Tham khảo tài liệu 'model predictive control part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Robust Adaptive Model Predictive Control of Nonlinear Systems 53 Proof of Proposition It can be shown that Assumption together with the compactness of Ex is sufficient for an analogue of Claim to hold . with J interpreted in a min max sense . In other words the cost J x 0 satisfies ai x Ex 0 J x 0 Kh x sx 0 for some functions Ki Kh which are class-C . x and whose parameterization in 0 satis- fies Kị x 01 Ki x 02 01 c 02. We then define the compact set Xj x I min0Ecov 0o J x 0 maxx0eX0 Kh llx0IIsỉ 0O . By a simple extension of Khalil 2002 the ISS property follows if it can be shown that there exists ac EC such that J x 0 satisfies xEX B L X Kc c maxf EFc . J x 0 0 minf ef D J x 0 0 36 where Fc B f x Kmpc x 0 t 0 t D c . To see this it is clear that J decreases until x t enters B zx ac c . While this set is not necessarily invariant it is contained within an invariant compact levelset Q c 0 x I J x 0 Kh ac c 0 . By the evolution of 0 t in 30b must approach some constant interior bound and thus limt . x t E Q c 0 . Defining Kd c maxxEQ c0 x sx completes the Proposition if c is sufficiently small such that B zx Kd c c Xj. Next we only prove decrease in the forward direction since the reverse direction follows analogously as it did in the proof of Theorem . Using similar procedure and notation as the Thm proof x 0 T denotes any worst-case prediction at t x 0 extended to T T via kf that is assumed to satisfy the specifications of Proposition . Following the proof of Theorem max .J x 0 maxliminf1 J x 6v 0 t 6 l ĩ LpdT Wp 0p Lph feFc . fEF 6 s TF T 6 J c J Sị0 maxliminf1 J x 6v 0 t 6 l ĩ LvdT W r v. LpL f EF 6 6 TF h 0 J 6ị0 1 f T v v Stvt_ f T pdr WR Sp 6 6 L dT WT0 0T6 6 L dT wT6 0T 37 where Lv Wv denote costs associated with a trajectory xV0 T satisfying the following initial conditions xv 0 x 0v 0 0. generated by the same worst-case ê and d - as xP0 T dynamics of form 30 on T E 0 6 and of form 25b 25c
