Assessment and Future Directions of Nonlinear Model Predictive ...

Towards the Design of Parametric Model Predictive Controllers 195The mathematical framework for the discrete-time nonlinear MPC can beshortly summarised as the following constrained non-linear programming (NLP)problem ([15, 18])z o (x t )=min t)U(2a)s.t h(U, x t ) ≤ b(2b)U L ≤ U ≤ U U ,x L ≤ x ≤ x U(2c)where x t is the state at the current time instant, U = {u t ,u t+1 ,...,u t+N−1 }is the sequence of control inputs over the prediction horizon N, J(U, x t )isascalar objective function, h(U, x t ) is a vector of non-linear functions, U L , U Uare lower and upper bounds for U and x L and x U are lower and upper boundsfor x. The functions J(U, x t )andh(U, x t ) are generally non-linear, althoughthe analysis that follows can be applied for the linear case as well, and mayinclude any terminal cost function and terminal constraints respectively to ensurestability ([15]).Transforming the NLP (2) to (1) can been done in two steps. First, if J(U, x t )is only a function of U then simply replace (2a) by simply J(U) and the objectivefunction of (2) is the same with (1). Otherwise, introduce a new scalar ɛ ∈ Rand transform (2) into the following NLPor simply to¯z(x t )=min U (3a)s.t. J(U, x t ) ≤ ɛ, h(U, x t ) ≤ b(3b)U L ≤ U ≤ U U ,x L ≤ x ≤ x U(3c)¯z(x t )=min U (4a)s.t. ¯h(U, x t ) ≤ ¯b , U L ≤ U ≤ U U ,x L ≤ x ≤ x U(4b)where ¯h(U, x t )=[J(U, x t ) h T (U, x t )] T and ¯b =[ɛb T ] T .A simple but conservative way to solve the above problem is by linearising theinequalities in (4) and solving off-line the linearized problem. More specifically,choose an initial x ∗ t and solve (4) to acquire U ∗ . Then linearize the inequalitiesin (4) over x ∗ t ,U∗ to obtain the following approximating, mp-LP problem overx t and U˘z(x t )=min U (5)¯h(U ∗ ,x ∗ )+∂¯h(U ∗ ,x ∗ t )t (U − U ∗ ) ≤∂U− ∂¯h(U ∗ ,x ∗ t )(x t − x ∗ t∂x ) t (6)U L ≤ U ≤ U U ,x L ≤ x ≤ x U (7)which now of form (1), where x is U and θ is x t . The solution to the mp-LP (5)is a linear, PWA function of x t ,˘z(x t ) ([8]). The control sequence U(x t )isalsoa