1. State Estimators or Observers

• One should note that although state feed back control is very attractive because of precise computation of the gain matrix K, implementation of a state feedback controller is possible only when all state variables are directly measurable with help of some kind of sensors.

• Due to the excess number of required sensors or unavailability of states for measurement, in most of the practical situations this requirement is not met.

• Only a subset of state variables or their combinations may be available for measurements. Sometimes only output y is available for measurement.

• Hence the need for an estimator or observer is obvious which estimates all state variables while observing input and output.

Full Order Observer : If the state observer estimates all the state variables, regardless of whether some are available for direct measurements or not, it is called a full order observer.

Reduced Order Observer : An observer that estimates fewer than ``n'' states of the system is called reduced order observer.

Minimum Order Observer : If the order of the observer is minimum possible then it is called minimum order observer.

2 Full Order Observers

Consider the following system

where $\mathbf{x} \in R^{n\times 1}$ , $\mathbf{u} \in R^{m\times 1}$ and $\mathbf{y} \in R^{p\times 1}$ .

Assumption : The pair (A,C) is observable.

Goal : To construct a dynamic system that will estimate the state vector based on the information of the plant input u and output y.

2.1 Open Loop Estimator

The schematic of an open loop estimator is shown in Figure 1.

Figure 1: Open Loop Observer

$\begin{figure}\centering \begin{pspicture}(0,0)(8,6) \pnode(-2,4){A} \rput(-2,... ...2.5){$\hat{\mathbf{y}}(k)$}\ncline{->}{M1}{Y1} \end{pspicture}\par \end{figure}$

The dynamics of this estimator are described by the following

where $\hat{\mathbf{x}}$ is the estimate of x and $\hat{\mathbf{y}}$ is the estimate of y