In this paper, adaptive regulation of block-oriented nonlinear systems, i.e., Hammerstein and Wiener systems, with binary-valued measurements of the regulation errors is considered. Compared with the classical framework for stochastic adaptive control, the new feature here is that only binary-valued observations of regulation errors are available to the controller. An adaptive regulator based on stochastic approximation algorithm is proposed and it is proved that the regulator is optimal in the sense that it minimizes the long-run average of the squared regulation errors almost surely. Numerical examples as well as real applications of the proposed algorithms to automotive engine control are given.

This paper develops a class of novel algorithms for online convex optimization. The key construct is a forgetting-factor regret. It introduces weights to the objective functions at each time instant $\mathbf {t}$ and allows the weights of the past objective functions decaying to zero. We establish the forgetting-factor regret bounds of classical algorithms including online gradient descent algorithms, online gradient-free algorithms, and online Frank-Wolfe algorithms. In addition, the paper introduces online gradient descent algorithm with a forgetting factor, and analyze its performance under the new regret. Sufficient conditions are obtained to guarantee the bounds of the forgetting-factor regret of the above algorithms being of the order $\mathbf {o(1)}$ , which guarantees the tracking performance for minimizers of time-varying objective functions. Finally, our results are tested through numerical demonstration.