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1. HOOGHLY ENGINEERING AND TECHNOLOGY COLLEGE
EE-593
STREAM-EE YEAR-3RD SEMESTER:5
DETERMINATION OF STEP RESPONSE FOR
FIRST ORDER & SECOND ORDER SYSTEM
WITH UNITY FEEDBACK ON CRO &
CALCULATION OF CONTROL SYSTEM
SPECIFICATION LIKE TIME CONSTANT, %
PEAK OVERSHOOT, SETTLING TIME ETC.
FROM THE RESPONSE.
DEBASISH BOSE
EE DEPARTMENT
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2. STEP SIGNAL
The step signal is a signal whose value
changes from zero to another level A in
negligible time.
The functional representation of a step
signal I given by
r(t)=Au(t)
Where
u(t)=1 when t > 0
=0 when t < 0
and u(t) represent a unit step function
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3. STEP RESPONSE
In electrical engineering and control system, step response is the time behavior of
the outputs of a general system when its inputs change from zero to one in a very
short time. The concept can be extended to the abstract mathematical notion of a
dynamic system using an evolution parameter.
From a practical standpoint, knowing how the system responds to a sudden input
is important because large and possibly fast deviations from the long term steady
state may have extreme effects on the component itself and on other portions of
the overall system dependent on this component. In addition, the overall system
cannot act until the component's output settles down to some vicinity of its final
state, delaying the overall system response. Formally, knowing the step response
of a dynamical system gives information on the stability of such a system, and on
its ability to reach one stationary state when starting from another.
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4. STEP RESPONSE IN R-C SERIES CIRCUIT
When something changes in a circuit, the
voltages and currents adjust to the new
conditions. If the change is an abrupt
step the response is called the step
response.
We apply an abrupt step in voltage to a
resistor-capacitor (RC) circuit and watch
what happens to the voltage across the
resistor and capacitor
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5. A TYPICAL STEP RESPONSE FOR A SECOND ORDER
SYSTEM
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6. 6

7. DELAY TIME( td ) is the time required for the response to reach 50 % of
the final value in first attempt.
RISE TIME ( tr ) is the time required for the response to rise from 10% to
90% of the final value for overdamped system and
0 to 100% for underdamped system.
PEAK TIME (tp) is the time required for the response to reach the peak of the
time response , i.e. the peak overshoot.
Rise time and peak time are intended as speed of response criteria.
The smaller values of these indicates faster time response.
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8. PEAK OVERSHOOT ( MP) is the peak value of the response
curve measured from unity.
If the final steady state value of the response differs from unity,
then it is common to use per cent peak overshoot. It is defined
by
per cent peak overshoot= Y(tp) -y(∞) X 100 %
y(∞)
Peak overshoot is mainly used for relative stability. Values in
excess of about 40 % may indicate that the system is
dangerously close to absolute instability.
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9. SETTLING TIME (ts) is the time required for the response to damp out all
transients .
Theoritically,the time taken to damp out all transients may be infinity.
In practice,however,the transient is assumed to be over when the error is
reduced below some acceptable value.
Typically , the acceptable level is set at 2% or 5% of the final value.
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10. MATLAB CODE FOR STEP
RESPONSE
For a control system defined in a state space form, the MATLAB command
step(A,B,C,D) will generate plots of unit step responses, with the time vector
automatically provided “t” is not explicitly provided in the step commands.
The MATLAB command step(sys) may also be used to obtain the unit-step
response of a system.
The command step(sys) can be used where the system is defined by
1.sys=tf(num,den)
2.sys=ss(A,B,C,D)
ss Construct state-space model or convert model to state space.
In order to obtain the response curves,plot command should be used
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11. sys = ss(A,B,C,D) creates an object SYS representing the continuous-
time state-space model
dx/dt = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
You can set D=0 to mean the zero matrix of appropriate size. SYS is
of type ss when A,B,C,D are numeric arrays, of type GENSS when
A,B,C,D depend on tunable parameters (see REALP and GENMAT), and
of type USS when A,B,C,D are uncertain matrices (requires Robust
Control Toolbox).
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12. sys = ss(A,B,C,D,Ts) creates a discrete-time state-space model with
sample time Ts (set Ts=-1 if the sample time is undetermined).
SYS = ss creates an empty ss object.
SYS = ss(D) specifies a static gain matrix D.
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13. CONVERSION
sys= ss(sys) converts any dynamic system SYS to the state-space
representation. The resulting model SYS is always of class ss.
sys= ss(sys,'min') computes a minimal realization of SYS.
sys= ss(sys,'explicit') computes an explicit realization (E=I) of SYS.
An error is thrown if SYS is improper.
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14. EXAMPLE 1
Consider any unity feedback
system with G(s)=(25)/( s2+5s+25)
Find the time response and time domain
specifications
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15. MATLAB PROGRAMME
% TIME RESPONSE AND TIME DOMAIN SPECIFICATIONS
num=input('ENTER NUMERATOR OF TRANSFER FUNCTION:')
den=input('ENTER DENOMINATOR OF TRANSFER FUNCTION:')
tf1=tf(num,den)
step(tf1)
omegan=sqrt(den(3))
zeta=den(2)/(2*omegan)
omegad=omegan*(sqrt(1-zeta^2))
peaktime=pi/omegad
pos=exp((-zeta*pi)/sqrt(1-zeta^2))*100
settlingtime=4/(zeta*omegan)
theta=atan(sqrt(1-zeta^2)/zeta)
risetime=(pi-theta)/omegad
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16. TO PUT INPUT IN COMMAND WINDOW
ENTER NUMERATOR OF TRANSFER FUNCTION:25
ENTER DENOMINATOR OF TRANSFER FUNCTION:[1 5 25]
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17. MATLAB PROGRAM AND OUTPUT IN COMMAND WINDOW 17

18. MATLAB PROGRAMME AND OUTPUT FROM COMMND WINDOW
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19. 19

20. OUTPUT IN FIGURE 20

21. omegan is a variable of type double.
omegad is a variable of type double.
pos is a variable of type double.
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22. EXAMPLE 2
Consider a system with unit step input of 12 units and G(S)=(10)/ (s2+2s+10)
with unity feedback. Obtain the time response and time domain specifications
HINTS:
WRITE THE MATLAB PROGRAM ON COMMAND WINDOW
ENTER NUMERATOR OF TRANSFER FUNCTION:10
ENTER DENOMINATOR OF TRANSFER FUNCTION:[1 2 10]
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23. MATLAB OUTPUT AT COMMAN WINDOW 23

24. 24

25. OUTPUT IN FIGURE 25

26. REFERENCES
1.Control Systems Principles and Design
M Gopal
Tata McGraw-Hill Publishing Company Limited
2.Analysis and Design of Control Systems Using MATLAB
Rao.V.Dukkipti
New Age International Publishers
3.Control Systems
Prithwiraj Purkait
Biplab Satpati
Galib Rahaman Mallik
Ujjwal Mondal
Mc Graw Hill Education
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