1̸ (φm − π ) = 1ej(φm −π )

)

C (jω) = tan−1

−KI sin λπ ω−λ + KD sin µπ ωµ 2 2 KP + KI cos

λπ 2

ω−λ + KD cos µπ ωµ 2

d B−C ̸ C (jω) = A × dω D where 1 A= ( 1+

−KI sin λπ ω−λ +KD sin µπ ωµ 2 2

(14)

)2

(15)

KP +KI cos λπ ω−λ +KD cos µπ ωµ 2 2

( B=

(13)

KP + KI cos

λπ 2

ω−λ +

Please cite this article as: O.W. Abdulwahhab, Design of a Complex fractional Order PID controller for a First Order Plus Time Delay system. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.10.010.

O.W. Abdulwahhab / ISA Transactions xxx (xxxx) xxx

µπ

KD cos

2

ωµ

)( −KI sin

λπ 2

(−λ) ω−λ−1 + KD sin

µπ 2

µωµ

)

KD cos

λπ

µπ 2

2

µπ

ω−λ + KD sin

2

ωµ )(KI cos

λπ 2

ω−λ + KD cos

2

µπ 2

2β π 2

π

π

= e0 × e−β 2 = e−β 2 (26)

Similarly (17)

λπ

jjβ = ejβ ln j = ejβ (ln 1+j 2 ) = ejβ ln 1 × ej

(−λ)ω−λ−1 +

µωµ )

D = (KP + KI cos

To find jjβ : π

(16) C = (−KI sin

3

ω−λ )2

(18)

Substituting (6), (7) and (11) in (4) yields the following two real-valued equations:

π

jjφ = e−φ 2 To find ω

(27)

jβ

ωjβ = ejβ ln ω = cos (β ln ω) + j sin(β ln ω)

(28)

Similarly KP +KI cos

λπ 2

ωgc +KD cos −λ

µπ 2

µ

cos(φm − π − ̸ P jωgc )

(

ωgc −

)

3.13

√

=0

2 +1 1.877×105 ωgc

and

−KI sin

λπ 2

µπ

−λ ωgc + KD sin

2

µ ωgc −

sin φm − π − ̸ P jωgc

(

3.13

√

)) =0

C (jω) = KP + 1 KI ( + ( απ )) −β π ( απ ) 2 e cos 2 + j sin 2 ωα (cos (β ln ω) + j sin (β ln ω))

( KD

2 +1 1.877×105 ωgc

Substituting (8) and (14) in (5) yields B−C D

−

433.33 2 +1 1.877 × 105 ωgc

− 50 = 0

The design of the ROPID controller is carried out done by first forming the objective function JR = |JR1 | + |JR2 | + |JR3 |

(22)

where JR1 , JR2 , and JR3 are the left hand sides of (19), (20), and (21), respectively, then, after substituting the values of ωgc and φm (that are given in the specifications) in (22), the optimal parameter vector [KP KI KD λ µ]T is found by minimizing the objective function JR . 3.2. COPID controller As stated in Section 1, this paper investigates a further extension to the ROPID controller, by letting the integration order λ and the differentiation order µ be complex numbers rather than real, i.e., λ = α + jβ and µ = θ + jφ , where α , θ ∈ [0, 2] (as in the ROPID controller) and β , φ ∈ R. The control action u(t) is given by u(t) = KP e(t) + KI D−(α+jβ ) e(t) + KD Dθ +jφ e(t)

(23)

The corresponding transfer function is C (s) =

U (s) E(s)

= KP + KI

1 sα+jβ

+ KD sθ +jφ

= KP + KI

1 (jω)α+jβ 1

+ KD jθ jjφ ωθ ωjφ

)

( + j sin

2

θπ

))

2

π

e−φ 2 ωθ (cos (φ ln ω) +

1

) ( )) + + β ln ω + j sin απ + β ln ω 2 2 ( ( ) ( )) θπ θπ π KD e−φ 2 ωθ cos + φ ln ω + j sin + φ ln ω 2 2 ( απ ) απ cos + β ln ω − j sin( + β ln ω) KI 2 2 ( απ ) + = KP + −β π 2 απ 2 + β ln ω + sin ( 2 + β ln ω) e 2 ωα cos 2 ( ( ) ( )) θπ θπ π KD e−φ 2 ωθ cos + φ ln ω + j sin + φ ln ω π

e−β 2 ωα cos

(

( απ

2

KI

2

απ

+ β ln ω)+ ( θπ KI απ −φ π2 θ ω cos( KD e + φ ln ω) + j − −β π sin( + β ln ω)+ α 2 2 2 e ω ) θπ π KD e−φ 2 ωθ sin( + φ ln ω) = R + jI (30) 2 √ |C (jω)| = R2 + I 2 (31) = KP +

̸

C (jω) = tan−1 d̸

C (jω) dω

cos(

π

e−β 2 ωα

I R 1

= ( )2 I R

2

+1

(32) RDI − IDR R2

(33)

where

( απ )β + β ln ω ω−α cos 2 ω ( απ )) − αω−α−1 sin + β ln ω 2 ( ( ) θπ φ π + KD e−φ 2 ωθ cos + φ ln ω 2 ω ( )) θπ + θωθ−1 sin + φ ln ω 2 ( ( απ )β β π2 −α D R = KI e −ω sin + β ln ω 2 ω ( απ )) −α−1 − αω cos + β ln ω 2 ( ( ) θπ φ π + KD e−φ 2 −ωθ sin + φ ln ω 2 ω (

DI = −KI e

(24)

+ KD (jω)θ +jφ

jα jjβ ωα ωjβ

θπ

β π2

The COPID controller has two extra parameters (β and φ ) compared to the ROPID controller. Introducing additional parameters into a control system adds complexity in its design and implementation; however, the extra parameters give more flexibility to the control system such that it can satisfy more design specifications and achieve more accurate results for these specifications. The sinusoidal transfer function of COPID controller is C (jω) = KP + KI

cos

KP + KI (21)

(

j sin (φ ln ω)) =

(20)

A×

(29)

Substituting (26)–(29) in (25) yields (19)

(

ωjφ = cos (φ ln ω) + j sin(φ ln ω)

(25)

(34)

Please cite this article as: O.W. Abdulwahhab, Design of a Complex fractional Order PID controller for a First Order Plus Time Delay system. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.10.010.

4

O.W. Abdulwahhab / ISA Transactions xxx (xxxx) xxx

+ θ ωθ −1 cos

(

θπ 2

+ φ ln ω

)) (35)

Substituting (6), (7) and (30) in (4) yields the following two real-valued equations:

απ KI θπ π cos( + β ln ω) + KD e−φ 2 ωθ cos( + φ ln ω) π 2 2 e−β 2 ωα ( ) cos(φm − π − ̸ G jωgc ) ⏐ ( )⏐ − =0 (36) ⏐G jωgc ⏐

Table 1 Design specifications of the control system.

ωgc

Design specification

φm

d dω

60◦ 70◦

0 0

rad/s Case 1 (simple) Case 2 (rigorous)

(

̸

C jωgc P jωgc

(

) (

))

s

0.01 0.05

KP +

Table 2 Parameters of the ROPID and COPID controllers. Case

Controller

and

−

( απ

KI

)

sin + β ln ω + KD e π 2 e−β 2 ωα ( ( )) sin φm − π − ̸ G jωgc ⏐ ( )⏐ − =0 ⏐G jωgc ⏐

−φ π

2

ωθ sin

(

θπ 2

+ φ ln ω

RDn − IDm R2

−

433.33 2 +1 1.877 × 105 ωgc

− 50 = 0

KD

α

1

0.09 0.76

0.02 0.16

5.22 21.54

0.88 0.04

2

ROPID COPID

17.40 0.94

0.07 20.65

29.30 31.92

1.76 0.48

(37)

β

θ

φ

1.19

0.31 0.21

1.56

−1.42

0.64 0.85

0.73

Table 3 Specifications of the designed ROPID and COPID controllers.

(38)

Case

Controller

Specifications of the closed loop system

ωgc

The design the COPID controller is carried out by first forming the objective function JC = |JC1 | + |JC2 | + |JC3 |

KI

ROPID COPID

)

Substituting (8) and (33) in (5) yields

Controller parameters KP

(39)

where JC1 , JC2 , and JC3 are the left hand sides of (36), (37), and (38), respectively, then, after substituting the values of ωgc and φm (that are given in the specifications) in (39), the optimal parameter vector [KP KI KD α β θ φ]T is found by minimizing the objective function JC .

φm

rad/s

d ̸ dω

C jωgc P jωgc

(

) (

)

s

1

ROPID COPID

0.01 0.01

59.99◦ 60◦

−0.0114 2 × 10−4

2

ROPID COPID

0.04 0.05

−125.60◦

99.88 9 × 10−4

70◦

4. Results and discussion To demonstrate the superiority of the COPID controller over the ROPID controller, the design specifications were taken as two cases, as shown in Table 1. In case 1, the design specifications are sufficiently simple so that the superiority of the COPID controller over the ROPID controller is not revealed. In case 2, the design specifications are made more rigorous by: (1) increasing the required gain crossover frequency to reduce the rise time of the system and (2) increasing the required phase margin to reduce the percentage overshoot of the system. The design specifications given in Table 1 were substituted in (22) and (39), and for fractional calculus, the definition of Grunwald– Letnikov was adopted [9]. The optimization problems were solved (using the Matlab function fmincon) and the resultant parameters of the ROPID and the COPID controllers are given in Table 2 (for convenience, for the ROPID controller, λ is renamed as α , and µ is renamed as θ ). The two extra parameters (β and φ ) of the COPID controller add complexity in the design (tuning) of this controller compared to the ROPID controller. The achieved design specification for both controllers are shown in Table 3. Figs. 1–4 show the bode diagrams of the two systems for both cases. These plots were obtained by substituting the optimal parameters of the ROPID controller in Eqs. (12) and (13) and substituting the optimal parameters of the COPID controller in Eqs. (31) and (32) and plotting the resultant functions with the Matlab plot command. From Table 3 and the bode plots of case 2, it is evident that the COPID controller satisfied the required design specification accurately, while the ROPID controller did not satisfy these specification, even worse, it did not stabilize the closed loop system (since the phase margin has a negative value). The COPID controller outperformed the ROPID controller since the two additional parameters gave more flexibility to the control system to achieve the rigorous design specifications of case 2.

Fig. 1. Open-loop bode diagram using ROPID controller: case 1.

Fig. 2. Open-loop bode diagram using COPID controller: case 1.

Please cite this article as: O.W. Abdulwahhab, Design of a Complex fractional Order PID controller for a First Order Plus Time Delay system. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.10.010.

O.W. Abdulwahhab / ISA Transactions xxx (xxxx) xxx

5

References

Fig. 3. Open-loop bode diagram using ROPID controller: case 2.

Fig. 4. Open-loop bode diagram using COPID controller: case 2.

5. Conclusions In this paper, a COPID controller has been analyzed by deriving its sinusoidal transfer function and frequency response. For comparison purposes, both ROPID and COPID controllers have been designed for a FOPTD system to achieve certain design specifications. From the results that have been obtained, it can be concluded that although introducing extra parameters to a control system rises the complexity of the design process, these additional parameters give flexibility to the control system to best fit the required specifications. Thus, the COPID controller outperforms the ROPID controller in satisfying the design specifications. Hence, there is a trade-off between the complexity of the design and the ability to achieve better performance. On the other hand, introducing the extra parameters in the COPID controller rises the complexity of its implementation. The implementation of the COPID controller and investigating the trade-off between implementation complexity and achieving better operation is suggested for future works. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[1] Tepljakov A, Belikov J, Petlenkov E, Gonzalez E. Design of retuning fractional PID controllers for a closed-loop magnetic levitation control system. In: 13th international conference on control automation robotics and vision. 2014. [2] Tepljakov A, Belikov J, Petlenkov E, Petlenkov E. FOPID controller tuning for fractional FOPDT plants subject to design specifications in the frequency domain. In: European control conference. [3] Tepljakov A, Belikov J, Petlenkov E, Petlenkov E. Robust FOPI and FOPID controller design for FFOPDT plants in embedded control applications using frequency-domain analysis. In: Proceedings of the American control conference. [4] Padula F, Visioli A. Optimal tuning rules for proportional-integral derivative and fractional-order proportional-derivative controllers for integral and unstable processes. IET Control Theory Appl 2012;6(6):776–86. [5] Padula F, Visioli A. Aadvances in robust fractional control. Springer; 2015. [6] Caponetto R, Dongola G, Fortuna L. Fractional order systems, modeling and control applications. World Scientific; 2010. [7] Podlubny I. Fractional-order systems and fractional-order controllers. Ins, Exp. Phys. Slovak Acad. Sci. UEF-03-94, Kosice 1994. [8] Podlubny I. Fractional-order systems and PIλ Dµ – controllers. IEEE Trans Autom Control 1999;44(1):208–14. [9] Monje CA, Chen YQ, Vinagre BM, Xue D, Feliu V. Fractional-order systems and controls, fundamentals and applications. London: Springer; 2010. [10] Badri V. Some analytical result on tuning fractional-order [proportionalintegral] controllers for fractional-order systems. IEEE Trans Control Syst Technol 2015;24(3). [11] Kesarkar AA, Novel Selvaganesan N. Tuning expressions for fractional order ([PD]β and [PI]α ) controllers using a generalized plant structure. Control Eng. Appl. Inform. 2015;17(1):70–80. [12] Zhong J, Li L. Tuning fractional-order PIλ Dµ controllers for a solidcore magnetic bearing system. IEEE Trans Control Syst Technol 2015;23(4):1648–56. [13] Nankar SH, Jain AM. Fractional order proportional-integral-differential based controller design for dc motor speed control. In: Proceedings of international conference on computing, communication and energy systems. 2016. [14] Isabela B, Cristina M, Silviu F, Ovidiu P. Fractional order controller design for vibration attenuation in an airplane wing. Int J Mech Aerosp Ind Mech Manuf Eng 2016;10(10). [15] Nagarajan M, Asokan A, Manikandan M, Sivakumar D. Design and implementation of fractional order controller for heat exchanger. ARPN J Eng Appl Sci 2016;11(3). [16] Takloo SD, Mozafari S, Mohamadi MR. Fractional order PID control mechanism for helicopter gearbox test control with internal and external disturbance. Bull Soc R Sci Liège 2017;86(special issue):127–38. [17] Ijaz S, Choudhry MA, Ali A, Javaid U. Application of fractional order control technique to an electro-hydraulic position servomechanism. J Eng Technol 2015;34(1):35–44. [18] Junyi L, Chen Q. Fractional order controller designing with firefly algorithm and parameter optimization for hydroturbine governing system. Math Probl Eng 2015;2015:1–11. [19] Xinsheng W, Huanqiang Z. Fractional order controller for satellite attitude control system with PWPF modulator. In: Proceedings of the 34th Chinese control conference. 2015. [20] Singh AP, Srivastava T, Agrawal H, Srivastava P. Fractional order controller design and analysis for crane system. Prog Fract Differ Appl 2017;3(2):155–62. [21] Folea S, Muresan CI, Keyser RD, Ionescu CM. Theoretical analysis and experimental validation of a simplified fractional order controller for a magnetic levitation system. IEEE Trans Control Syst Technol 2016;24(2):756–63. [22] Sharma R, Guar P, Mittal AP. Performance analysis of two-degree of freedom fractional order PID controllers for robotic manipulator with payload. ISA Trans 2015;58:279–91. [23] Abdulwahhab OW, Abbas NH. A new method to tune a fractional-order PID controller for a twin rotor aerodynamic system. Arab J Sci Eng 2017;42(12). [24] Ijaz S, Humayum MT, Yan L, Mumtaz MF. Fractional order modeling and control of twin rotor aero dynamical system using nelder mean optimization. J Electr Eng Technol 2016;11(6):1921–9. [25] Monje CA, Vinagre BM, Feliu V, Chen Y. Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng Pract 2008;16(7):798–812.

Please cite this article as: O.W. Abdulwahhab, Design of a Complex fractional Order PID controller for a First Order Plus Time Delay system. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.10.010.