Assessment and Future Directions of Nonlinear Model Predictive ...

Controlling Distributed Hyperbolic Plants 437stability in closed loop. For that sake, and inspired by [2], define the control lawgiven byu ∗ = αR∗ Lr ∗ (4)− T 0where r ∗ is the set-point of the outlet oil temperature x ∗ n.Letx ∗ be the equilibriumstate corresponding to u ∗ and consider the dynamics of the error e = x−x ∗ ,given byė = −AL u∗ e + −Ae− Ax∗ − Bx 0ũ + CαL˜R (5)where ũ = u − u ∗ and ˜R = R − R ∗ . As shown by applying the Gronwall-Bellmaninequality, for ũ =0and ˜R = 0 the error dynamics is stable whenever the matrixĀ = − A Lis stable. It is not easy to prove a general result concerning the stabilityof the matrix A generated by the OCM and hence its stability must be checkedfor each application.Define the RHC for the error dynamics byminuJ =∫t+Hwhere ρ ≥ 0, H>0, and subject tot(e T (τ)Pe(τ)+ρ ũ 2 (τ) ) dτ (6)ė = −AL u ∗ e + −Ae− Ax∗ − BT 0ũ (7)LV 0 (t + H) ≥ V rhc (t + H) (8)in which r ∗ is given by (4), V 0 (H) =e T (H) |ũ=0 Pe(H) |ũ=0 and V rhc (H) =e T (H)Pe(H) whereP is an arbitrary symmetric positive definite matrix.The constraint (8) is equivalent to impose to the RHC that, at each iteration,the norm of the error at the end of the optimal sequence is bounded by thesame norm resulting from the error when u = u ∗ . The existence of a control law,defined for u = u ∗ , which stabilizes the closed loop, allows to interpret V 0 as aControl Lyapunov Function [11] and, assuming complete plant knowledge, is asufficient condition to ensure Global Asymptotic Stability of the loop closed bythe RHC, when the controller is applied to (3) [8]. The constraint (3) is thereforea sufficient condition for stability. It has been observed in the simulations performedthat this condition is active in the initial period, depending on the initialconditions. The rationale for minimizing (7) under the constraint (8) consists inincreasing the performance while ensuring stability (by imposing the constraint).3.2 State ObserverTo (3) associate the state estimator with output error re-injection:˙ˆx = − u [ ]L (Aˆx + Bx 0)+C ˆαR(t)+K(t)D(x − ˆx) ŷ = Dˆx = 00··· 1 ˆx (9)