Assessment and Future Directions of Nonlinear Model Predictive ...

438 J.M. Igreja, J.M. Lemos, and R.N. da SilvaThe error dynamics e 1 := x − ˆx is given by:e˙1 (t) =A e e 1 + C ˜αR(t) (10)where A e := − u LA − K(t)D with K(t) the observer gain.3.3 Lyapunov Adaptation LawConsider the candidate Lyapunov functionV 1 = e T 1 Qe 1 + 1 γ ˜α2 (11)where γ>0 is a parameter, Q is a positive definite matrix and the parameterestimation error ˜α is defined as ˜α(t) :=α − ˆα(t) whereˆα is the estimate of α.Its derivative is given by:Stability holds if˙V 1 = e T 1 (A T e Q + QA e )e 1 +2˜αR(t)C T Qe 1 + 2 γ ˜α ˙˜α (12)−M(t) =(A T e Q + QA e) < 0 and ˙˜α = −γ(CR(t)) T Qe 1from which the following adaptation law follows:˙ˆα = γ(CR(t)) T Qe 1 (13)It is possible to prove that M(t) > 0 is ensured by the following choice of theobserver gain:K(t) = u L K 0 (14)with the matrix M 0 given byM 0 = −[(−A − K 0 D) T Q + Q(−A − K 0 D)] (15)that exists if the pair (A, D) is observable and choosing K 0 such that −A−K 0 Dis stable. With this choice, and remarking that u>0:˙V 1 = −e T 1 M(t)e 1 = − u L eT 1 Me 1 ≤− u maxL eT 1 Me 1 ≤ 0 (16)and, by La Salle’s Invariance Principle, it follows that lim t→0 e 1 (t) =0.Theparameter estimation error ˜α(t) will tend to zero if u satisfies a persistency ofexcitation condition.3.4 RHC Computational AlgorithmA computational efficient version of the the adaptive RHC is obtained by constrainingu in (6) to be a staircase function with N u steps u = seq{u 1 , ..., u Nu }and using x and α replaced by their estimates. The estimate of u ∗ is given by: