Feedback Control System Fundamentals

Transcription

1 unam online continuing education coure Feedback ontrol ytem Fundamental by eter ennedy

2 Feedback ontrol ytem Fundamental unam online continuing education coure Feedback ontrol ytem Fundamental. Introduction: hi coure dicue many fundamental concept aociated with claical feedback control theory. Feedback, in it mot general ene, refer to the interaction between interconnected dynamic ytem uch that the repone of one influence the dynamic of the other. In the context of feedback control; the tate of a phyical ytem or device i meaured by a ening ytem. he meaured tate or feedback i compared to a deired tate and the error ued by a controller to reduce the difference between the actual and deired tate. Uing the difference or error, equate to a negative feedback control loop; central to feedback control theory. he phyical device or ytem to be controlled i often called the plant, proce, or load. n example of a feedback control ytem i the central heating and air conditioning ytem for a home, or building. thermotat or temperature enor i the feedback enor that meaure the room temperature and compare it to the deired temperature or et point, calculating a difference or error. If the temperature i le than the et point, the error i ued by the controller to force more heat into the room. When the et point i reached, the error i zero or below an error threhold and the controller will top heating the room. nother example i the peed control in mot of today automobile. he peed of the vehicle i meaured and compared to a deired peed. Baed on the difference between actual peed and the et point, acceleration or braking i applied to the automobile drive to null the error and maintain the deired peed. laical control deal directly with the differential equation that decribe the dynamic of a plant or proce. hee equation are tranformed into frequency dependent tranfer function. he tranfer function i the ratio of two frequency dependent polynomial whoe root decribe the repone of the plant in a frequency domain. he controller or compenator hape the cloed feedback loop repone, given the plant repone, to achieve the control performance objective. Modern control theory i another approach to control loop deign and analyi; epecially ueful for ytem with multiple input and output. It utilize tate pace method to evaluate the repone of a ytem a well a generate the control for it. hi relie heavily on linear algebra and matrix theory. Modern control theory and deign method are not dicued further in thi coure. Even claical feedback control deign and analyi tend to require a good foundation in mathematic, however the purpoe of thi coure i not to dwell on the math, although example are provided, but to provide the baic deign and analyi concept.. Feedback ontrol Block iagram: he feedback control ytem block diagram provide both a viual a well a a bai for the mathematical repreentation of the feedback control ytem. It opyright 4 eter ennedy age of 44

3 Feedback ontrol ytem Fundamental unam online continuing education coure i an important part of the control ytem deign and analyi. In general, the feedback control block diagram will be complex incorporating many element and everal feedback loop. o decribe jut the baic element, a very imple block diagram i hown in Figure. md In - Σ ontroller or ompenator Σ lant, roce, or oad Output Σ N Figure Baic Feedback ontrol Block iagram F Feedback enor or randucer hi imple loop include a controller, plant, and feedback enor F. he input are the command input md In, diturbance and enor noie N. he command, which i the deired plant output, i applied to the input umming junction and compared with the actual plant output. he error between the command and actual output i calculated and fed to the controller. he controller generate the drive ignal to the plant that reduce the error until the command input equal the actual output or remain within a deired offet. he dynamic of each block can be expreed in the time or frequency domain. he time domain i very ueful for imulating the loop repone and evaluating performance a a function of time. et of differential equation which can include non-linear term are ued to decribe the dynamic for each block. he frequency domain i excellent for linear analyi. he gain of each element can be characterized a a function of frequency; effectively each element i a frequency dependent gain. he output repone can be evaluated a a function of frequency and the controller adjuted in the frequency domain to improve the loop performance. 3. ime and Frequency omain Repreentation of oop ynamic: he plant dynamic i normally decribed by a et of differential equation and non-linearity time and/or multivariable dependent. o decribe the repone in the frequency domain, the aplace tranform or the differential operator can be ued if the plant i linear or approximated by linearizing the ytem about an operating point. he definition of the aplace tranform of a time dependent variable ft i defined a: opyright 4 eter ennedy age 3 of 44

4 Feedback ontrol ytem Fundamental unam online continuing education coure F t f t e dt hi integral exit for an σj with any real part σ>. he variable f where f i frequency in hertz. ifferentiating thi expreion n time with repect to time, reult in the important relationhip: n n t n f t e dt F k differentiated function in time i algebraically related to it tranform a [f n t]-> n F where denote the aplace tranform. he ummation term on the right account for the initial condition of each differentiated term. For frequency analyi, the initial condition are often aumed to be zero o that: f n t e t dt n F for initial k f nk condition he invere aplace tranform i ued to convert a frequency domain function back to one in the time domain. he invere tranform i given by: c j t f t F e d j hi complex integral i evaluated along the path cj in the complex plane from c-j to cj where c i any real number for which the path cj lie in the region of convergence of the tranform F. lthough thi i a omewhat complicated definition; method uch a partial fraction expanion are available for evaluating the invere tranform. Mot control textbook and engineering reference book have very thorough table of both the aplace tranform and the invere aplace tranform olved for numerou function. o in many cae, one can obtain the tranform or it invere from a table and need not evaluate it directly. c j It i alo important to note that although the definition of the aplace variable i σj, when performing frequency domain analyi the variable will be equated to j where f. he σ term i aumed zero ince for frequency domain analyi a function i being evaluated baed upon it repone to periodic inuoidal ignal. he real axi σ in the -domain repreent an exponential decay or growth factor that i not relevant for thi analyi. he relationhip between the time and frequency domain can be hown, uing the aplace tranform, for a firt and econd order plant a follow. For a firt order plant, let xt equal the plant input and to horten notation let yt equal Outputt, then the plant can be decribed by the differential equation: opyright 4 eter ennedy age 4 of 44

5 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 5 of 44 t x t y t y he term i the plant time contant while i the caling factor. aking the aplace tranform, thi convert in the frequency domain denoted by the aplace variable to: ; ; and where X t x Y t y Y t y X X Y Output or X Y econd order ytem can be expreed a: t x t y t y t y ξ For thi plant ζ i the damping contant and the natural frequency of the plant. he aplace tranform of thi equation i: and where or Y t y X X Y Output X Y ξ ξ ξ hi general procedure can be applied to a differential equation of any order. 4. ey Feedback oop Relationhip: Each block in the feedback control loop block diagram can be expreed in the frequency domain uing the aplace tranform decribed in the lat ection, auming all contraint are met. In the frequency domain, block can be treated algebraically with each element of the loop in the block diagram treated a a frequency dependent function. Referring to the block diagram of Figure, an expreion for the Output can be obtained algebraically a: ] [ Output F N In md Output olving for the output:

6 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 6 of 44 N F F In md F Output he term in bracket that multiply each input are very important relative to the control loop deign. he firt term in bracket decribe the repone from the command input to the output and i termed the cloed loop tranfer function F with a magnitude called the cloed loop gain G or: F G F F he control compenator gain,, i normally a very high uch that magnitude of the product of the three term **F >> over the effective operating bandwidth of the loop. hi product i the open loop tranfer function OF and it magnitude called the open loop gain OG. F OG F OF he feedback term i often a unity gain or caled to provide unity gain in the feedback path a it i imply ening the plant output. hi being the cae, the cloed loop gain, G, i unity over the operating bandwidth of the loop and roll off toward zero at high frequency. If the feedback gain i not unity, it will change the cloed loop gain from one to the invere of the feedback gain magnitude i.e. /F. In the expreion for the loop output above, the econd term account for the impact of diturbance on the output repone. he tranfer function in bracket multiplying the diturbance i often termed the compliance, a in how compliant the loop i to a diturbance. F ompliance hi term attenuate the diturbance primarily due to the high gain control compenator,, in the open loop gain. herefore the higher the compenator gain, the more diturbance rejection and the le impact the diturbance will have on the output. However thi gain will be limited by the tability of the loop. Finally, a modeled, the lat term i enor noie which a hown effectively add directly to the output ince it i multiplied by the cloed loop gain. it i baically input to the ame umming junction a the command input, thi would be expected.

7 Feedback ontrol ytem Fundamental unam online continuing education coure he error between the command input and the output i given by: F Error md In Output md In F N F F he error i effectively the accuracy with which the output follow the command input. he firt term in bracket that i multiplying the command input i often called the loop enitivity function. For unity feedback F~ thi term i effectively one over the open loop gain, imilar to the compliance function dicued previouly, or: enitivit y for F he error will be a function of the attenuation provided by thi term a well a the magnitude of the command, diturbance, and noie. One final error relationhip of importance i the teady tate error which i defined a: E lim [ Error ] lim md In It i a meaure of the error repone or effectively the accuracy of a table ytem a time goe to infinity. It i normally evaluated againt three tandard input ignal; a tep, ramp or rate, and parabolic input or acceleration. hee ignal have the aplace tranform: tep ramp parobolic tep ; ramp ; parobolic 3 he teady tate error i often ued a part of the control loop performance pecification. he required repone to thee input ignal will determine what i termed the loop ype ; which refer to the open loop pole at or the number of integrator i.e. / term in the OF. If we decribe a general input to the ytem a x/ n which can be any of the above input depending on the value of n, then the teady tate error can be written a: x x E lim [ Error ] lim lim n n n From thi expreion, the following table can be generated that decribe the teady tate error a a function of loop type and input. opyright 4 eter ennedy age 7 of 44

8 Feedback ontrol ytem Fundamental unam online continuing education coure Input tep n Ramp n arabolic n3 able teady tate Error veru oop ype and Input teady tate Error ype ype ytem ype ytem More than two ytem one open loop pole two open loop pole open loop pole no open loop at at at pole at tep ramp V parobolic lim [ ] V lim [ ] From able it i oberved that with a ype loop and a tep input there will alway be an offet between the input and output which i a function of the loop gain. he ype loop cannot track a ramp or parabolic input a the error will continue to grow infinitely with time for both input type. ype loop will null the error to a tep input at a rate baed on the loop bandwidth. he ype loop will track a ramp with an error that i a function of the loop gain. It cannot track a parabolic input a the error will continue to grow with time. he ype loop track both a tep and ramp with zero error and a parabolic input with an offet between the input and output which i a function of the loop gain. oop ype greater than two can track all three input with zero error; however a loop with thi many integration term i difficult to tabilize. 5. ontrol oop tability: here are many method for analyzing feedback control loop tability. For a linear ytem, if the feedback control loop output remain bounded for any bounded input md In,, N, then the ytem i conidered to be bounded input- bounded output table. For non-linear ytem, more advanced method uch a the yapunov criteria are required. opyright 4 eter ennedy age 8 of 44

9 Feedback ontrol ytem Fundamental unam online continuing education coure Evaluating tability for linear ytem can be performed eaily in the frequency domain. he primary tranfer function,, F are decribed in the frequency domain by the ratio of two polynomial that are a function of. he root of thee polynomial in the numerator are termed zero and in the denominator termed pole. he cloed and open loop tranfer function gain and phae i a function of all thee polynomial. For example if ubcript N denote numerator and the denominator then the cloed loop tranfer function can be expreed a: N N F f F F N N FN FN N N F F F F One important obervation i that the zero of the OF will alo be the zero of the F. For a caual linear ytem to be table all of the pole of F mut have negative-real value. etermining the pole of the polynomial can be done with a root finder. tability method called the Routh criterion, baed upon evaluating only combination of the polynomial coefficient can alo be ued to determine if the pole all have negative real part. However thi doe not really quantify how table the ytem i or the margin of tability. Other method uing a frequency domain repreentation to evaluate tability while alo providing tability margin are: the Bode lot, Nyquit lot, and Nichol plot. 5. Bode tability riterion: he Bode lot i a relatively traightforward graphical method uing plot of the OF gain and magnitude to evaluate tability. he Bode criteria for tability, however, i ufficient only for feedback loop whoe F i minimum phae all zero and pole of F have negative real part or are in left half of -plane. he Bode plot i effectively two plot, one of the open loop gain OG or magnitude and the other the phae of the OF. he gain and phae of an arbitrary tranfer function G are determined by ubtituting j into the numerator polynomial, GNj, and the denominator polynomial, Gj, of Gj and then calculating the real and imaginary component of the numerator and denominator or: GN j RGN j IGN G j G j RG j IG where RGN Real[ GN j] ; IGN Imaginary GN j] RG Real[ G j] ; IG Imaginary G j] he magnitude and phae are then determined a: N N N opyright 4 eter ennedy age 9 of 44

10 Feedback ontrol ytem Fundamental unam online continuing education coure G RG j RG RI where RG RG IG RG IG N RG IG N ; IG IG Φ G j atan N IG RG N IG IGN atan RG RG N IG atan RG he magnitude of the product of everal tranfer function i the product of their magnitude and the phae i the um of their phae or: G j G j G j G j G j IG IG Φ G j Φ G j G j Φ G j Φ G j atan atan RG RG he baic tability criterion can be determined by examining the denominator of the F. F hi function goe to infinity if F the left ide of thi equation i the OF, intability will occur if the OF- or OF OG Φ OF where Φ OF phae of OF OG magnitude of OF hi condition will occur if: OG and Φ OF 8 With a negative feedback loop, if OF- thi i equivalent to a ignal input to the OF having the ame amplitude and phae a i fed back; which i a condition for utained ocillation. Bode plot are ued to determine jut how cloe, or the margin the feedback loop ha relative to an intability condition. he goal i to have ufficient margin to enure the ytem remain table. he Bode criteria define two critical frequencie; the gain croover frequency, fg, which i the frequency that the OG croe one and the phae croover frequency, f, which i the frequency at which the OF phae croe -8. table phae margin i the amount that the OF phae angle i > -8 when the OG at the gain croover frequency, or: hae Margin M 8 Φ OF fg Φ OF fg > 8 or M > for tability he gain margin i the magnitude of the OG relative to unity gain when the OF phae goe through -8 at the phae croover frequency or: Gain Margin GM OG f OG f < or GM > for tability opyright 4 eter ennedy age of 44

11 Feedback ontrol ytem Fundamental unam online continuing education coure he gain margin and phae margin are normally pecified in the performance requirement for a control loop. he gain margin i often expreed in decibel, or: GM db og GM og OG f It can be noted that given the F atifie the Bode criteria retriction, the loop i table if fg < f. If the retriction for the uing the Bode criteria are not met, other method uch a the Nyquit or Nichol plot hould be ued. couple of example of the Bode plot follow. plot i hown in Figure for an OF given by:.6 z OF p ξ r r where : z f z.59 p f p.59 f.6366 ; ξ.5 r r he olid line are for the open loop gain, which i caled in unit of db on the right hand ide of the plot and the dotted line are for the open loop phae caled in degree on the left hand ide of the plot. he plot of the OG i in olid black with a line at db in olid light blue. Each pole contribute - db/decade of lope to the magnitude plot and each zero db/decade of lope. he quadratic term contribute -4 db/decade of lope. he OF phae i the dotted red line and a line at the -8 reference in dotted dark blue. Each pole contribute -9 of phae lag to the phae plot and each zero 9 of phae lead. he quadratic term contribute -8 of phae lag. hown in the figure, the ytem ha a gain croover frequency at.5 Hz, a M~3 and a GM~3 db and i table baed upon the Bode tability criterion. he phae croover frequency i.6. hi i a low bandwidth ytem and the relatively low M indicating ome ocillatory repone can be expected. opyright 4 eter ennedy age of 44

12 Feedback ontrol ytem Fundamental unam online continuing education coure Figure Bode lot for Example nother example i for a ytem with a much higher bandwidth ytem. Figure 3 provide a Bode plot for an OF given by: 484 z OF p where : z f z 5.9 f 59. p gain the plot of the OG i in olid black with a line at db in olid light blue. he OF phae i the dotted red line and a line at the -8 reference in dotted dark blue at the bottom of the plot. he ytem i table with gain croover frequency at 4 Hz, a M~55 and a gain margin that i infinite phae i alway greater than and never croe -8. he M for thi ytem i better than the previou example and in the range that indicate a good repone. p opyright 4 eter ennedy age of 44

13 Feedback ontrol ytem Fundamental unam online continuing education coure Figure 3 Bode lot for Example 5. Nyquit tability riterion: he Nyquit tability criterion i another graphical technique for determining the ytem tability. he Nyquit plot i a graph of the OF imaginary veru real part a a function of j. he real part i plotted on x-axi and the imaginary on the y-axi. he plot can alo be implemented a a polar plot of the OF magnitude and phae a a function of j. It can be applied without explicitly computing the pole and zero of either the cloed-loop or open-loop ytem. a reult, it i applicable to ytem defined by non-rational function and delay. In contrat to Bode plot, it can handle minimum phae tranfer function a well a thoe with right half-plane ingularitie, or non-minimum phae plant. lthough it i one of the more generic tability tet, it i till retricted to linear, time-invariant ytem. Nonlinear ytem mut ue more complex tability criteria, uch a yapunov or the circle criterion. While Nyquit i a graphical technique, it provide only a limited amount of information about the reaon a ytem i table or untable, or how to modify an untable ytem to be table. echnique like the Bode plot previouly dicued, while le general, are ometime a more ueful deign tool. ement of the tability of a cloed-loop negative feedback ytem i done by applying the Nyquit tability criterion to the Nyquit plot. he plot i effectively a mapping of a contour with infinite radiu about the right half of the -plane to a contour of the OF magnitude and phae. tability i determined by looking at the number of encirclement, N, of the point at -, opyright 4 eter ennedy age 3 of 44

14 Feedback ontrol ytem Fundamental unam online continuing education coure. Uing the argument principle for contour integral, it can be hown that NN-NZ where N and NZ are the number pole and zero within the right half -plane contour. he pole of the OF, which are known, are the zero of the F, a wa hown in ection 5. by expanion of the cloed loop tranfer function. he number of zero, NZ that are in the right half -plane i known from the OF. he number of pole in the right half -plane i then obtained a NN-NZ. If the point -, i not encircled or interected, then the cloed loop ytem i table. he interection of the Nyquit contour with the negative real axi relative to the origin i the invere of the gain margin. For example, if thi ditance i a then the GM/a. he phae margin i obtained by drawing a line from the point of interection of the Nyquit contour with a circle of radiu one about the origin; to the origin and then calculating the angle relative to the negative real axi with poitive being counter-clockwie. he Nyquit plot for the firt example in ection 5. decribing Bode tability i hown in Figure 4. he top plot how the contour i for a frequency range out to Hz while the econd two plot are expanded o that the phae margin and gain margin can be oberved. opyright 4 eter ennedy age 4 of 44

15 Feedback ontrol ytem Fundamental unam online continuing education coure Nyquit lot Expanded View howing hae Margin Expanded View howing Gain Margin Unit circle -, a-.7 > GM/a ~3 db hae margin 3 Figure 4 Nyquit lot for Example 6. ontrol oop erformance pecification: he pecification define the deired feedback loop performance. From a practical perpective, the control loop deigner mut undertand the uer requirement to develop a atifactory pecification. ey characteritic often pecified by a uer are accuracy and repone time for a defined command ignal or range of ignal and the diturbance and noie environment. From thi information, the deigner mut develop the control requirement and pecification. dicued previouly, accuracy and diturbance rejection are improved by a high gain controller, a i repone time. gain i effectively proportional to bandwidth, thi implie a higher bandwidth improve performance. However a the bandwidth opyright 4 eter ennedy age 5 of 44

16 Feedback ontrol ytem Fundamental unam online continuing education coure increae, the loop tability margin begin to decreae, more overhoot reult, control element can aturate, and noie between loop element can be amplified. o the deign i a tradeoff between the uer requirement and control loop pecification that meet thee requirement with ufficient tability margin. Repone time can be pecified in term of rie time, overhoot, and ettling time all of which are a function of bandwidth and the control compenator tructure. In the frequency domain, the deigner can pecify the reonant peak, peak frequency, gain croover frequency, bandwidth which in ome cae are directly related, and the minimum allowable phae and gain margin. he reonant peak i a maximum of the gain at the peak frequency. he other important pecification i the control loop type which a dicued previouly refer to the number of integrator within the loop; either due to the controller or the plant dynamic. he loop type or number of integrator will determine the accuracy with which the output follow an input command in the long term. hi error i often referred to a the following error. he integrator alo provide very high gain at low frequency which i good for diturbance rejection and improved accuracy; however each pole at alo provide a -9 phae hift which can impact tability. he robutne of the controller deign mut alo be conidered. If the plant parameter change for whatever reaon, what will happen to the loop tability margin? he controller deign mut account for thee variation. hi can be accomplihed by deigning for the wort cae plant variation and enuring the ytem meet the tability margin over the full range of plant variation; meauring the plant repone i.e. plant identification and changing control parameter to account for the plant change either manually or on a cheduled bai, or a uing an adaptive controller that automatically adjut the control parameter a a function of the plant variation. 7. ontroller eign: here are many control loop deign technique; mot involve ome form of loop haping and/or algebraic pole-zero placement and cancellation. oop haping can be accomplihed uing Bode or Nyquit plot analyi. Viually, the Bode plot i eay to interpret and adjut the repone epecially if implemented in a mathematical computer aided deign program i.e. Matlab or Mathad. he Bode plot can be ued with meaured plant gain and phae repone data, the plant-pole zero tructure doe not have to be known exactly although the Bode deign i till ubject minimum phae criterion. lgebraic method appear traightforward but require knowledge of the plant pole-zero tructure and care mut be taken to inure table deign. For example, pole and zero cancellation mut be done with ome enitivity analyi becaue they are never known exactly and in ome cae can change or drift. hi i epecially true if the pole and zero are not table; direct cancellation can reult in an pole/zero pair that are both untable both root in right half -plane a oppoed cancellation of the untable root. ommon compenation element ued in a controller are proportional plu integral I and/or derivative I control, and firt or econd order lead and lag compenation a well opyright 4 eter ennedy age 6 of 44

17 Feedback ontrol ytem Fundamental unam online continuing education coure a a cacade of thee baic element. he loop ype can be adjuted uing the I or I controller. When uing any control term with a pure integrator, the output hould be limited or other meaure taken to inure windup of the integrator doe not occur. couple of imple deign example are provided. 7. Frequency omain oop haping: ume for a unity feedback loop the deired gain croover frequency i 4 Hz and the phae margin > 55. hi provide two deign criteria to atify with the compenator: OG fg M 8 Φ OF fg 55 or Φ OF fg 5 From the definition of the open loop tranfer function, thi can be expreed a: N jg N jg ; G fg jg jg Φ j j Φ j Φ j Φ j Φ j 5 G G N G N he plant i given by: he plant parameter i the caling factor. For a double integrator, the phae lag i -8 o it can be aumed that to obtain the deired phae repone i.e. > -8 a lead compenator i required. With a lead compenator, the zero frequency i le than the pole frequency while for a lag compenator the zero frequency i greater than the pole frequency. he compenator will have the form: 6 α 6 he pole frequency at 6 Hz i choen a the roll-off frequency for the loop. It effectively limit the frequency repone and reduce higher frequency noie. Repreenting the pole and zero in term of a frequency i.e f provide for better viualization of the deign proce and pole/zero location. It i alo ueful to cancel plant parameter with a caling factor o effectively the loop repone i haped only by the normalized plant repone and the compenator. he compenator i then expreed a: G G G opyright 4 eter ennedy age 7 of 44

18 Feedback ontrol ytem Fundamental unam online continuing education coure he OF i then: α α for 6 α OF α 6 hi will be a ype control loop. he value for the compenator gain and zero are obtained by firt obtaining the value of the zero from the phae margin condition and olving for the gain from the OG condition. From thi expreion, the phae component are given by: G G Φ N jg atan ; Φ jg atan α 6 Φ N jg ; Φ jg 8 ubtituting into the expreion for phae margin G G 5 atan atan 8 α 6 or atan atan atan 4 α 6 α olving for α 4 69 atan α or 4 α 97 f tan69 f 97 Z Z 5. 4 Hz he phae margin criterion i atified with the compenator pole at 5.4 Hz. he magnitude criterion i now ued to obtain the value of that atifie the open loop gain condition at the gain croover frequency. opyright 4 eter ennedy age 8 of 44

19 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 9 of G G G or j j j or j j j he compenator gain that atifie the open loop gain croover frequency condition i then: ompenator erivation from a eired F: nother method for controller deign i to pecify a cloed loop ytem model baed upon a deired performance. For example, repone time or bandwidth, overhoot, tability margin could be pecified. With the cloed loop tranfer function defined and the plant tranfer function known, the compenator can be determined. uming unity gain feedback and uing the definition for the cloed loop repone: or F F F F F F F F N N N he compenator cancel out the plant and ubtitute an open loop tranfer function that give the deired cloed loop tranfer function. here are ome important contraint: If the plant ha a pole in the right half -plane pole with a poitive real part then the deired F mut be choen uch the F-FN ha the ame root. If the plant ha a zero in the right half -plane then the deired F mut have the ame right half -plane zero

20 Feedback ontrol ytem Fundamental unam online continuing education coure o be realizable, the exce pole of the deired F mut be equal or greater than the exce pole of the plant. If the numerator polynomial of a tranfer function i of order nz and the denominator polynomial of order np then the exce pole are defined a np-nz. very imple example i provided with a deired F and plant of the form: a b F ; a b b a b a a b b b b a a he contant a, b define the deired repone of the F. Uing thi procedure baically reult in a proportional plu integral I controller to realize the form of a deired F. here were no right half plane pole or zero in thi example and the exce pole criterion i alo atified. Unfortunately, a real deign i eldom thi imple. o atify the exce pole criterion, it may be required to add more pole to the deired F than required to jut obtain the deired performance. In thi cae, the F i contructed a the cacade of a tranfer function that dominate the repone and define performance, and a tranfer function with pole ignificantly beyond the deired bandwidth that will atify the exce pole contraint. 8. Home Heating ytem: he firt example i a implified verion of the feedback control for heating a houe. he phyical configuration i illutrated in Figure 5. he houe i idealized a a box filled with air at a uniform temperature. he wall of the houe are conidered a pure reitance to heat tranfer with no energy torage capacity. he overall coefficient of heat tranfer i U and the heat tranfer area i. he outdoor environment temperature i e and varie with time thereby acting a a diturbance to the control ytem. he temperature i meaured by a temperature enor in a thermotat or temperature controller mounted inide the houe. he deired temperature can alo be et by thi device, or in today environment may even interface to a mart phone with an that let it be et remotely. It i aumed that the temperature i converted to a voltage or current with a caling contant V with unit Volt/ F and proceed by electronic, or more likely in today environment a micro-controller. opyright 4 eter ennedy age of 44

21 Feedback ontrol ytem Fundamental unam online continuing education coure Figure 5 oncept Feedback ontrol onfiguration for Heating a Houe he controller take the difference between the deired and actual temperature and generate a control ignal to the furnace. hi ignal control an actuator that increae or decreae the ga flow and effectively the amount of heat into the room. For thi imple example, the actuation and heat generation are modeled a a linear proce; in reality it i a complex proce. In developing a model of the ytem it i aumed that initially the houe i at equilibrium with contant value of and e. he furnace will then be upplying ufficient heat to balance the heat lo to the environment. ny diturbance or change in the deired temperature will reult in an increae or decreae in heat input from thi original value. ll variable hould be conidered deviation from their initial equilibrium condition. he imple firt order thermal dynamic model for the heat balance in the houe the control loop plant in the time domain i: thermal energy tored thermal energy in thermal energy leaving the houe or M t QM t U t e t BU where QM thermal energy in hr M ma of air in houe pecific heat of air at contant preure BU BU aume U 5 ; M 8 Hr F F onverting to the frequency domain and rearranging term: opyright 4 eter ennedy age of 44

22 Feedback ontrol ytem Fundamental unam online continuing education coure QM e U where M U. Hr 43 ec hi repreent the model of the plant; in term of previou notation: U 5 43 F BU / Hr feedback control ytem meaure the houe temperature and compare it to a deired temperature or the et point. he control loop i hown in Figure 6 with the yet undefined controller. e U* V cale volt/f V E Σ - controller volt/volt VQ furnace BU/hr/volt Q M Σ U*** houe F/BU/hr V enor volt/f Figure 6 Block diagram of home heating control ervo loop From the block diagram, the dynamic of the loop can be expreed a: [ U e V Volt aume V.67 F ] olving for : U e V V V opyright 4 eter ennedy age of 44

23 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 3 of 44 he error between the et point temperature and the actual temperature i given by: U V V e dicued previouly, the error i a function of the magnitude of the outide and et point temperature and the attenuation provided by the controller gain. he implet controller i a proportional gain or with unit of BU/Hr/Volt. ubtituting into the previou expreion and uing the plant definition: V e V V e U U U U U U λ λ λ λ where tability i not an iue, a the open loop ha only one real negative pole. he term λ i baically the attenuation factor whoe magnitude i baed upon the value of. he time contant of the ytem will alo be reduced by thi factor. thi i a ype ytem, there will be a teady tate error to a tep change. For a tep change in e and, the teady tate error i given by: e e E im im E δ δ λ δ λ λ δ λ λ he tep change in the input will mot likely come from a new et point, a the temperature of the environment would not be expected to change intantaneouly o δe: tep change E δ λ he gain i choen to provide an attenuation factor λ that i conitent with the amount of allowable offet that can be tolerated between the new et point and actual temperature. For example with the gain wa choen a: V U

24 Feedback ontrol ytem Fundamental unam online continuing education coure he reulting attenuation would be a factor of or -4 db. he achievable gain will alo depend upon the actual drive characteritic of the furnace. he gain croover can be determined by a Bode plot or etimated directly from the OF a fg~.4 Hz. With a proportional plu integral controller I the ytem i ype and the teady tate offet to a tep would be zero. he I compenator ha the form: α.95 U V α.45 he open loop tranfer function i now: α V V U loed loop ytem denominator i now a quadratic polynomial. ince all coefficient are poitive, the root will have negative real part o that the ytem will be table. he tability margin are obtained uing a Bode plot. he open loop gain and phae are hown in Figure 7. phae margin Figure 7 Bode plot for Example with I controller opyright 4 eter ennedy age 4 of 44

25 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 5 of 44 he gain croover frequency i ~.67 Hz, the M~64 and the gain margin infinite ince the I integrator lag i offet by the zero o the net loop lag i only the lag of the plant which being firt order i only 9. are mut be taken in an actual deign to enure the higher bandwidth doe not put too much demand on the furnace drive. he expreion for the error for thi compenator i: α α α λ λ α α V V V V r e r V V e U b U b U b b b b U U U U ; ; where For a tep change, the teady tate error will be zero ince: b b im im E e δ δ λ For a ramp or teadily increaing temperature change, the teady tate error i: e r e r E b b im im E δ δ λ δ δ λ he et point will normally not change a a ramp, however the temperature of the environment can o that mot likely the teady tate error for thi cae would be δd/dt : ramp e r E λ δ Uing the definition of for thi cae, λr.54 or -5 db with the I controller. 9. Motion ontrol Feedback ytem: oition and rate feedback control ytem are ued in many application; camera for urveillance and in the movie indutry, communication and radar antenna, poitioning concentrating olar collector, computer dik control, pointing laer, etc. typical pan tilt zoom camera aembly i hown in Figure 8. For thi application, the plant or load camera a hown in the figure i mounted on a rotary tage driven by a motor. ervo amplifier convert control command to a motor drive ignal, for example a control voltage to a motor drive current which i applied to the motor tator winding to generate a torque cauing the rotary tage on which the load i mounted to rotate. he baic element of the control loop are hown in Figure 9.

26 Feedback ontrol ytem Fundamental unam online continuing education coure. Figure 8 XI an ilt Zoom amera embly o md In Σ - ontrol ompenator F ervo mplifier M Motor M Σ J oad rate po F R F RE oition ignal onverter oition enor Figure 9 Baic Block iagram for a Motion ontrol Feedback oop he poition command i compared to the meaured poition, calculating the error between the two poition. he control compenator ue the error to calculate a drive command to the motor which i applied through the ervo amplifier. he output of the motor i a torque that i applied to the load along with any diturbance. he motor feedback to the ervo amplifier i a back electro-motive force EMF which will be dicued later. hown, the load i imply modeled a pure inertia and neglect the effect of friction which in general depend on the rate ignal. he opyright 4 eter ennedy age 6 of 44

27 Feedback ontrol ytem Fundamental unam online continuing education coure poition feedback enor and ignal converter work in tandem to generate a meaured ignal in the unit ued by the ervo loop; normally the net feedback gain i unity. 9. Motor and ervo mplifier Model: Many motor ued for poition control application are ervo motor, although drive can alo be ued. he motor aumed for the example i a ervo motor. imple model for a motor include three equation: Motor voltage to current: V t R I t I t rate t M where : R winding reitance ; It motor current winding inductance ; ratet motor rotor angular rate e back EMF voltage contant Motor current to motor torque Mt: t I t M where : motor torque contant Motor torque to load torque: J rate t B rate t t t where : J motor plu load inertia B load vicou damping contant he motor model i hown a a block diagram in Figure. e M Σ V M - motor admittance I motor torque contant M Σ oad rate e back EMF contant Figure Motor Block iagram opyright 4 eter ennedy age 7 of 44

28 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 8 of 44 he tranfer function from motor voltage and diturbance torque in to the rate out can be expreed a: winding admittance load repone where : R B J V rate e M e he electromechanical and electrical time contant, with vicou damping aumed negligible B~, can be derived approximately from thi expreion a: R J R E e M ; here are two type of ervo amplifier ued to drive the motor; voltage and current. With a voltage ervo amplifier the motor voltage i related to the command voltage by the caling factor or gain V a: V M V V he model of the voltage ervo amplifier and motor from a command voltage and diturbance in to the rate out can then be expreed a: V rate e e V he block diagram for the current feedback ervo amplifier configuration i hown in Figure. he current feedback loop i a high gain ervo loop that control the motor current. he amplifier GI ha very high gain and i effectively the caling contant mp/volt of the current amplifier.

29 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 9 of 44 Figure Motor Block iagram with urrent Feedback ervo mplifier with caling he tranfer function between a command voltage and diturbance in and the rate out i given by: GI GI V GI GI rate e e he gain of the current amplifier, GI, i choen uch that within loop bandwidth: e GI >> For thi condition, the tranfer function reduce to: V GI GI rate gain a GI i a high gain amplifier within the loop bandwidth: >> GI he final expreion can then be written a: V V rate Σ e V M - motor admittance motor torque contant oad rate Σ M I GI Σ - V back EMF contant

30 Feedback ontrol ytem Fundamental unam online continuing education coure With a current feedback ervo amplifier, the effect of the back EMF and motor admittance i nearly eliminated and the relationhip expreed by the ervo amplifier caling contant. 9. oition and Velocity Feedback enor: ypical rotary poition feedback enor are reolver and encoder. For rotary application, a feature of both thee device i that they can be mounted directly on the drive haft and are capable of providing a full 36 continuou meaurement range. here are everal unit of meaure that can be ued when decribing angular poition, degree, radian, arc-minute or arc-econd. here are 6 arc-minute per degree and /8 radian per degree which equate to /8*6 radian per arc-minute. here are 6 arc-econd per arc-minute or therefore /8*36 radian per arc-econd 4.85* -6. degree are a familiar unit of meaure, thi will be ued in all example, however actual application often ued thee other unit. Reolver provide abolute angular poition with electrical error ranging from. to a low a - arc-econd. ccuracy depend upon ize and angular range. Reolver conit of a tator and rotor, imilar to a motor. hi contruction make them very robut in the preence of hock and vibration diturbance and reduce their enitivity to temperature variation when compared to encoder. he reolver tator conit of two winding poitioned at right angle to each-other. he rotor ha a third winding that i energized with a inuoidal ignal and rotate relative to the tator. he ignal in the rotor winding induce a ignal in both tator winding whoe magnitude varie a a function of the rotation angle. he voltage induced in one tator winding i in quadrature to the voltage in the other winding. Quadrature refer to the 9 o phae relationhip between the ignal of the two winding. he output ratio of the two tator winding ignal i proportional to the abolute angular poition arc-tangent of the ratio. n electronic circuit, the reolver to digital converter R, i normally ued to convert the conditioned reolver ignal to a digital output that can be read by a computer. Encoder are fabricated in both optical and magnetic configuration. he mot baic encoder contain a rotating dik that interrupt tranmiion between a tranmitter and a receiver. With an optical encoder, the optical ource i typically a ight Emitting iode E and the receiver a photo detector. he dik ha coded pattern of tranparent and opaque ector that interrupt the light meaured by the photo detector. he number of count pule obtained a the dik rotate i a meaure of the angular poition. With a magnetic encoder, the E i replaced with a magnet, the photo detector by a magnetic pick-up ening element, and the dik i replaced by a rotating dik, imilar to a gear, made of ferrou metal. the gear rotate, the teeth diturb the magnetic flux emitted by the permanent magnet cauing the flux field to expand and collape. opyright 4 eter ennedy age 3 of 44

31 Feedback ontrol ytem Fundamental unam online continuing education coure hee change are again ened a pule by the magnetic pick-up detector. he implet and leat expenive encoder are incremental. hee device do not provide an abolute meaure of angular poition a with the reolver. hey have to be homed periodically to obtain a etimate of abolute poition and will loe the poition reference upon power reet. he mot common type of incremental encoder ha two output channel that ene poition. wo code track, referred to a channel and channel B, have ector poitioned 9 deg out of phae that provide quadrature ignal that can be ued to detect poition and direction of rotation. Each channel provide N count per revolution uually a quare wave level. hifting the ignal from the two channel by a quarter of a cycle enable the direction of rotation to be determined baed upon which channel i leading the other. he hift alo increae reolution a each encoder cycle can be divided into four quarter each called a quadrature count. n encoder with N cycle per revolution produce 4N quadrature count per revolution. Reolution le an arc-econd can be obtained with interpolation electronic/oftware. Optical encoder normally ue a gla ubtrate for the rotating dik, which make them le deirable for high diturbance environment. here alo are abolute encoder which are normally larger and have more complex dik pattern containing 4-6 track. hey generate a unique word i.e. B or gray code for every angular poition of the haft. Magnetic encoder alo come in both abolute and incremental verion. hey normally have le reolution than optical encoder but are more robut in a high diturbance environment. Other type of poition enor ometime ued for rotary angular poition meaurement include potentiometer, rotary and linear variable differential tranformer. However thee device normally provide only a limited angle meaurement range a oppoed to the reolver or encoder. hey do have one advantage in that they can generate a voltage directly proportional to angle requiring only a cale factor for converion. Velocity or angular rate can be meaured or derived from poition. Many reolver R card a well a encoder converion circuit can provide an etimate of velocity. imilarly it can be determined by differentiation of the poition ignal. If performed digitally at a high ample rate relative to the bandwidth of the rate ignal; a good rate etimate can often be generated. tachometer can alo be ued to meaure rate. he analog tachometer reemble a mall motor however the gauge of the wire i very fine. It i effectively a motor in revere. the haft i turned by a motor it generate a voltage proportional to the angular rate requiring only a cale factor for converion. ll the device decribed have an aociated frequency dependent bandwidth which i a function of the enor a well a the converion proce. In a detailed deign thi mut be accounted for. If the bandwidth of the feedback enor i high relative to the bandwidth of the feedback control loop, it will have minimal impact on the repone. If not, lead compenation may be required. dicued, each device ha a unique method of meauring angle and the actual output may be opyright 4 eter ennedy age 3 of 44

32 Feedback ontrol ytem Fundamental unam online continuing education coure volt, count, etc. However the feedback path i caled uch that it provide a ignal in the ame unit a the command input and i therefore a unity feedback path. In the example that follow, the poition feedback enor i aumed to be an ideal enor with effectively infinite bandwidth and caled to unity feedback gain. 9.3 Motion ontrol Feedback oop nalyi: poition feedback control loop i hown in Figure. It ue a current feedback ervo amplifier. o md In Σ - ontrol ompenator ervo mplifier Motor M Σ J oad rate po R F RE oition Feedback caling oition enor Figure Block iagram for a Motion ontrol Feedback oop with urrent Feedback ervo mp he contant and control compenator tranfer function are given a:. in-lb/amp.5 amp/volt *pi*4***pi*6**pi*6/*pi*6 volt/deg /J* ; J. in-lb-ec R *FRE deg/deg per aumption of ideal enor J/* hi i a lead compenator with a zero at 6 Hz to hape the loop and a roll off pole at 6 Hz for noie uppreion. here are already two integrator in the loop due to the phyical dynamic of converting acceleration to poition; which alo mean thi i a ype ervo loop. here are two caling contant, one in the forward path and the other in the feedback path. he caling contant in the feedback path provide for unity gain feedback or R*FRE. he caling contant in the forward path cancel the motor drive and plant caling parameter and allow the opyright 4 eter ennedy age 3 of 44

33 Feedback ontrol ytem Fundamental unam online continuing education coure control compenator to effectively et the loop bandwidth in the analyi. he open loop tranfer function i given by: OF J J It can be een from the Bode plot in Figure 3 that the gain cro over frequency i at ~4 Hz. he phae margin i about 54 with an infinite gain margin. loed loop repone hown olid green line, with unity gain to ~4 Hz and ome minor cloed loop peaking. he -3 db BW i 6 Hz. cloed loop repone phae margin Figure 3 Bode plot for Motion ontrol Feedback oop Example he compliance plot of a diturbance acceleration or torque a /J input to the control loop acceleration output i hown in Figure 4 and effectively provide -6 db of rejection at Hz. opyright 4 eter ennedy age 33 of 44

34 Feedback ontrol ytem Fundamental unam online continuing education coure Figure 4 orque ompliance lot for Motion ontrol Example he ame ytem i analyzed, however thi time a velocity or rate feedback loop i included in the deign. he rate loop i the inner loop while the actual poitioning loop i the outer loop; a hown in Figure 5. o md In rate cmd M Σ Σ R Σ - - oition ontrol Rate ontrol ervo Motor ompenator ompenator mplifier J oad rate po F Rate oop Rate Feedback caling Rate enor R F RE oition oop Figure 5 Motion ontrol with oition and Rate Feedback he contant are the ame a for the previou deign however the control compenator tranfer function have been modified a dp-degree per econd:. in-lb/amp oition Feedback caling oition enor opyright 4 eter ennedy age 34 of 44

35 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 35 of 44.5 amp/volt R*pi****pi*4**pi*35/[**pi*4] volt/dp J/* *pi***pi*6**pi*/[**pi*6] volt/dp /J* ; J. in-lb-ec *F dp/dp per aumption of ideal enor R*FRE deg/deg per aumption of ideal enor Both compenator now have the form of proportional plu integral compenator cacaded with a low pa roll off filter. he tranfer function for the inner rate loop from the rate command input to the rate output i given by: ; where J J ROM J J RF ROM cmd rate RF J J cmd rate J J rate R R R R R R he rate open loop tranfer function i given by: J J ROF R Bode plot of the rate open loop gain and phae i hown in Figure 6.

36 Feedback ontrol ytem Fundamental unam online continuing education coure phae margin Figure 6 Bode lot for Example Inner Rate oop he gain croover frequency i Hz, the M56.8 and the GM i infinite; effectively thi hould be a well behaved loop with a fat repone. he cloed loop repone i hown in green. he outer poition loop can be derived from the rate loop noting that rate *po and rate cmd *po cmd in po. Uing thee relationhip the expreion for the poition loop can be expreed a: o md In po po RF or olving for po RF po o md RF he poition open loop tranfer function i given by: ROM ROM In RF opyright 4 eter ennedy age 36 of 44

37 Feedback ontrol ytem Fundamental unam online continuing education coure OF RF 6 RF 6 It can be een from Figure 6 that over the poition loop bandwidth the cloed rate loop magnitude i ~, however there i a mall phae lag contribution from thi loop. he Bode plot for the poition loop i hown in Figure 7 which indicate the gain croover frequency i 4 Hz, the M i 54, the GM i 3 db, and the phae croover i 9 Hz. phae margin -8 gain margin Figure 7 Bode plot for outer poition loop he expreion for the torque compliance i obtained a: ROM ROF ROM OM RF ROF Noting that ROM J ROF he expreion for the compliance can be expreed a: opyright 4 eter ennedy age 37 of 44

38 Feedback ontrol ytem Fundamental unam online continuing education coure OM J ROF here are everal advantage to the inner/outer loop deign. he inner rate loop provide a damping effect for the poition loop repone. If poitioning i implemented with a enor uch a a camera a imple model being the outer loop umming junction to track a target, the loop i referred to a the track loop. Uing the inner/outer loop architecture allow the track enor or camera to operate over lower bandwidth reducing noie on the track ignal while the higher bandwidth rate loop primarily reject diturbance. ctually the effective gain and thereby diturbance rejection i a function of the product of the open loop gain for both loop, but the inner rate loop gain normally dominate. he compliance curve for thi deign i hown in Figure 8. omparing the rejection at Hz with the compliance for the firt example hown in Figure 4, thi deign provide ~5x 6 db v 3 db the rejection of the ingle loop deign. Figure 8 orque ompliance for Motion ontrol Example igital ontrol: he majority of control algorithm today are implemented in an embedded or digital ignal proceor. ignal to be meaured are ampled at a defined rate and command to control the plant or load are tranmitted at the ame rate. he motion control example dicued previouly will be ued to illutrate the digital deign procedure. hi ample data control configuration i illutrated for the Motion ontrol Example configuration in Figure 9. he block diagram i modified to include ample function which are analog to digital converter or digital to analog converter that ample ignal or generate command at a opyright 4 eter ennedy age 38 of 44

39 Feedback ontrol ytem Fundamental unam online continuing education coure pecified Δ time interval. Mathematically thee device can be modeled a zero order hold ZOH function which ample an input or apply an output at a pecified time increment; holding that value until the next time increment. Inherent with thi approach i internal ampling within the proceor. internal ampling Δ Σ o md Inz - omputer Δ z ontrol ompenator ervo mplifier Motor M Σ J oad rate po Δ R F RE oition Feedback caling oition enor Figure 9 irect O tabilization ontrol onfiguration with a igital ontroller With ampling, the infinite continuou frequency pectrum i mapped to a pectrum whoe maximum frequency i half the ampling frequency per Nyquit theorem. o the compenator mut be converted to a digital format; tranforming it from a continuou frequency dependent function into the ampled frequency pectrum limited at half the ampling frequency. Many textbook are available on the theory of ample-data ytem a i an excellent uncam oure EE 6 onverting Feedback ytem from nalog to igital ontrol. he z-tranform i often ued to repreent the frequency repone of tranfer function in the ample data pectrum. he bilinear tranformation with frequency pre-warping i one method for tranforming a tranfer function in the -domain to one in the z-domain. For a pecified continuou tranfer function G, the dicrete equivalent can be determined from: Gz G z z Frequency pre-warping trie to compenate for the frequency ditortion that occur when mapping the infinite continuou frequency pectrum into a ample limited frequency pectrum uing thi tranformation. ritical frequencie in the continuou domain are modified a: f ampled tan fcontinuou an example, the motion control example compenator wa given by: J 6 opyright 4 eter ennedy age 39 of 44

40 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 4 of 44 he ampling frequency i aumed to be 4 Hz Δ.5 ec. here are three critical frequencie, when pre-warped become: Becaue of the high ample rate relative to the critical frequencie, none really change ignificantly from their continuou frequency equivalent. In the z-domain uing the bilinear tranform the compenator i given by: ; ; with z z J z z z β α β α For digital implementation, thi can be expreed a: z z z β α Finally in digital format, for a compenator input X at ample interval k, the output Y at k can be expreed a: k X k X k Y k Y α α he Bode plot for the ampled ytem i hown in Figure and can be compared to Figure 3 for the continuou time verion. ue to the ampling, the phae margin i reduced to 5 while the gain margin i no longer infinite but ~8 db.

41 Feedback ontrol ytem Fundamental unam online continuing education coure phae margin gain margin Figure Bode plot for ampled Verion of Motion control Example he motion feedback control loop for example with rate plu poition feedback, implemented a a ampled controller, i hown in Figure. internal ampling Δ Σ o md Inz - internal ampling Δ z Σ rate cmd - oition ontrol ompenator omputer Δ Rz Rate ontrol ompenator ervo mplifier Motor M Σ J oad rate po Δ F Rate oop Rate Feedback caling R Δ oition Feedback caling Rate enor FRE oition enor oition oop Figure ample ata Motion ontrol with oition and Rate Feedback he rate open loop tranfer function and control compenator were given by: opyright 4 eter ennedy age 4 of 44

42 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 4 of where J J ROF R R Uing the ame bilinear tranformation with frequency pre-warping a decribed for the lat example and the ampling frequency of 4 Hz Δ.5 ec the three critical frequencie, when pre-warped are: gain becaue of the high ample rate relative to the critical frequencie, only the frequency at 4 Hz changed lightly from their continuou frequency equivalent. In the z-domain uing the bilinear tranform, the compenator i expreed a: ; ; with R R R R R R z z R R J z z z z z β α β α he Bode plot for the rate feedback loop i hown in Figure. he phae margin i ~48 and the gain margin db. he plot of the higher bandwidth loop alo how the effect of the 4 Hz ample function.

43 Feedback ontrol ytem Fundamental unam online continuing education coure opyright 4 eter ennedy age 43 of 44 phae margin gain margin -8 ampler hold function cloed loop Figure Bode plot for ample ata Rate Feedback oop he compenator for the poition feedback loop wa given by: 6 6 he ame proce i followed again with: he poition compenator in the z-domain can then be expreed a: ; ; with z z z z z z z β α β α

44 Feedback ontrol ytem Fundamental unam online continuing education coure he Bode plot for the poition plu rate feedback control i hown in Figure 3. he phae margin i 36 and gain margin.5 db. here i alo ignificant cloed loop peaking. dditional compenation hould be applied to the loop to improve the repone. phae margin -8 cloed loop peaking gain margin Figure 3 Bode plot for ample ata Motion ontrol ytem; oition and Rate Feedback oop ummary: hi complete coure material. he topic covered began with a decription of the baic block diagram in ection. he relationhip between time and frequency domain repreentation of the block diagram element were dicued in ection 3 followed by the key feedback relationhip derived from the block diagram algebra in ection 4. ontrol loop tability and method to determine tability margin were decribed in ection 5 followed by a dicuion of pecifying control loop performance in ection 6. couple of control loop deign method were provided in ection 7. he baic theory wa then applied to two example; a home heating ytem in ection 8 and motion control application in ection 9. onverting to a digital ample data controller wa dicued in ection ; a related to the motion control example in ection 9. opyright 4 eter ennedy age 44 of 44