Analysis of reset control systems
It is well-appreciated that Bode's gain-phase relationship places a hard limitation on performance tradeoffs in linear, time-invariant feedback control systems, Specifically, the need to minimize the open-loop high-frequency gain often competes with low-frequency loop gain and phase margin constraints. Our focus on reset control systems is motivated by recent work that showed its potential to improve this situation.
The basic concept in reset control is to reset the state of a linear controller to zero whenever its input meets a threshold. Typical reset controllers include the so-called Clegg integrator and the first-order reset element. The Clegg integrator has a describing function similar to the frequency response of a linear integrator but with only 38.1° phase lag. This was the original motivation.
Reset control is a hybrid control scheme. It resembles a number of popular non-linear control strategies including relay control, sliding-mode control and switching control. A common feature to these is the use of a switching surface to trigger change in control signal. Distinctively, reset control employs the same (linear) control law on both sides of the switching surface. Resetting occurs when the system trajectory impacts this surface. The reset action renders a discontinuity (a jump) in the system trajectory. This behavior is not invertible. The reset action can be alternatively viewed as the injection of judiciously-timed impulses of state-dependent magnitude into an otherwise linear, time-invariant feedback system.
In this dissertation we analyze stability and steady-state performance of reset control systems. First, we provide an example of reset control overcoming a limitation of linear feedback. Secondly, we present our main stability results: a necessary and sufficient condition for quadratic stability that under a certain assumption on reset controller dynamics implies uniform bounded-input bounded-state stability. We show that this condition is satisfied for a large and important class of reset control systems. We also introduce a passivity-based approach to stability. Next, we give our steady-state performance results. In doing this, we establish internal model and superposition principles, existence and local stability of periodic solutions due to sinusoidal sensor noise excitation. Finally, we suggest future research directions.