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.