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Model Predictive Control Part 6

Tham khảo tài liệu 'model predictive control part 6', 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 Algorithms for Nonlinear Systems an Input-to-State Stability Approach 93 It is now possible to define a prototype of the first one of two nonlinear MPC algorithms at every time instant t define x x t and find the optimal control sequence uOt N-1 by solving the FHOCP1. Then according to the Receding Horizon RH strategy define KMPC x u tt x where u t t x is the first column of u t t N-1 and apply the control law u kmpc x . 11 Although the FHOCP1 has been stated for nominal conditions under suitable assumptions and by choosing appropriately the terminal cost function Vf and the terminal constraint Xf it is possible to guarantee the ISS property of the closed-loop system formed by 1 and 11 subject to constraints 2 - 4 . Assumption 3. The function l x u is such that 1 0 0 0 l x u a x where a is a N-function. Moreover l x u is Lipschitz with respect to x and u in X X U with constant Ll and Llu respectively. Remark 3. Notice that if the stage cost l x u is a piece-wise differentiable function in X and U as for instance the standard quadratic cost l x u x Qx u Ru and X and U are bounded sets then the previous assumption is satisfied. Assumption 4. The design parameter Vf and the set 0 x Vf x a a 0 are such that given an auxiliary control law Kf 1. 0 c Xn-1 2. Kf x 6 U Vx 6 0 3. f x Kf x 6 0 Vx 6 0 4. Hy x Vf x PVf x Vx 6 0 where ty and fVf are K -functions 5. Vf f x Kf x Vf x l x Kf x Vx 6 0 6. Vf is Lipschitz in 0 with a Lipschitz constant Lv. Remark 4. The assumption above can appear quite difficult to be satisfied but it is standard in the development of nonlinear stabilizing MPC algorithms. Moreover many methods have been proposed in the literature to compute Vf 0 satisfying the Assumption 4 see for example Chen Allgower 1998 De Nicolao et al. 1998 Keerthi Gilbert 1988 Magni De Nicolao Magnani Scattolini 2001 Mayne Michalska 1990 . Assumption 5. The design parameter Xf x 6 Rn Vf x av av 0 is such that for all x 6 f x kf x 6 Xf. Remark 5.