Tham khảo tài liệu 'sliding mode control part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Multimodel Discrete Second Order Sliding Mode Control Stability Analysis and Real Time Application on a Chemical Reactor 479 Fig. 1. The structure of a multimodel discrete second order sliding mode control MM-2-DSMC . where md is the number of the partial models. The control applied to the system is given by the following relation U k V1 k U1eq k V2 k U2eq k . Vmd k Umdeq k Udis k 35 with Vi k the validity of the ith local state model Uieq k the partial equivalent term of the 2-DSMC calculated using the ith local state model Udis k the discontinuous term of the control. Ueqi k CTBt -1 aS k - CTAtx k CT xd k 1 Ai et Bi are the matrixes of the ith partial state model. The discontinuous term is given by the following expression Udis k Udis k - 1 - Msign a k The multimodel discrete second order sliding mode control MM-2-DSMC is then given by md U k L V k Ueqi k Udis k 36 i 1 A stability analysis of this last control law is proposed in the following paragraph. 480 Sliding Mode Control Stability analysis of the MM-2-DSMC Let s consider the following non stationary system x k 1 Adx k Bdu k r k y k Hx k 37 r k represents eventual non linearities and external disturbances. md md Considering the following notations Am E VjAj and Bm E UjBj i 1 i 1 we obtain the following model x k 1 Amx k Bmu k y k Hx k which is the multimodel approximation of the system 37 . This last system can be then written in the following form x k 1 Am A Am x k Bm ABm u k r k y k Hx k 39 such that Ad Am A Am Bd Bm ABm 40 We note A k A Amx k ABmu k r k 41 The system 37 can in this case be written as follows x k 1 Amx k Bmu k A k y k Hx k 42 The control law given by 36 is applied to the system. In the case of the multimodel the m equivalent term E Vj k ueqi k of 36 is written as follow i 1 Ueq k CTBm 1 S k - CTAmx k 43 In this case the sliding function dynamics are given by the following expression CTx k 1 S k 1 -fiS k CT A k CTBm udis k 44 The sliding function variation S k 1 S k is given by the .
