Adaptive Optimization with Periodic Dither Signals

Optimization methods in cyber-physical systems do not involve parameter uncertainties in most existing literature. This paper considers adaptive optimization problems in which searching for optimal solutions and identifying unknown parameters must be performed simultaneously. Due to the dual roles of the input signals on achieving optimization and providing persistent excitation for identification, a fundamental conflict arises. In this paper, a method of adding a small deterministic periodic dither signal to the input is deployed to resolve this conflict and provide sufficient excitation for estimating the unknown parameters. The designing principle of the dither is discussed. Under dithered inputs, the authors show that simultaneous convergence of parameter estimation and optimization can be achieved. Convergence properties and convergence rates of parameter estimation and optimization variable updates are presented under the scenarios of uncertainty-free observations and systems with noisy observation and unmodeled components. The fundamental relationships and trade-off among updating step sizes, dither magnitudes, parameter estimation errors, optimization accuracy, and convergence rates are further investigated.