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Variable structure control with sliding sector based on hybrid switching law

In general, a variable structure (VS) control system should be designed such that a sliding mode (or a boundary layer, or a high‐order sliding mode, or a sliding sector), where the reduced‐order system is stable, is reached in finite time, i.e. a reaching movement is necessary for VS control systems. This paper proposes a VS controller without the requirement of reaching movement. With the proposed controller, a system is stabilized by switching the system control input among a set of feedback control laws no matter whether a sliding mode happens or not. Each feedback control law is easy to be designed, because it uses only one of state variables with a fixed plus or minus feedback gain. For an n‐order system, 2n feedback control laws are needed to realize the proposed VS control. The concept of the sliding sector is used in the paper to show that there exists an area for each feedback control law, inside which a Lyapunov function decreases with the control law although the system may not be stabilized by it alone. A common Lyapunov function for all control laws exists if the system is controllable. Based on the extremum seeking control algorithm, a switching function is defined as the variation between the Lyapunov function and a pre‐determined decrease function. Then a VS control rule is designed such that a series of sliding modes happen with the switching function being a constant and the Lyapunov function decreases in every period after switching once through all control laws. The proposed VS control system is quadratically stable. Simulation results are given to show the convergence of the proposed VS control algorithm. Copyright © 2007 John Wiley & Sons, Ltd.