  This paper is a survey of extensions to finite automata theory to model
real-time systems as well as systems exhibiting mixed discrete-continuous
behavior. Real-time systems maintain a continuous and timely interaction with
the environment, often adhering to some timing constraints. Therefore, the
finite automata theory is extended to measure real-time values and accept or
reject runs on a class of automata known as timed-automata, upon satisfying
some timed properties. The automata modeling the mixed discrete-continuous
behavior of hybrid systems has its continuous-time dynamics described using
ordinary differential equations for the state space and discrete-time dynamics
describing the control decisions. Based on these dynamical system models, we
likewise extend the finite automata theory to describe the behavior of hybrid
systems using Hybrid Automata. We further study some applications of this class
of systems, sometimes referred to as Cyber-physical systems and perform a
case-study on Peterson's Mutual Exclusion protocol using Uppaal.
