561-70
ADAPTIVE CONTROL ODYSSEY
By E. H. Bristol
The Foxboro Company
Foxboro, Massachusetts 02035
Copyright 1970
Silver Jubilee
International
Conference & Exhibit
Oct. 26-29, 1970
Philadelphia, PA
Instrument Society of America
desired, it is necessary to provide means for predicting the effect of changes in parameters on performance.
ABSTRACT
The paper describes a 10-year pilgrimage in industrial adaptive control that began with the publications of Caldwell1 and Pessin,2 Reswick and
Goodman,3 and from some theoretical thinking on
Bayesian statistics and functional analysis. The
work ended with some recent papers by the author on
functional adaptive systems.
A more complex and formal approach to the problem,
and one which leads to greater understanding,
involves giving the computer the ability to use the
process measurement information to identify the
control characteristics of the process and control
disturbances. This information permits computation
of the settings which will give the best performance. It turns out, however, that this identification problem is not solvable as it stands; for any
set of measurement data, there will always be many
process disturbance combinations which could
explain the data. And yet controllers are easily
tuned manually.
Discussed are the philosophical underpinnings of
adaptive control, theoretical promise and practical
problems of identification, and the problems of
defining workable assumptions about the control
environment to give an adaptive system a starting
point. The consequences of failure of these assumptions are detailed, along with the problems of
guaranteeing safety, even when the adaptive system
is misused.
In practical adaptive systems this paradox is
resolved by supplying the missing information in
the form of explicit or implicit assumptions about
the process or the disturbances. These assumptions
may take a number of forms:
Where possible, the comments are illustrated by
experimental data, but since much of the data is
old and sketchy, the emphasis is on ideas. At the
odyssey's end, the important technical and economic
implications of adaptive control are discussed.
1
2.
INTRODUCTION
One may make statistical assumptions about
either the nature of process or the disturbances.
One may assume a model form (second order
process or deadtime and lag) for process or
disturbance.4
It would be a mistake to generally define adaptive
control. This paper will discuss adaptive control
in terms of automatic tuning of feedback controllers. Many people in the instrument industry contend that few users succeed in keeping controllers
anywhere near perfectly tuned, and many do not even
try. It is clear that if someone were able to provide an inexpensive way of automatically tuning
controllers and keeping them tuned, it would be of
value to everyone. The paper discusses an extended
effort to solve this problem.
One may assume a closed-loop model form.5
One may assume a frequency spectrum form
for either open or closed-loop response.
Each of these models or forms may be expressed in
terms of parameters which must be computed from
process data.
3.
4.
5.
PHILOSOPHY
A simple approach to the problem involves allowing
the adapting system (the computer in most cases) to
continually adjust the tunings to improve performance. A means of recognizing the quality of
response under varying control disturbances is necessary to the computer. If rapid adaptation is
Finally, assumptions may be made about process linearity, maximum settling time, or
about disturbance settling times and the
likelihood of occurrence of multiple disturbances within short spans of time.
SOME EXPERIMENTS ON IDENTIFICATIONI
The following experiments spanned the years 1962
561
In each of these experiments, to start with,
K = K’ = 1, τ1 = τ2 = 10, τ1’ = .625, τ2’ = 17.18
and 1964. The method used for identification is
similar in spirit to Kalman's method4,7 except that
an explicit attempt is made to identify load. The
method is described in Appendix I. It is assumed
the process is linear.
K
-------------------------------------------- ]. The second has eight lags where
( τ1 s + 1 ) ( τ2 s + 1 )
seven of the eight time constants are set identical. The remaining time constant and the process
gain may be set independently. [The model is of
K′
form -------------------------------------------------- ]. Thus, each process has
7
The control situation is shown in Figure 1. A hypothetical process was selected and controlled with a
digital controller to a zero set-point. A hypothetical upset was introduced to the process and the
values of manipulated variable M and controlled
variable C following the upset were recorded. Table
I shows a tabulation of the sampled values of C
going into the computer and the successive levels
of ∆M outputted to the process. Figure 2 is a
closed-loop analog record of C under the controlled
upset. From this data, the writer attempted to calculate the process response to step control action
and the upset response, which is the set of values
that C(t) would have taken under the upset in
absence of control. Figure 3 shows measured analog
records of the step response and upset response
recorded at a later date. The small x's on the
record show values calculated from the original
closed-loop data of Table I and Figure 2. Most of
these calculations are precise. The exception is
the last value in time on the record of the upset
response which is off by about ten percent. However, later examination showed that there was a
mistake in the gain settings for these recordings
and that the calculations were more accurate than
the measured record!
( τ′ 1 s + 1 ) ( τ′ 2 s + 1 )
three degrees of freedom. The two process outputs
can be compared in a mean square sense by the summing amplifier, squares, integrator combination.
Also switching allows tuning each process for optimum integral square error set-point change.
The purpose of the experiment is to demonstrate the
consequences of using models with limited degrees
of freedom to represent a process of differing
structures. This is important since none of the
usually recommended models (two lag, deadtime and
lags, etc.) is likely to give anywhere near exact
feedback control tuning results. Three consequences
will be apparent:
1
Modelling or identification arrived at
under one set of tunings or operating conditions may well give an extremely poor fit
under other tunings or operating conditions.
2. Repeated identification under successive
adapted settings will often give convergent
control settings and behavior which is reasonably good even when the initial identification gave poor control fit, as
discussed previously.
3. True optimum settings cannot generally be
inferred from an approximate model under
any circumstances, even though the resulting control is near optimum and the optimum
response itself can often be inferred from
the model.
In the experiment, one of the simulated processes
is considered to be the "real" process, the other
is considered to be the "model" for identification
and tuning purposes as if an on-line adaptation
were going on. For each of the experiments the twolag process was set initially with unity gain and
identical time constants, and the eight-lag process
was set to give a best mean square fit to the twolag process under an open-loop step. In the odd
numbered figures showing experimental results, the
first and third responses from the top correspond
to the two-lag process and the second and fourth
correspond to the eight-lag process. The even numbered figures show the results in the preceding
figure compared by superimposing the two- and
eight-lag responses.
The same method was used as a continuous adapting
scheme. Since the adaptation was continuously
active it was necessary to differentiate automatically between circumstances when data was meaningful and circumstances when the data could not give
valid identifications. The clearest case of the
latter occurs when the measurement is steady on
set-point, but there are instances where transient
measurements fail to give valid identification. The
method used was to calculate the sensitivity of the
identification and use this result to control the
use of information in the scheme. Figure 4 shows
the structure of the control system. The sensitivity calculation consists of a worst case analysis
of the effect of a unit amount of controlled variable noise on the identification. Readers interested in this type of technique might consult any
of the literature on the circuit analysis program
ECAP6. The identification is most credible when it
is least sensitive. Controller constants were averaged with past values weighted according to sensitivity.
The effects of this are shown in Figure 5 where
control measurements under set-point changes are
shown together with the identified process gain
(actually states as an unaveraged controller gain
K), the credibility estimate and the averaged gain
K (reflecting control settings used). This data
clearly shows the correlation between active signal, identification, and sensitivity-derived credibility calculation. Later experiments dealt with
situations in which the operating assumptions were
increasingly violated. (See Appendix I.) The
results gave good control even when the identification became erratic! This suggests that the model
may not be all that important.
In the first experiment (Figures 7 through 12), the
two-lag process is considered to be the "model."
The controller corresponding to it is tuned for
best performance and the "real" process (the eightlag) controller is set to the same settings. The
open-loop step responses are shown at the top of
Figures 7 and 8, and visually the fit is good, with
less than three percent error. But, the closed-loop
set-paint change responses, shown at the bottom of
Figures 7 and 8, resulted in unstable control of
the "real" process.
A second set of experiments amplify this position
(See Figure 6). In this experiment two independent
simulated control loops are shown each with two
term controller. The first is a two-lag simulated
process where the gain and the two time constants
are adjusted independently. [The model is of form
The data from the "real" process are then used to
arrive at a better "model" in a mean square sense.
The two-lag “model” is readjusted under the same
controller settings to give a best mean square fit
to this data as shown in the top of Figures 9 and
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with the desired response shape and the adjustment
of the controller settings to improve this transient. The scheme assumes no prior knowledge of the
process characteristics. Since the system is capable of adjusting the controller gains to achieve
the desired response, nonlinear systems can be
safely controlled under large control changes (even
where nonlinearity is apparent in one response). In
the following discussion the two-term controller is
in control all the time.
10. This is done by radically increasing the openloop model gain as shown in the bottom of Figures 9
and 10. The controller settings can now be readjusted to the new model giving better control to
both "model" and "real" process in the sequence
shown in the top of Figures 11 and 12. Continuing
in this vein, the experiment would converge to a
final set of settings.
The results made the identification look unnecessarily bad and show what happens if a particularly
naive modelling approach is taken. Any sophisticated designer would know of the danger in using a
second order model for these processes. But remember that the initial open-loop fits were good. Generally for process control, one would use a more
complex model (such as a deadtime on lag model)
here represented by the eight-lag process.
The user selects his desired response by inserting
only two values, B and C, which serve as comparison
levels during gain computations (Figure 18) and
serve to specify the desired response shape. (These
B and C levels can be set to correspond to any
desired control pattern, overdamped or underdamped.)
The pattern is evaluated by calculating the integrated difference of measurement normalized by setpoint change or upset magnitude from the B or C
level (Figure 18) over appropriate intervals of
time corresponding to overshoot and undershoot.
(The two intervals of time [one for B and one for
C] are chosen, based on time measurements for the
initial control recovery to correspond to the first
two half cycles in the dominant eigenfunction
[assuming that it is cyclic) after the initial
recovery. For most purposes, the three dominant
eigenfunctions accurately determine the pattern,
but not the control characteristics.) Thus, the
relationship between the initial recovery and the
position of these half cycles is sufficiently independent of deadtime, process order, and reasonable
degrees of nonlinearity that it does not adversely
affect pattern characterization. The pattern areas
thus evaluated are used to adjust control settings
to improve response using simple feedback. The
scheme works in a trouble-free and insensitive manner independent of process, initial settings, disturbances, or set-point magnitude.
In the second experiment (Figures 13 through 16)
the eight-lag process is the "model" and the twolag process is the "real" process. The tops of Figures 13 and 14 again show the open-loop responses.
The bottom of Figures 13 and 14 show the set-point
change responses resulting from optimum settings
derived from the model as before. Note that the
set-point response of the "real" process is now
fairly good, even though it differs noticeably from
the response of the "model." If the model is now
readjusted for best closed-loop fit, the resulting
closed- and open-loop responses are shown in Figures 15 and 16.
By generating conservative settings, the eight-lag
model is a safer basis for adaptation. It still
requires successive iterations to reach a final
"model" and each model is inaccurate except for the
control settings under which it was taken. The
quality of control is good because the control criterion is very insensitive near its optimum. This
is best seen by comparing the true optima closedloop responses where the two processes are matched
for open-loop step response (as in Figures 7, 8,
13, and 14). These are shown overlapped in Figure
17. Although the responses are similar in shape, if
not in time scale, the optimum controller gain for
the two-lag process is 13.16 as opposed to 3.85 for
the eight-lag process: a ratio of more than 3 to 1.
The reset times were respectively 34.12 and 27.7.
A slight modification permits adaptation under certain transient load conditions. Several possibilities exist to permit adaptation of the three-term
controller. The adaptive feedback equations are
given in Appendix II. This scheme has performed
safely and effectively in pilot plant tests and in
limited field tests.
In summary, the experiments indicate that an adaptation scheme based on such a modelling point of
view will give good practical results. There is
good cause to believe that much simpler methods
would work nearly as well. Even so, the identification approach should still be kept in mind for theoretical studies or situations where a formal
characterization of process is desired. The experiments took an extreme situation (two-lag versus
eight-lag), but this kind of difference does exist
in the real world. A more exact model with more
degrees of freedom in parameters would give better
results, but would be more sensitive to noise.
OTHER TECHNIQUES
The techniques presented have been those developed
by the author. Other authors have covered identification in equally effective ways. The techniques
using auto-correlation and cross-correlation3 filter out noise permitting data to be preprocessed
allowing methods like those already given without
requiring explicit identification of long noise
signals. These methods also provide one way of
dealing with nontransient signals. The identification methods previously discussed, as well as the
pattern scheme, all assume a transient disturbance.
The correlation methods3 can transform a continuously varying signal into a transient in an artificial time domain, thus permitting use of the
previously mentioned techniques, but they do so at
the cost of increased computation and signal time.
A PATTERN SCHEME
Note again the similarity of shapes in Figure 17.
Similar observations of the author and the observation by Paynter8 that different criteria usually
can be related to a fixed percentage of overshoot
as in Figure 17 (Integral square criterion corresponds to 20 percent overshoot, integral absolute
to 10 percent overshoot) led to the use of shape as
a basis for adaptation. A scheme was developed
based on the comparison of the actual process transient response under conventional two-term control
A much simpler technique due to Bakke for dealing
with continuously varying signals assumes that the
disturbances can be defined in terms of their spectrum. The scheme applies simple feedback to enforce
certain pretuned relationships on the closed-loop
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spectrum and is said to work well. The literature9
and work by Bakke and Inagaki indicates that the
adaptation takes a considerable amount of time to
converge.
2.
The adaptation will then process automatically on any set-point or load change
occurring in the loop.
3. Adaptation can prevent instability from
ever occurring in any loop.
4. For a loop with adaptation, storage far the
two or three additional adaptation parameters is required. Optionally, other specification parameters may be stored.
5. For any loop actually undergoing adaptation, the computational resources needed
may be as much as two or three times of the
unadapted two-term control algorithm
itself. However, adaptation is only
required when an appropriate transient has
occurred. This may occur infrequently. Thus
the additional load on the system may be
slight. And in a crisis decreased sampling
may be used.
This is not to say that initial applications should
not be approached carefully until operational characteristics are fully understood. But with reasonable care, it will be found that adaptation
simplifies, rather than complicates life.
This is to be expected since accurate experimental
evaluations of spectra, or other statistical parameters, require large amounts of data. One disadvantage with any scheme which requires data taken over
a long period of time is the increased likelihood
of unaccounted changes in surroundings during adaptation.
One of the earliest ideas in adaptation is that
patented by Caldwell and heavily experimented with
by Pessin.1,2 This scheme used spectrum-related
methods to evaluate natural frequency and damping.
Generally, schemes which calculate parameters of
the spectrum must infer their parameter from a
grossly sampled spectrum because every sample usually requires a filter. This limits their effectiveness.
CONCLUSIONS
Each of the schemes mentioned has its own special
assumptions which determine the applicability of
the scheme. The problem in designing adaptive control systems has been to come up with a scheme
which involves a minimum amount of computation,
requires little special understanding, and which
will adapt most processes under as many disturbance
situations as possible. It should not require pretuning or introduction of special tuning disturbances. Existing evidence indicates that the
pattern scheme should meet most of these criteria.
Some of the other schemes are also candidates for
generally or special purpose practical adaptive
control systems. Therefore, it follows that from
the point of view of control technology, adaptive
control is practical.
Adaptation will be an essential future characteristic of the fully automated plant. In addition to
giving the control system the means of monitoring
control performance for adaptation, it is also giving it a more sophisticated basis for alarming.
Thus instability or other dynamic conditions can be
alarmed. The simple improvement in control can be
significant in two ways: loops which we have been
unable to keep tuned will now be kept in tune. But
loops which would be unsafe if tightly tuned manually may now be kept tight since the adaptive system can prevent instability.
With respect to the field of adaptation itself, it
is now necessary to evolve means for adapting
larger and more complex process control systems as
a unit, automatically taking into account interactions between sections of the plant. The modeling
techniques and human set-up methods presently
available from the academic world are still too
cumbersome to be considered as useful to practical
operating personnel. The control system of the
future must automate all aspects of the adaptation
including any required set-up procedures.
What then stands in the way of its application in
industry? To begin with, most adaptation requires
digital computation. Since it has not been practical to date to produce digitally tuned analog controllers, it follows that adaptation presumes DDC.
This is likely to continue until adaptation or
other techniques justify development of more specialized controllers or adapting equipment. But
even this must await extensive adaptive experience.
This same experience is required to develop more
detailed design requirements for adaptations.
Within the context of DDC, both for the short and
long range, the industry will need to consider what
are the best operational philosophies to follow.
Taking the pattern scheme as a base, it is now possible technically to offer an adaptive DDC system
with the following characteristics:
1
TABLE I
Values of M, and C, in decimal equivalents of digital units
Sample times i
∆Mi in dig.
0
1
2
3
4
5
6
7
8
9
10
11
Scale 2047 dig. = 10 V
Human interference will not normally be
necessary in any loop except for an initial
pattern specification (two parameters specifying the desired quality of control) and
adaptation condition specifications
(whether or not set-point or load adaptation is allowed for the given loop). Occasionally measurement filtering may become
necessary.
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0
-225
-122
88
13
-56
4
12
0
-1
0
-1
Ci in dig.
0
307
321
39
22
77
33
-1
7
7
7
7
REFERENCES
where pj = ps(jT) with T the sampling time.
1. W. L Caldwell. "Control System with Automatic
Response Adjustment." U.S. Patent No.
2,517,081.
(Note that better model forms such as those of Kalman and others would be more efficient but less
clear for our results.)
2. D. W. Pessin. "Investigation of a Self-Adaptive
Three-Mode Controller." ISA September 1963.
Preprint 30.3.63.
This equation is exact as long as M(t) changes values only at each sampling time. If Ci and Mi-j are
known for all successive values of i and i-j the
equation becomes a set of linear equations in pj
and Ui:
3. T. P. Goodman and J. B. Reswick. "Determination
of System Characteristics from Normal Operating
Records." ASME No. 55-IRD-1. ASME-IRD Conference. April 1955.
C0 = ∆M0p0+ U0
4. E. Mishkin and L. Braun. Adaptive Control Systems. McGraw-Hill Book Company, Inc. 1961.
C1 = ∆M1p0 + ∆M0 p1+ U1
5. T. V. Osburn, H. P. Whitaker, A. Kezer. "New
Developments in the Design of Model Reference
Adaptive Control Systems." IAS 29th Annual
Meeting, January 1961. IAS Paper No. 61-39.
C2 = ∆M2 p0 + ∆M1p1 + ∆M0p2+ U2
n
6. R. W. Jensen and M. D. Lieberman. IBM Electronic
Circuit Analysis Program. Prentice-Hall, Inc.
1968. pp 322-330.
Cn =
∑ ∆Mn – j pj +
Un
j=0
7. R. E. Kalman. "Design of a Self-Optimizing Control System." Transactions of the ASME. February 1958.
There are n + 1 equations and 2(n + 1) unknowns. We
need n + 1 more equations which follow from experience with practical processes. In real life, all
processes have a finite settling time. This is
equivalent to a set of (n-k) equations for some k
of the form:
8. H. Paynter. "Self-Tuning Controller Studies."
December 1964. Unpublished Notes.
9. R. M. Bakke. "Adaptive Gain Tuning Applied to
Process Control." ISA October 1964. Preprint
3.2.-1-64.
pi = pk for k ≤ i ≤ n
10. E. H. 8ristol, G. F. Inaloglu, J. F. Steadman
"Adaptation of Process Controllers by Pattern
Recognition." JACC, August 1969. pp 605-606,
Similarly many real nonnoise disturbances and
upsets are isolated in time and of finite duration
giving n-l equations of form:
11. E. H. Bristol et al. "Adaptive Process Control
by Pattern Recognition." Instruments and Control Systems. March 1970. pp 101-105
uj. = ul for l ≤ j ≤ n
Also for most practical processes p0 = 0.
APPENDIX I
Applications of these equations to the original set
leaves n equations and k and l unknown. As long as k
+ l is less than n, the system is generally soluble.
It is possible to separate this set easily into a
set of k equations on pi, in which case the values
of Ui can be later calculated directly. In the
actual experiment l was assumed as 3, k as 5 and n
as 8. This assumption was based on the physical
feel of the calculator from looking at Figure 2 and
will be natural to experienced persons. The assumption is, of course, justified by the results and it
was made without collaboration of any kind between
experimenter and calculator. Such an assumption is
not critical and can be verified by taking several
values of k and calculating the matrix resulting.
The choice of l will be justified if changes in l do
not change the calculated values of pi and Ui
appreciably. The results may require that control
be poor enough so that meaningful measurements
occur for n greater than a reasonable l + k. This
limitation can be avoided by calculating on the
basis of several upsets or by recognizing that
identification is necessary only under poor control. It should be noticed that changing the sampling time changes the definition of the
identification but not the time necessary for the
identification. Thus n•T is a constant time chosen
appropriately to the process and upset. Under continuous adaptation these factors will not necessar-
The process of Figure 1 is defined by the convolution integral relationship where the upset and control begin at time t=0,
C(t) =
t
∫0 M(t – τ)p(τ) dτ + U(t)
where p(t) is the impulse response of the process
and U(t) is the upset response or disturbance
defined before. This same equation can be rearranged in terms of the step response and the derivative of M.
C(t) =
t
∫0 ddξ M(ξ) ξ = t – τ Ps(τ) dτ + U(t)
where Ps(t) is the step response of the process.
The sampled data equivalent of this equation is
i
Ci =
∑ ∆Mi – j pj + Ui
j=0
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ily cause trouble because any seriously incorrect
adaptation will cause control changes which degrade
control, but which lead to better identification
and thereby to better later control. This phenomenon was used in later adaptive experiments in which
k was given a value much smaller than that of the
process (up to 2 to l too small). Under these circumstances, identification was extremely erratic
but control was acceptable.
where:
∆S is the set-point step size, kp and kr are preset
scale factors which can be manually or automatically adjusted, if needed, to speed up the adaptation, ρ is a deadtime compensation factor based on
TD and is normally equal to unity. Note that the
error signal has been normalized by ∆S so that the
calculation depends not on set-point amplitude, but
only on response shape.
APPENDIX II10,11
At the end of T3 the scheme replaces the previous
two-term controller gains with the computed gains
P` and R'; adaptation stops and conventional twoterm control continues with these updated gains.
Consider Figure 18. Any set-point step change of
sufficient magnitude and duration applied by the
operator, or generated by the program, may trigger
the adaptation system. The respective times
required for the response to go from 0 percent to
25 percent and from 25 percent to 75 percent of the
set-point step are determined as TD and TL. The
latter is established as the time scale for the
adaptation and is used to establish three consecutive adaptation times:
If, on the next set-point change, the scheme still
sees a deviation from the desired B and C levels
during the adaptation intervals, the controller
gain values are updated again according to the
above formulae.
Procedures are included in the scheme to guarantee
universal convergence in any special situations.
The scheme considers only one transient at a time,
but safe provision is made in case of simultaneous
set-point or load changes. Extensions of the scheme
permit load adaptations for transients.
T1 = T D + T L + α • T L
T2 = T 1 + β • T L
T 3 = T2 + γ • T L
BIBLIOGRAPHY
where:
Bakke, R. M. "Adaptive Gain Tuning Applied to Process Control." ISA, October 1964. Preprint
3.2.-1-64
α, β, γ are preset constants selected on the basis
that the main transient is fully developed someβ + γ) • TL interval, and the first
where during the (β
half cycle in a resonant response develops during
the β • TL interval. This ensures that the relationship between the controller gains, and the corresponding transient pattern comparisons in the
adapting formulae are monotonic, assuring negative
adaptive feedback only.
Bellman, R.; Kalaba, R. E.; Lockett, Jo Ann. Numerical Inversion of the Laplace Transform: Application to Biology, Economics, Engineering, and
Physics. American Elsevier Publishing Company,
Inc. 1966.
Bristol, E. H. "A Simple Adaptive System for Industrial Control." Instrumentation Technology.
June 1467, pp 70-74.
The two-term controller has the DDC algorithm:
Bristol, E. H. et al. "Adaptive Process Control by
Pattern Recognition." Instruments and Control
Systems. March 1970. pp 101-105.
∆V = P•∆
∆E + P•R•E
where:
Bristol, E. H.; Inalaglu, G. F.; Steadman, J, F.
"Adaptation of Process Controllers by Pattern
Recognition." JACC, August 1969. pp 605-606.
∆V is the incremental output position, P and R are
initial proportional and reset gains, E is the
instantaneous error, and ∆E is the incremental
error.
Caldwell, W. I. "Control System with Automatic
Response Adjustment." U. S. Patent No.
2,517,081.
Goodman, T. P, and Reswick, J. B. "Determination of
System Characteristics from Normal Operating
Records." ASME No. 55-IRD-1. ASME-IRD Conference. April 1955.
During the intervals from T1 to T2 and from T1 to
T3, gain computations are made:
1
P′ = P 1 + ------------β ⋅ TL
T2
∑
Time = T 1
1
R′ = R 1 + ---------------------------⟨ β + γ ⟩ ⋅ TL
Inagaki, Tohru. "An Adaptive Direct Digital Control." Case Western Reserve University Systems
Research Center Report SRC-68-6.
E
k p ⟨ ------ – B⟩
∆S
T3
∑
Time = T 1
Kalman, R. E. "Design of a Self-Optimizing Control
System" Transactions of the ASME. February
1958.
E
k r ⟨ ------ – ρ ⋅ C⟩
∆S
Jensen, R. W.; Lieberman, M. D. IBM Electronic Circuit Analysis Program, Prentice-Hall Inc. 1968.
pp 322-330.
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Mishkin, E. and Braun, L. Adaptive Control Systems.
McGraw-Hill Book Company; Inc. 1961.
Osburn, T. V.; Whitaker, H. P.; Kezer, A. "New
Developments in the Design of Model Reference
Adaptive Control Systems." IAS 29th annual
meeting January 1961. IAS paper No. 61-39.
Pessin, D. W. "Investigation of a Self-Adaptive
Three-Mode Controller." ISA September 1963.
Preprint 30.3.63.
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