switchedsystem

A switched system, sometimes written as switched-system, is a dynamical system that can switch between a finite set of subsystems according to a switching signal. In continuous time, its state evolves according to x_dot = A_sigma(t) x + B_sigma(t) u, where sigma(t) selects one mode i from a finite index set. Each mode i has its own matrices (A_i, B_i, C_i, D_i, …). The switching signal can be time-driven, state-driven, or event-driven, and may be subject to constraints such as a minimum dwell time to limit rapid switching.

Stability and analysis for switched systems depend on both the individual subsystem dynamics and the switching pattern. It is possible for all subsystems to be stable yet for the overall switched system to be unstable under arbitrary switching. Common analytical tools include Lyapunov methods: a common (uniform) Lyapunov function that decreases for every mode guarantees stability under arbitrary switching; when such a function does not exist, multiple Lyapunov functions paired with dwell-time or average dwell-time conditions can ensure stability and performance.

Control design for switched systems aims to ensure stability or specified performance under switching. Techniques include mode-dependent state feedback, observer design, and switching laws that manage transitions. Practical considerations include robustness to modeling uncertainty, actuator limits, and measurement noise.

Applications are found in power electronics, automotive control, robotics, networked control systems, and process automation, where different modes model distinct operating regimes or fault conditions. Switched systems are a core concept within the broader framework of hybrid systems, capturing the interaction of discrete mode changes with continuous dynamics.