Time response

Chapter 4: Time Response1

Chapter 4

Time Response


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.


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.


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


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. =


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.


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.


Figure 4.2Effect of a real-axispole upon transientresponse


Figure 4.3System forExample 4.1


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


Figure 4.4a. First-order system;b. pole plot


Figure 4.5First-order systemresponse to a unitstep


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


Figure 4.6Laboratory resultsof a system stepresponse test


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.


Figure 4.7Second-ordersystems, pole plots,and stepresponses


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.


Figure 4.8Second-orderstep response componentsgenerated bycomplex poles


Figure 4.9System forExample 4.2


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


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


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


Figure 4.10Step responsesfor second-ordersystemdamping cases


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


Figure 4.11Second-orderresponse as a function of damping ratio


Figure 4.12Systems forExample 4.4


Figure 4.13Second-orderunderdampedresponses fordamping ratio values


Figure 4.14Second-orderunderdampedresponsespecifications


Figure 4.15Percentovershoot vs.damping ratio


Figure 4.16Normalized risetime vs. dampingratio for asecond-orderunderdampedresponse


Figure 4.17Pole plot for anunderdamped second-ordersystem


Figure 4.18Lines of constantpeak time,Tp , settlingtime,Ts , and percentovershoot, %OSNote: Ts2

< Ts1 ;

Tp2 < Tp1

; %OS1 <

%OS2


Figure 4.19Step responsesof second-orderunderdamped systemsas poles move:a. with constant real part;b. with constant imaginary part;c. with constant damping ratio


Figure 4.20Pole plot forExample 4.6


Figure 4.21Rotationalmechanical system for Example 4.7


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.


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)


Figure 4.24Step responsesof system T1(s),system T2(s), andsystem T3(s)


Figure 4.25Effect of addinga zero to a two-pole system


Figure 4.26Step responseof anonminimum-phase system


Figure 4.27Nonminimum-phaseelectrical circuit


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))


Figure 4.29a. Effect of amplifiersaturation on load angular velocityresponse;b. Simulink blockdiagram


Figure 4.30a. Effect ofdeadzone onload angulardisplacementresponse;b. Simulink blockdiagram


Figure 4.31a. Effect of backlashon load angulardisplacementresponse;b. Simulink blockdiagram


Figure 4.32Antenna azimuthposition controlsystem for angularvelocity:a. forward path;b. equivalentforward path


Figure 4.33UnmannedFree-SwimmingSubmersible(UFSS) vehicle

Courtesy of Naval Research Laboratory.


Figure 4.34Pitch control loop forthe UFSS vehicle


Figure 4.35Negative stepresponse of pitch control for UFSS vehicle


Figure 4.36A ship at sea,showing roll axis


Figure P4.1


Figure P4.2


Figure P4.3


Figure P4.4


Figure P4.5


Figure P4.6


Figure P4.7


Figure P4.8


Figure P4.9(figure continues)


Figure P4.9 (continued)


Figure P4.10Steps in determiningthe transfer functionrelating output physicalresponse to the inputvisual command


Figure P4.11Vacuum robot liftstwo bags of salt

Courtesy of Pacific Robotics, Inc.


Figure P4.12


Figure P4.13


Figure P4.14


Figure P4.15


Figure P4.16


Figure P4.17


Figure P4.18


Figure P4.19


Figure P4.20Pump diagram

© 1996 ASME.

Date post:	25-Jun-2015
Category:	Business
Upload:	sarah-krystelle
View:	3,550 times
Download:	1 times

Time response

Business