tailieunhanh - Model Predictive Control Part 2

Tham khảo tài liệu 'model predictive control part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Robust Model Predictive Control Design 13 Fig. 4. Dynamic behavior of unconstrained controlled system for u t . where Fki Rmyl k 2 3 .N2 i k 1 are output state feedback gain matrices to be determined so that cost function given below is optimal with respect to system variables. We would like to stress that y t k 1 y t 1 are predicted outputs obtained from predictive model 44 . Substituting control algorithm 36 to 33 we obtain x t 1 D1 j x t 38 where D1 j Aj BjK1 j K1 j I F12CBj 1 F11C F12 CAj j 1 2 .N For the first step of design procedure the cost function to be minimized is given as 1 Ể J1 t 39 t 0 1 t 4. 1 x m utfw and Q1 R1 are positive definite matrices of corresponding dimensions. For the case of k 2 u t 1 F22CD1 j x t F23 C AoD1 j x t Bou t 1 u t 1 K2 j x t and closed-loop system x t 2 AoD1 j BoK2 j x t D2 j x t j 1 2 .N 14 Model Predictive Control Fig. 5. Dynamic behavior of constrained controlled system for u t . Sequentially for k N2 2 step prediction we obtain the following closed-loop system x t k AoDk 1 j BoKk j x t Dk j x t 40 where D0 I Dk j AoDk 1 Ơ BoKk j k 2 3 . N2 j 1 2 .N Kk j 1 - Fkk 1CBo 1 FkkC Fkk 1 CAo Dk 1 j For the second step of robust MPC design procedure and k prediction horizon the cost function to be minimized is given as TO Jk E Jk t 41 t 0 where Jk t x t TQkx t u t k 1 TRku t k 1 and Qk Rk k 2 3 .N2 are positive definite matrices of corresponding dimensions. We proceed with two corollaries following from Definition 2 and Lemma 1. Corollary 1 The closed-loop system matrix of discrete-time system 1 is robustly stable if and only if there exists a symmetric positive definite parameter dependent Lyapunov matrix 0 P a P a T Iq such that P a D1 a TP a D1 a 0 42 where D1 a is the closed-loop polytopic system matrix for system 33 . The necessary and sufficient robust stability condition for closed-loop polytopic system with guaranteed cost is given by the recent result Rosinová et al. 2003 . Corollary 2 Consider the system 33 with control .