Assessment and Future Directions of Nonlinear Model Predictive ...

208 A.G. Wills and W.P. Heatharise at each iteration of an SCP approach. These algorithms are almost standardexcept that they are geared towards solving convex optimisation problems with aweighted barrier function appearing in the cost. This slight generalisation allowsa parsimonious treatment of both barrier function based model predictive control[14] and “standard” model predictive control, which can be identified as a speciallimiting case. The benefit of including a weighted barrier function is that iterationsstay strictly inside the boundary and fewer iterations are needed to converge. Thisbarrier approach is called r-MPC and has been successfully applied to an industrialedible oil refining process as discussed in [15].Note that due to page limitations all proofs have been omitted and can befound in [13].2 Nonlinear Model Predictive ControlIn what follows we describe NMPC and formulate an optimisation problem whichis convex except for the nonlinear equality constraints that represent the systemdynamics. This motivates a very brief discussion of SCP which leads to themain theme of this contribution being the two algorithms in Sections 3 and 4.The problem may be formulated as follows. Consider the following discrete-timesystem with integer k representing the current discrete time event,x(k +1)=f(x(k),u(k)). (1)In the above, u(k) ∈ R m is the system input and x(k) ∈ R n is the systemstate. The mapping f is assumed to be differentiable and to satisfy f(0, 0) = 0.Given some positive integer N let u denote a sequence of control moves givenby u = {u(0),u(1),...,u(N − 1)} and let x denote a state sequence given byx = {x(0),x(1),...,x(N)}.For the purposes of this contribution we require that the input sequence ushould lie within a compact and convex set U while the state sequence x shouldlie in the closed and convex set X. LetV N (x, u) denote the objective functionassociated with prediction horizon N. We assume that V N is a convex function.The control strategy for NMPC may be described as follows: at each time intervalk, given the state x(k), compute the following and apply the first control moveto the system.(MPC) : min V N (x, u), s.t. x 0 = x(k), x i+1 = f(x i ,u i ), x ∈ X, u ∈ U.x,uWe can associate with the sets X and U, respectively, gradient recentred selfconcordantbarrier functions B x and B u [14]. This allows (MPC) tobeexpressedas the limiting case when µ → 0 of the following class of optimisationproblems [3].(MPC µ ):minx,u V N (x, u)+µB x (x)+µB u (u) s.t.x 0 = x(k), x i+1 = f(x i ,u i ).The above class of optimisation problems (MPC µ ) have, by construction, a convexcost function and nonlinear equality constraints. If these equality constraints