Improving Performance and Robustness in the IMC-PID tuning method

Copyright © IFAC Robust Control Design Milan, Italy, 2003

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IMPROVING PERFORMANCE AND ROBUSTNESS IN THE IMC-PID TUNING METHOD Alberto Leva 1

Dipartimento di Elettronica e Injormazione, Politecnico di Milano Via Ponzio, 34/5 - 20133 Milano (Italy)

Abstract: This paper presents an extension of the IMC-PID tuning method. A more complex structure is adopted for the process model than in the original method, where a FOPDT (First-Order Plus Dead Time) model is used, the IMC procedure is applied, and the so obtained regulator is approximated with a (real) PID. Experimental results indicate that abandoning the FOPDT structure in the IMC approach is beneficial both in terms of performance and robustness. The selection of a more realistic model structure looks very promising for the synthesis of industrial regulators. Copyright © 2003 IFAC

Keywords: PID control, internal model control, robust autotuning, process control.

1. INTRODUCTION

the more the identification algorithm adopted is critical for the tuning results. To take profit of the IMC rationale for the (automatic) synthesis of industrial regulators, it is beneficial (and sometimes necessary) to abandon the FOPDT model structure, also while preserving the PID control law. Notice that in some cases, to maintain acceptable performance and robustness, the shortcomings of the FOPDT structure require to abandon that law (Ingimudarsson and Hiigglund, 2001). Extending the IMC-PID tuning method to non FOPDT structures is not an easy task, however. That is why this approach is not frequent in the literature, and practically absent in the applications (Leva, 2001), especially if analytical tuning formulce are required (Isaksson and Graebe, 1999).

In the last years, the Internal Model Control (IMC) principle has been extensively employed for the (automatic) synthesis of industrial regulators (Leva and Colombo, 2001b). Indeed, the numerous applications make that principle one of the most successful results of robust control. It is very straightforward to apply the IMC principle to the synthesis of industrial PI or PID regulators, as first proposed in (Doyle et al., 1992), but this requires that a specific structure be adopted for the process model. Therefore, the model structure is dictated by the type of regulator, not by the process dynamics. Most frequently, FOPDT (First-Order Plus Dead Time) models are used with satisfactory results. Nonetheless, there are some cases where the process-model mismatch caused by the limited descriptive possibilities of those models is keen to result in poor loop behaviour, especially in the presence of load disturbances, and also to reduce the significance and interpretability of design parameter(s). In addition, the less the model structure is close to reality,

This paper has two purposes. The first is to show, by means of an experimental example and some considerations, that the need for more complex model structure in IMC-based PID synthesis is a fact. The second is to propose a PID tuning method devised along this research line.

1 Voice (39) 022399 3410, Fax (39) 02 2399 3412, e-mail [email protected]

305

and adopts the FOPDT model (4)

The tuning formul~, see (Leva and Colombo, 2001a) for the derivation, are Fig. 1. The IMC control scheme.

L2

(5)

N

=

F(s)Q(s) 1 - F(s)Q(s)M(s)

1 =. 1 + S>.

=K

[1 + _1_ + STd] STi 1 + sTd/N

,d

=

>.LN 2(L + >') ,

The practical relevance of the problem addressed will now be shown with an experiment. The process is a laboratory setup for temperature control, in which a metal plate is heated by two transistors and cooled by a fan. One transistor is the control actuator, the other acts as a load disturbance source, and the fan modifies the process dynamics by altering the thermal exchanges with air. The controlled variable is the plate temperature. Applying the method of areas to the open-loop plant response to a heater command step leads to the FOPDT model M ( )

a S

0.1

= 1 + 195s'

(6)

while a better model can be identified easily as Mb(S)

= (1

0.1(1 + 80s) + 155s)(1 + 25s)(1 + 15s)(1 + 5s)'

( 7)

Figure 2(a) shows the measured response and those of the two models (6) and (7). Applying the formul~ (5) with A = 30 leads to a PI with K = 120 and T i = 195, while A = 10 gives K = 360 and T i = 195. A very simple spectral analysis shows that Mb(s) fits the data up to a frequency WM slightly greater than 1r/s, so the IMC-PID procedure can be applied taking

(1)

Q(s)

= (1 + 155s)(1 + 25s)(1 + 15s)(1 + Ss) 0.1(1

+ 80s)(1 + 2.5s)3

(8)

as approximate inverse of the plant model and F(s) _

(2)

-

The design parameter A can be interpreted as the desired closed-loop time constant of the control system, and determines the control system bandwidth and degree of robustness. Here a version of the IMC-PID method is used (Leva and Colombo, 2001a) that refers to the 1-d.oJ. ISA PlO control law (Astrom and Hagglund, 1995) RP/D(S)

-1 T

3. AN INTRODUCTORY EXPERIMENTAL EXAMPLE

The IMC synthesis method in the general case is a two-step procedure. First Q(s) is set to the inverse of the minimum-phase part of M(s) (Morari and Zafiriou, 1989). Then, the low-pass filter F(s) is introduced, see (Morari and Zafiriou, 1989) for a complete discussion. For simplicity, F(s) is very often chosen of the first order and (of course) with unity gain, i.e. F(s)

= T(L+>') >'Ti

The general IMC scheme is reported in figure 1: pes) is the transfer function of the process, assumed here asymptotically stable; M(s) is the process model available for tuningj Q(s) and F(s) are asymptotically stable transfer functions; yO, d and n are the set point, a load disturbance and a measurement noise; y and fj are the 'true' and 'nominal' controlled variables, Le., the outputs of the process and of its model, respectively. If M(s) = pes) and d = n = 0, the scheme is apparently open-loop, so that T(s) := Y(s)/YO(s) = F(s)Q(s)M(s). If it is also possible to set Q(s) = M-1(s) = P-1(s), then T(s) = F(s)M-l(S)M(s) = F(s), thus the transfer function from set point to controlled variable can be chosen arbitrarily. Finally, if d¥-O while still M(s) = pes), n = 0 and Q(s) = M-l(S), if F(s) = 1 d is rejected completely; otherwise, it is rejected asymptotically provided that F(O) = 1. When pes) i:- M(s), there is a limit on the acceptable model error. This limit can be quantified with the robust control theory, see e.g. (Doyle et al., 1992) and, more in detail, (Leva and Colombo, 2001bj Leva and Colombo, 2001a). The IMC regulator (the gray blocks in figure 1) is equivalent to a feedback controller R(s) given by R(s)

T,

T,=T+ K=' • 2(L+>.)' I-'(L+>')'

2. REVIEW OF THE IMC-PID TUNING METHOD

1 1 + 25s

(9)

as IMC filter. The rationale for choosing Q(s) is to invert the minimum-phase part of the model and add the poles required for properness at a frequency slightly greater than W M, while the time constant of F(s) is set below it. The magnitude plot of the so obtained regulator is depicted in figure 2(b). In the same plot are the results achieved with the formul~ (5). Moreover, the regulator obtained with the IMC procedure (Morari and

(3)

306

Model ideutiflCOliou

'"

DD

~"

«J

0.5

o

(b

Bl ...... IYC·PlD. 1=10

1.5

o

and noise insensitivity. It can be seen that A = 30 produces a slow loop, while with A = 10 the effects of noise are hardly acceptable. Moreover, with A = 10 the disturbance rejection is better due just to a quicker control response, while the settling remains slow as can be seen in figure 2(d). Finally, the set point responses with the two values of A are very similar - see figure 2(c) despite the parameter is expected to act on the response time significantly. To summarize, the IMCbased tuning of PID regulators can be improved significantly by abandoning the exclusive use of FOPDT models. This could be done by using alternative rules for different model structures, or by adopting nonparametric models. However, the former solution is difficult to automate because it requires structure selection capability, while the latter is conceptually neat but potentially very complex. In most industrial devices, especially in low-end ones, none of these solutions is applicable with acceptable effort and cost. Therefore, in the following a less powerful, yet much simpler alternative is presented. This consists of merely adopting another model structure instead of the FOPDT one. The structure is chosen based on a systematic analysis of the shortcomings of the former when used for IMC-based PID tuning. The other two mentioned solutions are at present being studied, and will be dealt with in future works.

Cou....l.... maguilll
2.5 Compla model 2

IMC

'MC.P'

(a)

:m «I] (DJ BQ) Closed·1oop sel poWI .....,.....

\

10-4

1(0)

.

.. .

""C·PlO.la'"

~'~ !~~~~~ ~ - ••.•..•

---

...

._~_:;:-:::::. 10·

100

Closed-loop Iced disllllbauce l~llpOII" 31.5

r'-=-=-::---,.,..-------,

IMC, PI.pr

IMC.PIO.,....

(c)

(d 100

2IlJ

]])

Cl)

Fig. 2. Introductory experimental example. Zafiriou, 1989) and Q(s), F(s) given by (8) and (9) is approximated with a PI (K = 40, T i = 40) and with a PID (K = 40, T i = 40, T d = 5, N = 2). In the latter case, attention is paid not to obtain an excessive high frequency gain. Also the magnitude plots of these two approximated IMC regulators are in figure 2(b). Figure 2(c-f) shows how all these regulators behave in presence of a set point and of a load disturbance step of amplitude 2°C and 35% respectively. Apparently, the PI and PID regulators obtained by approximating the results of the IMC procedure applied to an accurate model are better. Some considerations can now be made. First, the model (6) does not represent the process very precisely except for the dominant dynamics. Note that one may think to blame this on the method of areas, but a closer look at figure 2(a) immediately reveals that no reasonable identification method for FOPDT models would ever produce a time constant smaller than about lOOs. Therefore, the mismatch is yielded by the FOPDT structure intrinsically. Second, even if the dominant dynamics are caught, model mismatch caused by structural constraints can have highly undesirable effects. Asking the model to represent only the dominant dynamics and to have no zeros quite often causes the poles of its rational part to be at a lower frequency than needed. With a cancellationoriented approach like the IMC also the regulator zero(s) will be too slow, leading to sluggish load disturbance transients. But these are not due to cancellation per se, rather to canceling too crude an approximation of the process dynamics, as witnessed by the better behaviour of the PI and PID tuned by approximating the IMC results with model (7). Third, model mismatch adversely affects the significance of the design parameter A. In fact, once T i is stuck around the (too large) FOPDT model's time constant, in the IMC-PID formulre (5) A acts only on the regulator gain. This means that only the gain is used for trading stability and disturbance rejection versus performance

4. MAJOR SHORTCOMINGS OF THE FOPDT MODEL STRUCTURE In PID tuning, the process model's structure can be selected based on convenient features of the process response. IT a step response is employed (quite a reasonable stance and a frequently adopted solution), and limiting the scope to asymptotically stable processes, a set of such characteristics should include at least the presence of delay, overshoots, undershoots, and oscillations (Leva and Piroddi, 1996). There is not the space here to discuss in general the shortcomings of adopting the FOPDT structure a priori. Therefore, just a wide enough gallery of typical situations is reported. To show that structural inaccuracy also makes the identification method critical, two methods have been used in the presented tests: one is the well known method of areas in the 'standard' form, the other is the same method 'modified' as illustrated in figure 3, to improve (in a necessarily conservative manner) the results in the case of undershooting, overshooting or oscillatory responses. Figure 4 reports the identification results, in the time and frequency domain. Numbers are inessential at this level, but it is very easy to notice that in the various cases the FOPDT structure can give rise to very different frequency distributions of the model error, so that

307

Slip "SpOnsl poft) and Bodo pbls (ri!tlt) of process (dashld) Ind FOPDT modol (sdid)

90p response (lot) and Bodl plots (rilhl) d pro""s. (dashed) ,.,d FOPOT modll (soid)

f I'---_·_·.···,--' .. .j I'...' ···r I! I f I ..--1 ~ 1! . I f I --::=sJ bV ··..···.

["...

(b) l.Jlosely damped pro""s., shndanl method

P

(c) l.Jlosely damped pro""s., modified mlthod

L,· ..--"..

I

(d) OverstooDtllg process, standard method

fI :sd I! :::=;J I fI .:-s3 I! I rI ---- J I! '~ I f tI_~ ~

(f) UnderstooDtng ,roeas., standard m9thod

~

(g) UndolShooli~ process, modified method

i~

__. I

fI

······.1 V-"--'o\."shooti~

i 1-----·-·.-.~..-..-.-..-.1

I rl .~ :==~;;~;::=~. 11 ~ '..... --1

~-...

I

~

(e) OmstooDtllg process, rrodifoed method

~

(h) Undo> and onnholtong proc, std method

[tJ Und.,· and

proe, mod methOd!

···sa

~

r[

u

::_1. -.:::>~ ";.-~

"I Process with two RH" z.ros

Fig. 4. Structural shortcomings of the FOPDT model. •.'. Measured in

two facts. First, it is advisable to estimate the delay prior to the other parameters, for example with a thresholding mechanism applied to the response. This eases the subsequent identification of the rational part of (10). Then, (10) may be overparameterized for simple processes, but apparently (with a slight abuse of notation to lighten the discussion) "not too much". This means that virtually any algorithm can be employed for parameter identification, as the improvement with respect to the IMC-PID method is expected to come from the structure adopted, not from the identification. In fact, the identification method has proven not to be critical, in the sense that different methods produce similar model parameterizations and, above all, almost equal regulators. Here, simple LS ISE minimization was used, initializing the parameter estimator with a first order model obtained with the method of areas.

Measured

".<"","response '.

.....-.-...~......

Fig. 3. Modified method of areas. any tuning procedure has to be quite conservative in order to achieve closed-loop stability, 5. THE PROPOSED MODEL STRUCTURE AND TUNING FORMULAE The basic idea of the proposed method is to determine a complete IMC regulator, and then approximate this with a PID. In the experimental example presented, the model structure was selected without complexity bounds and its ability to describe the measured data was quantified by means of a spectral analysis. This has given very good results, but requires quite a bit of 'theoretical' skill and is almost impossible to automate. To devise an applicable method, the model structure cannot be completely free and there must be no need for an online spectral analysis. In one word, then, the structure of the identified model has to overcome the problems stated above, with the minimum number of parameters. On the basis of this intuitive reasoning, the structure employed is Mid () S

l+blS+b2S2 -sL = /.l 1 + alS + a2S 2 + a3s3 e ,

For tuning, first replace the model delay with its (1,1) Pade approximation (1 - s£/2)/(1 + sL/2), which leads to a rational model in the form M(s)

= /.l Mt(s)M;V(s) MD(S)

(11)

where MD(S) is Hurwitz, MD(O) = Mt(O) = MN(O) = 1, the roots of Mt(s) lie in the LHP, and those of MN(s) in the RHP. Then, make the empirical assumption that M(s) describes the process satisfactorily up to 'a bit more' (say twice) the maximum of the frequencies of its poles and zeros-quite reasonable, given that the model structure can fit the response better than any FOPDT model. This is of course a simplification, but avoids any spectral analysis or complex model evaluation. Note that in the case at hand the computation of Mt(s) and MN(s) is immediate. To obtain Q(s), set

(10)

that apparently can reproduce overdamped, oscillatory, overshooting and undershooting responses. The model is estimated from an open-loop step response. For space limitations, on the parameter identification phase suffice here to point out

308

Q(s)

=

MV(s) p.M"f{(s)Q'v(s)

Cn(s) = F(s)Q(s) being the nominal control sensitivity function. If the FOPDT model used in the IMC-PID method is a low-frequency approximation of (10), and if the same IMC filter is used, it turns out that the nominal control sensitivity functions of the proposed method and of the IMCPID are almost equal in the control band, their difference vanishing for W -+ o. However, in the proposed method the model error is computed with respect to (10), while in the IMC-PID it is computed with respect to the FOPDT model. Even from a so quickly sketched analysis, then, it is intuitive that the proposed method is more tolerant to model error than the IMC-PID, thus it takes profit of the increased model complexity also from the standpoint of robustness. This fact is confirmed by experience, as shown also in the example of the following section, but opens another problem. In the FOPDT case, there is the possibility of selecting parameter .x in the IMC filter based on quantitative model error information (Leva and Colombo, 2001a). This is more complex in the proposed method, so that precise clues for a case-specific selection of kw are not yet available. The problem is being studied.

(12)

where (13)

being such that all the poles and zeros of M (s) lie inside the circle centered in the origin and of radius wQ if M(s) is minimum-phase, or such that all the RHP zeros of M (s) lie outside the circle centered in the origin and of radius wQ in the opposite case. Notice that the degrees of MD(s) and Mj;(s) are at most 3 or 4 (but in the latter case there is a known factor 1 + sL/2) and 2, respectively, so that the frequencies and damping factors of their roots can be computed explicitly. This guarantees that Q(s) be an acceptable approximation of the inverse of the minimum-phase part of M(s) up to the reasonably achievable control bandwidth, accounting for non-minimum phase dynamics when required. The integer v is selected based on the degrees of MD (s) and Mj; (s) so that the relative degree of Q(s) be zero. Adopting the standard choice F(s) = 1/(1 + S/WF), the IMC regulator turns out to be

WQ

R(s)

=

Mv(s) (14) IlM"f{(s) (Q'v(s)(1 + S/WF) - MN(s))

6. EXPERIMENTAL RESULTS

where WF is set to kwwQ, kw E (0,1) being the design parameter of the proposed tuning method. The rationale of kw is to limit the bandwidth request, based on the model accuracy and/or non minimum-phase part; a value in the range 0.2-0.5 is advisable on the basis of experience, but this is just a first-cut clue.

The presented example refers to the same experimental setup employed for the introductory one, but in a different operating condition. Figure 5(a) shows an open-loop step experiment and two identified models: one has the proposed structure, and is Ml(S)

1 + 192.4s

In all the cases considered, the obtained IMC regulator can be approximated with a PID very effectively up to the necessary bandwidth. For this approximation, an ad hoc numeric procedure is employed. There is not the space here for describing this procedure; suffice to say that the underlying idea is to preserve the low-frequency aspect of the regulator's frequency response, and its mid-frequency phase lead (when there is one). In any case, this approximation lies in the deeply studied domain of model reduction, and there is plenty of literature on the subject. The key point of the approach proposed herein is that, by adopting the model structure (10), the IMC method produces a regulator that can be approximated with a PID, and is better than the one obtained with a FOPDT model.

+ 5180.4s 2 + 1231Os 3 .

(16)

the other has the FOPDT structure, and is M2(S)

=

0.106e- 2 . 4 • 1 + 126s '

(17)

see figure 5(a). Taking Q(s)

=

2

3

1 + 192.4s + 5180.4s + 1231Os 0.106(1 + 81.5s + 28.2s2)(1 + 1.5s) ,

(18)

i.e., wQ = 1/1.5, and F(s) = 1/(1+6s), i.e., kw = 0.25, leads to the approximating PID regulator

4S) .

1 Rl(S)=32.5 ( 1 + - + - 32s 1 + 2s

(19)

Figure 5(c,f) reports the experimental closed-loop transients obtained with the proposed method (RI), and figure 5(b,e) those of the IMC-PID rules (Morari and Zafiriou, 1989) with the FOPDT model (R z ) and a requested closed-loop dominant time constant of lOs, i.e.,

To compare the robustness properties of the proposed method with those of the IMC-PID, recall that in the scheme of figure 1 robust stability is guaranteed (Doyle et al., 1992) for any additive model error fulfilling Iw(jw)Cn (jw)1 < 1 'iw,

= 0.106(1 + 81.5s + 28.2s 2 )e- O 75.

R2(S) = 96.8 ( 1

(15)

309

+ -1- + -1.2S) -- . 127s 1 + 1.23

(20)

(a) Open-loop unrt step resp.

IMC·PIO. FOPOT model

PlO. proposed method

33 (b) Set point and output re)

32 .------. 31 : time (5

:ll L -_ _

o

2llJ 400 EOO ID)

time (5

time (5

~_---,

2llJ 400 EOO ID)

o

2llJ 400 EOO ID)

(d) Nyquist plots 0.5

solid: proposed dashed: IMC·PIO

o

\

'{).5

:\

"-

·1 -1.5 -1.5

(e) Conlrol (%)

100

.~

1--'

1. -1

'{).5

0

0.5

o o

time (s

200 400 EOO ID)

o

o

time (s)

200 400 EOO ID)

Fig. 5. Experimental tuning results. sented to back up the proposed approach. The presented research is continuing, with two major goals. One is to further extend the method by employing a set of possible model structures, among which the choice is to be made on-line. The second, and more ambitious, is to employ nonparametric models, so as to minimize the information loss on the process dynamics.

o dB FulllMC

-10 _20L---_ _ -3]

~

~

/ /'--, / IMC-PID (FOPD1) • - ••:"'._""'-.__-:-: __-:-: __c::..~.~ __

_511l-10~

Proposed melhod

/

",:,::

_:_l_

..,., 10'

Fig. 6. Magnitude plots of

radlsec .....,...-----.:.:=,--' 10

u

lien.

The response to a set point step of amplitude 2°C and to a load disturbance step of amplitude 30% are shown. Note the apparent lack of integral action on the part of R2 , which is structural, since no sensible identification algorithm for FOPDT models, starting from the measured step response of figure 5(a), would ever produce a time constant less than lOOs or so. The integral time of RI is one third of that of R 2 , and the transients of figure 5 are self-explanatory.

8. REFERENCES Astrom, K.J. and T. Hagglund (1995). PID controllers: theory, design and tuning - second edition. Instrument Society of America. Research Triangle Park, NY. Doyle, J.C., B.A. Francis and A.R. Tannenbaum (1992). Feedback control theory. MacMillan. Basingstoke, UK. Ingimudarsson, A. and T. Hagglund (2001). Robust tuning procedures of dead-time compensating controllers. Contr. Eng. Practice 9, 1195-1208. Isaksson, A.J. and S.F. Graebe (1999). Analytical pid parameter expression for higher order systems. Automatica 35, 1121-1130. Leva, A. (2001). Model-based tuning: the very basics and some useful techniques. JournalA 42(3), 14-22. Leva, A. and A.M. Colombo (2001a). !mc-based synthesis of the feedback block of isa-pid regulators. Proc. ECC 2001 pp. 125-247. Leva, A. and A.M. Colombo (2001b). Implementation of a robust pid autotuner in a control design environment. Trans. Inst. of Measurement and Control 23(1), 1-20. Leva, A. and 1. Piroddi (1996). Model-specific autotuning of classical regulators: a neural approach to structural identification. Contr. Eng. Practice 4(10), 1381-1391. Morari, M. and E. Zafiriou (1989). Robust process control. Prentice-Hall. Upper Saddle River, NJ.

In figure 5(d) the two PIDs (19) and (20) are compared in the frequency domain, by reporting the open-loop Nyquist plots with the (accurate) model (16). Finally, figure 6 shows the magnitude plots of the inverse of the nominal control sensitivity in the various cases. The improved robustness of the proposed method is apparent, especially in the vicinity of the cutoff frequency.

7. CONCLUDING REMARKS An extension of the IMC-PID tuning method has been presented. A reasoned choice of the process model structure, and an empirical quantification of the bandwidth where this model is reliable from the synthesis standpoint, allow the IMC procedure to yield a regulator that can be effectively approximated with a real PID, and produces better results than the one obtained with a FOPDT model (the most commonly used structure in IMC-based PID tuning) both in terms of performance and robustness. Experimental results have been pre-

310