## Description

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

Back-EMF Constant Km 0.00767 V.s/rad

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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