Ivancevic_Applied-Diff-Geom

566 Applied Differential Geometry: A Modern Introduction4.9.4.5 Lie–Adaptive ControlIn this subsection we develop the concept of machine learning in the frameworkof Lie derivative control formalism (see (4.9.2) above). Consider annD, SISO system in the standard affine form (4.44), rewritten here forconvenience:ẋ(t) = f(x) + g(x) u(t), y(t) = h(x), (4.57)As already stated, the feedback control law for the system (4.57) can bedefined using Lie derivatives L f h and L g h of the system’s output h alongthe vector–fields f and g.If the SISO system (4.57) is a relatively simple (quasilinear) system withrelative degree r = 1 it can be rewritten in a quasilinear formẋ(t) = γ i (t) f i (x) + d j (t) g j (x) u(t), (4.58)where γ i (i = 1, ..., n) and d j (j = 1, ..., m) are system’s parameters, whilef i and g j are smooth vector–fields.In this case the feedback control law for tracking the reference signaly R = y R (t) is defined as (see [Isidori (1989); Nijmeijer and van der Schaft(1990)])u = −L f h + ẏ R + α (y R − y), (4.59)L g hwhere α denotes the feedback gain.Obviously, the problem of reference signal tracking is relatively simpleand straightforward if we know all the system’s parameters γ i (t) and d j (t)of (4.58). The question is can we apply a similar control law if the systemparameters are unknown?Now we have much harder problem of adaptive signal tracking. However,it appears that the feedback control law can be actually cast in a similarform (see [Sastri and Isidori (1989); Gómez (1994)]):û = − ̂L f h + ẏ R + α (y R − y), (4.60)̂L g hwhere Lie derivatives L f h and L g h of (4.59) have been replaced by theirestimates ̂L f h and ̂L g h, defined respectively aŝL f h = ̂γ i (t) L fi h, ̂Lg h = ̂d j (t) L gi h,in which ̂γ i (t) and ̂d j (t) are the estimates for γ i (t) and d j (t).