ω r. Chapter 8 (c) K = 100 ω ζ M ω

Transcription

1 Chapter (a) K = 5 ω.4 ζ.46 M ω = 5 = rad / sec = 6.54 = 1 = 1 = rad / sec 4.48 n r r 1 (b) K = 1.39 ω = 139. = 4.6 rad / sec ζ = 6.54 =. 77 M = n r 9.4 ζ 1 ζ = 1 ω = ω 1 ς = 3.7 rad / sec r n (c) K = 1 ω ζ M ω 8- MATLAB cde: % Questin 8-, clear all; clse all; s = tf('s') %a) num_g_a= 5; den_g_a=s*(s+6.54); G_a=num_G_a/den_G_a; CL_a=G_a/(1+G_a) BW = bandwidth(cl_a) bde(cl_a) %b) figure(); num_g_b=1.38; den_g_b=s*(s+6.54); G_b=num_G_b/den_G_b; CL_b=G_b/(1+G_b) BW = bandwidth(cl_b) bde(cl_b) %c) figure(3); num_g_c=1; den_g_c=s*(s+6.54); G_c=num_G_c/den_G_c; = 1 rad / sec = 6.54 =. 37 = = 9.45 ra d / sec n r r Bde diagram (a) k=5: data pints frm tp t bttm indicate bandwidth BW, resnance peak M r, and resnant frequency ω r. Magnitude (db) Phase (deg) Bde Diagram Frequency (rad/sec) System: CL_a Frequency (rad/sec):.855 Magnitude (db): -3 System: CL_a Frequency (rad/sec):.5 Magnitude (db): -9.4 System: CL_a Frequency (rad/sec):.5 Phase (deg): -9. Bde diagram (b) k=1.38: data pints frm tp t bttm indicate bandwidth BW, resnance peak M r, and resnant frequency ω r.

2 CL_c=G_c/(1+G_c) BW = bandwidth(cl_c) bde(cl_c) - Magnitude (db) -4 System: CL_b Frequency (rad/sec): 4.6 Magnitude (db): -3.3 System: CL_b Frequency (rad/sec): 4.6 Bde Diagram Magnitude (db): Phase -45 (deg) System: CL_b Frequency (rad/sec): 4.6 Phase (deg): Frequency (rad/sec) Bde diagram (c) k=1: data pints frm tp t bttm indicate resnance peak M r, bandwidth BW, and resnant frequency ω r. Bde Diagram Magnitude (db) System: CL_c Frequency (rad/sec): 9.99 Magnitude (db): 3.6 System: CL_c Frequency (rad/sec): 14.3 Magnitude (db): Phase (deg) System: CL_c Frequency (rad/sec): 1 Phase (deg): Frequency (rad/sec) 8

3 8-3) If is the input, then where and Therefre: As a result: 8-4 (a) M r = ω =. 944 ( db) 3 rad / sec BW = rad / sec r (b) M r = ω = ( db) 4 rad / sec BW = 6.3 rad / sec r (c) M r = ω = 4.17 ( 1.4 db) 6.5 rad / sec BW = 9.18 rad / sec r (d) M r = ω = 1 ( db) rad / sec BW =.46 rad / sec r (e) M r = ω = 157. ( db). 8 rad / sec BW = 1.1 rad / sec r 8 3

4 (f) M r = = ( unstable) ω 15. rad / sec BW =.44 rad / sec r (g) M r = ω = 39. ( 98. db) 1.5 rad / sec BW =.7 rad / sec r (h) M r = = 4.1 ( 1. 3 db) ω r 3. 5 rad / sec BW = 5.16 rad / sec 8 5) Maximum versht =.1 Thus, ζ =.59 M r 1 = ζ 1 ζ 1 = t =.416 ζ ζ. = 1. sec r ω n Thus, minimum ω n = rad / sec Maximum M r = 15. Minimum BW = n ( ) 1/ 4 ( ) ω 1 ζ + 4ζ 4ζ + =.56 rad/sec 8 6) Maximum versht =. Thus,. = e πζ 1 ζ ζ = M r = ζ 1 ζ = 1.3 t r 1 =.416 ζ ζ =. ω n Thus, minimum ω n = rad/sec Maximum M r = 1.3 Minimum BW = ( ) 4 ( ζ ζ ζ ) = 18.7 rad/sec 1/ 8-7) Maximum versht =.3 Thus, 3. = e πζ 1 ζ ζ = M r = ζ 1 ζ = t r 1 =.416 ζ ζ =. ω n Thus, minimum ω n = rad/sec 8 4

5 Maximum M r = Minimum BW = ( ) 4 ( ζ ζ ζ ) = rad/sec 1/ 8-8) (a) At the gain crssver: Therefre: at ω = 1.5 (b) MATLAB cde: %slving fr k: syms kc mega=1.5 sl=eval(slve('.5*kc^=.779^*((-.5*mega^3+mega)^+(-.375*mega^+.5*kc)^)',kc)) %plting bde with K=1.37 s = tf('s') K=1.37; num_g_a=.5*k; den_g_a=s*(.5*s^+.375*s+1); G_a=num_G_a/den_G_a; CL_a = G_a/(1+G_a) BW = bandwidth(cl_a) bde(cl_a); Bde diagram: data pint shws -3dB pint at 1.5 rad/sec frequency which is the clsed lp bandwidth 8 5

6 Magnitude (db) Bde Diagram System: CL_a Frequency (rad/sec): 1.5 Magnitude (db): -3.1 Phase (deg) Frequency (rad/sec) 8-9) θ = α = 9 θ = 63 Therefre: As a result: 8 6

7 Therefre: T change the crssver frequency requires adding gain as: (b) MATLAB cde: s = tf('s') %(b) K =.95*; num_g_a =.5*K; den_g_a = s*(.5*s^+.375*s+1); G_a = num_g_a/den_g_a; CL_a = G_a/(1+G_a) bde(cl_a); figure(); sistl Peak mag =. can be cnverted t db units by: *Lg(.)= db 8 7

8 By using sistl and imprting the lp transfer functin, the verall gain (.5K) was changed until the magnitude f the resnance in Bde was abut 6.9 db. At.5K=~.95 r K=1.9, this resnance peak was achieved as can be seen in the BODE diagram f the fllwing figure: 6 4 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: 4.48 db Freq: rad/sec Stable lp P.M.: 59.1 deg Freq: 1.11 rad/sec Frequency (rad/sec) Frequency (rad/sec) 8-1) 1 M r = 1.4 = ζ 1 ζ Thus, ζ =.387 Maximum versht = e πζ 1 ζ =.675 (6.75%) 3 ω = 3 rad / sec = ω 1 ζ =. 8367ω rad/sec ω r n n n = = r ad/sec t max = ω n π = π 1 ζ (. 387) = 95. sec At ω =, M =

9 This indicates that the steady state value f the unit step respnse is.9. Unit step Respnse: 8-11) a) The clsed lp transfer functin is: as,which means ξ =.387 Accrding t the transfer functin: ξ ω n =.1 ω n =.19 rad/s As ω n =.1K ; then, K = 1 ω n =.1669 b) As K =.1664, then which means Accrdingly PM = 4 As, then 8 9

10 8-1) T BW (rad/sec) M r ) T BW (rad/sec) M r ) The Ruth array is: 8 1

11 S S.375.5K S 1 1-1/3 S.5K Therefre: As, if GH is rearranged as: then which gives (c) where therefre, MATLAB cde: Bde diagram: s = tf('s') %c) K = ; num_g_a =.5*K; den_g_a = s*(.5*s^+.375*s+1 ); %create clsed-lp system G_a = num_g_a/den_g_a; 8 11

12 CL_a = G_a/(1+G_a) bde(cl_a); Bde Diagram Ntes: 1 - BW is verified by finding - 3dB pint at Freq = 1.5 rad/sec in the Bde graph at calculated k. Magnitude (db) System: CL_a Frequency (rad/sec): 1.5 Magnitude (db): By cmparisn t diagram f typical nd rder ples with different damping ratis, damping rati is apprximated as: ξ = ~.77 Phase (deg) Frequency (rad/sec) 8 1

13 8 15 (a) Ls ( ) = s( s)( s) P ω = 1, P = When ω = : L( jω) = 9 L( jω) = When ω = : L( jω) = 7 L( jω) = ( ) ( ) ( ).6ω jω 1.5ω L( jω ) = = 4.6ω + jω 1.5ω.36ω + ω 1.5ω Setting Im L( jω ) = 1. 5ω = Thus, ω = ± 4.47 rad / sec L( j4.47) = Φ = 7 = ( Z.5P P) 18 = ( Z.5) 18 Thus, Z = = 11 ω 18 The clsed lp system is unstable. The characteristic equatin has tw rts in the right half s plane. MATLAB cde: s = tf('s') %a) figure(1); num_g_a= ; den_g_a=s*(.1*s+1)*(.5*s+1); G_a=num_G_a/den_G_a; nyquist(g_a) Nyquist Plt f L( jω ): 8 13

14 (b) Ls ( ) = 1 s( s)( s) Based n the analysis cnducted in part (a), the intersect f the negative real axis by the L( jω ) plt is at.8333, and the crrespnding ω is 4.47 rad/sec. δ ι Thus, The clsed lp system is stable. ω Φ 11 = 9 = Z. 5P P 18 = 18Z 9 Z =. MATLAB cde: s = tf('s') %b) figure(1); num_g_a= 1; den_g_a=s*(.1*s+1)*(.5*s+1); G_a=num_G_a/den_G_a; nyquist(g_a) Nyquist Plt f L( jω ): 8 14

15 (c) Ls ( ) = 1(1+ s) s( s)( 1+. s)( s) P = 1, P =. ω When ω = : L( j) = 9 L( j) = When ω = : L( j ) = 7 L( j ) = When ω = : L( jω) = 7 L( jω) = When ω = : L( jω) = 7 L( jω) = 4 ( ) j ( ) 4 jω ( ω ω ) jω( ω ) 4 ( ) ( ) 1(1 + jω ) 1(1 + ) L( jω ) = =.1ω.8ω + ω 1.17ω.1ω.8ω + ω 1.17ω 4 Setting Im L( jω) =. 1ω. 8ω ω = 4 ω 63ω 1 = Thus, ω = ω = ± 8. 3 rad/sec 4 ( ω ω ) + ω ( ω ) (.1ω.8ω ) + ω ( 1.17ω ) L( j8.3) = = 1 ω =

16 ( ) ( ) Φ = 7 = Z.5P P 18 = Z.5 18 Thus, Z = The clsed lp system is 11 ω unstable. The characteristic equatin has tw rts in the right half s plane. MATLAB cde: s = tf('s') %c) figure(1); num_g_a= 1*(s+1); den_g_a=s*(.1*s+1)*(.*s+1)*(.5*s+1); G_a=num_G_a/den_G_a; nyquist(g_a) Nyquist Plt f L( jω ): 8 16

17 (d) Ls ( ) = 1 P P ω s ( 1+. s)( s) = = When ω = : L( jω) = 18 L( jω) = When ω = : L( jω) = 36 L( jω) = ( ) 4 3 ( ω ω + j ω ) 4 6 ( ) L( jω ) = = 4 3.1ω ω j.7ω.1ω ω +.49ω Setting Im L( jω ) =, ω =. The Nyquist plt f L( jω ) des nt intersect the real axis except at the rigin where ω =. ( ) ( ) Φ = Z.5P P 18 = Z 1 18 Thus, Z =. 11 ω The clsed lp system is unstable. The characteristic equatin has tw rts in the right half s plane. MATLAB cde: s = tf('s') %d) figure(1); num_g_a= 1; den_g_a=s^* (.*s+1)*(.5*s+1); G_a=num_G_a/den_G_a; nyquist(g_a) Nyquist Plt f L( jω ): 8 17

18 8 15 (e) 3( s + ) Ls () = P = 1 P= ω s s 3 ( + 3s+ 1) When ω = : L( j) = 9 L( j) = When ω = : L( j ) = 7 L( j ) = 4 ( ) 4 ( ) 4 ( ) 3( jω + ) 3( jω + ) ω 3ω jω L( jω ) = = ω 3ω + jω 4 3ω + ω Setting Im L( jω ) =, 4 ω 3ω = r ω = ω = ± 189. rad/sec. L( j189. ) = 3 ( ) ( ) Φ = Z.5P P 18 = Z.5 18 = 9 Thus, Z = 11 ω MATLAB cde: The clsed lp system is unstable. The characteristic equatin has tw rts in the right half s plane. 8 18

19 s = tf('s') %e) figure(1); num_g_a= 3*(s+); den_g_a=s*(s^3+3*s+1); G_a=num_G_a/den_G_a; nyquist(g_a) Nyquist Plt f L( jω ): 8 15 (f).1 Ls () = P = 1 P= ω s s s s ( + 1)( + + 1) When ω = : L( j) = 9 L( j) = When ω = : L( j ) = 36 L( j ) = 4 ( ) j ( ) 4 ( ω ω ) jω( ω ) 4 ( ) ( ) L( jω ) = = ω ω + ω 1 ω ω ω + ω 1 ω Settiing Im L( jω ) = ω = r ω =. 5 ω =±. 77 rad / sec L( j. 77) = ( ) ( ) Φ = Z.5P P 18 = Z.5 18 = 9 Thus, Z = The clsed lp system is stable. 11 ω 8 19

20 MATLAB cde: s = tf('s') %f) figure(1); num_g_a=.1; den_g_a=s*(s+1)*(s^+s+1); G_a=num_G_a/den_G_a; nyquist(g_a) Nyquist Plt f L( jω ): 8 15 (g) 1 Ls () = P = 3 P= ω s s ( + 1)( s + ) When ω = : L( j) = 9 L( j) = When ω = : L( j ) = 36 L( j ) = The phase f L( jω ) is discntinuus at ω = rad/sec. ( ) ( Z ) Φ = = 9 Φ = = 9 Thus, P = = 18 The clsed lp system is unstable. The characteristic equatin has tw rts in the right half s plane. 8

21 MATLAB cde: s = tf('s') %g) figure(1); num_g_a= 1; den_g_a=s*(s+1)*(s^+); G_a=num_G_a/den_G_a; nyquist(g_a) Nyquist Plt f L( jω ): 8 15 (h) Ls ( ) = 1( s + 1) ss ( + 1)( s+ 1) P = 1 P = ω When ω = : L( j) = 9 L( j) = When ω = : L( j ) = 18 L( j ) = j ( ) ( ) 1( jω + 1) 1( jω + 1) 11ω ω 1 ω L( jω ) = = 4 11 ω + jω(1 ω ) 11ω + ω 1 ω 8 1

22 Setting Im L( jω ) =, ω = is the nly slutin. Thus, the Nyquist plt f L( jω ) des nt intersect the real axis, except at the rigin. ( ) ( ) Φ = Z.5P P 18 = Z.5 18 = 9 Thus, Z =. 11 ω MATLAB cde: The clsed lp system is stable. s = tf('s') %h) figure(1); num_g_a= 1*(s+1); den_g_a=s*(s+1)*(s+1); G_a=num_G_a/den_G_a; nyquist(g_a) 8

23 Nyquist Plt f L( jω ): 8-16 MATLAB cde: s = tf('s') %a) figure(1); num_g_a= 1; den_g_a=s*(s+)*(s+1); G_a=num_G_a/den_G_a; nyquist(g_a) %b) figure(); num_g_b= 1*(s+1); den_g_b=s*(s+)*(s+5)*(s+15); G_b=num_G_b/den_G_b; nyquist(g_b) %c) figure(3); num_g_c= 1; den_g_c=s^*(s+)*(s+1); G_c=num_G_c/den_G_c; nyquist(g_c) %d) figure(4); num_g_d= 1; den_g_d=(s+)^*(s+5); G_d=num_G_d/den_G_d; 8 3

24 nyquist(g_d) %e) figure(5); num_g_e= 1*(s+5)*(s+1); den_g_e=(s+5)*(s+)^3; G_e=num_G_e/den_G_e; nyquist(g_e) Nyquist graph, part(a): 1.5 Nyquist Diagram 1.5 Imaginary Axis Real Axis Nyquist graph, part(b): 8 4

25 4 Nyquist Diagram 3 1 Imaginary Axis Real Axis Nyquist graph, part(c):.5 Nyquist Diagram.4.3. Imaginary Axis Real Axis Nyquist graph, part(d): 8 5

26 .4 Nyquist Diagram.3..1 Imaginary Axis Real Axis Nyquist graph, part (e):.1 Nyquist Diagram Imaginary Axis Real Axis 8 6

27 8-17 (a) K Gs ( ) = ( s + 5) P = P = ω K G( j) = ( K > ) G( ) = 18 ( K < ) G( j) = 5 G( j ) = 18 (K > ) G( j ) = (K < ) G( j ) = Fr stability, Z =. ( P P) Φ = = 11 ω < K < Φ 11 = Stable K < 5 Φ 11 = 18 Unstable 5 < K < Φ 11 = Stable The system is stable fr 5 < K < (b) K Gs ( ) = ( s + 5) P 3 ω = P = K G( j) = ( K > ) G( ) = 18 ( K < ) G( j) = 15 G( j ) = 7 (K > ) G( j ) = 7 (K < ) G( j ) = 8 7

28 Fr stability, Z =. ( P P) Φ = = 11 ω < K < 1 Φ 11 = Stable K > 1 Φ 11 = 36 K < 15 Φ 11 = 18 Unstable Unstable 15 < K < Φ 11 = Stable The system is stable fr 15 < K < (c) K Gs ( ) = ( s + 5) P 4 ω = P = K G( j) = ( K > ) G( ) = 18 ( K < ) G( j) = 65 G( j ) = (K > ) G( j ) = 18 (K < ) G( j ) = 8 8

29 Fr stability, Z =. ( P P) Φ = = 11 ω < K < 5 Φ 11 = Stable K > 5 Φ 11 = 36 K < 65 Φ 11 = 18 Unstable Unstable 65 < K < Φ 11 = Stable The system is stable fr 65 < K < ) The characteristic equatin: r if, the crss real axis at s =.1. Fr stability if K <, the Nyquist crss the real axis at.s, fr stability, therefre, the rage f stability fr the system is MATLAB cde: s = tf('s') K=1 G= K/(s^+*s+); H=1/(s+1); GH=G*H; sistl 8 9

30 K=1.15 G= K/(s^+*s+); H=1/(s+1); GH=G*H; nyquist(gh) xlim[(-1.5,.5)] ylim[(-1,1)] After generating the feed-frward (G) and feedback (H) transfer functins in the MATLAB cde, these transfer functins are imprted t sistl. Nyquist diagram is added t the results f sistl. The verall gain f the transfer functin is changed until Nyquist diagram passes thrugh -1+j pint. Higher values f K resulted in unstable Nyquist diagram. Therefre K<1.15 determines the range f stability fr the clsed lp system. Nyquist at margin f stability: 4 Nyquist Diagram 3 1 Imaginary Axis Real Axis 8-19) 3 ( ) ( ) s s + s + s+ 1 + K s + s+ 1 = 8 3

31 K( s + s+ 1) 3 ( ) L () s = P = 1 P = eq ω s s s s L ( j) = 9 L ( j ) = 18 eq eq ( ) 4 ( ) j ( ) ( ) ( ) 4 ( ) ( ) ω + ω ω + ω ω ω ω + 1 K j K j L ( jω ) = = eq ω ω + ω 1 ω ω ω + ω 1 ω Setting Im L ( jω ) = eq 4 ω ω + 1= Thus, ω =±1 rad/sec are the real slutins. L ( j 1 ) = K eq Fr stability, Φ 11 = (. 5P + P) 18 = 9 ω When K = 1 the system is marginally stable. K > Φ 11 = 9 Stable K < Φ 11 =+ 9 Unstable 8 31

32 Ruth Tabulatin 4 s 1 K+ 1 K s s s 3 1 K K+ 1 K + 1 K K+ 1 ( K 1) = K + 1 K + 1 K > 1 s K K > When K = 1 the cefficients f the s 1 rw are all zer. The auxiliary equatin is s + 1= The slutins are ω =±1 rad/sec. Thus the Nyquist plt f L ( jω ) intersects the 1 pint when K = 1, when ω = ±1 eq rad/sec. The system is stable fr < K <, except at K = 1. 8-) Slutin is similar t the previus prblem. Let s use Matlab as an alternative apprach MATLAB cde: s = tf('s') figure(1); K=8.9 num_gh= K; den_gh=(s^3+3*s^+3*s+1); GH=num_GH/den_GH; nyquist(gh) xlim([-1.5,.5]) ylim([-1,1]) sistl; 8 3

33 After generating the lp transfer functin and analyzing Nyquist in MATLAB sistl, it was fund that fr values f K higher than ~8.9, the clsed lp system is unstable. Fllwing is the Nyquist diagram at margin f stability. Part(a), Nyquist at margin f stability: 1 Nyquist Diagram Imaginary Axis Real Axis Part(b), Verificatin by Ruth-Hurwitz criterin: Using Ruth criterin, the cefficient table is as fllws: S S 3 K+1 S 1 (8-K)/3 S K+1 The system is stable if the cntent f the 1 st clumn is psitive: (8-K)/3> K<8 8 33

34 K+1> K>-1 which is cnsistent with the results f the Nyquist diagrams. 8-1) Parablic errr cnstant ( ) K = lim s G( s) = lim1 K + K s = 1K = 1 Thus K = 1 a P D P P s s Characteristic Equatin: s + 1K s+ 1 = D G eq 1K s D ( s)= s + 1 Fr stability, P ω = P = Φ 11 = (. 5P + P) 18 = 18 ω The system is stable fr < K D <. 8 34

35 8 (a) The characteristic equatin is 1+ Gs ( ) Gs ( ) Gs ( ) = 1 Gs ( ) = G ( s ) K G ( s ) = = eq ( s+ 4) ( s+ 5) P ω 4 ( ) j ( ) = P = ( ω ω ) jω( ω ) ( ) ( ) K K G ( jω ) = = eq 4 1ω + ω + ω 36 18ω 4 1ω + ω + ω 36 18ω G eq K ( j) = 18 G ( j ) = 18 Setting Im G ( jω ) = eq eq ω = and ω = ± 4.47 rad / sec G ( j4.47 ) K = eq 8 Fr stability, ( P P) Φ = = 11 ω The system is stable fr K < r K < 8 35

36 8 (b) 4 3 Characteristic Equatin: s + 18s + 11s + 36s + 4 K = Ruth Tabulatin s K s s 11 4 K s K K > s 4 K K < Thus fr stability, K < 8 3 (a) Gs ( ) = N ss ( + )( s ) Nyquist Plt Fr stability, N < 3.89 Thus N < 3 since N must be an integer. 8 36

37 (b) 5 Gs ( ) = s(. 6s+. 76)( As+ 1) 3 Characteristic Equatin:. 6As + ( A) s s + 5 = G ( As s s ) ( = ) eq 6s s + 5 Since G eq ( s) has mre zers than ples, we shuld sketch the Nyquist plt f 1/ G ( s) fr stability study. eq ( 5 6ω ) + j7.6ω 3 ( ) 1 = G ( jω ) A.76ω j.6ω eq = 3 ( 5 6ω ) + j7.6ω (.76ω + j.6ω ) 4 6 A(.498ω +.36ω ) 1/ G ( j) = 18 1/ G ( j ) = 9 Setting eq ω = ω = ± rad/sec Fr stability, eq G ( j ) = A eq 1 Im G ( jω ) = eq ( P P) Φ = = ω Fr A > 4.3 Φ = 18 Unstable 11 Fr < A < 4.3 Φ 11 = 18 Stable The system is stable fr < A < 4.3. (c) 8 37

38 G( s) = 5 s(. 6s+. 76)( 5s+ K ) Characteristic Equatin: s(. 6s+. 76)( 5s+ K ) + 5 = G eq Ks(. 6s+. 76) ( s) = 3s s + 5 P 3 ω = P = G ( j) = 9 G ( j ) = 9 eq eq ( ) ( ) ( ) 6 ( ) 3 3 K (.6ω +.76 jω) K.6ω +.76 jω ω + j3ω G ( jω ) = = eq ω j3ω ω + 9ω 4 Setting Im G ( jω ) = ω ω = ω = 516. ω = ± rad/sec eq G ( j7. 18) =. 4 K eq Fr stability, ( P P) Φ = = 11 ω Fr stability, < K <

39 8 4 (a) K t = : Y( s) 1K 1K Gs () = = = Es s s s K s s ( t ) ( ) ( + 1) The ( 1, j) pint is enclsed fr all values f K. The system is unstable fr all values f K. (b) K t = 1. : ( s ) j ( ) ( ) 1K 1K 1ω ω 1 ω Gs () = G( jω ) = 4 s s ω + ω 1 ω Setting Im G( jω) = ω = 1 ω =±1 rad / sec G( j1) = 1K The system is stable fr < K <

40 (c) K t = 1.: ( s ) j ( ) ( ) 1K 1K 1ω ω 1 ω Gs () = G( jω ) = 4 s s ω + ω 1 ω Setting Im G( jω) = ω = 1 ω = ±31. 6 rad/sec G( j316. ) = K Fr stability, < K < 1 8 5) The characteristic equatin fr K = 1 is: 3 s + 1s + 1, K s+ 1, = t G eq ( s) = s 1,Ks t P 3 ω + 1s + 1, = P = 4 t ω + jω( ω ) ( ) 6 1, Kjω 1, K 1, 1 t G ( jω ) = = eq 3 1, 1ω jω 1, 1ω + ω Setting Im G ( jω ) = eq 8 4

41 ω =, ω = 1, ω =±1 rad / sec G ( j1 ) = K eq t Fr stability, ( P P) Φ = = ω The system is stable fr K t >. 8-6) a) b) c) MATLAB cde: s = tf('s') figure(1); J=1; B=1; K=1; Kf= G1= K/(J*s+B); CL1=G1/(1+G1*Kf); H = 1; G1G = CL1/s; L_TF=G1G*H; nyquist(l_tf) sistl 8 41

42 Part (a), Kf=: by pltting the Nyquist diagram in sistl and varying the gain, it was bserved that all values f gain (K) will result in a stable system. Lcatin f ples in rt lcus diagram f the secnd figure will als verify that. Nyquist Diagram Imaginary Axis Real Axis 8 4

43 1 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: Inf Freq: Inf Stable lp P.M.: 54.9 deg Freq:.74 rad/sec Frequency (rad/sec) Frequency (rad/sec) Part (b), Kf=.1: The result and apprach is similar t part (a), a sample f Nyquist diagram is presented fr his case as fllws: 15 Nyquist Diagram 1 5 Imaginary Axis Real Axis 8 43

44 Part (c), Kf=.: The result and apprach is similar t part (a), a sample f Nyquist diagram is presented fr his case as fllws: 15 Nyquist Diagram 1 5 Imaginary Axis Real Axis 8-7) MATLAB cde: s = tf('s') figure(1); J=1; B=1; K=1; Kf=. G1= K/(J*s+B); CL1=G1/(1+G1*Kf); H = 1; G1G = CL1/s; L_TF=G1G*H; nyquist(l_tf) After assigning K=1, different values f Kf has been used in the range f.1<k<1 4. The Nyquist diagrams shws the stability f the clsed lp system fr all <K<. A sample f Nyquist diagram is pltted as fllws: 8 44

45 5 Nyquist Diagram 15 1 Imaginary Axis Real Axis 8-8) a) K > system is stable b) < K < 1 and - < K < - < K < 1 system is stable MATLAB cde: s = tf('s') %a) figure(1); K=1 num_gh_a= K*(s+1); den_gh_a=(s-1)^; GH_a=num_GH_a/den_GH_a; nyquist(gh_a) %b) figure(); K=1 num_gh_b= K*(s-1); den_gh_b=(s+1)^; GH_b=num_GH_b/den_GH_b; nyquist(gh_b) 8 45

46 sistl Part(a): Using MATLAB sistl, the transfer functin gain can be iteratively changed in rder t btain different phase margins. By changing the gain s that PM= (margin f stability), K>~ resulted in stable Nyquist diagram fr part(a). Fllwing tw figures illustrate the sistl and Nyquist results at margin f stability fr part (a). Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) 1-1 G.M.: db Freq: 1.73 rad/sec Stable lp P.M.:.688 deg Freq: 1.73 rad/sec Frequency (rad/sec) 1 Frequency (rad/sec) 8 46

47 Nyquist Diagram Imaginary Axis Real Axis Part(b): Similar methdlgy applied as in part (a). K<1 results in clsed lp stability. Fllwing are sistl and Nyquist results at margin f stability (K=1): 1 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) Frequency (rad/sec) G.M.: -.36 db Freq: rad/sec Unstable lp P.M.: -4.1 deg Freq:.33 rad/sec Frequency (rad/sec) 8 47

48 1 Nyquist Diagram Imaginary Axis Real Axis 8-9) (a) Let Gs G se Ts d ( ) = ( ) 1 Then G () s = s s 1 1 ( + 1s+ 1) Let 1 1 = 1 r = 1 1/ 1ω + jω 1 ω ( ) 4 1ω + ω ( 1 ω ) 1ω 4 + ω 1 ω = 1, ω 6 1ω 4 + 1, ω 1, = Thus ( ) The real slutin fr ω are ω =±1 rad/sec. 8 48

49 1 ω G = = 1ω 1 ( j1) tan ω = 1 Equating ω Td ω = 1 = ( ) 18 π 84.3π = = 1.47 rad 18 Thus the maximum time delay fr stability is T d = 1.47 sec. (b) T d = 1 sec. s 1Ke 1Ke Gs () = G( jω ) = s s s j jω ( ) 1ω + ω( 1 ω ) At the intersect n the negative real axis, ω = 1.4 rad/sec. G( j1.4) =. 717K. The system is stable fr < K <

50 8 3 (a) K =.1 Ts d 1e Gs ( ) = = G( se ) 1 s s ( + 1s+ 1) Ts d Let 1 1 = 1 r = 1 1/ 1ω + jω 1 ω ( ) 4 1ω + ω ( 1 ω ) 6 4 Thus ω 1ω + 1, ω 1 = The real slutins fr ω is ω =±1. rad/sec. 1 ω G = = 1ω 1 ( j.1) tan ω =.1 Equate ω Td ω =.1 ( ) π = = 1.56 rad We have T d = sec. 18 We have the maximum time delay fr stability is 15.6 sec. 8 3 (b) T d = 1. sec. 1Ke 1Ke Gs () = G( jω ) = s s s j.1s.1 jω ( ) 1ω + ω( 1 ω ) At the intersect n the negative real axis, ω = 676. rad/sec. G( j6. 76) =. 176K 8 5

51 The system is stable fr < K < ) MATLAB cde: s = tf('s') %a) figure(1); K=1 num_gh_a= K; den_gh_a=s*(s+1)*(s+1); GH_a=num_GH_a/den_GH_a; nyquist(gh_a) %b) figure(); K= num_gh_b= K; den_gh_b=s*(s+1)*(s+1); GH_b=num_GH_b/den_GH_b; nyquist(gh_b) sistl; 8 51

52 By using sistl and imprting the lp transfer functin, different values f K has been tested which resulted in a stable system when K<, and unstable system fr K>. Fllwing diagrams crrespnd t margin f stability: 3 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: -.8 db Freq: 1 rad/sec Unstable lp Frequency (rad/sec) -5 P.M.: -.95 deg Freq: 1. rad/sec 1 Frequency (rad/sec) Part(a): small K resulted in stable system as shwn belw fr K=.19: 8 5

53 3 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: 19. db Freq: 1 rad/sec Stable lp Frequency (rad/sec) -5 P.M.: 66.3 deg Freq:.1 rad/sec 1 Frequency (rad/sec) 1 Nyquist Diagram Imaginary Axis Real Axis Part(b): Large K resulted in unstable system as shwn belw fr K=: 8 53

54 3 1-1 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: - db Freq: 1 rad/sec Unstable lp Frequency (rad/sec) P.M.: deg Freq:.59 rad/sec 1 Frequency (rad/sec) Nyquist Diagram 3 1 Imaginary Axis Real Axis 8 54

55 The system is stable fr small value f K, since there is n encirclement f the s = -1 The system is unstable fr large value f K, since the lcus encirclement the CCW; which means tw ples are in the right half s-plane. twice in 8-3) (a) The transfer functin (gain) fr the sensr amplifier cmbinatin is 1 V/.1 in = 1 V/in. The velcity f flw f the slutin is 3 1 in / sec v = = 1 in/sec 1. in The time delay between the valve and the sensr is T d = D/ v sec. The lp transfer functin is Gs ( )= s 1Ke Ts d + 1s

56 8 56

57 8-33) (a) The transfer functin (gain) fr the sensr amplifier cmbinatin is 1 V/.1 in = 1 V/in. The velcity f flw f the slutins is 1 in 3 / sec v = = 1 in / sec 1. in The time delay between the valve and sensr is T d = D/ v sec. The lp transfer functin is Gs ( )= s 1Ke Ts d + 1s + 1 (b) K = 1: Ts d Gs () = G() se G( jω ) = 1e ( ω ) 1 jωtd 1 + j1ω Setting 1 ( ω ) ω 1 ω + j1ω = + = ( ) , 4 Thus, ω 1ω = Real slutins: ω =, ω = ± 1 rad / sec 1ω G = = 1 (1) j tan ω ω = 1 Thus, 1 T d 9 π π = = rad 18 Thus, π T d = =. 157 sec Maximum D = vt d = = in 8 57

58 (c) D = 1 in. T d D 1 = = = 1. sec Gs ( ) = v 1 s 1Ke 1. s + 1s + 1 The Nyquist plt f G( jω ) intersects the negative real axis at ω = 1. 9 rad/sec. G( j) =. 773 K Fr stability, the maximum value f K is ) The system (GH) has zer ples in the right f s plane: P=. Accrding t Nyquist criteria (Z=N+P), t ensure the stability which means the number f right ples f 1+ GH= shuld be zer (Z=), we need N clckwise encirclements f Nyquist diagram abut -1+j pint. That is N=-P r in ther wrds, we need P cunter-clckwise encirclement abut -1+j. In this case, we need CCW encirclements. 8-34(a) Accrding t Nyquist diagrams, this happens when K<-1. The three Nyquist diagrams are pltted with K=-1, K=-1, K=1 as examples: MATLAB cde: s = tf('s') %a) figure(1); K=-1 num_g_a = K ; 8 58

59 den_g_a =(s+1); num_h_a = (s+); den_h_a = (s^+*s+); G_a=num_G_a/den_G_a; H_a = num_h_a/den_h_a; OL_a = G_a*H_a nyquist(ol_a) Case 1) Nyquist graph, K=-1: margin f stability K<-1 unstable figure(); K=-1 num_g_a = K ; den_g_a =(s+1); num_h_a = (s+); den_h_a = (s^+*s+); G_a=num_G_a/den_G_a; H_a = num_h_a/den_h_a; %CL_a = G_a/(1 + G_a*H_a); OL_a = G_a*H_a nyquist(ol_a) 8 59

60 Case ) Nyquist graph, K=-1: marginally unstable figure(3); K=1 num_g_a = K ; den_g_a =(s+1); num_h_a = (s+); den_h_a = (s^+*s+); G_a=num_G_a/den_G_a; H_a = num_h_a/den_h_a; %CL_a = G_a/(1 + G_a*H_a); OL_a = G_a*H_a nyquist(ol_a) 8 6

61 Case 3) Nyquist graph, K=1: stable case, -1<K n CCW encirclement abut -1+j pint 8 61

62 8-34 (b) Fr K<-1 (unstable), there will be 1 real ple in the right hand side f s-plane fr the clsed lp system, by running the fllwing cde. K=-1 num_g_a = K ; den_g_a =(s+1); num_h_a = (s+); den_h_a = (s^+*s+); G_a=num_G_a/den_G_a; H_a = num_h_a/den_h_a; OL_a = G_a*H_a nyquist(ol_a) CL=1/(1+OL_a) ple(cl) K = -1 Transfer functin: -1 s s^3 + 3 s^ + 4 s + Transfer functin: s^3 + 3 s^ + 4 s s^3 + 3 s^ - 6 s - 18 ans = Fr K=-1 (marginally unstable), there will be negative cmplex cnjugate ples and a ple at zer fr the clsed lp system, by running the fllwing cde. K=-1 num_g_a = K ; den_g_a =(s+1); num_h_a = (s+); den_h_a = (s^+*s+); 8 6

63 G_a=num_G_a/den_G_a; H_a = num_h_a/den_h_a; OL_a = G_a*H_a nyquist(ol_a) CL=1/(1+OL_a) ple(cl) K = -1 Transfer functin: -s s^3 + 3 s^ + 4 s + Transfer functin: s^3 + 3 s^ + 4 s s^3 + 3 s^ + 3 s ans = i i Nte: yu may als wish t use MATLAB sistl. See alternative slutin t

64 8-34(c) The Characteristic Equatin is: s s + (4+K) s + +K Using Ruth criterins, the cefficient table is as fllws: s K s 3 K+ s 1 K+1 s K+ The system is stable if the cntent f the 1 st clumn is psitive: K+1> K>-1 K+> K>-1 which is cnsistent with the results f the Nyquist diagrams. Fr K>-1 system is stable. 8-35) (a) M r = ω r =. 6, rad / sec, BW = 15. rad / sec (b) 1 M r = ζ 1 ζ 4 =. 6 ζ ζ = The slutin fr ζ <. 77 is ζ = ω 1 ζ = rad / sec Thus ω = = rad / sec r n ω n G () s = = = L s s+ s( s ) s(1 +.5 s) ( ζω ) n BW = 15.1 rad/sec 8 64

65 8-36) Assuming a unity feedback lp (H=1), G(s)H(s)=G(s) (a) Gs ( ) = 5 s( s)( s) 8 65

66 8 36 (b) Gs ( ) = 1 s( s)( s) (c) 5 Gs ( ) = ( s+ 1.)( s+ 4)( s+ 1) 8 66

67 (d) Gs ( ) = 1( s + 1) ss ( + )( s+ 1) 8 36 (e).5 Gs () = s s ( + s+ 1) 8 67

68 (f) Gs () = s s 1e s ( + 1s+ 5) 8 68

69 (g) Gs ( ) = 1e s ( + 1s+ 1) s s 8 36 (h) 1( s + 5) Gs () = s s ( + 5s+ 5) (a) 8 69

70 MATLAB cde: s = tf('s') num_g_a= 5; den_g_a=s*(.5*s+1)*(.1*s+1); G_a=num_G_a/den_G_a margin(g_a) Bde diagram: (b) MATLAB cde: s = tf('s') num_g_a= 1; den_g_a=s*(1+.5*s)*(1+.1*s); G_a=num_G_a/den_G_a margin(g_a) 8 7

71 (c) MATLAB cde: s = tf('s') num_g_a= 5; den_g_a=s*(s+1.)*(s+4)*(s+1); G_a=num_G_a/den_G_a margin(g_a) 8 71

72 (d) MATLAB cde: s = tf('s') num_g_a= 1*(s+1); den_g_a=s*(s+)*(s+1); G_a=num_G_a/den_G_a margin(g_a) 8 7

73 (e) MATLAB cde: s = tf('s') num_g_a=.5; den_g_a=s*(s^+s+1); G_a=num_G_a/den_G_a margin(g_a) 8 73

74 (f) MATLAB cde: s = tf('s') num_g_a= 1*exp(-s); den_g_a=s*(s^+1*s+5); G_a=num_G_a/den_G_a margin(g_a) 8 74

75 (g) MATLAB cde: s = tf('s') num_g_a= 1*exp(-s); den_g_a=s*(s^+1*s+1); G_a=num_G_a/den_G_a margin(g_a) 8 75

76 (h) MATLAB cde: s = tf('s') num_g_a= 1*(s+5); den_g_a=s*(s^+5*s+5); G_a=num_G_a/den_G_a margin(g_a) 8 76

77 8 77

78 8-37) MATLAB cde: s = tf('s') num_gh_a= 5*(s+1); den_gh_a=s*(s+)*(s^+*s+16); GH_a=num_GH_a/den_GH_a margin(gh_a) 8 78

79 8-38) MATLAB cde: s = tf('s') num_g_a= 5*(s+1); den_g_a=s*(s+)*(s^+*s+16); G_a=num_G_a/den_G_a margin(g_a) Bde diagram: PM=11 deg, GM= rad/sec Bde Diagram Gm = 3.91 db (at 4. rad/sec), Pm = 11 deg (at 1.8 rad/sec) Magnitude (db) Phase (deg) Frequency (rad/sec) 8-38 Alternative slutin MATLAB cde: s = tf('s') %a) figure(1); num_g_a = 1 ; den_g_a =(s+1); num_h_a = (s+); den_h_a = (s^+*s+); 8 79

80 G_a=num_G_a/den_G_a; H_a = num_h_a/den_h_a; CL_a = G_a/(1 + G_a*H_a); margin(cl_a) sistl Bde diagram: fr k=1 Bde Diagram Gm = Inf, Pm = Inf Magnitude (db) Phase (deg) Frequency (rad/sec) Using MATLAB sistl, the transfer functin gain can be iteratively changed in rder t btain different phase margins. By changing the gain K between very small and very big numbers, it was fund that the clsed lp system are stable (psitive PM) fr every psitive K in this system. K=

81 1 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) -1-1 G.M.: Inf Freq: Inf Stable lp P.M.: Inf Freq: NaN Frequency (rad/sec) Frequency (rad/sec) K= Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: Inf Freq: Inf Stable lp P.M.: 7.67 deg Freq: 7.89 rad/sec Frequency (rad/sec) Frequency (rad/sec) 8 81

82 In rder t test the negative range f K, 1 was multiplied t the lp transfer functin thrugh the fllwing cde, and sistl was used again. figure(1); num_g_a = -1 ; den_g_a =(s+1); num_h_a = (s+); den_h_a = (s^+*s+); G_a=num_G_a/den_G_a; H_a = num_h_a/den_h_a; CL_a = G_a/(1 + G_a*H_a); margin(cl_a) sistl at K= 1, margin f stability is bserved as PM~=: K= Rt Lcus Editr fr Open Lp 1 (OL1) -4-4 Bde Editr fr Clsed Lp 1 (CL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) G.M.: -.69 db Freq: rad/sec Unstable lp Frequency (rad/sec) 45 P.M.: deg Freq:.137 rad/sec Frequency (rad/sec) The system is stable fr K> 1 as fllws: K=.6 8 8

83 1 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: 4.37 db Freq: rad/sec Stable lp Frequency (rad/sec) 9 45 P.M.: Inf Freq: NaN Frequency (rad/sec) And the system is unstable fr K< 1: K= 3 1 Rt Lcus Editr fr Open Lp 1 (OL1) Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: db Freq: rad/sec Unstable lp Frequency (rad/sec) 9 45 P.M.: -19 deg Freq: 1.84 rad/sec Frequency (rad/sec) *Cmbining the individual ranges fr K, the system will be stable in the range f K>

84 8 39 See sample MATLAB cde in Part e. The MATLAB cdes are identical t prblem (a) Gs ( ) = K s( s)( s) The Bde plt is dne with K = 1. GM = 1.58 db Fr GM = db, K must be reduced by 1.58 db. Thus K =.8337 PM = 6.4. Fr PM = 45 K shuld be increased by 5.6 db. Or, K = 1.91 (b) K( s + 1) Gs ( ) = s( s)( 1+. s)( s) Bde plt is dne with K = 1. The GM = db. Fr GM = db, K 1. PM = Fr PM = 45 K shuld be increased by 8.9 db. Or, K =

85 8 39 (c) See the tp plt K Gs ( ) = ( s + 3) 3 The Bde plt is dne with K = 1. GM = db PM = infinity. Fr GM = db K can be increased by 6.69 db r K = 1.6. Fr PM = 45 deg. K can be increased by 8.71 db, r K = 7.6. (d) See the middle plt K Gs ( ) = ( s + 3) 4 The Bde plt is dne with K = 1. GM = 5.1 db PM = infinity. Fr GM = db K can be increased by 3.1 db r K = 3.4 Fr PM = 45 deg. K can be increased by 38.4 db, r K = (e) See the bttm plt The Bde plt is dne with K =

86 s Ke Gs ( ) = s s s ( ) GM=.97 db; PM = 6.58 deg Fr GM = db K must be decreased by 17.3 db r MATLAB cde: s = tf('s') num_g_a= exp(-s); den_g_a=s*(.1*s^+.1*s+1); G_a=num_G_a/den_G_a margin(g_a) K =.141. Fr PM = 45 deg. K must be decreased by.9 db r K =

87 8 39 (f) K (1 +.5 s) Gs () = s s ( + s+ 1) The Bde plt is dne with K = 1. GM = 6.6 db PM =.4 deg Fr GM = db K must be decreased by db r K =.55. Fr PM = 45 deg K must be decreased by 3.55 db r K = (a) Gs ( ) = 1K s( s)( s) The gain phase plt is dne with K = 1. GM = 1.58 db PM = 3.95 deg. Fr GM = 1 db, K must be decreased by 8.4 db r K =.38. Fr PM = 45 deg, K must be decreased by 14 db, r K =.. Fr M r = 1., K must be decreased t

88 Sample MATLAB cde: s = tf('s') num_g_a= 1; den_g_a=s*(1+.1*s)*(.5*s+1); G_a=num_G_a/den_G_a nichls(g_a) (b) 8 88

89 5K( s + 1) Gs ( ) = s( s)( 1+. s)( s) The Gain phase plt is dne with K = 1. GM = 6 db PM =.31 deg. Fr GM = 1 db, K must be decreased by 4 db r K =.631. Fr PM = 45 deg, K must be decrease by 5 db. Fr M r = 1., K must be decreased t (c) 1K Gs () = s s s ( ) The gain phase plt is dne fr K = 1. GM = db M r = PM = deg Fr GM = 1 db, K must be decreased by 1 db r K =.316. Fr PM = 45 deg, K must be decreased by 5.3 db, r K =.543. Fr M r = 1., K must be decreased t.13. (d) 8 89

90 Gs ( ) = Ke s ( ) s s s The gain phase plt is dne fr K = 1. GM =.97 db M r = 39. PM = 6.58 deg Fr GM = 1 db, K must be decreased by 7.3 db, K =.445. Fr PM = 45 deg, K must be decreased by.9 db, r K =.71. Fr M = 1., K = 61.. r 8 9

91 8-41 MATLAB cde: s = tf('s') %a) num_gh_a= 1*(s+1)*(s+); den_gh_a=s^*(s+3)*(s^+*s+5); GH_a=num_GH_a/den_GH_a; CL_a = GH_a/(1+GH_a) figure(1); bde(cl_a) %b) figure(); rlcus(gh_a) %c) num_gh_c= 53*(s+1)*(s+); den_gh_c=s^*(s+3)*(s^+*s+5); GH_c=num_GH_c/den_GH_c; figure(3); nyquist(gh_c) xlim([- 1]) ylim([ ]) %d) num_gh_d= (s+1)*(s+); den_gh_d=s^*(s+3)*(s^+*s+5); GH_d=num_GH_d/den_GH_d; CL_d = GH_d/(1+GH_d) figure(4); margin(cl_d) sistl Part (a), Bde diagram: 8 91

92 5 Bde Diagram Magnitude (db) Phase (deg) Part (b), Rt lcus diagram: Frequency (rad/sec) 3 Rt Lcus 1 Imaginary Axis Real Axis Part (c), Gain and frequency that instability ccurs: Gain=53, Freq = 4.98 rad/sec, as seen in the data pint in the figure: 8 9

93 1.5 Nyquist Diagram 1 Imaginary Axis.5 System: GH_a Real: -1 Imag:.174 Frequency (rad/sec): Real Axis Part (d), Gain and frequency that instability ccurs: Gain=.17 at PM = deg: By running sistl cmmand in MATLAB, the transfer functins are imprted and the gain is iteratively changed until the phase margin f PM= deg is achieved. The crrespnding Gain is K=.17. Part (e): The crrespnding gain margin is GM =.7 db is seen in the figure 8 93

94 8 4 (a) Gain crssver frequency =.9 rad/sec PM = deg Phase crssver frequency =.31 rad/sec GM = 1.13 db (b) Gain crssver frequency = 6.63 rad/sec Phase crssver frequency =.31 rad/sec PM = 7.8 deg GM = db (c) Gain crssver frequency = 19.1 rad/sec Phase crssver frequency =.31 rad/sec PM = 4.7 deg GM = 1.13 db (d) Fr GM = 4 db, reduce gain by (4 1.13) db = 18.7 db, r gain =.116 nminal value. (e) Fr PM = 45 deg, the magntude curve reads 1 db. This means that the lp gain can be increased by 1 db frm the nminal value. Or gain = 3.16 nminal value. (f) The system is type 1, since the slpe f G( jω ) is db/decade as ω. (g) GM = 1.7 db. PM = deg. (h) The gain crssver frequency is.9 rad/sec. The phase margin is deg. Set ωt d π = 9. T = =. rad d 18 Thus, the maximum time delay is T d = sec. 8 94

95 8 43 (a) The gain is increased t fur times its nminal value. The magnitude curve is raised by 1.4 db. Gain crssver frequency = 1 rad/sec Phase crssver frequency =.31 rad/sec PM = 46 deg GM = 9.9 db (b) The GM that crrespnds t the nminal gain is 1.13 db. T change the GM t db we need t increase the gain by 1.13 db, r times the nminal gain. (c) The GM is 1.13 db. The frward path gain fr stability is 1.13 db, r (d) The PM fr the nminal gain is deg. Fr PM = 6 deg, the gain crssver frequency must be mved t apprximately 8.5 rad/sec, at which pint the gain is 1 db. Thus, the gain must be increased by 1 db, r by a factr f (e) With the gain at twice its nminal value, the system is stable. Since the system is type 1, the steady state errr due t a step input is. (f) With the gain at times its nminal value, the system is unstable. Thus the steady state errr wuld be infinite. (g) With a pure time delay f.1 sec, the magnitude curve is nt changed, but the the phase curve is subject t a negative phase f 1. ω rad. The PM is PM = gain crssver frequency = = deg The new phase crssver frequency is apprximately 9 rad/sec, where the riginal phase curve is reduced by.9 rad r 51.5 deg. The magnitude f the gain curve at this frequency is 1 db. Thus, the gain margin is 1 db. (h) When the gain is set at 1 times its nminal value, the magnitude curve is raised by db. The new gain crssver frequency is apprximately 17 rad/sec. The phase at this frequency is 3 deg. Thus, setting ωt d 3 π = 17T = = d Thus T d =. 38 sec. 8 95

96 8-44 MATLAB cde: s = tf('s') %using pade cammand fr PADE apprximatin f expnential term num_g_a= pade((8*exp(-.1*s)),); den_g_a=s*(s+4)*(s+1); G_a=num_G_a/den_G_a; CL_a = G_a/(1+G_a) OL_a = G_a*1; %(a) figure(1) nyquist(ol_a) %(b) and (c) figure(); margin(cl_a) Part (a), Nyquist diagram: 8 Nyquist Diagram 6 4 Imaginary Axis Real Axis Part (b) and (c), Bde diagram: Using Margin cmmand, the gain and phase margins are btained as GM = 6.3 db, PM = 4.6 deg: 8 96

97 5 Bde Diagram Gm = 6.3 db (at 4. rad/sec), Pm = 4.6 deg (at.9 rad/sec) Magnitude (db) Phase (deg) Frequency (rad/sec) 8 97

98 8 45 (a) Bde Plt: Fr stability: 166 (44.4 db) < K < 779 (77 db) Phase crssver frequencies: 7 rad/sec and 85 rad/sec 8 98

99 Nyquist Plt: 8 99

100 8 45 (b) Rt Lci. 8 1

101 8 46 (a) Nyquist Plt Bde Plt 8 11

102 Rt Lcus 8 1

103 8-47 MATLAB cde: %a) k=1 num_gh= k*(s+1)*(s+5); den_gh=s*(s+.1)*(s+8)*(s+)*(s+5); GH=num_GH/den_GH; CL = GH/(1+GH) figure(1); bde(cl) figure(); OL = GH; nyquist(gh) xlim([-1.5.5]); ylim([-1 1]); sistl Part (a), Bde diagram: 8 13

104 Bde Diagram Magnitude (db) Phase (deg) Part (a), Nyquist diagram: Frequency (rad/sec) 1 Nyquist Diagram Imaginary Axis Real Axis Part (a), range f K fr stability: By running sistl cmmand in MATLAB, the transfer functins are imprted and the gain is iteratively changed until the phase margin f PM= deg is achieved (where K = ) which is the margin f stability. The stable rang fr K is K>8.16x1 4 : 8 14

105 Part (b), Rt-lcus diagram, K and ω at the pints where the rt lci crss the jω-axis: As can be seen in the figure at K=8.13x1 4 and ω =33.8 rad/sec, the ples crss the jω axis. Bth f these values are cnsistent with the results f part(a) frm sistl. 8 15

106 15 Rt Lcus 1 5 System: CL Gain: 8.13e+4 Ple: i Damping: -.33 Oversht (%): 11 Frequency (rad/sec): 33.8 Imaginary Axis -5-1 System: CL Gain: 7.99e+4 Ple: i Damping:.134 Oversht (%): 99.6 Frequency (rad/sec): Real Axis 8 16

107 8 48 Bde Diagram When K = 1, GM = db, PM = 9 deg. The critical value f K fr stability is

108 8 49 (a) Frward path transfer functin: Θ ( s) L Gs ( ) = = KG ( s) = a p E( s) K K ( Bs + K) a i Δ where ( )( ) 3 ( ) Δ =.1s s+.35 s s = s s + s + s ( s + 5) Gs () = s s s s ( ) (b) Bde Diagram: 8 18

109 Gain crssver frequency = 5.85 rad/sec PM =.65 deg. Phase crssver frequency = rad/sec GM = 1.51 db 8 19

110 8 49 (c) Clsed lp Frequency Respnse: M r = 17. 7, ω = rad / sec, BW = 9.53 rad / sec r 8 11

111 8-5 (a) (b) (c) t (e) MATLAB cde: s = tf('s') m =.11; r =.15; d =.3; g = 9.8; L = 1.; J = 9.99*1^-6 K=1 num_gh= K*m*g*d; den_gh=l*(j/r^+m)*s^; GH=num_GH/den_GH; CL = GH/(1+GH) %c) figure(1); nyquist(gh) xlim([-1.5.5]); ylim([-1 1]); %d) figure(); margin(cl) Part (c): since the system is a duble integratr (1/s ), the phase is always -18 deg, and the system is always marginally stable fr any K, leading t a cmplicated cntrl prblem

112 1 Nyquist Diagram Imaginary Axis Real Axis Part (d), Bde diagram: As explained in sectin (c), since the system is always marginally stable, GM=inf and PM =, as can be seen by MATLAB MARGIN cmmand, resulting in the fllwing figure: 8 11

113 Magnitude (db) Bde Diagram Gm = Inf db (at Inf rad/sec), Pm = deg (at.647 rad/sec) System: CL Frequency (rad/sec):.71 Magnitude (db): -3.5 System: CL Frequency (rad/sec):.458 Magnitude (db): 143 Phase (deg) Frequency (rad/sec) Part (e), M r,= 143 db, ω r =.458 rad/sec, and BW=.71 rad/sec as can be seen in the data pints in the abve figure (a) When K = 1, the gain crssver frequency is 8 rad/sec. (b) When K = 1, the phase crssver frequency is rad/sec. (c) When K = 1, GM = 1 db. (d) When K = 1, PM = 57 deg. (e) When K = 1, M r = 1.. (f) When K = 1, ω r = 3 rad/sec. (g) When K = 1, BW = 15 rad/sec. (h) When K = 1 db (.316), GM = db (i) When K = 1 db (3.16), the system is marginally stable. The frequency f scillatin is rad/sec. (j) The system is type 1, since the gain phase plt f G( jω ) appraches infinity at 9 deg. Thus, the steady state errr due t a unit step input is zer

114 8-5 When K = 5 db, the gain phase plt f G( jω ) is raised by 5 db. (a) The gain crssver frequency is ~1 rad/sec. (b) The phase crssver frequency is ~ rad/sec. (c) GM = 5 db. (d) PM = ~34.5 deg. (e) When K = 5, M r = ~ (smallest circle tangent t an M circle). (f) ω = 15 rad/sec r (g) BW = 3 rad/sec (h) When K = 3 db, the GM is 4 db (shift the graph f K=1, 3 dbs dwn). When K = 1 db, the gain phase plt f G( jω ) is raised by 1 db. (a) The gain crssver frequency is rad/sec. (b) The phase crssver frequency is rad/sec. (c) GM = db. (d) PM = deg. (e) When K = 1, M r = ~ 1.1 (smallest circle tangent t an M circle). (f) ω r = 5 rad/sec (g) BW = ~4 rad/sec (h) When K = 3 db, the GM is 4 db (shift the graph f K=1, 3 dbs dwn) Since the functin has expnential term, PADE cmmand has been used t btain the transfer functin. G( s) H ( s) =.1s 8e s( s + 4)( s + 1) MATLAB cde: s = tf('s') %a) 8 114

115 num_gh= pade(8*exp(-.1*s),); den_gh=s*(s+4)*(s+1); GH=num_GH/den_GH; CL = GH/(1+GH) BW = bandwidth(cl) bde(cl) %b) figure(); nichls(gh) Part(a), Nichlas diagram: 4 Nichls Chart Open-Lp Gain (db) Open-Lp Phase (deg) Part(b), Bde diagram: 8 115

116 5 Bde Diagram Magnitude (db) Phase (deg) Frequency (rad/sec) 8-54) Nte: G CL G = 1 + G T draw the Bde and plar plts use the clsed lp transfer functin, G CL, and find BW. Use G t btain the gain-phase plts and Gm and Pm. Use the Bde plt t graphically btain M r. Sample MATLAB cde: s = tf('s') %a) num_g= 1+.1*s; den_g=s*(s+1)*(.1*s+1); G=num_G/den_G figure(1) nyquist(g) figure() margin(g) GCL = G/(1+G) BW = bandwidth(gcl) figure(3) bde(gcl) 8 116

117 Transfer functin:.1 s s^ s^ + s Transfer functin:.1 s^ s^ s^ + s s^6 +. s^ s^ s^ s^ + s BW =

118 8 118

119 8-55) (a) The phase margin with K = 1 and T d = sec is apprximately 57 deg. Fr a PM f 4 deg, the time delay prduces a phase lag f 17 deg. The gain crssver frequency is 8 rad/sec. Thus, ω T d 17 π = 17 = =.967 rad / sec Thus ω = 8 rad / sec T d = =. 371 sec 8 (b) With K = 1, fr marginal stability, the time delay must prduce a phase lag f 57 deg. Thus, at ω = 8 rad/sec, ω T d 57 π = 57 = = rad T d = = 144. sec

120 8 56 (a) The phase margin with K = 5 db and T d = is apprximately 34.5 deg. Fr a PM f 3 deg, the time delay must prduce a phase lag f 4.5 deg. The gain crssver frequency is 1 rad/sec. Thus, ω T d 4.5 π. 785 = 4.5 = =. 785 rad Thus T d = =. 785 sec 18 1 (b) With K = 5 db, fr marginal stability, the time delay must prduce a phase lag f 34.5 deg. Thus at ω = 1 rad/sec, ω T d 34.5 π. 6 = 34.5 = =. 6 rad Thus T d = =. 6 sec ) Fr a GM f 5 db, the time delay must prduce a phase lag f 34.5 deg at ω = 1 rad/sec. Thus, ω T d 34.5 π. 6 = 34.5 = =. 6 rad Thus T d = =. 6 sec (a) Frward path Transfer Functin: s Y( s) e Gs ( ) = = Es ( ) ( 1+ 1s)( 1+ 5s) Frm the Bde diagram, phase crssver frequency =.1 rad/sec GM = 1.55 db 8 1

121 gain crssver frequency = rad/sec PM = infinite (b) Gs () = 1 ( 1+ 1s)( 1+ 5s)( 1+ s+ s ) Frm the Bde diagram, phase crssver frequency =.6 rad/sec GM = 5 db gain crssver frequency = rad/sec PM = infinite (c) Gs () = 1 s ( 1+ s)( 1+ 1s)( 1+ s) Frm the Bde diagram, phase crssver frequency =.6 rad/sec GM = 5.44 db gain crssver frequency = rad/sec PM = infinite Sample MATLAB cde s = tf('s') %a) num_g=exp(-*s); den_g=(1*s+1)*(5*s+1); G=num_G/den_G figure(1) margin(g) 8 11

122 8 58 (cntinued) Bde diagrams fr all three parts. 8 1

123 8 59 (a) Frward path Transfer Functin: s e Gs ( ) = ( 1+ 1s)( 1+ 5s) Frm the Bde diagram, phase crssver frequency =.37 rad/sec GM = 31.8 db gain crssver frequency = rad/sec PM = infinite (b) Gs () = 1 ( 1+ 1s)( 1+ 5s)( 1+ s+.5s ) Frm the Bde diagram, phase crssver frequency =.367 rad/sec gain crssver frequency = rad/sec GM = 3.7 db PM = infinite (c) ( 1 5. s) Gs ( ) = ( 1+ 1s)( 1+ 5s)( 1+. 5s) Frm the Bde diagram, phase crssver frequency =.3731 rad/sec GM = db gain crssver frequency = rad/sec PM = infinite 8 13

124 Plts 8 59 (a c) 8 14

125 8 6 Sensitivity Plt: 8 15

126 S G M max = ω = rad/sec max 8-61) (a) (b) (c)&(d) MATLAB cde: s = tf('s') %c) K = 1 num_gh= K*(1.151*s+.1774); den_gh=(s^3+.739*s^+.91*s); GH=num_GH/den_GH; CL = GH/(1+GH) sistl %(d) figure(1) margin(cl) Part (c), range f K fr stability: Sistl Result shws that by changing K between and inf., all the rts f clsed lp system remain in the left hand side plane and PM remains psitive. Therefre, the system is stable fr all psitive K. 8 16

127 3 x 14 Rt Lcus Editr fr Open Lp 1 (OL1) 1 Open-Lp Bde Editr fr Open Lp 1 (OL1) Bde Editr fr Clsed Lp 1 (CL1) G.M.: Inf Freq: Inf Stable lp P.M.:.149 deg Freq:.6e+4 rad/sec Frequency (rad/sec) Frequency (rad/sec) Part (d), Bde, GM & PM fr K=1: 8 17

128 Bde Diagram Gm = Inf db (at Inf rad/sec), Pm = 61.3 deg (at 1.59 rad/sec) Magnitude (db) Phase (deg) Frequency (rad/sec) 8 18