CA1115384A - Step-control of electrochemical systems - Google Patents
Step-control of electrochemical systemsInfo
- Publication number
- CA1115384A CA1115384A CA299,029A CA299029A CA1115384A CA 1115384 A CA1115384 A CA 1115384A CA 299029 A CA299029 A CA 299029A CA 1115384 A CA1115384 A CA 1115384A
- Authority
- CA
- Canada
- Prior art keywords
- response
- time
- signal
- input signal
- determining
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Expired
Links
Classifications
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
Abstract
ABSTRACT OF THE DISCLOSURE
The response of an automatic control system to a general input signal is improved by applying a test input signal, observing the response to the test input signal and deter-mining correctional constants necessary to provide a modified input signal to be added to the input to the system. A
method is disclosed for determining correctional constants.
The modified input signal, when applied in conjunction with an operating signal, provides a total system output exhibiting an improved response. This method is applicable to open-loop or closed-loop control systems. The method is also applicable to unstable systems, thus allowing con-trolled shut-down before dangerous or destructive response is achieved and to systems whose characteristics vary with tire, thus resulting in improved adaptive systems.
The response of an automatic control system to a general input signal is improved by applying a test input signal, observing the response to the test input signal and deter-mining correctional constants necessary to provide a modified input signal to be added to the input to the system. A
method is disclosed for determining correctional constants.
The modified input signal, when applied in conjunction with an operating signal, provides a total system output exhibiting an improved response. This method is applicable to open-loop or closed-loop control systems. The method is also applicable to unstable systems, thus allowing con-trolled shut-down before dangerous or destructive response is achieved and to systems whose characteristics vary with tire, thus resulting in improved adaptive systems.
Description
-3~4 STEP-CONTROL OF ELECTROMECHANICAL SYSTEMS
BACKGROUND OF THE INVENTION
The present invention relates to automatic control systems. It applies both to closed-loop (feedback) systems and to open-loop systems.
A basic problem in the design of automatic control systems is that of speed of response. The designer normally knows the idealized response that is desired from each par-ticular type of input and he frequently works with systems that are sufficiently close to linear to permit the super-~ ' .
. ~ ' . .
.j;~ .
... . .
., ~ - .
positlon of combined inputs to produce combined responses thereto. However, physical limitations upon the equlpment frequently lead to the introductlon of delays between the desired output signal and the observed output signal. This is especially true in the case of the linear second-order system which is typical of a wide range of automatic control systems and which provides an adequate descriptive approxi-mation to the performance of still more such systems. The second-order system is describable by a linear differential equation of the second order with constant coef~lcients.
Alternatively, it is characterized in the s-plane by a pair - of poles. Such a system may be in one of two conditions: it may be either overdamped or oscillatory, with a dividing line between the two comprising the case of critical damping. The overdamped conditlon is characterlzed by a step response that comprises the sum of the two exponentials wlth different time constants. In the s-plane the overdamped condition is represented by two poles on the negative real axis. The exponential having the longer time constant represents a delay between the actual value and the final value of the output that is often undesirable from the standpoint of the designer. Typically the designer would like to see a close approximation to a step output in response to a step input. However, if he varies system constants to vary the locatlon of the poles in an overdamped system so as to reduce the time constant of the dominant exponential term, he approaches more closely the oscilla-tory condition which may be undesirable for a number of
BACKGROUND OF THE INVENTION
The present invention relates to automatic control systems. It applies both to closed-loop (feedback) systems and to open-loop systems.
A basic problem in the design of automatic control systems is that of speed of response. The designer normally knows the idealized response that is desired from each par-ticular type of input and he frequently works with systems that are sufficiently close to linear to permit the super-~ ' .
. ~ ' . .
.j;~ .
... . .
., ~ - .
positlon of combined inputs to produce combined responses thereto. However, physical limitations upon the equlpment frequently lead to the introductlon of delays between the desired output signal and the observed output signal. This is especially true in the case of the linear second-order system which is typical of a wide range of automatic control systems and which provides an adequate descriptive approxi-mation to the performance of still more such systems. The second-order system is describable by a linear differential equation of the second order with constant coef~lcients.
Alternatively, it is characterized in the s-plane by a pair - of poles. Such a system may be in one of two conditions: it may be either overdamped or oscillatory, with a dividing line between the two comprising the case of critical damping. The overdamped conditlon is characterlzed by a step response that comprises the sum of the two exponentials wlth different time constants. In the s-plane the overdamped condition is represented by two poles on the negative real axis. The exponential having the longer time constant represents a delay between the actual value and the final value of the output that is often undesirable from the standpoint of the designer. Typically the designer would like to see a close approximation to a step output in response to a step input. However, if he varies system constants to vary the locatlon of the poles in an overdamped system so as to reduce the time constant of the dominant exponential term, he approaches more closely the oscilla-tory condition which may be undesirable for a number of
- 2 -, ~ - ' . . .
reasons. For example~ nonlinearities in the system may tend to sustain any oscillations that are started. The relocation of poles or, in other words, the variatlon of the time constants of exponentials in the step response of the linear second order system thus involves compromises that may have undesirable results. It would be useful to have a system for controlling response to a step input that does not require a change in the damping of the system.
It is not always desirable to have an overdamped system.
Sometimes a designer makes the choice of a system exhibiting an underdamped or oscillatory response. Such a system responds to a step lnput by overshooting the final value and oscillating with a damped oscillation about this final value. The amount of the overshoot and the frequency and rate of damping of the oscillation are functions of the system parametera. Two commonly applied figures of merit for systems that exhibit overshoot are the peak overshoot and the settling time. The peak overshoot is often expressed on a percent or a per unit basis as a measure of the ratio of the height of the first overshoot above the final value to the height of the final value for a step input. The settling time is defined as the time required for oscilla-tions to decrease to a specified absolute percentage of the final value and thereafter remain less than this value.
It is common to specify the allowable percentage as 2~ or 5%.
These figures are arbitraril~ chosen design criteria. Each provides a figure of merit in comparing systems for their ability to produce a desired output from a given input.
~, .
I.J.1.'~ 4 Generally speaking, rapid response of a system to a changing ~nput is assocated with oscillations about the final value following the change and slowness of response is associated with increased damping that removes such oscillations. In a linear system and, to a lesser degree, in nonlinear systems, part of the design problem includes a compromise between the desired degree of rapidity o~ response and the tolerable amount of oscillatlon about a final value.
The foregoing discussion has been cast primarily in terms o~ a so-called type-zero system. This is a system in which the steady-state response to a step input is a constant value. Such a system follows a ramp input with an error that increases in time and similarly produces an infinite steady-state error in response to a parabolic input. The same conclusions, though, hold for systems characterized by higher type numbers. For example, the type-one system has a step response that provides zero error in the steady state ~
and provides a constant error in the steady state in response -to a ramp input. The type-two system produces zero steady-state error to both a step and a ramp input and produces a constant error in response to a parabolic input. It can be seen by inspection that increasing the number of the system type increases the amount of intégration in the circuit and thereby enables the circuit to follow a higher-order input signal. These systems have in common the fact that selection of the system parameters and determination of whatever com-pensation may be necessary in ei~her a forward loop or a feedbacX loop determlnes the response of the system to any L~k ~lven class of signals. Improvement of this response in one area generally results ln deterioration in another area.
For example, improving the speed of response generally pro-duces an increase in any oscillation that exists about the final value and hence increases the time necessary to pro-duce settling within a given percent variation from the final value.
One method of overcoming the disadvantages inherent in dealing with a fixed-parameter control system is to condition the lnput si~nal to the system. The general result of such a process is to produce in the conditioning signal a correc-tion value that is 2 function of the signal itself. Such a signal is also a function of the system constants. The method of generating and applying sùch signals will require information as to current input signals and it must be based upon the parameters of the system.
It is an ob~ect of the present invention to improve the performance of control systems.
It is a further ob~ect of the present invention to produce a modified signal to replace the input signal to a control system so as to result in improved response.
It is a further ob~ect of the present invention to provide a corrective signal to add to the input signal to a system to provide an improved response to the original slgnal.
It is a further ob~ect of the ~resent invention to - provide an improved response ~or sampled-data control systems.
' i It is a further object of the present invention to pro-vide means for calculating and applying a corrective signal to be added to the signal present in a sample-data control system to provide an improved response of the system.
It is a further object of the present invention to provide means for calculating a signal correction that car be varied from time to time as system parameters change to produce a modified signal to supplement the input signal to the system and thus provide an adaptive version of improved system response.
It is a further object of the present invention to provide means allowing the use of underdamped systems and systems having high gain while maintaining stability of the system by calculating correction signals to be added to or used to supplement the input signals to the system to provide improved system response.
It is a further object of the present invention to provide an improvement in the response of an open-loop system by calcul-ating a correction signal to be used in conjunction with the input signal to the open-loop system.
Other objects of the invention will become apparent in the course of the detailed description of the invention.
; The response of an automatic control system to a general in-put signal that is a function of time, is improved by applying a ; test input signal to the system, determining a test response to the test input signal, and calculating therefrom a series of cor-rectional constants from the test response according to a prede-termined scheme. The correctional constants are applied to generate ., .
b from a random system input a correctional slgnal to be added to the random system input. The sum of the random system input and the correctional signal provides a system response that ls improved over the response obtained in the absence OL the correctional signal.
BRIEF DESCRIPTION OF THE DRAWINGS
Fig. 1 is a time plot of a step input signal and the response of an underdamped second order system to the stèp.
Fig. 2 is a time plot of a ramp input signal and representations of the responses of two different typical systems to a ramp input signal.
Fig. 3 ls a tlme plot of a parabolic input signal and the response of two control systems to the parabolic input signal.
Fig. 4 ls a time plot cf a number of input slgnals and an example of thelr combination into a more complicated signal.
Fig. 5 is a simplifled block diagram of a system that does not use the method of the present invention.
Fig. 6 ls a block diagram of an open-loop system including a signal processor for the practice of the present invention.
Fig. 7 is a block diagram representing a unity feed-back control system not embodying the present invention.
Fig. 8 ls a block diagram of a unity feedback control system employing a signal processor for the practice of the present invention.
Fig. 9 is a block diagram indicating a feedback system not embodyinc the principles of the pLesen~ ~nvention.
Fi~. 10 is a block diagr2m of a feedbac~ system including a signal processor according to the present invention.
Fig. 11 is an expanded bloc~ diagram of the signal processor of Fics. 6, 8 ~nd 10.
Fig. llA is a time plot of signals at various points ~n Fig. 11. -Fig. 12 is a time plot o~ a signal and of calculated correctional signals.
Fig. 13 is a block diagram of a circuit to perform certain calculations.
Fig. 13A is a succession of time plots of signals at various points of the circuit of Fig. 13.
Fig. 14 is a time plot of an error signal and its correction, appearing with Figs. 12 and 15.
Fig. 15 is a time plot OI' an error signal in an unstable system and of signals controlling this system, appear-ing with Figs. 12 and 14.
Fig. 16 is~a time plot of the step response of a system with overshoot.
Fig. 17 is a time plot of the uncorrected step response . . .
of a typical multiple-pole system.
Fig. 18 is a time plot of the correction signals and the corrected response for the typical multiple-pole system.
Fig. 19 is a block diagram showing the operational elements necessary to perform the operations required by the present invention using a digital computer, appearing with Figs. 9 and 10.
l. i.~.~io~
Fig. 20 is a block diagram of a computer arrangement for determining certain quantities (namely, N, L, M, and Ts).
Fig. 21 is a representation of a composite input and a corrected output signal in a typical control system.
Fig. 22 is a time plot of the corrective input signals and the corrected output signal of a system to which a ramp is applied. r Fig. 23 is a representation of the correction signals necessary to apply the principles of the present invention 10 to a system receiving an input that is a parabola.
DETAILED DESCRIPTION O~ T~E PREFERRED EMBODIMENT
The present invention comprises a method and means of improving the response of a control system to a general lnput signal. The problem to be overcome is indicated in various curves showing typical inputs to conventional systems and typical forms of the response thereto. Fig. l shows an inpu~ s~ep 30 having amplitude s. Curve 31 is a typical characteristic showing the response to such an - 8a -input step as observed at the output o~ a second-order system. Curve 31 reaches a peak value o~ amplitude '~sp at time Ts and thereafter oscillates with decreasing ampli-tude about flnal value Ks. One ~igure of merit applied to compare different designs o~ control systems is the time taken to achieve and maintain 2 response within a stated percentage of the ~inal value in response to an input step.
The present inventlon comprises a method ~or causing the system to achieve its ~inal value at time Ts in response to a step input. This can be seen to be an improvement over the response indicated as curve 31.
Fig. 2 is a curve showing a ramp signal and the response to it. In Fig. 2, curve 40 represents a typical ramp input signal. Typ$cal forms of the respon~e exhibited by control systems to an input signal such as curve 40 are indicated in response curves 42 and 46. Response 42 is that of a type-zero system. It indicates oscillation about a final value represented by displaced ramp 41. Response curve 46 is ' that of a type-one system. It exhibits an oscillation about ramp 45 which can be taken as the desired output ramp response. The principles of the present invention permit the achievement of the ideal ramp output in response to a ramp input.
Fig. 3 represents a parabolic input to a control system, curve 51. Two possible response curves are indi-cated. Curve 53 represents an output of a type-one system.
Curve 53 oscillates about a parabola wlth increasing steady-state error as time increases. Curve 52 represents a . , .
~].~ f~
response curve typical of a type-two system, oscillating with decreasing error about the parabola which represen~s ldeal response. The principles of the present invention permit the system to achieve the desired parabola without deviation.
Fig. 4 represents a composite general input signal, composite cùrve 65 and the components that go to make up such a signal. The components are step 61, ramp 62 and parabola 63. The sum of curves 61, 62 and 63 is composite curve 65, which can also be considered the first three terms of the Taylor's series for a general input curve. A system whlch gives improved response to composite curve 65 can be expected to provide a correspondingly improved response to the large maJorlty of signals that are likely to be applied to a physical system.
Turning now from signals to systems, we see in Fig 5 a conventional representation of an open-loop control system.
Input signal 71 is applied to system 72 which produces there-from output 73. The term "open-loop" is conventionally applied to lndicate the fact that there is no explicit connection between output signal 73 and input signal 71 in the reverse direction. The principles of the present inven-tion are applied in the form of a block diagram in Fig. 6.
Input signal 81 is there applled to signal processor 82.
Signal processor 82 generates modified input signal 83.
This is applied to system 84, producing therefrom output ~5.
The operations performed by signal processor 82 are dependent upon the parameters of system 84 and upon the form of input ~. '-signal 81. The result of the operation of slgnal processor 82 is to generate a modified signal 83 that causes a more nearly ideal response of system 84 than the rçsponse that would have been achieved by applying input signal 81 directly to system 84.
Figs. 7 and 8 comprise a conventional unity feedback system and the same system showing application of the principles of the present invention. Fig. 7 is the con-ventional unity feedback system. Summer 96 receives input signal 91 and feedback signal 92. The difference of these signals is error signal 93, which is applied to system 94 to achieve response 95. Fig. 8 comprises the modification o~ Fig. 7 according to the principles of the present inven-. . .
tion. In Fig. 8, lnput signal 101 and feedback signal 102are applied to summer 110 and the dif~erence is error slgnal 103. This is applied to signal processor 105 which applies the methods of the present invention to produce modified error signal 106. System 108 receives modified error signal 106 and generates therefrom output 109. Signal processor 105 produces an output that is a function both of the characteristics of system 108 and of error signal 103.
The modi~ied error slgnal 106 that is produced is capable of provldlng a response that is improved over the response that would be produced without such signal processor.
Figs. 9 and 10 show a general ~eedback system with and without the application of the methods of the present inventlon. Fig. 9 is a conventional representation of a general feedback control system. Summer 121 receives input signal 115 and feedback signal 120. Thelr dif~erence is error signal 11~, whlch is applied to signal 117. Output 118 is further modified in feedback elements 119 to produce feedback signal 120. The methods of the present invention are applied to such a system in Fig. 10, in w~ich input signal 131 and feedback signal 138 are combined in su~mer 139 to produce as a difference error signal 132. This is applied to signal processor 133, which operates according to the principles of the present invention to produce modified error signal 134. This, in turn, is applied to system 135 to produce output 136. A portion of output signal 136 is applied to feedback elements 137 to produce feedback signal 138. As earlier described, the signal processor, in this case signal processor 133, generates a modified error signal 134 that is a functlon both of the parameters of system 135 and feedback elements 137 and also of error signal 132. The various signal processors, namely 82 in Fig. 6, 105 in Fig. 8 and 133 in Fig. 10, perform the operation that effects the method of the present invention.
The mechanism of this operation will be described below.
Dynamical physical systems possess time constants and natural frequencies which cause the systems to fail exactly to follow the forms of input signals. Much of the design effort applied to control systems comprises a process of attempting to shape system parameters to improve in some measure the degree to which the form of the output approxi-mates that of the input. The present invention comprises a method of deriving a set of modified signals based upon the original signal and the parameters of the system to be controlled. The modified signal is applied as a replacement for the ori~inal input signal to the system or as a replace-ment for the original error signal in a feedback system.
Proper application o~ the method of this inven'cion allows ready calculation and generation of a signal that causes the system to achieve an improved response over that generated in response to the original input signal. The method comprises determining sets of five quantities. These are the system gain, K, the percent response at the time Ts, p, in response to a step input, the time Ts required to achieve the first peak of the output in response to a step input, the percent response U of the output at time Ts in response to an input ramp signal, and the percent response W at time Ts in response to a parabolic input signal. Three o~ the preceding quantities either appear in or can be obtained from the output signal 31 shown in Fig. 1. The time to the first peak is indicated in Fig. 1 as Ts, and the peak overshoot Ksp is seen in Fig. 1 as the peak value achieved by output slgnal 31 at time Ts. It is possible to measure or know amplitude s, the height of input step 30, and to measure the asymptotic final value Ks which is the height o~ curve 32. These quantities allow mechanical or other computation of quantities K, p, and Ts. Knowing Ts, it is a straightforward matter to obtain the amplitude of the appropriate curve 42 or 46 in Fig. 2 and to calculate therefrom the quantity U, the percent output at time Ts in response to a ramp input. Similarl~, knowing Ts, tne quantity W, the percent output at time Ts in response to a parabolic input, can also be determined.
The practice of the present invention now requires cal-culating correction factors to apply to the input signal to generate a modified input signal to replace the input signal.
This will be considered first from the standpoint of illus-trating the modifications that take place to the signal. To see this we consider three necessary correction factors defined as follows:
N =
P
P
2 (1 - W) - ULTs M = ~
The calculation of correctional signals for a general input requires the additional determination of the velocity V and acceleration a to derive a total correctional signal. ~his correctional signal, as described above, will add to the input to provide a composite signal that produces a response that ;: more closely approximates the ideal response to the signal.
These quantities are used to generate modified input signal R(t) as follows:
R(t) = r(t) + SN + VL + aM
where r(T) is the original input signal to the signal processor.
This modified signal in general represents the original input signal to a step modified by the factor N, added with a sum of steps and ramps and deltas modified by the factor L, added with a sum of steps and ramps modified by the factor M; the values of V and a are:
: '.
~ .
dr ~2r V = dt and a = ~ .
Fig. 11 is a block diagram of an analog version of the signal processor 133 of Fig. 10 for generating the signals necessary to practice the instant invention when it is applied to a system requiring constant values for N, L, T
and M. Fig. llA identifies signals at labeled points of Fig. 11. In Fig. 11, input signal 141 is applied to summer 142 and also to first differentiator 153~ The output of first differentiator 153 is applied in turn to second differentiator 156. Three scalers receive the signal and the differentiated outputs. Scaler 154 is set to a value equal to L to provide a scaled value of the once-differentiated signal from first differentiator 153. Scaler 155 is set to a value equal to 1 - N = l/P to provide a scaled value to the signal. Scaler 159 is set to a value M to be applied to the twice-differentiated signal produced by second differentiator 156. The symbols for scalers 154, 155, and 159 are those conventionally used for potentiometers which is appropriate if the values of these constants are less than or equal to 1.
If it becomes desirable to use values for these constants that ~; are greater than 1, the scalers can easily be combined with stages of amplification to produce the necessary gain.
The output of scaler 154, that of scaler 159 and the input signal 141 are combined in summing amplifier 142.
Its output in turn is fed to scaler 155, set to have a value :, ' 111.X.~4 equal to 1 - N = l/P. It is corrected by gate 150, which has the function of removing its output 157 after time Ts.
This is accomplished by the use of threshold detector 151 which receives first differentiated signal 145 and controls monostable 152 when first differentiated signal 145 exceeds a predetermined threshold limit. r~onostable 152 generates a gating square wave for application to gate 150. The result of this combination is that scaler 155 is dis-connected whenever there is no output from the threshold detector 151 in response to first differentiator signal 145.
- When signal 145 exceeds the predetermined threshold, scaler 155 is thus lnserted into the circuit to operate on the signal. This scales the input into the system 149 for a :
period selected by the length of the output of monostable 152. ~nis length will be chosen to eaual Ts.
Table I summarizes the response o~ the second-order system to the modl~ied signal. The weighting functlon of the system is W(t) = n e~ ~ t sin ~ t The response to a step s is s ~ ~ (x) dx The response to a ramp Vt is t at 2 The response to a parabola 2 is at2 ~ t ~ z ~ ~ (x) dx dy dz o o o . . ' ' : ' : '' -`3 r~
ul ~
+
~ o E~ c) ~a +
_ _~ + ~ b~
o ~ 3 o 3 +
E~ ~ ~ u~
~c ~ o _ .,. ...
U2 ~ _ Z
o ~ ~ ~ ~ U~
~ ,, . ~ , , o V~ ,_ I ~, _ a 1~1 -- I Q~
~: ~ . ~ X
X S~
t ~ ~ X
+ ~.~ I ~1 ~- 3 E~ ~ ~ C) u2 :, ~ ~U ~ ~ ~ , o q: Z ~ ~ ~ ~ ~ ~ ~
H ~ ~ ~ I td ~ 1_ E:l O
j~ m _ 'C:1 ~ E~
H :~ 11 li:, 0., .
Z
1:1 H
O ~_ ~: t~: +
N
.~
J~
P ~ 11 d 'C1 O S~
~I H
E~ 11 :~ ~ ~:
P~ J~ D;
Z `_ O
H S~ C~
., J~ _ ~: ~ ~?
E~ 02 ~: ~
+ ~q ~
+~a o E~ A
~ ~ ~ 3 V~o~
:Z o _ U~
o C~ ~ o _, _, X '~
_, X
J~
~ ~ ' 3 X
X ~ ~
Z
,, ~ ~ o m ~
~ H~ I ~:~ ~
m ~ E~ ~ ~ ;~
H :~11 h ~1 ~ Z
C~ H ~ ~1 O ~
~: ~: + +
~t ¢ ~ 11 H
E~
Il C~ ~1 E~
~ _~ ~; h Z _, O
H h , ~,J.~ 4 ,~
s ~ ) + ~
~: ~ +
~; ~ ~
E~ rA `~ ~ 3 ~:
C U~
o +
E~ 3 +
E~ ~ ~n ~
a~ ~ 3 xP- k ~ ~ O + a~
P~,~
o ~ o _~ +
' p~
~ X ~
reasons. For example~ nonlinearities in the system may tend to sustain any oscillations that are started. The relocation of poles or, in other words, the variatlon of the time constants of exponentials in the step response of the linear second order system thus involves compromises that may have undesirable results. It would be useful to have a system for controlling response to a step input that does not require a change in the damping of the system.
It is not always desirable to have an overdamped system.
Sometimes a designer makes the choice of a system exhibiting an underdamped or oscillatory response. Such a system responds to a step lnput by overshooting the final value and oscillating with a damped oscillation about this final value. The amount of the overshoot and the frequency and rate of damping of the oscillation are functions of the system parametera. Two commonly applied figures of merit for systems that exhibit overshoot are the peak overshoot and the settling time. The peak overshoot is often expressed on a percent or a per unit basis as a measure of the ratio of the height of the first overshoot above the final value to the height of the final value for a step input. The settling time is defined as the time required for oscilla-tions to decrease to a specified absolute percentage of the final value and thereafter remain less than this value.
It is common to specify the allowable percentage as 2~ or 5%.
These figures are arbitraril~ chosen design criteria. Each provides a figure of merit in comparing systems for their ability to produce a desired output from a given input.
~, .
I.J.1.'~ 4 Generally speaking, rapid response of a system to a changing ~nput is assocated with oscillations about the final value following the change and slowness of response is associated with increased damping that removes such oscillations. In a linear system and, to a lesser degree, in nonlinear systems, part of the design problem includes a compromise between the desired degree of rapidity o~ response and the tolerable amount of oscillatlon about a final value.
The foregoing discussion has been cast primarily in terms o~ a so-called type-zero system. This is a system in which the steady-state response to a step input is a constant value. Such a system follows a ramp input with an error that increases in time and similarly produces an infinite steady-state error in response to a parabolic input. The same conclusions, though, hold for systems characterized by higher type numbers. For example, the type-one system has a step response that provides zero error in the steady state ~
and provides a constant error in the steady state in response -to a ramp input. The type-two system produces zero steady-state error to both a step and a ramp input and produces a constant error in response to a parabolic input. It can be seen by inspection that increasing the number of the system type increases the amount of intégration in the circuit and thereby enables the circuit to follow a higher-order input signal. These systems have in common the fact that selection of the system parameters and determination of whatever com-pensation may be necessary in ei~her a forward loop or a feedbacX loop determlnes the response of the system to any L~k ~lven class of signals. Improvement of this response in one area generally results ln deterioration in another area.
For example, improving the speed of response generally pro-duces an increase in any oscillation that exists about the final value and hence increases the time necessary to pro-duce settling within a given percent variation from the final value.
One method of overcoming the disadvantages inherent in dealing with a fixed-parameter control system is to condition the lnput si~nal to the system. The general result of such a process is to produce in the conditioning signal a correc-tion value that is 2 function of the signal itself. Such a signal is also a function of the system constants. The method of generating and applying sùch signals will require information as to current input signals and it must be based upon the parameters of the system.
It is an ob~ect of the present invention to improve the performance of control systems.
It is a further ob~ect of the present invention to produce a modified signal to replace the input signal to a control system so as to result in improved response.
It is a further ob~ect of the present invention to provide a corrective signal to add to the input signal to a system to provide an improved response to the original slgnal.
It is a further ob~ect of the ~resent invention to - provide an improved response ~or sampled-data control systems.
' i It is a further object of the present invention to pro-vide means for calculating and applying a corrective signal to be added to the signal present in a sample-data control system to provide an improved response of the system.
It is a further object of the present invention to provide means for calculating a signal correction that car be varied from time to time as system parameters change to produce a modified signal to supplement the input signal to the system and thus provide an adaptive version of improved system response.
It is a further object of the present invention to provide means allowing the use of underdamped systems and systems having high gain while maintaining stability of the system by calculating correction signals to be added to or used to supplement the input signals to the system to provide improved system response.
It is a further object of the present invention to provide an improvement in the response of an open-loop system by calcul-ating a correction signal to be used in conjunction with the input signal to the open-loop system.
Other objects of the invention will become apparent in the course of the detailed description of the invention.
; The response of an automatic control system to a general in-put signal that is a function of time, is improved by applying a ; test input signal to the system, determining a test response to the test input signal, and calculating therefrom a series of cor-rectional constants from the test response according to a prede-termined scheme. The correctional constants are applied to generate ., .
b from a random system input a correctional slgnal to be added to the random system input. The sum of the random system input and the correctional signal provides a system response that ls improved over the response obtained in the absence OL the correctional signal.
BRIEF DESCRIPTION OF THE DRAWINGS
Fig. 1 is a time plot of a step input signal and the response of an underdamped second order system to the stèp.
Fig. 2 is a time plot of a ramp input signal and representations of the responses of two different typical systems to a ramp input signal.
Fig. 3 ls a tlme plot of a parabolic input signal and the response of two control systems to the parabolic input signal.
Fig. 4 ls a time plot cf a number of input slgnals and an example of thelr combination into a more complicated signal.
Fig. 5 is a simplifled block diagram of a system that does not use the method of the present invention.
Fig. 6 ls a block diagram of an open-loop system including a signal processor for the practice of the present invention.
Fig. 7 is a block diagram representing a unity feed-back control system not embodying the present invention.
Fig. 8 ls a block diagram of a unity feedback control system employing a signal processor for the practice of the present invention.
Fig. 9 is a block diagram indicating a feedback system not embodyinc the principles of the pLesen~ ~nvention.
Fi~. 10 is a block diagr2m of a feedbac~ system including a signal processor according to the present invention.
Fig. 11 is an expanded bloc~ diagram of the signal processor of Fics. 6, 8 ~nd 10.
Fig. llA is a time plot of signals at various points ~n Fig. 11. -Fig. 12 is a time plot o~ a signal and of calculated correctional signals.
Fig. 13 is a block diagram of a circuit to perform certain calculations.
Fig. 13A is a succession of time plots of signals at various points of the circuit of Fig. 13.
Fig. 14 is a time plot of an error signal and its correction, appearing with Figs. 12 and 15.
Fig. 15 is a time plot OI' an error signal in an unstable system and of signals controlling this system, appear-ing with Figs. 12 and 14.
Fig. 16 is~a time plot of the step response of a system with overshoot.
Fig. 17 is a time plot of the uncorrected step response . . .
of a typical multiple-pole system.
Fig. 18 is a time plot of the correction signals and the corrected response for the typical multiple-pole system.
Fig. 19 is a block diagram showing the operational elements necessary to perform the operations required by the present invention using a digital computer, appearing with Figs. 9 and 10.
l. i.~.~io~
Fig. 20 is a block diagram of a computer arrangement for determining certain quantities (namely, N, L, M, and Ts).
Fig. 21 is a representation of a composite input and a corrected output signal in a typical control system.
Fig. 22 is a time plot of the corrective input signals and the corrected output signal of a system to which a ramp is applied. r Fig. 23 is a representation of the correction signals necessary to apply the principles of the present invention 10 to a system receiving an input that is a parabola.
DETAILED DESCRIPTION O~ T~E PREFERRED EMBODIMENT
The present invention comprises a method and means of improving the response of a control system to a general lnput signal. The problem to be overcome is indicated in various curves showing typical inputs to conventional systems and typical forms of the response thereto. Fig. l shows an inpu~ s~ep 30 having amplitude s. Curve 31 is a typical characteristic showing the response to such an - 8a -input step as observed at the output o~ a second-order system. Curve 31 reaches a peak value o~ amplitude '~sp at time Ts and thereafter oscillates with decreasing ampli-tude about flnal value Ks. One ~igure of merit applied to compare different designs o~ control systems is the time taken to achieve and maintain 2 response within a stated percentage of the ~inal value in response to an input step.
The present inventlon comprises a method ~or causing the system to achieve its ~inal value at time Ts in response to a step input. This can be seen to be an improvement over the response indicated as curve 31.
Fig. 2 is a curve showing a ramp signal and the response to it. In Fig. 2, curve 40 represents a typical ramp input signal. Typ$cal forms of the respon~e exhibited by control systems to an input signal such as curve 40 are indicated in response curves 42 and 46. Response 42 is that of a type-zero system. It indicates oscillation about a final value represented by displaced ramp 41. Response curve 46 is ' that of a type-one system. It exhibits an oscillation about ramp 45 which can be taken as the desired output ramp response. The principles of the present invention permit the achievement of the ideal ramp output in response to a ramp input.
Fig. 3 represents a parabolic input to a control system, curve 51. Two possible response curves are indi-cated. Curve 53 represents an output of a type-one system.
Curve 53 oscillates about a parabola wlth increasing steady-state error as time increases. Curve 52 represents a . , .
~].~ f~
response curve typical of a type-two system, oscillating with decreasing error about the parabola which represen~s ldeal response. The principles of the present invention permit the system to achieve the desired parabola without deviation.
Fig. 4 represents a composite general input signal, composite cùrve 65 and the components that go to make up such a signal. The components are step 61, ramp 62 and parabola 63. The sum of curves 61, 62 and 63 is composite curve 65, which can also be considered the first three terms of the Taylor's series for a general input curve. A system whlch gives improved response to composite curve 65 can be expected to provide a correspondingly improved response to the large maJorlty of signals that are likely to be applied to a physical system.
Turning now from signals to systems, we see in Fig 5 a conventional representation of an open-loop control system.
Input signal 71 is applied to system 72 which produces there-from output 73. The term "open-loop" is conventionally applied to lndicate the fact that there is no explicit connection between output signal 73 and input signal 71 in the reverse direction. The principles of the present inven-tion are applied in the form of a block diagram in Fig. 6.
Input signal 81 is there applled to signal processor 82.
Signal processor 82 generates modified input signal 83.
This is applied to system 84, producing therefrom output ~5.
The operations performed by signal processor 82 are dependent upon the parameters of system 84 and upon the form of input ~. '-signal 81. The result of the operation of slgnal processor 82 is to generate a modified signal 83 that causes a more nearly ideal response of system 84 than the rçsponse that would have been achieved by applying input signal 81 directly to system 84.
Figs. 7 and 8 comprise a conventional unity feedback system and the same system showing application of the principles of the present invention. Fig. 7 is the con-ventional unity feedback system. Summer 96 receives input signal 91 and feedback signal 92. The difference of these signals is error signal 93, which is applied to system 94 to achieve response 95. Fig. 8 comprises the modification o~ Fig. 7 according to the principles of the present inven-. . .
tion. In Fig. 8, lnput signal 101 and feedback signal 102are applied to summer 110 and the dif~erence is error slgnal 103. This is applied to signal processor 105 which applies the methods of the present invention to produce modified error signal 106. System 108 receives modified error signal 106 and generates therefrom output 109. Signal processor 105 produces an output that is a function both of the characteristics of system 108 and of error signal 103.
The modi~ied error slgnal 106 that is produced is capable of provldlng a response that is improved over the response that would be produced without such signal processor.
Figs. 9 and 10 show a general ~eedback system with and without the application of the methods of the present inventlon. Fig. 9 is a conventional representation of a general feedback control system. Summer 121 receives input signal 115 and feedback signal 120. Thelr dif~erence is error signal 11~, whlch is applied to signal 117. Output 118 is further modified in feedback elements 119 to produce feedback signal 120. The methods of the present invention are applied to such a system in Fig. 10, in w~ich input signal 131 and feedback signal 138 are combined in su~mer 139 to produce as a difference error signal 132. This is applied to signal processor 133, which operates according to the principles of the present invention to produce modified error signal 134. This, in turn, is applied to system 135 to produce output 136. A portion of output signal 136 is applied to feedback elements 137 to produce feedback signal 138. As earlier described, the signal processor, in this case signal processor 133, generates a modified error signal 134 that is a functlon both of the parameters of system 135 and feedback elements 137 and also of error signal 132. The various signal processors, namely 82 in Fig. 6, 105 in Fig. 8 and 133 in Fig. 10, perform the operation that effects the method of the present invention.
The mechanism of this operation will be described below.
Dynamical physical systems possess time constants and natural frequencies which cause the systems to fail exactly to follow the forms of input signals. Much of the design effort applied to control systems comprises a process of attempting to shape system parameters to improve in some measure the degree to which the form of the output approxi-mates that of the input. The present invention comprises a method of deriving a set of modified signals based upon the original signal and the parameters of the system to be controlled. The modified signal is applied as a replacement for the ori~inal input signal to the system or as a replace-ment for the original error signal in a feedback system.
Proper application o~ the method of this inven'cion allows ready calculation and generation of a signal that causes the system to achieve an improved response over that generated in response to the original input signal. The method comprises determining sets of five quantities. These are the system gain, K, the percent response at the time Ts, p, in response to a step input, the time Ts required to achieve the first peak of the output in response to a step input, the percent response U of the output at time Ts in response to an input ramp signal, and the percent response W at time Ts in response to a parabolic input signal. Three o~ the preceding quantities either appear in or can be obtained from the output signal 31 shown in Fig. 1. The time to the first peak is indicated in Fig. 1 as Ts, and the peak overshoot Ksp is seen in Fig. 1 as the peak value achieved by output slgnal 31 at time Ts. It is possible to measure or know amplitude s, the height of input step 30, and to measure the asymptotic final value Ks which is the height o~ curve 32. These quantities allow mechanical or other computation of quantities K, p, and Ts. Knowing Ts, it is a straightforward matter to obtain the amplitude of the appropriate curve 42 or 46 in Fig. 2 and to calculate therefrom the quantity U, the percent output at time Ts in response to a ramp input. Similarl~, knowing Ts, tne quantity W, the percent output at time Ts in response to a parabolic input, can also be determined.
The practice of the present invention now requires cal-culating correction factors to apply to the input signal to generate a modified input signal to replace the input signal.
This will be considered first from the standpoint of illus-trating the modifications that take place to the signal. To see this we consider three necessary correction factors defined as follows:
N =
P
P
2 (1 - W) - ULTs M = ~
The calculation of correctional signals for a general input requires the additional determination of the velocity V and acceleration a to derive a total correctional signal. ~his correctional signal, as described above, will add to the input to provide a composite signal that produces a response that ;: more closely approximates the ideal response to the signal.
These quantities are used to generate modified input signal R(t) as follows:
R(t) = r(t) + SN + VL + aM
where r(T) is the original input signal to the signal processor.
This modified signal in general represents the original input signal to a step modified by the factor N, added with a sum of steps and ramps and deltas modified by the factor L, added with a sum of steps and ramps modified by the factor M; the values of V and a are:
: '.
~ .
dr ~2r V = dt and a = ~ .
Fig. 11 is a block diagram of an analog version of the signal processor 133 of Fig. 10 for generating the signals necessary to practice the instant invention when it is applied to a system requiring constant values for N, L, T
and M. Fig. llA identifies signals at labeled points of Fig. 11. In Fig. 11, input signal 141 is applied to summer 142 and also to first differentiator 153~ The output of first differentiator 153 is applied in turn to second differentiator 156. Three scalers receive the signal and the differentiated outputs. Scaler 154 is set to a value equal to L to provide a scaled value of the once-differentiated signal from first differentiator 153. Scaler 155 is set to a value equal to 1 - N = l/P to provide a scaled value to the signal. Scaler 159 is set to a value M to be applied to the twice-differentiated signal produced by second differentiator 156. The symbols for scalers 154, 155, and 159 are those conventionally used for potentiometers which is appropriate if the values of these constants are less than or equal to 1.
If it becomes desirable to use values for these constants that ~; are greater than 1, the scalers can easily be combined with stages of amplification to produce the necessary gain.
The output of scaler 154, that of scaler 159 and the input signal 141 are combined in summing amplifier 142.
Its output in turn is fed to scaler 155, set to have a value :, ' 111.X.~4 equal to 1 - N = l/P. It is corrected by gate 150, which has the function of removing its output 157 after time Ts.
This is accomplished by the use of threshold detector 151 which receives first differentiated signal 145 and controls monostable 152 when first differentiated signal 145 exceeds a predetermined threshold limit. r~onostable 152 generates a gating square wave for application to gate 150. The result of this combination is that scaler 155 is dis-connected whenever there is no output from the threshold detector 151 in response to first differentiator signal 145.
- When signal 145 exceeds the predetermined threshold, scaler 155 is thus lnserted into the circuit to operate on the signal. This scales the input into the system 149 for a :
period selected by the length of the output of monostable 152. ~nis length will be chosen to eaual Ts.
Table I summarizes the response o~ the second-order system to the modl~ied signal. The weighting functlon of the system is W(t) = n e~ ~ t sin ~ t The response to a step s is s ~ ~ (x) dx The response to a ramp Vt is t at 2 The response to a parabola 2 is at2 ~ t ~ z ~ ~ (x) dx dy dz o o o . . ' ' : ' : '' -`3 r~
ul ~
+
~ o E~ c) ~a +
_ _~ + ~ b~
o ~ 3 o 3 +
E~ ~ ~ u~
~c ~ o _ .,. ...
U2 ~ _ Z
o ~ ~ ~ ~ U~
~ ,, . ~ , , o V~ ,_ I ~, _ a 1~1 -- I Q~
~: ~ . ~ X
X S~
t ~ ~ X
+ ~.~ I ~1 ~- 3 E~ ~ ~ C) u2 :, ~ ~U ~ ~ ~ , o q: Z ~ ~ ~ ~ ~ ~ ~
H ~ ~ ~ I td ~ 1_ E:l O
j~ m _ 'C:1 ~ E~
H :~ 11 li:, 0., .
Z
1:1 H
O ~_ ~: t~: +
N
.~
J~
P ~ 11 d 'C1 O S~
~I H
E~ 11 :~ ~ ~:
P~ J~ D;
Z `_ O
H S~ C~
., J~ _ ~: ~ ~?
E~ 02 ~: ~
+ ~q ~
+~a o E~ A
~ ~ ~ 3 V~o~
:Z o _ U~
o C~ ~ o _, _, X '~
_, X
J~
~ ~ ' 3 X
X ~ ~
Z
,, ~ ~ o m ~
~ H~ I ~:~ ~
m ~ E~ ~ ~ ;~
H :~11 h ~1 ~ Z
C~ H ~ ~1 O ~
~: ~: + +
~t ¢ ~ 11 H
E~
Il C~ ~1 E~
~ _~ ~; h Z _, O
H h , ~,J.~ 4 ,~
s ~ ) + ~
~: ~ +
~; ~ ~
E~ rA `~ ~ 3 ~:
C U~
o +
E~ 3 +
E~ ~ ~n ~
a~ ~ 3 xP- k ~ ~ O + a~
P~,~
o ~ o _~ +
' p~
~ X ~
3 ' c~
- ~ u +
x ~ a X +
+
~ ~o I ~L 11 ~ 11 h !~1 ~ ~1 ~ o ~
~ ~q ~ :\ ll ~
o H _ q ~ ¢
+ ~
æ
E~ ~
~ ~ ~q ¢ ~ E~
Z ~ r~
cq H
E~ ~ I Q.
:~ ~ ~ ~1 ~ I
Z _~ O
H ~ O E-l ~`
The analysis shown in Table I shows that the response to the modi~ied input R(t) is perfectly obtained ~or ramp and parabollc inputs and is obtained in tlme Ts after applica-tion of a step.
Including the above signal modification unit within a ~eedback loop most o~ten will sause instability. This is because the open-loop transfer function is changed due to the presence of the signal processor and the ~eedback transfer function that originally yielded a stable system is no longer appropriate. A sampled signal modification unit may be inserted within a feedback loop without causing lnstability and lts presence will significantly reduce the output error o~ the system.
Fig. 12 is a time plot OL a signal and of calculated correctional signals.
In Fig. 12, an error signal such as that caused by a - switchlng transient is shown as curve 240, which is sampled at to and has a magnitude of ~ O and a rate of ~ O. With no slgnal modification the error set at to + Ts_will be ~ o(l - P) and ~ o(l - P) shown on curve 340. A ram~, curve 341, beglnnln~; at to and having the value ~Jt w~ 11 yield an output rate VP, curve 342, at to + Ts' (This is true because, since ~O(to + Ts) = SP; ~ O (to + Ts) ~ VP, ~ ~ (to + Ts) = aP, due to the time-integral relationships descrlbed immediately above.) For this ramp to exactly cancel the error rate at time to + Ts VP = ~o(l - P); V = ~o(l - P)/P
The ramp will cause an additional error, 346, o~ magnitude VUTS and to correct this to zero as well as the error ~ o(l - P), an input step of magnitude ~, curve 343, is also applied. The response to thls slgnal is shown as curve 344 and has a magnitude EP at t + T and has zero rate. For o s the slgnal magnitude to be zero at to + T requlres EP = VUTS + ~ P) (.1 - P) p - UTS + ~ o ( 1 - P ) whence E = ~ 1 - P U Ts + ~O l - ?
- 1 p P ~ ~ o Up 5 + ~ o}
( . T - LP
= N 'l ~ P T3 ~ ~ o ~
Shown in Fig. 13 is a circuit to perform this function, with ~ignals at various points shown in Fig. 13A. In Fig. 13, error signal 353 is fed to summing amplifier 354 and also to scaler 355, scaler 355 being ad~usted to a value of 1 - N.
Scaler output 368 is fed to sample-hold 356 and to time differentiator 357. The output of sample-hold 356 is fed to summing amplifier 354. The output of time differentiator 357 is ~ed to sample-hold 358 whose output 1s fed to scaler 359 and time integrator 360. Scaler 359 is set to a value T - LP
s T.s and its output as well as the output from time integrator 360 fed to summing amplifier 354. Points marked x in the drawing, namel~ at signal point 350 and at signal point 363, indicate where the system error path was opened to lnsert the sub~ect signal processor.
: .
x~
Clock 362 causes monostable 361 to generate a ~ulse train 370 whose pulses are separated by the time interval Ts. Signal 370 is fed to sample-hold units 356 and 358 and to electronic switch 369 that serves to dlscharge the integrator to zero every Ts seconds.
In Fig. 13, curve 371 is the output error resulting from an initial error, ~ 1' for operation of the system without use of the present invention. Curve 372 is the output error resulting from an initial error, ~ 1' sampled for the values ~ O, ~ O using the present invention. Curve 373 is the modified error signal R(t) which ls the sum of error signal 353 and correction signals 374, 375 and 376.
Use of the unit of Fig. 13 serves not only to improve system response to input signal 377 but also to rapidly ; correct for errors due to noise and disturbances both internal and external. This is illustrated in Fig. 14 for operation o~ a hi~h-voltage power supply. Step error 400 is caused by switching a control demanding a higher voltage.
Modi~ication signal 401 then causes the error to be smoothiy reduced to zero according to curve 402. The higher voltage causes an arc resulting in error signal 403 which is reduced smoothly to zero by auxiliary signal 404.
When a normally stab}e system becomes unstable due to changlng characteristics, the unit shown in Fig. 13 can keep lt under control. Fig. 15 depicts the situation. Curve 410 shows an initial error, cur~e 412, gradually ~uildlng up to a large oscillation amplitude. When the unit o~ Fig. 13 is used the error correction signal, curve 411, causes the ?~ 4 oscillation am~litude ~o be kept in control, as shown by curve 413. Tne unusuall~ large signal ~11 can be made to flash a warning light. This operation is potentially very important in the control of massive systems such as large aircraft during L light test, nuclear reactors, etc., where large-amplitude oscillation could lead to destruction or severe damage.
The method of the present invention can be used for improvement of multi-time-constant systems. It is usually advantageous to analyze test results from the measurements of P and Ts to determine the class or type of system.
Referring to Fig. 16, for a step input S, the system response overshoots to a value KSP, at time t1, undershoots to a value KSP2 at time t2, overshoots to a value KSP3 at t3, etc. The time intervals tl - to, t2 ~ tl, t3 - t2, --etc. can be measured as can the ratios (KSPl - KS)/KS, (KSP2 - KS)/(KSPl - KS), (KSP3 - KS)/(KSP2 - KS), etc. If these ratios are all the same and the time intervals equal, the system ls identified as a second-order system with no domlnant real poles or zeros.
When the ratios are the same but the time interval tl - to is shorter than the time intervals t2 ~ tl, t3 - t2, etc., and when the time intervals t2 ~ tl, t3 - t2, -- etc.
are equal, the system is identified as being a second-order system with a dominant real zero.
~ hen the ratios are not the same and when the time interval tl - to is longer than the succeeding time intervals the system is identified as being a second-order system with a dominant real pole.
These three classes of system cover ~he vast ma~ority of systems, although applicatlon of the present in~ention is not limited to these systems.
When used to improve the performance of second-order systems having a dominant simple zero, the schemes hereto-fore described are directly applicable wlth the only additional requirement being to redefine Ts. Fig. 16 s~ows the response of such a system. In Flg. 16 where the desired response to an input step S is KS, curve 420, the first response peak is at Tl, a time lnterval shorter than T2, T3, T4, etc., which are all equal. The time Ts is defined as the time interval T2 = T3 = T4 = ... NTS. ~his places the amplitude of the response measured to calculate P at the point x on curve 421. The value P' and values for U and W measured at Ts are then used to determine the proper ad~ustment of the signal modifier.
When used to improve the performance of second-order systems having a dominant simple pole, the scheme heretofore described can be used with some circuit modification.
Referrlng to Fig. 17, the response, curve 242, is not sgmmetrical about the deslred value, curve 430, due to the long exponential response of the simple pole. The time intervals between successive peaks and nulls are not quite equal. In Fig. 13, if the outputs from circuit elements 356, 359 and 360 are combined in an adder and its ouput fed through a lead circuit before being applied to the input of summing ampli~ier 354, the improved performance is achieved.
; - 24 _ .r~a~
The lead circuit should possess a value:
.~ P
where P is the value of the simple pole. Alternatively, where computer control is desirable and available, a sequential program can be used.
The practice of the present invention now r~quires calculating correction factors to apply to the inDut signal to generate a modified input signal to re~lace the input signal. Thls will be considered first from the standpoint of illustrating the modifications that taXe place to the signal. To see this we consider three necessary correction factors defined as follows:
1 - PN - NlPN ~ N2-PN_1 ~~~ NN-l 2 N
L , N(l UN) Ll N L2PN-1 LN-lP2 N . Pl M NZ (1 - WN) - ~ lPz _ ~ ZP3 --- -MlPN - LlTNUN
N
_-- -L2TN lUN_~ -LNTlUl Pl In these expressions, the subscript refers to the Nth lnterval of length Ts after time zero. Thus, substituting for the first few integers, N
1 - P2. - MlP2 Pl - 2~ -, . . ., . ~ , , .
- P3 - NlP M2 2 N3 = _ 3 , s 1_ 2 1 ( 2/ 1) where PN 13 the normalized amplltude of the signal at the Nth in~erval Ts in response to 2 unit step in~ut applied a~
time zero. The quantities LN and MN are calculated simi-larly. Thes~ quantities are used to generate modified input signals RN(t) as follows:
N N N
R (t) = r(t) ~ S n=l Nn + v n=l n n=l n where rtt) is the original input signal to the signal processor. This modifled signal in general represents the original input signal added to a sum of steps modified by the factor Nn, added with a sum of steps and ramps modified by the factor Ln, added with a sum of steps modified by the factor Mn; the values of v and a are: v = dt and a = 2 .
Each modified input RN(t) is to be applied between the times TN 1 and TN star~ing with R(t) from zero to Tl.
For the vast ma~ority of systems only two values for N ~ -and one each for L and M will be required, namely Nl, N2, ~ -Ll, and Ml. Instrumentation ~or control of these systems will thus be relatively simple to construct and ad~ust.
; Systems that possess these characteristlcs will yield only two values for N, i.e. Nl and N2, and values for L and M
that are constant for all values TN. The uncorrected step-function response of such a system is shown in Fig. 1, and the positional control of the system using only Ml and ~2 1~5;~4 is shown in Fig. 21. ~he test for the system characterlstic is that N2 = -Nl.
Step-positional control of a multi-time-constant system where Nn is not zero ~or n > 2 is shown in Figs. 17 and 18.
Figz 17 is a ~ime plot of the uncorrected step response of a typical multiple-pole system, and Fig. 1~ is a time plot of the correction signals and the corrected response for this system. Fig. 17 is taken from J. G. Truxal, "Auto-matic Feedback Control Synthesis," McGraw-~ill Book Company, Inc., 1955. For the s~stem described by the response of Fig. 17, tho application of a unit step at time zero pro-duced the response 240 with a ti~e Ts of approximately 0.4 second. The various values of PN are then observed from Fig. 17 as ~ollows. Pl occurs at point 242 at a value of 1.17. P2 ls at point 244, with a value of 0.79. P3 is at point 246, and has a value of 0.94. P4 is at point 248, with a value of 0.91. P5 i3 at point 252, with a value of 0.95. Further peaks are closer to the final value, and will be seen to be unnecessary to derive correction signals.
The present invention derives corrective signals as now calculated and as shown in Fig. 18 to provide a response that is better than the response shown in Fig. 17. ~he calculations are as follows. Only values of NN are calcu-lated, since the input signal is a step, and thus ramp and parabolic correction values are unnecessary. Substitution of the tabulated values for PN in the expression for NN
provides the ~ollowing values.
.
. , -:
r;~4 TABLE I
N NN
1 1.17 _0.15 2 0.79 +0.28 3 0,94 -0.02
- ~ u +
x ~ a X +
+
~ ~o I ~L 11 ~ 11 h !~1 ~ ~1 ~ o ~
~ ~q ~ :\ ll ~
o H _ q ~ ¢
+ ~
æ
E~ ~
~ ~ ~q ¢ ~ E~
Z ~ r~
cq H
E~ ~ I Q.
:~ ~ ~ ~1 ~ I
Z _~ O
H ~ O E-l ~`
The analysis shown in Table I shows that the response to the modi~ied input R(t) is perfectly obtained ~or ramp and parabollc inputs and is obtained in tlme Ts after applica-tion of a step.
Including the above signal modification unit within a ~eedback loop most o~ten will sause instability. This is because the open-loop transfer function is changed due to the presence of the signal processor and the ~eedback transfer function that originally yielded a stable system is no longer appropriate. A sampled signal modification unit may be inserted within a feedback loop without causing lnstability and lts presence will significantly reduce the output error o~ the system.
Fig. 12 is a time plot OL a signal and of calculated correctional signals.
In Fig. 12, an error signal such as that caused by a - switchlng transient is shown as curve 240, which is sampled at to and has a magnitude of ~ O and a rate of ~ O. With no slgnal modification the error set at to + Ts_will be ~ o(l - P) and ~ o(l - P) shown on curve 340. A ram~, curve 341, beglnnln~; at to and having the value ~Jt w~ 11 yield an output rate VP, curve 342, at to + Ts' (This is true because, since ~O(to + Ts) = SP; ~ O (to + Ts) ~ VP, ~ ~ (to + Ts) = aP, due to the time-integral relationships descrlbed immediately above.) For this ramp to exactly cancel the error rate at time to + Ts VP = ~o(l - P); V = ~o(l - P)/P
The ramp will cause an additional error, 346, o~ magnitude VUTS and to correct this to zero as well as the error ~ o(l - P), an input step of magnitude ~, curve 343, is also applied. The response to thls slgnal is shown as curve 344 and has a magnitude EP at t + T and has zero rate. For o s the slgnal magnitude to be zero at to + T requlres EP = VUTS + ~ P) (.1 - P) p - UTS + ~ o ( 1 - P ) whence E = ~ 1 - P U Ts + ~O l - ?
- 1 p P ~ ~ o Up 5 + ~ o}
( . T - LP
= N 'l ~ P T3 ~ ~ o ~
Shown in Fig. 13 is a circuit to perform this function, with ~ignals at various points shown in Fig. 13A. In Fig. 13, error signal 353 is fed to summing amplifier 354 and also to scaler 355, scaler 355 being ad~usted to a value of 1 - N.
Scaler output 368 is fed to sample-hold 356 and to time differentiator 357. The output of sample-hold 356 is fed to summing amplifier 354. The output of time differentiator 357 is ~ed to sample-hold 358 whose output 1s fed to scaler 359 and time integrator 360. Scaler 359 is set to a value T - LP
s T.s and its output as well as the output from time integrator 360 fed to summing amplifier 354. Points marked x in the drawing, namel~ at signal point 350 and at signal point 363, indicate where the system error path was opened to lnsert the sub~ect signal processor.
: .
x~
Clock 362 causes monostable 361 to generate a ~ulse train 370 whose pulses are separated by the time interval Ts. Signal 370 is fed to sample-hold units 356 and 358 and to electronic switch 369 that serves to dlscharge the integrator to zero every Ts seconds.
In Fig. 13, curve 371 is the output error resulting from an initial error, ~ 1' for operation of the system without use of the present invention. Curve 372 is the output error resulting from an initial error, ~ 1' sampled for the values ~ O, ~ O using the present invention. Curve 373 is the modified error signal R(t) which ls the sum of error signal 353 and correction signals 374, 375 and 376.
Use of the unit of Fig. 13 serves not only to improve system response to input signal 377 but also to rapidly ; correct for errors due to noise and disturbances both internal and external. This is illustrated in Fig. 14 for operation o~ a hi~h-voltage power supply. Step error 400 is caused by switching a control demanding a higher voltage.
Modi~ication signal 401 then causes the error to be smoothiy reduced to zero according to curve 402. The higher voltage causes an arc resulting in error signal 403 which is reduced smoothly to zero by auxiliary signal 404.
When a normally stab}e system becomes unstable due to changlng characteristics, the unit shown in Fig. 13 can keep lt under control. Fig. 15 depicts the situation. Curve 410 shows an initial error, cur~e 412, gradually ~uildlng up to a large oscillation amplitude. When the unit o~ Fig. 13 is used the error correction signal, curve 411, causes the ?~ 4 oscillation am~litude ~o be kept in control, as shown by curve 413. Tne unusuall~ large signal ~11 can be made to flash a warning light. This operation is potentially very important in the control of massive systems such as large aircraft during L light test, nuclear reactors, etc., where large-amplitude oscillation could lead to destruction or severe damage.
The method of the present invention can be used for improvement of multi-time-constant systems. It is usually advantageous to analyze test results from the measurements of P and Ts to determine the class or type of system.
Referring to Fig. 16, for a step input S, the system response overshoots to a value KSP, at time t1, undershoots to a value KSP2 at time t2, overshoots to a value KSP3 at t3, etc. The time intervals tl - to, t2 ~ tl, t3 - t2, --etc. can be measured as can the ratios (KSPl - KS)/KS, (KSP2 - KS)/(KSPl - KS), (KSP3 - KS)/(KSP2 - KS), etc. If these ratios are all the same and the time intervals equal, the system ls identified as a second-order system with no domlnant real poles or zeros.
When the ratios are the same but the time interval tl - to is shorter than the time intervals t2 ~ tl, t3 - t2, etc., and when the time intervals t2 ~ tl, t3 - t2, -- etc.
are equal, the system is identified as being a second-order system with a dominant real zero.
~ hen the ratios are not the same and when the time interval tl - to is longer than the succeeding time intervals the system is identified as being a second-order system with a dominant real pole.
These three classes of system cover ~he vast ma~ority of systems, although applicatlon of the present in~ention is not limited to these systems.
When used to improve the performance of second-order systems having a dominant simple zero, the schemes hereto-fore described are directly applicable wlth the only additional requirement being to redefine Ts. Fig. 16 s~ows the response of such a system. In Flg. 16 where the desired response to an input step S is KS, curve 420, the first response peak is at Tl, a time lnterval shorter than T2, T3, T4, etc., which are all equal. The time Ts is defined as the time interval T2 = T3 = T4 = ... NTS. ~his places the amplitude of the response measured to calculate P at the point x on curve 421. The value P' and values for U and W measured at Ts are then used to determine the proper ad~ustment of the signal modifier.
When used to improve the performance of second-order systems having a dominant simple pole, the scheme heretofore described can be used with some circuit modification.
Referrlng to Fig. 17, the response, curve 242, is not sgmmetrical about the deslred value, curve 430, due to the long exponential response of the simple pole. The time intervals between successive peaks and nulls are not quite equal. In Fig. 13, if the outputs from circuit elements 356, 359 and 360 are combined in an adder and its ouput fed through a lead circuit before being applied to the input of summing ampli~ier 354, the improved performance is achieved.
; - 24 _ .r~a~
The lead circuit should possess a value:
.~ P
where P is the value of the simple pole. Alternatively, where computer control is desirable and available, a sequential program can be used.
The practice of the present invention now r~quires calculating correction factors to apply to the inDut signal to generate a modified input signal to re~lace the input signal. Thls will be considered first from the standpoint of illustrating the modifications that taXe place to the signal. To see this we consider three necessary correction factors defined as follows:
1 - PN - NlPN ~ N2-PN_1 ~~~ NN-l 2 N
L , N(l UN) Ll N L2PN-1 LN-lP2 N . Pl M NZ (1 - WN) - ~ lPz _ ~ ZP3 --- -MlPN - LlTNUN
N
_-- -L2TN lUN_~ -LNTlUl Pl In these expressions, the subscript refers to the Nth lnterval of length Ts after time zero. Thus, substituting for the first few integers, N
1 - P2. - MlP2 Pl - 2~ -, . . ., . ~ , , .
- P3 - NlP M2 2 N3 = _ 3 , s 1_ 2 1 ( 2/ 1) where PN 13 the normalized amplltude of the signal at the Nth in~erval Ts in response to 2 unit step in~ut applied a~
time zero. The quantities LN and MN are calculated simi-larly. Thes~ quantities are used to generate modified input signals RN(t) as follows:
N N N
R (t) = r(t) ~ S n=l Nn + v n=l n n=l n where rtt) is the original input signal to the signal processor. This modifled signal in general represents the original input signal added to a sum of steps modified by the factor Nn, added with a sum of steps and ramps modified by the factor Ln, added with a sum of steps modified by the factor Mn; the values of v and a are: v = dt and a = 2 .
Each modified input RN(t) is to be applied between the times TN 1 and TN star~ing with R(t) from zero to Tl.
For the vast ma~ority of systems only two values for N ~ -and one each for L and M will be required, namely Nl, N2, ~ -Ll, and Ml. Instrumentation ~or control of these systems will thus be relatively simple to construct and ad~ust.
; Systems that possess these characteristlcs will yield only two values for N, i.e. Nl and N2, and values for L and M
that are constant for all values TN. The uncorrected step-function response of such a system is shown in Fig. 1, and the positional control of the system using only Ml and ~2 1~5;~4 is shown in Fig. 21. ~he test for the system characterlstic is that N2 = -Nl.
Step-positional control of a multi-time-constant system where Nn is not zero ~or n > 2 is shown in Figs. 17 and 18.
Figz 17 is a ~ime plot of the uncorrected step response of a typical multiple-pole system, and Fig. 1~ is a time plot of the correction signals and the corrected response for this system. Fig. 17 is taken from J. G. Truxal, "Auto-matic Feedback Control Synthesis," McGraw-~ill Book Company, Inc., 1955. For the s~stem described by the response of Fig. 17, tho application of a unit step at time zero pro-duced the response 240 with a ti~e Ts of approximately 0.4 second. The various values of PN are then observed from Fig. 17 as ~ollows. Pl occurs at point 242 at a value of 1.17. P2 ls at point 244, with a value of 0.79. P3 is at point 246, and has a value of 0.94. P4 is at point 248, with a value of 0.91. P5 i3 at point 252, with a value of 0.95. Further peaks are closer to the final value, and will be seen to be unnecessary to derive correction signals.
The present invention derives corrective signals as now calculated and as shown in Fig. 18 to provide a response that is better than the response shown in Fig. 17. ~he calculations are as follows. Only values of NN are calcu-lated, since the input signal is a step, and thus ramp and parabolic correction values are unnecessary. Substitution of the tabulated values for PN in the expression for NN
provides the ~ollowing values.
.
. , -:
r;~4 TABLE I
N NN
1 1.17 _0.15 2 0.79 +0.28 3 0,94 -0.02
4 0.90 -0.01 0.95 0 The values of NN are the am~plitudes of the steps that must be added to the input step signal to produce the response 253 of Fig. 18.` These steps occur at successive intervals of length Ts and are indicated in Fig. 18 as ::
follows. Step 254 corresponds to Nl, with an amplitude of Table I of -0.15. Step 256 corresponds to N2, with an amplitude from Table I of +0.28. Step 258 corresponds to N3, with an amplltude from Table I of -0.02. Step 260 corresponds to N4, with an amplitude from Table I of -0.01.
It is clearly unnecessary to calculate NN for higher values of N, and in fact the first three terms would probably suffice. These steps 254, 256, .258, and 260 comprise the corrective slgnal that must be added to the step that produced the response 240 of Fig. 17. When 'he corrective signal is added, the response 253 of Fig. 18 is seen to provide a much better approximation to a step than the uncorrected response 240 of Fig. 17. It is also evident from an examination of Table I that two values of NN, Nl and N2 would have sufficed to provide a corrected step response that is greatly improved over ~he uncorrected response 242 of Fig. 17.
The particular example shown in Figs. 17 and 18 necessi-tated measurement only of s and not of v and ~. The need for measurement is a function of the expected form of the input signal. The calculatlon of correctional signals for a general input requires the additional calculatlon of the quantities v, a, LN, and MN to derive a total correctional signal. This correctional signal, as described above, will add to the input to provide a composite signal that produces a response that more closely approximates the ideal response to the signal.
Flg. 19 is a block diagram showing the operational elements necessary to perform the operations required by the present invention using a digital computer. With the set-up shown in ~ig. 19, a system can be controlled requiring either constant values for N, L, and M or very complex systems requiring sequential values Nn, Ln, and Mn can be accommodated. In addition, the control can be expanded in Fig. 19 so as to accommodate systems where Nn, Ln, and Mn vary with time. It thus is the basis for an adaptive system. The input to Fig. 19 is error signal 171 and its output is modified error signal 177 which is pro-cessed for insertion into the system, as has been shown in earlier figures. Analysis unit 172 performs the necessary operations involved in differentiatlng error signal 171.
The assembly unit 173 applies multiplying factors N , Ln, Mn, and Tn under the control of memory and control unit 176.
The various outputs of assembly unit 173 are applied to summing unit 174 and the output of summing unit 174 is fed . , - ~ , .
to the system. It can be seen that the same operations performed by ~he analog elements of Fig. 11 are indicated in Fig. 19 as being performed by elements under the control of a digital computer with the additional ad~antage of being useful with more complicated systems and also with time-varying systems.
The embodiment o~ the principles of the present inven-tion described above have necessitated knowledge of the quantities N, L, M'g and Ts. Fig. 20 is a block diagram of a computer arrangement ~or determining these quantities. In Flg, 20, input signal 180 i3 the normal input to system 182 selected when switch 183 is in the upper position. Switch 183 is presently indicated in the lower position ~hich applies test step input 181 to system 182. It should be noted that system 182 may be an open-loop system or a closed-loop system and may include the elements o~ the present inventlon as a part; for exa~.ple, system 182 might be Fig. 11 in i,ts entiret~J. In this case, those portions of Fig. 11 relating to sig~al processing, namely differ-entiator 153 and threshold detector 151, would have to be disabled to prevent their operation when the switch was in position to receive test step input 181. t~ith such a test step input applied to the system 182, output 184 is applied to integrator 185, differentiator 190, and store and compute unit 193. Integrator 185 is connected to operational amplifier 186 which inverts the inte~rated out-put o~ integrator 185 for application to store and compute unit 193. Signal 188 is also applied to integrator 187 :l J ~ ri~
~hose output is also connected to store and compute unit 193. Output 184 is dif~erentiated in differentiator 190 and the di~ferentiated output is then applied to zero crossing - detector 191. This provides information in signal 189 which allows computation in store and compute unit 193, enabling the calculation of T~, L, M, and N according to the formulas listed above.
The operations per~ormed by the analog system of Fig. 20 can also be per~ormed by a digital computer with the advantages of rapid analysis and also of being able to more easily accommodate complex sys~ems, as well as time-varyln~ parameter systems. It can be combined with Fig. 19 to make up an adaptive system. T~ith such a set-up, an occasional step input would yield up-to-date values for Nn, Ln~ Mn, and T .
Flg. 21 is a representation of a composite input and a corrected output signal in a typical control system. In Flg. 21, step 201 is the correction slgnal that has been calculated according to the principles described above. It is applied to the system for a time Ts. After time Ts the input signal rises to its final level which is that of step 202. It has been shown earlier that, if the step had been applied initially at the level of step 202, the response would have exhiblted elther undershoot or over-shoot. The response signal 203 in Fig. 21 is seen to exhibit neither of these, but instead climbs to the height.
of step 202 at time Ts and thereafter remains at the level of step 202. It can be seen ~rom Fig. 21 that the - ~ .
~1.1.'l ' q~M4 uncorrected system to which steps 201 and 202 have been applied is an underdampled or oscillatory ~ystem. This follows because the calculated level o~ step 201 is below that of step 202. The undesired portion of the signal that is being corrected in Fig. 21 is thus an overshoot. If the system represented in Fig. 21 had instead been an over-damped system, then the necessary correction signals determined by the calculations described above would have caused step 201 to be hlgher in amplitude than ~tep 202.
In either case, the calculations determined as described above provide for the development of a composite correcting signal which provides an optimum response to a unit step as shown in Fig. 21. The correction is complete and the final value is achieved after a time T .
Fig. 22 i3 a time plot of the corrective input signals and the corrected output signal of a system to which a ramp is applied. The system whose response is shown in Fig. 22 has a characteristic time TS which is calculated as des-cribed earlier and is shown on the time a~is in Fig. 22.
In Fig. 22, ramp 211 and step 212 are appi ed as inputs to the system during the time up to Ts. Ramp 211 is itself the known varying portion of the signal applied to the system which comprises step 213 and ramp 211. Step 212 is calculated as described above to correct the response of the system to achieve its desired final state at time Ts.
Signal 214, a composite signal comprising the sum o~ ramp 211 and step 212, is actually applied to the system during the time up to Ts. A~ter time Ts, composite signal 215 1.11.~.~f~fl is applied to the system. Composite signal 215 comprises tne sum OI' step 213 and ramp 211 in that portion of its time period after Ts. The response of the system to the inputs described is curve 216 in the period up to Ts and curve 217 in the period after Ts. It can be seen from Fig. 22 that curve 210 departs considerably ~rom ramp 211 at portions of the interval over which both exist, but that as the time approaches TS curve 216 approaches and ~oins ramp 211 so that there is no error between c~rve 217 and ramp 211 after Ts. The signal has thus been corrected to eliminate error after the passage of time Ts.
For modest ramp rates the response of the system will also be perfect between application of the ramp and the time Ts as is detailed in Table IB. For steep ramps the output of second differentiator 156 o: Fig. 11 may be lnsufficient in amplitude, due to saturation of second differentiator 156 or to saturation of system, to provide the correct magnitude of delta-function, curve c, of Fig. llA demanded by the method. In the limit, when the steepness of the Z0 ramp approaches a step function, the response is as detalled in Table IC, and it is physically impossible ~o achieve perfect response for the time interval between application of the signal and Ts ~ithout overdriving the s~stem.
Fig. 23 is a representation of the correction signals necessary to apply the principles o the present invention to a system receiving an input that is a paraoola. In Fig. 23, s~ep 221 and ramp 222 are correctional signals , calculated according to the principles described above.Input parabola 22~ is the signal that it is desired that the system follow. Response signal 225 is seen to be coincident with input parabola 224 as a result of applica-tion of the calculated correctional signals, step 221 and ramp 222, during the period Ts (see Table IA).
A system for the practice of the present method has been used at the Argonne Natlonal Laboratory to achieve improved response of a control system for a calorimeter measuring the activity of nuclear-fuel rod3. The calorimeter is the sub-~ect of U. S. Patent 3,995,485, entitled "Dry, Portable Calorimeter for Nondestructive Measurement of the Activity of Nuclear Fuel." A control sy3tem responded to an error signal proportional to temperature to maintain the tempera-ture of an oven within 20 microK, and measurement of the power necessary to maintain this temperature gave a measure of the heat generated by radioactive materials. Such a control system must approach a final value rapidly to be .
of use, and lt must avoid overshoot to prevent the "lock-up"
characteristic of a thermal system that can be driven by heat but not actively cooled. The application of the method of the present invention reduced the time to obtain acceptable readings by a factor of three, and also reduced significantly the standard deviation of successive error readings.
follows. Step 254 corresponds to Nl, with an amplitude of Table I of -0.15. Step 256 corresponds to N2, with an amplitude from Table I of +0.28. Step 258 corresponds to N3, with an amplltude from Table I of -0.02. Step 260 corresponds to N4, with an amplitude from Table I of -0.01.
It is clearly unnecessary to calculate NN for higher values of N, and in fact the first three terms would probably suffice. These steps 254, 256, .258, and 260 comprise the corrective slgnal that must be added to the step that produced the response 240 of Fig. 17. When 'he corrective signal is added, the response 253 of Fig. 18 is seen to provide a much better approximation to a step than the uncorrected response 240 of Fig. 17. It is also evident from an examination of Table I that two values of NN, Nl and N2 would have sufficed to provide a corrected step response that is greatly improved over ~he uncorrected response 242 of Fig. 17.
The particular example shown in Figs. 17 and 18 necessi-tated measurement only of s and not of v and ~. The need for measurement is a function of the expected form of the input signal. The calculatlon of correctional signals for a general input requires the additional calculatlon of the quantities v, a, LN, and MN to derive a total correctional signal. This correctional signal, as described above, will add to the input to provide a composite signal that produces a response that more closely approximates the ideal response to the signal.
Flg. 19 is a block diagram showing the operational elements necessary to perform the operations required by the present invention using a digital computer. With the set-up shown in ~ig. 19, a system can be controlled requiring either constant values for N, L, and M or very complex systems requiring sequential values Nn, Ln, and Mn can be accommodated. In addition, the control can be expanded in Fig. 19 so as to accommodate systems where Nn, Ln, and Mn vary with time. It thus is the basis for an adaptive system. The input to Fig. 19 is error signal 171 and its output is modified error signal 177 which is pro-cessed for insertion into the system, as has been shown in earlier figures. Analysis unit 172 performs the necessary operations involved in differentiatlng error signal 171.
The assembly unit 173 applies multiplying factors N , Ln, Mn, and Tn under the control of memory and control unit 176.
The various outputs of assembly unit 173 are applied to summing unit 174 and the output of summing unit 174 is fed . , - ~ , .
to the system. It can be seen that the same operations performed by ~he analog elements of Fig. 11 are indicated in Fig. 19 as being performed by elements under the control of a digital computer with the additional ad~antage of being useful with more complicated systems and also with time-varying systems.
The embodiment o~ the principles of the present inven-tion described above have necessitated knowledge of the quantities N, L, M'g and Ts. Fig. 20 is a block diagram of a computer arrangement ~or determining these quantities. In Flg, 20, input signal 180 i3 the normal input to system 182 selected when switch 183 is in the upper position. Switch 183 is presently indicated in the lower position ~hich applies test step input 181 to system 182. It should be noted that system 182 may be an open-loop system or a closed-loop system and may include the elements o~ the present inventlon as a part; for exa~.ple, system 182 might be Fig. 11 in i,ts entiret~J. In this case, those portions of Fig. 11 relating to sig~al processing, namely differ-entiator 153 and threshold detector 151, would have to be disabled to prevent their operation when the switch was in position to receive test step input 181. t~ith such a test step input applied to the system 182, output 184 is applied to integrator 185, differentiator 190, and store and compute unit 193. Integrator 185 is connected to operational amplifier 186 which inverts the inte~rated out-put o~ integrator 185 for application to store and compute unit 193. Signal 188 is also applied to integrator 187 :l J ~ ri~
~hose output is also connected to store and compute unit 193. Output 184 is dif~erentiated in differentiator 190 and the di~ferentiated output is then applied to zero crossing - detector 191. This provides information in signal 189 which allows computation in store and compute unit 193, enabling the calculation of T~, L, M, and N according to the formulas listed above.
The operations per~ormed by the analog system of Fig. 20 can also be per~ormed by a digital computer with the advantages of rapid analysis and also of being able to more easily accommodate complex sys~ems, as well as time-varyln~ parameter systems. It can be combined with Fig. 19 to make up an adaptive system. T~ith such a set-up, an occasional step input would yield up-to-date values for Nn, Ln~ Mn, and T .
Flg. 21 is a representation of a composite input and a corrected output signal in a typical control system. In Flg. 21, step 201 is the correction slgnal that has been calculated according to the principles described above. It is applied to the system for a time Ts. After time Ts the input signal rises to its final level which is that of step 202. It has been shown earlier that, if the step had been applied initially at the level of step 202, the response would have exhiblted elther undershoot or over-shoot. The response signal 203 in Fig. 21 is seen to exhibit neither of these, but instead climbs to the height.
of step 202 at time Ts and thereafter remains at the level of step 202. It can be seen ~rom Fig. 21 that the - ~ .
~1.1.'l ' q~M4 uncorrected system to which steps 201 and 202 have been applied is an underdampled or oscillatory ~ystem. This follows because the calculated level o~ step 201 is below that of step 202. The undesired portion of the signal that is being corrected in Fig. 21 is thus an overshoot. If the system represented in Fig. 21 had instead been an over-damped system, then the necessary correction signals determined by the calculations described above would have caused step 201 to be hlgher in amplitude than ~tep 202.
In either case, the calculations determined as described above provide for the development of a composite correcting signal which provides an optimum response to a unit step as shown in Fig. 21. The correction is complete and the final value is achieved after a time T .
Fig. 22 i3 a time plot of the corrective input signals and the corrected output signal of a system to which a ramp is applied. The system whose response is shown in Fig. 22 has a characteristic time TS which is calculated as des-cribed earlier and is shown on the time a~is in Fig. 22.
In Fig. 22, ramp 211 and step 212 are appi ed as inputs to the system during the time up to Ts. Ramp 211 is itself the known varying portion of the signal applied to the system which comprises step 213 and ramp 211. Step 212 is calculated as described above to correct the response of the system to achieve its desired final state at time Ts.
Signal 214, a composite signal comprising the sum o~ ramp 211 and step 212, is actually applied to the system during the time up to Ts. A~ter time Ts, composite signal 215 1.11.~.~f~fl is applied to the system. Composite signal 215 comprises tne sum OI' step 213 and ramp 211 in that portion of its time period after Ts. The response of the system to the inputs described is curve 216 in the period up to Ts and curve 217 in the period after Ts. It can be seen from Fig. 22 that curve 210 departs considerably ~rom ramp 211 at portions of the interval over which both exist, but that as the time approaches TS curve 216 approaches and ~oins ramp 211 so that there is no error between c~rve 217 and ramp 211 after Ts. The signal has thus been corrected to eliminate error after the passage of time Ts.
For modest ramp rates the response of the system will also be perfect between application of the ramp and the time Ts as is detailed in Table IB. For steep ramps the output of second differentiator 156 o: Fig. 11 may be lnsufficient in amplitude, due to saturation of second differentiator 156 or to saturation of system, to provide the correct magnitude of delta-function, curve c, of Fig. llA demanded by the method. In the limit, when the steepness of the Z0 ramp approaches a step function, the response is as detalled in Table IC, and it is physically impossible ~o achieve perfect response for the time interval between application of the signal and Ts ~ithout overdriving the s~stem.
Fig. 23 is a representation of the correction signals necessary to apply the principles o the present invention to a system receiving an input that is a paraoola. In Fig. 23, s~ep 221 and ramp 222 are correctional signals , calculated according to the principles described above.Input parabola 22~ is the signal that it is desired that the system follow. Response signal 225 is seen to be coincident with input parabola 224 as a result of applica-tion of the calculated correctional signals, step 221 and ramp 222, during the period Ts (see Table IA).
A system for the practice of the present method has been used at the Argonne Natlonal Laboratory to achieve improved response of a control system for a calorimeter measuring the activity of nuclear-fuel rod3. The calorimeter is the sub-~ect of U. S. Patent 3,995,485, entitled "Dry, Portable Calorimeter for Nondestructive Measurement of the Activity of Nuclear Fuel." A control sy3tem responded to an error signal proportional to temperature to maintain the tempera-ture of an oven within 20 microK, and measurement of the power necessary to maintain this temperature gave a measure of the heat generated by radioactive materials. Such a control system must approach a final value rapidly to be .
of use, and lt must avoid overshoot to prevent the "lock-up"
characteristic of a thermal system that can be driven by heat but not actively cooled. The application of the method of the present invention reduced the time to obtain acceptable readings by a factor of three, and also reduced significantly the standard deviation of successive error readings.
Claims (11)
1. A method of improving the response of an automatic control system to a general input signal that is a function of time t, the method comprising:
a. applying a test input signal to the system;
b. obtaining a test output signal corresponding to the test input signal;
c. determining values of correctional quantities from the test output signal according to a pre-determined scheme;
d. generating correctional signals from the correctional quantities;
e. adding the correctional signals to the general input signal to obtain a modified input signal;
and f. applying the modified input signal to the system.
a. applying a test input signal to the system;
b. obtaining a test output signal corresponding to the test input signal;
c. determining values of correctional quantities from the test output signal according to a pre-determined scheme;
d. generating correctional signals from the correctional quantities;
e. adding the correctional signals to the general input signal to obtain a modified input signal;
and f. applying the modified input signal to the system.
2. The method of claim 1 wherein the test input signal comprises a step signal.
3. The method of claim 2 wherein calculating values of correctional signals comprises the following:
a. determining system gain K;
b. determining time Ts required to achieve a first peak of an output signal in response to the unit step input signal;
c. determining percent response P at the time Ts in response to the unit step input signal;
d. determining percent response U at the time Ts in response to a unit ramp input signal;
e. determining percent response W at the time Ts in response to a unit parabolic input signal;
f. calculating a quantity g. calculating a quantity h. calculating a quantity i. measuring amplitude S of the general input signal at a test time;
j. measuring time rate of change v of the general input signal at the test time;
k. measuring second time derivative a of the general input signal at the test time; and l. calculating the quantity SN + vL + aM, which quantity has a value equal to the value of the correc-tional signal.
a. determining system gain K;
b. determining time Ts required to achieve a first peak of an output signal in response to the unit step input signal;
c. determining percent response P at the time Ts in response to the unit step input signal;
d. determining percent response U at the time Ts in response to a unit ramp input signal;
e. determining percent response W at the time Ts in response to a unit parabolic input signal;
f. calculating a quantity g. calculating a quantity h. calculating a quantity i. measuring amplitude S of the general input signal at a test time;
j. measuring time rate of change v of the general input signal at the test time;
k. measuring second time derivative a of the general input signal at the test time; and l. calculating the quantity SN + vL + aM, which quantity has a value equal to the value of the correc-tional signal.
4. A method of improving the response of a feedback control system having an error signal to a general input signal that is a function of time t, the method comprising:
a. applying a test input signal to the system;
b. determining a test error response to the test input signal;
c. determining values of correctional signals from the test error response according to a pre-determined scheme;
d. generating correctional signals equal to the calculated values;
e. adding the correctional signals to the error signal to obtain a modified error signal; and f. applying the modified error signal to the system to replace the error signal.
a. applying a test input signal to the system;
b. determining a test error response to the test input signal;
c. determining values of correctional signals from the test error response according to a pre-determined scheme;
d. generating correctional signals equal to the calculated values;
e. adding the correctional signals to the error signal to obtain a modified error signal; and f. applying the modified error signal to the system to replace the error signal.
5. The method of claim 4 wherein the test input signal is a step.
6. The method of claim 5 wherein the step of calcu-lating values of correctional signals comprises:
a. determining a system gain K;
b. determining a time Ts required to achieve a first peak of an output signal in response to the test step input signal;
c. determining a percent response P at the time Ts in response to the test input signal;
d. determining percent response U at the time Ts in response to a unit ramp input signal;
e. determining percent response W at the time Ts in response to a unit parabolic input signal;
f. calculating a quantity N according to a formula g. calculating a quantity L according to a formula h. measuring a value ? o of the error signal at a time to;
i. measuring a value ? O of the time rate of change of the error signal at the time to;
j. calculating a quantity D according to the formula k. calculating a quantity E according to the formula ; and l. calculating a quantity F that is the sum of a step and 2 ramp according to the formula F = E + Dt, which quantity ls the desired correctional signal.
a. determining a system gain K;
b. determining a time Ts required to achieve a first peak of an output signal in response to the test step input signal;
c. determining a percent response P at the time Ts in response to the test input signal;
d. determining percent response U at the time Ts in response to a unit ramp input signal;
e. determining percent response W at the time Ts in response to a unit parabolic input signal;
f. calculating a quantity N according to a formula g. calculating a quantity L according to a formula h. measuring a value ? o of the error signal at a time to;
i. measuring a value ? O of the time rate of change of the error signal at the time to;
j. calculating a quantity D according to the formula k. calculating a quantity E according to the formula ; and l. calculating a quantity F that is the sum of a step and 2 ramp according to the formula F = E + Dt, which quantity ls the desired correctional signal.
7. The method of claim 5 wherein the step of calcu-lating values of correctional signals comprises:
a. determing a time T1 to achieve a first positive peak of an output signal in response to the test step input signal;
b. determining a percent response P1 to the test step input signal at time T1;
c. determing a time T2 to achieve a first relative minimum in response to the tast step input signal;
d. determining a percent response P2 to the test step input signal at time T2;
e. determining the quantities Ql = (P1 - 1), Q2 (P2 - 1)/Q1, and R1 = (T2 - T1); and f. comparing the values of Q1 with Q2 and T1 with R1.
a. determing a time T1 to achieve a first positive peak of an output signal in response to the test step input signal;
b. determining a percent response P1 to the test step input signal at time T1;
c. determing a time T2 to achieve a first relative minimum in response to the tast step input signal;
d. determining a percent response P2 to the test step input signal at time T2;
e. determining the quantities Ql = (P1 - 1), Q2 (P2 - 1)/Q1, and R1 = (T2 - T1); and f. comparing the values of Q1 with Q2 and T1 with R1.
8. The method of claim 7 comprising in addition the following:
a. finding that Q1 = Q2 and T1 = R1 to establish that the feedback control system is a second-order system without a dominant pole or zero;
b. determining a system gain K;
c. determining percent response U at time T1 in response to a unit ramp input signal;
d. determining percent response W at time T1 in response to a unit parabolic input signal;
e. calculating a quantity N1 according to a formula ;
f. calculating a auantity L1 according to a formula ;
g. measuring a value ? O of the error signal at a time to;
h. measuring a value ? O of the time rate of change of the error signal at the time to;
i. calculating a quantity D1 according to the formula ;
j. calculatlng a quantity E1 according to the formula ; and k. calculating a quantity F1 that is the sum of a step and a ramp according to the formula F1 = E1 + D1t, which quantity is the desired correctional signal to be applied for the time Ts.
a. finding that Q1 = Q2 and T1 = R1 to establish that the feedback control system is a second-order system without a dominant pole or zero;
b. determining a system gain K;
c. determining percent response U at time T1 in response to a unit ramp input signal;
d. determining percent response W at time T1 in response to a unit parabolic input signal;
e. calculating a quantity N1 according to a formula ;
f. calculating a auantity L1 according to a formula ;
g. measuring a value ? O of the error signal at a time to;
h. measuring a value ? O of the time rate of change of the error signal at the time to;
i. calculating a quantity D1 according to the formula ;
j. calculatlng a quantity E1 according to the formula ; and k. calculating a quantity F1 that is the sum of a step and a ramp according to the formula F1 = E1 + D1t, which quantity is the desired correctional signal to be applied for the time Ts.
9. The method of claim 7 comprising in addition the following:
a. determining a time T3 to achieve a second positive peak of the output signal in response to the test step input signal;
b. determining a percent response P3 to the test step input signal at time T3;
c. determining the quantities Q3 = (P3 - P2)/(P2 - P1) R2 = (T3 - T2); and d. comparing the values of Q3 with Q2 and of R2 with R1.
a. determining a time T3 to achieve a second positive peak of the output signal in response to the test step input signal;
b. determining a percent response P3 to the test step input signal at time T3;
c. determining the quantities Q3 = (P3 - P2)/(P2 - P1) R2 = (T3 - T2); and d. comparing the values of Q3 with Q2 and of R2 with R1.
10. The method of claim 9 comprising in addition:
a. finding that Q3 = Q2 < Q1 and that R2 = R1 > T1 to establish that the feedback control system is a second-order system with a dominant real zero;
b. determining a system gain K;
c. determining percent response U1 at the time R1 in response to a unit ramp input signal;
d. determining percent response U1 at the time R1 in response to a unit parabolic input signal;
e. calculating a quantity No according to a formula ;
f. calculating a quantity Lo according to a formula ;
g. measuring a value ?1 of the error signal at a lime t1;
h. measuring a value ?1 of the error signal at the lime t1;
i. calculating a quantity Do according to the formula ;
j. calculating a quantity Eo according to the formula ; and k. calculating a quantity F1 according to the formula F1 = Eo + Dot, which quantity is the desired correctional signal.
a. finding that Q3 = Q2 < Q1 and that R2 = R1 > T1 to establish that the feedback control system is a second-order system with a dominant real zero;
b. determining a system gain K;
c. determining percent response U1 at the time R1 in response to a unit ramp input signal;
d. determining percent response U1 at the time R1 in response to a unit parabolic input signal;
e. calculating a quantity No according to a formula ;
f. calculating a quantity Lo according to a formula ;
g. measuring a value ?1 of the error signal at a lime t1;
h. measuring a value ?1 of the error signal at the lime t1;
i. calculating a quantity Do according to the formula ;
j. calculating a quantity Eo according to the formula ; and k. calculating a quantity F1 according to the formula F1 = Eo + Dot, which quantity is the desired correctional signal.
11. The method of claim 9 comprising in addition:
a. finding that Q2 > Q1 and that R1 ? T1 to establish that the system is a second-order system with a dominant real pole;
b. inserting a lead circuit in series with the input signal to cancel the pole;
c. determining system gain K;
d. determining a time T11 to achieve a first positive peak of an output signal of the system with a lead circuit in response to the test step input signal;
e. determining a percent response P11 to the test step input signal at time T11;
f. determining a percent response U11 at time T11 in response to a unit ramp signal;
g. determining a percent response W11 in response to a unit parabolic input signal;
h. calculating a quantity N11 according to a formula ;
i. calculating a quantity L11 according to a formula ;
j. measuring a value ?11 of the error signal at a time t11;
k. measuring a value ?11 of the time rate of change of the error signal at the time to;
l. calculating a quantity D11 according to the formula ;
m. calculating a quantity E11 according to the formula ; and n. calculating a quantity F11 that is the sum of a step and a ramp according to the formula F11 = E11 + D11t, which quantity is the desired correctional signal for the time interval Ts.
a. finding that Q2 > Q1 and that R1 ? T1 to establish that the system is a second-order system with a dominant real pole;
b. inserting a lead circuit in series with the input signal to cancel the pole;
c. determining system gain K;
d. determining a time T11 to achieve a first positive peak of an output signal of the system with a lead circuit in response to the test step input signal;
e. determining a percent response P11 to the test step input signal at time T11;
f. determining a percent response U11 at time T11 in response to a unit ramp signal;
g. determining a percent response W11 in response to a unit parabolic input signal;
h. calculating a quantity N11 according to a formula ;
i. calculating a quantity L11 according to a formula ;
j. measuring a value ?11 of the error signal at a time t11;
k. measuring a value ?11 of the time rate of change of the error signal at the time to;
l. calculating a quantity D11 according to the formula ;
m. calculating a quantity E11 according to the formula ; and n. calculating a quantity F11 that is the sum of a step and a ramp according to the formula F11 = E11 + D11t, which quantity is the desired correctional signal for the time interval Ts.
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| US784,400 | 1977-04-04 | ||
| US05/784,400 US4169283A (en) | 1977-04-04 | 1977-04-04 | Step-control of electromechanical systems |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA1115384A true CA1115384A (en) | 1981-12-29 |
Family
ID=25132353
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA299,029A Expired CA1115384A (en) | 1977-04-04 | 1978-03-14 | Step-control of electrochemical systems |
Country Status (6)
| Country | Link |
|---|---|
| US (1) | US4169283A (en) |
| JP (1) | JPS53123785A (en) |
| CA (1) | CA1115384A (en) |
| DE (1) | DE2812851A1 (en) |
| FR (1) | FR2386861A1 (en) |
| GB (1) | GB1560721A (en) |
Families Citing this family (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4349869A (en) * | 1979-10-01 | 1982-09-14 | Shell Oil Company | Dynamic matrix control method |
| US4386397A (en) * | 1979-12-18 | 1983-05-31 | Honeywell Inc. | Method and apparatus for process control |
| US4616308A (en) * | 1983-11-15 | 1986-10-07 | Shell Oil Company | Dynamic process control |
| USRE33267E (en) * | 1983-12-12 | 1990-07-17 | The Foxboro Company | Pattern-recognizing self-tuning controller |
| US4602326A (en) * | 1983-12-12 | 1986-07-22 | The Foxboro Company | Pattern-recognizing self-tuning controller |
| JPH0736122B2 (en) * | 1984-07-24 | 1995-04-19 | 三菱電機株式会社 | Control method of servo control device |
Family Cites Families (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US3287615A (en) * | 1963-10-01 | 1966-11-22 | North American Aviation Inc | Self-adaptive control apparatus |
| US3697957A (en) * | 1968-12-23 | 1972-10-10 | Adaptronics Inc | Self-organizing control |
| US3671725A (en) * | 1970-10-01 | 1972-06-20 | Ibm | Dead time process regulation |
| US3798426A (en) * | 1971-12-20 | 1974-03-19 | Foxboro Co | Pattern evaluation method and apparatus for adaptive control |
-
1977
- 1977-04-04 US US05/784,400 patent/US4169283A/en not_active Expired - Lifetime
-
1978
- 1978-03-13 GB GB9788/78A patent/GB1560721A/en not_active Expired
- 1978-03-14 CA CA299,029A patent/CA1115384A/en not_active Expired
- 1978-03-23 DE DE19782812851 patent/DE2812851A1/en not_active Withdrawn
- 1978-04-04 JP JP3964978A patent/JPS53123785A/en active Pending
- 1978-04-04 FR FR7809940A patent/FR2386861A1/en not_active Withdrawn
Also Published As
| Publication number | Publication date |
|---|---|
| DE2812851A1 (en) | 1978-10-26 |
| GB1560721A (en) | 1980-02-06 |
| FR2386861A1 (en) | 1978-11-03 |
| US4169283A (en) | 1979-09-25 |
| JPS53123785A (en) | 1978-10-28 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Bartoszewicz | Discrete-time quasi-sliding-mode control strategies | |
| KR900005546B1 (en) | Adaptive process control system | |
| Utkin | Variable structure systems with sliding modes | |
| Amann et al. | An H∞ approach to linear iterative learning control design | |
| Coales et al. | An on-off servo mechanism with predicted change-over | |
| CA1115384A (en) | Step-control of electrochemical systems | |
| Hoff et al. | Extended comparison of the hilber-hughes-taylor α-method and the θ1-method | |
| EP1457846B1 (en) | Improving stability and response of control systems | |
| JPH077285B2 (en) | Plant control equipment | |
| US4245196A (en) | Highly-linear closed-loop frequency sweep generator | |
| Barnard | Continuous-time implementation of optimal-aim controls | |
| Das et al. | Periodic compensation of continuous-time plants | |
| US7054773B2 (en) | Dynamic model-based compensated tuning of a tunable device | |
| Piguet et al. | A minimax approach for multi-objective controller design using multiple models | |
| Garcia-Gabin et al. | Application of multivariable GPC to a four tank process with unstable transmission zeros | |
| Douce et al. | The development and performance of a self-optimizing system | |
| Hunt et al. | Driven dynamics of time-varying linear systems | |
| Mondal et al. | On Non-Integer Order Internal Model Control of Non-Minimum Phase Systems and Plants with Transport Delay | |
| CN112578668A (en) | Extended state observer design method capable of converging at any time | |
| Chai et al. | Global stabilization for a class of uncertain nonlinear time-delay systems by linear sampled-data output feedback | |
| CN116707162A (en) | Wireless power transmission system control method based on Hammerstein-IMC | |
| Wenhan et al. | A model reference robust control and its application to autopilot control law design | |
| Lucibello | Comments on" Nonlinear repetitive control" | |
| Sebakhy et al. | Optimal response in SISO ripple-free deadbeat systems | |
| Grossi et al. | PID, predictive and fuzzy temperature control for nuclear magnetic resonance spectroscopy experiments |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| MKEX | Expiry |