then, unknown inputs such as faults and attacks will be discussed. Since state estimators are usually implemented using digital processors, emphasis will be put on methods in which measurements are available at discrete sampling times. There are two classes of observers regarding the order of the observer as a dynamic system and the order of the system under study whose state is to be estimated:
Full‐order observers: The order of the observer is equal to the order of the main system.
Reduced‐order observers: The order of the observer is less than the order of the main system.
In the following, the symbol “ ” denotes the estimated variable. Since such variable is determined based on the measured outputs, for clarification, two time indices will be used: one for the estimate and the other for the measurement. For discrete‐time systems, denotes the estimate of the state, , at time instant , given output measurements up to time instant . Similarly, for continuous‐time systems, denotes the estimate of the state, , at time instant , given output measurements up to time instant . The estimate is
a smoothed state estimate, if ,
a filtered state estimate, if , and
a predicted state estimate, if .
For control applications, usually the control law uses the filtered state estimate, which is the current estimate based on all available measurements. The predicted estimate relies on the state transition model to extrapolate the filtered estimate into the future. Such estimates are usually used for computing the objective or cost function in model predictive control. Smoothed estimates are computed in an offline manner based on both past and future measurements. They provide a more accurate estimate than the filtered ones. They are usually used for process analysis and diagnosis [9].
3.2 Luenberger Observer
A deterministic discrete‐time linear system is described by the following state‐space model:
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.
