(702a) Using Prior Knowledge for Multivariable Model Identification for Integral Controllability

Integral
controllability (IC) is necessary and sufficient to achieve robustness of
decoupling multivariable control with integral action (Garcia and
Morari, 1985 ).
IC refers to the fact that the steady-state gain matrix model and actual plant
must satisfy the eigenvalue inequality

for
all plants belonging
to an uncertainty set U.  The design of experiments for identification
of the model, satisfying integral controllability is a challenge as the
identified model and actual plant must satisfy a cumbersome eigenvalue-based
inequality (eqn. ).  To alleviate
this problem, Darby and Nikolaou (2009) proposed a
sufficient condition for satisfaction of eqn. for all

Where

and
the set U is based on ellipsoidal uncertainty, resulting from standard
least-squares estimation:

Equations and can be used to
design experiments for an integral controllable model. 

               While
model identification relies on data used to identify the model, it is always beneficial
to use a priori knowledge while building such a model.  It may be difficult to
capitalize on this simple realization in practice, particularly when
experiments must be designed for identification of a system that is partially
known and whose model must satisfy integral controllability.  The objective of
this study is to propose an adaptive design of experiments which uses prior
knowledge about the system, to identify integral controllable model.  Prior
knowledge may come in a variety of forms.  The present study focuses on prior
knowledge expressed in terms of linear equality constraints on rows of the
steady-state gain matrix as

Using
eqn. , a new sufficient
condition for satisfaction of eqn. was derived as

The
adaptive design strategy proposed by (Darby, 2008a) was modified
using the above condition (eqn. ).  In summary,
a numerical solution of the problem minimize J(Q) was
developed, where the decision variable  is
a Cholesky

factor
of the input covariance matrix

               The
proposed modified adaptive design method was tested on a 5x5 fluidized
catalytic cracking (FCC) plant (Darby, 2008b).  The results
obtained from this modified adaptive experiment design (using prior knowledge) were
compared with the experiment design strategy proposed by (Darby, 2008a) .Figure 1 shows that the proposed
adaptive design produces inputs that generate data for identification of an
IC-compliant model much faster (in an order-of-magnitude shorter time) than a
comparable adaptive experiment design without use of prior knowledge.

               Additional
results and case studies as well as future directions will be included in the
presentation.

Figure 1: Satisfaction
of Sufficient condition (eqn. & eqn.) in adaptive design
References

Darby, M. L.
(2008a). Studies of online optimization methods for experimental test design
and state estimation. Chemical and Biomolecular Engineering. Houston,
University of Houston. PhD: 152-153.

Darby, M. L. (2008b).
"Studies of online optimization methods for experimental test design and
state estimation. ."

Darby, M. L. and
M. Nikolaou (2009). "Multivariable system identification for integral
controllability." Automatica 45(10): 2194-2204.

Garcia, C. E.
and M. Morari (1985 ). "Internal model control: 2. design procedure for
multivariate systems." Ind. Eng. Chem. Process Des. Dev. 24:
472-484.