linear control system(week-5)

7/28/2019 Linear Control System(Week-5)

1/5

Chapter-4 HW Exercises at the End

EL322near

Controlystems

Week-05

12th Mar 17th Mar 2012

Time Response (Contd. )

Chapter#04

Instructor: Engr. Shiraz Latif/ Engr. Atif Fareed/ EngrAreeb Ahmed

nd

Two Important quantities for 2nd order systems Natural Frequency n

The frequency of the oscillations of the system withoutdamping.

Damping Ratio

Ratio of exponential decay frequency to natural frequency

= exponential decay freq / natural freq (rad/sec)

&

Compare both these TF and get the formula for &n

Instructor:Shiraz Latif/ Atif Fareed/ Areeb Ahmed

3Instructor:Shiraz Latif/ Atif Fareed/ Areeb Ahmed


5Instructor:Shiraz Latif/ Atif Fareed/ Areeb Ahmed Instructor:Shiraz Latif/ Atif Fareed/ Areeb Ahmed


2/5

.

To analyze the step response ofunderdamped 2nd order systems.

Objectives: To define the transient s ecifications

associated with underdamped responses.

Relate these specifications to pole location,drawing an association between polelocation and response form.

Tie the pole location to system parameters.7Instructor:Shiraz Latif/ Atif Fareed/ Areeb Ahmed

General 2nd order sys:

For underdamped response, = ? ?

.

With step input,

Applying Partial fraction


Relationship between value ofand type of response

oscillations

Natural frequency dont cause any affect on the nature of the responseut is on y t e time sca e actor

econ or er un er ampe response orvarying values


Rise Time: Tr: Time required for the wave form to go from 10% of thefinal value to 90% of the final value.

Peak time: Tp : Time required to reach the first, or maximum, peak

Percentage Overshoot: %OS: The amount that the waveformovershoots thestead state orf inal valueat the eak time ex ressedas a

21

p

n

=

percentage of the steady-state value.

Settling time: Ts: Time required for the transients damped oscillations toreach and stay within 2% of the steady state value


Tr, Tp and Ts yields information about the speed of transient response.

s, r ave same e initions as in irst or er systems.

Same definition for order >2 as well.

These specifications (Tp, Tr, Ts)=

n. Tr

.

No precise expression exists for

Tr but can be obtained from plot & table Normalized rise time = n . Tr

11Instructor:Shiraz Latif/ Atif Fareed/ Areeb Ahmed 1Instructor:Shiraz Latif/ Atif Fareed/ Areeb Ahmed


3/5

If location of pole is s; S can be represented as s = d + j d

d is the real part of s exponential damping freq.

d is the imaginary part damped freq of oscillation

Tp, Ts and %OS can berelated to the location of


Tp is inversely proportion to theimaginary part of the pole

Consider the s-plane

l l l l(same y-value) constant Tp called lines ofconstant peak time

Ts is inversely proportion to the real part

Consider the s-plane

Vertical lines constant real part (same x-value) constant Ts called lines of constant

ll

Radial lines are lines of constant called lines of constant


%OS

Effect of movement of poles


Poles moves verticallyup

Real part same

Pole f butenvelop same

Ts remains same

Poles moveshorizontally left

Imaginary part same

Pole f same ,

damp rapidly

Poles moves atconstant angle (radiallines)

%OS remains same .

Pole away from originhave fast response.




4/5

..

35

2)(

2++

+=

ss

ssG

Num = [1 2];

Den = [1 5 3];

T = tf(num,den)

Zeros(T)Implement same

Pzmap(T)

Step (T)

19Instructor:Shiraz Latif/ Atif Fareed/ Areeb Ahmed Grid

xercises



so p o e s ep response usng



Transfer function shown above


5/5



linear control system(week-5)

Documents