Question Details

[answered] MMAE 443 HOMEWORK #10 Fall 2016 Due: Wednesday, November 16


i need help with the third problem .any help will be appreciated



MMAE 443 HOMEWORK #10 Fall 2016 Due: Wednesday, November 16, 2016

 

Problem #1 (30%)

 

You are given the following system 20

 

( )

 

+ 12 + 20

 

We now are interested in achieving a settling time that is less than 1 sec. You will have to use

 

feedback control to do this.

 

a. Draw the block diagram of the system with a proportional feedback controller (Kp) and

 

calculate the closed-loop-transfer function (CLTF).

 

b. Determine the poles of the CLTF with Kp=0, 0.2, 0.4, 0.6, 0.8, 1, 2, 4 , and plot them on

 

a complex plane.

 

c. Now determine the Kp value that will give you a 1 sec settling time (Make your slow

 

decaying pole settle at 1 sec). Plot the unit step response of the resulting system with this

 

Kp. The settling time should be fast enough, but you should see some steady state error.

 

d. To help mitigate the steady-state-error problem, look at your closed loop transfer function

 

and calculate the steady state response to a unit step using the final value theorem. What

 

do you need to do to Kp to ?help? with the steady state error problem?

 

e. Crank your Kp way up: make it 1000. Now plot the step response again. What happened

 

to the steady state error? Did using a large Kp cause any potential problems?

 

( )= Problem #2 (30%)

 

Consider the DC motor drive system shown in the block diagram. Instead of speed control, we are

 

interested in the position control of the motor. The block diagram of the system with a feedback

 

controller Gc (s ) is shown below: K LTL ( s )

 

- R(s) - Gc (s)

 

Controller U(s) Km

 

ms 1 ( s ) 1

 

s Y ( s) (s ) Motor

 

Fig3.1 Dynamics Assume K m 147, m 0.1 and the controller is a proportional plus derivative (PD) controller

 

given by Gc ( s ) K p K d s , which can be implemented in reality by applying the following

 

control input voltage:

 

u (t ) K p e(t ) K d e(t ), e(t ) r (t ) y (t )

 

a. Obtain the closed-loop transfer functions from the reference input R ( s) and the disturbance

 

input TL ( s) to the output Y ( s ) .

 

b. Determine the proportional gain K p and the derivative gain K d of the controller so that the

 

unit step response of the closed-loop system to the reference input has an overshoot of 15%

 

and a 2% setting time of 0.1 sec.

 

_1_ Problem #3 (40%)

 

Re-consider the DC motor drive system ( K m 147, m 0.1 ) that you examined in problem 2.

 

The goal is to design a feedback controller Gc ( s ) to meet the following performance

 

requirements:

 

P1. The steady-state error caused by a unit constant disturbance torque TL ( s) should be

 

zero (Steady-state performance specifications).

 

P2. The unit step response of the closed-loop system to the reference input should have

 

an overshoot of 15% and a 2% setting time of 0.1 sec (Transient performance

 

specifications).

 

a. Using the closed-loop transfer function GYT ( s) from the disturbance torque TL ( s) to the

 

output Y ( s ) , obtain the steady-state error caused by a unit constant disturbance torque

 

TL ( s) when a PD feedback controller Gc ( s ) K P K D s is used. If you are doing correctly,

 

you will find that the steady-state error obtained is not zero and thus a PD controller cannot

 

satisfy the steady-state performance requirement P1.

 

b. To satisfy the steady-state performance requirement P1, we can use the following

 

proportional plus integral and derivative (PID) feedback controller:

 

1

 

Gc ( s ) K P K D s K I

 

s

 

which can implemented by applying the following physical control input to the plant:

 

t

 

1 u (t ) K P e(t ) K D e(t ) K I e(t )dt or U ( s) K P K D s K I E ( s )

 

s 0

 

Obtain the closed-loop transfer functions GYR ( s) and GYT ( s ) when the above PID

 

feedback controller is used.

 

c. Show that with the above PID controller, as long as the closed-loop system is stable, the

 

steady-state performance requirement P1 will be satisfied.

 

d. To determine the PID gains K P , K D and K I to satisfy the transient performance requirement

 

P2, we will use the pole placement technique. Your denominator of the closed loop

 

transfer function should be third order so you will need to place three poles. Determine a

 

desired form for the denominator of the closed loop transfer function such that the

 

dominant poles will be second order with an overshoot of 15% and a 2% setting time of

 

0.1 sec (as prescribed by requirement P2). Place a third pole at -100.

 

e. Determine the PID gains K P , K D and K I which will produce the desired pole locations from

 

part d. _2_

 


Solution details:
STATUS
Answered
QUALITY
Approved
ANSWER RATING

This question was answered on: Sep 18, 2020

PRICE: $15

Solution~0001013948.zip (25.37 KB)

Buy this answer for only: $15

This attachment is locked

We have a ready expert answer for this paper which you can use for in-depth understanding, research editing or paraphrasing. You can buy it or order for a fresh, original and plagiarism-free copy from our tutoring website www.aceyourhomework.com (Deadline assured. Flexible pricing. TurnItIn Report provided)

Pay using PayPal (No PayPal account Required) or your credit card . All your purchases are securely protected by .
SiteLock

About this Question

STATUS

Answered

QUALITY

Approved

DATE ANSWERED

Sep 18, 2020

EXPERT

Tutor

ANSWER RATING

GET INSTANT HELP/h4>

We have top-notch tutors who can do your essay/homework for you at a reasonable cost and then you can simply use that essay as a template to build your own arguments.

You can also use these solutions:

  • As a reference for in-depth understanding of the subject.
  • As a source of ideas / reasoning for your own research (if properly referenced)
  • For editing and paraphrasing (check your institution's definition of plagiarism and recommended paraphrase).
This we believe is a better way of understanding a problem and makes use of the efficiency of time of the student.

NEW ASSIGNMENT HELP?

Order New Solution. Quick Turnaround

Click on the button below in order to Order for a New, Original and High-Quality Essay Solutions. New orders are original solutions and precise to your writing instruction requirements. Place a New Order using the button below.

WE GUARANTEE, THAT YOUR PAPER WILL BE WRITTEN FROM SCRATCH AND WITHIN YOUR SET DEADLINE.

Order Now