Page 131 - Distributed model predictive control for plant-wide systems

Cooperative Distributed Predictive Control 105

Table 6.1 Notations definition
Notations Explanations

diag {A} Diagonal block matrix made by a blocks equal to A
a
{A} The jth eigenvalue of a square matrix A
j
O(a) To be proportional to a
0 An a × b null matrix
a × b
0 a An a × a null matrix
I a An a × a identity matrix
̂ x (l|h) Predictions of x (l) computed by C at time h
i
i
i
ˆ y (l|h) Predictions of y (l) computed by C at time h
i
i
i
u (l|h) Input u (l) computed by C at time h
i
i
i
Δu (l|h) Input increment Δu (l) computed by C at time h
i
i
i
d
y (l|h) Set-point of y (l|h)
i i
d
y (l|h) Set-point of y(l|h)
i
̂ x (l|h) Predictions of x(l) computed by C at time h
i
i
ˆ y (l|h) Predictions of y(l) computed by C at time h
i
U (l, p|h) A complete input vector
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T
T
T
U (l, p|h)=[u (l|h) u (l + 1|h) ··· u (l + p|h)] T
i
i
i
i
ΔU (l, p|h) Input increment sequence vector,
i
[ T T T ] T
ΔU (l, p|h)= Δu (l|h) Δu (l + 1|h) · · · Δu (l + p|h)
i
i
i
i
U(l, p|h) Complete stacked input vector,
[ ] T
T
T
T
U(l, p|h)= u (l|h) u (l + p|h) · · · u (l + p|h)
̂ i
X (l, p|h) Sacked distributed state vector,
[ iT iT iT ] T
̂ i
X (l, p|h)= ̂ x (l |h) ̂ x (l + 1|h) · · · ̂ x (l + p|h)
X (l, p|h) Stacked distributed state vector,
̂
i [ ] T
T
T
T
X (l, p|h)= ̂ x (l |h) ̂ x (l + 1|h) · · · ̂ x (l + p|h)
̂
i i i i
X(l, p|h) Complete stacked state vector,
̂
[ ] T
T
T
T
X(l, p|h)= ̂ x (l|h) ̂ x (l + 1|h) · · · ̂ x (l + p|h)
̂
ˆ i
Y (l, p|h) Stacked distributed state vector,
[ ] T
iT
iT
iT
̂ i
Y (l, p|h)= ̂ y (l|h) ̂ y (l + 1|h) · · · ̂ y (l + p|h)
Y(l, p|h) Complete stacked state vector,
[ ] T
T
T
T
Y(l, p|h)= y (l|h) y (l + 1|h) · · · y (l + p|h)
d
Y (l, p|h) Set-point of Y(l, p|h);
̂ (l, p|h) Complete stacked state vector,
[ ] T
̂ T
̂ (l, p|h)= ̂ T ··· X (l, p|h)
X (l, p |h)
m
1
d
d
d
(l, p|h) (l, p|h)= diag (Y );
m
(l, p|h) Complete stacked state vector,
[ ] T
T
T
(l, p|h)= U (l, p |h) ··· U (l, p|h)
1 m
1. a and b are constants; p, l, h are positive integers, and h < l ;Ais amatrix.
• communication channel introduces a delay of a single sampling time interval, since an
instantaneous data transfer is not possible in real situations;
• controllers communicate only once within a sampling time interval;
• local states x (k), i = 1, 2, … , m, are accessible.
i