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Unformatted text preview: ME 132 Solutions # 11 1 Output feedback stabilization (a) The transfer function can be found by the standard formula, C ( sI- A )- 1 B + D , or by recognizing that the system is almost in controllable canonical form. H u y =- 1 s 3 + 4 s 2 + 3 s- 2 (b) Plugging in the control law into the state-space description of the plant gives x = ( A- BCk ) x + Br The output equation is not changed. This leads to the state-space realization of the closed loop system: x 1 x 2 x 3 y = 1 1 2 + k- 3- 4- 1 1 x 1 x 2 x 3 r (c) The closed loop characteristic polynomial is ( s ) = s 3 + 4 s 2 + 3 s 2- (2 + k ). From the Routh array, the conditions for stability are 4 >- (2 + k ) > 12 >- (2 + k ) It is possible to satisfy the three inequalities above when- 14 < k <- 2. 2 Pole-zero diagrams (a) CC. The pole-zero diagram shows that this is a first-order system without any zeros.(a) CC....
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- Spring '08