Flexible Beam. Objectives

Transcription

1 Flexile Bem Ojectives The ojective of this l is to lern out the chllenges posed y resonnces in feedck systems. An intuitive understnding will e gined through the mnul control of flexile em resemling lrge spce rootic rm. Control design will e performed in the frequency domin using led controller. A notch filter will e incorported in the feedck loop in order to reduce the excittion of resonnces. Introduction Systems with lightly-dmped, complex poles (resonnces), re encountered in mny pplictions. An exmple is lrge rootic rm in spce, whose trnsversl dimensions re mde smll to reduce weight. The rm will end nd oscillte if moved rpidly. In computer disk drive, red/write hed is ttched to the end of smll, rigid structure which is rotted rpidly to ccess vrious trcks. When the hed is positioned within frctions of microns, even such rigid structure ehves like flexile structure. The digrm of the flexile em is shown on Fig The ngle of the em t the shft is denoted θ, while φ is the ngle t the tip. If there ws no flexiility, the two ngles would e equl. Experimentl dt ws collected on flexile em of length 0.4m. Fig shows the frequency response tht ws mesured from the motor current to the ngulr ccelertion of the shft, while Fig shows the response mesured from the motor current to the ngulr ccelertion of the tip. The ccelertion t the shft ws otined y mesuring the position with n encoder (nd multiplying the frequency response y ω 2 to otin the ccelertion), while the ccelertion t the tip ws otined with n ccelerometer. The plots show the experimentl dt, s well s pproximte fits otined with fourth-order models (s dshed lines). The pproximte models shown on the plots re given y s 2 Θ(s) s 2 Φ(s) = k θ(s z 1 )(s z1)(s z 2 )(s z2) (s p 1 )(s p 1)(s p 2 )(s p 2) = k φ(s z 3 )(s z 4 )(s z 5 )(s z 6 ) (s p 1 )(s p 1)(s p 2 )(s p 2) (8.49) where p 1 = 3 +74j, p 2 = j, z 1 = j, z 2 = j, z 3 =100, z 4 = 120, z 5 =200,ndz 6 = 300 (note tht the poles re very lightly dmped). The input 263

2 φ θ Figure 8.33: Flexile Bem vrile is the current in the motor, mesured in A, nd the ngles θ nd φ re mesured in rdins. The constnts k θ nd k φ re such tht the DC gins of the trnsfer functions re equl, with k p = µ s 2 Θ(s) = s=0 µ s 2 Φ(s) =5.5 (8.50) s=0 The equlity for θ nd φ follows from the fct tht there is no ending of the em ner zero frequency. For low frequencies, the trnsfer functions re therefore pproximtely given y Θ(s) ' Φ(s) ' k p (8.51) s 2 This pproximtion of the system is the doule integrtor encountered with the ll nd em. The feedck design is more difficult thn for the ll & em, ecuse of dditionl poles close to the jω-xis, nd ecuse of zeros close to the jω-xis nd in the right hlf-plne. Mnul control The simultion file is clled flex.m. You should ply with the simultion nd move the em from the 45 line to the 45 line. You will encounter two difficulties: the 1/s 2 ehvior nd the flexiility of the em. Resonnces cn e voided y moving the em slowly, ut 264

3 Phse (degrees) Gin 10 4 Motor current (A) to shft ngulr ccelertion (rd/s ) Figure 8.34: Frequency response of the flexile em from motor current to shft ngulr ccelertion performnce will e unimpressive. It cn e enlightening(nd fun) to excite the resonnces y pplying commnds in the sme frequency rnge s the flexile modes. The em will end to lrge ngles. Once this mechnism is understood, you my return to the tsk of rpidly moving the em from side to side without exciting such resonnces. Note tht the simultion ws implemented differently from the previous ones. The continuoustime model ws discretized ssuming smpling period of 200Hz. Since the progrm runs t rte of pproximtely 20Hz, the visuliztion slows the dynmics y fctor of 10. This result is helpful, ecuse the dynmics of the ctul system re too fst to e controlled mnully. Led controller design The ojective is to design led controller C(s) = Φ ref (s) Φ(s) = k (s + ) c (s + ) (8.52) 265

4 Phse (degrees) Gin 10 3 Motor current (A) to tip 2 ngulr ccelertion (rd/s ) Figure 8.35: Frequency response of the flexile em from motor current to tip ngulr ccelertion Such controller ws designed for phse-locked loop in the dvnced PLL l. Here, the design will e performed in the frequency domin. The motor current i is the control signl, nd the tip ngle φ is the output to e regulted, with reference vlue φ ref. The Bode plots of the led controller re shown in Fig In the notes for the course, the vriles re shown tostisfy the following constrints ω p = m p = k c r = 1+sin(φ p) 1 sin(φ p ) (8.53) To determine the controller prmeters, we consider n pproximte model of the plnt with the two poles t the origin nd the first two resonnt modes. The trnfer function from 266

5 C(jω) (db) m mp mo 8 C(jω) (deg) 90 φp ωp ω (rd/s) ωp ω (rd/s) Figure 8.36: Bode plots of led controller CP(jω) (db) -40 db/dec CP(jω) (deg) ω p -20 db/dec -40 db/dec ωn ω (rd/s) ω p ω n ω (rd/s) -80 db/dec -360 Figure 8.37: Bode plots of plnt nd led controller (only one resonnt mode shown) i to φ is then P (s) = Φ(s) = k p s ωn 2 (8.54) 2 s 2 +2ζω n s + ωn 2 where k p =5.5, ω n = p 1, ζ = Re(p 1 )/ p 1,ndp 1 = 3 ± j73. Fig shows the Bode plots of the loop trnsfer function for the pproximte plnt nd the led compenstor. The mgnitude plot shows ω p s the crossover frequency, condition tht is to e enforced y proper choice of the prmeters. Assuming tht ω n À ω p, show using (8.53) tht C(jω p )P (jω p ) ' k ck p ω 2 p r C(jω n )P (jω n ) = k ck p 2ζω 2 n (8.55) 267

6 Then, find the vlues of the controller prmeters k c,, nd such tht φ p =60, C(jω p )P (jω p ) =1, C(jω n )P (jω n ) =0.1 (8.56) The ide is tht the first two conditions will ensure crossover frequency t ω p nd phse mrgin of 60. The third condition will yield gin mrgin of 10, given tht the phse crossover frequency will e close to ω n. Using the prmeters of the led controller, plot the step response of the closed-loop system (function step in Mtl) nd the Bode plots with the gin nd phse mrgins (function mrgin in Mtl). Mke sure to use the complete plnt trnsfer function for this step, not the pproximte one. The feedck system my e ssemled using the functions series nd feedck in Mtl, if desired. Also compute the loctions of the closed-loop poles (function pole or roots in Mtl). Automtic control Design #1 Implement the controller in the simultion with files flexc.m nd flexcinit.m. A discretetime equivlent of the control system should e computed, s ws done in the previous l. Discretiztion should e sed on the 200Hz smpling frequency, so tht visuliztion will show the system t rte slowed down y fctor of 10. Plot the responses of i, θ, φ, nd φ θ to step inputs. Let φ ref switch etween 45 nd 45 every 2 seconds, nd record the dt for 6 seconds (which will tke 60 seconds in the rel-time simultion). The visuliztion should show slow response, with lrge overshoot, due to the low frequency zero t s =. If you reduce the gin mrgin y incresing C(jω n )P (jω n ), youwillfind tht the crossover frequency increses nd tht the response speeds up, ut oscilltions re oserved due to excittion of the em s flexile modes. Design #2 Improve the design y cscding the led/lg controller with notch filter nd y prefiltering the reference input, so tht =C(s)C notch (s)(c F (s)φ ref (s) Φ(s)) (8.57) where C(s) is the led controller considered erlier. Let the prefilter e C F (s) = 1.2 s +1.2 (8.58) 268

7 where is the zero of the compenstor. Let the notch filter e C notch (s) = s 2 + ω 2 n s 2 +2ω n s + ω 2 n (8.59) where ω n is the nturl frequency of the first resonnt mode. The prefilter will eliminte the overshoot, nd the notch filter will llow you to increse the crossover frequency nd, s result, the speed of response. Re-design the led controller with ω p = 12 rd/s, keeping the condition tht φ p =60 nd the crossover frequency condition k c k p ω 2 p r =1 (8.60) Check the properties of this re-designed controller in comintion with the prefilter, notch filter, nd plnt trnsfer function: plot the step response, the Bode plots with gin nd phse mrgins, nd compute the vlues of the closed-loop poles. Implement the controller in the simultion, nd plot the responses of i, θ, φ, ndφ θ to step inputs in φ ref. Compre the responses to those of the previous controller. Show your code nd demonstrte the rel-time opertion to the TA. Report t glnce Be sure to include: Description of chllenges nd strtegies for mnul control. Design nd evlution of the led controller, with plots of the step response, Bode plots with gin nd phse mrgins, nd vlues of the closed-loop poles. Responses of i, θ, φ, nd φ θ to step inputs of φ ref in the simultion. Design nd evlution of the led controller with notch filter nd prefilter, with plots of the step response, Bode plots with gin nd phse mrgins, nd vlues of closed-loop poles. Responses of i, θ, φ, ndφ θ to step inputs of φ ref in the simultion. Oservtions nd comments. Listing of flexc.m nd flexcinit.m. 269