Tham khảo tài liệu 'optimal control with engineering applications episode 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 124 Solutions Reverting now to operator notation we have found the following results t t I d W t Ý t T UA t dT where the operator U has been defined on p. 71. Considering this result for A A PC and comparing it with the equations describing the costate operator Ao in Chapter establishes that Ao t is a positive operator at all times t G ta tb because Ỷ . . is a positive operator irrespective of the underlying matrix A . In other words infimizing the Hamiltonian H is equivalent to infimizing 5. Of course we have already exploited the necessary condition d51 dP 0 because the Hamiltonian is of the form H A 5 P . The fact that we have truly infimized the Hamiltonian and 51 with respect to the observer gain matrix P is easily established by expressing 5 in the form of a complete square as follows 5 A5 PC 5 5At 5C TPT BQBT PRPT A5 5At BQBt 5CTR-1C 5 P 5CtR-1 R P 5CtR-1 t . The last term vanishes for the optimal choice Po 5CTR-1 otherwise it is positive-semidefinite. This completes the proof. Chapter 3 1. Hamiltonian function H uY Aax Au Maximizing the Hamiltonian dH du uY-1 A 0 y 1 uY 2 0 . d2H du2 Since 0 Y 1 and u 0 the H-maximizing control is 1 u AT-1 0 . In the Hamilton-Jacobi-Bellman partial differential equation J H 0 for the optimal cost-to-go function J x t A has to be replaced by dx and u by the H -maximizing control . dJ u V dx Solutions 125 Thus the following partial differential equation is obtained J J 1 - 1 J 0 . dt dx 7 J x According to the final state penalty term of the cost functional its boundary condition at the final time tb is J x tb xY . Y Inspecting the boundary condition and the partial differential equation reveals that the following separation ansatz for the cost-to-go function will be successful J x t xY G t with G tb . Y The function G t for t e ta tb remains to be determined. Using J 1 x G and J x g dt Y dx the following form of the Hamilton-Jacobi-Bellman partial differential equation is obtained xY G YGa y 1 G 1 0 . Therefore the .
