Page 146 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

CHAPTER 5

Matrices and

Higher-Order

Process Models

n Chap. 4 the first-order with dead-time (FOWDT) process model
was presented. In this chapter higher-order models will be intro-
I duced. The simplest third-order model is constructed from three
cascaded first-order models which come from the water tank process.
The mathematical bookkeeping required by higher-order models
sometimes gets involved. To ameliorate this problem, matrices can
often provide aid. Appendix G contains an elementary introduction
to mabices in case the reader is a bit rusty in this area. Matrices form
the backbone of the state-space approach which will make its debut
in this chapter. All of the higher-order models covered in this chapter
will be written as differential equations in the time domain, as trans-
fer functions in the Laplace s-domain, as magnitudes and phases in
the frequency domain, and as matrix differential equations back in
the time domain.

5-1 Third-Order Process without Backflow
Figure 5-1 shows three independent tanks-independent in the sense
that each downstream tank does not influence its upstream neighbor.
Each tank in the series of three can be treated like the single tank we
treated earlier except that the outlet flow rate of the upstream tank
feeds into the next tank down the line. The single tank is described by
dL L
pAdt+ R =F

m ~
dL
-r-+L=RF -r= pAR
dt
m