2694
Ind. Eng. Chem. Res. 1992,31,2694-2699
Nonlinear Filter in Cascade Control Schemes Aleseandro Brambilla* and Daniele Semino Department of Chemical Engineering, University of Pisa, Via Diotisalui 2, 56100 Pisa, Italy
The present paper addresses the problem of using a cascade control structure in order to control system which have primary and secondary processes with comparable dynamics. As classical tuning techniques may be unable to give good performances, since they do not fully take into account the interaction between the two loops, a nonlinear filter is introduced between the two controllers in order to partially separate the dynamics of the loops. Such a filter, which involves the use of a single tuning constant, is meant to damp the action of the primary controller when it tends to amplify the variation of the manipulated variable. The nonlinear filter, besides having a stabilizing effect on the cascade structure, reduces the tuning difficulties. Significant improvements in robustness and performance are obtained in series cascade; moreover, with an appropriate choice of the secondary measurement, the benefits of the nonlinear filter structure can be extended to a parallel cascade scheme.
Introduction The idea of cascade control takes its origin from the problem of rejecting disturbances whose effects on a secondary output are much faster than the ones on the controlled variable. The use of cascade control is therefore recommended to control processes in which the secondary loop dynamics is much faster than the one of the primary process (Stephanopulos, 1984). In such conditions the interactions between the two controllers are nearly negligible due to the strong difference in dynamics and no tuning problems arise. However, a cascade control can be preferable to the usual feedback control also for processes with dynamic parameters of the two loops of the same order of magnitude, even if the expected improvements decrease as the dynamics of the secondary loop becomes slower, as has been shown by Krishnaswamy et al. (1990). The application of cascade control to systems with comparable dynamics of the secondary and primary processes is therefore under investigation. The interactions between the two loops can no longer be neglected, and appropriate design techniques have to be introduced. A possible approach involves advanced controllers (Morari and Zafiriou, 1989, Scali and Brambilla, 1990): the cascade control scheme (series cascade) is analyzed in an internal model control (IMC) or partly IMC structure. This approach is undoubtedly useful in order to examine which improvements can be obtained from a theoretical point of view and to develop effective advanced schemes, but it does not provide simple relations appliable to proportional integral derivative (PID)controllers if the common feedback structure is to be used. The usual practice for these conventional controllers is a sequential approach at first the secondary controller is tuned, ignoring the primary loop, and then the tuning of the primary controller is accomplished regarding the secondary loop + primary process as the system to be controlled. This method may be unable to give good performances when the dynamics of the two processes are comparable as it does not fully take into account the interactions between the loops. Such is the case of one of the peculiar problems of chemical process control: the composition control of a distillation tower product cascaded to the control of a tray temperature. In this case the dynamics of the two loops are not very different, except for the time delay, due to the presence of the sample and analysis system in the *Towhom correspondence should be addressed.
primary process. Moreover, it has to be taken into account that the manipulated variable (reflux flow rate, for instance) affecta composition and tray temperature through two parallel transfer functions (Luyben, 1973). Therefore the less usual parallel cascade scheme has to be used to describe the system and design the tuning. In the present work the introduction of a nonlinear filter between the two controllers is proposed. The filter, which depends on a single damping constant, partially separates the dynamics of the loops and makes the system stable even if an overtuned primary controller is used. Such an easily-tuned control structure has given good performances in series cascade control schemes, particularly when the dynamic parameters of the loops are comparable. As far as parallel cascade control is concerned, the performances of the nonlinear filter structure (NLFS) deteriorates when the gains of the transfer functions are such that a return to steady state of the secondary variable corresponds to a large offset of the primary one. However, an appropriate choice of the secondary measurement (peculiarlyof the tray location on the distillation column) addresses this problem and makes the filter structure work well even in parallel cascade control schemes.
Nonlinear Filter Structure Nonlinear Algorithm. Figure 1 shows the series cascade scheme with the introduction of the nonlinear fdter. The set-point that enters the secondary loop is denoted with rs while rs* refers to the set-point calculated by the primary controller. Its way of working is more clearly shown if discrete variables are used ( A t denotes the sampling time). The filter compares the sign of the output variation of the primary controller Ars*, as calculated by a control algorithm in velocity form (Ars* = k c p [ e p ( t )- ep(t - A t ) ] + k I e p ( t )for a PI algorithm), to the sign of the actual error of the secondary loop. The sign of their product is used to calculate the actual set-point change Ars = r&) - rs(t - At) as follows: if
> 0 then Ars = ArS*e-alesl esArs* < 0 then Ars = Ars*
esArs*
(1)
if (2) In order to explain the way the filter works in Figure 2, the two different situations are reported. Figure 2a shows a case of esArs* > 0: the primary loop calculates a change in the secondary loop set-point that would increase the actual error of the secondary loop; the filter reduces that increment by a factor that depends on the filter parameter and the amount of the secondary loop
08885885/92/2631-2694$03.00/0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 12,1992 2695 d
I
5
-
A
I
I
m
Figure 3. Nonlinear filter in parallel cascade block diagram.
A riIL’ ----rL---b A ps e,
The benefits obtained in terms of stability when model errors are present will be shown in the first case study by analyzing the robustness of the scheme to delay uncertainty. The use of the filter makes therefore the tuning of the primary controller not an issue, as even an overtuned controller can be stabilized by an appropriate choice of the parameter a. Different problems may arise if the filter is introduced into a parallel cascade control scheme (Figure 3). In this case the effectiveness of the structure depends on the value of the so-called “interactionmeasure” y introduced by Yu (1988). This parameter is defined as
, I
t
time
a>
t
[am/adlYs (3) [am/adlYP which, after some algebraic manipulation, corresponds to y =
PdS(0)PP(0) (4) PdP(0) PS(0) where P represents the transfer function according to Figure 3. The role of the parameter y in the parallel scheme can be usefully examined by analyzing the change of the setpoint of the secondary controller rs after a load disturbance d has been completely rejected. Reducing the block diagram of Figure 3, the following equation derives: y =
t
time
b> Figure 2. Effect of the nonlinear filter on the secondary set-point variation.
error. Figure 2b shows a case of eSArs* < 0 the change Ars* of the secondary loop set-point calculated by the primary loop leads to a reduction of the secondary loop error es or at least to an error es lower than the calculated set-point change Ars*; the filter does not damp the action of the primary loop. The stabilizing effect of the filter can be analyzed more clearly if the limiting case of high values of the filter parameter a is considered. In this case the damping effect of the filter is complete also for small values of the error es so that the primary loop in practice changes the secondary loop set-point only after the secondary loop action has been completed (es N 0). The action of the two controllers is no longer simultaneous: the primary loop is effective only when the secondary loop has completed ita duty. In order to understand the stabilizing effect of the nonlinear filter, one can moreover consider the primary controller as a common PI controller with a variable gain; such a parameter varies between zero and the nominal value it takea when the filter has no action. Within a linear framework a decrease in the controller gain for a stable process always acta in favor of stability, as a Nyquist diagram clearly shows. The effect of the variable gain can be described through an uncertainty region. The Nyquist diagram of the whole system lies therefore somewhere within a region that is delimited by the conventional structure diagram and extends toward a stabler area.
Then, the set-point change after a unit step disturbance (d(s) = l/s) has been rejected, assuming that both controllers Cs and Cp have integral action, is given by
It follows from eq 6 that 1. If y = 1: r, = 0; the set-point of the secondary controller remains unchanged. This is the case for which the use of the filter has been thought, and it is equivalent to the series cascade structure. 2. If y > 1: 0 < rs(O)/Pds(O) < 1; the change of the set-point of the secondary controller also when y becomes very large is bounded by the gain of the load transfer function PdS(0). In this case, to keep y constant a lower control action is required than to keep ys constant (see eq 3). 3. If 0 < y < 1: rs(o)/Pd,(o) < 0;the change of the set-point rs can assume large values when y deviates from 1. In this case, to keep y constant a larger control action is required than to keep ys constant. 4. If y < 0, the cascade control is not recommended, clearly shown by Yu (19881, because the effect of the disturbance on ys requires a control action in a direction
2696 Ind. Eng. Chem. Res., Vol. 31, No. 12, 1992
contrary to that required for y. The parameter y is then related to the work (rs(0))that the primary controller has to do in order to reject the load disturbance. As long as this work is small (i.e., y N l), the disturbance rejection is faced mainly by the secondary controller which, having a faster dynamics, can give better performances. In these circumstances the filter has the duty to reduce the interferences of the primary controller on the secondary loop. When the work that the primary controller has to do becomes significant (i.e,, y differs substantially from 1) the filter reduces progressively its effectiveness. In these latter conditions the performances do not improve much, even adapting the tuning of the secondary controller to the value of y (undertuning when y > 1 and overtuning when 0 < y < 1). Because the improvement of the performances obtained with a cascade scheme compared to the single loop increases when y is approaching 1(Yu, 198% an appropriate choice of the secondary measurement leading to values of y in a strict range around 1 makes the use of the filter effective even for parallel cascade schemes. With regard to this problem, the choice of the secondary measurement in a distillation compoeition control is d y z e d in the next section. Selection of the Secondary Measurement. As the most common process units, to which parallel cascade control is applied, are multistage types or distributed parameter systems, selection of the secondary measurement to be used in the scheme is an important problem. In this section, particular attention is paid to the selection of the tray temperature of a distillation column: the concepts explored here can also be extended to other similar cases. Let us consider a common distillation unit, subject to a feed composition disturbance, in which the overhead composition yp is controlled by cascade control of a tray temperature ys; let the reflux flow rate be the manipulated variable. T w o deviations of the temperature profile are considered the open-loop and the closed-loop deviations in presence of the disturbance. The tray temperature to be selected has to satisfy two conditions: 1. The open-loop deviation has to be significantly large. 2. The closed-loop deviation has to be as small as possible. The advantages of such a selection are the following: 1. Sensitivity to the disturbance is guaranteed by the first condition. 2. Better performance follows from the second condition, as the main disturbances are almost completely rejected by the temperature controller which has a more favorable dynamics. 3. Reliability in case of failure of the composition analyzer (which is a rather common occurrence) is assured by the second condition. In fact, the closed-loop deviation represents the change of the tray temperature, when the primary loop based on the analyzer measurement works; as a consequence, a small value assures the ability of the temperature controller to face the main perturbations by itself. As the open-loop and closed-loopsteady-state deviations of the secondary measurement are respectively (see eq 5)
it follows that
a)
I
A
d 10 V
i 5
a t 1
0.
0
n
--
-5
$)
1 1 1 1
1 1 1 1
I l l 1
1 1 1 1
d V
i a t 1 0
n -5
5
0.
10
15
20
25
30
tray
Figure 4. Temperature profile deviations for different composition disturbances. (a) Decreaee of the ratio C3/zC4;(b) increase of the ratio i-C4/n-C1. 1, open loop; 2, closed loop.
If both selection conditions are satisfied, the ratio ygL/ygLis very small and the value of the interaction
measure y is close to 1; therefore this choice makes effective the use of both the cascade and the NLFS. As an example, two typical feed composition disturbances for a LPG splitter, which separates propane (C3) from isobutane (i-C4) and n-butane (n-C4),are considered. a decrease of the ratio C3/CC4and an increase of the ratio i-C4/n-C4. The open-loop and closed-loop (composition) temperature profile deviations are shown in Figure 4. i the criterion of choice reported Trays 6 and 7, which M above for both disturbances, can be selected for the secondary measurement and show a value of y very close to 1. A value of the interaction measure close to 1and a strong sensitivity to the disturbance to be faced can be therefore considered the conditions to be fulfilled in order to have an effective use of the proposed filter. Tuning Rules. Once the secondary variable has been chosen and the transfer functions are available, simple tuning techniques are suggested in order to make the design of the regulator quick and straightforward. The secondary controller can be tuned according to classical techniques (Ziegler and Nichols, 1942);however, we suggest the criterion (which will be referred to as the AB criterion) described by Brambilla et al. (1990). According to this, if the process is represented by a simple first-order delay transfer function, the PI controller has the following parameters: T
YSL/YflL = (1 -
I I I I
1 1 1 1
+ 8/2
kp = kp8(C + 1)
e=
+ 8/2
(9)
where c is a given function of O / T and of the required
Ind. Eng. Chem. Rea., Vol. 31, No. 12, 1992 2697 robustness; these parameters will be referred to as AB parameters below. If the nonlinear filter is introduced into a parallel cascade scheme with y # 1,the following modifications are proposed:
1
0.5
Y 0.
which correspond to an undertuning if y > 1 and to an overtuning if 0 < y < 1. As previously stated, when the value of y gets far from 1,the performances deteriorate. The primary controller is designed according to the AB criterion for the global primary process (PSpp in series cascade and Ppin parallel cascade with reference to Figures 1 and 3) and overtuned by increasing the calculated integra~time (71 = .fB/2). The only parameter which has to be tuned on-line or through simulations is at this stage the filter constant a: a good value, in any case, can be usually found after only a few runs. In the usual case of dimensionless transfer functions with gains of the same order of magnitude, an acceptable rule of thumb is to select a in the range 2-10. This choice works well in most cases; however, if the time responses still remain oscillating, a higher damping constant can be selected.
-.5
-1
to point out the main properties of the nonlinear filter cascade control structure. The fvst case study shows the improvements in stability and performance obtained with the introduction of the nonlinear filter into a series cascade scheme. The second case study is meant to show the extension of the previous properties to a parallel cascade scheme (Figure 3). Case Study 1 (SeriesCascade). Let us refer to Figure 1 with the following transfer functions: ,-6s
,-lo8
The conventional cascade control structure is designed by tuning Cs for the process Ps (AB criterion) and Cp for the process Pp[PsCs/(l + PsCs)] according to Ziegler and Nichols (1942). Then the controllers have the following parameters: kcs = 0.93, 71s = 10.5, kCp= 0.57,and 7Ip = 28. The NLFS, designed according to the tuning rules proposed in the previous section, has the following parameters: k a = 0.93,7~= 10.5, kcp = 0.68,7p = 9.2, and a = 5. First the robustness of the two schemes is compared. Uncertaintieson a single process parameter are considered: the ratio of real to nominal value is assumed to stay in the range [l/A,A] and the value of A, for which marginal stability is reached, is found for each parameter. This analysis for the conventional control scheme by means of Nyquist diagrams indicates that uncertainties on the time delay 8%of Ps are the most critical for stability and that if Os = 2eS (A = 2), the system has a marginally stable behavior. Performing the same analysis for the NLFS by simulation, conditions of marginally stable response are reached only when A = 3.6. In Figure 5a and Figure 5b the responses to a step change of the disturbance d are reported respectively for the conventional control scheme and for the NLFS, with the uncertainty on the time delay Os as parameter. Similar results are obtained if uncertainties on a different process parameter are considered.
1
1
1
1
1
1
1
1
25
0.
\ )1 I I
l
I
50
,
,
I
I
I
75
I
100
e irm 1
0.5
Y 0.
-.5
Case Studies
T w o case studies are taken into examination in order
~
-1
1
I
0.
I
I
I
I
I
25
I
I
I
/I I '\ I I
50
I
1
75
1
I
I
I
100
t zrm
Figure 5. Case study 1: responses to a step disturbance for different values of 0s (Psdelay). (a) Conventional cascade; (b) NLFS. 1,Os = 5; 2, os = io; 3, es = 18.
A direct comparison of the responses for the nominal case and for a 20% uncertainty on time delays 8s and Op (Figure 6a and Figure 6b, respectively) shows the improvements in performance which are obtained with the NLFS in both cases. In Figure 6 the responses of the control scheme without cascade are added for comparison (controller parameters kc = 0.68, 71 = 18.5). In order to show that the proposed scheme gives good performances also for changes of the primary set-point, in Figure 6c the responses to a step change of set-point r are reported Case Study 2 (Parallel Cascade). Let us refer to Figure 3 with the following transfer functions:
Ps =
e-% 105 + 1
Pp
=
e-& 11s + 1
(12)
The secondary measurement has been selected in order to make y cloee to 1 (y = 1.15). The primary and secondary transfer functions differ mainly for the time delay as typically happens for composition control of a distillation tower product cascaded to a tray temperature controller. Three control schemes are considered: single primary loop control, conventional cascade control, and nonlinear filter cascade control. The conventional cascade control is tuned following the recommendation of Yu (1988), who suggesta the use of IMC-like tuning methods for both controllers. The AB criterion is used here, and the obtained parameters are the following: kcs = 1, 71s = 11.5, kcp = 0.84,and 71p = 15. The same primary controller is
2698 Ind. Eng. Chem. Res,, Vol. 31, No. 12,1992 1
0.8
A
0.4
Y
0
-.I 0.
25
50
75
0.
100
25
50
1
0.8
100
b)
-!
\
0.5
Y
4
0.
-.5
-.4
0.
25
50
75
100
0.
t imr
1.2
75
t imr
t inw
50
75
100
t imr
Figure 7. Case study 2 responses to a step disturbance. (a) y =
C)
4
25
1.15 1,single loop; 2, NLFS; 3, conventional cascade (Yu’s tuning). 0.6 1, single loop; 2, NLFS a = 5; 3, NLFS a = 10.
4
(b) y
return to steady state. However, the performance is still better than the one of the single primary loop. Y 0.6 0.4
0.2
0. 0.
100
50
150
t inw
Figure 6. Case study 1: dynamic behavior of different systems. (a) Responses to a step disturbance for the nominal case; (b) responses to a step disturbance for the worst case (A88 = 1, A$ = -2) of uncertainty;(c) reaponsea to a step set-point variation. 1, single loop; 2, NLFS;3, conventional cascade.
used in the single loop structure. The NLFS has instead the following parameters: k a = 0.87, ~p = 13.2, kCP= 0.84, TIP = 7.5, and a = 3. Figure 7a shows the responses to a step disturbance for the three schemes and indicates that the NLFS gives the most desirable response. It is interesting to note how the performances of the NLFS deteriorate if a wrong selection of the secondary measurement is made. A different value of the gain for the secondary load transfer function (Pa(0) = 0.6) is considered; the interaction measure y changes in the same way (y = 0.6) and the controller parameters are modified according to the proposed tuning rules. The NLFS has, in this case,an undesirable return to steady state. Figure 7b shows the responses for different values of the damping constant: the response is oscillating for a = 5 while an increase of the iilter action (a= 10) corresponds to a slower
Conclusions The use of a nonlinear fdter between the two controllers of a cascade scheme permita a reduction of the tuning difficulties and an increase in robustness to uncertainties on the system dynamics. The filter is meant to detune the action of the primary controller as long as it tends to amplify the error of the secondary controller. Cascade control of systems with comparable primary and secondary dynamics, typical of composition controllers cascaded to tray temperature controllers in distillation, is therefore made effective, as the actions of the controllers are decoupled and their interactions reduced. The analyzed examples have shown the improvements in robustness and performance in series cascade and also in parallel cascade, if an appropriate Selection of the secondary measurement is accomplished. Nomenclature Symbols
= robustness constant C = controller d = disturbance e = error F = filter k = gain m = manipulated variable P = process r = set-point Ar = set-point variation s = Laplace transformation variable c
Ind. Eng. Chem. Res. 1992,31,2699-2707
2699
y = controlled variable
Literature Cited
Greek Symbols = exponential constant y = interaction parameter X = uncertainty parameter 9 = delay 'e = nominal delay 7 = lag
Brambilla, A.; Chen, S.; Scali, C. Robust Tuning of Conventional Controllers. Hydrocarbon Process. 1990,69 (2), 53-68. Kriahnaewamy, P.R.;Fhngaiah, G. P.; Radha, K.J.; Deaphande, P. B. When to Uae Caecade Control. Z n d . Eng. Chem.Re8. ISSO,29, 2163-2166. Luyben, W.L. Parallel Cascade Control. Znd. Eng. Chem.Fundam. 1973,12 (4), 463-467. Morari, M.; Zafiiiou, E. Issues in SISO IMC Design. In Robuet Process Control; Prentice-Hak Englewood Cliffs, NJ, 1989; Chapter 6. Scali, C.; Brambilla, A. Analysis of Cascade Control Schemes for Chemical Processes. Chim. Znd. 1990,Q4620(in Italian). Stephanopuloe, G. Control System with Multiple Loops. In Chemical Process Control;Prentice-Hak Englewood Cliffs, NJ, 19&1; Chapter 20. Yu, C. C. Design of Parallel Cascade Control for Disturbance Rejection. AZChE J. 1988,34, 1833-1838. Ziegler, J. G.;Nichols, N. B. Optimum Settings for Automatic Controllers. Tram. ASME 1942,64,759-766.
a
Subscripts
C = controller d = disturbance I = integral P = primary 9 = secondary Superscripts
AB = AB criterion * = unfiltered value CL = closed loop OL = open loop
Received for review March 30, 1992 Accepted July 30,1992
Simulation and Optimization of a Fixed Bed Reactor Operating in Coking-Regeneration Cycles Daniel 0. Borio,+M. MenBndez, and J. Santamaria* Department of Chemical and Environmental Engineering, University of Zaragoza, 50009 Zaragoza, Spain
The optimization of the operation of a fixed bed catalytic reactor operating in deactivation-regeneration cycles for the dehydrogenation of butene into butadiene has been carried out. Unlike previous optimizations, in this work the influence of the regeneration stage has been taken into account by means of a detailed model which predicts the transient temperature profiles developed in the process. Also, maximum temperature restrictions during the regeneration stage were considered in the optimization scheme. The optimum values of the feed composition, duration of the production stage, and residual coke level a t the end of a coking-regeneration cycle have been obtained. The results of the optimization show that the different regeneration parameters (such as the maximum temperature allowable, the oxygen concentration, and the extent of the regeneration), strongly influence the global process optimum.
Introduction Catalyst deactivation by coke affects a large number of heterogeneous catalytic processes and poses important problems in industrial operation. Generally speaking, the activity of a catalyst will decrease with increasing coke deposition. Thus, in order to restore at least partially the catalytic activity, coke must be removed from deactivated catalysts after a certain time on stream, which varies for each process, and that is determined by economic considerations. There are several alternatives for coke removal, including the gasification of the coke deposita with mixtures containing steam, hydrogen, carbon dioxide, or oxygen. The last method, termed oxidative regeneration, involves the combustion of the coke deposita by mixtures of oxygen and a diluent. The implications of catalyst coking and of the subsequent regeneration on the behavior of single particles and of catalytic reactors have attracted a great deal of attention, as evidenced in recent years by reviews covering these subjects (e.g., Doraiswamy and Sharma, 1 9 W , Butt, 1984; Hughes, 1984;Lee, 1985; Butt and Petersen, 1988; Froment and Bischoff, 1990). 'Preaent address: Planta Piloto de Ingenieria Quimica, Bahia Blanca, Argentina.
Catalyst regeneration can be carried out either continuously or periodically, this being mainly determined by the active life of the catalyst. Thus, when the catalyst deactivates very rapidly (e.g., in the fluid catalytic cracking process), continuous regeneration is required, and moving bed reactors or fluidized bed reactors with continuous catalyst replacement are usually employed for the process. On the other hand, when catalysta with a sufficiently long active life are employed, periodic regeneration is an alternative to consider. The catalyst can then be unloaded and regenerated in a separate installation or regenerated in situ. This latter case is the subject of this work, which is concerned with operation cycles consisting of a production-deactivation period followed by a regeneration stage in the same reactor. The optimization of a catalytic reactor subjected to cyclic operation requires sufficiently detailed simulation models for both the deactivation and the regeneration stages. Concerning the fmt stage, a review of the works published on simulation of catalytic systems undergoing deactivation by coke is well beyond the scope of this introduction. As an example, the subject has been extensively addressed by Froment in several reviews (e.g., Froment, 1976, 1981, 1984,1991) concerning both kinetic modeling and reador operation aspects.
oaaa-5aa5/ 9212631-2699$03.00/0 0 1992 American Chemical Society
