st 1 HYCON PhD School on Hybrid Systems www.ist-hycon.org www.unisi.it Identification Algorithms for Hybrid Systems Giancarlo Ferrari-Trecate Politecnico di Milano, Italy Giancarlo.Ferrari-Trecate@inria.fr scimanyd suounitnoc enibmoc smetsys dirbyH lacipyt (snoitauqe ecnereffid ro laitnereffid) scimanyd etercsid dna stnalp lacisyhp fo fo lacipyt (snoitidnoc lacigol dna atamotua) fo senilpicsid gninibmoc yB .cigol lortnoc ,yroeht lortnoc dna smetsys dna ecneics retupmoc dilos a edivorp smetsys dirbyh no hcraeser ,sisylana eht rof sloot lanoitatupmoc dna yroeht fo ngised lortnoc dna ,noitacifirev ,noitalumis egral a ni desu era dna ,''smetsys deddebme`` ria ,smetsys evitomotua) snoitacilppa fo yteirav ssecorp ,smetsys lacigoloib ,tnemeganam ciffart .(srehto ynam dna ,seirtsudni HYSCOM IEEE CSS Technical Committee on Hybrid Systems 10 Siena, July 1 9-22, 2005 - Rectorate of the University of Siena
Modeling paradigms White box Black box Identification algorithms for Identification algorithms for Mechanics Chemistry System hybrid systems hybrid systems System Experimental data: ... Thermodynamics Giancarlo Ferrari-Trecate Drawbacks: Identification • Parameter values of components must be known algorithm Thanks to A. Juloski and S. Paoletti ! • Symplifying assumptions • Not feasible if first-principles are not available (e.g. economics, biology,...) Mathematical model Huge literature on identification of linear and smooth, nonlinear systems 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. Hybrid identification Motivating example Identification of an electronic component placement process What about identification of hybrid models ? Is it really a problem ? Fast component mounter (courtesy of Assembleon) First guess: • Each mode of operation is a linear/nonlinear system • Resort to known identification methods for each mode ! Not always feasible ! • 12 mounting heads working in parallel • Maximum throughput: 96.000 components per hour 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. Placement of the electronic component on Schematic representation the Printed Circuit Board (PCB) 2 basic modes of operation : • free mode • impact mode Experimental setup Mounting F head (M) F Input: Mounting head p p • Motor force F M M Moving impacting surface Output: • Head position p Elastic impact Ground connection surface Problem: the position of the impact surface is not measured The mode switch is not measured 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy.
Experimental data Hybrid identification Identification data Data could be naturally labeled Each mode has different, unknown according to finitely many modes of quantitative dynamics Head position operation Force Each mode has 3 modes Validation data a linear behavior Goals: Which are the data generated in the free and impact modes ? extract, at the same time, the switching mechanism and How to reconstruct the switching rule ? the mode dynamics 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. Outline of the lectures Hybrid identification: applications • Preliminaries: polytopes, PWA maps, PWARX models • Identification of PWARX models Domains: • Engineering: mechanical systems with contact phenomena • The key difficulty: classification of the data points • parenthesis: an introduction to pattern recognition • Computer vision: layered representation for motion analysis (Wang & Adelson, 1993) • Three identification algorithms : • Signal processing: signal segmentation (Heredia & Gonzalo, 1996) • Clustering-based procedure • Biology and medicine: sleep apneas detection form ECG, pulse detection • Algebraic procedure from hormone concentration, ... • Bounded-error procedure • Back to the motivating example: identification results 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. Preliminaries: polytopes Let Hyperplane: Half-space: Polyhedron: Preliminaries: polytopes, PWA maps, PWARX models - Polyhedra are convex and closed sets Polytope: bounded polyhedron Not Necessarily Closed (NNC) polyhedron: convex and s.t. is a polyhedron Polyhedral partition of the polyhedron : finite collection of NNC polyhedra such that and 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy.
Preliminaries: PWA maps PWARX models • is a polyhedral partition of the polytope AutoRegressive eXogenous (ARX) model of orders : • Switching function: ARX model (MISO) Vector of regressors: PieceWise ARX (PWARX) models of orders : Ingredients: PWARX model • is a PWA map • domain: (MISO) • number of modes: • modes: The interaction between logic/continuous components is modeled • Parameter Vectors (PVs): through discontinuities and the regions shape of the PWA map • Regions: 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. Identification of PWARX models Dataset (noisy samples of a PWARX model) : Identification: reconstruct the PWA map from Identification of PWARX models 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. Identification of PWARX models Dataset (noisy samples of a PWARX model) : Identification: reconstruct the PWA map from Standing assumptions: The key difficulty: data classification 1) Known model orders 2) Known regressor set (physical constraints) Estimate: • The number of modes • The PVs • The regions 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy.
Sub-problem: classification Sub-problem: classification Known switching sequence Maximal information about the modes Known switching sequence Maximal information about the modes -mode dataset: -mode dataset: • Pattern recognition algorithms Estimate the regions 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. Sub-problem: classification Sub-problem: classification Known switching sequence Maximal information about the modes Known switching sequence Maximal information about the modes -mode dataset: -mode dataset: • Pattern recognition algorithms Estimate the regions • Pattern recognition algorithms Estimate the regions • Least squares on Estimate the PVs 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. Sub-problem: classification Known switching sequence Maximal information about the modes -mode dataset: • Pattern recognition algorithms Estimate the regions • Least squares on Estimate the PVs An introduction to pattern recognition Classification problem: Estimate the switching sequence All algorithms for the identification of PWARX models solve, implicitly or explicitly, the classification problem ! 1st HYCON PhD School, 19-22 July 2005, Siena, Italy. 1st HYCON PhD School, 19-22 July 2005, Siena, Italy.
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