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engineering
modern control systems

Modern Control Systems 14th Global Edition Richard Dorf, Robert Bishop - Solutions

P9.3 (a) Find a suitable contour Γs in the s-plane that can be used to determine whether all roots of the characteristic equation have damping ratios greater than ζ1.(b) Find a suitable contour Γs in the s-plane that can be used to determine whether all the roots of the characteristic equation
P9.2 Sketch the Nyquist plots of the following loop transfer functions L s 1 1 ( ) = Gc ( )s G1( )s , and determine whether the system is stable by applying the Nyquist criterion: (a) L(s) =G(s)G(s) = (b) L(s) G(s)G(s) = K s($+2s+5)* K(s+2) s (s+5) If the system is stable, find the maximum value
P9.1 For the Nyquist plots of Problem P8.1, use the Nyquist criterion to ascertain the stability of the various systems. In each case, specify the values of N, P, and Z.
E9.33 Consider the system shown in Figure E9.33.Compute the loop transfer function L s( ), and sketch the Bode plot. Determine the phase margin and gain margin when the controller gain K = 5. FIGURE E9.33 Nonunity feedback system with proportional controller K. Controller R(s) K Process 4 Y(s)
E9.32 Consider the system described in state variable form by where A x(t) = Ax(t) + Bu(t) y(t)=Cx(t) 0 B 3.2 | C=[20] Compute the phase margin.
E9.31 A closed-loop feedback system is shown in Figure E9.31. Sketch the Bode plot, and determine the phase margin. R(3) I 2s(s+2) + FIGURE E9.31 Nonunity feedback system. Y(s)
E9.30 A system is represented in state variable form x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t), where -3 -2 A B= 10 C=[0 501], D = [0]. Sketch the Bode plot.
E9.29 A loop transfer function is L(s) = G(s)G(s) = 1 5+2 Using the contour in the s-plane shown in Figure E9.29, determine the corresponding contour in the F(s)-plane (B-1+ j). ja j2 B C D -1 A E 0 1 H G FIGURE E9.29 Contour in the s-plane.
E9.28 A unity feedback system has the loop transfer function 10 L(s) = G(s)G(s) = s(s +0.8) (a) Determine the phase margin for the system. (b) Use the phase margin to estimate the damping ratio, and predict the percent overshoot. (c) Calculate the actual response for this system, and compare the
E9.27 A unity feedback system has a loop transfer functionG M. . = 8 dB. (b) Determine the value of gain K and cross over frequency for marginal stability. K L(s) =G(s)G(s) = s(s+3) (a) Determine the maximum gain K for which the phase margin is P.M. 30, and the gain margin is
E9.26 For the system of E9.25, determine Mp r ω, ω , andωB for the closed-loop frequency response by using the Nichols chart. Magnitude M M 1 -19 K K L(s)-plane Frequency (rad/s) FIGURE 9.25 Closed-loop frequency response of T(jw) = Gc(jw)G(jw)/(1+G(jw)G(jw)). Note that K > K.
E9.25 A unity feedback system has a loop transfer function 11.7 L(s) =G(s)G(s) = s(1+0.05s) (1+0.1s) Determine the phase margin and the crossover frequency.
E9.24 A unity feedback system has a loop transfer function Imaginary Axis K L(s) = Ge(s)G(s) = (-1+TS) where K = 0.4 and 7 = 1. The Nyquist plot for G(jw)G(jw) is shown in Figure E9.24. Determine whether the system is stable by using the Nyquist criterion. Nyquist Diagram 0.5 0.4 0.3 0.2- 0.1 0
E9.23 Consider again the system of E9.21 when K = 100.Determine the closed-loop system bandwidth, resonant frequency, and Mpω.Data from in E9.21 A unity feedback control system has a loop transfer function K L(s)=G(s)G(s) = s(s+2)(s+50) Determine the phase margin, the crossover frequency, and
E9.22 A unity feedback system has a loop transfer function K L(s) = G(s)G(s) = (s+ 2) (a) Using a Bode plot for K = 40, determine the system phase margin. (b) Select a gain K so that the phase margin is P.M. 55.
E9.21 A unity feedback control system has a loop transfer function K L(s)=G(s)G(s) = s(s+2)(s+50) Determine the phase margin, the crossover frequency, and the gain margin when K = 1300.
9.20 Consider a simple model of an automobile driver following another car on the highway at high speed.The model shown in Figure E9.20 incorporates the driver’s reaction time, T. One driver has T = 1 s, and another has T = 1.5 s. Determine the time responsey t( ) of the system for both
E9.19 A unity feedback system with G s c( ) = K has e-0.12s G(s) = (s+15) Select a gain K so that the phase margin of the system is P.M. = 40. Determine the gain margin for the se- lected gain, K.
E9.18 An actuator for a disk drive uses a shock mount to absorb vibrational energy at approximately 60 Hz 14]. The Bode plot of the loop transfer function of the control system is shown in Figure E9.18. (a) Find the expected percent overshoot for a step input for the closed-loop system, (b)
E9.17 A unity feedback system has a loop transfer function(a) Obtain the Bode plot, and (b) determine the gainK required to obtain a phase margin of P M. . = 56°.What is the steady-state error for a ramp input for the gain of part (b)? Ge(s)G(s) = K(s+4) s (s + 3s+20)
E9.16 The pure time delay e−sT may be approximated by a transfer function as e-ST 1-Ts/2 1+ Ts/2 Obtain the Bode plot for the actual transfer function and the approximation for 7 = 0.05 for 0
E9.15 Consider a unity feedback system with the loop transfer function 100 L(s) = G(s)G(s) = s(s+20) Find the bandwidth of the closed-loop system.
E9.14 A Nichols chart is given in Figure E9.14 for a system with G j c( ) ω ω G j ( ). Using the following table, find(a) the peak resonance Mpω in dB; (b) the resonant frequency ωr; (c) the 3-dB bandwidth; and (d) the phase margin of the system. Loop gain G,G, in decibels -6 30 24 18 12 9
E9.13 A unity feedback system has a loop transfer function resonant frequency of this system. (c) Use these frequency measures to estimate the overshoot of the system to a step response. 200 Ge(s)G(s) = s(s+4) (a) Find the maximum magnitude of the closed-loop frequency response. (b) Find the
E9.12 A unity feedback system with the loop transfer function Ge(s)G(s) = K s(1718)(1 + T28)' where = 0.01 and T2 = 0.3 s. (a) Select a gain T1 K so that the steady-state error for a ramp input is 20% of the magnitude of the ramp function A, where r(t) = A(t), t0. (b) Obtain the Bode plot of loop
E9.11 Consider a unity feedback system with the loop transfer function L(s) = G(s)G(s) = 5(1+0.5s) s(1+2s)(2+0.25s +0.05s) (a) Obtain the Bode plot. (b) Find the gain margin and the phase margin.
E9.10 Consider a system with the loop transfer function L(s)=G(s)G(s) = 250(s+5) s(s+ 0.25)(s2+12.5s +120) Obtain the Bode plot, and show that the P.M. = 16.9 and that the G.M. = 9.63 dB. Also, show that the bandwidth of the closed-loop system is WB = 4.57 rad/s.
E9.9 Consider a unity feedlack system with loop transfer function L(s) = G(s)G(s) = 10 s(s+11s+10) Compute the phase margin and gain margin.
E9.8 Consider a unity feedback with the loop transfer function K L(s) = Ge(s)G(s) = s(s+2)(s+6) (a) For K=27, show that the gain margin is G.M. = 11 dB. (b) To achieve a gain margin G.M. = 28 dB, deter- mine the value of the gain K.
E9.7 A unity feedback system has a loop transfer function K L(s) = Ge(s)G(s) = s-5 Determine the range of K for which the system is stable using the Nyquist plot.
E9.6 A system has a loop transfer function L(s) = G(s)G(s) = K(s+10) s(s+6)(s+15) Determine the range of K for closed-loop stability. Find the gain margin and phase margin of the system with K = 40.
E9.5 An integrated CMOS digital circuit can be represented by the Bode plot shown in Figure E9.5. (a) Find the gain and phase margins of the circuit. (b) Estimate how much we would need to reduce the system gain (dB) to obtain a phase margin of P M. . = 60°. (a) Phase (deg) Magnitude (dB) 50 40
E9.4 Consider a system with a loop transfer function 100 L(s) = G(s)G(s) = s(s+10) We wish to obtain a resonant peak Mpw = 3.0 dB for the closed-loop system. The peak occurs between 6 and 9 rad/s and is only 1.25 dB. Plot the Nichols chart for the range of frequency from 6 to 15 rad/s. Show that
E9.3 An integrated circuit is available to serve as a feedback system to regulate the output voltage of a power supply. The Bode plot of the loop transfer function is shown in Figure E9.3 Estimate the phase margin of the regulator. Magnitude (dB) 80 60 40 40 20 20 Phase 0 Magnitude 0 -90 -180 -20
E9.2 A system has the loop transfer function = L(s) G(s)G(s) = K(1+s/20) s(1+s/8)(1+s+10)' where K 4. Show that the system crossover fre- quency is w=3.51 rad/s and that the phase margin is P.M. = 56.9.
E9.1 A system has the open-loop transfer function L(s) = G(s)G(s) = 3(1+5s) s(4+s)(1+2s+2s2) Obtain the Bode plot. Show that the phase margin is P.M. = 20.1 and that the gain margin is G.M. 6.61 dB.
15. Consider a control system with unity feedback as in Figure 9.69 with loop transfer function L(s) = G(s)G(s) = (s+4) s(s+1)(s+5) The gain and phase margin are: a. G.M. = dB, P.M. = 58.1 b. G.M. = 20.4 dB, P.M. = 47.3 c. G.M. = 6.6 dB, P.M. = 60.4 d. Closed-loop system is unstable
14. A feedback model of human reaction time used in analysis of vehicle control can use the block diagram model in Figure 9.69 with G(s)=es and G(s): s(0.2s+1) A typical driver has a reaction time of T = 0.3 s. Determine the bandwidth of the closed- loop system. a. wb = 0.5 rad/s b. wb = = 10.6
13. Consider the control system in Figure 9.69, where the loop transfer function is L(s) = G(s)G(s) = s(s+1) The value of the resonant peak, Mp and the damping factor, (, for the closed-loop system are: a. Mp=0.37, (= 0.707 b. Mp = 1.15, C = 0.5 Pw == c. Mp=2.55, (= 0.5 d. Mp = 0.55, C = 0.25
12. Consider the feedback system in Figure 9.69, where AI Edit Wipe Off G(s) =K_and_G(s) = $+5 Notice that the plant contains a time-delay of T = 0.2 seconds. Determine the gain K such that the phase margin of the system is P.M. = 50. What is the gain margin for the same gain K? a. K=8.36, G.M. =
11. Consider the closed-loop system in Figure 9.69, where the loop transfer function is L(s) = Ge(s)G(s) = K(s+4) $2 Determine the value of the gain K such that the phase margin is P.M. = a. K = 1.64 b. K = 2.15 c. K = 2.63 d. Closed-loop system is unstable for all K > 0 40%.
10. Determine whether the closed-loop system in Figure 9.69 is stable or not, given the loop transfer functionIn addition, if the closed-loop system is stable, compute the gain and phase margins.a. Stable, G M. . = 24 dB, P M. . = 2.5°b. Stable, G M. . = 3 dB, P M. . = 24°c. Stable, G M. . =
9. Using the value of K in Problem 8, compute the gain and phase margins when Td = 0.2.a. G M. . = 14 dB, P M. . = 27°b. G M. . = 20 dB, P M. . = 64.9°c. G M. . = ∞ dB, P M. . = 60°d. Closed-loop system is unstable For Problems 8 and 9, consider the block diagram in Figure 9.69 where
8. When Td = 0, the PD controller reduces to a proportional controller, G s c( ) = K. In this case, use the Nyquist plot to determine the limiting value of K for closed-loop stability.a. K = 0.5b. K = 1.6c. K = 2.4d. K = 4.3 For Problems 8 and 9, consider the block diagram in Figure 9.69
7. Consider the block diagram in Figure 9.69. The plant transfer function isUtilize the Nyquist stability criterion to characterize the stability of the closed-loop system.a. The closed-loop system is stable.b. The closed-loop system is unstable.c. The closed-loop system is marginally
6. Consider the closed-loop system in Figure 9.69 whereThe crossover frequency and the phase margin are:a. ω = 2.0 rad s, P M. . = 37.2°b. ω = 2.6 rad s, P M. . = 54.9°c. ω = 5.3 rad s, P M. . = 68.1°d. ω = 10.7 rad s, P M. . = 47.9° Controller Process R(s) G(s) G(s) Y(s) FIGURE
5. The phase margin of a second-order system (with no zeros) is a function of both the damping ratio ζ and the natural frequency, ωn.True or False
4. A Nichols chart displays curves describing the relationship between the open-loop and closed-loop frequency responses.True or False
3. The gain and phase margin are readily evaluated on either a Bode plot or a Nyquist plot.True or False
2. A conformal mapping is a contour mapping that retains the angles on the s-plane on the transformed F s( )-plane.True or False
1. The gain margin of a system is the increase in the system gain when the phase is −180° that will result in a marginally stable system.True or False
CP8.9 Design a filter, G s( ), with the following frequency response:1. For ω < 1 rad s, the magnitude 20 log10 G j ( ) ω < 0 dB2. For 1 1000 < < ω rad ,s the magnitude 20 log10G j ( ) ω ≥ 0 dB3. Forω > 1000 rad ,s the magnitude 20 log10G j ( ) ω < 0 dB Try to maximize the peak
CP8.8 pendulum on a moving base, as shown in Figure Consider the problem of controlling an inverted CP8.8(a). The transfer function of the system isThe design objective is to balance the pendulum(i.e., θ( )t ≈ 0 ) in the presence of disturbance inputs.A block diagram representation of the
CP8.7 A unity feedback system has the loop transfer functionGenerate a plot of the bandwidth versus the parameter p as 0.1 5. p L(s) = G(s)G(s) = P 2 (s+P)
CP8.6 Consider the feedback system in Figure CP8.6.Obtain the Bode plots of the loop transfer function and the closed-loop transfer function using an m-file. 40 R(s) Y(s) s+2 s+2s+40 FIGURE CP8.6 Closed-loop feedback system.
CP8.5 A block plot of a second-order system is shown in Figure CP8.5. (a) Determine the resonant peak Mpω the resonant frequency ωr, and the bandwidth ωB, of the system from the closed-loop Bode plot. Generate the Bode plot with an m-file for ω = 0.1 to ω = 1,000 rad s using the logspace
CP8.4 A unity negative feedback system has the loop transfer functionDetermine the closed-loop system bandwidth. Using the bode function, obtain the Bode plot and label the plot with the bandwidth. L(s) = G(s)G(s) = 150 s(s+10)
CP8.3 For each of the following transfer functions, sketch the Bode plot and determine the crossover frequency: (a) G(s) = 2500 (s+10)(s+100)
CP8.2 For the following transfer functions, sketch the Bode plots, then verify with the bode function: (a) G(s)= 1000 (s +10) (s +100)
CP8.1 Consider the closed-loop transfer functionDevelop an m-file to obtain the Bode plot, and verify that the resonant frequency is 5 rad/s and that the peak magnitude Mpω is 14 dB. T(s)= 25 +s+25
DP8.7 Consider the system of Figure DP8.7. Consider the controller to be similar to a proportional plus derivative (PD) given byDesign the PD controller gains to achieve (a) a velocity constant Kv ≥ 1 , (b) a phase margin ofP M. . ≥ 60 , and (c) a bandwidth ωb ≥ 2.0. Plot the response of
DP8.6 A single-input, single-output system is described by(a) Determine p and K such that the unit step response exhibits a zero steady-state error and the percent overshoot meets the requirement P O. . ≤ 5%.(b) For the values of p and K determined in part (a), determine the system damping
DP8.5 Consider the control system depicted in Figure DP8.5(a) where the plant is a “black box” for which little is known in the way of mathematical models. The only information available on the plant is the frequency response shown in Figure DP8.5(b).Design a controller G s c ( ) to meet the
DP8.4 Anesthesia can be administered automatically by a control system. To ensure adequate operating conditions for the surgeon, muscle relaxant drugs, which block involuntary muscle movements, are administered.A conventional method used by anesthesiologists for muscle relaxant administration is to
DP8.3 A table is used to position vials under a dispenser head, as shown in Figure DP8.3(a). The objective is speed, accuracy, and smooth motion in order to eliminate spilling. The position control system is shown in Figure DP8.3(b). Determine a K such that the bandwidth is maximized while
DP8.2 The unmanned exploration of planets requires a high level of autonomy because of the communication delays between robots in space and their Earthbased stations. This affects all the components of the system: planning, sensing, and mechanism. In particular, such a level of autonomy can be
DP8.1 Understanding the behavior of a human steering an automobile remains an interesting subject [14, 15, 16, 21]. The design and development of systems for four-wheel steering, active suspensions, active, independent braking, and “drive-by-wire” steering provide the engineer with considerably
CDP8.1 In this chapter, we wish to use a PD controller such thatG s c( ) = + K s( ) 2 .The tachometer is not used (see Figure CDP4.1).Obtain the Bode plot for the system when Mpω Determine the step response of this system and estimate the overshoot and settling time (with a 2%criterion). T(s)
AP8.7 An op-amp circuit is shown in Figure AP8.7. The circuit represents a lead compensator.(a) Determine the transfer function of this circuit.(b) Sketch the Bode plot of the circuit when R1 = Ω 10 kR1 = Ω 10 k , R2 = Ω 10 , C1 = 0.1 Fµ , and C2 = 1 mF. C R www V(5) C R www o+ Vo(s)
AP8.6 Consider the spring-mass system depicted in Figure AP8.6. Develop a transfer function model to describe the motion of the mass M = 2 kg, when the input is u t( ) and the output is x t( ). Assume that the initial conditions are x( ) 0 0 = and x( ) 0 0 = .Determine values of k and b such that
AP8.5 A closed-loop system with unity feedback has a transfer function(a) Determine the loop transfer function. (b) Plot the logarithmic-magnitude versus phase curve, and identify the frequency points for ω equal to 1, 10, 50, 110, and 500.(c) Is the open-loop system stable? Is the closed-loop
AP8.4 A helicopter with a load on the end of a cable is shown in Figure AP8.4(a). The position control system is shown in Figure AP8.4(b), where the visual feedback is represented by H s( ). Sketch the Bode plot of the loop transfer function. Determine the crossover frequency, that is, where 20
AP8.3 As an automobile moves along the road, the vertical displacements at the tires act as the motion excitation to the automobile suspension system [16]Figure AP8.3 is a schematic diagram of a simplified automobile suspension system, for which we assume the input is sinusoidal. Determine the
AP8.2 A system is shown in Figure AP8.2. The nominal value of the parameter b is 4.0 and K = 5.0.Determine the sensitivity Sb T, and plot 20 log S j b T ( ) ω . FIGURE AP8.2 System with parameter b and controller gain K. R(s) K 0.1 b Y(s) s+2
AP8.1 A high pass amplifier may be represented by the circuit model shown in Figure AP8.1. WhenR R 1 2 = Ω 100 , = Ω 100 k , C C 1 2 = = 100 Fµ µ , 10 F, and K = 1000, show that (a) Sketch the Bode plot of G j ( ) ω . (b) Find the pass band gain (in dB). (c) Find the low frequency −3
P8.27 A unity feedback system has the loop transfer functionSketch the Bode plot of the loop transfer function, and indicate how the magnitude 20 log L j ( ) ω plot varies as K varies. Develop a table for K = 10, 20, and 30, and for each K determine the crossover frequency( ω ω c for 20 log
P8.26 Determine the transfer function of the opamp circuit shown in Figure P8.26. Assume an ideal op-amp. Plot the frequency response whenR R = Ω 10 k , 1 2 = Ω 9 k , 1 R = Ω k , and C = 1 µF.
P8.25 A unity feedback closed-loop system has a steady-state error equal to A/10, where the input isr t( ) = / At 2 2. The Bode plot is shown in Figure P8.25 for G j ( ) ω . Determine the transfer function G s( ).
P8.24 The Bode plot of a closed-loop film transport system is shown in Figure P8.24 [17]. Assume that the system transfer function T s( ) has two dominant complex conjugate poles. (a) Determine the best secondorder model for the system. (b) Determine the system bandwidth. (c) Predict the
P8.23 The frequency response of a process G j ( ) ω is shown in Figure P8.23. Deduce the type number(number of integrations) for the system. Determine the transfer function of the system, G s( ). Calculate the error to a unit step input.
P8.22 The frequency response of a process G j ( ) ω is shown in Figure P8.22. Determine G s( ). Magnitude (dB) 10 5 0 40 20 0 -5 -10 -15 -20 Phase (deg) -20 -40 -60 -80 10-1 100 10 102 103 -100 104 105 10-1 10 10 Frequency (rad/s) FIGURE P8.22 Bode plot of G(s). 102 103 104 105 Frequency (rad/s)
P8.21 Low-altitude wind shear is a major cause of air carrier accidents in the United States. Most of these accidents have been caused by either microbursts (small-scale, low-altitude, intense thunderstorm downdrafts that impact the surface and cause strong divergent outflows of wind) or by the
P8.20 For the successful development of space projects,robotics and automation will be a key technology.Autonomous and dexterous space robots can reduce the workload of astronauts and increase operational efficiency in many missions. Figure P8.20 shows a concept called a free-flying robot [9, 13].
P8.19 A DC motor controller used extensively in automobiles is shown in Figure P8.19(a). The measured plot of Θ( )s I( )s is shown in Figure P8.19(b).Determine the transfer function of Θ( )s I( )s .
P8.18 Remote operation plays an important role in hostile environments. Research engineers have been trying to improve teleoperations by feeding back rich sensory information acquired by the robot to the operator with a sensation of presence. This concept is called tele-existence or telepresence
P8.17 The experimental Oblique Wing Aircraft (OWA)has a wing that pivots, as shown in Figure P8.17. The wing is in the normal unskewed position for low speeds and can move to a skewed position for improved supersonic flight [11]. The aircraft control system loop transfer function is(a) Sketch the
P8.16 A space shuttle was used to repair satellites. Figure P8.16 illustrates how a crew member, with her feet strapped to the platform on the end of the shuttle’s robotic arm, used her arms to stop the satellite’s spin. The control system of the robotic arm has a closed-loop transfer function
P8.15 To determine the transfer function of a processG s( ), the frequency response may be measured using a sinusoidal input. One system yields the data in the following table: w, rad/s 0.1 1 |G(jw)| Phase, degrees 50 -90 5.02 -92.4 245 2.57 -96.4 1.36 -100 1.17 -104 6.3 1.03 -110 00 8 0.97 -120 10
P8.14 A bandpass amplifier may be represented by the circuit model shown in Figure P8.14 [3].When R R 1 2 = = 10 k, C C 1 2 = = 1 µ µ F, 10 F, andK = 100, show that V R 5 C2 '+' V 1 KV2 R V FIGURE P8.14 Bandpass amplifier.
P8.13 A position control system may be constructed by using an AC motor and AC components, as shown in Figure P8.13. The syncro and control transformer may be considered to be a transformer with a rotating winding. The syncro position detector rotor turns with the load through an angle θ0. The
P8.12 The block diagram of a feedback control system is shown in Figure P8.12(a). The transfer functions of the blocks are represented by the frequency response curves shown in Figure P8.12(b). (a) When G3 is disconnected from the system, determine the damping ratio ζ of the system. (b) Connect G3
P8.11 Automatic steering of a ship is a particularly useful application of feedback control theory [20]. In the case of heavily traveled seas, it is important to maintain the motion of the ship along an accurate track.An automatic system is more likely to maintain a smaller error from the desired
P8.10 A linear actuator is used in the system shown in Figure P8.10 to position a mass M. The actual position of the mass is measured by a slide wire resistor, and thus H s( ) = 1.0. The amplifier gain is selected so that the steady-state error of the system is less than 1% of the magnitude of the
P8.9 Sketch the logarithmic-magnitude versus phase angle curve for the transfer functions (a) and (b) of Problem P8.1.
P8.8 A feedback control system is shown in Figure P8.8.The specification for the closed-loop system requires that the percent overshoot to a step input beP O. . ≤ 10%. (a) Determine the corresponding spec -ification Mpω in the frequency domain for the closed-loop transfer function. (b)
P8.7 Driverless vehicles can be used in warehouses, airports, and many other applications. These vehicles follow a wire embedded in the floor and adjust the steerable front wheels in order to maintain proper direction, as shown in Figure P8.7(a) [10]. The sensing coils, mounted on the front wheel
P8.6 The asymptotic log-magnitude curves for two loop transfer functions are given in Figure P8.6. Sketch the corresponding asymptotic phase shift curves for each system. Estimate the transfer function for each system. Assume that the systems have minimum phase transfer functions. Magnitude (dB)
P8.5 The global robot industry is growing rapidly [8]. A typical industrial robot has multiple degrees of freedom. A unity feedback position control system for a force-sensing joint has a loop transfer function. K Ge(s)G(s)= (1+s/4)(1+s/8)(1 + s/16)(1 + s/32)1 where K 20. Sketch the Bode plot of
P8.4 A control system for controlling the pressure in a closed chamber is shown in Figure P8.4. Sketch the Bode plot of the loop transfer function.
P8.3 A rejection network is the bridged-T network shown in Figure P8.3. The transfer function of this network is + to m L R C 1002 v(t) C R FIGURE P8.3 Bridged-T network. s + w G(s)= 2 +2(wn/Q)s + w where w=2/LC, Q=wL/R, and R is ad- justed so that R2 = (w,L)2/4R[3]. (a) Determine the poles and

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