Download Control Theoretic Splines: Optimal Control, Statistics, and by Magnus Egerstedt PDF

By Magnus Egerstedt

Splines, either interpolatory and smoothing, have an extended and wealthy background that has principally been software pushed. This e-book unifies those structures in a complete and available means, drawing from the most recent equipment and purposes to teach how they come up evidently within the conception of linear keep an eye on structures. Magnus Egerstedt and Clyde Martin are prime innovators within the use of regulate theoretic splines to compile many different functions inside of a standard framework. during this booklet, they start with a chain of difficulties starting from course making plans to stats to approximation. utilizing the instruments of optimization over vector areas, Egerstedt and Martin exhibit how all of those difficulties are a part of an identical normal mathematical framework, and the way they're all, to a definite measure, a end result of the optimization challenge of discovering the shortest distance from some degree to an affine subspace in a Hilbert house. They hide periodic splines, monotone splines, and splines with inequality constraints, and clarify how any finite variety of linear constraints might be further. This e-book unearths how the numerous traditional connections among keep watch over idea, numerical research, and records can be utilized to generate strong mathematical and analytical tools.

This booklet is a wonderful source for college students and pros up to speed idea, robotics, engineering, special effects, econometrics, and any zone that calls for the development of curves in keeping with units of uncooked data.

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Extra info for Control Theoretic Splines: Optimal Control, Statistics, and Path Planning (Princeton Series in Applied Mathematics)

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N. 13) There are, of course, infinitely many control laws that satisfy these constraints. The problem is to identify a scheme that will select a unique control law in some meaningful way. As already mentioned, linear quadratic optimal control provides a convenient tool for this selection and, in fact, the main objective of this book is to show that optimal control plays a natural role for this. For the sake of keeping things simple, we will here consider the energy cost functional J(u) = T 0 u2 (s)ds.

1) be a controllable and observable linear system, with initial data x(0) = x0 . We will think of this system as the curve generator. As will be seen, we achieve the smoothest approximation if we impose the conditions for n ≥ 2 cb = cAb = cA2 b = · · · = cAn−2 b = 0, where n is the dimension of the system. Now, let the data set be given as D = {(ti , αi ) : i = 1, . .

M , which satisfy the constraints of Problem 2. Then, for each i, we have ai = M σk ai ≤ k=1 M σk Lti (uk ) ≤ k=1 M σ k bi = b i , k=1 where M σk = 1, σk > 0, k = 1, . . , M. k=1 Now consider M k=1 σk Lti (uk ) = Lti M σ k uk . k=1 EditedFinal September 23, 2009 33 EIGHT FUNDAMENTAL PROBLEMS Thus, the convex sum of controls satisfies the constraints if the individual controls satisfy the constraints. On the other hand, assume that {uk } is a sequence of controls that each satisfy the constraints.

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