Sets of possible states of linear systems given perturbed observations

The state equation
x_{n} = A_{n}x_{n-1}+B_{n}W_{n}
and the output equation
y_{n} = C_{n}x_{n}+D_{n}W_{n}
relate the finite-dimensional vectors
x_{n}, y_{n}
and w
n
. The initial state x
0
is known to belong to a set X
0
and each w
n
. to a set W
n
, where
X_{0}, W_{1}, . . . , W_{N}
are compact and convex. Define
S_{n}(y_{1}, ... , y_{n})
as the set of all possible values of x
n
compatible with observation of outputs
y_{1}, ... , y_{n}
; it is compact and convex. S
n
plays the same role as the a posteriori distribution in the stochastic case and is determined recursively by
S_{n-1}(y_{1}, ... , y_{n-1})
and y
n
. The sets involved are completely characterized by their support functions. The law of evolution of the support function of S
n
is established. Some special cases and applications are pointed out.