USE OF PERFECT INDIRECT CONTROL TO MINIMIZE STATE DEVIATIONS Eduardo Shigueo Hori, Wu Hong Kwong Federal University of S

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USE OF PERFECT INDIRECT CONTROL TO MINIMIZE STATE
DEVIATIONSEduardo Shigueo Hori, Wu Hong
KwongFederal University of São Carlos São
Carlos/SP BrazilSigurd SkogestadNorwegian
University of Science and TechnologyN-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
• 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
  (cy2) such that we minimize the effect of
  disturbances (d) on the weighted states (y1Wx).
• 2. Previous work Perfect Indirect Control
• Consider that we have the following linear model

• In summary, the main result in this paper can be
  summarized as follows
• Theorem Let Pxd denote the steady-state transfer
  function from d to x with cHy kept constant.
  Then Pxd2 is minimized by selecting
  HGxTG1Gy, , 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.
• Combination 2 W was selected as being the
  transpose of Gx (WGxT).
• Combination 3 W was calculated solving minPx
  Pxd2.
• 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.

Indirect control control y1 indirectly by
controlling the secondary variables c
where H is the combination of measurements.
Making some algebraic manipulations
Pd - effect of disturbances with closed-loop
(partial) control of c Pc - effect on y1 of
changes in c
Ultimate goal Perfect disturbance rejection Pd
0 and Pc Pc0 Assumptions 1. c y1 u
2. y u d 3. The matrix Pc0 is
invertible. Solution
When Pc0I ? GG1 and GdGd1. 3. Extension
Minimum State Deviation In this case y1 gt u,
so Pd0 is not possible. Instead we want to
minimize the norm of Pxd. Consider the following
linear model
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). This
choice doesnt give good rejection of the
implementation error (see matrix Px in Table 2).
As expected (session 3), the results presented in
Table 1 confirm that the use of WGxT is an
optimum choice.
The effect of disturbance in the states is

• 5. Conclusions
• 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 doesnt give a good control of the states
  in the middle of the column.
• 4. The choice of WGxT 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 doesnt give
  perfect control of the top and bottom
  compositions.
• References
• 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.

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 (y1Wx)
What is the optimal choice of W that minimizes
the value of Pxd in Eq. 8? Assuming WGxT

• Gx(GxTGx)-1GxT is called projection matrix.
• The product Gx(GxTGx)-1GxTGxd is the closest
  point to Gxd.
• Thus, the choice WGxT gives the minimum value
  of Pxd
• WGxT is optimum for any choice of Pc0
  non-singular, i.e., result is not restricted to
  Pc0I.
• W can be arbitrarily chosen by the designer
  according to the objective of the process.