Page 153 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

128 Cha11ter File

The mabices Band C change to accommodate the two-dimensional
process, input and output, which have become two-dimensional col-
umn vectors. The matrix A and the state X are the same.

1
0 0
t't
X=(~:) A= ~_!_ t'2 0 B=[~ ;] U=(~)
1
~ t'2
~ 1 1
0
~ t'3 t'3
Y=CX
c=(o 1
0 0 ~) Y=(~:)

Using the "across the row and down the column" matrix multi-
plication rule, the reader should check that these equations do indeed
describe the process in Fig. 5-4.
In terms of the matrices and vectors, the state-space formula-
tion appears to be first order. This suggests that there is a solution
of the form

for the homogeneous form of the state-space equation which is

dXh =AX
dt ,,

Here, C is a column vector and a is a scalar. If this trial solution is
inserted into the homogeneous part of the matrix differential equa-
tion the following results:
d
dt Ceat =ACe"
(5-9)
Since the rule for differentiating a matrix is simply the derivative
of the elements in the matrix, Eq. (5-9) becomes

Caeat = ACeat
alC=AC
or (5-10)
{A-al)C= 0