The switched afine system considered in the paper comes to existence
when stabilizing the chain of three integrators subject to the additional
condition of asymptotic tracking a desired trajectory. The target trajectory is defined implicitly as that of the second-order integrator stabilized
by means of a feedback in the form of nested saturators. The control law
for the third-order integrator is suggested that ensures the fulfillment of
the additional condition, and the range of parameters of the proposed control guaranteeing global stability of the closed-loop system is determined.
Moreover, the problem of constructing ellipsoidal estimates of invariant
sets of the system is stated and solved. In particular, we consider the
problem of finding the best (in a certain sense) ellipsoidal estimate that
guarantees that the deviation of the system from the equilibrium point at
any time does not exceed a prescribed value as long as the initial state
vector falls into the ellipsoid.