Consider a dc motor driving a mechanical load as shown

Question

1. Consider a DC motor driving a mechanical load as shown below:

(a)Derive the Transfer Function G(s) = X_(s)/E_(s), where e_a (t) is the motor applied voltage

(b)Find the time domain solution for x(t) given the to the applied voltage: e_a(t) = 5, t > 0 and in Matlab, plot the function x(t) from t = 0_s to t = 10_s.

(c)Using transfer function objects, plot the step response in Matlab and on a single figure compare this plot with part (b).

2. For the following block diagram:

(a) Find the transfer function G _{( s )} =C_(s)/ R_(s)

(b) Plot the impulse and the step response on the same figure.

(c) Plot the Poles of the system on the Real/Imaginary Axis

3. A crane hoisting a load is pictured below:

(a) Determine the state space equations for this system using the following state definitions: {x₁, x₂} ={θ,θ}

(b)nom the state space equations derive the transfer function G(s) =θ_(s)/F_(s)

4. A high performance robotics motor schematic and values are pictured below:

(a)Determine the symbolic state space notation for the dc motor governed by the Figure and Tables Below. Define the states as:
{x1,x2,x3} =

(b)The motor manufacturer has specified the no-load speed with an input of 1.5V is 9600 RPM with a tolerance of 1%. Does the motor meet this specification?

(c)Using Matlab, Simulate the Motor Angular Displacement for the voltage profile below.

(d)Plot the Motor Current for the Voltage Profile in c).

(e)Solve for the transfer function G(s) = = C(sI - A)-1B + D where y(t) = θ and
u(t) = e_a(t).

(f)  Plot the step response of the closed loop system below with values of K = {1, 5, 10,20}. Label each one on a single figure and state which value of K has:

• Fastest Rise Time

• Least Amount of Overshoot