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Delft Center for Systems and Control Seoul, 8 July 2008 Crucial Aspects of Zero-Order Hold LPV State-Space System Discretization 17 th IFAC World Congress.

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Presentation on theme: "Delft Center for Systems and Control Seoul, 8 July 2008 Crucial Aspects of Zero-Order Hold LPV State-Space System Discretization 17 th IFAC World Congress."— Presentation transcript:

1 Delft Center for Systems and Control Seoul, 8 July 2008 Crucial Aspects of Zero-Order Hold LPV State-Space System Discretization 17 th IFAC World Congress Roland Tóth, Federico Felici, Peter Heuberger, and Paul Van den Hof

2 Delft Center for Systems and Control 8 July 2008 2/20 LPV systems and discretization The LPV Zero-Order Hold setting Performance analysis Example Conclusions Contents of the presentation

3 Delft Center for Systems and Control 8 July 2008 3/20 [Lockheed Martin] What is an LPV system? LPV systems and discretization

4 Delft Center for Systems and Control 8 July 2008 4/20 Continuous-time LPV framework, State-space representation I/O representation, LPV systems and discretization

5 Delft Center for Systems and Control 8 July 2008 5/20 Discrete-time LPV framework, State-space representation I/O representation, LPV systems and discretization

6 Delft Center for Systems and Control 8 July 2008 6/20 LPV systems and discretization

7 Delft Center for Systems and Control 8 July 2008 7/20 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 Delft Center for Systems and Control 8 July 2008 8/20 LPV systems and discretization The LPV Zero-Order Hold setting Performance analysis Example Conclusions Contents of the presentation

9 Delft Center for Systems and Control 8 July 2008 9/20 Zero-order hold discretization The LPV Zero-Order Hold setting 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 Delft Center for Systems and Control 8 July 2008 10/20 The LPV Zero-Order Hold setting Zero-order hold discretization methods

11 Delft Center for Systems and Control 8 July 2008 11/20 LPV systems and discretization The LPV Zero-Order Hold setting Performance analysis Example Conclusions Contents of the presentation

12 Delft Center for Systems and Control 8 July 2008 12/20 All methods are consistent Local Unit Truncation (LUT) error Consistency LUT error bound (Euler) Performance analysis

13 Delft Center for Systems and Control 8 July 2008 13/20 N-convergence implies: N-stability suff. small : (stability radius) Performance analysis

14 Delft Center for Systems and Control 8 July 2008 14/20 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. Performance analysis

15 Delft Center for Systems and Control 8 July 2008 15/20 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: Performance analysis

16 Delft Center for Systems and Control 8 July 2008 16/20 Overall comparison of the methods Performance analysis

17 Delft Center for Systems and Control 8 July 2008 17/20 LPV systems and discretization The LPV Zero-Order Hold setting Performance analysis Example Conclusions Contents of the presentation

18 Delft Center for Systems and Control 8 July 2008 18/20 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. ( ) Example

19 Delft Center for Systems and Control 8 July 2008 19/20 Example

20 Delft Center for Systems and Control 8 July 2008 20/20 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. Conclusions

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