Me 360 Second Order System Step Response

7/31/2019 Me 360 Second Order System Step Response

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ME 360 Control Systems

Second-Order System Step Response

The transfer function for a second-order system can be written in one of the two

general forms (depending on whether the system has a zero or not)

Case 1: ( ) 2K

G ss bs c

=+ +

Case 2: ( ) ( )2K s a

G ss bs c

+=

+ +.

Each of these cases can be broken into different types of response depending on whether the

poles of the system are real and unequal, real and equal, complex, or purely imaginary

We consider only the response ofstable systems in the discussion below. Stable systems are

systems whose poles havenon-positivereal parts.

Case 1: There arefour types of motion that are possible in this case. ( ) 2K

G ss bs c

=+ +

o If the poles are real and unequal, the response is over-damped with no overshoot.

If the poles are widely separated, the response may be dominatedby the smaller

(slower) pole.

o If the poles arereal and equal, the response is critically damped with no overshoot.

o If the poles arecomplex the system is under-damped with an overshoot.

o If the poles arepurely imaginary, then the system has no damping and the response

will beoscillatory with no reduction in amplitude (harmonic).

Fig. 1 shows the step response of a system with real, unequalpoles, and Fig. 2 shows the

step response of a system withcomplex poles.

( )( )( )52

10

++=

sssG

Fig. 1 Second Order, Case 1 Over-damped Step Response


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Fig. 3 shows the second-order system step response for various values of the damping factor

(0.1, 0.25, 0.5, 0.75, 1.0, 2.0).

Fig. 2 Second Order, Case 1, Under-damped Step Response

( )255

25

2 ++=

ss

sG

Fig. 3 Second Order, Case 1, Under-damped Step Responses

Time (sec.)

Amplitude

Step Response

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

02.=

01.=

750.=

50.= 250.

=

10.=

( )

(r/s)5

2 22

2

=

++

=

n

nn

n

ss

sG


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Case 2: ( )( )

2

K s aG s

s bs c

+=

+ +

o The motion of Case 2 systems is complicated somewhat over that of Case 1 systems

by the presence of the zero ( )s a+ in the transfer function. The zero has little effect

on the settling time of the system, butcan significantly affect the overshoot.o How much effect the zero has on overshoot depends on where it is (along the real

axis) compared to the poles of the system.

o Zero is far to the left of the poles: it has little affect on the motion of the

system. This makes the motion much like that of a Case 1 system.

o Zero located near or inside the poles: it will significantly affect the

overshoot of the system.

Im

Re

Zero may have little effect

oa

X

X

n

Im

Re

Zero has measurable effect

oa

X

X

n

Im

Re

Zero has substantial effect

oa

X

X

n


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Fig. 4 shows the step response of anover-damped, case 2, second order system. The presence

of the zero causes significant overshoot. Fig. 5 shows the step response of an under-

damped, case 2, second order system. Here, the zero increases the overshoot (over what you

would expect if it was a Case 1 system). Note that the only difference between these systems

and those Case 1 systems shown in Figs. 1 and 2 is the presence of the zero ( 1)s + in the

transfer function.

Fig. 4 Second Order, Case 2 Over-damped Step Response

( )( )

( )( )52

110

++

+=

ss

ssG

Fig. 5 Second Order, Case 2 Under-damped Step Response

( )( )

255

125

2 ++

+=

ss

ssG

Me 360 Second Order System Step Response

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