Hamzi - Eurandom

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

Controllability and Observability Energies of Nonlinear Systems in RKHSes • The domain of W † c is equal to the range of Wc, and so in general Kx may not be in the domain of W † c . We will therefore introduce the orthogonal projection W † c Wc mapping H ↦→ range(Wc) and define the nonlinear control energy on H as Lc(h) = ⟨ W † c (W † c Wc)h, h • Since we will consider finite sample approximations to the preceding expression, W † c Wc may not converge to W † c Wc in the limit of infinite data (taking the pseudoinverse is not a continuous operation), and W † c can easily be ill-conditioned in any event. One needs to impose regularization, and we replace the pseudoinverse A † with a regularized inverse (A + λI) −1 , λ > 0 throughout. ⟩ . . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 28 / 55
Controllability and Observability Energies of Nonlinear Systems in RKHSes • Intuitively, regularization prevents the estimator from overfitting to a bad or unrepresentative sample of data. We thus define the estimator ˆLc : X → R+ (that is, on the domain {Kx | x ∈ X } ⊆ H) to be with infinite-data limit ˆLc(x) = 1 ⟨ 2 ( Wc + λI) −2 ⟩ WcKx, Kx , x ∈ X L λ c (x) = 1 ⟨ 2 (Wc + λI) −2 ⟩ WcKx, Kx , where λ > 0 is the regularization parameter. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 29 / 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 27: Controllability and Observability E
Page 31 and 32: Controllability and Observability E
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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