Assessment and Future Directions of Nonlinear Model Predictive ...

Controlling Distributed Hyperbolic Plants 439û ∗ =ˆα(t) R(t) Lr(t) − T 0 (t)(17)Here, ˆx and ˆα are obtained using the state estimator (9), the adaptation law (13),u(¯t) is a sequence of step functions with amplitude u i (i =1,...,N u ) and durationTN u.Thevariable¯t represents time during the minimization horizon ¯t ∈ [0,H[. Theinitial condition ˆα(0) for the estimate of α is provided by the designer.Once the minimization result u(¯t) is obtained, according to a receding horizonscheme u 1 is applied to the plant at t + δ and the whole process is restarted,δ being an interval of time which is at least the time needed to compute thesolution. It is assumed that δ is much smaller than the sampling period. Thereare classes of plants, such as switched nonlinear systems [7], for which the choiceof δ has a bearing on the stability properties of the predictive controller. In thecase at hand, no such problems were found.The minimization is always feasible since ũ = 0 (corresponding to u = u ∗ )preservesthe closed loop stability while satisfying the constraint (8) with V 0 = V rhc .4 Simulation ResultsSimulation results of the proposed RHC have been performed in a detailed modelof the solar field obtained from first physical principles, according to [3], andcalibrated with plant data. Experimental sequences for R(t) andT 0 (t) areused.The reduced model (3) uses 3 interior collocation points z =[0.113 0.500 0.887]and leads to a matrix −A/L whose eigenvalues have all strictly negative part.In order to configure the controller, the following parameter choices have beenmade: γ =1× 10 5 , K 0 = [15 15 15 15], ρ =1.5 × 10 −10 , H = 180 s and N u = 26.Figures (1) through (3) show the results. Fig. (1) shows the time evolution of thereference (easily recognizable by being a sequence of steps) and of the temperature3002502001501005000 1 2 3 4 5 6 7Time [h]Fig. 1. Solar field with RHC. Outlet oil temperature and reference and temperatureestimates at the collocation points [ o C].