10 Design of Optimizing Control Structures for Chemical Plants
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on April 12, 2016 | http://pubs.acs.org Publication Date: May 30, 1980 | doi: 10.1021/bk-1980-0124.ch010
GEORGE STEPHANOPOULOS—Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, ΜΝ 55455 YAMAN ARKUN—Department of Chemical and Environmental Engineering, Rennselaer Polytechnic Institute, Troy, NY 12181 MANFRED MORARI—Department of Chemical Engineering, University of Wisconsin, Madison, WI 53706 Control objectives for a chemical process originate from certain regulation tasks (i.e. product quality control, material balance control, safety, environmental regulations, etc.) and economic objectives (i.e. optimizing the economic performance). Such a classification of control objectives automatically for mulates the different design activities for the regulatory and optimizing control structures. In the previous process control structure synthesis methods (1,2,3,4), the distinction between the different classes of control objectives and its impact on the design of the plant control structure have not been addressed. Articles representing the industrial views have recently indicated that the steady -state optimizing control constitutes the most fruitful control practice in the chemical process industry (5,6,7,8), whereas regulation of the chemical process units is accomplished by the practicing engineers with satisfactory degree of acceptance. Nevertheless, there has been no systematic approach towards the design and implementation of optimizing control structures. In the absence of any theoretical foundations and practical ramifi cations, the designer relies on his operating experience and intuition to select an optimizing control policy without exploring all the viable alternatives. In the present paper, we will lay down the theoretical foundations for the synthesis and design of steady-state optimizing control structures for chemical processes. The implementational problems will be also addressed to develop practical control strategies for the on-line application of the optimizing controllers. Optimum design of chemical processes dictates that the optimal operating point of a well-designed plant lies at the intersection of operating constraints (8-15). Furthermore, current industrial practice indicates that the optimal operating point switches from the intersection of one set of active constraints to another as process disturbances change with time (8-18). Such a dynamic evolution of the process operation
0-8412-0549-3/80/47-124-207$05.00/0 © 1980 American Chemical Society Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
208
COMPUTER
APPLICATIONS
TO CHEMICAL
ENGINEERING
constitutes the major thrust for the successful implementation of an optimizing control strategy. Formulating the Optimizing Control Problem for a Single Unit
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on April 12, 2016 | http://pubs.acs.org Publication Date: May 30, 1980 | doi: 10.1021/bk-1980-0124.ch010
The optimization problem for the single unit considered as a self-standing plant is to determine the optimal operating point when the values of a set of external disturbances change. The Steady State Optimization Problem. For a set of slowly varying external disturbances, we assume that the process is at pseudo-steady state. Then, the following static optimization problem can be formulated: Min m
Objective function
(x,m,d)
subject to (Pl)
h(x,m,d) g(x,m,d) r(x,m,d) -d + d*
=0 £b = r =0
System's State equations Process Design Constraints Regulatory Control Tasks Disturbance Specifications
d
where χ is the vector of states, m i s the vector of manipulated variables, and d is the vector of "slow" disturbances with major economic impact on the optimal process operation. Selection of Process Controlled and Manipulated Variables. At the calculated optimum X* = (x*,m*,d*) of the above problem (Ρχ), some of the inequality constraints w i l l be active. The regulatory control objectives and the active design constraints ( i . e . g ) at the current optimum w i l l constitute the class of primary controlled variables denoted by c , i . e . p
r (x,m,d)
p c (x,m,d)
g (x,m,d) A
'Ρ c reg c „ opt P
Some manipulated variables, m , w i l l be selected by the algorithm in (19) to control c , which w i l l partition the vector of the ~T\ manipulated variables: m ] with dim(m ) = dim c D D
p
D
[m
The available extra degrees of freedom at the optimum are: dim (in) = dim(m) - dim(c ) P
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
10.
STEPHANOPOULOS E T A L .
Optimizing Control Structures
Let us now define;
209
h(x,m)
χ d
f
and
c (x,m) -
=
- d + d* The Lagrangian Formulation and the Kuhn-Tucker Conditions. Formulate the Lagrangian function for the problem ( Ρ χ ) , L(x,m,X) = φ(χ,ίη) - À f(x,m) Downloaded by UNIV OF CALIFORNIA SAN DIEGO on April 12, 2016 | http://pubs.acs.org Publication Date: May 30, 1980 | doi: 10.1021/bk-1980-0124.ch010
T
A
g (x,m) IA
is the vector of Lagrange multipliers for the equality constraints f(x,in) = 0, is the vector of Kuhn-Tucker multipliers for the IA inactive design constraints g (x,m) < 0.
where;
IA
The Kuhn-Tucker conditions for the minimum are: IA m
dm
am
dm
W
V
λ
ΙΑ "
0
χ
" 3x
IA IA" IA (8
b
)
λ
3x
ΙΑ=°
r(x,in)-r = 0 -d + d* = 0 h(x,m) = 0 d
* 0; X
λ
g(x,m) S b
°
=
3x
IA
Hence at the design optimum X*:
8
= b A
A
>
g
IA IA'
