An Introduction to Iterative Learning Control - Inside Mines

(5) Nonlinear and Time-Varying Systems:
• Consider the affine nonlinear system:
Extensions (cont.)
xk(t + 1) = f(xk(t)) + g(xk(t))uk(t)
yk(t) = Cxk(t)
with the ILC algorithm uk+1(t) = uk(t) + γek(t + 1). Then we can derive the (previously
reported in the literature) convergence condition:
• Similarly, for the time-varying system:
max |1 − γCg(x)| < 1
x
xk(t + 1) = f(xk(t), t) + g(xk(t), t)uk(t)
yk(t) = Cxk(t)
with the time-varying ILC algorithm uk+1(t) = uk(t) + γtek(t + 1) the multi-loop interpretation
allows us to derive the convergence condition:
max
x |1 − γtCg(x, t)| < 1, t = 0, 1, · · · , tf
Using the same approach, with the time-invariant ILC rule uk+1(t) = uk(t) + γek(t + 1) we
get:
max |1 − γCg(x, t)| < 1
x,t