Hamzi - Eurandom

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

An Experiment Consider the 7 − D system (Nilsson, 2009) ˙x1 = −x 3 1 + u ˙x2 = −x 3 2 − x 2 1x2 + 3x1x 2 2 − u ˙x3 = −x 3 3 + x5 + u ˙x4 = −x 3 4 + x1 − x2 + x3 + 2u ˙x5 = x1x2x3 − x 3 5 + u ˙x6 = x5 − x 3 6 − x 3 5 + 2u ˙x7 = −2x 3 6 + 2x5 − x7 − x 3 5 + 4u y = x1 − x 2 2 + x3 + x4x3 + x5 − 2x6 + 2x7 . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 36 / 55
Experiment: Inputs ◮ Excite with impulses: inputs (Kc) and initial conditions (Ko, u = 0). ◮ Learn ˆ f, ˆ h using a 10Hz square wave input signal u. ◮ Reduce to a second-order system. ◮ Simulate the reduced system with a different input, u(t) = 1 ( ) 2 sin(2π3t) + sq(2π5t − π/2) and compare the output to that of the original system. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 37 / 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 33 and 34: Balanced Reduction of Nonlinear Con
Page 35: Balancing in RKHS • “Balancing
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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