SAMPLED-DATA, PROPORTIONAL-INTEGRAL CONTROL OF A CLASS OF STABLE PROCESSES HENRY A. MOSLER,' LOWELL B. KOPPEL, AND DONALD R. COUGHANOWR School of Chemical Engineering, Purdue University, Lufayette, Ind.
The analysis of sampled-data, feedback, proportional-integral control of processes adequately modeled as first-order with delay is presented. Controller settings are recommended as functions of the model parameters and sampling period. Control of an experimental higher-order system i s used to verify the suggested design. N EARLIER
work ( 5 ) , sampled-data, feedback, proportional
I control systems were studied, and design techniques were recommended. The primary deficiency of such control action is that steady-state offset will occur for deterministic loads which are long-term in nature. For certain systems, this offset is not of serious concern; however, for systems in \+hich steady-state offset is unacceptable, integral action must be added to the controller. In continuous-data systems, addition of integral action results in the transient response of the system having a slower response time, since phase lag has been added to the system. This same result will be shown to exist for sampled-data systems. This paper deals with the analysis of a class of stable processes under sampled-data, proportional-integral control, development of design techniques for such systems, and finally, experimental verification of the suggested design on a real system whose dynamics are not fully described by the model.
I
I
i
U a
Analysis of System for Sampling Period Equal to Delay Time (T = 4 7 )
T
The general class of processes to be considered includes those whose dynamics can be adequately modeled by a transfer function which is first-order with delay,
I
0
e -a.rs
G,(s) =
~
TS
+1
G,(s) G,(s)
EE
H(s)
G
E
controller transfer function process transfer function hold transfer function
The input and output of the sample-and-hold device are related as illustrated in Figure l b , which requires the hold transfer function to be
The transfer function for the P-I controller is G,(s)
=
K
[+ 1
-
(3)
+ G,G,H(z)
=
0
(4)
2 t/T b
3
4
Figure 1. Sampled-data feedback control system (a) and sample-and-hold device (b)
Because the process contains pure delay, the order of the characteristic equation is related to the ratio of sampling period to delay time and approaches infinity as sampling is performed infinitely often. In general, for P-I control of firstorder processes with delay, the order of the characteristic 3 where equation is n
+ u / ( n + 1) 6
T I T < a/n; n
= 0,
1, 2 , . .
(5)
,
We shall first analyze the system for a sampling period equal to the delay time in the loop ( T / T = a, n = 0), and then generalize for arbitrary sampling. For Z / T = a ,
TtJ]
The characteristic equation of the system described by the block diagram of Figure l a is 1
I
(1)
Unity process gain is assumed with no loss in generality, Sampling is performed on the output of the process in the manner suggested by the block diagram of Figure 1a, where
U
G,G,H(z) =
K
- [(TI TI
-
~ ) ( 1- b )
+ T]
2
t(2
- z1
- 1)(2
-
b)
(6)
where
1 Present address, Esso Research and Engineering Go., Florham Park, N. J.
VOL. 6
NO. 2
APRIL
1967
221
and
b =
e-TIT
Unit c i r c l e
(8)
z-plane
For this case, the order of the characteristic equation is three, with one open-loop pole at the origin, one pole a t unity, one pole at b, and an open-loop zero at the location described by Equation 7 . A comparison in the z-domain of the open-loop poles and zero for P-I control of sampled-data systems, with the poles for proportional control, illustrates the effect of addition of integral action on the root locus. For proportional control and sampling equal to the delay time of the process ( T / r = a) G,(s) = K
Unit c i r c l e
(9)
and
Comparing Equations 6 and 10, we see that the addition of the integral mode has placed an additional open-loop pole at unity, which is the stability boundary in the z-domain, and has given us a free open-loop zero with which to shape the root locus. The additional open-loop pole at unity results in a root lying further from the origin in the z-domain than any roots lie for proportional control. Hence, in sampled-data systems as well as in continuous-data systems, introduction of integral action increases the response time of the system. For the limiting case of no integral action, as T I + m , the open-loop zero of Equation 7 approaches unity, and Equation 6 reduces to Equation 10, as is expected. We next examine the different ranges within which the integral time constant can lie, to justify the particular design technique which is recommended in this paper. As integral action is introduced, the open-loop zero moves along the real axis in the z-domain from the open-loop pole at unity toward - m . From Equation 7 various ranges of T I are seen to affect z1 in the following manner.
7--
b I - b
r < r ~ < m
-+
b < t l < l
T < r ~ < r
+
O
