Uncertainty Bounds for Gramian-Based Interaction Measures Björn Halvarsson Uppsala University Miguel Castaño Arranz Luleå University of Technology Wolfgang Birk Luleå University of Technology
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
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
Quantification of system dynamics with gramians.
P and Q are the controllability and observabillity Gramians.
The eigenvalues of the product PQ quantifies the connection of the input and output spaces through the state space.
The Hankel matrix maps the past process inputs into the future process outputs.
The non-zero Hankel Singulat Values (HSV:s) are equal to the positive square root of the eigenvalues of PQ.
Conclusion: The eigenvalues of PQ and/or the HSV:s can be used to quantify process dynamics
Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide
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
Nominal Analysis of The Quadruple Tank Process Both measures indicate off-diagonal pairing. Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide
Improved bounds for Δ||G ij || H and for Δ tr(P j Q i ).
Previous bounds for the additive uncertainty in the Hankel-Norm were derived by Moaveni and Khaki-Sedigh:
where W co is the cross-Gramian matrix, obtained from
The new potentially tighter proposed bounds for the Hankel norm and tr(P j Q i ):
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
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:
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