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# Classl System and Cntrl Theory ECE 421

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This 19 page Class Notes was uploaded by Antonina Wuckert on Monday September 28, 2015. The Class Notes belongs to ECE 421 at George Mason University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/215035/ece-421-george-mason-university in ELECTRICAL AND COMPUTER ENGINEERING at George Mason University.

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Construction Rules for Root Locus K 2 0 The open loop system represented by GsHsithe product of the forward path transfer function 15 and the feedback path transfer function Hsi has n poles and m nite zeros with n 2 m The number of closed loop poles is equal to the number of open loop poles so there will be 71 closed loop poles Since each branch of the root locus shows the movement of one closed loop pole as K is varied from 0 to 00 there will be 71 branches lines of the root locus For a xed value of K 2 0 one point on each branch will be the location of one closed loop pole As K varies in value the point on each branch that is the closed loop pole also varies The governing rules for the root locus are the magnitude and phase criteria When 5 51 is a closed loop pole lG51Hsll 1 1G51H51 180 21 1 l 0i1i2 1 When 5 51 is not a closed loop pole either the magnitude criterion or the phase criterion is violated or both are violated If the phase criterion is satis ed then the point 51 is on the root locus and can be made a closed loop pole by proper choice of K the value of K that will satisfy the magnitude criterion at that point If the phase criterion is not satis ed then 51 is not on the root locus and there is no real value of K that will make 51 become a closed loop pole Examples of constructing the root locus and compensator design with root locus can be found at httpecegmuedugbealeece421examples421html 1 Each branch of the root locus begins at an open loop pole K 0 and ends at an open loop nite zero or at a zero at in nity K a 00 2 Real axis root locus Starting at 00 and moving along the real axis toward the left the root locus lies on the real axis to the left of an odd number of real axis open loop poles or zeros in any combination Multiplicities of poles and zeros must be taken into account when counting The locations of the open loop poles or zeros right half plane left half plane or origin does not matter only the cumulative number of poles or zerosiodd or eveniis important Poles or zeros off the real axis are not included in determining the real axis portion of the root locus 3 Asymptotes If n gt m there will be 71 7 m branches of the root locus going to in nity as K a 00 For large K they will follow asymptotes that meet at a common point on the real axis and make speci ed angles with respect to the positive real axis The angles of asymptotes gtA and the center of asymptotes 0A are given by V L m m e 21 1800 21 1 7 7 fp 1012n7m71 UAM 2 717m 717m where pl and 2139 are the open loop pole and zero locations respectively Complex poles and zeros are included in the calculation of 0A The angles of the asymptotes depend only on n 7 m not on the actual locations of the poles or zeros 4 Complex conjugate poles The root locus makes an angle 0 angle of departure with respect to the positive real axis as it leaves the complex conjugate pole with positive imaginary part given by 7L m 0D 7 180 7 90 24191 719 7 24191 7 2 3 i3 i1 where the complex pole with positive imaginary part is assumed to be p1 and pole p2 is its complex conjugate The angle from p2 to p1 is always 90 This equation comes from satisfying the phase criterion at a point near p1 Robust Performance Example 1 A nominal plant is given along with one extreme plant model The extreme model is as far away from the nominal model as any model in the family of plants that must be considered Robust stability of the family means that the compensator provides internal stability for each member of the family and robust performance means that each closedloop system in the family satis es the constraint on the sensitivity function 1 The nominal and extreme plants are given by Gs and G s respectively in l and 2 below 9 G l 5 55 2 5 9 2 5 55 1 A laglead compensator is designed for the nominal system to place the dominant closedloop poles at s 51 74ij2 and to make the steadystate error for a ramp input equal to ess 005 Details on the root locus design procedures for transient response 2 and steadystate error 3 are available on the ECE 421 web site and in most introductory controls texts 46 The compensator is 27745 s 004 s 22654 m m 3 s 64075 10 s 88284 This compensator will be placed in series with each of the plant models in the family The nominal and extreme loop gains are 24971s 004 s 22654 7 gt Us 7 4 L 7 13873 5 004 s 22654 5 S 7 s s 6407510 3s 1 s 88284 Performance will be de ned by speci cations imposed on the magnitude of the sensitivity function that is on 150w Nominal performance NP is de ned by NP Sjw lt Vw 3 w ijjw lt1 Vw 6 H 1 WM 1Plt 1 o where rm 5 is a speci ed weighting function The assumption here is that time domain speci cations such as overshoot and settling time or frequency domain speci cations such as phase margin and bandwidth can be achieved by proper choice of wp The performance weighting function in this example is isMerB 705s08 7 7 swBA 7 s which implies that lSjwl should have at least a 20 dbdecade slope at low frequencies since A 0 a maximum value of M 2 and pass through 73 db at w n23 08 radsec For robust performance RP each member of the plant family must satisfy an expression like 6 that is 10125 1 RP lSpJwl lt 10P Vw VSP 8 where Sp represents the sensitivities for the family of plants under consideration Figure 1 shows the sensitivity magnitudes for the nominal and extreme systems and the magnitude of 1 1w P jwl Since the sensitivity magnitudes are both below 1 1w p jwl for all frequencies each of those two systems satisfy the performance requirements This does not imply RP unless those are the only two plants in the family However before RP can be determined stability must be investigated Nominal stability NS is de ned by the nominal closedloop system being stable that is all the closedloop poles of the compensatornominal plant combination must be in the open lefthalf of the complex splane Under the assumption of an unstructured multiplicative uncertainty model robust stability RS is de ned by the following constraint on the magnitude of the nominal complementary sensitivity function 1T jwl These notes are lecture notes prepared by Prof Guy Beale for presentation in ECE 720 Multivariable and Roburt Control in the Electrical and Computer Engineering Department George Mason University Fairfax VA Additional notes can be found at the following website httpecegmueduNgbealeexarnpleshtrnl Nominal and Extreme Sensitivities and Performance Weighting Function I I I I I Magnitude db 70 I IIIi I IIZi I 10392 10quot 100 10 102 Frequency rs Fig 1 Sensitivity magnitudes for the nominal and extreme plants and the performance Weighting inction RS V02 3 iw1Tjwi lt 17 V02 9 lt 7 iwr Jwi Where the nominal complementary sensitivity function is given by Ts Ls 1 Ls 1 and 11215 is the uncertainty weighting function that overbounds the magnitude of the maximum relative uncertainty 1 w given by L W 7 LOW L W 7 LOW l w max p 10 I LP L W L W 13 s73gs0 04s2 2654 24 971s0 Elam 2554 7 ss5 4075 10 5s1s8 8284 ss5 4075 10 5s2s8 8284 11 7 24 971gs0 04 s2 2654 ss5 4075 10 5s2s8 8284 7 70444452 5 2 s L 025 7 704444 s L 025 I2 7 52s1s2 7 51 LP represents the loop gains for all the members of the family of plants being considered Since G s is an extreme plant and the only perturbed model given the relative uncertainty can be computed based on that transfer function Since only the frequency response magnitude 1w jw is needed 11215 can always be chosen to be a stable and minimumphase transfer function For this example 11215 is given by 04444 s 025 044445 01111 13 5 5 1 5 1 which provides an exact overbounding for 11 Since iTjwi lt 11101 jw is required for robust stability the low frequency magnitude restriction on iTjwi is 101111 9 191 db and the high frequency restriction on iTjwi is Nominal Complementary Sensitivity and Uncertainty Weighting Function Magnitude db 60 I IIIi I IIZi I fl 0 10 Frequency rs Fig 2 Nominal complementary sensitivity and uncertainty weighting function 104444 225 704 db Figure 2 shows the nominal complementary sensitivity lTjwl along with 1 lw jwl Since lTjwl lt 1 My jwl for all frequencies the family is robustly stable This means that any plantcompensator combination whose polar plot at each frequency falls within the closed disk of radius lw jw L jwl centered at L jw will be closedloop stable For robust performance the condition of 8 must be satis ed This means that at each frequency the distance from the 71 point to the nominal loop gain l1 L jwl must be greater than the sum of the radii of the circles representing uncertainty and performance that is lejwllw1jwLjwl lt i1Lltjwgtr m 14 lwpjwl lw1ltjwLjwl i1Lltjmi l1Ljwl 1 V 15 lejwSJ wllw1jwTjwl lt 1 m 16 Since lwp jwSjwl lt 1 at all frequencies corresponds to NP and lw1jwTjwl lt 1 at all frequencies corresponds to RS expression 16 shows that necessary conditions for RP are NP and RS Therefore the nominal system satisfying the performance requirements and every plant in the family being closedloop stable are necessary conditions for every plant in the family satisfying the performance requirements However 16 also shows that NP and RS are not suf cient conditions for RP The sum of those two terms must also be less than 1 at all frequencies in order to achieve robust performance Figure 3 shows the sum of the weighted sensitivity and weighted complementary sensitivity Since the peak value of that sum is shown to be less than 1 at all frequencies this family of plants has robust performance Figure 4 shows the closedloop step responses and the openloop polar plots for the nominal and extreme systems whose plant models are given in l and 2 respectively The nominal system clearly has better performance in terms of overshoot WPSWIT 1 09 07 f f 0 07 Magnitude 0 01 04 3 03 g g f39 g 02 012 l1 l0 10 10 10 10 Frequency rs Fig 3 Sum of weighted sensitivity and weighted complementary sensitivity inctions and it also has a shorter settling time that might be considered a bene t However the performance of the perturbed extreme system also satis es the speci cation on the sensitivity function so that performance must also be considered acceptable Figure 5 shows the maximum and minimum magnitudes for the sensitivity and complementary sensitivity functions for any plant model allowed in the family of systems de ned by w The maximum and minimum magnitudes for the sensitivity function are given by 1 1 ij max7 Sjw min 17 l l1LJwlelwIJwLJwl l l1LJwllw1JwLJwl and the maximum and minimum magnitudes for the complementary sensitivity function are given by L jw L jw T T 18 l mm 1 WW m l M 1 L M m lt gt The minimum and maximum sensitivity magnitudes are easy to compute based on the geometric relationship between sensitivity and the circles of uncertainty The corresponding complementary sensitivity magnitudes are less easy to visualize since they depend on the ratios of vectors drawn from the origin and from the 71 point to L jw Knowing that the family of systems is robustly stable from 1THoo lt 1 the fact that the maximum sensitivity magnitude for all systems in the family is below 1 lwp jw at all frequencies indicates that the family is also robustly stable Figure 6 shows the polar plots for L jw and L jw Circles of uncertainty are shown at several frequencies At frequency w mi the center of the circle is located at the point ReL jwi jIm L jwi and the radius of the circle is lw jwi L jwm The gure illustrates the fact that when the family of systems is robustly stable the circles of uncertainty do not touch or encircle the 71 point the Nyquist stability criterion is satis ed by each element of the family The four graphs in Fig 7 illustrate the relationship between the circle of uncertainty at a particular frequency centered on L jw with radius lw jw L jw and the performance weighting function at the same frequency centered at 71 jO Closed Loop Step Responses for Nominal and Perturbed Systems 08 B E lt06 04 02 0 I I 0 05 1 15 35 4 45 5 25 Time s Polar Plots for Nominal and Perturbed Systems 2 L me 282 rs PM 682 deg 15 LtmC 205 rs PM 541 deg g lmag Axis Real Axis Fig 4 Closedloop step responses and openloop polar plots for the nominal and extreme systems Maximum and Minimum Sensitivity Functions Maximum and Minimum Complementary Sensitivity Functions Magnitude db Frequency rs Fig 5 Worst case sensitivity and complementary sensitivity functions Polar Plots with Circles of Uncertainty 2 I I I 05 Imag Axis 1 Real Axis Fig 6 Illustration of robust stability with the circles of uncertainty added to the polar plot of Ljw with radius wp jw For the family of systems to be robustly stable the circles of uncertainty must not intersect the circles representing performance at any frequency This relationship cannot conveniently be shown on a single plot since the circles must be compared on a frequencybyfrequency basis At low frequencies wp jwl gtgt 1 and those circles would clearly enclose the circles of meertainty at high frequencies To show both sets of circles at all frequencies on a single plot and to indicate which pairs of circles should be compared would be dif cult The graphs in Fig 7 merely illustrate how the circles of meertainty move along the polar plot decreasing in radius and slide past the circles of the performance weighting fmetion when the family is robustly stable The plot of 10125 101T in Fig 3 is the easiest way to graphically show whether or not a family of systems is robustly stable REFERENCES 1 S Skogestad and I Postlethwaite Multivariable Feedback Control Chichester England John Wiley amp Sons 1996 2 G Beale Compensator design to improve transient performance using root locus Available from the author as a pdf le 3 G Beale Compensator design to improve steadystate performance using root locus Available from the author as a pdf le 4 R C Dorf and R H Bishop Modern Control Synems Upper Saddle River NJ Prentice Hall 10th ed 2005 5 J D Azzo and C Houpis Linear Control Synem Analyris and Den39gn New York McGrawHill 4th ed 1995 6 K Ogata Modern Control Engineering Upper Saddle River NJ Prentice Hall 4th ed 2002 Step Response For Type 2 Systems A System Model Assume that the system is modeled as a SingleInput SingleOutput SISO con guration with unity feedback and the following forward loop transfer function Nltsgt Ms 68 7 m8 7 82135 lt1 Nola e 0 lt2 The expressions in 1 and 2 indicate that the system is Type 2 having two openloop poles at the origin and there is no polezero cancellation of that terml With the openloop system Cs de ned in 172 the closedloop transfer function is N8 S Cltsgt Gltsgt NltSgt TCL RS 1GS 1 N5 S2b8NS ACL5 3 32bs It will be assumed that the closedloop system is internally stablel Therefore all roots of the closedloop character istic equation AOL 8 lie strictly in the lefthalf of the complex s plane LHP and there are no unstable polezero cancellations in Csithere are no unstable hidden modes in the systeml The reference input to the system will be the unit step function so the transform of the reference input is 1 33 7 4 s The transform of the system output is the product of the transform of the reference input signal and the closedloop transfer function Therefore the transform of the output signal is N s 5 SACL s The error in the system is de ned to be the difference between the reference input and the actual output The transform for the error signal is Cs T0L3Rs Eltsgt Rltsgt e 08 Rltsgt e T0LltsgtRltsgt 1 e Toms Rltsgt Rltsgt lt6 7 1 1 7 32bs 17 Sblt8gt W 39 Ams 392 Ame 52175 B Region of Convergence The singlesided unilateral Laplace transform for a signal 1t is de ned to be 2 1m Xe 0 zte dt 7 assuming that the integral can be evaluated at the upper and lower limits to yield wellde ned and bounded values Associated with this transform Xs and equivalent to the statement that the transform exists is the signals Region of Convergence ROG 1f the transform exists the ROC exists and vice versa The ROC is that open half of the s plane that lies to the right of all singularities poles of Xsl At any point 8 so that lies inside the ROC the following relationship is true X 30 Ooozte 5 dt 8 FirstOrder System Example 1 A Overview Two different rsteorder systems will be presented in this example The rst system G1s will have its one openeloop pole located at the origin of the seplane that is at s 0 The second system G2s will have its one openeloop pole located at some other place along the real axis Neither of these systems will have a zero in its transfer function Thus neither of these systems represents the most general rsteorder system model but the two of them together do represent the most general strictly proper rsteorder system models The reason for differentiating between G1s and G2s in terms of the location of the openeloop pole will be discussed in the following sections For each of these systems the corresponding closedeloop transfer function will be developed under the assumption of unity feedback that is with Hs l The closedeloop responses of these systems to a unit step input and to a unit ramp will be developed using partial fraction expansion Several transient response and steadyestate response characteristics will be de ned in terms of the parameters in the openeloop transfer functions These characteristics will be useful in comparing the timeedomain performance of different rsteorder systems and they will also serve as a basis for the more general characteristics of secondeorder systems to be studied later B System 1 Bl The System Models The rst system to be considered is given by the following transfer function which will be placed in the forward path of a unityefeedback closedeloop system G1s K gt 0 l where K is a positive real number serving as the gain of the openeloop system This transfer function can also be written in the following forms by simple algebraic manipulation l 1 mi 2 G1s where T lK is de ned as the time constant of the system All of the various timeedomain characteristics that will de ned for this rsteorder system will be expressed in terms of the time constant Using the last form for the expression in 2 the closedeloop system under unity feedback for this system is given by G1s 1 1T K i T L 3 45 1G1s n Ts1 s1T sK The closedeloop pole for this system is located at s ilT 7K Since K gt 0 the closedeloop system is guaranteed to be stable B2 The Unit Step Response Assume that the reference input to the closedeloop system is the unit step function which has the Laplace transform Rs 15 The transform of the output signal is 1T 17 1T 51T39E ss1T The output signal in the time domain 01t can be found from 4 using partial fraction expansion 015 TCL71S39RS 4 7 lT 7 A1 A2 015 ss1T s s1T 5 A1 S 01Slso 17 A2 5 1T 39 01Sls71T 1 6 Therefore the output signal is 7 71 i 7 1 7 7 itT TCL1S Step Response 01t 7 lT 7 1 e t 2 0 7 s s The output has an initial value 010 0 and the output asymptotically approaches 01t 1 as t 7gt 00 Since the value of the step input was assumed to be equal to 1 and the nal value of the output is also equal to 1 the error between input and output as t 7gt 00 is equal to 0 The error in the nal value of the step response output relative to the input is known as the steady75mm error For the open7loop system de ned by 2 the closed7loop step response has zero steady7state error for any value of T gt 0 Since the output signal never actually equals its nal value for nite values of i there need to be ways of expressing the speed of response of the system that allows different systems to be compared in a meaningful fashion The rst of the transient response characteristics that will be de ned is the settling time T3 Be careful to distinguish between the symbol TS for settling time and the product of the time constant T and the Laplace variable 5 namely T5 The settling time will be de ned in the following way De nition 1 The settling time TS is the length of time it takes the system output in response to a step input to reach and stay within a speci ed tolerance about the nal value of the output In this class the tolerance for settling time in the step response will always be i2 of the nal value assuming that the nal value is non7zero For the output signal given in 7 the nal value is limtnoo ct 1 so the settling time is determined by the amount of time it takes the output to reach the value Ct 098 and stay within the range 098 S Ct S 102 For this rst7order system the output is asymptotically approaching the value of 1 so the settling time is de ned as the time at which Ct reaches 098 To see what this time might be consider the following table which relates the value of Ct from 7 to the value of t It is clear from the table that the output signal gets close to its nal value in only a few time constants The exact value of time at which the output equals 098 can be computed from 7 by letting Ct 098 and solving for t 098 7 1 7 gitT 7 z 7 7T ln002 7 3912 T 8 Rather than using the multiplying factor of 3912 in the preceding equation the usual convention for settling timeiwhich is acceptable in this classiis to use the following conservative approximation for settling time TS 4T 9 Thus if the time constant of a system is 1 second then the settling time as given in 9 is 4 seconds If a system has a time constant of 32 microseconds then the settling time in response to a step input will be 128 microseconds or if the time constant is 16 days then the settling time will be 64 days The time constant of the open7loop system serves as a scale factor for the time7domain characteristics Regardless of the value of the time constant T the settling time will be approximately 4 times the value of T The other transient response characteristic that will be de ned for the rst7order system is rise time Tr The rise time is a measure of how quickly the system responds to the input in terms of getting close to the nal value the rst time The usual choice for rise time at least for rst7order systems is the time it takes the output to increase from 10 of its nal value to 90 of its nal value This is knownifor obvious reasonsias the 10790 rise time The rise time can be determined from 7 by setting 0 t 01 and solving for the resulting t tm and repeating that with Ct 09 which occurs at t tgo The rise time is T tgo 7t10 The calculations give the following results Czm 7 01 7 1 7 e tmT 7 10 7 7T1n09 7 01054 T 10 Czgo 7 09 7 1 7 hoT 7 90 7 7T1n01 7 23026 T 11 T 7 90 7 210 7 21972T m 22T 12 Just as with the settling time the rise time of the step response is scaled by the system time constant T Figure 1 graphically shows the de nitions of the settling time and rise time in the closed7loop step response of the rst7order system In both graphs in the gure the independent variable is the dimensionless normalized time tT Normalizing the time by the time constant allows the characteristics of the response to be shown without worrying about the actual time scale This will be particularly useful in the study of secondeorder systems where there are two design parameters in the transfer function Using normalized time for the rsteorder system is equivalent to letting the time constant be T 1 secon The de nition for settling time is shown in the top graph of Fig 1 Settling time for the rsteorder system is de ned to be the time at which the output reaches 098 actually 098168 From 9 the settling time is TS 4T so in terms of normalized time the settling time is TST 4 The de nition for rise time is shown in the bottom graph Figure 2 shows the step responses of the closedeloop system for four different values of T The independent variable in this gure is actual time t in seconds so the differences in the speeds of response can be seen It is clear from the curves plotted in the gure that the smaller the value of T the quicker the response of the system both in rise time and settling time This is also obvious from equations 9 and 12 B3 The Unit Ramp Response The second reference input signal that will applied to the closedeloop system is a unit ramp function whose transform is 135 152 The transform of the output signal is 1T 1 1T C T R 13 15 CL 15 5 51T 52 52s1T As before partial fraction expansion can be used to obtain the output signal 1T A1 A2 A3 0 14 15 52s1T 52 s S1T A1 7 5201mm 1 A3 lt5 1T01ltsgtsT T 15 d d 1 T 51 T 0 7 1 T 1 A2 d52015 lt lt 2 7T 16 5 30 d5 5 1T 30 5 1T 30 Therefore the unit ramp response is 1 T T 71 7 7 7tT gt TCL1S Ramp Response 01t 52 S S lT t T Te t 7 0 17 As i gt 00 the exponential term in 17 goes to 0 so the output signal increases linearly with timeijust as the reference input doesiwith an offset equal to the time constant T The slope of the output signal and the slope of the unit ramp input signal are both equal to 1 so after the transient part of the response decays the input and output signals are parallel The steadyestate error in the ramp response is the difference between input and output signals after the transient response has decayed This error is 6 731m 72 7 ct 72731111 2 7 T TENT 7 z 7 z 7 T T 18 Thus the system time constant T not only serves as a scale factor in the step response it also determines the accuracy in the ramp response for large values of t The smaller the value that T has the more rapidly the output responds to a step input and the more accurately the output follows a ramp input Figures 3 and 4 show the response of the closedeloop system to a unit ramp input The linearly increasing dashed lines in the gures represents the unit ramp input signal The steadyestate error in the ramp response is de ned in Fig 3 which is plotted in normalized time units The steadyestate error is the vertical distance between the input and output signals after the transient part of the response has decayed to zero For this system by tT 135 units the transient term 6 13 5 is negligibly small and the vertical distance between the curves is eSST 1 unit The ramp responses for four different values of T are illustrated in Fig 4 with a zoomed view of the curves shown in the bottom graph By t 15 seconds the output curves for T 025 T 05 and T 1 second essentially have reached their nal slopes the vertical distance from the reference input down to each of the curves is the corresponding value of T For T 2 seconds the error is within 001 of its steadyestate value at t 15 seconds so steadyestate has been reached for all practical considerations for that system also For larger values of T it would take correspondingly longer values of time before the ramp response reached steadyestate In general the settling time for the ramp response is longer than the settling time for the step response for a given value of T De nition of Set ing Time in the Step Response of 1Ts 1 1 I W 098168 099326 095021 1 7 1 086466 1 l 063212 l 39 T f1quot 4 I S I I I l I I I I 0 1 2 3 4 5 Normalized Time tT De nition of Rise Time in the Step Response of 1Ts 1 I I 7 l tgoT 23026 I TrT tgoT t 101quot 21972 I l ltwT01054 I I I I I I 0 1 4 5 3 Normalized Time tT Fig 1 De nitions of settling time and rise time in the step response for TOL15 1 T5 1 Amplitude Step Response oflTs 1 for T 025 05 1 2 seconds Time sec Fig 2 Step responses for TOL15 1 T5 1 Amplitude De nition ofSteadyState Error in the Ramp Response oflTs 1 15 I eT ss 7 7 7 7 7 7 7 7 7 7 7 7 n 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 77f 10 v 5 eT1 ss 0 I I 0 5 10 Normalized Time tT Fig 3 De nition of steadyistate error in the ramp response of TOL15 1 T5 1 Ramp Response of 1Ts 1 for T 025 05 1 2 seconds I 15 10 7 Q 1 43 E g 2 5 V 0 5 10 15 Time sec Zoomed View of the Ramp Responses 15 I I I I 145 14 43 g 135 13 125 e 7 12 I I 12 125 13 135 14 145 15 Time see Fig 4 Ramp responses for TOL15 1 T5 1 0 System 2 01 The System Models The second system model that will be discussed has the following forward7path transfer function K G2s57p Kgt0 p7E0 19 where the open7loop pole is located at s p lfp gt 0 the open7loop pole is in the right7half plane7G2s is open7loop unstable7and ifp lt 0 the open7loop pole is in the left7half plane and G2s is open7loop stable The closed7loop system with 19 and unity feedback is K G2s a K T 7 7 7 20 CL 25 MGM 13 Kim lt The closed7loop pole is located at s 7 K 717 If p lt 0 the closed7loop system is always stable with K gt 0 If p gt 0 the closed7loop system is stable if K gt p In the rest of these notes we will assume that p 71 so both the open7loop and closed7loop systems are stable Letting p 71 for purposes of illustration in these notes the closed7loop transfer function can be manipulated into a standard form in the following way K K KK1 TCL72ltSgt2ltK1gt1K1gtlt 1 gt7 ltK1gts11 K1 21 KT K T 7 25 Ts1 51T 22 where T 1 K 1 is de ned as the system time constant Note that the last term in 22 has exactly the same form as the system in 3 except now 1T K 1 rather than K In the general case when the open7loop pole is located as s p the expression for the time constant would be T 1 K 7p 02 The Unit Step Response If a unit step function is applied as the reference input signal to the closed7loop system the output signal can be derived using partial fraction expansion just as before K 1 K C T 1 R 23 25 CL 25 5 51T s ss1T l K 31 32 C 24 25 5s1T s s1T B1 S 02Slso KT B2 5 1T 39 025ls71T KT 25 and the output signal in the time domain is KT KT K TCL2S Step Response 02t 71 7 K I I 7 citT t2 0 26 The output has an initial value 020 0 and the output asymptotically approaches 02t KT K K 1 as t 7gt 00 Since the value of the step input was assumed to be equal to 1 and the nal value of the output is different from 1 the error between input and output as t 7gt 00 does not go to 0 there is a nonzero steady7state error in the step response for this system Since K gt 0 K K 1 lt 1 and the nal value of the output is less than the value of the reference input The larger the value of K the closer the nal value of the output will be to 1 but it will always be less than 1 1f the open7loop pole is in the right7half plane 17 gt 0 but the closed7loop system is stable K gt p then the nal value of the output signal will be greater than 1 The transient response of the output signal is still controlled by the time constant T just as with system TCL15 there is just a slightly different de nition for T in this second case The de nitions for settling time and rise time are exactly the same is given previously in 9 and 12 The only difference is that with 02t the nal value is K K 1 rather than 1 as it was for 01t Settling time is still the time required for the output to reach 098K K 1 and the rise time is the time required for the output to go from 10 to 90 of K K 1 Step Response of KTTs 1for K 05 1 2 5 10 1 I I I I I K 100 T 00909 sec K 50T 01667 sec K 20 T 03333 sec 0 m K 10 T 05000 sec Amplitude 0 Ln 0 4 I K 05 T 06667 sec 0 I I I 3 Time sec Fig 5 Step response plots for TOL25 KT Ts 1 with T 1K 1 and K 05 1 2 5 10 Figure 5 shows the step responses for the output signal in 26 for gain values of K 05 1 2 5 10 As the plots indicate the output signal gets closer to the reference step input value of 1 as the value of K increases The corresponding values for the time constant are T 1 K 1 06667 05 03333 01667 00909 The nal values for the output are K K 1 03333 05 06667 08333 09091 1f the openeloop pole was at some location other than 17 71 the values of T and the nal values of 02t would change 03 The Unit Ramp Response For a unit ramp reference input signal the transform of the output would be K 1 K 025 TCL2S Rs m 5 2 m which can be expanded in the partial fraction format K Bl B2 B3 02ltSlms 2m Bl 5 02sl0KT7 B31ST1T390251s71TKT2 is S 1 K lts1TltoeKlt1 2 32 dsi C2ls0 dss1T1S0 s1T2 lpo KT The ramp response is Ramp Response of KTTs 1 for K 05 1 2 5 10 15 I I 10 5 10 a 2 390 2 a E lt 1 5 05 0 r I 0 5 10 15 Time sec Fig 6 Ramp response plots for TOL25 KT T5 1 with T 1K 1 and K 05 1 2 5 10 KT KT2 KT2 TCL2S Ramp Response 02t 71 5 2 7 T KT t 7 T TaitT t2 0 31 The ramp response in 31 is seen to be identical to the response in 17 for TCL1S except for the scale factor KT K K 1 The same scale factor appeared in the step response for TCL2S having the effect of producing a nonzero steadyistate error The scale factor also affects the steadyistate error in the ramp response The slope of the output signal is no longer equal to 1 it is equal to K K 1 Thus the reference input signal and output signal will diverge with time The steadyistate error in this case will be tgtoo twawHartleylt gtlt gtw No matter what the value of K is the ramp response will have in nitely large steadyistate error Figure 6 shows the ramp responses for this system using the same values of K as for the step responses Clearly the output signal is diverging from the reference input in each case The output signal will stay close to the reference input for longer periods of time for larger K but the vertical distance between input and output will always become in nitely large for this system model 31330 W 7cm 2 lim punt 32

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