Presentation corfu (uncertain gramians)

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Presentation corfu (uncertain gramians)

  1. 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. 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
  3. 3. A complex process: Making paper <ul><li>From virgin and recycled fibers </li></ul><ul><li>Energy intensive process </li></ul><ul><li>More than 3000 control loops </li></ul><ul><li>Many storage vessels and return flows </li></ul><ul><li>Long process chain </li></ul>Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide
  4. 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
  5. 5. Quantification of system dynamics with gramians. <ul><li>P and Q are the controllability and observabillity Gramians. </li></ul><ul><li>The eigenvalues of the product PQ quantifies the connection of the input and output spaces through the state space. </li></ul><ul><li>The Hankel matrix maps the past process inputs into the future process outputs. </li></ul><ul><li>The non-zero Hankel Singulat Values (HSV:s) are equal to the positive square root of the eigenvalues of PQ. </li></ul><ul><li>Conclusion: The eigenvalues of PQ and/or the HSV:s can be used to quantify process dynamics </li></ul><ul><li> </li></ul>Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide
  6. 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. 7. Nominal Analysis of The Quadruple Tank Process Both measures indicate off-diagonal pairing. Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide
  8. 8. Improved bounds for Δ||G ij || H and for Δ tr(P j Q i ). <ul><li>Previous bounds for the additive uncertainty in the Hankel-Norm were derived by Moaveni and Khaki-Sedigh: </li></ul><ul><li>where W co is the cross-Gramian matrix, obtained from </li></ul><ul><li>The new potentially tighter proposed bounds for the Hankel norm and tr(P j Q i ): </li></ul>DON’T PANIC Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide
  9. 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. 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. 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
  12. 12. Conclusions and Future Work <ul><li>Previously existing bounds for two Gramian-Based interaction measures have been improved and evaluated. </li></ul><ul><li>These bounds have been found to be insufficiently tight in some examples. </li></ul><ul><li>A novel method for numerically obtaining the bounds of tr(PQ) is suggested. </li></ul><ul><li>Analyze which numerical approach for integrating the areas in the Nyquist array is more adequate. </li></ul><ul><li>Methods to create robust decisions in control structure selection form the Gramian-based interaction measures and their computed bounds. </li></ul><ul><li>Empirical estimation of interaction measures. </li></ul>Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide Conclusions Future Work
  13. 13. Other research topics in Control Structure Selection <ul><li>Analysis of complex industrial processes with signal flow graphs and methods based on brain connectivity theory. </li></ul>Miguel Castaño & Wolfgang Birk | MSC 2008| 2008-09-03 | Slide
  14. 14. Thanks for your attention Miguel Castaño | CSCC 2010 | 2010-07-24 | Slide [email_address]

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