Nonlinear Filters. Simon Haykin

      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) leads to:

      (3.55)

Hence, and is a free matrix. According to (3.46), we have:

      (3.56)

      (3.57)

      Defining , where has rows, we obtain:

      (3.58)

      Since is required to be a stable matrix, the pair must be detectable [35].

      From (3.35) and (3.36), we have:

(3.59)

      Assuming that has full column rank, there exists a matrix such that:

(3.60)

Left‐multiplying both sides of the equation (3.59) by , and then using (3.60), the input vector can be estimated based on the state‐vector estimate as:

      (3.61)

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