Observability - Definition

Definition

Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs (this definition is slanted towards the state space representation). Less formally, this means that from the system's outputs it is possible to determine the behaviour of the entire system. If a system is not observable, this means the current values of some of its states cannot be determined through output sensors. This implies that their value is unknown to the controller (although they can be estimated through various means).

For time-invariant linear systems in the state space representation, a convenient test to check if a system is observable exists. Consider a SISO system with states (see state space for details about MIMO systems), if the row rank of the following observability matrix

is equal to, then the system is observable. The rationale for this test is that if rows are linearly independent, then each of the states is viewable through linear combinations of the output variables .

A module designed to estimate the state of a system from measurements of the outputs is called a state observer or simply an observer for that system.

Observability index

The Observability index of a linear time-invariant discrete system is the smallest natural number for which is satisfied that, where

Detectability

A slightly weaker notion than observability is detectability. A system is detectable if and only if all of its unobservable modes are stable. Thus even though not all system modes are observable, the ones that are not observable do not require stabilization.