State Observation and Filtering

In the previous chapters, the discussion of problems (in particular, eigenvalue allocation in Chap. 7) was based on the assumption that the system state vector is known. In practice, however, only a part of its components or some of their functions may be measured directly. This gives rise to the problem ofdetermination or observation of the state vector through the information on the measured variables. Below, consideration will be given to the problem formulated in this way for the linear time-invariant system (14.1)$$\dot x = Ax + Bu,x \in {\mathbb{R}^n},u \in {\mathbb{R}^m},A,B = const.$$ and it will be assumed that one can measure the vector y that is a linear combination of the system state vector components: (14.2)$$y = Kx,y \in {\mathbb{R}^1},1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } 1 < n,K - const.$$