Page 82 - Distributed model predictive control for plant-wide systems

56 Distributed Model Predictive Control for Plant-Wide Systems

∗ 0 0 0
⎡ ⎤
⎢0 0 ∗ ∗⎥
B = 0 0 0 ∗ ⎥ (4.16)
⎢
⎢ ⎥
∗ ∗ 0 0
⎢ ⎥
⎣∗ 0 0 ∗⎦
[ ]
0 0 ∗ ∗ 0
C = (4.17)
0 0 0 0 ∗
Judge its input accessibility and output accessibility.
Here n = 5; according to
⎡∗ ∗ ∗ ∗ ∗⎤
T T 4 ⎢∗ ∗ ∗ ∗ ∗⎥
R = B (A + I) =
ux ⎢ 0 ∗ 0 0 0 ⎥
⎢ ⎥
⎣∗ ∗ ∗ ∗ ∗⎦
the system is input accessible. In addition, since

0 0
⎡ ⎤
⎢0 0⎥
T 4 T
R =(A + I) C = ∗ ∗ ⎥
⎢
ux
⎢ ⎥
∗ ∗
⎢ ⎥
⎣0 ∗⎦
the system is output inaccessible, where x and x cannot be reflected by any output.
1 2
Remark 4.1 In some literature works, the accessibility is called connectability. They all reflect
the characteristic of whether one unit could impact another one.

4.3.4 General Rank of the Structure Matrix
Considering that there is no quantity description of the structure matrix, and the rank of struc-
ture matrix cannot be defined as the numeric matrix; thus we define the maximum rank of its
corresponding numeric matrix rank as the rank of the structure matrix. To identify from the
rank of the numeric matrix, we call it the “generic rank” that is
gr(S)= max(rank(S))

The generic rank of the structure matrix can be calculated by a different method. See refer-
ence [82].

4.3.5 Structure Controllability

In the control of a dynamic large-scale system, to get a feasible control solution, the system
must be controllable. Thus, it is important to test the controllability of a system. Here, we will
introduce how to analyze the controllability of the dynamic system by the method of structure