Page 83 - Distributed model predictive control for plant-wide systems

Structure Model and System Decomposition 57

analysis. The analysis of structure controllability is not quantified and cannot give the results
of whether the controllability is weak or strong. However, it is very convenient since it only
involves some logical relationship.
Consider the structured linear system

x(k + 1)= Ax(k)+ Bu(k) (4.18)
n
n u
where x ∈ ℝ , u ∈ ℝ , and A, B are structure matrices. If there is a controllable general system
whose structure is equivalent to (A, B), then we say (A, B) is structure controllable.
It can be seen from the definition that if (A, B) is structure uncontrollable, then the numeric
system is definitely uncontrollable in spite of any values of the matrix’s elements.
For a general numeric system (A, B), Rosenbrock provided a condition related to the rank
of an extent matrix,
rank(kG )= n 2 (4.19)
c
2
to judge the controllability of (A, B), where n × n(n + m − 1)-dimensional matrix G is defined
c
as
⎡ B I 0 0 0 ··· 0 0 ⎤
⎢ 0 −A B −I 0 ··· 0 0 ⎥
⎢ 0 0 0 −A 0 ··· 0 0 ⎥
G = ⎢ ⎥ (4.20)
c ⎢··· ··· · ·· ··· ⋱ ··· ··· · ··⎥
⎢ 0 ··· · ·· 0 0 B I 0 ⎥
⎢ ⎥
⎣ 0 ··· · ·· 0 0 0 −A B ⎦
Then, this result is introduced to the study of structure controllability by Shields and
Pearson. They provided the following criterion.

Theorem 4.1 System (4.18) is controllable if and only if
2
gr(G )= n ,
c
where
⎡ B I 0 0 0 ··· 0 0 ⎤
⎢ ⎥
0 −A B −I 0 ··· 0 0
⎢ ⎥
⎢ 0 0 0 −A 0 ··· 0 0 ⎥
G = (4.21)
c ⎢ ⎥
··· · ·· ··· · ·· ⋱ ··· · ·· ···
⎢ ⎥
⎢ 0 ··· · ·· 0 0 B I 0 ⎥
⎣ 0 ··· · ·· 0 0 0 −A B ⎦
⎢ ⎥
In addition, combined with the input connectivity (accessibility) and the general rank crite-
rion, Division and Morari give another necessary and sufficient condition.
Theorem 4.2 System (4.18) is controllable if and only if the following two conditions are
satisfied:

1. System (4.18) is input accessible;
([ ])
2. gr A B = n.