x with A given by (6.2.1). The control ( ) ( )

Transcription

1 Homework 5 Sring 9 AerE33 Due 4/(F) SOLUTION PROBLEM (3t) In thi roblem we will invetigate the longitudinal dynamic of the B747 airlane a decribed in Etkin 66 The tate model i x Ax Bu where the tate vector i x with A given by (6) The conol [ u, w, q, ] inut i u [ e] with B given by (765) Becaue we are intereted in the entire tate, we et C=I and D All maice are given in the Matlab code below Problem in the Aendix (a)(4t) For any tate equation that the anformed tate equation i: x Ax Bu let x Tx be a anformation to a different tate, where T x TAT x TB u ( ) ( ) i invertible Show Since T i invertible, we have, hence Subtituting thee into the original tate equation give: T xat xbu Left-multilying thi equation by T give the deired reult x T x x T x (b)(6t) The value of (a) lie in our deire to obtain the tate equation for x [ u,, q, ] [ u,( w / u ), q, ] diag{,/ u,,} x Tx Verify thi anformation a follow For the elevator inut (#): (i) Ue the tf command to arrive at the 4x4 numerator maix, but where only the econd row i divided by u=774 (ii) Aly the above anformation directly in relation to A TAT and B TB (aociated with x Ax Bu ) to arrive at the aociated 4x4 numerator maix (iii) arrive at both anfer function maix denominator Finally, (iv) dicu the validity of the anformation Coy/ate your reult HERE] [See (b)] (i): Ne =[ , , ] (ii): NNe =[ , , ] (iii) Den = [ ], DDen =[ ] (iv) The two anfer function maice are identical (c)(3t) From the information in (c) arrive at the tatic gain for the anfer function ( ) / ( ) () / e() 39 /4 986 (d)(9t) For a te throttle oition of / 6: (i) Obtain the 4x4 anfer function numerator maix for the inut () (ii) Ue it to how that 3 o (iii) Verify thi value by lotting the reone () t to the above throttle te inut, and then comuting the error relative to the redicted value [See (d)] (i): NN = [ , - 7 3, 3] (ii): () / () 3/4 395 Hence, for / 6, 395 / o rad (iii): The ercent error i ( ) / , or -3% Figure (d) Reone () t to te / 6 e

2 (e)(3t) For the above throttle oition inut determine whether or not any noticeable dynamic would be oberved in relation to the AOA, er a lot of () t From the lot at right, we ee that the maximum AOA i le than o Hence, no noticeable dynamic would be oberved Figure (e) Reone (f)(5t) Your lot in (d) and (e) give the eak amlitude of the decaying inuoid aociated with () t to te () t / 6 reectively Thee ocillation occur at the hugoid damed natural frequency (i) Directly from your lot, etimate the magnitude of the firt eak in about it teady tate value divided by the magnitude of the firt eak in Give thi () t ratio directly and in decibel unit (ii) Arrive at a Bode lot of the ratio ( ) / ( ), and ue it to ee how well it redict the comuted ratio at thi frequency [Hint: Thi ratio i a ratio of numerator of related anfer function for () ] [See (f)] At the hugoid damed natural frequency d 67 the ratio of the eak amlitude in (d-e) i db Thi i what i redicted by the Bode lot at right That lot ha the anfer function: ( ) NN(,:) 7 ( ) NN(4,:) 3 and () t () t, Figure (f) Bode lot of ( ) / ( ) for inut ()

3 3 Problem (45t) Conider the lightly damed lant ingle inut-ingle outut (SISO) ytem Define the tate x x y y (a)(5t) Show that the defined tate give x Since have x y x x y x y G Y ( ) ( ) The ODE i: y y 9y u Thi i a 9 U( ) x 9 x u Ax Bu & y x u Cx Du, we have, o that x x 9x u Thi give x x 9x u () Since, we () Putting () and () into the maix form give the deired tate equation The ytem outut i y, y y y x u Thi i the deired outut equation which can be written a: x y (b)(3t) Ue the tf command to ut the SISO ytem into a tate ace form [Coy/ate your code/reult HERE] [A B C D]=tf(,[ 9]) A = [ - -9 ; ] B =[ ; ] C =[ ] D = (c)(4t) You hould have found that the form in (b) i not the form in (a) Show that the tate variable for thi rereentation i x x y y x 5 From y y 9y u we have: 5y 5y 9(5) y u Letting x 5yand x 5y, we can then roceed exactly a in (a) to obtain the deired rereentation (d)(5t) If in x Ax Bu we define the inut u to be u Kx [ K K][ y y], then the tate equation become x Ax BKx ( A BK) x Ax Taking the Lalace anform give: ( I A) X ( ) Hence, we ee that the cloed loo ole are the eigenvalue of A Ue the Matlab command lace to arrive at the gain maix, K K K ] that will lace ole at 5 5 [Note: Ue the tate ace rereentation in (a)] Coy/ate code/reult HERE, i =-5+i*5; =conj(); K=lace(A,B,[ ]) = [ 9 4] [ (e)(6t) Notice that in (d) it didn t matter if the conoller wa laced in the forward or the feedback loo It wa imly laced in the oen loo Here, we will aume that thi i a command (a ooed to a diturbance) feedback conol ytem Hence, the conoller hould be laced in the forward loo (i) Ue the feedback command to arrive at the CL anfer function W() (ii) Find the gain that will make W() unity tatic gain (iii) Plot the reulting te reone (i): W = (8 + 8)/(^ ) (ii) k 9/ 8 98 k Figure (e) Cloed loo unit te reone

4 4 (f)(7t) To reflect the fact that a rate gyro wa ued in an inner/outer loo feedback conol ytem, lace the P-conol in K () the forward loo, and the entire PD-conol in the feedback loo For let CL G WW () (i) Arrive at the ( K K ) G ( ) CL anfer function (ii) Ue the zk command to imlify it (iii) Find the gain (iv) Overlay the reulting te reone on your lot in (e) (i): WW = (8 ^ ) / ( ^4 + ^3 + 9 ^ ) (ii): zk(ww) = 8 (^ + + 9) / (^ ) (^ + + 9) = 8/(^ ) k d that will make W() unity tatic gain (iii): k 9/ 8 98 (g)(4t) Given that the tate vector i x x x y y, the tate feedback ole lacement method mut reult in a PD conoller Even o, the ue of PID conol i far more oular Conider the tate vector For thi tate, how that x 9 x u Ax Bu y x u Cx and Du x3 y x 3 y x The third row of A follow immediately, a doe the form for C x x x x y y y 3 (h)(4t) Ue the lace command to find the value for the PID conoller K K K ] that will lace CL ole at 5 5 and 3 5, i [See (h)] K = [ ] [ K3 (i)(7t) For the conoller in (h) lace the PI ortion in the forward loo and the entire PID conoller in the feedback ( K / ) ( ) loo For let Ki G WWW () (i) Arrive at the CL anfer function in the mot imlified form (ii) ( K K / K ) G ( ) i d Overlay the CL te reone on the lot in (e) (iii) Identify the caue of the undeirable element of the reone [See (i)] (i): zk(www) = 9 (+747) (^ + + 9) 9(+747) = (+5) (^ + + 9) (^ + + 5) (+5)(^ + + 5) (iii) The ignificant overhoot i caued by the CL zero

5 5 PROBLEM 3(5t) Conider again the continuou-time anfer function: (a)(9t)to obtain a dicrete-time aroximation, call it G () z T 34ec ue the anform air in Table 8 on 6 to comute command to obtain G () im z [See 3(a)] (i): G() (ii): 9 ( 5) 875 G Y ( ) ( ) 9 U( ) b, (i) find a and b uch thatg () ( a) b G () z (ii) For (iii) Ue the cd(g,t, imule ) (iv) Show that the reult from Table i off by a factor of T from the reult in (iii) b = 94 a = -963 a = 969 Hence: bz 677z G() z b z az a z 963 z 969 (iii): Gim = 94 z / ( z^ - 96 z + 969) (iv): 677*T= 9 Hence, the table doe NOT include T, while Matlab cd command with the im flag doe (b)(t) For T=34 ec ( r/ ) uing cd(g,t, zoh ) give: G zoh 976 z 966 ( z) By including z 96 z969 the zoh flag, Matlab include a zero-order-hold anfer function: G zoh ( ) ( z ) / ( z ) / z Conequently, z G() Gzoh ( z) Z[ Gzoh ( ) G( )] Z Ue eny in Table 8 to how that, indeed, thi i the cae z Hence: G() a b z( Az B) Z Z Z at at [( a) b ] a b [( a) b ] a b ( z )( z e co( bt ) z e ) Az B at ( z), where a at A e co( bt ) at at e in( bt ) 44 & a b z e co( bt) z e b G a in( ) co( ) 43 Hence: b a b 9 at at at B e e bt e bt 976 z 966 G ( z) z 96 z969 (c)(6t) From (b) we ee that, even though the cd command give G (z), thi i not ** a anfer function in dicrete time Rather, it i in continuou time due to the incluion of the ZOH digital-to-analog (D/A) circuit Overlay lot of the te reone for G(), G () z and Gzoh () z to ee thi viually im ** Thi i the cae if ZOH i a circuit If ZOH i an aroximate integration method then G (z) i in dicrete time In other word, the ue of zoh i ambiguou It can be ued for numerical integration, or it can rereent a circuit [See 3(c)] Figure 3(c) Ste reone (LEFT) and zoomed (RIGHT) for G(), G () z and Gzoh () im

6 Matlab Code %PROGRAM NAME: hw5m FROM Etkin (6) and(765) %PROBLEM %THE STATE IS: x = [u w q th] %The [A B C D] Maice: a=[ ]; a=[ ]; a3=[ ]; a4=[ ]; A=[a;a;a3;a4]; %State: [u w q th] b=[87;-785;-58;]; b=[966;;;]; B=[b,b]; %Inut: [de ; d] C=eye(4); D=zero(4,); u=774; %The lane eed i needed to convert w to a=w/u; % %(b): %TF for elevator inut %(i): [Ne,Den]=tf(A,B,C,D,); %TF for elevator inut Ne(,:)=Ne(,:)/u; Ne %(ii): T=eye(4); T(,)=/u; AA=T*A*T^-; BB=T*B; [NNe,DDen]=tf(AA,BB,C,D,); NNe %(iii) Den DDen % %(d): [N,Den]=tf(A,B,C,D,); [NN,DDen]=tf(AA,BB,C,D,); Gth=tf(NN(4,:),DDen); figure(3) t=::; theta=te((/6)*gth,t); thetad=(8/i)*theta; lot(t,thetad) title('pitch Angle Reone to Throttle Ste /6') xlabel('time (ec)') ylabel('degree') %(e): Ga=tf(NN(,:),DDen); alha=te((/6)*ga,t); alhad=alha*8/i; figure(4) lot(t,alhad) title('aoa Reone to Throttle Ste /6') xlabel('time (ec)') ylabel('degree') %(f): n=nn(,3:5); nth=nn(4,4:5); Gath=tf(n,nth); figure(5) bode(gath) title('a(w)/th(w) for Throttle Ste /6') aue %==================================== %PROBLEM %(b): [A B C D]=tf(,[ 9]) %(d): =-5+i*5; =conj(); 6

7 K=lace(A,B,[ ]) %(e): G=tf(,[ 9]); Gc=tf(K,); W=feedback(Gc*G,) k=9/8; W=k*W; figure() te(w) title('cl Ste Reone') %(f): K=K(); WW=K*G/(+Gc*G) WW=k*tf(8,[ 9 9]); hold on te(ww) title('cl Ste Reone in FRD and FRD/FDB Loo') legend('pdfwd','pfwd/pdfbk') %(g): A=[- -9 ; ; ]; B=[;;]; C=[ ]; 3=-5; vec=[ 3]; K=lace(A,B,vec) %(i): PID=tf(K,[ ]); PI=tf(K(:3),[ ]); WWW=PI*G/(+PID*G) te(www) legend('pdfwd','pfwd/pdfbk','pifwd/pidfbk') %=============================== %PROBLEM 3 G=tf(,[ 9]); z=tf('z',t); %(a): %(ii): T=34; a=5; %zeta*wn b=qrt(875); %wd b=ex(-a*t)*in(b*t); a=-*ex(-a*t)*co(b*t); a=ex(-*a*t); c=/b; Ga=c*b/(z^+a*z+a); Gim=cd(G,T,'im'); %(b): B=ex(-*a*T)+ex(-a*T)*((a/b)*in(b*T)-co(b*T)); A=-ex(-a*T)*(co(b*T)+(a/b)*in(b*T)); c=/(a^+b^); a=-*ex(-a*t)*co(b*t); a=ex(-*a*t); G_theory=c*(A*z+B)/(z^+a*z+a) Gzoh=cd(G,T,'zoh') %(c): figure(3) te(g) hold on [gim,t]=te(gim,'red'); lot(t,gim,'ro--') te(gzoh,'k') legend('g()','gim(z)','gzoh()') 7