- Email: [email protected]
Computer-aided process control system design using interactive graphics
Cornpvlrr~ &
Chico/ Eqinming Printed in Great Britain.
Vol. 5. No. 4, pp. 22~232,198l
COMPUTER-AIDED PROCESS CONTROL SYSTEM DESIGN USING INTERACTIVE GRAPHICS T. F. Eucim,t R. HEEBand J. 0. HOUGEN Department of Chemical Engineering, Universityof Texas,Austin,TX78712,U.S.A. (Received 15 November 1981)
Abstract-Design programs employing interactive computer graphics can simplify and greatly facilitate the synthesis of feedback controllers for dynamic systems. In this paper a program for design of three mode controllers for single input-singleoutput linear systems is described. Seven displays can be employed for analyzingeither time domainor frequency domaincharacteristics,as illustratedin the paper.
Scope-The purpose of the controller in a feedback control system is to impart desirable dynamic properties to the closed loop system. Desirablefeedback control system characteristicsmightinclude (1) rapid response to setpoint changes (2) adequate disturbanceor load rejection (3) insensitivityto model and measurementerrors (4) avoidance of excessive controller action (5) suitabilityover a wide range of operatingconditions. Attainingall of these goals may be impossiblefor a given process because of conthctingobjectives. Since it may be difficultin the design process to quantify accurately the trade-offs in performance objectives, the availability of interactive tools is crucial to efficient process controllerdesign. There are several approaches which are used in practice to design controllers: (1) correlations of tuning parameters (2) repetitive time domain simulation of candidate controllers (3) frequency response Correlations such as Ziegler-Nichols [ l] give approximate values for controller parameters, based on a specific type of dynamic model (the tirst order plus time delay transfer function). For other systems, such guidelines cannot be applied directly and do not attain the high performance realizable by more detailed design procedures and more complex controllers. Another disadvantage of tuning correlations is that they yield a highly oscillatory (although rapidly responding) controlled system. Repetitive time domain simulation of the closed loop response, where controller parameters are varied in a systematic fashion, overcomes some of the above disadvantages, but this approach can be very inefficient, sometimes requiring excessive iteration. Explicit stability and sensitivity information is also unavailable in this approach. A design procedure which overcomes most of the above objectives is based on the well-known concept of frequency response[l]. However, depending upon the experience of the designer, repetitive calculations requiring graph paper can be a large obstacle to use of frequency response methods in controller design. Computer-aided design facilities, especially those employing interactive computer graphics terminals, can simplify and facilitate greatly the design of feedback controllers. The use of interactive computer programs that plot graphical displays allows the designer to quickly examine results and make modifications so the final control system satisfies the constraints and design objectives. Design of three mode controllers for linear dynamic systems can be Coudusions and SIgnBIperformed rapidly using interactive computer graphics. High performance control systems can be synthesized with the frequency response method, thus eliminating the uncertainty when approximate tuning rules are employed in controller design. Explicit sensitivity information and stability characteristics can also be computed. The program discussed here is based on an ideal three mode servomechanism controller and thus is more directly applicable to digital control rather than electronic or pneumatic controllers. However, the program can be easily modified to treat either the electronic or pneumatic controller forms. The process model specified by the user can have up to third order dynamics with dead time. The user has the capability to recall any of seven displays (open or closed loop) at any time, thus reinforcing the correlation between time and frequency domain characteristics. After 1 hr of training using the interactive graphics package and the iterative procedure discussed in this paper, a typical undergraduate student can design a satisfactory three mode controller in less than 10 min of terminal time.
225
T. F. EDGAR d al.
226 USE OFFREQUENCYRESPONSEINPROCESS CONTROLLZRDESIGN
The analysis of a large body of experimental data obtained from full-scale chemical plants has led to the conclusion that the dynamic relationship between input and output process variables can be represented reasonably well by the transfer function form given in eqn (1):
e(s)=
K,(l t QS) eeDTS = G,(S) (1 t T*S)(lt 7*SHl+ 73s)
(1)
where the time constants (T,, 7*,Q and TV)and pure delay time, DZ’, are estimated based on testing of the process. The popular three term (or PID) controller usually is sufficient to provide satisfactory compensation for most process control applications. In the program discussed here, an ideal three mode controller of the form[l],
G,-(S)= Kc(l t Tdst I/TiS)
(2)
is used for design. This controller form is difficult to realize using analog devices; digital controllers, after discretization of eqn (2), can be programmed to give ideal behavior. For an electronic controller, usually the transfer function is expressed as:
K, (%)(e)=
G,(s).
(3)
Alpha (a), however, is normally not alterable in a fixed configuration electronic controller but usually lies between 10 and 30. Equation (3) can be interpreted as an ideal controller cascaded with a first order filter. Such a tilter could be incorporated into the design procedure for the ideal controller and implemented digitally if desired. Previously Kado [2] developed an interactive program similar to the one discussed here for an electronic controller. Based on Kado’s program, Hougen[3] has developed tuning correlations for this type of controller. When arranged in series compensation as shown in Fig. 1, the closed loop relationship becomes
C(s) -=ltG,G,., R
GG,
(4)
where C is the controlled variable and R is the set point. The closed loop frequency response can be found by setting s = io in eqn (4). The magnitude of this complex
‘dC Fig. 1. Typical series compensatednegativefeedback controltea system. GPcontainsall process componentsincludingsensor and finalcontrolelements.
t 1 dB = 1 de&e1 = 20 log,,AR
function is also called the closed loop amplitude ratio (AR). The objective in selecting controller parameters is to produce a closed loop frequency response which has an AR curve that possesses the following characteristics (Fig. 2): (1) The AR curve should be unity to as high a frequency as possible to ensure no offset and a quick return to steady state, after a setpoint change. (2) The peak magnitude ratio, M,, should be no greater than 1.25(2 dBt). (3) The AR curve following the peak should become unity at as high a frequency as possible, denoted here as the crossover frequency (o,,). The analogous transient response (normalized) for a step input is also shown in Fig. 2. When these conditions are met for the closed loop frequency responses, the response of the controlled system to a change in setpoint will resemble that of a second order system with a damping coefficient (0 of about 0.5 and an undamped natural frequency near 0,. Such a second order response exhibits desirable rise times and overshoot characteristics as observed in Fig. 2131. The synthesis procedure for an ideal three mode controller developed by Heeb[4] for a Tektronix 4014 graphics terminal employs the following steps to determine acceptable controller parameters (KC,TV,Td in eqn (2)): (1) Ti is chosen so that when the controller AR curve is combined with the process AR curve on the Bode plot, the slope of the resulting curve is - 1 to as high a frequency as possible. A rule of thumb which can be applied is to set Ti slightly less than the dominant time constant in the denominator. (2) Using the Nichols chart, KC is chosen so the open loop frequency response curve is tangent to M = 1.25 on the closed loop locus, indicating an overshoot of 2 dB on the closed loop Bode plot. (3) The peak frequency, o,, of the closed loop AR curve is obtained from the closed loop frequency response. A value of Td iS selected to that at this freqUenCy the controller contributes 45-60” phase lead to the open loop frequency response. A good first guess is 7d = 21% ]51. (4) Steps two and three are then repeated until the values of Td and K, which maximize o,, are determined. (5) Ti is modified by a small amount and step two is repeated. If o,, is not greatly decreased and the closed loop AR curve is unity to the highest obtainable frequency, this step is repeated until Ti is maximized. Periodic display of the closed loop time domain response may be desirable also. Description of the Interactive Program The CRT displays used in interactive controller design are discussed below; the designer has the ability to return to any previous display upon command. Typical displays are shown in subsequent figures for the process,
G&J=(1 t
*i;;*
t 5s)’
(5)
Each display is numbered for the purposes of interaction. (1) Confroffer parameters (Fig. 3) This display shows the parameter symbols along with
Computer-aided process control system design using interactive graphics
Desired Closed
Closed Loop
Loop
AR Curve
AR Curve
and
step
Step
221
Response
Response
Frequency
Fig. 2. Desired closed loop AR curve and step response.
are changedby keying in values for Tl, T2, 7’3,T4 and DT, as discussedabove.
the values entered KC=w
(KJ
TLI=y
(Td)
TZ= x
(Cl.
(2) Frequency response of individual functions This display shows the AR and phase angle curves for the controller and process. These displays are conventionalBode diagrams,i.e. log AR (in dB) vs log o and C$vs log o, where o = frequency in radiansper unit time.
This display shows default values of these parameters when it fnst appears (a default value of K, = 1.0 is employed until the Nichols chart is requested). To change the value of Ti to 10, e.g. the following statement is keyed in; TZ= 10. This change is entered upon depressingthe carriage return key. Similarlyother changes can be made in the controller parameters.When the last change has been entered, the carriage return is depressed again and the computer program and display are up-dated.
(3) Combined open loop with unity gain The previous sets of curves are combinedand presented as a Bode diagram(both AR and 0 are shown).The advantage of the Bode plot is that phase angles and log magnitude ratios are additive[l]. At this point an experienced designer can usually discern if grossly inadequate controller parameters have been selected and can return to Display 1 to revise the computations.
Systemparameters (Fig. 3) If a linear model of the system dynamics is available (eqn 1) parameters are entered here. The default values
(4) Nichols chart with combined open loop The Nichols chart is displayed covering the range of -90 to - 180”phase angle as abscissa. The 0 and 2 dB
PROCESS
CONTROLLER OC(a )= KC( l+TDa*l~tXm)
Fig. 3. Specilication of process and controller parameters.
T. F. EDGAR et al.
228
closed loop loci are explicitly shown as well as the combined open loop result derived from Display 4. At this point it is possibleto changethe controller gain from its default value of unity (0 dB) by keying in a new value for KT in dB. By visual inspection, the new KT is estimated so that tangency to the 2 dB closed loop locus is obtained. This step can be repeated if desired to improve upon KT. K, is computed from KT based on the decibel definition.
on a Bode diagram in Fig. 4. The combined frequency response is given in Fii. 5. Note that the phase marginis about 50’.This figure shows the compensatedAR curve havinga slope of - 1.0to a frequency of 0.1. Fiie 6 is the plot of the combined frequency response on the Nichols chart. Here we plot the magnitudein dB. Note that while the closed loop frequency response is not tangent to the 2dB curve, an increase in the open loop AR of about 3.5dB would cause the curve to be tangent. Therefore, a new gain of 3.5dB is selected (K, = 1.4%) (5) Comparison of fully compensated open loop based on the open loop AR in Fig. 6. Figure 7 showsthat frequency response with unity gain curve this value for the gain makes the frequency response This display shows the vertical shift of the AR in- curve tangent to the 2 dB closed loop locus. formation resultingfrom the changein gain from unity to Next we examine the closed loop frequency response, the chosen value of KT (dB). Once againan experienced which is plotted on a Bode diagramin Fig. 8; o, is equal designer can usually discern if less than optimal results to about 0.25.As a tirst choice, 7dis set equal to 2/oP or will be forthcoming. 8.0 and the component frequency response (Fig. 9) indicates that the controller adds around 45” phase lead (6) Closed loop frequency response near o = 0.25.A new gain equal to 5.079(14.1dB) is then The closed loop Bode plot is displayedfrom whichit is chosen to make the combined frequency response easy to estimate the values of o,, or o,, delined earlier tangent to the 2 dB locus. In Fig. 10, o, has increased to in Fig. 2. about 2 and the closed loop AR curve dips below the 0 dB line between o = 0.1 and o = 1. It is desirable to (7) Response of closed loop system to step input keep the AR curve flat alongthe 0 dB line whiletrying to This display shows the response of the controlled maximize o,,. Since o, now eqIdS 2, rd iS changed to variable c(t) subject to a unit step change in the set 2/@,, or 1.0, and the component frequency response point. At any point it is possibleto return to any previous showsthe controller adds around 60”phase lead at o = 2. display entering x(x = 1-7) followed by depressing the A smallergain of 6 dB makes the frequency response of carriage return key. the closed loop system tangent to the 2dB locus. The closed loop AR curve has a w, approximatelyequal to Example 0.3 and ?d must be increased to maximizeo,,. When 76 Let us illustrate this design procedure for the second is increased to 4, a gain of 9.441(19.5dB) is required to order plus dead time process mentioned earlier. The make the open loop frequency response curve tangent to steps of the controller design procedure are carried out the 2 dB locus (Fig. 11). The combined open loop and Figs.4-14 show the results for each step. In Fig. 4, 71 frequency response (G,G,) for this controller is shown is set equal to 9; note that this selection nearly cancels in Fig. 12. The adjusted frequency response AR curve the term (I + 10s) in G,,(s). This is slightlyless than the (Fig. 13) has a oCOnear 2 and is flat along AR = 1.0 to largest time constant of the process. The open loop about o = 0.7. Hence this controller appears to be satisprocess and controller frequency responses are plotted factory. The transient response to a step change in the
10-2
-100
10-3
-150
10-4
-200
10-5 10-4
-250 10-3
10-2
10“
1
101
102
103
Frequency
Fig. 4. Component frequency response, Kc = 1, r = 9, TD= 0.
104
229
Computer-aided process control system design using interactive graphics
AR
1 0 degrees
10-l 10-2 10-3 10-4 10‘5 10-4
10-3
10-Z
10-I FW"&
10'
102
103
104
Fig. 5. Combined frequency response, Kc = 1, z = 9.
3 b
-180
-170
El60
-150
-140
-130
-120
-110
-100
-90
Open Loop Phase Angle, degrees
Fig. 6. Nichols chart, Kc = l(0 dB), T = 9.
20 16
12 0 : P
8
kl
4
0P
0
A R
-4
;
-8 -12 -16 -20
-180
-170
-160
-150
Open Lwp
-140
-130
-120
-110
Phase Angle, degrees
Fig. 7. Nichols chart, Kc = 1.4% (3.5dB), T = 9. CACE vol. 5. No. 4-D
*
-100
-90
T.F.Eua~etaL
230
A R
10-q
10-J
10-2
10-l
1
10'
102
103
101
Fmqancy Fii.
8. Closed loop frequency response, K, = 1.4% (3.5dB), q = 9.
Frequency
Fig. 9. Component frequency response, Kc = 1.4% (3.5dB), T = 9, TO= 8.
A R
104
ma
103
150
102
1Oll
10.
50
1.0
0
d e 9 r e
10-l
-50
:
10-2
-100
lo-‘(
-150
10-4
-2ull
10-5. 10-4
-250 10-3
10-2
10-l
1
10'
102
103
Fig. 10. Closed loop frequency response, K, = 1.496, 4 = 9, TD= 8.
104
0
231
Computer-aided process control system design using interactive graphics
-160
-170
-160
-150
-14U
-130
-120
-110
-lw
-90
Open Loop Phase Angle, Degrees
Fig. 11.Nichols chart, Kc = 9.441(19.5dB), T = 9, TD= 4. 200 150 100 0 5o 0: -50 -100
-150 -2cil -250 10-4
16-3
10-2
10-l
1 Yreq"e"Cy
101
102
103
10"
Fig. 12. Combined open loop frequency response (Kc = 9.441, r = 9, TD= 4). 104
IO3 102
10 A R
Frequency
Fig. 13. Closed loop bode diagram (Kc = 9.441, F = 0, To = 4). . .
d c c :
T. F. EDGAR et al.
232
Fig. 14. Transient responsefor final control system (K, = 9.441,T = 9, 7” = 4).
set point is plotted in Fig. 14 and this response displays the desirable second order characteristics.
this topic here but rather refer the reader to the evolving
body of literature on this subject.
DESIGNOF MULTIV.4RlABLECONYROL SYsmM!3 The concepts presented in the previous sectionscan be extended to the case where multiple outputs are to be controlled by multiple inputs. In design practice, often the interactions among process variables are neglected, and it is assumed that a set of singleinput-singleoutput controllers will provide satisfactory regulation of the plant. Recently there has been a greater concern with treating rather than ignoring the multivariableinteractions. Rosenbrock[6] has developed a generalized computeraided approach for analyzingsuch multivariablesystems, and computer graphics are an integral part of the design procedure. Due to space limitations,we do not discuss,
1. D. A. Coughanowr& L. B. Koppel,ProcessSystems Analysis and ConfmL McGraw-Hi, New York (1%7). 2. T. Rado, Computer-AssistedDesign Proceduresfor Process Control Systems.M.S.E.Thesis. Universityof Texas, Austin, TX (1970). 3. J. 0. Hougen, Measumnents and Conhvl Applicatbns. Instrument Societyof America,Pittsburgh,PA (1979). 4. R. C. Heeb, Computer-AidedMuhivariaMeControllerDesign for DistillationColumns.M.S.E.Thesis, University of Texas, Austin, TX (1980). 5. M. Tyner & F. P. May, Pmcess EngineetingCon:roL Ronald Press, New YorkCity (1968). 6. H. H. Rosenbrock,Computer-Aided Control SystemDesign, AcademicPress, New York(1974).
Chico/ Eqinming Printed in Great Britain.
Vol. 5. No. 4, pp. 22~232,198l
COMPUTER-AIDED PROCESS CONTROL SYSTEM DESIGN USING INTERACTIVE GRAPHICS T. F. Eucim,t R. HEEBand J. 0. HOUGEN Department of Chemical Engineering, Universityof Texas,Austin,TX78712,U.S.A. (Received 15 November 1981)
Abstract-Design programs employing interactive computer graphics can simplify and greatly facilitate the synthesis of feedback controllers for dynamic systems. In this paper a program for design of three mode controllers for single input-singleoutput linear systems is described. Seven displays can be employed for analyzingeither time domainor frequency domaincharacteristics,as illustratedin the paper.
Scope-The purpose of the controller in a feedback control system is to impart desirable dynamic properties to the closed loop system. Desirablefeedback control system characteristicsmightinclude (1) rapid response to setpoint changes (2) adequate disturbanceor load rejection (3) insensitivityto model and measurementerrors (4) avoidance of excessive controller action (5) suitabilityover a wide range of operatingconditions. Attainingall of these goals may be impossiblefor a given process because of conthctingobjectives. Since it may be difficultin the design process to quantify accurately the trade-offs in performance objectives, the availability of interactive tools is crucial to efficient process controllerdesign. There are several approaches which are used in practice to design controllers: (1) correlations of tuning parameters (2) repetitive time domain simulation of candidate controllers (3) frequency response Correlations such as Ziegler-Nichols [ l] give approximate values for controller parameters, based on a specific type of dynamic model (the tirst order plus time delay transfer function). For other systems, such guidelines cannot be applied directly and do not attain the high performance realizable by more detailed design procedures and more complex controllers. Another disadvantage of tuning correlations is that they yield a highly oscillatory (although rapidly responding) controlled system. Repetitive time domain simulation of the closed loop response, where controller parameters are varied in a systematic fashion, overcomes some of the above disadvantages, but this approach can be very inefficient, sometimes requiring excessive iteration. Explicit stability and sensitivity information is also unavailable in this approach. A design procedure which overcomes most of the above objectives is based on the well-known concept of frequency response[l]. However, depending upon the experience of the designer, repetitive calculations requiring graph paper can be a large obstacle to use of frequency response methods in controller design. Computer-aided design facilities, especially those employing interactive computer graphics terminals, can simplify and facilitate greatly the design of feedback controllers. The use of interactive computer programs that plot graphical displays allows the designer to quickly examine results and make modifications so the final control system satisfies the constraints and design objectives. Design of three mode controllers for linear dynamic systems can be Coudusions and SIgnBIperformed rapidly using interactive computer graphics. High performance control systems can be synthesized with the frequency response method, thus eliminating the uncertainty when approximate tuning rules are employed in controller design. Explicit sensitivity information and stability characteristics can also be computed. The program discussed here is based on an ideal three mode servomechanism controller and thus is more directly applicable to digital control rather than electronic or pneumatic controllers. However, the program can be easily modified to treat either the electronic or pneumatic controller forms. The process model specified by the user can have up to third order dynamics with dead time. The user has the capability to recall any of seven displays (open or closed loop) at any time, thus reinforcing the correlation between time and frequency domain characteristics. After 1 hr of training using the interactive graphics package and the iterative procedure discussed in this paper, a typical undergraduate student can design a satisfactory three mode controller in less than 10 min of terminal time.
225
T. F. EDGAR d al.
226 USE OFFREQUENCYRESPONSEINPROCESS CONTROLLZRDESIGN
The analysis of a large body of experimental data obtained from full-scale chemical plants has led to the conclusion that the dynamic relationship between input and output process variables can be represented reasonably well by the transfer function form given in eqn (1):
e(s)=
K,(l t QS) eeDTS = G,(S) (1 t T*S)(lt 7*SHl+ 73s)
(1)
where the time constants (T,, 7*,Q and TV)and pure delay time, DZ’, are estimated based on testing of the process. The popular three term (or PID) controller usually is sufficient to provide satisfactory compensation for most process control applications. In the program discussed here, an ideal three mode controller of the form[l],
G,-(S)= Kc(l t Tdst I/TiS)
(2)
is used for design. This controller form is difficult to realize using analog devices; digital controllers, after discretization of eqn (2), can be programmed to give ideal behavior. For an electronic controller, usually the transfer function is expressed as:
K, (%)(e)=
G,(s).
(3)
Alpha (a), however, is normally not alterable in a fixed configuration electronic controller but usually lies between 10 and 30. Equation (3) can be interpreted as an ideal controller cascaded with a first order filter. Such a tilter could be incorporated into the design procedure for the ideal controller and implemented digitally if desired. Previously Kado [2] developed an interactive program similar to the one discussed here for an electronic controller. Based on Kado’s program, Hougen[3] has developed tuning correlations for this type of controller. When arranged in series compensation as shown in Fig. 1, the closed loop relationship becomes
C(s) -=ltG,G,., R
GG,
(4)
where C is the controlled variable and R is the set point. The closed loop frequency response can be found by setting s = io in eqn (4). The magnitude of this complex
‘dC Fig. 1. Typical series compensatednegativefeedback controltea system. GPcontainsall process componentsincludingsensor and finalcontrolelements.
t 1 dB = 1 de&e1 = 20 log,,AR
function is also called the closed loop amplitude ratio (AR). The objective in selecting controller parameters is to produce a closed loop frequency response which has an AR curve that possesses the following characteristics (Fig. 2): (1) The AR curve should be unity to as high a frequency as possible to ensure no offset and a quick return to steady state, after a setpoint change. (2) The peak magnitude ratio, M,, should be no greater than 1.25(2 dBt). (3) The AR curve following the peak should become unity at as high a frequency as possible, denoted here as the crossover frequency (o,,). The analogous transient response (normalized) for a step input is also shown in Fig. 2. When these conditions are met for the closed loop frequency responses, the response of the controlled system to a change in setpoint will resemble that of a second order system with a damping coefficient (0 of about 0.5 and an undamped natural frequency near 0,. Such a second order response exhibits desirable rise times and overshoot characteristics as observed in Fig. 2131. The synthesis procedure for an ideal three mode controller developed by Heeb[4] for a Tektronix 4014 graphics terminal employs the following steps to determine acceptable controller parameters (KC,TV,Td in eqn (2)): (1) Ti is chosen so that when the controller AR curve is combined with the process AR curve on the Bode plot, the slope of the resulting curve is - 1 to as high a frequency as possible. A rule of thumb which can be applied is to set Ti slightly less than the dominant time constant in the denominator. (2) Using the Nichols chart, KC is chosen so the open loop frequency response curve is tangent to M = 1.25 on the closed loop locus, indicating an overshoot of 2 dB on the closed loop Bode plot. (3) The peak frequency, o,, of the closed loop AR curve is obtained from the closed loop frequency response. A value of Td iS selected to that at this freqUenCy the controller contributes 45-60” phase lead to the open loop frequency response. A good first guess is 7d = 21% ]51. (4) Steps two and three are then repeated until the values of Td and K, which maximize o,, are determined. (5) Ti is modified by a small amount and step two is repeated. If o,, is not greatly decreased and the closed loop AR curve is unity to the highest obtainable frequency, this step is repeated until Ti is maximized. Periodic display of the closed loop time domain response may be desirable also. Description of the Interactive Program The CRT displays used in interactive controller design are discussed below; the designer has the ability to return to any previous display upon command. Typical displays are shown in subsequent figures for the process,
G&J=(1 t
*i;;*
t 5s)’
(5)
Each display is numbered for the purposes of interaction. (1) Confroffer parameters (Fig. 3) This display shows the parameter symbols along with
Computer-aided process control system design using interactive graphics
Desired Closed
Closed Loop
Loop
AR Curve
AR Curve
and
step
Step
221
Response
Response
Frequency
Fig. 2. Desired closed loop AR curve and step response.
are changedby keying in values for Tl, T2, 7’3,T4 and DT, as discussedabove.
the values entered KC=w
(KJ
TLI=y
(Td)
TZ= x
(Cl.
(2) Frequency response of individual functions This display shows the AR and phase angle curves for the controller and process. These displays are conventionalBode diagrams,i.e. log AR (in dB) vs log o and C$vs log o, where o = frequency in radiansper unit time.
This display shows default values of these parameters when it fnst appears (a default value of K, = 1.0 is employed until the Nichols chart is requested). To change the value of Ti to 10, e.g. the following statement is keyed in; TZ= 10. This change is entered upon depressingthe carriage return key. Similarlyother changes can be made in the controller parameters.When the last change has been entered, the carriage return is depressed again and the computer program and display are up-dated.
(3) Combined open loop with unity gain The previous sets of curves are combinedand presented as a Bode diagram(both AR and 0 are shown).The advantage of the Bode plot is that phase angles and log magnitude ratios are additive[l]. At this point an experienced designer can usually discern if grossly inadequate controller parameters have been selected and can return to Display 1 to revise the computations.
Systemparameters (Fig. 3) If a linear model of the system dynamics is available (eqn 1) parameters are entered here. The default values
(4) Nichols chart with combined open loop The Nichols chart is displayed covering the range of -90 to - 180”phase angle as abscissa. The 0 and 2 dB
PROCESS
CONTROLLER OC(a )= KC( l+TDa*l~tXm)
Fig. 3. Specilication of process and controller parameters.
T. F. EDGAR et al.
228
closed loop loci are explicitly shown as well as the combined open loop result derived from Display 4. At this point it is possibleto changethe controller gain from its default value of unity (0 dB) by keying in a new value for KT in dB. By visual inspection, the new KT is estimated so that tangency to the 2 dB closed loop locus is obtained. This step can be repeated if desired to improve upon KT. K, is computed from KT based on the decibel definition.
on a Bode diagram in Fig. 4. The combined frequency response is given in Fii. 5. Note that the phase marginis about 50’.This figure shows the compensatedAR curve havinga slope of - 1.0to a frequency of 0.1. Fiie 6 is the plot of the combined frequency response on the Nichols chart. Here we plot the magnitudein dB. Note that while the closed loop frequency response is not tangent to the 2dB curve, an increase in the open loop AR of about 3.5dB would cause the curve to be tangent. Therefore, a new gain of 3.5dB is selected (K, = 1.4%) (5) Comparison of fully compensated open loop based on the open loop AR in Fig. 6. Figure 7 showsthat frequency response with unity gain curve this value for the gain makes the frequency response This display shows the vertical shift of the AR in- curve tangent to the 2 dB closed loop locus. formation resultingfrom the changein gain from unity to Next we examine the closed loop frequency response, the chosen value of KT (dB). Once againan experienced which is plotted on a Bode diagramin Fig. 8; o, is equal designer can usually discern if less than optimal results to about 0.25.As a tirst choice, 7dis set equal to 2/oP or will be forthcoming. 8.0 and the component frequency response (Fig. 9) indicates that the controller adds around 45” phase lead (6) Closed loop frequency response near o = 0.25.A new gain equal to 5.079(14.1dB) is then The closed loop Bode plot is displayedfrom whichit is chosen to make the combined frequency response easy to estimate the values of o,, or o,, delined earlier tangent to the 2 dB locus. In Fig. 10, o, has increased to in Fig. 2. about 2 and the closed loop AR curve dips below the 0 dB line between o = 0.1 and o = 1. It is desirable to (7) Response of closed loop system to step input keep the AR curve flat alongthe 0 dB line whiletrying to This display shows the response of the controlled maximize o,,. Since o, now eqIdS 2, rd iS changed to variable c(t) subject to a unit step change in the set 2/@,, or 1.0, and the component frequency response point. At any point it is possibleto return to any previous showsthe controller adds around 60”phase lead at o = 2. display entering x(x = 1-7) followed by depressing the A smallergain of 6 dB makes the frequency response of carriage return key. the closed loop system tangent to the 2dB locus. The closed loop AR curve has a w, approximatelyequal to Example 0.3 and ?d must be increased to maximizeo,,. When 76 Let us illustrate this design procedure for the second is increased to 4, a gain of 9.441(19.5dB) is required to order plus dead time process mentioned earlier. The make the open loop frequency response curve tangent to steps of the controller design procedure are carried out the 2 dB locus (Fig. 11). The combined open loop and Figs.4-14 show the results for each step. In Fig. 4, 71 frequency response (G,G,) for this controller is shown is set equal to 9; note that this selection nearly cancels in Fig. 12. The adjusted frequency response AR curve the term (I + 10s) in G,,(s). This is slightlyless than the (Fig. 13) has a oCOnear 2 and is flat along AR = 1.0 to largest time constant of the process. The open loop about o = 0.7. Hence this controller appears to be satisprocess and controller frequency responses are plotted factory. The transient response to a step change in the
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Fig. 4. Component frequency response, Kc = 1, r = 9, TD= 0.
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Computer-aided process control system design using interactive graphics
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Fig. 5. Combined frequency response, Kc = 1, z = 9.
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Fig. 6. Nichols chart, Kc = l(0 dB), T = 9.
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Fig. 7. Nichols chart, Kc = 1.4% (3.5dB), T = 9. CACE vol. 5. No. 4-D
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Fmqancy Fii.
8. Closed loop frequency response, K, = 1.4% (3.5dB), q = 9.
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Fig. 9. Component frequency response, Kc = 1.4% (3.5dB), T = 9, TO= 8.
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Fig. 10. Closed loop frequency response, K, = 1.496, 4 = 9, TD= 8.
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Computer-aided process control system design using interactive graphics
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Fig. 11.Nichols chart, Kc = 9.441(19.5dB), T = 9, TD= 4. 200 150 100 0 5o 0: -50 -100
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Fig. 12. Combined open loop frequency response (Kc = 9.441, r = 9, TD= 4). 104
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Frequency
Fig. 13. Closed loop bode diagram (Kc = 9.441, F = 0, To = 4). . .
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Fig. 14. Transient responsefor final control system (K, = 9.441,T = 9, 7” = 4).
set point is plotted in Fig. 14 and this response displays the desirable second order characteristics.
this topic here but rather refer the reader to the evolving
body of literature on this subject.
DESIGNOF MULTIV.4RlABLECONYROL SYsmM!3 The concepts presented in the previous sectionscan be extended to the case where multiple outputs are to be controlled by multiple inputs. In design practice, often the interactions among process variables are neglected, and it is assumed that a set of singleinput-singleoutput controllers will provide satisfactory regulation of the plant. Recently there has been a greater concern with treating rather than ignoring the multivariableinteractions. Rosenbrock[6] has developed a generalized computeraided approach for analyzingsuch multivariablesystems, and computer graphics are an integral part of the design procedure. Due to space limitations,we do not discuss,
1. D. A. Coughanowr& L. B. Koppel,ProcessSystems Analysis and ConfmL McGraw-Hi, New York (1%7). 2. T. Rado, Computer-AssistedDesign Proceduresfor Process Control Systems.M.S.E.Thesis. Universityof Texas, Austin, TX (1970). 3. J. 0. Hougen, Measumnents and Conhvl Applicatbns. Instrument Societyof America,Pittsburgh,PA (1979). 4. R. C. Heeb, Computer-AidedMuhivariaMeControllerDesign for DistillationColumns.M.S.E.Thesis, University of Texas, Austin, TX (1980). 5. M. Tyner & F. P. May, Pmcess EngineetingCon:roL Ronald Press, New YorkCity (1968). 6. H. H. Rosenbrock,Computer-Aided Control SystemDesign, AcademicPress, New York(1974).