Projection-based integrators for improved motion control: Formalization, well-posedness and stability of hybrid integrator-gain systems

In this paper we formally describe the hybrid integrator-gain system (HIGS), which is a nonlinear integrator designed to avoid the limitations typically associated with linear integrators. The HIGS keeps the sign of its input and output equal, thereby inducing less phase lag than a linear integrator, much like the famous Clegg integrator. The HIGS achieves the reduced phase lag by projection of the controller dynamics instead of using resets of the integrator state, which forms a potential benefit of this control element. To formally analyze HIGS-controlled systems, we present an appropriate mathematical framework for describing these novel systems. Based on this framework, HIGS-controlled systems are proven to be well-posed in the sense of existence and forward completeness of solutions. Moreover, we propose two approaches for analyzing (input-to-state) stability of the resulting nonlinear closed-loop systems: (i) circle-criterion-like conditions based on (measured) frequency response data, and (ii) LMI-based conditions exploiting a new construction of piecewise quadratic Lyapunov functions. A motion control example is used to illustrate the results.