Incremental Input-to-State Stability for Discontinuous Dynamical Systems With First-Order Projection Elements

The notion of incremental stability provides a powerful tool for the analysis and design of nonlinear systems. This paper is concerned with establishing incremental stability properties for a class ofdiscontinuousdynamical systems that are formed by the feedback interconnection of a linear time-invariant plant (possibly including linear control elements) and afirst-order projection-based element. This class of control systems has received a lot of attention due to its abilities to overcome fundamental performance limitations of linear control and recent successes in industrial motion control. By deriving and exploiting dissipativity properties of the projection element's discontinuous vector field that hold only in certain subregions of its incremental input-output space, a tailored piecewise quadratic incremental Lyapunov function is constructed by appropriately “connecting” local incremental quadratic storage functions. Based on this result, conditions for guaranteeing incremental input-to-state stability of the considered class of discontinuous dynamical systems are formulated in the form of numerically tractable linear matrix inequalities. The benefits of these results are demonstrated through a numerical case-study showing “nonlinear” Bode-plots for accurate steady-state performance analysis.