S I X SOLUTIONS TO CASE STUDIES CHALLENGES. Antenna Control: Stability Design via Gain K s s s+76.39K. T(s) =

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1 S I X Stability SOLUTIONS TO CASE STUDIES CHALLENGES Antenna Control: Stability Design via Gain From the antenna control challenge of Chapter 5, Make a Routh table: 76.39K s s s+76.39K s s K K s K 0 From the row, K< From the s 0 row, 0<K. Therefore, 0<K< UFSS Vehicle: Stability Design via Gain G 3 = K 1 G 2 = (0.25s )K G 1 3 s s s s G (s) G 3 (s) = (0.25s )K 1 s s s (K )s K 1

2 204 Chapter 6: Stability s K 1 s (K ) 0 s (K ) K (K )(K ) K s K For stability : 0 < K 1 < ANSWERS TO REVIEW QUESTIONS 1. Natural response 2. It grows without bound 3. It would destroy itself or hit limit stops 4. Sinusoidal inputs of the same frequency as the natural response yield unbounded responses even though the sinusoidal input is bounded. 5. Poles must be in the left-half-plane or on the jω axis. 6. The number of poles of the closed-loop transfer function that are in the left-half-plane, the right-halfplane, and on the jω axis. 7. If there is an even polynomial of second order and the original polynomial is of fourth order, the original polynomial can be easily factored. 8. Just the way the arithmetic works out 9. The presence of an even polynomial that is a factor of the original polynomial 10. For the ease of finding coefficients below that row 11. It would affect the number of sign changes 12. Seven 13. No; it could have quadrantal poles. 14. None; the even polynomial has 2 right-half-plane poles and two left-half-plane poles. 15. Yes 16. Det (si-a) = 0

3 Solutions to Design Problems 205 SOLUTIONS TO PROBLEMS 1. s s s s s rhp; 3 lhp s s s 3 ε ε s 2 ε ε ε 1 4ε 0 0 s rhp, 2 lhp s s s ROZ s /3 s 0-4

4 206 Chapter 6: Stability Even (4): 4 jω; Rest(1): 1 rhp; Total (5): 1 rhp; 4 jω 4. s s s ROZ s Even (2): 2 jω; Rest (2): 2 lhp; Total: 2 jω; 2 lhp s s s s ROZ s 2 ε /ε 0 s Even (4): 2 rhp; 2 lhp; Rest (2): 1 rhp; 1 lhp; Total: 3 rhp; 3 lhp Program: den=[ ] A=roots(den) Computer response: den = A = i i i i 7. Program:

5 Solutions to Design Problems 207 %-det([si() si();sj() sj()])/sj() %Template for use in each cell. syms e %Construct a symbolic object for %epsilon. %%%%%%%%%%%%%%%%%%%%%%%%%%$$$$$$$$$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s5=[ ] %Create s^5 row of Routh table. %%%%%%%%%%%%%%%%%%%%%%%%%%%$$$$$$$$$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s4=[ ] %Create s^4 row of Routh table. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if -det([s5(1) s5(2);s4(1) s4(2)])/s4(1)==0 s3=[e... -det([s5(1) s5(3);s4(1) s4(3)])/s4(1) 0 0]; %Create s^3 row of Routh table %if 1st element is 0. else s3=[-det([s5(1) s5(2);s4(1) s4(2)])/s4(1)... -det([s5(1) s5(3);s4(1) s4(3)])/s4(1) 0 0]; %Create s^3 row of Routh table %if 1st element is not zero. end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if -det([s4(1) s4(2);s3(1) s3(2)])/s3(1)==0 s2=[e... -det([s4(1) s4(3);s3(1) s3(3)])/s3(1) 0 0]; %Create s^2 row of Routh table %if 1st element is 0. else s2=[-det([s4(1) s4(2);s3(1) s3(2)])/s3(1)... -det([s4(1) s4(3);s3(1) s3(3)])/s3(1) 0 0]; %Create s^2 row of Routh table %if 1st element is not zero. end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if -det([s3(1) s3(2);s2(1) s2(2)])/s2(1)==0 s1=[e... -det([s3(1) s3(3);s2(1) s2(3)])/s2(1) 0 0]; %Create s^1 row of Routh table %if 1st element is 0. else s1=[-det([s3(1) s3(2);s2(1) s2(2)])/s2(1)... -det([s3(1) s3(3);s2(1) s2(3)])/s2(1) 0 0]; %Create s^1 row of Routh table %if 1st element is not zero end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s0=[-det([s2(1) s2(2);s1(1) s1(2)])/s1(1)... -det([s2(1) s2(3);s1(1) s1(3)])/s1(1) 0 0]; %Create s^0 row of Routh table. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 's3' s3=simplify(s3); pretty(s3) 's2' s2=simplify(s2); pretty(s2) 's1' s1=simplify(s1); pretty(s1) 's0' s0=simplify(s0); pretty(s0) %Display label. %Simplify terms in s^3 row. %Pretty print s^3 row. %Display label. %Simplify terms in s^2 row. %Pretty print s^2 row. %Display label. %Simplify terms in s^1 row. %Pretty print s^1 row. %Display label. %Simplify terms in s^0 row. %Pretty print s^0 row.

6 208 Chapter 6: Stability Computer response: s5 = s4 = s [e 1 0 0] s2 [ e ] [ ] [ e ] s1 [ 2 ] [ 2 e e ] [ ] [ e ] s s s s s [ ] 9. Program: numg=240; deng=poly([ ]); 'G(s)' G=tf(numg,deng) 'Poles of G(s)' pole(g) 'T(s)' 2 rhp, 2 lhp

7 Solutions to Design Problems 209 T=feedback(G,1) 'Poles of T(s)' pole(t) Computer response: G(s) Transfer function: s^ s^ s^ s + 24 Poles of G(s) T(s) Transfer function: s^ s^ s^ s Poles of T(s) i i i i System is unstable, since two closed-loop poles are in the right half-plane s 4 + 4s Even (4): 4 jω 84 s 8 + 5s 7 +12s s s s s s + 84

8 210 Chapter 6: Stability s s s s ROZ s s s s 0 4 Even (6): 2 rhp, 2 lhp, 2 jω; Rest (2): 0 rhp, 2 lhp, 0 jω ; Total: 2 rhp, 4 lhp, 2 jω s 4 + 5s 3 + s 2 + 2s +1 s s s s 0 5 Total: 2 lhp, 2 rhp s 7 2 s 6 s 5 + 2s 4 + 4s 3 8s 2 4s + 8

9 Solutions to Design Problems 211 s s s ROZ s s s s Even (6): 3 rhp, 3 lhp; Rest (1): 1 rhp; Total: 4 rhp, 3 lhp Program: numg=8; deng=[ ]; 'G(s)' G=tf(numg,deng) 'T(s)' T=feedback(G,1) 'Poles of T(s)' pole(t) Computer response: G(s) Transfer function: s^7-2 s^6 - s^5 + 2 s^4 + 4 s^3-8 s^2-4 s T(s) Transfer function: s^7-2 s^6 - s^5 + 2 s^4 + 4 s^3-8 s^2-4 s + 8

10 212 Chapter 6: Stability Poles of T(s) i i i i Thus, there are 4 rhp poles and 3 lhp poles. Even (6): 1 rhp, 1 lhp, 4 jω; Rest (1): 1 lhp; Total: 1 rhp, 2 lhp, 4 jω 18 s 5 + s 4 7s 3 7s 2 18s 18 s s s ROZ s s Even (4): 1 rhp, 1 lhp, 2 jω; Rest (1): 1 lhp; Total: 1 rhp, 2 lhp, 2 jω G s = 507 s 4 3 s 3 10 s 2 ; H s = 1. Therefore, s s T s G = = 1+ G H 507 s s s s s s s s s ROZ s

11 Solutions to Design Problems 213 s Even (4): 2 rhp, 2 lhp, 0 jω; Rest (1): 0 rhp, 1 lhp, 0 jω; Total (5): 2 rhp, 3 lhp, 0 jω K(s 2 +1) (1+K)s 2 + 3s + (2+K) will be in the lhp. Thus, K > -1. K(s+6) s 3 + 4s 2 + (K+3)s + 6K. For a second-order system, if all coefficients are positive, the roots s K s 2 4 6K K 0 s 0 6K Stable for 0 < K < 6 K(s + 3)(s + 5) (1 + K)s 2 + (8K 6)s + (8 +15K) For 1 st column negative For 1 st column positive s 2 1+K 8+15K K < -1 K > -1 8K-6 0 K < 6/8 K > 6/8 s K 0 K < -8/15 K > -8/ Stable for K > 6/8 Program: K=[-6: :0]; for i=1:length(k); dent=[(1+k(i)) (8*K(i)-6) (8+15*K(i))]; R=roots(dent); A=real(R); B=max(A); if B>0 R K=K(i) break end end K=[6: :0]; for i=1:length(k); dent=[(1+k(i)) (8*K(i)-6) (8+15*K(i))]; R=roots(dent);

12 214 Chapter 6: Stability A=real(R); B=max(A); if B>0 R K=K(i) break end end Computer response: R = 1.0e+005 * K = R = i i K = Program: %-det([si() si();sj() sj()])/sj() %Template for use in each cell. syms K %Construct a symbolic object for %gain, K. s2=[(1+k) (8+15*K) 0]; %Create s^2 row of Routh table. s1=[(8*k-6) 0 0]; %Create s^1 row of Routh table. s0=[-det([s2(1) s2(2);s1(1) s1(2)])/s1(1)... -det([s2(1) s2(3);s1(1) s1(3)])/s1(1) 0 0]; %Create s^0 row of Routh table. 's2' %Display label. s2=simplify(s2); %Simplify terms in s^1 row. pretty(s2) %Pretty print s^1 row. 's1' %Display label. s1=simplify(s1); %Simplify terms in s^1 row. pretty(s1) %Pretty print s^1 row. 's0' %Display label. s0=simplify(s0); %Simplify terms in s^0 row. pretty(s0) %Pretty print s^0 row. Computer response: s2 [1 + K K 0]

13 Solutions to Design Problems 215 s1 [8 K ] s0 [ K 0 0 0] 23. K(s + 2)(s 2) (K +1)s 2 + (3 4K). For positive coefficients in the denominator, 1 < K < 3 4. Hence marginal stability only for this range of K K(s+1) s 5 + 2s 4 + Ks + K. Always unstable since s3 and s 2 terms are missing. K(s 2)(s + 4)(s + 5) Ks 3 + (7K +1)s 2 + 2Ks + (3 40K) s 3 K 2K 54K 2 K 7K +1 0 s K K(s+2) s 4 + 3s 3-3s 2 + (K+3)s + (2K-4) For stability, 1 54 < K < 3 40 s K - 4 s 3 3 K+ 3 0 s 2 - (K+12) 3 2K K(K+33) K s 0 2K Conditions state that K < -12, K > 2, and K > -33. These conditions cannot be met simultaneously. System is not stable for any value of K.

14 216 Chapter 6: Stability 27. K s 3 +80s s + (K ) s s 2 80 K K s 0 K There will be a row of zeros at row if K = The previous row, s 2, yields the auxiliary equation, 80s 2 + ( ) = 0. Thus, s = ± j Hence, K = yields an oscillation of rad/s. Ks 4 - Ks 2 + 2Ks + 2K (K+1)s 2 + 2(1-K)s + (2K+1) Since all coefficients must be positive for stability in a second-order polynomial, -1 < K < ; - < K < 1; -1 < 2K <. Hence, < K < 1. (s + 2)(s + 7) s 4 +11s 3 + (K + 31)s 2 + (8K + 21)s +12K Making a Routh table, s 4 1 K K s K K s K 0 24K K K s 0 12K 0 0 s 2 row says < K. row says K < and 6.64 < K. s 0 row says 0 < K.

15 Solutions to Design Problems K(s + 4) s 3 + 3s 2 + (2 + K)s + 4K Making a Routh table, s K s 2 3 4K 6 - K 0 s 0 4K 0 a. For stability, 0 < K < 6. b. Oscillation for K = 6. c. From previous row with K = 6, 3s = 0. Thus s = ± j 8, or ω = 8 rad/s. 31. a. G s K s 1 s 2 =. Therefore, s + 2 s 2 T s + 2 s + 2 s 2 s 1 K =. s 3 + K + 4 s K s + 2 K + 2 Making a Routh table, s K s 2 4+K 4+2K - (3K 2 +8K-20) K+4 0 s 0 4+2K From row: K = 1.57, -4.24; From s 2 row: - 4 < K; From s 0 row: - 2 < K. Therefore, - 2 < K < b. If K = 1.57, the previous row is 5.57s Thus, s = ± j1.13. c. From part b, ω = 1.13 rad/s. Applying the feedback formula on the inner loop and multiplying by K yields Thus, Making a Routh table: T G e s s = = K s s s + 7 K s s s + K

16 218 Chapter 6: Stability s s 2 5 K 35 - K 5 0 s 0 K 0 For oscillation, the row must be a row of zeros. Thus, K = 35 will make the system oscillate. The previous row now becomes, 5s Thus, s = 0, or s = ± j 7. Hence, the frequency of 33. oscillation is 7 rad/s. Ks 2 + 2Ks s 3 + (K-1)s 2 + (2K-4)s + 24 s 3 1 2K-4 s 2 K K 2 6K 20 K 1 0 s For stability, K > 5; Row of zeros if K = 5. Therefore, 4s = 0. Hence, ω = 6 for oscillation. Program: K=[0:0.001:200]; for i=1:length(k); deng=conv([1-4 8],[1 3]); numg=[0 K(i) 2*K(i) 0]; dent=numg+deng; R=roots(dent); A=real(R); B=max(A); if B<0 R K=K(i) break end end Computer response: R = i

17 Solutions to Design Problems i K = a. From the computer response, (a) the range of K for stability is 0 < K < 5. b. The system oscillates at K = 5 at a frequency of rad/s as seen from R, the poles of the closed-loop system. K(s+2) s 4 + 3s 3-3s 2 + (K+3)s + (2K-4) s K-4 s 3 3 K+3 0 s 2 - K K-4 0 K(K+33) K s 0 2K For K < -33: 1 sign change; For 33 < K < -12: 1 sign change; For 12 < K < 0: 1 sign change; For 0 < K < 2: 3 sign changes; For K > 2: 2 sign changes. Therefore, K > 2 yields two right-half-plane poles. K s 4 + 7s s s + (4+K) s K+4 s s K K s 0 K a. System is stable for - 4 < K < b. Row of zeros when K = Therefore, 92 7 s Thus, s = ± j1.3628, or ω = rad/s. K s s s + (K+50)

18 220 Chapter 6: Stability a. System is stable for -50 < K < 580. s s 2 14 K K 14 0 s 0 K b. Row of zeros when K = 580. Therefore, 14s Thus, s = ± j 45, or ω = 6.71 rad/s. K s 4 + 8s s s + K s K s s K K s 0 K 0 0 a. For stability 0 < K < b. Row of zeros when K = Therefore, ω = rad/s. s Thus, s = ± j 1.25, or 39. c. Denominator of closed-loop transfer function is s 4 + 8s s s + K. Substituting K = and solving for the roots yield s = ± j1.118, -4.5, and K(s 2 + 2s + 1) s 3 + 2s 2 + (K+1)s - K s 3 1 K+1 s K 3K s 0 - K 0 Stability if < K < 0.

19 Solutions to Design Problems s4 + (K+2)s 3 + Ks 2 s 3 + s 2 + 2s + K s s 2 1 K 2 - K 0 s 0 K 0 Row of zeros when K = 2. Therefore s and s = ± j 2, or ω = rad/s. Thus K = 2 will yield 41. the even polynomial with 2 jω roots and no sign changes. 1 K 2 1 s 3 K K 1 K 2 5 s 2 K K 12 5K 1 K K 1 K s For stability, K 1 K 2 > 5; K < 5K 1 K 2 ; and K 1 > 0. Thus 0 < K 1 2 < 5K 1 K 2-25, 42. or 0 < K 1 < 5K 1 K s s 3 K 1 K 2 0 K 1 -K 2 s 2 K K K 1 K 2 + K 2 2 K 2 - K s For two jω poles, K K 1 K 2 + K 2 2 = 0. However, there are no real roots. Therefore, there is no relationship between K 1 and K 2 that will yield just two jω poles.

20 222 Chapter 6: Stability 43. s E E E s E E E+03 0 s s s s s s a. From the first column, 1 rhp, 7 lhp, 0 jω. b. G(s) is not stable because of 1 rhp. Eigenvalues are the roots of the following equation: si A = s s = 0 0 s s s 2 4 = s3 5s 2 15s s Hence, eigenvalues are , , Therefore, 1 rhp, 2 lhp, 0 jω. Program: A=[0 1 0;0 1-4;-1 1 3]; eig(a) Computer response: i I Writing the open-loop state and output equations we get,

21 Solutions to Design Problems 223 x 1 x 2 x 3 = x 2 = x 2 + 3x 3 = 3x 1 4x 2 5x 3 + u y = x 2 + x 3 Drawing the signal-flow diagram and including the unity feedback path yields, 1 1 r s 3 1 x x x s c = y Writing the closed-loop state and output equations from the signal-flow diagram, x 1 x 2 x 3 = x 2 = x 2 + 3x 3 = 3x 1 4x 2 5x 3 + r c = 3x 1 4x 2 5x 3 + r (x 2 + x 3 ) = 3x 1 5x 2 6x 3 + r y = x 2 + x 3 In vector-matrix form, x = X + 0 r y = [ 0 1 1]x Now, find the characteristic equation.

22 224 Chapter 6: Stability s si A = 0 s s = s 3 + 5s 2 + 9s + 9 s 1 0 = 0 (s 1) 3 3 5` (s + 6) Forming a Routh table to determine stability s s s Since there are no sign changes, the closed-loop system is stable. Program: A=[0,1,0;0,1,3;-3,-4,-5]; B=[0;0;1]; C=[0,1,1]; D=0; 'G' G=ss(A,B,C,D) 'T' T=feedback(G,1) 'Eigenvalues of T' ssdata(t); eig(t) Computer response: G a = x1 x2 x3 x x x b = u1 x1 0 x2 0 x3 1 c = x1 x2 x3 y d = u1

23 Solutions to Design Problems 225 y1 0 Continuous-time model. T a = x1 x2 x3 x x x b = u1 x1 0 x2 0 x3 1 c = x1 x2 x3 y d = u1 y1 0 Continuous-time model. Eigenvalues of T i i SOLUTIONS TO DESIGN PROBLEMS 48. K(s+1)(s+10) s 3 + (5.45+K)s 2 + ( K)s + ( K) s K s K K 0 s K 0

24 226 Chapter 6: Stability 49. For stability, < K <. Stable for all positive K. 0.7K(s+0.1) s s s s + (0.07K+0.01) s K+0.01 s s K K 0 0 s K T s For stability, < K < = 0.6 K + 10 K s K s s s s K s K s K s K s K K s 3-10K K K K E9 s2-10k K E3K E11K E15K E19K E K K E11K+3.299E s 0 0.6K 0 0 Note: s 3 row was multiplied by 130 From row after canceling common roots: K K K K K K K From s 0 row: K > 0 From s 3 row: K < From s 2 row: K < ; < K From row: < K, or K < ;

25 Solutions to Design Problems 227 Therefore, for stability, 0 < K < s (100K+1) s K s K s K 60000K (K )(K ) (K ) 0 0 s K 0 0 From s 2 row: K < From row: < K < ; K > From s 0 row: 0 < K Therefore, 0 < K < Find the closed-loop transfer function. G(s) = Make a Routh table. 63x10 6 K (s + 30)(s +140)(s + 2.5) G(s) 1 + G(s)H(s) = 63x10 6 K s s s + ( x10 6 K) s s x10 6 K K 0 s x10 6 K 0 The line says K < 1.25x10-2 for stability. The s 0 line says K > -1.67x10-4 for stability.

26 228 Chapter 6: Stability 53. Hence, -1.67x10-4 < K < 1.25x10-2 for stability. Find the closed-loop transfer function. 7570Kp(s +103)(s + 0.8) G(s) = s(s )(s 62.61) G(s) 1+ G(s)H(s) = 7570Kp(s +103)(s + 0.8) s K p s 2 + (785766K p )s K p Make a Routh table. s K p s K p K p s K p Desired force The line says Kp > 5.09x10-3 for stability. The s 0 line says Kp > 0 for stability. Hence, Kp > 5.09x10-3 for stability Input transducer Input voltage+ - K F up ( s ) (s s )( s s ) Controller Actuator Pantograph dynamics Y h -Y cat Spring displacement Spring F out Sensor Y h -Y cat Spring displacement Desired force + - K F up (s ) (s s )(s s ) Controller Actuator Pantograph dynamics Spring F out G(s) = Y h(s) Y cat (s) F up (s) = (s ) (s s )(s s ) Ge(s)=(K/100)*(1/1000)*G(s)*82.3e3

27 Solutions to Design Problems 229 G e (s) = K (s+53.85) (s s ) (s s ) K (s+53.85) s s s + ( K )s + (34.94 K ) s (0.3493e K) + s (0.6488K+81190) 0 + s 2 ( K+6343) (0.3493e K 0 K< ) K K e6.0275K <K< s 0 (0.3493e K) <K The last column evaluates the range of K for stability for each row. Therefore < K <