Assessment and Future Directions of Nonlinear Model Predictive ...

NLMPC: A Platform for Optimal Control 369Equations 2 and 3 define the process model in differential algebraic equation(DAE) form. x are the states, u are the manipulated inputs, v are the measureddisturbances or disturbance model variables, p are parameters, and y are theoutputs. In this problem statement x, u, v, andy are in absolute, not deviation,terms.Equations 4 and 5 are examples of specifications for the reference trajectoriesfor the desired closed loop behavior of the outputs. In this example, a first-orderresponse is requested with a time constant of τ c . SP h and SP l are the highand low setpoint (target) values for the outputs that defines the allowed settlingzone. P h and P l are 1-norm penalty variables on deviations of the output fromthe reference trajectory. S h and S l are the corresponding slack variables (i.e. notcosted in the objective function).Output feedback is incorporated in additive form in Eqn. 7. For scaling reasons,we sometimes incorporate output feedback in multiplicative form.Equations 8 and 9 impose absolute and incremental bounds on the manipulatedvariables. The subscripts k and k − 1 refer to adjacent zero-order holdvalues of the manipulated variables across the entire control horizon.Objective FunctionIn the objective function, the J 1 term is a weighted 1-norm of errors from thedesired closed loop output trajectory over the prediction horizon.n yn pJ 1 = 1 ∑ ∑(w hi P hi,k + w li P li,k ) (10)n pi=1 k=1Conceptually, this can be depicted as a conic section of unpenalized trajectorieswith linear penalties assigned to trajectories outside of the section. n p is thelength of the prediction horizon. n y is the number of outputs. The significanceof the 1-norm is a distinct relaxation of soft constraints from lowest ranked tohighest ranked. This one-at-a-time relaxation is more aligned with industrial expectationsas opposed to 2-norm behavior which spreads error across multipleoutputs when constraints become active [10].The J 2 term is an economic cost whose mean value over the prediction horizonis minimized to specify where within the allowed zone the system will settle. Auseful alternative explanation is that −J 2 is a net income (product value - costof feed and control) to be maximized.n yn pJ 2 = 1 ∑ ∑( c yi y i,k +n pi=1 k=1∑n un pm=1 k=1n p∑∑n v∑c um u m,k + c vi v j,k ) (11)j=1 k=1n u is the number of manipulated inputs. n v is the number of measured disturbancevariables.The J 3 term is the cost of incremental moves of the manipulated variables.J 3 =∑n u∑n cm=1 l=1c ∆um |∆u m,l | (12)