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The state‐space model of a discrete‐time LTV system is represented by the following algebraic and difference equations:

(2.49)

(2.50)

Before proceeding with a discussion on the observability condition, we need to define the discrete‐time state‐transition matrix, , as the solution of the following difference equation:

(2.51)

with the initial condition:

(2.52)

The reason that is called the state‐transition matrix is that it describes the dynamic behavior of the following autonomous system (a system with no input):

(2.53)

with being obtained from

(2.54)

Following a discussion on energy of the system output similar to the continuous‐time case, we reach the following definition for the discrete‐time observability Gramian matrix:

(2.55)

As before, the system (2.49) and (2.50) is observable, if and only if the observability Gramian matrix is full‐rank (nonsingular) [9].