Reset control is concerned with how to reset a system when it is disturbed to overcome the inherent limitations of linear feedback control and to improve robustness. It has found applications in many practical systems including flexible mechanical systems, tapespeed control systems and high precision positioning systems.This book provides an introduction to the theory of reset control, and draws on the authors' own research and others' to explore the application of reset control in a variety of settings, with an emphasis on hard disk drive servo systems.Topics covered include the motivation and basic concepts of reset control systems; derivation of the describing function of reset systems; how the reset matrix affects the frequency domain property of a system; recent developments on stability analysis of reset control systems; robust stability of reset control systems with uncertainties; reset control systems with discrete-time reset conditions; optimal reset control design under fixed reset time instants; and reset control systems with conic jump sets. This book is essential reading for researchers, postgraduates and advanced students in control theory, and for research-based engineers who are interested in the theory of hybrid control systems and their engineering applications.
Inspec keywords: control system synthesis; control system analysis; discrete time systems; stability
Other keywords: reset control systems stability; discrete-time reset conditions; conic jump sets; reset control systems analysis; reset control systems design
Subjects: General and management topics; Control system analysis and synthesis methods; Discrete control systems; Stability in control theory
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.
One of the major difficulties in stability analysis of reset systems comes from uncertain reset time instants due to uncertain parameters. For instance, consider the case when a reset action is triggered whenever the tracking error crosses zero. If there are uncertain (constant or time-varying) parameters in the output matrix of a plant, then the time instants of zero-crossings also become uncertain. In practice, a well-designed reset control system (RCS) should be expected to be robust with respect to such uncertainty. This chapter is devoted to robust stability of RCSs with uncertainties in output matrices. The focus will be on single-output systems. Two kinds of uncertainties will be discussed. The first kind is bounded uncertain time-varying parameters and the second kind is uncertain constant parameters. Robust stability of RCSs with time-delay will also be discussed.
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.
The moving horizon optimal reset control and the infinite horizon optimal reset control design of this chapter are, respectively, based on References 3 and 8. The application of the moving horizon ORC to PZT-positioning stage is based on Reference 9. Propositions 6.1 and 6.2 are based on the common Lyapunov function condition and the Lie-algebraic criteria for stability of switched system in References 1, 10, 11, and 12. The solvability of the LQR problem for discrete-time systems can be found in Reference 13, for instance. The ORL designs proposed in this chapter are closely related to the optimal impulsive system design in Reference 14 where a nonlinear model was considered and a numerical algorithm for optimal impulse time instants and values has been given aiming to save computation time. There are many other techniques which have been developed and applied to improve track-seeking performance of HDD systems. They include time optimal control [15], [16], sliding mode control [17], and CNFC [4], [18], etc.
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.