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Analysis of a predictor—regulator system
Chemical Engine&g
Science, 1576, Vol. 31, pp. 1-S.
Pagmon
Press.
Printed in Great Britain
ANALYSIS OF .A PREDICTOR-REGULATOR
SYSTEM
C. CHANDRA PRASAD and P. R. KRISHNASWAMY Department of Chemical Engineering, Indian Instituteof Technology,Madras,India (Received
21April 1975;accepted 16May 1!275)
Abstract-A high-performance control system called “predictor-regulator” is proposed, its governing laws are derived, and it is applied to simulated processes to evaluate its characteristics, and to real processes to demonstrate its feasibility. The proposed system is made up of a two-loop configuration[4]: a fast auxiliary loop consisting of a controller and a simple lag element, and a main loop provided with another controller operating on the process. Both the controllers are conventional PID type commercial controllers. Theoretical study reveals that, by suitable choice of the compensator parameters, the control arrangement could result in remarkably superior performance. While, for the key loads, both steady state and transient errors become practically zero with the aid of only two P-controllers, inclusion of integral action aids in achieving control against secondary (unmeasured) disturbances also. The design and performance of the regulator system are evaluated via simulation. Several practical aspects of the control problem are examined, and criteria for choosing the auxiliary lag, the modes of control, and the controller settings are established. Applicability of the technique to a wide range of processes (e.g. complex higher order and dead time processes) is also discussed. The results obtained coniirm that the control approach could match combined feedforward-feedback regulation in its performance, and classical feedback in design simplicity and ease of implementation.
The control scheme suggested here includes a “fast feedback” loop, which is similar in principle to the one employed by Roots arid Giinenc [ l] for on-off control of a simple electroheat process, and subsequently extended by Krishnaswamy and Chandraprasad[2,3] to account for transient response design considerations as well as higher order process dynamics. Their arrangement [ l-31 essentially consists &--a controller and a lag element in the feedback loop with &process itself left in the open loop in such a way that for measured loads, it permits a considerably faster (and hence superior) control action compared to conventional control. However, unmeasured disturbances render this control arrangement useless as the process is located outside the control loop. The present work, as distinct from the previous ones, provides, in addition to the fast control action mentioned above, a second slow-responding feedback around the process such that the composite dual loop system (involving PID type control) becomes a practical technique, matching tombined feedforward-feedback in its performance and classical feedback in design simplicity and hardware realization.
INTRODUCTION Whenever standard feedback control proves inadequate, close regulation of chemical process variables is normally achieved by a multiloop control arrangement, such as, cascade control or feedforward control coupled with feedback. Though the master-slave cascade system represents a remarkable improvement over ordinary feedback, it cannot achieve perfect regulation because, like feedback, it ignores the load and bases its correction on an error occurring either at the outlet or at an intermediate point in the process. However, it is the load that wields powerful influence on regulation of a loop and hence the load must be measured and used for manipulation if absolute control is to be achieved. This is the basic concept of feedforward strategy. In order to make the feedforward correction fully effective, the process has to be modelled in toto to include both the steady state and the dynamic components-an obvious impossibility except for a few simple cases. Further, the feedforward scheme often needs special control hardware for its implementation and these may not be readily available. Therefore, although in principle feedforward arrangement achieves perfect control, in actual practice this is never fully realised owing to realization and implementation difficulties. The objective of this study is to develop an alternate high performance control called scheme, “predictor-regulator control”, which virtually retains all the benefits of feedforward control and at the same time is free from some of its limitations. Development of the regulator proposed herein involves (i) devising load compensators, suitable for chemical process control, on the basis of theoretical analysis and simulation study, (ii) evolving criteria for choosing appropriate values for the adjustable compensator parameters, and (iii) applying the control scheme to engineering processes to establish its feasibility.
THEORY
Structure
I-A
system
A block diagram embodying the principle of the
proposed control approach is given in Fig. 1. It may be seen from the figure that the technique employs a two loop configuration: (i) a main loop, or process loop, made up of the usual elements, namely, controller G,, manipulator G,, process G,, and measurement H,,and (ii) an auxiliary loop composed of a second controller G,., and a dynamic element G.. The auxiliary loop output, C., acts as common feedback to both the control loops. The main controller operates on the process through the manipulator, G,, on the basis of 1
CESVol.31,No.
of the control
C. C. PRASADand P. R. KRISHNASWAMY
2
CP
Hm
Fig. 1.
Blockdiagramfor predictor-regulatorcontrol.
the difference between the feedback, C,, and set point, R (held constant for regulator operation). The process output, C,, adjusts the set point of the auxiliary controller thereby placing the process in closed loop. The element, G,,, can be so chosen[4] as to make the dynamics of the auxiliary loop considerably faster than that of the process loop. The basic disturbance, U,, entering the process creates an equivalent disturbance, U., through load measurement, H., and this serves as an imput to the auxiliary loop as indicated by the broken line in Fig. 1. It is the essence of this control arrangement that, by use of the fast auxiliary loop, the control effort required by the process controller to counteract major load changes can be predicted even before an error appears in the controlled variable. Thus, the auxiliary loop as a whole will be referred to as the predictor loop. The feedback originating from the process output takes care of any unmeasurable or unpredictable upsets that may enter the system. It is important to realize that all the control elements in Fig. 1 including G. are physically realizable and may be made up of standard electronic or pneumatic control hardware. To aid in visualizing the effects of the auxiliary loop on the regulatory action of the control system, the block diagram of Fig. 1 is reduced to an alternate form as shown in Fig. 2. The figure reveals that the suggested control technique has a striking similarity to the standard feedforward-feedback arrangement. However, there are also some important differences between the two of which the following are of special interest. Unlike in the case of the feedforward-feedback control, (i) both measured and unmeasured disturbances entering the process are penalised here by a single controller, G,, operating on G,
(ii) with proportional control action alone, the load compensator generates the behaviour of a high pass filter and also eliminates steady state error (iii) derivative action, if needed for compensation, can be accomodated in the process controller itself. The first point is evident from Fig. 2 and the remaining ones will be discussed in subsequent sections. It may be seen from the figure that the. behaviour of the main feedback loop is altered with respect to the classical feedback by the presence of the block G,.G,/l + G,.G. in the feedback path. It should be an aim of the control system design to choose appropriate values for the parameters of G., G,, and G,, so that this additional lag in the feedback path becomes negligible or is adequately compensated. Derivation of control laws
Referring to Fig. 1, and using the standard block diagram algebra, the relationship between process output and load disturbance is derived as:
c, = Gp[l + Ga(Gc. Up
GvG& )I 1+ G,G,. (1 + G,GvGc,Hm 1’
(1)
The numerator of eqn (1) can be made zero if G
=
”
l+G.G,, G.G,Hu ’
In physical terms this means that, if the process controller is designed according to eqn (2), then perfect control will result, that is, steady state as well as transient errors will be zero at all times regardless of load disturbances. It is of interest to note that this equation does not contain the process transfer function, Gp’ Equation (2) involves an yet undefined element G,,. Based on certain physical and operational considerations, it has been shown[4] that this element assumes the form
G.=A.
TJ + 1
and satisfies the conditions T. # 0 and T. 4 T,,, where T, is a measure of the dominant process dynamics. In order to make this discussion more specific, let us assume that (i) the auxiliary controller is a standard commercially available PID controller. If the interaction between the modes is negligible, then (4)
(ii) the measurement and manipulation dynamics are negligible compared to the dominant process dynamics, which is the case with many chemical process systems. Then, H,,,=A, H,=A.
Fig. 2. Alternateformof Fig. 1.
G,=A,
(5)
Analysisof a predictor-regulatorsystem where the A’s represent the static gains of the elements involved. Substituting the last three equations in the controller expression (2), we get l+
G
cp T, +(l+A.K,)’
L&K. 1, (l+A.K,)Ti.s
A.,Ka (l+A,K.)
1’
G,=K,
1
w
(7)
and
(11)
Equations (8)-(11) reveal that the process and auxiliary controllers are related through the steady state gains of various elements in the control loop. Further, eqn (11) indicates that the process controller contains additional derivative action, T&,to compensate for the auxiliary lag, T,. If the auxiliary controller is provided with only PI action, i.e. for 1
(
>
the corresponding process controller expression, obtained as above, is Gcp=Kp
(
1 l+=+T&s.
>
(12)
Similarly for G,. = K,, we have G, = K,,(l t T;s).
>
K,.
(16)
1+ A,K. = A,A,A.K,
(17)
K.TI, = A,A.K,Ti,
(18)
K,Td. = A,AA.K,Td,.
(19)
The above equalities can be easily satisfied in any given physical situation.
(10)
Gc.=K.!+=
1
(6)
1 t A,K,, Kp = A,A,A.
T
l+%
Under this condition, eqns (8)-(10) become the governing laws for the proposed control system. These equations may be restated for convenience as
TdaS
where
T:,=1+A.K..
i
Gc, =
This equation is of the standard form l+&t(T,+Tb)s
3
(13)
The additional derivative term, T&,in the expressions for G,, viz. eqns (7), (12) and (13) need be satisfied only when the process is characterised by a fast dynamics. For sluggish processes, T,, (and hence T@ can in reality be made negligible as required in eqn (3), with the result the three expressions for Gep (i.e. eqns (7), (12) and (13)) simplify to (14)
Choice of controller modes Since the foregoing analysis reveals that all the standard combinations of control modes, namely, P, PI and PID, satisfy the conditions governing perfect regulator, the question naturally arises as to whether the simplest arrangement, i.e. proportional mode for both the controllers, would suffice to meet the control objectives. Equation (l), with G,. = K. and G, = K, reduces to
c, _ G,,[l t G,(Ka -A&K,)1 U,-
1 t G,K,,(l+ GpA&Am)’
(20)
The final value theorem and the controller relationship (17), when applied to the above equation, result in zero offset demonstrating that the steady state error can be eliminated with only P control action. Another distinct feature of this control structure is that it generates a lead term with only proportional action. This has been illustrated elsewhere[4] by reducing the numerator term inside the square bracket of eqn (20) to the form T,s /( T.s t 1). As T. is the only dynamic term inside the square bracket under consideration, the process transient to be tolerated on account of this control arrangement is limited by the relatively small magnitude of T,. To sum up, for the primary disturbance, U,, for which the system is designed, mere proportional action is sufficient to produce excellent closed loop response with negligible transient deviation and zero offset. However, in actual practice, there exists a real threat of unmeasurable disturbances (besides the key load, U,) entering the process. With respect to such disturbances, the system behaves as a single feedback loop (see Fig. 2), and consequently use of proportional control action will result in offset. This can be seen by applying the final value theorem to the appropriate closed loop transfer function relating unmeasured process load to the controlled variable. Since the prime purpose of feedback from process output is to eliminate offset even when load misses the predictor, proportional mode will not be sufficient; reset is obviously necessary. Although addition of reset will have a deteriorating effect on the transient response caused by the main load, U,, it will, on the other hand, provide adequate control against unmeasurable disturbances. For
4
C. C. PRASAD and P. R.
this later case, the system behaviour is that of a normal single loop augmented by an additional lag, G,G,,/(l + G.G,.), due to the auxiliary loop elements. In order to approach normal feedback performance, either this lag can be made negligible by keeping G, small and G,, large, or an appropriate lead compensation, T&,be added to the process controller as suggested earlier. Considering hardware simplicity, wherever possible, a small auxiliary lag that satisfies eqn (3) and two PI controllers tuned to obey eqns (17) and (18) may be employed. It should be realised that dissimilar controllers (say, for example, P controller for the auxiliary loop and PI for the process loop) cannot be recommended because they do not obey the specified control laws (eqn (18) in this case), and consequently do not meet the control objectives. The question of using the three mode controllers primarily depends on the control efficiency required against secondary (unmeasured) disturbances. Since for such disturbances the system acts as a single feedback loop, whatever criterion is employed for adding rate action to ordinary feedback may be applied here also. This and other aspects of the control problem are further explained in the simulation study that follows. SIMULATION mY Extensive simulation study was carried out to evaluate the performance of the predictor-regulator system and to observe the effects of various system parameters on control quality. First order, higher order as well as pure delay processes were controlled using the proposed configuration. All the simulations (except the ones involving pure delay) were carried out on a Pneumatic Process Control Simulator made up of control hardware and components that are actually found in industry. In other words, no attempt was made to idealise simulation, and the simulator was allowed to contain certain nonidealities found in real control systems, such as, interaction between control modes, restricted accuracy, and a certain amount of deviation from linearity. This permits the simulation study to be more compatible with reality.
KRISHNASWAMY
Several aspects of the control problem were analysed; these include the effect of auxiliary time constant on closed loop response, the type of control modes (namely P, PI or PID) needed for the twin controllers, optimum controller settings, applicability of the technique to complex higher order and dead time processes, and comparison with established methods of control. Certain practical problems which might arise during implementation, viz. load missing the predictor loop partly or completely, and unexpected disturbances entering the process were also examined.
Effects of auxiliary lag on closed loop response For the simulation study, a fourth order system was taken as a typical example of the process to be controlled by the proposed technique. The valve and measurements time constants were neglected and, for convenience, the static gains involved were all assumed to be unity. The numerical values used for the process and the auxiliary time constants (expressed in minutes throughout this study) and for the controller settings are included in the block diagram in Fig. 3. The figure also gives the results obtained when proportional-proportional (P-P) controller configuration was employed to control the process. The response curves for a step change in V,,,corresponding to several different values of the auxiliary lag (curves 1, 2 and 3), show no deviation at all from the preload conditions. For the chosen controller gains of 2.5 and 3.5, this represents a dramatic improvement in control system response to load upsets and signifies perfect control. Besides, the results clearly prove that the derivative term, TL, in the process controller (see eqn (13)) can be neglected for relatively small values of the auxiliary lag. The figure also provides a comparison with optimum PI feedback control (curve 4) set for minimum ITAE (integral of absolute error multiplied by time). This single loop control result is very poor by comparison with predictive control performance. It is of interest to note that no process modelling was required here for the successful application of the proposed control scheme.
Fig. 3. Effectof auxiliarylagon systemresponse.
Analysis of a predictor-regulator system
Time,Mins
5
-
Fig. 4. P-P control system-Effect of controllergains.
Controllersettingsfor the P-P control system For a given process and auxiliary lag, the closed loop response depends on the settings used for the controllers, G, and G,,. The controller tuning relationship, eqn (17), for the system under consideration with unity static gains, reduces to
Responses for three sets of proportional gains satisfying the above equality are shown in Fig. 4 for a fixed T, of 0.5 min. As predicted by the theory, in all the cases the final offset is zero. In addition, the use of larger controller gain practically eliminates all the transient errors as evident from the response curve coinciding with the x-axis for I&, = 3.5. The adverse effect of decreasing the controller gains is also evident from Fig. 4 where curves 2 and 3 show how the system dynamics progressively deteriorate, although only to a small extent, with lowering of the gain settings. This suggests that for best results higher gains should be employed. In order to aid in visualising the dependence of the system response on controller gains, a recording of the outputs of the two controllers and the auxiliary loop are shown in Fig. 5 on an expanded time scale for a specific case. The equation governing the auxiliary output, C.(t), for a step input of magnitude, U,, is
which at steady state (assuming an auxiliary element gain of unity) reduces to
ca=u”
1tK;
The corresponding steady state process controller output is xKp = - U,.
Fig. 5. Controller and auxiliary loop outputs for the P-P control system.
Thus, exact corrective action is eventually taken by the process controller prior to which, however, there exists a transient period (as represented by the shaded area in the figure). The duration of this transient, as may be seen from the equation for C,,(t), varies inversely with K., and hence by increasing the controller gains, the process output transients may be made negligible.
of load measurement on process response For proper control to be achieved, the load should be appropriately measured and transmitted to the auxiliary loop. Figure 6 compares the responses obtained (curves 1 and 2) when the entire load, VP,and when only part (70%) of the load were measured and transmitted to the predictor at U.. The 30% reduction in the load signal results in lowering of the predictor gain so as to satisfy eqn (17). Consequently the transient deviation increases as may be seen from curve 2, and although the resulting error is small ( < 2%), it nonetheless indicates the need for Efects
C. C. PRASAD and P. R.
i
35
Tune,Mins. Fig.
6. Effect of load measurement on process response.
proper sensing of the load. If the load is altogether missed by the auxiliary loop (this may happen if there is a failure in measurement or if unmeasurable loads ehter the process), then the present arrangement results in offset (curve 3) as predicted by the theory. For this reason, incorporation of reset action into the proposed control scheme becomes desirable. Control using twin PI controllers
Figure 7 shows the same 4th order process being controlled by two PI controllers. The controllers are tuned according to a modified version of the open loop ITAE criterion used in ordinary feedback control. Details on the tuning procedure adopted may be found in [5]. Step response curves obtained for different values of the auxiliary lag, G,, are labelled as curves 1,2 and 3 in the figure. The figure shows that by keeping G. sufficiently small, it is possible to achieve perfect regulation (see curve 1) with PI-PI control action. However, a comparison with Fig. 3, which describes P-P control results under identical conditions, indicates that the addition of integral mode results in slight deterioration in the dynamics at higher values of G,. Curve 4 in the present figure,
KRISHNASWAMK
representing feedback control, is the same as curve 4 in Fig. 3. Since P-P control arrangement fails to eliminate offset when unmeasurable disturbances enter the process, and since the prime purpose of feedback from process output is to eliminate offset for all loads (measured and unmeasured), proportional mode is not quite sufficient; reset is obviously necessary. Figure 8 illustrates the behaviour of the PI-PI control system previously considered when a step disturbance enters only the process without entering the predictor. A comparison with single loop PI control response tuned for minimum ITAE shows that the present system is nearly as good as ordinary feedback system with respect to unmeasurable upsets. This will be so as long as care is taken to ensure that the predictor loop dynamics is small compared to the process dynamics. Thus, for measured loads, PI-PI control achieves perfect regulation (Fig.7), and for unmeasured loads, it compares with the performance of ordinary feedback (Fig. 8). The outputs of the two PI controllers and that of the predictor loop are shown in Fig. 9 under conditions of simulation represented by the accompanying block diagram. It may be seen from the figure that, as expected, the process controller exactly cancels the effect of the disturbance after a transient period. During this period (resulting from a unit step change in U,), the manipulative signal, i.e. the process controller output, M, assumes the form
M=-
(
The nature of the equation shows that, depending on the numerical values used for the parameters of the auxiliary loop elements, the resulting manipulation could exhibit an oscillatory behaviour as in Fig. 9. In that case, the process output may also oscillate before reaching steady state (e.g. curve 2 in Fig. 7).
Fig. 7. Control using twin PI controllers.
Analysisof a predictor-regulatorsystem
Fig.8. PI-PI controlperformancewhenloadentersonly the process.
1.0
‘Dirt”rb.ncc
0.80.6O.L0.2
0 -0.2 -
*ux I
5
Imp
cJ”tp”l
I
I
I
10
15
20
Time,Mins
--_)
Fig. 9. Controller and auxiliary loop outputs for the PI-PI control system.
Control using twin PID controllers
Figure 10 records the recovery curves obtained when rate mode was added to the control system corresponding to curve 2 of Fig. 7. The controller tuning procedure mentioned earlier[5] was adopted here also. If the disturbance enters both the process as well as the predictor loops, then inclusion of derivative action does not result in any significant additional benefit as may be seen by examining curve 1 in Fig. 10 and curve 2 in Fig. 7. For secondary loads, i.e. for loads entering only the process, the predictor-regulator approaches the corresponding optimum single loop PID control in its performance as evident from curves 2 and 3 in Fig. 10. Thus, for unmeasured loads, the derivative action gives the predictor-regulator the type of benefit that it gives to standard single loop control, and this may be appreciated by references to Figs. 8 and 10. The need for two PID controllers is, therefore, determined mainly by the type of regulation required for secondary upsets, as no real benefit accrues from their use for measured loads if the predictor is sufficiently fast-acting. Thus it appears from the foregoing analysis that the
method developed here merits consideration for implementation whenever rigorous control is desired. APPLICATIONS
The proposed dual loop control policy was applied to two real processes in order to demonstrate its easy implementability and to confirm its superiority. The processes involved were (i) a pure delay process [4] with a dead time of 2.7 min, and (ii) a stirred tank described by second order-dead time transfer function and with provision for pH control [5]. These processes were chosen for application purposes on the basis of the fact that they represent particularly difficult control situations, with pure delay endangering stability and pH control involving parametric forcing and nonlinearity. Therefore, if the present technique could control these processes adequately, then it may be well assumed that the control would work for other types of processes as well. The control hardware used [4,5] essentially consisted of two commercially available pneumatic controllers, a pneumatic RC circuit, a control valve, and standard measuring and recording instruments. Excellent results
C. C. PRASAD and P. R. KRISHNASWAMY
Fig. 10. Control usingtwin PID controllers.
were obtained in the case of dead time control with the proposed system showing 14 fold improvement over other well known special control arrangements (viz. Smith’s linear predictor and Buckley’s feedforward control). Equally good results were also achieved in the case of pH control of a stirred tank where the use of the predictor system kept down the maximum deviation (between the process output and the set point) to O-03pH, which would have otherwise registered a deviation of 2.0 pH. Further details on applications to pure delay control and pH control may be found in [4] and [5] respectively.
it results in predictable success for a wide range of processes. The results also demonstrate that the technique achieves effective regulation even in the face of process parametric variations. (viii) Whenever load measurements are possible, the proposed technique is potentially capable of replacing other multiloop control arrangements such as combined feedback-feedforward. NOTATION A
steady state gain
C output variable CONCLUSIONS
A dual loop predictor-regulator concept was developed to obtain superior quality of control capable of meeting stringent tolerance on the controlled variable. Based on the theoretical, simulation, and experimental studies carried out to describe the characteristics and to test the applicability of the predictive system, the following conclusions could be drawn: (i) The theory and the simulation study demonstrate that the proposed predictor-regulator is capable of achieving perfect regulation. (ii) For achieving this, simple proportional action would s&ice. Reset mode is required only to account for secondary (unmeasured) disturbances. Rate action may be used for compensation in the case where the auxiliary lag is significant compared to the dominant process lag. (iii) This single control configuration affords both dynamic as well as steady state compensation for a variety of processes. (iv) For unmeasured loads, the performance does not generally differ from that of optimum single loop corrective action. (v) Translation of the regulator scheme from theory to practice is straightforward. Ordinary commercial controllers are adequate, controller tuning is simple, and process modelling may be avoided. (vi) Unlike feedforward regulation, this control approach does not demand special control hardware. (vii) The experimental evidence that resulted from applications to pure delay and CFSTR systems confirms that the control scheme is readily implementable, and that
G
H K M R s t T
Th Td x u
transfer function transfer function (measurement only) controller gain process controller output set point Laplace operator time time constant see eqn (11) derivative time integral time disturbance
Subscripts a auxiliary loop
c
controller measurement process u load v valve
m p
REFERENCES [l] Roots W. K. and Gijnenc G., IEEE Trans. Ind. Gen. Appl. 1970 IGA-6 52. [2] Krishnaswamy P. R. and Chandra Prasad C., IEEE Trans. Ind. Appl. in press. [3] Char&a Prasad C. and Krishnaswamy P. R., An improved On-Off Control Strategy, 4th Int. Gong. of Chem. Engng, Equipment Design and Automation, Prague, September 1972. [4] Chandra Prasad C. and Krishnaswamy P. R., Chem. Engng Sci. 197530 207. [5] Chandra Prasad C. and Krishnaswamy P. R., Instr.Tech. 1975 22 (7) 53.
Science, 1576, Vol. 31, pp. 1-S.
Pagmon
Press.
Printed in Great Britain
ANALYSIS OF .A PREDICTOR-REGULATOR
SYSTEM
C. CHANDRA PRASAD and P. R. KRISHNASWAMY Department of Chemical Engineering, Indian Instituteof Technology,Madras,India (Received
21April 1975;accepted 16May 1!275)
Abstract-A high-performance control system called “predictor-regulator” is proposed, its governing laws are derived, and it is applied to simulated processes to evaluate its characteristics, and to real processes to demonstrate its feasibility. The proposed system is made up of a two-loop configuration[4]: a fast auxiliary loop consisting of a controller and a simple lag element, and a main loop provided with another controller operating on the process. Both the controllers are conventional PID type commercial controllers. Theoretical study reveals that, by suitable choice of the compensator parameters, the control arrangement could result in remarkably superior performance. While, for the key loads, both steady state and transient errors become practically zero with the aid of only two P-controllers, inclusion of integral action aids in achieving control against secondary (unmeasured) disturbances also. The design and performance of the regulator system are evaluated via simulation. Several practical aspects of the control problem are examined, and criteria for choosing the auxiliary lag, the modes of control, and the controller settings are established. Applicability of the technique to a wide range of processes (e.g. complex higher order and dead time processes) is also discussed. The results obtained coniirm that the control approach could match combined feedforward-feedback regulation in its performance, and classical feedback in design simplicity and ease of implementation.
The control scheme suggested here includes a “fast feedback” loop, which is similar in principle to the one employed by Roots arid Giinenc [ l] for on-off control of a simple electroheat process, and subsequently extended by Krishnaswamy and Chandraprasad[2,3] to account for transient response design considerations as well as higher order process dynamics. Their arrangement [ l-31 essentially consists &--a controller and a lag element in the feedback loop with &process itself left in the open loop in such a way that for measured loads, it permits a considerably faster (and hence superior) control action compared to conventional control. However, unmeasured disturbances render this control arrangement useless as the process is located outside the control loop. The present work, as distinct from the previous ones, provides, in addition to the fast control action mentioned above, a second slow-responding feedback around the process such that the composite dual loop system (involving PID type control) becomes a practical technique, matching tombined feedforward-feedback in its performance and classical feedback in design simplicity and hardware realization.
INTRODUCTION Whenever standard feedback control proves inadequate, close regulation of chemical process variables is normally achieved by a multiloop control arrangement, such as, cascade control or feedforward control coupled with feedback. Though the master-slave cascade system represents a remarkable improvement over ordinary feedback, it cannot achieve perfect regulation because, like feedback, it ignores the load and bases its correction on an error occurring either at the outlet or at an intermediate point in the process. However, it is the load that wields powerful influence on regulation of a loop and hence the load must be measured and used for manipulation if absolute control is to be achieved. This is the basic concept of feedforward strategy. In order to make the feedforward correction fully effective, the process has to be modelled in toto to include both the steady state and the dynamic components-an obvious impossibility except for a few simple cases. Further, the feedforward scheme often needs special control hardware for its implementation and these may not be readily available. Therefore, although in principle feedforward arrangement achieves perfect control, in actual practice this is never fully realised owing to realization and implementation difficulties. The objective of this study is to develop an alternate high performance control called scheme, “predictor-regulator control”, which virtually retains all the benefits of feedforward control and at the same time is free from some of its limitations. Development of the regulator proposed herein involves (i) devising load compensators, suitable for chemical process control, on the basis of theoretical analysis and simulation study, (ii) evolving criteria for choosing appropriate values for the adjustable compensator parameters, and (iii) applying the control scheme to engineering processes to establish its feasibility.
THEORY
Structure
I-A
system
A block diagram embodying the principle of the
proposed control approach is given in Fig. 1. It may be seen from the figure that the technique employs a two loop configuration: (i) a main loop, or process loop, made up of the usual elements, namely, controller G,, manipulator G,, process G,, and measurement H,,and (ii) an auxiliary loop composed of a second controller G,., and a dynamic element G.. The auxiliary loop output, C., acts as common feedback to both the control loops. The main controller operates on the process through the manipulator, G,, on the basis of 1
CESVol.31,No.
of the control
C. C. PRASADand P. R. KRISHNASWAMY
2
CP
Hm
Fig. 1.
Blockdiagramfor predictor-regulatorcontrol.
the difference between the feedback, C,, and set point, R (held constant for regulator operation). The process output, C,, adjusts the set point of the auxiliary controller thereby placing the process in closed loop. The element, G,,, can be so chosen[4] as to make the dynamics of the auxiliary loop considerably faster than that of the process loop. The basic disturbance, U,, entering the process creates an equivalent disturbance, U., through load measurement, H., and this serves as an imput to the auxiliary loop as indicated by the broken line in Fig. 1. It is the essence of this control arrangement that, by use of the fast auxiliary loop, the control effort required by the process controller to counteract major load changes can be predicted even before an error appears in the controlled variable. Thus, the auxiliary loop as a whole will be referred to as the predictor loop. The feedback originating from the process output takes care of any unmeasurable or unpredictable upsets that may enter the system. It is important to realize that all the control elements in Fig. 1 including G. are physically realizable and may be made up of standard electronic or pneumatic control hardware. To aid in visualizing the effects of the auxiliary loop on the regulatory action of the control system, the block diagram of Fig. 1 is reduced to an alternate form as shown in Fig. 2. The figure reveals that the suggested control technique has a striking similarity to the standard feedforward-feedback arrangement. However, there are also some important differences between the two of which the following are of special interest. Unlike in the case of the feedforward-feedback control, (i) both measured and unmeasured disturbances entering the process are penalised here by a single controller, G,, operating on G,
(ii) with proportional control action alone, the load compensator generates the behaviour of a high pass filter and also eliminates steady state error (iii) derivative action, if needed for compensation, can be accomodated in the process controller itself. The first point is evident from Fig. 2 and the remaining ones will be discussed in subsequent sections. It may be seen from the figure that the. behaviour of the main feedback loop is altered with respect to the classical feedback by the presence of the block G,.G,/l + G,.G. in the feedback path. It should be an aim of the control system design to choose appropriate values for the parameters of G., G,, and G,, so that this additional lag in the feedback path becomes negligible or is adequately compensated. Derivation of control laws
Referring to Fig. 1, and using the standard block diagram algebra, the relationship between process output and load disturbance is derived as:
c, = Gp[l + Ga(Gc. Up
GvG& )I 1+ G,G,. (1 + G,GvGc,Hm 1’
(1)
The numerator of eqn (1) can be made zero if G
=
”
l+G.G,, G.G,Hu ’
In physical terms this means that, if the process controller is designed according to eqn (2), then perfect control will result, that is, steady state as well as transient errors will be zero at all times regardless of load disturbances. It is of interest to note that this equation does not contain the process transfer function, Gp’ Equation (2) involves an yet undefined element G,,. Based on certain physical and operational considerations, it has been shown[4] that this element assumes the form
G.=A.
TJ + 1
and satisfies the conditions T. # 0 and T. 4 T,,, where T, is a measure of the dominant process dynamics. In order to make this discussion more specific, let us assume that (i) the auxiliary controller is a standard commercially available PID controller. If the interaction between the modes is negligible, then (4)
(ii) the measurement and manipulation dynamics are negligible compared to the dominant process dynamics, which is the case with many chemical process systems. Then, H,,,=A, H,=A.
Fig. 2. Alternateformof Fig. 1.
G,=A,
(5)
Analysisof a predictor-regulatorsystem where the A’s represent the static gains of the elements involved. Substituting the last three equations in the controller expression (2), we get l+
G
cp T, +(l+A.K,)’
L&K. 1, (l+A.K,)Ti.s
A.,Ka (l+A,K.)
1’
G,=K,
1
w
(7)
and
(11)
Equations (8)-(11) reveal that the process and auxiliary controllers are related through the steady state gains of various elements in the control loop. Further, eqn (11) indicates that the process controller contains additional derivative action, T&,to compensate for the auxiliary lag, T,. If the auxiliary controller is provided with only PI action, i.e. for 1
(
>
the corresponding process controller expression, obtained as above, is Gcp=Kp
(
1 l+=+T&s.
>
(12)
Similarly for G,. = K,, we have G, = K,,(l t T;s).
>
K,.
(16)
1+ A,K. = A,A,A.K,
(17)
K.TI, = A,A.K,Ti,
(18)
K,Td. = A,AA.K,Td,.
(19)
The above equalities can be easily satisfied in any given physical situation.
(10)
Gc.=K.!+=
1
(6)
1 t A,K,, Kp = A,A,A.
T
l+%
Under this condition, eqns (8)-(10) become the governing laws for the proposed control system. These equations may be restated for convenience as
TdaS
where
T:,=1+A.K..
i
Gc, =
This equation is of the standard form l+&t(T,+Tb)s
3
(13)
The additional derivative term, T&,in the expressions for G,, viz. eqns (7), (12) and (13) need be satisfied only when the process is characterised by a fast dynamics. For sluggish processes, T,, (and hence T@ can in reality be made negligible as required in eqn (3), with the result the three expressions for Gep (i.e. eqns (7), (12) and (13)) simplify to (14)
Choice of controller modes Since the foregoing analysis reveals that all the standard combinations of control modes, namely, P, PI and PID, satisfy the conditions governing perfect regulator, the question naturally arises as to whether the simplest arrangement, i.e. proportional mode for both the controllers, would suffice to meet the control objectives. Equation (l), with G,. = K. and G, = K, reduces to
c, _ G,,[l t G,(Ka -A&K,)1 U,-
1 t G,K,,(l+ GpA&Am)’
(20)
The final value theorem and the controller relationship (17), when applied to the above equation, result in zero offset demonstrating that the steady state error can be eliminated with only P control action. Another distinct feature of this control structure is that it generates a lead term with only proportional action. This has been illustrated elsewhere[4] by reducing the numerator term inside the square bracket of eqn (20) to the form T,s /( T.s t 1). As T. is the only dynamic term inside the square bracket under consideration, the process transient to be tolerated on account of this control arrangement is limited by the relatively small magnitude of T,. To sum up, for the primary disturbance, U,, for which the system is designed, mere proportional action is sufficient to produce excellent closed loop response with negligible transient deviation and zero offset. However, in actual practice, there exists a real threat of unmeasurable disturbances (besides the key load, U,) entering the process. With respect to such disturbances, the system behaves as a single feedback loop (see Fig. 2), and consequently use of proportional control action will result in offset. This can be seen by applying the final value theorem to the appropriate closed loop transfer function relating unmeasured process load to the controlled variable. Since the prime purpose of feedback from process output is to eliminate offset even when load misses the predictor, proportional mode will not be sufficient; reset is obviously necessary. Although addition of reset will have a deteriorating effect on the transient response caused by the main load, U,, it will, on the other hand, provide adequate control against unmeasurable disturbances. For
4
C. C. PRASAD and P. R.
this later case, the system behaviour is that of a normal single loop augmented by an additional lag, G,G,,/(l + G.G,.), due to the auxiliary loop elements. In order to approach normal feedback performance, either this lag can be made negligible by keeping G, small and G,, large, or an appropriate lead compensation, T&,be added to the process controller as suggested earlier. Considering hardware simplicity, wherever possible, a small auxiliary lag that satisfies eqn (3) and two PI controllers tuned to obey eqns (17) and (18) may be employed. It should be realised that dissimilar controllers (say, for example, P controller for the auxiliary loop and PI for the process loop) cannot be recommended because they do not obey the specified control laws (eqn (18) in this case), and consequently do not meet the control objectives. The question of using the three mode controllers primarily depends on the control efficiency required against secondary (unmeasured) disturbances. Since for such disturbances the system acts as a single feedback loop, whatever criterion is employed for adding rate action to ordinary feedback may be applied here also. This and other aspects of the control problem are further explained in the simulation study that follows. SIMULATION mY Extensive simulation study was carried out to evaluate the performance of the predictor-regulator system and to observe the effects of various system parameters on control quality. First order, higher order as well as pure delay processes were controlled using the proposed configuration. All the simulations (except the ones involving pure delay) were carried out on a Pneumatic Process Control Simulator made up of control hardware and components that are actually found in industry. In other words, no attempt was made to idealise simulation, and the simulator was allowed to contain certain nonidealities found in real control systems, such as, interaction between control modes, restricted accuracy, and a certain amount of deviation from linearity. This permits the simulation study to be more compatible with reality.
KRISHNASWAMY
Several aspects of the control problem were analysed; these include the effect of auxiliary time constant on closed loop response, the type of control modes (namely P, PI or PID) needed for the twin controllers, optimum controller settings, applicability of the technique to complex higher order and dead time processes, and comparison with established methods of control. Certain practical problems which might arise during implementation, viz. load missing the predictor loop partly or completely, and unexpected disturbances entering the process were also examined.
Effects of auxiliary lag on closed loop response For the simulation study, a fourth order system was taken as a typical example of the process to be controlled by the proposed technique. The valve and measurements time constants were neglected and, for convenience, the static gains involved were all assumed to be unity. The numerical values used for the process and the auxiliary time constants (expressed in minutes throughout this study) and for the controller settings are included in the block diagram in Fig. 3. The figure also gives the results obtained when proportional-proportional (P-P) controller configuration was employed to control the process. The response curves for a step change in V,,,corresponding to several different values of the auxiliary lag (curves 1, 2 and 3), show no deviation at all from the preload conditions. For the chosen controller gains of 2.5 and 3.5, this represents a dramatic improvement in control system response to load upsets and signifies perfect control. Besides, the results clearly prove that the derivative term, TL, in the process controller (see eqn (13)) can be neglected for relatively small values of the auxiliary lag. The figure also provides a comparison with optimum PI feedback control (curve 4) set for minimum ITAE (integral of absolute error multiplied by time). This single loop control result is very poor by comparison with predictive control performance. It is of interest to note that no process modelling was required here for the successful application of the proposed control scheme.
Fig. 3. Effectof auxiliarylagon systemresponse.
Analysis of a predictor-regulator system
Time,Mins
5
-
Fig. 4. P-P control system-Effect of controllergains.
Controllersettingsfor the P-P control system For a given process and auxiliary lag, the closed loop response depends on the settings used for the controllers, G, and G,,. The controller tuning relationship, eqn (17), for the system under consideration with unity static gains, reduces to
Responses for three sets of proportional gains satisfying the above equality are shown in Fig. 4 for a fixed T, of 0.5 min. As predicted by the theory, in all the cases the final offset is zero. In addition, the use of larger controller gain practically eliminates all the transient errors as evident from the response curve coinciding with the x-axis for I&, = 3.5. The adverse effect of decreasing the controller gains is also evident from Fig. 4 where curves 2 and 3 show how the system dynamics progressively deteriorate, although only to a small extent, with lowering of the gain settings. This suggests that for best results higher gains should be employed. In order to aid in visualising the dependence of the system response on controller gains, a recording of the outputs of the two controllers and the auxiliary loop are shown in Fig. 5 on an expanded time scale for a specific case. The equation governing the auxiliary output, C.(t), for a step input of magnitude, U,, is
which at steady state (assuming an auxiliary element gain of unity) reduces to
ca=u”
1tK;
The corresponding steady state process controller output is xKp = - U,.
Fig. 5. Controller and auxiliary loop outputs for the P-P control system.
Thus, exact corrective action is eventually taken by the process controller prior to which, however, there exists a transient period (as represented by the shaded area in the figure). The duration of this transient, as may be seen from the equation for C,,(t), varies inversely with K., and hence by increasing the controller gains, the process output transients may be made negligible.
of load measurement on process response For proper control to be achieved, the load should be appropriately measured and transmitted to the auxiliary loop. Figure 6 compares the responses obtained (curves 1 and 2) when the entire load, VP,and when only part (70%) of the load were measured and transmitted to the predictor at U.. The 30% reduction in the load signal results in lowering of the predictor gain so as to satisfy eqn (17). Consequently the transient deviation increases as may be seen from curve 2, and although the resulting error is small ( < 2%), it nonetheless indicates the need for Efects
C. C. PRASAD and P. R.
i
35
Tune,Mins. Fig.
6. Effect of load measurement on process response.
proper sensing of the load. If the load is altogether missed by the auxiliary loop (this may happen if there is a failure in measurement or if unmeasurable loads ehter the process), then the present arrangement results in offset (curve 3) as predicted by the theory. For this reason, incorporation of reset action into the proposed control scheme becomes desirable. Control using twin PI controllers
Figure 7 shows the same 4th order process being controlled by two PI controllers. The controllers are tuned according to a modified version of the open loop ITAE criterion used in ordinary feedback control. Details on the tuning procedure adopted may be found in [5]. Step response curves obtained for different values of the auxiliary lag, G,, are labelled as curves 1,2 and 3 in the figure. The figure shows that by keeping G. sufficiently small, it is possible to achieve perfect regulation (see curve 1) with PI-PI control action. However, a comparison with Fig. 3, which describes P-P control results under identical conditions, indicates that the addition of integral mode results in slight deterioration in the dynamics at higher values of G,. Curve 4 in the present figure,
KRISHNASWAMK
representing feedback control, is the same as curve 4 in Fig. 3. Since P-P control arrangement fails to eliminate offset when unmeasurable disturbances enter the process, and since the prime purpose of feedback from process output is to eliminate offset for all loads (measured and unmeasured), proportional mode is not quite sufficient; reset is obviously necessary. Figure 8 illustrates the behaviour of the PI-PI control system previously considered when a step disturbance enters only the process without entering the predictor. A comparison with single loop PI control response tuned for minimum ITAE shows that the present system is nearly as good as ordinary feedback system with respect to unmeasurable upsets. This will be so as long as care is taken to ensure that the predictor loop dynamics is small compared to the process dynamics. Thus, for measured loads, PI-PI control achieves perfect regulation (Fig.7), and for unmeasured loads, it compares with the performance of ordinary feedback (Fig. 8). The outputs of the two PI controllers and that of the predictor loop are shown in Fig. 9 under conditions of simulation represented by the accompanying block diagram. It may be seen from the figure that, as expected, the process controller exactly cancels the effect of the disturbance after a transient period. During this period (resulting from a unit step change in U,), the manipulative signal, i.e. the process controller output, M, assumes the form
M=-
(
The nature of the equation shows that, depending on the numerical values used for the parameters of the auxiliary loop elements, the resulting manipulation could exhibit an oscillatory behaviour as in Fig. 9. In that case, the process output may also oscillate before reaching steady state (e.g. curve 2 in Fig. 7).
Fig. 7. Control using twin PI controllers.
Analysisof a predictor-regulatorsystem
Fig.8. PI-PI controlperformancewhenloadentersonly the process.
1.0
‘Dirt”rb.ncc
0.80.6O.L0.2
0 -0.2 -
*ux I
5
Imp
cJ”tp”l
I
I
I
10
15
20
Time,Mins
--_)
Fig. 9. Controller and auxiliary loop outputs for the PI-PI control system.
Control using twin PID controllers
Figure 10 records the recovery curves obtained when rate mode was added to the control system corresponding to curve 2 of Fig. 7. The controller tuning procedure mentioned earlier[5] was adopted here also. If the disturbance enters both the process as well as the predictor loops, then inclusion of derivative action does not result in any significant additional benefit as may be seen by examining curve 1 in Fig. 10 and curve 2 in Fig. 7. For secondary loads, i.e. for loads entering only the process, the predictor-regulator approaches the corresponding optimum single loop PID control in its performance as evident from curves 2 and 3 in Fig. 10. Thus, for unmeasured loads, the derivative action gives the predictor-regulator the type of benefit that it gives to standard single loop control, and this may be appreciated by references to Figs. 8 and 10. The need for two PID controllers is, therefore, determined mainly by the type of regulation required for secondary upsets, as no real benefit accrues from their use for measured loads if the predictor is sufficiently fast-acting. Thus it appears from the foregoing analysis that the
method developed here merits consideration for implementation whenever rigorous control is desired. APPLICATIONS
The proposed dual loop control policy was applied to two real processes in order to demonstrate its easy implementability and to confirm its superiority. The processes involved were (i) a pure delay process [4] with a dead time of 2.7 min, and (ii) a stirred tank described by second order-dead time transfer function and with provision for pH control [5]. These processes were chosen for application purposes on the basis of the fact that they represent particularly difficult control situations, with pure delay endangering stability and pH control involving parametric forcing and nonlinearity. Therefore, if the present technique could control these processes adequately, then it may be well assumed that the control would work for other types of processes as well. The control hardware used [4,5] essentially consisted of two commercially available pneumatic controllers, a pneumatic RC circuit, a control valve, and standard measuring and recording instruments. Excellent results
C. C. PRASAD and P. R. KRISHNASWAMY
Fig. 10. Control usingtwin PID controllers.
were obtained in the case of dead time control with the proposed system showing 14 fold improvement over other well known special control arrangements (viz. Smith’s linear predictor and Buckley’s feedforward control). Equally good results were also achieved in the case of pH control of a stirred tank where the use of the predictor system kept down the maximum deviation (between the process output and the set point) to O-03pH, which would have otherwise registered a deviation of 2.0 pH. Further details on applications to pure delay control and pH control may be found in [4] and [5] respectively.
it results in predictable success for a wide range of processes. The results also demonstrate that the technique achieves effective regulation even in the face of process parametric variations. (viii) Whenever load measurements are possible, the proposed technique is potentially capable of replacing other multiloop control arrangements such as combined feedback-feedforward. NOTATION A
steady state gain
C output variable CONCLUSIONS
A dual loop predictor-regulator concept was developed to obtain superior quality of control capable of meeting stringent tolerance on the controlled variable. Based on the theoretical, simulation, and experimental studies carried out to describe the characteristics and to test the applicability of the predictive system, the following conclusions could be drawn: (i) The theory and the simulation study demonstrate that the proposed predictor-regulator is capable of achieving perfect regulation. (ii) For achieving this, simple proportional action would s&ice. Reset mode is required only to account for secondary (unmeasured) disturbances. Rate action may be used for compensation in the case where the auxiliary lag is significant compared to the dominant process lag. (iii) This single control configuration affords both dynamic as well as steady state compensation for a variety of processes. (iv) For unmeasured loads, the performance does not generally differ from that of optimum single loop corrective action. (v) Translation of the regulator scheme from theory to practice is straightforward. Ordinary commercial controllers are adequate, controller tuning is simple, and process modelling may be avoided. (vi) Unlike feedforward regulation, this control approach does not demand special control hardware. (vii) The experimental evidence that resulted from applications to pure delay and CFSTR systems confirms that the control scheme is readily implementable, and that
G
H K M R s t T
Th Td x u
transfer function transfer function (measurement only) controller gain process controller output set point Laplace operator time time constant see eqn (11) derivative time integral time disturbance
Subscripts a auxiliary loop
c
controller measurement process u load v valve
m p
REFERENCES [l] Roots W. K. and Gijnenc G., IEEE Trans. Ind. Gen. Appl. 1970 IGA-6 52. [2] Krishnaswamy P. R. and Chandra Prasad C., IEEE Trans. Ind. Appl. in press. [3] Char&a Prasad C. and Krishnaswamy P. R., An improved On-Off Control Strategy, 4th Int. Gong. of Chem. Engng, Equipment Design and Automation, Prague, September 1972. [4] Chandra Prasad C. and Krishnaswamy P. R., Chem. Engng Sci. 197530 207. [5] Chandra Prasad C. and Krishnaswamy P. R., Instr.Tech. 1975 22 (7) 53.