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HW3-Solution(2) - Flight Control Systems Homework#3...

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Flight Control Systems Homework #3 Solutions Dr. Bishop - Fall 2007 Reading Assignment: Modern Control Systems: Chapter 3 1-2. Exercises: E3.5, E3.20 3-4. Problems: P3.15, P3.29 5. Advanced Problems: AP3.7 6. Computer Problems: CP3.5 E3.5 From the block diagram we determine that the state equations are ˙ x 2 = - ( fk + d ) x 2 + ax 1 + fu ˙ x 1 = - kx 2 + u and the output equation is y = bx 2 . Therefore, ˙ x = Ax + Bu y = Cx + Du , where A = bracketleftbigg 0 - k a - ( fk + d ) bracketrightbigg , B = bracketleftbigg 1 f bracketrightbigg , C = bracketleftbig 0 b bracketrightbig and D = [ 0 ] . E3.20 The linearized equation can be derived from the observation that sin θ θ when θ 0. In this case, the linearized equations are ¨ θ + g L θ + k m ˙ θ = 0 . Let x 1 = θ and x 2 = ˙ θ . Then in state variable form we have ˙ x = Ax y = Cx where A = bracketleftbigg 0 1 - g/L - k/m bracketrightbigg , C = bracketleftbig 1 0 bracketrightbig , and x ( 0 ) = bracketleftbigg θ (0) ˙ θ (0) bracketrightbigg . P3.15 A state variable representation is ˙ x = 0 1 0 0 0 1 - 20 - 31 - 10 x + 0 0 1 r y = [20 5 0] x .
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R ( s ) Y ( s ) 20 x 1 x 2 s 1 20
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