Nonlinear Filters. Simon Haykin

alt="bold z Superscript o"/>, and unobservable modes, :

      (2.13)

Then, the state‐space model of (2.3) and (2.4) can be rewritten based on the transformed state vector, , as follows:

(2.14)

(2.15)

      or equivalently as:

(2.16)

(2.17)

Any pair of equations (2.14) and (2.15) or (2.16) and (2.17) is called the state‐space model of the system in the observable canonical form. For the system to be detectable (to have stable unobservable modes), the eigenvalues of must have negative real parts ( must be Hurwitz).

      2.4.2 Discrete‐Time LTI Systems

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

(2.18)

(2.19)

      where , , and are the state, the input, and the output vectors, respectively, and , , , and are the system matrices. Starting from the initial cycle, the system output vector at successive cycles up to cycle can be written based on the initial state vector and input vectors as follows:

      (2.20)

      The aforementioned equations can be rewritten in the following