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

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

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

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

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

LPV systems and discretization 8 July 2008

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

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

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

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

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

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

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

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

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

Performance analysis Overall comparison of the methods 8 July 2008

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

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

Example 8 July 2008

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