Assessment and Future Directions of Nonlinear Model Predictive ...

Towards the Design of Parametric Model Predictive Controllers 201to obtain the optimal profiles of ˆξ(t k ,x 0 ), λ(t, tˆk ,x 0 ), ˆx(t, t k ,x 0 ), ˆµ(t, t k ,x 0 )andû(t, t k ,x 0 ). The vector t k (x 0 ) is calculated in the step 4 by solving symbolicallythe non-linear, algebraic equalities of the Jump conditions (28), (30) and (32). Instep 5 the optimal parametric control profile is obtained by substituting t k (x 0 )into û(t, t k ,x 0 ). Finally the critical region in which the optimal control profile isvalid, is calculated in step 6, following the procedure which will be described inthe following. The algorithm then repeats the procedure until the whole initialregion CR IR is covered.A critical region CR in which the optimal control profiles are valid, is theregion of initial conditions x 0 where the active and inactive constraints, obtainedin step 2 of algorithm 3.1, remain unaltered ([20]). Define the set of inactiveconstraints ğ , the active constraints ˜g and ˜ˆµ > 0 the Lagrange multipliersassociated with the active constraints ˜g; obviously the Lagrange multipliers µassociated with the inactive constraints are 0. The critical region CR is thenidentified by the following set of inequalitiesCR {x 0 ∈ R n | ğ(ˆx(t, t k (x 0 ),x 0 ), û(t, t k (x 0 ),x 0 )) < 0 , ˜ˆµ(t, t k (x 0 ),x 0 ) > 0˜ν(t, t k (x 0 ),x 0 ) > 0} (33)In order to characterize CR one has to obtain the boundaries of the set describedby inequalities (33). These boundaries obviously are obtained when each of thelinear inequalities in (33) is critically satisfied. This can be achieved by solvingthe following parametric programming problems, where time t is the variableand x 0 is the parameter.• Take first the inactive constraints through the complete time horizon andderive the following parametric expressions:Ğ i (x 0 )=max{ğ i (ˆx(t, t k (x 0 ),x 0 ), û(t, t k (x 0 ),x 0 ))|t ∈ [t 0 ,t f ]}, i=1,...,˘qt(34)where ˘q is the number of inactive constraints.• Take the path constraints that have at least one constrained arc [t i,˜kt,t i,˜kx]and obtain the following parametric expression˜G i (x 0 )=max{˜g i (ˆx(t, t k (x 0 ),x 0 ),û(t, t k (x 0 ),x 0 ))|t∈ [t 0 ,t f ]}∧{t∉[tti,˜kt,t i,˜kx]]}(35)k =1, 2,...,n i,˜kt,i=1, 2,...,˜qwhere n i,˜ktis the total number of entry points associated with the ith activeconstraint and ˜q is the number of active constraints.• Finally, take the multipliers of the active constraints and obtain the followingparametric expressions˘µ(x 0 )=mint{˜ˆµ(t, t k (x 0 ),x 0 )|t = t i,kt =t i,kx ,k =1, 2,...,n i,kt },i=1, 2,...,˜q(36)