Hamzi - Eurandom

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$Continuous-time random walks, fractional calculus and ... - Eurandom$

Reproducing Kernel Hilbert Spaces • In learning theory, the minimization is taken over functions from a hypothesis space often taken to be a ball of a RKHS HK associated to Mercer kernel K, and the function fs that minimizes the empirical error Es is m∑ fs = cjK(x, xj), where the coefficients (cj) m j=1 λ m ci + j=1 is solved by the linear system m∑ K(xi, xj)cj = yi, i = 1, · · · m, j=1 and fs is taken as an approximation of the regression function fρ. • We call learning the process of approximating the unknown function f from random samples on Z. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 26 / 55
Controllability and Observability Energies of Nonlinear Systems in RKHSes • We consider a general nonlinear system of the form { ˙x = f(x, u) y = h(x) with x ∈ R n , u ∈ R m , y ∈ R p , f(0, 0) = 0, and h(0) = 0. • Assume that the system is linear when lifted into an RKHS. • In the linear case, Lc(x0) = 1 2xT0 W −1 c x0 and Lo(x0) = 1 ⟨ ⟩ rewritten as Lc(x0) = 1 2 W † c x0, x0 2xT0 Wox0 can be and Lo(x0) = 1 2 ⟨Wox0, x0⟩. • In the nonlinear case, it may be tempting to write, in H, Lc(h) = 1 ⟨ 2 W † ⟩ c h, h complications... and Lo(h) = 1 2 ⟨Woh, h⟩. However, there are some . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 27 / 55
Page 1 and 2: Preliminary Results on Control and
Page 3 and 4: Outline • Review of Some Concepts
Page 5 and 6: Review of Some Concepts for Linear
Page 11 and 12: Controllability and Observability E
Page 13 and 14: Balancing of Linear Control Systems
Page 15 and 16: Balancing of Nonlinear Systems •
Page 17 and 18: Empirical Estimates of the Gramians
Page 19 and 20: Reproducing Kernel Hilbert Spaces
Page 21 and 22: Reproducing Kernel Hilbert Spaces T
Page 23 and 24: Reproducing Kernel Hilbert Spaces
Page 25: Reproducing Kernel Hilbert Spaces
Page 33 and 34: Balanced Reduction of Nonlinear Con
Page 35 and 36: Balancing in RKHS • “Balancing
Page 37 and 38: Experiment: Inputs ◮ Excite with
Page 39 and 40: Experiment diag(TKcT T ) 10 −1 10
Page 45 and 46: Nonlinear SDEs in RKHSes • Assump
Page 47 and 48: Nonlinear SDEs in RKHSes • The st
Page 49 and 50: Parameter Estimation for SDEs • I
Page 51 and 52: Parameter Estimation for SDEs • I
Page 53 and 54: Example (A Fokker-Planck Integrable
Page 55: Open Questions • Derivation of da

Hamzi - Eurandom

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