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)
