On Observers for Linear Multivariable Systems

Abstract

In the design of feedback control systems, the values of all the state-variables are necessary at each time to form a feedback control law. Unfortunately, in many situations, all state-variables are not available by direct measurement. So, it is necessary to construct estimates of some state-variables from the outputs available by direct measurement. If the system is linear, its state vector can be approximately reconstructed by building an observer which is itself a linear system driven by the available outputs and inputs of the plant.
For time-invariant systems, such an observer is defined by D.G. Luenberger, and for time-variant systems by W.A. Wolovich as an exponential estimator. But this exponential estimator is not effective for some time-invariant systems.
In this paper, it is aimed to define an observer which is effective both for time-invariant systems and time-variant systems. At first is given the definition of compatibility of observers to plants, and next are discussed the necessary condition and the sufficient condition for compatibility. Especially, in the case of time-invariant systems, a necessary and sufficient condition exists. Finally is given the construction of observers for time-variant systems with one output.