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# AMATH 455/655 Lab 1 Cart Position solution

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Consider a cart on a track, forced by a rotary motor. Let x(t) indicate the position
of the cart at time t, V (t) the motor voltage at time t, ηg the gearbox efficiency, Kg the
gearbox gear ratio, ηm the motor efficiency, Kt the motor torque constant, Km the back-EMF
constant, Rm the motor armature resistance, M is the mass of the cart, ν i the coefficient of
viscous friction, and rmp the motor pinion radius. (Except for x and V , all these quantities
are constant.)

The governing equation is
Mx¨(t) = ηgKgηmKt(rmpV (t) − KgKmx˙(t))
Rmr
2
mp
− νx˙(t),
The voltage can be controlled by the operator. The object of the control is to drive the cart
to a particular position.

1. Only the cart position is measured. Put this equation into state-space form, indicating
the state, input and output.
2. What are the eigenvalues of the system matrix?
3. (*) For any initial conditions, and desired position r, the required voltage so that the
steady-state value of x is r can be found by solving the differential equation.
(a) (*) Calculate the required constant voltage Vreq and time period T so that for the
given parameter values and input voltage
V (t) = 
Vreq 0 ≤ t ≤ T,
0 t > T,
we get
lim

t→∞
x(t) = 1, lim
t→∞
x˙(t) = 0.
Use zero initial conditions: x(0) = 0, ˙x(0) = 0. (There are different values of T
and Vreq that will give a correct answer, but the value of Vreq will change with the
choice of T. You may choose T = 0.9, for instance. Then, Vreq can be obtained in
terms of the system parameters only.)

(b) (*) The steady-state value of x(t) is limt→∞ x(t), if it exists. What happens to
the steady state position if there is an error in the system parameter values?
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(c) (*) What happens to the steady state value if because of some disturbance d, the
actual voltage to the motor is Vreq + d?
4. The procedure described in the previous step for calculating the voltage to drive the
cart to a particular position is complicated and has other issues; in particular it is
affected by errors in the parameter values. Consider instead for some constant Kp > 0
V (t) = Kp(r − x(t)), (1)

where r is the given desired position.
(a) What is the steady state (or final) position x by applying this voltage to the
system?
(b) How does the final value of x change if there are errors in some parameter values?
(c) Is the final position dependent on the value of Kp? If so, how?
(d) How does the final position change if because of some disturbance d, the actual
voltage to the cart is Kp(r − x(t)) + d instead of that in (1)? How can the effect
of the disturbance on the final position be reduced?

## Simulation

In this part, you will need to include an screenshot of your Simulink diagrams and
accompanied codes. If you are using subsystems, you should include their screenshots
as well.
5. Using the parameter values in Table 1, implement the system in Simulink, and by using
a “Signal Generator” block, simulate the effect of square-wave inputs of frequencies 0.5
and 0.1 Hertz and amplitude 1V. It should be noted that this square-wave signal is
directly applied as input V (t) to the system and there is no feedback in this part of
simulations.

(a) Plot the position and velocity over a range of 30 seconds. Make sure your
plots include enough information. For this purpose, plotting data from workspace is
recommended.
6. In this step, replace the square-wave input in the previous step with the feedback loop,
as in (1), in your simulink diagram. Then, in different scenarios, set Kp to 0.5, 1, 5.
Use a square wave with frequency 0.1 Hz and amplitude 1 as the reference input r in
(1).

(a) Discuss the behavior of the system with the closed feedback and the effect of
different values of the feedback gain Kp.
(b) Repeat the simulation with feedback loop with a reference input r with frequency 0.5 Hertz.
(c) What happens if Kp < 0?
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Cart Mass M 0.455 kg
Coefficient of Friction ν 10.682 kg/s
Gearbox Efficiency ηg 1.00
Gearbox Gear Ratio Kg 3.71
Motor Efficiency ηm .88
Motor Torque Constant Kt 0.00767 Nm/A
Motor Armature Resistance Rm 2.6 Ω
Motor Pinion Radius rmp 0.00635 m
Table 1: Parameter Values

7. Consider both configurations of the simulations, with and without feedback, with the
square-wave input of frequency 0.1 Hertz and amplitude 1, and Kp = 1. For both
cases, perturb M and ν values within ± 30% of their nominal values to consider the
effect of model uncertainty.

(a) How do the outputs of the system change compared to the results with no
perturbation?
(b) By comparing the results from (a) for both open-loop and closed-loop (with
feedback) cases, discuss their robustness against perturbations in the model parameters. Support your conclusion with the simulation results.
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