Nonlinear Filters. Simon Haykin

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(3.46)

Existence of a matrix that satisfies condition (3.45) is guaranteed by the following theorem [35].

      Theorem 3.1 There exists a matrix that satisfies (3.45), if and only if

(3.47)

Equation (3.47) can be interpreted as the inversion condition of the inputs with a known initial state and delay , which is a fairly strict condition. In the design phase, starting from , the delay is increased until a value is found that satisfies (3.47). However, is an upper bound for . To be more precise, if (3.47) is not satisfied for , then asymptotic state estimation will not be possible using the observer in (3.41).

      In order to satisfy condition (3.45), matrix must be in the left nullspace of the last columns of given by . Let be a matrix whose rows form a basis for the left nullspace of :

      (3.48)

      then we have:

      (3.49)

      Let us define:

      (3.50)

      where is an invertible matrix. Then, we have:

      (3.51)

      To choose , note that:

      (3.52)

From Theorem 3.1, the first columns of must be linearly independent of each other and of the other columns. Now, is chosen such that:

      (3.53)

      Regarding (3.45), can be expressed as:

      (3.54)

      where has columns. Then, equation (3.45)

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