Page 221 - Distributed model predictive control for plant-wide systems

Cooperative Distributed Predictive Control with Constraints 195

f
solution to Problem 9.1. Here, u (⋅|k) is the reminder of previous control concatenating with
i
a feedback control, that is,
{
u (k + l − 1|k − 1) l = 1, … , N − 1
f
i
u (k + l − 1|k)=
i f
K x (k + N − 1|k, i) l = N
i
f
x (k + l|k, i), l = 1, 2, … , N, equals to the solution of (9.6) under the initial state of x(k) and the
f
control sequence u (k + l − 1|k) and u (k + l − 1| k − 1), j ∈ P , and can be expressed as
j
i
i
l
∑ l−h f
f
l
x (k + l|k, i)= A x(k)+ A B u (k + h − 1|k)
i i
h=1
l
∑∑
+ A l−h B u (k + h − 1|k − 1)
j j
h=1
j∈P i
Substituting (9.13) into (9.14), we have
f f f
x (k + l|k, i)= x (k + l|k, j)= x (k + l|k),
∀i, j ∈ P, l = 1, 2, … , N
and
f
f
x (k + N|k)= A x (k + N − 1|k)
c
f
The control u (⋅|k) is a feasible solution to Problem 9.1 for any S and at any update k ≥ 1
i
i
f
refers that the control u (⋅|k) satisfies Equation (9.8) and the control constraints (9.9), and the
i
f
corresponding state x (k + N|k) satisfies the terminal-state constraint (9.10). To establish this
feasibility result, define that the state ̂ x(k + N|k − 1, i) be the closed-loop response of
̂ x(k + N|k − 1, i)= A ̂ x(k + N − 1|k − 1, i)
c
Here, the state ̂ x(k + N|k − 1, i) is not equal to the result of substituting u (k + N − 1| k − 1),
i
defined in (9.5), into system Equation (9.6). It is because that ̂ x(k + N|k − 1, i) is only a middle
variable used in the proof of feasibility, and has no impact on the optimization problem and
stability. Thus we can assume it as (9.15).
f
Figure 9.1 shows the basic idea of how to guarantee that x (k + N|k) satisfies the terminal
constraint (9.10). If the difference between the input sequences of two neighboring update time
is bounded, then the discrepancy between the presumed state sequence {̂ x(k + 1|k, i), ̂ x(k +
f
f
2|k, i), …} and the state sequence {x (k + 1| k, i), x (k + 2|k, i), … } will be limited. When an
f
appropriate bound is selected, ̂ x(k + N|k, i) and x (k + s|k) can be sufficiently close to each
f
other such that x (k + N|k) is within the indicated ellipsoid Ω( ).
In this section, Lemma 9.2 identifies sufficient conditions to ensure that
‖ f ‖ ≤ , for every i ∈ P. Lemma 9.3 establishes the con-
x (k + l|k) − ̂ x(k + l|k, i)
‖ ‖P
trol constraint feasibility. Lemma 9.4 presents the terminal constraint feasibility. Finally,
Theorem 9.1 combines the results in Lemmas 9.2–9.4 to arrive at the conclusion that, for any
f
i ∈ P, the control u (⋅|k) is a feasible solution to Problem 9.1 at any update k ≥ 1.
i
Consider that Ω( )isthe -level set of the Lyapunov function of the closed-loop dynamics
′
x(k + 1) = A x(k). Therefore under the condition (9.12), it is trivial to show x(k + N) ∈Ω( ),
c
′
provided that ̂ x(k + N − 1|k − 1, i)∈Ω( ) and = (1 − ) .