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practice-midterm3

practice-midterm3 - T> 0 3 Compute e At for A = 1-2 Is...

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P RACTICE M ID - TERM E XAM L INEAR S YSTEMS Jo˜ao P. Hespanha Please explain all you answers. 1. The following equation models the motion of a pendulum: ¨ θ + k sin = τ where R is the angle of the pendulum with the vertical, R an applied torque, and k a positive constant. (a) Compute a state-space model for the system when u : = is viewed as the input and y : = as the output. (b) Compute the linearization of the system around the solution ( t ) = ( t ) = ˙ ( t ) = 0, t 0. 2. Consider the homogeneous linear time-varying system ˙ x = A ( t ) x , x ( 0 ) = x 0 with state transition matrix Φ ( t , ) . Consider also the non-homogeneous system ˙ z = A ( t ) z + x ( t ) , z ( 0 ) = z 0 . whose input x ( t ) is the state of the homogeneous system. (a) Compute x ( t ) and z ( t ) as a function of x 0 , z 0 , and Φ . Hint: no integrals should appear in your answer. (b) What must be true of z 0 to have z ( T ) = 0 for some particular time
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Unformatted text preview: T > 0. 3. Compute e At for A = 1-2 . Is the system ˙ x = Ax asymptotically stable? What about marginally stable? 4. Compute the transfer function of the system ˙ x = Ax + bu y = cx + u , where A : = -2-1 , b : = 1-1 , c : = b 1 1 1 B . Is this system BIBO stable? 5. Can the inverse of a n × n nonsingular matrix A be written as a linear combination of the matrices I , A , A 2 ,..., A n-1 ? Carefully justify your answer. Hint: Mr. Cayley and his friend Mr. Hamilton would know how to do this with their hands tied behind their backs. 1...
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