LECTURE FOUR Time Domain Analysis Transient And Steady .

3rd Year-Computer Communication Engineering-RUCControl TheoryLECTURE FOURTime Domain AnalysisTransient and Steady-State ResponseAnalysis4.1 Transient Response and Steady-State ResponseThe time response of a control system consists of two parts: the transient response and thesteady-state response. By transient response, we mean that which goes from the initial stateto the final state. By steady-state response, we mean the manner in which the system outputbehaves as it approaches infinity. Thus the system response c(t) may be written as(4.1)4.2 First-Order SystemsConsider the first-order system shown in Figure 4.1-a below. Physically, this system mayrepresent an RC circuit, thermal system, or the like. A simplified block diagram is shown inFigure 4.1-b. The input-output relationship is given byDr. Mohammed Saheb KhesbakPage 31

3rd Year-Computer Communication Engineering-RUCControl TheoryIn the following, we shall analyze the system responses to such inputs as the unit-step, unitramp, and unit-impulse functions. The initial conditions are assumed to be zero.4.2.1 Unit-Step Response of First-Order SystemsSince the Laplace transform of the unit-step function is 1/s, substituting R(s) 1/s intoEquation (4.1), we obtainExpanding C(s) into partial fractions givesTaking the inverse Laplace transform, we obtain(4.2)Equation (4.2) states that initially the output c(t) is zero and finally it becomes unity. Oneimportant characteristic of such an exponential response curve c(t) is that at t T the value ofc(t) is 0.632, or the response c(t) has reached 63.2% of its total change. This may be easilyseen by substituting t T in c(t). That is,Note that the smaller the time constant T, the faster the system response. Another importantcharacteristic of the exponential response curve is that the slope of the tangent line at t 0 is1/T, sinceDr. Mohammed Saheb KhesbakPage 32

3rd Year-Computer Communication Engineering-RUCControl TheoryThe output would reach the final value at t T if it maintained its initial speed of response.From Equation above we see that the slope of the response curve c(t) decreasesmonotonically from 1/T at t 0 to zero at t .The exponential response curve c(t) given by Equation (4.2) is shown in Figure 4.2. In onetime constant, the exponential response curve has gone from 0 to 63.2%of the final value. Intwo time constants, the response reaches 86.5%of the final value. At t 3T, 4T, and 5T, theresponse reaches 95%, 98.2%, and 99.3%, respectively, of the final value. Thus, for t 4T,the response remains within 2% of the final value. As seen from Equation (4.2), the steadystate is reached mathematically only after an infinite time. In practice, however, a reasonableestimate of the response time is the length of time the response curve needs to reach and staywithin the 2%line of the final value, or four time constants.Dr. Mohammed Saheb KhesbakPage 33

3rd Year-Computer Communication Engineering-RUCControl Theory4.2.2 Unit-Ramp Response of First-Order SystemsSince the Laplace transform of the unit-ramp function is 1/s2, we obtain the output of thesystem of Figure 1(a) asExpanding C(s) into partial fractions givesTaking the inverse Laplace transform of above equation, we obtainThe error signal e(t) is thenAs t approaches infinity, e–t/T approaches zero, and thus the error signal e(t) approachesT orThe unit-ramp input and the system output are shown in Figure 4.3. The error in followingthe unit-ramp input is equal to T for sufficiently large t. The smaller the time constant T, thesmaller the steady-state error in following the ramp input.Dr. Mohammed Saheb KhesbakPage 34

3rd Year-Computer Communication Engineering-RUCControl Theory4.2.3 Unit-Impulse Response of First-Order SystemsFor the unit-impulse input, R(s) 1 and the output of the system of Figure 4.1(a) can beobtained asThe inverse Laplace transform of above Equation(4.3)The response curve given by Equation (3) is shown in Figure 4.4.Dr. Mohammed Saheb KhesbakPage 35

3rd Year-Computer Communication Engineering-RUCControl Theory4.3 Second-Order SystemsAny second order system can be represented by the following typical form and systemdiagram:(4.4)This is called the standard form of the second order system, where; n undamped natural frequency (rad/sec). (zeta) damping ratioDr. Mohammed Saheb KhesbakPage 36

3rd Year-Computer Communication Engineering-RUCControl TheoryExample 4.1:For the system shown below, find n and .Comparing with the standard form of the 2nd order system, obtaining; (rad/sec)also ; Example 4.2:For the system shown below, find n and .First of all, the transfer function should be rearranged as;and by comparing with the standard form, get;(rad/sec)Dr. Mohammed Saheb KhesbakPage 37

3rd Year-Computer Communication Engineering-RUCControl Theoryalso ; We shall now solve for the response of the system shown in equation (4.4) to a unit-stepinput. We shall consider three different cases: the underdamped (0 1), criticallydamped ( 1), and overdamped ( 1) cases.Unit Step input:(1) Underdamped case (0 1): In this case, C(s)/R(s) can be written asWhere . The frequencyis called the damped natural frequency.For a unit-step input, C(s) can be written(4.5)The inverse Laplace transform of Equation (4.5) can be obtained easily if C(s) is written inthe following form:Dr. Mohammed Saheb KhesbakPage 38

3rd Year-Computer Communication Engineering-RUCControl TheoryReferring to the Laplace transform, it can be shown thatHence the inverse Laplace transform of Equation (4.5) is obtained as(4.6)From Equation (4.6), it can be seen that the frequency of transient oscillation is the dampednatural frequency d and thus varies with the damping ratio . The error signal for thissystem is the difference between the input and output and isDr. Mohammed Saheb KhesbakPage 39

3rd Year-Computer Communication Engineering-RUCControl TheoryThis error signal exhibits a damped sinusoidal oscillation. At steady state, or at t , no errorexists between the input and output.(2) Undamped case ( 0):If the damping ratio is equal to zero, the response becomes undamped and oscillationscontinue indefinitely. The response c(t) for the zero damping case may be obtained bysubstituting 0 in Equation (6), yielding(4.7)Therefore, the time of oscillation (T) isThus, from Equation (4.7), we see that n represents the undamped natural frequency ofthe system. That is, n is that frequency at which the system output would oscillate if thedamping were decreased to zero. If the linear system has any amount of damping, theundamped natural frequency cannot be observed experimentally. The frequency that may beobserved is the damped natural frequency d, which is equal toDr. Mohammed Saheb Khesbak. This frequencyPage 40

3rd Year-Computer Communication Engineering-RUCControl Theoryis always lower than the undamped natural frequency. An increase in would reduce thedamped natural frequency d. If is increased beyond unity, the response becomesoverdamped and will not oscillate.(3) Critically damped case ( 1):If the two poles of C(s)/R(s) are equal, the system is said to be a critically damped one. For aunit-step input, R(s) 1/s and C(s) can be written(4.8)The inverse Laplace transform of Equation (8) may be found as(4.9)This result can also be obtained by letting approach unity in Equation (4.6) and by usingthe following limit:(4) Overdamped case ( 1):In this case, the two poles of C(s)/R(s) are negative real and unequal. For a unit-step input,R(s) 1/s and C(s) can be writtenDr. Mohammed Saheb KhesbakPage 41

3rd Year-Computer Communication Engineering-RUCControl Theory(4.10)The inverse Laplace transform of Equation (10) is(4.11)WhereThus, the response c(t) includes two decaying exponential terms. When is appreciablygreater than unity, one of the two decaying exponentials decreases much faster than theother, so the faster-decaying exponential term (which corresponds to a smaller time constant)may be neglected. That is, if –s2 is located very much closer to the j axis than –s1 (whichmeans s2 s1 ), then for an approximate solution we may neglect –s1.This is permissiblebecause the effect of –s1 on the response is much smaller than that of –s2, since the terminvolving s1 in Equation (4.11) decays much faster than the term involving s2 . Once thefaster-decaying exponential term has disappeared, the response is similar to that of a firstorder system, and C(s)/R(s) may be approximated by;Dr. Mohammed Saheb KhesbakPage 42

3rd Year-Computer Communication Engineering-RUCControl TheoryThis approximate form is a direct consequence of the fact that the initial values and finalvalues of the original C(s)/R(s) and the approximate one agree with each other. With theapproximate transfer function C(s)/R(s), the unit-step response can be obtained asThe time response c(t) is thenThis gives an approximate unit-step response when one of the poles of C(s)/R(s) can beneglected.A family of unit-step response curves c(t) with various values of is shown in Figure (4.5),where the x-axis is the dimensionless variable nt. The curves are functions only of . Thesecurves are obtained from Equations (4.6), (4.9), and (4.11). The system described by theseequations was initially at rest. Note that two second-order systems having the same butdifferent n will exhibit the same overshoot and the same oscillatory pattern. Such systemsare said to have the same relative stability. From Figure (4.5), we see that an underdampedDr. Mohammed Saheb KhesbakPage 43

3rd Year-Computer Communication Engineering-RUCControl Theorysystem with between 0.5 and 0.8 gets close to the final value more rapidly than acritically damped or over damped system. Among the systems responding withoutoscillation, a critically damped system exhibits the fastest response. An over damped systemis always sluggish in responding to any inputs. It is important to note that, for second-ordersystems whose closed-loop transfer functions are different from that given by Equation (4.4),the step-response curves may look quite different from those shown in Figure 4.5.4.4 Definitions of Transient-Response SpecificationsThe transient response of a practical control system often exhibits damped oscillations beforereaching steady state. In specifying the transient-response characteristics of a control systemto a unit-step input, it is common to specify the following:Dr. Mohammed Saheb KhesbakPage 44

3rd Year-Computer Communication Engineering-RUC1. Delay time,td2. Rise time,trControl Theory3. Peak time, tp4. Maximum over shoot, Mp5. Settling time, tsThese specifications are defined in what follows and are shown graphically in previousFigures.1. Delay time, td: The delay time is the time required for the response to reach half the finalvalue the very first time.2. Rise time, tr : The rise time is the time required for the response to rise from 10% to 90%,5% to 95%, or 0% to 100% of its final value. For underdamped second order systems, the0%to 100%rise time is normally used. For over damped systems, the 10% to 90% rise time iscommonly used.3. Peak time, tp: The peak time is the time required for the response to reach the first peak ofthe overshoot.4. Maximum (percentage) overshoot, Mp: The maximum overshoot is the maximum peakvalue of the response curve measured from unity. If the final steady-state value of theresponse differs from unity, then it is common to use the maximum percentage overshoot. Itis defined by:The amount of the maximum (percentage) overshoot directly indicates the relative stabilityof the system.Dr. Mohammed Saheb KhesbakPage 45