Comparative Analysis of Some Approaches to the Autotuning of

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Ind. Eng. Chem. Res. 2009, 48, 5708–5718

PROCESS DESIGN AND CONTROL Comparative Analysis of Some Approaches to the Autotuning of Cascade Controls Alberto Leva* and Alberto Marinelli† Politecnico di Milano, Dipartimento di Elettronica e Informazione Piazza Leonardo Da Vinci, 32s20133 Milano, Italy

Some approaches to the autotuning of cascade controls are comparatively analyzed on the basis of a conveniently characterized test batch. The reported analysis results provide some remarks and possible guidelines for the design of cascade autotuners. Peculiar to the presented simulation study is that the “identification experiment” is considered as an integral part of the autotuning proceduresan aspect of the overall matter that is seldom addressed in the literature. 1. Introduction The importance of the cascade control structure is witnessed by a great amount of literature that features more than 200 papers from 1990 to the present, just to give a figure taken from the ISI database. There are general treatises such as ref 4, surveylike works on various domains, see e.g. ref 7, and specialized applications as described in ref 5. Classical synthesis techniques are used,13,9,14 but also soft-computing methods come into play,8 and sometimes the diagnosis of cascade control behavior 12,6 is addressed. In such a heterogeneous scenario, many ways of classifying the attempted approaches can be devised. The purpose of this manuscript is first to identify the main approaches that a cascade tuning procedure may follow, and then to evaluate the strengths and weaknesses of those approaches in a wide enough set of possible control problems. It is the authors’ hope that this research will be helpful for people involved in both the use and the development of autotuning cascade controls. Users may benefit of the presented discussion to select which tuning approach to use in a given problem, by comparing the characteristics of that problem with those of the problems included in the test batch used herein. Developers may take profit of the same discussion to decide which tuning approach is advised for inclusion in an autotuner devoted to a particular class of problems. It is worth stressing right from the beginning that the objects discussed herein are not particular tuning procedures, but rather the main ideas that ground the rationale of such procedures. A discussion structured in that way can possibly be more difficult to map onto the literature than one organized by analyzing in detail some tuning procedures; however, in the authors’ opinion, the former discussion structure is less keen to disperse the reader’s attention amidst the details of each particular procedure, therefore providing a more clear picture at the overall level. 2. The Comparison Test Batch The cascade control scheme considered here is shown in Figure 1, where the transfer functions PI(s) and PE(s) are the process blocks, RI(s) and RE(s) are, respectively, the regulators of the internal and the external loop (also termed the “primary” * To whom correspondence should be addressed. E-mail: leva@ elet.polimi.it. † Former student at the Dipartimento di Elettronica e Informazione.

and “secondary” loop), wI and wE are the corresponding set points, uI is the control signal, dI and dE represent the disturbances, referred (as usual) to the outputs of the process blocks. Note that the scheme is a series cascade one: a similar study could be performed on the parallel cascade structure14sa task however left to future works. The scope of this work excludes unstable process blocks, in accordance with the majority of the autotuning literature. Also, the focus is set mainly on process control applications. Given this, cascade controls can first be divided in two groups: in the former, both PI(s) and PE(s) are asymptotically stable, in the latter PE(s) is of the “runaway” or “non self-regulating” type, that is, it has a pole in the origin of the s-plane. Since the information on whether PE(s) is asymptotically stable or runaway is normally available, effective tuning techniques are typically specialized to one case or the other, and consistently, in this study the two cases are treated separately. This section presents the batch of processes used to comparatively evaluate the procedures illustrated in section 4 (or, better, the tuning approaches they exemplify). The batch includes various combinations of internal and external process transfer functions, to test the procedures thoroughly, and to have them confront a sufficiently wide range of situations to cover (ideally) all of the cases of interest. 2.1. Asymptotically Stable PE. With an asymptotically stable PE(s), it is sensible to take into account three main facts that are summarized in the following. First, in many cases PI(s) represents “the actuator”, for example, a valve positioner. Actuators in process control typically behave as low order systems, and in most situations a first-order model is adequate. There are however actuators with a high inertia, and for them a first-order model may not provide the required phase lag; in such applications, a second-order PI(s) is needed. A second-order PI(s) is required also when that block does not represent the actuator alone but rather some part of the process dynamics that responds to the control signal faster than the external controlled variable does. This is frequently the case, for example, when both yE and yI (see Figure 1) are

Figure 1. Cascade control scheme.

10.1021/ie801139n CCC: $40.75  2009 American Chemical Society Published on Web 04/30/2009

Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009

temperatures, for example, those of a reactor and a cooling jacket, respectively. If the process sizing is carried out reasonably, higher orders for PI(s) are not of practical interest in process control. Apparently, if a second-order PI(s) is required, its two time constants have to be of comparable entity. The second point to consider is given by the main characteristics of the dynamics of PE(s). That block can in the first place exhibit a delay (e.g., when transport phenomena in pipelines are involved) or not. Then, it can be of low order (e.g., when it describes an energy storage that dominates all other dynamics) or of high order, for example when there are secondary storages the dynamic effect of which is comparable to that of the main one. In the latter case, it is quite frequent to encounter dynamics that cannot be effectively represented if not by high order modelssthink, for instance, of a temperature measured by a probe inserted in a multilayer shield. As such, the test batch presented here considers a “low-” (say first-) and a “high-” (say fifth-)order PE(s), both with and without delay. The third point is the band separation between the (main) dynamics of PI(s) and PE(s). If PI is “the actuator” and PE “the process”, that separation is typically high. If conversely the two blocks represent cascaded phenomena of essentially the same nature and time scale (think again for example of the temperatures of a cooling jacket and the enclosed vessel) the same separation is apparently smaller. Therefore, in the used batch the band separation between PI and PE can be “high” (one decade or more, say a factor of 20) or “low” (less than one decade, say a factor of 5). Having structured the test batch as stated above, the transfer functions used for the internal process are PI1(s) )

1 , 1+s

PI2(s) )

1 (1 + s)(1 + 0.5s)

(1)

while for the external process the four possible structures are PE1n(s) )

1 , 1 + sT1

1 PE5n(s) ) , (1 + sT5)5

PE1d(s) )

e-sT1/2 , 1 + sT1

e-s3T5/2 PE5d(s) ) (1 + sT5)5

(2)

where the subscript “1” or “5” denotes the order, and the subscript “n” or “d” the absence or presence of delay, respectively. In the tests presented later on, to obtain “high” band separation T1 is set to 20 and T5 to 15, so that the lowand high-order external processes more or less share the settling time, and in that case the transfer function used for PE has a further “H” subscript, that is, PE1nH(s)-PE5dH(s). In the “low” band separation tests, conversely, T1 is set to 5 and T5 to 3.5, with the same rationale as above, and in that case the transfer function used for PE has a further “L” subscript, that is, PE1nL(s)-PE5dL(s). 2.2. Runaway PE. As anticipated, in some cases the external process contains an integrator. A typical example, in the process context, is when the external loop controls a level, and the primary controls a flow rate. To achieve a wide enough variety for the comparison campaign, the cases considered herein are the same of section 2.1, where however the transfer function PE(s) is augmented by a factor 1/s. 2.3. A Final Remark on the Test Batch. As can be noticed from sections 2.1 and 2.2, the test batch is a collection of “abstract” dynamical systems, not of physical cases. Structuring a test batch in that way is common practice in the literature, see for example ref 3, as it eases the task of drawing conclusions based on the dynamic characteristics of a control problem, not

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the physics of a given type of system. In some sense, the adopted structuring tries to foster a “conceptual” use of the cascade structure, the suitability of which for a particular problem should be judged precisely from the aspect of the involved dynamics. In sections 2.1 and 2.2, some brief comments were included for the batch elements that admit a quite direct physical interpretation. However, from the control-theoretical standpoint, also the elements that do not reveal such an interpretation immediately are worth including, at least to complete the set of situations against which any possible “new” control problem can be classified. Further comments on the physical meaning of (some) batch elements, in the authors’ opinion, would stray from the scope of the manuscript, and are left to the interested reader. 3. The Tuning Approaches to Be Compared A key point of this research is that the results of any tuning procedure are significantly influenced by how the necessary process information is obtained, while in the literature many papers do not focus at all on that matter. With reference to Figure 1, such papers simply assume that “models” PI(s) and PE(s), whatever is meant for “models” depending on the particular approach followed, are known a priori: examples are refs 29, 28. Other works do address the problem of determining PI(s) and PE(s), but regard that task as ancillary with respect to the control synthesis activity, that is, they discuss the latter by just mentioning one possible way to perform the former. Doing so corresponds to the (implicit) hypothesis that the method used to find PI(s) and PE(s) has little or no influence on the synthesis result: an example here is ref 13. A small minority of papers pay a relevant attention to the problem of gathering process information (see the discussion reported in ref 24). Interestingly enough, if the scope is limited to that minority of works, a definite prevalence of relay-based identification schemes is immediately observed (see e.g., refs 10, 27 and the review ref 9. Hence, a first characteristic of the tuning approaches to be considered here is how process information is obtained. Restricting the scope to ways that can be realistically used in an industrial autotuner, there are two possibilities: the collection of a response in the time domain, and the measurement of some frequency response data (typically, for example, by inducing a limit cycle with relay feedback). In this work the first possibility is represented by the use of a step test, since the other solutions taken in the applications would not lead to different conclusions. A second characteristic is whether the information to tune the two cascade loops is collected separately, or at once with a single test. A third one is whether the tuning of the external regulator assumes the internal closed loop to be ideal (i.e., to have unity transfer function) or not. Having established the characteristics above as the significant ones for this work, the next question is how to select the tuning procedures to compare. A deep analysis of the literature (omitted here for brevity) reveals that the identified characteristics appear in the various tuning methods in virtually any combination, but given the significant amount of heuristics involved in those methods, it is not clear how to find out a “representative” one for each possible combination of characteristics. As such, it was decided to set up some tuning procedures especially for the purpose of this comparison, so as to be sure that relevant tuning approach characteristics are compared, and not the peculiar heuristics of one method or another. The following comparative analysis, to summarize, is therefore targeted at determining which characteristics of a tuning

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policy are best suited to which type of tuning problem (in abstracto, recall section 2): mapping the results onto the methods presented in the literature is left as an exercise to the interested reader. Of course no generality is formally claimed for the presented results, if not within the considered set of procedures, and the test batch. Nonetheless, some of the remarks stemming from the extensive simulation campaign performed are definitely of interest, and may serve as useful guidelines when setting up and applying cascade autotuning procedures. The considered procedures (that incidentally may be taken as tuning methods’ proposals, though this is not the purpose of this manuscript) are presented in section 4, with some brief motivation of their inclusion in the compared set, and above all stating which characteristic of a tuning approach each procedure specifically realizes. Both the cases in which PE(s) is asymptotically stable and runaway are covered, in accordance with the structuring of the batch of section 2. 4. The Considered Tuning Procedures In the first place, any procedure that makes two tests (one to tune the internal regulator and another to tune the external one with the internal loop closed) is de facto equivalent to two single-loop tuning procedures applied in sequence. The only additional fact is that the specifications for the external regulator can be made dependent on those of the internal one, but apart from that, in such procedures there is hardly anything to analyze for the purpose of this research. As such, it is sufficient that the set of compared procedures contain one that makes two tests: the choice was to include one with two step tests, since step tests are more critical in practice than relay ones, as they require the process to be at an equilibrium prior to the application of the stimulus, to prevent residual free motions to affect the identification resultssa relevant problem in control-oriented identification, methodologically addressed by recent works such as refs 1, 2 but not yet by the typical autotuning procedures found in industrial applications. As for one-test procedures, the most relevant characteristics are whether a time or frequency-domain attitude is taken, and whether or not the internal loop is considered ideal while tuning the external one. 4.1. Asymptotically Stable PE. In this case, given the considerations above, the set of procedures to be compared can be formed by considering the following ones: • one with a single step test, taking the internal loop as ideal while tuning the external one • one with a single step test, not taking the internal loop as ideal while tuning the external one • one with two step tests • one with a single relay test To better adhere to the literature, and also for the sake of simplicity, procedures involving PI or PID regulators are here considered. The rationale of the presented analysis could be seamlessly extended to other structures for the internal and/or external regulator, however. 4.1.1. Single Step Test, Ideal Internal Loop. This tuning procedure (TP1s for short in the following) starts with a step test recording the outputs of both process blocks, as shown in Figure 2. The identification of the model for the internal process (very often representing “the actuator”) normally does not pose serious problems. Here the method of areas is applied to the input ui and the output s1 to obtain a delay-free first-order model PÎ(s), and the IMC-PI tuning formulæ22 are then applied to determine

Figure 2. Single step experiment.

Figure 3. Backward filtering to determine PÊ(s).

RI(s). Since the role of the internal loop is to quickly recover the effects of dI on yI but above all on yE (see Figure 1), and since the dynamics of PI are faster than those of PE in any nonpathological case, there is normally no need to require for the internal loop a dominant time constant much shorter than that of PÎ(s) (a ratio of 0.8-1.5 is reasonable). That consideration allows us to select the IMC parameter λ (i.e., the required closedloop dominant time constant) in a straightforward manner, and hardly ever yields any stability or robustness problems. This is therefore the choice adopted herein. Once PÎ(s) is available, the assumed process linearity is exploited as in Figure 3, obtaining y by back-filtering s2 through PÎ. The method of areas is then applied to uI and y to obtain the model PÊ(s), still of the first order but, for intuitive reasons, including a time delay. Finally, RE(s) is determined with the IMC-PI or IMC-PID rules,22 depending on which type of controller is required. Hence, with respect to the purpose of this work, TP1s is taken as the representative of tuning approaches that (1) identify in open loop with a wide-band signal (the step) so as to excite all of the potentially relevant process dynamics and (2) implicitly assume a significant band separation between PI(s) and PE(s), so that one can suppose that low-order models are sufficient to describe both PI(s) and PE(s), provided each model is precise enough in the band it is targeted to represent and locating the critical frequency ωcI of the internal loop near (or “not too far above”) the dominant dynamics of PI(s) can be done reliably based on PIs, since (see above) that model is correct in the band required and results in residual dynamics of the closed internal loop that can be safely neglected for the tuning of RE(s) (recall that those dynamics will be practically unmodelled owing to the simple models used). 4.1.2. Single Step Test, Nonideal Internal Loop. This tuning procedure (TP2s in the following) is analogous to TP1s, except for the fact that the model error estimate used to tune RE(s) (see again ref 25) includes the dynamics of the (closed) internal loop, in turn estimated based on RI(s) and PÎ(s). With respect to the purpose of this work, TP2s is taken as the representative of tuning approaches that are conceptually similar to the TP1s case, but do not implicitly assume a significant band separation between PI(s) and PE(s), so that (1) locating the critical frequency ωcI of the internal not too far above the dominant dynamics of PI(s) can still be done reliably based on PIs, (2) but can result in residual dynamics of the closed internal loop that cannot be safely neglected for the tuning of RE(s), and will most likely turn into some performance reduction caused by increased robustness requirements, given again the modest descriptive capabilities of PEs. Parameter λ for the external loop is computed by employing the procedure proposed in ref 25. A more precise determination of λ could use, for example, its minimum value based on an estimate of the additive error committed by PÊ(s), as shown in


Figure 4. Single relay experiment.

Figure 5. Typical signals during the relay experiment.

ref 21, but the obtained results are very similar, proving that the comparison setup chosen here is representative and general enough. 4.1.3. Two Step Tests. Procedures similar to this (TP3s) are probably the most widely used in cascade autotuners. A first step test yields PÎ(s) from uI and s1, see Figure 2. Then RI(s) is tuned, the internal loop is closed, and a second test is performed, allowing us to compute PÊ(s) and to tune RE(s). Note that in TP3s the model PÊ(s) inherently includes the closed internal loop. For apparent reasons, the same model structures and tuning rules are used here as in TP1s and TP2s. IMC-base rules are used here too for the PI/PID synthesis. In this work, TP3s represents tuning approaches similar to the TP1s and TP2s cases, but that for the tuning of RE(s) “trust” the information provided by RI(s) and PI(s) on the closed internal loop even less than TP2s, and therefore “re-identify” that closed loop in cascade with PE(s). 4.1.4. Single Relay Test. The last procedure (TP4s) employs a different process description, obtained with the relay experiment of Figure 4, that provides the point of the frequency response PI(jω)PE(jω) with phase -90°. Assuming that the phase lag introduced by PE at the oscillation frequency is larger than the lag introduced by PI (which is very reasonable wherever a cascade structure is fit to the control problem), the typical aspect of the signals, when the permanent oscillation condition is reached, is synthetically shown in Figure 5. On the basis of the measured quantities Au, AyI, AyE, Tox, and TIE, it is then possible to determine two approximate models PÎ(s) and PÊ(s); RI(s) and RE(s) can be subsequently tuned with IMC-based rules, having as tuning parameters the two values of λ for the internal and the external loop (or, equivalently, the internal loop λ and the bandwidth separation factor between the two loops). Describing the involved procedure in detail would be too long; the interested reader can find all the information in ref 17. Suffice here to say that TP4s is qualitatively different from TP1sTP3s, but shares the way control specifications are entered (thanks to the use of IMC-like rules), allowing for meaningful comparisons. The motivation of TP4s within the scope of this work is quite different from the TP1s-TP3s cases. In fact, discussing the efficacy of step-based identification is essentially a matter of deciding (based on data) whether or not there is frequency separation between PI(s) and PE(s), and how the tuning of RI(s) enhances or reduces the effects of that separation. As witnessed by a vast literature,30 the success of relay-based identification in the general context of autotuning resides in its capability of

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obtaining frequency domain information that automatically “collapses” into the control-relevant band. Based on relayinduced oscillations, one can obtain some information on the process blocks’ band separation, thus on how much a cascade control is suited to the particular control problem at hand. TP4s is thus here as an alternative proposal with more standard procedures. 4.2. Runaway PE. For the type of procedures to use here, the same reasoning of the beginning of section 4.1 holds. Similarly, when it is to identify PÎ(s) and tune RI(s), things are completely analogous. Hence, in the following we just show the differences with respect to section 4.1. In single-test time domain cases, like in the two-test case for the identification of PÊ(s), a step experiment is clearly infeasible. A two-step stimulus is then employed, leading the process output back to (more or less) the same value as before the test with the second step, of opposite sign with respect to the first one. Trivial operations allow us to obtain the required process step response from the measured one (which is the sum of two step response, the second delayed). Once the step response is available, it is lowpass filtered and numerically differentiated, then the method of areas is used, and the result of that method is augmented with a pole in the origin to yield the required model. For the relay experiment case, one can still consider the scheme of Figure 4, with the sole difference that no integrator is added since PE(s) already has one. Therefore, the measured output of the (runaway) block PE can still play the role of yE in Figure 5, while integrating the measured output of PI provides the counterpart for yi. Accounting for the presence of the integrator in determining the frequency response points for PI and PE, based on that of the response of PIPE with phase -180° (recall again that PE here has the integrator) is absolutely straightforward. Finally, in the runaway case, a P or PD regulator needs to use RE in lieu of a PI or PID, respectively. The IMC rules can of course still be used, since their modification for type 1 processes is immediate (ref 11 and some papers quoted therein). The resulting four procedures are then analogous to the four of section 4.1 and are termed in the following TP1r-TP4r, the “r” standing for “runaway” (PE) as opposite to “s” for (asymptotically) “stable” in section 4.1. 5. Results of the Simulation Campaign This section reports the general remarks stemming from the analysis of the simulation campaign, based on the evaluation of the obtained results. For space reasons the focus is set mainly on the case of an asymptotically stable PE, and only some words are spent on the runaway case. Given the scope of this work, the “tuning quality” achieved in the various cases is assessed looking at synthetic indicators obtained from the control system responses: (i) the maximum overshoot and the settling time of the controlled variables’ responses to a set point stepsa couple of quite common tuning quality indicators, (ii) the ratio between the settling time of the external loop and the time when the internal loop’s controlled variable “catches” the corresponding set point, with the same threshold (1-5%) used to define the external settling time (this is apparently a band separation indicator, more directly related to the transients’ quality than the traditional ratio of the external and internal cutoff frequencies),

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Figure 6. Overall results of the simulation campaign.

(iii) the peak error and the settling time of the controlled variables’ responses to a disturbance step entering the feedforward path of the internal or the external loop, (iv) the peaks and peak rates of the control signal, to measure also the control upset required by a given tuning, (v) the overbound on the additive model error frequency response magnitude provided by the nominal control sensitivity function, particularly in the vicinity of the cutoff frequencies (see ref 21) for the significance of that indicator as a robustness measure. One could of course set up similar comparisons based for example, on integral indices, such as the ISE, ISTE, and so forth. It is however sometimes unclear how to relate those indices to the aspect of the obtained closed-loop transients, so that questionable results may be obtained (see for example ref 23, section 4.2.2) for a discussion on the matter, relative to the PI versus PID control comparison in some relevant cases. 5.1. Asymptotically Stable PE. 5.1.1. Overall Results. In this section we first present a quantitative overview of the simulation campaign results. The considered data are the 95% and 99% settling time of the responses of yE and yI to unit steps applied to the corresponding set points wE and wI; the time required to recover the controlled variables yE and yI to the set point, within 5% and 1%, in the presence of a unit step applied to the disturbances dE and dI, respectively; the maximum overshoot of yE caused by a unit step on wE (or, equivalently if not for the sign, the maximum undershoot of yE caused by a unit step on dE); the maximum absolute value of the response of the control variable uI to a step on wE (or, equivalently, on dE); and the maximum absolute value of the response of the control variable uI to a step on dI. As can be observed from the numerical results summarized in Figure 6, the tuning approaches considered in the campaign actually show different behaviours, thus more or less attitude

to the various tuning problems. More detailed remarks, supported by specific simulation tests, are reported in the following sections. In addition, both to give a further picture of the results and to show how similar comparative campaigns could be used qualitatively, the data collected from simulations of the test batch were organized so as to allow an intuitive evaluation. In other words, based on those data, a qualitative analysis was done, by subjectively assigning a “score” to the following capabilities of the various procedures: • set point tracking speed of the external loop (SPTS for short in the following tables), which also means rejection speed of the external loop output disturbance • rejection of internal loop disturbance in terms of amplitude of the external controlled variable transient (DRTA) • rejection of internal loop disturbance in terms of duration of the external controlled variable transient (DRTD) • achieved band separation of the two loops based on the settling time ratio (ALBS) • overshooting behavior of yE in response to wE and dE (YEOB) • peak absolute value of uI, considered in the totality of the possible inputs (UIPV) • uniformity of the results achieved by the procedure within the batch (URES) Of course one could quantify the capabilities above in a number of different ways, compute the weighted sum of them, and so forth. In the authors’ opinion, however, there is no rigorous way to overcome the inherently subjective character of comparisons like the one addressed here, and it is better to concentrate efforts in providing uniform and quantitative inputs for such comparisons. That is why a qualitative approach is taken in this part of the analysis, and the interested reader is encouraged to set up similar evaluations on his/her own: if


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Figure 7. Procedure TP1s for the PI structure applied to PI1 and PE1nL: set points and controlled variables (top row), and control signal (bottom row) in response to steps on wE (left column) and dI (right column). Table 1. Average Qualitative Performance of the Procedures on the Entire Batch SPTS DRTA DRTD ALBS YEOB UIPV URES TP1s-TP2s (PI) TP1s-TP2s (PID) TP3s (PI) TP3s (PID) TP4s (PI) TP4s (PID)

4 5 4 5 3 4

5 5 5 5 4 4

5 5 5 5 3 4

4 5 4 5 3 3

4 5 4 5 4 4

5 5 5 5 3 1

3 4 3 4 2 3

specialized for instance to a specific class of applications, the results can easily be turned into some selection guidelines for plant operators, to help them select the best procedure to use for a particular tuning problem. For example, if a unity acceleration factor is employed, Table 1 lists the average performance of each procedure on the entire batch, rated from 1 (worst) to 5 (best). Needless to say, the qualitative comparisons above are worth further research, aimed for example at formalising them within some soft-computing framework such as fuzzy logic. This is not within the scope of this work, however. On the same front, it is worth repeating, one could “score” the various tuning results in many ways. However, even if the coarse scoring used here were replaced by some finer one, possibly based on quantitative measurements such as an overshoot, in the authors’ opinion this would not make a so obtained selection method more useful. In the end, is there any real difference, in practice, between a 5% and a 6% overshoot? Both are “slight”, and for a generalpurpose selection procedure that is enough. Having shown the relevance of the presented comparative campaign, both with overall numeric results and with some qualitative interpretation, it is now time to state and briefly discuss the most relevant general remarks that can be drawn. 5.1.2. Ideality of the Internal Loop. This matter obviously concerns the first two procedures. It is important to emphasize that the results of this comparison are worked out from error analysis instead of closed-loop response investigation. In fact, the difference between TP1s and TP2s concerns robustness considerations only and does not affect, under normal circum-

stances (that is the case of our simulations), the overall performances. The performed study indicates that, in general terms, acceptable results are obtained with TP1s (see Figures 6 -10): in fact, most of the times, the additive error bound on the identification of the external process is greater than the nominal additive error caused by the internal closed loop. Exceptions may arise only in the case of a first-order external process and low bandwidth separation, since the modeling error of PE is limited and the internal closed loop modulus in the external loop bandwidth differs from 1 more evidently. 5.1.3. One versus Two Steps for the Model Identification Experiment. Another objective of interest in this work is to determine whether or not the procedures based on a single step test for model identification are “equivalent” to the one involving two step tests. In the following, two of the simulations performed to work out the results are shown. In the first one (Figures 7 and 8), involving the processes PI1 and PE1nL, shows that the TP1s procedure produces a more aggressive control as far as the set-point step response is concerned, while the load disturbances responses (in particular for the internal loop) are practically equivalent. The second one (Figures 9 and 10) confirms the equivalence of the control systems worked out with the two different procedures. In this case the used processes are PI2 and PE5dL. In general, TP3s produces more conservative a tuning, particularly when the close internal loop introduces a larger relative error (if taken as ideal). Quite surprisingly, if the processes to be controlled are well represented within the test batch used herein, the use of two-steps procedures basically allows the inherent conservatism of most modelbased rules to reduce the obtained performances by larger an amount than in one-step procedures. Clearly no generality is (nor could be) claimed for the above statement if the scope is widened with respect to the used test batch, but at least in that batch, the statement is definitely confirmed by simulations, and testifies that the process upset can safely be limited to one step.

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Figure 8. Procedure TP3s for the PI structure applied to PI1 and PE1nL: set points and controlled variables (top row), and control signal (bottom row) in response to steps on wE (left column) and dI (right column).

Figure 9. Procedure TP1s for the PI structure applied to PI2 and PE53dL: set points and controlled variables (top row), and control signal (bottom row) in response to steps on wE (left column) and dI (right column).

5.1.4. Step- versus Relay-Based Identification. As a general picture, the performed simulations suggest that the stepbased procedures TP1s-TP3s obtain better results than the relay-based one TP4s, especially if control signal moderation and robustness are taken as the main issues. In some situations, anyway, TP4s is practically as good as the others, particularly when the external controller is a PID, and the desired degree of stability is not too high. In general, TP4s tends to produce more “nervous” behaviours of the control signal, owing to typically large values of the external regulator’s high frequency gain. In synthesis, step-based procedures appear to be superior to relay-based ones, the only practical caVeat being the well-known sensitivity of the former to possible residual free motion of the controlled variable at the beginning of the experiment. Further studies on step- versus relay-based identification can be found by the interested reader in refs 15, 1 and 2.

5.2. Runaway PE. This section presents a few tests in the runaway PE case. To avoid too heavy a notation, the batch element names are here the same as in section 5.1: just recall that here PE is augmented by a factor 1/s. Basically, the following considerations can be made. Here too, the first two procedures are very similar. One can thus conclude that it is acceptable, at the generality level of the used batch and with consistent tuning specifications, to consider the internal loop ideal when synthesizing RE. In fact, it is not strictly necessary to distinguish between TP1(s,r) and TP2(s,r), but such a statement can only be done after the presented analysis, since it could not be apparent a priori. Then, here too the third procedure is similar to the first two, but exhibits a visibly more conservative attitudesactually, slightly less visible here than in the case of an asymptotically stable PE, probably owing to the necessity of a narrower control bandwidth in the presence of a runaway one. Finally, here the fourth procedure tends to


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Figure 10. Procedure TP3s for the PI structure applied to PI2 and PE5dL: set points and controlled variables (top row), and control signal (bottom row) in response to steps on wE (left column) and dI (right column).

Figure 11. Procedure TP1r for the PD structure applied to PI1 and PE5nL: set points and controlled variables (top row), and control signal (bottom row) in response to steps on wE (left column) and dI (right column).

produce more oscillatory results than the previous ones. The sample simulations of Figures 11 and 12 witness the last aspect. For further aspects, the same remarks of section 5.1 can be largely applied here. Also a similar scoring could be set up, which is omitted here for brevity. 6. A Brief Comparative Application Example To witness the usefulness of the presented considerations, in this section we apply them to the selection of a tuning procedure for a particular cascade control problem, and then compare the results with a literature reference. The problem addressed is described in ref 26, p 238 as “example 1”, and corresponds with the notation of this manuscript to PI(s) )

e-0.1s , 1 + 0.1s

PE(s) )

e-s . (1 + s)2

(3)

Notice in the first place the different results yielded by the various procedures, which evidence directly on the transients’ aspect the differences among the various approaches that were previously presented in a synthetic way in Figure 6. Furthermore, observe that the obtained transients’ characteristics are in accordance with the strength and weaknesses of the considered tuning approaches as summarized in Table 1. Finally, compare Figure 13 with Figures 3 and 4 in ref 26 to see, as a useful byproduct of the presented research, that the proposed procedures (though conceived more to evidence relevant tuning policy characters than to propose new tuning methods) provide de facto results comparable with literature references, and more specifically that the tuning procedure suggested by the proposed method (TP1s with a PI structure) approaches very closely the good results of ref 26.

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Figure 12. Procedure TP4r for the PD structure applied to PI1 and PE5nL: set points and controlled variables (top row), and control signal (bottom row) in response to steps on wE (left column) and dI (right column).

Figure 13. Comparative example results (the organization of the plots is analogous to that of the previous figures).

7. How to Use the Presented Results in Practice In section 5 some indicators were used to assess the achieved tuning quality, giving some further details and a scoring proposal

for the sole case of asymptotically stable PE in section 5.1.1. On the basis of the characteristics of a particular tuning problem, one can derive from that some guidelines to choose the “best” cascade tuning approach.

Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009 Table 2. Some Clues To Choose a Tuning Approach tracking of wE is relevant

yes

rejection of dI is relevant

no

yes

no

yes

no

effort of uI is relevant

yes

no

yes

no

yes

no

yes

procedure to use advised RE type

TP3 PI

TP3 PID

TP1 PI

TP1 PID

TP1 PI

TP4 PID

TP1 PI

In the following we sketch out what such guidelines could look like. For space reasons we do not exhaust all the possible cases, since completing the following picture is a straightforward exercise for the interested reader. First, one may or may not be interested in the tracking of the set point wE. Second, there may or may not be a significant disturbance dI. Third, the peak effort of the control uI may or may not be an issue. Based on scorings like those of table 1, the introduced indicators and the reported considerations on the simplicity and ease of use of the various approaches, the following clues are obtained (note that not all the procedures representing the approaches appear because the set of cases is partial). In view of a practical implementation, one could then further map the problem characteristics of Table 2 onto “physical” cases. For example, most speed/position controls are of the “yes/ yes/yes” type, or of the “yes/no/yes” one if the overall mechanical payload is known and constant, temperature regulations with an internal flow loop tend to be “no/yes/yes” or “no/ yes/no” if possible actuator wear is not an issue, and so forth. Given the scope of this work we do not delve into further details, but the practical use of the presented study should be clear enough. 8. Conclusions A set of approaches for the autotuning of the cascade control structure was compared, based on a conveniently defined test batch, and on some procedures that conveniently substantiate each tuning approach. As explained, analyzing the heterogeneous scenario of cascade tuning with the approach used herein, in the authors’ opinion, should help both the user and the designer of cascade (auto)tuning regulators to better concentrate on the essence of a tuning approach, so as to decide on the suitability of that approach for a given class of tuning problems. Both approaches based on step experiments and methods based on relay experiments were considered. When dealing with those based on step test identification, some basic robust stability considerations were also introduced, which are based on the identified error bound on the external process identification, and on the additive error caused by the closed internal loop. An extensive simulation campaign was performed, so as to point out the advantages and disadvantages of the procedures considered, and thereforesit is worth stressing againsof the underlying approaches. The outcome of that campaign is that the considered approaches work satisfactorily in all the cases of interest, but even within a test batch composed of processes that are well suited for the application of cascade control, the achieved results may vary. Sometimes the reasons for such variations are not immediate to figure out a priori, but can quite easily be understood after the overall set of simulations is collectively analyzed.

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Apart from discussing some cascade autotuning approaches, the main goal of the manuscript was to evidence that interesting and relevant results can emerge from extensive simulations of (auto)tuning procedures, provided that the test batch is well chosen, which is obvious, and that the evaluation includes both the “tuning” and the “identification” phase, which is not equally evident from the published literature. As a byproduct of such activities, guidelines are quite easily devised to select the autotuning approach most suited to the particular tuning problem at hand. Further research will concentrate on possible generalizations of the proposed study, with reference, for example, to more advanced tuning rules, both of the traditional IMC type16 and providing better model fidelity around the cutoffs,20 and possibly also considering two-degree-of-freedom blocks for RI and RE18 and, jointly, richer model structures.19 Also, further work is in order on the way the qualitative analysis reported here can be turned into a complete evaluation mechanism, completing the study that here was just sketched, for example, by suitably introducing some soft-computing methodologies in order to automate the selection procedure to the maximum possible extent. Literature Cited (1) Ahmed, S.; Huang, B.; Shah, S. L. Novel identification method from step response. Control Eng. Pract. 2007, 15 (5), 545–556. (2) Ahmed, S.; Huang, B.; Shah, S. L. Identification from step responses with transient initial conditions. J. Process Control 2008, 18 (2), 121–130. (3) Åstro¨mK, J. Ha¨gglund, T. Benchmark systems for PID control. In IFAC Workshop on Digital ControlsPast, present, and future of PID Control, Terrassa, Spain, 2000. (4) Brosilow, C. Joseph, B. Techniques of Model Based Control; Prentice Hall PTR, Indianapolis, IN, 2002. (5) Buschini, L., Ferrarini, L., Maffezzoni, C. Selftuning cascade temperature control. Proceedings of the 3rd IEEE Conference on Control Applications, Glasgow, UK, 1994; Vol. 1, pp 753-758. (6) Chen, J.; Yea, Y.; Kong, C. K. Diagnosis of cascade control loop status using performance analysis based approach. Ind. Eng. Chem. Res. 2006, 45 (22), 7540–7551. (7) Gunnarsson, F.; Gustafsson, F. Control theory aspects of power control in UMTS. Control Eng. Practice 2003, 11 (10), 1113–1125. (8) Ha, Q. P., Negnevitsky, M., Palis F. Cascade PI-controllers with fuzzy tuning. Proceedings of the 6th IEEE International Conference on Fuzzy Systems, Barcelona, Spain, 1997; Vol. 1, pp 361-366. (9) Hang, C. C.; Astrom, K. J.; Wang, Q. G. Relay feedback auto-tuning of process controllerssa tutorial review. J. Process Control 2002, 12 (1), 143–162. (10) Hang, C. C.; Loh, A. P.; Vasnani, V. U. Relay feedback autotuning of cascade controllers. IEEE Trans. Control Syst. Technol. 1994, 2 (5), 42–45. (11) Kaya, I. Two-degree-of-freedom IMC structure and controller design for integrating processes based on gain and phase-margin specifications. IEE Proc., Control Theor. Appl. 2004, 151 (4), 481–487. (12) Ko, B. S.; Edgar, T. F. Performance assessment of cascade control loops. AIChE J. 2000, 46 (2), 281–291. (13) Lee, Y.; Park, S.; Lee, M. PID controller tuning to obtain desired closed loop responses for cascade control systems. Ind. Eng. Chem. Res. 1998, 37 (5), 1859–1865. (14) Lee, Y.; Skliar, M.; Lee, M. Analytical method of PID controller design for parallel cascade control. J. Process Control 2006, 16 (8), 809– 818. (15) Leva, A. Model-based Proportional-Integral-Derivative autotuning improved with relay feedback identification. IEE Proc., Control Theor. Appl. 2005, 152 (2), 247–256. (16) Leva, A. Performance and robustness improvement in the IMCPID tuning method. Eur. J. Control 2006, 12 (2), 195–204. (17) Leva A.; Donida, F. Autotuning in cascaded systems based on a single relay experiment. J. Proc. Control 2009, 19 (5), 896–905. (18) Leva, A.; Bascetta, L. Designing the feedforward part of 2-d.o.f. industrial controllers for optimal tracking. Control Eng. Pract. 2007, 15 (8), 909–921. (19) Leva, A.; Bascetta, L. Set point tracking optimisation by causal nonparametric modelling. Automatica 2007, 43 (11), 1984–1991.

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(20) Leva, A.; Bascetta, L.; Schiavo, F. Model-based ProportionalIntegral/Proportional-Integral-Derivative (PI/PID) autotuning with fast relay identification. Ind. Eng. Chem. Res. 2006, 45 (7), 4052–4062. (21) Leva, A.; Colombo, A. M. Estimating model mismatch overbounds for the robust autotuning of industrial regulators. Automatica 2000, 36 (12), 1855–1861. (22) Leva, A.; Colombo, A. M. On the IMC-based synthesis of the feedback block of ISA-PID regulators. Trans. Inst. Meas. Control 2004, 26 (5), 417–440. (23) Leva, A.; Cox, C.; Ruano, A. Hands-on PID autotuning: a guide to better utilisation. IFAC Professional Brief, 2002. (24) Leva A.; Piroddi L. On the parametrisation of simple process models for the autotuning of industrial regulators. Proceedings of the ACC 2007, New York, NY, 2007. (25) Maffezzoni, C.; Rocco, P. Robust tuning of PID regulators based on step-response identification. Eur. J. Control 1997, 3 (2), 125-136.

(26) Tan, K. K.; Lee, T. H.; Ferdous, R. Simultaneous online automatic tuning of cascade control for open loop stable processes. ISA Trans. 2000, 39 (2), 233–242. (27) Vivek, S.; Chidambaran, M. Cascade controller tuning by relay auto tune method. J. Ind. Inst. Sci. 2004, 84, 89–97. (28) Vivek, S.; Chidambaran, M. A simple method of tuning cascade controllers. J. Ind. Inst. Sci. 2004, 84, 233–242. (29) Wang F. S.; Juang, W. S.; Chan C. T. Optimal tuning of cascade PID control systems. Proceedings of the 2nd IEEE Conference on Control Applications, Vancouver, Canada, 1993; p 825-828. (30) Yu C. C. Autotuning of PID Controllers: Relay Feedback Approach; Springer-Verlag: London, 1999.

ReceiVed for reView July 24, 2008 ReVised manuscript receiVed March 10, 2009 Accepted April 8, 2009 IE801139N

Comparative Analysis of Some Approaches to the Autotuning of

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