Assessment and Future Directions of Nonlinear Model Predictive ...

612 W.B. Dunbar and S. Desabacklog is always the true backlog, i.e., b x r,as(t k ; t k )=b x (t k ) for each stage x andat any update time t k . Let the set X x (t k )={s d ,o x ud (t k; t k ), ..., o x ud (t k+P ; t k )} denotethe desired states associated with stage x and update time t k .Usingtheequationsfrom the previous section, the desired unfulfilled order prediction o x ud (·; t k)in X x (t k ) can be computed locally for each stage x given the MDs D x (t k ). By ourdistributed MPC implementation, stages update their control in parallel at eachupdate time t k . The optimal control problem and distributed MPC algorithm forany stage are defined as follows.Problem 1. For any stage x ∈{S,M,R}, and at any update time t k , k ∈ N:Given: the current state (s x (t k ),o x u(t k )), the MDs D x (t k ), the desired statesX x (t k ), the non-negative weighting constants (W s ,W ou ,W u ,W δu ), and a nonnegativetarget order rate o targr ,Find: the optimal control o x r,∗(k) {o x r,∗(t k ; t k ),o x r,∗(t k+1 ; t k ), ..., o x r,∗(t k+M−1 ; t k )}satisfying{ ∑Po x r,∗(k) =argmin W s [s x (t k+i ; t k ) − s d ] 2 +W ou [o x u(t k+i ; t k ) − o x ud(t k+i ; t k )] 2∑M−1+j=0i=1[W u oxr (t k+j ; t k ) − o targ ] } 2r +Wδu [o x r (t k+j; t k )−o x r (t k+j−1; t k )] 2 ,where o x r (t k−1; t k ) o x r,∗ (t k−1; t k−1 ), subject to the discrete-time version of theappropriate model (equation (3), (4) or (5)), and the constraints in equation (2). Algorithm 1. The distributed MPC law for any stage x ∈ {S,M,R} is asfollows:Data: Current state: (s x (t 0 ),o x u (t 0),b x (t 0 )). Parameters: δ, M, P ,(W s ,W ou ,W u ,W δu ), and o targr .Initialization: At initial time t 0 = 0, generate D x (t 0 ) as follows: (a) Choose anominal constant order rate o x,nomr ,seto x r,as (t i; t 0 )=o x,nomr ,fori =0, ..., P ,andif x = R or M, transmit o x r,as(0) to M or S, respectively; (b) Compute b x r,as(0),and if x = S or M, transmit to M or R, respectively. Compute X x (t 0 )andsolveProblem 1 for o x r,∗ (0).Controller:1. Between updates t k and t k+1 , implement the current control action o x r,∗ (t k; t k ).2. At update time t k+1 :a) Obtain (s x (t k+1 ),o x u(t k+1 ),b x (t k+1 )).b) Generate D x (t k+1 ) as follows:i. Set o x r,as(t j+k+1 ; t k+1 )= o x r,∗(t j+k+1 ; t k ), for j = 0, ..., M − 2ando x r,as(t j+k+1 ; t k+1 )=o x r,∗(t k+M−1 ; t k )fori = M − 1, ..., P .Ifx =Ror M, transmit o x r,as (k + 1) to M or S, respectively.ii. Compute b x r,as(k +1), and if x = S or M, transmit to M or R,respectively.c) Compute X x (t k+1 ) and solve Problem 1 for o x r,∗ (k +1).3. Set k = k +1andreturntostep1.