By Magnus Egerstedt
Splines, either interpolatory and smoothing, have an extended and wealthy historical past that has mostly been software pushed. This e-book unifies those buildings in a complete and obtainable method, drawing from the newest equipment and functions to teach how they come up obviously within the conception of linear keep watch over structures. Magnus Egerstedt and Clyde Martin are top innovators within the use of keep an eye on theoretic splines to collect many different functions inside a typical framework. during this booklet, they start with a chain of difficulties starting from course 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 normal mathematical framework, and the way they're all, to a undeniable measure, a final result of the optimization challenge of discovering the shortest distance from some extent to an affine subspace in a Hilbert area. They hide periodic splines, monotone splines, and splines with inequality constraints, and clarify how any finite variety of linear constraints may be extra. This e-book unearths how the numerous normal connections among keep watch over thought, numerical research, and data can be utilized to generate robust mathematical and analytical tools.This publication is a wonderful source for college kids and pros on top of things thought, robotics, engineering, special effects, econometrics, and any zone that calls for the development of curves according to units of uncooked information.
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Additional info for Control Theoretic Splines: Optimal Control, Statistics, and Path Planning (Princeton Series in Applied Mathematics)
N . What is remarkable about this is that the optimal control problem, which is an inherently infinite-dimensional problem, has been transformed into a finite-dimensional problem involving finding the parameters τ1 , . . , τN . 15), it is in fact a semiparametric problem. 15) into the equations defining the affine variety, we have a set of equations y(t1 ) = τ1 Lt1 ( t1 ) + · · · + τN Lt1 ( tN ), .. y(tN ) = τ1 LtN ( t1 ) + · · · + τN LtN ( tN ). As in Chapter 2, we now let yˆ = (y(t1 ), .
15) where x(t) ∈ Rn and u(t) ∈ Rm . However, rather than stressing the fact that u at a particular time is a vector in Rm , one can view u itself as a point in a functional space. For example, we can insist that u ∈ Lm 2 [0, T ], which is the (Hilbert) space of equivalent classes of square-integrable mdimensional functions. ) With a slight abuse of notation (ignoring the equivalence class issue), we can think of Lm 2 [0, T ] as Lm 2 [0, T ] = w : [0, T ] → Rm such that T 0 wT (t)w(t)dt < ∞ , with inner product v, w L2 T = 0 v T (t)w(t)dt.
Recall that the vector τ satisfies the matrix equation (DW + ρI)τ = W α, and hence the τi are functions of the as of yet nonoptimal ξ(ti ). However, this choice of u is optimal for any choice of ξ(ti ). Substituting u into the functional J, we reduce J to a function of v alone. We are then faced with EditedFinal September 23, 2009 50 CHAPTER 3 a highly nonlinear functional to be minimized subject to a differential equation constraint. That is, we now have an optimal control problem with a nonlinear cost functional and a very simple linear control system as the constraint.