Tuning PID controllers for integrating processes

L.Wang W.R. Cluett

Inclminx terms: Algorithnzs, PID controllers, Tuning, Integuution

Abstract: The paper extends the frequency domain PID controller design method, first proposed by the authors in an earlier paper, by presenting a comprehensive treatment of integrating processes. The general class of integrating processes is divided into two types, based on the sign of the second coefficient of the Taylor-series expansion applied to the stable part of the process transfer function. In both cases, the closed-loop performance is specified in terms of the desired control-signal trajectory scaled with respect to the magnitude of this coefficient. In addition, explicit PID tuning rules are given, for integrating plus delay processes along with their associated gain and phase margins and allowable time delay variations.

1 Introduction

Integrating processes are frequently encountered in the process industries. To our knowlcdge, they are most commonly associated with level control problems. Two distinct control objectives can be associated with these types of processes [I]. The first is referred to as ‘tight’ control, where the controlled variable (e.g. level) is very important and variation in the manipulated variable (e.g. adjustable flow in or out of the vessel) is not of great importance. The second is referred to as ‘averag- ing’ control, where variation in the controlled variable is not important and the manipulated variable must not experience rapid fluctuations. To meet either con- trol objective with a PTD controller, the user needs to be able to translate time-domain performance specifica- tions, e.g. closed-loop settling time, into appropriate values for the controller parameters.

For this class of systems, the well-known Ziegler- Nichols [2] PID tuning rules, based on ultimate gain and period information, could be applied. However, these rules lack a time-domain performance parameter for allowing the user to achieve a particular control objective, and can produce responses which are too oscillatory [3]. Unfortunately, many of the other tuning rules available in the literature are not applicable to integrating processes, with a few exceptions. Rivera et al. [4] have applied the internal model controller (IMC)

0 IEE, 1997 IEE Proceedings online no. 19971435 Paper first received 27th September 1996 and in revised form 25th April I997 The authors are with the Department of Chemical Engineering, Univer- sity of Toronto, Toronto, Canada M5S 3E5

design procedure to tune PID controllers for various types of integrating processes. This method does have a user-selected performance parameter that is directly related to the closed-loop response speed. In a more recent paper, Tyreus and Luyben [5] closely examined the PI-IMC design for the integrator plus delay case (Rivera et al.’s Case N). Their main conclusion was that the IMC approach can lead to poor control, even when the practical recommendation provided by [4] for this case is followed. Tyreus and Luyben [5] proposed an alternative approach using classical frequency- response methods where the desired maximum closed- loop log modulus is specified.

This paper proposes a comprehensive PID controller design algorithm for a wide class of integrating proc- esses. The original design method presented by Wang et al. [6] uses process frequency-response information at just two frequencies to solve for the PID controller parameters and, thus, is not limited to simple model structures. Another key feature of the original design lies in the way the closed-loop performance is specified via the desired response of the control signal. In this paper, the basic algorithm presented in [6] is extended on two main fronts. First, we distinguish between two types of integrating processes, namely lag dominant and lead dominant. The basis for this division is the sign of the second coefficient of the Taylor series expansion applied to the stable part of the process transfer function. This coefficient places constraints on the response speed of the control signal, and, therefore, also on the response speed of the closed-loop system. From these constraints, we provide guidelines on the choice of the time constant (z) and damping factor (5) associated with the control-signal specification for the two types of integrating systems.

The second contribution is the derivation of explicit tuning rules for integrating plus delay processes. This type of integrating process is the most frequently encountered in industry and, in addition, time-con- stant-dominant, first-order plus delay processes are often approximated as integrating plus delay processes for controller design [7]. These tuning rules, which con- tain a single closed-loop response speed parameter @> to be selected by the user, are presented along with their gain and phase margins and allowable time-delay variations.

2 PID controller design algorithm

2. I Specification of control signal trajectory We assume that the integrating process can be described by the transfer function

(1) 1

S G ~ ( s ) = -G(s)

385 IEE Proc.-Control Theory Appl.. Vol. 144, No 5, September 1997

where G(s) is stable with all poles strictly on the left half complex plane. Our objective is to design a feed- back PID controller for this integrating process in the following form:

5 1 0 -

0 . 8 -

0 . 6 -

a 2 1

ar - - c 0

: O L - Y E 0 . 2 - ._

U

where K, = cl, z, = cl/co, z, = c21cl. The loop transfer function C(s)GAs) for this closed-loop system contains a double integrator resulting in a closed-loop transfer function from the setpoint r to the process output y given by

0 -

where the first term on the right-hand side is always unity and the second term is always zero. To ensure that eqn. 4 is satisfied, the desired closed-loop transfer function between the setpoint Y and the controller out- put U will be chosen to have the form [6]:

-

where eqn. 1 has been rewritten as

and K(l + yls + ...) is the Taylor-series expansion of G(s) around s = 0. K and y1 represent the key process parameters in the design of PID controllers for inte- grating processes.

Now we will divide the general class of integrating processes described by eqn. 1 into two types based on the sign of the parameter yl. In both cases, the trajec- tory of the desired control signal will be scaled with respect to Iyll. A single tuning parameter (/3 > 0) is introduced such that the time constant of the desired control signal response is given by z = Biy i . This will make analysis and tuning straightforward.

Type A (y, < 0): We will refer to this type of integrat- ing process as lag dominant which represents the majority of integrating processes encountered in the process industries. The time constant of the desired control signal is chosen as

Then, eqn. 5 becomes 7 = PI71 I (7 )

We will define s = IyIls as a scaled Laplace-transform variable which allows us to rewrite eqn. 8 as

(9) 2 ( 2 / 3 < + 1 ) 2 + 1

GT+u(s) = IY1/h’v + apc; + 1 The scaling with (y l ( in the Laplace domain naturally leads to a scaling in the time domain with Z = t/lyll, where ? represents normalised time. The desired con- trol signal response for a given step setpoint change of magnitude r will have an initial change of (2@c + 1)r/ (/!l2Kly,l) and then will exponentially decay to zero, fol- lowing a second-order response with a normalised time constant f l and a damping factor E.

For a type-A integrating process with its correspond- ing values of K and Iyll, the choice of parameter /3 can be based on hard constraints on the maximum desired

allowable change in the control signal, and/or the desired response speed of the control signal. A smaller /3 corresponds to a higher performance specification (namely a faster closed-loop dynamic system response) leading to a larger initial change in the control signal. On the other hand, a larger /3 corresponds to a lower performance specification (i.e. a slower closed-loop dynamic system response) and a smaller initial change in the control-signal response. For this type of lag- dominant processes, we suggest the damping factor < be chosen as either 1 or 0.707. The difference between these two choices for < in terms of their effect on the desired control signal trajectory, for a unit-step set- point change, is shown in Fig. 1 with Klyll = 1. The effect of /3 on the control signal response is illustrated in Fig. 2 with < = 1.

0 2 L 6 8 10 normaLised time

Fig. 1 Desired control-signal trajectory With < = 0.707 (-), < = I 0 (- -) and /3 = 2

2 /

I

Type B (yl > 0): This type of integrating process will be referred to as lead dominant. Here, we select

7 = Pr1 (10) Then, eqn. 5 becomes

and its scaled form is given by

To avoid an undesirable inverse response in the control signal, the requirement that (28< ~ 1) 2 0 must be satis-

IEE Proc -Control Theory Appl , Vol 144, No 5, September 1997 386

fied. This condition dictates that for a smaller choice of @, a larger value for 1; will eventually have to be selected. The poles of the second order system given in eqn. 12 are distributed according to

/ / / 1 \

(1 - dl - ;) P and

Therefore, as 1; increases, one pole becomes larger in its magnitude while the other approaches zero. Under these conditions, the control signal response tends to first order with time constant equal to @I(< ~ ./c2 ~ 1). The desired control signal response for a given step set- point change of magnitude I' will have an initial change of (205 ~ l)d(@2Ky,). Our suggestion for choosing @ and 5 is to first select @ such that z is of the same order of magnitude as the larger of the dominant-process time constant or delay, and then to adjust < to satisfy (2@< ~ 1) 2 0 and also to determine the closed-loop response speed, with a larger < corresponding to a faster response. Fig. 3 shows the trajectories of the desired control signal for a unit-step setpoint change with < = 5 and 5.025, @ = 0.1 and Icyl = 1 . These two trajectories are almost identical except in their initial responses to the setpoint change. This difference has a significant effect on the response speed of the closed- loop system.

I I--- - T ---- - 7 - p i 7 . 2 1

-0.2: ' __ -2 -1 0 1 2 3 L 5 6

normalised time Fig. 3 With 5 = 5 (initial value 'o'), 5 = 5.025 (initial value '*') and 0 7 0.1

Desired control signal trqjectory

2.2 PID parameter solutions Our design objective is to find the PID controller parameters such that the actual closed-loop frequency response is close to the desired closed-loop frequency response G,,],, where G,.,,(s) = Gr+n(s)GI(,s). However, a direct approach to solving this problem would involve a nonlinear optimisation procedure. In [6], we have carried out the design by working with the equiv- alent open-loop transfer function because in this case the problem becomes linear in the controller parame- ters. The desired open-loop transfer function is given by

and the actual open-loop transfer f h c t i o n is given by

With our aim being to match GJja) with the actual open-loop transfer function C(ju)Gr(jw), we now define a complex function

Note that the frequency domain error between the desired open-loop frequency response Go[ ( jw) and the actual open-loop frequency response is zero, if Y(ja) is of the form

Y ( j w ) = e2 ( j w ) 2 + C l (jw) + CO (16) where CO, c', and c2 are real constants. If two straight lines are used to fit the real part of You) (YR(u)) against 3 and the imaginary part of Y(jo~) ( YI(u) ) against w through two frequencies, ul and a2, then the coefficients co, cI and c2 can be found analytically (see Fig. 4 for a graphical illustration of this result). These coefficients are related to the PIT) controller parame- ters in eqn. 3 according to K, = c,; z, = c,lco; z, = c2/c1.

intercept = C O I f -

Fig. 4 Illustration of thp PID parameter solutions

Another contribution made in [6] was to determine the best choice for w1 and w2 to be 2dT, and 4?tlT, in order to obtain a close match between the desired and actual closed-loop time domain performance, where T, is the desired closed-loop process output settling time estimated from the desired closed-loop transfer func- tion G'-)(s). With both type-A and type-B integrating processes, a simple way to choose T, is according to

which is based on a crude estimate of the desired closed-loop process output settling time. A more accurate method would be to simulate the desired process-output response from the transfer function G,,,(s)Gr(s). Note that this latter approach does not require the values of the controller parameters because it is based only on the desired performance.

The PID parameter solutions are summarised as follows:

IEE Pror. -Control Theovy Appl., Vol. 144, N o 5, Seprcmhu 1997 3x7

Step 1: Letting = 2dTJ and w2 = 204, compute the frequency response of You) at oil and w2 from eqn. 15. Step 2: Calculate the intermediate coefficients in eqn. 16 as

1 . L - .+a 3

:I 2-

! 0.8- a,

0

1 . 0 -

Q

Step 3: Compute the PID controller parameters as: Proportional gain:

K, = cl (21) Integral time constant:

CO

Derivative time constant: c.2

r l ~ = - c1

Note that, if z, turns out to have a negative value, then a PI controller must be used by setting zD = 0. If only a PI controller structure is desired, the values for K, and zI are still calculated according to eqns. 21 and 22. This feature of PI controller design is different from the original approach proposed in [6] where we used only one frequency to calculate the controller parameters. We have found that the approach suggested here pro- duces better values for the integral time constant zp

2.3 Simulation example The following model is a modified version of the trans- fer function presented in [8] for the dryer steam pres- sure associated with paper machine dryer cans. We have added 5 s of delay to this transfer function to make it more challenging and realistic:

(24) 0.005(300s + I) - s s

G ~ ( s ) = P S ( 2 O S + 1)

This is a type-B, lead-dominant integrating process with y, = 300 - 20 - 5 = 275 > 0. We will first apply our design method and then make some comparisons with the IMC-PID design.

First, the parameter f l is chosen to be 0.1, which makes -c = By, = 27.5 on the same order of magnitude as the dominant-process time constant of 20. For a slow closed-loop response, we have set the initial change in the desired control-signal response to a unit- step setpoint change equal to zero (28< = 1). This choice gives a value for 5 equal to 5. For a fast closed- loop response, we have set the initial change in the desired control-signal response to a unit-step setpoint change equal to 6/(Kyl) which gives a value for equal to 5.3. The desired closed-loop settling time T, was approximated as 62 = 165 for both cases. The PID con- troller parameters were then solved for using eqns. 18- 23. For 1; = 5 , the derivative time constant z, turns out to be a negative number and, therefore, is set equal to zero. The final controller is PI with K,. = 0.48 and zI = 18.5. For 5 = 5.3, the calculated PID controller param- eters are K,. = 1.6, zI = 26.0 and z, = 2.0. For all exam- ples presented in the paper, we have simulated the closed-loop system with the derivative action, including a first-order filter with time constant equal to O.lz,,

388

applied only to the measurement. Fig. 5 shows the closed-loop process output responses for these two designs with a unit-step setpoint change occurring at t = 0 and a unit-step disturbance occurring at t = 400 with the disturbance entering at the process input. Fig. 6 shows the corresponding controller output responses.

2.0,

1 6 ;i

:li. 0.2 0

-100 0 100 200 300 LO0 500 600 700 t time, s

0

For comparison, we have examined the IMC-PID design from [4] for this case. Rivera et al. [4] suggest that for processes with a left half plane zero such as in this example, the PID controller should be augmented with a first-order lag to cancel this zero. We have designed an IMC-PID controller using Rivera et ul.’s 141 Case R which is for a model of the form

where, for this examplc, K = 0.005, d = 5 , T = 20 and we have added a first-order lag with unit gain and time constant equal to 300 in series with the PID controller. We based our choices for the IMC filter time constant E on the approximate process-output settling times found in Fig. 5. To compare with the slow response design, we selected E = 40016 which gives PID parame- ters of K, = 6.2, zI = 158.3 and z, = 17.5. To compare with the fast-response design, we selected E = 10016 which gives PTD parameters of K, = 24.8, z, = 58.3 and z, = 13.1. Figs. 7 and 8 show the closed-loop process-

IEE 1’ro~:~’untroI Th,mivy Appl , Vu1 144. Vo 5, Si~pinnhei. I997

output and controller-output responses for these two designs with the same setpoint and disturbance inputs used previously.

In comparing the results, it is important to point out that the large discrepancy in the magnitude of the con- troller parameters results from the fact that two differ- ent process models have been used in the controller calculations, i.e. the new design uses the original proc- ess model given by eqn. 24 and the IMC design uses the process model given in eqn. 25. It also must be noted that a series filter is required to implement the IMC design but is not required with the new design. Comparing Figs. 5 and 6 with Figs. 7 and 8, we can see that the new design produces less overshoot to the set- point change and faster disturbance rejection without oscillation.

1.81

1 . L

0 .LI

-100 0 100 200 300 LOO 500 600 700 800

Ptotery-output ruponw fir diyei tun eunnipke nrth IMC d m g n time, s

Fig.7 c = 400/6 ( 0) dnd t = 100/6 (- )

* 1.51 -I

I\ I 4 e 8 -0.5-

-1.0-

-1.51 100 0 100 200 300 LOO 500 600 700 800

time,s C(Jntr[llker-oL~tl,Ut r tyonse ,for dryer can rxumpkt> with IMC Fig. 8

rksign c = 400/6 ((:<I<>) and c = IOW6 ( ---)

3 processes

Assume that the process has the following transfer function:

G l ( s ) = -epds

For this process, y1 = --d < 0, and, therefore, this is a type-A integrating process. The desired control-signal specification is chosen as

Tuning rules for integrating plus delay

(26) K S

By letting J' = ds, G,,,(s) is scaled in the Laplace domain as

And the desired closed-loop transfer function is given by

3. I Normalised PID parameter solutions From the desired closed-loop transfer function, the cor- responding desired open-loop transfer function is obtained as

This then gives

- - j2 (2CP+l)s+l K d Z p252+2pjs+1- [( '@C+l) 2+1]s- 5 (31 ) -

The two frequencies used in the controller parameter solutions are col = 2dT5 and w2 = 201, where T, is the desired closed-loop settling time chosen here as (6/3 + 1)d. The corresponding normalised frequencies are = d q = 2nl(6P + 1) and &, = 261~. We will now define

( X P + 1 ) P + l Y(J&) = (JV /32(3&)2+2BCJ&+I - [ ( 2 ~ ~ + l ) ~ d + l ] ~ - 7 w

( 3 2 ) With our choice of 6 , and (j2, P is only dependent on the performance parameters /3 and 5, and is independ- ent of the actual process parameters. Applying the ana- lytical solutions in eqns. 18-23 with eqn. 32, the PID controller parameters are given by the following. Proportional gain:

where K, = f'I(&,)/&l Integral time constant:

71 = d?l where

k 71 = Fz? (Cl) - ?n ( 6 2

Y E ( U i 1 ) + 3

Derivative time constant: TLj = d?o

where Y R ( 6 1 ) - YR(G2)

3Kcw; T D =

(34)

(35)

We refer to I?,, 2, and ZD as the normalised PID controller parameters, which only depend on the per- formance parameters /?J and c. 3.2 Stability margins To examine the stability margins of the integrating plus delay process under feedback control with a PID con- troller, the actual open-loop transfer function is formu- lated as follows:

389

As the parameters in the actual open-loop transfer function are only functions of the performance parame- ters, the gain and phase margins of the closed-loop sys- tem are independent of the actual process parameters. However, the actual critical frequencies will be obtained by dividing the normalised critical frequencies with the process delay. Another important measure of the robustness of the closed-loop system is the allowa- ble time-delay variation. This is defined as the maxi- mum increment in the process delay (Ad) that will bring the closed-loop system to the stability boundary. We will now define the relative delay margin as

(37) Ad - & d .jlj - __ -

where #,,? is the phase margin and hp is the correspond- ing critical phase margin frequency. Note that, the right-hand side of eqn. 37 is only dependent on the per- formance parameters /3 and <.

3.3 Presentation of the tuning rules For the development or tuning rules for this type of process, we have chosen the damping factor < to be equal to either 0.707 or 1, to produce two sets of rules with the parameter /? being used to adjust the closed- loop response speed. has been varied for both cases from 1 to 17 and the corresponding normalised PID controller parameters K,,, + I and Zn have been calcu- lated. From this information, polynomial functions have been fit to produce explicit solutions for the nornialised controllcr parameters as a function of /3. < = 0.707:

1 0.71388 + 0.3904

K , =

i, = 1.40203 + 1.2076 (39) 1

1.4167R + 1.6999 711 =

< = 1:

71 = 1.98854 + 1.2235 (42)

(43) 1

I .(I0435 + 1.8194 ru =

The final controller parameters are obtained using the scaling eqns. 33 -35. The gain, phase and relative delay margins for these PlD tuning rules are shown in Figs. 9, 1 1 and 13, In general, the choice of damping factor = 0.707 produces smaller proportional gains, integral time constants and derivative time constants in comparison to < = 1. The former choice also leads to larger gain and rclativc delay margins but smaller phase margins. A smaller phase margin implies a faster closed-loop response.

The settings for P1 control with these rules are obtained from the same set of rules for PID by simply setting ZD = 0. The gain, phase and relative delay mar- gins for PI control are shown in Figs. 10, 12 and 14.

n

Fig. 10 t: = I (---) and 5 = 0.707 ( c x )

G& margins for. PI tuning rules

P Fig. 11 < = 1 ( ) and = 0.707 (:>03)

Phase mur.gin.t fbr PID tuning r u b

3.4 Evaluation and comparison of the tuning rules Consider the integrating plus delay process used in [5] described by the transfer function

where K = 0.0506 and d = 6. The tuning rules will be evaluated using this process. The closed-loop time constant z is chosen to be equal to d, 2d, 3d and 4d cor-

Ihh Pia( - C o n t i u l T/z(,uij A p p i Vol 144 No J Sepiember 1997 190

responding to = 1, 2, 3 and 4. For the first two choices, a PID controller is used and, for the latter two choices, a PI controller is used. The normalised con- troller parameters are calculated from eqns. 38-43 and then the final parameters are obtained using the scaling eqns. 33-35. Figs. 15 and 16 show the process-output and controller-output responses for < = 0.707, and Figs. 17 and 18 show the process-output and control- let--output responses for < = 1.

I

l o b 2 1 6 8 i o 12 1’1 16 la P

Fig. 12 C = I (solid line) and

Phaw margins for PI tuning rules = 0 707 (solid line with ‘ 0 )

12

10

D !l c F

8

F x 6 A

W D

W L 2 1

0 W - L 2

‘0 2 L 6 8 10 12 11 16 18 P

Fig. 13 Relative delay margins for PID tuning rules = 1 (solid line) and 5 = 0.707 (solid liiie with ‘0 ’ )

Fig. 14 c = 1 (solid line) and 5 = 0.707 (solid line with ‘0 ’ )

Relative delay marginsfor PI tuning d e s

In Tyreus and Luyben’s work [5], their objective was to design a PI controller for integrating plus delay processes. Their performance specification was given in the frequency domain, where the peak value of the magnitude of the closed-loop transfer function Gr-)y was chosen to be +2dB. A numerical procedure was used to find the normalised PI controller parameters to achieve this performance specification. It is interesting to note that their final tuning rules for the PI parame-

1 . 6 -

1 . L -

-50 0 50 100 150 200 250 300 time,s

Fig. 15 Process-output responses for [ S ] exumple 0 = I (solid line), p = 2 (‘*’), /j = 3 (‘X’), /3 = 4 (10’) and = 0.707

5r--- 1 31

-2 1 I -50 0 50 100 150 200 250 300

time,s

Fig. 16 p = I (solid line), /j = 2 (‘*’), p = 3 (‘X’), p = 4 ( ‘0 ’ ) and r = 0 707

Controller-output responses for [S ] example

1.81 I

4 0 . 8 - 8

0.6-

0 . L -

0.2-

I -J -50 0 50 100 150 200 250 300

time,s

Fig. 17 /3 = 1 (solid linc), /3 = 2 (‘*’),

Process-output raponses fbu [S ] example = 3 (‘X’), = 4 (10’) and 5 = 1.0

IEE Proc.-Control Theory Appl., Vol. 144, No. 5 , September 1997 391

5’-1

-21 -50 0 50 100 150 200 250 300

time,s Fig. 18 Controller-outpul responses for [SJ example p = I (solid line). p = 2 (‘*’), 0 = 3 (‘X’), p = 4 ( (0 ’ ) and = 1.0

ters are presented in the same fashion as eqns. 33 and 34, except they are given for only a single performance specification. Their normalised proportional gain K , is equal to 0.487 and their normalised integral time con- stant ?, is equal to 8.75. With our tuning rules, if we choose 1; = 1 and /3 = 3, then K , = 0.466 and ?, = 7.19. Therefore, for this particular choice of closed-loop per- formance, our tuning rules correspond almost exactly to those presented by Tyreus and Luyben [5] . The key benefit with our tuning rules is that different values for /3, depending on the control objective and desired sta- bility margins, may be selected and used to easily calcu- late the required PID controller parameters.

4 Conclusions

This paper presents a general PID design method for integrating processes, which uses the desired control

signal trajectory as a performance specification and solves for the PID controller parameters in the fre- quency domain. Explicit tuning rules have been pre- sented for integrating plus delay processes with a single closed-loop response speed parameter to be selected by the user. The combination of a time-domain perform- ance specification with a frequency-domain design makes the method straightforward to apply, with mini- mal constraints imposed by the process model struc- ture.

5 Acknowledgments

The authors wish to acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada. We would also like to thank Mr. W.L. Bialkowski, President of EnTech Control Engi- neering Inc., Toronto, Canada, for motivating us to look more closely at this problem.

References

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Engineering Inc., Toronto, 1993

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