Assessment and Future Directions of Nonlinear Model Predictive ...

A Nonlinear Model Predictive Control Framework as Free Software 233( )p∑m∑Ψ i Y,U = e T ( ) TQuk ( )i+kQ yk e i+k + ui+k−1 − u r,i+k−1 ui+k−1 − u r,i+k−1k=1k=1where the subscript r stands for reference [17, 18], Q uk and Q yk are weightingdiagonal matrices, and e i+k = y sp,i+k −y i+k . The penalty term is used only whenconstraint relaxation is requested, and ɛ is a measure of the original constraintviolations on the states, outputs, inputs and control move rates, defined byɛ = [ ɛ T x ɛT y ɛT u ɛT ∆u] T.The problem formulation is coded such that it can handle either an exact ora quadratic penalty formulation. For instance, if the penalty term is definedaccording to the exact penalty formulation, it follows that P i (ɛ) =r T ɛ,wherer isthe vector of penalty parameters of appropriate size defined by: r =[ρ ···ρ] T ,ρ∈R + .Theaugmented vectors X, Y , U and ∆U are defined by⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤∆u ix i+1 y i+1u iX = ⎢⎣⎥. ⎦ ,Y= ⎢⎣⎥. ⎦ ,U= ⎢⎣⎥. ⎦ and ∆U = ∆u i+1⎢ . ⎥⎣ . ⎦ ,x i+p y i+p u i+m−1∆u i+m−1where ∆u i+k = u i+k − u i+k−1 , k =2,...,m− 1. Vectors ∆U min and ∆U maxin (10) are defined as follows:∆U min = [ ∆u T min ··· ∆uT min] T, ∆Umax = [ ∆u T max ··· ∆uT max] T,with ∆u min ,∆u max ∈ R nm . Although in this representation it is assumed thatvectors ∆U min and ∆U max are constant over the entire input predictive horizon,the implementation of a variable profile is straightforward. Equality constraints(5) result from the multiple shooting formulation and are incorporatedinto the optimization problem such that after convergence the state and outputprofiles are continuous over the predictive horizon. Note that φ(¯x i+k−1 , ū i+k−1 ),that is, x i+k , is obtained through the integration of (1) inside the sampling intervalt ∈ [t i+k−1 ,t i+k ] only, using as initial conditions the initial nominal statesand controls, ¯x i+k−1 and ū i+k−1 respectively. Equation (6) is the the outputterminal equality constraint.Finally, the actual implementation of the control formulation includes integralaction to eliminate the steady-state offset in the process outputs resulting fromstep disturbances and to compensate to some extent the effect due to the modelplantmismatch. This is achieved by adding in the discrete linearized model thestate equations [17, 18]z i+k = z i+k−1 + K I(yi+k − y sp,i+k), k =1,...,p (12)with z i = z 0 ,wherez i ∈ R no , K I ∈ R no×no ,andz 0 is the accumulated valueof steady state offset over all the past and present time instants. The constant