  This paper conducts a comprehensive study of a classical adaptive control
problem: adaptive control of a state-space plant model: $\dot{x}(t) = A x(t) +
B u(t)$ in continuous time, or $x(t+1) = A x(t) + B u(t)$ in discrete time, for
state tracking of a chosen stable reference model system: $\dot{x}_m(t) = A_m
x_m(t) + B_m r(t)$ in continuous time, or $x_m(t+1) = A_m x_m(t) + B_m r(t)$ in
discrete time. Adaptive state tracking control schemes for continuous-time
systems have been reported in the literature, using a Lyapunov design and
analysis method which has not been successfully applied to discrete-time
systems, so that the discrete-time adaptive state tracking problem has remained
to be open. In this paper, new adaptive state tracking control schemes are
developed for discrete-time systems, using a gradient method for the design of
adaptive laws for updating the controller parameters. Both direct and indirect
adaptive designs are presented, which have the standard and desired adaptive
law properties. Such a new gradient algorithm based framework is also developed
for adaptive state tracking control of continuous-time systems, as compared
with the Lyapunov method based framework.
