[403e] - Self-optimizing control: Optimal measurement
selection
- Vidar Alstad
- Norwegian University of Science
and Technology
- Department of Chemical
Engineering, NTNU
- Trondheim, 7491
- Norway
- Phone: +47 73 59 36
91
- Fax: +47 73 59 40
80
- Email: vidaral@chemeng.ntnu.no
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- Sigurd
Skogestad (speaker)
- Norwegian University of Science
and Technology (NTNU)
- Sem Sealands vei
4
- Trondheim, N7491
- Norway
- Phone:
+4773594154
- Fax:
- Email: skoge@chemeng.ntnu.no
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Abstract:
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"http://www.w3.org/TR/REC-html40/loose.dtd">
Demands
for more economical and environmental operation in the process
industries, has lead to an increased focus on optimal operation.
Typically, there are two sources of uncertainty in optimal control
problems. First, the effect of external disturbances (load
disturbances, etc.) must be handled and appropriate input usage must
be imposed in order to suppress the disturbances and to continue
operate in an optimal fashion. Second, the implementation error is
inherent in all control problems. Implementation error is the effect
of measurement error and control error (neglected here by assuming
integral action in the controller). This paper deals with the
selection of control structure in presence of such uncertainties.
Assuming pseudo steady-state, for a given disturbance d, the
optimization problem may be formulated as
|
| minx,u
J(x,u,d) |
|
|
|
| f(x,u,d)=0 |
|
|
| | where
u is the inputs, x the states, f=[f'
g'], f' is the equality constraints and g' is
the active inequality constraints (assumed active for all
disturbances). The search for the optimal control variables
correspond to solving the optimization problem over all d and
nc
|
| minx,c
J(c,d,nc)
|
|
|
(1) |
| f(x,c,d)=0 |
|
|
|
| c(x,d)-cs0+nc=0 |
|
|
| | where
nc is the
implementation error, c(x,d) is the control law
and c0 is the nominal
set-point. A possible solution approach to this problem is to
discretize in d and nc. This problem formulation may be
extended to also include the optimal set-point c0. The difficulties solving such problems,
especially for a general class of control laws c is not
straight forward and a simpler method is much appreciated.
A
common approach for optimal operation is implementation of Real-Time
optimization (RTO) systems, in which the optimization problem is
solved on-line or semi-online. Another and simpler approach is the
self-optimizing control approach, which is a control strategy where
by controlling carefully selected controlled variables at constant
set-point, the plant operation will be near optimal without the need
to re-optimize [6].
The idea of a self-optimizing control structure was proposed by
Morari et. al [5]
and Findeisen et. al. [4]
but has received little attention in the literature. Several
strategies for selecting control variables that have good
self-optimizing properties has been proposed in literature [6,
7,
3].
The disadvantages with the cited methods is that an extensive search
in all possible candidates are necessary or a rigorous analytical
differentiation are necessary.
A new and simple strategy for
selecting controlled variables is the null space method proposed by
[1,
2],
where control variables (c) are selected as linear
combinations of a subset of the available measurements (y),
c=Hy where H is selected in the null space of
the optimal sensitivity matrix (D
yopt=FD d), H Î
FT where F is
found by perturbation in the disturbances and solving eq.(1).
The null space method is a systematic method for selecting control
variables that ensure good self-optimizing properties of the
resulting control structure with respect to external disturbances.
In this paper the null space method is further explored and
extended, and a main results is the derivation of an explicit
expression for H. The derivation is based on a Taylor series
expansion of the reduced space optimization problem [6].
Another important result is that for MIMO problems the selection of
the basis for the null space give additional freedom in shaping the
resulting plant. This extra degree of freedom may be used to shape
the plant to have desired properties, such as steady state
decoupling which is preferable for a decentralized control structure
implementation.
As shown in Alstad & Skogestad [1],
the minimum number of measurement needed must be equal to the number
of ``unconstrained inputs'' and disturbances. In this work we assume
integral action in the controllers, thus restricting the source of
the implementation error to be measurement noise. In Alstad &
Skogestad[2],
a method for selecting the optimal measurements was prosed, based on
physical reasoning. The proposed method was to select measurements
that maximize the minimum singular value of the augmented plant
matrix, maxy s(G'y) where G'y=[Gy Gdy]
and D y=GyD u +
Gdy D d.
In this paper, we show that this is a reasonable but not
necessary optimal rule, and derive an expression for the optimal
selection rule.
The proposed ideas are illustrated on an
example.
References
- [1]
- V. Alstad and S. Skogestad. Robust operation by
controlling the right variable combination. Presented at AIChE
annual meeting, Indianapolis, USA, 2002.
- [2]
- V. Alstad and S. Skogestad. Combinations of
measurements as controlled variables;application to a petlyuk
distillation column. in the IFAC Symposium on Advanced Control
of Chemical Processes (ADCHEM) 2003, (Hong Kong),
2004.
- [3]
- Y. Cao. Self-optimizing control structure selection via
differentiation. in European Control Conference (ECC
2003), 2003. In CDROM.
- [4]
- W. Findeisen, P.Tatjewski, A. Bailey, M. Brdys,
K.Malinowski, and A.Wozniak. Control and Coordination in
Hierarchial Systems. John Wiley & Sons, 1980.
- [5]
- M. Morari, G. Stephanopoulos, and Y. Arkun.
Studies in the synthesis of control structures for chemical
processes. part i: Formulation of the problem. process
decomposition and the classification of the controller task.
analysis of the optimizing control structures. AIChE
Journal, 26(2):220--232, 1980.
- [6]
- S. Skogestad, I. Halvorsen, J.C. Morud, and
V.Alstad. Optimal selection of controlled variables. Ind. Eng.
Chem. Res., 42(14), 2003.
- [7]
- S. Skogestad and I. Postlethwaite. Multivariable
feedback control. John Wiley & Sons, 1996.
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