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

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

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

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

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

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

Delft Center for Systems and Control 8 July /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)

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

Delft Center for Systems and Control 8 July /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

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

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

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

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

Delft Center for Systems and Control 8 July /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

Delft Center for Systems and Control 8 July /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

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

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

Delft Center for Systems and Control 8 July /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

Delft Center for Systems and Control 8 July /20 Example

Delft Center for Systems and Control 8 July /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