fractional order phase shaper design with bode’s integral for iso-damped control system

ISA Transactions 49 (2010) 196–206

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ISA Transactions

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Fractional order phase shaper design with Bode’s integral for iso-dampedcontrol system

Suman Saha a, Saptarshi Das a, Ratna Ghosh b, Bhaswati Goswami b, R. Balasubramanian c, A.K. Chandra c,Shantanu Das d, Amitava Gupta a,∗a Power Engineering Department, Jadavpur University, Kolkata-700098, Indiab Instrumentation & Electronics Engineering Department, Jadavpur University, Kolkata-700098, Indiac R & D, Electronic System, Nuclear Power Corporation of India Ltd., Mumbai-400085, Indiad Reactor Control Division, Bhabha Atomic Research Centre, Mumbai-400085, India

a r t i c l e i n f o

Article history:Received 23 October 2009Received in revised form3 December 2009Accepted 7 December 2009Available online 1 January 2010

Keywords:Bode’s integralFractional order calculusIso-dampingPhase shaper

a b s t r a c t

The phase curve of an open loop system is flat in nature if the derivative of its phase with respect tofrequency is zero. With a flat-phase curve, the corresponding closed loop system exhibits an iso-dampedproperty i.e. maintains constant overshoot with the change of gain. This implies enhanced parametricrobustness e.g. to variation in system gain. In the recent past, fractional order (FO) phase shapers havebeen proposed by contemporary researchers to achieve enhanced parametric robustness. In this paper, asimplemethodology is proposed to design an appropriate FO phase shaper to achieve phase flattening in acontrol loop, comprising a plant controlled by a classical Proportional Integral Derivative (PID) controller.The methodology is demonstrated with MATLAB simulation of representative plants and accompanyingPID controllers.

© 2009 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

PID controllers account for over 90% of controllers used inthe process industry. While PID controllers produce closed loopsystems with satisfactory tracking, disturbance rejection and ro-bustness of varying degrees depending upon the tuning method-ology [1,2], enhanced closed loop performance in terms of noiseattenuation, for example, can only be achieved by specific loopshaping techniques e.g. [3]. As a further enhancement, amethodol-ogy to achieve increased parametric robustness through flatteningof phase curve with a PID controller has been proposed by Chenet al. in [4] and extended in [5] with the use of a FO phase shaper.[4,5] propose an analytical method to obtain the constants of aPID controller to make the phase derivative zero around the tan-gent frequency (the frequency at which the Nyquist curve touchesthe sensitivity circle tangentially) and a simple FO element of theform sq to adjust the width of the flat-phase region. The methodhas been extended by Monje et al. in [6] to tune a Fractional OrderPID (FOPID) controller or a PIλDµ controller to achieve a flat-phasesystemwith enhanced parametric robustness and iso-damped stepresponse under varying gains.

∗ Corresponding author. Tel.: +91 3323355813; fax: +91 3323357254.E-mail address: [email protected] (A. Gupta).

0019-0578/$ – see front matter© 2009 ISA. Published by Elsevier Ltd. All rights reservdoi:10.1016/j.isatra.2009.12.001

A major issue in the use of FOPID controllers is the physical re-alization of such controllers. Vinagre et al. deal with issues relatedto the use of FOPID controllers for industrial applications in [7].As reported in [7], a FO element can be realized by a lossy ca-pacitor based micro-electronic approach [8,9]; by analog circuitrealization [10,11] or by integer approximation using, for exam-ple, Carlson representation [12,13] or the Crone Toolbox [14–16].However, integer approximations are valid for specific frequencyranges. For the present work, Carlson representation is adoptedwhich, in the simplest case, approximates a FO differ–integrator(differentiator or integrator) as a rational transfer function with afirst order numerator and a first order denominator. The approachis independent of the method of realization of FO elements. Themethodology presented in the paper proposes the use of a FO dif-fer–integrator for phase shaping and automatically establishes theinteger approximated FO phase shaper as a rational transfer func-tion, which is effective for a frequency spread around a specifiedfrequency viz. the gain crossover frequency. The phase shaper de-signed by this methodology produces the widest flat-phase regionaround the gain crossover frequency of the system comprising theplant and its controller, with the phase margin fixed above a spec-ified value. Thus, the resultant closed loop system exhibits iso-damped step response, with constant overshoot for a variation ofsystem gain within a range. The plant may be assumed to be a FirstOrder Plus Time Delay (FOPTD) or a Second Order Plus Time

ed.

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S. Saha et al. / ISA Transactions 49 (2010) 196–206 197

Delay (SOPTD) system, which is a reasonable approximation formost process plants [17] and assumed to be tuned by any methodsuitable for tuning industrial PID controllers, for example, those re-ported in [2,18].The methodology is demonstrated using different FOPTD sys-

tems viz. balanced, lag dominant and delay dominant as reportedin [19] and with a SOPTD system, to cover a wide spectrum of pro-cess plants. The results have been repeated using higher order ap-proximation of FO phase shaper [12,13] with little difference insimulated results, implying that a simple phase shaper approxi-mated by first order transfer functions is sufficient. This feature,coupled with the fact that the methodology does not assume anyspecific characteristic of the PID controller used, makes it prac-tically usable in existing control loops. Enhanced parametric ro-bustness to gain variation makes it particularly usable for caseswhere the system gain varies depending on the regime of opera-tion e.g. nonlinear systems controlled using piece-wise linear ap-proximations, requiring switching of controllers [20].The paper is organized as follows. Section 2 presents the

methodology of phase flattening using a FO phase shaper. Simu-lation results are presented in Section 3, followed by conclusion inSection 4 and references.

2. Phase shaping with FO differ–integrator based on Bode’sintegral

In this section, the methodology of designing a phase shaperwith a FO differ–integrator is presented for a given system G(s)comprising a plant Gpl(s) and its PID controller Gc(s) i.e.

G(s) = Gc(s)× Gpl(s). (1)

The FO phase shaper Gph(s) is so designed that the resultant closedloop system exhibits iso-damped response to step changes in inputover a range of gain variations. The design methodology for thephase shaper maximizes the width in terms of frequency of theflat-phase region in the asymptotic phase curve around the gaincrossover frequency of G(s). This ensures a constant phase marginand hence gain-independent overshoot (iso-damped response) forthe time response of the system

Gol(s) = Gph(s)× Gc(s)× Gpl(s) (2)

in closed loop. A flat-phase curve around the gain crossover fre-quency also ensures enhanced parametric robustness.The methodology presented in this section assumes the

following:

(a) The plant transfer function Gpl(s) is can be approximated bya First Order plus Time Delay (FOPTD) or a Second Order plusTime Delay (SOPTD) model.

(b) The PID controller may be tuned by any standard method andthe closed loop system comprising the plant and the controlleris a stable system.

(c) The phase shaper does not add any additional gain, nor does itadd a net phase change to G(s).

The design methodology for a FO phase shaper presented inthis section, uses Bode’s phase and gain integral formulae whichprovide information regarding the gain and phase derivatives of asystemwith respect to frequencywithout the specific model of theplant. The methodology differs from the one proposed in [21–23]in the sense that the plant in this case is considered along with itsPID controller, and thus it can be applied to any existing controlloop without having to tune the controller again.The Bode’s phase integral formula is used to approximate the

derivative of the phase of G(s) around its gain crossover frequency

ωgc . Using Bode’s phase integral, it can be shown, that for a stable,minimal phase system in the neighborhood of any ω,

ωd6 G(jω)dω

= 6 G(jω)+2π[ln |kg | − ln |G(jω)|] (3)

where kg is the static gain of G(s).It is established in [21,23] that Eq. (3) is valid for both minimal

and non-minimal phase systems alike.Substituting ω = ωgc in Eq. (3) yields

d6 G(jω)dω

∣∣∣∣ω=ωgc

=φm − π

ωgc+

2πωgc

ln |kg | (4)

where φm is the phase margin of G(s).Now, if it is attempted to flatten the phase around ω = ωgc

using a phase shaper Gph(s), then the relationship

ddω6 G(jω)+

ddω6 Gph(jω) = 0 (5)

must be satisfied over a frequency band∆ω, around ωgc .

For a FOPTD or a SOPTD system, d6 G(jω)dω

∣∣∣ω=ωgc

is negative and

therefore, and therefore, for Eq. (5) to be valid in the neighborhoodof ωgc , ddω 6 Gph(jω)

∣∣ω=ωgc

should be positive. This can be achievedby using a phase shaper of the form

Gph(s) = (1+ asq) (6)

with1a≤ ωqgc (7)

and 0 ≤ q ≤ 1. (8)

Next, if it be assumed that the phase shaper Gph(s) does notintroduce any change in static gain or phase of the original plant,then the phase shaper described by Eq. (6) must be modified as

Gph(s) =(1+ asq)sq

. (9)

With the phase shaper represented by Eq. (9), using Eq. (4), Eq.(5) can be re-written as

φm − π

ωgc+

2πωgc

ln |kg | +aqωq sin qπ2

ω(1+ 2aωq cos qπ2 + a2ω2q)

= 0. (10)

Eq. (10) assumes that the slope of the phase curve of G(s) re-mains constant aroundωgc . The advantage of a FO differ–integratoras a phase shaper arises from the fact that it allows a flexibilityin making the phase curve of Gol(s) flat over a varying frequencyspread by suitable selection of a and q.However, the addition of Gph(s) also alters the phase of G(s) at

ω = ωgc and the net phase of Gol(s) at ωgc can be expressed as

φ|ω=ωgc = φm − π −qπ2+ tan−1

(aωq sin qπ2

1+ aωq cos qπ2

). (11)

From Eq. (11) it follows that the phase of Gol(s) at ωgc is lessthan the phase of G(s) at the same frequency. Since the phaseshaper flattens the phase curve of Gol(s) around ωgc , it follows thatthe phase margin may reduce with the introduction of the phaseshaper. Thus if the minimum desired phase margin with the phaseshaper be φmd, then it follows that the constraint

φmd − φm +qπ2− tan−1

(aωq sin qπ2

1+ aωq cos qπ2

)≤ 0 (12)

must also be satisfied.

198 S. Saha et al. / ISA Transactions 49 (2010) 196–206

80

60

40

20

–20

–40

–135

0M

agni

tude

(dB

)

10–2 10–1 100 101

Bode DiagramGm = 20 dB (at 257 rad/sec) , Pm = 35.8 deg (at 0.353 rad/sec)

–90

–180

Pha

se (

deg)

10–3 102

Frequency (rad/sec)

Fig. 1. Frequency response of plant (13) with controller (14) with and without phase shaper (15).

1.5

1

0.5

0

5 10 15 20 25 30 35

2

–0.5

Am

plitu

de

Time (sec)

Step Response

0 40

Fig. 2. Step responses of plant (13) with controller (14) with and without phase shaper (15), under varying loop gain.

Thus, the problem of designing a phase shaper of the form rep-resented by Eq. (9) that producesmaximum flatness in terms of fre-quency spread around gain crossover frequency can be viewed asone involving constrained optimization that finds (q, a)maximiz-ing the value of |ω−ωgc | and satisfying the constraints representedby Eqs. (7), (8), (10) and (12).This optimization problem is solvedusingMATLAB’s Optimization Toolbox function fmincon() [24]. Therationale behind use of fmincon() arises from the fact that this is anoptimization problem with nonlinear constraints and it has beenshown that for such optimization problems fmincon() can be usedeffectively, as reported in [6], for example. For the present work,fmincon() was customized to use an Active set algorithm [24] andis found to produce satisfactory results.

3. Simulation and results

In this section, the design methodology, presented in theprevious section is applied to different FOPTD plants with varying

relative dead-time τ = LL+T (where L is the delay and T is the

time constant) and a SOPTD plant to obtain a flat-phase region onthe phase curve around ωgc . In each case, the frequency responsecurves of the systems with and without the phase shaper arepresented. The plant with the phase shaper is then subjected togain variations, and the resultant step response is shown to exhibitiso-damping over the range of gain variation selected. The plantswith respective phase shapers in closed loop are then tested forstepped input and output load disturbances as reported in [1]and the results are presented. These results directly indicate thetracking and disturbance rejection ability of the system normallyinferred from sensitivity and complementary sensitivity.

3.1. FOPTD plant with balanced lag & delay (L ≈ T)

A balanced lag & delay FOPTD plant is chosen as

Gpl(s) =5

1.5s+ 1e−s (13)


1.5

1

0.5

0

10 20 30 40 50 60 70

–0.5

–1

–1.5

Am

plitu

de

Controller Output

Time (sec)

0 80

Fig. 3. Controller output signal for plant (13) with controller (14) with and without phase shaper (15).

4

3

2

10 20 30 40 50 60 70

1

0

Time (sec)

5

–1

Am

plitu

de

0 80

Step Input & Load Disturbance Response

Fig. 4. Step input response & load disturbance response for plant (13) with controller (14) with and without phase shaper (15).

with the transfer function of its accompanying PID controller as

Gc(s) = 0.364+0.22s+ 0.149s. (14)

The open loop transfer function system Gol(s) = Gc(s)× Gpl(s)is found to have a gainmargin of 6.08 dB and a phasemargin of 59◦at a gain crossover frequency of ωgc = 1.03 rad/s.Following, the methodology presented in Section 2, the phase

shaper is obtained with φmd ≥ 35◦ (desired phase margin) as

Gph =1+ 2.3945s0.8182

s0.8182. (15)

Fig. 1 shows the frequency response curves for the plant (13)and the controller (14)with andwithout the phase shaper (15). It isseen that the phasemargin reduces to 35.8◦ from59◦, as predicted,and the gain margin increases to 20.0 dB from 6.0 dB. The gaincrossover frequency reduces from 1.03 rad/s to 0.353 rad/s.Next, the phase shaper is augmented with a scalar factor k,

Gph = k(1+ asq

sq

)(16)

and the time response of the corresponding closed loop systemdueto a step input is plotted in Fig. 2 with k varying from 1 to 5 (500%)in steps of 1 to demonstrate iso-damping. The factor k varies theloop gain and the phase shaper with k = 1 may be viewed as thenominal phase shaper. If the time response is iso-damped, k maybe varied to alter the rise-time, keeping the overshoot constant.In Fig. 1, it is seen that the phasemargin reduces with the intro-

duction of the phase shaper, which is evident from the increasedovershoot due to a step input as shown in Fig. 2. Again, it is seenfrom Fig. 1, if the scalar gain k is increased, the gain crossover fre-quency shifts towards a higher frequency value, but the phasemar-gin remains constant. Fig. 2 shows that increase in scalar factor kreduces the rise-time while maintaining the same overshoot. Thescalar gain, as shown in Fig. 2 can be varied by 500% keeping theovershoot constant. The advantage of the phase shaper becomesevident considering the fact that the PID controller alone cannothandle such large variation in gain. The closed loop system, withthe PID controller alone becomes unstable when the system gainis increased by a factor more than two. Next, the respective closedloop systems described in Fig. 2 are subject to an input and an


20

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10–1 100 101

–90

–180

–225

102

30

–50

Mag

nitu

de (

dB)

–45

–270

Pha

se (

deg)

Frequency (rad/sec)

Bode DiagramGm = 14.8 dB (at 7.29 rad/sec) , Pm = 87.7 deg (at 0.248 rad/sec)

10–2 103


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2

–0.5

Am

plitu

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Time (sec)

Step Response

0


output disturbance in the form of a unit step. Fig. 3 shows the con-troller output with and without the phase shaper for a unit stepinput and output load disturbance and Fig. 4 shows the closed loopsystem outputs under these conditions.Fig. 3 shows that themaximum controller output becomes con-

siderably lower in magnitude with the introduction of the phaseshaper compared to the system controlled by only a PID controlleralone. Thus, with lower value of maximum controller output, theactuator size can be reduced considerably and the possibility of theactuator saturation is also reduced.This methodology is tested further for a delay dominated and

a lag dominated FOPDT system and finally a SOPDT system. Thesimulation results are presented in Sections 3.2–3.4.

3.2. Delay dominated FOPTD plant (L� T)

A delay dominant FOPTD plant is chosen with a transferfunction

G(s) =1

0.05s+ 1e−s (17)

and assumed to be tuned by a PI controller with

Kp = 0.41, Ki = 0.24. (18)

Application of the proposed methodology presented in Section 2yields a phase shaper

Gph =1+ 0.1439s0.666

s0.666. (19)

The frequency response curves of the plant (17) with the controller(18) with andwithout the phase shaper (19) are presented in Fig. 5and the corresponding closed loop time response to unit step,under varying gain are presented in Fig. 6. The controller output& system output responses of the closed loop system with andwithout the phase shaper, for input and load disturbances in theform of a unit step are shown in Figs. 7 and 8 respectively.


1.2

1

0.6

0.8

0.4

10 20 30 40 50 60 70

0.2

0

Controller Output

Time (sec)

0 80

1.4

–0.2

Am

plitu

de


1.5

1

10 20 30 40 50 60 70

0.5

0

Time (sec)

0 80

Step Input & Load Disturbance Response2

–0.5

Am

plitu

de

Fig. 8. Step input response & load disturbance response of plant (17) with controller (18) with and without phase shaper (19).

It is clear from the time response curves presented in Fig. 6 thatwithout the phase shaper (19), as the scalar loop gain is increasedby a factor of 2.25, the overshoot becomes very large. With theintroduction of the phase shaper, the closed loop system producesfairly acceptable response up to a loop gain variation of 3.5 or 350%,with steady decrease in the rise-time.In Fig. 7 the maximum value of controller output is lesser with

the proposed phase shaper, which enables to reduce actuator size& hence the associated cost, involved.

3.3. Lag dominated FOPTD plant (L� T)

G(S) =1

1.11s+ 1e−0.105s (20)

is controlled by a PI controller.

c(s) = 7.73+10.5s. (21)

Transfer function of the phase shaper obtained

G(s) =1+ 1.3104s0.7895

s0.7895. (22)

Figs. 9–12 represent the iso-damped step response under varyingk, the frequency response with phase flattening due to the phaseshaper, the output of the closed loop system due to an input vari-ation (unit step) and output disturbance and corresponding con-troller outputs for this system.It is observed that for all three kinds of FOPTDplants,with phase

shaper proposed in this paper, phase margin reduces, leading toslight increase in overshoot but in each case the response exhibitsiso-damping. Further, in each case, the controller output becomeslower than the system controlled by PI/PID controller, which ishighly desirable for reduction in actuator size. This, however, as-sumes that a nominal phase shaper is used.


1.5

1

0.5

0

2

–0.5

Am

plitu

de

0.5 1 1.5 2 2.5 3 3.5Time (sec)

Step Response

0 4


60

40

–20

0

–40

–135

20

10–1 100 101

–90

–180

–225

102–270

Pha

se (

deg)

Frequency (rad/sec)


10–2 103

80

–60

Mag

nitu

de (

dB)


3.4. SOPTD plant

Finally, a second order plant with time delay is considered

G(s) =2.25

s2 + 2.7s+ 2.25e−0.2s. (23)

With the transfer function of the PID controller

C(s) = 5.238+4.896s+ 1.965s. (24)

Phase shaper transfer function obtained is

Gph(s) =1+ 0.5s0.8182

s0.8182. (25)

As reported for the FOPTD plants considered, phase marginof the SOPDT plant also reduces (Fig. 13), which is evident fromthe increased overshoot in Fig. 14. The scalar gain k is increasedto shift the gain crossover towards higher frequency, but the

movement over constant phase makes the system iso-damped innature (Fig. 14). Similar to the case of FOPTD systems presentedin this section, the initial value of controller output reduces withphase shaper for the SOPTD system also (Fig. 15). In Fig. 16 thestep input and the load disturbance responses are shown for theconsidered SOPTD system.With the proposed phase shaper & PID controller, for the SOPTD

system also, the closed loop response is found to have constantovershoot with progressively reducing rise times for 200% increasein scalar gain, whereas the PID controller alone gives huge oscilla-tory response with that increased value of gain which is evidentfrom Fig. 14.It is well known that the Sensitivity function S(s) is an indi-

cation of the ability of the system to suppress load disturbances& achieve good set-point tracking. The complementary sensitiv-ity function T (s) indicates the robustness to measurement noise& other unmodeled system dynamics [19]. To have a good time re-sponse under these disturbed conditions, the sensitivity function


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Controller Output

Time (sec)

0 80


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plitu

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Step Input & Load Disturbance Response


Table 1Complementary sensitivity & sensitivity values of the test plants.

Plant T (jω) at ω = 10 rad/s S(jω) at ω = 0.01 rad/sWith PID & phase shaper With PID only With PID & phase shaper With PID only

Balanced FOPTD 0.0920 0.3347 0.0091 0.0090Delay dominated FOPTD 0.0251 0.2687 0.0400 0.0400Lag dominated FOPTD 0.0927 0.4117 0.000954 0.000951SOPTD 0.1027 0.3135 0.0020 0.0020

should have small values at lower frequencies & complementarysensitivity should have small values at higher frequencies [6].Withthe introduction of the phase shaper, in all the above four cases, sig-nificant improvements in robustness (in terms of gain variation),reduced controller outputs (less chance of actuator saturation &lesser actuator size)& complementary sensitivity at high frequency(better high frequency noise rejection) are obtained. But thesethree improvements are obtained at the cost of slight increase insensitivity function at low frequency for balanced FOPTD & SOPTDplant & hence poor load disturbance response. Table 1 lists the

sensitivity and complementary sensitivity functions computed foreach plant considered, with a nominal phase shaper in each case.This problem of inferior load disturbance response has been

tackled in [6] with the inclusion of additional constraints limitingthe values of S(jω), while designing the fractional order controlleritself. Also, in all the simulated examples, the overshoot is slightlyincreased due to the lagging nature of the phase shaper & hencedue to the reduction in phase margin. But as reported in (12),the optimization problem itself is designed to take care of theminimum desired phase margin (φmd) as a nonlinear constraint,


5040

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2030

10

10–1 100 101

–90

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–225

102–270

Pha

se (

deg)

Frequency (rad/sec)


10–2

60

–40

Mag

nitu

de (

dB)


1.5

1

0.5

0

1 2 3 4 5 6 7

2

Step Response2.5

–0.5

Am

plitu

de

Time (sec)

0 8


which effectively controls the overshoot of the overall closed loopsystem.For the presentwork, it is assumed that the PID controllers used

for simulation are assumed to be by the methodology describedin [18] which guarantees a certain minimum phase margin. Theconstraint (12) specifying the minimum acceptable phase margincan be used to get desired closed loop response. Tuning the PIDcontrol loops with a higher value of damping e.g. methods specify-ing gain and phase margins are likely to result in acceptable closedloop responsewith iso-dampingwhen the phase shaper is includedin the loop.While simulating the time & frequency response of the de-

lay term of the plants, Pade’s First Order approximation is usedthroughout the paper. For all the results presented, asmentioned inSection 2, sq is assumed to be represented by a first order Carlson’sapproximation which is a rational transfer function with a first or-der numerator and a first order denominator. The constrained op-timization technique automatically establishes the parameters of

the phase shaper {a, q} that produces themaximum flat-phase fre-quency spread around the specified gain crossover frequency ωgc .With the {a, q} obtained for each phase shaper, the results wererepeated using higher order Carlson’s representation and almostidentical results were obtained. Higher order Carlson’s representa-tions of a fractional differ–integrator use rational transfer functionswith higher order polynomials as integer order approximations ofa FO differ–integrator with a greater accuracy. This proves the ad-equacy of a first order Carlson approximated FO differ–integratorand shows that the methodology presented in this paper can beused to design a practically realizable phase shaper which can beused in conjunction with any PID controller for a process plant.

4. Conclusion

The methodology put forward in this paper, can be used for en-hancing the parametric robustness of PID control loops to gain vari-ations. It iswell known that PID controllers are predominantly used


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Am

plitu

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for process control applications. The enhanced parametric robust-ness of any PID control loop with a FO phase shaper proposed inthis paper makes it a very useful tool to be used for process con-trol applications where system gains tend to vary with time. Theiso-damped nature of the response allows design of extremely fastsystems, keeping overshoot constant, provided the actuator con-straints can be met.Further, the phase shaper proposed is of a low order and prac-

tically realizable. Thus, in practical terms the methodology allowsdesign of a simple hardware element which can be used with PIDcontroller, tuned by any standard method [17,2].The enhanced robustness allows use of the controller and the

phase shaper to control nonlinear plants, for example, with vary-ing system gain in different regimes of operation. However, the in-creased parametric robustness is achieved at the cost of reducedphase margin. The practical approach, therefore, would be to tunethe system to a high value of damping, and then use FO phaseshaper so that the phase margin remains appreciably high. Theminimum phase margin constraint allows the designer to spec-ify the maximum allowable overshoot while flattening the phase

curve around gain crossover frequency for enhanced robustness ofthe control loop.The extension of this methodology to analytically establish the

adequacy of a single stage FO differ–integrator and its expressionis left as a scope for future work.

Acknowledgement

The work presented in this paper has been supported by theBoard of Nuclear sciences (BRNS) of the Department of AtomicEnergy, India, sanction no 2006/34/34-BRNS dated March 2007.

References

[1] Astrom KarlJ, Hagglund Tore. PID controller: Theory, design, and tuning.Instrument Society of America; 1995.

[2] Cominos P, Munro N. PID controllers: Recent tuning methods and designto specification. IEE Proceedings: Control Theory & Application 2002;149(1):46–53.

[3] Garcia Daniel, Karimi Alireza, Longchamp Roland, Dormido Sebastian. PIDcontroller design with constraints on sensitivity functions using loop slopeadjustment. In: Proceedings of the 2006 American control conferenceminneapolis, 2006.


[4] Chen YangQuan, Hu ChuanHua, Moore KL. Relay feedback tuning of robustPID controllers with iso-damping property. In: Proceedings of the 42nd IEEEconference on decision and control, vol. 3, 2003. p. 2180–5.

[5] Chen YangQuan, Moore Kevin L, Vinagre Blas M, Podlubny Igor. Robust PIDcontroller with a phase shaper. In: Proceedings of the first IFAC symposium onfractional differentiation and its application, 2004.

[6] Monje CA, Vinagre BM, Chen YQ, Feliu V, Lanusse P, Sabatier J. Proposalsfor fractional PIλDµ tuning. In: Proceedings of the first IFAC symposium onfractional differentiation and its application, 2004.

[7] Vinagre Blas M, Monje Concepcion A, Caldesron Antonio J, Suarez JoseI.Fractional PID controllers for industry applications. A brief introduction.Journal of Vibration an Control 2007;13(9–10):1419–29.

[8] Bohannan Gary W. Analog realization of a fractional control element—revisited. In: Proceedings of the 41st IEEE international conference ondecision and control, Tutorial Workshop 2: Fractional calculus applications inautomatic control and robotics, 2002. p. 203–8.

[9] Bhaskaran Tripti, Chen YangQuan, Bohannan Gary. Practical tuning offractional order proportional and integral controller (2): Experiments.In: Proceedings of ASME 2007 international design engineering technicalconferences & computers and information in engineering conferences, 2007.

[10] Podlubny I, Petras I, Vinagre BM, O’Leary P, Dorcak L’. Analogue realizations offractional-order controllers. Nonlinear Dynamics 2002;29(July):281–96.

[11] Charef A. Analogue realization of fractional-order integrator, differentiator andfractional PIλDµ controller. IEE Proceedings: Control Theory & Applications2006;153(6):714–20.

[12] Carlson GE, Halijak CA. Approximation of fractional capacitors (1/s)̂(1/n) bya regular newton process. IEEE Transactions on Circuit Theory 1964;11(2):210–3.

[13] Das Shantanu. Functional fractional calculus for system identification &controls. Berlin: Springer; 2008.

[14] Oustaloup A, Melchior P, Lanusse P, Cois O, Dancla F. The CRONE toolbox forMatlab. In: IEEE international symposium on computer-aided control systemdesign, 2000.

[15] Melchior P, Orsoni B, Lavialle O, Oustaloup A. The CRONE toolbox for Matlab:fractional path planning design in robotics. In: Proceedings of 10th IEEEinternational workshop on robot and human interactive communication,2001.

[16] Oustaloup Alain, Melchior Pierre. The great principles of the CRONE control.In: Proceedings of international conference on systems, man and cybernetics,systems engineering in the service of humans, vol. 2, Oct. 1993. p. 118–29.

[17] Madhuranthakam CR, Elkamel A, Budman H. Optimal tuning of PID controllersfor FOPTD,SOPTD and SOPTD with lead processes. Chemical Engineering andProcessing: Process Intensification 2008;47(2):251–64.

[18] He JB, Wang Qing-Guo, Lee Tong-Heng. PI/PID controller tuning via LQRapproach. Chemical Engineering Science 2000;55(13):2429–39.

[19] Chen YangQuan, Bhaskaran Tripti, Xue Dingyu. Practical tuning rule develop-ment for fractional order proportional and integral controllers. Journal of Com-putational and Non-linear Dynamics 2008;3(2):021403.

[20] Liu Cheng, Peng Jin Feng, Zhao Fu Yu, Li Chong. Design and optimizationof fuzzy-PID controller for the nuclear reactor power control. NuclearEngineering and Design 2009;239(11):2311–6.

[21] Karimi A, Garcia D, Longchamp R. PID controller design using Bode’s integrals.Proceedings of the American Control Conference 2002;6(8–10):5007–12.

[22] Karimi Alireza. Daniel Garcia and Roland Longchamp, Iterative controllertuning using bode’s integrals. In: Proceedings of the 41st IEEE conference ondecision and control, vol. 4, 2002. p. 4227–32.

[23] Karimi Alireza, Garcia Daniel, Longchamp Roland. PID controller tuning usingBode’s integrals. IEEE Transactions on Control Systems Technology 2003;11(6):812–21.

[24] MATLAB optimization toolbox user’s guide, The Mathworks, Inc.

fractional order phase shaper design with bode’s integral for iso-damped control system

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