By Magnus Egerstedt
Splines, either interpolatory and smoothing, have a protracted and wealthy historical past that has mostly been program pushed. This publication unifies those buildings in a complete and obtainable approach, drawing from the newest tools and functions to teach how they come up obviously within the thought of linear keep watch over structures. Magnus Egerstedt and Clyde Martin are prime innovators within the use of keep watch over theoretic splines to compile many various purposes inside of a standard framework. during this e-book, they start with a sequence of difficulties starting from course making plans to statistical data 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 final result of the optimization challenge of discovering the shortest distance from some extent 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 might be additional. This e-book finds how the numerous common connections among regulate concept, numerical research, and facts can be utilized to generate strong mathematical and analytical tools.
This e-book is a superb source for college kids and execs up to the mark thought, 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 resources for Control theoretic splines: Optimal control, statistics, and path planning
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.
The problem is min J(u). u∈L2 The constants wi , i = 1, . . , N , are assumed to be strictly positive, as is the smoothing parameter ρ. The choice of the parameters wi and ρ is important. They control the rate of convergence of the optimal control signal as the number of data points goes to infinity. This is discussed in detail in [91]. The choice of the smoothing parameter ρ for fixed data sets is an important issue and is discussed at length in the monograph by Wahba [96]. Even though Problem 3 can be solved quite elegantly as a minimum norm problem in Hilbert spaces, the topic of the next chapter, we here follow the EditedFinal September 23, 2009 36 CHAPTER 3 variational method used in the previous section in order to stress that this important problem admits multiple solution methods.
This problem has a significant increase in difficulty over the problems we have considered to this point. Problem 5: Monotone Smoothing Splines Let J(u) = ρ T 0 u2 (t)dt + N wi (Lti (u) − αi )2 , i=1 and let a set of constraints be imposed as DLti (u) ≥ 0, i = 1, . . , N. The problem then becomes min J(u). u∈L2 We define H as 1 H(u, λ) = J(u) + 2 N i=1 2 wi (Lti (u) − αi ) − N λi DLti (u). 35) i=1 As before, the idea is to minimize H over u and to maximize H over all component-wise positive Lagrange multipliers λ ∈ RN .
