Questions and Answers of Modern Control Systems

Consider the control system in Figure CP10.1, WhereDevelop an m-file to show that the phase margin is approximately P.M. = 50° and that the percent overshoot to a unit step input is approximately
A unity feedback system has a loop transfer functionDetermine the phase margin and the crossover frequency. L(s) = G(s)G(s) 15 s(1 + 0.01s) (1 + 0.1s)
A model of an automobile driver attempting to steer a course is shown in Figure P9.18, where K = 6.0. (a) Find the frequency response and the gain and phase margins when the reaction time T =
A multiloop block diagram is shown in Figure AP9.10. (a) Compute the transfer function T(s) = Y(s)/R(s). (b) Determine K such that the steady-state tracking error to a unit step input R(s)
Consider a unity feedlach system with the loop transfer functionDevelop an m-file to plot the bandwidth of the closed- loop system as K varies in the interval 1 ≤ K ≤ 80.
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
Consider the vibration absorber in Figure P7.17 Using the root locus method, determine the effect of the parameters M2 and k12 .Determine the specific values of the parameters M2 and k12 so that the
Suppose that the dynamics of the motorcycle and rider can be represented by the loop transfer functionSketch the root locus for the system. Determine the damping ratio of the dominant roots when K =
Consider the closed-loop transfer functionDevelop an m-file to, obtain the Bode plot and verify that the resonant frequency is 7 rad/s and that the peak magnitude Mpω is 17 dB. T(s) = 50 s²
A unity feedback system has a loop transfer function Where K = 20. Find the roots of the closed-loop system’s characteristic equation. L(s): = K (s+1)(s+3)(s+ 6)'
A system with a transfer function Y(s)/R(s) is Determine the steady-state error to a unit step input. Is the system stable? Y(s) R(s) 15(s+2) s4 +85³ +2s²2 + 3s +1
To minimize vibrational effects, a telescope is magnetically levitated. This method also eliminates friction in the azimuth magnetic drive system. The photodetectors for the sensing system require
A closed-loop system as shown in Figure E13.10 has Calculate and plot y(kT) for 0 ≤ k ≤ 25 when T = 1s and the input is a unit step. Gp(s): = 1 s(s+1)
Let And D= [0]. Then design a controller using internalmodel methods so that the steady-state error for astep input is zero and the desired roots of the charac-teristic equation are s = -2 ± 2j
Consider the closed-loop sampled-data system shown in Figure AP13.5. Determine the acceptable range of the parameter K for closed-loop stability. FIGURE
An aircraft aileron can be modeled as a first-order system Where p depends on the aircraft. Obtain a family of step responses for the aileron system in the feedback configuration shown in Figure
A magnetically levitated train operated in Berlin, Germany from 1989–1991. Fully automated trains can run at short intervals and operate with excellent energy efficiency. The control system for the
An automobile suspension system has three physical state variables, as shown in Figure AP11.5. The state variable feedback structure is shown in the figure, with K1 = 1 Select K2 and K3 so that
A system has the model Add state variable feedback so that the closed-loop poles are s = −2 ± 2j and = -20. -5 -2 x (t) = -1 0 -1 0 0 x(t) + 1 0 16 မားမာ 0u(t) 0 y(t) = [0
The global robot industry is growing rapidly A typical industrial robot has multiple degrees of freedom. A unity feedback position control system for a force-sensing joint has a loop transfer
A large, braced robot arm for welding large structures is shown in Figure DP1.5. Sketch the block diagram of a closed-loop feedback control system for accurately controlling the location of the weld
For the open-loop control system described by the block diagram shown in Figure P2.12, determine the value of K such that y(t) →1 as to when r(t) is a unit step input. Assume zero initial
Space X has developed a very important system to allow for recovery of the first stage of their Falcon rocket at sea, as depicted in Figure AP1.7. The landing ship is an autonomous drone ship. Sketch
Consider the block diagram in Figure P3.37. Using the block diagram as a guide, obtain the state variable model of the system in the formUsing the state variable model as a guide, obtain a
A hydraulic servomechanism with mechanical feedback is shown in Figure P2.20 .The power piston has an area equal to A. When the valve is moved a small amount Δz, the oil will flow through to the
Figure P2.21 shows two pendulums suspended from frictionless pivots and connected at their midpoints by a spring . Assume that each pendulum can be represented by a mass M at the end of a massless
Consider the single-input, single-output system described by Where Assume that the input is a linear combination of the states, that is, Where r(t) is the reference input. The matrixK
A voltage follower (buffer amplifier) is shown in Figure P2.22. Show that T = Vo(s)/Vin (s) = Assume an ideal op-amp. FIGURE P2.22 A buffer amplifier. + o Vin (s) Vo(s)
Consider a crane moving in the x direction while the mass m moves in the z direction, as shown in Figure AP3.6. The trolley motor and the hoist motor are very powerful with respect to the mass of the
Consider the system in state variable formwith(a) Compute the transfer function G(s) = Y(s)/U (s(b) Determine the poles and zeros of the system.(c) Ifpossible, represent the system as a first-order
A digital audio system is designed to minimize the effect of disturbances as shown in Figure E4.1. As an approximation, we may represent G(s) = K?.(a) Calculate the sensitivity of the system due
A proposed hypersonic plane would climb to 80,000 feet, fly 3800 miles per hour, and cross the Pacific in 2 hours. Control of the aircraft speed could be represented by the model in Figure P4.14.(a)
Consider the closed-loop transfer function Obtain the impulse response analytically and compare the result to one obtained using the impulse function. T(s): = 20 s² +9s + 20
A second-order system is Consider the case where 1< z < 8. Obtain the par- tial fraction expansion, and plot the output for a unit step input for z = 2, 4, and 6. T(s): Y(s) (10/z)
A second-order system has the closed-loop transfer function (a) Estimate the percent overshoot P.O., the time to peak Tp, and the settling time Ts of the unit step response.(b) Obtain the system
Consider the systemDetermine if the system is controllable and observable. Compute the transfer function from u(t) to y(t). x(t) 0 = [ _ _ _!]x(1) + [9] (1). u(t), -5-8 y(t) = [10]x(t).
A system is described by the matrix equationsDetermine whether the system is controllable and observable. x(1) = [8]x(+ [2]4(0) x(t) u(t) y(t) = [02]x(t).
A system is described by the matrix equationsDetermine whether the system is controllable and observable. x (1) = - [ -8 -2]x (1) _-2]x(0) + [2](1) 0 y(t) = [10]x(t).
A system is described by the matrix equationsDetermine whether the system is controllable and observable. x(t) 0 -[-22] (0) _ 1] x(1) + [ 3 ]μ(1) = y(t) = [10]x(t).
Consider the system represented in state variable formSketch a block diagram model of the system. where x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t), 1 ¹] = [] B C = [24], and D= [0]. 0 -4
Consider the system represented in state variable formVerify that the system is observable and controllable. If so, design a full-state feedback law and an observer by placing the closed-loop system
Consider the third-order systemSketch a block diagram model of the system. X(t): 0 0 -8 y(t) = [2 8 || 10+ [2] 1 x(t) + 2 u(t) -1 10]x(t) + [1]u(t). 1 0 -3
A DC motor has the state variable modelDetermine whether this system is controllable and observable. -3 -2 -0.8 -3 0 0 2 0 0 0 0 y(t) = 0 0 0 0 3.2]x(1) x(t) = 0 0 0 0 0 0 1 0 0 2 2 0 0 x(t) + | 0
Consider the sampled data system with the loop transfer function(a) Plot the root locus using the rlocus function.(b) From the root locus, determine the range of K for stability. z² + 3z + 4 z²
A system the loop transfer function with unity feedback hasWe desire the steady-state error to a step input to be approximately 5% and the phase margin to be P.M. = 45°. Design a phase-lag
The acidity of water draining from a coal mine is often controlled by adding lime to the water. A valve controls the lime addition and a sensor is downstream. For the model of the system shown in
Consider the control system in Figure CP10.1, whereDevelop an m-file to show that the phase margin is P.M. = 60° and that the percent overshoot to a unit step input is P.O. = 8.8%. G(s) 1 s + 12 and
A simplified version of the attitude rate control for a supersonic aircraft is shown in Figure P10.3. When the vehicle is flying at four times the speed of sound (Mach 4) at an altitude of 100,000
High-performance tape transport systems are designed with a small capstan to pull the tape past the read/write heads and with take-up reels turned by DC motors. The tape is to be controlled at speeds
Consider the system with the loop transfer functionsWhen K = 10, find T(s) and estimate the expected percent overshoot and settling time (with a 2% criterion). Compare your estimates with the actual
Consider a unity feedback system with the loop transfer functionwhere z = 1 and p = 3.6. The actual percent overshoot of the compensated system will be P.O. = 46%. We want to reduce the percent
Consider a circuit with the transfer functionq where C1 = 0.1 μF, C2 = 1 mF, R1 = 10 kΩ, and R2 = 10 Ω. Plot the frequency response of the circuit. G(s) Vo(s) V;(s) 1 +
A unity feedback system has the loop transfer function(a) Determine the step response when Gc (s) = 1, and calculate the settling time and steady state for a ramp input r(t) = t, >
A feedback system has a plant transfer functionWe want the velocity error constant Kv to be 35 and the percent overshoot to a step to be P.O. = 4% so that ζ = 1/√2. The settling time (with a 2%
A system has a transfer functionDetermine a real value of a so that the system is either uncontrollable or unobservable. Y(s) R(s) s + a s4+ 13s³ + 54s² + 82s + 60
Consider a unity feedback system withwhere 1 ≤ a ≤ 3 and 2 ≤ K ≤ 4.Use a PID controller and design the controller for the worst-case condition. We desire
A photovoltaic system is mounted on a space station in order to develop the power for the station. The photovoltaic panels should follow the Sun with good accuracy in order to maximize the energy
Electromagnetic suspension systems for air cushioned trains are known as magnetic levitation (maglev) trains. One maglev train uses a superconducting magnet system. It uses superconducting coils, and
Consider the closed-loop system represented in state variable formThe nominal value of k = 2. However, the value of k can vary in the range 0.1≤ k ≤ 4. Plot the percent overshoot to a unit step
A unity feedback system has a plantWe want to attain a steady-state error for a step input. Select a compensator Gc (s) using the pseudo-QFT method, and determine the performance of the system
We have a functionUsing a partial fraction expansion of Y(s) and a table of z-transforms, find Y(z) when T = 0.1 s. Y(s) 4 s(s+ 1)(s + 8)
Plot the root locus for the systemFind the range of K for stability. Z z² − z + 0.75* - G(z)D(z) = K-
Find the z-transform ofwhen the sampling period is T = 1 s. X(s) = S+2 S² +65 +8
The transfer function of a plant and a zero-order hold is(a) Plot the root locus(b) Determine the range of gain K for a stable system G(z) = K(z + 0.8) Z(z − 2)
Consider the loop transfer function of a phase-lock loop systemSketch the root locus as a function of the gain Kv = KaK. Determine the value of Kv attained if the complex roots have a damping ratio
A unity feedback system has a loop transfer function(a) Sketch the root locus for K > 0(b) Find the roots when K = 2 and 3(c) Compute the rise time, percent overshoot, and settling time (with a 2%
The loop transfer function of a unity feedback system isThis system is called conditionally stable because it is stable only for a range of the gain K such that k1 < K < k2. Using the
A unity feedback system has a loop transfer functionSketch the root locus. Determine the gain K when the complex roots of the characteristic equation have a ζ = 0.6. L(s) = Gc(s) G(s) = Ks 5³ +85²
Compute the partial fraction expansion ofand verify the result using the residue function. Y(s) = 9 + S s($² +65 + 5)
Consider a unity feedback system with a loop transfer function(a) Find the breakaway point on the real axis(b) Find the asymptote centroid(c) Find the value of K at the breakaway point
Consider the closed-loop transfer functionDevelop an m-file to, obtain the Bode plot and verify that the resonant frequency is 5.44 rad/s and that the peak magnitude Mpω is 14.8 dB. T(s) || 30 s² +
A unity feedback system has a loop transfer function(a) Sketch the root locus for 0 ≤ K < ∞ . (b) Determine the range of the gain K for which the system is stable. (c) For what value
A unity feedback system has a loop transfer functionSketch the root locus as a function of K. Calculate where the segments of the locus enter and leave the real axis. L(s) = Ge(s)G(s) = K(s² +
A tendon-operated robotic hand can be implemented using a pneumatic actuator. The actuator can be represented byPlot the frequency response of G (jω). Show that the magnitude of G (jω) is -3 dB at
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. (s)5(S)'
Several studies have proposed an extravehicular robot that could move around in a NASA space station and perform physical tasks at various worksites. The arm is controlled by a unity feedback control
In this chapter, we wish to use a PD controller such thatThe tachometer is not used. Obtain the Bode plot for the system when K = 40. Determine the step response of this system and estimate the
The 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
Consider a unity negative feedback control system withVerify that the gain margin is G.M. = ∞ and that the phase margin is P.M. = 9.8°. L(s) Ge(s)G(s): = 150 s² + 2s + 15
A system has a loop transfer functionWhen K ≥ 342.8, the closed-loop system is unstable. Find the gain margin and phase margin of the system with K = 25. L(S) = Ge(s)G(s) = K(s + 75) s(s+ 10) (s
A unity feedback system has a loop transfer function Determine the range of K for which the system is stable using the Nyquist plot. L(s) = Ge(s)G(s) = K S-4
Consider a unity feedback system withand determine the gain KP that provides the maximum phase margin. and Let G(s) = = 1 s(s² + 3s +3.5) K₁ Ge(s) = K₂ + S K₁ Kp = 0.2,
Consider a system with the loop transfer functionObtain the Bode plot and show that the P.M. = 23ºand that the G.M. = 13 dB. Also, show that the bandwidth of the closed-loop system is ωB = 5.8
A unity feedback system has a loop transfer functionDetermine the phase margin and the crossover frequency. L(s) = = Gc(s) G(s) = 15 s(1 + 0.01s) (1 + 0.1s)
Consider a unity feedback system with the loop transfer functionFind the bandwidth of the closed-loop system. L(s) = Ge(s)G(s) 90 s(s+15) ✓
A unity feedback control system has a loop transfer functionDetermine the phase margin, the crossover frequency, and the gain margin when K = 50. L(s) = Ge(s) G(s) K s(s+ 2) (s + 10)
Consider the system described in state variable form byCompute the phase margin. where x(t) = Ax(t) + Bu(t) y(t) = Cx(t) 0 1 A = [44].B=[9].C=D В -4
Describe a feedback control system in which a user utilizes a smart phone to remotely monitor and control a washing machine as illustrated in Figure DP1.3. The control system should be able to start
A system is represented by Figure P2.36. (a) Determine the partial fraction expansion and y1t2 for a ramp input, r(t) = t, t ≥ 0. (b) Obtain a plot of y(t) for part (a), and find y(t) for
Consider the system(a) Find the state transition matrix Φ(t). (b) For the initial conditions x1(0) = x2(0) = 1, find x(t). X(t) || 0 0 0 x(t).
In the past 50 years, over 20,000 metric tons of hardware have been placed in Earth’s orbit. During the same time span, over 15,000 metric tons of hardware returned to Earth. The objects remaining
Consider the systemFind the roots of the characteristic equation. x(1) = 0 0 0 1 0 -6-3 0 1 x(t).
Consider the state variable model with parameter K given byPlot the characteristic values of the system as a function of K in the range 0 ≤ K ≤ 100. Determine that range of K for which all the
The dynamics of a controlled submarine are significantly different from those of an aircraft, missile, or surface ship. This difference results primarily from the moment in the vertical plane due to
A magnetic disk drive requires a motor to position a read/write head over tracks of data on a spinning disk, as shown in Figure E4.4. The motor and head may be represented by the transfer
A unity feedback system has the loop transfer functionDetermine the relationship between the steady-state error to a ramp input and the gain K and system parameter b. For what values of K and b can
A single-input, single-output system has the matrix equationsandDetermine the transfer function G(s) = Y(s) /U(s). 0 x (1) - [ [1] 17x² = x(t) + -3 -5
Extreme temperature changes result in many failures of electronic circuits. Temperature control feedback systems reduce the change of temperature by using a heater to overcome outdoor low
Consider a unity feedback system with loop transfer functionDetermine the value of the gain K such that the percent overshoot to a unit step is minimized. L(s) = Ge(s)G(s) = K(s + 2) (s + 5) (s² + s
Consider the closed-loop system in Figure P5.22, where(a) If Ƭ = 2.43, determine the value of K such that the steady-state error of the closed-loop system response to a unit step input, is zero.(b)

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