Observability

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Observability in control theory is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals.

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 and, consequently, that it will be unable to fulfil the control specifications referred to these outputs.

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

$O=\begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{n-1} \end{bmatrix}$

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

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 $v$ of a linear time-invariant discrete system is the smallest natural number for which is satisfied that $r a n k (O v) = r a n k (O v + 1)$ , where

$O_v=\begin{bmatrix} C \\ CA \\ CA^2 \\ \vdots \\ CA^{v-1} \end{bmatrix}$

Detectability

A system is detectable if and only if all of its unstable modes are observable

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