Modern Control System Theory and Design, 2nd Edition

8.7.  ESTIMATOR DESIGN IN CONJUNCTION WITH THE POLE PLACEMENT APPROACH USING LINEAR-STATE-VARIABLE FEEDBACK

In the discussion of Sections 8.2 and 8.3 on linear-state-variable feedback, it was assumed that all of the states are observable and measurable, and available to accept control signals (controllable). As Sections 8.4 and 8.5 on controllability and observability have shown, some states of a feedback control system may not always be controllable and/or observable. In some systems, the system may be observable, but all of the states may not be measured due to physical limitations (e.g., chemical process control systems), or it may be due to cost restrictions that limit the use of costly sensors needed to measure all of the states. It is assumed in this section that the system is observable (no part of the system is disconnected physically from the output), but measurements are being made only on some of the states, and we wish to estimate all of the states.

Let us focus attention on the process portion of the system illustrated in Figure 8.3. We wish to use the closed-loop estimator system shown in Figure 8.18 for determining an estimate of the state vector x(t) and output c(t) [8]. This estimator feeds back the difference between the measured output c(t) and the estimated output ĉ(t) that is obtained from a model of the process. Therefore,

where M defines the gain factors ...