Ind. Eng. Chem. Res. 1988,27, 2323-2331
2323
Feedforward Control in the Presence of Uncertainty Daniel R. Lewin*vt and Claudio Scalil Chemical Engineering, California Institute of Technology, Pasadena, California 91125
In this paper, a new systematic approach to the design of feedforward-feedback (FF-FB) control systems is proposed, applicable in cases where bounds on plant and disturbance uncertainties are known. The robust performance of this control scheme is compared with that of two alternative feedback designs, on the basis of a test case in which the superiority of pure feedforward control is guaranteed for perfect models. On the basis of the results, the limits to the advantages of a combined feedforward-feedback controller over the optimal feedback compensator are determined as a function of plant and disturbance uncertainties. Design guidelines are laid for both F B and FB-FF designs where the significant disturbance can be measured and where model and disturbance uncertainties can be quantified. Feedforward (FF) control is considered as an alternative or complement to a feedback (FB) scheme in cases when its inclusion can eliminate the effect of measurable disturbances faster and more efficiently than would be possible by the use of feedback alone. However, feedforward control design demands that the disturbance be detected and that at least approximate models be available to describe the effect of disturbance and control action on the process. In order to compensate for unmeasured disturbance and because of model inaccuracies, feedback action will usually be retained for robustness, and the implementation of feedforward control thus leads to more complex structures. Most published papers on feedforward control refer to their application to industrial plants. They commonly compare performance of commercial FB controllers (e.g., PID) with FF controllers designed according to short-cut methods (e.g., lead-lag compensators), and thus their results cannot be generalized (for examples see Soule (1970) and Bavadas (1984). To date, nothing has been published that deals quantitatively with sensitivity of FF design to plant uncertainty. A comprehensive approach is therefore sought in order to identify the cases where the complexity of FF design is justified. In this study, we shall systematically deal with the salient design issues involved in robust feedforward control: 1. The analysis for nominal design (i.e., assuming perfect models) and for design in the presence of uncertainty will be dealt with separately. 2. A distinction will be made between minimum-phase (MP) and nonminimum-phase (NMP) systems. 3. The comparison between schemes with or without FF will be made on the basis of an optimal design procedure in order to set a clear upper bound on achievable performance. A disturbance model is required as the optimality of a given controller, holding only for a specific disturbance. Two examples serve to illustrate the importance of these points in light of the current status on FF design as reported in the literature: 1. For an open-loop stable minimum-phase( M P ) system without uncertainty, the optimal feedback controller will perform equally well as the optimally designed feedforward compensator in rejecting measured disturbances. Here, the whole question of a comparison between feedforward Present address: Chemical Engineering, Technion I.I.T., Haifa 32000, Israel. Present address: Chemical Engineering, University of Pisa, Via Diotisalvi 2, 56100 Pisa, Italy.
*
0888-5885/88/2627-2323$01.50/0
and feedback is irrelevant. Yet in well-accepted control textbooks (e.g., Luyben (1973) and Stephanopoulos(1984)), we find case studies in which feedforward schemes are compared with suboptimal feedback schemes for MP systems. 2. In a paper dealing with the control of processes exhibiting inverse response, Luyben (1969) recognizes that perfect feedforward control is physically unrealizable when the plant response is nonminimum phase (the term NMP indicates systems exhibiting pure time delay or inverse response) with right half-plane zeros and reverts to an ad hoc lead-lag design for such cases. A systematic approach which covers both MP and NMP plants is called for. Figure 1 serves both to illustrate the control problem and also to define symbols used. The process model consists of two transfer functions: p , describing the effect of controller action u on output y; and Pd, describing the effect of a measurable load disturbance d on y ( d is assumed to be a generic step, d ( s ) = l/s). The effective disturbance, d ’, is the effect of d in the process output (d’ = fidd). It should be noted that the FF-FB scheme is a 2 degrees of freedom structure which allows two different objectives to be attained (one can design for both set-point tracking and disturbance rejection). Although there are no stability problems associated with the open-loop structure of a FF controller, one would expect it to add sensitivity to model error, both in the plant and in the disturbance. Thus, the issue then becomes one of evaluating to what extent will model uncertainty affect the potential improvements of FF over FB control. A related matter is the relevance of accurate disturbance modeling when there is significant parametric uncertainty. A valuable result would be a limit on the usefulness of control designs based on detailed disturbance models, compared with those assuming generic step disturbance descriptions, as indicated by the relative sensitivities to model uncertainty of the alternative designs. In this study, we shall compare three alternative control strategies for the rejection of d’: (A) a “suboptimal” FB design based on the plant model, fi, but independent of the actual disturbance (i.e., we design for steps in the output, d’ = d = l/s); (B) a “optimal” FB design based both on the plant model and on the actual disturbance, d’= Pad; (C) a combined FF-FB structure where the FB controller is designed for set-point tracking and the FF controller to reject the actual disturbance (d’ = fidd). T h e Formulation of the Problem A well-posed control problem requires the following elements: (1) a model of the process describing the response of the process to actuator and disturbance changes 0 1988 American Chemical Society
2324 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1388 -
p-
4
i-_
d
-0
_-J
A* i
_ . .
-
-7
P .
f
Y
+ -~
Figure 1. Effect of disturbance and actuator on output.
r
(as in Figure 1);(2) a measure of confidence in the model, the uncertainty description; and (3) the performance objectives. The solution of the control synthesis problem is a controller of the chosen type, based on the process model, which satisfies desired closed-loop performance despite the (known) model imperfections. The design approach should compensate for model uncertainty, thus guaranteeing that the performance specifications are met even when the real process does not behave exactly like the model (robust control). We shall now consider the elements of the design formulation indicated above in more detail. Describing Model Uncertainty. The control system methodology to be applied requires that the process be defined as a linear system. However, since the true process is in general nonlinear, this implies an uncertain linear description, because there may be uncertainty in the real parameters affecting the actual (nonlinear) process and because the Laplace transfer function obtained by linearizing the process model in the vicinity of a chosen operating point may be different if this point is changed. Both of these sources of uncertainty can be modeled by defining model uncertainty regions in the complex plane. Let x ( w ) be a simply connected region of model uncertainty in the complex plane a t each frequency, w. The set, II, of all possible process models is defined by
n
= IP(S)lP(iW)f d w ) ,
v4
(1)
We note that, since there is an infinite number of transfer functions p(s) that can lie inside the regions x(w), set II is in general infinite. A subset of II, n', is defined as
I
11' = p(s)(p(s)= a,
b,sn
(2)
+ ... + b,s + 1
E @,,aL),h, E ( h h ) , 0 E (j',@
where t ( i w ) is the upper bound on multiplicative uncertainty, l(iw). In this representation, uncertainty is modeled in the complex plane by disk-shaped regions of radius IP(iw)lI(w), centered on g(iw). l ( w ) can be estimated from description (1) by the best fit of disks to a ( w ) a t each frequency as discussed by Laughlin et al. (1986). The development carried out here relating the uncertain plant p ( i w ) to a nominal plant mogel p(iw) and a bound on the multiplicative uncertainty l ( w ) also applies to the disturbance, p d ( i w ) : =
Pd(W)(l
+ MiW))?
Ild(iW)l
< td(W),
v
d
I r
w
(4)
z
Pd]
I
pd \
I
1
I
Figure 3. IMC FF-FB structure.
Closed-Loop Performance Specifications. The aim of the synthesis method is to design a controller which will achieve desired performance for all plants within uncertainty set 11 (robust performance). However, since what constitutes acceptable performance is a qualitative issue, we shall insist that at the very least all plants in II be closed-loop stable with the designed controller (robust stability). Uncertainty in the disturbance model, has no effect on stability but will, of course, affect performance. The classical feedforward-feedback structure is shown in Figure 2, where the output, y, can be described in terms of pd and p, controllers e and cF, and the load disturbance and set-point signals d and r: CP y = F F p
+-Pd-CFP d 1+cp
(5)
In the context of IMC (Internal Model Control), the structure shown in Figure 3 is recommended. The output, y , is then given by P4
The variations in the coefficients of the linear model p(s) are the result of the factors discussed above. Clearly, in order to satisfy performance requirements for all models in set II', it is sufficient to satisfy them for all models in II. A method has been developed by Laughlin et al. (1986) which translates parameter uncertainty described by formulation (2) to regions x(w) at each frequency. It is mathematically convenient to approximate the uncertainty with a single multiplicative perturbation. The uncertain process, p ( i w ) , is related to the nominal process model, P ( i w ) , by p(iw1 = PCiw)(l + I ( i w ) ) , Il(iw)l e t(U), v w (3)
Pd(i")
Figure 2. Classical FF-FB structure.
Y = 1+4(P-PIr+
(Pd - qFP) + (I?& - P&)qd 1 + d P -17)
(6)
where p and I j d are the assumed model for the plant and disturbance, respectively, and q and q F are the IMC feedback and feedforward controllers. The two structures become completely equivalent when the classical controllers (c and cF)are related to the IMC controllers (q and qF): e=-
4
1 - Ps
(7)
In the discussion which follows, we shall utilize either the classical or the IMC structures for analysis, with the knowledge that controllers of either structure are convertable into the other using eq 7 and 8. Robust Stability. We require that these control structures be stable with the designed controller e(s) for any p(s) in set II. Robust stability is guaranteed if and only if a nominal system with p ( s ) = p(s) E II is stable and lii(iw)l2(w) < 1, v w (9)
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2325
RHP zeros and delays and has ita gain set to unity for all
Figure 4. Nyquist plot showing the uncertainty disk at frequency w satisfying robust stability.
where the nominal complementary sensitivity function, +j(iw),is given by p ( i w )C (iw) q(iw) = 1 + P(iw)c(iw) = P(iw)q(iw) (10) Expression 9 is equivalent to the condition that in the complex plane the disk-shaped regions centered at p(iw)c(iw) with radius IP(io)c(iw)lZ(w) exclude the point (-1,O) for all frequencies, as illustrated schematically for one frequency in Figure 4. Nominal Performance. In addition to imposing robust stability, the designer will also require that the closed-loop system satisfy desired disturbance rejection and set-point tracking specifications. For the nominal case (P = I?, Pd = pd), the errors (e = r - y ) for the feedforward scheme for the classical and IMC structures are given respectively as
(12) eFF(1MC) = (1 - Pq)r - ( P d - PqF)d In both structures, perfect disturbance rejection (e(d) = 0) can be achieved by selecting
= q F = Pd/P (13) If the feedforward controller given by eq 13 is stable and causal, the design is equally trivial in both casey. When this is not so, CF and qF must be designed in order to minimize the effect of the disturbance on the output. In this case, the advantage of the IMC structure becomes evident. The optimal choice for the feedforward controller, qF, depends only on the disturbance and plant, while in the classical structure CF is also dependent on the feedback controller, c. The IMC parametrization thus generates a 2 degree of freedom structure, permitting the independent design of the two controllers in which q can be designed for set-point tracking and q F for disturbance rejection. The advantage of FF over FB control can be evaluated for the nominal case (i.e., no uncertainty), by comparing the error, e@), which is obtained by the two optimal controllers, designed for disturbance rejection. The H,optimal controller minimizes the 2-norm of the error cF
min, Ile(d)l12= min, min LJ+mle(d)12 2T -dw (14) and is given by the two controllers; for feedback leFB(d)l =
- P4)PddllZ
(15)
and for the feedforward design leFF(d)l = II(Pd - PqF)dIh (16) For each transfer function, we identify the minimum-phase and nonminimum-phase components. The nonminimumphase component of a transfer function includes all the
frequencies. Thus, we may define P = PMPA Pd = PdAPdM d = dAdM where the subscript A indicates the all pass and M the minimum-phase componentsof the given transfer function. The H,-optimal controllers minimize the objective function (14):
4 = (Ph@dMdM)-'bA-'PdMdM)*
(17)
q F = @MdM)-'bA-'PddM~*
(18) where (G), denotes that, after the partial fraction of the transfer function G, the unstable poles have been dropped to ensure causal, stable controllers. For development of eq 17 and 18, the reader is referred to Morari et al. (1988). For the two cases, the magnitudes of the error become leFB(d)l
=
[PdMdM - PAbA-'PdMdM)*l
(19)
(20) leFF(d) I = bddM - PAbA-lPddM]*l We note that the difference between the above expressions is that, whereas the FF scheme accounts for nonminimum-phase componentsof the disturbance, the FB scheme does not. Hence, if p d is minimum phase, the errors shown in the above equations are in general non-zero but equal. Furthermore, it can be trivially shown that ifp is minimum phase, the errors will be zero for both schemes, irrespective of the nature of Pd. Thus, the only case for which there is potential advantage to the employment of FF action is that in which both plant and disturbance have nonminimum-phase components. The improvement possible can be computed using eq 19 and 20. We note that the feedback controller designed by using eq 17 assumes knowledge of the actual disturbance (d' = Pdd). An alternative procedure (design A) generates FB control for generic step disturbances. For the nominal case, it can be shown that the error, leA(d)l,is greater than the error for the feedback case accounting for pd((em(d)l); the controller and the error are given, respectively, by QA leAl
=
(PM~M)-'IPA-'&I*
IPd,lldM
- PAbA-ldM)*I
(21) (22)
Example. If both p and Pd can be described as first order lags with time delays
it can easily be shown that lem(d)l < lem(d)l 5 leA(d)l, since
2326 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988
Robust Performance. Apart from demanding robust stability, the designer will also require that the closed-loop system satisfy desired disturbance rejection and/or setpoint tracking specifications. A fair comparison between FB and combined FF-FB designs will evaluate their performance in disturbance rejection alone. The disturbance rejection error, e ( d ) , for the two cases of FB only and FF-FB for uncertain plant and disturbance models can be described in terms of the multiplicative uncertainty expressions introduced in eq 3 and 4:
”il
r, = I p c ( i w ) l i ( w ) 1
Figure 5. Nyquist plot showing the uncertainty disk a t frequency w satisfying robust performance as brought by the weight W ~ computed at the same frequency.
(29)
Since the error depends on both q and qF, it is not possible to generalize about the relative merits of one control scheme over the other without referring to a specific design procedure. However, perfect disturbance rejection is possible with FF control, irrespective of model uncertainties for a particular case, when p = Pd, i.e., when the disturbance acts on the plant input. Then, leFF(d)l= 0 for any p and P d , when q F = CF = 1, while for the same case, leFB(d)l = 0 can only be achieved if q = j5-l is stable and causal. This is an important and useful result. It tells us that the nominal design rules already discussed hold for uncertain systems when we wish our design to reject disturbances entering at the same locations as the control signal does. Using the classical control structure, the disturbance rejection error is related to the disturbance load by the expressions, for FB alone
If uncertainties are expressed in- terms of the single pertubations 1 and &, bounded by 1 and &, respectively (as expressed in eq 3 and 4), eq 34 can be rewritten as
Ifi(iw)lt(w) + 1(1- 3(iw))wdw2(iw)l < 1,
v
(36)
where w d can now be expressed in terms of plant and disturbance multiplicative uncertainty: wd={
fid(1
+ ld)
(for FB alone)
fid(1
+ ld) - cFfi(1 -t I )
(for FF-FB)
(37)
Expression 36 is equivalent to the condition that, in the complex plane, the disk-shaped regions centered at @(iw)c(iw)with radius I@(iw)ll(o)do not approach the point (-1,O) closer than Iwd(iw)llwz(iw)l for all frequencies, as shown schematically in Figure 5 at one frequency. We note that, if condition 36 is satisfied, we guarantee both robust stability and robust performance. This expression is equivalent to a special case of the robust performance condition derived by Doyle (1982), who defines the “structured singular value” as the supremum in (36).
(30) and for the FF-FB scheme
Qualitatively, “good” response is equivalent to minimizing e. Thus, mathematically, robust performance is imposed by placing a bound on the magnitude of the sensitivity functions, t g ( i w ) and ~ ( i w ) :
The appropriate choice of the performance weighting function wz(iw) will express the designer’s requirements regarding bandwidth and maximum allowed magnitude of the sensitivity function. Physically, such specificationscan be related to desired speed of response and underdampedness in the time domain trajectory. Expressions 32 and 33 can be more conveniently expressed as 11 + P(iw)c(iw)l > Iwdw2(iw)I, v a, v P E n (34) Here the additional term wd accounts for the affect of the particular disturbance (d’ = P d d ) as observed in the process output: (35)
Systematic FF-FB Design in the Presence of Uncertainty The IMC parametrization suggests the following approach to the systematic design of combined FF-FB schemes (designated design C ) : (1A) A nominal H,-optimal design is performed to generate a feedforward controller, qF, to reject the actual disturbance using eq 18. All uncertainty in both plant and disturbance models is ignored at this stage. (1B) A nominal feedback controller, (7, is independently designed based on step disturbances using eq 17. The purpose of this controller is to perform set-point tracking or to reject additional disturbances. Again, this design will also be independent of uncertainties. (2) The nominal feedback controller is then detuned by augmenting an low-pass filter ( q = Q f ) in order to make the design robust. The filter is designed in order to minimize an H , objective function derived from the robust stability constraint (eq 36): minf sup,
p =
([(I - 3(iw))wdz+l
+ lij(iw)tl)
(38)
where ?(io)= @(iw)(7(iw)f(iw). This two-step FF-FB design is suboptimal in that it does not carry out the design of the two controllers in a truly simultaneous fashion. We are, in effect, freezing both the feedforward and the feedback controllers at their nominal values and imparting robustness through the addition of a filter to 0. On the other hand, the proposed approach accesses the power of H2-optimal design tools in a systematic fashion, leaving only a search for the optimal filter tuning.
W
~
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2327 For the case of the FF-FB scheme, the disturbance weight, w d = wd,c, accounts for the global error resulting from incorrect feedforward action (incurred from either imperfect nominal feedforward compensation or through modeling error). An upper bound can be derived for wd,C based on eq 37 and the known uncertainty bounds: (39) wd,C 5 wd,C = lfid - CFPl + [I?didl + IcFPil This bound separates the contributions to error of imperfect nominal FF design, disturbance uncertainty, and plant uncertainty and can be assumed for wd,C without introducing excessive conservatism. For no uncertainty, perfect FF design (cF = P d / P ) makes wd,C equal zero. The performance and sensitivity to model uncertainty of this FF-FB design should be compared to that of pure feedback schemes. As mentioned previously, we will look at two possible feedback designs: (A) an Hz-optimalFB design for a generic (step) disturbance, detuned in order to satisfy best possible performance for all plants within the uncertainty set (i.e., for robust performance); (B) an H,-optimal FB design for the particular disturbance k d d ) , detuned for robust performance. These two alternative FB designs also involve the minimization of the objective function given in eq 38. For the two FB designs, the bounds on w d are for A Wd = Wd,A = 1 (40) for B wd = wd,B 5 wd,B = P d ( 1 + id) (41) One should recall that in designs B and C filters fB and fc are designed to minimize the objective function p, while in design A filter fA is designed to minimize it without accounting for the disturbance, P d . After a value for fA has been thus determined, the value of p is then recomputed for comparison with the other designs using wd = Wd,B. For p C 1, robust stability is guaranteed since this condition is inbedded in the definition of p. However, because the minimized value of p may be higher than unity, the search for the optimum filter value must be constrained to those which guarantee robust stability (eq 9).
Case Study (Example Design) For perfect modeling, it has been shown that leFF(d)lC leFB(d)I when p and P d are both nonminimum phase. For an example system in which this is so, we shall now investigate the relative sensitivity to model uncertainty of designs A, B, and C described in the previous section. Plant and Disturbance Models and Associated Uncertainties. Let and p d be models describing the plant and disturbance as first-order lags with time delays:
(43) where all coefficients and s’ are given in their physical units. The models can be made di-mensionless by defining dimensionless frequency as s = Dps‘. Then (44) (45)
where 8 = jp/dP, p = i d l i p , and I$ = d d / d p . The model uncertainties are expressed in terms of bounds on the model parameters, ai:
ai = bi
+6
~ i
16ai/dil
< Ai
(46)
where bi and 6a; are nominal parameter values and perturbations of a; respectively, and Ai are the upper bounds on parametric fractional uncertainty. Bounds 1 and id defined in eq 3 and 4 can be constructed from parametric uncertainties as stated in eq 46 by using the methods described by Laughlin et al. (1986, 1987). Performance Weight Specification. In order to make the comparison between the three schemes as objective as possible, we opt here for a performance weight which reflects the best possible nominal performance achievable by a FB scheme. H,-optimal design of the feedback controller for step disturbances (i.e., P d = 1)gives Q = pM-’, and thus the nominal sensitivity function would be 7 = 1 - PA. Thus, condition 32 at the limit generates the performance weight w2
= 11 - P A [ - ’
which for our example becomes wz(s) = 11 - e-SI-l
(47) (48)
which is independent of model parameters. The Design of H2-OptimalControllers. Feedback. For strategy B (feedback designed for rejection of the disturbance d r= P d d , where d is assumed to be the generic step), the Hz-optimal controller derived by using eq 17 is (8s + 1)(A8ps + 1) (49) 6B = K P
where A = 1 - e-(ljsP), A filter, fB, must be appended in order that QB be proper. For simplicity, the one parameter filter, fB = (AB + I)-’, is adopted. For the FF-FB scheme (design C), and in the FB scheme of design A, the nominal controller is designed for step inputs and is given by (8s 1) QA = Qc = (50)
+
EP
In this case, a first-order filter, f = (Xis + l)-’, i = A or C, is appended to make q A and qc proper. Feedforward. The feedforward controller is designed for rejection of the disturbance d = p d d and is computed from eq 18 as
where B = 1- e-(l-O)/sp. The controller given by eq 51 may be made proper by the addition of a first-order filter of arbitrarily large bandwidth in order that its dynamics be unaffected.
Results For the example problem defined by eq 44 and 45, the H2-optimal controllers were designed as in the previous section, with the values of the filter tuning parameters, XA, AB, or hc, chosen in order to minimize the objective function (38) subject to the robust stability constraint (9). The sensitivity to uncertainties in plant and disturbance parameters, A and Ad, together with the effect of various nominal parameter values was sought. Figures 6-10 show the sensitivity to the following two parameters: 4. This parameter is a ratio of the delay times of the disturbance and the plant. For 4 > 1, perfect nominal disturbance rejection is possible with a FF controller, since
2328 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 0 - 2 . 0
p - 2 . 0
0
0.3
0.2
0.1
0.4
0 - 2 . 0
p - 0 . 5
0.5
0.6
0
0.1
0.2
0.5
P
p - 0 . 5
0 - 0 . 5
A=O
-2
.O
e - 0 . 5
.6
.4
0.2
2.0
A=O
0.3
0.4
0.5
0
0.6
0.1
0.2
0.3
*d
P =2.0 @
0.5
0.4
.2
.O
.4
.6
A =
A P = 0.5 @ =
0.5
A, -6
0.1
=
0.6
4
4
0
0.5 @
=
*d
Ad p - 2 . 0
0.4
0.3
P = 2 . 0 (0 = 2 . 0
0.6
0.5
Ad -6
.4
.4
.2
-2
Ad
Figure 6. p for designs A, B, and C with A = 0.0 and 0 I4 I0.6 for the four system cases.
.o -2
.O
-4
-0 . .O
.6
.
.
.
.
.
.4
.2
a 0 - 2 . 0
p - 2 . 0
I
A
0 - 2 . 0
$ 7 - 0 . 5
6
. .6
6
I
c superior,
next B, worst A.
B equivalent t o A. C superior to both. All three designs equally bad.
0
0.1
0.3
0.2
0.4
0.5
0.6
0
0.1
0.5
9 ' 0 . 5
p - 0 . 5
I
1
0
0.1
0.2
0.6
Ad 0 - 0 . 5
p - 2 . 0
0.4
0.3
0.2
*d
0.3
0.4
0.5
0.6
Figure 8. Maps in uncertainty space A - 4 showing the limits at which pB = pA and = pB (to within 5%)for each of the four system cases. The curves indicate the levels of uncertainty on plant and disturbance models a t which there is no advantage in disturbance modeling (B A) or where there is no advantage in the FF-FB design over the FB designed for the correct disturbance (C B).
0
0.1
0.2
0.3
d'
0.4
0.5
0.6
'
Ad
0
'
'
1
'
'
2
'
,
,
'
'
,
'
4
,
'
P e r f o r r a n c r
o f
"
'
6
D i ~ e n s i o n l e s st i l e ,
Figure 7. p for designs A, B, and C with A = 0.3 and 0.6 and 0 I & I0.6 for the four system cases. Table I. Classification of Parameter Values system 4 I 2.0 I1 2.0 I11 0.5 IV 0.5
'
F B
~ e
I
I
I
I
10
t / D ,
c o n t r o l l e r
d e s i g n
B .
P
2.0 0.5 2.0 0.5
-1 '
D i m e n S i O h l e s s
P e r f o r m 8 n c e
em(d) = 0 for no uncertainty in plant and disturbance models. When this is not so, as we have seen in the previous discussion, the "nominal" error incurred using the FF structure is still smaller than that using the optimal FB controller. p. This parameter is a ratio of time constants of the disturbance and the plant. Unity value represents cases when the dynamic lag of the plant and disturbance are the same. When p is small, the disturbance is almost pure time delay. The values of 8, I?lp, and I?ld were left at unity. We shall investigate four systems characterized by four value pairs in (4,p) as shown in Table I. Systems I and I1 represent cases where the disturbance delay is twice that of the plant. For the nominal system, these represent the easiest examples for FF design: The H2optimal feedforward controllers are lead-lags with time delays of (4 - 1)time units. Systems I11 and IV, on the other hand, represent cases where the plant delay is larger than that of the disturbance,
of
6
10
time,
F F - F B
t / D p
D e 8 i p n
C .
1.5
-
-1
0
D i r a n s i o n l e s 8
10 5
.
t i m e ,
t / D p
Figure 9. Simulation of the performance of the three designs for the system C#J = 2 and p = 2 and for varying degrees of uncertainty, A = Ad = 0.0 (-), 0.2 (---), arid 0.4 (---).
so perfect nominal disturbance rejection cannot be effected using feedforward action (4 < 1). System IV is inherently the most difficult of the four cases to control, since not only is perfect feedforward compensation not possible, but also the disturbance acts on the output faster than the effect of the actuator ( p < 1). Figure 6 shows plots of p as a function of Ad for A = 0. The three curves for each plot show the sensitivity of each
Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988 2329
Discussion The results shown in Figures 6-8 (the performance index as a function of uncertainties A and Ad) and in Figures 9 and 10 (simulations of responses to unit step in d ) indicate that over a wide range of uncertainty the FF-FB design (strategy C) is superior to either of the FB designs (strategies A and B). Of the two feedback strategies, design B, accounting for the true disturbance (pdd),gives better nominal performance. However, as uncertainty increases, the performance of all three schemes deteriorates, with greater sensitivity exhibited by the more sophisticated of the control schemes. Thus, the performance of two FB schemes tends to coincide at intermediate uncertainty levels and, eventually, similar (poor) results are obtained for all designs. To better illustrate these findings, the effect of uncertainties in the models of the process (A) and of the disturbance (Ad) will be presented for different values of the parameters 4 and p describing the family of systems analyzed in the case study. The comparison of the limiting advantages of the three design strategies is summarized in Figure 8; here the limit of usefulness of a particular design strategy relative to the next step down in sophistication is delineated. We shall discuss the results in terms of the effects of the two design parameters, p and 4. The value of p serves to compare disturbance dynamics with that of the plant. The relative advantage of accurate disturbance modeling for the family of systems under study is closely related to this parameter. From Figure 8, it is clear that, for p = 2, there is no advantage to be gained in designing simple feedback based on accurate disturbance modeling (A B) for model uncertainty greater than 40%, independent of disturbance uncertainty. For p = 0.5, the limit becomes 20%, and this is further reduced as disturbance uncertainty increases. These findings are confirmed by the simulation results given in parts a and b of Figures 9 and 10. Clearly, the H,-optimal strategy (design B) would be expected to be more sensitive to uncertainty than the “slower” one (design A) if it is trying to “match” faster disturbances. The parameter 4 distinguishes between designs where perfect nominal FF disturbance rejection is possible (4I 1) and situations when it is not (4 < 1). For 4 > 1,design C gives significant improvement over either of the feedback designs for low A. Designs for 4 < 1 have lower margins of improvement over the FB designs for low uncertainty but are less sensitive to error in the disturbance model than FF designs for 4 > 1. This can be seen in Figures 6 and 7 by comparing the relative gradients dpc/dAd for 4 = 2.0 and 4 = 0.5 for each of the two values of p. It can be seen that, over the range of plant uncertainties 0 5 A 5 0.6, the deterioration in pc with increasing disturbance uncertainty is about 50% greater for the cases when 4 = 2 than for 4 = 0.5. The frequency domain results based on p plots are corroborated by the simulation results, although it should be noted that since the former compute the optimal worst case they are expected to be more pessimistic than the worst realizable time domain simulation. However, comparing the predictions of Figure 8 with the simulation results in Figures 9 and 10 we find the following: I. The uncertainty maps predict that design C should be superior for uncertainties up to 50% and that design B is superior to A for uncertainties up to 40%. The simulation confirms the prediction on C, but there is no significant difference in the two feedback designs for uncertainties greater than 20%. IV. The predictions suggest that all three designs are equivalent at uncertainties of 40% and that B is superior
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of the three optimal designs to uncertainty in the disturbance model. Figure 7 brings the same measure for two values of plant model uncertainty, A = 0.3 and A = 0.6. Crossovers in any pair of curves pertaining to two different design strategies indicate the level of disturbance uncertainty at which the worst case responses are equivalent; i.e., there is no advantage to be gained by opting for the most sophisticated of the two designs. Figure 8 summarizes the information brought by Figures 6 and 7 over the entire range of uncertainties in both plant and disturbance tested. We show maps in uncertainty space (A, Ad) where the limits at which p~ = p~ and pc pB (to within 5%) are delineated for each of the four system cases. The curves indicate the levels of uncertainty on plant and disturbance models at which there is no advantage in disturbance modeling (B A) or where there is no advantage in the FF-FB design over the FB designed for the correct disturbance (C = B). The responses of the three design schemes to step disturbances in d are shown in Figures 9 and 10. The two extreme combinations of the stated parameter values for 4 and for p (systems I and IV) have been tested for uncertainties A = Ad = 0.0, 0.2, and 0.4. The descriptions for plant and disturbance allow for an infinite number of possible transfer functions within the uncertainty bounds as defined. However, for the particular form of the example, a combination of extreme parameter values can be identified, which represent the greatest perturbation from the nominal values over most of the relevant frequency range: gain at maximum, time constant, and delay time at minimum. Thus, the “worst case” system to be used in time domain simulation is
where clearly the nominal plant and disturbance models are eq 52 and 53 with A and Ad set to zero.
2330 Ind. Eng. Chem. Res., Vol. 27, No. 12, 1988
to A for uncertainties up to 10%. The simulation results bear this out. In the FB designs, the optimal feedback filter is chosen to minimize the objective function, and in general, increasing uncertainty implies more detuning of the H,-optimal controller (increasing AB). On the other hand, in design C (the FF-FB scheme), the optimal feedback filter setting, Xc, generally decreases as uncertainty increases. The decreasing trend in X , results from the fact that, when the plant and disturbance is well-known, the feedforward controller removes most of the disturbance and only a small component of feedback is required. As uncertainty increases, the feedback action must be applied more vigorously in order to compensate for errors in feedforward disturbance rejection. The lower bound of the value of Xc is usually the limit of robust stability: where 6 is some arbitrarily small constant. Conclusions The following two questions can now be answered: 1. When is it justified to accurately model disturbances? 2. When is the added complexity of feedforward control justified? For the nominal case (no uncertainty in plant and disturbance), the answer to both questions is clear. It has been shown that, for nonminimum-phase plant and disturbance, the relative performance of the three design schemes proposed gives related errors; thus, ec C eB I eA. For minimum-phase plants, the optimal feedback design gives equally good performance as the best feedforward control, making the latter superfluous. In the presence of uncertainty, we can make only one general conclusion. The nominal design rules outlined above will also hold for uncertain systems if p = P d , that is, if we wish to reject disturbances entering at the same location as the control signal. For p # p d , we can only draw qualitative conclusions and present numerical results for a specific system studied. However, the choice of lag-delay transfer functions to depict plant and disturbance characteristics is not unduly constrictive, since many typical processes can be well modeled by this form. It is apparent from the study that the sensitivity to uncertainty of the three design strategies depends more on p than on 4. The following numerical results are specific to the particular case study investigated but draw light on accepted practice: 1. When the response to disturbance is slower than the actuator, for example when p = 2, there are advantages to designing feedback based on accurate modeling of the disturbance if uncertainty on plant parameters is less than 40%. When the disturbance acts faster than the actuator, for example for p = 0.5, the models must be accurate to within 20%. These results indicate that there is indeed justification for the industrial practice of designing feedback for generic steps if the process uncertainty is large. 2. On the other hand, even up to quite high levels of uncertainty, the FF-FB strategy (design C) appears to be more tolerant of inaccuracies in plant or disturbance modeling than either of the FB designs. This suggests that, having gone to the trouble of accurately modeling the disturbance, the best strategy would be to introduce a FF-FB scheme, since this will be far more tolerant of uncertainty. 3. At high levels of uncertainty (60% for system I, 40% for system IV), all three designs perform equally poorly. Therefore, in such cases, the correct choice is the least sophisticated design strategy. So, feedback can be de-
signed based on a generic step with no loss of performance over more “optimal” designs. The second result goes against the currently accepted belief concerning sensitivity of feedforward schemes to uncertainty. However, we recall that design C incorporates both feedforward and feedback, and the role of each of the two components to the successful operation of the scheme should be understood. The role of the feedforward component is to remove as much of the effect of the specific disturbance (pdd)as possible. If uncertainty is low, this will easily be achieved, and any (small) residual signal resulting in either the effect of uncertainty or because of imperfect FF action will be dealt with by the FB component. If uncertainty is high, the FF controller will still have some beneficial effect (up to some limit in uncertainty), and the feedback controller, acting on the residual disturbance, would then not do any worse than the best nominal feedback would, acting alone. Thus,the combined FF-FB controller constitutes a robust package. Acknowledgment The authors are grateful to Prof. Manfred Morari for many useful comments. Daniel Lewin acknowledges the generous support of the Weizmann Foundation. Nomenclature c = classical feedback controller CF = classical feedforward controller d = disturbance load (d(s) = l/s) d_’ =_actualdisturbance (d’(s) = pdd) D,, D d = nominal plant and disturbance delay time e = error (e = y - r) f = robustness filter G = generic transfer function K p ,Kd = nominal plant and disturbance gain 1, 1d = multiplicative uncertainty for plant and disturbance 1, l d = bounds on plant and disturbance uncertainty p, fi = uncertain and nominal plant transfer functions pd, fid = uncertain and nominal disturbancetransfer functions Q, q = nominal and detuned IMC feedback controllers q F = IMC feedforward controller r = set point u = controller action w 2 = performance specification weighting function w d = disturbance weighting function Greek Symbols nominal plant and disturbance time constant B = dimensionless plant time constant p = dimensionless disturbance time constant 4 = dimensionless disturbance time delay e = sensitivity function +j= nominal complementary sensitivity function A = robustness filter constant I.L = objective function for robust performance A, 4 = fractional parameter uncertainty for plant, disturbance w = frequency i,, i d =
Literature Cited Bavadas, P. C. “Feedforward Methods for Process Control Systems”. Chem. Eng. 1984, 91(21), 103-108. Doyle, J. C. “Analysis of Feedback Systems with Structured Uncertainty”. ZEEE Proc. 1982, 129(D(6)),242-250. Laughlin, D. L.; Jordan, K. G.; Morari, M. “Internal Model Control and Process Uncertainty: Mapping Uncertainty Regions for SISO Controller Design”. Znt. J. Control 1986, 44, 1675-1698. Laughlin, D. L.; Rivera, D. E.; Morari, M. “Smith Predictor Design for Robust Performance-. Znt. J . Control 1987, 46,477-504. Luyben, W. L. “Feedback and Feedforward Control of Distillation Columns with Inverse Response”. Znst. Chem. Eng. Symp. Ser. 1969, 32, 6:39-48.
Ind. Eng. Chem. Res. 1988,27, 2331-2341 Luyben, W. L. Process Modeling, Simulation, and Control for Chemical Engineers; McGraw-Hill: New York, 1973; 429-451. Morari, M.; Economov, C. G.; Zafiriov, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1988, in press. Soule, L. M. “Feedforward Control Improves System Response”. Chem. Eng. 1970, 77(24), 113-116.
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Stephanopoulos, G. Chemical Process Control; Prentice Hall: Englewood Cliffs, NJ, 1984; pp 411-427.
Received for review August 6 , 1987 Revised manuscript received June 17, 1988 Accepted July 5, 1988
SEPARATIONS Entrainment from Sieve Trays in the Froth Regime Henry Z. Kister* and Joe R. Haas+ C F Braun, Znc., 1000 South Fremont Avenue, Alhambra, California 91802
The effects of tray geometry and operating parameters on sieve tray entrainment in the froth regime were investigated. The study was based on published entrainment data for the air-water system a t tray spacings exceeding 300 mm. Several distinct modes in which the above variables affect entrainment are described. A new correlation is presented for predicting entrainment as a function of tray geometry and flow rates for the air-water system. The correlation was shown to give reliable predictions of the effects of various design and operating parameters on entrainment and a good fit to experimental data. The sieve tray is one of the most extensively used vapor-liquid contactors in distillation and absorption operations because of its simplicity and low cost. One of the most common phenomena to adversely affect the capacity and efficiency of sieve trays is excessive entrainment. When entrainment appears to exceed about 5% of the liquid flow rate, remedial steps were recommended at the design stage (Lockett, 1986). These steps include increasing column diameter, tray spacing, or judicious variation of tray geometry. Entrainment recycles liquid in the wrong way through a column, thus reducing the driving force for mass transfer. This lowers tray efficiency. Quantitatively, the effect of entrainment on tray efficiency is complex; a detailed analysis is described by Lockett (1986). Entrainment from sieve trays has been extensively studied in the past 2 or 3 decades, yet factors affecting it are poorly understood. Several conflicting trends have been reported for the effect of operating and design variables on entrainment (e.g., Friend et al. (1960), Bain and Van Winkle (1961), and Kister et al. (1981b)). Recent investigations (e.g., Lockett et al. (1976) and Porter and Jenkins (1979))have shown that some of these trends can be explained if the nature of the dispersions formed on the sieve tray is taken into account. Using this approach, it has been demonstrated (Kister et al., 1981a; Kister and Haas, 1987) that the various data sources describing entrainment in the spray regime are in good agreement with each other and can be successfully correlated. However, the behavior of entrainment in the froth regime is not well understood. This paper is concerned with entrainment in the froth regime. The spray and froth regimes are the most common regimes on commercial-scale sieve trays handling nonfoaming *Author t o whom correspondence should be directed. Present address: UOP, Inc., 25 E. Algonquin Rd, Des Plaines, IL 60017-5017.
systems. In the spray regime, gas is the continuous phase, while liquid is present in the form of drops dispersed in the gas. In the froth regime, liquid is the continuous phase, while gas is distributed as bubbles in the liquid. Between these two regimes of dispersion there exists a transition during which an inversion of the continuous phase occurs. The transition from the fully developed froth regime to the fully developed spray regime occurs gradually over a mixed frothlspray region. At a constant gas velocity, tray dispersion changes from froth to spray as the clear liquid height decreases. The froth regime commonly occurs at moderate and high liquid flow rates, moderate and high pressures, and low and moderate gas velocities. Kister et al. (1981b) carried out extensive entrainment measurements in the froth regime and the partially developed spray region. While they investigated and correlated the behavior of entrainment in the partially developed spray region, no attempt was made to investigate or correlate the trends shown by the froth regime data. In this paper, we extend the previous investigation and study the behavior of entrainment and its dependence on tray geometry and operating parameters in the froth regime.
Background A change in entrainment behavior near the transition from the froth regime to the spray regime has been observed by several investigators (e.g., Shakhov et al. (1964), Lockett et al. (1976), Stichlmair (1978), and Kister et al. (1981b)). This change can usually be recognized by a minimum in a plot of entrainment versus liquid flow rate at constant gas flow rate. Work by Kister et al. (1981b) suggests that the minimum specifically marks the transition from the froth regime to the partially developed spray region. In the partially developed spray region, entrainment dependence on tray geometry and operating parameters was shown to be similar to that observed in the fully developed spray regime and can be correlated by 0 1988 American Chemical Society
