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href="#fb3_img_img_3b78f05e-6f91-5c00-8e7b-828d2a533e02.png" alt="bold e Subscript bold z Sub Subscript k minus 1 Sub Superscript l"/> is obtained from (3.32). Given , the discrete‐time sliding‐mode observer provides the state estimate as [37]:

      (3.34)

3.5 Unknown‐Input Observer

      The unknown‐input observer (UIO) aims at estimating the state of uncertain systems in the presence of unknown inputs or uncertain disturbances and faults. The UIO is very useful in diagnosing system faults and detecting cyber‐attacks [35, 39]. Let us consider the following discrete‐time linear system:

(3.35)

(3.36)

where , , and . It is assumed that the matrix has full column rank, which can be achieved using an appropriate transformation. Response of the system (3.35) and (3.36) over time steps is given by [35]:

(3.37)

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)

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