[Luyben] Process Mod.. - Student subdomain for University of Bath

ANALYSIS OF MULTIVARIABLE SYSTEMS 591
controllers should be designed so that the minimum dip in the Doyle-Stein criterion
curves (the minimum singular value of the matrix [1; + Q--l]) occurs in a
frequency range where the uncertainty is not significant. z
If the uncertainty has a known structure, “structured singular values” are
used. In the book by Morari and Zafiriou (Robust Process Control, 1989,
Prentice-Hall) this topic is discussed in detail.
PROBLEMS
16.1. Wardle and Wood (I. Chem. E. Symp. Series, 1969, No. 32, p. 1) give the following
transfer function matrix for an industrial distillation column:
GM$iEi$ ‘.~~~~;l,j
The empirical PI controller settings reported were:
K, = 18/-24 7, = 19124
(a) Use a multivariable Nyquist plot and characteristic loci plots to see if the system
is closedloop stable.
(b) Calculate the values of the RGA, the Niederlinski index, and the Morari
resiliency index.
16.2. A distillation column has the following transfer function matrix:
G =
hi
34 -44.7
(54s + 1)(0.5s + 1)2 (114s + l)(OSs + l)*
31.6 -45.2
I
(78s + l)(OSs + 1)’ (42s + 1)(0.5s + 1)’ J
Empirical PI diagonal controller settings are:
K, = 1.6/- 1.6 7, = 2019 min
(a) Check the closedloop stability of the system using a multivariable Nyquist plot
and characteristic loci plots.
(b) Calculate values of the RGA, the Niederlinski index, and the Morari resiliency
index.
16.3. A distillation column is described by the following linear ODES:
4
- = -4.14x, + 5.99x, + 0.708R - 0.472V
dt
dx,
- = 10.84x, - 18.24x, + 1.28R - 1.92V + 42
dt
(a) Use state-variable matrix methods to derive the openloop transfer function
matrix.
(b) What are the openloop eigenvalues of the system?