Page 220 - Distributed model predictive control for plant-wide systems

194 Distributed Model Predictive Control for Plant-Wide Systems

Assumption 9.2 At initial time k , there exists a feasible control u (k + l)∈ U ,
0
0
i
i
l ∈ {1, … , N} , for each S , such that the solution to the full system x(l + 1 + k ) = Ax(l + k ) +
i
0
0
Bu(l + k ), denoted ̂ x(⋅|k , i), satisfies ̂ x(N + k |k , i)∈Ω( ) and results in a bounded cost
0 0 0 0
J (k ). Moreover, each subsystem has access to u (⋅|k ).
i
0
0
i
Assumption 9.2 bypasses the task of actually constructing an initially feasible solution in a
distributed way. In fact, finding an initially feasible solution for many optimization problems is
often a primary obstacle, whether or not such problems are used in a control setting [69, 115].
One possible way to obtain an initially feasible solution can be to solve the corresponding
centralized MPC solution at the initial time instant.
The dual mode C-DMPC law for any S , which communicates once every update, is as
i
follows.
Algorithm 9.1
Step 1: Initialization at time k
0
• Initialize x(k ), u(k + l − 1| k ), where l = 1, 2, … , N, to satisfy Assumption 9.1
0
0
0
• Transmit u (k + l|k ) and x (k ) to all other subsystems, receive u (k + l − 1| k ) and
0
0
i
0
j
j
0
0
x (k ), j ∈ P from all S j
i
0
j
• At time k ,if x(k ) ∈Ω( ), then apply the terminal controller u (k ) = K x(k ), for all
i
0
0
0
i
0
k ≥ k ,else
0
• Solve Algorithm 9.1 for u (k + l − 1| k ) and apply u (k |k )to S
i 0 0 i 0 0 i
Step 2: Update control law at time k
• Measure x (k), transmit x (k) and u (k + l|k) to all other subsystems, receive x (k) and
i i i j
u (k + l − 1| k − 1) , j ∈ P from all S
j i j
•If x(k) ∈Ω( ), then apply the terminal controller u (k) = K x(k), else
i
i
• Solve Algorithm 9.1 for u (k + l − 1| k) and apply u (k|k)to S i
i
i
Step 3: Update control at time k + 1
•Let k + 1 → k repeat Step 2
Algorithm 9.1 presumes that C , ∀i ∈ P have access to the full state x(k). In the next section,
i
it is shown that the C-DMPC policy drives the state x(k + l)to Ω( ) in a finite number of updates
and the state remains in Ω( ) for all future time. And the analysis of the feasibility and stability
of C-DMPC algorithm is preceded as follows.
9.4 Analysis
In this section, feasibility is analyzed in the first subsection, followed by stability in the second
subsection.
9.4.1 Feasibility
The main result of this section is that, provided an initial feasible solution is available and
f
Assumption 9.2 holds true, for any S and at any time k ≥ 1, u (⋅|k)= u (⋅|k) is a feasible control
i
i
i