Linearization modeling for non-smooth dynamical systems with approximated scalar sign function

Traditionally, one typical way to deal with the sign function constrained non-smooth system is to divide the state space into two or more subspaces based on the direction of sign function. The resulting piecewise model is with complicated structure, leading to the difficulty in controller design as well as stability analysis. In this paper, an innovative concept of approximated scalar sign function is introduced. Through the proposed methodology, the non-smooth dynamical system model can be transformed into a universal and smooth model. Thus, the following optimal linearization can be applied to obtain the local linear model at any operating point. Finally, two illustrative examples, representing hysteresis and friction, respectively, are given to demonstrate the effectiveness of the proposed method.