### Abstract:

This project paper is aimed to explain observability of linear time invariant dynamical
system. We develop a linear systems theory that coincides with the existing theories for
continuous and discrete dynamical systems. We explore observability in terms of both
Gramian and rank conditions and establish related realizability results.
An observable system is one in which the latent variables can be reconstructed from
the manifest variables (in sate space system, the manifest variables are input and output
and the latent variable is the state). In order to reconstruct the state at any time from
the input and the output, due to the property of state, it su ces to reconstruct state at
a speci c time to, then, we know it every where in the future, i.e, for all t t0. Thus, we
only need to reconstruct x(0). We will also state necessary and su cient conditions for
the recostructiblity of the state x(0) or observability of the system, namely, Kalman
observability test, Hautus observability test and observability test using the Gramian
matrix of the system.
In addition, if the system is not observable, i.e, if the state x(0) is not reconstructible,
using Kalman observability decomposition, we will identify which components of x(0)
are reconstructible and which are not. Finally we will give a test for observability of a
behaviour. Some examples are included to show the utility of these results.
The rst chapter of the paper mainly discusses basic preliminaries for the discussion
of the main topic "Observability of linear time invariant dynamical system". In here we
will dene several terminologies both verbally and mathematically. we will also study
and proof some basic theorems. The later chapter discusses observability for linear time
invariant dynamical system. Several system properties will be developed and used in
checking observability of a given system and prooing system related theorems.