Ind. Eng. Chem. Process Des. Dev. 1083, 22, 571-576
571
Stability and Response Properties of the Analytical Predictor Ram Srlnlvasan' and Duncan A. Melllchamp D e p a ~ m e n tof Chemical and Nuclear Engineerlng, University of Californh. Santa Barbara, California 93 106
The use of computers for process control has kindled interest in the application of dead-time compensating techniques within the process Industries. The analytical predictor technique, one of a class of dead-time compensation methods, uses a firstorder plus delay process model which is subject to the usual modeling errors. The effect of these model discrepancies on the stability properties of the analytical predictor is considered in this paper. The characteristic equation obtalned for the proportionably mode of the analytical predictor is analyzed for several loworder cases, Le., where the order of the characteristic equation is 3 or less. The order of the system equation is solely determined by the relative magnitude of the system dead time ( T ~to ) the sampling interval (T). This approach establishes ultimate gains which are then verified by simulation. For higher-order characteristic equations, the inverse z-transform technique can be used for analysis. From all of these results a design rule-of-thumb is established to obtain an acceptable value of the proportional controller gain. This design criterion is then tested on a bench-scale process. Results of the test runs are presented as verification of the method.
Introduction A frequently encountered problem in the process industries is that of controlling processes with a significant time-delay (alternatively known as "dead time", "transportation lag", or "distance velocity lag"). This characteristic feature results either from the presence of an inherent delay, e.g., as from a gas chromatograph measurement, or from the use of a low-order plus delay model for industrial processes which are innately of higher order. Dealing with this problem serves then as a starting point for the design of almost any process control system, regardless of ita configuration. "Dead-time compensators" were developed in an attempt to overcome the detrimental effects of the time-delay. The Smith Linear Predictor (Smith, 1957) was the first of the compensation techniques. This technique utilizes an inner feedback compensation loop based on a first-order plus delay model of the process. The main drawback with the approach is that, though conceptually sound, hardware realizations are difficult for the dead-time generator if only analog techniques are available. Certain direct digital control algorithms are developed so that they contain inherent (i.e., built-in) dead-time compensation. A classic example is the minimal prototype (or dead beat) algorithm. Mosler et a1 (1966)carried out early developments related to chemical process control using this approach. Of recent advances in this class of compensating techniques, several algorithms should be cited (Doss, 1974), (Mutharason, 1978),and (Edgar and Vogel, 1980). Some work has been directed at evaluating stability properties of digital algorithms. Particularly to be noted are recent studies by Palmor and Shinnar (1979) and by Berg and Edgar (1980). The former study was directed toward sampled data controllers, in general, including a discussion of the effect of perturbing the process parameters on the stability of the closed-loop system. The latter study dealt particularly with stability properties of the discrete proportional-integral, Dahlin, and deadbeat (Kalman) algorithms. For the discrete PI and Dahlin algorithms, stability regions were mapped as a function of errors in the assumed parameters of the process model; however, there was no investigation of response properties.
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The Analytical Predictor developed by Moore (1969)and extended by Doss (1974)is something of a combination of the above techniques. The technique may be represented schematically as shown in Figure 1 where a predictor, utilizing past values of the process input and the present value of the process output, is used to estimate the future process response for purposes of compensation. With a particularly simple choice of process model, e.g., first-order plus delay, the prediction of process response can be carried out analytically; hence the choice of name for the method. The analytical predidor would appear to have a number of practical advantages. In particular, it is based on a first-order plus delay model which is the most commonly used model in the process industries. Also, for digital computer implementations the associated dead-time compensation technique can be easily implemented as compared to using analog methods, e.g., with the original Smith predictor. Nevertheless it has received relatively little attention in the literature (Meyer et al., 1978,1979)compared to the earlier Smith predictor (Buckley, 1960;Eisenberg, 1967;Giles and Bartley, 1977;Lupfer and Oglesby, 1962; Marshall, 1974; Meyer et al., 1976, 1978, 1979; Nielsen, 1969). An investigation of some of the properties of the analytical predictor is the subject of this paper. T h e Analytical Predictor As noted above, the analytical predictor is based on a firsborder plus delay model given by the following transfer function kpe-'Ds
G,(s) = 7s + 1
Following Moore's choice of notation, the time delay rD can be treated as two components, i.e., as ( N f)T,where N is the largest integer multiple of the sampling interval contained in the dead time and f is the fractional part that is left over. In order to eliminate the effect of dead time on the process, at least approximately, one might choose to use the estimated value of the process output rD time units into the future instead of the actual process output. Since a digital implementation is to be used, the prediction should also compensate for the approximate delay introduced by sampling, i.e., T/2. Hence the predicted process output to be used as input to the controller comparator at the kth sampling instant should be x [ ( k + +N +
+
0 1983 American Chemical Society
572
Ind. Eng. Chem. Process Des. Dev., Vd. 22, No. 4, 1983
Figure 1. Use of a predictor in the feedback path to compensate for process time delay.
f)T'j. This predicted value (2) can be obtained analytically (Moore, 1969) as zk = mN(kpuk.N.l(l- c) + cxk} + kpuk(1 - A) + N
Akp(1 - B)CBi-lUk-i (1) i= 1
where A = exp(-T/2~),B = exp(-T/r), and C = exp(fT/+ Thus the equation governing a proportional feedback controller would be (2) uk = k c b k - R k ) where k, = controller gain, Pk = [ ( 1 + k& )rkl/kckp, r k = set point at the kth sampling instant, an2 uk = vdue of the input a t the kth sampling instant. Note that the multiplier on the set point is chosen to eliminate steady-state offset resulting from set point changes for the proportional-only controller. Substituting (1)in (2) and solving yields an analytical result for the process input
ABN[kp(l- c)uk-N-1 + C X k ] ( 3 ) Moore also extended the algorithm to handle both measurable and unmeasurable load disturbances; however, we will deal only with the basic control algorithm here. In order to make a formal analysis of the properties of the analytical predictor, including an analysis of its stability properties, the characteristic equation of the system must be developed. This may be done by converting eq 1 from the discrete time domain to the z-domain by appropriate transformation. The predicted value can be written as
R ( z ) = (1 - A)kPu(z) + A ( l
N
- B ) k , u ( ~ CB'-'Z-' ) + i=l
Figure 2. Block diagram representation of the analytical predictor.
Notice that the order of the characteristic equation depends fundamentally on the value of N, the maximum integer multiple of the sampling period contained in the time delay. It is, in fact, N + 2 for the case of proportional-only control. Initially, we will concentrate on the stability Properties of the analytical predictor and how they are affected by errors in the assumed process model. The stability analysis can be considered in two classifications, i.e. (1)slower sampling (equivalently,small time delay) for which the low-order characteristic equation can be analyzed analytically, and (2) faster sampling (large time delay) for which analytical methods cannot be used. Below we show that conclusions concerning the effects of parameter errors on system stability which can be deduced readily from analysis of the low-order cases are verified through simulation of high-order cases.
Lower-Order Cases (N = 0 or 1) The simplest of the low-order cases occurs when the predictor is used to compensate a process time delay with magnitude 0 ITD < T plus the time delay (T/2) approximately introduced by sampling. In this case N = 0. If the predictor is based on a first-order plus delay model of the process, which itself is assumed to be described by a first-order plus delay transfer function, then we can account for mismatches between any of the three model/ process parameter pairs as follows. Let 6k = k,/k,, 6, = T,/T, and 6D = ( T D ) ~ / T D Now, for the case when the model exactly represents the process, all of the 6 quantities will be equal to 1. By selectively varying the 6 quantities, we can explore the effect of modeling errors on the characteristics of the closed-loop system. In this case the predictor elements for the algorithm would be given by
ABN((l- C)K,U(Z)Z-(~+') + C X ( Z )(4) ~
P ~ ( z=) (1- Aqk,
i.e. 2(z) =
U(Z)Pl(Z)
+ x(z)P&)
- C')~,Z-'
P~(z) = A%' (5)
where Pl(Z) =
(1- A)k,
+ A'(l
N
+ A ( l - B)k, i=l B'-'z-' + ABN(l - C ) ~ , Z - ( ~ + ' ) (6)
and P2(z)= ABNC (7) It is now possible to represent the closed-loop control system in terms of a block diagram containing pulsetransfer functions as shown in Figure 2. From this block diagram we can derive a pulse-transfer function relating the output X ( z )to the input R(z) (Srinivasan, 1980) in the form
(9)
(10)
and the process-plus-hold transfer function by k,(z(l - D) - B + D] G,,H(z) = Z(Z - B )
(11)
The primes on the parameters A'and C'indicate that they now depend on the appropriate model/process parameter ratios, i.e.
A' = exp(-T/27,)
= exp(-T/2~6,)
C' = exp(-f'T/r)
(12a) (1%)
Also
D = exp
I
Finally, for a proportional-only controller
(13)
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 4, 1983 573
S,=0.9 0.1
0.I
I
10 0.1
I
IO
sk
Figure 3. Plot of constraints for the N = 0 case ( T / r = 0.6, a = 0.5). [An ultimate gain exists in the region above either or both curves.]
After substituting eq 9, 10, 11, and 14 in the characteristic equation 1 + G,(z)f'i(z)
+ G#(z)G,(z)P=Az) = 0
(15)
making the bilinear transformation, and performing a Routh-Hurwitz analysis on the resulting equation, we obtain the following constraints 1 + k&,{bkfl - A'C')
+ A'C? > 0
k&,,{A%'(D - B) - 6kA'B(1 - C') - 6 k ( l - A'))
(16)
