Page 227 - Distributed model predictive control for plant-wide systems

Cooperative Distributed Predictive Control with Constraints 201

Consider
√
‖ f ‖
2‖x (k + N − 1|k)‖ − ‖̂ x (k + N − 1|k − 1, i)‖ P
‖ ‖P
‖ f ‖
≤ 0.42‖x (k + N − 1|k)‖
‖ ‖P
‖ f ‖
+ ‖x (k + N − 1|k) − ̂ x(k + N − 1|k − 1, i)‖
‖ ‖P
≤ 0.42 e + e
and substitute Equations (9.33)–(9.35) into Equation (9.32), it yields

V − V k−1,i
k,i
′
≤ − e +(N − 1) e + 0.42 e + e
′
=−e( − (0.42 +((N − 1) + 1) ))
which, in view of (9.28), implies that
V − V < 0
k,i k−1,i
Thus, for any k ≥ 0, if x(k)∈ X∖Ω( ), then there is a constant ∈ (0, ∞) such that
i ∑ m
V ≤ V − . Furthermore, we have the inequality of V ≤ V − , where =
k,i k − 1,i i k k − 1 i=1 i
since m is limited. From this inequality, it follows by contradiction that there exists a finite
′
′
time k such that x(k ) ∈Ω( ). If this is not the case, the inequality implies V → −∞ as
k
′
′
k → ∞. However, V ≥ 0, therefore, there exists a finite time k such that x(k ) ∈Ω( ). This
k
concludes the proof.
So far, the feasibility and stability of proposed C-DMPC are analyzed. Provided an initially
feasible solution could be found, subsequent feasibility of the algorithm is guaranteed at every
update, and the resulting closed-loop system is asymptotically stable at the origin.
9.5 Simulation
The multizone building temperature regulation systems are a class of typical spatially dis-
tributed systems, as shown in Figure 9.2, which are composed of many physically interacted
subsystems (rooms or zones) labeled with S , S , …, respectively. The thermal influences
1 2
between rooms of the same building occur through internal walls (the internal walls isolation
is weak) and/or door openings. A thermal-meter and a heater (or air condition) are installed in
each zone, which is used to measure and adjust the temperature of the multizone building.
For simplicity, the seven-zones building is taken as the example. The relationship among
these seven zones is also shown in Figure 9.2, where zone S is impacted by zone S and zone
1 2
S ; zone S is impacted by zone S , S , and zone S ; zone S is impacted by zone S , S , and
7
2
7
4
2
3
3
1
zone S ; zone S is impacted by zone S , S , and zone S ; zone S is impacted by zone S ,
5
7
4
7
4
5
3
S , and zone S ; zone S is impacted by zone S and zone S ; zone S is impacted by all the
7
4
6
5
7
6
other zones. Let U be defined to reflect both the constraint on the input u ∈ [u , u ] and the
i,U
i,L
i
i
constraint on the increment of the input Δu ∈[Δu , Δu ].
i
i,L
i,U