In linear feedback control design, performance specifications can be given in both the time and the frequency domains. In the frequency domain, specifications are usually given in terms of the gain and phase properties of open-loop transfer functions over certain frequency range. For instance, consider a standard linear feedback system where P(s) and C(s) are the plant and the controller, respectively, d _i, d _o, and n are the input disturbance, output disturbance, and measurement noise, respectively, r, y, and e are the reference input, output, and tracking error, respectively. In this book, we focus on a special kind of hybrid techniques called reset control whose original motivation is to overcome the inherent limitation imposed by the Bode's gain-phase constraint. The study of reset control can be traced several decades back. The first reset element is the so-called Clegg integrator (CI) proposed by Clegg in 1958. The CI is described by the impulsive differential equation which consists of a linear integrator (LI) and a reset mechanism. When the input e and the output z of the integrator have the same sign, then it evolves according to the LI. On the other hand, if the input and the output have opposite signs, then the state is reset to zero. The notation z ⁺ denotes the state of the integrator after the reset. The condition ze ≤ 0 is called the reset condition which determines when the state of the integrator should be reset.

Stability is a basic requirement for reset control systems (RCSs). However, for RCSs with zero-crossing type of reset, because of the dependence of the reset time instants on the system state and input, the stability analysis of RCSs is challenging. This chapter presents some major developments concerning the stability of RCSs with zero-crossing type of reset in recent years. These results include quadratic stability of RCSs with and without time-delays, reset time-dependent stability criteria, and finite L₂-gain stability. These results were originally developed for RCSs with zero reset matrices, i.e., part of the state is reset to zero whenever the tracking error crosses zero, but most of them are slightly modified to adapt to arbitrarily reset matrices case in this chapter.

Note that the input-dependent reset mechanism brings possibility of Zeno behavior and beating phenomenon. One approach to avoid these phenomena is to limit the reset rate by time-regularization. Another way is to sample the triggering signal at a pre-specified rate and reset the controller states only when the discretized triggering signal crosses zero. Precisely, let the tracking error e(t) be the triggering signal and Ts the sampling period.

This chapter discusses reset control systems (RCSs) with conic jump sets. Different from the zero-crossing type reset where a jump set is a subset with zero measure, a conic jump set is a closed conic in the state space or the input/output space which contains a nonempty interior. The key role a conic jump set plays is to remove part of the state space and force the trajectories of a system to evolve within the remaining part. This kind of reset leads to less conservative Lyapunov stability conditions and is capable of making a system passive. This chapter only briefly introduces the main idea and collects some of the main results reported in recent literature including passification via reset and L₂-gain analysis.