DESIGN OF CONTINUOUS LAG COMPENSATORS J. Pulusová, L. Körösi, M. Dúbrvská Institute of Robotics nd Cybernetics, Slovk University of Technology, Fculty of Electricl Engineering nd Informtion Technology Abstrct In this pper design of continuous lg compenstors with specified structure for liner system is ddressed, proposed nd tested. The design of led compenstors is proposed by using Bode digrm. The pper dels with theoreticl nd prcticl methodology, nd its successful ppliction for controlled systems. Advntges nd disdvntges of phse lg compenstors re summrized. Introduction Automtion lgorithms re instlled in vrious res of industril production equipment, which llow self-control of processes without the involvement of humns. One of such devices is the correcting element with specified structure []. The phse lg compenstor with specified structure is dynmic system, which djusts sttic nd dynmic chrcteristics of controlled system. In this pper seril connection of bove mentioned controllers is discussed, see Fig.. Figure : Control structure with compenstor G C (s) w(t) is reference input signl, e(t) is error signl, u(t) is input signl to the plnt model with trnsfer function G(s), y(t) is plnt output The simplest structure of the phse lg compenstor is the first order system with trnsfer function Ts G C ( s) () Ts where K, T, computed prmeters, <. 2 Design of Phse Lg Compenstor The stedy-stte error cn be reduced (not eliminted) by using phse lg compenstor. The / 2,. Trnsfer function of compenstor is phse φ(ω) is negtive nd is from intervl given by () nd Bode digrm is shown in Fig. 2.
Figure 2: Bode digrm of lg compenstor Algorithm of design consists of the following steps [2]:. Determining of the vlue of gin K for open control loop. It is chosen to stisfy stedy-stte performnce requirement. 2. For expressed vlue of the gin is plotted the Bode digrm of uncompensted open control loop KG(s). 3. For required vlue of phse mrgin Δφ Z is deduced vlue of the future mplitude intersect ω from the Bode digrm. The phse curve is chnged round ω. So ω is compensted (pproximtely bout 3- rd/sec) to the new gin crossover frequency, where the compensted mgnitude curve psses through the db xis. 4. Angulr frequency will be new mgnitude crossover of mgnitude curve of open control loop. Is ssumed tht mgnitude Bode chrcteristic of compenstor in point will hve mount -2log db A 2log KG( j ) 2log (2) from where A 2 (3) 5. The vlidity of reltionship (2) will be ensured by moving of mgnitude curve to the left. Upper point /T will be choosen one decde below s [Hrsányi et l., 998]. (4) T from where T (5) 6. Constnt T is now known. Design of phse led compenstor is completed. The design is checked by simultion of the step response. Plot lso the Bode digrm of compensted open control loop. If performnce requirements re met, stop. Otherwise go bck to step. 3 Advntges nd disdvntges of phse lg compenstor - The gin is moved crossover to lower frequency while the phse curve is unchnged [3].
Phse (deg) Mgnitude (db) - The rise time nd the settling time of the system re usully slower. - The bndwidth of the closed-loop system is decresed. 4 Cse Study nd Simultion Results Consider the following system G ( s) (6) s( T s ) s(.726s ) The tsk is to design continuous compenstor tht stisfies the following requirements:. Stedy-stte error e () due to unit rmp function input w(t)=t, where ε=.. 2. Mximum percent overshoot η mx <3% or phse mrgin Δφ OZ >45. Solution of the exmple: The gin K= is expressed from the stedy-stte error for w(t)=t s follows e( ) lim se( s) lim s W ( s) s s G ( s) e s s W lim s s KG 2 lim s s s K s.726s. K K 99 The controller is now P (proportionl) controller (K=). The trnsfer function of openloop system will be s follows G s KGs (8) s.726s Bode digrm for the uncompensted system G (s) is shown in Fig. 3. (7) 8 Bode Digrm Gm = Inf db (t Inf rd/sec), Pm = 2 deg (t 35.9 rd/sec) 6 4 2 Frequency (rd/sec): Mgnitude (db): 8. -2-4 -6-9 KG(s) -35 Frequency (rd/sec): Phse (deg): -26 Frequency (rd/sec): 3.6 Phse (deg): -35-8 - 2 3 Frequency (rd/sec) Figure 3: Bode digrm of (8) The uncompensted gin crossover frequency (with K) is 35.9 rd/sec nd the phse mrgin is Δφ OZ =2. From the Bode digrm in Fig. 3 is deduced the frequency of the future mplitude intersect ω for the desired vlue of phse mrgin Δφ Z =45. For this system is ω =3.6 rd/sec. The phse curve is chnged round ω. Therefore, ω is compensted (pproximtely bout 3- rd/sec) to the new frequency 3.6 3.6 rd/sec. This vlue will be the new crossover of mgnitude curve of
Amplitude Phse (deg) Mgnitude (db) compensted open-loop system. From the Bode digrm is deduced the vlue of mgnitude A=8. db for frequency. The corresponding vlue of is clculted by using (3), nd the result is =.35. The prmeter T is 8.8 ccording eqution (5). The trnsfer function of phse lg compenstor from () is s G C ( s) (9) 8.8s nd Bode digrm is shown in Fig. 4. 5 Bode Digrm Gm = Inf db (t Inf rd/sec), Pm = 5.3 deg (t 9.42 rd/sec) 5-5 - -9 KG(s) KG(s)G C (s) -35-8 -3-2 - 2 3 Frequency (rd/sec) Figure 4: Bode digrm for compensted (green) nd uncompensted system (blue) In Fig. 4 the blue curves re for uncompensted system nd green for system with the phse led compenstor. The compensted gin crossover frequency of 9.42 rd/sec nd the phse mrgin Pm=5.3. Time response of the output vrible with led compenstor is shown in Fig. 5..4 System: gw Time (sec):.32 Amplitude:.22 Step Response y.2.8.6.4.2.5.5 2 2.5 3 Time (sec) Figure 5: Time responses of the output vrible with led compenstor Mximum percent overshoot η mx =22%.
5 Conclusions The specifictions of the system (6) were stisfied for compenstor the phse lg (9). The phse mrgin, stedy-stte error nd mximl percent overshoot specifictions were stisfied. The phse lg compenstor dds gin t low frequencies. This compenstor increses dmping of system, rise time nd settling time. ACKNOWLEDGEMENT This pper hs been supported by the Slovk Scientific Grnt Agency, Grnt No. /2256/2. References [] L. Hrsányi, J. Murgš, D. Rosinová, A. Kozáková. Theory of Automtic Control. Brtislv STU, 26 pges, 998. [2] I. Holič, M. Dúbrvská, J. Pulusová. Design of Continuous Compenstors with Specified Structure. 9th Interntionl Scientific-Technologicl Conference Process Control 2. Kouty nd Desnou, Czech Republic: 7- June 2, s. C3. [3] B. C. Kuo, F. Golnrghi. Automtic Control Systems (8th Edition), Wiley, 22. Jn Pulusová, Ldislv Körösi nd Mári Dúbrvská Institute of Robotics nd Cybernetics, Fculty of Electricl Engineering nd Informtion Technology, Slovk University of Technology, Ilkovičov 3, 82 9 Brtislv, Slovk Republic e-mil: jn.pulusov@stub.sk, ldislv.korosi@stub.sk, mri.dubrvsk@stub.sk