Nonlinear Filters. Simon Haykin

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The matrix is the observability matrix for the pair , and is the invertibility matrix for the tuple . The matrices and can also be expressed as [35]:

      (3.38)

      (3.39)

Equation (3.37) can be rewritten in the following compact form:

(3.40)

      Then, the dynamic system

(3.41)

      is a UIO with delay , if

(3.42)

regardless of the values of . Since the input is unknown, the observer equation (3.41) does not depend on the input. Moreover, the system outputs up to time step are used to estimate the state at time step . Hence, the observer given by (3.41) is a delayed state estimator. Alternatively, it can be said that at time step , the observer estimates the state at time step [35].

      In order to design the observer in (3.41), the matrices and are chosen regarding the state estimation error:

      (3.43)

Using (3.40), the state estimation error can be rewritten as:

      (3.44)

To force to go to zero, regardless of the values of and , must be a Hurwitz matrix (its eigenvalues must be in the left‐half of the complex plane), and must simultaneously satisfy the following conditions:

(3.45)СКАЧАТЬ