hwk4 - the rate of change of this angle. Finally, u is the...

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MAE 143 A: Signals and Systems. Homework #4. Assigned Jan 27. Due Feb 3. 1. A circuit with two resistors, an inductor and a capacitor element, is modeled by the following linear ODE: ± dV dt di L dt ² = ± 1 / 3 0 - 1 / 3 0 ²± V i L ² + ± 1 / 3 1 / 3 ² i S Here, V is the potential difference at the capacitor, i L is the current going through the inductor, and i S is the current source we apply to the circuit through some batteries. The outputs for the system coincide with V, i L . Check that the system is not stable. Is the system controllable? ( Observation: Instability here means that there are some current values i S that are unsafe ; that is, they can give rise to large i L and V which can burn out the circuit.) 2. A linearized model of a ball and beam system (see figure below) is modeled through the following system of ODEs: ˙ x 1 ˙ x 2 ˙ x 3 ˙ x 4 = 0 1 0 0 0 0 - mg J m 2 + m 0 0 0 0 1 0 0 0 0 x 1 x 2 x 3 x 4 + 0 0 0 1 u Here, x 1 = r the position of the ball over the beam, x 2 = ˙ r , its velocity, x 3 = α the angle that the beam makes with the horizontal, x 4 = ˙
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Unformatted text preview: the rate of change of this angle. Finally, u is the torque applied to the lower end of the beam. Suppose that the mass of the ball m = 1 kg , g = 10 m/s 2 , J = 1 kg m 2 . Check the controllability properties of this system and explain what it means in your own words. Figure 1: By rotating the beam, we can change the position of the ball on it. 3. The impulse response of an LTI system is measured to be h ( t ) = e 2 t u ( t ), where u ( t ) is the unit step signal. What is the response of the system to x ( t ) = ramp( t )? (Zero initial conditions are assumed.) 4. Suppose that the impulse response of an LTI physical system is approximately measured to be h ( t ) = e-7 t cos(2 t ) u ( t ), where u ( t ) is the unit step signal. What is the response of the system to the signal x ( t ) = 7 u ( t )? (Zero initial conditions are assumed.) 1...
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