# The Frequency-response Design Method

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1 Chapter 6 The Frequency-response Design Method Problems and Solutions for Section 6.. (a) Show that α 0 in Eq. (6.2) is given by α 0 = G(s) U 0ω = U 0 G( jω) s jω s= jω 2j and α 0 = G(s) U 0ω = U 0 G(jω) s + jω s=+jω 2j. (b) By assuming the output can be written as derive Eqs. (6.4) - (6.6). Solution: (a) Eq. (6.2): y(t) =α 0 e jωt + α 0e jωt, Y (s) = α s p + α 2 s p Multiplying this by (s + jω) : α n s p n + α o s + jω o + α o s jω o Y (s)(s+jω) = α (s+jω)+...+ (s+jω)+α o + s + a s + a n s jω (s+jω) 283 α n α o

2 284 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD α o = Y (s)(s + jω) α (s + jω)... s + a Y (s)(s + jω) α (s + jω)... s + a α o = α o s= jω = U o ω = Y (s)(s + jω) s= jω = G(s) s 2 + ω 2 (s + jω) s= j ω = G(s) U oω s jω s= j ω = U o G( jω) 2j Similarly, multiplying Eq. (6.2) by (s jω) : α n s + a n (s + jω) α o α o (s + jω) s jω (s + jω) s jω s= j ω (b) Y (s)(s jω) = α (s jω) α n (s jω)+ α o s + a s + a n s + jω (s jω)+α o α o = α U o ω o s=jω = Y (s)(s jω) s=jω = G(s) s 2 + ω 2 (s jω) s=j ω = G(s) U oω s + jω s=j ω = U o G(jω) 2j y(t) = α o e jωt + α oe jωt y(t) = U o G( jω) 2j e jωt + U o G(jω) 2j ejωt G(jω)e jωt G( jω)e jωt = U o 2j n G(jω) = Re [G(jω)] 2 +Im[G(jω)] 2o 2 = A Im [G(jω)] G(jω) = tan Re [G(jω)] = φ n G( jω) = Re [G( jω)] 2 +Im[G( jω)] 2o 2 = G(jω) n = Re [G(jω)] 2 +Im[G(jω)] 2o 2 = A Im [G( jω)] Im [G(jω)] G( jω) = tan =tan = φ Re [G( jω)] Re [G(jω)] G(jω) =Ae jφ,g( jω) =Ae jφ Thus, Ae jφ e jωt Ae jφ e jωt ej(ωt+φ) e j(ωt+φ) y(t) = U o = U o A 2j 2j y(t) = U o A sin(ωt + φ)

3 285 where Im [G(jω)] A = G(jω), φ =tan Re [G(jω)] = G(jω) 2. (a) Calculate the magnitude and phase of G(s) = s +0 by hand for ω =, 2, 5, 0, 20, 50, and 00 rad/sec. (b) sketch the asymptotes for G(s) according to the Bode plot rules, and compare these with your computed results from part (a). Solution: (a) G(s) = G(jω) = s +0,G(jω) = = 0 jω 00 + ω jω, G(jω) = ω 00 + ω 2 tan 0 ω G(jω) G(jω)

4 286 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (b) The Bode plot is : 0 0 M agnitude Phase (deg) Sketch the asymptotes of the Bode plot magnitude and phase for each of the following open-loop transfer functions. After completing the hand sketches verify your result using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales (a) L(s) = s(s + 400) 00 (b) L(s) = s(0.s +)(0.5s +) (c) L(s) = s(s +)(0.02s +) (d) L(s) = (s +) 2 (s 2 +2s +4) 0(s +4) (e) L(s) = s(s +)(s 2 +2s +5) 000(s +0.) (f) L(s) = s(s +)(s 2 +8s + 64) (s +5)(s +3) (g) L(s) = s(s +)(s 2 + s +4) 4s(s + 0) (h) L(s) = (s + 00)(4s 2 +5s +4) s (i) L(s) = (s +)(s + 0)(s 2 +2s )

5 287 Solution: 0 (a) L(s) = h s s i M agnitude Phase (deg) (b) L(s) = s s 0 + s M agnitude Phase (deg)

6 288 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) L(s) = s(s +) s M agnitude Phase (deg) (d) L(s) = h i (s +) 2 s s M agnitude Phase (deg)

7 289 8 s 4 (e) L(s) = ³ + + s s(s +) s 0 5 M agnitude 0 0 Phase (deg) (f) L(s) = 25 6 (0s +) s(s +)h s i s M agnitude Phase (deg)

8 290 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (g) L(s) = 5 4 s 5 + s 3 + s(s +)h s i s M agnitude Phase (deg) (h) L(s) = 0 s s 0 + s 00 + s s M agnitude Phase (deg)

9 29 (i) L(s) = s (s +) s 0 + h s s + i 0-4 M agnitude Phase (deg) Real poles and zeros. Sketch the asymptotes of the Bode plot magnitude and phase for each of the following open-loop transfer functions. After completing the hand sketches verify your result using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. (a) L(s) = (b) L(s) = (c) L(s) = (d) L(s) = s(s +)(s +5)(s + 0) (s +2) s(s +)(s +5)(s + 0) (s +2)(s +6) s(s +)(s +5)(s + 0) (s +2)(s +4) s(s +)(s +5)(s + 0) Solution:

10 292 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (a) L(s) = 50 s(s +) s 5 + s M agnitude Phase (deg) (b) L(s) = s s(s +) s 5 + s M agnitude Phase (deg)

11 293 (c) L(s) = 6 s s 6 + s(s +) s 5 + s M agnitude Phase (deg) (d) L(s) = 4 s s 4 + s(s +) s 5 + s M agnitude Phase (deg)

12 294 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD 5. Complex poles and zeros Sketch the asymptotes of the Bode plot magnitude and phase for each of the following open-loop transfer functions and approximate the transition at the second order break point based on the value of the damping ratio. After completing the hand sketches verify your result using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. (a) L(s) = s 2 +3s +0 (b) L(s) = s(s 2 +3s + 0) (c) L(s) = (s2 +2s +8) s(s 2 +2s + 0) (d) L(s) = (s2 +2s + 2) s(s 2 +2s + 0) (e) L(s) = (s2 +) s(s 2 +4) (f) L(s) = (s2 +4) s(s 2 +) Solution: (a) L(s) = ³ 0 s s M agnitude Phase (deg)

13 295 (b) L(s) = ³ s 0 s s M agnitude Phase (deg)

14 296 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) L(s) = 4 5 ³ ³ s s s + + s s 0 2 M agnitude Phase (deg)

15 (d) L(s) = 6 5 ³ ³ s s s + + s s 297 M agnitude Phase (deg)

16 298 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD 4 (e) L(s) = (s2 +) h s s i M agnitude Phase (deg) (f) L(s) = h 4 s i s(s 2 +) 0 2 M agnitude Phase (deg)

17 Multiple poles at the origin Sketch the asymptotes of the Bode plot magnitude and phase for each of the following open-loop transfer functions. After completing the hand sketches verify your result using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. (a) L(s) = s 2 (s +8) (b) L(s) = s 3 (s +8) (c) L(s) = s 4 (s +8) (s +3) (d) L(s) = s 2 (s +8) (s +3) (e) L(s) = s 3 (s +4) (f) L(s) = (g) L(s) = Solution: (a) L(s) = (s +)2 s 3 (s +4) (s +)2 s 3 (s + 0) 2 8 s 2 s M agnitude Phase (deg)

18 300 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (b) L(s) = 8 s 3 s M agnitude Phase (deg) (c) L(s) = 8 s 4 s M agnitude Phase (deg)

19 30 (d) L(s) = 3 s s 2 s M agnitude Phase (deg) (e) L(s) = 3 s s 3 s M agnitude Phase (deg)

20 302 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (f) L(s) = 4 (s +)2 s 3 s M agnitude Phase (deg) (g) L(s) = 00 (s +)2 s 3 s M agnitude Phase (deg)

21 Mixed real and complex poles Sketch the asymptotes of the Bode plot magnitude and phase for each of the following open-loop transfer functions. After completing the hand sketches verify your result using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. (a) L(s) = (b) L(s) = (s +2) s(s + 0)(s 2 +2s +2) (s +2) s 2 (s + 0)(s 2 +6s + 25) (s +2) 2 (c) L(s) = s 2 (s + 0)(s 2 +6s + 25) (d) L(s) = (s +2)(s2 +4s + 68) s 2 (s + 0)(s 2 +4s + 85) (e) L(s) = [(s +)2 +] s 2 (s +2)(s +3) Solution: (a) L(s) = s s ³ s s s M agnitude Phase (deg)

22 304 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (b) L(s) = s s 2 s 0 + h s s + i 0 0 M agnitude Phase (deg) (c) L(s) = 2 s s 2 s 0 + h s s + i M agnitude Phase (deg)

23 (d) L(s) = 4 s 25 2 s 2 s 0 + ³ + ³ s s + s s M agnitude Phase (deg)

24 306 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (e) L(s) = 3 ³ s s + s 2 s 2 + s M agnitude Phase (deg) Right half plane poles and zeros Sketch the asymptotes of the Bode plot magnitude and phase for each of the following open-loop transfer functions. Make sure the phase asymptotes properly take the RHP singularity into account by sketching the complex plane to see how the L(s) changesas s goes from 0 to +j. After completing the hand sketches verify your result using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. (a) L(s) = s +2 s +0s 2 ; The model for a case of magnetic levitation with lead compensation. (b) L(s) = s +2 s(s + 0) (s 2 ; The magnetic levitation system with integral control and lead compensation. ) (c) L(s) = s s 2 s 2 +2s + (d) L(s) = s(s + 20) 2 (s 2 2s +2) (s +2) (e) L(s) = s(s )(s +6) 2 (f) L(s) = (s )[(s +2) 2 +3]

25 307 Solution: (a) L(s) = s s +0 s M agnitude Phase (deg) (b) L(s) = s s(s + 0) s M agnitude Phase (deg)

26 308 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) L(s) = s s M agnitude Phase (deg) (d) L(s) = (s2 +2s +) s ³ s s 2 2 s M agnitude Phase (deg)

27 309 (e) L(s) = s s(s ) s M agnitude Phase (deg) (f) L(s) = ³ (s ) 7 s s M agnitude Phase (deg)

28 30 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.89: Magnitude portion of Bode plot for Problem 9 9. A certain system is represented by the asymptotic Bode diagram shown in Fig Find and sketch the response of this system to a unit step input (assuming zero initial conditions). Solution: By inspection, the given asymptotic Bode plot is from Therefore, G(s) = 0(s/0 + ) s = s +0 s The response to a unit step input is : Y (s) = G(s)U(s) = s +0 s s = s + 0 s 2 y(t) = [Y (s)] = (t)+0t (t 0)

29 3 25 UnitStep Response 20 5 y(t) Time (sec) 0. Prove that a magnitude slope of in a Bode plot corresponds to 20 db perdecadeor-6dbperoctave. Solution: The deþnition of db is db = 20 log G () Assume slope = d(log G ) d(log ω) = (2) (2) = log G = log ω + c (c is a constant.) (3) ()and(3)= db = 20 log ω +20c Differentiating this, d (db) d (log ω) = 20 Thus, a magnitude slope of - corresponds to -20 db per decade. Similarly, d (db) d (log 2 ω) = d (db) + 6 d ³ log ω log 2 Thus, a magnitude slope of - corresponds to -6 db per octave.. A normalized second-order system with a damping ratio ζ = 0.5 and an additional zero is given by G(s) = s/a + s 2 + s +. Use MATLAB to compare the M p fromthestepresponseofthesystem for a =0.0, 0.,, 0, and 00 with the M r from the frequency response of each case. Is there a correlation between M r and M p?

30 32 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Solution: α Resonant peak, M r Overshoot, M p As α is reduced, the resonant peak in frequency response increases. This leads us to expect extra peak overshoot in transient response. This effect is signiþcant in case of α =0.0, 0.,, while the resonant peak in frequency response is hardly changed in case of α = 0. Thus, we do not have considerable change in peak overshoot in transient response for α 0. The response peak in frequency response and the peak overshoot in transient response are correlated. U nit Step Response 50 a= y(t) a=0. 0 a= a=00 a= Tim e (sec)

31 33 M agnitude 0 0 a=0.0 a=0. a= a=0 a= Phase (deg) 00 a=0.0 a=0. 0 a= a=0 a= A normalized second-order system with ζ =0.5 and an additional pole is given by. G(s) = [(s/p)+](s 2 + s +) Draw Bode plots with p =0.0, 0.,, 0 and 00. What conclusions can you draw about the effect of an extra pole on the bandwidth compared to the bandwidth for the second-order system with no extra pole? Solution: p Additional pole ( p) Bandwidth, ω Bw As p is reduced, the bandwidth decreases. This leads us to expect slower time response and additional rise time. This effect is signiþcant in case of p =0.0, 0.,, while the bandwidth is hardly changed in case of p = 0. Thus, we do not have considerable change in rise time for p 0. Bandwidth is a measure of the speed of response of a system, such as rise time.

32 34 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD U nit Step Response.2 a=00 a=0 0.8 y(t) a= a= a= Tim e (sec) M agnitude a= a=0. a=0.0 a=00 a= a=0. a= Phase (deg) a=0.0 a=0 a= For the closed-loop transfer function ω 2 n T (s) = s 2 +2ζω n s + ω 2, n

33 35 derive the following expression for the bandwidth ω BW of T (s) interms of ω n and ζ: q ω BW = ω n r 2ζ ζ 4 4ζ 2. Assuming ω n =,plotω BW for 0 ζ. Solution : The closed-loop transfer function : ω 2 n T (s) = s 2 +2ζω n s + ω 2 n s = jω, T (jω) = ³ 2 ³ ω ω n +2ζ ω ω n j T (jω) = {T (jω)t (jω)} 2 = ³ 2 ω ω n ζ 2 + n 2ζ ³ 2 o 2 ω ω n Let x = ω BW ω n : " T (jω) ω=ωbw = ( x 2 ) 2 +(2ζx) 2 = x 4 +(4ζ 2 2)x 2 =0 ( 2ζ 2 )+ = x = ω BW ω n = = ω BW = ω n r 2ζ 2 + # 2 =0.707 = 2 q ( 2ζ 2 ) q 2+4ζ 4 4ζ 2 ζ µ x = ω BW ω n ω BW.5ω n.27ω n 0.87ω n

34 36 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD.6 Bandwidth vs zeta Bandwidth, M BW Damping ratio, 4. Consider the system whose transfer function is A 0 ω 0 s G(s) = Qs 2 + ω 0 s + ω 2 0 Q. This is a model of a tuned circuit with quality factor Q. (a)computethe magnitude and phase of the transfer function analytically, and plot them for Q =0.5,, 2, and 5 as a function of the normalized frequency ω/ω 0. (b) DeÞne the bandwidth as the distance between the frequencies on either side of ω 0 where the magnitude drops to 3 db below its value at ω 0 and show that the bandwidth is given by BW = 2π µ ω0. Q (c) What is the relation between Q and ζ? Solution : (a) Let s = jω, G(jω) = = G(jω) = A o ω o jω Qω 2 + ω o jω + ω 2 oq A o + Qω2 o Qω2 jω o ω r ³ +Q 2 A o 2 ω ω o ωo ω φ = tan µ ω ω o ω o ω

35 Phase (deg) QG(jw)/Ao ω o 37 ³ The normalized magnitude QG(jω) A o and phase are plotted against ³ ω normalized frequency for different values of Q.. 0 Bode Diagrams 0 0 Q=5 Q=2 Q= Q= w/wo w/wo Q=5 Q=0.5 Q= Q=2 (b) There is symmetry around ω o. For every frequency ω < ω o,there exists a frequency ω 2 > ω o which has the same magnitude G(jω ) = G(jω 2 ) We have that, ω ω µ o ω2 = ω o ω o ω ω o ω 2 which implies ω 2 o = ω ω 2. Let ω < ω o and ω 2 > ω o be the two frequencies on either side of ω o for which the gain drops by 3db from its value of A o at ω o. BW = ω 2 ω 2π Now ω 2 is found from, = µ ω 2 ω2 o 2π ω 2 G(jω) A o = 2 ()

36 38 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD or µ +Q 2 ω2 ω 2 o =2 ω o ω 2 which yields µ ω2 Q ω µ o == Qωo ω 2 ω2 o ω o ω 2 ω 2 (2) Comparing () and (2) we Þnd, BW = 2 µ Q ω o (c) G(s) = = = A 0 ω 0 s Qs 2 + ω 0 s + ω 2 0 Q A 0 ω 0 s ³ Q s 2 + ω 0 Q s + ω2 0 A 0 ω 0 s Q (s 2 +2ζω 0 s + ω 2 0 ) Therefore Q =2ζ 5. A DC voltmeter schematic is shown in Fig The pointer is damped so that its maximum overshoot to a step input is 0%. (a) What is the undamped natural frequency of the system? (b) What is the damped natural frequency of the system? (c) Plot the frequency response using MATLAB to determine what input frequency will produce the largest magnitude output? (d) Suppose this meter is now used to measure a -V AC input with a frequency of 2 rad/sec. What amplitude will the meter indicate after initial transients have died out? What is the phase lag of the output with respect to the input? Use a Bode plot analysis to answer these questions. Use the lsim command in MATLAB to verify your answer in part (d). Solution : Theequationofmotion:I θ + bθ ú + kθ = T = K m v,whereb is a damping coefficient.

37 39 Figure 6.90: Voltmeter schematic Taking the Laplace transform with zero initial conditions: Θ(s) = Km K m Is 2 + bs + k V (s) = I s 2 +2ζω n s + ω 2 V (s) n Use I = Kg m 2, k =4 0 6 Kg m 2 /s 2, K m =4 0 6 N m/v (a) Undamped natural frequency: ω 2 n = k I = ω n = r k I =0.36 rad/sec πζ (b) Since M p =0. andm p = e ζ 2, log 0. = p πζ = ζ =0.59 (' 0.6 from Figure 2.44) ζ 2 Damped natural frequency: ω d = ω n q ζ 2 =0.255 rad/sec

38 320 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) T (jω) = Θ(jω) V (jω) = K m /I (jω) 2 +2ζω n jω + ω 2 n K m /I T (jω) = [(ω 2 n ω 2 ) 2 +(2ζω n ω) 2 ] 2 d T (jω) dω = µ Km I 2ω ω 2 n ω 2 2ζ 2 ω 2 nª [(ω 2 n ω 2 ) 2 +(2ζω n ω) 2 ] 3 2 When d T (jω) dω =0, ω 2 ( 2ζ 2 )ω 2 n = 0 ω = 0.549ω n =0.73 Alternatively,thepeakfrequencycanbefoundfromtheBodeplot: ω = 0.73 rad/sec (d) With ω = 2 rad/sec from the Bode plot: Amplitude = rad Phase = M agnitude G(jw) =0.025 w= Phase (deg) w=2-69 (deg) Problems and Solutions for Section 6.2

39 32 6. Determine the range of K for which each of the following systems is stable by making a Bode plot for K = and imagining the magnitude plot sliding up or down until instability results. Verify your answers using a very rough sketch of a root-locus plot. (a) KG(s) = (b) KG(s) = (c) KG(s) = K(s +2) s +20 K (s + 0)(s +) 2 K(s + 0)(s +) (s + 00)(s +5) 3 Solution : (a) K(s +2) KG(s) = s +20 = K s s M agnitude Phase (deg)

40 Phase (deg) Magnitude 322 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD The gain can be raised or lowered on the Bode gain plot and the phase will never be less than -80 o, so the system is stable for any K>0. (b) K KG(s) = (s +0)(s +) 2 = K 0 s 0 + (s +) Bode Diagrams K=/

41 323 Thebodeplotsshowthatthegain,K, would equal 242 when the phase crosses 80 o. So, K<242 is Stable and K>242 is Unstable. The phase crosses the 80 o at ω =4.58 rad/sec. The root locus below veriþes the situation. (c) KG(s) = K(s + 0)(s +) (s + 00)(s +5) 3 = K 250 s 0 + (s +) s 00 + s M agnitude Phase (deg)

42 324 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD The phase never crosses -80 o so it is stable for all K>0, as con- Þrmed by the root locus. 7. Determine the range of K for which each of the following systems is stable by making a Bode plot for K = and imagining the magnitude plot sliding up or down until instability results. Verify your answers using a very rough sketch of a root-locus plot. (a) KG(s) = (b) KG(s) = (c) KG(s) = (d) KG(s) = K(s +) s(s +5) K(s +) s 2 (s + 0) K (s +2)(s 2 +9) K(s +)2 s 3 (s + 0) Solution : (a) KG(s) = K(s +) s(s +5) = K 5 (s +) s s 5 +

43 325 0 M agnitude Phase (deg) The phase never crosses -80 o so it is stable for all K>0, as con- Þrmed by the root locus. (b) K(s +) KG(s) = s 2 (s + 0) = K 0 s + s 2 s 0 +

44 326 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD 0 2 M agnitude Phase (deg) The phase never crosses -80 o so it is stable for all K>0, as con- Þrmed by the root locus. The system is stable for any K>0. (c) K KG(s) = (s +2)(s 2 +9) = K 8 s 2 + s 2 9 +

45 M agnitude Phase (deg) The bode is difficult to read, but the phase really dropped by 80 o at the resonce. (It appears to rise because of the quadrant action in Matlab) Furthermore, there is an inþnite magnitude peak of the gain at the resonance because there is zero damping. That means that no matter how much the gain is lowered, the gain will never cross magnitude one when the phase is -80 o. Soitcannotbemade stable for any K. This is much clearer and easier to see in the root locus below..

46 Phase (deg) Magnitude 328 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (d) K(s +)2 KG(s) = s 3 (s + 0) = K 0 (s +) 2 s 3 s Bode Diagrams 0 0 Gain=/ This is not the normal situation discussed in Section 6.2 where increasing gain leads to instability. Here we see from the root locus that K must be 6.7 in order for stability. Note that the phase is increasing with frequency here rather than the normal decrease we saw on the previous problems. It s also interesting to note that the margin command in Matlab indicates instability! (which is false.) This problem illustrates that a sketch of the root locus really helps understand what s going on... and that you can t always trust Matlab, or at least that you need good understanding to interpret what Matlab is telling you.

47 329 K<6.25 : Unstable K>6.25 : Stable ω =.2 rad/sec for K =6.7. Problems and Solutions for Section (a) Sketch the Nyquist plot for an open-loop system with transfer function /s 2 ; that is, sketch s 2 s=c, where C is a contour enclosing the entire RHP, as shown in Fig.??. (Hint: AssumeC takes a small detour around the poles at s =0,as shown in Fig.??.) (b) Repeat part (a) for an open-loop system whose transfer function is G(s) =/(s 2 + ω 2 0 ). Solution : (a) G(s) = s 2 Note that the portion of the Nyquist diagram on the right side below that corresponds to the bode plot is from B to C. The large loop

48 330 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD from F to A to B arises from the detour around the 2 poles at the origin. (b) G(s) = s 2 + ω 2 0 Note here that the portion of the Nyquist plot coming directly from a Bode plot is the portion from A to E. That portion includes a 80 o arc that arose because of the detour around the pole on the imaginary axis. 9. Sketch the Nyquist plot based on the Bode plots for each of the following systems, then compare your result with that obtained using the MATLAB command nyquist: K(s +2) (a) KG(s) = s +0 K (b) KG(s) = (s + 0)(s +2) 2

49 (c) KG(s) = K(s + 0)(s +) (s + 00)(s +2) 3 33 (d) Using your plots, estimate the range of K for which each system is stable, and qualitatively verify your result using a rough sketch of a root-locus plot. Solution : (a) N =0,P =0= Z = N + P =0 The closed-loop system is stable for any K>0. (b) The Bode plot shows an intial phase of 0 o hence the Nyquist starts on the positive real axis at A. The Bode ends with a phase of o hence the Nyquist ends the bottom loop by approaching the

50 332 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD origin from the positive imaginary axis (or an angle of -270 o ). The magnitude of the Nyquist plot as it crosses the negative real axis is It will not encircle the /K point until K = / = 576. i. 0 <K<576 N =0,P =0= Z = N + P =0 The closed-loop system is stable. ii. K>576 N =2,P =0= Z = N + P =2 The closed-loop system has two unstable roots as veriþed by the root locus. (c) The Bode plot shows an intial phase of 0 o hence the Nyquist starts onthepositiverealaxisata. TheBodeendswithaphaseof- 80 o hence the Nyquist ends the bottom loop by approaching the

51 333 origin from the negative real axis (or an angle of -80 o ). It will never encircle the -/K point, hence it is always stable. root locus below conþrms that. The N =0,P =0= Z = N + P =0 The closed-loop system is stable for any K> Draw a Nyquist plot for K(s +) KG(s) = () s(s +3) choosing the contour to be to the right of the singularity on the jω-axis. and determine the range of K for which the system is stable using the Nyquist Criterion. Then redo the Nyquist plot, this time choosing the contour to be to the left of the singularity on the imaginary axis and again check the range of K for which the system is stable using the Nyquist Criterion. Are the answers the same? Should they be?

52 334 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Solution : If you choose the contour to the right of the singularity on the origin, the Nyquist plot looks like this : From the Nyquist plot, the range of K for stability is K < 0(N = 0,P =0= Z = N + P =0). So the system is stable for K>0. Similarly, in the case with the contour to the left of the singularity on the origin, the Nyquist plot is: From the Nyquist plot, the range of K for stability is K < 0(N =,P == Z = N + P =0). So the system is stable for K>0. The way of choosing the contour around singularity on the jω-axis does not affect its stability criterion. The results should be the same in either way. However, it is somewhat less cumbersome to pick the contour to the right of a pole on the imaginary axis so that there are no unstable poles within the contour, hence P=0.

53 335 Figure 6.9: Control system for Problem 2 2. Draw the Nyquist plot for the system in Fig Using the Nyquist stability criterion, determine the range of K for which the system is stable. Consider both positive and negative values of K. Solution : The characteristic equation: +K (s 2 +2s +2) (s +) =0 G(s) = (s +)(s 2 +2s +2) For positive K, note that the magnitude of the Nyquist plot as it crosses the negative real axis is 0., hence K<0 for stability. For negative K, the entire Nyquist plot is essentially ßipped about the imaginary axis, thus the magnitude where it crosses the negative real axis will be 0.5 and the stability limit is that K < 2 Therefore, the range of K for stability is 2 <K<0.

54 336 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD 22. (a) For ω =0. to 00 rad/sec, sketch the phase of the minimum-phase system s + G(s) = s +0 s=jω and the nonminimum-phase system s G(s) = s +0 s=jω, noting that (jω ) decreases with ω rather than increasing. (b) Does a RHP zero affect the relationship between the encirclements on a polar plot and the number of unstable closed-loop roots in Eq. (6.28)? (c) Sketch the phase of the following unstable system for ω =0. to 00 rad/sec: G(s) = s + s 0 s=jω. (d) Check the stability of the systems in (a) and (c) using the Nyquist criterion on KG(s). Determine the range of K for which the closedloop system is stable, and check your results qualitatively using a rough root-locus sketch. Solution : (a) Minimum phase system, G (jω) = s + s +0 s=jω

55 M agnitude Phase (deg) Non-minimum phase system, G 2 (jω) = s s +0 s=jω 0 0 M agnitude Phase (deg) (b) No, a RHP zero doesn t affect the relationship between the encirclements on the Nyquist plot and the number of unstable closed-loop roots in Eq. (6.28).

56 338 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) Unstable system: G 3 (jω) = s + s 0 s=jω 0 0 M agnitude Phase (deg) i. Minimum phase system G (jω): For any K>0, N =0, P =0= Z =0= The system is stable, as veriþed by the root locus being entirely in the LHP. ii. Non-minimum phase system G 2 (jω): the /K point will not be encircled if K<. 0 <K< N =0,P=0= Z =0= Stable <K N=,P=0= Z == Unstable

57 339 This is veriþed by the Root Locus shown below where the branch ofthelocustotheleftofthepoleisfromk<. iii. Unstable system G 3 (jω): The /K point will be encircled if K>0, however, P =,so 0 <K<0 : N =0,P== Z == Unstable 0 <K : N =, P== Z =0= Stable This is veriþed by the Root Locus shown below right, where the locus crosses the imaginary axis when K = 0, and stays in the LHP for K>0. Problems and Solutions for Section The Nyquist plot for some actual control systems resembles the one shown in Fig What are the gain and phase margin(s) for the system of

58 340 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.92: Nyquist plot for Problem 23 Fig given that α =0.4, β =.3, and φ =40 o. Describe what happens to the stability of the system as the gain goes from zero to a very large value. Sketch what the corresponding root locus must look like for such a system. Also sketch what the corresponding Bode plots would look likeforthesystem. Solution : The phase margin is deþned as in Figure 6.33, PM = φ (ω = ω ), but now there are several gain margins! If the system gain is increased (multiplied) by α or decreased (divided) by β, then the ststem will go unstable. This is a conditionally stable system. See Figure 6.39 for a typical root locus of a conditionally stable system. gain margin = -20 log α db (ω = ω H ) gain margin = +20 log β db (ω = ω L ) For a conditionally stable type of system as in Fig. 6.39, the Bode phase plot crosses -80 twice; however, for this problem we see from the Nyquist plot that it crosses 3 times! For very low values of gain, the entire Nyquist plot would be shrunk, and the - point would occur to the left of thenegativerealaxiscrossingatω o, so there would be no encirclements and the system would be stable. As the gain increases, the - point occurs between ω o and ω L so there is an encirclement and the system is unstable.

59 34 Further increase of the gain causes the - point to occur between ω L and ω H (as shown in Fig. 6.92) so there is no encirclement and the system is stable. Even more increase in the gain would cause the - point to occur between ω H and the origin where there is an encirclement and the system is unstable. The root locus would look like Fig except that the very low gain portion of the loci would start in the LHP before they loop out into the RHP as in Fig The Bode plot would be vaguely like that drawn below: 24. The Bode plot for is shown in Fig G(s) = 00[(s/0) + ] s[(s/) ][(s/00) + ] (a) Why does the phase start at -270 o at the low frequencies? (b) Sketch the Nyquist plot for G(s). (c) Is the closed-loop system shown in Fig stable? (d) Will the system be stable if the gain is lowered by a factor of 00? Make a rough sketch of a root locus for the system and qualitatively conþrm your answer Solution : (a) From the root locus, the phase at the low frequencies (ω =0+)is calculated as :

60 342 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.93: Bode plot for Problem 24

61 343 Thephaseatthepoint {s = jω(ω =0+)} = 80 (pole : s =) 90 (pole : s =0)+0 (zero : s = 0) + 0 (pole : s = 00) = 270 Or, more simply, the RHP pole at s =+causesa 80 o shift from the 90 o that you would expect from a normal system with all the singularities in the LHP. (b) The Nyquist plot for G(s) : (c) As the Nyquist shows, there is one counter-clockwise encirclement of -. = N = We have one pole in RHP = P = Z = N + P = +=0= The closed-loop system is stable. (d) The system goes unstable if the gain is lowered by a factor of Suppose that in Fig. 6.94, G(s) = 25(s +) s(s +2)(s 2 +2s + 6). Use MATLAB s margin to calculate the PM and GM for G(s) and,based on the Bode plots, conclude which margin would provide more useful information to the control designer for this system.

62 344 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.94: Control system for Problem 25 Solution : 40 G m =3.905 db (at rad/sec),pm =00.55 deg.(at.0799 rad/sec) 20 Phase (deg);m agnitude (db ) FromtheBodeplot, PM = deg, GM=3.9db =.57 Since both PM and GM are positive, we can say that the closed-loop of this system is stable. But GM is so small that we must be careful not to increase the gain much, which leads the closed-loop system to be unstable. Clearly, the GM is the more important margin for this example. 26. Consider the system given in Fig (a) Use MATLAB to obtain Bode plots for K = and use the plots to estimate the range of K for which the system will be stable.

63 345 Figure 6.95: Control system for Problem 26 (b) Verify the stable range of K by using margin to determine PM for selected values of K. (c) Use rlocus and rlocfind to determine the values of K at the stability boundaries. (d) Sketch the Nyquist plot of the system, and use it to verify the number of unstable roots for the unstable ranges of K. (e) Using Routh s criterion, determine the ranges of K for closed-loop stability of this system. Solution : (a) The Bode plot for K =is: 0 Magnitude Phase (deg) From the Bode plot, the closed-loop system is unstable for K =. But we can make the closed-system stable with positive GM by

64 346 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD increasing the gain K up to the crossover frequency reaches at ω =.44 rad/sec (K = 2), where the phase plot ccrosses the -80 line. Therefore : <K<2 = The closed-loop system is stable. (b) For example, PM =6.66 deg for K =.5. 0 G m = db (at.442 rad/sec),pm = deg.(at.0806 rad/sec) 0-0 Phase (deg);m agnitude (db ) (c) Root locus is : jω-crossing : jω +2 +K (jω) 3 +(jω) 2 2 =0

65 347 ω 2 2K +2 = 0 ω(ω 2 K) = 0 K =2, ω = ± 2, or K =, ω =0 Therefore, <K<2 = The closed-loop system is stable. (d) i. 0 <K< N =0,P= = Z = One unstable closed-loop root. ii. <K<2 N =, P= = Z =0 Stable. iii. 2 <K N =,P= = Z =2 Two unstable closed-loop roots. (e) The closed-loop transfer function of this system is : y(s) r(s) = = k s +k s s +2 (s +) 2 + K(s 2 +2s +2) s 3 + s 2 + Ks +2K 2 So the characteristic equation is : = s 3 + s 2 + Ks +2K 2=0

66 348 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Using the Routh s criterion, For stability, s 3 : K s 2 : 2K 2 s : 2 K 0 s 0 : 2k =2 27. Suppose that in Fig. 6.94, 2 K > 0 2K 2 > 0 = 2 >K> 0 <K< Unstable <K<2 Stable 2 <K Unstable 3.2(s +) G(s) = s(s +2)(s s +6). Use MATLAB s margin to calculate the PM and GM for G(s) and comment on whether you think this system will have well damped closed-loop roots. Solution : MATLAB s margin plot for the given system is : 40 G m =.0027 db (at rad/sec),pm = deg.(at rad/sec) 20 0 Phase (deg);m agnitude (db )

67 349 From the MATLAB margin routine, PM =92.8 o. Based on this result, Fig suggests that the damping will be = ; that is, the roots will be real. However, closer inspection shows that a very small increase in gain would result in an instability from the resonance leading one to believe that the damping of these roots is very small. Use of MATLAB s damp routine on the closed loop system conþrms this where we see that there are two real poles (ζ = ) and two very lightly damped poles with ζ = This is a good example where one needs to be careful to not use Matlab without thinking. 28. For a given system, show that the ultimate period P u and the corresponding ultimate gain K u for the Zeigler-Nichols method can be found using the following: (a) Nyquist diagram (b) Bode plot (c) root locus. Solution : (a) See sketch below. P u = 2π ω u

68 350 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (b) See sketch below. (c) P u = 2π ω u +K u G(jω u )=0 +K u Re[G(jω u )] + K u j Im[G(jω u )] = 0 K u = Re[G(jω u )] Im[G(jω u )] = 0; or P u = 2π ω u

69 If a system has the open-loop transfer function G(s) = ω 2 n s(s +2ζω n ) with unity feedback, then the closed-loop transfer function is given by ω 2 n T (s) = s 2 +2ζω n s + ω 2. n Verify the values of the PM shown in Fig for ζ =0., 0.4, and 0.7. Solution : ω 2 n G(s) = s(s +2ζω n ),T(s) = G(s) +G(s) = ω 2 n s 2 +2ζω n s + ω 2 n ζ PM from Eq PM from Fig PM from Bode plot (ω =0.99 rad/sec) (ω =0.85 rad/sec) (ω =0.65 rad/sec) 30. Consider the unity feedback system with the open-loop transfer function G(s) = K s(s +)[(s 2 /25) + 0.4(s/5) + ]. (a) Use MATLAB to draw the Bode plots for G(jω) assumingk =. (b) What gain K is required for a PM of 45? What is the GM for this value of K? (c) What is K v when the gain K is set for PM = 45? (d) Create a root locus with respect to K, and indicate the roots for a PM of 45. Solution : (a) The Bode plot for K = is shown below and we can see from margin

70 352 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD that it results in a PM = 48 o. 0 5 Magnitude Phase (deg) (b) Although diffult to read the plot above, it is clear that a very slight increase in gain will lower the PM to 45 o,sotryk =.. The margin routine shows that this yields PM =45 o and GM =5db. (c) K v = lim s 0 {skg(s)} = K =. whenk is set for PM=45 K v =. (d) The characteristic equation for PM of 45 : +. s(s +)h s s i =0 5 + = s 4 +3s 3 +27s 2 +25s = 0 = s =.03 ± j4.78, 0.47 ± j0.97

71 For the system depicted in Fig. 6.96(a), the transfer-function blocks are deþned by G(s) = (s +2) 2 (s +4) and H(s) = s +. (a) Using rlocus and rlocfind, determine the value of K at the stability boundary. (b) Using rlocus and rlocfind, determine the value of K that will produce roots with damping corresponding to ζ = (c) What is the gain margin of the system if the gain is set to the value determined in part (b)? Answer this question without using any frequency response methods. (d) Create the Bode plots for the system, and determine the gain margin that results for PM = 65. What damping ratio would you expect for this PM? (e) Sketch a root locus for the system shown in Fig. 6.96(b).. How does it differ from the one in part (a)? (f) For the systems in Figs. 6.96(a) and (b), how does the transfer function Y 2 (s)/r(s) differ from Y (s)/r(s)? Would you expect the step response to r(t) bedifferent for the two cases? Solution : (a) The root locus crosses jω axis at s 0 = j2. K = H(s 0 )G(s 0 ) s 0 =j2 = j2+ j2+4 j2+2 2 = K =80

72 354 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.96: Block diagram for Problem 3: (a) unity feedback; (b) H(s) in feedback (b) ζ =0.707 = = sin θ = θ =45 From the root locus given, s = j0.9 K = H(s )G(s ) s = 0.9+j0.9 = j j j0.9 2 = K =5.9 (c) GM = K a = 80 K b 5.9 =3.5 (d) From the Root Locus : G(s)H(s) = (s +)(s +2) 2 (s +4)

73 355 PM=65 when K = 30. Instability occures when K =80.0. = GM =2.67 We approximate the damping ratio by ζ ' PM 00 ζ ' = Magnitude Phase (deg) (e) The root locus for Fig.??(a) is the same as that of Fig.??(b). (f) Y (s) R(s) Y 2 (s) R(s) = = KG(s)H(s) +KG(s)H(s) = K (s +)(s +2) 2 (s +4)+K KG(s) +KG(s)H(s) = K(s +) (s +)(s +2) 2 (s +4)+K Y (s) R(s) and Y 2(s) R(s) havethesameclosed-looppoles. However, Y 2(s) R(s) has doesn t have a zero. We would therefore expect a zero, while Y (s) R(s) more overshoot from system (b). 32. For the system shown in Fig. 6.97, use Bode and root-locus plots to determine the gain and frequency at which instability occurs. What gain (or gains) gives a PM of 20? What is the gain margin when PM = 20?

74 356 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.97: Control system for Problem 32 Solution : 0 5 Magnitude Phase (deg)

75 357 Figure 6.98: Magnetic tape-drive speed control The system with K =gives, GM = 52 (ω = 5 rad/sec) PM = 0 (ω =0.65 rad/sec) Therefore, instability occurs at K 0 =52andω = 5 rad/sec. From the Bode plot, a PM of 20 is given by, K = 3.9 (ω =0.33 rad/sec), GM = =3 K 2 = 49 (ω =4.6 rad/sec), GM = = A magnetic tape-drive speed-control system is shown in Fig The speed sensor is slow enough that its dynamics must be included. The speed-measurement time constant is τ m =0.5 sec; the reel time constant is τ r = J/b = 4 sec, where b = the output shaft damping constant = N m sec; and the motor time constant is τ = sec. (a) Determine the gain K required to keep the steady-state speed error to less than 7% of the reference-speed setting. (b) Determine the gain and phase margins of the system. Is this a good system design? Solution : (a) From Table 4.3, the error for this Type system is e ss = +K Ω c Since the steady-state speed error is to be less than 7% of the reference speed, +K p 0.07

76 358 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.99: Control system for Problems 34, 6, and 62 (b) and for the system in Fig with the numbers plugged in, Eq shows that K p = K. Therefore, K 3. G(s) = 0.79 at GH = 80 = GM = GH =.3 GH = 73 at G(s) == PM = G + 80 =7 GM is low = The system is very close to instability. PM is low = The damping ratio is low. = High overshoot. We see that to have a more stable system we have to lower the gain. With small gain, e ss will be higher. Therefore, this is not a good design, and needs compensation. 34. For the system in Fig. 6.99, determine the Nyquist plot and apply the Nyquist criterion (a) to determine the range of values of K (positive and negative) for which the system will be stable, and (b) to determine the number of roots in the RHP for those values of K for which the system is unstable. Check your answer using a rough root-locus sketch. Solution : (a) & b. 3 KG(s) =K s(s +)(s +3)

77 359 From the Nyquist plot above, we see that: i. < K < 4 = 0 <K<4 There are no RHP open loop roots, hence P =0forallcases. For 0 <K<4, no encirclements of - so N =0, N =0,P=0 = Z =0 The closed-loop system is stable. No roots in RHP. ii. 4 < K < 0 = 4 <K< Two encirclements of the - point, hence N =2,P=0 = Z =2 Two closed-loop roots in RHP. iii. 0 < K = K<0 N =,P=0 = Z = One closed-loop root in RHP.

78 360 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.00: Control system for Problem 35 Therootlocibeowshowthesameresults. 35. For the system shown in Fig. 6.00, determine the Nyquist plot and apply the Nyquist criterion. (a) to determine the range of values of K (positive and negative) for which the system will be stable, and (b) to determine the number of roots in the RHP for those values of K for which the system is unstable. Check your answer using a rough root-locus sketch. Solution : (a) & b. KG(s) =K s + (s ) 2

79 36 From the Nyquist plot we see that: i. < K < 2 = 0 <K<2 N =0,P=2 = Z =2 Two closed-loop roots in RHP. ii. 2 < K < 0 = 2 <K N = 2, P=2 = Z =0 The closed-loop system is stable. iii. 0 < K < = K< N =, P=2 = Z = One closed-loop root in RHP. iv. < K < = <K<0 N =0,P=2 = Z =2 Two closed-loop roots in RHP.

80 362 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.0: Control system for Problem 36 These results are conþrmed by looking at the root loci below: 36. For the system shown in Fig. 6.0, determine the Nyquist plot and apply the Nyquist criterion. (a) to determine the range of values of K (positive and negative) for which the system will be stable, and (b) to determine the number of roots in the RHP for those values of K for which the system is unstable. Check your answer using a rough root-locus sketch. Solution : (a) & b. KG(s) =K s (s +) 2

81 363 From the Nyquist plot we see that: i. < K < = 0 <K< N =0,P=0 = Z =0 The closed-loop system is stable. ii. < K < 0 = <K N =,P=0 = Z = One closed-loop root in RHP. iii. 0 < K < 2 = K< 2 N =2,P=0 = Z =2 Two closed-loop roots in RHP. iv. 2 < K = 2 <K<0 N =0,P=0 = Z =0 The closed-loop system is stable. TheseresultsareconÞrmed by looking at the root loci below:

82 364 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.02: Nyquist plots for Problem The Nyquist diagrams for two stable, open-loop systems are sketched in Fig The proposed operating gain is indicated as K 0, and arrows indicate increasing frequency. In each case give a rough estimate of the following quantities for the closed-loop (unity feedback) system: (a) phase margin (b) damping ratio (c) range of gain for stability (if any) (d) system type (0,, or 2).

83 365 Solution : For both, with K = K 0 : N =0,P=0 = Z =0 Therefore, the closed-loop system is stable. Fig.6.02(a) Fig.6.02(b) a. PM '7 '45 b. Damping ratio 0.7(' 7 00 ) (' 00 ) c. To determine the range of gain for stability, call the value of K where the plots cross the negative real axis as K. For case (a), K > K for stability because gains lower than this amount will cause the - point to be encircled. For case (b), K<K for stability because gains greater than this amount will cause the - point to be encircled. d. For case (a), the 360 o loop indicates two poles at the origin, hence the system is Type 2. For case (b), the 80 o loop indicates one pole at the origin, hence the system is Type. 38. The steering dynamics of a ship are represented by the transfer function V (s) δ r (s) = G(s) = K[ (s/0.42) + ] s(s/ )(s/0.0362) + ), where v is the ship s lateral velocity in meters per second, and δ r is the rudder angle in radians. (a) Use the MATLAB command bode to plot the log magnitude and phase of G(jω) fork =0.2 (b) On your plot, indicate the crossover frequency, PM, and GM, (c) Is the ship steering system stable with K =0.2?

84 366 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (d) What value of K would yield a PM of 30 o crossover frequency be? and what would the Solution : (a) The Bode plot for K =0.2 is: 0 5 Magnitude 0 0 GM= rad/sec Phase (deg) deg PM =-23.7 deg (b) From the Bode plot above : ω c = rad/sec PM = 23.7 deg GM =.95 (c) Since PM < 0, the closed-loop system with K =0.2 is unstable. (d) From the Bode plot above, we can get better PM by decreasing the gain K. Then we will Þnd that K =0.042 yields PM =30 at the crossover frequency ω c =0.032rad/sec. TheBodeplotwith

85 367 K =0.042 is : 00 G m = db (at rad/sec),pm =30.07 deg.(at rad/sec) 50 Phase (deg);m agnitude (db ) For the open-loop system KG(s) = K(s +) s 2 (s + 0) 2. Determine the value for K at the stability boundary and the values of K at the points where PM =30. Solution :

86 368 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD ThebodeplotofthissystemwithK =is: 50 G m = db (at rad/sec),pm = deg.(at rad/sec) 0 Phase (deg);m agnitude (db ) Since GM =64. db(' 600), the range of K for stability is : K<600 FromtheBodeplot,themagnitudeatthefrequencywith 50 phase is ( 34.5 db) at rad/sec and ( 54. db) at 4.44 rad/sec. Therefore, the values of K at the points where PM =30 is : K = K = =53.2, = 505 (a) Problems and Solutions for Section The frequency response of a plant in a unity feedback conþguration is sketched in Fig Assume the plant is open-loop stable and minimum phase. (a) What is the velocity constant K v for the system as drawn? (b) What is the damping ratio of the complex poles at ω = 00? (c) What is the PM of the system as drawn? (Estimate to within ±0 o.)

87 369 Figure 6.03: Magnitude frequency response for Problem 40 Solution : (a) From Fig. 6.03, K v = lim s 0 sg = Low frequency asymptote of G(jω) ω= = 00) (b) Let G (s) = ³ 2 ³ s ω n +2ζ s ω n + For the second order system G (s), G (jω) ω= = 2ζ () From Fig : G (jω) ω=00 = From()and(2)wehave: G(jω) ω= = =2 (2) Asysmptote of G(jω) ω= ζ =2 = ζ =0.25

88 370 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) Since the plant is a minimum phase system, we can apply the Bode s approximate gain-phase relationship. When G =,theslopeof G curve is = -2. = G(jω) = 2 90 = 80 PM = G(jω) + 80 =0 Note : Actual PM by Matlab calculation is 6.4, so this approximation is within the desired accuracy. 4. For the system 00(s/a +) G(s) = s(s +)(s/b +), where b =0a, Þnd the approximate value of a that will yield the best PM by sketching only candidate values of the frequency response magnitude. Solution : 0 2 Magnitude 0 0 a= U ncom pensated a= Phase (deg) a=3.6 a=0-80 U ncom pensated Without the zero and pole that contain the a & b terms, the plot of G shows a slope of 2 atthe G = crossover at 0 rad/sec. We clearly need to install the zero and pole with the a & b terms somewhere at frequencies greater rad/sec. This will increase the slope from -2 to - between the zero and pole. So the problem simpliþes to selecting a so that the - slope region between the zero and pole brackets the crossover frequency. That scenario will maximize the PM. Referring to the plots

89 Magnitude 37 above, we see that 3.6 <a<0, makes the slope of the asymptote of G be at the crossover and represent the two extremes of possibilities for a - slope. The maximum PM will occur half way between these extremes on a log scale, or = a = = 5.6 Note : Actual PM is as follows : PM = 46.8 for a =3.6 (ω c =25.0 rad/sec) PM = 58. for a =5.6 (ω c =7.8 rad/sec) PM = 49.0 for a =0(ω c =2.6 rad/sec) Problem and Solution for Section For the open-loop system K(s +) KG(s) = s 2 (s + 0) 2. Determine the value for K that will yield PM 30 and the maximum possible closed-loop bandwidth. Use MATLAB to Þnd the bandwidth. Solution : From the result of Problem 6.39., the value of K that will yield PM 30 is : 53.2 K 505 The maximum closed-loop bandwidth will occur with the maximum gain K within the allowable region; therefore, the maximum bandwidth will occur with K = 505. TheBodeplotoftheclosedloopsystemwithK = 505 is 0 4 Closed-loop Frequency Response rad/sec :

90 372 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Looking at the point with Magnitude 0.707(-3 db), the maximum possible closed-loop bandwidth is : ω BW, max ' 7.7 rad/sec. Problems and Solutions for Section For the lead compensator where α <. D(s) = Ts+ αts+, (a) Show that the phase of the lead compensator is given by φ =tan (T ω) tan (αt ω). (b) Show that the frequency where the phase is maximum is given by ω max = T α, and that the maximum phase corresponds to sin φ max = α +α. (c) Rewrite your expression for ω max to show that the maximum-phase frequency occurs at the geometric mean of the two corner frequencies on a logarithmic scale: log ω max = µ log 2 T +log. αt (d) To derive the same results in terms of the pole-zero locations, rewrite D(s) as D(s) = s + z s + p, and then show that the phase is given by such that φ =tan µ ω z tan µ ω p ω max = p z p. Hence the frequency at which the phase is maximum is the square root of the product of the pole and zero locations.,

91 373 Solution : (a) The frequency response is obtained by letting s = jω, Tjω + D(jω) =K αtjω + The phase is given by,φ =tan (T ω) tan (αt ω) (b) Using the trigonometric relationship, then and since, then tan(a B) = tan(a) tan(b) +tan(a)tan(b) tan(φ) = T ω αt ω αt 2 ω 2 sin 2 (φ) = tan2 (φ) +tan 2 (φ) s ω sin(φ) = 2 T 2 ( α) 2 +α 2 ω 4 T 4 +(+α 2 )ω 2 T 2 To determine the frequency at which the phase is a maximum, let us set the derivative with respect to ω equal to zero, d sin(φ) =0 dω which leads to 2ωT 2 ( α) 2 ( αω 4 T 4 )=0 The value ω = 0 gives the maximum of the function and setting the second part of the above equation to zero then, ω 4 = α 2 T 4 or ω max = αt The maximum phase contribution, that is, the peak of the D(s) curve corresponds to, sin φ max = α +α or α = sin φ max +sinφ max tan φ max = ω maxt αω max T +ω 2 maxt 2 = α 2 α

92 374 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) The maxmum frequency occurs midway between the two break frequencies on a logarithmic scale, log ω max = log T αt = log +log T αt = µ log 2 T +log αt as shown in Fig (d) Alternatively, we may state these results in terms of the pole-zero locations. Rewrite D(s) as, (s + z) D(s) =K (s + p) then and or (jω + z) D(jω) =K (jω + p) µ µ ω ω φ =tan tan z p ω z ω p ω tan φ = + ω z p Setting the derivative of the above equation to zero we Þnd, µ ω z ω µ+ ω2 2ω µ ω p z p z p z ω =0 p and and ω max = p z p log ω max = (log z +log p ) 2 Hence the frequency at which the phase is maximum is the square root of the product of the pole and zero locations. 44. For the third-order servo system G(s) = 50, 000 s(s + 0)(s + 50).

95 377 Figure 6.04: Control system for Problem 45 TheBodeplotofthegivensystemis: 00 G m =.5836 db (at rad/sec),pm =3.943 deg.(at rad/sec) 50 Phase (deg);m agnitude (db ) Since PM =3.9, let s add phase lead 60.FromFig.6.53, ' 20 = choose α =0.05 α To apply maximum phase lead at ω = 0 rad/sec, ω = αt =0= T =2.2, αt =45 Therefore by applying the lead compensator : D(s) = s s 45 +

96 378 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD we get the compensated system with : PM =40, ω c =2.5 The Bode plot with designed compensator is : 00 G m =24.3 db (at rad/sec),pm = deg.(at rad/sec) 50 0 Phase (deg);m agnitude (db ) From Fig. 6.50, we see that ω BW ' 2 ω c ' 5 rad/sec. 46. Derive the transfer function from T d to θ for the system in Fig Then apply the Final Value Theorem (assuming T d = constant) to determine whether θ( ) is nonzero for the following two cases: (a) When D(s) has no integral term: lim s 0 D(s) =constant; (b) When D(s) has an integral term: D(s) = D0 (s), s where lim s 0 D 0 (s) =constant. Solution : The transfer function from T d to θ : where T d (s) = T d /s. 0.9 Θ(s) T d (s) = s s 2 s+2 D(s)

97 379 (a) Using the Þnal value theorem : θ( ) = lim θ(t) = limsθ(t) = lim t s 0 s s 2 s 2 (s+2)+.8d(s) s 2 (s+2) T d s T d = lim D(s) = T d constant 6=0 s 0 Therefore, there will be a steady state error in θ for a constant T d input if there is no integral term in D(s). (b) θ( ) = lim θ(t) = limsθ(t) =lim t s 0 s 0 = 0.8lim s 0 D 0 (s) =0 0.9 s 2 s 3 (s+2)+.8d 0 (s) s 3 (s+2) T d s So when D(s) contains an integral term, a constant T d input will result in a zero steady state error in θ. 47. The inverted pendulum has a transfer function given by Eq. (2.35), which is similiar to G(s) = s 2. (a) Design a lead compensator to achieve a PM of 30 using Bode plot sketches, then verify and reþne your design using MATLAB. (b) Sketch a root locus and correlate it with the Bode plot of the system. (c) Could you obtain the frequency response of this system experimentally? Solution : (a) Design the lead compensator : D(s) =K Ts+ αts+ such that the compensated system has PM ' 30 & ω c ' rad/sec. (Actually, the bandwidth or speed of response was not speciþed, so any crossover frequency would satisfy the problem statement.) α = sin(30 ) +sin(30 ) =0.32 To apply maximum phase lead at ω =rad/sec, ω = αt == T =0.57, αt =.77

98 380 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Therefore by applying the lead compensator : s D(s) =K s.77 + By adjusting the gain K so that the crossover frequency is around rad/sec, K =.3 results in : PM =30.8 TheBodeplotofcompensatedsystemis: 5 G m =-.066 db (at 0 rad/sec),pm = deg.(at rad/sec) Phase (deg);m agnitude (db ) (b) Root Locus of the compensated system is :

99 38 and conþrms that the system yields all stable roots with reasonable damping. However, it would be a better design if the gain was raised some in order to increase the speed of response of the slow real root. A small decrease in the damping of the complex roots will result. (c) No, because the sinusoid input will cause the system to blow up because the open loop system is unstable. In fact, the system will blow up even without the sinusoid applied. Or, a better description would be that the pendulum will fall over until it hits the table. 48. The open-loop transfer function of a unity feedback system is G(s) = K s(s/5+)(s/50 + ). (a) Design a lag compensator for G(s) using Bode plot sketches so that the closed-loop system satisþes the following speciþcations: i. The steady-state error to a unit ramp reference input is less than 0.0. ii. PM =40 (b) Verify and reþne your design using MATLAB. Solution : Let s design the lag compensator : D(s) = Ts+ αts+, α > From the Þrst speciþcation, Steady-state error to unit ramp = lims D(s)G(s) s 0 +D(s)G(s) s 2 s 2 < 0.0 = K < 0.0 = Choose K = 50 Uncompensated, the crossover frequency with K = 50 is too high for a good PM. With some trial and error, we Þnd that the lag compensator, D(s) = s s will lower the crossover frequency to ω c ' 4.46 rad/sec where the PM =

100 382 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD G m =8.873 db (at rad/sec),pm =40.7 deg.(at rad/sec) Phase (deg);m agnitude (db ) The open-loop transfer function of a unity feedback system is K G(s) = s(s/5+)(s/200 + ). (a) Design a lead compensator for G(s) using Bode plot sketches so that the closed-loop system satisþes the following speciþcations: i. The steady-state error to a unit ramp reference input is less than 0.0. ii. For the dominant closed-loop poles the damping ratio ζ 0.4. (b) Verify and reþne your design using MATLAB including a direct computation of the damping of the dominant closed-loop poles. Solution : Let s design the lead compensator : D(s) = Ts+ αts+, α < From the Þrst speciþcation, Steady-state error to unit ramp = lims D(s)G(s) s 0 +D(s)G(s) = K < 0.0 = Choose K = 50 s 2 s 2 < 0.0

101 383 From the approximation ζ ' PM, the second speciþcation implies PM After trial and error, we Þnd that the compensator, D(s) = s 0 + s 00 + results in a PM =42.5 and a crossover frequency ω c ' 5.2 rad/sec as shown by the margin output: 80 G m =3.292 db (at rad/sec),pm = deg.(at 52.7 rad/sec) Phase (deg);m agnitude (db ) and the use of damp veriþes the damping to be ζ =0.42 for the complex closed-loop roots which exceeds the requirement. 50. A DC motor with negligible armature inductance is to be used in a position control system. Its open-loop transfer function is given by G(s) = 50 s(s/5+). (a) Design a compensator for the motor using Bode plot sketches so that the closed-loop system satisþes the following speciþcations: i. The steady-state error to a unit ramp input is less than /200. ii. The unit step response has an overshoot of less than 20%. iii. The bandwidth of the compensated system is no less than that of the uncompensated system.

102 384 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (b) Verify and/or reþne your design using MATLAB including a direct computation of the step response overshoot. Solution : The Þrst speciþcation implies that a loop gain greater than 200 is required. Since the open loop gain of the plant is 50, a gain from the compensator, K, is required where K>4= so choose K =5 From Figure 3.28, we see that the second speciþcation implies that : Overshoot < 20% = ζ > 0.5 = PM > 50 A sketch of the Bode asymptotes of the open loop system with the required loop gain shows a crossover frequency of about 30 rad/sec at a slope of -2; hence, the PM will be quite low. To add phase with no decrease in the crossover frequencey, a lead compensator is required. Figure 6.53 shows that a lead ratio of 0: will provide about 55 o of phase increase and the asymptote sketch shows that this increase will be centered at the crossover frequencyifweselectthebreakpointsat D(s) = s 20 + s Use of Matlab s margin routine shows that this compensation results in a PM =59 and a crossover frequency ω c ' 60 rad/sec.

103 Phase (deg); Magnitude (db) 385 Bode Diagrams 80 Gm = Inf, Pm= deg. (at rad/sec) and using the step routine on the closed loop system shows the step response to be less than the maximum allowed 20%. 5. The open-loop transfer function of a unity feedback system is G(s) = K s( + s/5)( + s/20). (a) Sketch the system block diagram including input reference commands and sensor noise. (b) Design a compensator for G(s) using Bode plot sketches so that the closed-loop system satisþes the following speciþcations: i. The steady-state error to a unit ramp input is less than 0.0. ii. PM 45 iii. The steady-state error for sinusoidal inputs with ω < 0.2 rad/sec is less than /250. iv. Noise components introduced with the sensor signal at frequencies greater than 200 rad/sec are to be attenuated at the output by at least a factor of 00,.

104 Phase (deg) Magnitude 386 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) Verify and/or reþne your design using MATLAB including a computation of the closed-loop frequency response to verify (iv). Solution : a. The block diagram shows the noise, v, entering where the sensor would be: b. The Þrst speciþcation implies K v 00 and thus K 00. The bode plot with K =andd = below shows that there is a negative PM but all the other specs are met. The easiest way to see this is to hand plot the asymptotes and mark the constraints that the gain must be 250 at ω 0.2 rad/sec and the gain must be 0.0 for ω 200 rad/sec. 0 5 uncompensated Bode Plot

105 387 In fact, the specs are exceeded at the low frequency side, and slightly exceeded on the high frequency side. But it will be difficult to increase the phase at crossover without violating the specs. From a hand plot of the asymptotes, we see that a combination of lead and lag will do the trick. Placing the lag according to D lag (s) = (s/2+) (s/0.2+) will lower the gain curve at frequencies just prior to crossover so that a - slope is more easily achieved at crossover without violating the high frequency constraint. In addition, in order to obtain as much phase at crossover as possible, a lead according to D lead (s) = (s/5+) (s/50 + ) will preserve the - slope from ω =5rad/sectoω = 20 rad/sec which will bracket the crossover frequency and should result in a healthy PM. A look at the Bode plot shows that all specs are met except the PM =44. Perhaps close enough, but a slight increase in lead should do the trick. So our Þnal compensation is D(s) = (s/2+) (s/4+) (s/0.2+)(s/50 + ) with K = 00. This does meet all specs with PM =45 o exactly, as can be seen by examining the Bode plot below.

106 Phase (deg) Magnitude 388 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD 0 5 compensated Bode Plot Consider a type I unity feedback system with G(s) = K s(s +). Design a lead compensator using Bode plot sketches so that K v =20sec and PM > 40. Use MATLAB to verify and/or reþne your design so that it meets the speciþcations. Solution : Use a lead compensation : D(s) = Ts+ αts+, α > From the speciþcation, K v =20sec, = K v = limsd(s)g(s) =K =20 s 0 = K =20

107 Phase (deg); Magnitude (db) 389 From a hand sketch of the uncompensated Bode plot asymptotes, we see that the slope at crossover is -2, hence the PM will be poor. In fact, an exact computation shows that Adding a lead compensation PM =2.75 (at ω c =4.42 rad/sec) D(s) = s 3 + s 30 + will provide a - slope in the vicinity of crossover and should provide plenty of PM. The Bode plot below veriþes that indeed it did and shows that the PM =62 at a crossover frequency = 7 rad/sec thus meeting all specs. Bode Diagrams 00 Gm = Inf, Pm=6.797 deg. (at rad/sec) Consider a satellite-attitude control system with the transfer function G(s) = 0.05(s + 25) s 2 (s 2 +0.s +4).

108 Phase (deg) Magnitude 390 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Amplitude-stabilize the system using lead compensation so that GM 2 (6 db), and PM 45, keeping the bandwidth as high as possible with asinglelead. Solution : The sketch of the uncompensated Bode plot asymptotes shows that the slope at crossover is -2; therefore, a lead compensator will be required in order to have a hope of meeting the PM requirement. Furthermore, the resonant peak needs to be kept below magnitude so that it has no chance of causing an instability (this is amplitude stabilization). This latter requirement means we must lower gain at the resonance. Using the single lead compensator, D(s) = (s +0.06) (s +6) will lower the low frequency gain by a factor of 00, provide a - slope at crossover, and will lower the gain some at the resonance. Thus it is agoodþrst cut at a compensation. The Matlab Bode plot shows the uncompensated and compensated and veriþes our intent. Note especially that the resonant peak never crosses magnitude for the compensated (dashed) case. 0 5 Bode Diagrams

109 39 Figure 6.05: Control system for Problem 54 The Matlab margin routine shows a GM =6.3 dbandpm =48 thus meeting all specs. 54. In one mode of operation the autopilot of a jet transport is used to control altitude. For the purpose of designing the altitude portion of the autopilot loop, only the long-period airplane dynamics are important. The linearized relationship between altitude and elevator angle for the longperiod dynamics is G(s) = h(s) δ(s) = 20(s +0.0) s(s s ) ft deg. The autopilot receives from the altimeter an electrical signal proportional to altitude. This signal is compared with a command signal (proportional to the altitude selected by the pilot), and the difference provides an error signal. The error signal is processed through compensation, and the result is used to command the elevator actuators. A block diagram of this system is shown in Fig You have been given the task of designing the compensation. Begin by considering a proportional control law D(s) =K. (a) Use MATLAB to draw a Bode plot of the open-loop system for D(s) =K =. (b) What value of K would provide a crossover frequency (i.e., where G = ) of 0.6 rad/sec? (c) For this value of K, would the system be stable if the loop were closed? (d) What is the PM for this value of K? (e) Sketch the Nyquist plot of the system, and locate carefully any points where the phase angle is 80 or the magnitude is unity. (f) Use MATLAB to plot the root locus with respect to K, andlocate the roots for your value of K from part (b).

110 392 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (g) What steady-state error would result if the command was a step change in altitude of 000 ft? For parts (h)and (i), assume a compensator of the form D(s) =K Ts+ αts+. (h) Choose the parameters K, T,andα so that the crossover frequency is 0.6 rad/sec and the PM is greater that 50. Verify your design by superimposing a Bode plot of D(s)G(s)/K on top of the Bode plot you obtained for part (a), and measure the PM directly. (i) Use MATLAB to plot the root locus with respect to K for the system including the compensator you designed in part (h). Locate the roots for your value of K from part (h). (j) Altitude autopilots also have a mode where the rate of climb is sensed directly and commanded by the pilot. i. Sketch the block diagram for this mode, ii. deþne the pertinent G(s), iii. design D(s) so that the system has the same crossover frequency as the altitude hold mode and the PM is greater than 50 Solution : The plant transfer function : ³ s + h(s) 80 δ(s) = ½ 0.0 ³ ¾ s 2 0. s s +

111 393 (a) SeetheBodeplot: 0 6 Magnitude Phase (deg) (b) Since G = 865 at ω =0.6, K = G ω=0.6 =0.002 (c) The system would be stable, but poorly damped. (d) PM =0.39 (e) The Nyquist plot for D(jω)G(jω) : The phase angle never quite reaches 80.

112 394 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (f) See the Root locus : The closed-loop roots for K =0.002 are : (g) The steady-state error e : s = 0.009, ± j0.6 e ss = lim = 0 s 0 s +K h(s) δ(s) as it should be for this Type system. (h) Phase margin of the plant : 000 s PM =0.39 (ω c =0.6 rad/sec) Necessary phase lead and α : necessary phase lead = ' 50 From Fig : = α =8 Set the maximum phase lead frequency at ω c : ω = = ω c =0.6 = T =8 αt so the compensation is 8s + D(s) =K 2.2s +

113 395 For a gain K, wewant D(jω c )G(jω c ) =atω = ω c =0.6. So evaluate via Matlab D(jω c )G(jω c ) K ωc=0.6 and Þnd it = = K = = Therefore the compensation is : 4 8s + D(s) = s + which results in the Phase margin : PM =52 (ω c =0.6 rad/sec) 0 6 Magnitude K=4.09x0-4 DG/K G w = Phase (deg) deg G DG/K 5.4 deg

114 396 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (i) See the Root locus : The closed-loop roots for K = are : s = 0.27, , ± j0.099 (j) In this case, the referance input and the feedback parameter are the rate of climb. i. The block diagram for this mode is : ii. DeÞne G(s) as: ³ s + h(s) G(s) = ú 80 δ(s) = 0.0 ³ s 2 0. s s + iii. By evaluating the gain of G(s) atω = ω c =0.6, and setting K equal to its inverse, we see that proportional feedback : D(s) =K = satisþes the given speciþcations by providing: PM =90 (ω c =0.6 rad/sec)

115 397 TheBodeplotofthecompensatedsystemis: 30 Gm = Inf,Pm = deg.(at rad/sec) 20 0 Phase (deg);m agnitude (db ) For a system with open-loop transfer function c 0 G(s) = s[(s/.4) + ][(s/3) + ], design a lag compensator with unity DC gain so that PM 40.Whatis the approximate bandwidth of this system? Solution : Lag compensation design : Use D(s) = Ts+ αts+ K= so that DC gain of D(s) =. (a) Find the stability margins of the plant without compensation by plotting the Bode, Þnd that: PM = 20 (ω c =3.0 rad/sec) GM = 0.44 (ω = 2.05 rad/sec) (b) The lag compensation needs to lower the crossover frequency so that a PM ' 40 will result, so we see from the uncompensated Bode that we need the crossover at about = ω c,new =0.8

116 398 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD where G(jω c ) =0.4 so the lag needs to lower the gain at ω c,new from 0.4 to. (c) Pick the zero breakpoint of the lag to avoid inßuencing the phase at ω = ω c,new by picking it a factor of 20 below the crossover, so T = ω c,new 20 = T =25 (d) Choose α : Since D(jω) = α for ω >> T,let α = G(jω c,new ) α = G(jω c,new ) =0.4 (e) Compensation : s 0.04 D(s) = + s (f) Stability margins of the compensated system : PM = 42 (ω c =0.8 rad/sec) GM = 4.4 (ω = 2.0 rad/sec) Approximate bandwidth ω BW : PM = 42 = ω BW = 2ωc =.6 (rad/sec) 0 4 Magnitude U ncom pensated U ncom pensated Phase (deg)

117 For the ship-steering system in Problem 38, (a) Design a compensator that meets the following speciþcations: i. velocity constant K v =2, ii. PM 50, iii. unconditional stability (PM > 0 for all ω ω c,thecrossover frequency). (b) For your Þnal design, draw a root locus with respect to K, and indicate the location of the closed-loop poles. Solution : The transfer function of the ship steering is V (s) δ r (s) = G(s) = K[ (s/0.42) + ] s(s/ )(s/0.0362) + ). (a) Since the velocity constant, K v must be 2, we require that K =2. i. The phase margin of the uncompensated ship is PM = (ω c =0.363 rad/sec) which means it would be impossible to stabilize this system with one lead compensation, since the maximum phase increase would be 90 o. There is no speciþcation leading to maintaining a high bandwidth, so the use of lag compensation appears to be the best choice. So we use a lag compensation: D(s) = Ts+ αts+ ii. The crossover frequency which provides PM ' 50 is obtained by looking at the uncompensated Bode plot below, where we see that the crossover frequency needs to be lowered to ω c,new =0.07, where the uncompensated gain is G(jω c,new ) = 07 iii. Keep the zero of the lag a factor of 20 below the crossover to keep the phase lag from the compensation from fouling up the PM, so we Þnd: T = ω c,new 20 = T =.2 0 3

118 400 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD iv. Choose α so that the gain reduction is achieved at crossover : α = G(jω c,new ) = 07 (D(jω) ' α for ω À T ) v. So the compensation is : s 200s + D(s) = 2.6s + = s vi. Stability margins of the compensated system : PM = 52. (ω c =0.07 rad/sec) GM = 5.32 (ω = rad/sec) and the system is unconditionally stable since the phase > 0for all ω < ω c.as can be seen by the plot below. (b) See the root locus. (Note that this is a zero degree root locus.) The closed-loop roots for K =2are: s = 0.33, , 0.04 ± j U ncom pensated Magnitude Phase (deg) -300 U ncom pensated -80 deg PM =52. deg

119 For a unity feedback system with G(s) = s( s s2 20 +)( s ) (2) (a) A lead compensator is introduced with α =/5 andazeroat/t = 20. How must the gain be changed to obtain crossover at ω c = 3.6 rad/sec, and what is the resulting value of K v? (b) With the lead compensator in place, what is the required value of K for a lag compensator that will readjust the gain to the original K v value of 00? (c) Place the pole of the lag compensator at 3.6 rad/sec, and determine the zero location that will maintain the crossover frequency at ω c = 3.6 rad/sec. Plot the compensated frequency response on the same graph. (d) Determine the PM of the compensated design. Solution : (a) From a sketch of the asymptotes with the lead compensation (with K =): s 20 D (s) =K + s 00 + in place, we see that the slope is - from zero frequency to ω = 00 rad/sec. Therefore, to obtain crossover at ω c =3.6 rad/sec, the gain K =3.6 is required. Therefore, K v =3.6 (b) To increase K v to be 00, we need an additional gain of 3.6 from the lag compensator at very low frequencies to yield K v = 00.

120 Phase (deg) Magnitude 402 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) For a low frequency gain increase of 3.6, and the pole at 3.6 rad/sec, the zero needs to be at 0 in order to maintain the crossover at ω c =3.6 rad/sec. So the lag compensator is and D 2 (s) =3.6 D (s)d 2 (s) = 00 s 0 + s s 20 + s 00 + s 0 + s The Bode plots of the system before and after adding the lag compensation are 0 2 Bode Diagrams Lead only Lead and Lag (d) By using the margin routine from Matlab, we see that PM =49 (ω c =3.6 deg/sec)ec) 58. Golden Nugget Airlines had great success with their free bar near the tail of the airplane. (See Problem 5.4) However, when they purchased a much

121 403 larger airplane to handle the passenger demand, they discovered that there was some ßexibility in the fuselage that caused a lot of unpleasant yawing motion at the rear of the airplane when in turbulence and was causing the revelers to spill their drinks. The approximate transfer function for the dutch roll mode (See Section 9.3.) is r(s) δ r (s) = 8.75(4s s +) (s/0.0 + )(s s +) where r is the airplane s yaw rate and δ r is the rudder angle. In performing a Finite Element Analysis (FEA) of the fuselage structure and adding those dynamics to the dutch roll motion, they found that the transfer function needed additional terms that reßected the fuselage lateral bending that occurred due to excitation from the rudder and turbulence. The revised transfer function is r(s) δ r (s) = 8.75(4s s +) (s/0.0 + )(s s +) ( s2 +2ζ s ω 2 ω b b +) where ω b is the frequency of the bending mode (= 0 rad/sec) and ζ is the bending mode damping ratio (= 0.02). Most swept wing airplanes have a yaw damper which essentially feeds back yaw rate measured by a rate gyro to the rudder with a simple proportional control law. For the new Golden Nugget airplane, the proportional feedback gain, K =, where δ r (s) = Kr(s). (3) (a) Make a Bode plot of the open-loop system, determine the PM and GM for the nominal design, and plot the step response and Bode magnitude of the closed-loop system. What is the frequency of the lightly damped mode that is causing the difficulty? (b) Investigate remedies to quiet down the oscillations, but maintain the same low frequency gain in order not to affectthequalityofthe dutch roll damping provided by the yaw rate feedback. SpeciÞcally, investigate one at a time: i. increasing the damping of the bending mode from ζ =0.02 to ζ = (Would require adding energy absorbing material in the fuselage structure) ii. increasing the frequency of the bending mode from ω b = 0 rad/sec to ω b = 20 rad/sec. (Would require stronger and heavier structural elements) iii. adding a low pass Þlter in the feedback, that is, replace K in Eq. (3) with KD(s) where D(s) = s/τ p +. (4) Pick τ p so that the objectionable features of the bending mode are reduced while maintaing the PM 60 o.

122 404 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD iv. adding a notch Þlter as described in Section Pick the frequency of the notch zero to be at ω b with a damping of ζ = 0.04 and pick the denominator poles to be (s/00 + ) 2 keeping thedcgainoftheþlter =. (c) Investigate the sensitivity of the two compensated designs above (iii and iv) by determining the effect of a reduction in the bending mode frequency of -0%. SpeciÞcally, re-examine the two designs by tabulating the GM, PM, closed loop bending mode damping ratio and resonant peak amplitude, and qualitatively describe the differences in the step response. (d) What do you recommend to Golden Nugget to help their customers quit spilling their drinks? (Telling them to get back in their seats is not an acceptable answer for this problem! Make the recommendation in terms of improvements to the yaw damper.) Solution : (a) The Bode plot of the open-loop system is : 20 G m =.2833 db (at rad/sec),pm = deg.(at rad/sec) 0-20 Phase (deg);m agnitude (db ) PM = 97.6 (ω = rad/sec) GM =.28 (ω = 0.0 rad/sec) The low GM is caused by the resonance being close to instability.

123 405 The closed-loop system unit step response is : 0.9 Unit Step R esponse 0.6 Amplitude Tim e (sec) TheBodeplotoftheclosed-loopsystemis: 0 0 Magnitude Phase (deg) From the Bode plot of the closed -loop system, the frequency of the lightly damped mode is : ω ' 0 rad/sec

124 406 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD and this is borne out by the step response that shows a lightly damped oscillation at.6 Hz or 0 rad/sec. i. The Bode plot of the system with the bending mode damping increased from ζ =0.02 to ζ =0.04 is : 20 G m = db (at rad/sec),pm =97.64 deg.(at rad/sec) 0-20 Phase (deg);m agnitude (db ) PM = 97.6 (ω = rad/sec) GM = 7.3 (ω = 0.0 rad/sec) and we see that the GM has increased considerably because the resonant peak is well below magnitude ; thus the system will be much better behaved. ii. The Bode plot of this system (ω b = 0 rad/sec = ω b =20

125 407 rad/sec) is : 20 G m =7.348 db (at rad/sec),pm = deg.(at rad/sec) 0-20 Phase (deg);m agnitude (db ) PM = 97.6 (ω = rad/sec) GM = 7.34 (ω = 20.0 rad/sec) and again, the GM is much improved and the resonant peak is signiþcantly reduced from magnitude. iii. By picking up τ p =, thebodeplotofthesystemwiththelow

126 408 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD pass Þlter is : 50 G m = db (at rad/sec),pm = deg.(at rad/sec) 0 Phase (deg);m agnitude (db ) PM = 92.9 (ω =0.083 rad/sec) GM = (ω = 8.62 rad/sec) which are healthy margins and the resonant peak is, again, well below magnitude.

127 409 iv. The Bode plot of the system with the given notch Þlter is : 20 G m = db (at rad/sec),pm = deg.(at rad/sec) Phase (deg);m agnitude (db ) PM = 97.6 (ω = rad/sec) GM = 55.3 (ω = 99.7 rad/sec) which are the healthiest margins of all the designs since the notch Þlter has essentially canceled the bending mode resonant peak. (b) Generally, the notch Þlter is very sensitive to where to place the notch zeros in order to reduce the lightly damped resonant peak. So if you want to use the notch Þlter, you must have a good estimation of the location of the bending mode poles and the poles must remain at that location for all aircraft conditions. On the other hand, the low pass Þlter is relatively robust to where to place its break point. Evaluation of the margins with the bending mode frequency lowered by 0% will show a drastic reduction in the margins for the notch Þlter and very little reduction for the low pass Þlter. Low Pass Filter Notch Filter GM (ω = 8.62 rad/sec) 55.3 (ω = 99.7 rad/sec) PM 92.9 (ω =0.083 rad/sec) 97.6 (ω = rad/sec) Closed-loop bending mode damping ratio ' 0.02 ' 0.04 Resonant peak

128 40 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD The magnitude plots of the closed-loop systems are : M agnitude (Low Pass Filter) Magnitude (N otch Filter) The closed-loop step responses are : 0.9 Unit Step R esponse Low Pass Filter N otch Filter 0.6 Amplitude Tim e (sec) (c) While increasing the natural damping of the system would be the best solution, it might be difficult and expensive to carry out. Likewise, increasing the frequency typically is expensive and makes the

129 4 Figure 6.06: Control system for Problem 59 structure heavier, not a good idea in an aircraft. Of the remaining two options, it is a better design to use a low pass Þlter because of its reduced sensivity to mismatches in the bending mode frequency. Therefore, the best recommendation would be to use the low pass Þlter. Problems and Solutions for Section A feedback control system is shown in Fig The closed-loop system is speciþed to have an overshoot of less than 30% to a step input. (a) Determine the corresponding PM speciþcation in the frequency domain and the corresponding closed-loop resonant peak value M r.(see Fig. 6.37) (b) From Bode plots of the system, determine the maximum value of K that satisþes the PM speciþcation. (c) Plot the data from the Bode plots (adjusted by the K obtained in part (b)) on a copy of the Nichols chart in Fig and determine the resonant peak magnitude M r. Compare that with the approximate value obtained in part (a). (d) Use the Nichols chart to determine the resonant peak frequency ω r and the closed-loop bandwidth. Solution : (a) From Fig : M p 0.3 = PM 40 o = M r.5 resonant peak : M r.5 (b) A sketch of the asymptotes of the open loop Bode shows that a PM of = 40 o is obtained when K =8. A Matlab plot of the Bode can be used to reþne this and yields for PM =40 o. K =7.8

130 42 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (c) The Nichols chart below shows that M r =.5 which agrees exactly with the prediction from Fig. 6.37: (d) The corresponding frequency where the curve is tangent to M r =.5 is: ω r =2.4 rad/sec as can be determined by noting the frequency from the Bode plot that corresponds to the point on the Nichols chart. The bandwidth ω BW is determined by where the curve crosses the closed-loop magnitude of 0.7 and noting the frequency from the Bode plot that corresponds to the point on the Nichols chart ω BW =3.94 rad/sec 60. The Nichols plot of an uncompensated and a compensated system are shown in Fig (a) What are the resonance peaks of each system? (b) What are the PM and GM of each system? (c) What are the bandwidths of each system?

131 Figure 6.07: Nichols plot for Problem 60 43

132 44 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (d) What type of compensation is used? Solution : (a) Resonant peak : Uncompensated system : Resonant peak =.5 (ω r = 50 rad/sec) Compensated system : Resonant peak =..05 (ω r = 20 rad/sec) (b) PM, GM : Uncompensated system : PM =42,GM= 0.2 =5 Compensated system : PM =64,GM= 0. =0 (c) Bandwidth : Uncompensated system : Bandwidth = 70 rad/sec Compensated system : Bandwidth = 30 rad/sec (d) Lag compensation is used, since the bandwidth is reduced.

133 45 6. Consider the system shown in Fig (a) Construct an inverse Nyquist plot of [Y (jω)/e(jω)]. (b) Show how the value of K for neutral stability can be read directly from the inverse Nyquist plot. (c) For K = 4, 2, and, determine the gain and phase margins. (d) Construct a root-locus plot for the system, and identify corresponding points in the two plots. To what damping ratios ζ do the GM and PM of part (c) correspond? Solution : (a) See the inverse Nyquist plot. (b) Let G(jω) = Y (jω) E(jω) The characteristic equation with s = jω : +K u G(jω) =0 From the inverse Nyquist plot, = G = K u K u = 4 = K u =4 (c) K GM PM = =2 8.3 =4 38.

134 46 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (d) K closed-loop poles ζ ±j ± j ± j An unstable plant has the transfer function Y (s) F (s) = s + (s ) 2. A simple control loop is to be closed around it, in the same manner as the block diagram in Fig (a) Construct an inverse Nyquist plot of Y/F. (b) Choose a value of K to provide a PM of 45. What is the corresponding GM?

135 47 (c) What can you infer from your plot about the stability of the system when K<0? (d) Construct a root-locus plot for the system, and identity corresponding points in the two plots. In this case, to what value of ζ does PM = 45 correspond? Solution : (a) The plots are : (b) From the inverse Nyquist plot, K =3.86 provides a phase margin of 45. Since K =2gives G(jω) = 80, GM = =0.58 Note that GM is less than, but the system with K =3.86 is stable. K =3.86, GM=0.58 (c) We can apply stability criteria to the inverse Nyquist plot as follows : Let N = Net number of clockwise encirclement of K P = Number of poles of G in RHP ( = Number of zeros of G in RHP) Z = Number of closed-loop system roots in RHP

136 48 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Then, K < 2 2 < K < K > K>2= N =0,P =0 = Z =0= Stable >K>2= N =2,P =0 = Z = 2 = Two unstable closed-loop roots K< = N =,P =0 = Z == Oneunstableclosed-looproot Then, we can infer from the inverse Nyquist plot the stability situation when K is negative. In summary, when K is negative, there are either one or two unstable roots, and the system is always unstable. (d) The stability situation seen in the root locus plot agrees with that obtained from the inverse Nyquist plot. They show : K>2 Stable <K<2 Two unstable closed-loop roots K< Oneunstableclosed-looproot For the phase margin 45, closed-loop roots = ±.999j ζ = Consider the system shown in Fig. 6.08(a). (a) Construct a Bode plot for the system. (b) Use your Bode plot to sketch an inverse Nyquist plot. (c) Consider closing a control loop around G(s), as shown in Fig. 6.08(b). Using the inverse Nyquist plot as a guide, read from your Bode plot thevaluesofgmandpmwhenk =0.7,.0,.4, and 2. What value of K yields PM = 30?

137 49 Figure 6.08: Control system for Problem 63 (d) Construct a root-locus plot, and label the same values of K on the locus. To what value of ζ does each pair of PM/GM values correspond? Compare the ζ vs PM with the rough approximation in Fig Solution : (a) The Þgure follows : Magnitude Phase (deg)

138 420 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (b) The Þgure follows : (c) K GM PM K = (ω =2.00) 54.7 (ω c =0.64) K = 4.00 (ω =2.00) 44. (ω c =0.85) K = (ω =2.00) 33. (ω c =.08) K = (ω =2.00) 2.4 (ω c =.36) For PM =30 K = (ω =2.00) 30.0 (ω c =.5) Magnitude 0 0 K=0.7 K= K=.4 K=.54 K=2 K= Phase (deg) deg PM =54.7 PM =44. PM =33. PM =30 PM =

139 42 (d) K closed-loop roots ζ K =0.7 K = K =.4 K = ± 0.82j ±.04j ±.25j ±.49j Problems and Solutions for Section Consider a system with the open-loop transfer function (loop gain) G(s) = s(s +)(s/0 + ). (a) Create the Bode plot for the system, and Þnd GM and PM.

140 422 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (b) Compute the sensitivity function and plot its magnitude frequency response. (c) Compute the Vector Margin (VM). Solution : (a) The Bode plot is : Magnitude Phase (deg) (b) Sensitivity function is : S(s) = = +G(s) + s(s +)( s 0 +) The magnitude frequency frequency response of this sensitivity func-

141 423 tion is : F re q u e n c y R e s p o n s e o f S e n s itiv ity F u n c tio n 0 0 M agnitude F re q u e n c y (ra d / s e c ) (c) Vector Margin is deþned as : VM = min ω s(jω) =.6 = Prove that the sensitivity function S(s) has magnitude greater than inside a circle with a radius of centered at the point. Whatdoes this imply about the shape of the Nyquist plot if closed-loop control is to outperform open-loop control at all frequencies? Solution : S(s) = +G(s) Inside the unit circle, +G(s) < which implies S(s) >. Outside the unit circle, +G(s) > which implies S(s) <. On the unit circle, +G(s) =whichmeans S(s) =. If the closed-loop control is going to outperform open-loop control then S(s) for all s. This means that the Nyquist plot must lie outside the circle of radius one centered at.

142 424 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.09: Control system constraints for Problem Consider the system in Fig with the plant transfer function G(s) = 0 s(s/0 + ). We wish to design a compensator D(s) that satisþes the following design speciþcations: (a) i. K v = 00, ii. PM 45, iii. sinusoidal inputs of up to rad/sec to be reproduced with 2% error, iv. sinusoidal inputs with a frequency of greater than 00 rad/sec to be attenuated at the output to 5% of their input value. (b) Create the Bode plot of G(s), choosing the open-loop gain so that K v = 00. (c) Show that a sufficient condition for meeting the speciþcation on sinusoidal inputs is that the magnitude plot lies outside the shaded regions in Fig Recall that Y R = KG +KG and E R = +KG. (d) Explain why introducing a lead network alone cannot meet the design speciþcations. (e) Explain why a lag network alone cannot meet the design speciþcations.

143 425 (f) Develop a full design using a lead-lag compensator that meets all the design speciþcations, without altering the previously chosen low frequency open-loop gain. Solution : (a) To satisfy the given velocity constant K v, K v = lim skg(s) =0K = 00 s 0 = K =0 (b) The Bode plot of G(s) with the open-loop gain K =0is: 0 2 Magnitude Phase (deg) (c) From the 3rd speciþcation, E R = +KG < 0.02 (2%) = KG > 49 (at ω < rad/sec) From the 4th speciþcation, Y R = KG +KG < 0.05 (5%) = KG < (at ω > 00 rad/sec) which agree with the Þgure.

144 426 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (d) A lead compensator may provide a sufficient PM, but it increases the gain at high frequency so that it violates the speciþcation above. (e) A lag compensator could satisfy the PM speciþcation by lowering the crossover frequency, but it would violate the low frequency speciþcation, W. (f) One possible lead-lag compensator is : s D(s) = s s s which meets all the speciþcation : K v = 00 PM = 47.7 (at ω c =2.9 rad/sec) KG = (at ω = rad/sec) > 49 KG = (at ω = 00 rad/sec) < The Bode plot of the compensated open-loop system D(s)G(s) is: 0 2 Magnitude Phase (deg) Problems and Solutions for Section Assume that the system G(s) = e T ds s +0,

146 428 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Magnitude 0 0 D a(s)g (s) w =8.6 w =. K G (s) D a(s)g (s) Phase (deg) -50 K G (s) PM =40 deg PM =40 deg -80 deg (b) Two lead sections : With b/a = 0, the lead compensator can add the maximum phase lead : φ max = sin a b + a b = 54.9 deg(atω = 0a rad/sec) By trial and error, one of the possible compensators is : (s + 70)2 K = 359, a=70= D b (s) = 359 (s + 700) 2 PM = 40 (at ω c =9.6 rad/sec)

147 429 The Bode plot is shown below. Magnitude 0 0 D b(s)g (s) w =8.6 w =9.57 K G (s) D b(s)g (s) Phase (deg) -50 K G (s) PM =40 deg -80 deg (c) The statement in the text is that it should be difficult to stabilize a system with time delay at crossover frequencies, ω c & 3/T d. This problem conþrms this limit, as the best crossover frequency achieved was ω c =9.6 rad/sec whereas 3/T d = 5 rad/sec. Since the bandwidth is approximately twice the crossover frequency, the limitations imposed on the bandwidth by the time delay is veriþed. 68. Determine the range of K for which the following systems are stable: (a) G(s) =K e 4s s (b) G(s) =K e s s(s+2) Solution : (a) G(jω) K =2.54, when G(jω) K = 80 range of stability : 0 <K< 2.54

148 430 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD 0 2 Magnitude G/K = Phase (deg) deg (b) G(jω) K =0.409 =, when G(jω) 2.45 K = 80 rangeofstability:0<k< Magnitude G/K =/ Phase (deg) deg

149 Phase (deg) In Chapter 5, we used various approximations for the time delay, one of which is the Þrst order Padé e T ds = H (s) = T ds/2 +T d s/2. Using frequency response methods, the exact time delay H 2 (s) =e T ds. can be used. Plot the phase of H (s) andh 2 (s) and discuss the implications. Solution : The approximation H (jω) andthetruephaseh 2 (jω) are compared in the plot below: 0 Time Delay Phase Lag H H Normalized Frequency, M T (rad) d H (jω) closely approximates the correct phase of the delay (phase of H 2 (s)) for ωt d. π and progressively worsens above that frequency. The 2 implication is that the H (s) approximation should not be trusted for crossover frequencies ω c & π. Instead, one should use the exact phase 2T d for the time delay given by H 2 (s). 70. Consider the heat exchanger of Example 2.7 with the open-loop transfer function e 5s G(s) = (0s +)(60s +). (a) Design a lead compensator that yields PM 45 and the maximum possible closed-loop bandwidth.

150 Phase (deg) Magnitude 432 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD (b) Design a PI compensator that yields PM 45 and the maximum possible closed-loop bandwidth. Solution : (a) First, make sure that the phase calculation includes the time delay lag of -T d ω = 5ω. A convenient placement of the lead zero is at ω =0. because that will preserve the - slope until the lead pole. We then raise the gain until the speciþed PM is obtained in order to maximize the crossover frequency. The resulting lead compensator, D(s) = 90(s +0.) (s +) yields PM =46 asseenbythebodebelow. Bode Diagrams with Lead Compensation Also note that the crossover frequency, ω c =0.5 rad/sec, which can be read approximately from the plot above, and veriþed by using the margin command in Matlab with the phase adjusted by the time delay lag. (b) The brealpoint of the PI compensator needs to be kept well below 0. in order to maintain a positive phase margin at as high a crossover frequency as possible. In Table 4., Zeigler-Nichols suggest a breakpoint at ω =/7, so we will select a PI of the form : µ D(s) =K + 20s

151 Phase (deg) Magnitude 433 and select the gain so that the PM speciþcation is met. For K =0.55 the phase margin is 46 as shown by the Bode below: 0 Bode Diagrams with PI compensation Note with this compensation that ω c =0.02 rad/sec, which is considerably lower than that yielded by the lead compensation. Problems and Solutions for Section The Bode plot in Fig. 6.0 is for a transfer function of the form G(s) = Ks ( + T s)( + T 2 s) 2, where K, T,andT 2 are positive constants. Determine values for these three constants from the Bode plot. Solution : By drawing asymptotes to a) the low frequency + slope portion, b) the high frequency -2 slope portion, then c) drawing a zero slope asymptote that is tangent to the curve, we see that the break from the + portion to thezeroslopeportionisat5rad/secandthebreakfromthezeroslopeto

152 434 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD Figure 6.0: Bode plot for Problem 7

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