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Periodic and Recursive Control Theoretic Smoothing Splines
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. (Optimeringslära och systemteori)
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. (Optimeringslära och systemteori)ORCID iD: 0000-0003-0177-1993
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, a recursive control theoretic smoothing spline approach is proposed forreconstructing a closed contour.Periodic splines are generated by minimizing a cost function subjectto constraints imposed by a linear control system. The optimal controlproblem is shown to be proper, andsufficient optimality conditions are derived for a special case of the problem using Hamilton-Jacobi-Bellman theory.

The filtering effect of the smoothing splines allows for usageof noisy sensor data. An important feature of the method is thatseveral data sets for the same closed contour can be processedrecursively so that the accuracy is improved stepwiseas new data becomes available.

Keyword [en]
Smoothing splines, optimal control, Hamilton-Jacobi-Bellman theory, periodic solutions
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-11147OAI: oai:DiVA.org:kth-11147DiVA: diva2:236374
Note
QC 20100722Available from: 2009-10-01 Created: 2009-09-22 Last updated: 2010-07-22Bibliographically approved
In thesis
1. Data Filtering and Control Design for Mobile Robots
Open this publication in new window or tab >>Data Filtering and Control Design for Mobile Robots
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we consider problems connected to navigation and tracking for autonomousrobots under the assumption of constraints on sensors and kinematics. We study formation controlas well as techniques for filtering and smoothing of noise contaminated input. The scientific contributions of the thesis comprise five papers.In Paper A, we propose three cascaded, stabilizing formation controls for multi-agent systems.We consider platforms with non-holonomic kinematic constraints and directional rangesensors. The resulting formation is a leader-follower system, where each follower agent tracksits leader agent at a specified angle and distance. No inter-agent communication is required toexecute the controls. A switching Kalman filter is introduced for active sensing, and robustnessis demonstrated in experiments and simulations with Khepera II robots.In Paper B, an optimization-based adaptive Kalman filteringmethod is proposed. The methodproduces an estimate of the process noise covariance matrix Q by solving an optimization problemover a short window of data. The algorithm recovers the observations h(x) from a system˙ x = f (x), y = h(x)+v without a priori knowledge of system dynamics. The algorithm is evaluatedin simulations and a tracking example is included, for a target with coupled and nonlinearkinematics. In Paper C, we consider the problem of estimating a closed curve in R2 based on noisecontaminated samples. A recursive control theoretic smoothing spline approach is proposed, thatyields an initial estimate of the curve and subsequently computes refinements of the estimateiteratively. Periodic splines are generated by minimizing a cost function subject to constraintsimposed by a linear control system. The optimal control problem is shown to be proper, andsufficient optimality conditions are derived for a special case of the problem using Hamilton-Jacobi-Bellman theory.Paper D continues the study of recursive control theoretic smoothing splines. A discretizationof the problem is derived, yielding an unconstrained quadratic programming problem. Aproof of convexity for the discretized problem is provided, and the recursive algorithm is evaluatedin simulations and experiments using a SICK laser scanner mounted on a PowerBot from ActivMedia Robotics. Finally, in Paper E we explore the issue of optimal smoothing for control theoretic smoothingsplines. The output of the control theoretic smoothing spline problem is essentially a tradeoff between faithfulness to measurement data and smoothness. This tradeoff is regulated by the socalled smoothing parameter. In Paper E, a method is developed for estimating the optimal valueof this smoothing parameter. The procedure is based on general cross validation and requires noa priori information about the underlying curve or level of noise in the measurements.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. xii, 30 p.
Series
Trita-MAT. OS, ISSN 1401-2294
Keyword
formation control, tracking, nonlinear control, optimal smoothing, adaptive filtering
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-11011 (URN)978-91-7415-432-0 (ISBN)
Public defence
2009-10-22, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
QC 20100722Available from: 2009-10-01 Created: 2009-09-08 Last updated: 2010-07-22Bibliographically approved

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Hu, Xiaoming

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