State-estimators for continuous time linear systems

The state space state and output equations of continuous time linear systems is of the form?

x = Ax + Bu + v

and

y = Cx + Du + w

where u is the control input (signal to actuator), v is the disturbance on the state and w is the disturbanceon the output.

 When we can measure all the components of the state vector x then we can use state feedback to control the system. We cannot always measure states however (usually we can only measure a few and possibly only one output variable) due to practical infeasibility or cost of installing and maintaining transducers. To use state feedback control methods we can then estimate the states from the input and output of the system and a knowledge of the system dynamics (that is using the known values of A , B,C and D ). Reconstructing, observing or estimating the states can also useful for condition monitoring of the system.

 If the system we are to observe has the state and output equation above then to observe or estimate the state x we attach a system called an observer or state estimator (implemented in the flight control computer) as in with the negative feedback u_e = -Ge_y so that it is governed by the equations?

where x is the additional state of the attached observer which as will be seen represents an estimate of the original state x. As can be seen from ?gure the output error term term

is a measure of the error in the estimate of the output y and is called the innovation term because it is the