USE OF PERFECT INDIRECT CONTROL TO MINIMIZE STATE DEVIATIONS
Eduardo Shigueo Hori, Wu Hong Kwong
Federal University of São Carlos
São Carlos/SP – Brazil
Sigurd Skogestad
Norwegian University of Science and Technology
N-7491 Trondheim - Norway
Abstract
An important issue in control structure selection is plant ”stabilization”. The paper presents a way to select measurement
combinations c as controlled variables, such that when c is controlled at a constant setpoint, the effects of disturbances
on the states are minimized.
1. Introduction
In summary, the main result in this paper can be summarized as follows:
Regulatory control layer:
•
main objective: ”stabilize” the plant
”Stabilization”: Includes both modes which are mathematically unstable (modes with
RHP poles) as well as ”drifting” modes which need to be kept within limits to avoid
operational problems.
By avoiding “drift” (keeping all states close to their nominal values we are able to avoid
problems resulting from nonlinear effects.
Goal: Select secondary controlled variables (c=y2) such that we minimize the effect of
disturbances (d) on the weighted states (y1=Wx).
2. Previous work: Perfect Indirect Control
Consider that we have the following linear model:
y1  G1u  Gd1d
(1)
y  G u  G d  n
y
y
d
y
Theorem: Let Pxd denote the steady-state transfer function from d to x with c=Hy kept constant.
T
†
x
x
y
Then ||P d||2 is minimized by selecting H=G Ğ1Ğ , , where † indicates the pseudo-inverse.
4. Application to Distillation
Distillation column: 82 states (41 compositions and 41 liquid holdups).
Manipulated variables: reflux flow rate (L) and vapor boilup (V)
Disturbances: feed flow rate (F) and fraction of liquid in the feed (qF)
Measurements: flow rates (L, V, D, and B).
Combinations of states used:
-Combination 1: Bottom and top compositions are the primary variables. Most common choice.
T
x
x
-Combination 2: W was selected as being the transpose of G (W=G ).
-Combination 3: W was calculated solving min||[Px Pxd]||2.
For each combination, matrix H was determined using Eq. 5. The resulting values of the 2-norm
of ||Px||, ||Pxd||, and ||[Px Pxd]|| are presented in Table 1.
(2)
Table 1 - Values of ||Px*||, ||Px*d||, and ||Px* Px*d|| for all 3 combinations.
||Px*||
||Px*d||
||Px* Px*d||
1
48.8289
2.5182
48.8817
2
0.0252
1.0886
1.0886
3
0.2560
1.0886
1.0886
Indirect control: control y1 indirectly by controlling the ”secondary” variables c:
c  Hy  HGy u  HGdy d  Hny
G
(3)
c
n
Gd
where H is the combination of measurements. Making some algebraic manipulations:


1
1
y1  Gd1  G1G Gd d  G1G n
c
Table 2 - Values of the matrices Px* and Px*d for the 3 combinations.
(4)
Pc
Pd
Combination 1
Pd - effect of disturbances with closed-loop (partial) control of c
Pc - effect on y1 of changes in c
Px*
0 
 1.0000
 1.4851 0.0000 






12.4987
2.7164






 2.8924 13.0048




 0
1.0000 
Ultimate goal: Perfect disturbance rejection: Pd = 0 and Pc = Pc0
Assumptions: 1. #c = #y1 = #u;
HP
-1
c0
Solution:
G1
Gd1  G
2. #y = #u + #d; 3. The matrix Pc0 is invertible.
G 
y
G1
y
d
Gy
1
(5)
-1
Combination 2
Px*d
-0.0000
-0.0000



 -0.0001


 -0.0001


-0.0000
0 
0.0000 


0.3994 


0.7381 


-0.0000 
Px*
 0.0005
 0.0007



 0.0046


-0.0040


-0.0016
0.0005 
0.0007 


0.0045 


-0.0040 


-0.0016 
Combination3
Px*d
0.0000
0.0000



0.0000


0.0000


0.0000
-0.0315
-0.0468


-0.0877 


0.4134 


-0.0342
Px*
 0.0070
 0.0104



 0.0824


-0.0056


-0.0020
-0.0018 
-0.0027 


-0.0051


0.0797 


0.0065 
Px*d
0.0000
0.0000



0.0000


0.0000


0.0000
-0.0315
-0.0468


-0.0877 


0.4134 


-0.0342
When Pc0=I → G=G1 and Gd=Gd1.
3. Extension: Minimum State Deviation
In this case #y1 > #u, so Pd=0 is not possible. Instead we want to minimize the
norm of Pxd.
This choice doesn’t give good rejection of the implementation error (see matrix Px in Table 2).
T
As expected (session 3), the results presented in Table 1 confirm that the use of W=Gx is an
optimum choice.
Consider the following linear model:
x  Gx u  Gdx d
Although the choice of top and bottom compositions as primary variables (combination 1) is
able to control perfectly these two variables (the closed-loop gains relating the disturbances to
the bottom and top compositions are zero), the gains of the states in the middle of the column
are very large (above 0.7) (see Table 2).
(6)
The effect of disturbance in the states is:

5. Conclusions

x  Gdx  GxG-1Gd d  GxG-1 nc
(7)
Px
Pdx
Px and Pxd represent the effect of the disturbances and implementation errors in the states
Define the primary variables as linear combinations of the states (y1=Wx):


x  G  G WG
x
d
x
x

-1


WG d  G WG
x
d
x
x

-1
nc
(8)
Px
Pdx
What is the optimal choice of W that minimizes the value of Pxd in Eq. 8? Assuming W=GxT:
-
T x -1 xT
x
x
G (G G ) G
References
is called projection matrix.
T
T
- The product Gx(Gx Gx)-1Gx Gxd is the closest point to Gxd.
- Thus, the choice
-
T
x
W=G
1. It is possible to control perfectly (having perfect disturbance rejection and minimizing the
implementation error effects) any combination of the states if we have enough measurements
available.
2. It is shown the importance of the use of the combination of states as primary variables.
3. Although the choice of the top and bottom compositions of a distillation column is good to
reject perfectly the disturbances, it fails in the rejection of the implementation error and also it
doesn’t give a good control of the states in the middle of the column.
T
x
4. The choice of W=G proved to be the best choice if the objective is to keep the states as
close as possible to their desired (nominal) values.
5. It rejects very well both disturbances and implementation errors, although it doesn’t give
perfect control of the top and bottom compositions.
T
x
W=G
gives the minimum value of Pxd
is optimum for any choice of Pc0 non-singular, i.e., result is not restricted to Pc0=I.
- W can be arbitrarily chosen by the designer according to the objective of the process.
•Skogestad, S, 2004, Control structure design for complete chemical plants. Comp. Chem. Eng. 28, 219.
•Skogestad, S., and I. Postlethwaite, 1996, Multivariable Feedback Control. John Wiley & Sons, London.
•Strang, G, 1980, Linear Algebra and its Applications. Academic Press, New York.
Acknowledgments
The financial support of The National Council for Scientific and Technological Development (CNPq/Brasil) and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES/Brasil) is gratefully acknowledged.