Theory and practical considerations in reset control design
In the past three decades, linear time-invariant (LTI) control design techniques have been developed which can achieve stringent performance specifications in the presence of process uncertainties. However, all such LTI techniques are limited by the Bode gain-phase relation. Specifically, an LTI controller's high-frequency magnitude can not be designed arbitrarily but depends on the low-frequency specifications and stability constraints. Qualitatively speaking, "large" low-frequency magnitudes have to go with "large" high-frequency magnitudes, which make the closed-loop system more sensitive to sensor noise and high-frequency modeling errors. This phenomenon is called "cost of feedback", for which LTI control theory can not offer a remedy.
In this dissertation, a reset control design method is studied, which is based on ideas originated in the 1950's. The nonlinear controller is composed of a reset network, whose states are reset to zero when its input crosses zero, cascaded with a linear network. Using describing function analysis, the low-frequency performance of this nonlinear system is similar to that of a linear system where the resetting mechanism is not used. However, the control bandwidth is reduced, thereby reducing the "cost of feedback" beyond the limitation imposed by the LTI system's gain-phase relation. In this research, a theoretical framework for the reset control systems is developed, which has connection with the recently developed framework for systems with impulse effects. In this framework, the uniform exponential stability and uniform bounded-input-bounded-output stability for reset control systems are studied and a small-gain stability condition is derived. An optimal matrix norm search algorithm is developed to sharpen this small-gain condition. Based on these theoretical study, a set of engineering design guidelines for reset control are developed and are applied to a tape-drive servo control system. The simulations and experiments of the reset control for this tape-drive servo control system show the potential of the reset control design.