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

2. Contents of the presentation
LPV systems and discretization
The LPV Zero-Order Hold setting
Performance analysis
Example
Conclusions
8 July 2008

3. LPV systems and discretization
What is an LPV system?
[Lockheed Martin]
8 July 2008

4. LPV systems and discretization
Continuous-time LPV framework,
State-space representation
I/O representation,
8 July 2008

5. LPV systems and discretization
Discrete-time LPV framework,
State-space representation
I/O representation,
8 July 2008

6. LPV systems and discretization
8 July 2008

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 July 2008

8. Contents of the presentation
LPV systems and discretization
The LPV Zero-Order Hold setting
Performance analysis
Example
Conclusions
8 July 2008

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
8 July 2008

10. The LPV Zero-Order Hold setting
Zero-order hold discretization methods
8 July 2008

11. Contents of the presentation
LPV systems and discretization
The LPV Zero-Order Hold setting
Performance analysis
Example
Conclusions
8 July 2008

12. Performance analysis Local Unit Truncation (LUT) error Consistency
LUT error bound (Euler)
All methods are consistent
8 July 2008

13. Performance analysis N-convergence implies: N-stability suff. small :
(stability radius)
8 July 2008

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.
8 July 2008

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:
8 July 2008

16. Performance analysis
Overall comparison of the methods
8 July 2008

17. Contents of the presentation
LPV systems and discretization
The LPV Zero-Order Hold setting
Performance analysis
Example
Conclusions
8 July 2008

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. ( )
8 July 2008

19. Example
8 July 2008

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.
8 July 2008