We derive a discrete-time adaptive stabilization algorithm and prove closed-loop attractivity with respect to the plant states. In this paper, we use a new method of analysis based on a modified Lyapunov technique and an adaptive step size. We begin by considering a one-step backward-horizon cost function, whose gradient provides an update direction for modifying the feedback gain matrix. The step size in the gradient direction is chosen to minimize the cost function along that direction. Finally, we use a modified Lyapunov technique to prove convergence of the plant states to the origin. We present the main results of Goodwin et al. (1980). An unstable and abruptly varying plant was simulated. Implementation issues are discussed and some results from simulation studies are presented.