(124e) Dynamic Model Development for Integral Controllability

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

for all plants G in set U where U is an uncertainty set [1], such as the following ellipsoidal uncertainty set

Therefore, a plant model G_m that does not satisfy eqn. is not useful for such a controller, even if G_m is close to the true plant G.  This may easily happen, particularly for ill-conditioned systems.

                The idea proposed in this work is that if a model G_m violates eqn. , then an alternative model X satisfying IC may be found that is still as close as possible to G_m.  The idea can be cast as a multi-objective optimization problem, namely

                Objective 1:                                                                                                                                                                                                           

where

and the right-hand side in eqn. is the singular-value decomposition of X.

                The rationale for Objective 2 is in replacing eqn. by the much more direct condition

which was developed in [2].

                A Pareto front is developed for eqns. - using a genetic algorithm (GA).

                An example of the resulting Pareto front is shown in Figure 1.  Of the proposed models X, the ones corresponding to eqn. are guaranteed to satisfy IC.  Furthermore, models violating eqn. may satisfy IC while also being closer to G_m; therefore they would be preferable for use in the corresponding controller. 

                A number of additional simulation examples are used in the presentation to illustrate the approach proposed above.

Figure 1 Pareto front for multi-objective optimization problem

References

1.            Garcia, C.E. and M. Morari, Internal Model Control .2. Design Procedure for Multivariable Systems. Industrial & Engineering Chemistry Process Design and Development, 1985. 24(2): p. 472-484.

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