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 
.
, 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 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:
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 
:
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 
:
), 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.
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:
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 
).
by the pair (resp by ).