Seoul, 8 July 2008

Delft Center for Systems and Control

Crucial Aspects of Zero-Order HoldLPV State-Space System

Discretization

17th IFAC World Congress

Roland Tóth, Federico Felici, Peter Heuberger, and Paul Van den Hof

8 July 2008 2/20

Delft Center for Systems and Control

• LPV systems and discretization• The LPV Zero-Order Hold setting• Performance analysis• Example• Conclusions

Contents of the presentation

8 July 2008 3/20

Delft Center for Systems and Control

[Lockheed Martin]

• What is an LPV system?

LPV systems and discretization

8 July 2008 4/20

Delft Center for Systems and Control

• Continuous-time LPV framework,• State-space representation

• I/O representation,

LPV systems and discretization

8 July 2008 5/20

Delft Center for Systems and Control

• Discrete-time LPV framework,• State-space representation

• I/O representation,

LPV systems and discretization

8 July 2008 6/20

Delft Center for Systems and Control

LPV systems and discretization

8 July 2008 7/20

Delft Center for Systems and Control

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 8/20

Delft Center for Systems and Control

• LPV systems and discretization• The LPV Zero-Order Hold setting• Performance analysis• Example• Conclusions

Contents of the presentation

8 July 2008 9/20

Delft Center for Systems and Control

• 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

8 July 2008 10/20

Delft Center for Systems and Control

The LPV Zero-Order Hold setting

• Zero-order hold discretization methods

8 July 2008 11/20

Delft Center for Systems and Control

• LPV systems and discretization• The LPV Zero-Order Hold setting• Performance analysis• Example• Conclusions

Contents of the presentation

8 July 2008 12/20

Delft Center for Systems and Control

All methods are consistent

• Local Unit Truncation (LUT) error

• Consistency

• LUT error bound (Euler)

Performance analysis

8 July 2008 13/20

Delft Center for Systems and Control

• N-convergence

implies:

• N-stability

suff. small : (stability radius)

Performance analysis

8 July 2008 14/20

Delft Center for Systems and Control

• 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

8 July 2008 15/20

Delft Center for Systems and Control

• 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

8 July 2008 16/20

Delft Center for Systems and Control

• Overall comparison of the methods

Performance analysis

8 July 2008 17/20

Delft Center for Systems and Control

• LPV systems and discretization• The LPV Zero-Order Hold setting• Performance analysis• Example• Conclusions

Contents of the presentation

8 July 2008 18/20

Delft Center for Systems and Control

• 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

8 July 2008 19/20

Delft Center for Systems and Control

Example

8 July 2008 20/20

Delft Center for Systems and Control

• 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