Hamzi - Eurandom

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

Controllability and Observability Energies of Nonlinear Systems in RKHSes • Similarly, letting x now denote the collection of m = Np observability samples, we can approximate the future output energy by ˆLo(x) = 1 ⟨ ⟩ WoKx, 2 Kx = 1 2m ⟨ ⟩ ∗ SxSxKx, Kx = 1 2m ko(x) ⊤ ko(x) = 1 2m ∥ko(x)∥ 2 2 where ko(x) := ( K(x, dµ) ) Np is the Np-dimensional column vector µ=1 containing the kernel products between x and the observability samples. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 32 / 55 (1)
Balanced Reduction of Nonlinear Control Systems in RKHS • 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. We assume that the system is zero-state observable, and that the linearization of around the origin is controllable. We also assume that the origin of ˙x = f(x, 0) is asymptotically stable. Proposed Data-Driven Approach: ◮ Assume that the system behaves linearly when lifted to a high dimensional feature space. ◮ Carry out balancing and truncation (linear techniques) implicitly in the feature space (discard unimportant states). ◮ Construct a nonlinear reduced-order model by learning approximations to f, h defined directly on the reduced state. space. . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 33 / 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 31: Controllability and Observability E
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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