Projeto de Compensador com Base nos Métodos de Ogata

Projeto por LGR Sistemas de Controle Prof Julio Cesar Ceballos Aya 13 de novembro de 2023 Prof Julio Cesar Ceballos Aya Projeto por LGR 13 de novembro de 2023 1 7 Avanco e Atraso de fase y 4 3 pag 303 304 Livro Ogata O compensador tém o seguinte formato 1 1 Sr STs cte te 7 T ST aR St an Determine os pdlos em MF com base nas especificades do projeto 51 Determine a contribuiao angular do compensador Assumindo que 7T é alto para que s 5 sp a St an escolha Ty e 7 sti s t oS5K Gs 1 str st yt Ti si BY FSG Shiversiéro Avanco e Atraso de fase y 4 3 pag 303 304 Livro Ogata se K é especificada Ky lim sGsGs i sth st Keo Ii lim sKe se Gs Ker lim sGs podese determinar G Escolha T tal que stt 1 SR 8 eb oo 1 se ST BT Is ae BY FSG CE dria Avanco e Atraso de fase γ β pag 303 304 Livro Ogata Exemplo Seja Gs 4 ss 05 os polos em MF sao s 025 198j ζ 0125 ωn 2rads e Kv 8s1 Projeto ζ 05 ωn 5rads Kv 80s1 Prof Julio Cesar Ceballos Aya Projeto por LGR 13 de novembro de 2023 4 7 Avanco e Atraso de fase y 3 pag 304 305 Livro Ogata O compensador tém o seguinte formato 1 1 S Tr S Db Ges Ke 7 sta St an Determine os pdlos em MF com base nas especificagdes do projeto s1 Determine K com a seguinte equacdo Ky lim KGs s0 Determine a contribuiao angular do compensador y Assuma que T é grande para que 7 x1 Sit BT 1 ate sate Ke7Gs1 7h o sit Tr sity Com determinado escolha T para que att sth F1 8 ask 0 BY FSG Si BT 1 Bt ato Passo 4 IEEE TRANSACTIONS ON EDUCATION VOL 37 NO I FEBRUARY 1994 63 Direct Expressions for Ogatas LeadLag Design Method Using Root Locus Marcel0 Carvalho Minhoto Teixeira Member ZEEE Abstruct4irect expressions for the design of a leadlag contin uous compensator using the root locus method and the procedure described in the 1970 and 1990 books by Ogata are presented These results are useful in the Ogata design method because they avoid the geometrical determination of poles and zeros making it easier to create a computerbased design I INTRODUCTION LTHOUGH control theory offers a great number of A results in practical applications often a simple compen sator such as PID or leadlag is sufficient For example in the control of dc motors and in magnetic suspension systems PID and leadlag controllers are sufficient The design of PID and leadlag compensators are usually made by the root locus or frequency response methods using a graphic trial and error procedure With commercially available control softwares such as MATLAB and Program CC the design and performance verification of a compensator becomes easier because they make for example fast simulations and root locus and Bode plots which make it possible to avoid the geometrical work on paper However in some designs the determination of poles and zeros of the compensator are made using a graphic mean or a computer solution For example the lead and leadlag com pensators created using the methods described in l and 2 The determination of direct expressions for these parameters makes it easier to create a computerbased design without geometrical determinations In 3 equations are presented for the design of a lead compensator with minimum attenuation and without geometrical constructions or computer solutions In this note we present equations for the analytical determi nation of a leadlag compensator using the method described by Ogata in l and 2 also without the geometrical deter mination of parameters 11 THEANALYTICAL DETERMINATION OF A LEAD LAG COMPENSATOR Fig lb presents a leadlag compensator It is known l 2 that Manuscript received July 1991 revised October 1991 M C Minhoto Teixeira is with the Department of Electrical Engineering IEEE Log Number 9214219 FEISUNESP Ilha SolteiraSP Brazil c Fig 1 Compensation of systems with leadlag compensators a Configura tion of the compensated system b the leadlag compensator with electrical elements c geometrical description of 4 where TI R1 C1 T2 R2C2 and P 1 is such that References I and 2 describe a design method for the system of Fig la with the leadlag compensator 1 presented in Fig lb for a convenient modification of the transient and steadystate responses of this system when this is not possible using only one proportional lead or lag compensator Gs t w n J m w n j 0 f j w for an adequate transient response of the feedback system in Fig la for example for a desired overshoot and settling time 2 Calculate the angle q5 for s1 0 j w in such a way that it belongs to the root locus of the feedback system described in Fig la 1 Choose two dominant poles s12 4 IGcsi IGsl 2h llSOo h E l701 3 Choose one value of h such that 0 5 q5 5 360 00189359940400 0 1994 IEEE 64 IEEE TRANSAaIONS ON EDUCATION VOL 31 NO I FEBRUARY 1994 3 Choose the gain K in 1 for an adequate steady state error of the feedback system in Fig la Note that GO K and this error is independent of TlT2 and 3 for usual inputs such as step ramp and parabolic 4 Determine 71 and p such that 4 Fig Ic shows the problem of these determinations given slab IKGsll and 4 determine the points Ti and Note that from Fig lc and 0 71 90 it is necessary that 0 5 4 180 7 Remark If 4 180 7 one leadlag controller described in Fig lb is not sufficient to compensate the feedback system of Fig la In this case we can for instance use two or more cascade leadlag sections or one leadlag and some lead sections to supply the phase angle needed 4 The design of each leadlag section would also have the same problems of the determination of parameters such as TI and in 4 given SI and ab Assume that in 3 In l and 2 the determination of TI and P in 4 given s1 ab and 4 is made by geometrical construction the author of this note has found analytical expressions for this design From Fig lc considering SI ab and 4 given we have lT1 00 4 180 7 W cos cos 4 W E b and so cos 4 E a cos 4 COS sin 4 sin E cos b cos cos 4 tan E sin 4 6 Then 1 l a tan tan4 sin4 b Fig lc and 7 we can determine and 7 a tan 4 tan E 1 tan 4 tan E 0 1 a w t a n 4 c Tl Hence dividing 8 by 9 we obtain 3 and from 9 TI 5 Using 3 obtained in Step 4 choose T2 such that Note that it is possible for large values of T2 6 Verify the dominance of the poles s12 by the rootlocus andor simulations If s12 are not dominant then remake the design with other compensators for instance by cancelling inadequate zeros or poles of the plant G s Example In I an example is presented where Gs 4s s 05 it can be a transfer function of a dc motor for a position control We need a compensator Gs with the structure given in Fig la such that 05 and w 5 rads for the dominant poles and a steadystate error ess 002 for an input ut tt 2 0 Then from step 1 s1 250j433 from step 2 4 55O and from step 3 K 625 Now IKGsll 5477 ab Hence from 7 tan 058 from 8 and 9 PIT1 501 and 1T1 049 thus 3 10 and TI 2 In l Ti and P are obtained geometrically Choosing T2 10 10 is satisfied and the leadlag compensator is Gs 625 s 05 s Ols 5s 001 In l simulation results showing the success of this design are presented In 2 the same example is presented but considering a desired steadystate error ess 180 for an input ut tt 2 0 similar results were obtained 111 CONCLUSION Direct expression for the design of a leadlag controller using the root locus method and the procedure described by Ogata are presented These results avoid the geometrical determination or computer solutions for the obtainment of poles and zeros which is made in l and 2 enabling a computerbased design to be made faster and with greater accuracy REFERENCES K Ogata Modern Conrrol Engineering Englewood Cliffs NJ PrenticeHall 1970 K Ogata Modern Control Engineering 2nd Ed Englewood Cliffs NJ PrenticeHall 1990 R Unnikrishnan Design of a lead compensator with minimum atten uation Int J Elect Eng Educ vol 17 pp 8588 1980 Marcclo C M Tcixcira S86ASS received the BScEE degree from Lins College of Engineering EEL Brazil in 1979 the MScEE degree from the Federal University of Rio de Janeiro COPPEUFKJ Brazil in 1982 and the DSc degree in contcol engineering from Ihe Catholic University PUC of Rio de Janeiro Brazil in 1989 Sincc 1982 he has been with thc Iaulista State University UNESP Depament of Electrical Enginecring llha Soltcira Campus SI Brazil In 1990 and 1991 he W S the vicehcad of Department of Electrical Engineering and he is currently the coordinator of the Electrical Engineering Course at UNESI llha Solteira Brazil His current interests include variable structure with sliding motions adaptive systcms control with neural networks ccntrol cducation and control applications Prof Julio Cesar Ceballos Aya Projeto por LGR 13 de novembro de 2023 6 7 Passo 4 Exemplo Gs 4 ss 05 s1 25 433j Kv 80s1 1 s1 25 433j 2 Kv lims0 Kc 4 05 8Kc 80 Kc 10 3 Φ 55o 4 Usa a equacao 7 para calcular tanϵ as equacoes 8 e 9 para encontrar β e T1 estas equacoes estao no artigo DIRECT EXPRESSIONS FOR OGATA LEADLAG DESIGN METHOD USING ROOT LOCUS Prof Julio Cesar Ceballos Aya Projeto por LGR 13 de novembro de 2023 7 7