INVERSE RESPONSE IN PROCESS CONTROL KOlCHl IlNOYA
ROGER J. ALTPETER
An ordinary thermometer is heated. It expands
mih'enb, bejore the measuring @id can expand. The result is a dip in the reading, and control ystem time lag-thds
is an examQle of inuerse
response ystems which have inverse responses are difficult to control. Many types of process show inverse response over a part of the normal range of operation-among these is the recycling system so common in engineering practice. Control can be .greatly improved by introducing a compensating feedback element (4) into the control loop. An analog computer has been used to simulate processes having inverse response and to demonstrate the operation of the controlling instruments. The linear three-mode controller is the system most commonly used in chemical processing. It is very effective if the process has stable dynamic characteristics such as low order delay with a short dead time. This controller runs into trouble with the inverse r e s p o n s e t h e correction made initially would be in the wrong direction. The total effect would be that the controller would force the process to the desired
S
conditions only very slowly. Also, the closed loop responses are apt to be unstable at a low proportional sensitivity. To decrease the initial inverse shoot, and thereby decrease time required to bring the process to the d e s i i d point, a compensating feedback element can be used. Such a method was fmt introduced by Smith to control a process with dead time (5). This causes a marked improvement. Further improvement in control results when the feedback element is adapted for inverse response, and the Smith method is modified as shown below. We have found that an optimum value exists for the compensating feedback element. This value varies, depending upon the transfer functions of the inverse responses in the process. These values can be calculated, or their ranges determined by computer simulation. Tmnshr Functions of I n w m Responw
In order to study process control of invene responses, the transfer functions should 6rst be developed. The transfer functioni.e., the ratio of the output vmiablc X to the input signal Y in the Laplace trmjwmation-detines the performance of the process. The transfer functions for several inverse responses given by Equations 1 to 8 are easily developed (3, 7). These equations represent a variety of typical combined processes or their approximations. The conditions are cited for which these equations represent inverse responscs.
-r
(Continued on mxtpagc)
II
G
rn
X
1 . main feedback
loop
Closed control Joop with compensating feedback element
Y O L 5 4 N a . 7 JULY I962
39
.
An anaIvg cirndt of a rccirnrlatiue froce'
'vcrsi& of Wisconsinfrom stadard components 40
INDUSTRIAL AND ENGINEERING C H E M l S T P Y
'
IF THE PROCESS HAS THIS TRANSFER FUNCTION..
.
1ESPONSE WILL BE NVERSE I F . .
.
1. Integral response minus first order response (Figure 1):
MT.
> TI
2. D i f f m c e batween two flrst order responses (Figure 2):
MI Gz=---=
TIS+^
- ( M z T I - M1Tn)s
Mi Ts+1
+ MI - MI
+ 1) ( T z f + 1 )
(TIS
TI MI ->->1 Tz Mz
3. Second order minus first order response
4. DiRerence batween two first o r d r responses with dead times:
M I > Mz and L1 > Lz 1 0
M$-Lv
M,~-L~s
G4=--Tis+ 1
Tns
+1
5. Dihrence h w e e n two second order responses:
G5 =
(Tis
+
MI 1) (TU
+ 1)
-
M2
(Tas
+ 1) (TQ+ 1 )
but the derivative (inclination) of step response at initial condition (t = 0) is zero (horizontal), and inverse response may not be evident. 6. Second order with a leading element minus fire order response:
7. Diffsnnce between two second order responses with leading elements:
8. Dihrence batween two second order responses with leading elements and dead times:
M I 7 Ma and LI>Lt>O
9. Oeneral case (Figure 3):
G, = M I ( b 3 a.s"
++6iP-l + . . . + l)e-L" ais"-' + . . . + 1 Mn(b,'P' a,'?'
+ b,'P'-' + . . . -E l)e-" + a,'s*'-' + . . . + 1
All coefficientsare positive, usually n 2 rn 2 1 and n'
2 m' 2
1.
(i) i f L1 = Le = 0,
L I > Lz 2 0 and M I > Mn 10. A recirculahve process (Figures 4 and 5):
Tn > T I and MiMa > 1 regardless of dead time, L (Continwd on ne.ttpagl) V O L 5 4 NO. 7 J U L Y 1 9 6 2
41
Control of an Inverse Response Process
A common linear (three mode) controller can be applied to control an inverse response process, but the controlled responses are very sloiv and inverse at any appropriate setting of the controller. The Smith method (5), in which a compensating feedback element is introduced for dead time, improves the controlled response; and a modified method, proposed here, again improves control of inverse response processes. -4s an example, consider the transfer function of the process to be Gz (Equation 2) : and the transfer function of the controller to be of the general form :
K - ( T D T I s 2 T,s
+
TIS
+ 1)
(11)
The transfer function of the compensating feedback element in the proposed modified Smith method is defined in this example as :
where T i / T z > M I / M z> 1 and the condition
corresponds to the original Smith method. The transfer function of the closed loop is
X =
1
+
GzHi R Hl(G2 Hs)
+
+
+
Gn(1 ffiN2)U 1 H I ( G Hz)
+
+
(1 3 )
If a proportional and integral controller ( T D= 0) is used in this system, the closed loop transfer functions for a change in the reference value R and for an external disturbance U are given in Equations 14 and 1 5 . From Equation 14, necessary conditions for the stable response are found by the Rourh-Hurwitz theorem and are given in Equation 17.
42
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
If the Smith method, in which A = 0, is applied, the system becomes more stable than for the conditions Hz = 0. The condition for Hz = 0 corresponds to a linear controller without the compensating feedback element-Le., k = 0 and il < 0. With the modified method, A > 0, the system is more stable than that when A = 0. However, an excessively large value of A produces no overshoot and a smaller inverse shoot, but a slower response at the same values of K and T I . Furthermore, a higher proportional sensitivity K and a smaller integral time T I of the controller cause a larger inverse shoot. Therefore, an optimum value of A or k exists that yields acceptable responses to external disturbances. When derivative control ( T , > 0) is introduced, the initial response to a step input has a steep inverse inclination. If a first-order element with a large time constant, such as the so-called inverse rate action, is added in the main closed loop, the initial inclination for a step input becomes zero and response becomes slower. Numerical Example--Difference First-Order Responses
between Two
The transfer function of the process is represented by Equation 2, and the numerical values substituted into = 1, T I = 3, and Tz = 1 Equation 14 are: M I= 2, IM:! at T , = 0 from which MlTa - MzTl = - 1, .MI - -M2= 1. T I - Ti = 2, and A = 2k - 1. The condition k = 1/2 corresponds to the Smith method with A = 0. If k = 1, A becomes unity and the response becomes faster and stable with no critical cycling for any values of K and T I . The results of an analog simulation showed that an optimum condition existed for 1 Q k < 2 where 1 < -4 < 3. This condition corresponds to the modified method. These values promote faster stable responses with smaller overshoots and inverse shoots than for k < 1. For values of k B 2 andA 3 the responses are slower. The optimum condition permitted larger values of proportional sensitivity and rest, resulting in shorter cycling periods and faster responses. From Equation I ? , the following theoretical conditions for stability are obtained by the Routh-Hurwitz relationship. They can be compared with the experimental results at critical cycling.
Control of a Recirculative Process Simulated by a n Analog Computer (PI Control)
25y0 Damped Cycling
Proportional Sensitivity K
Reset Rate I / T I k = O k = l
-
0.72 0.32
1/ 6 1/3 1/2 2/3 1 2
At k
=
2.00 1.50 1.37 1.27 1.20 1.10
12.5 7.2
< 4, and TI>
(7 - K ) K
+ K ) (4 - K )
(1
At k = 1/2; A = 0 (original Smith method)
K is not limited, but
TI
>
3K 4(1
+ K)
At k 2 7/8; A 2 3/4 (modified Smith method) K and T I are not limited. Numerical Example-Circulative
- 3 ( - 2 ~ + 1) -3s + 1.5 + 1.5s + 0.5 - (2s + l)(s + 1) 1
3ks
=
(s
+ 0.5)(s + 1)
= 6k [
s
+
-2s ~ 1
From the Routh-Hurwitz relationship the theoretical conditions for stability are : K(k - 1)
[(i
4- K ) T I
>
+ 2K(k - 1)1X
-1/2
60 75 90 100 120 170
35
=-I
55
,
~
45 70 90 105 140 210
8 10
5 5
8 10 10 15
The authors are grateful to Dr. 0. A. Hougen for his encouragement and to the Wisconsin Alumni Foundation for financial support.
Process
and the compensating feedback element is:
Hz
60 60
ACKNOWLEDGMENT
The transfer function of the process is given by Equation 10; and the numerical substitutions are: M I = 1.5, M z = 1, T I = 0.5, T z = 2, and L = 0. The equation of the process becomes = s2
9.2 7.4 6.2 5.5 4.6 3.3
Inverse shoot, yo Offset, yo k = O k = ? k = O k - l
too large in order to obtain a quick response and a small inverse shoot. The theoretical values determined from Equations 19 and 20 can be compared with experimental results obtained at critical cycling on an analog computer. Results at approximately 25% damping are given in the table. Optimum control is atthined at a value, k > 1, producing a faster response and eliminating cycling.
0, A = -1 (PI controller)
K
Overshoot, 70 k = O k = ?
Cycling Period k = O k = l
(19)
NOMENCLATURE
A
= constant of a compensating feedback element
a, b
= coefficients
G HI
=
H1
=
K k L M m, n
= proportional sensitivity of a controller = parameter of a compensating feedback element
R,r
= =
transfer function of a process
= transfer function of a controller
=
transfer function of a compensating feedback element dead time of a process element
= static gain of a process element = power or suffix
reference input (desired value) parameter of Laplace transformation Ti = time constant of a process element T D = derivative time of a controller TI = integral time of a controller U = external disturbance X , x = controlled variable (output of a process) y = manipulating signal (input of a process) s
LITERATURE CITED
I n this case k = 1 corresponds to the original Smith method. The values of K , l/TI, and k should not be A U T H O R Roger J . Altpeter is a Professor of Chemical Engineering at the University of Wisconsin. H i s major interest is process measurement and control. While this article was prepared, Koichi Iinoya was a Postdoctoral Fellow in Chemical Engineering, also at Wisconsin. H e has now returned to his position as Professor of Chemical Engineering, Nagoya University, Japan.
(1) Eckman, D. P., “Industrial Instrumentation,” p. 55, Wiley, New York, 1950. (2) Gille, J. C., others, “Feedback Control System,” p. 155, McGraw-Hill, New York, 1959. (3) Kaneshige, K., “Automatic Control Handbook,” p. 902, Corona Co., Tokyo, 1957. (4) Newton, G. C., Gould, L. A., Kaiser, J. F., “Analytical Design of Linear Feedback Controls,” pp. 35, 326, Wiley, New York,
1957. (5) Smith, 0. J. M., 2, 6’.A . Journal 6 , No. 2, 28 (1959). ( 6 ) Solheim, 0. A., Rept. 56-2, Pt. I, p. 36, Dept. of Applied Physics, Chr. Michelsen Institute, Bergen, Norway, October 1959. (7) Takahashi, Y., “Theory of Automatic Control,” p. 62, Yuwanami, Tokyo, 1954. VOL. 5 4
NO. 7 J U L Y 1 9 6 2
43
