Page 197 - Distributed model predictive control for plant-wide systems

Local Cost Optimization Based Distributed Predictive Control with Constraints 171

where x ∈ ℝ , u ∈ U ⊂ ℝ n ui and y ∈ ℝ n yi are, respectively, the local state, input and output
n xi
i i i i
vectors, and U is the feasible set of the input u , which is used to bound the input according
i i
to the physical constraints on the actuators, the control requirements, or the characteristics
of the plant. A nonzero matrix A , that is, j ∈ P , indicates that S is affected by S .Inthe
i
+i
j
ij
concatenated vector form, the system dynamics can be written as
{
x (k + 1) = Ax(k)+ Bu(k)
(8.2)
y(k)= Cx(k)
where
[ ] T
T
T
T
x(k)= x (k) x (k) ··· x (k) ∈ R n x
1 2 m
[ T T T ] T
u(k)= u (k) u (k) ··· u (k) ∈ R ⊂ R n u
1 2 m
[ T T T ] T
y(k)= y (k) y (k) ··· y (k) ∈ R n y
m
1 2
are, respectively, the concatenated state, control input, and output vectors of the overall
system S.Also, u(k) ∈ U = U × U ×· · · × U . A, B, and C are the constant matrices of
m
2
1
appropriate dimensions and are defined as follows:
T
⎡ A 11 A 12 ··· A 1m ⎤
A 21 A 22 ··· A 2m
⎢ ⎥
A = ⎢ ⎥
⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎢ ⎥
⎣A A ··· A
m1 m2 mm ⎦
T
⎡ B 11 B 12 ··· B 1m ⎤
⎢B B ··· B ⎥
B = 21 22 2m
⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎢ ⎥
⎣B m1 B m2 ··· B mm ⎦
T
⎡ C 11 C 12 ··· C 1m ⎤
⎢C 21 C 22 ··· C 2m ⎥
C =
⎢ ⋮ ⋮ ⋱ ⋮ ⎥
⎢ ⎥
⎣C m1 C m2 ··· C mm ⎦
The control objective is to stabilize the overall system S in a distributed predictive control
framework.
8.3 Stabilizing Dual Mode Noncooperative DMPC with
Input Constraints
8.3.1 Formulation

In this section, m separate optimal control problems, one for each subsystem, and the
LCO-based DMPC algorithm with communicating once a control period are defined. In every
distributed optimal control problem, the same constant prediction horizon N, N ≥ 1, is used.
And every distributed MPC law is updated globally synchronously. At each update, every