Tham khảo tài liệu 'optimal control with engineering applications episode 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 82 3 Optimal State Feedback Control has the following H-minimizing control u -R-1 BtX NTx -R-1 BtVx J NTx . Thus the resulting Hamilton-Jacobi-Bellman partial differential equation is 0 ìJ H x u x Vx J t Vx J t at dj 2 xTQx - xTNR-1NTx - VxJtBR-1BtVxJ Vx J A-BR-1Nt x xt A-BR-1Nt TVx J with the boundary condition J x tb 2xTFx . Obviously an ansatz for the optimal cost-to-go function J x t which is quadratic in x should work. This results in a partial differential equation in which all of the terms are quadratic in x. The ansatz J x t 2xTK t x leads to VxJ x t K t x and x -xTK t x . dt 2 The following final form of the Hamilton-Jacobi-Bellman partial differential equation is obtained 1 xT K t Q - NR-1NT - K t BR-1BTK t K t A-BR-1Nt A-BR-1Nt TK t x 0 J x tb 2xTFx . Therefore we get the following optimal state feedback control law u t -R-1 t BT t K t Nt t x t where the symmetric and positive- semi definite matrix K t has to be computed in advance by solving the matrix Riccati differential equation K t - A t -B t R-1 t NT t TK t - K t A t -B t R-1 t NT t - K t B t R-1 t BT t K t - Q t N t R-1 t NT t with the boundary condition K tb F . Hamilton-Jacobi-Bellman Theory 83 The Time-Invariant Case with Infinite Horizon In this section time-invariant optimal control problems with the unconstrained control vector u t G Rm an infinite horizon and a free final state x tb at the infinite final time tb to are considered. The most general statement of this optimal control problem is Find a piecewise continuous control u 0 to Rm such that the completely controllable dynamic system x t f x t u t is transferred from the given initial state x 0 xa to an arbitrary final state x to G Rn at the infinite final time and such that the cost functional J u L x t u t dt J0 is minimized and attains a finite optimal value. In order to have a well-posed problem the variables of the problem should be chosen in such a way that the intended stationary equilibrium state is at x 0 and that it