CO M M UN ICATlO N COMPUTING T H E T R A N S I E N T RESPONSE OF DEAD-TIME CONTROL S Y S T E M S The closed loop transient response of linear feedback control systems in which the plant contains a dead-time lag may b e calculated by a simple technique in which a Laplace transformation is approximated by a z-transformation. This technique, which eliminates the mathematical difficulties introduced by the dead time, is applied to a typical problem.
AUTHORS (2, 3, 5, 7) have presented methods which can be used to calculate the closed loop transient response of a linear feedback control system containing a dead-time lag. I n general, all of these techniques suffer from the fact that they are tedious and difficult to apply even to transfer functions of moderate complexity. T h e z-form, developed by Boxer and appears to be the best technique which is available Thaler (I), for solving the problem. This technique consists in approximating a Laplace transformation with a z-transformation. T h e major advantage of this method for the problem under consideration is that the resulting z-transformation is considerably easier to invert than the original Laplace transformation. Although the z-form technique is not new, it has largely been ignored in the chemical engineering literature. The purpose of this communication is to demonstrate both the power and the simplicity of the z-form method. SEVERAL
Calculating Transient Response by Using the z-Form
T h e type of control system which is considered in this communication is a simple feedback regulator with the controller transfer function given by
and the plant transfer function given by
Table 1. 5 -k
S -1
S -2
z-Forms
2-form T 1 +z-' -.__ 2 1-2-1 T2 1 102-' + z V a 12 (1 - t - 1 ) 2
+
-.
s -3 s -4 s -5
-T.4
2-l
6
-. ~5
+ 4 ~ +- ~ - _ T4 (1 720 + 112-2 + 1 1 ~ +3 2-3
~-1)4
2-1
(1
24
2-4
- 2-1)s
First, the numerator and denominator of the transform are divided by the highest power of s which appears. Second, the z-forms are substituted for the resulting s - ~terms. Third, a value of T is chosen so that the ratio T / T is a positive integer and a direct substitution of t - ' I T is made for the term e - r 5 . Fourth, after collecting terms the resulting z-transform approximation is rewritten as the ratio of two polynomials in z. Fifth, this ratio is multiplied by the factor 1 / T . Sixth, a simple long division technique is used to expand the z-transformation into the following series ( 7 ) : m
e&) where g(s), h ( s ) ,p ( s ) , and q ( s ) are polynomials. The relationship between the controlled variable, Bc, and the upset variable, eu, is given by
(3) T o obtain B,(t), it is necessary to invert Equation 3. The zform method of obtaining this inverse consists in substituting for s, the Laplace variable, a function of z , the z-transform variable. T h e exact definition of z as a function of s is given as (6) z = es'
(4)
Boxer and Thaler ( 7 ) developed polynomials in z, called zforms, which approximate negative integral powers of s. These z-forms are tabulated in Table I. T o use the z-forms to invert a Laplace transformation one proceeds as follows. 440
l h E C P R O C E S S D E S I G N AND DEVELOPMENT
=
(5)
unZ-n
n=O
The coefficients of this series, a,, are approximately equal to B,(nT), the values of the desired transient response a t integral multiples of the sampling time. These coefficients can be plotted and connected with a smooth curve to give a very good picture of the actual response. The z-form technique can be applied to multiple loop control systems. More than one of the plants in such a system can contain a dead time. However, if such a case arises, all of the dead times must possess a least common divisor.
E xa m ple As a n example of the principles which were presented in the preceding section, the approximate inverse Laplace transform of Equation 3 with G(s) given as G(s) =
e-'
4s2
+ 8s f
1
m
I
G,(s) given as .20
GCb)
6.756 (s f 0.175) =
1
1
(7)
S
and &(s) given as 1
&(s)
will be obtained.
=
;
T h e resulting closed loop transfer function is
= s(4s2 f
8s
+ 1) f
e-' 6.756 e-'(s
+ 0.175)
(9)
m
Dividing the numerator and the denominator by s + ~ ,substituting the z-forms given in Table I, and setting T = 1 yields ec
(z) =
+
0.500 (2 z) (10) 8 . 0 8 ~-~ 1 4 . 6 9 ~f~ 1 2 . 9 1 ~-~ 4 . 5 6 ~- 0.563
____
The denominator of Equation 10 can be divided into the numerator to give a n expansion in powers of z-". This calculation is shown in Table 11. The resulting coefficients are plotted in Figure 1 against the actual response. T h e agreement which is obtained is good enough for most engineering purposes. T h e increase in accuracy which can be gotten by halving the sampling time is also shown in Figure 1. I n this case, the z-transform approximation is given by
e,b)
=
0.125(z2 f z ) 6.0225 - 1 3 . 8 1 ~ f~ 9 . 9 5 ~-~ 0.681~' - 1.192 - 0.141
(11)
and the same inverse procedure can be used. Since T = 0.5 for Equation 11, the coefficients of z-l, z-*, z - ~ , . . . are the approximate transient response a t t = 0.5, 1, 1.5, . . . as opposed to the case where T = 1 and the response a t t = 1, 2, 3: . . . is obtained. As a result, it takes about twice as much calculation to determine the transient response for large values of time with T = 0.5 than \vith T = 1. Determining a Suitable Value of T
The problem of determining a suitable value of T can be divided into tlvo cases depending on whether the inverse ztransform calculations are to be carried out by hand or by machine. If a computer is to be used, then T can be chosen small enough to achieve any desired accuracy. A quantitative estimate of the accuracy of the results obtained for a particular choice of T can be made by comparing the approximate transient response a t t = 2 r with the exact response determined from Tyner's series technique (7). This technique can be used to show that for 7 5 t 5 27 the inverse of Equation 9 is the open loop response of the plant to step forcing. Thus,
2
0
4
I
I
I
I
6
8 t
10
12
I
14
16
Figure 1. Comparison of exact transient response with that predicted by the z-form
The exact value of O , ( t ) for t = 2 can be determined from Equation 12 to be 0.070. For T = 1 the z-form predicts a value of 0.062. Similarly for T = 0.5 the z-form predicts a value of 0.068. As can be seen, as T becomes smaller the Lform results converge to the exact response. A computer can keep decreasing T until the desired accuracy a t t = 2 7 is achieved. The resulting value of T can then be chosen assuming that the accuracy of the z-form approximation will be maintained throughout the remainder of the transient response. If the computation of the inverse z-transformation is to be carried out by hand then the largest value of T which is consistent with accuracy requirements should be used in order that the calculations d o not become too tedious. Boxer and Thaler have suggested that a value of T , which is 10% of the fundamental frequency of the closed loop system, should be used. Hoivever. it is often impossible to estimate this frequency. McAvoy ( 4 ) has presented a criterion for choosing a large value of T for the case where the plant transfer function is given as ..--rs
I ;
G(s) =
w2s2
+ 2 {ws + 1
(13)
and the controller possesses any combination of proportional, integral, and compensated derivative modes of operation. According to this criterion, the following values of T should be used:
T
= r
for
>1
(14)
w / ~
T = 0.57 for
5
w / ~
1
(15)
For higher order plant and controller transfer functions, 7. The z-form results can then be compared with the first term in Tyner's technique in order to determine whether or not T should be decreased.
T should be set equal to
for
l 5 t 5 2
Table II. 8.082
- 14.692'
+ 12.919 -
Example of Long Division Method for Obtaining Inverse z-Transformations 0.062t-* 0 . 1 7 4 ~ - ~ 0.2182-4 . . . . . . , .4.562 - 0.563 0 . 5 0 0 ~ ' 0.5002 0 . 5 0 0 ~ '- 0.9082 4- 0.798 - 0.2822-' - 0 . 0 3 5 ~ - ~ 1 ,4082 - 0.798 0.282r' 0.035~-~ 1 ,4082 - 2.559 2,2492-' - 0 . 7 9 5 P - 0 .0 9 W 3 1,761 - 1 ,9672-' O.829~-~ 0 . 0 9 8 ~ - ~ 1.761 - 3.199~-' 4-2.8112-' - 0 . 9 9 3 ~ --~ 0 . 1 2 3 ~ - ~
+ +
+
+
+ +
+
+
+
~
VOL. 5
NO. 4
OCTOBER 1 9 6 6
441
Conclusion
The z-form technique possesses several advantages which make it a n ideal tool for determining the closed loop transient response of dead-time control systems. The method is straightforward and easy to apply. T h e inverse of the resulting ztransformation only requires a knowledge of long division but it yields fast and good approximations to the transient response. The amount of labor involved is not very great, and the calculations can be carried out easily by hand. The method is easy to program for digital computation and does not require a large amount of computer storage. The calculating speed is extremely fast even for small values of the sampling time, and by choosing a small T , the z-form results can be made as accurate as desired. Acknowledgment
One of the authors held a National Science Foundation Fellowship during the course of this work. The numerical calculations were carried out a t the Princeton University Computation Center, which is supported in part by the National Science Foundation. Part of Figure 1 was drawn on a n Electronic Associates Dataplotter through the courtesy of that company.
n = integer variable = polynomial function of s q = polynomial function of s s = Laplace transform variable t = time T = sampling time t = t-transform variable
p
GREEKLETTERS 8, = controlled variable 8, = upset variable ( = parameter 7 = dead time w = parameter
MATHEMATICAL SYMBOLS = Laplace transformation
-
Literature Cited (1) Boxer, R., Thaler, S., Prod. I. R. E. 45, 89 (1956). (2) Chu, Y., Elec. Eng. 70, Pt. 11, 291 (1951). (3) Hull, T., Wolfe, W., Can. J . Phys. 32, 72, 80 (1954). (4) McAvoy, T., Ph.D. dissertation, p. 43, Princeton University,
Princeton, N. J., 1964. (5) Oldenbourg, R., Sartorius, H., “The Dynamics of Automatic Control,” Chap. 111-B-3, Am. SOC.Mech. Engrs., New York, 1948. ( 6 ) Tou, J., “Digital and Sampled Data Control Systems,” p. 145, McGraw-Hill, New York, 1959. (7) Tyner, M., Control Eng. 81,78, 81 (1964).
T. J. McAVOY‘ E. F. JOHNSON
N o menclature a, = constant = polynomial function of s
g G G, h
k
Princeton University Princeton, N . J .
= plant transfer function = controller transfer function = polynomial function of s = integer variable
442
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
RECEIVED for review November 2, 1965 ACCEPTED July 19, 1966 1
Present address, University of Massachusetts,‘Amherst, Mass.
