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The PI+p controller structure and its tuning
Journal of Process Control 19 (2009) 1451–1457
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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
The PI+p controller structure and its tuning Alberto Leva *, Martina Maggio Politecnico di Milano, Dipartimento di Elettronica e Informazione Via Ponzio 34/5, 20133 Milano, Italy
a r t i c l e
i n f o
Article history: Received 27 June 2008 Received in revised form 8 May 2009 Accepted 16 May 2009
Keywords: Autotuning PI control Process control
a b s t r a c t This manuscript is part of a long-term research, aimed at establishing methodologically grounded relationships between model- and relay-based tuning of industrial regulators, and at consequently deriving synthesis procedures that couple the advantages of model-based tuning to the simplicity and clarity of relay-based identification. In this work, the addressed controller structure is the ‘‘PI+p”, i.e., a PI augmented with an additional, stable pole. The advantages of using the combined model-relay-based approach on that structure are evidenced, by means of both simulation and experimental results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In the autotuning of industrial regulators, the importance of the model-based approach is witnessed by a vast literature, see e.g. [1]. Two are the main strengths of that approach. First, a model of the dynamics seen by the regulator is made available to forecast the tuning results. Second, the design parameters can be interpreted easily, since the control specifications can be expressed with reference to that model [15]. As recently shown, see e.g. [18], model-based tuning has also an inherent problem, however. The model structure is typically chosen a priori, and using complex ones is beneficial [8] but practically difficult; as a result, very simple model structures are typically adopted. In such a scenario, the particular model parametrisation method employed - the term ‘‘identification” would be questionable here, see e.g. [17] – has a very relevant influence on the tuning results: for example, it is possible to alter some rankings of modelbased methods by just changing the parametrisation procedure— see again [17]. Relay-based tuning, instead, has an inherent advantage: it is maybe the only framework where there is no ambiguity about how the process data (in the simplest case, one point of its frequency response) is obtained and used [27,18]. This is probably one of the main reasons for its success, and widespread use. Obviously, also relay-based tuning has some shortcomings. The most widely recognised is the limited information conveyed by a few points of an unknown Nyquist curve, as noted in works such as [19,22,24,21]. Another one, less frequently addressed, is that in the relay-based context the cutoff frequency is typically a result
of the relay experiment, although said experiment can be driven so as to achieve a prescribed cutoff [10]. Therefore, defining the control specifications is not always easy and intuitive [10,2]. The phase margin is frequently used as design parameter, for example, but it is sometimes difficult to relate a priori that parameter to the desired closed-loop behaviour, especially in a manner suitable for an industrially acceptable product. In [13], relay-based identification was used for model-based PI/ PID autotuning. Here, extending the preliminary results of [12], a similar approach is applied to a slightly different structure, namely the ‘‘PI+p”, composed of a PI regulator augmented with one additional, stable pole. The advantages of that structure (of course not proposed here for the first time, see e.g. [9]) are briefly evidenced, and then an autotuning procedure for it is devised along the approach formalised in [13]. 2. The PI+p structure Consider a PI regulator in the so-called 1-d.o.f. (one degree of freedom) ISA form, i.e., the transfer function from the error to the control signal expressed as
1 RPI ðsÞ ¼ K 1 þ : sT i
Irrespective of the tuning method adopted to synthesise (1), a relevant problem is exhibited by that structure. Denoting by C PI ðsÞ the corresponding control sensitivity function, defined as
C PI ðsÞ :¼ * Corresponding author. Tel.: +39 02 2399 3410; fax: +39 02 2399 3412. E-mail addresses: [email protected] (A. Leva), [email protected] (M. Maggio). 0959-1524/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2009.05.007
ð1Þ
RPI ðsÞ 1 þ RPI ðsÞPðsÞ
ð2Þ
where the transfer function PðsÞ represents the process under control, and assuming (as is standard in autotuning) that the process
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(model) is strictly proper, the limit value of the magnitude of the frequency response C PI ðjxÞ for the frequency x going to infinity turns out to be
lim jC PI ðjxÞj ¼ K:
x!1
ð3Þ
Hence, with (1) the control sensitivity magnitude does not roll off at high frequencies, potentially leading to a control signal that is very sensitive to measurement noise, which in turn can easily cause unnecessary actuator wear. In many applications this is a significant problem, especially when the actuator has moving mechanical parts—a frequent case for example in process control. The PI+p regulator structure, conversely, is defined as
RPIþp ðsÞ ¼
K 1 : 1þ 1 þ sT p sT i
ð4Þ
with T i ; T p > 0, and leads to a control sensitivity function frequency response the magnitude of which rolls off for x ! 1. A first reason to employ the PI+p regulator structure in all the situations where a PI is advised (see e.g. [5,4] and many other works for extensive explanations of the statement ‘‘a PI is advised”) concerns exactly the control sensitivity frequency response magnitude. Most model-based tuning methods operate, more or less implicitly, by cancellation. In that vast category of techniques, the integral time is set to a value near to the dominant time constant observed for the dynamics to be controlled, and the regulator gain is used to trade response speed against stability degree, or to privilege one of them, or with similar criteria. In so doing, it is quite easy to end up with a high control sensitivity magnitude at frequencies where typical measurement noise exhibit significant signal power, to the apparent detriment of control signal moderation. Given that the model structure is decided a priori and ‘‘simple”, it is obviously necessary to concentrate its (limited) descriptive capabilities on the control-relevant dynamics. However, what is ‘‘control-relevant” depends on both the process and the control problem. That is the (obvious) reason why, for example, tuning a PI by cancellation results in a good tracking performance, while if load disturbance rejection is the issue, smaller integral times than the dominant process time constant are normally preferable [23,26,11]. If both the model structure and the tuning policy (to stick to the example, think again to a cancellation one) are fixed, it is then required that the model parametrisation phase be driven to capture not the dominant dynamics, but the process behaviour at slightly higher frequencies than those dynamics. Doing so means abandoning the idea that the fidelity of the model to the data used for its parametrisation is of value per se, and treating the model parametrisation and the regulator tuning jointly. Many attempts to do so can be found in the literature, for example under the well known title of ‘‘identification for control” [7]. However, the results of such neat a theory can often prove of scarce usefulness in autotuning, since in that field there is hardly any room for any ‘‘experiment design”, and the stimuli that can be realistically used, given potentially severe technological limitations, make lack of excitation an ubiquitous problem. In addition, to obtain a process model that can be reliably used to forecast the closed-loop transients, it is required that the model be precise near the cutoff frequency, which however is a product of the tuning. Indeed, this is another reason why the model parametrisation and the regulator tuning should be contextual. The use of relay-based identification done herein is an attempt to solve the problem sketched above, possibly at a cost of a slightly increased duration of the model parametrisation phase. If one uses that model parametrisation technique in conjunction with the regulator tuning as done in this work, then the obtained model is inherently ‘‘exact” at the nominal cutoff frequency, which is not in general guaranteed with other methods, like for example those
based on the step response. As a by-product, and consistently with the remarks above, in the case of a FOPDT (First Order Plus Dead Time) model, the time constant (thus the PI integral time if a cancellation policy is adopted) is not stuck to the dominant open-loop process time constant, which can be too high e.g. for an effective disturbance rejection. Finally, a model ‘‘precise at the cutoff’ makes it simpler to extend PI-based tuning rules to the PI+p structure, avoiding the unpleasant effects that may arise otherwise (as shown in the examples reported later on). A second reason for using a PI+p is that the nominal control sensitivity magnitude (i.e., the one computed with the tuned regulator and the model used for the tuning) provides a frequency domain overbound for the magnitude of the additive model error [6,14– 16], thus being very useful in the autotuning context at large. Apparently, a control sensitivity that rolls off as sharply as possible after the cutoff leads to more robust a system in the presence of unmodelled dynamics above the cutoff, which a very frequent situation in autotuning. Also the robustness issue is very relevant in model-based tuning, as witnessed by many works such as the recent example [25]; here too, the proposed approach can yield some advantages, as will be shown in the examples. Notice that the additional pole of the PI+p with respect to the PI apparently results in sharper a magnitude roll-off above the cutoff frequency also for the complementary sensitivity function, hence improving robustness also in the presence of model errors and/or perturbations that are more naturally described in a multiplicative manner. 2.1. A tuning method Consider the PI+p regulator (4), and assume that a FOPDT process model
MðsÞ :¼ lM
esLM 1 þ sT M
ð5Þ
is available, where T M > 0; LM P 0, and for simplicity (but without loss of generality) in the following it is also assumed lM > 0. Supposing to tune the regulator zero by cancellation, i.e., to set T i ¼ T M , (4) and (5) allow to define a nominal open-loop transfer function as
Ln;PIþp ðsÞ ¼ RPIþp ðsÞMðsÞ ¼
K lM esLM : sT M ð1 þ sT p Þ
ð6Þ
It is now sensible to have the cutoff frequency lie in the range where the regulator frequency response is decaying with slope 1 before its zero with corner frequency 1=T M , and to place the pole 1=T p at a frequency higher than the cutoff, that in the previously introduced hypothesis equals K lM =T M . In other words, a reasonable aspect of the involved frequency responses is that depicted in Fig. 1 where, with reference to (4),
Tp ¼
aT M K lM
ð7Þ
and a is a design parameter in the range 0–1, used to specify the frequency distance between the (nominal) cutoff frequency and that of the additional PI+p pole. If the above choice is taken, it is correspondingly possible to compute a nominal cutoff frequency xcn and a nominal phase margin umn , recalling (7), as
K lM ; TM p K lM L M p K lM LM ¼ arctan xcn T p ¼ arctan a: 2 TM 2 TM
xcn ¼ umn
ð8Þ
Now, bring in the relay-based identification as suggested in b xox Þ [13]. Suppose that a relay test has provided an estimate Pðj of the point of the process frequency response corresponding to
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0 < a < 1, since it would make little sense to put the second PI+p pole below the cutoff. The necessity of having a nonnegative model delay LM therefore turns into
a < tan
p 2
um
ð12Þ
which constrains the required stability degree ðum Þ once the highfrequency control sensitivity roll-off ðaÞ is decided, or vice versa. Then, in the hypothesis of an asymptotically stable process model implicitly assumed here (and most frequent in PI autotuning), T M has to be positive, whence
tan ðxox LM þ uox Þ < 0:
Fig. 1. Rationale of the proposed PI+p tuning method summarised by means of Bode magnitude diagrams.
the frequency xox of the permanent oscillation induced in the test, i.e., assume the availability of
b xox Þ ¼ Pox ejuox : Pðj
ð9Þ
It is now straightforward to write an equation system involving b xox Þ, the PI+p regulator (4), and the FOPDT process model (5), so Pðj that The nominal cutoff frequency xcn equals xox , b xox Þ, i.e., ‘‘the The frequency response of the model contain Pðj model be exact at the (nominal, of course) cutoff frequency”, And the nominal phase margin umn equals a desired value um . Doing so, the system
8 K lM ¼ xox > > > TM > > < p K lM LM arctan a ¼ um 2 TM lM > ffi ¼ P ox > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 1þðxox T M Þ2 > : arctanðxox T M Þ xox LM ¼ uox
ð10Þ
is readily written. Now, if (10) is solved for ðlM ; T M ; LM ; KÞ, one simultaneously obtains The PI+p tuning via the solution of system (10), Eq. (7), and the cancellation relationship T i ¼ T M , And a process model (nominally) exact at the cutoff frequency, useful to simulate the closed-loop system to assess the tuning results. The computations sketched above yield the PI+p (and nominal process model) tuning rules
1 p 1 tan ðxox LM þ uox Þ; um arctan a ; T M ¼ xox 2 xox qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TM aT M ¼ Pox 1 þ ðxox T M Þ2 ; K ¼ xox ; T ¼ TM; Tp ¼ ; lM i K lM
LM ¼
lM
ð11Þ to be used in sequence, having the couple ðum ; aÞ as design parameters. The formul(11) are apparently subject to some validity limits. For example, ðum ; aÞ must be bound to produce a nonnegative LM , the regulator phase at the cutoff frequency must be negative, or an inconsistent tuning arises, and so on. All the necessary checks are quite immediate to do, rejecting as a consequence possibly incorrect choices of the design parameters. Omitting full details for space limitations, a few words on the matter are however in order. First, recalling Fig. 1, it is obvious that
ð13Þ
To ensure that a priori, one can simply drive the relay experiment so that uox > um p, i.e., the regulator phase at the cutoff be negative, yielding a positive integral time, as is done in the example procedure presented in the following. Some other checks may be required, for example to avoid that the phase lag introduced by the regulator at the cutoff frequency be too small (resulting in a very large sensitivity of T i to the required phase margin), that the gain has the proper sign if the hypothesis lM > 0 is released (as would be necessary in practical realisations, and was not done here for brevity), and so forth. None of those practical necessities is difficult to manage, however, as the previous considerations should have clarified. A simple and effective tuning procedure is obtained with the b xox Þ the point of the process proposed approach by taking as Pðj frequency response with phase equal to 90°, that is easily found by closing the feedback loop with the cascade of a relay with no hysteresis (or, better, with so small a hysteresis to avoid spurious toggles, yet still allowing to take the real negative semiaxis as precise enough an approximation of its critical points’ locus) and an integrator. Doing so is common practice, see e.g. [27] for extensive discussions and explanations. The above procedure is tested in the following (Section 3). The proposed tuning method is very simple to implement, and computationally light. It can therefore be ported also to low-end industrial regulators, considering that in many of them the functionalities required for relay tests are already present. The sensitivity of the control results to the typical errors introduced by relaybased identification procedures when finding the required process frequency response points are tolerated by the proposed procedure much like they are by any other relay-based method in the literature (there is not the space here to document that with experiments, since a lot of tests are involved in such a check). As for the use, the method has only two parameters. The meaning of the required phase margin is obvious, and that parameter could also be fixed to some reasonable default (say 50°–70°) in industrial implementation. Also a, that of course has to be in the range (0–1), is quite simple to understand. An experienced user could think of it as a way to have the PI approach an integral controller as much as allowed, since a larger a moves the PI+p pole toward the zero, but the position of that pole is constrained to be above the (nominal) cutoff frequency, thus. In other words, such a user could think of a as a means to take profit of the phase lead introduced by the zero, but at the same time to trade that lead against high-frequency control sensitivity. A less educated operator could simply be told to choose higher values of a if the measurement of the controlled variable is very noisy. One could also think to choose two or three default values, based on a qualitative estimation of that noise (say for example ‘‘negligible”, ‘‘average”, or ‘‘high”). The only possible problem is numeric, and could arise if the chosen a puts the pole with time constant T p at so high a frequency to introduce possible representation errors in the discrete time domain, assuming – as usual – that the implementation of the regu-
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lator is digital. However, since the machine precision is known, it is straightforward to detect such problems, limiting as a consequence the admissible values for a. 3. Simulation examples Suppose to tune the PI (1) with a model-based method, taking the IMC-PI formul[20] as a representative of that class of tuning methods—a choice that is easily justified by their widespread use. Suppose also, to get rid a priori of possible effects of the identification procedure, that the tuning model is exact. If [16] are used on (5), the resulting PI has
K¼
TM
lM ðLM þ kÞ
;
Ti ¼ TM;
ð14Þ
where k is the method’s design parameter, and can be interpreted as the (dominant) desired closed-loop time constant [3,15,16]. To set up a meaningful comparison with the relay-based procedure devised herein, one can take a ‘‘reasonable default” for k as the inverse of the frequency where the phase of the process frequency response is more or less 90°. The tuning results then depend on the process (model) normalised delay, defined here as
hM ¼
LM : LM þ T M
ð15Þ
and obviously ranging from 0 to 1. Normalising for convenience the treatise to the case with T M ¼ 1 (the generalisation is just a matter of frequency rescaling), k is readily obtained from
hM 1 p þ arctg ¼ : k 2 kð1 hM Þ
ð16Þ
Solving (16) numerically is straightforward, and computing the K=lM ratio based on (14) leads to the results of Fig. 2, that relate the high-frequency value of the control sensitivity to the process normalised delay. From Fig. 2, it can be observed that for high-order processes, which lead to high normalised delays when described by FOPDT models, the use of cancellation-based methods often results in large values of the high-frequency control sensitivity. Some modifications to PI tuning rules were introduced to mitigate that effect, see e.g. [21], for example choosing an integral time slightly smaller than the model time constant and adapting the PI gain in accordance, but the fact that the control sensitivity does not roll off at high frequency is structural. In the following examples, the results of (14) and (11) are therefore compared, using for the latter the point of the process frequency response with phase -90° found by a relay test with cascaded integrator, and choosing k in the former as discussed above. The first example is made with the process described by
PðsÞ ¼
1 ð1 þ sÞ3
ð17Þ
Fig. 2. Relationship between log 10 ðK=lM Þ and hM in IMC-PI tuning, assuming arg ðRPI ðj=kÞPðj=kÞÞ ¼ 90 .
that is often taken as a benchmark for PI autotuners. Fig. 3 reports the results. The left plot shows the inverse of the nominal control sensitivity function with a PI+p tuned with the proposed procedure and with a PI tuned with the IMC-PI rules. The right plot shows the response of the controlled variable yðtÞ to a unit load disturbance step (a) with the PI+p tuned by the proposed procedure and the real process, (b) with the PI+p tuned by the proposed procedure and as forecast with the FOPDT model, (c) with the PI obtained by means of the IMC-PI rules and the real process, and (d) with the PI obtained by means of the IMC-PI rules and as forecast with the FOPDT model. The proposed procedure was here applied with um ¼ 70 and a ¼ 0:2. As far as the load disturbance responses in the time domain are considered, the behaviour of the two control systems is more or less identical. Moreover, in both cases the tuning results forecast with the FOPDT model obtained together with the regulator are sensible. Of course it would be unrealistic that with such a simple model the dynamics of the real control loop were reproduced exactly. Nevertheless, the major characteristics of the obtained transients (in particular, the peak deviation of the controlled variable from the set point and the transient duration) are caught precisely enough. On the other hand, things are much different from the point of view of the control sensitivity function. Even if the proposed procedure results in a slightly more strict (just a few dB, however) overbound for the additive model error near the cutoff, as frequency increases the same overbound becomes very soon much looser than with the IMC-PI rules. A second (and deliberately extreme) example shows the advantages of the PI+p structure and the proposed tuning method in terms of control sensitivity roll-off. In this example a PI is tuned on the nominal process
Pnom ðsÞ ¼
1 ð1 þ 60sÞð1 þ 10sÞ
ð18Þ
that is then subject to the severe additive perturbation
PD ðsÞ ¼
s 1 þ 0:48
1 3 s2 1 þ 2 0:24 s þ 0:6 2 0:6
ð19Þ
The proposed method, applied to Pnom with um ¼ 60 and a ¼ 0:2, produces lM ¼ 1:11; T M ¼ 73:81; LM ¼ 8:15; K ¼ 2:65; T i ¼ 73:81; k ¼ 24:97. Fig. 4 reports the simulation results, where the tuned PI+p is compared to a two PIs obtained from the SIMC rules [23]: the first of those PIs was tuned with the same FOPDT model and k used for the PI+p, the second was synthesised with a FOPDT model obtained (always from the Pnom step response) with the method of areas (yielding lM ¼ 0; 98; T M ¼ 56:77; LM ¼ 10:34) and k selected with the widely used rule k ¼ maxð0:25LM ; 0:2 T M Þ, resulting here in k ¼ 11:35. The considered transient is the closed-loop response of the controlled variable yðtÞ to a unit load disturbance step—a purpose for which the SIMC is particularly suited. Briefly, Fig. 4a–b shows how relevant the perturbation is. Moreover, it can be seen that the inverse nominal control sensitivities of the PI+p and the SIMC PID tuned with the same model are similar in the vicinity of the cutoff, while the PI+p allows for a larger model error at high frequency (Fig. 4c). Correspondingly, the PI+p obtains (slightly) better results than the SIMC PI with the same model (Fig. 4d, top and middle plot), and the tuning results of both are caught quite well by the model forecasts. Conversely, the SIMC PI tuned with the method of areas performs well in the nominal case, but produces an unstable loop in the perturbed one (Fig. 4d, bottom plot). Recall that the example is extreme, and in no sense a criticism to the SIMC: the purpose was to show that the PI+p, if tuned with a proper procedure, can achieve more or less the same results of a very good PI.
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Fig. 3. Results of simulation example 1.
Fig. 4. Results of simulation example 2.
Incidentally, examining also the magnitude of the nominal complementary sensitivity function’s frequency response (omitted here for brevity) reveals that in this case - and in most cases of practical interest, by the way – the PI+p structure provides looser an overbound also for the multiplicative model error, which suggests that the robustness improvement given by said control scheme is quite general with respect to the type of model perturbation. The third example focuses specifically on the tuning method, and shows that it actually allows to overcome the load disturbance response sluggishness typical of cancellation-based policies. A lag dominated process is considered, namely
PðsÞ ¼
e0:1s 1 þ 10s
ð20Þ
and three PIs are tuned by setting k to 10, 5, and 1. Then, three relay experiments are made as explained in [10], driving the oscillation frequency to match the three corresponding desired cutoff frequencies (1/10, 1/5 and 1), and the so measured points are used to tune three PI+p controllers with the proposed method, um ¼ 70 , and a ¼ 0:2. All the IMC PIs have an integral time of 9.9 s, while the three PI+p controllers have T i equal, respectively, to 5.5, 5.4, and
3.3 s. The effect on the load disturbance responses of the controlled variable, depicted on the left in Fig. 5, is self-explanatory. A fourth example is reported to deal with a possible - and reasonable - objection, centred again on the tuning method. One may in fact think to extend PI tuning techniques to the PI+p structure not by using the proposed approach, but by tuning the PI part with a standard procedure (say again the IMC) selecting the design parameter ðkÞ for robustness, and then introduce the additional pole by simply setting it to a ‘‘high” frequency with respect to the cutoff. Incidentally, doing so in comparative tests with the method proposed here seems to yield even more fair a comparison, since the two regulators have exactly the same structure. The process considered, for a meaningful test, allows the FOPDT structure to evidence its limitations owing to a (not excessive) overshooting behaviour, and is
PðsÞ ¼
1 þ 3s ð1 þ sÞ4
:
ð21Þ
The proposed PI+p tuning method is used with the -90° point,
um ¼ 50 , and a set to 0.01, 0.05, 0.10, and 0.15 (recall that the rationale is to put the additional pole at frequency xcn =a). Then,
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Fig. 5. Results of simulation examples 3 (left) and 4 (right).
a FOPDT model for the process is found with the method of areas, and four IMC PIs are tuned with (14) and augmented with a pole at frequency xcn =a, and k selected so that the magnitude of the frequency response of the inverse of the nominal control sensitivity function be greater than the additive model error (which in this case is known, since the process is known) at all frequencies. Fig. 5, right column, shows the controlled variable’s responses to a unit step load disturbance (top) and the inverse of the nominal control sensitivities compared to the additive model error (bottom). As can be seen, even in an idealised case like this, the idea of introducing the additional PI+p pole a posteriori can have unpleasant effects, although stability is of course preserved. In the case of a process the dynamics of which cannot be represented ‘‘well” by a FOPDT model, example 4 has thus shown the advantages of the PI+p structure over an IMC PI with a second pole added a posteriori, essentially in terms of (actually achieved) stability degree, or equivalently of robustness. One could however further object that the PI+p advantages may be less clear in the case of a process that can be represented ‘‘well” by a FOPDT model. To briefly address such an issue, it is useful to reconsider example 3 above (Fig. 5, left column), and compare the tuned PI+p regulators with IMC PI ones augmented with a second pole at an angular frequency equal to kxa times the nominal cutoff frequency, that the IMC rules allow to estimate. Said comparison is shown in Fig. 6, that replicates the left column of Fig. 5
Fig. 6. Comparison with an augmented IMC PI with reference to example 3.
and has the PI+p results evidenced by a thick line: the values of kxa adopted, for each choice of k the IMC PIs, are 2, 5, 10, and 100, apparently covering any reasonable range. One can observe that in some cases the so obtained IMC PIs exhibit invariantly lower performance with respect to the PI+p, while in others the IMC performance may be either better or worse. It is therefore possible to conclude that, in the case of a ‘‘FOPDT-like” process, the PI+p regulator and the proposed tuning rule provide some objectiveness in the choice of the additional pole, which is not observed if said pole is introduced a posteriori, even based on the same data used to tune the rest of the regulator. Other examples on the advantages of the proposed scheme and tuning method are omitted for space reasons.
4. A laboratory application This section presents an application of the PI+p tuning procedure to a laboratory tank level control system, where the control variable is the command to an inlet volumetric pump, the controlled variable is the level, and a pulse-like output disturbance is realised by pouring into the tank the whole content of a half-litre bottle, which takes about six seconds—a short time with respect to the main dynamics of the system. Fig. 7, where the signals are expressed in volts in the range 0–10 as read and written by the A/D and D/A interfaces, shows the whole experiment. The system is initially led to a convenient steady state by a ‘‘best practice” PI controller, having a gain of 5 and an integral time of 20 s. Then, the presented PI+p tuning procedure is applied twice, the first time with a required phase margin of 70°, obtaining K ¼ 4:24; T i ¼ 37:27; T p ¼ 1:92 (Pi+p 1 in the figure), and the second time with 50°, resulting in K ¼ 3:79; T i ¼ 19:43; T p ¼ 1:63 (PI+p 2 in the figure); a was set to 0.15 in both cases. For each of the three regulators, the response to the disturbance described above and to a couple of set point steps is obtained. Some reference straight lines were added to Fig. 7, to help compare the results obtained by the various regulators. In synthesis, one can say that the proposed procedure obtains comparable results with respect to a well tuned PI, that the phase margin acts as design variable in a clear way (smaller phase margins, within a reasonable range, result in shorter but more overshooting and/ or slightly oscillatory transients), and above all that the advantages of the PI+p in terms of high-frequency control sensitivity are evident, since the noise spikes observed on the controlled variable are completely eliminated in PI+p control.
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Fig. 7. Application of the presented PI+p tuning method to a laboratory level control apparatus: set point and controlled variable (above), and control signal (below); pulselike output disturbances are applied at the instants marked with ‘‘Dist.”.
5. Conclusions The tuning of the ‘‘PI+p” controller structure was herein addressed. The PI+p is simply a PI augmented with an additional, stable pole. Its main advantage is that it can make the magnitude of the control sensitivity frequency response roll off after the cutoff frequency, therefore yielding a reduction of the control signal’s sensitivity to measurement noise, and also a more robust tuning in the presence of model errors and/or perturbations exerting their effect slightly above the cutoff frequency itself. After briefly discussing the advantage of the PI+p structure, a relay-based tuning procedure was derived for it, by coupling relaybased identification with a convenient use of model-based tuning relationships. In so doing, both the PI+p tuning was achieved, and a process model was obtained that is ‘‘precise around the cutoff”, i.e., not aimed at representing the open-loop process behaviour, but well suited to assess the main characteristics of the obtained closed-loop system. It is worth stressing that, thanks to the adopted approach coupling relay-based identification and model-based tuning, both the regulator and the model come from a single relay test. The purpose of the obtained model is specific, as said above, but there is no need, for example, to ensure that the identification and tuning procedure starts from a steady state condition—quite significant an advantage for the real-life use of an autotuner. Some implementation-related issues were discussed, and the interpretability of the involved design parameters was also considered. The presented procedure can be considered quite easy to implement, also on low-end devices, and to use, which is in favour of its acceptance in the application domain. Future work will be aimed at studying also the PID+p case – the meaning of the structure name is obvious – and possibly extending the idea to more complex regulators. References [1] K.J. Åström, T. Hägglund, PID Controllers: Theory, Design and Tuning, second ed., Instrument Society of America, Research Triangle Park, NY, 1995. [2] A. Besançon-Voda, H. Roux-Buisson, Another version of the relay feedback experimentm, Journal of Process Control 7 (4) (1997) 303–308. [3] R.D. Braatz, Internal model control, in: S. Levine (Ed.), The Control Handbook, CRC Press, Boca Raton, FL, 1996, pp. 215–224. [4] C. Brosilow, B. Joseph, Techniques of Model Based Control, Prentice Hall PTR, Indianapolis, IN, 2002.
[5] R.C. Dorf, H. Bishop, Modern Control Systems, Addison-Wesley, Reading, UK, 1995. [6] J.C. Doyle, B.A. Francis, A.R. Tannenbaum, Feedback Control Theory, MacMillan, Basingstoke, UK, 1992. [7] M. Gevers, Identification for control: from the early achievements to the revival of experiment design, European Journal of Control 11 (4–5) (2005) 335–352. [8] A.J. Isaksson, S.F. Graebe, Analytical PID parameter expression for higher order systems, Automatica 35 (1999) 1121–1130. [9] K. Ezra Kwok, M. Chong Ping, P. Li, A model-based augmented PID algorithm, Journal of Process Control 10 (1) (2000) 9–18. [10] A. Leva, PID autotuning algorithm based on relay feedback, IEE Proceedings-D 140 (5) (1993) 328–338. [11] A. Leva, Autotuning process controller with improved load disturbance rejection, Journal of Process Control 55 (2) (2005) 223–234. [12] A. Leva, Autotuning of PI+p controllers, in: Proc. CACSD08, San Antonio, TX, 2008. [13] A. Leva, L. Bascetta, F. Schiavo, Model-based proportional-integral/ proportional-integral-derivative (PI/PID) autotuning with fast relay identification, Industrial & Engineering Chemistry Research 45 (7) (2006) 4052–4062. [14] A. Leva, A.M. Colombo, Estimating model mismatch overbounds for the robust autotuning of industrial regulators, Automatica 36 (12) (2000) 1855– 1861. [15] A. Leva, A.M. Colombo, Implementation of a robust PID autotuner in a control design environment, Transactions of the Institut of Measurment and Control 23 (1) (2001) 1–20. [16] A. Leva, A.M. Colombo, On the IMC-based synthesis of the feedback block of ISA-PID regulators, Transactions of the Institute of Measurement and Control 26 (5) (2004) 417–440. [17] A. Leva, L. Piroddi, On the parametrisation of simple process models for the autotuning of industrial regulators, in: Proc. ACC 2007, New York, NY, 2007, pp. 125–247. [18] A. Leva, F. Schiavo, On the role of the process model in model-based autotuning, in: Proc. 16th IFAC World Congress, Praha, Czech Republic, 2005. [19] W.L. Luyben, Getting more information from relay feedback tests, Industrial & Engineering Chemistry Research 40 (20) (2001) 4391–4402. [20] M. Morari, E. Zafiriou, Robust Process Control, Prentice-Hall, Upper Saddle River, NJ, 1989. [21] A. O’Dwyer, Handbook of PI and PID Controller Tuning Rules, World Scientific Publishing, Singapore, 2003. [22] R.C. Panda, C.C. Yu, Analytical expressions for relay feed back responses, Journal of Process Control 13 (2003) 489–501. [23] S. Skogestad, Simple analytic rules for model reduction and PID controller tuning, Journal of Process Control 13 (4) (2003) 291–309. [24] T. Thyagarajan, C.C. Yu, Improved autotuning using the shape factor from relay feedback, Industrial & Engineering Chemistry Research 43 (2003) 4425– 4440. [25] R. Vilanova, IMC based robust PID design: tuning guidelines and automatic tuning, Journal of Process Control 18 (1) (2008) 61–70. [26] D. Vranc˘ic´, S. Strmc˘nik, J. Kocijan, Improving disturbance rejection of PI controllers by means of the magnitude optimum method, ISA Transactions 43 (1) (2004) 73–84. [27] C.C. Yu, Autotuning of PID Controllers: Relay Feedback Approach, SpringerVerlag, London, 1999.
Contents lists available at ScienceDirect
Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
The PI+p controller structure and its tuning Alberto Leva *, Martina Maggio Politecnico di Milano, Dipartimento di Elettronica e Informazione Via Ponzio 34/5, 20133 Milano, Italy
a r t i c l e
i n f o
Article history: Received 27 June 2008 Received in revised form 8 May 2009 Accepted 16 May 2009
Keywords: Autotuning PI control Process control
a b s t r a c t This manuscript is part of a long-term research, aimed at establishing methodologically grounded relationships between model- and relay-based tuning of industrial regulators, and at consequently deriving synthesis procedures that couple the advantages of model-based tuning to the simplicity and clarity of relay-based identification. In this work, the addressed controller structure is the ‘‘PI+p”, i.e., a PI augmented with an additional, stable pole. The advantages of using the combined model-relay-based approach on that structure are evidenced, by means of both simulation and experimental results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In the autotuning of industrial regulators, the importance of the model-based approach is witnessed by a vast literature, see e.g. [1]. Two are the main strengths of that approach. First, a model of the dynamics seen by the regulator is made available to forecast the tuning results. Second, the design parameters can be interpreted easily, since the control specifications can be expressed with reference to that model [15]. As recently shown, see e.g. [18], model-based tuning has also an inherent problem, however. The model structure is typically chosen a priori, and using complex ones is beneficial [8] but practically difficult; as a result, very simple model structures are typically adopted. In such a scenario, the particular model parametrisation method employed - the term ‘‘identification” would be questionable here, see e.g. [17] – has a very relevant influence on the tuning results: for example, it is possible to alter some rankings of modelbased methods by just changing the parametrisation procedure— see again [17]. Relay-based tuning, instead, has an inherent advantage: it is maybe the only framework where there is no ambiguity about how the process data (in the simplest case, one point of its frequency response) is obtained and used [27,18]. This is probably one of the main reasons for its success, and widespread use. Obviously, also relay-based tuning has some shortcomings. The most widely recognised is the limited information conveyed by a few points of an unknown Nyquist curve, as noted in works such as [19,22,24,21]. Another one, less frequently addressed, is that in the relay-based context the cutoff frequency is typically a result
of the relay experiment, although said experiment can be driven so as to achieve a prescribed cutoff [10]. Therefore, defining the control specifications is not always easy and intuitive [10,2]. The phase margin is frequently used as design parameter, for example, but it is sometimes difficult to relate a priori that parameter to the desired closed-loop behaviour, especially in a manner suitable for an industrially acceptable product. In [13], relay-based identification was used for model-based PI/ PID autotuning. Here, extending the preliminary results of [12], a similar approach is applied to a slightly different structure, namely the ‘‘PI+p”, composed of a PI regulator augmented with one additional, stable pole. The advantages of that structure (of course not proposed here for the first time, see e.g. [9]) are briefly evidenced, and then an autotuning procedure for it is devised along the approach formalised in [13]. 2. The PI+p structure Consider a PI regulator in the so-called 1-d.o.f. (one degree of freedom) ISA form, i.e., the transfer function from the error to the control signal expressed as
1 RPI ðsÞ ¼ K 1 þ : sT i
Irrespective of the tuning method adopted to synthesise (1), a relevant problem is exhibited by that structure. Denoting by C PI ðsÞ the corresponding control sensitivity function, defined as
C PI ðsÞ :¼ * Corresponding author. Tel.: +39 02 2399 3410; fax: +39 02 2399 3412. E-mail addresses: [email protected] (A. Leva), [email protected] (M. Maggio). 0959-1524/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2009.05.007
ð1Þ
RPI ðsÞ 1 þ RPI ðsÞPðsÞ
ð2Þ
where the transfer function PðsÞ represents the process under control, and assuming (as is standard in autotuning) that the process
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(model) is strictly proper, the limit value of the magnitude of the frequency response C PI ðjxÞ for the frequency x going to infinity turns out to be
lim jC PI ðjxÞj ¼ K:
x!1
ð3Þ
Hence, with (1) the control sensitivity magnitude does not roll off at high frequencies, potentially leading to a control signal that is very sensitive to measurement noise, which in turn can easily cause unnecessary actuator wear. In many applications this is a significant problem, especially when the actuator has moving mechanical parts—a frequent case for example in process control. The PI+p regulator structure, conversely, is defined as
RPIþp ðsÞ ¼
K 1 : 1þ 1 þ sT p sT i
ð4Þ
with T i ; T p > 0, and leads to a control sensitivity function frequency response the magnitude of which rolls off for x ! 1. A first reason to employ the PI+p regulator structure in all the situations where a PI is advised (see e.g. [5,4] and many other works for extensive explanations of the statement ‘‘a PI is advised”) concerns exactly the control sensitivity frequency response magnitude. Most model-based tuning methods operate, more or less implicitly, by cancellation. In that vast category of techniques, the integral time is set to a value near to the dominant time constant observed for the dynamics to be controlled, and the regulator gain is used to trade response speed against stability degree, or to privilege one of them, or with similar criteria. In so doing, it is quite easy to end up with a high control sensitivity magnitude at frequencies where typical measurement noise exhibit significant signal power, to the apparent detriment of control signal moderation. Given that the model structure is decided a priori and ‘‘simple”, it is obviously necessary to concentrate its (limited) descriptive capabilities on the control-relevant dynamics. However, what is ‘‘control-relevant” depends on both the process and the control problem. That is the (obvious) reason why, for example, tuning a PI by cancellation results in a good tracking performance, while if load disturbance rejection is the issue, smaller integral times than the dominant process time constant are normally preferable [23,26,11]. If both the model structure and the tuning policy (to stick to the example, think again to a cancellation one) are fixed, it is then required that the model parametrisation phase be driven to capture not the dominant dynamics, but the process behaviour at slightly higher frequencies than those dynamics. Doing so means abandoning the idea that the fidelity of the model to the data used for its parametrisation is of value per se, and treating the model parametrisation and the regulator tuning jointly. Many attempts to do so can be found in the literature, for example under the well known title of ‘‘identification for control” [7]. However, the results of such neat a theory can often prove of scarce usefulness in autotuning, since in that field there is hardly any room for any ‘‘experiment design”, and the stimuli that can be realistically used, given potentially severe technological limitations, make lack of excitation an ubiquitous problem. In addition, to obtain a process model that can be reliably used to forecast the closed-loop transients, it is required that the model be precise near the cutoff frequency, which however is a product of the tuning. Indeed, this is another reason why the model parametrisation and the regulator tuning should be contextual. The use of relay-based identification done herein is an attempt to solve the problem sketched above, possibly at a cost of a slightly increased duration of the model parametrisation phase. If one uses that model parametrisation technique in conjunction with the regulator tuning as done in this work, then the obtained model is inherently ‘‘exact” at the nominal cutoff frequency, which is not in general guaranteed with other methods, like for example those
based on the step response. As a by-product, and consistently with the remarks above, in the case of a FOPDT (First Order Plus Dead Time) model, the time constant (thus the PI integral time if a cancellation policy is adopted) is not stuck to the dominant open-loop process time constant, which can be too high e.g. for an effective disturbance rejection. Finally, a model ‘‘precise at the cutoff’ makes it simpler to extend PI-based tuning rules to the PI+p structure, avoiding the unpleasant effects that may arise otherwise (as shown in the examples reported later on). A second reason for using a PI+p is that the nominal control sensitivity magnitude (i.e., the one computed with the tuned regulator and the model used for the tuning) provides a frequency domain overbound for the magnitude of the additive model error [6,14– 16], thus being very useful in the autotuning context at large. Apparently, a control sensitivity that rolls off as sharply as possible after the cutoff leads to more robust a system in the presence of unmodelled dynamics above the cutoff, which a very frequent situation in autotuning. Also the robustness issue is very relevant in model-based tuning, as witnessed by many works such as the recent example [25]; here too, the proposed approach can yield some advantages, as will be shown in the examples. Notice that the additional pole of the PI+p with respect to the PI apparently results in sharper a magnitude roll-off above the cutoff frequency also for the complementary sensitivity function, hence improving robustness also in the presence of model errors and/or perturbations that are more naturally described in a multiplicative manner. 2.1. A tuning method Consider the PI+p regulator (4), and assume that a FOPDT process model
MðsÞ :¼ lM
esLM 1 þ sT M
ð5Þ
is available, where T M > 0; LM P 0, and for simplicity (but without loss of generality) in the following it is also assumed lM > 0. Supposing to tune the regulator zero by cancellation, i.e., to set T i ¼ T M , (4) and (5) allow to define a nominal open-loop transfer function as
Ln;PIþp ðsÞ ¼ RPIþp ðsÞMðsÞ ¼
K lM esLM : sT M ð1 þ sT p Þ
ð6Þ
It is now sensible to have the cutoff frequency lie in the range where the regulator frequency response is decaying with slope 1 before its zero with corner frequency 1=T M , and to place the pole 1=T p at a frequency higher than the cutoff, that in the previously introduced hypothesis equals K lM =T M . In other words, a reasonable aspect of the involved frequency responses is that depicted in Fig. 1 where, with reference to (4),
Tp ¼
aT M K lM
ð7Þ
and a is a design parameter in the range 0–1, used to specify the frequency distance between the (nominal) cutoff frequency and that of the additional PI+p pole. If the above choice is taken, it is correspondingly possible to compute a nominal cutoff frequency xcn and a nominal phase margin umn , recalling (7), as
K lM ; TM p K lM L M p K lM LM ¼ arctan xcn T p ¼ arctan a: 2 TM 2 TM
xcn ¼ umn
ð8Þ
Now, bring in the relay-based identification as suggested in b xox Þ [13]. Suppose that a relay test has provided an estimate Pðj of the point of the process frequency response corresponding to
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0 < a < 1, since it would make little sense to put the second PI+p pole below the cutoff. The necessity of having a nonnegative model delay LM therefore turns into
a < tan
p 2
um
ð12Þ
which constrains the required stability degree ðum Þ once the highfrequency control sensitivity roll-off ðaÞ is decided, or vice versa. Then, in the hypothesis of an asymptotically stable process model implicitly assumed here (and most frequent in PI autotuning), T M has to be positive, whence
tan ðxox LM þ uox Þ < 0:
Fig. 1. Rationale of the proposed PI+p tuning method summarised by means of Bode magnitude diagrams.
the frequency xox of the permanent oscillation induced in the test, i.e., assume the availability of
b xox Þ ¼ Pox ejuox : Pðj
ð9Þ
It is now straightforward to write an equation system involving b xox Þ, the PI+p regulator (4), and the FOPDT process model (5), so Pðj that The nominal cutoff frequency xcn equals xox , b xox Þ, i.e., ‘‘the The frequency response of the model contain Pðj model be exact at the (nominal, of course) cutoff frequency”, And the nominal phase margin umn equals a desired value um . Doing so, the system
8 K lM ¼ xox > > > TM > > < p K lM LM arctan a ¼ um 2 TM lM > ffi ¼ P ox > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 1þðxox T M Þ2 > : arctanðxox T M Þ xox LM ¼ uox
ð10Þ
is readily written. Now, if (10) is solved for ðlM ; T M ; LM ; KÞ, one simultaneously obtains The PI+p tuning via the solution of system (10), Eq. (7), and the cancellation relationship T i ¼ T M , And a process model (nominally) exact at the cutoff frequency, useful to simulate the closed-loop system to assess the tuning results. The computations sketched above yield the PI+p (and nominal process model) tuning rules
1 p 1 tan ðxox LM þ uox Þ; um arctan a ; T M ¼ xox 2 xox qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TM aT M ¼ Pox 1 þ ðxox T M Þ2 ; K ¼ xox ; T ¼ TM; Tp ¼ ; lM i K lM
LM ¼
lM
ð11Þ to be used in sequence, having the couple ðum ; aÞ as design parameters. The formul(11) are apparently subject to some validity limits. For example, ðum ; aÞ must be bound to produce a nonnegative LM , the regulator phase at the cutoff frequency must be negative, or an inconsistent tuning arises, and so on. All the necessary checks are quite immediate to do, rejecting as a consequence possibly incorrect choices of the design parameters. Omitting full details for space limitations, a few words on the matter are however in order. First, recalling Fig. 1, it is obvious that
ð13Þ
To ensure that a priori, one can simply drive the relay experiment so that uox > um p, i.e., the regulator phase at the cutoff be negative, yielding a positive integral time, as is done in the example procedure presented in the following. Some other checks may be required, for example to avoid that the phase lag introduced by the regulator at the cutoff frequency be too small (resulting in a very large sensitivity of T i to the required phase margin), that the gain has the proper sign if the hypothesis lM > 0 is released (as would be necessary in practical realisations, and was not done here for brevity), and so forth. None of those practical necessities is difficult to manage, however, as the previous considerations should have clarified. A simple and effective tuning procedure is obtained with the b xox Þ the point of the process proposed approach by taking as Pðj frequency response with phase equal to 90°, that is easily found by closing the feedback loop with the cascade of a relay with no hysteresis (or, better, with so small a hysteresis to avoid spurious toggles, yet still allowing to take the real negative semiaxis as precise enough an approximation of its critical points’ locus) and an integrator. Doing so is common practice, see e.g. [27] for extensive discussions and explanations. The above procedure is tested in the following (Section 3). The proposed tuning method is very simple to implement, and computationally light. It can therefore be ported also to low-end industrial regulators, considering that in many of them the functionalities required for relay tests are already present. The sensitivity of the control results to the typical errors introduced by relaybased identification procedures when finding the required process frequency response points are tolerated by the proposed procedure much like they are by any other relay-based method in the literature (there is not the space here to document that with experiments, since a lot of tests are involved in such a check). As for the use, the method has only two parameters. The meaning of the required phase margin is obvious, and that parameter could also be fixed to some reasonable default (say 50°–70°) in industrial implementation. Also a, that of course has to be in the range (0–1), is quite simple to understand. An experienced user could think of it as a way to have the PI approach an integral controller as much as allowed, since a larger a moves the PI+p pole toward the zero, but the position of that pole is constrained to be above the (nominal) cutoff frequency, thus. In other words, such a user could think of a as a means to take profit of the phase lead introduced by the zero, but at the same time to trade that lead against high-frequency control sensitivity. A less educated operator could simply be told to choose higher values of a if the measurement of the controlled variable is very noisy. One could also think to choose two or three default values, based on a qualitative estimation of that noise (say for example ‘‘negligible”, ‘‘average”, or ‘‘high”). The only possible problem is numeric, and could arise if the chosen a puts the pole with time constant T p at so high a frequency to introduce possible representation errors in the discrete time domain, assuming – as usual – that the implementation of the regu-
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lator is digital. However, since the machine precision is known, it is straightforward to detect such problems, limiting as a consequence the admissible values for a. 3. Simulation examples Suppose to tune the PI (1) with a model-based method, taking the IMC-PI formul[20] as a representative of that class of tuning methods—a choice that is easily justified by their widespread use. Suppose also, to get rid a priori of possible effects of the identification procedure, that the tuning model is exact. If [16] are used on (5), the resulting PI has
K¼
TM
lM ðLM þ kÞ
;
Ti ¼ TM;
ð14Þ
where k is the method’s design parameter, and can be interpreted as the (dominant) desired closed-loop time constant [3,15,16]. To set up a meaningful comparison with the relay-based procedure devised herein, one can take a ‘‘reasonable default” for k as the inverse of the frequency where the phase of the process frequency response is more or less 90°. The tuning results then depend on the process (model) normalised delay, defined here as
hM ¼
LM : LM þ T M
ð15Þ
and obviously ranging from 0 to 1. Normalising for convenience the treatise to the case with T M ¼ 1 (the generalisation is just a matter of frequency rescaling), k is readily obtained from
hM 1 p þ arctg ¼ : k 2 kð1 hM Þ
ð16Þ
Solving (16) numerically is straightforward, and computing the K=lM ratio based on (14) leads to the results of Fig. 2, that relate the high-frequency value of the control sensitivity to the process normalised delay. From Fig. 2, it can be observed that for high-order processes, which lead to high normalised delays when described by FOPDT models, the use of cancellation-based methods often results in large values of the high-frequency control sensitivity. Some modifications to PI tuning rules were introduced to mitigate that effect, see e.g. [21], for example choosing an integral time slightly smaller than the model time constant and adapting the PI gain in accordance, but the fact that the control sensitivity does not roll off at high frequency is structural. In the following examples, the results of (14) and (11) are therefore compared, using for the latter the point of the process frequency response with phase -90° found by a relay test with cascaded integrator, and choosing k in the former as discussed above. The first example is made with the process described by
PðsÞ ¼
1 ð1 þ sÞ3
ð17Þ
Fig. 2. Relationship between log 10 ðK=lM Þ and hM in IMC-PI tuning, assuming arg ðRPI ðj=kÞPðj=kÞÞ ¼ 90 .
that is often taken as a benchmark for PI autotuners. Fig. 3 reports the results. The left plot shows the inverse of the nominal control sensitivity function with a PI+p tuned with the proposed procedure and with a PI tuned with the IMC-PI rules. The right plot shows the response of the controlled variable yðtÞ to a unit load disturbance step (a) with the PI+p tuned by the proposed procedure and the real process, (b) with the PI+p tuned by the proposed procedure and as forecast with the FOPDT model, (c) with the PI obtained by means of the IMC-PI rules and the real process, and (d) with the PI obtained by means of the IMC-PI rules and as forecast with the FOPDT model. The proposed procedure was here applied with um ¼ 70 and a ¼ 0:2. As far as the load disturbance responses in the time domain are considered, the behaviour of the two control systems is more or less identical. Moreover, in both cases the tuning results forecast with the FOPDT model obtained together with the regulator are sensible. Of course it would be unrealistic that with such a simple model the dynamics of the real control loop were reproduced exactly. Nevertheless, the major characteristics of the obtained transients (in particular, the peak deviation of the controlled variable from the set point and the transient duration) are caught precisely enough. On the other hand, things are much different from the point of view of the control sensitivity function. Even if the proposed procedure results in a slightly more strict (just a few dB, however) overbound for the additive model error near the cutoff, as frequency increases the same overbound becomes very soon much looser than with the IMC-PI rules. A second (and deliberately extreme) example shows the advantages of the PI+p structure and the proposed tuning method in terms of control sensitivity roll-off. In this example a PI is tuned on the nominal process
Pnom ðsÞ ¼
1 ð1 þ 60sÞð1 þ 10sÞ
ð18Þ
that is then subject to the severe additive perturbation
PD ðsÞ ¼
s 1 þ 0:48
1 3 s2 1 þ 2 0:24 s þ 0:6 2 0:6
ð19Þ
The proposed method, applied to Pnom with um ¼ 60 and a ¼ 0:2, produces lM ¼ 1:11; T M ¼ 73:81; LM ¼ 8:15; K ¼ 2:65; T i ¼ 73:81; k ¼ 24:97. Fig. 4 reports the simulation results, where the tuned PI+p is compared to a two PIs obtained from the SIMC rules [23]: the first of those PIs was tuned with the same FOPDT model and k used for the PI+p, the second was synthesised with a FOPDT model obtained (always from the Pnom step response) with the method of areas (yielding lM ¼ 0; 98; T M ¼ 56:77; LM ¼ 10:34) and k selected with the widely used rule k ¼ maxð0:25LM ; 0:2 T M Þ, resulting here in k ¼ 11:35. The considered transient is the closed-loop response of the controlled variable yðtÞ to a unit load disturbance step—a purpose for which the SIMC is particularly suited. Briefly, Fig. 4a–b shows how relevant the perturbation is. Moreover, it can be seen that the inverse nominal control sensitivities of the PI+p and the SIMC PID tuned with the same model are similar in the vicinity of the cutoff, while the PI+p allows for a larger model error at high frequency (Fig. 4c). Correspondingly, the PI+p obtains (slightly) better results than the SIMC PI with the same model (Fig. 4d, top and middle plot), and the tuning results of both are caught quite well by the model forecasts. Conversely, the SIMC PI tuned with the method of areas performs well in the nominal case, but produces an unstable loop in the perturbed one (Fig. 4d, bottom plot). Recall that the example is extreme, and in no sense a criticism to the SIMC: the purpose was to show that the PI+p, if tuned with a proper procedure, can achieve more or less the same results of a very good PI.
A. Leva, M. Maggio / Journal of Process Control 19 (2009) 1451–1457
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Fig. 3. Results of simulation example 1.
Fig. 4. Results of simulation example 2.
Incidentally, examining also the magnitude of the nominal complementary sensitivity function’s frequency response (omitted here for brevity) reveals that in this case - and in most cases of practical interest, by the way – the PI+p structure provides looser an overbound also for the multiplicative model error, which suggests that the robustness improvement given by said control scheme is quite general with respect to the type of model perturbation. The third example focuses specifically on the tuning method, and shows that it actually allows to overcome the load disturbance response sluggishness typical of cancellation-based policies. A lag dominated process is considered, namely
PðsÞ ¼
e0:1s 1 þ 10s
ð20Þ
and three PIs are tuned by setting k to 10, 5, and 1. Then, three relay experiments are made as explained in [10], driving the oscillation frequency to match the three corresponding desired cutoff frequencies (1/10, 1/5 and 1), and the so measured points are used to tune three PI+p controllers with the proposed method, um ¼ 70 , and a ¼ 0:2. All the IMC PIs have an integral time of 9.9 s, while the three PI+p controllers have T i equal, respectively, to 5.5, 5.4, and
3.3 s. The effect on the load disturbance responses of the controlled variable, depicted on the left in Fig. 5, is self-explanatory. A fourth example is reported to deal with a possible - and reasonable - objection, centred again on the tuning method. One may in fact think to extend PI tuning techniques to the PI+p structure not by using the proposed approach, but by tuning the PI part with a standard procedure (say again the IMC) selecting the design parameter ðkÞ for robustness, and then introduce the additional pole by simply setting it to a ‘‘high” frequency with respect to the cutoff. Incidentally, doing so in comparative tests with the method proposed here seems to yield even more fair a comparison, since the two regulators have exactly the same structure. The process considered, for a meaningful test, allows the FOPDT structure to evidence its limitations owing to a (not excessive) overshooting behaviour, and is
PðsÞ ¼
1 þ 3s ð1 þ sÞ4
:
ð21Þ
The proposed PI+p tuning method is used with the -90° point,
um ¼ 50 , and a set to 0.01, 0.05, 0.10, and 0.15 (recall that the rationale is to put the additional pole at frequency xcn =a). Then,
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Fig. 5. Results of simulation examples 3 (left) and 4 (right).
a FOPDT model for the process is found with the method of areas, and four IMC PIs are tuned with (14) and augmented with a pole at frequency xcn =a, and k selected so that the magnitude of the frequency response of the inverse of the nominal control sensitivity function be greater than the additive model error (which in this case is known, since the process is known) at all frequencies. Fig. 5, right column, shows the controlled variable’s responses to a unit step load disturbance (top) and the inverse of the nominal control sensitivities compared to the additive model error (bottom). As can be seen, even in an idealised case like this, the idea of introducing the additional PI+p pole a posteriori can have unpleasant effects, although stability is of course preserved. In the case of a process the dynamics of which cannot be represented ‘‘well” by a FOPDT model, example 4 has thus shown the advantages of the PI+p structure over an IMC PI with a second pole added a posteriori, essentially in terms of (actually achieved) stability degree, or equivalently of robustness. One could however further object that the PI+p advantages may be less clear in the case of a process that can be represented ‘‘well” by a FOPDT model. To briefly address such an issue, it is useful to reconsider example 3 above (Fig. 5, left column), and compare the tuned PI+p regulators with IMC PI ones augmented with a second pole at an angular frequency equal to kxa times the nominal cutoff frequency, that the IMC rules allow to estimate. Said comparison is shown in Fig. 6, that replicates the left column of Fig. 5
Fig. 6. Comparison with an augmented IMC PI with reference to example 3.
and has the PI+p results evidenced by a thick line: the values of kxa adopted, for each choice of k the IMC PIs, are 2, 5, 10, and 100, apparently covering any reasonable range. One can observe that in some cases the so obtained IMC PIs exhibit invariantly lower performance with respect to the PI+p, while in others the IMC performance may be either better or worse. It is therefore possible to conclude that, in the case of a ‘‘FOPDT-like” process, the PI+p regulator and the proposed tuning rule provide some objectiveness in the choice of the additional pole, which is not observed if said pole is introduced a posteriori, even based on the same data used to tune the rest of the regulator. Other examples on the advantages of the proposed scheme and tuning method are omitted for space reasons.
4. A laboratory application This section presents an application of the PI+p tuning procedure to a laboratory tank level control system, where the control variable is the command to an inlet volumetric pump, the controlled variable is the level, and a pulse-like output disturbance is realised by pouring into the tank the whole content of a half-litre bottle, which takes about six seconds—a short time with respect to the main dynamics of the system. Fig. 7, where the signals are expressed in volts in the range 0–10 as read and written by the A/D and D/A interfaces, shows the whole experiment. The system is initially led to a convenient steady state by a ‘‘best practice” PI controller, having a gain of 5 and an integral time of 20 s. Then, the presented PI+p tuning procedure is applied twice, the first time with a required phase margin of 70°, obtaining K ¼ 4:24; T i ¼ 37:27; T p ¼ 1:92 (Pi+p 1 in the figure), and the second time with 50°, resulting in K ¼ 3:79; T i ¼ 19:43; T p ¼ 1:63 (PI+p 2 in the figure); a was set to 0.15 in both cases. For each of the three regulators, the response to the disturbance described above and to a couple of set point steps is obtained. Some reference straight lines were added to Fig. 7, to help compare the results obtained by the various regulators. In synthesis, one can say that the proposed procedure obtains comparable results with respect to a well tuned PI, that the phase margin acts as design variable in a clear way (smaller phase margins, within a reasonable range, result in shorter but more overshooting and/ or slightly oscillatory transients), and above all that the advantages of the PI+p in terms of high-frequency control sensitivity are evident, since the noise spikes observed on the controlled variable are completely eliminated in PI+p control.
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Fig. 7. Application of the presented PI+p tuning method to a laboratory level control apparatus: set point and controlled variable (above), and control signal (below); pulselike output disturbances are applied at the instants marked with ‘‘Dist.”.
5. Conclusions The tuning of the ‘‘PI+p” controller structure was herein addressed. The PI+p is simply a PI augmented with an additional, stable pole. Its main advantage is that it can make the magnitude of the control sensitivity frequency response roll off after the cutoff frequency, therefore yielding a reduction of the control signal’s sensitivity to measurement noise, and also a more robust tuning in the presence of model errors and/or perturbations exerting their effect slightly above the cutoff frequency itself. After briefly discussing the advantage of the PI+p structure, a relay-based tuning procedure was derived for it, by coupling relaybased identification with a convenient use of model-based tuning relationships. In so doing, both the PI+p tuning was achieved, and a process model was obtained that is ‘‘precise around the cutoff”, i.e., not aimed at representing the open-loop process behaviour, but well suited to assess the main characteristics of the obtained closed-loop system. It is worth stressing that, thanks to the adopted approach coupling relay-based identification and model-based tuning, both the regulator and the model come from a single relay test. The purpose of the obtained model is specific, as said above, but there is no need, for example, to ensure that the identification and tuning procedure starts from a steady state condition—quite significant an advantage for the real-life use of an autotuner. Some implementation-related issues were discussed, and the interpretability of the involved design parameters was also considered. The presented procedure can be considered quite easy to implement, also on low-end devices, and to use, which is in favour of its acceptance in the application domain. Future work will be aimed at studying also the PID+p case – the meaning of the structure name is obvious – and possibly extending the idea to more complex regulators. References [1] K.J. Åström, T. Hägglund, PID Controllers: Theory, Design and Tuning, second ed., Instrument Society of America, Research Triangle Park, NY, 1995. [2] A. Besançon-Voda, H. Roux-Buisson, Another version of the relay feedback experimentm, Journal of Process Control 7 (4) (1997) 303–308. [3] R.D. Braatz, Internal model control, in: S. Levine (Ed.), The Control Handbook, CRC Press, Boca Raton, FL, 1996, pp. 215–224. [4] C. Brosilow, B. Joseph, Techniques of Model Based Control, Prentice Hall PTR, Indianapolis, IN, 2002.
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