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engineering
modern control systems

Modern Control Systems 13th Edition Richard C. Dorf, Robert H. Bishop - Solutions

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 electrical connections. The system block diagram is shown in Figure AP12.1.Design a PID controller so
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 and s = -20. A -1 2 01 B= 1 C = [1 0],
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 AP13.5 R(s) T=0.1 Zero-order hold Go(s) Gp(s) 1 s(s+5) Y(s)
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 CP12.2. G(s) = P s+ p
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 levitation of the car is shown in Figure P10.34.Select a compensator so that the phase margin of
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 the roots of the characteristic equation are three real roots lying between s = −3 and s = −6.
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 0_10]x(t).
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 function Where K = 20. Sketch the Bode plot of this system. Ge(s)G(s) = K (1 + s/2)(1+s)(1+ s/30)(1+
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 tip.Data in Figure DP1.5. Workpiece Weld tip
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 conditions.Data in Figure P2.12 R(s) Controller K Process 1 s +50 Open-loop control system. Y(s)
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 a block diagram describing a control system that would control the pitch and roll of the landing
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 third-order differential equation model for the system. x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t).
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 cylinder at a rate p. Δz, where p is the port coefficient. The input oil pressure is assumed to be
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 bar of length L. Also assume that the displacement is small and linear approximations can be used for
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 = [K1 K2] is known as the gain matrix. Substituting u(t) into the state variable equation gives
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 trolley, the hoist wire, and the load m. Consider the input control variables as the distances D(t)
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 systemWhere a, b, c, and d are scalars such that the transfer function is the same as obtained in
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 toK?. (b) Calculate the effect of the disturbance noiseTa (s) on Vo(s). (c) What value would
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) Find the sensitivity of the closed-loop transfer function T(s) to a small change in the parameter
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) (s+z) R(s) (s+1)(s+8)
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 response to a unit step and verify the results in part (a). T(s) = Y(s) R(s) = wh/1 s² + 25wns +
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 poles at s1, 2 = -1 + j and the observer poles at s1, 2 = -12. where x(t) = Ax(t) + Bu(t) y(t) =
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 |u(t) 0 0 0 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 -0.1z - 2 G(z)D(z) = K-
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 compensator to meet these specifications. K (s + 6)² L(s) = Ge(s)G(s) = Ge(s)
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 Figure AP9.6, determine K and the distance D to maintain stability. We require D > 2 meters in order
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 Ge(s) 96 S
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 ft, the parameters areDesign a compensator Gc (s) so that the response to a step input has a
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 up to 200 inches per second, with start-up as fast as possible, while preventing permanent
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 percent overshoot of P.O. = 47.5% and a settling time of Ts = 32.1 s. L(s) = Ge(s)G(s) = K(s +
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 overshoot to P.O. = 32%. Using a m-file script, determine an appropriate value for the zero of Gc(s).
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 + R₂C₂8 1 + R₁C₁s
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, > 0. (b) Design a phase-lag compensator using the root locus method so that the velocity constant is
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% criterion) desired is Ts = 0.11 s. Design an appropriate state variable feedback system for r (t)
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 that the settling time (with a 2% criterion) be Ts ≤ 0.8 s with an ITAE performance. G(s) = K s² +
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 from the panels. The unity feedback control system uses a DC motor, so that the transfer function of
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 the levitation distance x(t) is inherently unstable. The model of the levitation iswhere V(s) is
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 input as k varies from 0.1 to 4. where B = x(t) = Ax(t) + Br(t) y(t) = Cx(t) +
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 when all the poles of G(s) change by -50%. Describe the robust nature of the system. G(s) 1 (s + 2)
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 equal to 0.60. 10(s + 10) s(s+ 1)(s + 100) L(s) = Ge(s) G(s) = KK-
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% criterion) of the system for a unit step input when K = 2 and 3. L(s) = KG(s) = K(s + 1) S² + s +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 Routh–Hurwitz criteria and the root locus method, determine the range of the gain for which the system is
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 + 12
Compute the partial fraction expansion ofand verify the result using the residue function. Y(s) = 9 + S s($² +65 + 5)
Consider a unity feedback system with a loop transfer function(a) Find the breakaway point on the real axis(b) Find the asymptote centroid(c) Find the value of K at the breakaway point L(s) = Ge(s)G(s) = 1 2 S³ + 50s² + 500s + 1000
Consider the closed-loop transfer functionDevelop an m-file to, obtain the Bode plot and verify that the resonant frequency is 5.44 rad/s and that the peak magnitude Mpω is 14.8 dB. T(s) || 30 s² + s + 30
A unity feedback system has a loop transfer function(a) Sketch the root locus for 0 ≤ K < ∞ . (b) Determine the range of the gain K for which the system is stable. (c) For what value of K in the range K ≥ 0 do purely imaginary roots exist? What are the values of these
A unity feedback system has a loop transfer functionSketch the root locus as a function of K. Calculate where the segments of the locus enter and leave the real axis. L(s) = Ge(s)G(s) = K(s² + 0.1) s(s² + 1)
A tendon-operated robotic hand can be implemented using a pneumatic actuator. The actuator can be represented byPlot the frequency response of G (jω). Show that the magnitude of G (jω) is -3 dB at ω = 10 and -33 dB at ω = 200. Show also that the phase is -171º at ω = 700. G(s) 1000 (s + 100)
A unity negative feedback system has the loop transfer functionDetermine the closed-loop system bandwidth. Using the bode function obtain the Bode plot and label the plot with the bandwidth. (s)5(S)' = ($)7 = 60 s(s+ 6)*
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 with loop transfer functionSketch the Bode plot for K = 20 and determine the frequency when 20
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 overshoot and settling time (with a 2% criterion). Ge(s) K(s + 2). =
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 the satellite’s spin. The control system of the robotic arm has a closed-loop transfer function(a)
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 +30)
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 rad/s. L(s) = Ge(s)G(s) = 300(s + 4) s(s+ 0.16) (s² + 14.6s+ 149)
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 and stop the wash cycle, control the amount of detergent and the water temperature, and provide
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 t = 1.0 s. (c) Determine the impulse response of the system y(t) for t ≥ 0. (d) Obtain
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 in Earth’s orbit range in size from large operational spacecraft to tiny flecks of paint. There
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 characteristic values lie in the left half-plane. 0 1 0 0 -2 -K y(t) = [1 0 0]x(t). X(t) || 0 0 1 x(t)
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 the buoyancy effect. Therefore, it is interesting to consider the control of the depth of a
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 functionwhere Ƭ = 0.001 s. The controller takes the difference of the actual and desired positions and
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 we guarantee that the magnitude of the steady-state error to a ramp input is less than 0.1? L(s) =
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 temperatures. A block diagram of one system is shown in Figure P4.8. The effect of a drop in environmental
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 + 10)
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) Determine the percent overshoot and the time to peak of the unit step response when K is as in part
A motor control system for a computer disk drive must reduce the effect of disturbances and parameter variations, as well as reduce the steady-state error. We want to have no steady-state error for the head-positioning control system. (a) What type number is required? (How many
Consider the case of a navy pilot landing an aircraft on an aircraft carrier. The pilot has three basic tasks. The first task is guiding the aircraft’s approach to the ship along the extended centerline of the runway. The second task is maintaining the aircraft on the correct glideslope. The
A unity feedback system has a loop transfer functionwhere K = 20. Find the roots of the closed-loop system’s characteristic equation. L(S) K (s + 1)(s + 3) (s + 6)'
Consider a feedback system with closed-loop transfer functionIs the system stable? T(S): 4 S³+ 45² + s + 4
A system has a characteristic equation s3 + 15s2 + 2s + 40 = 0. Using the Routh–Hurwitz criterion, show that the system is unstable.
A system with a transfer function Y(s) / R(s) isDetermine the steady-state error to a unit step input. Is the system stable? Y(s) R(S) 10(s + 1) 54 + 65³ +5² + s +3²
A feedback control system has a characteristic equationDetermine whether the system is stable, and determine the values of the roots. 56 +255 + 1354 + 16s³ + 56s² + 32s + 80 = 0.
The matrix differential equation of a state variable model of a system isFind the range of k where the system is stable. X(t) 0 1 0 0 -k -8 0 1 x(t). -4
Determine whether the systems with the following characteristic equations are stable or unstable:(a) s3 + 3s2 + 5s + 75 = 0(b) s4 + 5s3 + 10s2 + 10s + 80 = 0(c) s2 + 6s + 3 = 0
Find the roots of the following polynomials:(a) s3 + 5s2 + 8s + 4 = 0(b) s3 + 9s2 + 27s + 27 = 0

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