Journal of Process Control 19 (2009) 349–352

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Journal of Process Control

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Short communication

Guaranteed dominant pole placement with PID controllers

Qing-Guo Wang a,*, Zhiping Zhang a, Karl Johan Astrom b, Lee See Chek a

a Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singaporeb Department of Automatic Control, Lund Institute of Technology, Sweden

a r t i c l e i n f o

Article history:Received 27 February 2008Received in revised form 18 April 2008Accepted 25 April 2008

Keywords:PID controllersDominant polesPole placementRoot locusNyquist plot

0959-1524/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.jprocont.2008.04.012

* Corresponding author. Tel.: +65 874 2282; fax: +6E-mail address: elewqg@nus.edu.sg (Q.-G. Wang).

a b s t r a c t

Pole placement is a well-established design method for linear control systems. Note however that with anoutput feedback controller of low-order such as the PID controller one cannot achieve arbitrary poleplacement for a high-order or delay system, and then partially or hopefully, dominant pole placementbecomes the only choice. To the best of the authors’ knowledge, no method is available in the literatureto guarantee dominance of the assigned poles in the above case. This paper proposes two simple and easymethods which can guarantee the dominance of the two assigned poles for PID control systems. They arebased on root locus and Nyquist plot respectively. If a solution exists, the parametrization of all the solu-tions is explicitly given. Examples are provided for illustration.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Pole placement in the state space and polynomial settings isvery popular. For SISO plants, the equivalent output feedback con-trol should be at least of the plant order minus one to achieve arbi-trary pole placement. Arbitrary pole placement is otherwisedifficult to achieve if one has to use a low-order output feedbackcontroller for a high-order or time-delay plant. One typical exam-ple is that in process control, the PID controller is used to regulate aplant with delay. To overcome this difficulty, the dominant poledesign was proposed. It is to choose and position a pair conjugatepoles which represent the requirements on the closed-loop re-sponse, such as overshoot and settling time. The dominant pole de-sign was first introduced by Persson [1] and further explained in[2]. The methods they proposed are based on a simplified modelof plants and thus cannot guarantee the chosen poles are indeeddominant in reality. In the case of high-order plants or plants withtime-delay, this conventional dominant pole design, if not wellhandled, could result in sluggish response or even instability ofthe closed-loop system. To the best of the authors’ knowledge,no method is available in the literature to guarantee the domi-nance of the assigned poles in the case above.

Thus, it is desirable to find out ways to ensure the dominance ofchosen poles and the closed-loop stability. This paper aims to solvethe problem. The idea behind our methods is that the chosen pairof poles give rise to two real equations which are solved for I and Dterms via the proportional gain and the locations of all other

ll rights reserved.

5 67791103.

closed-loop poles can then be studied with respect to this singlevariable gain by means of root locus or Nyquist plot techniques.Hence, two methods for guaranteed dominant pole placement withPID controller are naturally developed. The root locus method isused for systems without time-delay and the Nyquist plot methodis developed to handle time-delay systems.

The rest of the paper is organized as follows. Section 2 states theproblem and preliminary. Sections 3 and 4 each present one meth-od along with illustrating examples. Section 5 is the conclusion.

2. Problem statement and preliminary

Consider a plant described by its transfer function,

GðsÞ ¼ NðsÞDðsÞ e�sL; ð1Þ

where NðsÞDðsÞ is a proper and co-prime rational function. A PID control-

ler in the form of

CðsÞ ¼ KP þK I

sþ KDs

is used to control the plant in the conventional unity output feed-back configuration as depicted in Fig. 1. The closed-loop character-istic equation is

1þ CðsÞGðsÞ ¼ 0: ð2Þ

And the closed-loop transfer function is

HðsÞ ¼NðsÞ KDs2 þ KPsþ K I

� �DðsÞsþ NðsÞe�Ls KDs2 þ KPsþ K Ið Þ e

�Ls:

C s G sR(s) Y(s)

( ( ( (

Fig. 1. Unity output feedback control system.

350 Q.-G. Wang et al. / Journal of Process Control 19 (2009) 349–352

Suppose that the requirements of the closed-loop control perfor-mance in frequency or time domain are converted into a pair of con-jugate poles [2]: q1,2 = � a ± bj. Their dominance requires that theratio of the real part of any of other poles to �a exceeds m (m is usu-ally 3–5) and there are no zeros nearby. Thus, we want all otherpoles to be located at the left of the line of s = �ma, that is, the de-sired region as hatched in Fig. 2. The problem of the guaranteeddominant pole placement is to find the PID parameters such thatall the closed-loop poles lie in the desired region except the domi-nant poles, q1,2.

Substitute q1 = � a + bj into (2):

KP þK I

�aþ bjþ KDð�aþ bjÞ ¼ � 1

Gðq1Þ;

which is a complex equation. The complex equations can be decom-posed into two sub-equations, one from the real part and the otherfrom the imaginary part. Solving the two equations for KI and KD interms of KP yields

K Ia2þb2

2a Kp � ða2 þ b2ÞX1;

KD1

2a Kp þ X2;

(ð3Þ

where X1 ¼ 12b Im½ �1

Gðq1Þ� þ 1

2a Re½ �1Gðq1Þ�, X2 ¼ 1

2b Im½ �1Gðq1Þ� � 1

2a Re½ �1Gðq1Þ�. This

simplifies the original problem to a one-parameter problem forwhich well known methods are applicable now. The root locusmethod can be used for systems without time-delay, but it cannothandle time-delay systems. Hence the Nyquist plot method is pro-posed for time-delay systems.

3. Root locus method

The root locus method is used to show movement of the roots ofthe characteristic equation for all values of a system parameter.Here we plot the roots of the closed-loop characteristic equationfor all the positive values of KP and determine the range of KP such

-ma

Im

Re

A

A

2ρ

1ρ

'

Fig. 2. Desired region (hatched) of other poles.

that the roots except the chosen dominant pair are in the desiredregion as hatched in Fig. 2.

Substituting (3) into (2) yields

1þ X2NðsÞe�Ls

DðsÞ s� a2 þ b2� �

X1NðsÞe�Ls

DðsÞs

þ KPNðsÞe�Ls

DðsÞs2 þ 2asþ ða2 þ b2Þ

2as¼ 0: ð4Þ

Dividing both sides by the terms without KP, after some manipula-tion, gives

1þ KPGðsÞ ¼ 0; ð5Þ

where

GðsÞ ¼NðsÞ s2 þ 2asþ ða2 þ b2Þ

h ie�Ls

2aDðsÞsþ 2aX2NðsÞs2e�Ls � 2a a2 þ b2� �

X1NðsÞe�Ls: ð6Þ

It can be easily verified that the manipulation does not change theroots. If G(s) has no time-delay term, GðsÞ is a proper rational trans-fer function since the degrees of its numerator and denominator ofGðsÞ equal those of the closed-loop transfer function’s numeratorand denominator respectively. The design procedure is simply:

Step 1. Draw the root locus of (5) as KP varies;Step 2. Determine the interval of KP for guaranteed dominant

pole placement from the root locus;Step 3. Choose KP, KI and KD accordingly.

Example 1 shows the design procedure in detail.

Example 1Consider a fourth-order process,

GðsÞ ¼ 1

sþ 1ð Þ2 sþ 5ð Þ2:

If the overshoot is to be no larger than 8% and the rise time less than3 s, the corresponding dominant poles are q1,2 = � 0.4849 ± 0.6031j.Eq. (3) becomes

K I ¼ 0:6175KP þ 2:6037;KD ¼ 1:0312KP � 15:3910:

�

And it follows from (6) that

GðsÞ ¼ s2 þ 0:9698sþ 0:59890:9698s5 þ 11:64s4 þ 44:61s3 þ 43:26s2 þ 24:24sþ 2:525

:

The root locus of GðsÞ is exhibited in Fig. 3 with the solid lines whilethe edge of the desired region with m = 3 is indicated with dottedlines. Note that GðsÞ is of 5th order and has five branches of root loci,of which two are fixed at the dominant poles while the other threemove with the gain. From the root locus, two intersection pointscorresponding to root locus entering into and departing from thedesired region are located and give the gain range of KP 2 (25,83),which ensures all other three poles in the desired region. Besides,the positiveness of KD and KI requires KP > 14.9253. Taking the jointsolution of these two, we still have KP 2 (25,83). If KP = 45 is chosen,the PID controller is

CðsÞ ¼ 45þ 30:3931s

þ 31:0110s:

The zeros of the closed-loop system are at s = � 0.7255 ± 0.6735j.Fig. 4 shows the step response of the closed-loop system. The spec-ifications are met, whereas if the approximate dominant pole place-ment method in [2] is used the step response has an overshoot morethan 8%. As the dominance of the chosen poles are guaranteed, one

–12 –10 –8 –6 –4 –2 0–6

–4

–2

0

2

4

6

Kp = 25Kp = 25

Kp = 83

Fig. 3. Root locus for Example 1.

Time (sec)

Out

put

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

Rise Time: 1.46s

Overshoot: 6%

Fig. 4. Closed-loop step response for Example 1.

Q.-G. Wang et al. / Journal of Process Control 19 (2009) 349–352 351

would expect the proposed method always works better than theapproximate dominant pole placement method.

4. Nyquist plot method

If G(s) has time-delay, so will be GðsÞ. Then, drawing the rootlocus for it could be difficult and checking locations of infinitepoles is a forbidden task. Hence we propose the Nyquist plot basedmethod since the Nyquist plot works well for time-delay systems.The Nyquist stability criterion determines the number of unstableclosed-loop poles based on the Nyquist plot and the open-loopunstable poles. We use the same idea but have to modify theconventional Nyquist contour. The Modified Nyquist contour is ob-tained by shifting the conventional Nyquist contour to the left byma, as Fig. 2 shows. The image of G(s) when s traverses the modi-fied Nyquist contour is called the modified Nyquist plot. Hence thenumber of poles located outside the desired region plays the samerole as that of unstable poles in the standard Nyquist criterion.

Rewrite (5) as

1KPþ GðsÞ ¼ 0: ð7Þ

Eq. (7) always has q1,2 as its two roots by our construction. Thesetwo poles lie outside the desired region (shown in Fig. 2). We want

no more pole outside the desired region to ensure dominant poleplacement. Equivalently, we want the modified Nyquist plot ofGðsÞ to have the net number of clockwise encirclements with re-spect to ð� 1

KP;0Þ equal to 2 minus the number of poles of GðsÞ out-

side the desired region. This condition determines the interval of KP

such that roots of (7) other than two dominant poles are in the de-sired region, which can guarantee the two dominant poles are in-deed dominant. Note the condition comes from the criterion thatthe net number of clockwise encirclement equals the number ofclosed-loop poles outside the desired region (in this case 2) minusthe number of open-loop poles (poles of GðsÞ) outside the desiredregion. It is analog to the conventional Nyquist stability criterion;the only difference is that here we are concerned about poles out-side the desired region, not poles in RHP.

To find the number of poles of GðsÞ located outside the desiredregion is not easy at first glance. The poles of GðsÞ are the roots ofits denominator. The denominator has time-delay terms whosedimension is infinite. To solve the problem we construct anothercharacteristic equation from the denominator of GðsÞ in (6) asfollows:

1þ GoðsÞ ¼ 0; ð8Þ

where GoðsÞ ¼ X2NðsÞs2�ða2þb2ÞX1NðsÞDðsÞs e�Ls. GoðsÞ has its rational part with

the degrees of its numerator and denominator being equal to thoseof the original open-loop transfer function’s nominator and denom-inator respectively. The number of the roots of (8), i.e., poles of GðsÞ,lying outside the desired region, equals the number of clockwiseencirclements of the modified Nyquist plot of GoðsÞ with respectto (�1,0), plus the number of poles of GoðsÞ located outside the de-sired region. We note that the poles of GoðsÞ are easy to find fromthe denominator of GoðsÞ, which is just D(s)s. The theoretical basisfor applying the Nyquist stability criterion in this special case canbe found in [3–6].

The design procedure is summarized as follows:

Step 1. Find the poles of GoðsÞ (the roots of D(s)s) outside thedesired region and name its total number as Pþ

Go;

Step 2. Draw the modified Nyquist plot of GoðsÞ, count the num-ber of clockwise encirclements with respect to the�1 + j0 point as Nþ

Go, and obtain the number of poles of

GðsÞ outside the desired region as PþG¼ Nþ

Goþ Pþ

Go;

Step 3. Draw the modified Nyquist plot of GðsÞ and find the rangeof KP during which the clockwise encirclements withrespect to the ð� 1

KP; 0Þ is 2� Pþ

G.

We provide Example 2 here to illustrate the design procedure indetail.

Example 2Consider a highly oscillatory process,

GðsÞ ¼ 1s2 þ sþ 5

e�0:1s:

If the overshoot is to be not more than 10% and the settling time tobe less than 15 s, the dominant poles are q1,2 = � 0.2751 ± 0.3754j.Eq. (3) becomes

K I ¼ 0:3937KP þ 1:8773;KD ¼ 1:8173KP þ 7:7760:

�

We have

GoðsÞ ¼7:776s2 þ 1:877

sðs2 þ sþ 5Þ e�0:1s:

Take m = 3. We have ma = 0.8253 and all three poles of GoðsÞ outsidethe desired region and Pþ

Go¼ 3. Fig. 5 is the modified Nyquist plot of

GoðsÞ and there is one anti-clockwise encirclement of the point

–14 –12 –10 –8 –6 –4 –2 0 2–8

–6

–4

–2

0

2

4

6

8

Real axis

Imag

inar

y ax

is

Fig. 5. Modified Nyquist plot of Go for Example 2.

–0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

Real axis

Imag

inar

y ax

is

Fig. 6. Modified Nyquist plot of G for Example 2.

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Out

put

Overshoot: 10%

Settling Time: 13s

Fig. 7. Closed-loop step response for Example 2.

352 Q.-G. Wang et al. / Journal of Process Control 19 (2009) 349–352

(�1,0), that is, NþGo¼ �1. Therefore, GðsÞ has two poles located in the

desired region since PþG¼ Nþ

Goþ Pþ

Go¼ 2. It means the modified

Nyquist plot of GðsÞ should have its clockwise encirclement withrespect to the point (�1/KP,0), equal to 2� Pþ

G¼ 0, that is zero net

encirclement, for two assigned poles to dominate all others. Fig. 6shows the modified Nyquist plot of GðsÞ, from which � 1/KP 2(�1, � 0.2851) is determined to have zero clockwise encirclement.A positive KP could always make KD and KI positive. Therefore, wehave the joint solution as KP 2 (0,3.5075). If KP = 1 is chosen, thePID controller is

CðsÞ ¼ 1þ 2:2709s

þ 9:5933s:

The zeros of the closed-loop system are at s = � 0.0521 ± 0.4837j.Fig. 7 shows the step response of the closed-loop system. The spec-

ifications are met. The approximate dominant pole placementmethod in [2] gives an unstable closed-loop system in this case.

5. Conclusion

Two simple yet effective methods have been presented for guar-anteed dominant pole placement by PID controllers, based on rootlocus and Nyquist plot, respectively. Each method is demonstratedwith examples. It should be noted that due to the closed-loop zerosand other effects the specifications cannot always be met exactly.Obviously it is the limitation of the system, not the problem ofthe proposed method. In that case one can try changing m or loos-ing the specifications and re-do the design. Obviously, the methodsare not limited to PID controllers. They can be extended to othercontrollers where one controller parameter is used as the variablegain and all other parameters are solved in terms of this gain tomeet the fixed pole requirements.

Extension of the method to multi-loop PID controllers ispossible, but it would create an essentially different problem. Interms of controller parameters, the MIMO case is a nonlinear prob-lem whereas the SISO case discussed in the paper is a linearproblem.

References

[1] P. Persson, K.J. Astrom, Dominant pole design – a unified view of PID controllertuning, in: L. Dugard, M. M’Saad, I.D. Landau (Eds.), Adaptive Systems in Controland Signal Processing 1992: Selected Papers from the Fourth IFAC Symposium,Grenoble, France, 1–3 July 1992, Pergamon Press, Oxford, 1993, pp. 377–382.

[2] K.J. Astrom, T. Hagglund, PID Controllers Theory Design and Tuning, second ed.,Instrument Society of America, Research Triangle Park, North Caorlina, 1995.

[3] G.F. Franklin, J.D. Powell, A. Emami-Naeini, Feedback Control of DynamicSystems, Pearson Prentice Hall, Upper Saddle River, New Jersey, 1995.

[4] S. Thompson, Control System Engineering and Design, Harlow, Essex, England,1989.

[5] Charles L. Phillips, Royce D. Harbor, Feedback Control Systems, EnglewoodCliffs, New Jersey, 1994.

[6] P.K. Stevens, A Nyquist criterion for distributed systems subject to a one-parameter family of feedback compensators, IEEE Transaction on AutomaticControl 27 (3) (1982) 700–702.