Incremental Stability of Discrete-Time First-Order Projection Elements

Abstract

In this paper, we present and analyze a class of discrete-time and sampled-data controllers named first-order projection elements (FOPEs). FOPEs form a class of hybrid controllers that have an underlying linear base dynamics, which are combined with a projection operator that ensures that the input-output (I/O) pair is contained in a well-designed sector. The I/O sector-constraint enables performance benefits in various application domains including motion control of high-precision machines. After the introduction of FOPEs, we focus on incremental stability analysis of these hybrid controllers. We show that the properties of the projection operator ensure incremental stability of the open-loop FOPE dynamics in case of an underlying linear dynamics that is exponentially stable. In this case, we compute tight bounds on the incremental gain. Under an appropriate small-gain condition, we show incremental input-to-state stability of the sampled-data interconnection of a FOPE with a continuous-time linear time-invariant plant. For the case of projected integrator dynamics, we provide a counterexample showing that incremental stability can be compromised. We provide additional, necessary and sufficient conditions under which FOPEs with integrator dynamics are incrementally asymptotically stable. Numerical simulations provide a practical example of the established incremental properties for the trajectories of a dynamical system coming from an industrial motion control application.