Control Theoretic Splines: Optimal Control, Statistics, and by Magnus Egerstedt
By Magnus Egerstedt
Splines, either interpolatory and smoothing, have an extended and wealthy background that has mostly been software pushed. This publication unifies those structures in a accomplished and obtainable method, drawing from the most recent equipment and purposes to teach how they come up obviously within the thought of linear keep an eye on structures. Magnus Egerstedt and Clyde Martin are top innovators within the use of keep an eye on theoretic splines to assemble many various purposes inside of a typical framework. during this ebook, they start with a chain of difficulties starting from direction making plans to statistical data to approximation. utilizing the instruments of optimization over vector areas, Egerstedt and Martin display how all of those difficulties are a part of an identical common mathematical framework, and the way they're all, to a undeniable measure, a outcome of the optimization challenge of discovering the shortest distance from some degree to an affine subspace in a Hilbert house. They disguise periodic splines, monotone splines, and splines with inequality constraints, and clarify how any finite variety of linear constraints could be further. This booklet finds how the various common connections among keep watch over conception, numerical research, and data can be utilized to generate strong mathematical and analytical tools.This booklet is a wonderful source for college students and pros up to the mark concept, robotics, engineering, special effects, econometrics, and any sector that calls for the development of curves in keeping with units of uncooked facts.
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Extra resources for Control Theoretic Splines: Optimal Control, Statistics, and Path Planning (Princeton Series in Applied Mathematics)
The last problem concerned trajectory planning where, given a set of target points and target times, we wanted to be close to the points at times close to the target times. We were able to formulate these prob- EditedFinal EIGHT FUNDAMENTAL PROBLEMS September 23, 2009 51 lems within our optimal control framework in order to unite them with the previous problems into one unified theory. In the coming chapters, we will return to a number of these problems and see how they can be viewed directly as minimum norm problems in certain Hilbert spaces, as well as discuss extensions and potential applications.
Problem 7: Model Following min u∈L2 0 T (u2 (t) + (z(t) − y(t))2 )dt, subject to the constraint that x(t) ˙ = f (x(t)) + u(t)g(x(t)), y(t) = h(x(t)). It is clear that Problem 6 is a special case of Problem 7 by taking z˜ = x x˙ , and then noting that z˜˙ = 0 1 0 0 z˜ + u 0 0 0 1 z˜, with z = (10)˜ z. The solution to this problem is given by the solution to the corresponding Euler-Lagrange equations, which in this case is a nonlinear two-point EditedFinal September 23, 2009 48 CHAPTER 3 boundary value problem.
14) It should be noted that it is possible to increase the complexity of the cost functional, which is the case in references ,. 1 Problem 1 As seen in the previous chapter, in order for the optimal control problem to be well posed, we must specify from what set the control is to be chosen, and here we insist that u ∈ L2 [0, T ]. The first of the fundamental problems (the interpolation problem) then becomes Problem 1: Interpolating Splines min J(u) u∈L2 subject to the N constraints αi = Lti (u), i = 1, .