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 heritage that has principally been software pushed. This publication unifies those structures in a complete and obtainable manner, drawing from the most recent equipment and functions to teach how they come up clearly within the thought of linear keep an eye on structures. Magnus Egerstedt and Clyde Martin are top innovators within the use of keep watch over theoretic splines to assemble many varied functions inside a typical framework. during this e-book, they start with a sequence of difficulties starting from direction 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 basic 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 conceal periodic splines, monotone splines, and splines with inequality constraints, and clarify how any finite variety of linear constraints could be extra. This publication unearths how the various common connections among regulate idea, numerical research, and statistics can be utilized to generate robust mathematical and analytical tools.

This booklet is a superb source for college students and execs up to speed idea, robotics, engineering, special effects, econometrics, and any sector that calls for the development of curves according to units of uncooked data.

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Read or Download Control Theoretic Splines: Optimal Control, Statistics, and Path Planning (Princeton Series in Applied Mathematics) PDF

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

Example text

As in the proof that T is one-to-one, we look for a solution of the form u(t) = N τi ti (t). 22), we have upon equating coefficients of ti (t), i = 1, . . , N , the system of linear equations (W G+ρI)τ = W α. 3, the coefficient matrix is invertible, and hence the solution exists and is unique. The resulting curve y(t) is a spline. The major difference between classic, interpolating splines and smoothing splines is that the nodal points are determined by the optimization instead of being predetermined.

Under this assumption, all that follows in this chapter reduces to the case of polynomial splines. 2 The Data Sets The data sets we consider in this chapter are of two basic types, namely deterministic and random. For trajectory planning problems, we usually consider the data to be given in a deterministic form, that is, the coordinates of the locations and times are given exactly. We denote this ”deterministic data” set by DD = {(αi , ti ), i = 1, . . 5) where DD stands for deterministic data, and where we are interested in the behavior of the system over the time interval [0, T ].

Proof. Variations of the proof can be found in any number of textbooks. Suppose that, for some finite number M , we have controls uk (s), k = 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.

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