Hamzi - Eurandom

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

Review of Some Concepts for Linear Control Systems •If (A, B) is controllable, then ΨcΨ∗ c =: Wc is nonsingular, and the answer to the preceding question is uopt := Ψ∗ cW −1 c x0. The input energy is given by ∥uopt∥2 2 = x∗0 W −1 c x0. • The reachable set through uopt, i.e. the final states x0 = Ψcu that can be reached given an input u ∈ L2(−∞, 0] of unit norm, {Ψcu : u ∈ L2(−∞, 0] and ∥u∥2 ≤ 1} may be defined as R := {W 1 2 c xc : xc ∈ C n and ∥xc∥ ≤ 1}. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 8 / 55
Review of Some Concepts for Linear Control Systems • For the autonomous system ˙x = Ax, x(0) = x0 ∈ Cn ; y = Cx where A is Hurwitz, the observability operator is defined as Ψo : C n { CeAtx0, for t ≥ 0 → L2(0, ∞); x0 ↦→ 0, otherwise • The corresponding observability ellipsoid is given by E := {W 1 2 o x0 : x0 ∈ C n and ∥x0∥ = 1}. • The energy of the output signal y = Ψox0, for x0 ∈ C n can then be computed as ∥y∥ 2 2 = ⟨Ψox0, Ψox0⟩ = ⟨x0, Ψ ∗ oΨox0⟩ = ⟨x0, W0x0⟩ where Ψ ∗ o : L2[0, ∞) → C n is the adjoint of Ψo. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 9 / 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 7: 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

Hamzi - Eurandom

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