Elgersburg Workshop 2010, März Path-Following for Nonlinear Systems Subject to Constraints Timm Faulwasser

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1 # Photos.com, ein Unternehmensbereich von Getty Images. Alle Rechte vorbehalten. Steering a Car as a Control Problem Path-Following for Nonlinear Systems Subject to Constraints Chair for Systems Theory andcontrol Institute for Automation Engineering Otto von Guericke University Magdeburg Elgersburg Workshop, Stabilisation control distance velocity? Trajectory Tracking determine reference offline design trajectory tracking controller robustness? Path-Following geometric reference determine reference velocity online Path-Following Problem Existing Results linear systems [Aguiar et al. `05; Dacic & Kokotovic `06] path regular, 1D curve nonlinear systems [Aguiar et al. `08] final point =origin given initial point robust path-following [Skjetne, Kokotovic et al. `05; Do et al. `06 ] passive systems [El-Hawwary `08] path convergence feedforward control of robots [Shin & McKay `85, Verscheure et al. `09] Usually, no consideration of constraints Elgersburg Workshop 2010, März

2 Trajectory Tracking and Path-Following Principle of Model Predictive Control predictive control = repeated optimisation 1. State measurement at 2. Solve reference trajectory tracking error reference path path-following error 3. Apply time variant problem performance limits? Consideration of constraints? Optimization-based approaches time invariant problem additional design parameter stability by choice of Applicable to path-following? Principle of Predictive Path-Following Stability expanded system dynamics path parameter and additional input costs penalize path-following error Theorem i. exists s.t. Optimal Control Problem ii. Optimal control problem solvable at Optimal control problem solvable for all [ Faulwasser & Findeisen `09] Stability? Robustness? Existence of solutions? direct formulation of path-following problem suff. stability conditions, through and calculation of terminal weights? Elgersburg Workshop 2010, März

3 # Photos.com, ein Unternehmensbereich von Getty Images. Alle Rechte vorbehalten. Calculation of Stabilising Terminal Weights Example: Path-Following for an Autonomous Helicopter Idea path as terminal region path exactly followable terminal weight = costs along path Prototype system ARTIS on-board image processing I-/O-linearising flight stabilisation task: accurate path follwoing Corollary quadratic cost input signals if an exists, s.t. joint project with Institute for Flight Systems, DLR (S. Lorenz) Comparision existing controller and path-following absolute velocity error Then stability and guarantees [Faulwasser & Findeisen `09] Steering a Car as a Control Problem Corridor Path-Following Predictive Path-Following 1D paths consideration of constraints path = regular, kd surface endpoint = origin Extensions multi-dimensional path corridors? path-following for output paths? convergence to corridor given initial point Elgersburg Workshop 2010, März

4 Predictive Corridor Path-Following Corridor Path-Following: Stability expanded system dynamics k path parameters, k add. inputs Theorem i. exist s.t. Optimal Control Problem ii. optimal control solvable at optimal control problem solvable for all [Faulwasser & Findeisen `09] Stability? sufficient stability conditions spacial deivation from 1d path online trajectory planning on path corridor Output Path-Following Idea of Predictive Output Path-Following Challenge natural cost function positive semi-definite output path regular, 1d curve state space output space convergence of output path consistent state set Path convergence instead of stability Elgersburg Workshop 2010, März

5 Example: Ship Control Summary new approach to constrained path-following problems nonlinear path-following problems subject ot constraints rigorous stability conditions extensions to path corridors and output paths outlook: optimal control on manifolds, robustness, 2D Output Path Corridor Possible Applications CNC-machines, robotics autonomous vehicles & planes crystallisation processes Many applications path-following problems Literatur Example: Path-Following for an Autonomous Helicopter Aguiar et al. (2005). Path-following for nonminimumphase systems removes performanc e limitations. IEEE Transactions on Automatic Control 50, Dacic & Kokotovic (2006). Path-following for linear systems with unstable zero dynamics. Automatica 42, Aguiar et al. (2008). Performance limitations in reference tracking and path-following for nonlinear systems. Automatica 44, Skjetne et al. (2005). Robust output maneuvering for a class of nonlinear systems. Automatica 40, Do & Pan (2006). Global robust adaptive path followng of under actuated ships. Automatica 42, Shin & McKay (1985). Minimum-time control of robotic manipulators with geometric path constraints. IEEE Transactions on Automatic Control 30, Verscheure et al. (2009). Time-optimal oath tracking for robots: A convex optimization approach. IEEE Transactions on Automatic Control 54, Faulwasser & Findeisen (2009). Nonlinear model predictive path-following control. In Magni et al. (ed.) Assesment and future directions of nonlinear model predictive control, Springer. Faulwasser et al. (2009). Model predictive path-following for constrained nonlinear systems. Proc. of 48th CDC, Faulwasser & Findeisen (2010). Constrained output path-following for nonlinear systems using predicitve control. Submitted to NOLCOS joint project with Institute for Flight Systems, DLR (S. Lorenz) Elgersburg Workshop 2010, März