Problems 213 4-3. Reduce the block diagram shown in Fig. 4P-3 and find the Y/X. H2 + + G1 G2 G3 + + X + Y – H1 H3 Figure 4P-3 4-4. Reduce the block diagram shown in Fig. 4P-4 to unity feedback form and find the Y/X. + + +– H3 Y X G1 G2 G3 – – H2 H1 Figure 4P-4 4-5. The aircraft turboprop engine shown in Fig. 4P-5a is controlled by a closed-loop system with block diagram shown in Fig. 4P-5b. The engine is modeled as a multivariable system with input vector E(s), which contains the fuel rate and propeller blade angle, and output vector Y(s), consisting of the engine speed and turbine-inlet temperature. The transfer function matrices are given as 2 10 G(s) s(s + 2) 1 (s) 1 0 = 5 s +1 H = 0 1 s Find the closed-loop transfer function matrix [I + G(s)H(s)]−1G(s).
214 Chapter 4. Block Diagrams and Signal-Flow Graphs COMBUSTION COMPRESSOR TURBINE Y(s) PROPELLER E(s) G(s) R(s) + – H(s) Figure 4P-5 4-6. Use MATLAB to solve Prob. 4-5. 4-7. The block diagram of the position-control system of an electronic word processor is shown in Fig. 4P-7. (a) Find the loop transfer function Θo(s)/Θe (s) (the outer feedback path is open). (b) Find the closed-loop transfer function Θo(s)/Θr (s). Kb Gear ratio Sensor Preamp – K K1 + θr θe Ks E +– Ea 1 Ia Ki Tm 1 ωm 1 θm N θo +– +– Ra + Las Jts + Bt s Power amplifier Current feedback K2 Tachometer feedback Kt Figure 4P-7 4-8. The block diagram of a feedback control system is shown in Fig. 4P-8. Find the following trans- fer functions: (a) Y(s) R(s) N=0 (b) Y(s) E(s) N=0 (c) Y(s) N(s) R=0 (d) Find the output Y(s) when R(s) and N(s) are applied simultaneously.
Problems 215 2 N(s) + + + R(s) E(s) s + 2 10 Y(s) s(s + 1) + + –– 0.5s Figure 4P-8 4-9. The block diagram of a feedback control system is shown in Fig. 4P-9. (a) Apply the SFG gain formula directly to the block diagram to find the transfer functions: Y(s) Y(s) R(s) N=0 N(s) R=0 Express Y(s) in terms of R(s) and N(s) when both inputs are applied simultaneously. (b) Find the desired relation among the ttrhaendsfiestrufrubnacntcioenssigGn1a(ls )N, (Gs)2 (ast),aGll.3 (s ), (s ), (s ), and (s ) so that the output Y(s) is not affected by G4 H1 H 2 G4(s) R(s) + G1(s) + + G3(s) + + – H1(s) – G2(s) H2(s) Y(s) – + + N(s) Figure 4P-9 4-10. Figure 4P-10 shows the block diagram of the antenna control system of the solar-collector field shown in Fig. 1-5. The signal N(s) denotes the wind gust disturbance acted on the antenna. The feedforward Ytr(asn)s/fNer(fsu)nRc=t0i.oDneGtedr(ms)inise used to eliminate the effect of N(s) on the output Y(s). Find the transfer function the expression of Gd (s) so that the effect of N(s) is entirely eliminated. N(s) Gd(s) – E(s) s + 5 10 + Y(s) R(s) s + 10 s(s + 5) + + – Figure 4P-10
216 Chapter 4. Block Diagrams and Signal-Flow Graphs 4-11. Figure 4P-11 shows the block diagram of a dc-motor control system. The signal N(s) denotes the frictional torque at the motor shaft. (a) Find the transfer function H(s) so that the output Y(s) is not affected by the disturbance torque N(s). (b) With H(s) as determined in part (a), find the value of K so that the steady-state value of e(t) is equal to 0.1 when the input is a unit-ramp function, r (t ) (t ), R(s) 1/s 2 , and N(s) 0. Apply the final-value theorem. = tus = = N(s) R(s) + E (s) G(s) + Y(s) __ + H(s) G(s) = K(s + 3) 2) s(s + 1)(s + Figure 4P-11 4-12. The block diagram of an electric train control is shown in Fig. 4P-12. The system parameters and variables are er(t) = voltage representing the desired train speed, v(t) = speed of train, ft/sec M =Mass of train = 30,000 lb/sec K = amplifier gain Kt = gain of speed indicator = 0.15 V/ft/sec er(t) e(t) AMPLIFIER ec(t) CONTROLLER f(t) 1 v(t) + K Gc(s) Ms – Train speed SPEED DETECTOR Kt Figure 4P-12 To determine the transfer function of the controller, we apply a step function of 1 V to the input of the controller, that is, (t ) (t ). The output of the controller is measured and described by the fol- lowing equation: ec = us f (t) = 100(1− 0.3e−6t − 0.7e−10t )us (t) (a) Find the transfer function Gc(s) of the controller. (b) Derive the forward-path transfer function V(s)/E(s) of the system. The feedback path is opened in this case. (c) Derive the closed-loop transfer function V(s)/Er (s) of the system. (d) Assuming that K is set at a value so that the train will not run away (unstable), find the steady- state speed of the train in feet per second when the input is er (t) = us(t)V. 4-13. Use MATLAB to solve Prob. 4-12. 4-14. Repeat Prob. 4-12 when the output of the controller is measured and described by the follow- ing expression: f (t) =100(1− 0.3e−6(t−0.5) )us (t − 0.5) when a step input of 1 V is applied to the controller.
Problems 217 4-15. Use MATLAB to solve Prob. 4-14. 4-16. A linear time-invariant multivariable system with inputs r1 (t ) and r2 (t ) and outputs y1 (t ) and y 2 (t ) is described by the following set of differential equations. d 2 y1(t ) + 2 dy1(t ) + 3 y 2 (t ) = r1(t ) + r2 (t ) dt 2 dt d2 y2(t ) dy1(t ) dr1(t ) dt 2 + 3 dt + y1(t ) − y2 (t ) = r2 (t ) + dt Find the following transfer functions: Y1(s) Y2 (s ) Y1(s) Y2 (s ) R1(s) R2=0 R1(s) R2=0 R2 (s) R1=0 R2 (s) R1=0 PROBLEMS FOR SEC. 4-2 4-17. Find the state-flow diagram for the system shown in Fig. 4P-4. 4-18. Draw a signal-flow diagram for the system with the state-space of −5 −6 3 0.5 0 X 1 0 −1 0 0.5 = −0.5 1.5 0.5 X + 0.5 0.5 U Z = 0.5 0.5 0 X 0.5 0 0.5 4-19. Find the state-space of a system with the following transfer function: G(s ) = s 2 B1s + B0s s + A1s + A0 4-20. Draw signal-flow graphs for the following sets of algebraic equations. These equations should first be arranged in the form of cause-and-effect relations before SFGs can be drawn. Show that there are many possible SFGs for each set of equations. (a) x1 = −x2 − 3x3 + 3 (b) 2x1 + 3x2 + x3 = −1 x2 = 5x1 − 2x2 + x3 x1 − 2x2 − x3 = 1 x3 = 4x1 + x2 − 5x3 + 5 3x2 + x3 = 0 4-21. The block diagram of a control system is shown in Fig. 4P-21. (a) Draw an equivalent SFG for the system. (b) Find the following transfer functions by applying the gain formula of the SFG directly to the block diagram. Y(s) Y(s) E(s) E(s) R(s) N=0 N (s) R=0 R(s) N=0 N (s) R=0 (c) Compare the answers by applying the gain formula to the equivalent SFG. G1(s) N(s) R(s) + E(s) G2(s) + ++ + G3(s) Y(s) –– H1(s) Figure 4P-21
218 Chapter 4. Block Diagrams and Signal-Flow Graphs 4-22. Apply the gain formula to the SFGs shown in Fig. 4P-22 to find the following transfer functions: Y5 Y4 Y2 Y5 . Y1 Y1 Y1 Y2 G4 1 G1 G2 G3 1 Y1 Y2 Y3 Y4 Y5 Y5 –H1 –H2 –H3 (a) –H4 G4 1 G1 G2 G3 1 Y1 Y2 Y3 Y4 Y5 Y5 –H1 –H2 –H3 (b) G4 1 G1 G2 G3 1 Y1 Y2 Y3 Y4 Y5 Y5 –H1 –H2 (c) –H3 G4 1 G1 G2 G3 1 Y5 Y1 Y2 Y3 Y4 Y5 –H1 –H2 –H3 (d) –H4 G4 Y6 G5 1 G1 G2 G3 1 Y1 Y2 Y3 Y4 Y5 Y5 –H1 –H2 –H3 (e) Figure 4P-22
Problems 219 4-23. Find the transfer functions Y7 /Y1 and Y2 /Y1 of the SFGs shown in Fig. 4P-23. G6 –H6 1 G1 G2 G3 G4 G5 1 Y1 Y2 Y3 Y4 Y5 Y7 Y6 Y7 –H1 –H2 –H3 –H4 –H5 (a) G6 1 G1 G2 G3 G4 G5 1 Y7 Y1 Y2 Y3 Y4 Y5 Y6 Y7 –H1 –H2 –H3 (b) –H4 Figure 4P-23 4-24. Signal-flow graphs may be used to solve a variety of electric network problems. Shown in baFnfaiefgntew.cc4teoePrvd-ko2b.l4tTyaihesgidets(h.tiTen).hveTeoqoluovsibevojsaelvalceetcnitovthemceiibrspcirtnuooaibtftilioneofmdna,tnohifteenilsevocabdtleruesoetanntoiodcf ftclihoriresoctpcuwoietnrq.istTuteahaanteitsoveknotslsot.oafTgcthheaeausntosetuch-oraecnneosdteur-dteu(pftcuf)tetcravetnopeSlrqteFausGgaeetniueotssoni(anst)gdfoiitssrhtntuehosree-t equations. Find the gain eo /ed with all other inputs set to zero. For ed not to affect eo, set eo /ed to zero. Rs e2 R2 i2 e3 i3 + e1 _ + _ ed(t) + ke1(t) + R1 R3 _ es(t)_ + is(t) R4 R5 eo(t) _ Figure 4P-24
220 Chapter 4. Block Diagrams and Signal-Flow Graphs 4-25. Show that the two systems shown in Fig. 4P-25a and b are equivalent. –GH 1G 1 Y1 Y2 Y3 Y3 (a) 1G 1 Y1 Y2 Y3 Y3 –H (b) Figure 4P-25 4-26. Show that the two systems shown in Fig. 4P-26a and b are not equivalent. 1 G1 1 G2 1 G3 1 Yi Yo –H1 –H2 –H3 (a) 1 G1 G2 G3 1 Yi Yo –H1 –H2 –H3 Figure 4P-26 (b)
4-27. Find the following transfer functions for the SFG shown in Fig. 4P-27. Problems 221 Y6 Y6 Y6 Y Y1 Y7=0 7 Y1=0 Y7 G5 1 1 G1 G2 G3 G4 1 Y1 Y2 Y3 Y4 Y5 –H2 Y6 Y6 –H1 –H3 (a) Y7 G5 1 1 G1 G2 G3 G4 1 Y1 Y2 Y3 Y4 –H2 Y5 Y6 –H1 –H4 –H3 (b) Figure 4P-27 4-28. Find the following transfer functions for the SFG shown in Fig. 4P-28. Comment on why the results for parts (c) and (d) are not the same. (a) Y7 Y1 Y8=0 (b) Y7 Y8 Y1=0 (c) Y7 Y4 Y8=0 (d) Y7 Y4 Y1=0 Y8 G6 1 1 G1 G2 G3 G4 G5 1 Y1 Y2 Y3 Y4 Y5 Y6 –H2 Y7 Y7 –H1 –H3 Figure 4P-28
222 Chapter 4. Block Diagrams and Signal-Flow Graphs 4-29. The coupling between the signals of the turboprop engine shown in Fig. 4P-4a is shown in Fig. 4P-29. The signals are defined as R1(s) = fuel rate R2(s) = propeller blade angle Y1(s) = engine speed Y2(s) = turbine inlet temperature (a) Draw an equivalent SFG for the system. (b) Find the Δ of the system using the SFG gain formula. (c) Find the following transfer functions: Y1(s) Y1(s) Y2 (s ) Y2 (s ) R1(s) R2=0 R2 (s) R1=0 R1(s) R2=0 R2 (s) R1=0 (d) Express the transfer functions in matrix form, Y(s) = G(s)R(s). R1(s) – G(s) Y1(s) + + + G(s) Y2(s) R2(s) + + Figure 4P-29 4-30. Figure 4P-30 shows the block diagram of a control system with conditional feedback. The transfer function (s) denotes the controlled process, and (s) and H(s) are the controller transfer functions. Gp Gc (a) Derive the transfer functions Y (s)/R(s)|N=0 and Y (s)/N(s)|R=0. Find Y(s)/R(s)| N = 0 when Gc (s) = Gp(s). (b) Let G p (s ) = Gc (s ) = (s + 100 + 5) 1)(s Find the output response y(t) when N(s) = 0 and r(t) = uHs ((ts)). among the following choices such that w(ocnh)e enWannist(wht )eG=r.)pu(ss()t )aannddGrc((ts))=a0s, given in part (b), select may be more than the steady-state value of y(t) is equal to zero. (There H (s) = 10 H(s) = (s + 10 + 2) s(s +1) 1)(s H (s) = 10(s +1) H(s) = K (n = positive integer) Select n. s+2 sn Keep in mind that the poles of the closed-loop transfer function must all be in the left-half s-plane for the final-value theorem to be valid.
Problems 223 N(s) R(s) + Gp(s) Y(s) Gc(s) + + + H(s) +– Figure 4P-30 4-31. Use MATLAB to solve Prob. 4-30. PROBLEMS FOR SEC. 4-3 4-32. Consider the following differential equations of a system: dx1(t ) = −2x1 (t ) + 3x 2 (t ) dt dx 2 (t ) dt = −5x1 (t ) − 5x 2 (t ) + 2r (t ) (a) Draw a state diagram for the following state equations. (b) Find the characteristic equation of the system. (c) Find the transfer functions X1(s)/R(s) and X2(s)/R(s). 4-33. The differential equation of a linear system is d 3 y(t ) + 5 d 2 y(t ) + 6 dy(t ) + 10 y(t ) = r (t ) dt 3 dt 2 dt where y(t) is the output, and r(t) is the input. (a) Draw a state diagram for the system. (b) Write the state equation from the state diagram. Define the state variables from right to left in ascending order. (c) Find the characteristic equation and its roots. Use any computer program to find the roots. (d) Find the transfer function Y(s)/R(s). (e) Perform a partial-fraction expansion of Y(s)/R(s). (f) Find the output y(t) for t ≥ 0 when r(t) = us (t). (g) Find the final value of y(t) by using the final-value theorem. 4-34. Consider the differential equation given in Prob. 4-33. Use MATLAB to (a) Perform a partial-fraction expansion of Y(s)/R(s). (b) Find the Laplace transform of the system. (c) Find the output y(t) for t ≥ 0 when r(t) = us (t). (d) Plot the step response of the system. (e) Verify the final value that you obtained in Prob. 4-33 part (g).
224 Chapter 4. Block Diagrams and Signal-Flow Graphs 4-35. Repeat Prob. 4-33 for the following differential equation: d 4 y(t ) + 4 d 3 y(t ) + 3 d 2 y(t ) + 5 dy(t ) + y(t ) = r(t ) dt 4 dt 3 dt 2 dt 4-36. Repeat Prob. 4-34 for the differential equation given in Prob. 4-35. 4-37. The block diagram of a feedback control system is shown in Fig. 4P-37. (a) Derive the following transfer functions: Y(s) Y(s) E(s) R(s) N=0 N (s) R=0 R(s) N=0 (b) The controller with the transfer ftuontacltlyioinndGe4p(se)nidsefnotrotfhNe r(es)d.uction of the effect of the noise N(s). Find G4(s) so that the output Y(s) is (c) Find the characteristic equation and its roots when G4(s) is as determined in part (b). (d) F ind the steady-state value of e(t) when the input is a unit-step function. Set N(s) = 0. (e) Find y(t) for t ≥ 0 when the input is a unit-step function. Use G4(s) as determined in part (b). N(s) G4(s) R(s) + – + Y(s) – + E(s) 100 s + 1 10 – – s+2 s(s + 20) + Figure 4P-37 4-38. Use MATLAB to solve Prob. 4-37. ADDITIONAL PROBLEMS 4-39. Assuming P1 = 2S6 + 9S5 +15S4 + 25S3 + 25S2 +14s + 6 P2 = S6 + 8S5 + 23S4 + 36S3 + 38S2 + 28s +16 (a) Use MATLAB to find roots of P1 and P2. (b) Use MATLAB to calculate P3 = P2 − P1, P4 = P2 + P1, and P5 = (P1 − P2 )∗ P1. 4-40. Use MATLAB to calculate the polynomial. (a) P6 = (s +1)(s2 + 2)(s + 3)(2s2 + s +1) (b) P7 = (s2 +1)(s + 2)(s + 4)(s2 + 2s +1)
Problems 225 4-41. Use MATLAB to perform partial-fraction expansion to the following functions: (a) G1(s) = (s +1)(s2 + 2)(s + 4)(s +10) s(s + 2)(s2 + 2s + 5)(2s2 + s + 4) (b) G2(s) = 4s6 + 28s5 s3 +12s2 + 47s + 60 + 62s +12 + 83s4 +135s3 +126s2 4-42. Use MATLAB to calculate unity feedback closed loop transfer function for Prob. 4-41. 4-43. Use MATLAB to calculate (a) G3(s) = G1(s) + G2(s) (b) G4(s) = G1(s) − G2(s) (c) G5 (s ) = G4 (s) G3 (s) (d) G6(s) = G4 (s ) G1 (s ) ∗ G2 (s )
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