Assessment and Future Directions of Nonlinear Model Predictive ...

630 J.J. Arrieta–Camacho, L.T. Biegler, and D. Subramaniansuch integration is achieved by coupling a Finite State Machine (FSM) with anNMPC regulator. The FSM receives the current state of the environment andoutputs a collection of sets, which is used to alter a nominal optimal controlproblem (OCP) in the NMPC regulator. For instance, the detection of a newobstacle leads the FSM to add a new element to the relevant set. The updatethen alters a nominal OCP by adding the constraints pertinent to the obstaclejust detected, thus leading to an optimal avoidance maneuver.In the next section we derive the multi-stage NMPC problem formulation.Within this framework the NMPC regulator incorporates a 3 degree-of-freedomnonlinear dynamic model of each aircraft, and considers a path constrained OCPthat minimizes a performance index over a moving time horizon. In addition,we describe characteristics of the NMPC formulation that allow the aircraft tomeet their targets. Stability properties for NMPC are discussed and adapted tothe particular characteristics of this application in Section 3. In Section 4, ouroverall approach is applied to a small case study which demonstrates collisionavoidance as well as implementation of the NMPC controller within the FSMframework. Finally, Section 5 concludes the paper and presents directions forfuture work.2 Optimization Background and FormulationWe begin with a discussion of the dynamic optimization strategy used to developour NMPC controller.Optimization of a system of Differential Algebraic Equations (DAEs) aimsto find a control action u ∈ U ⊆ R nu such that a cost functional is minimized.The minimization is subject to operational constraints and leads to the followingOptimal Control Problem (OCP):minJ[z d (t F ),t F ]usubject to: z˙d = f d [z d ,z a ,u], t ∈ T H0=z d (t I ) − z d,I(1)0=f a [z d ,z a ,u], t ∈ T H0 ≤ g[z d ,z a ,u,t],u(t) ∈ U, t ∈ T Hwhere z d ∈ R n dand z a ∈ R na are the vectors of differential and algebraic variables,respectively. Given u(t), a time horizon of interest T H := [t I ,t F ]andappropriateinitial conditions z d (t I )=z d,I , the dynamic behavior of the aircraft canbe simulated by solving the system of DAEs: z˙d = f d [z d ,z a ,u], f a [z d ,z a ,u]=0,with this DAE assumed to be index 1. Notice that some constraints are enforcedover the entire time interval T H . In this study, we solve this problemwith a direct transcription method [4, 5], which applies a simultaneous solutionand optimization strategy. Direct transcription methods reduce the originalproblem to a finite dimension by applying a certain level of discretization. Thediscretized version of the OCP, a sparse nonlinear programming (NLP) problem,can be solved with well known NLP algorithms [5] like sequential quadratic