Fractional Order PI Controller Design for Non-Monotonic Phase Systems

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Abstract: In this paper, a fractional order proportional-integral (FOPI) controller is proposed for controlling the non-monotonic decreasing phase DC-buck ...
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IFAC-PapersOnLine 49-1 (2016) 236–240 Fractional Order PI Controller Design for Non-Monotonic Fractional Order PI Controller Design for Non-Monotonic Fractional Order PI Controller Design for Non-Monotonic Phase Systems Fractional Order PI Controller Design for Non-Monotonic Fractional Order PI Controller Design for Non-Monotonic Phase Systems Phase Systems *Phase Systems ** Santosh Kumar Verma Phase , ShekharSystems Yadav , Shyam Krishna Nagar***

*** Santosh Kumar Verma***, Shekhar Yadav** **, Shyam Krishna Nagar*** *** Santosh , Shyam Krishna Nagar Santosh Kumar Kumar Verma Verma*,, Shekhar Shekhar Yadav Yadav** , Shyam Krishna Nagar *** Santosh Kumar Verma , Shekhar Yadav , ShyamofKrishna Nagar Department of Electrical Engineering, Indian**Institute Technology (BHU), Department of ElectricalVaranasi-221005, Engineering, Indian Institute of Technology (BHU), U.P., India of Technology Department Engineering, Institute Department of of Electrical ElectricalVaranasi-221005, Engineering, Indian Indian Institute Technology (BHU), (BHU), U.P., India of Department of Electrical Engineering, Indian Institute of Technology (BHU), e-mail: [email protected]*; Varanasi-221005, U.P., India Varanasi-221005, U.P., India e-mail: [email protected]*; Varanasi-221005, U.P., India [email protected]**; [email protected]*** e-mail: e-mail: [email protected]*; [email protected]*; [email protected]**; [email protected]*** e-mail: [email protected]*; [email protected]**; [email protected]*** [email protected]**; [email protected]*** [email protected]**; [email protected]*** Abstract: In this paper, a fractional order proportional-integral (FOPI) controller is proposed for Abstract: In this paper, a fractional order proportional-integral (FOPI) is proposed for controlling the non-monotonic decreasing phase DC-buck regulator system.controller The parameters of FOPI Abstract: In this paper, a fractional order proportional-integral (FOPI) controller is for Abstract: In this paper, a fractional order proportional-integral (FOPI) controller is proposed proposed for controlling the non-monotonic decreasing phase DC-buck regulator system. The parameters of FOPI Abstract: In this paper, a fractional order proportional-integral (FOPI) controller is proposed for controller are optimized by Nelder’s-Mead (NM) method. The FOPI controller provides fast closed-loop controlling the the non-monotonic non-monotonic decreasing decreasing phase phase DC-buck DC-buck regulator regulator system. system. The The parameters parameters of of FOPI FOPI controlling controller are optimized by Nelder’s-Mead (NM) method. The FOPI in controller provides fastdomain. closed-loop controlling the non-monotonic decreasing phase DC-buck regulator system. The parameters of FOPI performance as well as improves the robust properties of the system time and frequency The controller are by (NM) method. The FOPI controller provides fast closed-loop controller areasoptimized optimized by Nelder’s-Mead Nelder’s-Mead (NM) method. Thesystem FOPI in controller provides fastdomain. closed-loop performance well the as improves the robust properties of the time frequency The controller are optimized by Nelder’s-Mead (NM) method. Thesystem FOPI in controller provides fastcharacteristic closed-loop preserves monotonic phase between the desired bandwidth andand improves the performance as well as improves the robust properties of the time and frequency domain. The performance as well as improves the robust properties of the system in time and frequency domain. The controller preserves the monotonic phase between the desired bandwidth and improves the characteristic performance as the wellsystem. as improves the robust properties of the system in time and frequency domain. of The proposed method is validated by comparing the time domain as wellThe as controller preserves the monotonic phase between the desired bandwidth and improves the characteristic controller preserves the monotonic phase between the desired bandwidth and improves the characteristic performance of the characteristics system. The proposed method isorder validated by comparing the time domain as well as controller preserves the monotonic phase between the desired bandwidth and improves the characteristic frequency domain with the integer proportional-integral (IOPI) controllers tuned performance of system. method is validated by time as as performance of the the characteristics system. The The proposed proposed method isorder validated by comparing comparing the the(IOPI) time domain domain as well well as frequency domain with the integer proportional-integral controllers tuned performance of the characteristics system. The proposed method isorder validated by comparing the(IOPI) time domain as well as using various techniques. frequency domain with the integer proportional-integral controllers tuned frequency domain characteristics with the integer order proportional-integral (IOPI) controllers tuned using various techniques. frequency domain characteristics with the integer order proportional-integral (IOPI) controllers tuned Keyword: Fractional Order PI controller, non-monotonic phase system, DC-buck regulator, Nelder’susing various techniques. using various techniques. © 2016, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. Allregulator, rights reserved. Keyword: Fractional OrderFederation PI controller, non-monotonic phase system, DC-buck Nelder’susing techniques. Mead various optimizer. Keyword: Fractional Order Keyword: Fractional Order PI PI controller, controller, non-monotonic non-monotonic phase phase system, system, DC-buck DC-buck regulator, regulator, Nelder’sNelder’sMead optimizer. Keyword: Fractional Order PI controller, non-monotonic phase system, DC-buck regulator, Nelder’sMead Mead optimizer. optimizer. Mead optimizer. 2015; H. Shayeghi et al. 2015; I. Podlubny 1999a, b). 1. INTRODUCTION 2015; H. Shayeghi et al. 2015; I.thePodlubny 1999a, b). These fractional terms increases complexity of the 2015; H. H. Shayeghi et al. al. 2015; I. I. Podlubny Podlubny 1999a, b). 1. INTRODUCTION 2015; Shayeghi et 2015; 1999a, b). These fractional terms increases the complexity of the 1. INTRODUCTION 1. INTRODUCTION 2015; H. Shayeghi et al. 2015; I.thePodlubny 1999a, b). controller but moreterms powerful than the conventional integer Feedback control system maintains a prescribed These fractional increases complexity of the These fractional terms increases the complexity of the 1. INTRODUCTION controller but more powerful than the conventional integer Feedback control system maintains a the prescribed These fractional terms increases the complexity of the order controllers and supple design methods to satisfy relationship between the reference input and desired controller but more powerful than the conventional integer Feedback control system maintains a prescribed controller but more powerful than the conventional integer Feedback control system maintains a the prescribed order controllers and supple design to (1999) satisfy the relationship between the reference inputtheand desired controller but more powerful thanInthe conventional integer controlled system specifications. I.methods Podlubny two Feedback control system maintains a input-output prescribed output. The difference between order controllers and supple design methods to satisfy the relationship between the reference input and the desired order controllers and supple design methods to satisfy the relationship between the reference input and the desired controlled system specifications. In I. Podlubny (1999) two output. The difference between the input-output order controllers and supple design methods to satisfy the additional tuning knobs given which are able to make relationship between the reference input andofinput-output the desired relationships of the system is noted as error the system controlled system specifications. In I. Podlubny (1999) two output. The difference between the controlled system specifications. In I. Podlubny (1999) two output. The difference between the input-output additional tuning knobs given which are able to make relationships of the system is noted as error of the system controlled system specifications. In I. Podlubny (1999) two balance between settling time and maximum overshoot of output. The difference between the input-output and minimized using controller. Inas frequency response additional tuning knobs given which are to relationships of system is error system additional tuningsettling knobs time givenand which are able able to make make relationships of the the system is noted notedIn as frequency error of of the theresponse system balance between maximum overshoot of and minimized using controller. additional tuning knobs given which are able to make the system. relationships of the system is noted as error of the system compensation of a continuous (or discrete) linear time balance between and minimized controller. frequency response balance between settling settling time time and and maximum maximum overshoot overshoot of of and minimizedofusing using controller.(orIn Indiscrete) frequency response the system. compensation a continuous linear time balance between settlingfractional time and order maximum overshoot of In the last few years, control strategies and minimized using controller. frequency invariant (LTI)of system is done (or byIndiscrete) applying a response negative the system. compensation aa continuous linear time the system. compensation ofsystem continuous (or discrete) linear time In the last few years, fractional order control strategies invariant (LTI) is done by applying a negative the system. have been successfully applied to many systems like compensation of a continuous (or discrete) linear time feedback control system. In classical control design the In the years, order control strategies invariant (LTI) is by aa negative In the last last few few years, fractional fractional order control strategies invariant (LTI) system system is Indone done by applying applying negative have successfully applied to many systems like feedback control system. classical control design the In thebeen last of few years, fractional order control strategies controlling smart wheel via internet with variable delay invariant (LTI) system done by applying negative phase margin of a is In closed-loop systemadesign having have been successfully applied to many systems like feedback control system. classical control the have been successfully applied to many systems like feedback control system. In classical control design the controlling of smart wheel via internet with variable delay phase margin of a closed-loop system having have been successfully applied to many systems like (Inés Tejado et al. 2015), Automatic voltage regulator feedback control system. In classical control design the monotonically decreasing phase inside the bandwidth is controlling of smart wheel via internet with variable delay phase margin of a closed-loop system having controlling of smart wheel via internet with variable delay phase margin of a closed-loop system having (Inés Tejado et al. wheel 2015), voltage regulator monotonically decreasing phase inside at thethe bandwidth is the controlling of smart viaAutomatic internet with variable delay system (H. Remezanian et al. 2013) and many other (D.Y. phase margin of open-loop a phase closed-loop system having distance between the phase gain crossover (Inés Tejado et al. 2015), Automatic voltage regulator monotonically decreasing inside the bandwidth is the (Inés Tejado et al. 2015), voltageother regulator monotonically decreasing phase phase inside at thethe bandwidth is the system (H.2006; Remezanian et al.Automatic 2013) and many (D.Y. distance between the open-loop ° gain crossover (Inés et S.al.Das 2015), Automatic voltage regulator Xue etTejado al. 2012; G. Q. Zeng et al. other 2015; Guomonotonically phase inside thethe bandwidth is the . gain However, frequencybetween and decreasing thethe stability limitphase of −180 system (H. Remezanian et al. 2013) and many (D.Y. distance open-loop at crossover system (H. Remezanian et al. 2013) and many other (D.Y. distance between the open-loop phase at the crossover ° gain Xue et al. 2006; S. Das 2012; G. Q. Zeng et al. 2015; Guofrequency and the stability limit of −180 . However, the ° system (H. Remezanian et al. 2013) and many other (D.Y. Qiang Zeng et al. 2015). The FOPI controller is designed distance having between the open-loop phase at the crossover system a left half-plane zero located near the Xue et al. 2006; S. Das 2012; G. Q. Zeng et al. 2015; Guo° . gain However, the frequency and the stability limit of −180 Xue et al. 2006; S. Das 2012; G. Q. Zeng et al. 2015; Guofrequency and the stability limit of −180 . However, the Qiang Zeng et al. 2015). The FOPI controller is designed system having a left half-plane zero located nearnonthe ° Xue et al. 2006; S. 2015). Das 2012; G. Q. Zeng et al. work. 2015; Guofor DC-buck regulator system in this The . However, the frequency and the stability limit of −180 dominant-poles (i.e minimum-phase system) shows Qiang Zeng et al. The FOPI controller is designed system having a left half-plane zero located near the Qiang Zeng et al. 2015). The FOPI controller is designed system having a left half-plane zero located near the for DC-buck regulator system in this work. The dominant-poles (i.e minimum-phase system) shows nonQiang Zeng et al. 2015). The FOPI controller is designed optimization of the FOPI parameters can be done by any system having a left half-plane zero located near the monotonic phase behaviour inside the bandwidth (Caio for DC-buck regulator system work. dominant-poles (i.e minimum-phase system) shows nonfor DC-buckof the regulator system in incanthis this work.by The The dominant-poles (i.ebehaviour minimum-phase system) shows (Caio nonoptimization FOPI parameters be done any monotonic phase inside the bandwidth for DC-buck regulator system in this work. The optimization algorithm like Genetic Algorithm, Particle dominant-poles (i.e minimum-phase system) shows nonF.de Paula et al. 2012). For a closed loop system to be of the parameters can be by monotonic phase behaviour inside the bandwidth (Caio optimization of the FOPI FOPIlike parameters can be done done Particle by any any monotonic phase behaviour inside theloop bandwidth (Caio optimization algorithm Genetic Algorithm, F.de Paula et al. 2012). For a closed system to be optimization of the FOPI parameters can be Nelder-Mead done Particle by any Swarm Optimization etc. InGenetic this paper monotonic phase behaviour theloop bandwidth (Caio stable the phase must positive. In system addition, optimization algorithm like Algorithm, F.de Paula Paula et al. al.margin 2012). For inside abeclosed closed system to the be optimization algorithm like Genetic Algorithm, Particle F.de et 2012). For a loop to be Swarm Optimization InGenetic this paper Nelder-Mead stable the phase must positive. addition, optimization like Algorithm, Particle algorithm has algorithm been usedetc. for this purpose (Nelder, John A. F.de Paula et al.margin 2012). For abeclosed loopIn system to the be phase margin a closed-loop system the Swarm Optimization etc. In paper Nelder-Mead stable the phase margin must In addition, Swarm Optimization etc. In this this paper Nelder-Mead stable the phase of margin must be be positive. positive. In estimate addition, the the algorithm has been used for this purpose (Nelder, John A. phase margin of a closed-loop system estimate the Swarm Optimization etc. Inapplying this paper Nelder-Mead and R. Mead (1965). Before in the closed-loop stable the phase margin must be positive. In addition, the robustness and informs how much the open-loop system algorithm has been used for this purpose (Nelder, John phase margin of a closed-loop system estimate the algorithm has been used for this purposein(Nelder, John A. A. phase margin of a closed-loop system estimate the and R. the Mead (1965). Before applying thecontroller closed-loop robustness and informs how much the open-loop system algorithm has been used forterms this purpose (Nelder, John are A. system fractional order of the FOPI phase may margin ofwhile a closed-loop system estimate the vary the closed-loop system remains and R. Mead (1965). Before applying in the closed-loop robustness and informs how much the open-loop system and R. Mead (1965). Before applying in the closed-loop robustness and informs how much the open-loop system system the fractional order terms of the FOPI controller are phase may vary while the closed-loop system remains and R. Mead (1965). Before applying in the closed-loop approximated into integer order using Oustaloup’s robustness and informs much the open-loop system stable (S. Skogestad and I.how Postlethwaite system order terms of FOPI are phase may vary the closed-loop system system the the fractional fractional order termsorder of the the using FOPI controller controller are phase may vary while while closed-loop2005). system remains remains approximated into integer Oustaloup’s stable (S. Skogestad and I.the Postlethwaite system the fractional order terms of athefrequency FOPI controller are approximation algorithm within range of phase may vary while the closed-loop2005). system remains approximated into integer order using Oustaloup’s stable (S. Skogestad and I. Postlethwaite 2005). approximated into integer order using Oustaloup’s stable (S. regulator Skogestadtaken and I.inPostlethwaite 2005). DC-buck this paper also shows the nonapproximation algorithm within a frequency range of �� �� approximated into integer order using Oustaloup’s − 10 ) rad/sec (A. Oustaloup, 1991; A. ω ∈ (10 stable (S. Skogestad and I. Postlethwaite 2005). approximation algorithm within aa frequency range of DC-buck regulator taken in this paper also shows the non-a �� �� approximation algorithm within frequency 1991; range A. of monotonic phase behaviour inside the bandwidth. Hence − 10 ) rad/sec (A. Oustaloup, ω ∈ (10 DC-buck regulator taken in this paper also shows the non�� �� approximation algorithm within aOustaloup, frequency 1991; range A. of DC-buck regulator taken in this paper also shows the non-a Oustaloup, 1981). �� − �� ) rad/sec 10 (A. ω ∈ (10 monotonic phase behaviour inside the bandwidth. Hence − 10 ) rad/sec (A. Oustaloup, 1991; A. ω ∈ (10 DC-buck regulator taken in this paper also showstothe controller isphase needed for non-monotonic system getnontheaa Oustaloup, �� 1981). �� monotonic behaviour inside the bandwidth. Hence − 10 ) rad/sec (A. Oustaloup, 1991; A. ω ∈ (10 monotonic phase behaviour inside the bandwidth. Hence Oustaloup, 1981). The detailed organization of this paper is as follows: in controller isphase needed for time non-monotonic system get thea Oustaloup, 1981). monotonic behaviour inside the bandwidth. Hence wider bandwidth, faster response and moreto controller is for non-monotonic system to get The detailed organization of of this paper isregulator as follows: in Oustaloup, 1981). controller is needed needed for time non-monotonic system tosensitive get the the Section 2 the circuit diagram DC-buck and its wider bandwidth, faster response and more sensitive The detailed organization of this paper as in controller is needed for non-monotonic system to get the to noisebandwidth, and parameter variations. The detailed organization of of this paper is isregulator as follows: follows: in wider faster time response and more sensitive Section 2 the circuit diagram DC-buck and its wider bandwidth, faster time response and more sensitive The detailed organization of this paper is as follows: in nonmonotonically decreasing phase behaviour is to noisebandwidth, and parameter variations. Section 2 the circuit diagram of DC-buck regulator and its wider faster time response and more sensitive Section 2 the circuit diagram of DC-buck regulator and its to noise and parameter variations. nonmonotonically decreasing phase behaviour is to noisefew and decades, parameterthe variations. Since fractional order controllers are Section 2 the circuit diagram of DC-buck regulator and its addressed. Section 3 explains detailed study of FOPID nondecreasing phase behaviour is to noisefew and decades, parameterthe variations. non- monotonically monotonically decreasing phasestudy behaviour is Since fractional order controllers are addressed. Section 3 explains detailed of FOPID being the part of the control application due controllers to having extra non- monotonically decreasing phasestudy is controller design. 4 detailed shows the behaviour results and Since few decades, the fractional order are addressed. Section 33Section explains of FOPID Since few decades, the fractional order controllers are addressed. Section explains detailed study of FOPID being the part of theThe control application due to having extra controller design. Section 4 detailed shows the results and Since few decades, theintegral fractional order controllers are degree of freedom. and the derivative term of addressed. Section 3 explains study of FOPID comparative study of FOPI and other conventional IOPI being the part of the control application due to having extra controller design. Section 4 shows the results and being the of theThe control application due to having extra controller design. 4 shows the results IOPI and degree of part freedom. integral andinthe derivative of comparative study ofSection FOPIconclusion and other conventional being the part of theThe control application due to having controller design. Section 4 shows the results and Dμμextra ) (I. techniques. Finally, the followed by IOPI the the fractional order controller are fractions (PI λλterm degree of freedom. integral and the derivative term of comparative study of FOPI and other conventional degree of freedom. The integral and the derivative term of comparative study of FOPI and other conventional IOPI Dμμet) al. (I. techniques. Finally, followed by IOPI the the fractional order controller fractions (PI λλterm degree ofS.freedom. The integralare andin the derivative of comparative study of the FOPIconclusion and other conventional Pan and Das 2012; Shantanu Das 2008; G. Q. Zeng references. D ) (I. techniques. Finally, the conclusion followed by the fractional order controller are in fractions (PI ) al. (I. techniques. the conclusion followed by the the the fractional fractions λ Dμet Pan and S. Dasorder 2012;controller Shantanu are Dasin 2008; G. Q. (PI Zeng references. Finally, D ) (I. techniques. Finally, the conclusion followed by the the fractional order controller are in fractions (PI Pan references. Pan and and S. S. Das Das 2012; 2012; Shantanu Shantanu Das Das 2008; 2008; G. G. Q. Q. Zeng Zeng et et al. al. references. Pan and S. Das 2012; Shantanu Das 2008; G. Q. Zeng et al. references.

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2.

237

DC-BUCK REGULATOR SYSTEM

The DC-buck regulator is the most commonly used dc-dc converter topology and have a very large application area like in power management and microprocessor voltage regulator applications (Caio F.de Paula, and Luís H. C. Ferreira, 2012). This is very popular because of its smaller size and efficiency compared to the linear regulators. In this paper the design of controller is made on the simplest dc-dc converter circuit i.e. the DC-buck regulator circuit. The DC-buck regulator system is a combination of the power stage (i.e a LC low-pass filter) and a pulse-width modulation (PWM)-based controller (International Rectifier, 2002). The circuit diagram of DC-buck converter with voltage controller is given in Fig. 1. Fig. 2. Bode plot of DC-buck regulator system

Fig. 1. The DC-Buck Converter with a voltage controller

The transfer function of DC-buck regulator system can be written as ratio of the output (regulated voltage) to the input (PWM modulator input voltage) as: 𝐺=

��� (����� �)

� �

���� ��� � ����� �� ���

(1) Fig.3. Step response of closed-loop DC-buck regulator

where C is output capacitance, L is output Inductance, R is load resistance, R � is the output capacitor intrinsic resistance, V�� the power stage input voltage and V��� is the PWM oscillator reference voltage.

3.

FOPID is presented in generalized form of IOPID controller and represented as PI � D� (I. Podlubny, 1999). The FOPID controller is very powerful and provides flexible control design approach. The transfer function for the FOPID controller is given as:

For a typical application of DC-buck regulator taken in International Rectifier, (2002) with R � = 40 mΩ, the transfer function is given as: 𝐺=

�(���.�����

����� � � ��.����� ���

FRACTIONAL ORDER PID CONTROLLER

�

𝐶����� (𝑠) = 𝐾� + ��� + 𝐷𝑠 �

(2)

(3)

where for λ and μ are the fractional power of integral and differential control respectively.

The Bode plot of the DC-buck regulator is shown in Fig. 2, where the non-monotonic phase behaviour close to the undammed frequency can be seen in the phase plot. The closed-loop step response without any controller is also shown in Fig. 3, which shows that the system is underdamped. Hence a controller is needed to compensate the non-monotonic phase actions of the system.

For FOPI controller the transfer function will be: �

𝐶���� (𝑠) = 𝐾� + ���

(4)

Theses FOPI controller variables (K � , K � , and λ) are optimized by using Nelder-Mead algorithm (Nelder, John A. and R. Mead, 1965).

237

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The fractional terms of this optimized FOPI controller is needed to approximate into integer order before applying in the closed-loop system. Oustaloup’s approximation algorithm within a frequency range of ω ∈ (10�� − 10�� ) rad/sec is used in this paper.

with a new vertex. Thus a new triangle is formed and the search is continued. The process generates a chain of triangles (it may have different shapes depending on number of variables), for which the function values at the vertices get lesser and lesser. The size of the triangles is reduced and the coordinates of the smallest point are obtained.

There are mainly two methods in the literature for integer order approximation of fractional order terms in continuous domain as:

4.

 Approximations using continued fraction expansions and interpolation techniques. The techniques are based on this approximation are: a) General CFE method for approximation of fractional integro-differential operators. b) Carlson's method. c) Matsuda's method.  Approximations using curve fitting or identification techniques. The techniques are based on this approximation are given as: a) Oustaloup Recursive Approximations. b) Chare's method. c) Modified Oustaloup Filter.

As shown in Fig. 3 the closed-loop response of the DCbuck regulator system is under-damped and shows the nonmonotonic phase inside the bandwidth. Therefore, a controller is required to control the output voltage of the system. A FOPI controller is designed to improve the performance of the system. The parameters of the FOPI controller are optimized using NM-method and before implementing it into the system the fractional order terms are approximated into integer order by using Oustaloup’s approximation algorithm. The performance criterion considered for designing the controller is ISE. The objective function is given by following equation: �

𝐽 = ∫� 𝑒 � (𝑡) 𝑑𝑡

The Oustaloup’s approximation algorithm is one of the most popular methods used for integer order approximation of fractional order systems within a specified frequency band (A. Oustaloup, 1991; A. Oustaloup, 1981). Suppose that the frequency range to be fit by an integer order filters to fractional order derivative are given by [ω� , ω� ], the term s/ω� can be substituted with. ���/�

(7)

where 𝑒 is the error of the system and 𝑡 is the time period. The parameters of the FOPI controller obtained using Nelder-Mead optimization areK � = 175, K � = 97 and λ = 0.81. The step response of the FOPI controller is compared with the closed-loop response of the system in Figure 4.

3.1 Oustaloup’s Approximation Algorithm

𝐶� ���/� �

RESULT AND DISCUSSION

(5)

�

�

�

where �𝜔� 𝜔� = 𝜔� and 𝐶� = � � = �� �

�

The Oustaloup’s approximation of a fractional order differentiator s∝ can be written as

G ( s )  (C0 )

N

1  s  'k

 1 s 

k N

��

(6)

k

� ∝ ���� � � � ����

��

� ∝ ���� � � � ����

and 𝜔� = 𝜔� �� � where, 𝜔�′ = 𝜔� �� � � � are respectively the zeros and poles of rank 𝑘. The total number of zeros or poles are given as (2𝑁 + 1).

Fig. 4. Comparison of step response of FOPI controller and closed-loop system

The step response of the DC-buck regulator system with FOPI controller is also compared with PI controller designed by using various conventional methods like ZN (J. G. Ziegler and N. B. Nichols; 1943), Approximated Mconstrained integral gain optimization (MIGO) (K.J. Astrom and T. Hagglund; 2004), Chen-Hrones-Reswick (Kun Li Chien et al. 1952) and Skogested Internal model control (IMC) (Rivera D.E. et al. 1986) in Figure 5. The

3.2 Nelder-Mead Optimization Algorithm Nelder and Mead presented a very simple method in 1965 to find a local minimum of a function of many variables (Nelder, John A. and R. Mead, 1965). The method is a pattern search which compares function values at the three vertices of a triangle for two variables. The worst vertex of the triangle, where f(x, y) is largest, is rejected and replaced 238

IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, India Santosh Kumar Verma et al. / IFAC-PapersOnLine 49-1 (2016) 236–240

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the fastest and better control as compared to the all other methods.

FOPI controller provides better response in terms of performance characteristics like Rise-time, Settling-time and Peak-overshoot as compared to the other controllers.

Table 1: Comparison of Performance characteristics Control Techniques NM-FOPI ZN-IOPI Approx-MIGOIOPI Chen-HronesReswick-IOPI Skogested IMCIOPI Closed-loop Open-loop

Rise time (sec) 7.2848 × 10�� 2.7524 × 10�� 5.8402 × 10�� 3.6435 × 10�� 4.0378 × 10�� 2.8388 × 10�� 7.1775 × 10�� 5.

Fig. 5. Comparison of step response of FOPI controller and other conventional techniques

Peak Settling time Overshoot (sec) (%) 4.5028 2.2699 × 10�� 8.2769 52.0312 × 10�� 12 × 10�� 8.1506 × 10�� 8.6556 × 10�� 2.6031 × 10�� 5.9789 × 10��

30.8267

41.4965 38.4285 36.2644 34.3293

CONLUSION

A FOPI controller is designed to control the non-monotonic decreasing phase actions and output voltage of the DCbuck regulator system. The parameters of FOPI controller are optimized using Nelder-Mead method and the approximation is done by the use of Oustaloup’s approximation algorithm. The performance characteristics of FOPI are compared with various conventional techniques. The proposed technique offers enviable improvements which are not achievable when the classical design methods are used. Hence the proposed FOPI controller enhanced the system performance as compare to other conventional control techniques.

REFERENCES [1]

[2] [3]

[4]

Fig. 6. Comparison of Bode plots of FOPI controller and other conventional techniques

[5]

Comparison of bode plot of FOPI with other conventional methods are shown in Figure 6. It is very clear that using FOPI controller we get monotonic phase of the system whether in case of other controller the non-monotonic behaviour of the system remains same. The performance characteristics like Rise-time, Settlingtime and Peak-overshoot of all the methods are compared in Table 1, which reports that the FOPI controller provides

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