Page 115 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 89

The integral Nash optimal solution of the whole system, provided that the algorithm is
convergent at each sampling time, can be written as
−1
N
ΔU (k)=(I − D ) D [R(k)− F X(k)] (5.61)
1
0
2
This is the state feedback control law. The instant control law of the whole system is
N
N
Δu (k) = LΔU (k), with
⎛ ⎞
⎜ ⎟ [ ]
L = block-diag L ··· L 0⎟ , L = 10 ··· 0 1×M
⎜ 0
0
⎜⏟⏞⏞⏟⏞⏞⏟⎟
⎝ m ⎠
F = block-diag (CS, … , CS)
2
⏟⏞⏞⏞⏟⏞⏞⏞⏟
m
[( ) T
N
T T
ΔU (k)= Δu N (k) ··· (Δu N (k)) ]
1,M m,M
[ T T ] T
(k)= (k) ··· (k)
1 m
[ T T ] T
X(k)= x (k) ··· x (k)
m
1
Without loss of generality, let the expected outputs be (k) = 0,(i = 1, … , m), then the
i
state-space model of the whole system at the time instant k can be expressed as
−1
N
X(k + 1)= F X(k)+ BLΔU (k)=[F − BL(I − D ) D F ]X(k) (5.62)
1 1 0 1 2
with
⎡a ··· a
11 1m ⎤
⎢ ⎥
F = block-diag (S, … , S), B = ⋮ ⋱ ⋮
1 ⎢ ⎥
⏟⏟⏟
⎢ ⎥
a
m ⎣ m1 ··· a mm⎦
Expression (5.62) shows the state mapping relationship of the distributed system between the
time instants k and k+1. According to the contraction mapping principle [96], the nominal
stability of the whole distributed system can be guaranteed, if and only if
‖ −1 ‖
‖ [F − BL(I − D ) D F ]‖ < 1 (5.63)
1 2
1
0
‖ ‖
That is, the norms of eigenvalues of state mapping are less than 1.
5.3.5 Performance Analysis with Single-step Horizon Control Under
Communication Failure
In distributed control, each controller can work independently to achieve its local objective,
but cannot accomplish the whole task on its own. These autonomous agents can communicate
and coordinate with each other, and can exchange information through a network in order to
accomplish the whole task or objective. If a distributed system is subjected to communica-
tion failure, does this strategy work well and what does the performance of the whole system