Time response
-
Upload
sarah-krystelle -
Category
Business
-
view
3.554 -
download
1
Transcript of Time response
Chapter 4: Time Response1
Chapter 4
Time Response
Chapter 4: Time Response2
POLES,ZEROS,AND SYSTEM RESPONSE
The output response of a system is the sum of two responses: the forced response and natural response
The concept of poles and zeros , fundamental to the analysis and design of control systems, simplifies the evaluation of a system’s response.
Chapter 4: Time Response3
POLES OF TRANSFER FUNCTION
The poles of transfer function areThe value of Laplace transform variable, s,
that cause the transfer function become infinite. Or
Any roots of the denominator of the transfer function that are common to roots of the numerator.
Chapter 4: Time Response4
ZEROS OF A TRANSFER FUNCTION
The zeros of a transfer function are the values of the Laplace transform
variable, s, that cause the transfer function to become zero.
Any roots of the numerator of the transfer function that are common to roots of denominator
Chapter 4: Time Response5
POLES AND ZEROS OF A FIRST-ORDER SYSTEM
Given the function G(s) a pole exists at s= -5 and a zero exists at-2. These values are plotted on the complex s-plane, using an x for the pole and 0 for the zero. =
Chapter 4: Time Response6
From the example, we draw the following conclusion.
1. A pole of the input functions generate the form of the forced response (that is, the pole at the origin generated a step function at t5he output).
2. A pole of the transfer function generates the form of the natural response (that is, the pole at -5 generated e^-5t).
3. The pole on the real axis generates an exponential response of the form e^-at where –a is the pole location on the real axis.
4. The zeros and poles generate the amplitudes for both the forced and natural responses.
Chapter 4: Time Response7
Figure 4.1a. System showinginput and output;b. pole-zero plotof the system;c. evolution of asystem response.Follow blue arrowsto see the evolutionof the responsecomponent generatedby the pole or zero.
Chapter 4: Time Response8
Figure 4.2Effect of a real-axispole upon transientresponse
Chapter 4: Time Response9
Figure 4.3System forExample 4.1
Chapter 4: Time Response10
FIRST ORDER SYSTEMS
A first order systems without zeros can be described by the transfer function shown in the figure4.4(a). If the input is a unit step, where R(s)=1/s, the Laplace transform of the step response is C(s), where.
Taking the inverse transform, the step response given by
Chapter 4: Time Response11
Figure 4.4a. First-order system;b. pole plot
Chapter 4: Time Response12
Figure 4.5First-order systemresponse to a unitstep
Chapter 4: Time Response13
TIME CONSTANTWe. The call 1/a the time constant of the response. The time constant can be described as the time for to decay to 37%.of its initial value.reciprocal of the time constant has the units (1/seconds), or frequency. Thus we call the parameter a the exponential frequency
RISE TIME Trdefined as the time for the waveform to go from 0.1 to 0.9 of its initial value. Rise time is found by solving Eq. 4.6 for the difference in time at c(t)= 0.9 and c(t)= 0.1
SETTLING TIME Tssettling time is defined as the time for the response to reach and stay within 2% of its final value. Letting c(t)= 0.98 in eq. 4.6 and solving the time t, we find the settling time to be
Chapter 4: Time Response14
Figure 4.6Laboratory resultsof a system stepresponse test
Chapter 4: Time Response15
SECOND ORDER SYSTEMS
Second order system exhibits a wide range of responses that must be analyzed and described. Whereas varying a first order system's parameter simply changes the speed of response, changes in the parameters of a second order system can change the form of the response. For example a second order system can display characteristics much like a first order system or depending on component values, display damped or pure oscillations for its transient response.
Chapter 4: Time Response16
Figure 4.7Second-ordersystems, pole plots,and stepresponses
Chapter 4: Time Response17
OVERDAMPED RESPONSE (4.7b)
UNDERDAMPED RESPONSE (4.7c)
This function has a pole at the origin that comes from the unit step input and two complex poles that come from the system.
Chapter 4: Time Response18
Figure 4.8Second-orderstep response componentsgenerated bycomplex poles
Chapter 4: Time Response19
Figure 4.9System forExample 4.2
Chapter 4: Time Response20
Solution: First determine that the form of the forced response is a step. Next we find the form of the natural response. Factoring the denominator of the transfer function, we find the poles to be s=- 5 ± j13.23. The real part -5, is the exponential frequency for the damping. It is also the reciprocal of the time constant of the decay of the oscillations. The imaginary part, 13.23, is the radian frequency for the sinusoidal oscillations
Chapter 4: Time Response21
UNDAMPED RESPONSE (4.7d)
This function has a pole at the origin that comes from the unit step input and two imaginary poles that come from the system. The input pole at the origin generates the constant forced response and the two system poles on the imaginary axis at ±j3 generate a natural response whose frequency is equal to the location of the imaginary poles. Hence the output can be estimated as c(t)=K1+K4 cos(3t-ø).
CRITICALLY UNDAMPED RESPONSE (4.7e)
This function has a pole at the origin that comes from the unit step input and two multiple real poles that come from the system. The input pole at the origin generates the constant forced response and the two poles on the real axis at -3 generate a natural response consisting of an exponential and an exponential multiplied by time where the exponential frequency is equal to the location of real poles. Hence the output can
Chapter 4: Time Response22
1. Overdamped responses
Poles : two real at –ø1,-ø2
Natural response: two exponentials with time constant equal to the reciprocal of the pole locations or
2. Underdamped response
poles: two complex at ød±jwd
natural responses: damped sinusoid with an exponential enveloped whose time constant is equal to the reciprocal of the poles part. The radian frequency of the sinusoid, the damped frequency of oscillation, is equal to the imaginary part of the poles or
3.Undamped responses
Poles: Two imaginary at ±jwt
Natural response: Undamped sinusoid with radian frequency equal to the imaginary part of the poles or
Chapter 4: Time Response23
Figure 4.10Step responsesfor second-ordersystemdamping cases
Chapter 4: Time Response24
THE GENERAL SECOND ORDER SYSTEM
NATURAL FREQUENCY, WN
the natural frequency of a second order system is the frequency of oscillation of the system without damping.
DAMPING RATIO
Chapter 4: Time Response25
Figure 4.11Second-orderresponse as a function of damping ratio
Chapter 4: Time Response26
Figure 4.12Systems forExample 4.4
Chapter 4: Time Response27
Figure 4.13Second-orderunderdampedresponses fordamping ratio values
Chapter 4: Time Response28
Figure 4.14Second-orderunderdampedresponsespecifications
Chapter 4: Time Response29
Figure 4.15Percentovershoot vs.damping ratio
Chapter 4: Time Response30
Figure 4.16Normalized risetime vs. dampingratio for asecond-orderunderdampedresponse
Chapter 4: Time Response31
Figure 4.17Pole plot for anunderdamped second-ordersystem
Chapter 4: Time Response32
Figure 4.18Lines of constantpeak time,Tp , settlingtime,Ts , and percentovershoot, %OSNote: Ts2
< Ts1 ;
Tp2 < Tp1
; %OS1 <
%OS2
Chapter 4: Time Response33
Figure 4.19Step responsesof second-orderunderdamped systemsas poles move:a. with constant real part;b. with constant imaginary part;c. with constant damping ratio
Chapter 4: Time Response34
Figure 4.20Pole plot forExample 4.6
Chapter 4: Time Response35
Figure 4.21Rotationalmechanical system for Example 4.7
Chapter 4: Time Response36
Figure 4.22The CybermotionSR3 security roboton patrol. Therobot navigates byultrasound and pathprograms transmittedfrom a computer,eliminating the needfor guide strips onthe floor. It has videocapabilities as well astemperature, humidity,fire, intrusion, and gas
sensors.
Courtesy of Cybermotion, Inc.
Chapter 4: Time Response37
Figure 4.23Component responses of a three-pole system:a. pole plot;b. componentresponses: nondominant pole is neardominant second-order pair (Case I), far from the pair (Case II), andat infinity (Case III)
Chapter 4: Time Response38
Figure 4.24Step responsesof system T1(s),system T2(s), andsystem T3(s)
Chapter 4: Time Response39
Figure 4.25Effect of addinga zero to a two-pole system
Chapter 4: Time Response40
Figure 4.26Step responseof anonminimum-phase system
Chapter 4: Time Response41
Figure 4.27Nonminimum-phaseelectrical circuit
Chapter 4: Time Response42
Figure 4.28Step response of the nonminimum-phasenetwork of Figure 4.27 (c(t)) and normalized step response of anequivalent networkwithout the zero(-10co(t))
Chapter 4: Time Response43
Figure 4.29a. Effect of amplifiersaturation on load angular velocityresponse;b. Simulink blockdiagram
Chapter 4: Time Response44
Figure 4.30a. Effect ofdeadzone onload angulardisplacementresponse;b. Simulink blockdiagram
Chapter 4: Time Response45
Figure 4.31a. Effect of backlashon load angulardisplacementresponse;b. Simulink blockdiagram
Chapter 4: Time Response46
Figure 4.32Antenna azimuthposition controlsystem for angularvelocity:a. forward path;b. equivalentforward path
Chapter 4: Time Response47
Figure 4.33UnmannedFree-SwimmingSubmersible(UFSS) vehicle
Courtesy of Naval Research Laboratory.
Chapter 4: Time Response48
Figure 4.34Pitch control loop forthe UFSS vehicle
Chapter 4: Time Response49
Figure 4.35Negative stepresponse of pitch control for UFSS vehicle
Chapter 4: Time Response50
Figure 4.36A ship at sea,showing roll axis
Chapter 4: Time Response51
Figure P4.1
Chapter 4: Time Response52
Figure P4.2
Chapter 4: Time Response53
Figure P4.3
Chapter 4: Time Response54
Figure P4.4
Chapter 4: Time Response55
Figure P4.5
Chapter 4: Time Response56
Figure P4.6
Chapter 4: Time Response57
Figure P4.7
Chapter 4: Time Response58
Figure P4.8
Chapter 4: Time Response59
Figure P4.9(figure continues)
Chapter 4: Time Response60
Figure P4.9 (continued)
Chapter 4: Time Response61
Figure P4.10Steps in determiningthe transfer functionrelating output physicalresponse to the inputvisual command
Chapter 4: Time Response62
Figure P4.11Vacuum robot liftstwo bags of salt
Courtesy of Pacific Robotics, Inc.
Chapter 4: Time Response63
Figure P4.12
Chapter 4: Time Response64
Figure P4.13
Chapter 4: Time Response65
Figure P4.14
Chapter 4: Time Response66
Figure P4.15
Chapter 4: Time Response67
Figure P4.16
Chapter 4: Time Response68
Figure P4.17
Chapter 4: Time Response69
Figure P4.18
Chapter 4: Time Response70
Figure P4.19
Chapter 4: Time Response71
Figure P4.20Pump diagram
© 1996 ASME.