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Replacing for from (2.39) in the aforementioned equation, we will have:

      (2.44)

      whose energy is obtained from:

      (2.45)

      In the aforementioned equation, the matrix in the parentheses is called the continuous‐time observability Gramian matrix:

      (2.46)

      From its structure, it is obvious that the observability Gramian matrix is symmetric and nonnegative. If we apply a transformation, , to the state vector, , such that , the output energy:

      (2.47)

can be rewritten as:

      (2.48)

If the transformation is chosen in a way that the transformed observability Gramian matrix, , is diagonal, then, its diagonal elements can be viewed as the contribution of different state variables in the initial state vector to the energy of the output. The continuous‐time LTV system of (2.36) and (2.37) is observable, if and only if the observability Gramian matrix is nonsingular [9, 14].

      2.5.2 Discrete‐Time LTV Systems

      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)Скачать книгу