Page 53 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

44 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

Figure 3.4 Closed-loop control diagram composed by two-inertia system, the ECDSC
and FC.

3.3 CONTROLLER DESIGN AND STABILITY ANALYSIS

This section will present the control design by using Luenberger observer
to address the state estimation and modifying the DSC design. The overall
control structure can be found in Fig. 3.4.

3.3.1 Luenberger Observer

To address unmeasurable system states (e.g., torsional torque m s and the
load speed ω m)in(3.3), a Luenberger state observer is designed, which is
given by

˙ ˆ x = Aˆx + Bu + Bˆz + L(y − y)
(3.20)
ˆ y = Cˆx

where ˆx and ˆy represent the estimations of x and y, respectively. L =
T
[l 1 ,l 2 ,l 3 ] is the feedback matrix which should be set to make the matrix
A − LC stable. ˆz =− ˆ F with ˆ F being the estimation of the lumped dynam-
ics including frictions and other uncertainties, which will be approximated
by an ESN as shown in the following Section.
Define ˜x = x − x as the estimation error, then we have the observer
error dynamics as

˙
˜ x =˙x − ˆx = (A − LC)˜x + B˜z (3.21)
˙