Assessment and Future Directions of Nonlinear Model Predictive ...

MPC for Stochastic Systems 261form based on the solution of a pair of Lyapunov equations, and shows that theLyapunov-like property:E k[L(k + N +1|k +1)+l(k + N|k +1)]≤ L(k + N|k) (15)holds whenever predictions at time k + 1 are generated by the sequenceu(k +1)={u(k +1|k),...,u(k + N − 1|k),Kx(k + N|k +1)} (16)where u(k) ={u(k|k),u(k +1|k),...,u(k + N − 1|k)} is the predicted inputsequence at time k and u(k) =u(k|k).To simplify notation, let x δ = x − ¯x, where¯x(k + j|k) =E k x(k + j|k), anddefine[Z 1 (k + j|k) =E k x(k + j|k)x T (k + j|k) ] ,[Z 2 (k + j|k) =E k xδ (k + j|k)x T δ (k + j|k)] .Lemma 3. If the terminal penalty in (4) is defined byL(k + N|k) =Tr ( Z 1 (k + N|k)S 1)+(κ21 − 1)Tr ( Z 2 (k + N|k)S 2)(17)where S 1 = S T 1 ≻ 0 and S 2 = S T 2 ≻ 0 are the solutions of the Lyapunov equations¯Φ T S 1 ¯Φ + L ∑i=1Φ T i¯Φ T S 2 ¯Φ + c1 c T 1 = S 2(18a)(S1 +(κ 2 1 − 1)S )2 Φi + c 1 c T 1 = S 1 (18b)then L(k+N|k) is the cost-to-go: L(k+N|k) = ∑ ∞j=Nl(k+j|k) for the closed-loopsystem formed by (3) under the terminal control law u(k + j|k) =Kx(k + j|k).Proof. With u(k + j|k) =Kx(k + j|k), it is easy to show that, for all j ≥ N:Z 1 (k + j +1|k) = ¯ΦZ 1 (k + j|k) ¯Φ T +Z 2 (k + j +1|k) = ¯ΦZ 2 (k + j|k) ¯Φ T +L∑Φ i Z 1 (k + j|k)Φ T ii=1L∑Φ i Z 2 (k + j|k)Φ T i .Using these expressions and (18a,b) to evaluate L(k + j +1|k), we obtaini=1(19a)(19b)L(k + j +1|k)+l(k + j|k) =L(k + j|k), (20)which can be summed over all j ≥ N to giveL(k + N|k) − lim L(k + j|k) = ∑∞ l(k + j|k),j→∞but x(k +1)=Φx(k) is necessarily mean-square stable [16] in order that thereexist positive definite solutions to (18a,b), and it follows that lim j→∞ L(k +j|k) =0.j=N