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Crucial Aspects of Zero-Order Hold LPV State-Space System Discretization

Crucial Aspects of Zero-Order Hold LPV State-Space System Discretization. Roland T ó th, Federico Felici, Peter Heuberger, and Paul Van den Hof. 17 th IFAC World Congress. Contents of the presentation. LPV systems and discretization The LPV Zero-Order Hold setting Performance analysis

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Crucial Aspects of Zero-Order Hold LPV State-Space System Discretization

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  1. Crucial Aspects of Zero-Order HoldLPV State-Space System Discretization Roland Tóth, Federico Felici, Peter Heuberger, and Paul Van den Hof 17th IFAC World Congress

  2. Contents of the presentation • LPV systems and discretization • The LPV Zero-Order Hold setting • Performance analysis • Example • Conclusions

  3. LPV systems and discretization • What is an LPV system? [Lockheed Martin]

  4. LPV systems and discretization • Continuous-time LPV framework, • State-space representation • I/O representation,

  5. LPV systems and discretization • Discrete-time LPV framework, • State-space representation • I/O representation,

  6. LPV systems and discretization

  7. LPV systems and discretization • Here we aim to compare the available dicretization methods of LPV state-space representations with static dependency in terms of these questions. Preliminary work: Apkarian (1997), Hallouzi (2006)

  8. Contents of the presentation • LPV systems and discretization • The LPV Zero-Order Hold setting • Performance analysis • Example • Conclusions

  9. The LPV Zero-Order Hold setting • Zero-order hold discretization • To compute , variation of and must be restricted to a function class inside the interval • We choose here this class to be the piece-wise constant • No switching effects

  10. The LPV Zero-Order Hold setting • Zero-order hold discretization methods

  11. Contents of the presentation • LPV systems and discretization • The LPV Zero-Order Hold setting • Performance analysis • Example • Conclusions

  12. Performance analysis • Local Unit Truncation (LUT) error • Consistency • LUT error bound (Euler) All methods are consistent

  13. Performance analysis • N-convergence implies: • N-stability suff. small : (stability radius)

  14. Performance analysis • Preservation of stability For LPV-SS representations with static dependency, all 1-step discretization methods have the property that • N-convergence and N-stability are implied by the property of preservation of uniform local stability.

  15. Performance analysis • Choice of discretization step-size: • N-stability (preservation of local stability) e.g. Euler method: • LUT performance (for a given percentage) e.q. Euler method:

  16. Performance analysis • Overall comparison of the methods

  17. Contents of the presentation • LPV systems and discretization • The LPV Zero-Order Hold setting • Performance analysis • Example • Conclusions

  18. Example • LPV discretization and quality of the bounds • Asymptotically stable LPV system with state-space representation ( ): • Discretize the system with the complete and approximate methods by choosing the step size based on the previously derived criteria. ( )

  19. Example

  20. Conclusions • The zero-order hold setting can be successfully used for the discretization of LPV state-space representations with static dependency. • Approximative methods can be introduced to simplify the resulting scheduling dependency of the DT representation. • The quality of approximation can be analyzed from the viewpoint of the LUT error, N-stability, and preservation of local stability. • Based on the analysis computable criteria can be given for sample-interval selection.

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