Assessment and Future Directions of Nonlinear Model Predictive ...

Nonlinear Model Predictive Control: An Introductory Review 11order to reduce the number of Quadratic Problems needed to be solved by theoptimization algorithm.5 Stability and Nonlinear Model Predictive ControlThe efficient solution of the optimal control problem is important for any applicationof nmpc to real processes, but stability of the closed loop is also ofcrucial importance. Even in the case that the optimization algorithm finds a solution,this fact does not guarantee closed-loop stability (even with perfect modelmatch). The use of terminal penalties and/or constraints, Lyapunov functionsor invariant sets has given rise to a wide family of techniques that guaranteethe stability of the controlled system. This problem has been tackled from differentpoints of view, and several contributions have appeared in recent years,always analyzing the regulator problem (drive the state to zero) in a state spaceframework. The main proposals are the following:• infinite horizon. This solution was proposed by Keerthi and Gilbert [18] andconsists of increasing the control and prediction horizons to infinity, P, M →∞. In this case, the objective function can be considered a Lyapunov function,providing nominal stability. This is an important concept, but it cannot bedirectly implemented since an infinite set of decision variables should becomputed at each sampling time.• terminal constraint. The same authors proposed another solution consideringa finite horizon and ensuring stability by adding a state terminal constraintof the form x(k+P )=x s . With this constraint, the state is zero at the end ofthe finite horizon and therefore the control action is also zero; consequently(if there are no disturbances) the system stays at the origin. Notice that thisadds extra computational cost and gives rise to a restrictive operating region,which makes it very difficult to implement in practice.• dual control. This last difficulty made Michalska and Mayne [28] look for aless restrictive constraint. The idea was to define a region around the finalstate inside which the system could be driven to the final state by means ofa linear state feedback controller. Now the constraint is:x(t + P ) ∈ ΩThe nonlinear mpc algorithm is used outside the region in such a way thatthe prediction horizon is considered as a decision variable and is decreasedat each sampling time. Once the state enters Ω, the controller switches to apreviously computed linear strategy.• quasi-infinite horizon. Chen and Allgöwer [8] extended this concept, using theidea of terminal region and stabilizing control, but only for the computation ofthe terminal cost. The control action is determined by solving a finite horizonproblem without switching to the linear controller even inside the terminalregion. The method adds the term ‖x(t + T p )‖ 2 P to the cost function. Thisterm is an upper bound of the cost needed to drive the nonlinear system to