Stability Properties of the Impulsive Goodwin’s Oscillator in 1-cycle

The Impulsive Goodwin’s Oscillator (IGO) is a mathematical model that represents a hybrid closed-loop system. It arises by closing a specific of type continuous positive linear time-invariant system with impulsive feedback, incorporating both amplitude and frequency pulse modulation. The structure of the IGO precludes the existence of equilibria and thus ensures that all of its solutions, whether periodic or non-periodic, are oscillatory. Originating in mathematical biology, the IGO also constitutes a control paradigm applicable to a wide range of fields, particularly to closed-loop dosing of chemicals and medicines. The pulse modulated feedback introduces strong nonlinearity and non-smoothness into the closed-loop dynamics thus rendering conventional controller design methods not applicable. However, the hybrid dynamics of IGO reduce to a nonlinear discrete-time system, exhibiting a one-to-one correspondence between solutions of the original hybrid IGO and those of the discrete-time system. The paper proposes a design approach that leverages the linearization of the equivalent discrete-time dynamics in the vicinity of a fixed point. An efficient local stability condition of the 1-cycle in terms of the characteristics of the amplitude and frequency modula- tion functions is obtained. Unlike the conventional Schur-Cohn and Jury stability conditions applied to the Jacobian matrix, the obtained criterion requires checking a single inequality that is linear in the slopes of the modulation characteristics.