Hamzi - Eurandom

Controllability and Observability Energies of Nonlinear
Systems in RKHSes
• 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.
• Assume that the system is linear when lifted into an RKHS.
• In the linear case, Lc(x0) = 1
2xT0 W −1
c x0 and Lo(x0) = 1
⟨ ⟩
rewritten as Lc(x0) = 1
2
W † c x0, x0
2xT0 Wox0 can be
and Lo(x0) = 1
2 ⟨Wox0, x0⟩.
• In the nonlinear case, it may be tempting to write, in H,
Lc(h) = 1
⟨
2 W † ⟩
c h, h
complications...
and Lo(h) = 1
2 ⟨Woh, h⟩. However, there are some
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 27 / 55