  In this paper, we present an output feedback based design of event-triggered
sliding mode control for delta operator systems. For discrete-time systems,
multi-rate output sampling based state estimation technique is very useful if
the output information is available. But at high sampling rates, the
discrete-time representation of the system using shift operator becomes
numerically ill-conditioned and as a result, the observability matrix becomes
singular as the sampling period tends to zero. Here, a new formulation of
multi-rate state estimation (MRSE) for a small sampling period is presented. We
first propose a new observability matrix and then discuss its relationship with
the observability matrix defined in the conventional sense. For the delta
operator system with matched uncertainty, we have presented the design of MRSE
based sliding mode control (SMC). Additionally, to make the control efficient
in terms of resource utilization, MRSE based event-triggered SMC is proposed.
The absence of Zeno phenomenon is guaranteed as the control input is inherently
discrete in nature. Finally, the effectiveness of the proposed method is
illustrated through numerical simulations, considering a ball and beam system
and a general linear system as a numerical example.
