SYNTHESIS OF ROBUST PID CONTROLLERS FOR TIMEDELAY SYSTEMS

N. Hohenbichler, J. Ackermann Institute of Automatic Control (IRT), RWTH Aachen University, 52056 Aachen, Germany,

N.Hohenbichler@irt.rwth-aachen.de Institute of Robotics and Mechatronics, German Aerospace Centre (DLR) Oberpfaffenhofen, P.O. Box 1116, 82230 Weling,

Germany, juergen.ackermann@dlr.de

Keywords: PID control, time delay system, robust control, pa-rameter space approach, quasipolynomials.

AbstractTuning rules are frequently used to choose the parameters ofPID controllers. However, they are often based on heuristics,are limited to systems of a certain class, use dead timeapproximations and ignore parameter uncertainties in themodeled system. This work develops a systematic, universaland transparent method to design a robust PID controller basedon the parameter space approach, which is extended to copewith quasipolynomials.

1 IntroductionPID is the most common controller in industrial practice [2].But a high percentage of PID control systems seems to be tunedbadly [6]. One major reason may be that todays tuning meth-ods are limited to very restricted conditions on the plant (con-cerning model order, pole and zero location, neglected param-eter uncertainty) [12]. Time delay systems, especially with un-certain and immeasurable dead time, present one of the mostchallenging problems for tuning a PID controller.

This paper develops a PID tuning method based on the param-eter space approach [1]. So far, in [3] the synthesis step isextended to time delay systems, but important results for thepractical application are still missing. Also, the analysis step isnot developed in the literature and results have not been com-pared with existing tuning methods.

In [15] an analytical solution to nd the Hurwitz stable PIDparameters for a rst order system plus time delay is published;

rPID(s,k)

u

d

A(s,q)R(s,q) e

sL y

Figure 1: Single loop with PID controller and time delay sys-tem

in [11] formulas to calculate some stability boundaries in thedead time for certain classes of quasipolynomials are given.However, both approaches are limited to very restricted cases.

2 Problem formulationConsider a single loop containing a PID controller and a lineartime delay system (see Fig. 1), given by the transfer functions

PID(s,k) =KI + KP s + KDs

2

s (1 + TRs), (1)

G(s, L, q) =A(s, q)

R(s, q)esL, (2)

where k = (KI ,KP ,KD)T are the controller parameters. (TRensures feasibility of the controller and lters noise ; it is as-sumed to be xed prior to the controller design, e.g. by addinga non dominant pole to R(s, q)). The unknown but constantplant parameters are the dead time L > 0 and the parametersin the vector q. They lie in an operation domain

Q = { (L, q)T | L [L;L+], qi [qi ; q

+i ] }, (3)

where qi and q+i are specied as the lower and upper limits of

parameter qi in q (analog L and L+).The problem of designing a robust PID controller is to nd a setof controller parameters k = k, that meets the specicationfor all values of (L, q)T Q. Specications are assumed in theform ofHurwitz stability (all roots of the characteristic functionare in the open left half plane (LHP)) and -stability (all rootshave a real part smaller then a real number 0).The characteristic function of the loop in Fig. 1

P (s,k, L, q) = (KI + KP s + KDs2)A(s, q)+

+ s (1 + TRs)R(s, q) B(s,q)

esL, (4)

with polynomials

A(s, q) = a0(q)+a1(q) s + . . . + am(q) sm, (5)

B(s, q) = b0(q) +b1(q) s + . . . + bn(q) sn, (6)

with am(q) = 0, bn(q) = 0 belongs to the class of quasipoly-nomials [5, 13] due to the dead time. (Note that b0(q) = 0 for

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*4#/

basic case of a PID controller (1). However, later a b0(q) = 0may appear through transformations, see section 7.)The principal term condition [13] requires for Hurwitz stabilitythat in the case of PID control (KD = 0) the degrees fulln m + 2. In the sequel we treat only this case (i.e. weassume a proper A(s, q)/R(s, q) for TR = 0.)

3 Parameter space approachThe parameter space approach is used to solve the problem intwo main steps. In the controller synthesis step, we computethe stable (either Hurwitz or -stable) region in the space ofcontroller parameters k for several representatives (L, q)Tout of Q (usually the vertices). A candidate for a robust con-troller k is chosen from the intersection of stable regions.

This controller satises the specication for the representatives.The second step, the control loop analysis, is applied to testthe robust stability for the continuum of all values in Q. Nowwe compute the stable region in the space of plant parameters(L, q)T with xed controller k. If Q lies entirely in the stableregion, then a solution of the problem is found.

The calculation of a Hurwitz stable region in a parameter space(either k or (L, q)T ) is based on the fact that the roots ofthe quasipolynomial (4) with continuous coefcient functionsai(q), bi(q) do not jump when the parameters are changed con-tinuously. Thus, a stable quasipolynomial, whose roots all liein the LHP, becomes unstable if and only if at least one rootcrosses the imaginary axis. The parameter values of the rootcrossings form the stability boundaries in the parameter space,which can be classied into three cases: the real root bound-ary (RRB), where a root crosses the imaginary axes at the ori-gin (substitute s = 0 in the quasipolynomial), the innite rootboundary (IRB), where a root leaves the LHP at innity (set|s| ) and the complex root boundary (CRB), where a pairof conjugate complex roots crosses the imaginary axes (substi-tute s = j and sweep over all real > 0). These stabilityboundaries separate different regions in the parameter space.To classify a region as Hurwitz stable it sufces to prove sta-bility for one inner test point (e.g. by the Nyquist criterion).

4 Controller design algorithmThe proposed controller design procedure is summarized in thefollowing steps:

1. Specify the maximum real part from closed-loop sett-ling time requirements.

2. Compute the -stable regions in controller parameterspace for representatives (usually the vertices) of the Q-domain.

3. Determine the intersection of the -stable regions.

4. Choose a candidate controller out of the intersection.

5. Compute the -stable region in plant parameter space forthe candidate controller.

6. If the Q-domain lies entirely in the -stable region, thenthe problem is solved.

-stability can be reduced to the Hurwitz case by the substitu-tion s = v + 0 which leads to a transformation in parametersand polynomials (see section 7). So Hurwitz stability is con-sidered rst in the next paragraphs.

5 Controller synthesisFor each xed representative (L, q)T the Hurwitz stabilityboundaries of (4) in the k-space are determined. The RRBturns out to be simply a straight line given by the equation

P (0,k) = KI A(0) + B(0) = 0 KI = b0a0

. (7)

(In the basic case we have b0 = 0 and the RRB is KI = 0.)More theoretical difculties arise when calculating the IRB.Quasipolynomials possess an innite number of roots, whichcan not be calculated analytically in the general case. How-ever, the asymptotic location of roots far from the origin is wellknown [5]. It turns out that innite root boundaries only ex-ist, if the degree equation n = m + 2 is fullled (in case ofKD = 0). These are two straight lines

KD = bnam

. (8)

The calculation of the CRB starts analog to the delay free caseof polynomials [1]. The root condition P (j,k) = 0 can beseparated into a system of two equations for real and imaginarypart

(RP (,k)IP (,k)

)=

(RA

2RAIA

2IA

)(KIKD

)+

+

(RB KPIAIB + KPRA

)=

(00

), (9)

where B(j) = B(j) ejL and R, I denote the real andimaginary parts of A, B and P at (j).

Clearly, the matrix multiplying (KI ,KD)T is singular. Thus,the key idea is to xKP = KP and to evaluate the CRB in the(KD,KI)-plane. A solution of (9) exists and only exists forthe real zeros gi of

g() = det

(RA RB K

PIA

IA IB + KPRA

)

= KP (R2A + I

2A) + RAIB IARB .

(10)

The zeros of g() are called singular frequencies. For each iappears a straight line as CRB in the (KD,KI)-plane, ruled bythe equation

KI = 2gi KD + K

0I (g), (11)

2 4 6 8 10 12 14 16 18 2020

15

10

5

0

5

10

15

20

KP() KP=1

Figure 2: Function KP () (solid), its limit function (dashed),singular frequencies for KP = 1 (x-marks) and stabilizingKP -interval (dash-dotted) of G1(s).

where K0I (g) can be easily determined from the rst or sec-ond row of (9).Thus, the stability boundaries RRB, IRB and CRB are straightlines in the (KD,KI)-plane and partition the plane into convexpolygons. (Additionally, for each boundary line the side can bedetermined which possesses the lower number of stable poles,see [3].)The singular frequencies may be determined by a graph of

KP () =RAIB + IARB

(R2A + I2A)

. (12)

Graphically, the singular frequencies for a xed KP are theabscissa values of the intersections between the KP ()-plotand the (KP = KP )-line. Due to the dead time the number ofsingular frequencies is innite. Algorithms for the automaticcalculation of the singular frequencies can be found in [8].The function KP () and the resulting boundaries in the(KD,KI)-plane for a xedKP = 1 are demonstrated in Fig. 2and Fig. 3 for the example system (with ideal PID controllerTR = 0)

G1(s) =1

s + 1es. (13)

Stability checks of test points prove that the polygon aroundP1 is stable (and it is the only one). Two questions arise: Howmany singular frequencies have to be evaluated and whichKP -values lead to a stable polygon?

1. For high frequencies, function (12) tends to a much sim-pler limit function of the form

KP () trig(L), (14)

where and are real constants ( 1) and trig is eitherthe sin or cos function. It can be shown, that singular

1.5 1 0.5 0 0.5 1 1.51

0

1

2

3

4

5

P1

KD

KI

RRBIRBCRB

Figure 3: Stability boundaries in (KD,KI)-plane for KP = 1of G1(s). The side of the lines with more unstable poles isshaded.

frequencies which lie in a frequency range where (14) and(12) are practically identical result in CRB lines which donot contribute to boundaries of the stable polygon. So, itsufces to compute and evaluate only a nite number oflow singular frequencies, given by the comparison of (14)and (12).

2. The question of determining the interval of KP whichleads to a stable polygon is analytically solved only fora rst order system plus time delay [15]. Following workhypothesis for the general system (2) is stated:Work hypothesis. Provided that there is a stabiliz-ing PID controller, a Hurwitz stable polygon exists in the(KD,KI)-plane, ifKP lies in the interval ofKP that pro-duces the highest possible number of singular frequencies.The result of the hypothesis forG1(s) is depicted in Fig. 2.

The entire stable region in k-space can be computed by grid-ding KP in its stabilizing interval and extracting the stablepolygon for each gridded KP (see Fig. 4).

6 Control loop analysisNowwe determine the Hurwitz stable region in plant parameterspace by mapping the stability boundaries of the quasipolyno-mial

P (s, L, q) = D(s, q) + B(s, q) esL, (15)where

D(s, q) = (KI + KP s + K

D s

2)A(s, q)

= d0(q) + d1(q) s + . . . + dm(q) sm

(16)

The case of mapping the boundaries to a plane of L and oneadditional parameter q is treated. If there are more uncertainparameters, then they have to be gridded.

1

0

10 0.5 1 1.5 2 2.5 3 3.5

1

0

1

2

3

KI KD

KP

Figure 4: Hurwitz stable controller parameters k of G1(s).

The RRB are lines q = q, where q are the solutions of theequation

d0(q) + b0(q) = 0. (17)

If the degrees of B(s, q) and D(s, q) fulll n = m, then thesolutions of the following equation lead to IRB

dm(q) bn(q) = 0. (18)

The CRB condition P (j,L, q) = 0 can be split into a modulusand a phase equation D(j, q)B(j, q)

= 1, (19)L(, q, k) =

1

(arg(D(j, q)) arg(B(j, q))+

+ (2k 1)), k

. (20)

The modulus equation does not depend on L and can be trans-formed to a polynomial in 2. Thus, the equation can be solvedby nding the roots of a polynomial, but it is especially con-venient if (19) can be solved symbolically for q (e.g. if q isthe dc-gain). With (20) the corresponding values for L can beadded to plot the CRB curves. The phase equation carries theuncertainty k (only k 0 generate nonnegative L), so there isan innite number of CRB. However, as k grows, the curvesmove due to (20) to higher dead times.Fig. 5 shows the boundaries and the stable region for the exam-ple quasipolynomial

P2(s, L,K) = K + (s2 + 0.25 s + 1) esL. (21)

In [11] this example serves for the calculation of some deadtime stability limits (corresponding to k = 1), which agreewith these presented here. However, Fig. 5 presents the stableregion completely for L 22.

5 0 5 10 15 20 25 301.5

1

0.5

0

0.5

1

1.5

L

KP1

P2

stable

k=0k=1k=2k=3k=4

Figure 5: Stability boundaries RRB (dashed) and CRB (solid,k = 0, .. , 4) of P2(s, L,K).

7 PID design using -stabilityThe concept of -stability can be used to speed up the transientresponses robustly. Concerning the synthesis step, this casecan be reduced to the Hurwitz case by substituting s = v + 0.Following transformations result

K I = KI + KP0 + KD02,

K P = KP + 2KD0,

K D = KD,

A(v) = A(v + 0),

B(v) = e0L

B(v + 0).

(22)

Following the parameter space approach presented in section 3,the smallest possible value for 0 is determined for a given sys-tem (2) and an operation domain (3), s. t. the intersection of thestable regions belonging to the vertices ofQ is not empty. This0 is approximated by an iterative approach: Beginning withzero, 0 is stepwise reduced and the function KP () is plot-ted for each vertex, until the work hypothesis reveals that thereis no interval of KP that stabilizes simultaneously all vertices.With the last 0 having such a interval, the stable k-regions arecomputed for all vertices, and a controller k is taken out of theintersecting region.

The analysis step can be reduced to he Hurwitz case, if the sec-ond parameter q enters in form of a dc-gain into the quasipoly-nomial. In that case, the transformations are

q = q e0L,

L = L,

D(v) = D(v + 0),

B(v) = B(v + 0).

(23)

0 1 2 3 4 5 6 70

0.20.40.60.8

11.21.4

Robust PID controller

y(t)L=1.0 ISE=1.34L=0.7 ISE=1.11L=1.3 ISE=1.64

0 1 2 3 4 5 6 70

0.20.40.60.8

11.21.4

t

Robust IMC controller

y(t)L=1.0 ISE=1.17L=0.7 ISE=0.946L=1.3 ISE=1.55

Figure 6: Step responses of PID and IMC controlled loop ofG3(s, L) for nominal, maximal and minimal L.

8 Application to literature examplesWe apply the proposed method to design PID controllers forexample systems of the literature and compare the results withother PID tuning methods or control concepts. Whenever pos-sible, we respect the following comparison laws: Copy theplant model and the controller with all parameters. Prefer ro-bust control methods and use the same operation domain. Ap-ply the same criteria to evaluate the results. Take only literatureexamples whose results can be reconstructed.

First the comparison is illustrated by the following system andoperation domain

G3(s) =1

1 + sesL, Q3 : L [0.7 ; 1.3]. (24)

which serve as an example in [10] to develop an Internal ModelControl (IMC) controller.The robust PID controller parameters are KI = 0.58, KP =0.82, KD = 0.26 with TR = 0.1 and 0 = 0.8. For acomparison, in Fig. 6 the step responses for nominal, maximaland minimal dead times are shown for the robust PID and IMCcontroller. Clearly, by using the PID controller the overshoot,settling time and oscillations are reduced, while the IntegralSquare Error (ISE) is slightly higher.The second example,

G4(s, L,K) =K

(1 + 5s)3esL,

Q4 : L [12 ; 18], K [0.9 1.1], (25)

is used to compare the proposed method with tuning rulesthat are applied to the system in [14, 4]. Notice the intersec-tions of the -stable regions and the tuning rule controllers

0 5 10 150.02

0.04

0.060.4

0.6

0.8

1

1.2

1.4

1.6

KDKI

KP

Modulus optimumChien track. aperiodicLatzel 20%Latzel 10%Ziegler oscillationTsumrule normalChien track. 20%Chien disturb. aperiodicPessenChien disturb. 20%CohenZiegler step response

I

Figure 7: Intersection (I) of the -stable regions (0 =0.05) of the vertices of Q4 for G4(s, L,K) and tuning rulecontrollers. The tuning rules are sorted by the settling time af-ter a reference step for the nominal plant, see [14, 4].

in Fig. 7. (The tuning rules are applied to the nominal plantL = 15, K = 1). Clearly, the the tuning rules leading tothe lowest settling time lie close to the intersection region. In[14, 4] it is stated that it depends heavily on the plant whichtuning rule produces the fastest settling time. However, theproposed method nds a fast controller universally for all sys-tems in the class (2) and guarantees robustness in the wholeoperating domain.

Additionally, in [8] the proposed method is applied to a greatvariety of systems (up to fourth order, stable and unstable,with real and complex poles, with zeros) in different parameterdependencies, which serve as benchmark problems for singleloop control strategies in the literature (IMC [10], tuning rulesetc. [14, 4, 9] and genetic optimization [7]). In all consid-ered cases we achieve superior or at least similar results: Theovershoot of the robust PID controller is low, the transient re-sponses are robustly fast, there are only small oscillations andthe responses for different operation points resemble largely.

9 ConclusionsThe parameter space approach offers convincing results in thesynthesis of robust PID controllers for time delay systems. Thedeveloped tuning method is systematic, universal and transpar-ent and leads to superior or similar results than literature exam-ples. Exact stability (Hurwitz or -stability) regions can be de-termined in the space of controller and plant parameters whiletreating the dead time without approximation.

The development of an interactive graphical software packagebased on the stated algorithm seems very promising to be ahelpful tool in daily engineers work. So the engineer would be

able to re-tune the great amount of existing PID loops at lowcost in industry.

References[1] J. Ackermann, P. Blue, T. Bunte, L. Guvenc, D. Kaes-

bauer, M. Kordt, M. Muhler, and D. Odenthal. RobustControl. Springer, New York, 2002.

[2] K. Astrom and T. Hagglund. PID Controllers: Theory,Design and Tuning. Instrument Society of America, 1995.

[3] N. Bajcinca, R. Koeppe, and J. Ackermann. Design ofrobust stable master-slave systems with uncertain dynam-ics and time-delay. In Proceedings IFAC (InternationalFederation of Automatic Control) 15th World Congress,Barcelona, 2002.

[4] N. Becker, W. M. Grimm, and U. Piechottka. Vergleichverschiedener PID-Regler. atp, (41):3946, 1999.

[5] R. E. Bellman and K. L. Cooke. Differential-DifferenceEquations. Academic Press, New York, 1963.

[6] David B. Ender. Process control performance: Not asgood as you think. Control Engineering, page 180,September 1993.

[7] K. Hirata, Y. Yanase, T. Kawabe, and T. Katayama.A minimax design of robust I-PD controller for time-delay systems with parametric uncertainty. In Proceed-ings IFAC (International Federation of Automatic Con-trol) 14th World Congress, volume C, pages 259264,Peking, 1999.

[8] N. Hohenbichler. Auslegung robuster PID-Regler fur Tot-zeitsysteme. TU Munchen, Lehrstuhl fur Steuerungs- undRegelungstechnik, 2002. Diplomarbeit.

[9] Yongho Lee, Jeongseok Lee, and Sunwon Park. PID con-troller tuning for integrating and unstable processes withtime delay. Chemical Engineering Science, 55(17):34813493, 2000.

[10] M. Morari and E. Zariou. Robust Process Control.Prentice-Hall, Englewood Cliffs, 1989.

[11] S.-I. Niculescu. Delay Effects on Stability. Number 269in LNCIS. Springer, 2001.

[12] A. ODwyer. PI and PID controller tuning rulesfor time delay processes: A summary. Techni-cal report, School of Control Systems and Electri-cal Engineering, Dublin Institute of Technology, 2000.http://citeseer.nj.nec.com/dwyer00pi.html.

[13] L. S. Pontryagin. On the zeros of some elementary trans-cendental functions, volume 2, pages 95110. Ameri-can Mathematical Society Translation, 1955. EnglischeUbersetzung.

[14] T. Schaar. Vergleich von verschiedenen Einstellregeln furPID-Regler. Fachhochschule Koln, Fachbereich Elek-trische Energietechnik, 1998. Diplomarbeit.

[15] G. J. Silva, A. Datta, and S. P. Bhattacharyya. New resultson the synthesis of PID controllers. IEEE Transactions onAutomatic Control, 47(2):241252, Februar 2002.

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