Simon Haykin - Nonlinear Filters

To simplify the notation, let us define such that

(3.12)

then we have:

Defining

(3.13)

the filtered state estimate is computed from the following discrete‐time system [9]:

(3.14)

(3.15)

where is the observer gain, , and is the discrete nonlinear observability matrix, which is computed at sample time as:

(3.16)

The observer gain is determined in a way to guarantee local stability of the perturbed linear system for the reconstruction error in :

(3.17)

where and

This approach for designing nonlinear observers is applicable to systems that are observable for every bounded input, , with and being uniformly Lipschitz continuous functions of the state:

(3.18)

(3.19)

where and denote the corresponding Lipschitz constants. However, convergence is guaranteed only for a neighborhood around the true state [36].

The equivalent control approach allows for designing the discrete‐time sliding‐mode realization of a reduced‐order asymptotic observer [37]. Let us consider the following discrete‐time state‐space model:

(3.20)

(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: