  The Impulsive Goodwin's Oscillator (IGO) is a mathematical model of a hybrid
closed-loop system. It arises by closing a special kind of continuous linear
positive time-invariant system with impulsive feedback, which employs both
amplitude and frequency pulse modulation. The structure of IGO precludes the
existence of equilibria, and all its solutions are oscillatory. With its origin
in mathematical biology, the IGO also presents a control paradigm useful in a
wide range of applications, in particular dosing of chemicals and medicines.
Since the pulse modulation feedback mechanism introduces significant
nonlinearity and non-smoothness in the closedloop dynamics, conventional
controller design methods fail to apply. However, the hybrid dynamics of IGO
reduce to a nonlinear, time-invariant discrete-time system, exhibiting a
one-to-one correspondence between periodic solutions of the original 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. A simple and efficient local stability condition of
the 1-cycle in terms of the characteristics of the amplitude and frequency
modulation functions is obtained.
