TAILIEUCHUNG - Advanced Model Predictive Control Part 6

Tham khảo tài liệu 'advanced 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 | Fuzzy-neural Model Predictive Control of Multivariable Processes 139 . 1 T-minJ x 2xTHx fTx subject to Ax b 50 where H and f are the Hessian and the gradient of the Lagrange function x is the decision variable. Constraints on the QP problem 50 are specified by Ax b according to 49 . The Lagrange function is defined as follows N L x Ằ J x X A i 1 2 .N i 1 51 where Xi are the Lagrange multipliers ai are the constraints on the decision variable x N is the number of the constraints considered in the optimization problem. Several algorithms for constrained optimization are described in Fletcher 2000 . In this chapter a primal active set method is used. The idea of active set method is to define a set S of constraints at each step of algorithm. The constraints in this active set are regarded as equalities whilst the rest are temporarily disregarded and the method adjusts the set in order to identify the correct active constraints on the solution to 52 min J x 2 xTHx fTx subject to aix bi aix bi 52 At iteration k a feasible point x k is known which satisfies the active constraints as equalities. Each iteration attempts to locate a solution to an equality problem EP in which only the active constraints occur. This is most conveniently performed by shifting the origin to x k and looking for a correction 8 k which solves nun I Hx ỊA subject to a 0 0 dị e S 53 where f k is defined by f k f Hx k and is VJ x k for the function defined by 52 . If 8 k is feasible with regard to the constraints not included in S then the feasible point in next iteration is taken as x k 1 x k ổ k . If not a line search is made in the direction of 8 k to find the best feasible point. A constraint is active if the Lagrange multipliers Xi 0 . it is at the boundary of the feasible region defined by the constraints. On the other hand if there exist Xi 0 the constraint is not active. In this case the constraint is relaxed from the active constraints set S and the algorithm continues as before by .

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