3.32 In aircraft control systems, an ideal pitch response ( qo) versus a pitch command ( qc) is described by the transfer function

Transcription

1 . In acaft contol ytem, an deal ptch epone ( qo) veu a ptch command ( qc) decbed by the tanfe functon Q () τωn ( + / τ ) Qc() + ζωn+ ωn The actual acaft epone moe complcated than th deal tanfe functon; nevethele, the deal model ued a a gude fo autoplot degn. Aume that t the deed e tme, and that.789 ωn t.6 τ t ζ.89 Show that th deal epone poee a fat ettlng tme and mnmal ovehoot by plottng the tep epone fo t.8,.,., and.5 ec. Soluton: The followng pogam tatement n MATLAB poduce the followng plot: % Poblem. t [.8...5]; t[:4]/; tbackflpl(t); clf; fo I:4, wn(.789)/t(i); %Rad/econd taut(i)/(.6); %tau zeta.89; % btau*(wn^)*[ /tau]; a[ *zeta*wn (wn^)]; ytep(b,a,t); ubplot(,,i); plot(t,y); ttletextpntf('t%.f econd',t(i)); ttle(ttletext); xlabel('t (econd)'); ylabel('qo/qc'); ymax(max(y)-)*; mgpntf('max ovehoot%.f%%',ymax); text(.5,.,mg); ybackflpud(y); yndfnd(ab(yback-)>.); ttback(mn(ynd)); mgpntf('settlng tme %.f ec',t); text(.5,.,mg); gd; end

2 .4 t.8 econd.4 t. econd.. Qo/Qc.8.6 Qo/Qc Max ovehoot.%. Settlng tme. ec t (econd).4 Max ovehoot.%. Settlng tme.9 ec t (econd).4 t. econd.4 t.5 econd.. Qo/Qc.8.6 Qo/Qc Max ovehoot.%. Settlng tme.5 ec t (econd).4 Max ovehoot.%. Settlng tme 4.4 ec t (econd).5 Conde the two nonmnmum phae ytem, ( ) G ( ) ( + )( + ) ( )( ) G ( ) ( + )( + )( + ) (a) Sketch the unt tep epone fo G ( ) and G ( ), payng cloe attenton to the tanent pat of the epone. (b) Explan the dffeence n the behavo of the two epone a t elate to the zeo locaton. (c) Conde a table, tctly pope ytem (that, m zeo and n pole, whee m< n). Let yt ( ) denote the tep epone of the ytem. The tep epone ad to have an undehoot f t ntally tat off n the "wong" decton. Pove that a table, tctly pope ytem ha an undehoot f and only f t tanfe functon ha an odd numbe of eal RHP zeo.

3 Soluton: ( a) Fo G ( ) : ( ) Y( ) G( ) ( + )( + ) H( ) k ( z ) ( p ) R lm[( p ) H( )] lm k p l p p l ( z ) ( p z ) k ( p ) ( p p ) Each facto ( p z ) o ( p p ) can be thought of a a complex numbe l l l l l (a magntude and a phae) whoe pctoal epeentaton a vecto pontng to p and comng fom z o p epectvely. l The method fo calculatng the edue at a pole p () Daw vecto fom the et of the pole and fom all the zeo to the pole. () Meaue magntude and phae of thee vecto. () The edue wll be equal to the gan, multpled by the poduct of the vecto comng fom the zeo and dvded by the poduct of the vecto comng fom the pole. In ou poblem: Y( ) ( ) R R R 4 ( + )( + ) t y ( t) 4e + e t Step Repone fo G : p y(t) Tme (ec)

4 Fo G ( ) : ( )( ) 9 8 Y ( ) ( + )( + )( + ) y ( t) 9e + 8e e t t t Step Repone fo G y(t) Tme (ec) (b) The ft ytem peent an "undehoot". The econd ytem, on the othe hand, tat off n the ght decton. The eaon fo th ntal behavo of the tep epone wll be analyzed n pat c. t In y( t): domnant at t the tem 4e t In y( t): domnant at t the tem 8e (c) The followng conce poof fom [] (ee alo []-[]). Wthout lo of genealty aume the ytem ha unty DC gan ( G() ). Snce the ytem table, y( ) G(), and t eaonable to aume y( ). Let u denote the pole-zeo exce a n m. Then, y( t) and t devatve ae zeo at t, and y () the ft non-zeo devatve. The ytem ha an undehoot f y () y( ) <. The tanfe functon may be e-wtten a m ( ) z G ( ) m+ ( ) p The numeato tem can be clafed nto thee type of tem: (). The ft goup of tem ae of the fom ( α) wth α >. (). The econd goup of tem ae of the fom ( + α ) wth α >. 4

5 (). Fnally, the thd goup of tem ae of the fom, ( + β+ α ) wth α >, and β could be negatve. Howeve, β < 4 α, o that the coepondng zeo ae complex. All the denomnato tem ae of the fom (), (), above. Snce, y () lm G( ) t een that the gn of y () detemned entely by the numbe of tem of goup above. In patcula, f the numbe odd, then y () negatve and f t even, then y () potve. Snce y( ) G(), then we have the deed eult. [] Vdyaaga, M., "On Undehoot and Nonmnmum Phae Zeo," IEEE Tan. Automat. Cont., Vol. AC-, p. 44, May 986. [] Clak, R., N., Intoducton to Automatc Contol Sytem, John Wley, 96. [] Mta, T. and H. Yohda, "Undehootng phenomenon and t contol n lnea multvaable evomechanm, " IEEE Tan. Automat. Cont., Vol. AC-6, pp. 4-47, Suppoe that unty feedback to be appled aound the followng open-loop ytem. Ue Routh' tablty cteon to detemne whethe the eultng cloed-loop ytem wll be table. 4( + ) ( a) KG( ) ( + 4) ( b) KG( ) ( + ) ( c) KG( ) Soluton: ( 4) ( ) 4( ) + KG( ) 4( + ) ( a) T( ), a( ) + KG( ) a b : 8 : 8 : : c : d 8 8 whee a, b 8, b 8a c 4, d b 8 a gn change n ft column oot not n LHP untable

6 KG( ) ( + 4) b T a + KG( ) ( ) ( ), ( ) The Routh' aay, : : 8 : 6 : 8 Thee ae two gn change n the ft column of the Routh aay. Theefoe, thee ae two oot not n the LHP..4 Ue Routh' tablty cteon to detemne how many oot wth potve eal pat the followng equaton have. 5 4 ( b) ( d) Soluton: 5 4 ( b) : 44 4 : 8 48 : 48 ( ) : 5 44 ( ) 5 : ( ) 4 : ( ) 44 oot not n LHP. ( d) : : 78 : ( ) 58 : ( ) 78 oot not n LHP. 6

7 .4 Fnd the ange of K fo whch all the oot of the followng polynomal ae n the LHP K Ue MATLAB to vefy you anwe by plottng the oot of the polynomal n the -plane fo vaou value of K K 5 4 Soluton: : 5 : 5 K : a a : b K : c : K ( ) 5() ( K) 5(5) 5 K Whee a 8, a a a 55+ K b, a 8 Ka ab K + 5K 75 c, b 5( K + 55) Fo tablty: all tem n ft column > 55 + K () b > 8 K > 55 K + 5K 75 ( K.89)( K + 54) () c >, < 5( K + 55) 5( K + 55) 55 < K <.89 () K > Combnng (), (), and () < K <.89. If we plot the oot of the polynomal fo vaou value of K we obtan the followng oot locu plot (ee Chapte 5), 7

8 A: Ue Matlab to compute the ovehoot, e tme, ettlng tme (wth epect to tep epone) ( +.5α ) fo ytem H ( ) when α, 4,,, epectvely. Plot the tme.5α ( + + ) epone fo each cae and compae the eult. Chooe the fnal tme a 5 econd. Be caeful that when mall, the tep epone may co the lne y.9 eveal tme. Soluton: % Poblem 5.A lnepec['','g','b','k']; alpha [ 4 ]; mp[]; t[]; t[]; t[:.:5]; clf; fo :4 b[.5*alpha()]; a.5*alpha()*[ ]; ytep(b,a,t); plot(t,y,lnepec()); hold on; % Ovehhoot 8

9 mp[mp (max(y)-)*]; % Rng Tme yndfnd(y>.); tt(mn(ynd)); yndfnd(y>.9); tt(mn(ynd)); t[t t-t]; % Settlng Tme yndfnd(ab(y-)>.); t[t t(max(ynd))]; end xlabel('t (econd)'); ylabel('step Repone y(t)'); legend(pntf('\\alpha %d, mp %.f%%, t %.f, t %.f', alpha(), mp(), t(), t()),... pntf('\\alpha %d, mp %.f%%, t %.f, t %.f', alpha(), mp(), t(), t()),... pntf('\\alpha %d, mp %.f%%, t %.f, t %.f', alpha(), mp(), t(), t()),... pntf('\\alpha %d, mp %.f%%, t %.f, t %.f', alpha(4), mp(4), t(4), t(4))); gd; α, mp 6.7%, t.6, t 8.6 α 4, mp 9.%, t.4, t 8. α, mp 9.8%, t.9, t 7.9 α, mp 69.9%, t.5, t. Step Repone y(t) t (econd) 9