Control Engineering An introduction with the use of Matlab

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2 Derek Atherton An introdution with the ue of Matlab

3 : An introdution with the ue of Matlab nd edition 3 Derek Atherton ISBN

4 Content Content Prefae 9 About the author Introdution What i? Content of the Book 3 Referene 4 Mathematial Model Repreentation of Linear Dynamial Sytem 5 Introdution 5 The Laplae Tranform and Tranfer Funtion 6 3 State pae repreentation 9 4 Mathematial Model in MATLAB 5 Interonneting Model in MATLAB 4 6 Referene 6 4 Clik on the ad to read more

5 Content 3 Tranfer Funtion and Their Repone 7 3 Introdution 7 3 Step Repone of Some Speifi Tranfer Funtion 9 33 Repone to a Sinuoid 35 4 Frequeny Repone and Their Plotting 38 4 Introdution 38 4 Bode Diagram Nyquit Plot Nihol Plot 47 5 The Bai Feedbak Loop 48 5 Introdution 48 5 The Cloed Loop Sytem Speifiation 5 54 Stability 54 6 More on Analyi of the Cloed Loop Sytem 58 6 Introdution 58 6 Time Delay 58 5

6 Content 63 The Root Lou 6 64 Relative Stability M and N Cirle 66 7 Claial Controller Deign 68 7 Introdution 68 7 Phae Lead Deign Phae Lag Deign PID Control Referene 89 8 Parameter Optimiation for Fixed Controller 9 8 Introdution 9 8 Some Simple Example 9 83 Standard Form Control of an Untable Plant Further Comment 86 Referene 6

7 Content 9 Further Controller Deign Conideration 9 Introdution 9 Lag-Lead Compenation 93 Speed Control 5 94 Poition Control 6 95 A Tranfer Funtion with Complex Pole 7 96 The Effet of Parameter Variation 97 Referene 7 State Spae Method 8 Introdution 8 Solution of the State Equation 8 3 A State Tranformation 4 State Repreentation of Tranfer Funtion 5 State Tranformation between Different Form 6 6 Evaluation of the State Tranition Matrix 7 7 Controllability and Obervability 8 8 Caade Connetion 3 7

8 Content Some State Spae Deign Method 3 Introdution 3 State Variable Feedbak 3 3 Linear Quadrati Regulator Problem 34 4 State Variable Feedbak for Standard Form 35 5 Tranfer Funtion with Complex Pole 4 Appendix A 4 3 Appendix B 48 4 Appendix C 49 5 Appendix D 5 8

9 Prefae Prefae It i almot four year ine the firt edition of thi book o it eemed appropriate to reread it arefully again and make any uitable hange Alo during the intervening period I have added two further bookboon book one on An Introdution to Nonlinearity in Control Sytem and another very reently on Control Engineering Problem with Solution Thi later book ontain worked example and ome problem with anwer only, whih over the material in thi book and An Introdution to Nonlinearity in Control Sytem It i hoped that the relevant hapter of Problem with Solution will help the reader gain a better undertanding and deeper knowledge of the topi overed in thi textbook Minor hange have been made to thi eond edition mainly with repet to a few hange in wording, but adly depite repeated reading a few minor tehnial error were found and orreted, for whih I apologie Thee were Figure 36 whih had ome inorret marking and wa not very lear due to the number hoen giving line almot on top of eah other Thi ha been orreted by hooing a different frequeny for illutrating the frequeny repone alulation proedure Further, ome negative ign were omitted from equation (4), the unit of H on page 5 were given inorretly a were the ubript on the a and a matrix in the material in etion 5, page 3, on tranforming to the ontrollable anonial form Finally the over page ha been hanged to ontain a piture whih i more relevant to the book Derek P Atherton Brighton, June 3 9

10 About the author About the author Profeor Derek P Atherton BEng, PhD, DS, CEng, FIEE, FIEEE, HonFIntMC, FRSA Derek Atherton tudied at the univeritie of Sheffield and Manheter, obtaining a PhD in 96 and DS in 975 from the latter He pent the period from 96 to 98 teahing in Canada where he erved on everal National Reearh Counil ommittee inluding the Eletrial Engineering Grant Committee He took up the pot of Profeor of at the Univerity of Suex in 98 and i urrently retired but ha an offie at the univerity, give ome leture, and ha the title of Emeritu Profeor and Aoiate Tutor He ha been ative with many profeional engineering bodie, erving a Preident of the Intitute of Meaurement and Control in 99, Preident of the IEEE Control Sytem Soiety in 995, being the only non North Amerian to have held the poition, and a a member of the IFAC Counil from He erved a an Editor of the IEE Proeeding on Control Theory and Appliation (CTA) for everal year until 7 and wa formerly an editor for the IEE Book Serie He ha erved EPSRC on reearh panel and a an aeor for reearh grant for many year and alo erved a a member of the Eletrial Engineering Panel for the Reearh Aement Exerie in 99 Hi major reearh interet are in non-linear ontrol theory, omputer aided ontrol ytem deign, imulation and target traking He ha written two book, i a o-author of two other and ha publihed more than 35 paper in Journal and Conferene Proeeding Profeor Atherton ha given invited leture in many ountrie and upervied over 3 Dotoral tudent Derek P Atherton February 9

11 Introdution Introdution What i? A it name implie ontrol engineering involve the deign of an engineering produt or ytem where a requirement i to aurately ontrol ome quantity, ay the temperature in a room or the poition or peed of an eletri motor To do thi one need to know the value of the quantity being ontrolled, o that being able to meaure i fundamental to ontrol In priniple one an ontrol a quantity in a o alled open loop manner where knowledge ha been built up on what input will produe the required output, ay the voltage required to be input to an eletri motor for it to run at a ertain peed Thi work well if the knowledge i aurate but if the motor i driving a pump whih ha a load highly dependent on the temperature of the fluid being pumped then the knowledge will not be aurate unle information i obtained for different fluid temperature But thi may not be the only pratial apet that affet the load on the motor and therefore the peed at whih it will run for a given input, o if aurate peed ontrol i required an alternative approah i neeary Thi alternative approah i the ue of feedbak whereby the quantity to be ontrolled, ay C, i meaured, ompared with the deired value, R, and the error between the two, E = R C ued to adjut C Thi give the laial feedbak loop truture of Figure In the ae of the ontrol of motor peed, where the required peed, R, known a the referene i either fixed or moved between fixed value, the ontrol i often known a a regulatory ontrol, a the ation of the loop allow aurate peed ontrol of the motor for the aforementioned ituation in pite of the hange in temperature of the pump fluid whih affet the motor load In other intane the output C may be required to follow a hanging R, whih for example, might be the required poition movement of a robot arm The ytem i then often known a a ervomehanim and many early textbook in the ontrol engineering field ued the word ervomehanim in their title rather than ontrol Figure Bai Feedbak Control Struture

12 Introdution The ue of feedbak to regulate a ytem ha a long hitory [, ], one of the earliet onept, ued in Anient Greee, wa the float regulator to ontrol water level, whih i till ued today in water tank The firt automati regulator for an indutrial proe i believed to have been the flyball governor developed in 769 by Jame Watt It wa not, however, until the wartime period beginning in 939, that ontrol engineering really tarted to develop with the demand for ervomehanim for munition fire ontrol and guidane With the major improvement in tehnology ine that time the appliation of ontrol have grown rapidly and an be found in all walk of life Control engineering ha, in fat, been referred to a the uneen tehnology a o often people are unaware of it exitene until omething goe wrong Few people are, for intane, aware of it ontribution to the development of torage media in digital omputer where aurate head poitioning i required Thi tarted with the magneti drum in the 5 and i required today in dik drive where poition auray i of the order of µm and movement between trak mut be done in a few m Feedbak i, of oure, not jut a feature of indutrial ontrol but i found in biologial, eonomi and many other form of ytem, o that theorie relating to feedbak ontrol an be applied to many walk of life Content of the Book The book i onerned with theoretial method for ontinuou linear feedbak ontrol ytem deign, and i primarily retrited to ingle-input ingle-output ytem Continuou linear time invariant ytem have linear differential equation mathematial model and are alway an approximation to a real devie or ytem All real ytem will hange with time due to age and environmental hange and may only operate reaonably linearly over a retrited range of operation There i, however, a rih theory for the analyi of linear ytem whih an provide exellent approximation for the analyi and deign of real world ituation when ued within the orret ontext Further, imulation i now an exellent mean to upport linear theoretial tudie a model error, uh a the affet of negleted nonlinearity, an eaily be aeed There are a total of hapter and ome appendie, the major one being Appendix A on Laplae tranform The next hapter provide a brief deription of the form of mathematial model repreentation ued in ontrol engineering analyi and deign It doe not deal with mathematial modelling of engineering devie, whih i a huge ubjet and i bet dealt with in the diipline overing the ubjet, ine the devie or omponent ould be eletrial, mehanial, hydrauli et Suffie to ay that one hope to obtain an approximate linear mathematial model for thee omponent o that their effet in a ytem an be invetigated uing linear ontrol theory The mathematial model diued are the linear differential equation, the tranfer funtion and a tate pae repreentation, together with the notation ued for them in MATLAB

13 Introdution Chapter 3 diue tranfer funtion, their zero and pole, and their repone to different input The following hapter diue in detail the variou method for plotting teady tate frequeny repone with Bode, Nyquit and Nihol plot being illutrated in MATLAB Hopefully uffiient detail, whih i brief when ompared with many textbook, i given o that the reader learly undertand the information thee plot provide and more importantly undertand the form of frequeny repone expeted from a peifi tranfer funtion The material of hapter 4 ould be overed in other oure a it i bai ytem theory, there having been no mention of ontrol, whih tart in hapter 5 The bai feedbak loop truture hown in Figure i ommented on further, followed by a diuion of typial performane peifiation whih might have to be met in both the time and frequeny domain Steady tate error are onidered both for input and diturbane ignal and the importane and propertie of an integrator are diued from a phyial a well a mathematial viewpoint The hapter onlude with a diuion on tability and a preentation of everal reult inluding the Mikhailov riterion, whih i rarely mentioned in Englih language text Chapter 6 firt introdue the propertie of a time delay before ontinuing with further material relating to the analyi and propertie of the loed loop Briefly mentioned are the root lou and it plotting uing MATLAB and variou onept of relative tability Thee inlude gain and phae margin, enitivity funtion, and M and N irle Chapter 7 i a relatively long hapter dealing with laial ontroller deign method The bai onept of laial ontrol deign i that one deide on a uitable ontrol trategy and then the deign problem beome one of obtaining appropriate parameter for the ontroller element in order to meet peified ontrol performane objetive Typially a ontroller with a peified truture i plaed in either the forward or feedbak path, or even both, of the loed loop The firt point diued i therefore the differene between a feedforward and a feedbak ontroller on the loed loop tranfer funtion The deign of lead and lag ontroller i then diued followed by a long etion on PID ontrol, a topi on whih far too muh ha probably been written in the literature in reent year due in no part to it extenive ue in pratie The early work of Ziegler and Nihol i the tarting point whih largely foue on the ontrol of a plant with a time ontant plu time delay By dealing with thi plant in o alled normalied form, where it behaviour i expreible in term of the time delay to time ontant ratio, new reult are preented omparing variou uggeted parameter etting, uually known a tuning, for PID ontroller It i pointed out that if a mathematial model i obtained for the plant then the priniple and poibilitie for obtaining parameter for a PID ontroller are no different to thoe whih may be ued for any other type of ontroller However a major ontribution of Ziegler and Nihol in their loop yling method wa to how how the PID ontroller parameter might be hoen without a mathematial model, but imply from knowledge of the o alled plant tranfer funtion ritial point, namely the magnitude and frequeny of the tranfer funtion for 8 phae hift It modern equivalent i known a relay autotuning and thi topi i overed in ome detail at the end of the hapter 3

14 Introdution The ontroller deign onept preented in the previou hapter baed on open loop frequeny repone ompenation were regularly ued in the early day of ontrol engineering by deigner who were adept at kething Bode diagram, o that the ue of modern oftware ha imply brought more effiieny to the deign proe Some ignifiant theoretial work on optimiing ontroller parameter to meet peifi performane riteria wa alo done in the early day but here the limitation wa the diffiulty of uing the theory to obtain reult of ignifiane With modern omputation tool numerial approahe an be ued to olve thee problem either by writing MATLAB program baed on linear ytem theory or writing optimiation program around digital imulation in program uh a SIMULINK Thee are appropriate indutrial deign method whih appear to reeive little attention in textbook, poibly beaue they are not uitable for traditional examination Chapter 8 over parameter optimiation baed on integral performane riteria beaue it allow ome imple reult to be obtained and onept undertood Further it lead to a deign approah baed on loed loop tranfer funtion ynthei, known a tandard form, preented at the end of the hapter Chapter 9 diue further apet of laial ontroller deign and highlight the diffiulty of trying to deign erie ompenator for, o alled unertain plant, plant whoe parameter may vary or not be aurately known Thi lead to onideration of ome elegant reent reult on unertain plant but whih unfortunately appear too onervative for pratial ue in many intane The final two hapter are onerned with the ue of tate pae method in ontrol ytem analyi and deign Chapter provide bai overage of tate pae onept overing tate equation and their olution, tate tranformation, tate repreentation of tranfer funtion, and ontrollability and obervability Some tate pae deign method are overed in Chapter, inluding tate variable feedbak, LQR deign and tate variable feedbak deign to ahieve the loed loop tandard form of hapter 8 3 Referene Bennett, S A hitory of, 8 93 IEE ontrol engineering erie Peter Peregrinu, 979 Bennett, S A hitory of, IEE ontrol engineering erie Peter Peregrinu, 993 4

15 Mathematial Model Repreentation of Linear Dynamial Sytem Mathematial Model Repreentation of Linear Dynamial Sytem Introdution Control ytem exit in many field of engineering o that omponent of a ontrol ytem may be eletrial, mehanial, hydrauli et devie If a ytem ha to be deigned to perform in a peifi way then one need to develop deription of how the output of the individual omponent, whih make up the ytem, will reat to hange in their input Thi i known a mathematial modelling and an be done either from the bai law of phyi or from proeing the input and output ignal, in whih ae it i known a identifiation Example of phyial modelling inlude deriving differential equation for eletrial iruit involving reitane, indutane and apaitane and for ombination of mae, pring and damper in mehanial ytem It i not the intent here to derive model for variou devie whih may be ued in ontrol ytem but to aume that a uitable approximation will be a linear differential equation In pratie an improved model might inlude nonlinear effet, for example Hooke Law for a pring in a mehanial ytem i only linear over a ertain range; or aount for time variation of omponent Mathematial model of any devie will alway be approximate, even if nonlinear effet and time variation are alo inluded by uing more general nonlinear or time varying differential equation Thu, it i alway important in uing mathematial model to have an appreiation of the ondition under whih they are valid and to what auray Starting therefore with the aumption that our model i a linear differential equation then in general it will have the form:- A(D)y(t) = B(D)u(t) () where D denote the differential operator d/dt A(D) and B(D) are polynomial in D with D i = d i / dt i, the i th derivative, u(t) i the model input and y(t) it output So that one an write n A( D) = D a a D a () n n n D an D m B( D) = D b D b (3) m m m D bm D b where the a and b oeffiient will be real number The order of the polynomial A and B are aumed to be n and m, repetively, with n m 5

16 Mathematial Model Repreentation of Linear Dynamial Sytem Thu, for example, the differential equation (4) with the dependene of y and u on t aumed an be written ( D 4D 3) y = (D ) u (5) In order to olve an n th order differential equation, that i determine the output y for a given input u, one mut know the initial ondition of y and it firt n- derivative For example if a projetile i falling under gravity, that i ontant aeleration, o that D y= ontant, where y i the height, then in order to find the time taken to fall to a lower height, one mut know not only the initial height, normally aumed to be at time zero, but the initial veloity, dy/dt, that i two initial ondition a the equation i eond order (n = ) Control engineer typially tudy olution to differential equation uing either Laplae tranform or a tate pae repreentation The Laplae Tranform and Tranfer Funtion A hort introdution to the Laplae tranformation i given in Appendix A for the reader who i not familiar with it ue It i an integral tranformation and it major, but not ole ue, i for differential equation where the independent time variable t i tranformed to the omplex variable by the expreion 6

17 Mathematial Model Repreentation of Linear Dynamial Sytem (6) Sine the exponential term ha no unit the unit of are eond -, that i uing mk notation ha unit of - - If denote the Laplae tranform then one may write [f(t)] = F() and [F()] = f(t) The relationhip i unique in that for every f(t), [F()], there i a unique F(), [f(t)] It i hown in Appendix A that when the n- initial ondition, D n- y() are zero the Laplae tranform of D n y(t) i n Y() Thu the Laplae tranform of the differential equation () with zero initial ondition an be written A ( ) Y ( ) = B( ) U ( ) (7) or imply A ( ) Y = B( ) U (8) with the aumed notation that ignal a funtion of time are denoted by lower ae letter and a funtion of by the orreponding apital letter If equation (8) i written Y ( ) U ( ) B( ) = = G( ) (9) A( ) then thi i known a the tranfer funtion, G(), between the input and output of the ytem, that i whatever i modelled by equation () B(), of order m, i referred to a the numerator polynomial and A(), of order n, a the denominator polynomial and are from equation () and (3) n A( ) = a a a a n n n n () m B( ) = b b () m m m bm b Sine the a and b oeffiient of the polynomial are real number the root of the polynomial are either real or omplex pair The tranfer funtion i zero for thoe value of whih are the root of B(), o thee value of are alled the zero of the tranfer funtion Similarly, the tranfer funtion will be infinite at the root of the denominator polynomial A(), and thee value are alled the pole of the tranfer funtion The general tranfer funtion (9) thu ha m zero and n pole and i aid to have a relative degree of n-m, whih an be hown from phyial realiation onideration annot be negative Further for n > m it i referred to a a tritly proper tranfer funtion and for n m a a proper tranfer funtion 7

18 Mathematial Model Repreentation of Linear Dynamial Sytem When the input u(t) to the differential equation of () i ontant the output y(t) beome ontant when all the derivative of the output are zero Thu the teady tate gain, or ine the input i often thought of a a ignal the term d gain (although it i more often a voltage than a urrent!) i ued, and i given by G ( ) = b a () / If the n root of A() are α i, i = n and of B() are β j, j = m, then the tranfer funtion may be written in the zero-pole form G( ) K m j= = n i= ( β ) ( α ) i j (3) where in thi ae G() K m j= = n i= β α i j (4) When the tranfer funtion i known in the zero-pole form then the loation of it zero and pole an be hown on an plane zero-pole plot, where the zero are marked with a irle and the pole by a ro The information on thi plot then ompletely define the tranfer funtion apart from the gain K In mot intane engineer prefer to keep any omplex root in quadrati form, thu for example writing (5) rather than writing ( 5 j866)( 5 j866) for the quadrati term in the denominator Thi tranfer funtion ha K = 4, a zero at -, three pole at -, -5 ± 866 repetively, and the zero-pole plot i hown in Figure Figure Zero-pole plot 8

19 Mathematial Model Repreentation of Linear Dynamial Sytem 3 State pae repreentation Conider firt the differential equation given in equation (4) but without the derivative of u term, that i (6) To olve thi equation, a mentioned earlier, one mut know the initial value of y and dy/dt, or put another way the initial tate of the ytem Let u hooe therefore to repreent y and dy/dt by x and x the omponent of a tate vetor x of order two Thu we have x = x, by hoie, and from ubtitution in the differential equation x 4x x u The two equation an be written in the matrix form = 3 x = x u (7) 3 4 and the output y i imply, in thi ae, the tate x and an be written y = ( )x (8) 9

20 Mathematial Model Repreentation of Linear Dynamial Sytem For thi hoie of tate vetor the repreentation i often known a the phae variable repreentation The olution for no input, that i u =, from an initial tate an be plotted in an x -x plane, known a a phae plane with time a parameter on the olution trajetory Equation (7) i a tate equation and (8) an output equation and together they provide a tate pae repreentation of the differential equation or the ytem deribed by the differential equation Sine thi ytem ha one input, u, and one output, y, it i often referred to a a ingle-input ingleoutput (SISO) ytem The hoie of the tate variable x i not unique and more will be aid on thi later, but the point i eaily illutrated by onidering the imple R-C iruit in Figure If one derive the differential equation for the output voltage in term of the input voltage, it will be a eond order one imilar to equation (6) and one ould hooe a in that equation the output, the apaitor voltage, and it derivative a the omponent of the tate variable, or imply the tate, to have a repreentation imilar to equation (7) From a phyial point of view, however, any initial non zero tate will be due to harge tored in one or both of the two apaitor and therefore it might be more appropriate to hooe the voltage of thee two apaitor a the tate Figure Simple R-C iruit In the tate pae repreentation of (7) and (8) x i the ame a y o that for the tate equation (8) the tranfer funtion between U() and X () i obviouly X ( ) U ( ) = 4 3 (9) That i x replaing y in the tranfer funtion orreponding to the differential equation (6) Now the tranfer funtion orreponding to equation (5) i Y ( ) U ( ) = 4 3 () whih an be written a Y ( ) = X U ( ) () X () ()

21 Mathematial Model Repreentation of Linear Dynamial Sytem Sine in our tate repreentation x x =, whih in tranform term i X () X (), thi mean in thi ae with the ame tate equation the output equation i now y = x x Thu a tate pae repreentation for equation (5) i u x x = 4 3, ( )x y = () It i eay to how that for the more general ae of the differential equation () a poible tate pae repreentation, whih i known a the ontrollable anonial form, illutrated for m < n-, i u x a a a a x n = (3) ( )x b b b y m = (4) In matrix form the tate and output equation an be written (5) where the tate vetor, x, i of order n, the A matrix i nxn, B i a olumn vetor of order, n, and C i a row vetor of order, n Beaue B and C are vetor for the SISO ytem they are often denoted by b and T, repetively Alo in the ontrollable anonial form repreentation given above the A matrix and B vetor take on peifi form, the former having the pole polynomial oeffiient in the lat row and the latter being all zero apart from the unit value in the lat row If m and n are of the ame order, for example if they are both and the orreponding tranfer funtion i , then thi an be written a 3 4 3, whih mean there i a unit gain diret tranmiion between input and output, then the tate repreentation take the more general form (6) where D i a alar, being unity of oure in the above example A tate pae repreentation an be ued for a mathematial model of a ytem with multiple input and output, denoted by MIMO, and in thi ae B, C and D will be matrie of appropriate dimenion whih aount for the ue of apital letter

22 Mathematial Model Repreentation of Linear Dynamial Sytem Thu, in onluion, a mathematial model of a linear SISO dynamial ytem may be a differential equation, a tranfer funtion or a tate pae repreentation A tate pae repreentation ha a unique tranfer funtion but the revere i not the ae 4 Mathematial Model in MATLAB MATLAB, although not the only language with good failitie for ontrol ytem deign, i eay to ue and very popular A well a tool for analyi it alo ontain a imulation language, SIMULINK, whih i alo very ueful If it ha a weakne it i probably with regard to phyial modelling but for the ontent of thi book, where our tarting point i a mathematial model, thi i not a problem Model of ytem omponent an be entered into MATLAB either a tranfer funtion or tate pae repreentation A model i an objet defined by a ymbol, ay G, and it tranfer funtion an be entered in the form G=tf(num,den) where num and den ontain a tring of oeffiient deribing the numerator and denominator polynomial repetively MATLAB tatement in the text, uh a the above for G, will be entered in bold itali but not in program extrat uh a that below The oeffiient are entered beginning with the highet power of Thu the tranfer funtion G ( ) =, an be entered by typing:- 4 3 >>num=[ ]; >> den=[ 4 3];

23 Mathematial Model Repreentation of Linear Dynamial Sytem >> G=tf(num,den) Tranfer funtion: ^ 4 3 The >> i the MATLAB prompt and the emiolon at the end of a line uppree a MATLAB repone Thi ha been omitted from the expreion for G o MATLAB repond with the tranfer funtion G a hown Alternatively, the entry ould have been done in one expreion by typing:- >>G=tf([ ],[ 4 3]) The root of a polynomial an be found by typing root before the oeffiient tring in quare braket Thu typing:- >> root(den) an = -3 - Alternatively the tranfer funtion an be entered in zero, pole, gain form where the ommand i in the form G=zpk(zero,pole,gain) Thu for the ame example >> G=zpk([-5],[-;-3],) Zero/pole/gain: (5) () (3) where the value of zero or pole in a tring are eparated by a emiolon Alo to enter a tring with a ingle number, here ued for the value of K but not for the ingle zero, the quare braket may be omitted 3

24 Mathematial Model Repreentation of Linear Dynamial Sytem A tate pae model or objet formed from known A,B,C,D matrie, often denoted by (A,B,C,D),an be entered into MATLAB with the ommand G=(A,B,C,D) Thu for the ame example by entering the following ommand one define the tate pae model >> A=[,;-3,-4]; >> B=[;]; >> C=[,]; >> D=; >> G=(A,B,C,D); And aking afterward for the tranfer funtion of the model by typing >> tf(g) One obtain Tranfer funtion: ^ 4 3 Obviouly the above have been very imple example but hopefully they have overed the bai of putting the mathematial model of a linear dynamial ytem into MATLAB The only way to learn i by doing example and ine MATLAB ha an exellent help faility the reader hould not find thi diffiult For a more extenive overage of MATLAB routine and example of their ue in ontrol engineering the reader i referred to the book given in referene 5 Interonneting Model in MATLAB Control ytem are made up of everal omponent, o a well a deribing a omponent by a mathematial model, one need to deal with the mathematial model for interonneted omponent Typially a omponent i repreented a a blok with input and output ignal and labelled, uually with a tranfer funtion, ay G (), a hown in Figure 3 Stritly peaking if the blok i labelled with a tranfer funtion the input and output ignal hould alo be in the domain, a the blok in Figure 3 implie Y() = G ()U() (7) but it i uually aepted that the time domain notation, y(t) and u(t) for the ignal, may alo be ued 4

25 Mathematial Model Repreentation of Linear Dynamial Sytem Figure 3 Blok repreentation of a tranfer funtion When a eond blok, with tranfer funtion G (), i onneted to the output of the firt blok, to give a erie onnetion, then it i aumed that in making the onnetion of Figure 4 that the eond blok doe not affet the output of the firt one In thi ae the reultant tranfer funtion of the erie ombination between input u and output y i G ()G (), whih i obtained diretly by ubtitution from the individual blok relationhip X()=G ()U() and Y()=G ()X() where x i the output of the firt blok Figure 4 Serie (or aade) onnetion of blok If two ytem objet G and G are provided to MATLAB then the ytem objet orreponding to the erie ombination an be obtained by typing G=G *G 5

26 Mathematial Model Repreentation of Linear Dynamial Sytem If two tranfer funtion model, G () and G () are onneted in parallel, a hown in Figure 5, then the reultant tranfer funtion between the input u and output y i obtained from the relationhip X () = G ()U(), X () = G ()U() and Y() = X ()X () and i G ()G () It an be obtained in MATLAB by typing G=G G Figure 5 Parallel onnetion of blok Another onnetion of blok whih will be ued i the feedbak onnetion hown in Figure 6 For the negative feedbak onnetion of Figure 6 the relationhip i Y() = G()[U() H()Y()], where the expreion in the quare braket i the input to G() Thi an be rearranged to give a tranfer funtion between the input u and output y of Y ( ) U ( ) G( ) = (8) G( ) H ( ) If thi tranfer funtion i denoted by T() then the MATLAB ommand to obtain T() i T=feedbak(G,H) If the poitive feedbak onfiguration i required then the tatement T=feedbak(G,H,ign) an be ued where the ign = Thi an alo be ued for the negative feedbak with ign = - Figure 6 Feedbak onnetion of blok 6 Referene Xue D, Chen Y and Atherton DP Linear Feedbak Control: Analyi and Deign in MATLAB, Siam, USA, 7 6

27 Tranfer Funtion and Their Repone 3 Tranfer Funtion and Their Repone 3 Introdution A mentioned previouly a major reaon for wihing to obtain a mathematial model of a devie i to be able to evaluate the output in repone to a given input Uing the tranfer funtion and Laplae tranform provide a partiularly elegant way of doing thi Thi i beaue for a blok with input U() and tranfer funtion G() the output Y() = G()U() When the input, u(t), i a unit impule whih i onventionally denoted by δ(t), U() = o that the output Y() = G() Thu in the time domain, y(t) = g(t), the invere Laplae tranform of G(), whih i alled the impule repone or weighting funtion of the blok The evaluation of y(t) for any input u(t) an be done in the time domain uing the onvolution integral (ee Appendix A, theorem (ix)) (3) but it i normally muh eaier to ue the tranform relationhip Y() = G()U() To do thi one need to find the Laplae tranform of the input u(t), form the produt G()U() and then find it invere Laplae tranform G()U() will be a ratio of polynomial in and to find the invere Laplae tranform, the root of the denominator polynomial mut be found to allow the expreion to be put into partial fration with eah term involving one denominator root (pole) Auming, for example, the input i a unit tep o that U() = / then putting G()U() into partial fration will reult in an expreion for Y() of the form C Y ( ) = C n i i= α i (3) where in the tranfer funtion G() = B()/A(), the n pole of G() [zero of A()] are α i, i = n and the oeffiient C and C i, i = n, will depend on the numerator polynomial B(), and are known a the reidue at the pole Taking the invere Laplae tranform yield y( t) = C n i= C e αit i (33) 7

28 Tranfer Funtion and Their Repone The firt term i a ontant C, ometime written C u (t) beaue the Laplae tranform i defined for t, where u (t) denote the unit tep at time zero Eah of the other term i an exponential, whih provided the real part of α i i negative will deay to zero a t beome large In thi ae the tranfer funtion i aid to be table a a bounded input ha produed a bounded output Thu a tranfer funtion i table if all it pole lie in the left hand ide (lh) of the plane zero-pole plot illutrated in Figure The larger the negative value of α i the more rapidly the ontribution from the i th term deay to zero Sine any pole whih are omplex our in omplex pair, ay of the form α,α = σ ± jω, then the orreponding two reidue C and C will be omplex pair and the two term will ombine to give a term of the form Ce σt in(ωt φ) Thi i a damped oillatory exponential term where σ, whih will be negative for a table tranfer funtion, determine the damping and ω the frequeny [tritly angular frequeny] of the oillation For a peifi alulation mot engineer, a mentioned earlier, will leave a omplex pair of root a a quadrati fator in the partial fatorization proe, a illutrated in the Laplae tranform inverion example given in Appendix A For any other input to a table G(), a with the tep input, the pole of the Laplae tranform of the input will our in a term of the partial fration expanion (3), [a for the C / term above], and will therefore produe a bounded output for a bounded input 8

29 Tranfer Funtion and Their Repone 3 Step Repone of Some Speifi Tranfer Funtion In ontrol engineering the major determiniti input ignal that one may wih to obtain repone to are a tep, an impule, a ramp and a ontant frequeny input The purpoe of thi etion i to diu tep repone of peifi tranfer funtion, hopefully imparting an undertanding of what an be expeted from a knowledge of the zero and pole of the tranfer funtion without going into detailed mathemati 3 A Single Pole Tranfer Funtion K A tranfer funtion with a ingle pole i G() =, whih may alo be written in the o-alled time a ontant form G() =, where K= K /a and T = /a The teady tate gain G() = K, whih i the final value of the repone to a unit tep input, and T i alled the time ontant a it determine the peed of the repone K will have unit relating the input quantity to the output quantity, for example C/V, if the input i a voltage and the output temperature T will have the ame unit of time a -, normally eond The output, Y(), for a unit tep input i given by (34) Taking the invere Laplae tranform give the reult t / T y( t) = K( e ) (35) The larger the value of T (ie the maller the value of a), the lower the exponential repone It an eaily be hown that or in word, the output reahe 63% of the final value after a time T, the initial lope of the repone i T and the repone ha eentially reahed the final value after a time 5T The tep repone in MATLAB an be obtained by the ommand tep(num,den) The figure below how the tep repone for the tranfer funtion with K = on a normalied time ale 9

30 Tranfer Funtion and Their Repone Figure 3 Normalied tep repone for a ingle time ontant tranfer funtion 3 Two Complex Pole Here the tranfer funtion G() i often aumed to be of the form o G( ) = ω (36) ζω ω o o It ha a unit teady tate gain, ie G() =, and pole at, whih are omplex when ζ < For a unit tep input the output Y(), an be hown after ome algebra, whih ha been done o that the invere Laplae tranform of the eond and third term are damped oinuoidal and inuoidal expreion, to be given by (37) Taking the invere Laplae tranform it yield, again after ome algebra, y( t) = e ζω ot ζ in( ζ ω t ϕ) o where ϕ = o ζ ζ i known a the damping ratio It an alo be een that the angle to the negative / real axi from the origin to the pole with poitive imaginary part i tan ( ζ ) / ζ = o ζ = ϕ Meaurement of the angle φ and thi relationhip i often ued to refer to the damping of omplex pole even when not dealing with a eond order ytem The repone on the normalied time ale ω o t an be found from Matlab by taking ω o equal to one The damping of the repone then depend on ζ and the oillatory behaviour on the normalied damped frequeny, that i ω / ω = ζ Figure 3 how a normalied plot for everal value of ζ o (38) 3

31 Tranfer Funtion and Their Repone The repone an be hown to have the following propertie:- ) For ζ = the repone i undamped and ontinue to oillate with frequeny ω o (ω o = on the normalied plot) ) The overhoot and underhoot our at half period of the damped frequeny, ω, that i time of nπ/ω, for integer n greater and equal to 3) The firt overhoot i, then the underhoot i Δ, the next overhoot i Δ 3 and o on 4) The overhoot i often given a a perentage, ie Δ, and i hown in Figure 33 a a funtion of ζ 5) For ζ > the tranfer funtion ha two real pole and the repone ha no overhoot 6) For ζ = both pole are at ω o and the repone i the fatet with no overhoot 3

32 Tranfer Funtion and Their Repone Figure 3 Normalied tep repone of eond order ytem for different ς Figure 33 Graph of % overhoot a a funtion of the damping ratio 33 The Effet of a Zero Conider the general tranfer funtion G() = B()/A() again, and alo G () = /A(), that i G() with B() = A outlined above the effet of a non-unity B() will be to give different value of the C oeffiient in the partial fration expanion of equation (3) Thu one an find the new partial fration expanion when B() i not a ontant and invert to find the time repone There i another way, however, whih alo help in undertanding the repone and that i to reognie that an be regarded a a derivative operator Thu, for example, uppoe the repone of G () to a unit tep input i y (t) then the repone of G() to a unit tep input an be written a (39) 3

33 Tranfer Funtion and Their Repone where the b are the oeffiient of B() in equation () To illutrate thi onider t t then the olution for y (t) i y ( t) = 5( e e ), whih annot have an overhoot a the exponential dereae with inreae in time Uing the above reult y(t) i given by It i eay to how mathematially that the repone will have an overhoot for T > The repone for T = 5, T = and T = are hown in Figure 34, obtained uing the following MATLAB tatement >> G=tf([],[ 3 ]); >> tep(g) >> hold Current plot held >> G=tf([5 ],[ 3 ]); >> G=tf([ ],[ 3 ]); >> G3=tf([ ],[ 3 ]); >> tep(g) >> tep(g) >> tep(g3) where the hold tatement keep the plot allowing the repone to be ompared Figure 34 Step repone for variou value of T 33

34 Tranfer Funtion and Their Repone The unit impule, δ(t), i the derivative of the unit tep and ha a Laplae tranform of unity Thu the repone to a unit impule i the derivative of the repone to a unit tep 34 A 3 Pole Tranfer Funtion In order to appreiate the repone from multiple pole onider the tep repone of two tranfer funtion eah with three pole, a real pole and a omplex pair The example tranfer funtion are written in fatored form, whih of oure orrepond to tranfer funtion in parallel, and are:- and 5 G( ) = 5 5 G ( ) = 5 Both tranfer funtion when written with a ommon denominator have two zero and eah term in G and G ontribute a final value of 5, with the repone from the omplex pole the ame The tep repone are hown in Figure 35 The time ontant of the ingle pole in G i 5 eond but only eond in G Thu for the tep repone of G the time ontant low the repone down and the overhoot i not a large a it would be for the omplex pole alone, although the repone till oillate The maller time ontant of G i evident in the rapid initial hange in the tep repone 34

35 Tranfer Funtion and Their Repone Figure 35 Step repone of 3 pole tranfer funtion 33 Repone to a Sinuoid The Laplae tranform of in ωt i ω/( ω ), o that when the partial fration expanion i ued to get Y() it will now be of the form (3) For a table tranfer funtion, G(), all the exponential term in the ummation will eventually go to zero and the invere Laplae tranform of the firt term will be expreible a M in(ωt φ), a inuoidal ignal of magnitude (or amplitude) M and phae lag φ relative to the input inuoid, where M and φ will be funtion of ω Thi i known a the teady tate frequeny repone of G(), often imply hortened to frequeny repone To determine it value it i not neeary to go through the partial fration and Laplae tranform proe indiated above a it an be hown that it an be obtained from the omplex number G(jω), where M i the modulu and φ the argument of G(jω) Thi i a very bai property of a linear ytem that for a ine wave input the output, in the teady tate, will alo be a ine wave of the ame frequeny with the magnitude and phae hift dependent on the frequeny The value of a tranfer funtion G() for a peifi value of = i G( ) and from onideration of the zero-pole repreentation of equation (3) it an be een that it i given by (3) 35

36 Tranfer Funtion and Their Repone where P i the point in the plane and PB j and PA i are the ditane from P to the m zero, β j and n pole, α i Alo the argument φ i given by m j= n ϕ ( ) = θ j ψ i (3) i= where θ j and ψ i are the zero and pole angle repetively, that i the angle meaured from the diretion of the poitive real axi to the line drawn from zero j to the point P and from pole i to the point, P, repetively Evaluating the frequeny repone a ω goe from to mean evaluating the above a goe from to on the imaginary axi of the -plane The value of undertanding thi i that it enable one to appreiate how M and φ of a frequeny repone will vary a ω i inreaed A a imple example onider again the tranfer funtion of equation (5) that i (33) It zero-pole plot, hown in Figure, i repeated below a Figure 36 but with the edition of line joining the one zero and three pole to the point P = 3j on the imaginary axi The length of the line and angle are marked from whih it an be een that the frequeny repone of G at ω = 3, ha and (34) ϕ = θ ψ ψ ψ = tan 3 3 tan 5 tan 468 tan 773 giving (35) P 3j X A ψ 866j X - A ψ B θ - X A3 ψ3-866j Figure 36 Graphial evaluation of a frequeny repone from the zero-pole plot 36

37 Tranfer Funtion and Their Repone Magnitude and phae of the output for a inuoidal input have a very phyial meaning but mathematially they are a polar repreentation of the output, whih an therefore be written in the retangular form for a omplex number, that i G(jω) = M(ω)e jϕ(ω) = X(ω) jy(ω) (36) The relationhip between the polar and retangular repreentation are M(ω) = [X (ω) Y (ω)] / (37) φ = a tan (Y (ω), X(ω)) (38) a tan i the artangent funtion ued in MATLAB whih orretly give the phae φ between and 36 Mot book write φ = tan (Y(ω) / X(ω)) whih i imply inorret without further qualifiation a the mathematial funtion tan only exit between -9 and 9 37

38 Frequeny Repone and Their Plotting 4 Frequeny Repone and Their Plotting 4 Introdution The frequeny repone of a tranfer funtion G(jω) wa introdued in the lat hapter A G(jω) i a omplex number with a magnitude and argument (phae) if one wihe to how it behaviour over a frequeny range then one ha 3 parameter to deal with the frequeny, ω, the magnitude, M, and the phae φ Engineer ue three ommon way to plot the information, whih are known a Bode diagram, Nyquit diagram and Nihol diagram in honour of the people who introdued them All portray the ame information and an be readily drawn in MATLAB for a ytem tranfer funtion objet G() One diagram may prove more onvenient for a partiular appliation, although engineer often have a preferene In the early day when omputing failitie were not available Bode diagram, for example, had ome popularity beaue of the eae with whih they ould, in many intane, be rapidly approximated All the plot will be diued below, quoting many reult without going into mathematial detail, in the hope that the reader will obtain enough knowledge to know whether MATLAB plot obtained are of the general hape expeted 4 Bode Diagram A Bode diagram onit of two eparate plot the magnitude, M, a a funtion of frequeny and the phae φ a a funtion of frequeny For both plot the frequeny i plotted on a logarithmi (log) ale along the x axi A log ale ha the property that the midpoint between two frequenie ω and ω i the frequeny ω = ω ω A deade of frequeny i from a value to ten time that value and an otave from a value to twie that value The magnitude i plotted either on a log ale or in deibel (db), where db = log M The phae i plotted on a linear ale Bode howed that for a tranfer funtion with no right hand ide (rh) -plane zero the phae i related to the lope of the magnitude harateriti by the relationhip (4) where φ(ω ) i the phae at frequeny ω, u = log e ( ω / ω) and A( ω) = log e G( jω) It an be further hown from thi expreion that a relatively good approximation i that the phae at any frequeny i 5 time the lope of the magnitude urve in db/otave Thi wa a ueful onept to avoid drawing both diagram when no omputer failitie were available 38

39 Frequeny Repone and Their Plotting For two tranfer funtion G and G in erie the reultant tranfer funtion, G, i their produt, thi mean for their frequeny repone G jω) = G ( jω) G ( j ) (4) ( ω whih in term of their magnitude and phae an be written M = M M and ϕ = ϕ ϕ (43) Thu ine a log ale i ued on the magnitude of a Bode diagram thi mean Bode magnitude plot for two tranfer funtion in erie an be added, a alo their phae on the phae diagram Hene a tranfer funtion in zero-pole form an be plotted on the magnitude and phae Bode diagram imple by adding the individual ontribution from eah zero and pole It i thu only neeary to know the Bode plot of ingle root and quadrati fator to put together Bode plot for a ompliated tranfer funtion if it i known in zero-pole form 4 A ingle time ontant The ingle pole tranfer funtion i normally onidered in time ontant form with unit teady tate gain, that i (44) It i eay to how that thi tranfer funtion an be approximated by two traight line, one ontant at db, a G() =, until the frequeny, /T, known a the break point, and then from that point by a line with lope -6dB/otave The atual urve and the approximation are hown in Figure 4 together with the phae urve The differene between the exat magnitude urve and the approximation are ymmetrial, that i a maximum at the breakpoint of 3dB, db one otave eah ide of the breakpoint, 3 db two otave away et The phae hange between and -9 again with ymmetry about the breakpoint phae of -45 Note a teady lope of -6 db/otave ha a orreponding phae of -9 39

40 Frequeny Repone and Their Plotting Figure 4 Bode exat and approximate magnitude urve, and phae urve, for a ingle time ontant The Bode magnitude plot of a ingle zero time ontant, that i G() = T (45) 4

41 Frequeny Repone and Their Plotting i imply a refletion in the db axi of the pole plot That i the approximate magnitude urve i flat at db until the break point frequeny, /T, and then inreae at 6 db/otave Theoretially a the frequeny tend to infinity o doe it gain o that it i not phyially realiable The phae urve goe from to 9 4 An Integrator The tranfer funtion of an integrator, whih i a pole at the origin in the zero-pole plot, i / It i ometime taken with a gain K, iek/ Here K will be replaed by /T to give the tranfer funtion (46) On a Bode diagram the magnitude i a ontant lope of -6 db/otave paing through db at the frequeny /T Note that on a log ale for frequeny, zero frequeny where the integrator ha infinite gain (the tranfer funtion an only be produed eletronially by an ative devie) i never reahed The phae i -9 at all frequenie A differentiator ha a tranfer funtion of T whih give a gain harateriti with a lope of 6 db/otave paing through db at a frequeny of /T Theoretially it produe infinite gain at infinite frequeny o again it i not phyially realiable It ha a phae of 9 at all frequenie 43 A Quadrati Form The quadrati fator form i again taken for two omplex pole with ζ < a in equation (37), that i o G( ) = ω ζω ω (47) o o Again G() = o the repone tart at db and an be approximated by a traight line at db until ω o and by a line from ω o at - db/otave However, thi i a very oare approximation a the behaviour around ω o i highly dependent on ζ It an be hown that the magnitude reahe a maximum value of M =, whih i approximately /ζ for mall ζ, at a frequeny of ω = ω p o ζ Thi ζ ζ frequeny i thu alway le than ω o and only exit for ζ < 77 The repone with ζ = 77 alway ha magnitude, M < The phae urve goe from to -8 a expeted from the original and final lope of the magnitude urve, it ha a phae hift of -9 at the frequeny ω o independent of ζ and hange more rapidly near ω o for maller ζ, a expeted due to the more rapid hange in the lope of the orreponding magnitude urve Figure 4 how Bode plot for variou value of ζ againt normalied frequeny ω/ω o For the quadrati zero ζ ω ωo G( ) = (48) ω o o the Bode plot are jut refletion in the db and zero phae axe of the graph for the quadrati pole 4

42 Frequeny Repone and Their Plotting Figure 4 Normalied Bode plot for the quadrati pole form for different ς 44 An Example Bode Plot Conider again the one zero, three pole tranfer funtion (49) Dividing numerator and denominator by, it an be written in the form (4) For plotting the Bode diagram it an be thought of a 4 tranfer funtion:- ) a ontant gain of ) a ingle zero with a breakpoint of 3) a ingle pole with a breakpoint of 4) a quadrati pole with natural frequeny and damping ratio, ζ = 5 The intrution in MATLAB to obtain the Bode plot of a tranfer funtion objet G i imply bode(g) The reultant Bode magnitude plot, marked (R), i hown in Figure 43 together with the individual plot of it four ontituent, marked () to (4) a given above The grid i added to the plot by typing grid 4

43 Frequeny Repone and Their Plotting Figure 43 Bode plot of G() of equation (4) and it ontituent 43

44 Frequeny Repone and Their Plotting 43 Nyquit Plot Sine G(jω) = M(ω)e jφ(ω) = X(ω) jy(ω) (4) every hoie of ω give a point in a omplex plane either plotted in polar oordinate for the M, φ form or in retangular oordinate in X, Y form Joining the point together a ω i varied produe a lou with ω a a parameter whih i known a a polar or Nyquit plot To obtain analytial reult one need to be able to work in both polar and retangular oordinate, ine one may be more appropriate than the other for a partiular evaluation From onideration of the individual element of a tranfer funtion in the Bode approah of the previou etion one hould be able to etimate the hape of a Nyquit plot Important point in thi repet are the high and low frequeny limit, that i the value of G(jω) a ω and For large the limit of the general tranfer funtion G() of equation (9) will tend to / (n-m), that i over to the relative degree Thu for a tritly proper tranfer funtion the gain will tend to zero and the phae to -9(n m) a ω For a proper tranfer funtion with n = m the gain will tend to a finite value and the phae to zero At low frequenie the tranfer funtion, G() will tend either to a ontant or to the power of the number of differentiation term minu integration term in the tranfer funtion Typially only integration term exit in tranfer funtion for ontrol ytem o the behaviour at low frequenie depend on the number of integrator and G() tend to / i where i i the number of integrator Thu at low frequenie for i > the magnitude tend to infinity and the phae to -9i Thi phae reult doe not mean that the lou tart on an axi a kethe in many book inorretly how A a imple example of thi point onider the tranfer funtion G ( ) = (4) ( ) ( ω ) then putting = jω, and writing G(jω) in the form X(ω) jy(ω) give X(ω) = and Y(ω) = ( ω ) ( ω ) Clearly a ω, X(ω), not the imaginary axi, although the phae doe tend to -9 Thi will alway be the ae that the lou for a tranfer funtion with one or more integrator will tend to an aymptote whih in priniple an be alulated The Nyquit plot of thi tranfer funtion i obtained with the intrution nyquit(g) It i hown in Figure 44, whih i obtained by the following ingle intrution defining the tranfer funtion of G in the Nyquit tatement:- >> nyquit(tf(,[ ])) 44

45 Frequeny Repone and Their Plotting Information about where a Nyquit plot ut the axe an be obtained from the fat that the real axi i ut when Y(ω) = or arg G(jω) = or 8, and the imaginary axi when X(ω) = or arg G(jω) = -9 or 9 Whih are the eaiet alulation an depend on the tranfer funtion For the above example it i eaily een from Y(ω) that the real axi i ut when ω = and the imaginary axi i only reahed a ω tend to infinity However for G() = /( ) 6 then where it ut the axe i bet obtained uing arg G(jω), whih i imply equal to 6 tan - ω Figure 44 Nyquit plot of /( ) Three further omment mut be made here about the plot of Figure 44:- ) For reaon to be explained later the graph i drawn for both poitive and negative frequenie The labelling of thee ha been added to the plot afterward ) It an be hown for all tranfer funtion that X(ω) i an even funtion and Y(ω) an odd funtion of ω Thu the negative frequeny part of the plot i a refletion of the poitive frequeny plot in the real axi 3) MATLAB doe not label the frequenie automatially on the plot but they an be eleted by ue of the uror a ha been done to obtain one frequeny point on thi plot 4) The frequeny repone plot intrution bode(g) and nyquit(g) in MATLAB automatially elet the frequeny range Thi an be done by the uer by eleting a vetor ω, typially on a log ale uing the intrution ω = logpae(a,b,n), whih generate n point on a log ale between a and b If n i omitted the default i 5 point The plot intrution are then bode(g, ω) and nyquit(g, ω) 45

46 Frequeny Repone and Their Plotting The lat intrution in (4) ha been ued with the ω vetor generated by logpae for a = -5 and b = to how a more detailed plot near the origin in Figure 45 From the two plot it an be learly een that at low frequenie, where the gain tend to infinity beaue of the ingle integration, the lou tart from the aymptote at X(ω) = - with a phae of -9, roe the real axi, that i ha a phae of -8, at X = -5 and tend to the origin (zero gain) at high frequenie with a phae of -7 (relative degree of 3 time -9 ) The real axi roing our at a frequeny of unity Figure 45 Nyquit plot with new ω vetor 46

47 Frequeny Repone and Their Plotting A final omment on Nyquit plot i that ometime Invere Nyquit plot are drawn, thee are imply the Nyquit plot of the invere of the tranfer funtion, ie a Nyquit plot of G(jω) - 44 Nihol Plot The Nihol plot i imilar to the Nyquit plot in that it i a lou a a funtion of ω, the differene being the hoen axe On a Nihol plot thee are the magnitude in db on the ordinate and the phae in degree on the abia The origin i hoen, for reaon whih will be explained, later a db and -8 The Nihol plot for the ame tranfer funtion a the Nyquit plot of Figure 44 i obtained by the intrution nihol(g) and i hown in Figure 46 The grid i obtained by typing ngrid A expeted the plot how the magnitude dereaing monotonially with inreae in frequeny, the arrow for whih wa added to the plot, and the phae hanging from -9 to -7 Figure 46 Nihol plot of /( ) 47

48 The Bai Feedbak Loop 5 The Bai Feedbak Loop 5 Introdution The bai onept, of feedbak ontrol, a mentioned in the firt hapter i to meaure the quantity to be ontrolled, uually alled the ontrolled variable and denoted by C, and to ompare it with the deired or referene value, uually denoted by R, and to ue any error to adjut C to the deired value Thu a bai feedbak loop ha the truture hown in the diagram of Figure 5 where the variou phyial element are repreented by their mathematial model in tranfer funtion form The proe being ontrolled, denoted G(), i uually referred to a the proe or plant tranfer funtion Meaurement of it output, C, i obtained by a enor of tranfer funtion H, whih i alo known a the feedbak tranfer funtion In many ae the dynami of H may be negleted, o that H i jut a ontant with unit onverting the output to appropriate unit for ue in the ontrol ytem For example, if the output ontrolled variable i a temperature and the ontrol ytem error hannel ue voltage then H will have unit of V/ C The importane of H annot be underetimated ine if the enor i uppoed to give 5V at 5 C but atually give 5V at 48 C the perfet ontrol ytem with it referene et to 5V for 5 C will ontrol the temperature at 48 C In other ae H will ontain dynami of the enor and/ or loop ompenation dynami Some enor introdue a ignifiant noie level into the loop and thi an be repreented by the ignal, N, hown The error ignal i normally proeed through a ontroller of tranfer funtion G (), a hown, before providing the plant input ignal U The tranfer funtion of the forward path of the loop i G ()G() The loop i often ubjet to diturbane input, for intane in a poition ontrol ytem for a large antenna dih, varying wind peed impating the dih will produe a torque diturbane A diturbane ignal D i therefore hown in Figure 5 Sine the loop i linear the effet of all of the three input ignal, R, D and N at a partiular point an be found independently and then ummed Figure 5 Bai Blok Diagram of Feedbak Control Sytem 48

49 49 The Bai Feedbak Loop 5 The Cloed Loop It an be hown for the loed loop of Figure 5 that ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( H G G N H G G H G G D G H G G R G G C = (5) The numerator term are the loop tranfer funtion from the peifi input to the output C() and the denominator term i plu the produt of the tranfer funtion in the loop, whih i known a the open loop tranfer funtion, G ol () That i ) ( ) ( ) ( ) ( H G G G ol = (5) The negative feedbak i alway aumed o in atual fat if the loop were opened and a ignal, V(), injeted, it would return a G ol ()V() From here, unle otherwie tated, our onern will be with the repone to the input R, o that D and N will be aumed to be zero The tranfer funtion from R to C, often denoted by T(), i given by ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( H G G G G R C T = = (53)

50 The Bai Feedbak Loop It pole are the root of F( ) = G ( ) G( ) H ( ) = Gol ( ) = (54) whih i known a the harateriti equation of the loed loop ytem, and the loed loop will be table if all it root are in the left hand ide (lh) of the -plane Denoting eah of the individual element tranfer funtion in term of their numerator and denominator polynomial, that i N ( ) G ( ) =, D ( ) N( ) G ( ) = and D( ) N h ( ) H ( ) = (55) D ( ) h then the loed loop tranfer funtion N ( ) N( ) Dh ( ) T ( ) = (56) N ( ) N( ) N ( ) D ( ) D( ) D ( ) h h The important point to note i that the zero of T() are the zero of G () and G(), but the pole of H() 53 Sytem Speifiation The deigner of a loed loop ontrol ytem will be given peifiation whih the reulting ytem ha to meet Deign i invariably an iterative proe and begin with the eletion and modelling of the variou ytem omponent before the performane of the loed loop ytem an be evaluated It may be after ome analyi, ay for a poition ontrol ytem, it i found that the required peed of repone an only be ahieved with a larger motor, o the deigner return to the omponent eletion and modelling proe Here it i aumed that the plant tranfer funtion i fixed and G and poibly H have to be hoen to try and meet the peifiation The atual deign peifiation, whih, for example, may involve a limit on the ue of energy, may have to be tranlated into appropriate quantifiable propertie of the loed loop, whih i all that an be atified with analytial ontrol tehnique To make the deign eaier it i often aumed that there are a limited number of tranfer funtion that might be ued in G and H and the deign objetive then beome one of eleting uitable parameter for thee fixed form ontroller The feedbak loop of Figure 5 an be redrawn, a in Figure 5, with H in the forward path of the loop and the referene, R, going through /H Figure 5 Equivalent blok diagram to Figure 5 5

51 The Bai Feedbak Loop The input to H in the forward path of the loop i then the error in output unit, ay C for a temperature ontrol, and it output the error in the enor output unit, ay voltage V For loop analyi the gain of H an then be inluded in G, o for thi reaon many reult are derived with H= Some typial loed loop peifiation are therefore diued below:- ) Stability Obviouly the prime requirement for a feedbak loop i that it be table Method for invetigating tability are diued in the next etion ) Steady tate error In many intane the input, partiularly R and D, may be aumed ontant and it i often required that in the teady tate they do not produe an error Mathematially the teady tate error an be found by applying the final value theorem to the domain expreion E() for the error, whih for the feedbak loop of Figure 5 with H =, for the input R() a tep of amplitude R i R E( ) = lim (57) G ( ) G( ) whih will only be zero if lim ( ) G( G ), and thi will only be true if G ()G() ontain an integrator Obviouly if thi i not the ae then equation (57) allow the error magnitude to be alulated However, if one jut need to determine if the teady tate error i zero then thi an be done imply from a onideration of the d gain of the loop element Thi i infinite for an integrator, a in the teady tate it an have a finite output with zero input For example, for the above ae a finite output C ha to be obtained in the teady tate for zero E, whih i only poible with an integrator in the forward loop If E were required to be zero for a ramp input the forward loop would require two integrator, the firt produing a ontant output for no input in the teady tate and the eond integrating the ontant to produe a ramp In the ae of a diturbane D the error it produe i the value it ha at C, (ie C hould not be affeted by it) For D not to affet C then the forward path ignal at the point where D enter the loop, U, hould be equal and oppoite to it Thu if D i a ontant G () mut ontain an integrator for the output not to be affeted in the teady tate 3) Step repone Charateriti of the loed loop repone to a tep input are often peified Thee are baed on the typial repone with zero teady tate error to a unit tep hown in Figure 53 and are:- a) Rie time,t r Thi i the time taken to reah the teady tate value of unity for the firt time If the repone ha no overhoot the time i often given for the repone to go from to 9, that i from to 9% b) Peak time,t p Thi i the time taken to reah the firt overhoot of the repone 5

52 The Bai Feedbak Loop ) Overhoot,%O Thi i the magnitude of the firt overhoot in the repone, normally expreed a a perentage If the peak value i then the overhoot i % d) Settling time,t Thi i a meaure of the time for the repone to have approximately reahed the teady tate of unity It i normally defined a the time to reah within a % band of the teady tate (between 98 and ) and remain there Sometime a 5% band i ued a in the figure illutration Figure 53 Typial tep repone for peifiation 5

53 The Bai Feedbak Loop For the repone illutrated in the figure t r i approximately eond, t p approximately 4 eond with a %O of 5% and t of 35 eond for a 5% band 4) Frequeny repone Sometime peifiation are given with repet to the loed loop frequeny repone requirement of the ytem The ideal requirement i for C to follow R exatly but thi annot be ahieved a the input frequeny i inreaed but i normally the ae at low frequenie Thu the loed loop frequeny repone T(jω) typially tart at unit gain (db) and zero phae hift The magnitude repone may have one or more peak, the uual ae a hown in Figure 54, and then dereae a) Bandwidth, Bw Thi i defined from zero frequeny to the firt time (often the only time) the magnitude goe through -3dB (value of 77) b) Frequeny peak, M p Thi i the maximum of the frequeny repone, provided it exeed db (unit gain) For the repone hown in the figure M p i approximately 6dB at 9 rad/ and Bw i around 3 rad/ The frequeny repone peifiation are related to the tep repone one dependent on the peifi tranfer funtion For a imple tranfer funtion like the eond order one of equation (36) thee relationhip an eaily be found a in etion 3 it wa hown how the overhoot wa related to ς and in etion 43 how M p wa related to ς Thu if one i given time domain and frequeny domain peifiation one mut look at their oniteny A rie time of eond, for example, would require a bandwidth ignifiantly greater than rad/ Figure 54 Typial frequeny repone for peifiation 53

54 The Bai Feedbak Loop 54 Stability The requirement for tability of the loed loop i that all the pole of the loed loop tranfer funtion T() of equation (53) lie in the lh -plane The pole are the zero of the harateriti equation (54), whih will be a polynomial in If thi polynomial i denoted by F( ) = f f n ( n ) n f ( n ) f with > f (58) then it root an eaily be found uing Matlab by the ommand root (poly), where poly i entered like num or den a a tring of oeffiient with the highet power of firt For example >> root([ 6 6]) an = Routh Hurwitz Criterion Finding the root of a polynomial of large order wa very diffiult before the advent of modern omputational tehnique and in 876 a major ontribution wa made by Routh who obtained ondition whih had to be atified for all root of a polynomial to lie in the lh -plane A polynomial whih atifie thi ondition i known a a table polynomial The riterion wa later modified by Hurwitz to give the Routh-Hurwitz reult preented in Appendix B Two imple reult whih prove ueful are a) A neeary but not uffiient ondition, apart from the eond order polynomial where it i both neeary and uffiient, i that all the oeffiient of mut be poitive that i f j > for all j b) For the third order polynomial a neeary and uffiient ondition i all the oeffiient mut be poitive and f f > f f 3 54

55 The Bai Feedbak Loop 54 Mikhailov Criterion The Mikhailov riterion i a imple graphial approah only normally mentioned in Ruian textbook If the polynomial F(jω) i plotted for ω inreaing from zero on a omplex plane, then all it root will lie in the lh -plane if from tarting on the poitive real axi at f it move in a ounter lokwie diretion paing ueively through the poitive imaginary axi, negative real axi et in turn until it ut no further axe but head for infinity a illutrated in Figure 55 The number of axe ut will be n- Figure 55 Illutration of Mikhailov Criterion for Stable Fourth Order F(jω) 55

56 The Bai Feedbak Loop 543 Nyquit Criterion In the early day when ontrol engineering wa developing a a diipline it wa very deirable to try and develop onept to predit apet of the loed loop ytem behaviour baed on propertie of the open loop tranfer funtion There were three major reaon for thi:- a) When a ompenator (ontroller) i within the loop it i muh eaier to ee how hange in it parameter will affet the open loop propertie, for example the frequeny repone, than the loed loop propertie b) Plant model were often obtained by frequeny repone teting o that G(jω) wa then available a a plot from experimental data ) Even when all the loop tranfer funtion were known alulating a loed loop tep repone wa a laboriou proedure For thee reaon the Nyquit tability riterion, whih i baed on the open loop frequeny repone, wa thu not only ueful but alo very pratial The derivation of the riterion, whih ue the mathemati of funtion of a omplex variable, i relatively eay to explain in priniple It i baed on Cauhy mapping theorem whih tate that if a omplex funtion, F(), i mapped around a loed ontour in a lokwie diretion in the -plane (that i it value alulated at point on the ontour and plotted in it own omplex plane) the origin will be enirled N o time in the lokwie diretion where N o i the differene between the number of zero and pole of F() enloed by the hoen -plane ontour When the ontour i taken a the imaginary axi, thi mean taking ω from to, and then the infinite emiirle in the right hand ide (rh) -plane (around thi ω remain infinite), known a the Nyquit D ontour, hown in Figure 56, then the origin will be enirled by F(jω) N o time in a lokwie diretion, where N o i given by:- N o = [zero of F() pole of F()] in rh -plane (59) The zero of F() are required to ae tability o the equation may be written zero of F() in rh = N o pole of F() in rh (5) From equation (54) it an be een that the pole of F() are the ame a the pole of G ol () and that the only differene between a mapping of F(jω) and G ol (jω) i that the latter i hifted from the former by - along the real axi Thu equation (5) an be written zero of F() in rh = N pole of G ol () in rh (5) 56

57 The Bai Feedbak Loop where N now denote the lokwie enirlement of the (-,) point by the plot of G ol (jω) Thu if the number of pole of G ol in the rh -plane i known, whih will of oure be zero for a table G ol, the number of zero of F() in the rh -plane an be found from equation (5) to determine the tability of the feedbak loop Thi equation give the Nyquit tability riterion whih may be formally tated a the loed loop ytem will be table if the number of lokwie enirlement by the frequeny repone lou G ol (jω) of the (-,) point plu the number of rh -plane pole of G ol () i zero Showing the Nyquit plot for both negative and poitive frequenie allow the enirlement to be found So that the D ontour doe not pa through them when ingularitie (pole or zero) exit on the imaginary axi, it ha to be modified o that they lie outide by indentation of infiniteimal radiu, ε, a hown dotted in figure 56 In many intane the plant to be ontrolled will be table o that G ol () will have no rh -plane pole o for tability the Nyquit plot of G ol (jω) mut have N = Control engineer are, however, required to ontrol plant whih are untable, a modern fighter airraft being a good example Figure 56 Nyquit D ontour 57

58 More on Analyi of the Cloed Loop Sytem 6 More on Analyi of the Cloed Loop Sytem 6 Introdution In the previou hapter the bai feedbak loop wa diued and typial peifiation that might be required for it performane introdued Before going on to diu analytial method that an be ued for deigning the ontroller to try and meet given peifiation it i neeary to preent ome further analytial onept ued for feedbak loop analyi Alo up to thi point it ha been aumed that the tranfer funtion repreentation for blok deription i a ratio of polynomial in There i, however, one linear element whih often exit in a ontrol ytem for whih thi i not the ae, namely a time delay Thi i therefore overed firt in the next etion 6 Time Delay A time delay a it name ugget i an element whih produe an output whih i a time delayed verion of it input It i alo known a dead time or tranport delay The latter name reflet the fat that a ommon ourrene i due to ay, a temperature meaurement being made on a moving fluid down tream from where it ha been heated It i normally aumed that it initial output i zero Thu for example, if the time delay i τ eond the input, v(t), and output will be a hown in Figure 6 for the linear input v(t) = t Mathematially the input, v(t) an be defined a either v(t) = for t < and t for t >, or tu (t) where u (t) i the unit tep at t = Uing the unit tep notation the output an be written a (t τ)u (t τ), that i a unit ramp beginning at time τ The Laplae tranform of the input i / and of the output e -τ / (ee theorem (vi) Appendix A) Thu the tranfer funtion for the time delay blok, the ratio of the output to the input in the -domain, i e -τ Figure 6 Illutration of a time delay 58

59 More on Analyi of the Cloed Loop Sytem The time delay tranfer funtion i eaily handled when uing frequeny domain method a with = jω, it i e -jωτ, whih ha a unit magnitude at all frequenie and a phae lag of ωτ Thu, for example, it Nyquit plot i a unit irle whih ha frequeny point of value π/τ, π/τ, 3π/τ, π/τ, 5π/τ et, at -9, -8, -7, -36, -45, et Figure 6 how Nyquit plot of the tranfer funtion G() = /( ) and G()e, that i the former with an additional time delay of eond Figure 6 Nyquit plot of G() = /( ) with (dotted) and without time delay of eond 59

60 More on Analyi of the Cloed Loop Sytem Although a time delay an be approximated by the tandard erie for an exponential a better approximation i to ue a ratio of polynomial, a reult due to Pade Thi allow hoie of the order of the numerator and denominator polynomial The Pade Table of approximation i given in Appendix C 63 The Root Lou Deign of a imple ontrol loop may ometime jut involve the hoie of a uitable gain, K, in whih ae the harateriti equation will be KG ol () = (6) and the pole of the loed loop tranfer funtion, the root of equation (6), will vary with K Evan in 948 found a diagrammati method for howing how thee root would vary a K hanged, known a a root lou, by reogniing that, ine i omplex, equation (6) ould be written a the two equation Arg(G ol ()) = -8 ~ (6) and K G ol () = (63) Baed on equation (6) he wa able to prove everal reult indiating where the root would be and then ued equation (63) to mark the orreponding value of gain on the lou MATLAB plot a root lou with the ommand rlou(g) Some imple rule whih enable a quik hek of a root lou, auming G ol i in the form of G() given in equation (9) to (), and K i poitive are:- The number of root lou path will be n, auming n m The loi tart at the pole of the open loop tranfer funtion, G ol, with K = 3 The loi finih at the zero of the open loop tranfer funtion, G ol, a K 4 A number of loi equal to the relative degree, (n-m), or the o-alled number of zero at infinity, of the open loop tranfer funtion will tend to infinity a K tend to infinity 5 Loi exit on the real axi to the left of an odd number of ingularitie (pole plu zero) A a imple example the ommand >> rlou(tf([],[ 3 ])) 6

61 More on Analyi of the Cloed Loop Sytem produe the root lou hown in Figure 63 Thi tranfer funtion for, G ol, ha no zero and 3 pole at,- and -, and it an eaily be een that the plot atifie the above five rule for the loi The harateriti equation ha three root all of whih, initially for low gain, are real A gain inreae the root moving from the origin move toward that from the one at -, they meet at the o alled breakaway point where they are equal and then form a omplex pair for further inreae in gain The third root move all the time to the left on the negative real axi from - Information for a peifi point on the lou an be obtained by pointing at it and uing a left ide uror lik, with the reult hown by the label on the Figure The value of the Gain, K, and the oordinate of the point a Pole, are given The other information given on the damping i baed on the aumption, a explained in etion 3, that thi omplex pole wa one of the two omplex pole of a eond order ytem with the tranfer funtion of equation (36) Thi i obviouly only indiative ine, for example, the loed loop repone to a tep input will, a indiated earlier depend upon any zero; and in thi ae when there are no zero on the relative weighting of the repone from the real pole beyond -, atually at -5 for K = 89, and the omplex pair, a illutrated in etion 34 The overhoot of the loed loop tep repone for thi third order tranfer funtion with the gain value of 89 i 37 not 394% Figure 63 Illutrative root lou plot The gain range for the root lou plot i eleted automatially but a for ω in a frequeny repone may be eleted if preferred by the ommand K=linpae(a,b,n), whih generate n point linearly paed from a to b, inluive, and then ue of the plotting ommand rlou(g,k) When a root pae into the rh -plane the loed loop beome untable For the above ae interetion with the imaginary axi our for K = 6 at a frequeny of / = 44, the value of the imaginary axi oordinate Thi ondition i often referred to a being neutrally table, ine mathematially a ontant amplitude oillation exit 6

62 More on Analyi of the Cloed Loop Sytem Although the root lou method i normally ued for a varying gain in the harateriti equation it an be ued for any variable parameter Conider for example (64) The loed loop harateriti equation i (65) Thi an be written, by dividing by the term independent of a, a (66) Thu omparing with equation (6) a replae K and the equivalent G ol i ( ) Plotting the root 3 4 lou of thi tranfer funtion will then how how the loed loop pole vary a a funtion of a Note that to find the pole a ubi ha to be olved, whih i a mentioned previouly eaily done in MATLAB with root 6

63 More on Analyi of the Cloed Loop Sytem 64 Relative Stability Relative tability, whih may alo be alled robutne, i a meaure of how near a ytem i to being untable Robutne, however, i ued with repet to many propertie and it ue i bet qualified by uing it in the form robutne of property X with repet to property Y if the ontext i not lear There are everal meaure whih are ued to indiate relative tability and ome are diued below 64 Pole poition Obviouly ine intability reult from a pole entering the rh -plane, the nearer a pole in the lh -plane i to the imaginary axi the nearer the ytem being tudied will be to intability Thu in the previou root lou example the nearer the gain to the value of 6 the nearer the two omplex pole are to the imaginary axi The information in the panel of the root lou plot therefore give an indiation of the relative tability of the ytem 64 Gain and Phae Margin If the open loop ytem tranfer funtion i table then from the Nyquit riterion given in etion 543 the loed loop ytem will be table if it Nyquit plot doe not enirle the Nyquit point (-, ) Paing of the lou through thi point orrepond to neutral tability like the roing of the imaginary axi in a root lou plot For the G() onidered in the root lou plot thi would our at a frequeny of 44 rad/ with an additional gain of K = 6 In gain-phae term the Nyquit point ha a gain of unity and a phae of -8, hene the hoie of thi point a the origin for a Nihol plot Obviouly therefore meaure of how near the open loop frequeny repone lou i to thi ritial point, on the Nyquit or Nihol plot, are indiator of relative tability Figure 64 how a typial open loop frequeny repone Nyquit plot, it i in fat (67) the plant tranfer funtion a ued in the root lou plot with an additional gain of 3 The (-, ) point i labelled N, the origin O, and the point where the lou ut the negative real axi a P ) Gain Margin The gain margin i the amount by whih the gain need to be inreaed for the loed loop to beome untable It i uually given in db and i log (ON/OP) For the example plot the negative real axi i ut at -5, o for the lou to pa through N the gain ha to be inreaed by a fator of, whih i 6dB A the phae hift i -8 the frequeny at thi point i uually known a the phae roover frequeny, whih will be denoted by ω p and i 44 rad/ in the example 63

64 More on Analyi of the Cloed Loop Sytem ) Phae Margin The phae margin i the amount by whih the loop phae need to be hanged for the loop to beome untable The point G on the frequeny repone ha a gain of unity, that i OG =, o for thi point to pa through N the phae need to be hanged by the amount of the angle GON marked in the figure Mathematially the phae margin i 8 arg (G(jω g ), where ω g i the frequeny at G and i known a the gain roover frequeny ine (G(jω g ) = In the example ω g = 969 rad/ and the phae margin i Figure 64 Nyquit plot illutrating gain and phae margin Figure 65 Bode plot illutrating gain and phae margin 64

65 More on Analyi of the Cloed Loop Sytem Figure 65 give the Bode plot of the ame open loop tranfer funtion of equation (67) and how how the loed loop gain and phae margin are found from it Figure 64 alo illutrate another point that for a table open loop tranfer funtion the loed loop will be table if the open loop frequeny repone traed with inreaing frequeny pae the ritial point to it left Sometime Nyquit loi are muh more ompliated than the imple mooth one hown, for example with multiple roing of the negative real axi, and in uh ae further larifiation may be neeary when uing gain and phae margin term Note alo that with a non mooth lou it would be poible to have a large gain margin and a mall phae margin or via vera 643 Senitivity funtion The loed loop tranfer funtion, with H =, i (68) If one regard G ol () a a variable and wihe to deribe the enitivity, S, of T to hange in G ol then thi may be written a (69) 65

66 More on Analyi of the Cloed Loop Sytem whih on evaluating the differentiation give (6) The omplimentary enitivity funtion i defined a S = T (6) whih i the loed loop tranfer funtion Sine the vetor G ol (jω) i a meaure of the ditane of G ol at any frequeny from the Nyquit point {ie in figure 64 the length NG i G ol (jω g )}, when thi i mall S(jω) and T(jω) will have large peak magnitude, o their maximum magnitude may be ued a relative tability indiator That of T(jω) i diued further in the next etion 65 M and N Cirle The idea of M and N irle again relate to the day when omputer oftware wa not available and deigner were intereted in finding out about the loed loop frequeny repone behaviour from the open loop frequeny repone The omputation from open loop to loed loop propertie are now eaily done but the onept are till of ome value, partiularly that of M irle If the feedbak tranfer funtion H = then, G ol (jω) = G (jω)g(jω), and the loed loop frequeny repone funtion i (6) If the magnitude of thi funtion i required to remain ontant, at a value M, a ω varie then it an be hown that G ol (jω) hould move on a irular path on a Nyquit plot Thu by uperimpoing thi grid, known a M irle, on a Nyquit plot of the open loop frequeny repone one an ee how the magnitude of the loed loop frequeny repone will vary with frequeny The magnitude value of M are normally labelled in db It an further be hown that if the loed loop phae i to remain ontant then thi alo produe a grid of irle, known a N irle The M irle an be overlaid on a Nyquit plot in MATLAB by uing the right hand moue button and eleting grid from the reulting option Thi i hown in Figure 66 for the ame tranfer funtion a ued in Figure 64and 65 66

67 More on Analyi of the Cloed Loop Sytem Figure 66 Nyquit plot with M irle The larget M value irle whih the plot reahe i een to be approximately db at the point P Thu the loed loop frequeny repone hould have a maximum magnitude M p of db at the frequeny of the point P, whih i approximately rad/, a i een to be the ae in the MATLAB plot of Figure 67 The obervant reader may alo have notied that the db (unit gain) M irle i in fat a traight line (irle with entre at infinity and infinite radiu) through the point (-5,) Thi mean for a typial open loop Nyquit plot it mut alway tay to the right of thi line if the loed loop frequeny repone mut not have a gain greater than unity Obviouly peifying the M p of a frequeny repone i another meaure of relative tability Figure 67 Cloed loop frequeny repone for G ol (jω) of equation (67) with H = 67

68 Claial Controller Deign 7 Claial Controller Deign 7 Introdution Claial ontroller deign involve the hoie of a uitable tranfer funtion in the ontroller G, or poibly H, of Figure 5 o that the loed loop performane meet the required peifiation Thi an often be ahieved with quite imple tranfer funtion with three ommon one being the phae lead ontroller, the phae lag ontroller and the PID (Proportional, Integral and Derivative) ontroller Sine the ytem peifiation often inlude that there hould be no teady tate error to a tep input, the phae lead and lag ontroller, whih do not inlude an integral term, are normally ued with plant tranfer funtion with an integral term Many plant tranfer funtion in proe ontrol, for example temperature ontrol, do not inlude an integral term o that PID ontroller, or ometime jut PI ontroller, are often ued to ontrol them PID ontroller are alo ued on plant with an integration term to eliminate teady tate error aued by a ontant diturbane, D, in Figure 5; a topi whih will be diued in hapter 9 68

69 Claial Controller Deign Mot textbook diu the deign of phae lead and lag ontroller uing both frequeny domain and root lou method but here only frequeny domain method will be overed The main reaon for thi i that frequeny domain method involve loop haping whih i ued in reent approahe to multivariable ontrol Both method invariably involve iteration a in the frequeny domain approah one i haping the open loop frequeny repone and in the root lou approah one i eleting the loed loop pole A explained earlier the relationhip between thee propertie and the reulting loed loop tep repone, whih i often a ytem peifiation, i baed on qualitative onept For example, it an be een from equation (56) that if the ompenator i moved from the forward path to the feedbak path, the loed loop tranfer funtion, T() hange, but both the open loop frequeny repone and loation of the loed loop pole are unhanged 7 Phae Lead Deign A phae lead ontroller a tated above i normally ued when the plant tranfer funtion G() ha an integration Auming thi to be the ae then, with G = K and H =, it will be found that lim G ( ) G( ) = K v /, where K v i a ontant and the error to a ramp input will be maller the larger the value of K v Thi onideration often affet the hoie of the ontroller gain o that the phae lead ontent of the ontroller i normally determined auming G () = o a not to affet K v The tranfer funtion of the phae lead ontroller i therefore taken a (7) whih produe a lead when α < A ommon frequeny domain approah for eleting the parameter α and T for a phae lead ontroller i to do the deign o that the ompenated open loop frequeny repone lou ahieve a preeleted phae margin, φ Thi i baed on the aumption that for a mooth open loop frequeny repone inreaing the phae margin will redue the overhoot in the tep repone There are everal way of trying to ahieve thi and dependent on the hoie of j for a given G() they may or may not be ueful Poible method are:- ) The laial method, whih will be deribed below, and i overed in mot textbook ) Chooing the ontroller zero to anel the dominant pole of the plant, auming of oure one exit 3) Deigning for a hoen phae or gain roover frequeny, ie where the open loop gain i unity or the phae -8 4) Fix a, uually baed on bandwidth or noie onideration, and find a uitable value of T 69

70 Claial Controller Deign Before outlining the proedure of (i) a few fat regarding the phae lead network are needed The derivation are traightforward and an be found in many textbook on ontrol The network give a maximum phae lead, ϕ m at the frequeny of and the orreponding gain i Note that on the Bode diagram with the logarithmi ale for frequeny, the frequeny ω m lie half way between the two break point, /T and /αt, and the orreponding gain in db i half the gain of the tarting and finihing gain Value of the phae lead and the orreponding gain for hoie of α are given in Table 7 and Fig 7 how a Bode diagram for the phae lead network for T = and α= /8 α / /3 /4 /5 /6 /8 / Max Lead φ m Gain at φ m G m db Table 7 Phae lead network parameter Figure 7 Bode diagram of phae lead with T = and α = /8 The proedure for (i), if the deired phae margin i φ, i then a follow:- Evaluate the unompenated ytem phae margin ϕ Allowing for a mall amount of afety, e, etimate the required phae lead, ϕ m = φ ϕ ε (ε typially 5-8 ) 3 Evaluate α for thi value of ϕ m from the above equation (or ue Table 7) 7

71 Claial Controller Deign 4 Evaluate log α (or take from Table 7 where G m db = -log a db) and determine the frequeny where the unompenated Bode frequeny magnitude urve i equal to log α db Thi frequeny i the etimated new db roover frequeny and ω m imultaneouly (if the gue for e i orret), beaue the ompenation network provide a gain of -log α at ω m 5 Draw the ompenated frequeny repone, hek the reulting phae margin, and repeat the tep from if neeary (ie hange ε) 6 If the deign doe not meet the peifiation repeat for a different hoie of phae margin, ie inreae if the overhoot i too high The problem with the approah i in etimating e although it i muh eaier interating with Bode plot in MATLAB than it wa with kethed Bode plot and penil and paper Critial to the ue of the method i the rate of hange of phae of the plant tranfer funtion beyond the frequeny of the unompenated ytem phae margin If thi i too high the method will fail Thi alo affet the etimate that hould be taken for e A guide, i 5 - for a plant of relative degree, and -8 for one of relative degree 3 A an example onider a plant with tranfer funtion (7) 7

72 Claial Controller Deign If after doing the Bode plot the ommand margin(g) i given in Matlab then the value of the gain and phae margin for the tranfer funtion will be given on the Bode diagram a hown in Figure 7 The phae margin i een to be 56 For a eond order ytem a phae margin of 4 orrepond to an overhoot of lightly over 5%, o let u aume the phae lead ompenator i required to produe a phae margin of around 4 Figure 7 Bode diagram for the tranfer funtion G() The extra phae required to inreae the phae margin to 4 i (4-56ε) i 4 if ε i taken equal to 56 whih i reaonable for a tranfer funtion with relative degree 3 From Table 7 thi ugget an α of either /4 or /5 Taking the latter value mean ε = 74 and from Table 7 the db for α = /5 i 699 From Figure 7 the gain plot i approximately 7dB down at a frequeny of 473, whih i lightly higher than the phae roover frequeny Thu 473 = /T, giving T = 47 The ompenator i therefore taken a G () = Figure 73 how the ompenated open loop frequeny repone together with that of the plant alone It an be een that the gue for e wa very good with the reulting gain roover frequeny being 47 and the phae margin 396 Ue of a phae lead ompenator i een to inreae the gain and phae roover frequenie and the open loop bandwidth It therefore an be expeted that the loed loop tep repone of the ompenated ytem will be fater than that for the plant with a unit gain ompenator Thee tep repone are hown in Figure 74 where the overhoot of the phae lead ompenated ytem i ignifiantly le a expeted beaue of it inreaed phae margin The overhoot of the ompenated ytem i, however, lightly more than 5% It i of interet to ompare thi loed loop repone with that obtained with the phae advane ompenator plaed in the feedbak path and thi i hown in Figure 75 The ompenator in the feedbak path i een to reult in a repone whih ha a longer rie time, a omparable ettling time, and no overhoot Another intereting apet of plaing the ompenator in the feedbak path i een by looking at the ontroller output ignal, normally alled the ontrol ignal, for tep input Thee are hown in Figure 76 7

73 Claial Controller Deign Figure 73 Open loop frequeny repone of the ompenated and unompenated ytem Figure 74 Cloed loop tep repone of the ompenated and unompenated ytem A the relative degree of the phae advane ompenator tranfer funtion i zero, it initial output in repone to a tep input i /α, in thi ae 5 Thu the loed loop repone for the ompenator in the forward loop produe a derivative kik of 5 Thi aount for the fater rie time but alo mean that the ontrol ignal may aue aturation for large tep input In ontrat, with the ompenator in the feedbak path the ontrol ignal tart from zero and reahe, in thi ae, a maximum of around unity 73

74 Claial Controller Deign Before onluding phae lead ontroller deign a few omment on the other uggeted approahe (ii) to (iv) are appropriate Firt with repet to method (ii), the dominant plant pole in the example G() of equation (7), ie the larget time ontant, i 5, o uing thi method the phae advane ontroller tranfer funtion would be G () =, where T would be evaluated to give the required phae margin The reult for thi partiular example, hooing a phae margin of 4, give T = 9, whih make the ompenator almot the ame a in (i) due to the nearne of 5 to 47 With method (iii), whih i uually done for the gain roover frequeny, the problem i eleting how muh higher thi frequeny hould be than for the plant 74

75 Claial Controller Deign Figure 75 Comparion of tep repone for forward path and feedbak path loation of the ompenator Figure 76 Correponding ontrol ignal for the different ompenator poition 75

76 Claial Controller Deign alone If it i eleted too high a deign will not be poible Method (iv) i traight forward, a value of α, uually in the range /8 to /6, i eleted and then T found to give the deired phae margin If in the above example α i eleted a /8, then T an be found by iteration in MATLAB to be about 9, with a orreponding gain roover frequeny of 34 rad/ Interetingly thi open loop frequeny repone ha a higher gain margin but a maller phae margin than the deign uing method (i) a hown in Figure 77 Lead ompenation may not be poible in ome ae a it depend very muh how rapidly the phae of the plant tranfer funtion hange beyond the exiting unompenated gain roover frequeny Figure 77 Comparion of Bode diagram for method (i) with a deign uing an alpha of /8 73 Phae Lag Deign A phae lag ompenator i ahieved with the tranfer funtion of equation (7) with α > In doing a phae lag deign one ue the fat that the ompenator gain hange from db at low frequenie to log (/α)db = -log (α)db at high frequenie The Bode diagram for the phae lag tranfer funtion G() = i hown in Figure 78 for α = and T= The phae lag, δ, a deade above the eond break point i δ = tan tan α, whih depend upon α, with value of 85, 47, 499 and 53 for α =, 4, 8 and repetively The orreponding gain differ by le than 5dB from the aymptoti value of -log α The idea i to have thi point a the gain roover frequeny of the ompenated lou Thu if the required phae margin of the ompenated ytem i φ, then one need to find the frequeny ω, where arg G(jω) = -(8- φ) δ and G(j ω) = log α 76

77 Claial Controller Deign A an example the ame plant a previouly, given by equation (7), i taken for a phae lag ompenator deign with again the requirement for the ompenated ytem to have a phae margin of 4 Auming δ = 4 the frequeny where G(jω) ha a phae of -36 i required From the Bode diagram thi i approximately at 5rad/ where the orreponding gain i 989dB, a gain of 3 Thu, /T = 5, α = 3 and the required ompenator tranfer funtion i G () = 66 7 Figure 78 Bode diagram of lag ompenator 77

78 Claial Controller Deign The Bode diagram for the plant alone and the lag ompenated ytem are hown in Figure 79 The bandwidth of the ompenated ytem with the lag network i lower and the loed loop tep repone i lower Thi repone and that for the phae lead ompenated ytem with the ompenator in the forward path are hown in Figure 7 Note the % overhoot for both plot i roughly the ame The lag ompenator doe produe a mall initial jump in the ontrol ignal to /3 whih i roughly it peak value Unlike lead ompenation the ue of lag ompenation in the feedbak path produe very poor reult due to the delay it aue to the feedbak ignal Figure 79 Bode diagram for the plant alone and the lag ompenated ytem For thi example a very muh lower repone, with an overhoot of over 5%, reult Unlike lead ompenation lag ompenation i uually poible, however ine it low down the repone and redue the bandwidth it may not be deirable Figure 7 Cloed loop tep repone for lead and lag ompenated ytem 78

79 Claial Controller Deign 74 PID Control Mot plant in the proe indutrie do not ontain an integration term in their tranfer funtion It ha been een that it i neeary to have an integration term in the forward path to ahieve zero error in the teady tate repone to a referene tep input, o an integration term i normally required in the ontroller for thee plant The ideal phae lead ontroller with α =, i a PD, that i proportional plu derivative, ontroller Thu the ue of a PID ontroller ontaining proportional, integral and derivative term i a logial form of fixed term ontroller for plant without an integration term in their tranfer funtion PID ontroller have thu been ued extenively in the proe ontrol indutrie for many year PID ontrol wa firt implemented with pneumati ontroller and ubequently went through the ue of vauum tube, tranitor, integrated iruit to today ituation where it i typially oftware in a miroproeor There are variou way in whih the ontroller may be implemented with mot aademi paper onidering it repreentation by the ideal tranfer funtion (73) with the loop error a input An alternative form with real zero only whih i alo frequently ued i (74) The above an be onverted to the former uing, K T = K T i, K = K (T T )/T, and K T d = K T Sometime the derivative term i fed from the output rather than the error, whih will be denoted PI-D Thi avoid the derivative kik diued earlier with repet to the phae lead ompenator Alo in pratie the derivative term ha an additional time ontant being of the form T d /( αt d ), with α typially around 74 The Ziegler Nihol Approah The earliet work uually referened on PID ontrol i that of Ziegler and Nihol (Z-N) [7], whih related to identifiation and ontrol, the idea being to preent tehnique whih ould be ued to et the parameter of the PID ontroller by proe ommiioning engineer Thu the proedure i often known a ontroller tuning and Z-N uggeted the following two method Method An open loop tep repone identifiation of the plant wa uggeted with the reulting repone modelled ( fit ) by a firt order plu dead time (FOPDT) tranfer funtion 79

80 Claial Controller Deign Baed on the FOPDT model (75) they uggeted the ontroller parameter be et aording to Table 7 Type K T i T d P T/τK p PI 9T/τK p 333τ PID T/τK p τ 5τ Table 7 Z-N Method Parameter Method They uggeted that with the ontroller in itu in the loop it hould be put into the P mode and the gain turned up until an oillation took plae The gain, of the P term, known a the ritial (or ultimate) gain, K, and the frequeny of the oillation, ω =π/t, known a the ritial (ultimate) frequeny were then reorded 8

81 Claial Controller Deign Baed on thee value the ontroller parameter hould then be et aording to Table 7 Type K T i T d P 5K PI 45K 8T PID 6K 5T 5T Table 7 Z-N Method Parameter The firt method i a laial identifiation and then ontroller deign approah Step repone were regularly ued in the early day of ontrol for identifiation and trying to fit to another model, if felt appropriate, ay from phyial onideration ould be done The FOPDT model, however, i often a good etimate for many proee Other that have been ued are eond order plu time delay (SOPDT) and a ingle time ontant to a large power, ay 6 or higher There i no reaon why after identifying an FOPDT model another deign method hould not be ued to et the ontroller parameter The Z-N deign method wa baed on ahieving around a 5% overhoot to a et point tep repone The eond method i baed on the fat that if a imple linear feedbak loop beome untable it will do o at a frequeny where the phae hift i -8 Thu, in priniple, the approah give the frequeny, ω, and the gain, /K, of the point at -8 on the plant frequeny repone From a theoretial viewpoint the method i fine but it uffer from many pratial problem Thee inlude:- Even if the loop were linear the fat that many proee have very long time ontant make it extremely diffiult and time onuming to try and find the gain, K It an be dangerou if there i no atifatory limiting effet in the loop a an adjutment to an over etimated value of K an reult in a large oillation 3 Many pratial loop are nonlinear Saturation i helpful in limiting the amplitude of the oillation but dead zone effet make finding K even more diffiult 74 Time Saling and the FOPDT Plant Time aling and amplitude aling were very familiar to uer of an analogue omputer Time aling wa een to be ueful in plotting tep repone in hapter 3 ine it baially redue the parameter dependene by one Here it relevane to eleting (or tuning) ontroller parameter for a PID ontroller ontrolling an FOPDT plant i diued If for the tranfer funtion of equation (75) a normalied, n, i taken equal to T and with ρ = τ/t the tranfer funtion beome G( n ) = K p e nρ n (76) whih an be referred to a a normalied FOPDT tranfer funtion 8

82 Claial Controller Deign The normalied tranfer funtion ha a unit time ontant and only two parameter K p and ρ The atual ytem ha a tep repone whih i T time lower and marking on it frequeny repone T time maller If thi normalied plant i ontrolled by an ideal PID ontroller in the error hannel with the tranfer funtion of equation (73) then thi beome (77) where T d = T d /T and T i =T i /T and the normalied open loop tranfer funtion an be written (78) where K = K p K Thi mean that if the ontroller parameter are deigned baed on ome property of the open or loed loop tranfer funtion, the reult will be of the form:- K = f (ρ), T i = f (ρ), and T d = f 3 (ρ) Thu for any FOPDT plant the ontroller parameter mut be of the form K = f (ρ) /K p, T i = Tf (ρ), and T d = Tf 3 (ρ) for the ame performane property to be maintained for all plant whih an be time aled to the ame normalied plant Thi will be referred to a onitent tuning Method of Z-N i eaily een to be onitent with a imple hoie for the funtion f, f and f 3 of being inverely proportional, proportional, and proportional to ρ, repetively It i alo eay to how that Method i alo onitent Table 73 lit the funtion of ρ for everal onitent tuning formula whih have been uggeted Apart from Z-N method the other are Cohen and Coon (C-C) [7], Zhuang and Atherton (Z-A) [73] and Wang, Juang and Chan (W-J-C) [74] Figure 7 to 73 how thee relationhip graphially for both Z-N method and the other The C-C reult are omitted from f a it value at mall value of ρ beome large The a and b parameter in the Z-A method depend on the integral performane tuning riterion ued The book by O Dwyer [75] give a large number of o-alled tuning rule for PID ontroller but they are unfortunately not given in the normalied form whih ha been demontrated here for the FOPDT plant Two other plant tranfer funtion whih an alo be normalied in term of the parameter ρ are (79) and (7) 8

83 Claial Controller Deign For time aling oniteny the required ontroller parameter mut again be in the form Table 73 Funtion of Rho for Different Tuning Formula 83

84 Claial Controller Deign Figure 7 Gain K (= K K p ) a a funtion of rho 743 Relay Autotuning and Critial Point Deign The priniple of the Z-N eond method i very ueful a it an be ued in loed loop but the diffiulty of adjuting the P, a mentioned earlier wa a pratial diffiulty With the advent of miroproeor ontroller, however, Atrom and Hagglund [76] uggeted a muh more uitable method for pratial implementation for etimating the ritial point Figure 7 Graph of normalied integral time againt rho 84

85 Claial Controller Deign Figure 73 Graph of normalied derivative time againt rho Thi involved replaing the P term by an ideal relay funtion to obtain a limit yle It an then eaily be hown uing a deribing funtion (DF) analyi that the frequeny of the limit yle, ω o, i approximately the ritial frequeny, ω, and the ritial gain, K, i given approximately by K = 4h/aπ, where h i the peak to peak amplitude of the relay output and, a, i the fundamental frequeny amplitude of the limit yle It an be hown that the etimate for ω will be better than for K and that the reult will be better the nearer the limit yle at the relay input i to a inuoid The error introdued by replaing a by half the peak to peak amplitude of the limit yle i uually quite mall and i normally done in pratie beaue of the eae of meaurement Thu with a more pratial approah for etimating the ritial point it i appropriate to omment further on ue of the ritial point in PID tuning Firt what i the priniple of the Z-N method in uing the ritial point for tuning? Sine all that i known about the plant i it ritial point then all one an do in eleting the ontroller parameter i to plae thi frequeny at a known point on the ompenated open loop frequeny repone lou Sine in the Z-N method T i i taken equal to 4T d, whih orrepond to the two zero of the PID ontroller tranfer funtion being real and equal, it i eay to how that for the PI ontroller thi point i 46 arg -9 and for the PID ontroller it i 66 arg

86 Claial Controller Deign Thi onept i a ueful deign approah and if felt appropriate a different point an be hoen, within the allowable range Alo ine one only ha freedom to adjut two ontroller parameter the T i /T d ratio may be eleted to be other than 4 It i eay to how with T i /T d = 4, that for the FOPDT plant the ontroller parameter are obtained from the following equation for moving the ritial point to g arg (8 φ) (7) ' tan( ϕ / ) T d = (7) ω ( tan( φ / ) o ' / 4gωoTd ( ω ) K = (73) 4ω T o o ' d Critial point deign i a ueful onept ine loed loop performane i very dependent on the open loop frequeny repone in the region of the Nyquit ritial point (-, ) Certainly if one ha little knowledge of the plant dynami it an be very ueful a illutrated here for a plant with tranfer funtion G() and a poible redued order model G r (), for whih the tep repone are hown in Figure 74 The differene between the tep repone i very mall and it would be diffiult to detet with noiy meaurement However, the frequeny repone hown in Figure 75 are quite different around the Nyquit ritial point, indeed G() ha a finite gain margin whilt that of G r () i infinite 86

87 Claial Controller Deign Thu deigning a ontroller baed on what appear to be a reaonable redued order model from tep repone information would be extremely poor ompared with ue of an approximate ritial point The trouble with a tep repone i that the higher frequeny information i ontained in the early part of the repone For the intereted reader the two tranfer funtion are (74) Figure 74 Step repone of G() and G r () Figure 75 Frequeny repone of G() and G r () 87

88 Claial Controller Deign and (75) Before onluding thi etion it i probably worth mentioning that although relay autotuning ha been primarily aoiated with PID ontroller it an alo be ued for other imple ontroller, uh a phae lead and phae lag [77] Alo by inluding known network with the relay, additional information an be obtained, for intane inluding a tuned filter at the etimated limit yle frequeny will make the limit yle almot inuoidal and yield more aurate ritial point information 744 Further Deign Apet When a tranfer funtion model i available for a plant for whih a PID ontroller i to be ued then a frequeny repone approah to ahieve ertain propertie, ay a phae margin a ued for the lead and lag network deign, an be ued Often to make it eaier thi i done with a fixed ratio for T i /T d, typially 4 ine the approximate Bode amplitude diagram for thi i a V hape Pole plaement deign are alo often uggeted but thee have a major diffiulty that for the ideal PID ontroller in the error hannel one ha two zero in the loed loop tranfer funtion Their effet on the loed loop repone i not eay to predit and their loation i affeted by the hoie of the pole In general the bet method of deign for eleting parameter of fixed form ontroller i to ue optimiation method, whih will be diued in the next hapter Pratial PID ontroller alway have a faility to prevent integral windup that i a mehanim, and many algorithm are ued, for topping the integrator integrating when plant input or atuator aturation our Alo it i quite ommon for PID ontroller to be old in pair a they are often ued in aade in proe ontrol, a illutrated in the blok diagram of Figure 76 The et point for the inner loop ontroller ome from the outer loop ontroller and two meaurement are available a feedbak from the proe The main advantage i to obtain a fater reation to the inner loop diturbane D But often an improved input-output repone an alo be ahieved Alo the inner loop ontroller i often et in jut the P or PI mode Figure 76 Blok Diagram of Caade Control 88

89 Claial Controller Deign 75 Referene 7 Ziegler JG, Nihol NB Optimum etting for automati ontroller Tranation of ASME, 94, 64: Cohen GH, Coon GA Theoretial onideration of retarded ontrol Tranation of ASME, 953, Zhuang M, Atherton DP Automati tuning of optimum PID ontroller Proeeding of IEE, Part D, 993, 4: Wang FS, Juang WS, Chan CT Optimal tuning of PID ontroller for ingle and aade ontrol loop Chemial Engineering Communiation, 995, 3: O Dwyer, A Handbook of PI and PID ontroller tuning rule, 6, Imperial College Pre, UK 76 Atrom, KJ and Hagglund, T Automati tuning of imple regulator with peifiation on phae and amplitude margin Automatia 984 Vol, pp Atherton DP and Boz AF, Autotuning of Phae Lead Controller 4th IFAC Workhop on Algorithm and Arhiteture for Real-Time Control (AARTC 97), Vilamoura, Portugal, 9 April 997, pp 6 89

90 Parameter Optimiation for Fixed Controller 8 Parameter Optimiation for Fixed Controller 8 Introdution The bai onept here i to optimie the ontroller parameter to meet a performane riterion Before the prevalene of digital omputer riteria were put forward for whih analytial reult ould poibly be found, or omputation ould be done uing analogue imulation A logial hoie wa to hooe a riterion baed on minimiation of the error over time, a the objetive of good ontrol i to maintain a minimum error between the deired and atual output, Thu integral performane riteria of the form (8) where f(e(t),t) i a funtion of time and the time varying error, were uggeted Typial riteria ued are ummaried in Table 8 Funtion f e(t) t e(t) t e(t) e (t) [te(t)] Name Integral abolute error IAE Integral time abolute error ITAE Integral time quared abolute error IT AE Integral quared error ISE Integral quared time error ISTE [t n e(t)] Integral quared time to n error IST n E Table 8 Lit of Some Error Funtion It i poible in priniple to obtain analytial olution for the lat three ine the integral quared error, denoted by J, an be found in the -domain from (8) whih i known a Pareval integral It an be evaluated when E() = ()/d() i a ratio of polynomial in, a given in Table D in Appendix D for low order polynomial d(), and for higher order polynomial, d(), an be evaluated uing reurion relationhip a given by Atrom [8] For the higher time weighting (83) 9

91 Parameter Optimiation for Fixed Controller an again, in priniple, be evaluated in the -domain by utiliing the time multipliation formula of the Laplae tranform (Theorem (v) Appendix A) The diffiulty for hand alulation i that the order of d() double with eah differentiation, however, it i eay to write a omputer program to ompute the reult for ome low value of n An exellent treatie on the analytial approah whih alo onider a weighting on the ontrol effort in the performane riterion and poible atifation of ontraint i referene [8] 8 Some Simple Example Here ome imple analytial example are given to illutrate the approah and bring out ome bai idea For realiti pratial problem, however, reult will normally have to be obtained omputationally Example Conider a feedbak loop with G = /, ie an integrator, and G = K a gain For a unit tep input R = / and E = /( K) Clearly e = exp(-kt) and ine the expreion for e i o imple it integral quared value an be found from either the time domain or - domain integral to give the ISE = /K Thi i a expeted, ine the maximum phae lag of the loop i 9 it remain table no matter how high the gain However, the initial value of the ontrol ignal at the input to the plant, given by u = Kexp(-Kt), inreae a K inreae One way to find a finite gain value i to put a ontraint on ome funtion of u A imple olution i to minimie the time domain integral (84) whih i eaily hown, by ubtituting u = Ke, to have the value I = ( λ K ) / K (85) By differentiation, it i found that K = /λ yield the minimum value for I of λ Thi example although trivial bring out the point that are ha to be taken in obtaining olution to optimiation problem It i important to undertand the problem o a to know whether a olution will only exit if ome ontraint are impoed and alo when a minimum ha been found that it i realiti Sytemati approahe may be neeary, for example if a ontroller ha two variable parameter it may be deirable to fix one initially and jut look at the effet of varying the other 9

92 Parameter Optimiation for Fixed Controller Example Conider G = /( ) and G = K In thi ae, it i eaily hown that if K i inreaed the ytem will go untable for K > o there hould be a value of K < whih minimie the ISE For a unit tep input one obtain E = 3 K Uing Table D to evaluate I 3 give (86) 3K I3 = (87) K( K) Differentiating to find the minimum yield K = /3 and the orreponding minimum value of the performane index i I 3min = 5 Cheking the tep repone for thi value of K how it to have an overhoot of 36% Inidentally, a eond order tranfer funtion orreponding to the dominant omplex pair of pole ha an overhoot of 4% 9

93 Parameter Optimiation for Fixed Controller Example 3 A another example onider the ontrol of a double integrator plant, G() = /, by a phae lead ontroller with tranfer funtion ( T)/( at), in both the forward, G, and feedbak, H, path With the ontroller in G, the value of, E, for R a unit tep i (88) and uing Table D for I 3, give for the ISE, (89) Differentiating with repet to T, how that the optimum value of T = a -/, and the orreponding minimum value of I 3 i a / /( a) Thi an be een to be infinite when a =, a the ytem i neutrally table, and tend to zero a a tend to zero, whih reult in the derivative kik of the ontrol ignal tending to infinity On the other hand if the ontroller i plaed in H, the loed loop tranfer funtion i (8) The error ignal i E = R C (8) o that (8) giving for a unit tep input R = / (83) Again uing the Table D one obtain T 3αT 3α T I 3 = (84) T ( α) Differentiating to find the optimal value of T yield T = ( 3 3 ) / (85) α α 93

94 Parameter Optimiation for Fixed Controller and the orreponding value of the ISE i / ( 3α 3α ) ISE = (86) min α Differentiation of thi expreion how that the abolute minimum value obtainable i 866 when a = /3 and the value of T = 73 Three imple example have been taken to illutrate the analytial approah to minimiing the ISE, whih orrepond to n = in the general riterion of equation (83) Thi ha been done to illutrate the proedure whilt keeping the algebra relatively imple If, for example, one wihed to invetigate the lat example for the ISTE riterion, that i n =, then one ha to differentiate E() with repet to Thi inreae the order of the denominator from 3 to 6, the algebra for the integral beome horrendou and differentiation of the reult i then required for the optimum value Computationally, however, the minimum an be found very quikly, one elet value for T and a, evaluate the ISTE and ha an optimiation algorithm built around to adjut T and a to onverge to the optimum value, whih will of oure exit if the ompenator i in H One reaon for having overed the laial optimization approah in ome detail i that it lead naturally to the onideration of tandard form Thee provide an intereting loed loop diret ynthei approah for obtaining ontroller parameter 94

95 Parameter Optimiation for Fixed Controller 83 Standard Form Baed on the approah of the previou etion it i poible to obtain normalied loed loop tranfer funtion whih atify error performane riteria Their value i that they indiate good pole loation for the loed loop tranfer funtion To illutrate the approah onider a feedbak ytem with G = /( a), G = K and H = For a unit tep input E = ( a)/( a K) The ISE an be found from Table D and ine the denominator of E i eond order it i denoted, I, and i given by (87) Thi an be hown to be a minimum for a = K and the orreponding loed loop tranfer funtion i K T ( ) = (88) K K Comparing thi with the tandard form for the eond order equation of how that o T ( ) = ω ζω ω (89) o o K = ω o, o the natural frequeny inreae with K but the damping ratio V = 5 Thi value of V give a tep repone overhoot of around 6% The value i le than the unit value required for no overhoot in the tep repone and the value of 77 required for no peak in the frequeny repone Time aling the tandard eond order equation (89), that i replaing /ω o by n, give the tranfer funtion T ( n ) = n ζ n (8) Thi equation i known a the time normalied form and a explained in hapter 3 ha exatly the ame time repone a eqn(89) but eqn (89) i a fator ω o fater Eqn (8) with ζ = 5 i referred to a the tandard form of the eond order all pole loed loop tranfer funtion whih minimie the ISE, J, for H = Note alo that the forward loop tranfer funtion, GG, mut ontain an integrator to enure zero teady tate error to a tep input Standard form for any order of the denominator polynomial and for variou integral performane riteria an be found and written, with the ubript n dropped from, a T ( ) j j (8) d d = j 95

96 Parameter Optimiation for Fixed Controller They have been derived in referene [83] for the more general performane index IST n E for different value of n and are denoted a:- T ( ) = j (8) j j d d j The required oeffiient value a well a the reulting value of the performane index are given in Table 8 Note the value of the index inreae for larger n beaue of the time weighting fator a the ettling time i greater than unity It i intereting to look at the oeffiient of thee tranfer funtion Firt, purely of aademi interet, i the fat that for the ISE, that i n =, all the oeffiient are integer value More important, however, i the fat that a n i inreaed the oeffiient inreae in value and the tep repone have le overhoot with only a mall hange in ettling time Thi an be een for the eond order ytem a the damping ratio V, equal to d / in Table 8, inreae a n inreae Thee point are further demontrated by Fig 8 whih how the tep repone for j = 4 for n = to 3 4 Step repone of T () 4 Magnitude ISE ISTE IST E IST 3 E Time() Figure 8 Step repone for j = 4 96

97 Parameter Optimiation for Fixed Controller Table 8 All Pole Standard Form for T j (From referene 84) Sine adding a ompenator often reult in a loed loop tranfer funtion having a zero, a uggeted ue for thee tandard form in ontroller deign ha been to add a prefilter with a pole to anel the zero To avoid the ue of a prefilter tudie have been done to invetigate tandard form with a zero, that i for T j ( ) = (83) j j d d j 97

98 Parameter Optimiation for Fixed Controller In early work on thi topi reult were given to minimie the ITAE riterion but with the requirement that the ytem hould alo have zero teady tate error to a ramp input, whih require the ontraint that = d Reently [84] reult have been derived without thi ontraint but the required d oeffiient vary with the hoie of, a illutrated in Fig 8 for T 4 () It i alo poible to plot how the pole of the tranfer funtion vary with the zero parameter, and ome of thee an be found in referene [84] Thee plot how that the optimal pole poition vary appreiably with the value of the zero ISE ISTE IST E IST 3 E d oeffiient of T () 4 6 d d, d, d3 5 4 d 3 d C Figure 8 d oeffiient for T 4 for 3 value of n The onept of trying to deign a ontroller o that the loed loop tranfer funtion ha a peifi form i ueful, a it addree the loed loop performane diretly, and will be illutrated by an example in the next etion, a well a being given further onideration in hapter There are many loed loop tandard form that might be hoen apart from thoe baed on the riteria of Table 8 For example, the Butterworth filter form ould be eleted 84 Control of an Untable Plant A imple linearied model for a magneti upenion i often taken a G( ) = K p λ (84) It i required to ontrol thi plant tranfer funtion with a PID type ontroller If the laial PID ontroller of equation (73) i ued in the error hannel then the loed loop tranfer funtion, T(), i given by T = ( ) 3 Ti T T i K d K K p K ( p ( K K T T i p d ) T λ) K i K p (85) 98

99 99 Parameter Optimiation for Fixed Controller It an be een that the 3 pole of the tranfer funtion an be alloated by the hoie of the three ontroller parameter, but the two zero annot then be loated independently a their loation i dependent on the parameter hoen to loate the pole If, however a PI-PD ontroller i ued, that i a ontroller whoe output i obtained from the error feeding the PI and the plant output the PD, then the open loop tranfer funtion i (86) where the ontroller parameter are K and T i for the PI term and K f and T d for the PD term Thi give the loed loop tranfer funtion (87) In thi ae the 3 pole and the zero an be adjuted independently by the ontroller parameter To deign the ontroller uing the tandard form approah equation (87) an be written in normalied form a (88) where α i the timeale fator (K K p /T i ) /3 by whih the ytem i fater than the normalied one

100 Parameter Optimiation for Fixed Controller In priniple the time ale, α, an be eleted by the hoie of K, and the oeffiient for the hoen tandard form; d by the hoie of T d, d by the hoie of K f, and by the hoie of T i In pratie K will normally be ontrained to an upper value, poibly to limit the initial ontrol effort, and T i will involve a trade off between the value hoen for α and Conider, for example, the ae of the plant parameter K p = and λ = 4 and ontraining K to a maximum of Then two poible deign ould be:- ) Time aling by Thi mean eleting α = whih give T i = 5 and = 5 For thi value of the value of d and d, repetively, to minimie the ISTE are 595 and Thi give T d = 595 and K f = 54 ) No time aling Thi mean eleting α = whih give T i = and = For thi value of the value of d and d, repetively, to minimie the ISTE are 5 and 99 Thi give T d = 7 and K f = 495 Beaue both deign are baed on minimiation of the ISTE the loed loop tep repone are quite imilar in hape with around % overhoot, and the firt twie a fat a the eond The fater time aled repone in thi ae ha been ahieved not by inreaing the ontroller gain K but by variation of T i and 85 Further Comment The topi of optimiing the parameter of a fixed form ontroller ha been diued in thi hapter Some imple analytial example baed on the ISE integral performane riterion were firt diued and then a imple algebrai approah baed on tandard form wa given, whih will be onidered further in hapter It i very ommon to ue fixed form ontroller in indutrial deign and with today omputation failitie optimiation of the parameter to meet the deign peifiation i an exellent pratial approah Referene [85, 86] deribe oftware that ha been developed to do deign uing optimiation tehnique and to how how the variou performane ontraint might be traded off againt eah other When a large number of riteria need to be onidered thee program an beome very ompliated Optimiation approahe, uh a the ue of integral performane riteria when there are only a few variable parameter, an often be ahieved uing imulation, a the performane riteria lited in Table 8 an eaily be found from a imulation run One may then either interat with the imulation manually to obtain the optimal parameter or do o with an optimiation program ontrolling the imulation run

101 Parameter Optimiation for Fixed Controller 86 Referene 8 Atrom, KJ Introdution to Stohati Control Theory Aademi Pre, 97 pp Newton, GC, Gould, LA and Kaier, JF Analytial Deign of Linear Feedbak Control Wiley, New York, Atherton, DP and Boz, AF Uing Standard Form for Controller Deign Proeeding UKACC International Conferene on Control 998 (Control 98), Univerity of Wale, Swanea, September 998, pp Boz, AF Computational Approahe to and Comparion of Deign Method for Linear Controller DPhil thei, Univerity of Suex, Fleming PJ: Managing Competing objetive in ontrol ytem engineering deign: Pro UKACC International Control Conferene, Control 6, Glagow, 6 86 Fleming PJ, Purhoue RC, Lygoe RJ: Many-objetive optimization: An engineering deign perpetive, Leture Note in Computer Siene 34, pp 4 3, 5

102 Further Controller Deign Conideration 9 Further Controller Deign Conideration 9 Introdution Additional apet related to ompenator deign are overed in thi hapter The firt topi diued in the next etion i lag-lead ompenator deign an extenion of the lead and lag ompenator diued in hapter 7 In the next two etion ome apet of peed and poition ontrol are diued with partiular emphai on the rejetion of teady tate diturbane It i hown that thi require an integration in the ontroller whih ompliate the deign for obtaining a good tep repone Simple rigid body type plant tranfer funtion are ued in thee etion, wherea in many ae it i required to ontrol the peed or poition of a haft whih i driven through a flexible link Typially thi reult in a tranfer funtion ontaining not only omplex pole but alo omplex zero To illutrate the diffiultie of ontrolling uh ytem with a erie ompenator the next etion onider the ontrol of a plant tranfer funtion with omplex pole The final etion diue the problem of the effet of parameter variation on ontrol ytem performane Although thi i a topi of major interet in deign it i a very diffiult theoretial one and few reult of pratial ignifiane have been obtained However modern alulation and imulation method are now o fat that the inreae in time required for doing tudie with different et of parameter i uually eonomially jutifiable 9 Lag-Lead Compenation A mentioned in etion 7 it may not be poible to ahieve a atifatory phae lead deign and the bandwidth ahievable by a phae lag deign may be le than deired It may be poible to improve the loop performane by a lag-lead deign Thi i illutrated by taking a ytem with the ame tranfer funtion dynami but with a higher gain in the numerator, whih might be required to redue the teady tate error to a ramp input, K v, a mentioned in etion 7 Conider therefore (9) The loed loop ytem with thi G() and H() = G () = i neutrally table o that the phae margin i zero ompared with a value of 56 for the previouly onidered tranfer funtion (9)

103 Further Controller Deign Conideration To add a phae lead network to G() to ahieve the ame phae margin of 4 will require a lead of around 6 whih i very high and the deign may not be ahievable An alternative i to ue a lag-lead deign where the gain i redued by a lag network before the gain roover frequeny i reahed If after adding the lag network the frequeny repone around the gain roover frequeny i imilar to that of G () then the phae lead network of etion 7 will be uitable Thu, hooing a lag network with tranfer funtion (93) and plotting the Bode diagram of the erie ombination G G, it i een to be almot idential to G, in the required region, a hown in Figure 9 Figure 9 Bode diagram for the example Adding the phae lead ompenator of etion 7 the lag-lead ompenator i (94) The reulting ytem ha a phae margin of 39 at a frequeny of 47 rad/ The loed loop tep repone i hown in Figure 9 together with that uing a lag network deign, and a expeted an appreiable inreae in the peed of repone ha been ahieved 3

104 Further Controller Deign Conideration Figure 9 Comparion of tep repone 4

105 Further Controller Deign Conideration 93 Speed Control Control of peed i a ommon problem enountered by many ontrol engineer, perhap the mot ommon well known ituation being the ruie ontrol fitted to many automobile Here it will be aumed that the peed i rotary and the tranfer funtion from the torque to the load, where the peed ha to be ontrolled, i (95) In pratie there may be more than one time ontant but quite often there i one dominant one a aumed here A problem whih often arie, however, i when the oupling from the drive torque to the load i not rigid and a more ompliated tranfer funtion reult with both omplex pole and zero Thi preent a muh more diffiult ontrol problem whih will be ommented on further in etion 95 The ontrol loop i typially a hown in Figure 5, where N i aumed zero and H will onvert peed, ay radian per eond to voltage with a time ontant whih i probably mall enough to be negleted In order that the peed hould remain ontant at the required value with a fixed referene input, R, auming H i alibrated orretly, the ontroller G mut ontain an integration Thi an alo be een from Figure 5 to be a requirement for a ontant diturbane D to have no effet on the load peed in the teady tate a the output of the ontroller mut have a ignal equal and oppoite to D Thu, G, i typially a PI ontroller and the open loop tranfer funtion i (96) where K P and K i are repetively the ontroller proportional and integral gain and k = KK i H and T = K P / K i Moving H inide the loop a explained in etion 53, the loed loop tranfer funtion an be written a (97) One approah to the deign i to hooe T = T, o that T() beome a ingle time ontant tranfer funtion, although uh a zero-pole anellation will never be orret in pratie due to unertainty in the ytem parameter Thi doe provide a imple analytial approah and k an in priniple be inreaed, by inreaing K i and onequently K P for a given T, a muh a required to peed up the ytem repone In pratie k will be retrited a inreaing k inreae the maximum magnitude of the ontroller output ignal Another approah i to do a deign baed on the open loop tranfer funtion of equation (96) with T T, uing ay frequeny repone or root lou tehnique One point to note i that if the loed loop tranfer funtion i written a (98) 5

106 Further Controller Deign Conideration that i with ritial damping for the eond order denominator, where ω = k T and ω o = ( kt )/T, o / then an overhoot may till exit in the loed loop tep repone due to the zero It i eaily hown that the tep repone of equation (97) i (99) and that it ha a maximum when t = T /(ω o T ) Thu an overhoot will exit when thi i poitive, that i for ω o T > 94 Poition Control Many of the early appliation of ontrol engineering were involved with poition ontrol due to the requirement for aurate poition ontrol of gun and other devie during the 94 Indeed everal of the early textbook written in ontrol engineering ued the word ervomehanim in the title to aount for the fat that muh of their overage wa related to poition or peed ontrol Today there remain many requirement for aurate poition ontrol from large drive and roboti to head for reading or writing to rotary torage media Again if flexure in the drive dynami an be negleted the imple rigid body tranfer funtion for the plant in Figure 5 of a poition ontrol may often be taken a G() = K / ( T) (9) There will be no teady tate error to a tep input a G() ontain an integration term, o that a atifatory loed loop tep repone may be ahieved with G () a ontant gain, phae lead or lag network Alo veloity feedbak may be ued whih mean that the tranfer funtion H() i of the form T and the loed loop tranfer funtion will beome (9) Apart from the dynami repone requirement more tringent teady tate requirement are often required of a poition ontrol ytem uh a being able to follow a ramp input with no poition error or rejet the effet of any ontant diturbane D on the output Both thee require the ontroller to have an integration term If H = and G i a PI ontroller G = (K p K i ) then the loed loop tranfer funtion i (9) On the other hand if veloity feedbak i ued H = T and with G () = K / the loed loop tranfer funtion i (93) 6

107 Further Controller Deign Conideration Thi avoid the zero and therefore i omewhat eaier to deign for a required dynami repone One imple way i to make ue of tandard form The tranfer funtion in normalied form i (94) / 3 where the time ale fator α = ( K C K / T ) Sine there are only two variable ontroller parameter K C and T one an hooe the value o that the denominator oeffiient fit a tandard form but in doing o the time ale fator, α, i fixed Figure 93 how the loed loop tep repone obtained uing thi deign proedure to ahieve the performane indie J n for n = to 3 Figure 93 Repone to a unit tep at unit time for the tandard form deign 95 A Tranfer Funtion with Complex Pole A mentioned earlier plant tranfer funtion may involve omplex pole whih may be lightly damped Deigning of atifatory erie ompenator for uh ytem i not eay o thi i a problem whih will be examined again in hapter when tate feedbak ompenation i diued To ee the diffiultie onider the tranfer funtion G ( ) = (95) ( ) where the omplex pole have a natural frequeny of unity and a damping ratio of The phae of the Bode plot hange rapidly near the reonant frequeny of unity a een in Figure 94 and with a unit gain ontroller the loed loop ha a gain margin of 6dB and a phae margin of 89 The tep repone, however, ha oillation on it due to the omplex pole (ee urve K = in Figure 95) 7

108 Further Controller Deign Conideration Figure 94 Bode diagram of tranfer funtion of equation (95) The poor tep repone i due to the relatively low gain margin, not low phae margin, and a imple phae lead deign to peed up the repone i not poible A poible approah i to ue a ompenator with two zero and two pole with the former being hoen to anel the omplex pole of the plant Chooing K ( G( ) = ) (96) 8

109 Further Controller Deign Conideration K an be hoen equal to 5 whih give a gain margin of db, a phae margin of 44 and a tep repone with around 5% overhoot The problem i if the parameter of the plant are not a aumed Figure 95 how tep repone for a ontroller with gain and 5, the lower repone, and with the tranfer funtion of equation (96) with K = 5 for the three ae of the plant omplex pole having the nominal damping ratio of and alo 5 and 5 The tep repone i thu hardly affeted by an inorret aumption for the damping ratio On the other hand the open loop Bode diagram are hown in Figure 96 for the ompenator plu the plant with reonant frequenie for the pole of 8 and, not the nominal value of unity It an be een that the gain margin for the reonant frequeny of 8 i very mall and therefore not urpriingly the tep repone i highly oillatory a hown in Figure 97, together with that for a natural frequeny of It i thu een from thi example that the loed loop tep repone i very enitive to an over etimation of the reonant frequeny of the plant pole Figure 95 Step repone for different ontroller and plant pole damping ratio 9

110 Further Controller Deign Conideration Figure 96 Bode diagram for ompenating ontroller plu plant for different natural frequenie of the plant pole Figure 97 Step repone for loed loop with ontroller and plant for different natural frequenie of the plant pole

111 Further Controller Deign Conideration 96 The Effet of Parameter Variation Mot method of ontroller deign, a ha been een, require the ue of a mathematial model for the proe In pratie thi model, whih may be alled the nominal model, i alway an approximation of the real ituation Further the model may hange dependent on environmental hange or with age The effet of inauraie in a model on the ytem performane ha therefore alway been a onern of the deign engineer The omment in thi etion will aume that the form of the model i not in doubt but unertainty exit in the etimate for ome of it parameter From a pratial viewpoint today imulation failitie are o fat that for the majority of ituation after a deign i ompleted multiple imulation an be done to ae the effet of hange in model parameter However ome theoretial reult are available whih will be ommented on here Muh of the reent interet in thi topi wa tarted a a reult of the work on the tability of interval polynomial by Kharitonov onidered in the next etion 96 Stability of Interval Polynomial Muh reent work on ytem with unertain parameter ha been baed on Kharitonov reult [9] on the tability of interval polynomial Kharitonov howed that for the interval polynomial 3 4 n P ( ) = a a a a3 a4 a n (97)

112 Further Controller Deign Conideration where ], [ i i i a a a, n i,,, =, the tability of the et ould be found by applying the Routh-Hurwitz riterion to only the following four polynomial = = = = ) ( ) ( ) ( ) ( a a a a a p a a a a a p a a a a a p a a a a a p (98) Although thi may eem a urpriing reult it i eaily proved from the Mikhailov riterion of etion 54 It an be eaily hown that the value et of an interval polynomial at a fixed frequeny i a retangle (Kharitonov retangle) a hown in Figure 98, that i the value of every polynomial of the family at that frequeny lie within or on the retangle, whoe ide are parallel to the real and imaginary axe Sine the ide of the retangular value et are parallel to the real and imaginary axe, it an eaily be hown that the exluion of the origin from the retangular value et at all frequenie, whih will be required for all the polynomial to atify the Mikhailov riterion, an be heked by uing the orner point whih orrepond to the four Kharitonov polynomial The Kharitonov theorem i only appliable to interval unertain parameter but unfortunately the harateriti equation of even imple ontrol ytem do not normally have an interval unertainty truture For example, to take a imple ae, onider a plant tranfer funtion model of the form (99) where unertainty may exit in K, T and T If the plant i in the feedbak loop of Figure 5 with G = H = the harateriti equation for aeing tability i ) ( ) ( 3 = = K T T T T δ (9)

113 Further Controller Deign Conideration whih i not an interval polynomial The only imple way to ue the Kharitonov reult i Fig98 : Kharitonov box and the Mikhailov lou for p () to overbound and underbound the parameter of 3 and, whih produe a very onervative reult Auming Ti [ Ti, Ti ] the gain required to atify the four equation i K < ( T T ) T T For thi peifi example the diret appliation of the Routh-Hurwitz riterion give / K ( T T ) / TT = (/ T ) (/ T the reult < ), whih obviouly ha a minimum value when T and T have their maximum value For the peifi ae of the bound T [, ] and T [,4] the exat reult of the Routh-Hurwitz riterion i K < 3/4 wherea the Kharitonov reult i K < 3/8, whih i onervative by a fator of The fat that one an obtain an exat olution from the Routh-Hurwitz riterion i beaue thi i one of a few unique ituation It doe, however, erve to how the onervativene of reult that an be expeted from the Kharitonov riterion when applied to pratial ontrol ituation, beaue of the parameter dependene typial of the term in the loed loop harateriti equation 96 Envelope on Bode Plot It i poible to obtain bound on Bode plot for tranfer funtion with variable parameter [9] To ee thi onider a general tranfer funtion fatoried into zero-pole form a given below (9) N where in the denominator repreent a pole of multipliity N at the origin and a m N n b It i aumed that the parameter and τ are not known exatly but vary within the following interval (9) 3

114 Further Controller Deign Conideration The maximum gain and maximum phae lag at a partiular frequeny will be obtained from the produt of the maximum gain and the um of the maximum phae lag at that frequeny of the individual element, with a imilar reult for the minimum value Therefore onidering the individual element in turn beginning with the time delay, ine it gain i alway unity, the maximum (minimum) phae lag i obtained with the maximum (minimum) value of τ Alo from kethe of Bode gain and phae diagram for a ingle time ontant tranfer funtion it i obviou that the urve for T give the maximum gain and minimum phae hift and thoe for T give the minimum gain and maximum phae hift, repetively The urve for all other value of T lie within the repetive gain and phae boundarie of thee plot Finally the Bode gain diagram for the eond order omplex pole tranfer funtion (93) only ha a peak in the repone if ς i le than 77 Thu, if the Bode gain and phae diagram are onidered for thi tranfer funtion with ς [ ς, ς ] and ω [ ω, ω ], and are drawn for the four ae of ω with ς and ς, and ω with ς and ς it an eaily be een that: 4

115 Further Controller Deign Conideration The minimum magnitude if (a) ς > 77 i given for all ω by ω G ( jω) = (94) ω ςω jω ω and if (b) ς < 77 then from ω = to ω x = (ω ω ( ς ) /( ω ω ) it i given by ω G ( jω) = (95) ω ςω jω ω and from ω x to by ω G ( jω) = (96) ω ςω jω ω The maximum magnitude if (a) ς > 77 i given for all ω by ω G ( jω) = (97) ω ςω jω ω and if (b) ς < 77 then from ω = to ω p min = ω ς it i given by ω G ( jω) = (98) ω ςω jω ω The maximum value of the gain at ω p min i ς ς (99) and the maximum poible gain remain ontant at thi value independent of ω until max = ω ς ω p and then for ω [ ω p max, ) it i given by ω G ( jω) = (93) ω ςω jω ω 3 The maximum phae for ω [, ω ) i given by ω arg[ G ( jω)] = arg[ ] (93) ω ςω jω ω 5

116 Further Controller Deign Conideration and for ω [ ω, ) by ω arg[ G ( jω)] = arg[ ] (93) ω ςω jω ω 4 The minimum phae for ω [, ω ) i given by ω arg[ G ( jω)] = arg[ ] ~ (933) ω ςω jω ω and for ω [ ω, ) by ω arg[ G ( jω)] = arg[ ] (934) ω ςω jω ω Unfortunately it i not poible to derive imilar reult for Nyquit plot Some reult have been obtained but they do not provide aurate bound on the plot [93] Alo although aurate bounding of the Bode plot i obtained by the above approah reult obtained uing them are till onervative beaue the link between the gain plot and the phae plot of a peifi tranfer funtion i lot To how thi, onider the loed loop tability problem for the open loop tranfer funtion of equation (99) with the ame bound on the time ontant The bound of the Bode plot are hown in Figure 99 and to enure tability one ha to ue the lower bound of the phae plot and the upper bound of the gain plot Thi give a value for tability of K < 46, an improvement on the Kharitonov reult but till very onervative 5 Gain db -5 - Phae deg Frequeny(rad/e) -3 - Frequeny(rad/e) Figure 99 Stability from the Bound of the Bode plot 6

117 Further Controller Deign Conideration 97 Referene 9 Kharitonov, VL: Aymptoti tability of an equilibrium poition of a family of ytem of linear differential equation, Differential Equation, 979, 4, pp Tan, N and Atherton, DP: New Approah to Aeing the Effet of Parametri Variation in Feedbak Loop IEE Proeeding Control Theory and Appliation, Vol 5, No Marh 3, pp 93 Hollot, CV and Tempo, R: On the Nyquit envelope of an interval plant family, IEEE Tran Automat Contr, 994, 39, (), pp

118 State Spae Method State Spae Method Introdution State pae modelling wa briefly introdued in hapter Here more overage i provided of tate pae method before ome of their ue in ontrol ytem deign are overed in the next hapter A tate pae model, or repreentation, a given in equation (6), i denoted by the two equation () () where equation () and () are repetively the tate equation and output equation The repreentation an be ued for both ingle-input ingle-output ytem (SISO) and multiple-input multiple-output ytem (MIMO) For the MIMO repreentation A, B, C and D will all be matrie If the tate dimenion i n and there are r input and m output then A, B, C and D will be matrie of order, n n, n r, m n and m r, repetively For SISO ytem B will be an n olumn vetor, often denoted by b, C a n row vetor, often denoted by T, and D a alar often denoted by d Here the apital letter notation will be ued, even though only SISO ytem are onidered, and B, C, and D will have the aforementioned dimenion A mentioned in hapter the hoie of tate i not unique and thi will be onidered further in etion 3 Firt, however, obtaining a olution of the tate equation i diued in the next etion Solution of the State Equation Obtaining the time domain olution to the tate equation i analogou to the laial approah ued to olve the imple firt order equation (3) The proedure in thi ae i to take u =, initially, and to aume a olution for x(t) of e at x() where x() i the initial value of x(t) Differentiating thi expreion give o that the aumed olution i valid Now if the input u i onidered thi i aumed to yield a olution of the form x(t) = e at f(t), whih on differentiating give Thu the differential equation i atified if 8

119 State Spae Method whih ha the olution t t a f ( t) = [ e τ ] u( τ ) dτ, giving at aτ x( t) = e [ e ] u( τ ) dτ, where τ i a dummy variable Thi olution an t a( t τ ) be written x( t) = e u( τ ) dτ o that the omplete olution for x(t) onit of the um of the two olution, known a the omplimentary funtion (or initial ondition repone) and partiular integral (or fored repone), repetively and i t at a( t τ ) x( t) = e x() e u( τ ) dτ (4) For equation () x i an n vetor and A an n n matrix not a alar a and to obtain the omplimentary funtion one aume i now a funtion of a matrix, whih i defined by an infinite power erie in exatly the ame way a the alar expreion, o that (5) where I i the n n identity matrix Term by term differentiation of equation (5) how that the derivative of e At i Ae At and that atifie the differential equation with u = e At i often denoted by φ(t) and i known a the tate tranition matrix Uing the ame approah a for the alar ae to get the fored repone the total olution i found to be (6) Aτ It i eaily hown that the tate tranition matrix ϕ( τ ) = e ha the property that ϕ( t τ ) = ϕ( t) ϕ ( τ ) o that equation (6) an be written alternatively a (7) Thi time domain olution of equation () i ueful but mot engineer prefer to make ue of the Laplae tranform approah Taking the Laplae tranform of equation () give (8) whih on rearranging a X() i an n vetor and A a n n matrix give (9) Taking the invere Laplae tranform of thi and omparing with equation (7) indiate that () 9

120 State Spae Method Alo taking the Laplae tranform of the output equation () and ubtituting for X() give () o that the tranfer funtion, G(), between the input u and output y i () Thi will, of oure, be the ame independent of the hoie of the tate 3 A State Tranformation Obviouly there mut be an algebrai relationhip between different poible hoie of tate variable Let thi relationhip be x = Tz (3) where x i the original hoie in equation () and () and z i the new hoie Subtituting thi relationhip in equation () give whih an be written (4)

121 State Spae Method Alo ubtituting in the output equation () give (5) Thu under the tate tranformation of equation (3) a different tate pae repreentation (T AT, T B, CT, D ) i obtained If the new A matrix i denoted by A z = T AT then it i eay to how that A and A z have the following propertie ) The ame eigenvalue ) The ame determinant 3) The ame trae (Sum of element on the main diagonal) There are ome peifi form of the A matrix whih are often ommonly ued in ontrol engineering and not unurpriingly thee relate to how one might onider obtaining a tate pae repreentation for a tranfer funtion, the topi of the next etion 4 State Repreentation of Tranfer Funtion Thi topi wa introdued in etion 3 where the ontrollable anonial form for a differential equation wa onidered Here thi and ome other form will be onidered by making ue of blok diagram where every tate will be an integrator output To develop ome repreentation onider the tranfer funtion (6) 4 Controllable Canonial Form A een from equation () the firt n- tate variable are integral of the next tate, that i, or a hown in the equation by x ( j ) = x j, for j = to n Thu the blok diagram to repreent thi i n integrator in erie The input to the firt integrator i x n and it value i given by x n = a x a x a x u, the lat row of the matrix repreentation of equation () The 3 numerator term are provided by feeding forward from the tate to give the required output Thu, for our imple example, thi an be hown in the blok diagram of Figure, done in SIMULINK, where ine the tranfer funtion i third order n = 3, there are three integrator, blok with tranfer funtion /, in erie Feedbak from the tate, where the integrator output from left to right are the tate x 3, x, and x, repetively, i by the oeffiient -8, -4 and -7 (negative and in the revere order of the tranfer funtion denominator) The numerator oeffiient provide feedforward from the tate, with the term from x 3

122 State Spae Method Figure Controllable Canonial Form Diagram for the Example The matrie for the tate repreentation are MATLAB ha a ompanion form, whih for any tate pae ytem G=(A,B,C,D), will be returned on typing y=anon(g, ompanion ) The ompanion repreentation y will have an A matrix whih i the tranpoe of the above A matrix and a B = ( ) T 4 Obervable Canonial Form The obervable anonial form i related to the ontrollable form by the following relationhip, T A o = A T, B o = C and C o = B T The ubript and o relate to the ontrollable and obervable form matrie repetively and T denote the tranpoe It i left to the intereted reader to develop a blok diagram imilar to Figure for thi form 43 Diagonal (or Modal) Form If the impule repone of G() i required then it evaluation by invere Laplae tranform require a partial fration expanion of G() Thi i G() = / 3 4 / 3 4, whih i imply a parallel onnetion of three firt order tranfer funtion The firt order tranfer funtion K/( a) an be modelled with one integrator a hown in Figure If the output of the integrator i denoted by the tate variable x then it tate and output equation are

123 State Spae Method Figure Diagram of Model for One State Variable Note that there i no unique value for b and a all that i required i that their produt hould equal K Thu a tate repreentation for G() ha A =, 4 = B and C = ( / 3 4 / 3) 3

124 State Spae Method where we have hoen to take all the B value a unity Thi form of the A matrix i known a a diagonal form and an alway be found if the denominator of the tranfer funtion ha real root To keep the matrix real for omplex root, ay σ ± jω, the orreponding row around the diagonal are replaed σ ω by the matrix For example if G() =, the ontrollable anonial form, with the ω σ ( ) matrie ubripted with, i A =, B = and C = ( ) A diagonal form i A Λ = 5 866, Λ = 73 B and = ( 5774 ) C Λ Thi i the one given by MATLAB if the intrution y=anon(g, modal ) i ued, where y i the new tate pae repreentation in the hoen anonial form modal A peial ae i when G() ha a repeated root, for example if G() =, whih ha a tate pae repreentation of A J =, ( ) B = and C J J = ( ) Thi an be een from the partial fration expanion of G(), whih i G() = ( ) The numerator oeffiient are in C, the three root, -, - remain on the diagonal but the off-diagonal unit term in A and the zero in B are due to the fat that the lat term of the partial fration expanion ha a input the output from the eond tate, not the input u Thi form of A matrix i known a a Jordan form and due to the numerial method ued annot be found with MATLAB 44 Guillemin Form Another imple way of obtaining a tate pae repreentation of a tranfer funtion i to make repeated ue of the tate repreentation of Figure 3 for the one-zero one-pole tranfer funtion (d )/( a) whih ha a tate pae repreentation (a,,e-ad,d) for the ingle tate x 4

125 State Spae Method Figure 3 Diagram for One State Variable for Guillemin Form Thu for a tranfer funtion with real pole and zero in fatored form given by G() = ( ) ( )/ ( 3) ( 5) one an plit it into one of everal poible erie (aade) ombination uh a and then ue the repreentation of Figure 3 for eah tranfer funtion to ontitute the overall model a in Figure 4, whih ha, auming the output of the integrator from left to right are x 3, x and x, repetively, the equation 3 x 3 = u x = 3x x x = 5x x x and y = x x Subtituting appropriately for the derivative give the tate repreentation 5 A = 3, B = and C = ( 3 ) where the A matrix i upper triangular Figure 4 Diagram for Guillemin Form State Repreentation 5

126 State Spae Method 5 State Tranformation between Different Form Given a tate pae repreentation then one an evaluate the orreponding tranfer funtion and ue thi to obtain a different tate pae repreentation In ome ae, however, it i more onvenient if one an obtain the peifi tate tranformation, T, diued in etion 3, that will do thi diretly It an be hown that any A matrix an be tranformed to a diagonal form by it own eigenvetor matrix Eigenvetor only define diretion, however, o that uh a matrix i not unique with a alar multiplier being allowed on any olumn vetor t i of T The eigenvetor, t i, orreponding to a partiular eigenvalue, i, of a matrix A i found from ( i I A) ti = for i = n For example, the A matrix ha a harateriti equation of, giving, o ha eigenvalue of - and - The orreponding eigenvetor are obtained from ( I A) t = and ( I A ) t =, yielding t ( a 3a)T = where a and b are any ontant Thu a b taking T = then the tranformation T - AT will yield the diagonal matrix A Λ = 3a 4b whatever the hoie of a and b However, if the tranformation wa applied to a tate pae repreentation = and t ( b 4b) T (A,B,C,D) the reulting (A Λ,B Λ,C Λ,D), would have different reult for B Λ and C Λ dependent on the hoie of a and b When the given A matrix i in ontrollable anonial form then it an be hown ( n ) that the olumn eigenvetor t i of T are given by ( ) T Vandermonde matrix t i =, whih i known a a i i i 6

127 State Spae Method 5 Tranforming to Controllable Canonial Form If it i required to find the ontrollable anonial form of a tate pae repreentation (A,B,C,D) then thi an be ahieved by a unique tranformation a not only i the A matrix of a peifi form but o alo i the vetor B From the two equation it an be hown that the olumn vetor t i of T are given, for i = n, by, et Here the a i, i = (n-), are the oeffiient of the harateriti equation of A, whih of oure form the lat row of A Some algebrai manipulation on thee equation how that the tranformation matrix T an be written a (7) 6 Evaluation of the State Tranition Matrix There are everal way to evaluate the tate tranition matrix below and ome of thee are outlined 6 Diret Expanion Thi i tediou and involve alulating power of A, ubtituting them in equation (5) and finding the exponential erie whih give eah term in the ummed matrix expreion 6 The Invere Laplae Tranform Thi involve finding φ(t) from the invere of equation (), that i φ(t) = - (I-A) - Thi i traightforward but very laboriou for alulating the required matrix inverion, exept for low order matrie, A One then ha to find the invere Laplae tranform of the individual matrix term whih are funtion of 7

128 8 State Spae Method 63 Ue of a Diagonal Tranformation If the matrix T i a tranformation whih diagonalie the A matrix to Λ then it an be hown that Thu, one T and it invere have been found thi approah require evaluation of the produt of three n x n matrie 64 Ue of the Cayley Hamilton Theorem Thi theorem tate that a matrix atifie it own harateriti equation Thu, if the matrix A ha a harateriti equation ) ( = = a a a n n n, then ) ( = = a A a A a A A n n n Thi mean that A n an be alulated in lower power of A and that any infinite erie of A,, an be expreed a 3 3 ) ( = n n A A A A I A f γ α γ γ γ for a matrix of order n Further all eigenvalue of A mut alo atify thi equation Thu when the funtion of the matrix A of order n i the exponential it eigenvalue and the matrix mut atify (8) The firt equation when ued for all the eigenvalue provide n equation whih an be olved for the n oeffiient α Subtituting thee in the eond equation enable the tate tranition matrix to be found 7 Controllability and Obervability Conider a tate pae repreentation (A,B,C,) with = 4 3 A, = B and ( ) C =

129 State Spae Method Then, in blok diagram form thi onit of the four mode at -, -, -3, -4 whih are onneted repetively to the input and output, output only, input only and to neither input or output The tranfer funtion from input to output i imply /() a the - mode i the only one onneted to both the input and output Sine the - mode i onneted to both input and output it i aid to be both ontrollable and obervable The - mode i aid to be unontrollable and obervable being onneted to the output only; the -3 mode ontrollable and unobervable being onneted to the input only; and the -4 mode i aid to be unontrollable and unobervable being onneted to neither the input or the output Given a tate pae deription it i deirable, a will be een in the next hapter, to know whih mode are in the different ituation exemplified by the four above mode A ytem i aid to be ontrollable if all the tate are ontrollable, and obervable if all the tate are obervable The formal definition are given below From the above example it i lear that only thoe mode whih are ontrollable and obervable appear in the tranfer funtion between input and output Thu, if a ytem with an n n A matrix i ontrollable and obervable the denominator of it tranfer funtion will be of order n (ie it will have n pole) 9

130 State Spae Method 7 Controllability A ytem i ontrollable if there exit an input u whih tranfer the initial tate x() to the zero tate x(t) = in a finite time t Given any SISO ytem, A (n n) and B (n ) matrie then it an be hown that the ytem will be ontrollable if the (n n) ontrollability matrix X = (B AB A B A n B) ha rank n It will be notied that thi matrix i the firt part of the tranformation matrix for T in equation (7) and, a a onequene, a ytem an only be put into ontrollable anonial form if it i ontrollable Or, alternatively, a ytem whih ha a ontrollable anonial form tate pae repreentation i ontrollable 7 Obervability A ytem i obervable if the initial tate x() an be uniquely determined by oberving the output over a finite time t Given any SISO ytem, A (n n) and C ( n) matrie then it an be hown that the ytem will be obervable if the (n n) obervability matrix ha rank n Again it an be hown that a ytem an only be put into obervable form if it i obervable 8 Caade Connetion In previou hapter on ontrol ytem deign ignifiant attention ha been given to aade ompenation and the effet on the open loop frequeny repone lou of adding a ompenator If the ompenator and plant are given in tate pae form then it may be deirable to obtain a tate pae repreentation for their aade ombination Thu, let the ompenator G () with tate z, input e, and output u have the tate pae repreentation (A,B,C,D ) and the plant G() with tate x, have input u, and output have the tate pae repreentation (A,B,C,D ), then z = A z Be, u = C z De and = A x B u, = C x D u x z A B Writing a ombined tate vetor (z, x) T z one an write = e x B C A x B D and z = ( D C C ) DDe whih give a tate pae repreentation (A,B,C,D) with x A B A =, B = BC A, C = ( DC C ) and BD D = D D (9) 3

131 Some State Spae Deign Method Some State Spae Deign Method Introdution The previou hapter on ontroller deign have mainly onentrated on introduing the ompenator in the forward path, but ue of a imple ompenator in the feedbak path ha been diued Alo feedbak ompenation ha been mentioned with repet to the PI-PD ontroller and veloity feedbak in a poition ontrol ytem Both thee two ae an be regarded a feedbak of two tate, namely, the output to form the error and the derivative of the output It i therefore appropriate to look in general at how the performane of a ontrol ytem an be hanged by the feedbak of tate variable If thi i to be done in pratie then the tate variable have to be available either a meaured value or etimate Obtaining meaurement an be otly beaue of the requirement for additional enor o in many ae the variable are etimated uing etimation method Thi i a topi outide the ope of thi book but it will uffie to ay that etimation method have beome relatively eay to implement with the ue of modern tehnologie employing miroproeor with ignifiant oftware inluded to do the required omputation In the next etion reult are derived for full tate variable feedbak and thi i followed by a diuion of the linear quadrati regulator problem The problem of diret loed loop tranfer funtion ynthei, or tandard form, i looked at again in term of uing tate variable feedbak to ahieve uh a deign Finally a an example of the benefit of uing a tate variable feedbak deign the problem of ontrolling a plant having a tranfer funtion with lightly damped omplex pole, onidered initially in etion 95 i reonidered State Variable Feedbak Conider a SISO ytem, G, with a tate pae repreentation (A,B,C,) Aume tate feedbak i ued T o that the ytem input u = K ( v k x), a hown in Figure Here the row vetor k T, i given by T k = ( k k k3 kn ), whih mean that the ignal fed bak and ubtrated from v i k x k x k x n n The thik line i ued to how that it repreent more than one ignal, in thi ae the tate x whih ha n omponent 3

132 Some State Spae Deign Method Figure Blok diagram of tate feedbak The new ytem, with input v, i () whih an be written x = A x B v () f f 3

133 33 Some State Spae Deign Method where the matrie (3) Now uppoe the original ytem wa in ontrollable anonial form o that = = n a a a a A A = = B B (4) then = n n f k K a k K a k K a k K a A 3 (5) a the matrix i all zero apart from the lat row The gain vetor ha been ubripted by to denote that it ha tate input from the ontrollable anonial form Thu the harateriti equation of the ytem with tate feedbak i ) ( ) ( ) ( ) ( = n n n n n n n k K a k K a k K a k K a (6)

134 Some State Spae Deign Method T and in priniple the pole an be plaed anywhere by hoie of the omponent of k Larger value of T the omponent of k will peed up the ytem repone but in pratie thi will not be poible due to phyial limitation on the magnitude of ignal for linear operation The gain K i baially redundant, however, it i ueful to inlude it a the truture might, a i lear from Figure, be a reultant loed loop ytem with K the ontroller gain In thi ae the ontroller input will inlude the error and for thi to be the ae when the tate x i the output, k will be equal to one If the ytem i not in ontrollable anonial form then the oeffiient term in the harateriti equation will not eah involve a ingle feedbak gain Thi mean that imultaneou equation need to be olved to find the required feedbak gain to give a peifi harateriti equation One way to avoid thi i to tranform the original ytem to ontrollable anonial form, determine the required feedbak gain for thi repreentation and then tranform thee gain bak to the required feedbak value from the original tate The ytem mut be ontrollable to do thi tranformation and it an be hown that thi i a required ondition to be able T to plae the pole in deired loation Thu, if the alulated tate feedbak gain vetor i k from the ontrollable form tate x and the tranformation from the original tate x i x = Tx then the required vetor k T for the original tate, x, i obtained from the relationhip k T = k T T - Several algorithm are available in MATLAB whih alulate the required feedbak gain vetor k T for a given ytem (A,B) to give peified pole loation The feedbak ignal k T x an be written in tranfer funtion term a and the output o that in term of the laial blok diagram of Figure 5 the tate feedbak i T k Φ( ) equivalent to a feedbak tranfer funtion of H ( ) = CΦ( ) 3 Linear Quadrati Regulator Problem It an be hown [] for a tate pae repreentation with matrix A and olumn vetor B that if a performane index (7) T i to be minimied then the required ontrol ignal, u(t), i given u( t) = k x( t), a linear funtion of the tate variable Further the value of the feedbak gain vetor i given by k T = R B P where P i T the unique poitive definite ymmetrial matrix olution of the algebrai Riati equation (ARE) (8) 34

135 Some State Spae Deign Method Obviouly the olution for k T depend upon the hoie of the poitive alar, R, and the matrix Q whih mut be at leat emi-poitive definite Although thi i a very elegant theoretial reult, the olution depend on the hoie of the weighting matrix Q and alar R No imple method i available for hooing their value o that the loed loop performane meet a given peifiation A further point i that whatever value are hoen then the open loop frequeny repone will avoid the irle of unit radiu entred at (-,) on the Nyquit plot [] Thi mean a phae margin of at leat 9 for a typial ontrol ytem open loop tranfer funtion, whih make the deign very onervative The ommand [x,l,g] = are(a,b, T *,R) in MATLAB will return the olution P for the ARE in x, where the vetor define the matrix Q by Q = T * 4 State Variable Feedbak for Standard Form To how how tate variable feedbak an be ued to ahieve a tandard form tep repone deign onider a fourth order all-pole ytem tranfer funtion G() with one integrator given in phae variable anonial tate pae form with A =, a a a3 a4 B = and C = (K p ) (9) 35

136 36 Some State Spae Deign Method Uing tate variable feedbak the new tate pae deription ha a tate variable repreentation with = f K k a K k a K k A 5 4 * *, = f B, and ) ( p f K C = () The orreponding loed loop tranfer funtion i p K k K k a K k a K K ) ( ) ( () Dividing by K p K and etting k = K p, give the tandard form n n n n n d d d d, () where the time ale fator by whih the tranfer funtion i fater than the normalied one i α = (K p K ) /5 and d i = (a i k (i) K ) /α (5-i) for i = to 4 Thu time aling i ahieved by varying K and then the tandard form aomplihed by hooing the value of k to k 5 to give the required value of d to d 4 Speeding up of the repone an be done by inreaing K or by inreaing the feedbak gain but thi alo inreae the magnitude of the ontrol ignal Thu, in pratie limiting value will normally exit for thee quantitie Trade off are of oure poible if there i ome flexibility in the allowable repone time It may, for example, be poible to hooe the time ale o a to require no feedbak from one tate or to realie an almot tandard form with no feedbak from more than one tate If the plant ha a zero then C f will be of the form C f = (K p K ) Proeeding a above the only hange i that the numerator of the tranfer funtion will be K K p K K whih in normalied form i n, with = αk /K p and the parameter d i now need to be hoen for the hoen value of, whih depend on the time ale fator For a plant tranfer funtion with no integration term the deign an be ahieved uing a PI ontroller In thi ae again auming the plant tranfer funtion i in ontrollable anonial form = 4 3 a a a a a A, = B and C = (K p ) (3)

137 37 Some State Spae Deign Method If tate feedbak i applied aording to u = r-k T x, where r i the output of the PI ontroller with tranfer funtion (K K )/, then the loed loop tranfer funtion i ) ( ) ( ) ( ) ( K K K k a k a K K K p p (4) The tranfer funtion i normalied by dividing by K p K to give n n n n n n n d d d d d (5) where d i = [a (i-) k i ]/α (6-i) for i = to 5, d = (a k K p K )/α 5, = αk /K, and the time ale fator α = (K p K ) /6 Thu, in priniple K an be hoen to elet the time ale, K to elet the zero and the feedbak gain to get the orret value of the d oeffiient for the hoen zero Again trade off are poible if there i flexibility in the hoie of the repone peed To illutrate the proedure two example are given below Example A ytem with a plant tranfer funtion having the tate pae repreentation i onidered If x denote the tate for the ontrollable anonial form and x that for the original ytem, then the required tranformation x=tx ha = T and the ontrollable anonial form i = 5 6 A, = B and C = ( )

138 Some State Spae Deign Method The orreponding ytem tranfer funtion i ()/( 3 5 6), whih i een to ontain an integration term Deign are arried out for the ISTE and IST E riteria, with α= and Firt for the T ISTE ae with α =, K =, =, d = 3, d = 8 giving K k = ( ) and K k T = ( -688 T -) and eondly with α =, K = 8, =, d = 95, d = giving K k = (8 58-6) and K k T = (8 5 74) For the IST E ae with α =, K =, =, d = 43, d = 4 giving K k T = ( ) and K k T T =( ) and eondly with α =, K = 8, =, d = 39, d = 6 giving K k = (8 636 ) and K k T = ( ) The repone for the four ae are hown in Fig Fig Repone for example for ISTE (i) and (ii); IST E (iii) and (iv) 38

139 Some State Spae Deign Method Example Conider a plant with tate pae repreentation The tranformation to put the repreentation into ontrollable anonial form ha Uing the tate feedbak u = r k T x and the PI ontroller (K K )/ give the loed loop tranfer funtion whih an be put into normalied form by dividing by K, The time ale fator α = (K ) /4 and the oeffiient of the normalied form are = αk /K, d 3 = (3k 3 )/α, d = (7k )/α and d 3 = (k K )/α 3 From examination of thee value it an be een that time aling by two hould give reaonable feedbak gain value Fig 3 how the oeffiient required a funtion of for ISE, ISTE and IST E deign, from whih it an again be een that the oeffiient inreae with inreaing value of and are larger for a given the higher the time weighting in the performane index Deign are done to minimie the ISTE with α = and = and 4 For thee two T ae the required value of the feedbak vetor k are ( ) and (-8 6 4), repetively From the original ytem tate the required vetor are k T T - and are ( ) and ( ), repetively The reulting output repone for the two ae are hown in Fig 4, where the one with the fater rie time i for = 4 39

140 Some State Spae Deign Method 7 6 ISE ISTE IST E T 4 () d 5 d d d, d, d d d d 3 d 3 d d Fig 3 Optimum oeffiient for T 4 () Fig 4 Cloed loop tep repone for example 5 Tranfer Funtion with Complex Pole The deign of a tate feedbak ompenator i onidered for the plant tranfer funtion with omplex pole, G() = ( ) diued in etion 95 So that the enitivity of the repone to hange in the damping and natural frequeny of the lightly damped pole an be een the tranfer funtion i K pωo taken a G() =, where the nominal value of K ( ζ ωo ) p, ω o, and ς are, and, repetively In ontrollable anonial form, with a aling on x and the plant gain taken at the input, the tate pae matrie are A = ω o ω o ζω o, B = K p and C = ( ) (6) The loed loop tranfer funtion with the tate feedbak i then (7) 4

141 Some State Spae Deign Method It an be een from the tranfer funtion that without feedbak the oeffiient of the term an be muh maller than that of the term With k =, the nominal parameter ubtituted and the ontroller gain hoen a the loed loop tranfer funtion i G() = 3 ( k3) ( k ), whih i in normalied form Doing an ISTE deign require k 3 = 7and k = 4 Cloed loop tep repone are hown for thi deign in Figure 5 The repone marked 8 and are obtained with the natural frequeny of the plant at thee value rather than the nominal one of unity The three repone hown, all marked, are for the nominal natural frequeny of unity and with damping ratio equal to 5, the nominal value, and 5, repetively, for the plant It an be een from thee reult that a muh better performane an be ahieved by tate feedbak than by the erie ompenator ued in etion 95 if the plant parameter hange Fig 5 Cloed loop tep repone for the ytem having a plant with omplex pole 4