Simon Haykin - Nonlinear Filters

A deterministic discrete‐time linear system is described by the following state‐space model:

(3.1)

(3.2)

where , , and denote state, input, and output vectors, respectively, and are the model parameters, which are matrices with appropriate dimensions. Luenberger observer is a sequential or recursive state estimator, which needs the information of only the previous sample time to reconstruct the state as:

(3.3)

where is a constant gain matrix, which is determined in a way that the closed‐loop system achieves some desired performance criteria. The predicted estimate, , is obtained from ( 3.1) as:

(3.4)

with the initial condition .

The dynamic response of the state reconstruction error, , from an initial nonzero value is governed by:

(3.5)

The gain matrix, , is determined by choosing the closed‐loop observer poles, which are the eigenvalues of . Using the pole placement method to design the Luenberger observer requires the system observability. In order to have a stable observer, moduli of the eigenvalues of must be strictly less than one. A deadbeat observer is obtained, if all the eigenvalues are zero. The Luenberger observer is designed based on a compromise between rapid decay of the reconstruction error and sensitivity to modeling error and measurement noise [9]. Section 3.3provides an extension of the Luenberger observer for nonlinear systems.

A class of nonlinear observers is designed based on observer error linearization. State estimation by such observers involves multiplication of a matrix gain by the difference between predicted and measured outputs. After designing the observer (selecting the observer gain), this calculation can be performed fairly quickly, which is an advantage from the computational complexity perspective. However, the domain of applicability of such observers may be restricted to a specific class of nonlinear systems. A discrete‐time nonlinear Luenberger‐type observer has been proposed in [36]. A deterministic discrete‐time nonlinear system is described by the following state‐space model:

(3.6)

(3.7)

where and are nonlinear vector functions. At time instant , let denote the sequence of the past inputs:

(3.8)

where is the dimension of the state vector. Given an initial state and a sequence of previous inputs, a series of predicted outputs can be generated using the state equation ( 3.6) as:

(3.9)

Let denote the th composite of the system function as:

(3.10)

Given an initial state and the sequence , the corresponding sequence of predicted outputs is obtained as:

(3.11)