Nonlinear Filters. Simon Haykin

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(3.21)

where , , and . It is assumed that the pair is observable, and is full rank. The goal is to reconstruct the state vector from the input and the output vectors using the discrete‐time sliding‐mode framework. Using a nonsingular transformation matrix, , the state vector is transformed into a partitioned form:

      (3.22)

      such that the upper partition has the identity relationship with the output vector:

      (3.23)

Then, the system given in (3.20) and (3.21) is transformed to the following canonical form:

(3.24)

      where

      (3.25)

      (3.26)

      The corresponding sliding‐mode observer is obtained as:

(3.27)

      where , and is the observer gain. Defining the estimation errors as:

      (3.28)

and subtracting (3.27) from (3.24), we obtain the following error dynamics:

(3.29)

According to (3.29), the state estimation problem can be viewed as finding an auxiliary observer input in terms of the available quantities such that the observer estimation errors and are steered to zero in a finite number of steps. To design the discrete‐time sliding‐mode observer, the sliding manifold is defined as . Hence, by putting in (3.29), the equivalent control input, СКАЧАТЬ