Tham khảo tài liệu 'optimal control with engineering applications episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 92 3 Optimal State Feedback Control Controller with a Progressive Characteristic For a linear time-invariant dynamic system of first order we want to design a time-invariant state feedback control u x the characteristic of which is super-linear . u x is progressive for larger values of the state x. In order to achieve this goal we formulate a cost functional which penalizes the control quadratically and the state super-quadratically. As an example let us consider the optimal state feedback control problem described by the following equations x t ax t u t J u i qcosh x t q -u2 t dt Jo 2 where a and q are positive constants. Using the series expansion . . x2 x4 x6 x8 cosh x 1 2T 4T 6T 8r for the hyperbolic cosine function we get the following correspondences with the nomenclature used in Chapter A a B 1 f x u 0 fu x u 0 R 1 N 0 Q q e x u q T 6 8 . t x u 0 1st Approximation LQ-Regulator x t ax u J u Ị p qx 2 2u dt uo 1 Kx Approximatively Optimal Control 93 where K a ựa2 q is the positive solution of the Riccati equation K2 2aK q 0 . The resulting linear control system is described by the differential equation x t a K x t Ao x t Ự a2 q x t and has the cost-to-go function J 2 x 1 Kx2 1 a ự a2 q x2 with the derivative x Kx a y2 a2 q x . 2nd Approximation From 0 j A x J2 f 2 t 3 we get J 0 . Since fu x u 0 u x u 0 B 1 and R 1 we obtain the following result for all k 2 uo k J fc i Hence uo 2 J3 0 . 3rd Approximation 0 J Aox J Buo 2 j 5-j f j 1 uo 2 TRuo 2 t 4 j 2 7 4 qx3 Jx 4 a2 q uo 3 J 4 3 qx 4 a2 q 94 3 Optimal State Feedback Control 4th Approximation 3 4 2 0 j Ax 2 Jx j Bu j 2 J6-j f j 2 uo j Ruo 5-j 5 J 0 uo 4 -J5 0 5th Approximation 0 J A x J-i Bu j J- f j uo j Ruo 6-j 1 uo 3 Ruo 3 t 6 j 2 7 6 qq2 A x5 Jx 6 2 4 2 a2 q J ựãrq uo 5 -J6 - q q2 A 2 4 2 a2 q J x5 Võ q 6th Approximation 0 J A x J 8-j Bu j J8-j f j u j Ru 7-j e 7 j 2 j 2 j 2 J 7 0 uo 6 -J 7 0 7th Approximation 0 j Aox j 9-j Buo j j 9-j f j j 2 j 2 uo j Ruo 8-j 1 uo
