Читать книгу Nonlinear Filters - Simon Haykin - Страница 34

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 3.3 provides an extension of the Luenberger observer for nonlinear systems.