Robust Control Indicators in Process Design and Control

ROBUST CONTROL INDICATORS IN PROCESS DESIGN ·» m CONTROL

J. Figueroa, J. Romagnoli and G. Barton Department of Chemical Engineering University of Sydney NSW 2006 Australia

INTRODUCTION -i, Increasingly, compani~s will come to expect qua~antees on both "~ the .steady-state and the dynamic performance , of process ' designs. As part of · this integration of steady-state and ,dynamic requirements, the process designer will be faced with ·the question of how to .reach ' an acceptable trade-off between ··!tJhe 'f ollowing: . :- (a)

Model uncertainty. Most chemical engineering operations do not have accurate d.ynamicmodels available for them, so some ' degree of plant model mismatch is inevitable.

. (b)

Controller complexity. As a general rule, t .h e designer is happy with the simplest controller that does the job.

~ (c)

Performance specifications . The tighter these are made, the "better" the controller has to be to achieve them.

,It i 's fair to . say that most previous ' approaches treat the integration . of flowsheet design and control considerations very much from the steady-state design point of view. FOr example, i:f ' a measure of (closed':"'loop) dynamic flowsheet behaviour is required, this maybe provided by open-loop indicators based on a lineaiised model of the process (Bartoh et aI', ' 1991). Such indicators do not specifically address the issue of the trade-off between the three criteria introduced above, although some of them do provide the designer with some information. A low process condition number, for example, is variously taken to indicat.e that simple multiloop control is adequate and . that any control scheme designed on the basis of the process model will be relatively insensitive to model uncertainties. . :rn this paper" we present a methodology for analysing these trade-offs within a robust controller design framework which is described in detail elsewhere (Figueroa et al, . 1991). :iFRAMEWORK FOR TRADE-OFF BETWEEN CRITERIA.

,Consider a closed-loop system where G (s) is , the plant transfer , function and Kn (s) is a controller designed to achieve various closed-loop . characteristics. In . order to incorporate the previous criteria into a closed-loop control scheme, we may add in three "uncertainty blocks" as shown in Figl,.lre 1. Each block is modelled ina similar fashion, as outlined below.

139

· Medel Uncertainty: The transfer functienefthe actual plant may be described as,

where ' Gn (s) is the neminal plant medel ~nd AG (s) multiplicative uncertainty characterised as, 40 E Do(u,s)=

fAG =

di.ag(t,:L);

A~

is an input

< uPG }

where PG is a pesitive c;iiagonal matrix and a is a scalar • .A value efa=O cerrespends te ne medel uncertainty, . while a=l gives the neminal (er base case) medel uncertainty. Netethat the superscript "+" indicates that each element .of a matrix has been replaced by the medulusef that element. Centreller ' Cemplexity: ' The difference between · the neminal (high-erder) centreller.K(s) that meets all required stability and perfermance criteria and its (lewer-erder) implementatien Kn(s) may be medelled as an additive uncertainty: K(s)

= Kn(S) + A K(S}

Thecentreller uncertainty L1K (s) can be expressed as, Il K En K «(3,S)

= {ilK:

A~~ (3K;)

where we have assumed that the magnitude .of this uncertainty is prepertienal(via the scalar ~) te the magnitude .of the neminal centreller.A value ef~=O cerrespends te using the neminal (high-erder) contreller, while use efalewer-erder centreller cerrespends te positive values ef~ . . Perfermance Specificatiens: Atypical perfermance requirement ~in the frequency demain is given by ,

By wpSD.

~ 1

where Wp is a weighting functien designed te achieve seme desiredclesed-leep perfermance by shaping the sensitivity function. This perfermance specificatien may be intreduced inte the preblem by means .of a "virtual uncertainty" Ap, as shewn in Figure 1. Ap may be used te previde ' the desired .output sensitivity te disturbances by setting, Ap

E

np(Y.,s) = {A p :lw;lApD.

$

y}

A value .of 1=0 cerrespends te requiring specificatiens, while 1=1 gives the neminal perfermance specificatiens.

ne perfermance (er base case)

We may transferm the uncertain system shewn in Figure 1 inte an equivalent M-A structure (Merari andZafirieu, 1989). The value .of this M-A fermat is that a structured singular value analySis yields a sufficient cendi,tien te guarantee beth ' the robust stability and perfermance .of the system (Figuerea et

140

" al, 1991). This condition may · be expressed as,

r(u,p,y) = where r is the maximum singular value (J1).

max .. 6> f1(6),U,p,y) I

value

~

1

attained · by .the

structured

I.LLUSTRATIVE EXAMPLE He.r e we will apply . our trade-'off methodology to provide a ·· comparison . between three possible sets of input variables for f the · dual composition control of a high · purity column studied . previously by Tsagas and McAvoy (1981): Conventional Control: The inputs are the reflux flowrate (L)and the vapour boilup rate (V). Material Balance Control: '; The inputs are the distillate flowrate (0) and the vapour boiluprate (V).

.. "

~:~

Ryskamp's Control · Scheme: The inputs are the distillate to vapour flowrate ratio . (O/V) and the vapour boilup (V) •

The . nominal plant models (ie the transfer functions) cases are gi venin ,the original reference.

~·three

for all

In order to provide a comparison that was ' independent · of the type of controller. that was employed, we applied our analysis using an "ideal" (Agarnennoniet aI, 1989) nominal controller Kn (s). This ideal controller is obtained by multiplying the inverse of the plant model by a filter, designed to ensure nominal stability and provide acceptable closed-loop performance. The plant uncertainty PG (s) and performance specification Wp (s) were · not taken from the6riginal reference but were based on similar functions u~ed in studies into the robust contr6l of distill.a tion columns. P (8) = 10.018+1 . ·1· G

..

.

w:p (s) = 1

0.018+1

208 1

+ 1 408

Figures 2-4 show plots of r as a function of ~ parametrized for three values of
141

,.p ossible and there is no real scope ' for any trade-offs. By . comparison, both the material balance and Ryskamp control ,schemes exhibit improved ,r obustness characteristics. There is some (relatively small) scope for trade-offs in the material balance and Ryskamp control scheme cases. Figure 3, for example, shows that at fixed model uncertainty (a=1) with material balance ' control it is po·s sil:>le to . trade-off reduced performance against reduced controller complexity. Much larger trade-offs, however, were found to be possible for an industrial (C 3 -splitter) column example. INTEGRATION OF PROCESS DESIGN AND CONTROL One consequence of the almost certain availability of a modern distributed control scheme for any new process plant; is that the possibility . of employing a wide . range '.of control schemes can now be considered. The consequences 'Of .such a range of control optioris should also be considered simultaneously with the process design. The approach outlined in this paper offers one possible way of considering the trade-offs between what the designer wants from the process,what the designer knows about the process and what level of control the designer has . .at his disposal . . The outstanding . challenge with the use of this trade-off analysis in ' tne integration of flowsheet design and control considerations is determining the economic value of changes in the a, J3 and "( parameters. To be useful, we need a ready means ·o·f answering questions . such as , ~Whatis the overall economic impact of achieving tighter control ,(ie a larger "( value) by employing a more sophisticated control scbeme (ie a lower J3 - value)?". REFERENCES Agamennoni, O.E., Desages, A.C. and Romagnoli, Eng.Sci., Vol. 43, pp. 2937-295U, 198B. Barton, G.W., Chan, W.K. and Perkins, Vol. 1, pp. 161-170, 1991.

J.D.,

J

.A.,

Chem.

J. Proc. Cont.,

Figueroa, J~L., Desages, A.C., Palazoglu,A. and Romagnbli, J .A., "Trade-offs " in robust controller design", AIChE Annual Meeting, Los Angeles, November 1991. . . Morari, M. and Zafiriou, E., Robust Process Control, Hall,1989.

Prentice~

Tsagas, A. and McAvoy, T. J., "Dynamic simulation of a nonlinear dual composition control scheme", Second World Congress of Chemical Engineering, Montreal, October, 1981.

142

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n

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Nominal controller

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