An impulsive dynamical systems framework for reset control systems
For control theorists and engineers, this work provides rigorous mathematical foundations for reset control systems, addressing fundamental issues of well-posedness and sensitivity, though it is incremental as it applies existing impulsive systems theory to a specific class of systems.
This paper uses impulsive dynamical systems theory to analyze well-posedness of reset control systems with linear time-invariant base systems and zero-crossing resetting laws. It provides necessary and sufficient conditions for existence and uniqueness of solutions, shows absence of Zeno solutions for strictly proper plants, and establishes conditions for continuous dependence on initial conditions, which is then used to analyze sensitivity to sensor noise.
Impulsive dynamical systems is a well-established area of dynamical systems theory, and it is used in this work to analyze several basic properties of reset control systems: existence and uniqueness of solutions, and continuous dependence on the initial condition (well-posedness). The work scope is about reset control systems with a linear and time-invariant base system, and a zero-crossing resetting law. A necessary and sufficient condition for existence and uniqueness of solutions, based on the well-posedness of reset instants, is developed. As a result, it is shown that reset control systems (with strictly proper plants) do no have Zeno solutions. It is also shown that full reset and partial reset (with a special structure) always produce well-posed reset instants. Moreover, a definition of continuous dependence on the initial condition is developed, and also a sufficient condition for reset control systems to satisfy that property. Finally, this property is used to analyze sensitivity of reset control systems to sensor noise. This work also includes a number of illustrative examples motivating the key concepts and main results.