Page 63 - Distributed model predictive control for plant-wide systems

Model Predictive Control 37

V(k)− V(k − 1)
N−1
∑ 2 2
2
= ‖x(k + N|k)‖ + (‖x(k + l|k)‖ + ‖u(k + l|k)‖ )
P Q R
l=0
N−1
∑ 2 2
2
− ‖x(k − 1 + N|k − 1)‖ − (‖x(k − 1 + l|k − 1)‖ + ‖u(k − 1 + l|k − 1)‖ )
P Q R
l=0
N−1
∑ f 2 f 2 (2.47)
2
f
≤ ‖x (k + N|k)‖ + (‖x (k + l|k)‖ + ‖u (k + l|k)‖ )
P Q R
l=0
N−1
∑ 2 2
2
− ‖x(k − 1 + N|k − 1)‖ − (‖x(k − 1 + l|k − 1)‖ + ‖u(k − 1 + l|k − 1)‖ )
P Q R
l=0
f 2 f 2 f 2
= ‖x (k + N|k)‖ + ‖x (k + N − 1|k)‖ + ‖u (k + N − 1|k)‖
P Q R
2 2 2
− ‖x(k − 1 + N|k − 1)‖ − ‖x(k − 1|k − 1)‖ − ‖u(k − 1|k − 1)‖
P Q R
Substituting (2.44), (2.45), and (2.39) into (2.47) yields
V(k)− V(k − 1)
2 2 2
= ‖A x(k − 1 + N|k − 1)‖ + ‖x(k − 1 + N|k − 1)‖ + ‖Kx(k − 1 + N|k − 1)‖
c P Q R
2 2 2
− ‖x(k − 1 + N|k − 1)‖ − ‖x(k − 1|k − 1)‖ − ‖u(k − 1|k − 1)‖
P Q R
2
= ‖x(k − 1 + N|k − 1)‖ (2.48)
P
2 2 2
− ‖x(k − 1 + N|k − 1)‖ − ‖x(k − 1|k − 1)‖ − ‖u(k − 1|k − 1)‖
P Q R
2 2
=−‖x(k − 1|k − 1)‖ − ‖u(k − 1|k − 1)‖
Q R
< 0
Thus, for any k ≥ 0, if x(k)∈ X∖Ω( ), there is a constant ∈ (0, ∞) such that
′
′
V(k) ≤ V(k − 1) − . It then follows that there exists a finite time k such that x(k ) ∈Ω ( ).
This concludes the proof of stability of dual mode predictive control.
We have now established the feasibility and the stability of the resulting closed-loop system.
That is, if 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.
2.5 Conclusion
In this chapter, we introduced three MPC algorithms: dynamic matrix control, state
space-based MPC, and dual mode MPC. These three algorithms are very important and
are the fundamental of the distributed predictive controls which will be introduced in the
following chapters.