Simon Haykin - Nonlinear Filters

(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, , can be found as [37, 38]:

(3.30)

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 :

(3.32)

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)

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]: