Assessment and Future Directions of Nonlinear Model Predictive ...

Nonlinear Model Predictive Control: A Passivity-Based Approach 161[Ṡ(x) ≤ − a 3√2gx3 + (1 − γ ] [2)u 2 x 3 + − a 4√2gx4 + (1 − γ ]1)u 1 x 4 ,A 3 A 3 A 4 A 1≤ x 3 u 2 + x 4 u 1 .Consequentely, y =[x 3 ,x 4 ] T is a passive output for the quadruple tank systemwith the storage function S(x) = 1 2 x2 3 + 1 2 x2 4. Since the quadruple tank system iszero state detectable with respect to the output y =[x 3 ,x 4 ] T , the passivity-basednonlinear model predictive control (12) can be used to asymptotically stabilizethe quadruple tank system. In the following, the control task is to stabilize the thequadruple tank system at the equilibrium point x s =[14cm 14 cm 15.3 cm20.3 cm] T . The steady state control input u s at the equilibrium point x s is u s =[43.2 mls38.2 mls ]T . The performance index (5) was chosen as 1000([x 1 − x 1s ] 2 +[x 2 − x 2s ] 2 )+(u 1 − u 1s ) 2 +(u 2 − u 2s ) 2 . Furthermore, the nonlinear model predictivecontrol schemes were implemented with a prediction horizon T =60sand a sampling time δ =1s in the nonlinear model predictive control toolbox[13]. Figure 1 to Figure 6 show the dynamic behavior with the initial conditionx 0 =[5cm 5.6 cm 4 cm 4.5 cm] T of the passivity-based nonlinear model predictivecontroller (12) and the model predictive controller (9), i.e., a model predictivecontroller without guaranteed closed loop stability. As it can be seen fromthe figures, the closed loop system is unstable with the nonlinear model predictivecontroller (9). This instability is not a new fact in model predictive control [1] andhas motivated the development of model predictive control schemes with guaranteedclosed loop stability. In contrast to the model predictive controller (9), thepassivity-based nonlinear model predictive controller (12) asymptotically stabilizesthe quadruple system. Hence, this examples illustrates nicely that the proposedapproach achieves closed loop stability while improving the performance.6 ConclusionsIn this paper a nonlinear model predictive control scheme based on the conceptof passivity was developed. It was shown that by using a specific passivitybasedstate constraint closed loop stability is guaranteed. The basic idea ofthe passivity-based nonlinear model predictive control scheme is to unify optimalcontrol, passivity and nonlinear model predictive control based on theirrelationships. Since passivity and stability are closely related, the proposed approachcan be seen as an alternative to the control Lyapunov function basednonlinear model predictive control scheme [16]. Finally, the passivity-based nonlinearmodel predictive control scheme was applied to control a quadruple tanksystem in order to demonstrate its applicability.References[1] R. R. Bitmead, M. Gevers, I. R. Petersen, and R. J. Kaye, “Monotonicity andStabilizability Properties of Solutions of the Riccati Difference Equation: Propositions,Lemmas, Theorems, Fallacious Conjectures and Counterexamples”, Systemsand Control Letters , pages 309-315, (1985).