Applying the equivalent control, , sliding mode occurs at the next step, and the governing dynamics of will be described by:
(3.31)
which converges to zero by properly choosing the eigenvalues of through designing the observer gain matrix, . In order to place the eigenvalues of at the desired locations, the pair must be observable. This condition is satisfied if the pair is observable. This leads to a sliding‐mode realization of the standard reduced order asymptotic observer [37].
In (3.30), is unknown. Therefore, the following recursive equation is obtained from (3.29) to compute :
Now, the equivalent observer auxiliary input can be calculated as:
(3.33)
where 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:
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]:
v Subscript k Superscript e q"/>, can be found as [37, 38]:
