## [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

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_

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