Observability of Linear Time Invariant Dynamical System

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Addis Ababa University


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.



Observability of Linear Time