The document discusses methods for quantifying system dynamics and interactions using gramians to inform control structure selection for complex processes. It proposes improved uncertainty bounds for existing gramian-based interaction measures and evaluates them on examples. Future work is identified to further tighten the bounds and develop robust control structure selection methods based on empirical gramian estimates.

1. Uncertainty Bounds for Gramian-Based Interaction Measures Björn Halvarsson Uppsala University Miguel Castaño Arranz Luleå University of Technology Wolfgang Birk Luleå University of Technology

2. A procedure for control design of complex processes Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide Model structure Model Control structure Subset of sensors and actuatos Performance specification Controller Parameters Implemented Controller

4. Control Structure Selection & Interaction Measures. e u - y + r Gramian-based Interaction Measures quantify the importance of the input-output channels Decentralized Controller Plant Sparse Controller Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide

6. Quantification of system dynamics with gramians. Hankel Norm ( ||G|| H ) We consider two different factors to quantify process dynamics Sum of the eigenvalues of PQ ( tr(PQ) ) Both factors have important implications in time domain . tr(PQ) has an interesting relationship with the frequency domain. Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide

7. Nominal Analysis of The Quadruple Tank Process Both measures indicate off-diagonal pairing. Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide

9. Example 1 Performed Monte-Carlo simulations revealed that both bounds are still too loose. Previous bound: New bound: Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide Uncertainty level

10. Approximating the analytical bounds for tr(PQ) Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide Uncertainty Description: Normal vector: Minimum area is enclosed by: Maximum area is enclosed by:

11. Example 2. The Quadruple Tank Process. Nominal Case Bounds obtained from the inequality (loose bound) : Bounds obtained from integrating the Nyquist Diagram (numerical approach): Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide Uncertainty level

14. Thanks for your attention Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide [email_address]