Observing the State of a Linear System

TLDR

Modern control theory assumes the state vector is measurable, but in practice only a few outputs are available, limiting the applicability of such theories. This paper demonstrates that the state vector of a linear system can be reconstructed from observations of its inputs and outputs. The observer can be integrated into the control of systems lacking measurable state vectors. The observer is a linear system whose complexity decreases with more outputs, but it adds poles to the overall system.

Abstract

In much of modern control theory designs are based on the assumption that the state vector of the system to be controlled is available for measurement. In many practical situations only a few output quantities are available. Application of theories which assume that the state vector is known is severely limited in these cases. In this paper it is shown that the state vector of a linear system can be reconstructed from observations of the system inputs and outputs. It is shown that the observer, which reconstructs the state vector, is itself a linear system whose complexity decreases as the number of output quantities available increases. The observer may be incorporated in the control of a system which does not have its state vector available for measurement. The observer supplies the state vector, but at the expense of adding poles to the over-all system.