CONTROL OF INTEGRAL PROCESSES WITH DEAD TIME:

PRACTICAL ISSUES AND EXPERIMENTAL RESULTS

Antonio Visioli ∗,1 and Qing-Chang Zhong ∗∗,2

∗ Dipartimento di Elettronica per l’Automazione, University of Brescia

Via Branze 38, I-25123 Brescia, Italy

E-mail: antonio.visioli@ing.unibs.it∗∗ Dept. of Electrical Eng. and Electronics, The University of Liverpool

Brownlow Hill, Liverpool L69 3GJ, UK

E-mail: Q.Zhong@liv.ac.uk

Abstract The problem of controlling an integral process with dead time is ad-dressed in this paper. In particular, various practical issues concerned with thecontroller implementation are discussed and then verified with experiments carriedout on a laboratory-scale setup where a level control problem is concerned. Acomparison with a standard Proportional-Integral (PI) controller is also performed.

Keywords: Integral processes, dead-time compensators, disturbance observer,two degree-of-freedom control

1. INTRODUCTION

Integral processes are frequently encountered inthe process industries and their dynamics canbe often well described by an integrator plusdead time (IPDT) transfer function [Chien andFruehauf, 1990]. The control of such systemshas been investigated widely in the last years.By considering the standard solution in indus-trial settings, namely the use of Proportional-Integral-Derivative (PID) controllers, many tun-ing rules and methods for the improvement of theperformance have been proposed; see for exam-ple [Åström and Hägglund, 2006, Visioli, 2006,O’Dwyer, 2006]. Moreover, the stability region ofa PI controller parameters has been characterizedin [Wang et al., 2006], where the limitations onthe achievable gain and phase margin have beenalso analyzed.

From a slightly different viewpoint, it is well-known that a two degree-of-freedom control schemeis an effective way to achieve satisfactory per-

1 A. Visioli’s research was partially supported by the

Ministry for University and Research, Italy.2 Q.-C. Zhong’s research was supported by the EPSRC,

UK under Grant No. EP/C005953/1.

formance both in set-point tracking and in loaddisturbance rejection. In this context, differentsolutions have been proposed in the literature,developing the basic Smith predictor scheme; see,for example, [Åström et al., 1994, Matausek andMicic, 1999, 1996, Normey-Rico and Camacho,1999, Zhang and Sun, 1996, Normey-Rico andCamacho, 2002]. In particular, a methodologybased on the concept of disturbance observer, i.e.,the use of the inverse of the nominal model toobserve the disturbances with the aim of can-celing their effects directly in the control signal,has been presented in [Zhong and Normey-Rico,2002, Zhong and Mirkin, 2002]. A two degree-of-freedom control scheme, where the feedback con-troller is inherently a PID controller which retainsthe Smith predictor principle, has been proposedin [Zhong and Li, 2002]. The robustness issue canbe explicitly analyzed and a tuning parameterthat handles the trade-off between aggressivenessand robustness is available.

However, despite the development of these tech-niques, surprisingly, no experimental results arereported in the literature related to the control ofintegral processes, to the best knowledge of theauthors. Practical issues are often not discussed

thoroughly and this implies that these techniquesare not fully characterized from the industrialpoint of view. While trying to implement someof the techniques into practice, it turns out thatthe controller implementation is not straightfor-ward. If the controller is not implemented prop-erly, the system might be internally unstable orthere might be a non-zero static error etc. In thispaper, these issues are discussed and verified withexperimental results obtained with a laboratory-scale level control system, in which the water levelis controlled by regulating the difference betweenthe input and output flow rate.

The paper is organized as follows. The controlstructure under consideration is reviewed in Sec-tion 2. The practical issues in controller imple-mentation are discussed in Section 3 and the ex-perimental results are presented in Section 4.

2. THE CONTROL STRUCTURE UNDERCONSIDERATION

The integral processes with dead time (IPDT) canbe described as,

G(s) = Gp(s)e−τs =

K

se−τs, (1)

where the pure dead time τ and the static gain K

are all positive.

In this paper, the attention is paid to the dis-turbance observer-based control scheme proposedin [Zhong and Normey-Rico, 2002]. Similar issuesexist with other control schemes for integral pro-cesses with dead time. The scheme is revisited inFigure 1(a) for the readers’ convenience, where τm

is the estimated dead time and Gm(s) is the nom-inal delay-free part. The controller C(s) = 1

KT isdesigned to be proportional and the low-pass filter

Q(s) =(2λ + τm)s + 1

(λs + 1)2(2)

is designed to reject step disturbances with nostatic error, where λ is a design parameter tocompromise the disturbance rejection, robustnessand, possibly, the control action bound [Zhongand Normey-Rico, 2002].

Under the nominal condition, Gm(s) = Gp(s) =Ks and τm = τ. The setpoint response and distur-bance response are respectively:

Gyr(s) =1

Ts + 1e−τms,

Gyd(s) =K

s

(

1 − Q(s)e−τms)

e−τms. (3)

It is quite easy to evaluate the achievable specifi-cations of the set-point response. The disturbance

g g g

g g

C(s)

Gm(s) e−τms

Gp(s)e−τs

G−1m (s)

Q(s)

-

�

�

- ��- �

6 6

?

?

d

r u y

n

−−

−

Disturbance Observer

c

d̂

- - - --

(a) Structure for design

g g

g

CGm

(1+CGm)Q(s)Q(s)

Gm(1−Qe−τms) Gp(s)e−τs

�

6

?

?

M(s)

d

r u y

n

−

e- - - - --

(b) Equivalent structure for implementation

Figure 1. Disturbance observer-based controlscheme

response obtained using (2), which is sub-ideal ashaving been shown in [Mirkin and Zhong, 2003],can be obtained with several different schemes[Mirkin and Zhong, 2003]. The achievable per-formance specifications and the robust stabilityregions of the system can be found in [Zhong andNormey-Rico, 2002].

The scheme shown in Figure 1(a) is not internallystable nor causal. An algebraically equivalent butcasual and internally stable structure is shown inFigure 1(b), as proposed in [Zhong and Mirkin,2002]. The implementation of the block

M(s) =Q(s)

Gm(s)(1 − Qe−τms)

needs to be very careful. If not properly imple-mented, then problems may occur.

3. PRACTICAL ISSUES

3.1 Zero static error

The block diagram shown in Figure 2(a) forM(s) is implementable because the block Q(s)

Gm(s) iscausal. However, if the block M(s) is implementedin this way, then the system will not be able toreject step disturbances completely. There will bea non-zero static error because the block

Q(s)

Gm(s)=

(2λ + τm)s2 + s

K(λs + 1)2

contains a differentiator and any constant error e

will be ignored by the controller. This causes anon-zero static error. As a matter of fact, sincethe static gain of Q(s) is unity, the local positivefeedback loop in Figure 2(a) contains an integralaction. The differential action in the previous

fQ(s)Gm(s)

Q(s)e−τs �

6

u

+

e- - -

(a) Non-zero static error

f

f

1K

(2λ+τm)s+1λ2s−τm

1se−τms

-

��

�6

6

u

−

−

e

ui

-- -

(b) Internally unstable

Figure 2. Wrong implementation schemes forM(s)

block cancels this integral action. This should beavoided in practical implementation.

3.2 Internal stability

One special property of the control plant is the in-stability due to the pole at the origin. This makesthe structure shown in Figure 1(a) internally un-stable because of the pole-zero cancellation ats = 0 between the block G−1

m (s), the inverse of thedelay-free model, and the plant Gp(s)e

−τs. Theequivalent structure for implementation shown inFigure 1(b) guarantees the internal stability if im-plemented properly. As discussed in the previoussubsection, the implementation of M(s) shown inFigure 2(a) causes non-zero static errors when thesystem is subject to step a disturbance. The can-cellation between the differential action in Q(s)

Gm(s)

and the integral action in the local loop should beremoved in implementation. The block M(s) canbe algebraically equivalent to

M(s) =Q(s)

Gm(s)(1 − Qe−τms)

=1

K

(2λ + τm)s + 1

λ2s − τm + ((2λ + τm)s + 1) 1−e−τms

s

=1

K

(2λ+τm)s+1λ2s−τm

1 + (2λ+τm)s+1λ2s−τm

·1−e−τms

s

.

This can be described as shown in Figure 2(b)as a local feedback loop. The signal ui in Figure2(b), i.e., the state of the integrator, is unboundedand hence the system is not internally stable evenwhen the output of the whole system is stablebecause the block 1−e−τms

s is split into the cascadeof an integrator and 1 − e−τms, which causes azero-pole cancellation at s = 0. One way to avoidthis is to use the approach proposed in [Zhong,2004, 2006] to approximate 1−e−τms

s as

f1K

(2λ+τm)s+1λ2s−τm

Cǫ

s+ǫD(s)

-

��

6

u

−

e

u′

i

-- -

Figure 3. Correct implementation scheme of M(s)

1 − e−τms

s≈

Cǫ

s + ǫ· D(s),

with

Cǫ =τm

N ǫ

1 − e−ǫτm/N,

and

D(s) = (1 − e−τm

N(s+ǫ)) ·

N−1∑

i=0

e−i τm

Ns,

where ǫ > 0 is a small enough number and N is anatural number. The longer the delay, the biggerthe N is needed. Then, the block M(s) can beimplemented as shown in Figure 3. With this im-plementation, the closed-loop system is internallystable. Moreover, there is no static error. The onlydownside is that the performance will be slightlyworse than the designed performance due to theapproximation. This can be improved by choosinga small ǫ and a big N .

4. EXPERIMENTAL RESULTS

4.1 The experimental setup

The laboratory experimental setup employed toverify the findings in the previous sections isshown in Figure 4. This is a small perspex tower-type tank with a section area of A = 40cm2,of which the water level is regulated by a PC-based controller. The tank is filled with water bymeans of a pump whose speed is controlled by aDC voltage, in the range 0-5 V, through a PWMcircuit. Then, an outlet at the base allows thewater to return to the reservoir. The measure ofthe level of the water is given by a capacitive-typeprobe that provides an output signal between 0(empty tank) and 5 V (full tank). The process canbe modeled by the following differential equation:

dy(t)

dt=

1

A(qi(t) − qo(t))

where y is the water level (i.e., the process vari-able), qi is the input flow rate and qo is the outputflow rate. The input flow rate qi is linearly depen-dent on the voltage applied to the DC pump. Theoutput flow rate can be calculated by measuring

Figure 4. The experimental setup (only one tankhas been employed in the experiments).

the level y(t) because qo(t) = α√

2gy(t), whereg is the gravitational acceleration constant and α

is the cross-sectional area of the output orifice.This is then added to the control variable u via alocal feedback loop to obtain the input flow rateqi, i.e., u(t) + q0(t) = qi(t), which is then usedto determine the speed of the DC motor (pump).Hence, the control variable u is

u(t) = qi(t) − qo(t).

We have now obtained an integral process

dy(t)

dt=

1

Au(t).

Dead time also occurs in the process because ofthe length of the pipe. The transfer function ofthe process, from the control variable u to thevoltage corresponding to the water level, has beenestimated as follows by means of a simple open-loop step response experiment:

G(s) =0.026

se−1.5s.

4.2 The scheme shown in Figure 2(a)

The control parameters are chosen as T = 5 andλ = 8. A set-point change from 1.5V to 2.3Vhas been applied to the control system and, after100s, a disturbance d = −2.5V has been appliedto the control variable. The resulting processvariable together with the set-point signal areshown in Figure 5. As expected, non-zero staticerror appears.

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50

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Figure 6. Experimental results with the controlscheme of Figure 2(b)

4.3 The scheme shown in Figure 2(b)

The same experiment as described in the previ-ous subsection has been repeated for the schemeshown in Figure 2(b). The resulting process andcontrol variables are plotted in Figure 6(a). Thestatic error has disappeared but, as expected, thesystem is internally unstable as can be seen fromthe plot of the variable ui (i.e., the state of theintegrator) shown in Figure 6(b). This variablebecomes unbounded eventually.

4.4 The scheme shown in Figure 3

Then, the scheme of Figure 3 has been employedfor the same control task. The control parametersare selected as T = 5, λ = 8, ǫ = 0.01 and N = 1.Results are shown in Figure 7(a). It appears thatthe control scheme is effective in both the set-point response and disturbance rejection. In orderto evaluate the physical meaning of the parame-ter λ, the same experiment as before has beenrepeated with different values of λ = 4 and λ = 12.The resulting process and control variables areplotted in Figures 7(b) and 7(c) respectively. Thecontrol system is more aggressive when the valueof λ is decreased and it is more sluggish when λ

is increased, as expected.

4.5 Comparison with a PI controller

In order to evaluate better the performance of thedisturbance observer-based control scheme, com-parison with a standard PI controller in the unity-feedback control scheme has been performed. Thecontroller is

CPI(s) = Kp

(

1 +1

Tis

)

,

where the proportional gain Kp and the integraltime constant Ti have been selected by applyingthe tuning rule proposed in [Chien and Fruehauf,1990], i.e., Kp = 0.3306

KL = 8.48, Ti = 10L = 15.The experimental results are shown in Figure 8.The disturbance rejection is very good but theset-point response has a large overshoot. Thisclearly shows the benefit of the two degree-of-freedom control scheme. It is able to providegood performance for both the set-point responseand disturbance rejection. Obviously, being of asingle degree-of-freedom, the PI controller is ableto obtain satisfactory performance in disturbancerejection at the expense of a significant overshootin the set-point response. In general, it is not easyto select the most appropriate tuning rule to meetthe requirement for both.

4.6 Experiments with an additional delay

The same experiments have also been performedon the system as before but with an additional in-put delay of 1s, which is implemented via software.The controller for the scheme shown in Figure 3has been re-designed with λ=8 and λ=12, respec-tively, and the PI controller is tuned as Kp=5.09and Ti=25 using the same tuning rule as before.The results are shown in Figures 9. It can be seenthat the observer-based control scheme performsbetter than the PI controller.

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Figure 7. Experimental results with the controlscheme of Figure 3.

5. CONCLUSIONS

In this paper, some practical issues in controllerimplementation for integral processes with deadtime are reported and verified with experimentalresults. It is shown that with proper implementa-tion, these problems can be eliminated.

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Figure 9. Experimental results for the system withan additional 1s-delay

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