Stability and performance analysis of hybrid integrator–gain systems: A linear matrix inequality approach

Hybrid integrator–gain systems (HIGS) are nonlinear elements designed for overcoming fundamental limitations of linear time-invariant integrators. This paper presents numerically robust conditions for time-domain stability and performance of HIGS-controlled systems. In particular, using piecewise quadratic (PWQ) Lyapunov functions defined over polyhedral subregions of the state-space, conditions for stability and computations of upper-bounds on the L₂-gain and H₂-norm are formulated as convex optimization problems in terms of numerically tractable linear matrix inequalities (LMIs). In order to improve accuracy and robustness, the LMIs are constructed in a manner to eliminate explicit equality constraints typically related to continuity of the PWQ functions. Novel conditions are presented that guide further refinement of the subregions over which the PWQ Lyapunov functions are defined in order to increase the accuracy of the approach. The effectiveness of the presented analysis tools is demonstrated through a numerical case-study on a motion system.