Assessment and Future Directions of Nonlinear Model Predictive ...

156 T. Raff, C. Ebenbauer, and F. Allgöwernonlinear model predictive control based on their relationship to optimal control[16], passivity is merged with nonlinear model predictive control in the samespirit. Suppose that the system (1) is passive with a continuously differentiablestorage function S and zero-state detectable. Then the passivity-based nonlinearmodel predictive control scheme for the system (1) is given byminu(·)s.t.∫t+Tt(q(x(τ)) + u T (τ)u(τ) ) dτẋ = f(x)+g(x)uy = h(x)u T (t)y(t) ≤−y T (t)y(t).(12)The passivity-based state constraint in the last line in the setup (12) is motivatedby the fact that in case the system (1) is passive and zero-state detectable,it can be stabilized with the feedback u = −y. Hence, the passivity-based stateconstraint is a stability constraint which guarantees closed loop stability. Furthermore,if the storage function S is radially unbounded, and all solutions ofthe system are bounded, then the closed loop system is globally asymptoticallystable. In contrast to many other nonlinear model predictive control schemes [11]which achieve stability by enforcing a decrease of the value function along thesolution trajectory, the stability of the proposed nonlinear model predictive controlscheme is achieved by using directly a state constraint. Hence, one obtainsthe following stability theorem of the passivity-based nonlinear model predictivecontrol scheme (12):Theorem 1. The passivity-based nonlinear model predictive control scheme (12)locally asymptotically stabilizes the system (1) if it is passive with a continouslydifferentiable storage function S and zero-state detectable.Proof. The proof of Theorem 1 is divided into two parts. In the first part it isshown that the nonlinear model predictive control scheme (12) is always feasible.In the second part it is then shown that the scheme (12) asymptotically stabilizesthe system (1).Feasibility: Feasibility is guaranteed due to the known stabilizing feeback u =−y. Stability: Let S be the storage function of the passive system (1). With thedifferentiable storage function S and the state constraint in the model predictivecontrol scheme (12), one obtainsṠ(x(t)) ≤ u T (t)y(t) ≤−y T (t)y(t).Using the fact that the system (1) is zero-state detectable, the same argumentspresented in Theorem 2.28 of [17] can be used in order to show asymptotic stabilityof the origin x =0. Hence, the passivity-based nonlinear model predictivecontrol scheme (12) asymptotically stabilizes system (1) if it is passive and zerostatedetectable.