PID tuning for integrating

E. Poutin A. Pomerleau

Indexing terms: Tuning metlrods, PI and PID controllers, Nichols chart

Abstract: A sysitematic PI and PID tuning method is developed for integrating and unstable processes. The method, based on a maximum peak-resonance specification, leads to simple tuning parameter expressions and is graphically supported by the Nichols chart. The common characteristic of' integrating and unstable processes is that their open-loop frequency response, with the controller in cascade, has a phase maximum. The controller parameters are adjusted such that this maximum is located on the right-most po mt of the ellipse, corresponding to the maximum peak resonance on the Nichols chart. For these types of processes, making the open-loop frequency-response curve tangent to a specified ellipse is an efficient method of controlling the overshoot, the stability and the dynamics of the system. Charts that give the optimal peak resonance, according to the ITAE criterion, are given. The results obtained with this method are presented for typical design examples. A coimparison of the performances of the proposed design settings with those given by earlier workers for unstable processes is presented.

1 Introduction

Different approachems have been taken to de:velop tun- ing methods for integrating and unstable processes. Morari and Zafiriou [I] used the internal model control (IMC) tlo derive ]'ID tuning methods. The IMC approach was also used by Rotstein and Lewin [2] in the context of parameter uncertainty. They compared their tuning methods to different adaptive schemes for the control of an unstable batch chemical reactor (Rot- stein and Lewin [3]:1. The design of pole placing con- trollers for unstable systems with time delay was presented1 by Stahl and Hippe [4]. Modified Smith pre- dictors were used by De Paor [5], De Paor and Egan [6] and Astrom et al. [7], to cope with unstable and inte- grating processes with long time delay. De Paor and Egan [8] also used a Luenberger observer augmented by PI and PID contirol for unstable processe:s with time delay. Quinn and S'anathanan [9] presented a model- 0 IEE, 1996 IEE Proceedings online no. 19960442 Paper first received 31st October 1995 and in revised form 13th February 1996 The author,s are with the Departement de Genie Electricpe, Universite Laval, Sdinte-Foy, Qdbec. Canada G l K 7P4

and unstable pirocesses

matching method in the frequency domain to approxi- mate low-order controllers. De Paor and O'Malley [lo] derived tuning methods for unstable processes with time delay based on gain and phase margins for P, PI and PID controllers. From this concept, approximate solutions for calculating the controller gain and the integral time constant were proposed by Venkatashankar and Chidanibaram [ 1 11 for first-order with time-delay systems. Shafiei and Shenton [12] pro- posed a graphical technique for tuning PID controllers for unstable and stable systems with time delay.

This article presents a systematic and simple approach for the design of €'I and PID controllers for integrating and unstable processes. It has been imple- mented as part of a complete set of tuning methods in an auto-tuning and adaptive PID controller (Poulin et al. [13]). The method is based on the analysis of the open-loop frequency response of the process in series with the controller, conveniently represented on the Nichols chart. The controller parameters are adjusted to satisfy the specification on the maximum peak reso- nance (Mr) of the closed-loop system. The method leads to simple expressions for PI and PID parameters. These expressions are given for first and second-order systems with time delay, since most industrial processes can easily be represented by this type of model. How- ever, the method can be extended to higher-order sys- tems. Stability conditions are also discussed for unstable processes.

2 Integrating processes

The transfer function of a second order integrating process with time delay is given by

K e-OS s(T,s + 1)

G p ( s ) = 2-

Since the process includes an integrator, there is no static error to a set-point change with a proportional controller. This is not the case, however, when nonzero mean disturbances act at the process input. In order to ensure that there will be no static error, a controller with an integrator must be used. Thus, only PI and PID controllers are considered.

2. I PI controller The transfer function of the PI controller and the open- loop system are given, respectively, by

and

IEE Proc-Cbnrrol Theory Appl. , Vol. 143, No. 5, September 1996 429

Typical open-loop frequency responses GCjw) with K,? = 1, 8 = 0.2s, TI = Is, T, = 7.8s (calculated according to the method described later) are presented on the Nichols chart by Fig. 1 for different proportional gains IC,,, = 0.43, KC2 = 0.86 and Kc3 = 0.215. The Nichols chart overlaps GCjw) and the closed-loop frequency response HCjw) = G(jw)/(l + GQw)). The concentric ellipses represent constant amplitudes of H(j03). The phase of the open-loop systems goes from LGQ.0) = -180" to LG6.m) = -MO, and there is a maximum at wi?7,,y. When the proportional gain K,. = Kcl is used, the point GQw,,,,) is located on the right most point of the 3dB ellipse, and the curve GCjw) never crosses higher concentric ellipses. Increasing the proportional gain (K, = Kc2) to increase the closed-loop bandwidth, or decreasing it (K,. = Kc3) to have larger phase and gain margins, results in a higher maximum peak resonance M,.. A higher M,. indicates that the system is less damped and has larger overshoots. It is not surprising that higher proportional gains give more oscillatory responses, but it is worth noting that lower gains and even larger phase and gain margins have the same effect.

30

20 m U

5- 10

a 0

; -10

0 ol

0 - a a 0 -20

- 30

-40 -300 -200 -100 0

open-loop phase, deg Fig. 1 ing processes iz;itiz time de ay w'ith a PI controller (*I K,, (0) K z (+I K ?

Typical open-loof frequency responses for .second-ordei. intgrat-

With this type of process, M,. is more representative of the desired response than traditional stability indica- tors. The maximum peak resonance is thus used as a specification. The controller parameters are adjusted such that GCjw,,,,,) is tangent to the ellipse specified by M,. As an example, for a M , specification of 3dB, GCjw) must have a maximum phase LGCjo,,,,) = -135" with an amplitude IGCjw,,,)I = 3.02dB (see the curve (*) on Fig. 1) . The design procedure to calculate the PI parameters follows three steps. (1) The open-loop frequency response GCjo) is first analysed. The phase LGCjo) has a maximum if Ti > T,. The frequency at which it occurs (o,,,,) and the value LG(~U,~, ,J are calculated. (2) Afterwards, the integral time constant Ti of the PI controller is calculated to give the desired phase value at wino,L according to the specified M,. (3) Finally, the proportional gain K, is adjusted such that the point G(~W~~~,J is located on the right-most point of the ellipse Mr on the Nichols chart. The method is detailed for a second-order model with time delay. The phase equation of the system (eqn. 3) is

(4) i G ( j w ) = -T - arctan(T1w) - W Q + arctan(T,w)

410

The frequency wmax at which the phase maximum occurs, i.e. a t which dLG(jo)/dw = 0, is given by

wmaz = [ l / m T l T , ] [T:T, - T,"Q - TIT," ~ T:O

2 ," H 2 - ZTPT? H + T : T : - 2T2TZ!+ 1 (5) +

i - Z T f Tk 8' - 2 r: T: + 2 T f TP Y i T f T," +2TiT? H+TP O 4

This expression is reduced to eqns. 6 or 7 when T, = Os or 8 = Os, respectively:

Knowing that the maximum is located between UT, and l/Tl, a simpler expression than eqn. 5 can be obtained by taking the following approximation of arctan(x) (Bower and Schulteiss [14]):

arctan(z) "7r/2 - 1/z z > 1 5 2 5 1 (8)

In the present case, both approximations are used since T l w < 1 and T p > 1 at om,, Then the phase equation is given by

(9) The frequency om,, becomes

i G ( j w ) = -7r - T l w - w0 + [7r/2 - ~/(T+.J)]

and the maximum phase value is given by

i G ( j w m a z ) = -: - (11)

Afterwards, using the specification M,. that determines LGCjwmn.J and eqii. 11, Ti can be calculated. The rela- tion between LGCjo,,,) and M y can be visualised on the Nichols chart and is given analytically, when M, is expressed in dB, by

i G(jwnbaz) = arccos[ d m / 10°,05"i 1 -7.r

T, = 16(T1 + i9) / (2LG(jwmaz) + T)'

(12) The integral time constant TI that gives the desired L m % , , , Y > is

(13) Finally, the proportional gain K, is adjusted so that the point G(jomu,J is located on the right-most point of the ellipse specified by M,.. The relation between M,. and IG(jwmon)i can be visualised on the Nichols chart and is given, when Mi. is expressed in dB, by

IG(jumaz)l = 10 o ~ o ~ ~ ~ ~ / ~ ~ (14)

The gain of the system (eqn. 3) is

thus

It is important to note that the closed-loop transfer function H(s) has a zero since Ti + TI.

IEE Proc.-Cor?trol Theory Appl . , Vol. 143, N o 5. September 1996

This zero produces large overshoots since the value of T, is typically large compared to T I . For set-point changes, the probleni can be easily resolved, without changing the closed-l oop properties, by cancelling the zero with a first-order set-point filter with T5r = Ti.

h I

5.0 I m U

f 1 4.5

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

BIT, Fig.2 Optimal M,, according to the ITAE criterion for output (-) and input (- -) step load disturbances, for second-order integrating processes with time delay as afunctioiz of the ratio SIT,

It may also be useful for the user to be oriented with respect to the selection of the specification[ M,,. The closed-loop peak resonance can be chosen to minimise the ITAE criterion (eqn. 18) for a step load disturbance applied ai. the output or the input of the process. Fig. 2 presents !he M , values that minimise the ITAE crite- rion as a function of the ratio O/T1. As the ratio 0/T, - 00 (purely first-order system with time delay), the opti- mal specificdtion for output and input disturbdnces asymptotically goes i o A4, = 4.52dB and M,, = 5.55dB, respectively .

ITAE = lerror(t)/ t dt (18) 7 0

Example 1: A PI controller is designed for the second- order integrating process with time delay given by

,-0.2s

G,(,s) = 2 s ( s + 1)

According to Fig. 2., the optimal M , for a disturbance acting at the process output is 4dB. This means that the G@J) curve must be tangent to the right-most point of the 4clB ellipse on the Nichols chart. Eqns. 12 and 14 give the coordinates of this point: ~G~(jwmax)~ = 1.2890 and L C ( ~ O ~ ~ . J = -2.4588 rad. The integral time constant T, is calculated using eqn. 13, such that G(jwmax) has the desired maximum phase value, and the proportional gain K , is adjusted according to eqn. 16, such that G(jwmax> has the desired open-loop gain. The obtained values are: K, = 0.4647 and T, = 6.0921s. For a disturbance acting at the process input, the optimal M , is 5.4dB. Proceeiding in the same way, this gives K, = 0.4808 and T, = 41.7348s. In both cases, a first-order set-point filter with Tsp = Ti is used. The frequency responses are presented on the Nichols charl, by Fig. 3. Figs. 4-9 present, respectively, the set-point change responses and the output and input step load distur- bance responses.

IEE Pvoc.-Control Theury Appl., Vol. 143, No. 5 , Septemhrr 1996

4 0

3 0

m 2 0 U

c- 1 0

a 0

. _ 0 cn

0 - 2 -10 al a 0 -20

- 30

-40

open-loop phase, deg

Fig.3 (*j M , = 4dB (0) M , = 5.4dB

Example I , frequency responst:r G(jco)

t ime. s Fig.4 (-) M , = 4dB (- -) M , = 5.4dB

Example I , set-point change responses: process output

time, s

Fig. 5 (-j M, = 4dB (- -) M , = 5.4dB

Example I , set-point change responses: controller output

i I 3

b-0 5 0 5 10 15 20 25 30 35 40

time, s Fig. 6 put (-1 M , = 4dB (- -) M , = 5.4dB

Exumple 1, output step load disturbance responses: process out-

- - 0 . 4 I / ; I V , , , , , , \ E - 0 . 4

0 5 10 15 20 25 30 35 40 time, s

Exumple I , output ,step load disturbance responses: controller out- Fig. 7 put (-) M ) = 4dB (- -) M, = 5.4dB

43 1

0 5 10 15 20 25 30 35 4 0 t ime, s

Fig. 8 (-) M, = 4dB (- - J M , = 5.4dB

Exunzple I , inpul step loud cli.stui.bunce responses: process output

t irne, s Fig. 9

(-J M , = 4dB (- -J M , = 5.4dB

E,uuinpk~ I , i i z p t . s t q load &sturhtiiicc re.spoizse.s: controller out- j J U f

The set-point filter eliminates most of the overshoot for set-point changes. With the specification Mr = 4dB, there is no overshoot, and with M,. = 5.4dB, it is less than 5%. The output step load disturbance responses are similar for both specifications. The large excursion of the process output in the negative direction is caused by the zero of H(s) that cannot be cancelled in regula- tion. This can also be explained in the time domain by analysing the controller output. When the disturbance is applied, the error between the set point and the proc- ess output is negative. The proportional action is nega- tive and the integral action takes on negative values. In the steady state, the controller output must be zero. Thus, to bring back the integrator output to zero, the error between the set point and the process output must be positive for a certain period. For the input dis- turbance responses, this excursion is not necessary, since the controller output in the steady state is nega- tive.

2.2 PID controller A PID controller can be used to accelerate the system if the process is well known and the model sufficiently accurate. In order to be operational in a noisy environ- ment, only a controller having an equal number of poles and zeros is used. The interacting (series) form of the PID controller (Astrom et ul. [15]) is used, since most commercial PID controllers fit under this classifi- cation (Besharati Rad and Gawthrop [16]). Its equation is given by

This form is equivalent to the noninteracting (parallel) one given by eqn. 21 with proper parameter conver- sions (eqns. 22 to 25).

Ki[1 + l/(Tis) + TAs] (1 + T i s )

GL(s) =

432

The derivative time constant is used to cancel the time constant T, of the model given by eqn. 1. The choice of T, relies on the desired noise attenuation and the uncer- tainties on the parameters of the model. Afterwards, the tuning method is exactly the same as the one used in the PI case, but with the model

KpecsS (26) Gb(s) =

S ( T f S + 1) since, when T, = T,

2.3 Higher-order models The methodology developed in this Section can easily be applied to higher-order models. Only the phase and gain expressions (eqns. 9 and 15) will change. The equations for the controller parameters will be more complicated, and it could be more convenient to apply the method graphically.

3 Unstable processes

The transfer function of a second-order unstable proc- ess with time delay is given by

The open-loop step response of the process is unbounded, since it has a pole in the right-half plane. In order to have a stable closed-loop system, the Nyquist criterion must be satisfied. A closed-loop sys- tem is stable if the point (-1, j0) in the complex plane is encircled by GCjo) a number of times equal to the number of unstable poles of the process in the anti- clockwise direction. The point (-1, j0) corresponds to the (OdB, -180") point on the amplitude versus phase plane.

3. I PI controller The open-loop transfer function is given by

The phase goes from LGCj.0) = -270" to LGCj.a) = -ao and there is a phase maximum at omax if Ti> T I , and if T, and 0 are small compared to TI . In fact, with PI control, the condition

LG(jwTnaZ) = arctan(T%w,,,) + arctan(Tlw,,,) - arctan(Tzw,,,) - Owmaz ~ 3 ~ 1 2 > -T

(30) must be satisfied to respect the Nyquist criterion. Since the PI controller always reduces LGCjo), the process itself must satisfy:

arctari(T1wm,,) - arctan(T2wm,,) - Qw,,, > 0 (31)

Eqn. 31 means that if the phase of the process is always smaller than -180°, it is impossible to use a PI control- ler. For first-order models with time delay (T, = Os), eqn. 31 is reduced to 8/T, < 1 as mentioned by De Paor and O'Malley [lo] It is worth noting that for OlT, > 0.6, the system is highly oscillatory with unacceptable overshoots in practice (Venkatashankar and Chidam- baram [ll]). For second-order models with no time delay (0 = Os), the limit for stability is T, < TI .

IEE Proc.-Control Theory Appl., Vol 143, No. 5, Septembes 1996

The design procediire follows the three-steps method presented for integrating processes. Firstly tlhe charac- terisation of the phase maximum, secondly the calcula- tion of the integral time constant T, that gives the desired open-loop phase at (I),,,, and finally {.he adjust- ment of Kc such that the open-loop curve G(jo) is tan- gent to the right-most point of the ellipse specified by M,..

The phase equation of the open-loop transfer func- tion (eqn. 29) is obtained with the arctan(x) approxi- mation given by eqn. 8. Both approximations are used since Tlw > 1, T20 < 1 and T,o > 1 at o,,,,,. The phasc equation is iG(jw) = -3n/2-we-i-[?r/2-1/(~*~)]+[1;/21/( TL&)]-TJu

The frequency at which the maximum phase 'occurs, i.e. at which dLG(jo)/do = 0, and its valuc are given, respectively, by

(32)

(33)

and 7

(34) Using the specification M,. and eqns. 12 and 34, the integral time const ant Ti that gives the desired LGC~CO,~,,) can be calculated as follows: T, ~ T ~ ( Q + T ~ ) / [ T I i iG(jw, , , )+-7i /2)2-4(Q-t~~)] (35)

Finally, iusing the specification A4,. and eqn. 14, the proportional gain K, that gives the desired ~G(jco,~,~,~J is evaluated. The gain of the system (eqn. 29) i:s

thus

h-<, = __

Since T, > TI, the closed-loop transfer function has a large zero and this typically produces large overshoots. As for integrating processes, the overshoot for set-point changes can be eliminated without changing the closed- loop properties, by using a first-order set-point filter with Typ = T,. Chartis that give the optimal specification of Mr according to the ITAE criterion (eqn. 18) for dif- ferent ratios (0 + T2)/T, are presented by Fig. 10 for output and input step load disturbances. Exumple 2: The tuning method is applied to the first- order process, with a time delay given by

T2wm(Lz IG(jcvVLas-) I J[l + (Tt~7r, ,z)2][l + (T24nnT)21

(37) h j J 1 + ( T i ~ m , n x ) 2

e - 0.2 s

(--s + 1)

-

GP(:?) = ~ (38 )

The PI parameters are calculated using eqns. 33, 35 and 37 for M,, = 8tlB (output disturbance) and M,. = 8.8dB (input disturbance), evaluated from Fig. 10. The results are compared with those obtained by the method proposed by De Paor and O'Malley [lo] and by Venkatashankar and Chidambaram [l I] for F = WT, = 0.2. N o set-point filter was used in order to present a fair comparison. The tuning parameters are given in Table 1, and the open-loop frequency responses are presented on the Niizhols chart by Figs. 11 and 12. The set-point change re:sponses and the output and input step load disturbance responses are presented by Figs. 13 to 18.

20

l a -

m 1 6 - U

;I

1

1

I

I , ' ,/ 4

40

30

20 ?3 g- 10 0 0 a 0 0 - ; - 1 0 a Q 0 - 2 0

open-loop phase, deg. Fig. 12 ,%ziiwpk 2, jiequeiicy srsponscx G(jo),, (x) De Paor and O'Malley (+) Vcnkatashankar and Chidambardm

In the case of set-point changes, the settling time (* 5% of the final value) obtained with the present method is much smaller. The output and input distur- bance responses are not oscillatory, and the system is stabiliscd in a short time compared to the other meth- ods. The oscillations and the large overshoot obtained with the quasi-optimum phase-margin design of De Paor and O'Malley [lo] is easily explained by the Nichols chart presented in Fig. 12. Even if their method gives a similar phase margin to those obtained with the present method and a larger gain margin, the

433 IEE Proc.-Control Theory Aiipl., Vol. 143, No. 5. Septenihci. 1996

Table 1: Example 2, PI parameters

PI parameters M r = 8dB

Kc 3.39 3.42 1.70 2.47

T, (s) 1.46 1.25 1.35 19.60

Mr = 8.8dB De Paor and Venkatashankar and Present method Present method O’Malley [I01 Chidambaram [I I ]

Goo) curve crosses ellipses with corresponding M, much greater than 8.8dB. Conversely, the results obtained by the method of Venkatashankar and Chidambaram [ 1 11 have no oscillations but longer time responses, since GCjo) only crosses the 5dB ellipse. A similar comment to the one made for integrating proc- esses, concerning the negative excursion for output dis- turbance, is appropriate here. Initially, the integral action takes negative values and, in the steady state, the controller output (that is produced by the integral action only) must be positive. The excursion of the process output to the negative region is thus necessary to bring the integral action from negative to positive values. - 2.51

o p i 0 5 10 15

time, s Fig. 13 Example 2, set-point change responses: process output (-) M,. = XdB (- -) M , = X.8dB (-.) De Paor and O’Malley ( . ) Venkatashankar and Chidambaram

i- 3 4 a 1

i

” -4 I I 0 5 10 15

time, s Example 2, set-point change responses. controller output Fig. 14

(-) M, = XdB (- -) Mr = 8.8dB (-,) De Paor and O’Malley (-) Venkatashankar and Chidambaram

time, s Fig. 15 Example 2, output step load disturbance responses: process out- put (-) M, = 8dB (- -) Mr = 8.XdB (-,) De Paor and O’Malley ( ) Venkatashankar and Chidambaram

c

6 - 4 0 5 10 15

time, s Fig. 16 Example 2, output step loud disturbance responses: controller output (-) M, = 8dB (- -) M , = 8.8dB (-) De Paor and O’Malley (..) Venkatashankar and Chidambaram

434

. I k-o.5L , I , I I I I , , 0 2 4 6 8 10 1 2 14 16 18 20

time, s Fig. 17 Example 2, input step load disturbance responses: process out- put (-) Mr = 8dB (- -) M, = 8.8dB (-,) De Paor and O’Malley (-) Venkatashankar and Chidambaram

0 2 4 6 8 10 12 14 16 18 20 time, s

Fig. 18 Example 2, input step load disturbance responses: controller out- put (-) M , = 8dB (- -) M , = 8.XdB (-1 De Paor and O’Malley (..) Venkatashankar and Chidambaram

3.2 PlD controller The PID controller gives the possibility of overcoming the phase reduction brought by the second time con- stant T2, even if T2 2 T I . The derivative time constant T, is used to cancel T2. Then, the PI tuning method is applied to the model

-Kpe-6s G;(s) =

( - T I S + l)(TfS + I) (39)

since, when Td = T2,

-KcKp(T,s + l)ecsS K,(Tts + 1) T, S

Gb(s) - G(s ) = -

T,s(-TIs + ~ ) ( T s s + 1) (40 1

The stability conditions are obtained by replacing T2 by Tf in eqns. 30 and 31.

3.3 Higher-order models The methodology developed in this Section can be applied to higher-order models if LGOw) (PI control) or LG’Cjw) (PID control) reaches values greater than -180”. Only the phase and gain expressions (eqns. 32 and 36) will change. It could be more convenient to apply the method graphically.

4 Conclusion

A systematic and simple tuning method for integrating and unstable processes has been presented. The method, based on a maximum peak resonance specifi- cation, is graphically supported by the Nichols chart. The PI and PID parameters are adjusted such that the point Gfiwn2ax) is tangent to the right-most point of the specified ellipse. For these types of processes, the M,. specification is more representative of the stability of

IEE Proc -Control Theory A p p l . Vol 143, No 5, September I996

the system and the desired response than phase or gain margins. In the case of unstable processes, stability conditions have been given. Charts that give the opti- mal specification according to the ITAE criterion for output and input step load disturbances were pre- sented. The tuning method gave good responses for integrating processes and generally better results than those obtained by earlier workers for unstable proc- csses.

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