Incremental Stability and Convergence Properties of Discrete-Time Projected Control Systems

Abstract

Projection-based controllers can overcome fundamental limitations of classical linear time-invariant control by modifying the controller's input-output behavior via projection. A key example is given by the hybrid integrator-gain system, a projected integrator, which has recently found successful application in several industrial systems. While prior work on analysis and design of projection-based control systems has primarily focused on the continuous-time setting and non-incremental analysis, a more refined incremental analysis in discrete-time is needed to better reflect actual digital implementation and obtain more accurate (robust) performance assessment. To address this need, this paper considers incremental stability and convergence analysis of discrete-time projection-based control systems. Our first methodology is based on showing that such controllers preserve the quadratic incremental stability of their nominal (unprojected) dynamics, if the projection metric is well-designed. Building on this, we derive a small-gain condition guaranteeing incremental input-to-state stability for interconnections of projected controllers with general nonlinear plants. A second approach is grounded in a direct Lyapunov-based method for verifying incremental stability in input-affine piecewise-smooth systems, which can be seen as an extension of the classical discrete-time Demidovic conditions. We illustrate our results through several examples, and demonstrate performance quantification via nonlinear Bode plots, with a special focus on first-order projection elements.