- Email: [email protected]
The Normal-Mode-Inaction Adaptive PID Controller
Copyright © IFAC Adaptive Systems in Control
and Signal Processing, Grenoble, France, 1992
THE NORMAL-MODE-INACTION ADAPTIVE PID CONTROLLER Chang Chieh Hang, Shi-Zhong Hel and Tong Heng Lee Department o/Electrical Engineering, National University o/Singapore,10 Kent Ridge Crescent, Singapore 0511
is carried out using a recursive least squares estimator. Unlike [1], it is proposed that two sets of filters in parallel are used to track two points on the Nyquist curve. This will be used to compute the ultimate period and ultimate gain using interpolation or extrapolation and linear approximation of the phase and magnitude response curves. It also allows the rapid change of the centre frequency of the filter when necessary. To facilitate the adaptation of PID controller over a wide range of process dynamics we shall use the refined Ziegler-Nichols tuning formula [7], [8]. Robustness in the detection of parameter change is further ensured by checking whether there is a significant change of the cross-over point before and after the occurrence of an "abnormality".
Abstract
This paper presents a new adaptive PID controller that we call the normal-mode-inaction (NMI) adaptive PID controller_ Unlike the conventional adaptive PID controller, its control parameters are normally fixed, and adaptation only occurs when there is a significant change in the magnitude or phase of certain point on the Nyquist plot that is being tracked continuously. In the proposed NMI adaptive PID controller, a relay tuning procedure is used to initialize the adaptive controller. The adjustment of the PID controller parameters is based OIl the refined Ziegler-Nichols tuning formula. Simulation results and analysis have confirmed that the NMI adaptive PID controller has improved robustness properties.
1
This paper is organized as follows. The initialization of the NMI adaptive PID controller using a relay auto-tuner is briefly described in Section 2. In Section 3 the design of the NMI adaptive PID controller is presented. Section 4 contains the supervision of the PID controller. In Section 5 the NMI adaptive PID controller is illustrated. Conclusions are given in Section 6.
Introduction
Adaptive control has been one of the major research areas in automatic control for 30 years [1]-[4]. In current practice, the usual philosophy of adaptive control is that in its normal mode of operation, the adaptive controller will be continuously updated. Based on our earlier findings of the unpredictable behavior of adaptive control systems upon the onset of unmeasurable load disturbance [5], a proposal is made in this paper to adopt a different philosophy in adaptive control, which we call "normal- mode-inaction (NMI) adaptive control". The main idea of this approach is that the normal mode of operation of this class of adaptive controller should be one of inaction, i.e., the controller parameters are normally fixed, and adaptation or updating only occurs during instances of a well-defined "abnormality". In this paper the "abnormality" is defined as a significant change in plant parameters. The adaptation or updating is only activated upon confirmation of this abnormality. Thus, when an unmeasurable load disturbance occurs, the NMI controller parameters will not be updated unless the disturbance also causes a significant parameter change. As in conventional adapt.ive control, we assume that the process parameters are slowly time-varying so that the controller effectively sees a constant but unknowlI process [3], [,1].
2
Initialization Of The Adaptive Controller
It is now well know in the literature and in practice that the initialization of any adaptive controller is an important issue [4]. The initialization of the proposed NMI adaptive PID controller can be simply achieved by using relay auto-tuning with self-corrective bias [6]. The greatest merits of this relay feedback method are that a priori information about the time scale or the structure of the process is not needed, and that a good excitation signal that is tuned to the process is generated automatically. By using a relay auto-tuner, we obtain initial values of the PID parameters to control the process; we also obtain valuable process information for the initialization of the identification procedure in the adaptive I'll) controller, sllch as the centre frequency of the BPF. This mclhod is based 011 the observation that a process with the dynamics typically encollntered in process control will exhibit limit cycle oscillation under relay control [2], [6J. The relay auto-tuner principle is shown in Fig.l. The frequency of the limit cycle is approximately the ultimate frequency where the process has a phase Jag of 180°. If d is the relay amplitude and a is the process output amplitude, the ultimate gain [2J is approximately given by
In the NMI adaptive PID controller the change of the phase and magnitude of a tracked point on the Nyquist plot is used as an indication of whether there are significant changes of the process parameters. This point on the Nyquist plot is tracked by approximating the process as a second order plus time delay model that is rather accurate at a certain frequency. Signals for parameter estimation are obtained by filtering the process input and output using narrow band-pass filters with centre frequency being initially determined by a relay experiment [I]. The parameter estimation
IOn leave from Department of Automation, Tsinghua University, Beijing 100084, China.
383
3.1
y.
PLANT
The narrow band-pass filters
The narrow band-pass filters (BPFs) are of the form 3
GBP(3) =
(1)
'Ira
The period of the oscillation is easily obtained by measuring the time between zero crossings. The amplitude may be determined by measuring the peak to peak values of the output.
t - t
tl
1
d+ k () + t2 p tt + t2
_1_ _ 2
1+.'+.,
3.2
The least-squares estimators
The filtered control and measurement signals by using BPFs can be approximated by two sine waves of different amplitudes and phases. The ratio a between the amplitudes and the phase shift cp between the two signals gives the tracked Nyquist point [l].
(4) The process is modeled as a delayed second oeder model at the centre frequ encies of the BPFs [1]:
T
(2)
edt
(3)
able choice of the damping ratio ( is 0.3. The choice of the frequency w will be discussed in section 4.
The accuracy of the relay auto-tuning method depends on the relay oscillation being symmetrical. In practice, especially in process control application where the tuning duration is long, this condition may be easily violated owing to cOl1trol interaction or unexpected load disturbances. In this paper a relay auto-tuning scheme with self-correction as proposed in [6] is used. If tt and t2 are the respective intervals of positive and negative relay output, d is the relay amplitude, y is the process output, and kp is the process static gain, then the corrective bias for the relay is given by
corrective bias =
+ 2(ws + w2
This filter will give a relatively high gain at the frequency w, and suppress the signal at other frequencies. The control and measurement signals are filtered through narrow bandpass filters at the frequency w. By using this filter, low frequency signals will be filtered out and this reduces the influence of load disturbance on the parameter estimation. Likewise high frequency signals will be filtered out to minimize the effect of high frequency noise on the parameter estimates. It has been found from experience that a suit-
Figure 1: Block diagram of the relay auto-tuner
ku = 4d
32
T
(5)
A large static load disturbance greater than the relay amplitude will quench the relay output. In this case the corrective bias can be increased in the same direction as the current relay output until the limit cycle reappears.
where Uj and Yj are the filtered control and measurement signals respectively and h is the sampling period of the parameter estimator given by
From the relay auto-tuning experiment we can obtain the ultimate period tu and the ultimate gain ku used in the initialization of the PID controller and the centre frequencies of the BPFs used in the process identification. A small hysteresis may be added to the relay if the measurement is very noisy.
(6) The estimated parameters bl and b2 are easily determined by means of the recursive least squares estimation method. Using estimated parameters, the amplitude and phase of the tracked point on Nyquist plot can be obtained according to
[I]: btsin(wuh)
'P
= arctan(b ICOS (Wu!') + b) - 2wu h 2
(7) (8)
3.3
The method of computing PID controller parameters using the Ziegler-Nichols tuning formula is based on the automatic measurement of the ultimate gain and period. Various techniques such as the relay-feedhack, approximate system identification and cross-correlation [8] have been developed to determine the ultimate gain and ultimate period without the need to drive the system to the verge of instability. The automatic measurement of the ultimate gain and period in the NMI adaptive PID controller is made possible from the information on the amplitudes and phases of the tracked points and piece-wise linear approximation of the amplitude and phase response curves, which is shown in the following.
y
Figure 2: Block diagram describing the identification process
3
PID controller design
The NMI Adaptive PID Controller
In the proposed NMI adaptive PID controller two points on the Nyquist curve will be tracked to detect possible changes in the process dynamics. It involves the use of narrow bandpass filters and least squares estimators. Such an identification technique was first proposed by Astram and Hagglund [1], and is illustrated in Fig.2.
It is assumed that the two tracked points are Pt and P2 and they have magnitudes at and a2 and phases 'PI and 'P2 respectively. The magnitude of the critical point is assigned to be aCT while the phase of the critical point 'PCT is -'lr. The magnitude and phase of the tracked points can be determined by using equations 7 and 8. For a small variation in frequency, the phase and magnitude curves can be approxi-
384
mated to be linear for that small range of frequencies . Then the critical frequency Wc can be obtained from the linear approximation of the phase response curve using the following equation
Iog (Wc ) =
log(w2) -log(wd ( <.p2 -
-7r -
<.pI
<.pI
) + Iog (WI )
simulation study has indicated that the transient period of a stable process which is used in this paper is about 0.8 times to 1.8 times of the ultimate period tu.
3.) Accurate parameter estimation requires that the process is sufficiently excited. Therefore adaptation is only allowed when both the control signal and the measurement signal have had a transient recently. Parameter estimation will need to be frozen when the process is not sufficiently excited. Furthermore, the PID parameters in the NMI adaptive PID controller are only updated when both points PI and P2 are identified.
(9)
After knowing Wc we can determine the magnitude of the critical point from equation 9 which uses linear approximation of the magnitude response curve within that range of frequencies .
log(a2) - log(at) log(acr ) = I ( ) I ( ) (log(wc) -log(wtJ) og W2 - og WI
+ log(ad
4.) Additional checks are made on the estimated parameters to ensure that the estimates obtained have reasonable values. This is achieved by using the equations 7 and 8 which relate the estimated parameters to the physical parameters a and <.p. Reasonable values for the parameter estimates should result in values for <.p that lie inside a sector of the third quadrant, and for a which should not vary by more than a specified factor from the initial value given by the auto-tuner.
(10) The ultimate period, tu, and ultimate gain, ku, are determined by
(11) (12) The updating of the PID controller using the latest estimates of tu and ku is straightforward using the refined ZieglprNichols formula [A.3] - [A. 15] as outlined in Appendix I. Finally, the refined formula requires t.he knowledge of B or k that can be easily computed using formula [A.l] and [A.2] for B'or k.
4
5.) When a change in the process characteristics has occurred, the approximate adjustments in the centre frequency of the BPFs and the normalized process gain are made according to the estimated process parameters. This is to ensure that the tracked points are always within the range where equations 9 and 10 can be applied; also. I j,;s ensures that the normalized process gain is appropriately updated so that equations (A.12) - (A.l5) can be applied.
Supervision of the NMI Adaptive PID Controller
It is well known [4] that robust adaptive controller in prac-
tice requires supervision. In the NMI adaptive PlO controller the proposed supervision is essentially based on the measure of certainty that an abnormal event has occurred . When these "abnormalities" are confirmed the mechanism for updating the controller parameters in the NMI adaptive PlO controller is activated. Though narrow band-pass filters that attenuate effects of high frequency noise and low frequency load disturbance are used, we have found that this is not always enough, especially when the load disturbance is large. To overcome the effect of a load disturbance on the controller and ensure correct adaptation or updating, the following supervisory measures are recommended for the NMI adaptive PlO controller:
5
The performance of the NMI Adaptive PID Controller
After the relay auto-tuner commissioning, the adaptive control system is initialized with the identified period, tu, and gain, ku. The relay auto-tuner is then disconnected from the process. The NMI adaptive PlO control system is activated. Without loss of generality, the following process is used throughout the simulation study.
(13)
1.) We have used the change in the phase or magnitude of the tracked point as an indication of the change in process parameters, i.e., as a definition of "abnormality". When the change of the phase and magnitude of the tracked point is larger than some preset tolerance the PlO paramet.ers of the NMI adaptive PID controller will be updated subject to some further conditions that are discussed below .
The sampling periods of the estimators for each tracked point are different where hi is for EST1 and h2 for EST2. They can be calculated using equation 6. The forgetting factor used in the recursive least squares estimation is 0.99. The input to the ESTl and EST2 are the averaged values of the output from BPFl and BPF2 respectively. For EST1, we use a moving average of kl inputs. For example,
2.) The change in the values of the estimated crossover frequencies from the beginning of the transient period 1.0 the end is used to confirm the change in the process parameters. The definition of this "abnormality" is based on the following observation. When a load disturbance is introduced into the process the value of the estimated crossover frequency at the beginning of the transient period and the cnd is not changed unless the process parameters have also changed. Thus, if the difference in values of the estimated crossover frequency from the beginning of the transient period to the end is smaller than the preset tolerance the PlO parameters of the NMI adaptive PID controller will not be adjusted. The ability of the adaptive PID controller to overcome the effect of an unmeasurable load disturbance is improved by checking of the estimated crossover frequency from the beginning of the transient period to the end. An extensive
and Yal = [Yf1(k - kd
+ Yf1(k -
kl - 1)
+ ... + Y/I(k)]/k l
(15)
where Ual and Yal are the averaged values of the outputs U/I and Y/I of BPFl respectively. EST2 uses moving averages of k2 . Values of kl = 9, k2 = 10 have been used. In the simulation the set-point response is first made before a load disturbance is introduced into the input of he process and the parameters Tb T2 and Kp of the process are changed.
385
5.1
a change in the process parameters. The simulation results also show that the PID parameters of the NMI adaptive PlO controller are adjusted when the process parameters are changed. In the closed-loop response of the process, it is observed that the output of the process is quickly adjusted to the appropriate set-point value as a consequence of the updating of the controller parameters.
Set-point response and load response
The change of the magnitude and phase of the tracked points on the Nyquist plot is used as indication of the change in the process parameters in this section. The initial value Wu is found using the relay auto-tuner. The sampling periods of the estimators are determined as follows.
h2 = 0.9758
(16)
It is found that the magnitude and phase of the tracked points are not changed and the controller parameters are not adjusted when a load disturbance (I = 0.1) is introduced into the input of the process. This results from the filtering of the narrow band pass filters. But a change of the controller parameters occurs when a simulated load disturbance is increased to more than 0.2. For example, when the value of a load disturbance is -1, the magnitude and phase of the tracked points and controller parameters are obviously changed (at t = 958 in Fig.5). The simulation results indicate that the ability of the NMI adaptive PID controller in overcoming problems arising from unmeasurable load disturbances is limited when the technique of detection of the transient period is not incorporated.
(17) The relay tuning is made before t = 158. The set-points are introduced into the system at t = 158 and t = 558 respectively and a load disturbance is introduced at t = 958. The simulation result of the set-point response of the NMI adaptive PID controller is shown in Fig.3. To make the comparison of the performances of the NMI adaptive PlO controller and the normal-mode· action (NMA) adaptive PID controller, the set-point response of the NMA adaptive PID controller that have same conditions with NMI adaptive PID controller is shown in FigA.
5.2
Performance without the detection of the transient period
3.) The ability to overcome problem arising from load disturban ce is improved after incorporating the technique described in section 5.3. From the load disturbances where magnitudes are larger than 0.5, and where the estimated crossover point has changed considerably, load regulation response remains good as the normal-mode-inaction feature has kept the PlO parameters unchanged in the absence of a conclusive "abno rrn~lit.y" condition. The relevant simu lation r('sults are sholl 11 " ' Fig.6 (at t = 958).
A load disturbance is introduced into the input of the process at t = 958 and the process parameters are changed at t = 1758. Before t = 1758, the process parameters are kp = 1, TJ = 1 a nd T2 = 2. After the process parameters are changed, k,> = 1.2, TJ = 1 and T2 = 4. The set-points are introduced into the system at t = 158, 558, 1358, and 2158 respectively. The performances of the NMI adaptive PID controller without the detection of the transient period arc shown in Fig.5.
5.3
4.) After a load disturbance is introduced into the system and the parameters of the process are changed the set-point responses are made again. We can obviously observe from Fig.5 and Fig.6 that the set-point response of the NMI adaptive PlO controller with the detection of the transient period is superior to the one without the detection of the transient period.
Performace with the detection of the transient period
In this section, we consider the use of the technique where the values of the estimated crossover point at the beginning of the transient period, and at the end, are monitored for possible changes. Monitored changes in both the phase and magnitude of the tracked points on the Nyquist plot and in the estimated crossover frequency are collecti vely used as an indication of a change in process parameters. The PID parameters are adjusted when the change "r . II!' phase and magnitude of the tracked points and the cl,"nge of th e estimated crossover frequency are all greater than the respective pre-set tolerances. The times of the int.roduction of the set-point and a load disturbance and time of the chang(, of the process parameter in the simu lation are the same as in sect ion 5.2. The performances of the N MI adaptive l' llJ con· troller with the detection of the t ransient period are shown in Fig.6.
5.4
6
Conclusoins
The key idea of the proposed NMI adaptive PID controller is that the normal mode of operation of this class of adaptive controllers is one of inaction , i.e., the adjustment, of the controller parameters on ly occurs during instances of a welldellned "abnormality". The initialization of the controller is achieved by using relay auto-tuning with self-corrective bias . The adjustment of the I'll) parameters of the controller is basC'd on the refined Ziegler-Nichols tuning form ulilllsing the n ormal i ~ed process gain. 1'0 overcome the effect of a load disturbance on the controller and ensure corr('ct. adapt.ation, sOJlle supe rvisory llleasur('s a.re used for th e NM I adaptive PlO controller, espC'ciall y rheckillgo f the estimated crossove r frequency froll. the transient period to the ('nd.
Summary of performance evaluation
Th(' following obsen'ations have been made from the simu· lation plots Fig.3 to Fig.6:
This paper shows that it is possible to realize the NMI adaptive PIU controller using the conc.ept of detectable "abnormality". The simulation resu lts have shown that the NMI adaptive PID controller can provide a solution for overcoming load disturbance problems in adapt ive cont rol. The ability to overcome problems arising from load disturbance is obviously improved when changes in the magnitude and phase of the tracked points are used as the indication of the c.hange of the process parameters and the techn ique of the transient period checking is incorporated.
1.) The results of the Fig.3 and Fig.4 clearly show that the set-point response and load response of the NMl adaptive PlO controller are supe rior to the NMA adaptive PIIJ controller. 2.) From the simulation plots of Fig.5 and Fig.6, we observe that changes of the magnitude and phase of the tracked points on the Nyquist plot occur when the process parameters are changed. The simulation results indicate that change in the magnitude and phase of the tracked points on the Nyquist plot can be used as the indication of
386
References
[5] T. H. Lee and C.C .Hang, "The effect of an unmeasurable step disturbance 011 Recursive Parameter Estimation and Adaptive Control", Proc. Int. Workshop 011 Application of Adaptive SystelTl Theory, Yale Ulliv. JUlle 1983, pp.227-233 .
[1] K. J. Astrom and T. Hagglund, "An industrial adaptive PID controller", Proc. IFAC Symphosium on Adaptive System in Control and Signal Processing, April 1989, pp.293-298. [2] K. J. Astrom and T. Hagglund, "Automatic Tuning of Simple Regulators with Specification on Phase and Amplitude Margins", Automatica, Vo1.20, No, 5, pp.645651, 1984.
[6] C. C. Hang and K. J. AstrOID, "Practical Aspects og PID Auto-tuners Based on Relay Feedback", Froc. 1nl. Symp. on Adaptive Control of Chemical Processes, Demark, August 1988, pp .149-154.
[3] K. J. Astrom and B. J. Wittenmark, "Computer Controlled System: Theory and Design", PRETICE-HALL, INC. , 1984.
[7J C. C. Hang and K. J. AstrOID, "Refinements of the Zeigler-Nichols Tuning Formula for PID Auto-tuner", Proc . of ISA 88 International conference and Exhibit, Houston , Texas, October 1988, pp. 1021-1030.
[4] K. J . Astrom, "Adaptive Control - A perspective", Proc. IFAC Symphosium on Adaptive System in Control and Signal Processing, April 1989, pp.1-6.
[8J C. C. Hang, K. J. Astrom and W. K. Ho, "Refinements of the Zeigler-N ichols Tuning Formula", lEE Proc. Pa7'/. IJ, March 1991 , pp.11l-1l8.
1.5 r-_ _~--~C""lo,.,sed"'-"'Lo"'o;"'_"R""es"""'ns""eT0"-f",-roc""",es... s~--~--_.
O,S
.{).SO'----2~0--~4O---6O~--8~0---1~00---12~0---'14O yp(-) vs time
The ou
1.5 r -_ _~--~C"' lo"'sed=-Lo "'o~R"'es"""ns"'eT°"-f"'r""oc""es"-s~--~--_.
.{).S 0
-I
20
40
60
lOO
80
120
0
140
20
I
40
-I
0
20
140
120
140
120
140
PlO parameter variation
1.5
V
60
120
a1 of the controUer
lr 40
lOO
80
uc( -) vs time
2.S
:~
al of the controller
60
yp (-). uc (-.) vs time The ou ut si
si
80
100
V
r---'l V
--
O.S 120
140
0 0
uc \'s time
20
40
60
- -
-" ..
--------_ ..
80
100
Kp(- ).Td( -·) vs time Phase variat ion of Point Traded
\
.{).S -I -1.5
-- ..
·2 -2.5
0
20
.{).S -I -1.5
-_ .. ".--------- ... _------------------- 40
·2
--------------
100 80 phasel (-). phase2 (--) vstime 60
Phase variation of Point Tracked
0
120
\
:------ --- ---------------'-- .-.
;'"
-2.5
140
0
20
40
60
80
100
phase 1(-), phase2( --) vs time
Figure 4: The response of the NMA adaptive PlO controller
Figure 3: The response of the NMI adaptive PlO controller
387
Closed·Loo Res onse of rocess
so
200
ISO 100 yp C-), uc C-) vs time The 0
ut si
250 50
100
150
200
250
yp C-), uc C-) vs lime
a1 of the controUer
",e out
t si
n:Ji
or the controller
2
_IL-______~______~______~~----~~----~ o SO 100 150 200 250
250
uc vs time
parameters change
a load disturbance PlO :lrtmeter v:lri:ltion I.S
0.5
\
....~ ..................................... ................: ................... .
0.5
0 0
50
100
150
200
250
0
Kp (-). Td C--) vs time
0.5 0 0
r50
SO
100
200
ISO
250
Kp C-), Td C-) vs lime
The estimated crossover freouencv
1.5
0
I.S
n,e estimated crossover r
uenc
\'---0.5 100
150
200
250
frequency vs time
°OL--L------S~O---------100~--------IS~O--------2~00~------~2S0 frequency vs time
Figure 5: The perfonnance of the NMI adaptive PID controller
388
Figure 6: The perfonnance of the NMI adaptive PID controlle
and Signal Processing, Grenoble, France, 1992
THE NORMAL-MODE-INACTION ADAPTIVE PID CONTROLLER Chang Chieh Hang, Shi-Zhong Hel and Tong Heng Lee Department o/Electrical Engineering, National University o/Singapore,10 Kent Ridge Crescent, Singapore 0511
is carried out using a recursive least squares estimator. Unlike [1], it is proposed that two sets of filters in parallel are used to track two points on the Nyquist curve. This will be used to compute the ultimate period and ultimate gain using interpolation or extrapolation and linear approximation of the phase and magnitude response curves. It also allows the rapid change of the centre frequency of the filter when necessary. To facilitate the adaptation of PID controller over a wide range of process dynamics we shall use the refined Ziegler-Nichols tuning formula [7], [8]. Robustness in the detection of parameter change is further ensured by checking whether there is a significant change of the cross-over point before and after the occurrence of an "abnormality".
Abstract
This paper presents a new adaptive PID controller that we call the normal-mode-inaction (NMI) adaptive PID controller_ Unlike the conventional adaptive PID controller, its control parameters are normally fixed, and adaptation only occurs when there is a significant change in the magnitude or phase of certain point on the Nyquist plot that is being tracked continuously. In the proposed NMI adaptive PID controller, a relay tuning procedure is used to initialize the adaptive controller. The adjustment of the PID controller parameters is based OIl the refined Ziegler-Nichols tuning formula. Simulation results and analysis have confirmed that the NMI adaptive PID controller has improved robustness properties.
1
This paper is organized as follows. The initialization of the NMI adaptive PID controller using a relay auto-tuner is briefly described in Section 2. In Section 3 the design of the NMI adaptive PID controller is presented. Section 4 contains the supervision of the PID controller. In Section 5 the NMI adaptive PID controller is illustrated. Conclusions are given in Section 6.
Introduction
Adaptive control has been one of the major research areas in automatic control for 30 years [1]-[4]. In current practice, the usual philosophy of adaptive control is that in its normal mode of operation, the adaptive controller will be continuously updated. Based on our earlier findings of the unpredictable behavior of adaptive control systems upon the onset of unmeasurable load disturbance [5], a proposal is made in this paper to adopt a different philosophy in adaptive control, which we call "normal- mode-inaction (NMI) adaptive control". The main idea of this approach is that the normal mode of operation of this class of adaptive controller should be one of inaction, i.e., the controller parameters are normally fixed, and adaptation or updating only occurs during instances of a well-defined "abnormality". In this paper the "abnormality" is defined as a significant change in plant parameters. The adaptation or updating is only activated upon confirmation of this abnormality. Thus, when an unmeasurable load disturbance occurs, the NMI controller parameters will not be updated unless the disturbance also causes a significant parameter change. As in conventional adapt.ive control, we assume that the process parameters are slowly time-varying so that the controller effectively sees a constant but unknowlI process [3], [,1].
2
Initialization Of The Adaptive Controller
It is now well know in the literature and in practice that the initialization of any adaptive controller is an important issue [4]. The initialization of the proposed NMI adaptive PID controller can be simply achieved by using relay auto-tuning with self-corrective bias [6]. The greatest merits of this relay feedback method are that a priori information about the time scale or the structure of the process is not needed, and that a good excitation signal that is tuned to the process is generated automatically. By using a relay auto-tuner, we obtain initial values of the PID parameters to control the process; we also obtain valuable process information for the initialization of the identification procedure in the adaptive I'll) controller, sllch as the centre frequency of the BPF. This mclhod is based 011 the observation that a process with the dynamics typically encollntered in process control will exhibit limit cycle oscillation under relay control [2], [6J. The relay auto-tuner principle is shown in Fig.l. The frequency of the limit cycle is approximately the ultimate frequency where the process has a phase Jag of 180°. If d is the relay amplitude and a is the process output amplitude, the ultimate gain [2J is approximately given by
In the NMI adaptive PID controller the change of the phase and magnitude of a tracked point on the Nyquist plot is used as an indication of whether there are significant changes of the process parameters. This point on the Nyquist plot is tracked by approximating the process as a second order plus time delay model that is rather accurate at a certain frequency. Signals for parameter estimation are obtained by filtering the process input and output using narrow band-pass filters with centre frequency being initially determined by a relay experiment [I]. The parameter estimation
IOn leave from Department of Automation, Tsinghua University, Beijing 100084, China.
383
3.1
y.
PLANT
The narrow band-pass filters
The narrow band-pass filters (BPFs) are of the form 3
GBP(3) =
(1)
'Ira
The period of the oscillation is easily obtained by measuring the time between zero crossings. The amplitude may be determined by measuring the peak to peak values of the output.
t - t
tl
1
d+ k () + t2 p tt + t2
_1_ _ 2
1+.'+.,
3.2
The least-squares estimators
The filtered control and measurement signals by using BPFs can be approximated by two sine waves of different amplitudes and phases. The ratio a between the amplitudes and the phase shift cp between the two signals gives the tracked Nyquist point [l].
(4) The process is modeled as a delayed second oeder model at the centre frequ encies of the BPFs [1]:
T
(2)
edt
(3)
able choice of the damping ratio ( is 0.3. The choice of the frequency w will be discussed in section 4.
The accuracy of the relay auto-tuning method depends on the relay oscillation being symmetrical. In practice, especially in process control application where the tuning duration is long, this condition may be easily violated owing to cOl1trol interaction or unexpected load disturbances. In this paper a relay auto-tuning scheme with self-correction as proposed in [6] is used. If tt and t2 are the respective intervals of positive and negative relay output, d is the relay amplitude, y is the process output, and kp is the process static gain, then the corrective bias for the relay is given by
corrective bias =
+ 2(ws + w2
This filter will give a relatively high gain at the frequency w, and suppress the signal at other frequencies. The control and measurement signals are filtered through narrow bandpass filters at the frequency w. By using this filter, low frequency signals will be filtered out and this reduces the influence of load disturbance on the parameter estimation. Likewise high frequency signals will be filtered out to minimize the effect of high frequency noise on the parameter estimates. It has been found from experience that a suit-
Figure 1: Block diagram of the relay auto-tuner
ku = 4d
32
T
(5)
A large static load disturbance greater than the relay amplitude will quench the relay output. In this case the corrective bias can be increased in the same direction as the current relay output until the limit cycle reappears.
where Uj and Yj are the filtered control and measurement signals respectively and h is the sampling period of the parameter estimator given by
From the relay auto-tuning experiment we can obtain the ultimate period tu and the ultimate gain ku used in the initialization of the PID controller and the centre frequencies of the BPFs used in the process identification. A small hysteresis may be added to the relay if the measurement is very noisy.
(6) The estimated parameters bl and b2 are easily determined by means of the recursive least squares estimation method. Using estimated parameters, the amplitude and phase of the tracked point on Nyquist plot can be obtained according to
[I]: btsin(wuh)
'P
= arctan(b ICOS (Wu!') + b) - 2wu h 2
(7) (8)
3.3
The method of computing PID controller parameters using the Ziegler-Nichols tuning formula is based on the automatic measurement of the ultimate gain and period. Various techniques such as the relay-feedhack, approximate system identification and cross-correlation [8] have been developed to determine the ultimate gain and ultimate period without the need to drive the system to the verge of instability. The automatic measurement of the ultimate gain and period in the NMI adaptive PID controller is made possible from the information on the amplitudes and phases of the tracked points and piece-wise linear approximation of the amplitude and phase response curves, which is shown in the following.
y
Figure 2: Block diagram describing the identification process
3
PID controller design
The NMI Adaptive PID Controller
In the proposed NMI adaptive PID controller two points on the Nyquist curve will be tracked to detect possible changes in the process dynamics. It involves the use of narrow bandpass filters and least squares estimators. Such an identification technique was first proposed by Astram and Hagglund [1], and is illustrated in Fig.2.
It is assumed that the two tracked points are Pt and P2 and they have magnitudes at and a2 and phases 'PI and 'P2 respectively. The magnitude of the critical point is assigned to be aCT while the phase of the critical point 'PCT is -'lr. The magnitude and phase of the tracked points can be determined by using equations 7 and 8. For a small variation in frequency, the phase and magnitude curves can be approxi-
384
mated to be linear for that small range of frequencies . Then the critical frequency Wc can be obtained from the linear approximation of the phase response curve using the following equation
Iog (Wc ) =
log(w2) -log(wd ( <.p2 -
-7r -
<.pI
<.pI
) + Iog (WI )
simulation study has indicated that the transient period of a stable process which is used in this paper is about 0.8 times to 1.8 times of the ultimate period tu.
3.) Accurate parameter estimation requires that the process is sufficiently excited. Therefore adaptation is only allowed when both the control signal and the measurement signal have had a transient recently. Parameter estimation will need to be frozen when the process is not sufficiently excited. Furthermore, the PID parameters in the NMI adaptive PID controller are only updated when both points PI and P2 are identified.
(9)
After knowing Wc we can determine the magnitude of the critical point from equation 9 which uses linear approximation of the magnitude response curve within that range of frequencies .
log(a2) - log(at) log(acr ) = I ( ) I ( ) (log(wc) -log(wtJ) og W2 - og WI
+ log(ad
4.) Additional checks are made on the estimated parameters to ensure that the estimates obtained have reasonable values. This is achieved by using the equations 7 and 8 which relate the estimated parameters to the physical parameters a and <.p. Reasonable values for the parameter estimates should result in values for <.p that lie inside a sector of the third quadrant, and for a which should not vary by more than a specified factor from the initial value given by the auto-tuner.
(10) The ultimate period, tu, and ultimate gain, ku, are determined by
(11) (12) The updating of the PID controller using the latest estimates of tu and ku is straightforward using the refined ZieglprNichols formula [A.3] - [A. 15] as outlined in Appendix I. Finally, the refined formula requires t.he knowledge of B or k that can be easily computed using formula [A.l] and [A.2] for B'or k.
4
5.) When a change in the process characteristics has occurred, the approximate adjustments in the centre frequency of the BPFs and the normalized process gain are made according to the estimated process parameters. This is to ensure that the tracked points are always within the range where equations 9 and 10 can be applied; also. I j,;s ensures that the normalized process gain is appropriately updated so that equations (A.12) - (A.l5) can be applied.
Supervision of the NMI Adaptive PID Controller
It is well known [4] that robust adaptive controller in prac-
tice requires supervision. In the NMI adaptive PlO controller the proposed supervision is essentially based on the measure of certainty that an abnormal event has occurred . When these "abnormalities" are confirmed the mechanism for updating the controller parameters in the NMI adaptive PlO controller is activated. Though narrow band-pass filters that attenuate effects of high frequency noise and low frequency load disturbance are used, we have found that this is not always enough, especially when the load disturbance is large. To overcome the effect of a load disturbance on the controller and ensure correct adaptation or updating, the following supervisory measures are recommended for the NMI adaptive PlO controller:
5
The performance of the NMI Adaptive PID Controller
After the relay auto-tuner commissioning, the adaptive control system is initialized with the identified period, tu, and gain, ku. The relay auto-tuner is then disconnected from the process. The NMI adaptive PlO control system is activated. Without loss of generality, the following process is used throughout the simulation study.
(13)
1.) We have used the change in the phase or magnitude of the tracked point as an indication of the change in process parameters, i.e., as a definition of "abnormality". When the change of the phase and magnitude of the tracked point is larger than some preset tolerance the PlO paramet.ers of the NMI adaptive PID controller will be updated subject to some further conditions that are discussed below .
The sampling periods of the estimators for each tracked point are different where hi is for EST1 and h2 for EST2. They can be calculated using equation 6. The forgetting factor used in the recursive least squares estimation is 0.99. The input to the ESTl and EST2 are the averaged values of the output from BPFl and BPF2 respectively. For EST1, we use a moving average of kl inputs. For example,
2.) The change in the values of the estimated crossover frequencies from the beginning of the transient period 1.0 the end is used to confirm the change in the process parameters. The definition of this "abnormality" is based on the following observation. When a load disturbance is introduced into the process the value of the estimated crossover frequency at the beginning of the transient period and the cnd is not changed unless the process parameters have also changed. Thus, if the difference in values of the estimated crossover frequency from the beginning of the transient period to the end is smaller than the preset tolerance the PlO parameters of the NMI adaptive PID controller will not be adjusted. The ability of the adaptive PID controller to overcome the effect of an unmeasurable load disturbance is improved by checking of the estimated crossover frequency from the beginning of the transient period to the end. An extensive
and Yal = [Yf1(k - kd
+ Yf1(k -
kl - 1)
+ ... + Y/I(k)]/k l
(15)
where Ual and Yal are the averaged values of the outputs U/I and Y/I of BPFl respectively. EST2 uses moving averages of k2 . Values of kl = 9, k2 = 10 have been used. In the simulation the set-point response is first made before a load disturbance is introduced into the input of he process and the parameters Tb T2 and Kp of the process are changed.
385
5.1
a change in the process parameters. The simulation results also show that the PID parameters of the NMI adaptive PlO controller are adjusted when the process parameters are changed. In the closed-loop response of the process, it is observed that the output of the process is quickly adjusted to the appropriate set-point value as a consequence of the updating of the controller parameters.
Set-point response and load response
The change of the magnitude and phase of the tracked points on the Nyquist plot is used as indication of the change in the process parameters in this section. The initial value Wu is found using the relay auto-tuner. The sampling periods of the estimators are determined as follows.
h2 = 0.9758
(16)
It is found that the magnitude and phase of the tracked points are not changed and the controller parameters are not adjusted when a load disturbance (I = 0.1) is introduced into the input of the process. This results from the filtering of the narrow band pass filters. But a change of the controller parameters occurs when a simulated load disturbance is increased to more than 0.2. For example, when the value of a load disturbance is -1, the magnitude and phase of the tracked points and controller parameters are obviously changed (at t = 958 in Fig.5). The simulation results indicate that the ability of the NMI adaptive PID controller in overcoming problems arising from unmeasurable load disturbances is limited when the technique of detection of the transient period is not incorporated.
(17) The relay tuning is made before t = 158. The set-points are introduced into the system at t = 158 and t = 558 respectively and a load disturbance is introduced at t = 958. The simulation result of the set-point response of the NMI adaptive PID controller is shown in Fig.3. To make the comparison of the performances of the NMI adaptive PlO controller and the normal-mode· action (NMA) adaptive PID controller, the set-point response of the NMA adaptive PID controller that have same conditions with NMI adaptive PID controller is shown in FigA.
5.2
Performance without the detection of the transient period
3.) The ability to overcome problem arising from load disturban ce is improved after incorporating the technique described in section 5.3. From the load disturbances where magnitudes are larger than 0.5, and where the estimated crossover point has changed considerably, load regulation response remains good as the normal-mode-inaction feature has kept the PlO parameters unchanged in the absence of a conclusive "abno rrn~lit.y" condition. The relevant simu lation r('sults are sholl 11 " ' Fig.6 (at t = 958).
A load disturbance is introduced into the input of the process at t = 958 and the process parameters are changed at t = 1758. Before t = 1758, the process parameters are kp = 1, TJ = 1 a nd T2 = 2. After the process parameters are changed, k,> = 1.2, TJ = 1 and T2 = 4. The set-points are introduced into the system at t = 158, 558, 1358, and 2158 respectively. The performances of the NMI adaptive PID controller without the detection of the transient period arc shown in Fig.5.
5.3
4.) After a load disturbance is introduced into the system and the parameters of the process are changed the set-point responses are made again. We can obviously observe from Fig.5 and Fig.6 that the set-point response of the NMI adaptive PlO controller with the detection of the transient period is superior to the one without the detection of the transient period.
Performace with the detection of the transient period
In this section, we consider the use of the technique where the values of the estimated crossover point at the beginning of the transient period, and at the end, are monitored for possible changes. Monitored changes in both the phase and magnitude of the tracked points on the Nyquist plot and in the estimated crossover frequency are collecti vely used as an indication of a change in process parameters. The PID parameters are adjusted when the change "r . II!' phase and magnitude of the tracked points and the cl,"nge of th e estimated crossover frequency are all greater than the respective pre-set tolerances. The times of the int.roduction of the set-point and a load disturbance and time of the chang(, of the process parameter in the simu lation are the same as in sect ion 5.2. The performances of the N MI adaptive l' llJ con· troller with the detection of the t ransient period are shown in Fig.6.
5.4
6
Conclusoins
The key idea of the proposed NMI adaptive PID controller is that the normal mode of operation of this class of adaptive controllers is one of inaction , i.e., the adjustment, of the controller parameters on ly occurs during instances of a welldellned "abnormality". The initialization of the controller is achieved by using relay auto-tuning with self-corrective bias . The adjustment of the I'll) parameters of the controller is basC'd on the refined Ziegler-Nichols tuning form ulilllsing the n ormal i ~ed process gain. 1'0 overcome the effect of a load disturbance on the controller and ensure corr('ct. adapt.ation, sOJlle supe rvisory llleasur('s a.re used for th e NM I adaptive PlO controller, espC'ciall y rheckillgo f the estimated crossove r frequency froll. the transient period to the ('nd.
Summary of performance evaluation
Th(' following obsen'ations have been made from the simu· lation plots Fig.3 to Fig.6:
This paper shows that it is possible to realize the NMI adaptive PIU controller using the conc.ept of detectable "abnormality". The simulation resu lts have shown that the NMI adaptive PID controller can provide a solution for overcoming load disturbance problems in adapt ive cont rol. The ability to overcome problems arising from load disturbance is obviously improved when changes in the magnitude and phase of the tracked points are used as the indication of the c.hange of the process parameters and the techn ique of the transient period checking is incorporated.
1.) The results of the Fig.3 and Fig.4 clearly show that the set-point response and load response of the NMl adaptive PlO controller are supe rior to the NMA adaptive PIIJ controller. 2.) From the simulation plots of Fig.5 and Fig.6, we observe that changes of the magnitude and phase of the tracked points on the Nyquist plot occur when the process parameters are changed. The simulation results indicate that change in the magnitude and phase of the tracked points on the Nyquist plot can be used as the indication of
386
References
[5] T. H. Lee and C.C .Hang, "The effect of an unmeasurable step disturbance 011 Recursive Parameter Estimation and Adaptive Control", Proc. Int. Workshop 011 Application of Adaptive SystelTl Theory, Yale Ulliv. JUlle 1983, pp.227-233 .
[1] K. J. Astrom and T. Hagglund, "An industrial adaptive PID controller", Proc. IFAC Symphosium on Adaptive System in Control and Signal Processing, April 1989, pp.293-298. [2] K. J. Astrom and T. Hagglund, "Automatic Tuning of Simple Regulators with Specification on Phase and Amplitude Margins", Automatica, Vo1.20, No, 5, pp.645651, 1984.
[6] C. C. Hang and K. J. AstrOID, "Practical Aspects og PID Auto-tuners Based on Relay Feedback", Froc. 1nl. Symp. on Adaptive Control of Chemical Processes, Demark, August 1988, pp .149-154.
[3] K. J. Astrom and B. J. Wittenmark, "Computer Controlled System: Theory and Design", PRETICE-HALL, INC. , 1984.
[7J C. C. Hang and K. J. AstrOID, "Refinements of the Zeigler-Nichols Tuning Formula for PID Auto-tuner", Proc . of ISA 88 International conference and Exhibit, Houston , Texas, October 1988, pp. 1021-1030.
[4] K. J . Astrom, "Adaptive Control - A perspective", Proc. IFAC Symphosium on Adaptive System in Control and Signal Processing, April 1989, pp.1-6.
[8J C. C. Hang, K. J. Astrom and W. K. Ho, "Refinements of the Zeigler-N ichols Tuning Formula", lEE Proc. Pa7'/. IJ, March 1991 , pp.11l-1l8.
1.5 r-_ _~--~C""lo,.,sed"'-"'Lo"'o;"'_"R""es"""'ns""eT0"-f",-roc""",es... s~--~--_.
O,S
.{).SO'----2~0--~4O---6O~--8~0---1~00---12~0---'14O yp(-) vs time
The ou
1.5 r -_ _~--~C"' lo"'sed=-Lo "'o~R"'es"""ns"'eT°"-f"'r""oc""es"-s~--~--_.
.{).S 0
-I
20
40
60
lOO
80
120
0
140
20
I
40
-I
0
20
140
120
140
120
140
PlO parameter variation
1.5
V
60
120
a1 of the controUer
lr 40
lOO
80
uc( -) vs time
2.S
:~
al of the controller
60
yp (-). uc (-.) vs time The ou ut si
si
80
100
V
r---'l V
--
O.S 120
140
0 0
uc \'s time
20
40
60
- -
-" ..
--------_ ..
80
100
Kp(- ).Td( -·) vs time Phase variat ion of Point Traded
\
.{).S -I -1.5
-- ..
·2 -2.5
0
20
.{).S -I -1.5
-_ .. ".--------- ... _------------------- 40
·2
--------------
100 80 phasel (-). phase2 (--) vstime 60
Phase variation of Point Tracked
0
120
\
:------ --- ---------------'-- .-.
;'"
-2.5
140
0
20
40
60
80
100
phase 1(-), phase2( --) vs time
Figure 4: The response of the NMA adaptive PlO controller
Figure 3: The response of the NMI adaptive PlO controller
387
Closed·Loo Res onse of rocess
so
200
ISO 100 yp C-), uc C-) vs time The 0
ut si
250 50
100
150
200
250
yp C-), uc C-) vs lime
a1 of the controUer
",e out
t si
n:Ji
or the controller
2
_IL-______~______~______~~----~~----~ o SO 100 150 200 250
250
uc vs time
parameters change
a load disturbance PlO :lrtmeter v:lri:ltion I.S
0.5
\
....~ ..................................... ................: ................... .
0.5
0 0
50
100
150
200
250
0
Kp (-). Td C--) vs time
0.5 0 0
r50
SO
100
200
ISO
250
Kp C-), Td C-) vs lime
The estimated crossover freouencv
1.5
0
I.S
n,e estimated crossover r
uenc
\'---0.5 100
150
200
250
frequency vs time
°OL--L------S~O---------100~--------IS~O--------2~00~------~2S0 frequency vs time
Figure 5: The perfonnance of the NMI adaptive PID controller
388
Figure 6: The perfonnance of the NMI adaptive PID controlle