Rule-based Autotuning Based On Frequency ... - Semantic Scholar
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 1, JANUARY 1998
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Rule-Based Autotuning Based on Frequency Domain Identification Anthony S. McCormack and Keith R. Godfrey
Abstract— A frequency domain procedure is proposed for the automatic tuning of proportional integral derivative (PID) controllers. The nonparametric open-loop frequency response function is estimated on-line (in closed loop), and simple discriminatory parameters are obtained, which enable appropriate tuning of the controller. For processes with some prior knowledge of the dynamics, the method removes the need for relay tuning or openloop step response testing. If there is little or no prior knowledge of the process dynamics, relay or step response testing can be used initially, with the method then being used for the subsequent tuning. As part of the paper, a comprehensive comparison of existing rule-based tuning formulas is given using realistic plant examples. Index Terms—Automatic tuning, computer control, frequency response, identification, perturbation signals, PID control, threeterm control.
I. INTRODUCTION
T
HE automatic tuning of proportional integral derivative (PID) controller parameters is both an established feature of commercial controllers as well as a fruitful area of academic research [1]–[4]. The requirements of commercial auto tuners are two-fold. First, the controller must exhibit selfinitialization features in the face of no a priori knowledge of the process dynamics. The relay excitation method [5], is attractive in this sense, since a control-relevant excitation signal is generated automatically, and many tuning rules exist to utilize the resulting process information. The second requirement is recalibration in the face of significant changes in the process dynamics. This problem manifests itself regularly in many industrial feedback loops, e.g., due to changeable loading conditions and/or significant process nonlinearities. Retuning of the controller parameters in this case is usually achieved by performing another relay experiment, or alternatively an open-loop step response. The autotune approach detailed in this paper is based on the use of multiharmonic excitation signals for frequency response estimation. The use of such signals allows automated identification to be carried out, with only modest calculations required by the controller hardware. Inevitably, the scheme
Manuscript received April 2, 1996; revised July 7, 1997. Recommended by Associate Editor, S. A. Malik. This work was supported by Grant GR/K 09373 from the U.K. Engineering and Physical Sciences Research Council. A. S. McCormack is with Tensor Technologies, Dublin 16, Ireland. K. R. Godfrey is with the Department of Engineering, University of Warwick, Coventry, CV4 7AL, U.K. Publisher Item Identifier S 1063-6536(98)00577-6.
relies on the integrity of the estimated frequency response. In the face of significant unmeasurable disturbances, a number of solutions are possible. One solution makes use of nonparametric averaging schemes, hence preserving the simple nature of the identification procedure. Alternatively, the frequency response function can be estimated via a parametric model, with the ensuing problem of automating a procedure which is usually solved in an interactive fashion, e.g., structure selection/model validation. The former approach is concentrated upon here, with the proviso that stochastic disturbances can be countered somewhat with amplitude selection and nonparametric smoothing [6], together with an estimate of the coherence function. The layout of this paper is as follows. Section II describes the common rationale involved in the automatic tuning of three-term controllers. Section II also lists tuning rules which have been developed as alternatives to those of Ziegler and Nichols [7]. Section III discusses periodic signal design and the closed-loop identification of the process (open-loop) frequency response. The paper ends with the application of the techniques to a number of systems, which illustrate the practical nature of the approach.
II. RULE-BASED CONTROLLER DESIGN The tuning rules to be used in subsequent sections are introduced here to set the scene for the requirements of the system identification. Attention is restricted to tuning formulas, due to the acceptance of Ziegler–Nichols (ZN) type rules in industry, and also due to the proliferation of competing designs. The performance of the feedback loop is generally prescribed in terms of the transient response to setpoint changes [3], [8], [9], load disturbances [3], [10], or frequency domain criteria such as the symmetric optimum [4]. These will now be discussed in turn. A. Tuning Formulas A large number of control system design methods now exist which, unlike the majority of design methods, do not require a parametric transfer function model of the process. Instead a nonparametric representation of the system is assumed, and identification methods that can be easily automated are used to obtain the process information. Frequency-domain representations are most popular, with many control system tuning formulas available which require either one or two frequency response measurements.
1063–6536/98$10.00 1998 IEEE
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available for both setpoint and load changes. Note that ISTE tuning formulas are not available for simultaneous setpoint tracking and load rejection, i.e., for two degree of freedom controllers. The formulas require knowledge of and , along with the static gain of the system. 3) Pessen Integral of Absolute Error (PIAE): Pessen remarks in [10] that the ZN tuning formulas were developed for interacting controllers, i.e., PID controllers which do not have the ideal PID relationship
Fig. 1. General regulating and tracking PID controller.
Eight such tuning formulas are assessed in later sections. Although the main emphasis of the work reported here is directed toward improving the identification of the process dynamics, it is worthwhile here to give a brief summary of each tuning formula. The two degree of freedom controller shown in Fig. 1 allows each tuning rule to be encapsulated. The numerator polynomial, , of the prefilter differs from only in the weighting of the signals and , the reference and output, respectively. The tuning rules require knowledge of either one or two points of the process frequency response. Ideally, knowledge of the plant structure (not necessarily parameterized), time delays, and normalized quantities such as (1) for FOPDT models of the form (2) and the normalized gain (3) is the critical gain, are useful in PID design where [1]. The frequency domain identification procedure developed here, specifically sets out to deliver an accurate estimate of the system frequency response. This may then be used in a straightforward manner to estimate the model order [11] and time delay, as well as quantities such as and . Due to the restricted set of models typically found in autotune PID loops, this procedure can be easily automated. 1) Ziegler–Nichols (ZN): This formula for single degree of freedom controllers was initially designed to give a quarter decay ratio response to load disturbances [7]. It requires , and critical period, , knowledge of the critical gain, i.e., the inverse of the system gain and frequency at which the phase is 180 . Since the tuning formula is intended for load disturbances, oscillatory transient responses to setpoint changes typically result from its use. 2) Integral of Squared Time Weighted Error (ISTE): Following on from work carried out on the optimal design of PID controllers for transfer function models [12], Zhuang and Atherton proposed tuning formulas which are ISTE optimal for FOPDT models [3]. Using their formulation, tuning rules are
(4) and are the PID parameters and and represent the actuation and error signals, respectively. For this reason, Pessen claims that the ZN formulas have been misapplied whenever the PID controller has a form similar to (4). As an alternative, he proposes IAE formulas for FOPDT models. As with the ZN formulas, the PIAE rules apply for and is required. load disturbances. Knowledge of 4) Kessler Landau Voda (KLV): This rule attempts to achieve the “symmetric optimum,” originally proposed by Kessler [13]. The model used to represent the system contains a sum of “long compensatable time-constants,” along with a sum of small “parasitic time constants,” which may include a delay [4]. The design goal is to have the crossover frequency rad/s, where of the compensated system equal to is the sum of the parasitics, and for the PID controller to give a magnitude slope of 20 dB/dec two octaves to the left of the crossover and one octave to the right of the crossover. The KLV tuning formulas require the gain of the system and the frequency at which the phase is 135 . The authors also consider setpoint weighting to improve the transient response. 5) Some Overshoot Rule (SO-OV): This formula is a simple modification of the ZN tuning rule, in order to achieve reduced overshoots to setpoint changes [8]. 6) No Overshoot Rule (NO-OV): Similar to the SO-OV rule, with the intention of giving no overshoot in the response to setpoint changes [8]. The proportional gain of the ZN rule is reduced by a factor of three, with derivative time being increased by nearly the same factor. 7) Mantz–Tacconi Ziegler–Nichols (MT-ZN): Mantz and Tacconi propose in [14] a two degree of freedom controller to achieve the regulatory performance of the ZN rule, along with improved setpoint control through the use of a setpoint prefilter ˚ om [15], they (a static element). Using an interpretation by Astr¨ propose a weighting of the setpoint in the proportional and derivative terms of the controller. The weights are selected to cancel the under-damped complex conjugate poles which typically result from ZN tuning. Their rules are applicable to minimum phase systems, and their modification to the ZN and tuning rule is independent of the system dynamics. are required. 8) Refined Ziegler–Nichols (R-ZN): This rule incorporates knowledge of the normalized gain, , or normalized deadtime, , into the ZN tuning rule [9]. The rule results from a correlation between the overshoot (and undershoot) expected where
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(a)
(b)
Fig. 2. (a) Time-domain realization of multisine with snow signal specified to have a broad-band spectrum. (b) amplitude spectrum.
Fig. 3. Power spectra for 15 harmonic multiharmonic signal (with maximum absolute value of unity) and pseudo random binary signal of length 255 (with amplitude levels 1).
6
from using the ZN formulas with models typically associated with PID control. Two cases are considered. The first is appropriate for large values of the normalized gain , or small normalized dead-time, , while the second applies to small or large . The former case is treated using setpoint weighting (related to ) in the proportional term of the controller, while the latter requires both setpoint weighting and a reduction
of the ZN value for the integral time, again related to . and as well as the static gain of the Knowledge of system is required. and correspond to the gain of the In Table I, system and frequency at which the system phase response is 135 . is an acceleration factor taken between one and two, and is assumed to be 1.5 (as suggested by the authors)
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 1, JANUARY 1998
SUMMARY
OF
TABLE I FREQUENCY DOMAIN TUNING RULES POSTULATED
Fig. 4. Feedback loop showing location of disturbance signals.
in this work. and are weights applied to the setpoint in the proportional and derivative terms of the controller, respectively. III. FREQUENCY RESPONSE ESTIMATION The proposed identification procedure allows many points on the process Nyquist curve to be measured, simultaneously, with a high accuracy. A prerequisite for this to be achieved is an estimate of the bandwidth of the system. This is only required at the initial tuning stage (commissioning of the controller), and hence may be obtained using any conventional system identification technique (including the use of a relay). It
FOR
PID CONTROLLERS
should be noted that in the examples given in the paper, relay excitation is only used to measure the system bandwidth, when no prior knowledge of the system dynamics is available. Values for and are calculated from the resulting response to give initial controller settings, which are then updated using the frequency-domain identification technique. It is acknowledged that if either the system is low-order or the relay contains hysteresis, values for and are only approximations to their true values. The main advantage of the identification approach is the excitation signal used. The design of this signal will now be discussed. A. Signal Design Periodic excitations of the form (5) are used to excite the plant while in closed loop. The design of the signal in (5) involves first the selection of the power spectrum (components of ), followed by optimization of the harmonic phases, , to increase the signal-to-noise ratio [16]. In general a fixed signal (i.e., either the time-domain samples or the harmonic information amplitudes), is desirable,
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Fig. 5. Output of third-order system during tuning phase, followed by step response.
Fig. 6. Step responses for deterministic third-order system. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1),SO-OV (–13–1), NO-OV (–12–1), MT-ZN (– –o– –), R-ZN (– –3– –).
since the optimization of the phases is a computationally expensive procedure. A narrow-band excitation with (6) where , will mimic the action of a relay in the frequency domain, with harmonics centred around the a
priori estimate of , with bandwidth rad/s. This approach is not adopted here, first because there is a high dependence in the critical freon an assumed level of uncertainty quency, and also because broad-band knowledge of the system characteristics allows greater confidence to be placed in the feedback design. A signal containing equal power in the first 15 consecutive harmonics is used for the results obtained
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Fig. 7. Load responses for deterministic third-order system. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1),SO-OV (–13–1), NO-OV (–12–1).
in this work. This number of harmonics was found to be a practical compromise between achieving sufficient coverage of the process dynamics and ensuring that a reasonable amount of power is present in each harmonic. The phases have been method of Guillaume et al. [17]. The optimized using the ) and amplitude spectra of signal waveform (with the excitation is shown in Fig. 2. It can be seen that arbitrary nonzero amplitudes are present above harmonic 15. This effect is known as “snowing” the spectrum, and generally allows greater signal-to-noise ratios to be achieved at the desired harmonics (1–15). For comparison purposes, Fig. 3 shows the power spectrum of the multiharmonic signal as well as that of . This has a a pseudorandom binary signal of length fixed power spectrum of the form [18]
(7)
is the amplitude level. It is clear that the ability where to specify an arbitrary power spectrum and optimize the time-domain behavior of the signal has great benefits for nonparametric identification. Once the signal phases have been optimized, the fundaand number of samples are the only mental frequency remaining variables. In selecting these, the desired bandwidth, sampling rate and Nyquist conditions are the most important considerations. The maximum frequency of interest dictates the fundamental frequency. In this work, the maximum frequency (i.e., harmonic 15) is always placed one octave above the critical frequency. This is believed to give good coverage of the dynamic range of typical systems under PID control,
TABLE II REFERENCE CONTROLLER PARAMETERS
FOR
THIRD-ORDER SYSTEM
and it also allows for a reasonably fast signal period. The number of samples in the signal, , is selected to ensure that the sampling frequency is at least 20 Hz, and also that there is a minimum number of 240 samples. The limit on the sampling frequency is heuristic, but it does ensure that the systems treated in later sections are accommodated. If the output is sampled at the same frequency as the clock rate of the excitation, the bound on ensures that the Nyquist frequency is at least three times greater than the maximum frequency in the excitation. It is clear that the amplitude and phase information is the most flexible way to store the signal in the controller. In practice, the frequency range of the signal, and hence , has to be set based on some predetermined knowledge or estimate of the system crossover frequency. This procedure is straightforward once the initial commissioning of the controller has
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(a)
(b) Fig. 8. Empirical frequency response estimates for the third-order system Mean (—), 1:
been carried out, but during the commissioning, if no accurate knowledge of the system dynamics is available, a bandwidth estimation procedure is necessary. A simple technique which may be readily automated makes use of a relay excitation. The second application considered in Section IV makes use of this procedure. It should be stressed however that, at most, one relay experiment is needed during the operation of controller,
uncertainty region (— —) and nominal (—1—).
and may indeed be dispensed with if approximate knowledge of the system dynamics is available. B. Nonparametric Estimation During the identification, the feedback loop is assumed to be of the form shown in Fig. 4. The multifrequency signal is added to the reference, during the tuning phase. For this
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(c) Fig. 8. (Continued.) Empirical frequency response estimates for the third-order system Mean (—), 1:
uncertainty region (— —) and nominal (—1—).
Fig. 9. Step responses for the third-order system in the presence of stochastic disturbances. ZN (—o—), ISTE (—3—),NO-OV (–12–1), R-ZN (– –3– –).
periodic excitations, the estimate of
system, the spectral analysis estimate
is calculated as (9)
(8) yields an unbiased estimate of the frequency response of the process, , even with nonzero and/or [19]. With
and representing the Fourier coefficients with of the th period of the excitation and closed-loop response, respectively. The estimate is defined similarly.
MCCORMACK AND GODFREY: RULE-BASED AUTOTUNING
MEAN
AND
STANDARD DEVIATION
51
OF THE
TABLE III CONTROLLER PARAMETERS
FOR THE
THIRD-ORDER SYSTEM
Fig. 10. Step responses for deterministic first-order plus time delay system. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1), SO-OV (–13–1), NO-OV (–12–1), MT-ZN (– –o– –), R-ZN (– –3– –).
In [19], Wellstead shows that, for
estimated as in (8) (10)
is the number of periods of the perturbation signal where over which the data has been averaged, is the power spectral density of the noise signal , and is the closed-loop frequency response. An important point to note is that for a given noise level and a particular value of , the variance is low around the crossover frequency (i.e., the frequency at which the forward path phase is 180 )—a point brought out clearly by Gevers in [20]. This is a very helpful result for this application, because for the PID controllers considered, the phase of the controller is reasonably small at the frequency, , at which the phase of the process,
, is 180 . For example, for a ZN controller, the controller phase, , at this frequency is only 25 . Thus, in the region around which we are interested, the variance of is relatively low. The accuracy of the experiment may be easily assessed using the coherence function (11) The Fourier coefficients of the actuation and response signals are calculated recursively during the application of the excitation signal. It is assumed here that the amplitude of the excitation is selected to comply with the maximum level of disturbance that
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Fig. 11. Load responses for deterministic first-order plus time delay system. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1), SO-OV (–13–1), NO-OV (–12–1).
TABLE IV REFERENCE CONTROLLER PARAMETERS FOR FIRST-ORDER PLUS TIME-DELAY SYSTEM
may be present at the output of the process during the tuning phase, in a similar fashion to the procedure adopted for relay experiments. It is also assumed that the setpoint is nonzero during the identification which allows the dc component of the Fourier data to be used in estimating the dc gain of the system under control. If this is not the case, the period before or after the application of the signal may be used to estimate this information. The results presented in Section IV correspond to averaging the spectral terms over two periods of steady-state data. This figure is adopted to enable the use of the coherence function in assessing the accuracy of the frequency response estimates. The uncertainty on the coherence function after two periods is very high, but at the signal-to-noise ratios achievable with multiharmonic signals in typical PID feedback loops, this is
not deemed a problem. This assumption is vindicated by the examples of Section IV. In applications where significant disturbances affect the feedback loop, smoothing techniques may be necessary to reduce the scatter of the empirical frequency response. A natural solution is either to average the response over a greater number of periods of the excitation or to use a parametric identification scheme which can greatly reduce the resulting uncertainty. Both of these solutions have significant drawbacks. Increasing the number of periods of the excitation may lead to an excessively long tuning time, while using a parametric identification scheme considerably increases the required computations and effectively cancels the great advantage of using a periodic excitation with a low number of harmonics. Neither of these approaches has been adopted here. Once the frequency response function and coherence function have been calculated, all estimates with a coherence of less than 0.75 are rejected. Nonparametric smoothing is then employed with the use of a median filter. This reduces the scatter of the frequency response estimates significantly. The frequency response may be used to find the model order using the slope of the magnitude curve, and whether the system is nonminimum phase (and if so a representative value of the time-delay), in a similar fashion to that proposed in [11]. This knowledge is generally useful in tuning PID controllers [1]. IV. APPLICATIONS The results obtained from applying the frequency domain techniques to three systems are presented here. The systems are 1) a minimum-phase third-order system; 2) a first-order plus dead-time process; and 3) a hot-air flow device. In each case the accuracy of the frequency response is assessed, as
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Fig. 12.
Output of first-order plus time-delay system during tuning phase, followed by step response.
Fig. 13.
Step responses for the first-order plus time delay system in the presence of stochastic disturbances. ZN (—o—), KLV (—2—),NO-OV (–12–1).
well as each systems response to setpoint changes and load disturbances. All the results have been obtained using an automatic tuning virtual instrument created using the LabVIEW development system [21]. A. Third-Order System The system here is implemented on a hardware simulator (Feedback Instruments Ltd. PCS327), and has a nominal
transfer function of (12) For the ideal case where no disturbances are present in the feedback loop, Fig. 5 shows the output signal during the tuning phase. It is assumed that commissioning of the controller has been carried out and that an initial estimate of the critical gain
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(a)
(b) Fig. 14.
Empirical frequency response estimates for the first-order plus time delay system Mean (—), 1: uncertainty region (— —), and nominal (—1—).
and period are available from the nominal transfer function. The maximum useful harmonic of the excitation is placed one octave above the initial estimate (0.27 Hz), which gives 0.54 Hz. The number of samples of the signal is selected to give a sampling frequency of at least 20 Hz, with the minimum being 240 samples, as discussed in Section III-A. In this case the number of samples is 540. Two and a quarter periods of
the signal are added to the setpoint, with the initial quarter of the first period being discarded due to the presence of transients. Following the tuning period shown in Fig. 5 a ZN step-response occurs. The controller parameters for each of the rule-based designs and are shown in Table II. The values estimated for are 7.51 and 3.7 s, respectively. The PID parameters for the
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(c) Fig. 14. (Continued.) Empirical frequency response estimates for the first-order plus time-delay system Mean (—), 1: uncertainty region (— —), and nominal (—1—).
MEAN
AND
STANDARD DEVIATION
OF THE
TABLE V CONTROLLER PARAMETERS
Refined ZN rule are the same as the ZN and the MT-ZN and rules, with only the setpoint filtering parameters changing. Step and load responses for this system with the parameters given in Table II are shown in Figs. 6 and 7, respectively. With this system, the ISTE and KLV rules give the best tracking performance followed closely by the R-ZN rule. The advantage of the two degree of freedom controller is clear by the superior load response of the R-ZN rule. The SO-OV and NO-OV rules give sluggish load disturbance rejection and also do not lead to low overshoot when compared to the other rules. The PIAE rule leads to the best load rejection, which is its intended use. Although ISTE tuning formulas are available for disturbance rejection, both tracking and disturbance rejection
FOR THE
FIRST-ORDER PLUS TIME-DELAY SYSTEM
properties are felt to be important here. The MT-ZN rule reduces the overshoot obtained with the ZN rule slightly. Gaussian noise with a root mean square (rms) value of 0.1 V and a bandwidth of 15 Hz was then added to the process output while in the feedback path. The frequency response was repeated under the same conditions 25 times. The mean magnitude and phase of the frequency response estimates together with the coherence function are shown in Fig. 8. Estimated uncertainty regions (one standard deviation) for each of these quantities are also shown. The magnitude and phase plots include the frequency response function estimated when no noise was added to the feedback loop. As can be seen the frequency response estimates are effectively unbiased and generally have a high coherence. The mean and standard
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Output of hot-air flow device during the tuning phase, followed by step response.
deviation of the 25 sets of controller parameters are shown in Table III. Typical step responses are shown in Fig. 9. The number of rules shown is limited to four due to the scattered nature of the data. From this application it can be concluded that the identification method gives very accurate results in the presence of modest stochastic disturbances. The coherence function provides a useful indicator of the accuracy of each frequency response estimate, even when calculated from just two periods of the excitation. B. First-Order Plus Dead Time System The system under investigation in this section has a nominal transfer function of (13) This is implemented on a hardware simulator (Feedback Instruments Ltd. PCS327), in which a Pad´e approximation of unknown (low) order is used to represent the time-delay. It is assumed here that no a priori estimate of the system bandwidth is available, and so a limit cycle is used. A standard procedure in this situation is to monitor the process output to determine a suitable value for the relay hysteresis (to prevent relay “chattering”), and then to measure several periods of the limit cycle data [22]. With no disturbances affecting the feedback loop (and hence no hysteresis), tuning was carried out with the limit cycle yielding approximate values for the critical gain and frequency of 0.432 and 0.275 Hz, respectively. Using the estimate of the critical frequency, the maximum frequency is set to 0.551 Hz, with equal to 548. Two and a quarter periods of the excitation were used to estimate the frequency response, which resulted in the parameter values shown in Table IV.
The values estimated for and are 2.27 and 3.66 s, respectively. Step and load responses for this system with the parameters given in Table IV are shown in Figs. 10 and 11, respectively. With step changes in this system the R-ZN, the MT-ZN and ISTE rules produce very similar results with a low overshoot. The KLV rule produces a similar response but with a greater undershoot. This time the SO-OV and NO-OV rules produce low overshoot and no overshoot, respectively. The PIAE rule gives a similar response to that of the ZN rule. With the load disturbance, the PIAE rule again gives the fastest response, and the ISTE rule also performs well, even though both formulas are intended for optimum tracking performance. As with the third-order system the SO-OV and NO-OV rules produce slow disturbance rejection. A Gaussian noise source with bandwidth 15 Hz and rms value of 0.3 V was added to the output of the process inside the feedback loop. The tuning cycle is shown in Fig. 12. As can be seen, significant disturbances affect the output. The output is monitored for approximately 8 s, followed by a number of periods of a limit cycle. The limit cycle yielded values for the critical gain and frequency as 0.694 and 0.231 Hz, respectively. The error in the limit cycle estimates is mainly due to the significant hysteresis present in the relay to combat the disturbance. The multifrequency signal is added to the 23 s and is removed at 96 s. A sample of setpoint at the resulting step responses is shown in Fig. 13. The frequency response estimation was repeated 25 times, as in Section IV-A. The mean magnitude, phase, and coherence together with the uncertainty intervals are shown in Fig. 14. The frequency response estimate when the noise source is absent is also shown. The mean of the resulting controller parameters and their standard deviations is shown in Table V. The uncertainty of proportional gain has increased, but it is clear that the
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(a)
(b) Fig. 16.
Empirical frequency response estimates for the hot-air flow device. Mean (—), 1:
identification together with the median smoothing method has given very reliable estimates in the presence of very high disturbance levels. Again the use of the coherence function allows reliable gain and frequency estimates to be selected in a simple and reliable manner. This example has been included mainly to illustrate the accuracy of the identification results. The level of noise is thought to be higher than would be encountered in most practical situations.
uncertainty region (— —), and nominal (—1—).
C. Hot-Air Flow Device The system under investigation here is a PT 326 hot-air flow device available from Feedback Instruments Ltd. Air is blown through an electric heater, the power to which may be controlled. The temperature of the air is measured at the end of a plastic tube connected to the fan. This system represents quite a realistic example since it has a limited linear range and has an internal noise source due to turbulence within
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(c) Fig. 16.
(Continued.) Empirical frequency response estimates for the hot-air flow device. Mean (—), 1:
uncertainty region (— —), and nominal (—1—).
Fig. 17. Step responses for hot-air flow device. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1), SO-OV (–13–1), NO-OV (–12–1), MT-ZN (– –o– –), R-ZN (– –3– –).
the plastic tube. The same procedure is followed here as was adopted in Section IV-A. The output of the process during the initial tuning cycle is shown in Fig. 15. The manufacturer’s data suggested a value of 1.23 Hz for the 180 crossover frequency and so the bandwidth of the excitation is set to 2.46 Hz, resulting in , with a sampling frequency of 40.07
Hz. The period of the excitation signal is approximately 6 s. The frequency response estimation was carried out 25 times and controller parameters calculated for each rule. Fig. 16 shows the mean frequency response estimate together with the coherence function and their standard deviations. It can be seen that the frequency response estimate has a high accuracy,
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Fig. 18.
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Load responses for hot-air flow device. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1),SO-OV (–13–1), NO-OV (–12–1).
MEAN
AND
STANDARD DEVIATION
OF THE
TABLE VI CONTROLLER PARAMETERS
with a coherence of almost unity across the frequency range. The resulting controller parameters are given in Table VI. The and are 4.88 and 0.76 s, mean values estimated for respectively. Step and load responses for the system with each set of controller parameters are shown in Figs. 17 and 18, respectively. The step response obtained with the ISTE rule exhibits minimal overshoot and a fast rise-time, with the KLV being slightly more oscillatory. The ZN, PIAE, R-ZN, and MT-ZN rules produce a single overshoot with the setpoint weighting of the R-ZN and MT-ZN reducing the magnitude of the overshoot slightly. The SO-OV and NO-OV rules produce slow response times with large significant overshoots. The load responses of the ZN and PIAE rules exhibit the fastest settling times with, (as with the other systems), the SO-OV and NO-OV rules
FOR THE
HOT-AIR FLOW DEVICE
producing very sluggish responses. The ISTE rule results in a slower settling time than for the ZN and PIAE rules, but again ISTE rules are available for load rejection if this is the only design goal. The KLV rule also produces sluggish performance. V. CONCLUSION Most commercial autotuners use one of three methods for updating the PID controller parameters. In the first, normal control is interrupted, a relay is inserted in the loop in place of the controller, and the amplitude and period of the resulting limit cycle are measured. The controller parameters are updated and the controller is switched back into the loop. Some suggestions for using relay excitation with closed-loop ˚ om [23]. The second approach opens systems are given by Astr¨
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the loop, measures at least part of an open-loop step response of the process, and then closes the feedback loop again. The controller parameters are updated on the basis of the step response. The third approach involves switching off full power before the setpoint is reached and then estimating and from the resulting transient. All of these approaches involve interruption of the normal closed-loop control, which is an undesirable feature. In this paper, an alternative approach, in which closedloop operation is maintained throughout, has been investigated. The approach uses frequency domain system identification techniques to estimate the open-loop frequency response of the process from data obtained while the loop is closed. Periodic excitations have been used, enabling frequency response estimates to be obtained in closed loop at comparatively high signal-to-noise ratios. The identification experiment may be automated easily, and only a modest amount of computation is needed. The identification techniques have been applied to three realistic examples, and it has been shown that reliable results have been obtained in all three situations. An arbitrary number of frequency response estimates may be obtained, thereby allowing more confidence to be placed in the controller design and/or further processing of the frequency response. In all three examples, a comparatively broadband perturbation signal has been used. This consisted of a multifrequency (sum-of-harmonics) signal with the specified harmonics being one to 15 consecutively. It is, of course, essential that the bandwidth of the signal contains the frequency (or frequencies) of interest, and this may require some prior knowledge of the process dynamics. In the first example (Section IV-A), a hardware simulator was used to simulate a third-order process, and the highest frequency of interest (the negative real axis crossover point) was known with good accuracy. In the second example (Section IV-B), although the same hardware simulator was used, there was some uncertainty about the way in which the pure time-delay was approximated in the simulator. In this case, it was assumed that no prior knowledge of the dynamics was available, and to obtain an initial estimate of the system dynamics, a relay was inserted into the loop, although with level of noise present, this had to incorporate significant hysteresis to prevent relay “chattering.” It should be emphasised that the relay was only used in the first tune and not subsequently. Further, it is felt that, in most applications, prior knowledge of the process dynamics is likely to make the initial relay phase unnecessary. This was certainly the case in the third example (Section IV-C), where manufacturer’s data provided reasonably good prior knowledge of the process dynamics. In theory, a signal containing fewer harmonics, with more power in each harmonic, could have been used, but it is essential to trade off possible inaccuracies in prior knowledge, and changing process dynamics, against reducing the number of harmonics. This is certainly a topic for further research. A notable feature of recent research has been the interest in improving on the performance of the widely quoted ZN tuning rules, which date from 1942. As part of the present paper eight tuning rules (including those of Ziegler and Nichols) have been
compared, in terms of both setpoint (tracking) and disturbance (regulating) control. The availability of tuning rules for two degree of freedom controllers allows both criteria to be treated for a given set of PID parameters. In summary, a new approach to autotune controller design has been proposed. Except in the case of processes in which there is very little prior knowledge of the dynamics, the method removes the need for relay tuning or open-loop step response tuning. The dynamics are estimated in normal closedloop operation, with a small-amplitude signal added to the setpoint during the tuning phase. The techniques have been shown to work well on three examples, the first a hardware simulated process, the second a hardware simulated process with a considerable amount of noise present, and the third a laboratory-scale heating process. REFERENCES ˚ om, C. C. Hang, P. Persson, and W. K. Ho, “Toward intelligent [1] K. J. Astr¨ PID control,” Automatica, vol. 28, no. 1, pp. 1–9, 1992. ˚ om and T. H¨agglund, Automatic Tuning of PID Controllers, [2] K. J. Astr¨ Instrument Soc. Amer., Research Triangle Park, NC, 1988. [3] M. Zhuang and D. P. Atherton, “Automatic tuning of optimum PID controllers,” Inst. Elect. Eng. Proc., Pt. D, vol. 140, no. 3, pp. 216–224, 1993. [4] A. Voda and I. D. Landau, “A method for the auto-calibration of PID controllers,” Automatica, vol. 31, no. 1, pp. 41–53, 1995. ˚ om and T. H¨agglund, “Automatic tuning of simple regulators [5] K. J. Astr¨ with specifications on phase and amplitude margins,” Automatica, vol. 20, no. 5, pp. 645–651, 1984. [6] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal. Cambridge, U.K.: Cambridge Univ. Press, 1989. [7] J. G. Ziegler and N. B. Nichols, “Optimum settings for automatic controllers,” Trans. Amer. Soc. Mech. Eng., vol. 64, pp. 759–768, 1942. [8] D. E. Seborg, T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control. New York: Wiley, 1989. ˚ om, and W. K. Ho, “Refinements of the [9] C. C. Hang, K. J. Astr¨ Ziegler–Nichols tuning formula,” Inst. Elect. Eng. Proc., Pt. D, vol. 138, no. 2, pp. 111–118, 1991. [10] D. W. Pessen, “A new look at PID-controller tuning,” Trans. Amer. Soc. Mech. Eng., J. Dynamic Syst., Meas., Contr., vol. 116, pp. 553–557, 1994. ˚ om, “Automatic initialization of a robust [11] M. Lundh and K. J. Astr¨ self-tuning controller,” Automatica, vol. 30, no. 11, pp. 1649–1662, 1994. [12] D. P. Atherton and M. Zhuang, “Tuning PID controllers with integral performance criteria,” in Inst. Elect. Eng. Int. Conf. Contr. ’91, Edinburgh, U.K., pp. 481–486. [13] C. Kessler, “Das symmetrische optimum,” Regelungstetechnik, vol. 6, no. 11, pp. 395–400, 1958. [14] R. J. Mantz and E. J. Tacconi, “Complementary rules to Ziegler and Nichols’ rules for a regulating and tracking controller,” Int. J. Contr., vol. 49, no. 5, pp. 1465–1471, 1989. ˚ om, “Ziegler–Nichols auto-tuners,” Dept. Automat. Contr., [15] K. J. Astr¨ Lund Inst. Technol., Tech. Rep. TFRT-3167, 1982. [16] K. Godfrey, “Introduction to perturbation signals for frequency-domain system identification,” in Perturbation Signals for System Identification, K. Godfrey, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1993, ch. 2. [17] P. Guillaume, J. Schoukens, R. Pintelon, and I. Koll´ar, “Crest-factor minimization using nonlinear Chebyshev approximation methods,” IEEE Trans. Instrum. Meas., vol. 40, pp. 982–989, 1991. [18] K. Godfrey, “Introduction to perturbation signals for time-domain system identification,” in Perturbation Signals for System Identification, K. Godfrey, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1993, ch. 1. [19] P. E. Wellstead, “Nonparametric methods of system identification,” Automatica, vol. 17, no. 1, pp. 55–69, 1981. [20] M. Gevers, “Toward a joint design of identification and control?” in Essays on Control: Perspectives in the Theory and Its Applications, H. L. Trentelman and J. C. Willems, Eds. Boston, MA: Birkh¨auser, 1993, ch. 5. [21] National Instruments Corporation, LabVIEW for Windows User-Manual, 1993.
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[22] T. H¨agglund, Process Control in Practice. Bromley, U.K.: ChartwellBratt, 1991. ˚ om, “Tuning and Adaptation,” in 13th Triennial IFAC World [23] K. J. Astr¨ Congr., San Francisco, CA, 1996, pp. 1–18.
Anthony S. McCormack received the bachelor’s degree in electrical engineering from the University of Glamorgan in 1991 and the Ph.D. degree for work on system identification and signal processing from the Department of Engineering at the University of Warwick, Coventry, U.K., in 1995. From 1991 to 1995 he was a Research Fellow in the Department of Engineering at the University of Warwick working on experiment design for system identification and the application of frequency domain identification to autotune control systems. He is Operations Manager of the Real-time Systems Group at Tensor Technologies in Dublin, Ireland. His main research interests are in frequency domain system identification and process control.
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Keith R. Godfrey received the Doctor of Science degree from the University of Warwick, Coventry, U.K., in 1990 for publications with the collective title “Applications of Modeling, Identification, and Parameter Estimation in Engineering and Biomedicine.” He is Head of the Electrical and Control Group in the Department of Engineering at the University of Warwick. He is author of a book on compartmental modeling published by Academic Press in 1983 and is coeditor of Signal Processing for Control (New York: Springer, 1986). He is also editor (and author of the first two chapters) of Perturbation Signals for System Identification (Englewood Cliffs, NJ: PrenticeHall, 1993) (now available from the editor). He is author or coauthor of more than 150 papers.
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Rule-Based Autotuning Based on Frequency Domain Identification Anthony S. McCormack and Keith R. Godfrey
Abstract— A frequency domain procedure is proposed for the automatic tuning of proportional integral derivative (PID) controllers. The nonparametric open-loop frequency response function is estimated on-line (in closed loop), and simple discriminatory parameters are obtained, which enable appropriate tuning of the controller. For processes with some prior knowledge of the dynamics, the method removes the need for relay tuning or openloop step response testing. If there is little or no prior knowledge of the process dynamics, relay or step response testing can be used initially, with the method then being used for the subsequent tuning. As part of the paper, a comprehensive comparison of existing rule-based tuning formulas is given using realistic plant examples. Index Terms—Automatic tuning, computer control, frequency response, identification, perturbation signals, PID control, threeterm control.
I. INTRODUCTION
T
HE automatic tuning of proportional integral derivative (PID) controller parameters is both an established feature of commercial controllers as well as a fruitful area of academic research [1]–[4]. The requirements of commercial auto tuners are two-fold. First, the controller must exhibit selfinitialization features in the face of no a priori knowledge of the process dynamics. The relay excitation method [5], is attractive in this sense, since a control-relevant excitation signal is generated automatically, and many tuning rules exist to utilize the resulting process information. The second requirement is recalibration in the face of significant changes in the process dynamics. This problem manifests itself regularly in many industrial feedback loops, e.g., due to changeable loading conditions and/or significant process nonlinearities. Retuning of the controller parameters in this case is usually achieved by performing another relay experiment, or alternatively an open-loop step response. The autotune approach detailed in this paper is based on the use of multiharmonic excitation signals for frequency response estimation. The use of such signals allows automated identification to be carried out, with only modest calculations required by the controller hardware. Inevitably, the scheme
Manuscript received April 2, 1996; revised July 7, 1997. Recommended by Associate Editor, S. A. Malik. This work was supported by Grant GR/K 09373 from the U.K. Engineering and Physical Sciences Research Council. A. S. McCormack is with Tensor Technologies, Dublin 16, Ireland. K. R. Godfrey is with the Department of Engineering, University of Warwick, Coventry, CV4 7AL, U.K. Publisher Item Identifier S 1063-6536(98)00577-6.
relies on the integrity of the estimated frequency response. In the face of significant unmeasurable disturbances, a number of solutions are possible. One solution makes use of nonparametric averaging schemes, hence preserving the simple nature of the identification procedure. Alternatively, the frequency response function can be estimated via a parametric model, with the ensuing problem of automating a procedure which is usually solved in an interactive fashion, e.g., structure selection/model validation. The former approach is concentrated upon here, with the proviso that stochastic disturbances can be countered somewhat with amplitude selection and nonparametric smoothing [6], together with an estimate of the coherence function. The layout of this paper is as follows. Section II describes the common rationale involved in the automatic tuning of three-term controllers. Section II also lists tuning rules which have been developed as alternatives to those of Ziegler and Nichols [7]. Section III discusses periodic signal design and the closed-loop identification of the process (open-loop) frequency response. The paper ends with the application of the techniques to a number of systems, which illustrate the practical nature of the approach.
II. RULE-BASED CONTROLLER DESIGN The tuning rules to be used in subsequent sections are introduced here to set the scene for the requirements of the system identification. Attention is restricted to tuning formulas, due to the acceptance of Ziegler–Nichols (ZN) type rules in industry, and also due to the proliferation of competing designs. The performance of the feedback loop is generally prescribed in terms of the transient response to setpoint changes [3], [8], [9], load disturbances [3], [10], or frequency domain criteria such as the symmetric optimum [4]. These will now be discussed in turn. A. Tuning Formulas A large number of control system design methods now exist which, unlike the majority of design methods, do not require a parametric transfer function model of the process. Instead a nonparametric representation of the system is assumed, and identification methods that can be easily automated are used to obtain the process information. Frequency-domain representations are most popular, with many control system tuning formulas available which require either one or two frequency response measurements.
1063–6536/98$10.00 1998 IEEE
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available for both setpoint and load changes. Note that ISTE tuning formulas are not available for simultaneous setpoint tracking and load rejection, i.e., for two degree of freedom controllers. The formulas require knowledge of and , along with the static gain of the system. 3) Pessen Integral of Absolute Error (PIAE): Pessen remarks in [10] that the ZN tuning formulas were developed for interacting controllers, i.e., PID controllers which do not have the ideal PID relationship
Fig. 1. General regulating and tracking PID controller.
Eight such tuning formulas are assessed in later sections. Although the main emphasis of the work reported here is directed toward improving the identification of the process dynamics, it is worthwhile here to give a brief summary of each tuning formula. The two degree of freedom controller shown in Fig. 1 allows each tuning rule to be encapsulated. The numerator polynomial, , of the prefilter differs from only in the weighting of the signals and , the reference and output, respectively. The tuning rules require knowledge of either one or two points of the process frequency response. Ideally, knowledge of the plant structure (not necessarily parameterized), time delays, and normalized quantities such as (1) for FOPDT models of the form (2) and the normalized gain (3) is the critical gain, are useful in PID design where [1]. The frequency domain identification procedure developed here, specifically sets out to deliver an accurate estimate of the system frequency response. This may then be used in a straightforward manner to estimate the model order [11] and time delay, as well as quantities such as and . Due to the restricted set of models typically found in autotune PID loops, this procedure can be easily automated. 1) Ziegler–Nichols (ZN): This formula for single degree of freedom controllers was initially designed to give a quarter decay ratio response to load disturbances [7]. It requires , and critical period, , knowledge of the critical gain, i.e., the inverse of the system gain and frequency at which the phase is 180 . Since the tuning formula is intended for load disturbances, oscillatory transient responses to setpoint changes typically result from its use. 2) Integral of Squared Time Weighted Error (ISTE): Following on from work carried out on the optimal design of PID controllers for transfer function models [12], Zhuang and Atherton proposed tuning formulas which are ISTE optimal for FOPDT models [3]. Using their formulation, tuning rules are
(4) and are the PID parameters and and represent the actuation and error signals, respectively. For this reason, Pessen claims that the ZN formulas have been misapplied whenever the PID controller has a form similar to (4). As an alternative, he proposes IAE formulas for FOPDT models. As with the ZN formulas, the PIAE rules apply for and is required. load disturbances. Knowledge of 4) Kessler Landau Voda (KLV): This rule attempts to achieve the “symmetric optimum,” originally proposed by Kessler [13]. The model used to represent the system contains a sum of “long compensatable time-constants,” along with a sum of small “parasitic time constants,” which may include a delay [4]. The design goal is to have the crossover frequency rad/s, where of the compensated system equal to is the sum of the parasitics, and for the PID controller to give a magnitude slope of 20 dB/dec two octaves to the left of the crossover and one octave to the right of the crossover. The KLV tuning formulas require the gain of the system and the frequency at which the phase is 135 . The authors also consider setpoint weighting to improve the transient response. 5) Some Overshoot Rule (SO-OV): This formula is a simple modification of the ZN tuning rule, in order to achieve reduced overshoots to setpoint changes [8]. 6) No Overshoot Rule (NO-OV): Similar to the SO-OV rule, with the intention of giving no overshoot in the response to setpoint changes [8]. The proportional gain of the ZN rule is reduced by a factor of three, with derivative time being increased by nearly the same factor. 7) Mantz–Tacconi Ziegler–Nichols (MT-ZN): Mantz and Tacconi propose in [14] a two degree of freedom controller to achieve the regulatory performance of the ZN rule, along with improved setpoint control through the use of a setpoint prefilter ˚ om [15], they (a static element). Using an interpretation by Astr¨ propose a weighting of the setpoint in the proportional and derivative terms of the controller. The weights are selected to cancel the under-damped complex conjugate poles which typically result from ZN tuning. Their rules are applicable to minimum phase systems, and their modification to the ZN and tuning rule is independent of the system dynamics. are required. 8) Refined Ziegler–Nichols (R-ZN): This rule incorporates knowledge of the normalized gain, , or normalized deadtime, , into the ZN tuning rule [9]. The rule results from a correlation between the overshoot (and undershoot) expected where
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(a)
(b)
Fig. 2. (a) Time-domain realization of multisine with snow signal specified to have a broad-band spectrum. (b) amplitude spectrum.
Fig. 3. Power spectra for 15 harmonic multiharmonic signal (with maximum absolute value of unity) and pseudo random binary signal of length 255 (with amplitude levels 1).
6
from using the ZN formulas with models typically associated with PID control. Two cases are considered. The first is appropriate for large values of the normalized gain , or small normalized dead-time, , while the second applies to small or large . The former case is treated using setpoint weighting (related to ) in the proportional term of the controller, while the latter requires both setpoint weighting and a reduction
of the ZN value for the integral time, again related to . and as well as the static gain of the Knowledge of system is required. and correspond to the gain of the In Table I, system and frequency at which the system phase response is 135 . is an acceleration factor taken between one and two, and is assumed to be 1.5 (as suggested by the authors)
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SUMMARY
OF
TABLE I FREQUENCY DOMAIN TUNING RULES POSTULATED
Fig. 4. Feedback loop showing location of disturbance signals.
in this work. and are weights applied to the setpoint in the proportional and derivative terms of the controller, respectively. III. FREQUENCY RESPONSE ESTIMATION The proposed identification procedure allows many points on the process Nyquist curve to be measured, simultaneously, with a high accuracy. A prerequisite for this to be achieved is an estimate of the bandwidth of the system. This is only required at the initial tuning stage (commissioning of the controller), and hence may be obtained using any conventional system identification technique (including the use of a relay). It
FOR
PID CONTROLLERS
should be noted that in the examples given in the paper, relay excitation is only used to measure the system bandwidth, when no prior knowledge of the system dynamics is available. Values for and are calculated from the resulting response to give initial controller settings, which are then updated using the frequency-domain identification technique. It is acknowledged that if either the system is low-order or the relay contains hysteresis, values for and are only approximations to their true values. The main advantage of the identification approach is the excitation signal used. The design of this signal will now be discussed. A. Signal Design Periodic excitations of the form (5) are used to excite the plant while in closed loop. The design of the signal in (5) involves first the selection of the power spectrum (components of ), followed by optimization of the harmonic phases, , to increase the signal-to-noise ratio [16]. In general a fixed signal (i.e., either the time-domain samples or the harmonic information amplitudes), is desirable,
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Fig. 5. Output of third-order system during tuning phase, followed by step response.
Fig. 6. Step responses for deterministic third-order system. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1),SO-OV (–13–1), NO-OV (–12–1), MT-ZN (– –o– –), R-ZN (– –3– –).
since the optimization of the phases is a computationally expensive procedure. A narrow-band excitation with (6) where , will mimic the action of a relay in the frequency domain, with harmonics centred around the a
priori estimate of , with bandwidth rad/s. This approach is not adopted here, first because there is a high dependence in the critical freon an assumed level of uncertainty quency, and also because broad-band knowledge of the system characteristics allows greater confidence to be placed in the feedback design. A signal containing equal power in the first 15 consecutive harmonics is used for the results obtained
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Fig. 7. Load responses for deterministic third-order system. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1),SO-OV (–13–1), NO-OV (–12–1).
in this work. This number of harmonics was found to be a practical compromise between achieving sufficient coverage of the process dynamics and ensuring that a reasonable amount of power is present in each harmonic. The phases have been method of Guillaume et al. [17]. The optimized using the ) and amplitude spectra of signal waveform (with the excitation is shown in Fig. 2. It can be seen that arbitrary nonzero amplitudes are present above harmonic 15. This effect is known as “snowing” the spectrum, and generally allows greater signal-to-noise ratios to be achieved at the desired harmonics (1–15). For comparison purposes, Fig. 3 shows the power spectrum of the multiharmonic signal as well as that of . This has a a pseudorandom binary signal of length fixed power spectrum of the form [18]
(7)
is the amplitude level. It is clear that the ability where to specify an arbitrary power spectrum and optimize the time-domain behavior of the signal has great benefits for nonparametric identification. Once the signal phases have been optimized, the fundaand number of samples are the only mental frequency remaining variables. In selecting these, the desired bandwidth, sampling rate and Nyquist conditions are the most important considerations. The maximum frequency of interest dictates the fundamental frequency. In this work, the maximum frequency (i.e., harmonic 15) is always placed one octave above the critical frequency. This is believed to give good coverage of the dynamic range of typical systems under PID control,
TABLE II REFERENCE CONTROLLER PARAMETERS
FOR
THIRD-ORDER SYSTEM
and it also allows for a reasonably fast signal period. The number of samples in the signal, , is selected to ensure that the sampling frequency is at least 20 Hz, and also that there is a minimum number of 240 samples. The limit on the sampling frequency is heuristic, but it does ensure that the systems treated in later sections are accommodated. If the output is sampled at the same frequency as the clock rate of the excitation, the bound on ensures that the Nyquist frequency is at least three times greater than the maximum frequency in the excitation. It is clear that the amplitude and phase information is the most flexible way to store the signal in the controller. In practice, the frequency range of the signal, and hence , has to be set based on some predetermined knowledge or estimate of the system crossover frequency. This procedure is straightforward once the initial commissioning of the controller has
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(a)
(b) Fig. 8. Empirical frequency response estimates for the third-order system Mean (—), 1:
been carried out, but during the commissioning, if no accurate knowledge of the system dynamics is available, a bandwidth estimation procedure is necessary. A simple technique which may be readily automated makes use of a relay excitation. The second application considered in Section IV makes use of this procedure. It should be stressed however that, at most, one relay experiment is needed during the operation of controller,
uncertainty region (— —) and nominal (—1—).
and may indeed be dispensed with if approximate knowledge of the system dynamics is available. B. Nonparametric Estimation During the identification, the feedback loop is assumed to be of the form shown in Fig. 4. The multifrequency signal is added to the reference, during the tuning phase. For this
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(c) Fig. 8. (Continued.) Empirical frequency response estimates for the third-order system Mean (—), 1:
uncertainty region (— —) and nominal (—1—).
Fig. 9. Step responses for the third-order system in the presence of stochastic disturbances. ZN (—o—), ISTE (—3—),NO-OV (–12–1), R-ZN (– –3– –).
periodic excitations, the estimate of
system, the spectral analysis estimate
is calculated as (9)
(8) yields an unbiased estimate of the frequency response of the process, , even with nonzero and/or [19]. With
and representing the Fourier coefficients with of the th period of the excitation and closed-loop response, respectively. The estimate is defined similarly.
MCCORMACK AND GODFREY: RULE-BASED AUTOTUNING
MEAN
AND
STANDARD DEVIATION
51
OF THE
TABLE III CONTROLLER PARAMETERS
FOR THE
THIRD-ORDER SYSTEM
Fig. 10. Step responses for deterministic first-order plus time delay system. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1), SO-OV (–13–1), NO-OV (–12–1), MT-ZN (– –o– –), R-ZN (– –3– –).
In [19], Wellstead shows that, for
estimated as in (8) (10)
is the number of periods of the perturbation signal where over which the data has been averaged, is the power spectral density of the noise signal , and is the closed-loop frequency response. An important point to note is that for a given noise level and a particular value of , the variance is low around the crossover frequency (i.e., the frequency at which the forward path phase is 180 )—a point brought out clearly by Gevers in [20]. This is a very helpful result for this application, because for the PID controllers considered, the phase of the controller is reasonably small at the frequency, , at which the phase of the process,
, is 180 . For example, for a ZN controller, the controller phase, , at this frequency is only 25 . Thus, in the region around which we are interested, the variance of is relatively low. The accuracy of the experiment may be easily assessed using the coherence function (11) The Fourier coefficients of the actuation and response signals are calculated recursively during the application of the excitation signal. It is assumed here that the amplitude of the excitation is selected to comply with the maximum level of disturbance that
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Fig. 11. Load responses for deterministic first-order plus time delay system. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1), SO-OV (–13–1), NO-OV (–12–1).
TABLE IV REFERENCE CONTROLLER PARAMETERS FOR FIRST-ORDER PLUS TIME-DELAY SYSTEM
may be present at the output of the process during the tuning phase, in a similar fashion to the procedure adopted for relay experiments. It is also assumed that the setpoint is nonzero during the identification which allows the dc component of the Fourier data to be used in estimating the dc gain of the system under control. If this is not the case, the period before or after the application of the signal may be used to estimate this information. The results presented in Section IV correspond to averaging the spectral terms over two periods of steady-state data. This figure is adopted to enable the use of the coherence function in assessing the accuracy of the frequency response estimates. The uncertainty on the coherence function after two periods is very high, but at the signal-to-noise ratios achievable with multiharmonic signals in typical PID feedback loops, this is
not deemed a problem. This assumption is vindicated by the examples of Section IV. In applications where significant disturbances affect the feedback loop, smoothing techniques may be necessary to reduce the scatter of the empirical frequency response. A natural solution is either to average the response over a greater number of periods of the excitation or to use a parametric identification scheme which can greatly reduce the resulting uncertainty. Both of these solutions have significant drawbacks. Increasing the number of periods of the excitation may lead to an excessively long tuning time, while using a parametric identification scheme considerably increases the required computations and effectively cancels the great advantage of using a periodic excitation with a low number of harmonics. Neither of these approaches has been adopted here. Once the frequency response function and coherence function have been calculated, all estimates with a coherence of less than 0.75 are rejected. Nonparametric smoothing is then employed with the use of a median filter. This reduces the scatter of the frequency response estimates significantly. The frequency response may be used to find the model order using the slope of the magnitude curve, and whether the system is nonminimum phase (and if so a representative value of the time-delay), in a similar fashion to that proposed in [11]. This knowledge is generally useful in tuning PID controllers [1]. IV. APPLICATIONS The results obtained from applying the frequency domain techniques to three systems are presented here. The systems are 1) a minimum-phase third-order system; 2) a first-order plus dead-time process; and 3) a hot-air flow device. In each case the accuracy of the frequency response is assessed, as
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Fig. 12.
Output of first-order plus time-delay system during tuning phase, followed by step response.
Fig. 13.
Step responses for the first-order plus time delay system in the presence of stochastic disturbances. ZN (—o—), KLV (—2—),NO-OV (–12–1).
well as each systems response to setpoint changes and load disturbances. All the results have been obtained using an automatic tuning virtual instrument created using the LabVIEW development system [21]. A. Third-Order System The system here is implemented on a hardware simulator (Feedback Instruments Ltd. PCS327), and has a nominal
transfer function of (12) For the ideal case where no disturbances are present in the feedback loop, Fig. 5 shows the output signal during the tuning phase. It is assumed that commissioning of the controller has been carried out and that an initial estimate of the critical gain
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(a)
(b) Fig. 14.
Empirical frequency response estimates for the first-order plus time delay system Mean (—), 1: uncertainty region (— —), and nominal (—1—).
and period are available from the nominal transfer function. The maximum useful harmonic of the excitation is placed one octave above the initial estimate (0.27 Hz), which gives 0.54 Hz. The number of samples of the signal is selected to give a sampling frequency of at least 20 Hz, with the minimum being 240 samples, as discussed in Section III-A. In this case the number of samples is 540. Two and a quarter periods of
the signal are added to the setpoint, with the initial quarter of the first period being discarded due to the presence of transients. Following the tuning period shown in Fig. 5 a ZN step-response occurs. The controller parameters for each of the rule-based designs and are shown in Table II. The values estimated for are 7.51 and 3.7 s, respectively. The PID parameters for the
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(c) Fig. 14. (Continued.) Empirical frequency response estimates for the first-order plus time-delay system Mean (—), 1: uncertainty region (— —), and nominal (—1—).
MEAN
AND
STANDARD DEVIATION
OF THE
TABLE V CONTROLLER PARAMETERS
Refined ZN rule are the same as the ZN and the MT-ZN and rules, with only the setpoint filtering parameters changing. Step and load responses for this system with the parameters given in Table II are shown in Figs. 6 and 7, respectively. With this system, the ISTE and KLV rules give the best tracking performance followed closely by the R-ZN rule. The advantage of the two degree of freedom controller is clear by the superior load response of the R-ZN rule. The SO-OV and NO-OV rules give sluggish load disturbance rejection and also do not lead to low overshoot when compared to the other rules. The PIAE rule leads to the best load rejection, which is its intended use. Although ISTE tuning formulas are available for disturbance rejection, both tracking and disturbance rejection
FOR THE
FIRST-ORDER PLUS TIME-DELAY SYSTEM
properties are felt to be important here. The MT-ZN rule reduces the overshoot obtained with the ZN rule slightly. Gaussian noise with a root mean square (rms) value of 0.1 V and a bandwidth of 15 Hz was then added to the process output while in the feedback path. The frequency response was repeated under the same conditions 25 times. The mean magnitude and phase of the frequency response estimates together with the coherence function are shown in Fig. 8. Estimated uncertainty regions (one standard deviation) for each of these quantities are also shown. The magnitude and phase plots include the frequency response function estimated when no noise was added to the feedback loop. As can be seen the frequency response estimates are effectively unbiased and generally have a high coherence. The mean and standard
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Fig. 15.
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Output of hot-air flow device during the tuning phase, followed by step response.
deviation of the 25 sets of controller parameters are shown in Table III. Typical step responses are shown in Fig. 9. The number of rules shown is limited to four due to the scattered nature of the data. From this application it can be concluded that the identification method gives very accurate results in the presence of modest stochastic disturbances. The coherence function provides a useful indicator of the accuracy of each frequency response estimate, even when calculated from just two periods of the excitation. B. First-Order Plus Dead Time System The system under investigation in this section has a nominal transfer function of (13) This is implemented on a hardware simulator (Feedback Instruments Ltd. PCS327), in which a Pad´e approximation of unknown (low) order is used to represent the time-delay. It is assumed here that no a priori estimate of the system bandwidth is available, and so a limit cycle is used. A standard procedure in this situation is to monitor the process output to determine a suitable value for the relay hysteresis (to prevent relay “chattering”), and then to measure several periods of the limit cycle data [22]. With no disturbances affecting the feedback loop (and hence no hysteresis), tuning was carried out with the limit cycle yielding approximate values for the critical gain and frequency of 0.432 and 0.275 Hz, respectively. Using the estimate of the critical frequency, the maximum frequency is set to 0.551 Hz, with equal to 548. Two and a quarter periods of the excitation were used to estimate the frequency response, which resulted in the parameter values shown in Table IV.
The values estimated for and are 2.27 and 3.66 s, respectively. Step and load responses for this system with the parameters given in Table IV are shown in Figs. 10 and 11, respectively. With step changes in this system the R-ZN, the MT-ZN and ISTE rules produce very similar results with a low overshoot. The KLV rule produces a similar response but with a greater undershoot. This time the SO-OV and NO-OV rules produce low overshoot and no overshoot, respectively. The PIAE rule gives a similar response to that of the ZN rule. With the load disturbance, the PIAE rule again gives the fastest response, and the ISTE rule also performs well, even though both formulas are intended for optimum tracking performance. As with the third-order system the SO-OV and NO-OV rules produce slow disturbance rejection. A Gaussian noise source with bandwidth 15 Hz and rms value of 0.3 V was added to the output of the process inside the feedback loop. The tuning cycle is shown in Fig. 12. As can be seen, significant disturbances affect the output. The output is monitored for approximately 8 s, followed by a number of periods of a limit cycle. The limit cycle yielded values for the critical gain and frequency as 0.694 and 0.231 Hz, respectively. The error in the limit cycle estimates is mainly due to the significant hysteresis present in the relay to combat the disturbance. The multifrequency signal is added to the 23 s and is removed at 96 s. A sample of setpoint at the resulting step responses is shown in Fig. 13. The frequency response estimation was repeated 25 times, as in Section IV-A. The mean magnitude, phase, and coherence together with the uncertainty intervals are shown in Fig. 14. The frequency response estimate when the noise source is absent is also shown. The mean of the resulting controller parameters and their standard deviations is shown in Table V. The uncertainty of proportional gain has increased, but it is clear that the
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(a)
(b) Fig. 16.
Empirical frequency response estimates for the hot-air flow device. Mean (—), 1:
identification together with the median smoothing method has given very reliable estimates in the presence of very high disturbance levels. Again the use of the coherence function allows reliable gain and frequency estimates to be selected in a simple and reliable manner. This example has been included mainly to illustrate the accuracy of the identification results. The level of noise is thought to be higher than would be encountered in most practical situations.
uncertainty region (— —), and nominal (—1—).
C. Hot-Air Flow Device The system under investigation here is a PT 326 hot-air flow device available from Feedback Instruments Ltd. Air is blown through an electric heater, the power to which may be controlled. The temperature of the air is measured at the end of a plastic tube connected to the fan. This system represents quite a realistic example since it has a limited linear range and has an internal noise source due to turbulence within
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(c) Fig. 16.
(Continued.) Empirical frequency response estimates for the hot-air flow device. Mean (—), 1:
uncertainty region (— —), and nominal (—1—).
Fig. 17. Step responses for hot-air flow device. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1), SO-OV (–13–1), NO-OV (–12–1), MT-ZN (– –o– –), R-ZN (– –3– –).
the plastic tube. The same procedure is followed here as was adopted in Section IV-A. The output of the process during the initial tuning cycle is shown in Fig. 15. The manufacturer’s data suggested a value of 1.23 Hz for the 180 crossover frequency and so the bandwidth of the excitation is set to 2.46 Hz, resulting in , with a sampling frequency of 40.07
Hz. The period of the excitation signal is approximately 6 s. The frequency response estimation was carried out 25 times and controller parameters calculated for each rule. Fig. 16 shows the mean frequency response estimate together with the coherence function and their standard deviations. It can be seen that the frequency response estimate has a high accuracy,
MCCORMACK AND GODFREY: RULE-BASED AUTOTUNING
Fig. 18.
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Load responses for hot-air flow device. ZN (—o—), ISTE (—3—), KLV (—2—), PIAE (–1o–1),SO-OV (–13–1), NO-OV (–12–1).
MEAN
AND
STANDARD DEVIATION
OF THE
TABLE VI CONTROLLER PARAMETERS
with a coherence of almost unity across the frequency range. The resulting controller parameters are given in Table VI. The and are 4.88 and 0.76 s, mean values estimated for respectively. Step and load responses for the system with each set of controller parameters are shown in Figs. 17 and 18, respectively. The step response obtained with the ISTE rule exhibits minimal overshoot and a fast rise-time, with the KLV being slightly more oscillatory. The ZN, PIAE, R-ZN, and MT-ZN rules produce a single overshoot with the setpoint weighting of the R-ZN and MT-ZN reducing the magnitude of the overshoot slightly. The SO-OV and NO-OV rules produce slow response times with large significant overshoots. The load responses of the ZN and PIAE rules exhibit the fastest settling times with, (as with the other systems), the SO-OV and NO-OV rules
FOR THE
HOT-AIR FLOW DEVICE
producing very sluggish responses. The ISTE rule results in a slower settling time than for the ZN and PIAE rules, but again ISTE rules are available for load rejection if this is the only design goal. The KLV rule also produces sluggish performance. V. CONCLUSION Most commercial autotuners use one of three methods for updating the PID controller parameters. In the first, normal control is interrupted, a relay is inserted in the loop in place of the controller, and the amplitude and period of the resulting limit cycle are measured. The controller parameters are updated and the controller is switched back into the loop. Some suggestions for using relay excitation with closed-loop ˚ om [23]. The second approach opens systems are given by Astr¨
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the loop, measures at least part of an open-loop step response of the process, and then closes the feedback loop again. The controller parameters are updated on the basis of the step response. The third approach involves switching off full power before the setpoint is reached and then estimating and from the resulting transient. All of these approaches involve interruption of the normal closed-loop control, which is an undesirable feature. In this paper, an alternative approach, in which closedloop operation is maintained throughout, has been investigated. The approach uses frequency domain system identification techniques to estimate the open-loop frequency response of the process from data obtained while the loop is closed. Periodic excitations have been used, enabling frequency response estimates to be obtained in closed loop at comparatively high signal-to-noise ratios. The identification experiment may be automated easily, and only a modest amount of computation is needed. The identification techniques have been applied to three realistic examples, and it has been shown that reliable results have been obtained in all three situations. An arbitrary number of frequency response estimates may be obtained, thereby allowing more confidence to be placed in the controller design and/or further processing of the frequency response. In all three examples, a comparatively broadband perturbation signal has been used. This consisted of a multifrequency (sum-of-harmonics) signal with the specified harmonics being one to 15 consecutively. It is, of course, essential that the bandwidth of the signal contains the frequency (or frequencies) of interest, and this may require some prior knowledge of the process dynamics. In the first example (Section IV-A), a hardware simulator was used to simulate a third-order process, and the highest frequency of interest (the negative real axis crossover point) was known with good accuracy. In the second example (Section IV-B), although the same hardware simulator was used, there was some uncertainty about the way in which the pure time-delay was approximated in the simulator. In this case, it was assumed that no prior knowledge of the dynamics was available, and to obtain an initial estimate of the system dynamics, a relay was inserted into the loop, although with level of noise present, this had to incorporate significant hysteresis to prevent relay “chattering.” It should be emphasised that the relay was only used in the first tune and not subsequently. Further, it is felt that, in most applications, prior knowledge of the process dynamics is likely to make the initial relay phase unnecessary. This was certainly the case in the third example (Section IV-C), where manufacturer’s data provided reasonably good prior knowledge of the process dynamics. In theory, a signal containing fewer harmonics, with more power in each harmonic, could have been used, but it is essential to trade off possible inaccuracies in prior knowledge, and changing process dynamics, against reducing the number of harmonics. This is certainly a topic for further research. A notable feature of recent research has been the interest in improving on the performance of the widely quoted ZN tuning rules, which date from 1942. As part of the present paper eight tuning rules (including those of Ziegler and Nichols) have been
compared, in terms of both setpoint (tracking) and disturbance (regulating) control. The availability of tuning rules for two degree of freedom controllers allows both criteria to be treated for a given set of PID parameters. In summary, a new approach to autotune controller design has been proposed. Except in the case of processes in which there is very little prior knowledge of the dynamics, the method removes the need for relay tuning or open-loop step response tuning. The dynamics are estimated in normal closedloop operation, with a small-amplitude signal added to the setpoint during the tuning phase. The techniques have been shown to work well on three examples, the first a hardware simulated process, the second a hardware simulated process with a considerable amount of noise present, and the third a laboratory-scale heating process. REFERENCES ˚ om, C. C. Hang, P. Persson, and W. K. Ho, “Toward intelligent [1] K. J. Astr¨ PID control,” Automatica, vol. 28, no. 1, pp. 1–9, 1992. ˚ om and T. H¨agglund, Automatic Tuning of PID Controllers, [2] K. J. Astr¨ Instrument Soc. Amer., Research Triangle Park, NC, 1988. [3] M. Zhuang and D. P. Atherton, “Automatic tuning of optimum PID controllers,” Inst. Elect. Eng. Proc., Pt. D, vol. 140, no. 3, pp. 216–224, 1993. [4] A. Voda and I. D. Landau, “A method for the auto-calibration of PID controllers,” Automatica, vol. 31, no. 1, pp. 41–53, 1995. ˚ om and T. H¨agglund, “Automatic tuning of simple regulators [5] K. J. Astr¨ with specifications on phase and amplitude margins,” Automatica, vol. 20, no. 5, pp. 645–651, 1984. [6] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal. Cambridge, U.K.: Cambridge Univ. Press, 1989. [7] J. G. Ziegler and N. B. Nichols, “Optimum settings for automatic controllers,” Trans. Amer. Soc. Mech. Eng., vol. 64, pp. 759–768, 1942. [8] D. E. Seborg, T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control. New York: Wiley, 1989. ˚ om, and W. K. Ho, “Refinements of the [9] C. C. Hang, K. J. Astr¨ Ziegler–Nichols tuning formula,” Inst. Elect. Eng. Proc., Pt. D, vol. 138, no. 2, pp. 111–118, 1991. [10] D. W. Pessen, “A new look at PID-controller tuning,” Trans. Amer. Soc. Mech. Eng., J. Dynamic Syst., Meas., Contr., vol. 116, pp. 553–557, 1994. ˚ om, “Automatic initialization of a robust [11] M. Lundh and K. J. Astr¨ self-tuning controller,” Automatica, vol. 30, no. 11, pp. 1649–1662, 1994. [12] D. P. Atherton and M. Zhuang, “Tuning PID controllers with integral performance criteria,” in Inst. Elect. Eng. Int. Conf. Contr. ’91, Edinburgh, U.K., pp. 481–486. [13] C. Kessler, “Das symmetrische optimum,” Regelungstetechnik, vol. 6, no. 11, pp. 395–400, 1958. [14] R. J. Mantz and E. J. Tacconi, “Complementary rules to Ziegler and Nichols’ rules for a regulating and tracking controller,” Int. J. Contr., vol. 49, no. 5, pp. 1465–1471, 1989. ˚ om, “Ziegler–Nichols auto-tuners,” Dept. Automat. Contr., [15] K. J. Astr¨ Lund Inst. Technol., Tech. Rep. TFRT-3167, 1982. [16] K. Godfrey, “Introduction to perturbation signals for frequency-domain system identification,” in Perturbation Signals for System Identification, K. Godfrey, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1993, ch. 2. [17] P. Guillaume, J. Schoukens, R. Pintelon, and I. Koll´ar, “Crest-factor minimization using nonlinear Chebyshev approximation methods,” IEEE Trans. Instrum. Meas., vol. 40, pp. 982–989, 1991. [18] K. Godfrey, “Introduction to perturbation signals for time-domain system identification,” in Perturbation Signals for System Identification, K. Godfrey, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1993, ch. 1. [19] P. E. Wellstead, “Nonparametric methods of system identification,” Automatica, vol. 17, no. 1, pp. 55–69, 1981. [20] M. Gevers, “Toward a joint design of identification and control?” in Essays on Control: Perspectives in the Theory and Its Applications, H. L. Trentelman and J. C. Willems, Eds. Boston, MA: Birkh¨auser, 1993, ch. 5. [21] National Instruments Corporation, LabVIEW for Windows User-Manual, 1993.
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[22] T. H¨agglund, Process Control in Practice. Bromley, U.K.: ChartwellBratt, 1991. ˚ om, “Tuning and Adaptation,” in 13th Triennial IFAC World [23] K. J. Astr¨ Congr., San Francisco, CA, 1996, pp. 1–18.
Anthony S. McCormack received the bachelor’s degree in electrical engineering from the University of Glamorgan in 1991 and the Ph.D. degree for work on system identification and signal processing from the Department of Engineering at the University of Warwick, Coventry, U.K., in 1995. From 1991 to 1995 he was a Research Fellow in the Department of Engineering at the University of Warwick working on experiment design for system identification and the application of frequency domain identification to autotune control systems. He is Operations Manager of the Real-time Systems Group at Tensor Technologies in Dublin, Ireland. His main research interests are in frequency domain system identification and process control.
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Keith R. Godfrey received the Doctor of Science degree from the University of Warwick, Coventry, U.K., in 1990 for publications with the collective title “Applications of Modeling, Identification, and Parameter Estimation in Engineering and Biomedicine.” He is Head of the Electrical and Control Group in the Department of Engineering at the University of Warwick. He is author of a book on compartmental modeling published by Academic Press in 1983 and is coeditor of Signal Processing for Control (New York: Springer, 1986). He is also editor (and author of the first two chapters) of Perturbation Signals for System Identification (Englewood Cliffs, NJ: PrenticeHall, 1993) (now available from the editor). He is author or coauthor of more than 150 papers.
