265_adv_control_eng

Evaluate the coefficients of the state feedback gain

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Unformatted text preview: te equation in the controllable canonical form. Evaluate the coefficients of the state feedback gain matrix using: (a) The direct comparison method (b) The controllable canonical form method (c) Ackermann's formula such that the closed-loop poles have the values s ˆ À2, s ˆ À2 Solution From the open-loop transfer function  • y ‡ 4y ˆ u (8:105) x1 ˆ y (8:106) • x1 ˆ x2 • x2 ˆ À4x2 ‡ u (8:107) Let Then //SYS21/D:/B&H3B2/ACE/REVISES(08-08-01)/ACEC08.3D ± 252 ± [232±271/40] 9.8.2001 2:34PM 252 Advanced Control Engineering Equation (8.106) provides the output equation and (8.107) the state equation ! ! ! ! • x1 01 x1 0 ˆ ‡ u (8:108) • 0 À4 x2 1 x2 y ˆ [1 x 0] 1 x2 ! (8:109) The characteristic equation for the open-loop system is  ! ! s 0 01   ˆ s2 ‡ 4s ‡ 0 jsI À Aj ˆ  À 0s 0 À4  ˆ s2 ‡ a1 s ‡ a0 (8:110) Thus a1 ˆ 4, a0 ˆ 0 The required closed-loop characteristic equation is (s ‡ 2)(s ‡ 2) ˆ 0 or s 2 ‡ 4s ‡ 4 ˆ 0 (8:111) s2 ‡ 1 s ‡ 0 ˆ 0 (8:112) i....
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