Hamzi - Eurandom

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

Review of Some Concepts for Linear Control Systems • In the linear case, it can be shown that Lc(x0) = 1 2 x⊤ 0W −1 c x0, Lo(x0) = 1 2 x⊤ 0Wox0. • Moreover, Wc and Wo satisfy the following Lyapunov equations AWc + WcA ⊤ = −BB ⊤ , A ⊤ Wo + WoA = −C ⊤ C. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 6 / 55
Review of Some Concepts for Linear Control Systems • These energies are directly related to the controllability and observability operators. • Given a matrix pair (A, B), the controllability operator Ψc is defined as Ψc : L2(−∞, 0) → C n ; u ↦→ ∫ 0 −∞ e−Aτ Bu(τ)dτ. • The significance of this operator is made evident via the following optimal control problem: Given the linear system ˙x(t) = Ax(t) + Bu(t) defined for t ∈ (−∞, 0) with x(−∞) = 0, and for x(0) ∈ C n with unit norm, what is the minimum energy input u which drives the state x(t) to x(0) = x0 at time zero? That is, what is the u ∈ L2(−∞, 0] solving Ψcu = x0 with smallest norm ∥u∥2? . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 7 / 55
Page 1 and 2: Preliminary Results on Control and
Page 3 and 4: Outline • Review of Some Concepts
Page 5: Review of Some Concepts for Linear
Page 9 and 10: 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 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

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