Simon Haykin - Nonlinear Filters

(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)

regardless of the values of . Since the input is unknown, the observer equation ( 3.41) does not depend on the input. Moreover, the system outputs up to time step are used to estimate the state at time step . Hence, the observer given by ( 3.41) is a delayed state estimator . Alternatively, it can be said that at time step , the observer estimates the state at time step [35].

In order to design the observer in ( 3.41), the matrices and are chosen regarding the state estimation error:

(3.43)

Using ( 3.40), the state estimation error can be rewritten as:

(3.44)

To force to go to zero, regardless of the values of and , must be a Hurwitz matrix (its eigenvalues must be in the left‐half of the complex plane), and must simultaneously satisfy the following conditions:

(3.45)

(3.46)

Existence of a matrix that satisfies condition ( 3.45) is guaranteed by the following theorem [35].

Theorem 3.1 There exists a matrix that satisfies (3.45), if and only if

(3.47)

Equation (3.47) can be interpreted as the inversion condition of the inputs with a known initial state and delay , which is a fairly strict condition. In the design phase, starting from , the delay is increased until a value is found that satisfies ( 3.47). However, is an upper bound for . To be more precise, if ( 3.47) is not satisfied for , then asymptotic state estimation will not be possible using the observer in ( 3.41).

In order to satisfy condition ( 3.45), matrix must be in the left nullspace of the last columns of given by . Let be a matrix whose rows form a basis for the left nullspace of :

(3.48)

then we have:

(3.49)

Let us define:

(3.50)

where is an invertible matrix. Then, we have: