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A Relay-Based Autotuner Specifically Aimed at Load Disturbance Rejection
Copyright@ IFAC Advanced Control of Chemical Processes Pisa, Italy, 2000 '
A RELAY-BASED AUTOTUNER SPECIFICALLY AIMED AT LOAD DISTURBANCE REJECTION
AIberto Leva
Dipartimento di Elettronica e Informazione, Politecnico di Milano Via Ponzio, 34/5 - 20133 Milano (Italy) [email protected]
Abstract: This paper presents a relay-based autotuner aimed specifically at process control applications where the main goal of the regulator is to reject load disturbances. The regulator structure adopted is particularly suited for this purpose and not completely standard, yet it can be easily implemented in any industrial control environment. The tuning procedure is presented in detail and a simulation example is reported. Copyright © 2000IFAC Keywords: Autotuners, Disturbance rejection, Process control.
literature: normally these are based on specifications given on the set point response - see e.g. Hagglund and Astrom (1985), Hang et al. (1991), Leva (1993) while in many applications (and especially in process control, where most potential uses of autotuning methods reside) the set point is seldom changed and the main role of the controller is to reduce the effects of phenomena which are most naturally modeled as load disturbances. Thus, an industrial regulator with autotuning capability aimed specifically at rejecting load disturbances is undoubtedly desirable in applications; the conception, derivation and implementation of such an object is the scope of this paper. The paper is organized as follows : section 2 illustrates the proposed technique, motivating the choice of the regulator structure, and illustrates the synthesis procedure. Section 3 reports three simulation examples to demonstrate the autotuner's operation, and in section 4 some implementation remarks are given. Finally, in section 5 some conclusions are drawn.
1. INTRODUCTION Autotuning capabilities are nowadays a common feature in industrial regulators. Several applications make use of these functionalities and, as they become more and more common also in quite low-cost and general-purpose devices and architectures, users are progressively becoming confident in them (Astrom et al., 1992;Astrom and Hagglund, 1995; Gawthrop, 1986). As a result, the success of an autotuner now comes basically from two factors: the methodological and algorithmic issues behind it, which determine its flexibility and robustness in the various situations it has to face, and - which is even more important - its ability to fit the needs of control engineers. This is particularly evident in mid- or large-scale process control applications, where a significant fraction of the effort required for the overall application is devoted to tuning the regulators (Astrom and Hagglund, 1995). In such cases, a control design system combining flexibility and good autotuning capabilities can shorten the time required for setting up a complete control system significantly (Leva, 1998; Skogestad and Postlethwaite, 1996). As expected, this results in a remarkable research effort in the domain. However, a quite peculiar fact hinders the practical application of several autotuning methods developed in the academic community and presented in the
2. THE PROPOSED TECHNIQUE To introduce the problem consider the block diagram of figure I, where yO is the set point, y the controlled variable, u the control signal, d the load disturbance, P(s) and R(s) the transfer function of the process
557
under control (assumed stable and of type 0, as typical in process control applications) and of the regulator to be synthesized respectively.
ljL-_R(_S)---,~,--_p(_S)---,r Fig. 1. The main block diagram. From figure 1 it comes immediately that the two relevant transfer functions are
G (s)
= Yes) =
o
YO(s )
R(s)P(s) 1+ R(s)P(s)
compromise with the degree of stability is in order, and automating this compromise is the key point in deriving the required autotuning policy. This compromise depends on the regulator structure, which - deliberately - has not been fixed a priori. In fact most autotuners do fix it, or are even conceived for a prespecified one (e.g. typically a PID). However, whilst a fixed-structure regulator can be easily accepted and "worked around" if only the set point response is subject to specifications, this is not the case if load disturbance rejection is the issue, simply because for respecting requirements on the former the magnitude of R(jw) at Wc counts almost nothing while for requirements on the latter it is crucial. On the basis of these considerations, the idea proposed herein for obtaining an autotuning method aimed specifically at load disturbance rejection can be characterized by the following key aspects:
and Gd(s) = Yes) D(s)
=
a. The regulator structure is chosen so as to be able of providing a large phase lead (i.e. a lead significantly bigger than the 90° maximum given by a PID), which is necessary if its magnitude at the cutoff frequency has to be kept as high as possible; moreover - which is even more important - the structure is chosen so that its parameters allow to determine the maximum regulator lead and the regulator magnitude at the cutoff frequency easily.
pes) 1+ R(s)P(s)
Moreover, it is possible to state that a requirement on the rejection of d can be quite naturally expressed as an upper bound on the magnitude of Gd(s) . Denoting now by L(s) the open-loop transfer function R(s)P(s), one can write
IR(jW)1
IL(jW)I» 1
b. The tuning is performed by assigning one point of the open-loop Nyquist diagram on the basis of one point of the process frequency response, identified by means of a relay experiment at a frequency as close as possible to the desired cutoff. Details are omitted in this paper for brevity, but in (Leva, 1993) the interested reader can find a complete explanation of the relay-based identification method adopted and, in particular, of how an oscillation frequency close to a desired one can be obtained. This tuning policy, anyway, allows to guarantee the phase margin for the closed-loop system.
IGd(jw)1 = !p(jw)1
!L(jw)!« 1
and the situation, with respect to the posed problem, is depicted in figure 2: in order to achieve a good rejection of d, the regulator R(s) must be designed so that its magnitude be as high as possible near the cutoff frequency wc.
dB
c. The tuning is aimed at keeping the regulator magnitude at the cutoff frequency as large as possible, which - as pointed out above - is the key point of the problem.
(J)
Note, incidentally, that it is also possible to employ the same rationale for constructing a set of tuning rules based on a parametric model of the process, typically a transfer function identified on the basis of an VO experiment. This is still being studied and will be reported in future papers. Another important remark is that the regulator structure adopted in this work is not a completely
Fig. 2. Requirements for load disturbance rejection. Of course this effect can be obtained by conveniently placed zeros, but doing so in turn requires to introduce the necessary number of poles so as to preserve the regulator properness; hence a
558
standardized one, yet it can be easily implemented in any control environment. It encompasses one or more integrators for static precision purposes, a gain, a variable number of zeros and the corresponding number of poles to preserve properness. Since the tuning policy is meant to use a single point of the process Nyquist curve, the number of regulator parameters is kept to the minimum necessary. In detail, the structure is
R(s) = K
(l + sTY sm (1 + saTy-m
which respectively represent the maximum phase lead achievable by the regulator and the regulator magnitude at the frequency corresponding to this maximum lead. Given now a required phase margin
(I)
and exhibits its maximum at the frequency
-( T) =1 a(n-m)-n ma,n,m, T a(an+m-n) under the condition (note that the denominator of the fraction in the square root is always negative) n
O
ROw) resemble those depicted in figure 3 (obtained with m=l and n=3; for better readability, since the figure is aimed at illustrating qualitative facts, the frequency axis is normalized - i.e. graduated in wT and the magnitude plot is scaled as if K1'"=l) .
Fig. 4. Region of points which the regulator can move to t/(rpm- J800).
lI£gulator magnitude (dB) 60
The synthesis procedure consists then of finding a suitable point of the process Nyquist curve - i.e. finding a couple (a, T) so that the point nearest to Wc found on POw) falls in the illustrated region, once n and m have been chosen - and tuning the regulator by moving this point to t/(rpm-J 800). In detail, the steps are the following :
40 20
OL-______~______~______~______~ O.o! 0.1 10 lOO Regulator phase (deg) ISO
·IOO!:===::::::===--_~
0.01
0.1
Fig. 3. Typical behaviour magnitude and phase.
I. Given the desired cutoff frequency wc, identify by a relay experiment a point on the process Nyquist curve as close as possible to wc. Since the following steps are not very sensitive to identification errors at this stage, as discussed e.g. in (Leva, 1993) and in some of the papers quoted there, assume that the computed point is POwJ.
_ _ _",--_ _~ 10 lOO Normalized j rr!qu£7ICy
of the
regulator'S
2. Choose m (normally 1) and n. This is done by a heuristic rule too long to explain here, based essentially on how much lead the regulator has to introduce to achieve a reasonable phase margin (say at least 45°). The rationale of this rule, anyway, can be summarized as follows so that the reader can appreciate it without delving in details: the rule uses a number
On the basis of figure 3, then, it is possible to define two functions
qf(a, n,m,T) = arg(R(jm(a,n,m ,T»)) and
R(a,n,m ,T) =IR(jm(a,n,m,T)~
559
of zeros sufficiently large to achieve the necessary lead without having to locate these zeros at a frequency too low (say not more than two decades below) with respect to the control bandwidth (i.e. to wc). Also, the rule assumes that a lowpass filter is already present on the process output for noise reduction, as typical in industrial applications, and takes its phase into account.
pes) _ _1----:- (1+S)4 and the presented technique, applied with a requested Wc of 2.4 and m=1 , has produced a regulator in the form (1) characterized by n=4 K=O.577 a=O.102
3. Compute a and T so that the maximum regulator lead occur at wc, which gives the best phase margin, and so as to avoid excessively small time constants in the regulator: this means solving (recall that n and m are fixed) max arg(R(jwc(a,n , m,T»)) a ,T
T=1 .572
The set point filter is first-order, with time constant equal to 5.3s. The noise filter on y too is first-order, with time constant equal to 3 times the sampling time employed (0.25s). The cutoff frequency actually obtained is slightly smaller than the desired one (about 1.8), while the transfer function from d to y exhibits its magnitude maximum at a frequency approximately equal to 0.46; what is more important, however, is that this maximum is limited to -8dB. The corresponding behaviour of the control system is shown in figure 5, which in fact illustrates three subsequent experiments: the response to a unit step variation of the set point, that to a unit-step load disturbance and the rejection of an additive output noise with amplitude 0.2 f.s. It is worth noting that disturbance rejection is particularly good (the maximum error peak is less than 0.2) and the sensitivity to output noise remains satisfactory.
(2)
subject to
n
a min
where a",in is typically set to 0.05. If this is not feasible change (i.e. reduce) Wc and identify a new point of the process Nyquist curve, otherwise compute K so that IL(jwJI =1. 4. Compute the cutoff frequency of a filter to be applied to the set point, outside the loop. This is because load disturbance rejection calls for a "strong" feedback whatever policy is adopted for achieving it, and this can make the set point response a bit too nervous. Note, anyway, that in most of the applications this autotuner is meant for the set point response is not the main issue. As such, for this purpose too in this work a very simple, heuristic rule is adopted and the time constant of this filter is merely computed as 10/ wc ' Of course to this end much more sophisticated techniques could be employed, but this would stray from the scope of the paper.
y.u
Control
.... .
_. ; . .
Controlled variable
timers)
120
36 I
Load disubance
(uniISlep)
Fig. 5. Simulation results of example I.
3.2. Example 2 3. THREE SIMULATION EXAMPLES
In the second example the same process transfer function has been used, but with a delay of 3s cascaded to it. The required cutoff frequency too is the same. This time the increased need for phase lead has caused the algorithm to produce
To illustrate the autotuner's operation, three simple simulation examples are reported.
3.1 . Example 1
n=5
The process considered in the first example is described by the transfer function
K=O.187
560
time constant equal to 2s. The sampling time employed is again 0.25s. The corresponding behaviour of the control system is shown in figure 7, which illustrates the usual three subsequent experiments:
a=O.JOJ T=J.5JO
The corresponding behaviour of the control system is shown in figure 6, which illustrates the same three subsequent experiments of figure 5:
Y,U
Y,U
Controlled variable
Control signal
Controlled variable time (s)
timers) 110
55 I
I
(lIIJitslepJ
nois£
60
ISO
I Load disturlxra
160
(uniJs"P)
Addia.._
Loadtlislwt>:Jra
Fig. 7. Simulation results of example 3. Fig. 6. Simulation results of example 2.
The transfer function from d to y has its magnitude maximum is at a frequency approximately equal to 0.6 and with a value of about 8dB: as a consequence disturbance rejection properties are not stunning, but the overall performance of the control system especially with respect to the transients of the control variable - remains fairly good.
The cutoff frequency obtained in this case is almost equal to that of example 1. As for the transfer function from d to y, its magnitude maximum is at a frequency approximately equal to 0.46. However, this time its value is slightly less than 2dB, and this can be appreciated considering the response to the load disturbance at 55s (the time scale is different from the first example for better readability) . The noise rejection characteristics in these two examples are almost the same.
4. IMPLEMENTATION REMARKS The first remark to be done is that the proposed tuning procedure is not very simple and can result in a significant computational effort. In fact its core is represented by the maximization problem expressed by eqn. (2), which - if considered in the most general case - is not simple to be solved. However, though this fact is true in general, practical applications of the method show that in the majority of relevant real-world cases things are not so difficult. The real problem is that, even restricting the possible situations (i.e. the possible regulator structures) to a prespecified subset, it is not easy to give problem (2) a standardized form nor is it possible to compute its solution analytically. This means that any implementation of the algorithm will have to resort to sequential search for finding the required couple (a, T). Of course the consequent burden can be reduced by tabulating some convenient quantities. This has been explored only partially and will be further investigated in future works. Another remark is that the proposed method might "reject" several points identified by the relay experiments, thus increasing the tuning time. The importance of this problem is difficult to forecast, since no general results can be derived and conducting enough experiments to form a statistically significant knowledge base, and remains as another open question to devote future work to. On the basis of present experiences, however, the points to be
3.3. Example 3 The process considered in this example is described by the transfer function
P(s) =
e- SS 1 + O.Ss +s
2
and is then characterized by a dominant delay and a loosely damped rational dynamics. The aim of this test is to show the algorithm's performance also in a case where relay-based identification can experience some difficulties. With a required cutoff frequency of 3 and m=J, the proposed technique has produced
n=4 K=O.J29 a=O.J08 T=J .257
The set point filter is first-order, with time constant equal to 3s. The noise filter on y is first-order, with
561
rejected (Le. the performance reduction to be accepted for preserving stability) is generally small. Finally, it is worth recalling that the choice of the regulator structure has been made essentially in view of its ease of implementation: leaving more free parameters would have been conceptually more neat, but it would have increased the overall complexity. The question is then how much the potentialities of the general idea (making /ROco,) / large by taking profit of the regulator's maximum phase lead) have been turned into real advantages with such structural restrictions. This aspect too is at present being investigated.
5. CONCLUSIONS An autotuner specifically aimed at rejecting load disturbances, which is a quite common need in process control applications, has been presented. The regulator structure adopted is particularly suited for this purpose: as a consequence it is not completely standard, consisting of one or more integrators and zeros plus the correspondinglky necessary poles to preserve properness, yet it can be easily implemented in any industrial control environment. The tuning is done by assigning one point of the open-loop Nyquist diagram on the basis of one of the process frequency response (identified as close as possible to the desired cutoff by a conveniently driven relay experiment), keeping the regulator magnitude at the cutoff frequency as large as possible. The proposed autotuner has been tested in a number of simulation cases (two of which have been reported) exhibiting good performance.
562
ACKNOWLEDGEMENTS This work has been partially supported by MURST (Ministero dell'Universita e della Ricerca Scientifica e Tecnologica) and CNR (Consiglio Nazionale delle Ricerche).
REFERENCES Astrom, KJ. , c.c. Hang, P. Persson and W.K. Ho (1992). Towards Intelligent PID Control. Automatica, 28 (1), pp. 1-9. Astrom, K.J. and T. Hiigglund (1995). PID Controllers: Theory, Design and Tuning - Second Edition. Instrument Society of America. Gawthrop, PJ. (1986). Self-Tuning PID Controllers: Algorithms and Implementation. IEEE Trans.aC, 31 (3), pp. 201-209. Hiigglund, T. and KJ. Astrom (1985). Automatic Tuning of PID Controllers Based on Dominant Pole Design. Proc. IFAC Workshop on Adaptive Control of Chemical Processes, Frankfurt. Hang, c.c., KJ. Astrom and W.K. Ho (1991). Refinements of the Ziegler-Nichols Tuning Formula. lEE Proceedings-D, 138 (2), pp. 111 118. Leva, A. (1993). PID Autotuning Algorithm based on Relay Feedback. lEE Proceedings-D, 140 (5), pp. 328-338. Leva, A. (1998). How to Construct PID Synthesis Rules for Structurally Complex Process Models. Proc. SSC '98, Nantes. Skogestad, S. and 1. Postlethwaite (1996). Multivariable Feedback Control - Analysis and Design.Wiley.
A RELAY-BASED AUTOTUNER SPECIFICALLY AIMED AT LOAD DISTURBANCE REJECTION
AIberto Leva
Dipartimento di Elettronica e Informazione, Politecnico di Milano Via Ponzio, 34/5 - 20133 Milano (Italy) [email protected]
Abstract: This paper presents a relay-based autotuner aimed specifically at process control applications where the main goal of the regulator is to reject load disturbances. The regulator structure adopted is particularly suited for this purpose and not completely standard, yet it can be easily implemented in any industrial control environment. The tuning procedure is presented in detail and a simulation example is reported. Copyright © 2000IFAC Keywords: Autotuners, Disturbance rejection, Process control.
literature: normally these are based on specifications given on the set point response - see e.g. Hagglund and Astrom (1985), Hang et al. (1991), Leva (1993) while in many applications (and especially in process control, where most potential uses of autotuning methods reside) the set point is seldom changed and the main role of the controller is to reduce the effects of phenomena which are most naturally modeled as load disturbances. Thus, an industrial regulator with autotuning capability aimed specifically at rejecting load disturbances is undoubtedly desirable in applications; the conception, derivation and implementation of such an object is the scope of this paper. The paper is organized as follows : section 2 illustrates the proposed technique, motivating the choice of the regulator structure, and illustrates the synthesis procedure. Section 3 reports three simulation examples to demonstrate the autotuner's operation, and in section 4 some implementation remarks are given. Finally, in section 5 some conclusions are drawn.
1. INTRODUCTION Autotuning capabilities are nowadays a common feature in industrial regulators. Several applications make use of these functionalities and, as they become more and more common also in quite low-cost and general-purpose devices and architectures, users are progressively becoming confident in them (Astrom et al., 1992;Astrom and Hagglund, 1995; Gawthrop, 1986). As a result, the success of an autotuner now comes basically from two factors: the methodological and algorithmic issues behind it, which determine its flexibility and robustness in the various situations it has to face, and - which is even more important - its ability to fit the needs of control engineers. This is particularly evident in mid- or large-scale process control applications, where a significant fraction of the effort required for the overall application is devoted to tuning the regulators (Astrom and Hagglund, 1995). In such cases, a control design system combining flexibility and good autotuning capabilities can shorten the time required for setting up a complete control system significantly (Leva, 1998; Skogestad and Postlethwaite, 1996). As expected, this results in a remarkable research effort in the domain. However, a quite peculiar fact hinders the practical application of several autotuning methods developed in the academic community and presented in the
2. THE PROPOSED TECHNIQUE To introduce the problem consider the block diagram of figure I, where yO is the set point, y the controlled variable, u the control signal, d the load disturbance, P(s) and R(s) the transfer function of the process
557
under control (assumed stable and of type 0, as typical in process control applications) and of the regulator to be synthesized respectively.
ljL-_R(_S)---,~,--_p(_S)---,r Fig. 1. The main block diagram. From figure 1 it comes immediately that the two relevant transfer functions are
G (s)
= Yes) =
o
YO(s )
R(s)P(s) 1+ R(s)P(s)
compromise with the degree of stability is in order, and automating this compromise is the key point in deriving the required autotuning policy. This compromise depends on the regulator structure, which - deliberately - has not been fixed a priori. In fact most autotuners do fix it, or are even conceived for a prespecified one (e.g. typically a PID). However, whilst a fixed-structure regulator can be easily accepted and "worked around" if only the set point response is subject to specifications, this is not the case if load disturbance rejection is the issue, simply because for respecting requirements on the former the magnitude of R(jw) at Wc counts almost nothing while for requirements on the latter it is crucial. On the basis of these considerations, the idea proposed herein for obtaining an autotuning method aimed specifically at load disturbance rejection can be characterized by the following key aspects:
and Gd(s) = Yes) D(s)
=
a. The regulator structure is chosen so as to be able of providing a large phase lead (i.e. a lead significantly bigger than the 90° maximum given by a PID), which is necessary if its magnitude at the cutoff frequency has to be kept as high as possible; moreover - which is even more important - the structure is chosen so that its parameters allow to determine the maximum regulator lead and the regulator magnitude at the cutoff frequency easily.
pes) 1+ R(s)P(s)
Moreover, it is possible to state that a requirement on the rejection of d can be quite naturally expressed as an upper bound on the magnitude of Gd(s) . Denoting now by L(s) the open-loop transfer function R(s)P(s), one can write
IR(jW)1
IL(jW)I» 1
b. The tuning is performed by assigning one point of the open-loop Nyquist diagram on the basis of one point of the process frequency response, identified by means of a relay experiment at a frequency as close as possible to the desired cutoff. Details are omitted in this paper for brevity, but in (Leva, 1993) the interested reader can find a complete explanation of the relay-based identification method adopted and, in particular, of how an oscillation frequency close to a desired one can be obtained. This tuning policy, anyway, allows to guarantee the phase margin for the closed-loop system.
IGd(jw)1 = !p(jw)1
!L(jw)!« 1
and the situation, with respect to the posed problem, is depicted in figure 2: in order to achieve a good rejection of d, the regulator R(s) must be designed so that its magnitude be as high as possible near the cutoff frequency wc.
dB
c. The tuning is aimed at keeping the regulator magnitude at the cutoff frequency as large as possible, which - as pointed out above - is the key point of the problem.
(J)
Note, incidentally, that it is also possible to employ the same rationale for constructing a set of tuning rules based on a parametric model of the process, typically a transfer function identified on the basis of an VO experiment. This is still being studied and will be reported in future papers. Another important remark is that the regulator structure adopted in this work is not a completely
Fig. 2. Requirements for load disturbance rejection. Of course this effect can be obtained by conveniently placed zeros, but doing so in turn requires to introduce the necessary number of poles so as to preserve the regulator properness; hence a
558
standardized one, yet it can be easily implemented in any control environment. It encompasses one or more integrators for static precision purposes, a gain, a variable number of zeros and the corresponding number of poles to preserve properness. Since the tuning policy is meant to use a single point of the process Nyquist curve, the number of regulator parameters is kept to the minimum necessary. In detail, the structure is
R(s) = K
(l + sTY sm (1 + saTy-m
which respectively represent the maximum phase lead achievable by the regulator and the regulator magnitude at the frequency corresponding to this maximum lead. Given now a required phase margin
(I)
and exhibits its maximum at the frequency
-( T) =1 a(n-m)-n ma,n,m, T a(an+m-n) under the condition (note that the denominator of the fraction in the square root is always negative) n
O
ROw) resemble those depicted in figure 3 (obtained with m=l and n=3; for better readability, since the figure is aimed at illustrating qualitative facts, the frequency axis is normalized - i.e. graduated in wT and the magnitude plot is scaled as if K1'"=l) .
Fig. 4. Region of points which the regulator can move to t/(rpm- J800).
lI£gulator magnitude (dB) 60
The synthesis procedure consists then of finding a suitable point of the process Nyquist curve - i.e. finding a couple (a, T) so that the point nearest to Wc found on POw) falls in the illustrated region, once n and m have been chosen - and tuning the regulator by moving this point to t/(rpm-J 800). In detail, the steps are the following :
40 20
OL-______~______~______~______~ O.o! 0.1 10 lOO Regulator phase (deg) ISO
·IOO!:===::::::===--_~
0.01
0.1
Fig. 3. Typical behaviour magnitude and phase.
I. Given the desired cutoff frequency wc, identify by a relay experiment a point on the process Nyquist curve as close as possible to wc. Since the following steps are not very sensitive to identification errors at this stage, as discussed e.g. in (Leva, 1993) and in some of the papers quoted there, assume that the computed point is POwJ.
_ _ _",--_ _~ 10 lOO Normalized j rr!qu£7ICy
of the
regulator'S
2. Choose m (normally 1) and n. This is done by a heuristic rule too long to explain here, based essentially on how much lead the regulator has to introduce to achieve a reasonable phase margin (say at least 45°). The rationale of this rule, anyway, can be summarized as follows so that the reader can appreciate it without delving in details: the rule uses a number
On the basis of figure 3, then, it is possible to define two functions
qf(a, n,m,T) = arg(R(jm(a,n,m ,T»)) and
R(a,n,m ,T) =IR(jm(a,n,m,T)~
559
of zeros sufficiently large to achieve the necessary lead without having to locate these zeros at a frequency too low (say not more than two decades below) with respect to the control bandwidth (i.e. to wc). Also, the rule assumes that a lowpass filter is already present on the process output for noise reduction, as typical in industrial applications, and takes its phase into account.
pes) _ _1----:- (1+S)4 and the presented technique, applied with a requested Wc of 2.4 and m=1 , has produced a regulator in the form (1) characterized by n=4 K=O.577 a=O.102
3. Compute a and T so that the maximum regulator lead occur at wc, which gives the best phase margin, and so as to avoid excessively small time constants in the regulator: this means solving (recall that n and m are fixed) max arg(R(jwc(a,n , m,T»)) a ,T
T=1 .572
The set point filter is first-order, with time constant equal to 5.3s. The noise filter on y too is first-order, with time constant equal to 3 times the sampling time employed (0.25s). The cutoff frequency actually obtained is slightly smaller than the desired one (about 1.8), while the transfer function from d to y exhibits its magnitude maximum at a frequency approximately equal to 0.46; what is more important, however, is that this maximum is limited to -8dB. The corresponding behaviour of the control system is shown in figure 5, which in fact illustrates three subsequent experiments: the response to a unit step variation of the set point, that to a unit-step load disturbance and the rejection of an additive output noise with amplitude 0.2 f.s. It is worth noting that disturbance rejection is particularly good (the maximum error peak is less than 0.2) and the sensitivity to output noise remains satisfactory.
(2)
subject to
n
a min
where a",in is typically set to 0.05. If this is not feasible change (i.e. reduce) Wc and identify a new point of the process Nyquist curve, otherwise compute K so that IL(jwJI =1. 4. Compute the cutoff frequency of a filter to be applied to the set point, outside the loop. This is because load disturbance rejection calls for a "strong" feedback whatever policy is adopted for achieving it, and this can make the set point response a bit too nervous. Note, anyway, that in most of the applications this autotuner is meant for the set point response is not the main issue. As such, for this purpose too in this work a very simple, heuristic rule is adopted and the time constant of this filter is merely computed as 10/ wc ' Of course to this end much more sophisticated techniques could be employed, but this would stray from the scope of the paper.
y.u
Control
.... .
_. ; . .
Controlled variable
timers)
120
36 I
Load disubance
(uniISlep)
Fig. 5. Simulation results of example I.
3.2. Example 2 3. THREE SIMULATION EXAMPLES
In the second example the same process transfer function has been used, but with a delay of 3s cascaded to it. The required cutoff frequency too is the same. This time the increased need for phase lead has caused the algorithm to produce
To illustrate the autotuner's operation, three simple simulation examples are reported.
3.1 . Example 1
n=5
The process considered in the first example is described by the transfer function
K=O.187
560
time constant equal to 2s. The sampling time employed is again 0.25s. The corresponding behaviour of the control system is shown in figure 7, which illustrates the usual three subsequent experiments:
a=O.JOJ T=J.5JO
The corresponding behaviour of the control system is shown in figure 6, which illustrates the same three subsequent experiments of figure 5:
Y,U
Y,U
Controlled variable
Control signal
Controlled variable time (s)
timers) 110
55 I
I
(lIIJitslepJ
nois£
60
ISO
I Load disturlxra
160
(uniJs"P)
Addia.._
Loadtlislwt>:Jra
Fig. 7. Simulation results of example 3. Fig. 6. Simulation results of example 2.
The transfer function from d to y has its magnitude maximum is at a frequency approximately equal to 0.6 and with a value of about 8dB: as a consequence disturbance rejection properties are not stunning, but the overall performance of the control system especially with respect to the transients of the control variable - remains fairly good.
The cutoff frequency obtained in this case is almost equal to that of example 1. As for the transfer function from d to y, its magnitude maximum is at a frequency approximately equal to 0.46. However, this time its value is slightly less than 2dB, and this can be appreciated considering the response to the load disturbance at 55s (the time scale is different from the first example for better readability) . The noise rejection characteristics in these two examples are almost the same.
4. IMPLEMENTATION REMARKS The first remark to be done is that the proposed tuning procedure is not very simple and can result in a significant computational effort. In fact its core is represented by the maximization problem expressed by eqn. (2), which - if considered in the most general case - is not simple to be solved. However, though this fact is true in general, practical applications of the method show that in the majority of relevant real-world cases things are not so difficult. The real problem is that, even restricting the possible situations (i.e. the possible regulator structures) to a prespecified subset, it is not easy to give problem (2) a standardized form nor is it possible to compute its solution analytically. This means that any implementation of the algorithm will have to resort to sequential search for finding the required couple (a, T). Of course the consequent burden can be reduced by tabulating some convenient quantities. This has been explored only partially and will be further investigated in future works. Another remark is that the proposed method might "reject" several points identified by the relay experiments, thus increasing the tuning time. The importance of this problem is difficult to forecast, since no general results can be derived and conducting enough experiments to form a statistically significant knowledge base, and remains as another open question to devote future work to. On the basis of present experiences, however, the points to be
3.3. Example 3 The process considered in this example is described by the transfer function
P(s) =
e- SS 1 + O.Ss +s
2
and is then characterized by a dominant delay and a loosely damped rational dynamics. The aim of this test is to show the algorithm's performance also in a case where relay-based identification can experience some difficulties. With a required cutoff frequency of 3 and m=J, the proposed technique has produced
n=4 K=O.J29 a=O.J08 T=J .257
The set point filter is first-order, with time constant equal to 3s. The noise filter on y is first-order, with
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rejected (Le. the performance reduction to be accepted for preserving stability) is generally small. Finally, it is worth recalling that the choice of the regulator structure has been made essentially in view of its ease of implementation: leaving more free parameters would have been conceptually more neat, but it would have increased the overall complexity. The question is then how much the potentialities of the general idea (making /ROco,) / large by taking profit of the regulator's maximum phase lead) have been turned into real advantages with such structural restrictions. This aspect too is at present being investigated.
5. CONCLUSIONS An autotuner specifically aimed at rejecting load disturbances, which is a quite common need in process control applications, has been presented. The regulator structure adopted is particularly suited for this purpose: as a consequence it is not completely standard, consisting of one or more integrators and zeros plus the correspondinglky necessary poles to preserve properness, yet it can be easily implemented in any industrial control environment. The tuning is done by assigning one point of the open-loop Nyquist diagram on the basis of one of the process frequency response (identified as close as possible to the desired cutoff by a conveniently driven relay experiment), keeping the regulator magnitude at the cutoff frequency as large as possible. The proposed autotuner has been tested in a number of simulation cases (two of which have been reported) exhibiting good performance.
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ACKNOWLEDGEMENTS This work has been partially supported by MURST (Ministero dell'Universita e della Ricerca Scientifica e Tecnologica) and CNR (Consiglio Nazionale delle Ricerche).
REFERENCES Astrom, KJ. , c.c. Hang, P. Persson and W.K. Ho (1992). Towards Intelligent PID Control. Automatica, 28 (1), pp. 1-9. Astrom, K.J. and T. Hiigglund (1995). PID Controllers: Theory, Design and Tuning - Second Edition. Instrument Society of America. Gawthrop, PJ. (1986). Self-Tuning PID Controllers: Algorithms and Implementation. IEEE Trans.aC, 31 (3), pp. 201-209. Hiigglund, T. and KJ. Astrom (1985). Automatic Tuning of PID Controllers Based on Dominant Pole Design. Proc. IFAC Workshop on Adaptive Control of Chemical Processes, Frankfurt. Hang, c.c., KJ. Astrom and W.K. Ho (1991). Refinements of the Ziegler-Nichols Tuning Formula. lEE Proceedings-D, 138 (2), pp. 111 118. Leva, A. (1993). PID Autotuning Algorithm based on Relay Feedback. lEE Proceedings-D, 140 (5), pp. 328-338. Leva, A. (1998). How to Construct PID Synthesis Rules for Structurally Complex Process Models. Proc. SSC '98, Nantes. Skogestad, S. and 1. Postlethwaite (1996). Multivariable Feedback Control - Analysis and Design.Wiley.