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

A Small-Gain Approach to Incremental Input-to-State Stability Analysis of Hybrid Integrator-Gain Systems

“…, and where f h : R × R → R is a nonlinear function. Note that (2a) is equivalent to the description used in [27] and generalizes [9], where f h was restricted to be linear. Allowing f h to be nonlinear provides additional freedom in controller design, and allows to include a broader class of controllers, e.g., variable-gain or anti-windup integrators.…”

“…Intuitively, such sign equivalence helps in reducing the phase lag typically induced by LTI integrators/low-pass filters as a consequence of Bode's gain-phase relationship, and may benefit robust performance and stability properties when placing a hybrid integrator in closed-loop with an LTI plant. A detailed discussion on the construction of the sets in ( 3) and ( 4), along with a visualization, can be found in [9,Sec. 3], whereas examples (both industrial and academic) motivating and demonstrating the performance potential of the hybrid integrator are given in, e.g., [9], [10], [14].…”

“…A detailed discussion on the construction of the sets in ( 3) and ( 4), along with a visualization, can be found in [9,Sec. 3], whereas examples (both industrial and academic) motivating and demonstrating the performance potential of the hybrid integrator are given in, e.g., [9], [10], [14].…”

“…T HE IDEA of developing nonlinear control strategies that can overcome some of the fundamental limitations of linear time-invariant (LTI) control for LTI systems already dates back to the 1950's when J.C. Clegg introduced its celebrated integrator with resetting mechanism [1]. Since its introduction, the Clegg integrator has inspired many alternative strategies including generalized reset elements [2], [3], [4], [5], split-path integrators [7], [8], switching controllers [6], and hybrid integrator-gain systems (HIGS) [9], [10], [11]. HIGS recently gained much attention due to its ability to overcome fundamental limitations of linear feedback control [10] and various engineering successes were reported in industrial applications such as wafer scanners [14] and atomic force microscopes [11].…”

“…As stability is a prerequisite for control system performance, a particular challenge to be solved for the non-smooth and even discontinuous control strategies mentioned above lies in the development of constructive tools for stability analysis. Over the years, many tools for stability analysis have been proposed, ranging from signal-based approaches [16], [31] to Lyapunov techniques [3], [9] and graphical methods [4], [12]. Most of these methods, however, primarily focus on stability of some equilibrium point, typically the zero equilibrium, or an equilibrium set.…”