# Observability:

## 1. General result:

A linear continuous-time time-invariant system: , is observable iff: from the observation of for a given final time , it is possible possible to determine the initial value of the state . Let n be the system order ( ). The observabilty property depends only on matrices and . In the sequel, we will consider the pair instead of the 4 matrices of the system. The pair is observable iff: 𝒪 is called the observability matrix .

## 2. Proof:

Let us consider the first time-derivative of the free () response of the output : ⋮

or:

In the (non-restrictive) single output case (), 𝒪 is a matrix. Thus, at any time , on can determine from , , ⋯, iff : Remark: in this approach, it is assumed that one can derive at any time t in using perfect non causal derivators (i.e.: knowing the whole trajectory ). In a real-time implementation, we must keep in mind that such a perfect derivation is not realizable. ## 3. Duality:

The dual system of the primal system defined by the 4 state-space matrices , , , also noted: is:

The controllablity (resp. observability) conditions can be converted into observability (resp. controllabiity) conditions by changing the pair by the pair (resp by ).