On Convergence of Systems with Sector-Bounded Hybrid Integrators

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.…”

“…It essentially shows that (10) inherits the incremental dissipativity property (5) in a subregion of the incremental input-output (δz, δu)-space. 2 Property 1 [27]: Suppose Assumption 2 is satisfied. Then, the incremental system in (10) satisfies for all (δz, δx h ) ∈ (δẋ h )δx h ≤ (−c 1 δx h + c 2 δz)δx h (11) where…”

“…Notable exceptions are [23], [25], [26], where sufficient conditions for discontinuous piecewise affine (PWA) systems are formulated in terms of linear matrix inequalities (LMIs). Recently, we have started to address this shortcoming by deriving sufficient conditions for δ-ISS of HIGS, focussing on the complete closed-loop dynamics as a whole [27]. The first main contribution of the current paper is to demonstrate that under appropriate assumptions, the hybrid integrator possesses a δ-ISS property.…”

“…The second main contribution is a novel and "composite" condition that guarantees the feedback interconnection of such hybrid integrator and an LTI plant to be δ-ISS. Here, we do not use as in [27] a Lyapunov-based dissipativity mechanism that requires certain passivity properties of the plant. Instead, our new conditions are based on a small-gain argument and guarantee δ-ISS for plants that possibly violate the earlier passivity requirements [27].…”

“…Here, we do not use as in [27] a Lyapunov-based dissipativity mechanism that requires certain passivity properties of the plant. Instead, our new conditions are based on a small-gain argument and guarantee δ-ISS for plants that possibly violate the earlier passivity requirements [27]. This is demonstrated in this letter through a numerical example.…”