New Math Keeps Precision Machines From Wobbling Off Course

A new paper on arXiv sets out incremental stability and convergence analysis for discrete-time projection-based control systems. Projection-based controllers, the authors write, can overcome fundamental limitations of classical linear time-invariant control by modifying the controller's input-output behavior via projection. A key example is the hybrid integrator-gain system, a projected integrator, which has recently found successful application in several industrial systems. Prior work on the analysis and design of projection-based control systems has primarily focused on the continuous-time setting and non-incremental analysis, and the authors argue a more refined incremental analysis in discrete-time is needed to better reflect actual digital implementation and obtain more accurate, robust performance assessment. The paper presents two methodologies. The first is based on showing that such controllers preserve the quadratic incremental stability of their nominal, unprojected dynamics if the projection metric is well-designed, and building on that, derives a small-gain condition guaranteeing incremental input-to-state stability for interconnections of projected controllers with general nonlinear plants. The second 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. The results are illustrated through several examples, with performance quantification via nonlinear Bode plots and a special focus on first-order projection elements.