State Estimation for Generalized First-Order Systems

Many practical applications give rise to state-space systems of the form

Here , , , and .

Once more, the state feedback control law

is considered. Feeding this information back into the system gives the closed-loop system

The concepts of controllability and observability of the system can be defined in terms of the associated standard state-space system.

The analogue of the Sylvester-observer equation for the generalized system is

where and are given and the matrices , and are to be found. Last equation will be referred to, by analogy, as the generalized Sylvester-observer equation.

The Luenberger-observer for the generalized system is the same as that of the standard system. That is, it is given by a system of differential equations

where any initial condition can be taken.

It can be shown that, if is a stable matrix, then approaches zero as time increases.

If a full-order observer is constructed (), then an estimate to the state vector is obtained by solving the system

once a solution is computed.

However, if a reduced-order observer is constructed (), an estimate of the state vector is obtained by solving the system

carvalho 2003-08-14