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- Physics
- Classical Dynamics Of Particles

b. Evaluate the unit-step response of each design.

c. Evaluate the unit-ramp response of each design.

Figure P11.30

Q:Compare the performance of the critically damped controllers shown inCompare the performance of the critically damped controllers shown in Figure P11.30 with the plant Gp(s) = 1 / I s having the following inputs:

a. A unit-ramp disturbance

b. A sinusoidal disturbance

c. A sinusoidal command input

Q:A certain field-controlled de motor with load has the followingA certain field-controlled de motor with load has the following parameter values.

L = 2 × 10-3 HR = 0.6 Ω

KT = 0.04N · m/AC = 0

I = 6 × 10-5 kg · m2

Compute the gains for a state variable feedback controller using P action to control the motor’s angular position. The desired dominant time constant is 0.5 s. The secondary roots should have a time constant of 0.05 s and a damping ratio of ( = 0.707.

Q:In Figure P11.33 the input u is an acceleration providedIn Figure P11.33 the input u is an acceleration provided by the control system and applied in the horizontal direction to the lower and of the rod. The horizontal displacement of the lower and is y. The linearized from of Newton’s law for small angles gives

a. Put this model into state variable form by letting x1 = ( and x2 = .

b. Construct a state variable feedback controller by letting u = k1x1 + k2 x2. Over what ranges of values of k1 and k2 will the controller stabilize the system? What does this formulation imply about the displacement y?

Figure P11.33

m1 1 = k1(y – x1) – k2 (x1 – x2) – c(1 – 2) – f

m2 2 = k2(x1 – x2) + c(1 – 2) + f

Figure P11.34

Q:Figure P11.35 a is the circuit diagram of a speed-controlFigure P11.35 a is the circuit diagram of a speed-control system in which the dc motor voltage va is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not so sensitive to speed and thus can be used to drive the locomotive at various speeds. The motor voltage va is varied by changing the generator input voltage vf. The voltage va is related to the generator field current if by va = Kjif.

Figure P11.35 b is a diagram of a feedback system for controlling the speed by measuring it with a tachometer and varying the voltage uf. Use the following values in SI units.

Lf = 0.2 Rf = 2 Kt = 1

La = 0.2Ra = 1Kb = KT = 0.5

Kf = 50I = 10c = 20

Develop a state variable model of the plant that includes the generator, the motor, and the load. Include the load torque TL as a disturbance.

Develop a proportional controller assuming all the state variables can be measured. Analyze its steady-state error for a step command input and for a step disturbance.

Figure P11.35

Q:The following equations are the model of the roll dynamicsThe following equations are the model of the roll dynamics of a missile ([Bryson, 1975]). See Figure P11.36.

Figure P11.36

WhereÎ´ = aileron deflection

b = aileron effectiveness constant

u = command signal to the aileron actuator

( = roll angle, ( = roll rate

Using the specific value b = 10 s-1 and ( = 1 s, and assuming that the state variables Î´,(, and ( can be measured, develop a linear state-feedback controller to keep ( near 0. The dominant roots should be s = – 10 ( 10j, and the third root should be s = – 20.

dT / dt = k (ur – up)

For a paper thickness d, the rate of change of the roll radius is

dR/dt = d/2 W

The inertia of the windup roll is I = ( ( W R4/2, where ( is the paper mass density and W is the width of the roll. So R and I are functions of time.

The viscous damping constant for the roll is c. For the armature-controlled motor driving the windup roll, neglect its viscous damping and armature inertia.

Assuming that the paper thickness is small enough so that ( 0 for a short time, develop a state-variable model with the motor voltage e and the paper speed up as the inputs.

Modify the model developed in part (a) to account for R and I being functions of time.

Figure P11.37

Q:An electro-hydraulic positioning system is shown in Figure P11.38. UseAn electro-hydraulic positioning system is shown in Figure P11.38. Use the following values.

Ka = 10 V/A Ki = 10-2 in./V

K2 = 3 Ã 105 sec-3 K3 = 20 V/in.

( = 0.8 (n = 100 rad/sec ( = 0.01 sec

a. Develop a state-variable model of the plant with the controller current ic as the input and the displacement y as the output.

b. Assuming that proportional control is used so that Gc(s) = Kp, develop a state model of the system with y as the input and y as the output. Draw the root locus and use it to determine whether or not the system can be made stable with an appropriate choice for the value of KP.

Figure P11.38

Q:A) The equations of motion of the inverted pendulum modela) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use the linearized model developed in part (b) to design a state variable feedback controller to stabilize the pendulum angle near ( = 0. It is required that the 2% settling time be no greater that 4 s and that the response be non-oscillatory. This means that the dominant root should be real and no greater than – 1. No restriction is placed on the motion of the base. Assume that (, (, x, and can be measured.

Q:Sketch the root locus plot of ms2 + 12s +Sketch the root locus plot of ms2 + 12s + 10 = 0 for m ≥ 2. What is the smallest possible dominant time constant, and what value of m gives this time constant?

Q:The following table gives the measured open-loop response of aThe following table gives the measured open-loop response of a system to a unit-step input. Use the process reaction method to find the controller gains for P. PL and PID control.

Time (min)Response

0………………………………………..0

0.5………………………………………4

1.0…………………………………….20

1.5…………………………………….32

2.0…………………………………….56

2.5…………………………………….84

3.0…………………………………..1 16

3.5…………………………………..140

4.0…………………………………..160

4.5…………………………………..172

5.0…………………………………..184

5.5…………………………………..190

6.0…………………………………..194

7.0…………………………………..196

Q:A liquid in an industrial process must be heated withA liquid in an industrial process must be heated with a heat exchanger through which steam passes. The exit temperature of the liquid is controlled by adjusting the rate of flow of steam through the heat exchanger with the control valve. An open-loop test was performed in which the steam pressure was suddenly changed from 15 to 18 psi above atmospheric pressure. The exit temperature data are shown in the following table. Use the

Time (min) Temperature ( I)

0………………………………….156

1………………………………….157

2………………………………….159

3………………………………….162

4………………………………….167

5………………………………….172

6………………………………….175

7………………………………….179

8………………………………….181

9………………………………….182

10…………………………………183

11…………………………………184

12…………………………………184

Q:Use MATLAB to obtain the root locus plot of 5s2Use MATLAB to obtain the root locus plot of 5s2 + cs + 45 = 0 for c ≥ 0.

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable k ≥ 0. Use the values m = 4 and c = 8. What is the smallest possible dominant time constant and the associated value of k?

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable c ≥ 0. Use the values m = 4 and k = 64. What is the smallest possible dominant time constant and the associated value of c?

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.45 in terms of the variable k2 â¥ 0. Use the values m = 2, c = 8, and k1 = 26. What is the value of k2 required to give ( = 0.707?

Figure P11.43

Figure P11.45

Figure P11.46

Q:Use MATLAB to obtain the root locus plot of s3Use MATLAB to obtain the root locus plot of s3 + 13s2 + 52s + 60 + K = 0 for K ≥ 0. Is it possible for any dominant roots of this equation to have a damping ratio in the range 0.5 ≤ ( ≥ 0.707 and an un-damped natural frequency in the range 3 ≤ (n Q:(a) Use MATLAB to obtain the root locus plot of(a) Use MATLAB to obtain the root locus plot of 2s3 + 12s2 + 16s + K = 0 for K ≥ 0. (b) Obtain the value of K required to give a dominant root pair having ( = 0.707. (c) For this value of K. obtain the unit-step response and the maximum overshoot, and evaluate the effects of the secondary root. The closed-loop transfer function is K/(2s3 + 12s2 + 16s + K).

Q:Use MATLAB to obtain the root locus of the armature-controlledUse MATLAB to obtain the root locus of the armature-controlled dc motor model in terms of the damping constant c, and evaluate the effect on the motor time constant. The characteristic equation is

LaIs2 + (Ra I + cLa)s + cRa + Kb KT = 0

Use the following parameter values:

Kb = KT = 0.1 N · m/AI = 4 × 10-5 kg · m2

Ra = 2 ΩLa = 3 × 10-3 H

Q:In the following equations, identify the root locus plotting parameterIn the following equations, identify the root locus plotting parameter K and its range in terms of the parameter p, where p ≥ 0.

6s2 + 8s + 3p = 0

3s2 + (6 + p)s + 5 + 2p = 0

4s3 + 4ps2 + 2s + p = 0

Q:Consider the two-mass model shown in Figure P11.50. Use theConsider the two-mass model shown in Figure P11.50. Use the following numerical values: m1 = m2 = 1, k1 = 1, k2 = 4, and c2 = 8.

a. Use MATLAB to obtain the root locus plot in terms of the parameter c1.

b. Use the root locus plot to determine the value of c1 required to give a dominant root pair having a damping ratio of ( = 0.707.

c. Use the root locus plot to determine the value of c1 required to give a dominant root that is real and has a time constant equal to 4.

d. Using the value of c1 found in part (c), obtain a plot of the unit-step response.

s3 + 10s2 + 24s + K = 0

a. Use MATLAB to obtain the value of K required to give dominant roots with ( = 0.707. Obtain the three roots corresponding to this value of K.

b. Use MATLAB to obtain the value of K required to give a dominant time constant of ( = 2/3. Obtain the three roots corresponding to this value of K.

Q:Consider the equation s3 + 9s2 + (8 + K)s +Consider the equation

s3 + 9s2 + (8 + K)s + 2 K = 0

a. Use MATLAB to obtain the value of K required to put the dominant root at the breakaway point. Obtain the three roots corresponding to this value of K.

b. Investigate the sensitivity of the dominant root when K varies by ± 10% about the value found in part (a).

Q:Consider the equation s3 + 10s2 + 24s + K =Consider the equation

s3 + 10s2 + 24s + K = 0

Use the sgrid function to determine if it is possible to obtain a dominant root having a damping ratio in the range 0.5 ≤ ( ≥ 0.707, and an un-damped natural frequency in the range 2 ≤ (n ≤ 3. If so, use MATLAB to obtain the value of K required to give the largest possible value of ( (n in the ranges stated.

Q:In Example 10.7.4 the steady-state error for a unit-ramp disturbanceIn Example 10.7.4 the steady-state error for a unit-ramp disturbance is 1/KI.For the gains computed in that example, this error is 1 /25. We want to see if we can make this error smaller by increasing KI. Using the values given for Kp and KD in that example, obtain a root locus plot with KI as the variable. Discuss what happens to the damping ratio and time constant of the dominant root as KI is increased.

Q:In Example 10.8.3 the steady-state error for a unit-ramp commandIn Example 10.8.3 the steady-state error for a unit-ramp command is -4/KI. For the gains computed in that example, this error is 1 /1000. We want to see if we can make this error smaller by increasing KI. Using the values given for Kp and Kd in that example, obtain a root locus plot with K1 as the variable. Discuss what happens to the damping ratio and time constant of the dominant root as K1 is increased.

Q:With the PI gains set to Kp = 6 andWith the PI gains set to Kp = 6 and K1 = 50 for the plant

Gp (s) = 1 / s + 4

The time constant is ( = 0.2 and the damping ratio is ( = 0.707.

a. Suppose the actuator saturation limits are ( 5. Construct a Simulink model to simulate this system with a unit-step command. Use it to plot the output response, the error signal, the actuator output, and the outputs of the proportional term and the integral term versus time.

b. Construct a Simulink model of an anti-windup system for this application. Use it to select and appropriate value for KA and to plot the output response and the actuator output versus time.

Q:Consider a unity feedback system with the plant Gp(s) andConsider a unity feedback system with the plant Gp(s) and the controller Gc(s). PID control action is applied to the plant

Gp(s) = s + 10 / (s + 1) (s + 2)

The PID controller has the transfer function

Gc(s) = Kp (1 + 1/TIs + TDs

Use the values TI = 0.2 and TD = 0.5.

Identify the open-loop poles and zeros.

Identify the root locus parameter K in terms of Kp.

Identify the closed-loop poles and zeros for the case Kp = 10.

Q:With the PI gains set to Kp = 6 andWith the PI gains set to Kp = 6 and K1 = 50 for the plant

Gp (s) = 1 / s + 4

The time constant is ( = 0.2 and the damping ratio is ( = 0.707. Suppose there is a rate limiter of ± 0.1 between the controller and the plant. Construct a Simulink model of the system and use it to determine the effect of the limiter on the speed of response of the system. Use a unit-step command.

Q:A certain dc motor has the following parameter values: L =A certain dc motor has the following parameter values:

L = 2 Ã 10-3 HR = 0.6 Î©

KT = 0.04 N Â· m/Ac = 0

I = 6 Ã 10-5 kg Â· m2

Figure P11.61

Figure P11.61 shows an integral controller using state-variable feedback to control the motor’s angular position.

a. Compute the gains to give a dominant time constant of 0.5 s. The secondary roots should have a time constant of 0.05 s and a damping ratio of ( = 0.707. The fourth root should be s = -20.

b. Construct a Simulink model of the system and use it to plot the response of the system to a step disturbance of magnitude 0.1.

Suppose the motor current is limited to Â± 2 A. Modify the Simulink model to include this saturation, and use the model to obtain plots of the responses to a unit-step command and a step disturbance of magnitude 0.1. Discuss the results.

Q:In parts (a) through (f), sketch the root locus plotIn parts (a) through (f), sketch the root locus plot for the given characteristic equation for K ≥ 0.

s(s + 5) + K .= 0

s(s + 7)(s + 9) + K = 0

s2 + 3s + 5 + K(s + 3) = 0

s(s + 4) + K(s + 5) = 0

s(s2 + 3s + 5) + K = 0

s{s + 3)(s + T) + K(s + 4) = 0

Q:PID control action is applied to the plant Gp(s) = sPID control action is applied to the plant

Gp(s) = s + 10/ (s + 2) (s + 5)

The PID controller has the transfer function

Gc(s) = Kp (1 + 1/TIs + TDs)

Use the values TI = 0.2 and TD = 0.5. Plot the root locus with the proportional gain Kp as the parameter.

Q:Consider the following equation where the parameter p is nonnegative. 4s3Consider the following equation where the parameter p is nonnegative.

4s3 + (25 + 5p)s2 + (16 + 30p)s + 40p = 0

Put the equation in standard root locus form and define a suitable root locus parameter K in terms of the parameter p.

Obtain the poles and zeros, and sketch the root locus plot.

Q:Control of the attitude 0 of a missile by controllingControl of the attitude 0 of a missile by controlling the fin angle cp. as shown in Figure P12.1, involves controlling an inherently unstable plant. Consider the specific plant transfer function

Gp(s) = ( (s) / ( (s) = 1 / 5s2 – 6

Design a PD compensator for this plant. The dominant roots of the closed loop system must have ( = 0.707 and (n = 0.5.

Q:Consider a plant whose open-loop transfer function is G(s) H (s)Consider a plant whose open-loop transfer function is

G(s) H (s) = 1 / s [(s + 2)2 + 9]

The complex poles near the origin give only slightly damped oscillations that are considered undesirable. Insert a gain Kc and a compensator Gc (s)in series to speed up the closed-loop response of the system. Consider the following for Gc(s):

a. The lead compensator

b. The lag compensator

c. The so-called reverse-action compensator

Gc (s) = 1 – T1s / T2s + 1

Obtain the root locus plots for the compensated system using each compensator. Use the compensator parameters. Determine which compensator gives the best response.

Q:A) The equations of motion of the inverted pendulum modela) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that ( is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use the linearized model developed in part (b) to design a series compensator to stabilize the pendulum angle near ( = 0. It is required that the 2% settling time be no greater that 4 s and that the response be nonoscillatory. This means that the dominant root should be real and no greater than -1. No restriction is placed on the motion of the base. Assume that only ( can be measured.

Q:A certain unity feedback system has the following open-loop systemA certain unity feedback system has the following open-loop system transfer function.

G(s) = 5K / s3 + 6s2 + 5s

Obtain the Bode plots and compute the phase and gain margins for

a. K = 2

b. K =20

c. Use the Bode plots to determine the upper limit on K for the system to be stable. Which is the limiting factor: the phase margin or the gain margin?

Q:Figure P12.13 shows a pneumatic positioning system, where the displacementFigure P12.13 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume that

Figure P12.13

The pressure p2 is constant, and consider the specific plant

Gp(s) = X (s)/P1 (s) = 1 / 100s2 + s

With the following series PD compensator is used to control the pressure.

Gc (s) = 2 + 19s

Obtain the Bode plots for this system, and determine the phase and gain margins.

Gp (s) = H2 (s) / Q1 (s) = 25 / 5s2 + 6s + 5

With the following series PI compensator is used to control it.

Gc (s) = 7s + 64 / s

Obtain the Bode plots for this system, and determine the phase and gain margins.

Figure P12.14

Q:Rolling motion of a ship can be reduced by usingRolling motion of a ship can be reduced by using feedback control to vary the angle of the stabilizer fins, much like ailerons are used to control aircraft roll. Figure PI2.15 is the block diagram of a roll control system in which the roll angle is measured and used with proportional control action. Determine the phase and gain margins of the system if (a) Kp = 1, (b) KP = 10, and (c) Kp = 100. Determine the stability properties for each case. If the system is unstable, what effect will this have on the ship roll?

Q:The following transfer functions are the forward transfer function G(s)The following transfer functions are the forward transfer function G(s) and the feedback transfer function H(s) for a system whose closed-loop transfer function is

G (s) / 1 + G (s) H (s)

For each case determine the system type number, the static position and velocity coefficients, and the steady-state errors for unit-step and unit-ramp inputs.

a. G (s) H (s) = 20/s

b. G (s) H (s) = 20 / 5s + 1

c. G (s) H (s) = 7 / s2

Q:Remote control of systems over great distance, such as requiredRemote control of systems over great distance, such as required with robot space probes, may involve relatively large time delays in sending commands and receiving data from the probe. Consider a specific system using proportional control, where the total dead time is D = D1 + D2 = 100 sec (Figure P12.18). The plant is

Gp(s) = 1 / 100s + 1

How large can the gain KP be without the system being unstable?

Q:Hot-air heating control systems for large buildings may involve significantHot-air heating control systems for large buildings may involve significant dead time if there is a large distance between the furnace and the room being

Heated (Figure P12.19). Proportional control applied to the specific plant

Gp(s) = 1 / 0.1s + 1

Has the gain KP = 10. The time units are minutes. How large can the dead time D between the controller and the plant be if the phase margin must be no less than 40o?

Gp (s) = X (s) / P1 (s) = K / s2 + 2s

With K = 4, the damping ratio is ( = 0.5, the natural frequency is (n = 2 rad/s, and the steady-state ramp error is 0.5.

Figure P12.2

a. Design an electrical compensator to obtain (n = 4 while keeping ( = 0.5. Obtain the compensator’s resistances if C= 1 Î¼F.

b. Suppose that with K= 4, the original system gives a satisfactory transient response, but the ramp error must be decreased to 0.05. Design a compensator to do this.

Gp (s) = 1 / s(s2 + 3s + 2)

So that the static velocity error coefficient will be Cu = 5/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 40o.

Figure P12.20

Q:The speed wi of the load is to be controlledThe speed wi of the load is to be controlled with the torque T acting through a fluid coupling (see Figure P12.5). Design a compensator for the specific plant

Gp (s) = Ω2 (s) / T (s) = 4 / s2 + 2s

So that the static velocity error coefficient will be Cu = 20/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:Design a compensator for the plant Gp (s) = 2 /Design a compensator for the plant

Gp (s) = 2 / s2 + 2s

So that the static velocity error coefficient will be Cu = 20/sec and the phase margin at least 45o.

Q:Figure P12.2 shows a pneumatic positioning system, where the displacementFigure P12.2 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume that the pressure p2 is constant, and consider the specific plant

Gp (s) = X (s) / P1 (s) = 1 / s2 + s

Design a compensator for the plant so that Cu = 20/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:The block diagram of a position control system is shownThe block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant

Gp (s) = 1 / s (s + 5) (s + 1)

That will give a static velocity error coefficient of Cu = 50/sec and closed loop roots with a damping ratio of ( = 0.5.

Q:The block diagram of a position control system is shownThe block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant

Gp (s) = 1 / s (s2 + 3s + 2)

So that the static velocity error coefficient will be Cu = 10/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:Consider a unity-feedback system having the open-loop transfer function G(s) =Consider a unity-feedback system having the open-loop transfer function

G(s) = (2n / s(s + 2 ( (n)

Derive the following expression for this system’s phase margin

Q:Automatic guided vehicles are used in factories and warehouses toAutomatic guided vehicles are used in factories and warehouses to transport materials. They require a guide path in the floor and a control system for sensing the guide path and adjusting the steering wheels. Figure P12.28 is a block diagram of such a control system. Obtain the transfer function Gc (s) so that the step response has an overshoot no greater than 20% with a 2%

Figure P12.28

Settling time of no more than 1 s, and a steady-state unit-ramp error of no more than 0.1 m.

NOTE:

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- Physics
- Classical Dynamics Of Particles

b. Evaluate the unit-step response of each design.

c. Evaluate the unit-ramp response of each design.

Figure P11.30

Q:Compare the performance of the critically damped controllers shown inCompare the performance of the critically damped controllers shown in Figure P11.30 with the plant Gp(s) = 1 / I s having the following inputs:

a. A unit-ramp disturbance

b. A sinusoidal disturbance

c. A sinusoidal command input

Q:A certain field-controlled de motor with load has the followingA certain field-controlled de motor with load has the following parameter values.

L = 2 × 10-3 HR = 0.6 Ω

KT = 0.04N · m/AC = 0

I = 6 × 10-5 kg · m2

Compute the gains for a state variable feedback controller using P action to control the motor’s angular position. The desired dominant time constant is 0.5 s. The secondary roots should have a time constant of 0.05 s and a damping ratio of ( = 0.707.

Q:In Figure P11.33 the input u is an acceleration providedIn Figure P11.33 the input u is an acceleration provided by the control system and applied in the horizontal direction to the lower and of the rod. The horizontal displacement of the lower and is y. The linearized from of Newton’s law for small angles gives

a. Put this model into state variable form by letting x1 = ( and x2 = .

b. Construct a state variable feedback controller by letting u = k1x1 + k2 x2. Over what ranges of values of k1 and k2 will the controller stabilize the system? What does this formulation imply about the displacement y?

Figure P11.33

m1 1 = k1(y – x1) – k2 (x1 – x2) – c(1 – 2) – f

m2 2 = k2(x1 – x2) + c(1 – 2) + f

Figure P11.34

Q:Figure P11.35 a is the circuit diagram of a speed-controlFigure P11.35 a is the circuit diagram of a speed-control system in which the dc motor voltage va is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not so sensitive to speed and thus can be used to drive the locomotive at various speeds. The motor voltage va is varied by changing the generator input voltage vf. The voltage va is related to the generator field current if by va = Kjif.

Figure P11.35 b is a diagram of a feedback system for controlling the speed by measuring it with a tachometer and varying the voltage uf. Use the following values in SI units.

Lf = 0.2 Rf = 2 Kt = 1

La = 0.2Ra = 1Kb = KT = 0.5

Kf = 50I = 10c = 20

Develop a state variable model of the plant that includes the generator, the motor, and the load. Include the load torque TL as a disturbance.

Develop a proportional controller assuming all the state variables can be measured. Analyze its steady-state error for a step command input and for a step disturbance.

Figure P11.35

Q:The following equations are the model of the roll dynamicsThe following equations are the model of the roll dynamics of a missile ([Bryson, 1975]). See Figure P11.36.

Figure P11.36

WhereÎ´ = aileron deflection

b = aileron effectiveness constant

u = command signal to the aileron actuator

( = roll angle, ( = roll rate

Using the specific value b = 10 s-1 and ( = 1 s, and assuming that the state variables Î´,(, and ( can be measured, develop a linear state-feedback controller to keep ( near 0. The dominant roots should be s = – 10 ( 10j, and the third root should be s = – 20.

dT / dt = k (ur – up)

For a paper thickness d, the rate of change of the roll radius is

dR/dt = d/2 W

The inertia of the windup roll is I = ( ( W R4/2, where ( is the paper mass density and W is the width of the roll. So R and I are functions of time.

The viscous damping constant for the roll is c. For the armature-controlled motor driving the windup roll, neglect its viscous damping and armature inertia.

Assuming that the paper thickness is small enough so that ( 0 for a short time, develop a state-variable model with the motor voltage e and the paper speed up as the inputs.

Modify the model developed in part (a) to account for R and I being functions of time.

Figure P11.37

Q:An electro-hydraulic positioning system is shown in Figure P11.38. UseAn electro-hydraulic positioning system is shown in Figure P11.38. Use the following values.

Ka = 10 V/A Ki = 10-2 in./V

K2 = 3 Ã 105 sec-3 K3 = 20 V/in.

( = 0.8 (n = 100 rad/sec ( = 0.01 sec

a. Develop a state-variable model of the plant with the controller current ic as the input and the displacement y as the output.

b. Assuming that proportional control is used so that Gc(s) = Kp, develop a state model of the system with y as the input and y as the output. Draw the root locus and use it to determine whether or not the system can be made stable with an appropriate choice for the value of KP.

Figure P11.38

Q:A) The equations of motion of the inverted pendulum modela) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use the linearized model developed in part (b) to design a state variable feedback controller to stabilize the pendulum angle near ( = 0. It is required that the 2% settling time be no greater that 4 s and that the response be non-oscillatory. This means that the dominant root should be real and no greater than – 1. No restriction is placed on the motion of the base. Assume that (, (, x, and can be measured.

Q:Sketch the root locus plot of ms2 + 12s +Sketch the root locus plot of ms2 + 12s + 10 = 0 for m ≥ 2. What is the smallest possible dominant time constant, and what value of m gives this time constant?

Q:The following table gives the measured open-loop response of aThe following table gives the measured open-loop response of a system to a unit-step input. Use the process reaction method to find the controller gains for P. PL and PID control.

Time (min)Response

0………………………………………..0

0.5………………………………………4

1.0…………………………………….20

1.5…………………………………….32

2.0…………………………………….56

2.5…………………………………….84

3.0…………………………………..1 16

3.5…………………………………..140

4.0…………………………………..160

4.5…………………………………..172

5.0…………………………………..184

5.5…………………………………..190

6.0…………………………………..194

7.0…………………………………..196

Q:A liquid in an industrial process must be heated withA liquid in an industrial process must be heated with a heat exchanger through which steam passes. The exit temperature of the liquid is controlled by adjusting the rate of flow of steam through the heat exchanger with the control valve. An open-loop test was performed in which the steam pressure was suddenly changed from 15 to 18 psi above atmospheric pressure. The exit temperature data are shown in the following table. Use the

Time (min) Temperature ( I)

0………………………………….156

1………………………………….157

2………………………………….159

3………………………………….162

4………………………………….167

5………………………………….172

6………………………………….175

7………………………………….179

8………………………………….181

9………………………………….182

10…………………………………183

11…………………………………184

12…………………………………184

Q:Use MATLAB to obtain the root locus plot of 5s2Use MATLAB to obtain the root locus plot of 5s2 + cs + 45 = 0 for c ≥ 0.

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable k ≥ 0. Use the values m = 4 and c = 8. What is the smallest possible dominant time constant and the associated value of k?

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable c ≥ 0. Use the values m = 4 and k = 64. What is the smallest possible dominant time constant and the associated value of c?

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.45 in terms of the variable k2 â¥ 0. Use the values m = 2, c = 8, and k1 = 26. What is the value of k2 required to give ( = 0.707?

Figure P11.43

Figure P11.45

Figure P11.46

Q:Use MATLAB to obtain the root locus plot of s3Use MATLAB to obtain the root locus plot of s3 + 13s2 + 52s + 60 + K = 0 for K ≥ 0. Is it possible for any dominant roots of this equation to have a damping ratio in the range 0.5 ≤ ( ≥ 0.707 and an un-damped natural frequency in the range 3 ≤ (n Q:(a) Use MATLAB to obtain the root locus plot of(a) Use MATLAB to obtain the root locus plot of 2s3 + 12s2 + 16s + K = 0 for K ≥ 0. (b) Obtain the value of K required to give a dominant root pair having ( = 0.707. (c) For this value of K. obtain the unit-step response and the maximum overshoot, and evaluate the effects of the secondary root. The closed-loop transfer function is K/(2s3 + 12s2 + 16s + K).

Q:Use MATLAB to obtain the root locus of the armature-controlledUse MATLAB to obtain the root locus of the armature-controlled dc motor model in terms of the damping constant c, and evaluate the effect on the motor time constant. The characteristic equation is

LaIs2 + (Ra I + cLa)s + cRa + Kb KT = 0

Use the following parameter values:

Kb = KT = 0.1 N · m/AI = 4 × 10-5 kg · m2

Ra = 2 ΩLa = 3 × 10-3 H

Q:In the following equations, identify the root locus plotting parameterIn the following equations, identify the root locus plotting parameter K and its range in terms of the parameter p, where p ≥ 0.

6s2 + 8s + 3p = 0

3s2 + (6 + p)s + 5 + 2p = 0

4s3 + 4ps2 + 2s + p = 0

Q:Consider the two-mass model shown in Figure P11.50. Use theConsider the two-mass model shown in Figure P11.50. Use the following numerical values: m1 = m2 = 1, k1 = 1, k2 = 4, and c2 = 8.

a. Use MATLAB to obtain the root locus plot in terms of the parameter c1.

b. Use the root locus plot to determine the value of c1 required to give a dominant root pair having a damping ratio of ( = 0.707.

c. Use the root locus plot to determine the value of c1 required to give a dominant root that is real and has a time constant equal to 4.

d. Using the value of c1 found in part (c), obtain a plot of the unit-step response.

s3 + 10s2 + 24s + K = 0

a. Use MATLAB to obtain the value of K required to give dominant roots with ( = 0.707. Obtain the three roots corresponding to this value of K.

b. Use MATLAB to obtain the value of K required to give a dominant time constant of ( = 2/3. Obtain the three roots corresponding to this value of K.

Q:Consider the equation s3 + 9s2 + (8 + K)s +Consider the equation

s3 + 9s2 + (8 + K)s + 2 K = 0

a. Use MATLAB to obtain the value of K required to put the dominant root at the breakaway point. Obtain the three roots corresponding to this value of K.

b. Investigate the sensitivity of the dominant root when K varies by ± 10% about the value found in part (a).

Q:Consider the equation s3 + 10s2 + 24s + K =Consider the equation

s3 + 10s2 + 24s + K = 0

Use the sgrid function to determine if it is possible to obtain a dominant root having a damping ratio in the range 0.5 ≤ ( ≥ 0.707, and an un-damped natural frequency in the range 2 ≤ (n ≤ 3. If so, use MATLAB to obtain the value of K required to give the largest possible value of ( (n in the ranges stated.

Q:In Example 10.7.4 the steady-state error for a unit-ramp disturbanceIn Example 10.7.4 the steady-state error for a unit-ramp disturbance is 1/KI.For the gains computed in that example, this error is 1 /25. We want to see if we can make this error smaller by increasing KI. Using the values given for Kp and KD in that example, obtain a root locus plot with KI as the variable. Discuss what happens to the damping ratio and time constant of the dominant root as KI is increased.

Q:In Example 10.8.3 the steady-state error for a unit-ramp commandIn Example 10.8.3 the steady-state error for a unit-ramp command is -4/KI. For the gains computed in that example, this error is 1 /1000. We want to see if we can make this error smaller by increasing KI. Using the values given for Kp and Kd in that example, obtain a root locus plot with K1 as the variable. Discuss what happens to the damping ratio and time constant of the dominant root as K1 is increased.

Q:With the PI gains set to Kp = 6 andWith the PI gains set to Kp = 6 and K1 = 50 for the plant

Gp (s) = 1 / s + 4

The time constant is ( = 0.2 and the damping ratio is ( = 0.707.

a. Suppose the actuator saturation limits are ( 5. Construct a Simulink model to simulate this system with a unit-step command. Use it to plot the output response, the error signal, the actuator output, and the outputs of the proportional term and the integral term versus time.

b. Construct a Simulink model of an anti-windup system for this application. Use it to select and appropriate value for KA and to plot the output response and the actuator output versus time.

Q:Consider a unity feedback system with the plant Gp(s) andConsider a unity feedback system with the plant Gp(s) and the controller Gc(s). PID control action is applied to the plant

Gp(s) = s + 10 / (s + 1) (s + 2)

The PID controller has the transfer function

Gc(s) = Kp (1 + 1/TIs + TDs

Use the values TI = 0.2 and TD = 0.5.

Identify the open-loop poles and zeros.

Identify the root locus parameter K in terms of Kp.

Identify the closed-loop poles and zeros for the case Kp = 10.

Q:With the PI gains set to Kp = 6 andWith the PI gains set to Kp = 6 and K1 = 50 for the plant

Gp (s) = 1 / s + 4

The time constant is ( = 0.2 and the damping ratio is ( = 0.707. Suppose there is a rate limiter of ± 0.1 between the controller and the plant. Construct a Simulink model of the system and use it to determine the effect of the limiter on the speed of response of the system. Use a unit-step command.

Q:A certain dc motor has the following parameter values: L =A certain dc motor has the following parameter values:

L = 2 Ã 10-3 HR = 0.6 Î©

KT = 0.04 N Â· m/Ac = 0

I = 6 Ã 10-5 kg Â· m2

Figure P11.61

Figure P11.61 shows an integral controller using state-variable feedback to control the motor’s angular position.

a. Compute the gains to give a dominant time constant of 0.5 s. The secondary roots should have a time constant of 0.05 s and a damping ratio of ( = 0.707. The fourth root should be s = -20.

b. Construct a Simulink model of the system and use it to plot the response of the system to a step disturbance of magnitude 0.1.

Suppose the motor current is limited to Â± 2 A. Modify the Simulink model to include this saturation, and use the model to obtain plots of the responses to a unit-step command and a step disturbance of magnitude 0.1. Discuss the results.

Q:In parts (a) through (f), sketch the root locus plotIn parts (a) through (f), sketch the root locus plot for the given characteristic equation for K ≥ 0.

s(s + 5) + K .= 0

s(s + 7)(s + 9) + K = 0

s2 + 3s + 5 + K(s + 3) = 0

s(s + 4) + K(s + 5) = 0

s(s2 + 3s + 5) + K = 0

s{s + 3)(s + T) + K(s + 4) = 0

Q:PID control action is applied to the plant Gp(s) = sPID control action is applied to the plant

Gp(s) = s + 10/ (s + 2) (s + 5)

The PID controller has the transfer function

Gc(s) = Kp (1 + 1/TIs + TDs)

Use the values TI = 0.2 and TD = 0.5. Plot the root locus with the proportional gain Kp as the parameter.

Q:Consider the following equation where the parameter p is nonnegative. 4s3Consider the following equation where the parameter p is nonnegative.

4s3 + (25 + 5p)s2 + (16 + 30p)s + 40p = 0

Put the equation in standard root locus form and define a suitable root locus parameter K in terms of the parameter p.

Obtain the poles and zeros, and sketch the root locus plot.

Q:Control of the attitude 0 of a missile by controllingControl of the attitude 0 of a missile by controlling the fin angle cp. as shown in Figure P12.1, involves controlling an inherently unstable plant. Consider the specific plant transfer function

Gp(s) = ( (s) / ( (s) = 1 / 5s2 – 6

Design a PD compensator for this plant. The dominant roots of the closed loop system must have ( = 0.707 and (n = 0.5.

Q:Consider a plant whose open-loop transfer function is G(s) H (s)Consider a plant whose open-loop transfer function is

G(s) H (s) = 1 / s [(s + 2)2 + 9]

The complex poles near the origin give only slightly damped oscillations that are considered undesirable. Insert a gain Kc and a compensator Gc (s)in series to speed up the closed-loop response of the system. Consider the following for Gc(s):

a. The lead compensator

b. The lag compensator

c. The so-called reverse-action compensator

Gc (s) = 1 – T1s / T2s + 1

Obtain the root locus plots for the compensated system using each compensator. Use the compensator parameters. Determine which compensator gives the best response.

Q:A) The equations of motion of the inverted pendulum modela) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that ( is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use the linearized model developed in part (b) to design a series compensator to stabilize the pendulum angle near ( = 0. It is required that the 2% settling time be no greater that 4 s and that the response be nonoscillatory. This means that the dominant root should be real and no greater than -1. No restriction is placed on the motion of the base. Assume that only ( can be measured.

Q:A certain unity feedback system has the following open-loop systemA certain unity feedback system has the following open-loop system transfer function.

G(s) = 5K / s3 + 6s2 + 5s

Obtain the Bode plots and compute the phase and gain margins for

a. K = 2

b. K =20

c. Use the Bode plots to determine the upper limit on K for the system to be stable. Which is the limiting factor: the phase margin or the gain margin?

Q:Figure P12.13 shows a pneumatic positioning system, where the displacementFigure P12.13 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume that

Figure P12.13

The pressure p2 is constant, and consider the specific plant

Gp(s) = X (s)/P1 (s) = 1 / 100s2 + s

With the following series PD compensator is used to control the pressure.

Gc (s) = 2 + 19s

Obtain the Bode plots for this system, and determine the phase and gain margins.

Gp (s) = H2 (s) / Q1 (s) = 25 / 5s2 + 6s + 5

With the following series PI compensator is used to control it.

Gc (s) = 7s + 64 / s

Obtain the Bode plots for this system, and determine the phase and gain margins.

Figure P12.14

Q:Rolling motion of a ship can be reduced by usingRolling motion of a ship can be reduced by using feedback control to vary the angle of the stabilizer fins, much like ailerons are used to control aircraft roll. Figure PI2.15 is the block diagram of a roll control system in which the roll angle is measured and used with proportional control action. Determine the phase and gain margins of the system if (a) Kp = 1, (b) KP = 10, and (c) Kp = 100. Determine the stability properties for each case. If the system is unstable, what effect will this have on the ship roll?

Q:The following transfer functions are the forward transfer function G(s)The following transfer functions are the forward transfer function G(s) and the feedback transfer function H(s) for a system whose closed-loop transfer function is

G (s) / 1 + G (s) H (s)

For each case determine the system type number, the static position and velocity coefficients, and the steady-state errors for unit-step and unit-ramp inputs.

a. G (s) H (s) = 20/s

b. G (s) H (s) = 20 / 5s + 1

c. G (s) H (s) = 7 / s2

Q:Remote control of systems over great distance, such as requiredRemote control of systems over great distance, such as required with robot space probes, may involve relatively large time delays in sending commands and receiving data from the probe. Consider a specific system using proportional control, where the total dead time is D = D1 + D2 = 100 sec (Figure P12.18). The plant is

Gp(s) = 1 / 100s + 1

How large can the gain KP be without the system being unstable?

Q:Hot-air heating control systems for large buildings may involve significantHot-air heating control systems for large buildings may involve significant dead time if there is a large distance between the furnace and the room being

Heated (Figure P12.19). Proportional control applied to the specific plant

Gp(s) = 1 / 0.1s + 1

Has the gain KP = 10. The time units are minutes. How large can the dead time D between the controller and the plant be if the phase margin must be no less than 40o?

Gp (s) = X (s) / P1 (s) = K / s2 + 2s

With K = 4, the damping ratio is ( = 0.5, the natural frequency is (n = 2 rad/s, and the steady-state ramp error is 0.5.

Figure P12.2

a. Design an electrical compensator to obtain (n = 4 while keeping ( = 0.5. Obtain the compensator’s resistances if C= 1 Î¼F.

b. Suppose that with K= 4, the original system gives a satisfactory transient response, but the ramp error must be decreased to 0.05. Design a compensator to do this.

Gp (s) = 1 / s(s2 + 3s + 2)

So that the static velocity error coefficient will be Cu = 5/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 40o.

Figure P12.20

Q:The speed wi of the load is to be controlledThe speed wi of the load is to be controlled with the torque T acting through a fluid coupling (see Figure P12.5). Design a compensator for the specific plant

Gp (s) = Ω2 (s) / T (s) = 4 / s2 + 2s

So that the static velocity error coefficient will be Cu = 20/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:Design a compensator for the plant Gp (s) = 2 /Design a compensator for the plant

Gp (s) = 2 / s2 + 2s

So that the static velocity error coefficient will be Cu = 20/sec and the phase margin at least 45o.

Q:Figure P12.2 shows a pneumatic positioning system, where the displacementFigure P12.2 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume that the pressure p2 is constant, and consider the specific plant

Gp (s) = X (s) / P1 (s) = 1 / s2 + s

Design a compensator for the plant so that Cu = 20/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:The block diagram of a position control system is shownThe block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant

Gp (s) = 1 / s (s + 5) (s + 1)

That will give a static velocity error coefficient of Cu = 50/sec and closed loop roots with a damping ratio of ( = 0.5.

Q:The block diagram of a position control system is shownThe block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant

Gp (s) = 1 / s (s2 + 3s + 2)

So that the static velocity error coefficient will be Cu = 10/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:Consider a unity-feedback system having the open-loop transfer function G(s) =Consider a unity-feedback system having the open-loop transfer function

G(s) = (2n / s(s + 2 ( (n)

Derive the following expression for this system’s phase margin

Q:Automatic guided vehicles are used in factories and warehouses toAutomatic guided vehicles are used in factories and warehouses to transport materials. They require a guide path in the floor and a control system for sensing the guide path and adjusting the steering wheels. Figure P12.28 is a block diagram of such a control system. Obtain the transfer function Gc (s) so that the step response has an overshoot no greater than 20% with a 2%

Figure P12.28

Settling time of no more than 1 s, and a steady-state unit-ramp error of no more than 0.1 m.

Just from $13/Page

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b. Evaluate the unit-step response of each design.

c. Evaluate the unit-ramp response of each design.

Figure P11.30

Q:Compare the performance of the critically damped controllers shown inCompare the performance of the critically damped controllers shown in Figure P11.30 with the plant Gp(s) = 1 / I s having the following inputs:

a. A unit-ramp disturbance

b. A sinusoidal disturbance

c. A sinusoidal command input

Q:A certain field-controlled de motor with load has the followingA certain field-controlled de motor with load has the following parameter values.

L = 2 × 10-3 HR = 0.6 Ω

KT = 0.04N · m/AC = 0

I = 6 × 10-5 kg · m2

Compute the gains for a state variable feedback controller using P action to control the motor’s angular position. The desired dominant time constant is 0.5 s. The secondary roots should have a time constant of 0.05 s and a damping ratio of ( = 0.707.

Q:In Figure P11.33 the input u is an acceleration providedIn Figure P11.33 the input u is an acceleration provided by the control system and applied in the horizontal direction to the lower and of the rod. The horizontal displacement of the lower and is y. The linearized from of Newton’s law for small angles gives

a. Put this model into state variable form by letting x1 = ( and x2 = .

b. Construct a state variable feedback controller by letting u = k1x1 + k2 x2. Over what ranges of values of k1 and k2 will the controller stabilize the system? What does this formulation imply about the displacement y?

Figure P11.33

m1 1 = k1(y – x1) – k2 (x1 – x2) – c(1 – 2) – f

m2 2 = k2(x1 – x2) + c(1 – 2) + f

Figure P11.34

Q:Figure P11.35 a is the circuit diagram of a speed-controlFigure P11.35 a is the circuit diagram of a speed-control system in which the dc motor voltage va is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not so sensitive to speed and thus can be used to drive the locomotive at various speeds. The motor voltage va is varied by changing the generator input voltage vf. The voltage va is related to the generator field current if by va = Kjif.

Figure P11.35 b is a diagram of a feedback system for controlling the speed by measuring it with a tachometer and varying the voltage uf. Use the following values in SI units.

Lf = 0.2 Rf = 2 Kt = 1

La = 0.2Ra = 1Kb = KT = 0.5

Kf = 50I = 10c = 20

Develop a state variable model of the plant that includes the generator, the motor, and the load. Include the load torque TL as a disturbance.

Develop a proportional controller assuming all the state variables can be measured. Analyze its steady-state error for a step command input and for a step disturbance.

Figure P11.35

Q:The following equations are the model of the roll dynamicsThe following equations are the model of the roll dynamics of a missile ([Bryson, 1975]). See Figure P11.36.

Figure P11.36

WhereÎ´ = aileron deflection

b = aileron effectiveness constant

u = command signal to the aileron actuator

( = roll angle, ( = roll rate

Using the specific value b = 10 s-1 and ( = 1 s, and assuming that the state variables Î´,(, and ( can be measured, develop a linear state-feedback controller to keep ( near 0. The dominant roots should be s = – 10 ( 10j, and the third root should be s = – 20.

dT / dt = k (ur – up)

For a paper thickness d, the rate of change of the roll radius is

dR/dt = d/2 W

The inertia of the windup roll is I = ( ( W R4/2, where ( is the paper mass density and W is the width of the roll. So R and I are functions of time.

The viscous damping constant for the roll is c. For the armature-controlled motor driving the windup roll, neglect its viscous damping and armature inertia.

Assuming that the paper thickness is small enough so that ( 0 for a short time, develop a state-variable model with the motor voltage e and the paper speed up as the inputs.

Modify the model developed in part (a) to account for R and I being functions of time.

Figure P11.37

Q:An electro-hydraulic positioning system is shown in Figure P11.38. UseAn electro-hydraulic positioning system is shown in Figure P11.38. Use the following values.

Ka = 10 V/A Ki = 10-2 in./V

K2 = 3 Ã 105 sec-3 K3 = 20 V/in.

( = 0.8 (n = 100 rad/sec ( = 0.01 sec

a. Develop a state-variable model of the plant with the controller current ic as the input and the displacement y as the output.

b. Assuming that proportional control is used so that Gc(s) = Kp, develop a state model of the system with y as the input and y as the output. Draw the root locus and use it to determine whether or not the system can be made stable with an appropriate choice for the value of KP.

Figure P11.38

Q:A) The equations of motion of the inverted pendulum modela) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use the linearized model developed in part (b) to design a state variable feedback controller to stabilize the pendulum angle near ( = 0. It is required that the 2% settling time be no greater that 4 s and that the response be non-oscillatory. This means that the dominant root should be real and no greater than – 1. No restriction is placed on the motion of the base. Assume that (, (, x, and can be measured.

Q:Sketch the root locus plot of ms2 + 12s +Sketch the root locus plot of ms2 + 12s + 10 = 0 for m ≥ 2. What is the smallest possible dominant time constant, and what value of m gives this time constant?

Q:The following table gives the measured open-loop response of aThe following table gives the measured open-loop response of a system to a unit-step input. Use the process reaction method to find the controller gains for P. PL and PID control.

Time (min)Response

0………………………………………..0

0.5………………………………………4

1.0…………………………………….20

1.5…………………………………….32

2.0…………………………………….56

2.5…………………………………….84

3.0…………………………………..1 16

3.5…………………………………..140

4.0…………………………………..160

4.5…………………………………..172

5.0…………………………………..184

5.5…………………………………..190

6.0…………………………………..194

7.0…………………………………..196

Q:A liquid in an industrial process must be heated withA liquid in an industrial process must be heated with a heat exchanger through which steam passes. The exit temperature of the liquid is controlled by adjusting the rate of flow of steam through the heat exchanger with the control valve. An open-loop test was performed in which the steam pressure was suddenly changed from 15 to 18 psi above atmospheric pressure. The exit temperature data are shown in the following table. Use the

Time (min) Temperature ( I)

0………………………………….156

1………………………………….157

2………………………………….159

3………………………………….162

4………………………………….167

5………………………………….172

6………………………………….175

7………………………………….179

8………………………………….181

9………………………………….182

10…………………………………183

11…………………………………184

12…………………………………184

Q:Use MATLAB to obtain the root locus plot of 5s2Use MATLAB to obtain the root locus plot of 5s2 + cs + 45 = 0 for c ≥ 0.

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable k ≥ 0. Use the values m = 4 and c = 8. What is the smallest possible dominant time constant and the associated value of k?

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable c ≥ 0. Use the values m = 4 and k = 64. What is the smallest possible dominant time constant and the associated value of c?

Q:Use MATLAB to obtain the root locus plot of theUse MATLAB to obtain the root locus plot of the system shown in Figure P11.45 in terms of the variable k2 â¥ 0. Use the values m = 2, c = 8, and k1 = 26. What is the value of k2 required to give ( = 0.707?

Figure P11.43

Figure P11.45

Figure P11.46

Q:Use MATLAB to obtain the root locus plot of s3Use MATLAB to obtain the root locus plot of s3 + 13s2 + 52s + 60 + K = 0 for K ≥ 0. Is it possible for any dominant roots of this equation to have a damping ratio in the range 0.5 ≤ ( ≥ 0.707 and an un-damped natural frequency in the range 3 ≤ (n Q:(a) Use MATLAB to obtain the root locus plot of(a) Use MATLAB to obtain the root locus plot of 2s3 + 12s2 + 16s + K = 0 for K ≥ 0. (b) Obtain the value of K required to give a dominant root pair having ( = 0.707. (c) For this value of K. obtain the unit-step response and the maximum overshoot, and evaluate the effects of the secondary root. The closed-loop transfer function is K/(2s3 + 12s2 + 16s + K).

Q:Use MATLAB to obtain the root locus of the armature-controlledUse MATLAB to obtain the root locus of the armature-controlled dc motor model in terms of the damping constant c, and evaluate the effect on the motor time constant. The characteristic equation is

LaIs2 + (Ra I + cLa)s + cRa + Kb KT = 0

Use the following parameter values:

Kb = KT = 0.1 N · m/AI = 4 × 10-5 kg · m2

Ra = 2 ΩLa = 3 × 10-3 H

Q:In the following equations, identify the root locus plotting parameterIn the following equations, identify the root locus plotting parameter K and its range in terms of the parameter p, where p ≥ 0.

6s2 + 8s + 3p = 0

3s2 + (6 + p)s + 5 + 2p = 0

4s3 + 4ps2 + 2s + p = 0

Q:Consider the two-mass model shown in Figure P11.50. Use theConsider the two-mass model shown in Figure P11.50. Use the following numerical values: m1 = m2 = 1, k1 = 1, k2 = 4, and c2 = 8.

a. Use MATLAB to obtain the root locus plot in terms of the parameter c1.

b. Use the root locus plot to determine the value of c1 required to give a dominant root pair having a damping ratio of ( = 0.707.

c. Use the root locus plot to determine the value of c1 required to give a dominant root that is real and has a time constant equal to 4.

d. Using the value of c1 found in part (c), obtain a plot of the unit-step response.

s3 + 10s2 + 24s + K = 0

a. Use MATLAB to obtain the value of K required to give dominant roots with ( = 0.707. Obtain the three roots corresponding to this value of K.

b. Use MATLAB to obtain the value of K required to give a dominant time constant of ( = 2/3. Obtain the three roots corresponding to this value of K.

Q:Consider the equation s3 + 9s2 + (8 + K)s +Consider the equation

s3 + 9s2 + (8 + K)s + 2 K = 0

a. Use MATLAB to obtain the value of K required to put the dominant root at the breakaway point. Obtain the three roots corresponding to this value of K.

b. Investigate the sensitivity of the dominant root when K varies by ± 10% about the value found in part (a).

Q:Consider the equation s3 + 10s2 + 24s + K =Consider the equation

s3 + 10s2 + 24s + K = 0

Use the sgrid function to determine if it is possible to obtain a dominant root having a damping ratio in the range 0.5 ≤ ( ≥ 0.707, and an un-damped natural frequency in the range 2 ≤ (n ≤ 3. If so, use MATLAB to obtain the value of K required to give the largest possible value of ( (n in the ranges stated.

Q:In Example 10.7.4 the steady-state error for a unit-ramp disturbanceIn Example 10.7.4 the steady-state error for a unit-ramp disturbance is 1/KI.For the gains computed in that example, this error is 1 /25. We want to see if we can make this error smaller by increasing KI. Using the values given for Kp and KD in that example, obtain a root locus plot with KI as the variable. Discuss what happens to the damping ratio and time constant of the dominant root as KI is increased.

Q:In Example 10.8.3 the steady-state error for a unit-ramp commandIn Example 10.8.3 the steady-state error for a unit-ramp command is -4/KI. For the gains computed in that example, this error is 1 /1000. We want to see if we can make this error smaller by increasing KI. Using the values given for Kp and Kd in that example, obtain a root locus plot with K1 as the variable. Discuss what happens to the damping ratio and time constant of the dominant root as K1 is increased.

Q:With the PI gains set to Kp = 6 andWith the PI gains set to Kp = 6 and K1 = 50 for the plant

Gp (s) = 1 / s + 4

The time constant is ( = 0.2 and the damping ratio is ( = 0.707.

a. Suppose the actuator saturation limits are ( 5. Construct a Simulink model to simulate this system with a unit-step command. Use it to plot the output response, the error signal, the actuator output, and the outputs of the proportional term and the integral term versus time.

b. Construct a Simulink model of an anti-windup system for this application. Use it to select and appropriate value for KA and to plot the output response and the actuator output versus time.

Q:Consider a unity feedback system with the plant Gp(s) andConsider a unity feedback system with the plant Gp(s) and the controller Gc(s). PID control action is applied to the plant

Gp(s) = s + 10 / (s + 1) (s + 2)

The PID controller has the transfer function

Gc(s) = Kp (1 + 1/TIs + TDs

Use the values TI = 0.2 and TD = 0.5.

Identify the open-loop poles and zeros.

Identify the root locus parameter K in terms of Kp.

Identify the closed-loop poles and zeros for the case Kp = 10.

Q:With the PI gains set to Kp = 6 andWith the PI gains set to Kp = 6 and K1 = 50 for the plant

Gp (s) = 1 / s + 4

The time constant is ( = 0.2 and the damping ratio is ( = 0.707. Suppose there is a rate limiter of ± 0.1 between the controller and the plant. Construct a Simulink model of the system and use it to determine the effect of the limiter on the speed of response of the system. Use a unit-step command.

Q:A certain dc motor has the following parameter values: L =A certain dc motor has the following parameter values:

L = 2 Ã 10-3 HR = 0.6 Î©

KT = 0.04 N Â· m/Ac = 0

I = 6 Ã 10-5 kg Â· m2

Figure P11.61

Figure P11.61 shows an integral controller using state-variable feedback to control the motor’s angular position.

a. Compute the gains to give a dominant time constant of 0.5 s. The secondary roots should have a time constant of 0.05 s and a damping ratio of ( = 0.707. The fourth root should be s = -20.

b. Construct a Simulink model of the system and use it to plot the response of the system to a step disturbance of magnitude 0.1.

Suppose the motor current is limited to Â± 2 A. Modify the Simulink model to include this saturation, and use the model to obtain plots of the responses to a unit-step command and a step disturbance of magnitude 0.1. Discuss the results.

Q:In parts (a) through (f), sketch the root locus plotIn parts (a) through (f), sketch the root locus plot for the given characteristic equation for K ≥ 0.

s(s + 5) + K .= 0

s(s + 7)(s + 9) + K = 0

s2 + 3s + 5 + K(s + 3) = 0

s(s + 4) + K(s + 5) = 0

s(s2 + 3s + 5) + K = 0

s{s + 3)(s + T) + K(s + 4) = 0

Q:PID control action is applied to the plant Gp(s) = sPID control action is applied to the plant

Gp(s) = s + 10/ (s + 2) (s + 5)

The PID controller has the transfer function

Gc(s) = Kp (1 + 1/TIs + TDs)

Use the values TI = 0.2 and TD = 0.5. Plot the root locus with the proportional gain Kp as the parameter.

Q:Consider the following equation where the parameter p is nonnegative. 4s3Consider the following equation where the parameter p is nonnegative.

4s3 + (25 + 5p)s2 + (16 + 30p)s + 40p = 0

Put the equation in standard root locus form and define a suitable root locus parameter K in terms of the parameter p.

Obtain the poles and zeros, and sketch the root locus plot.

Q:Control of the attitude 0 of a missile by controllingControl of the attitude 0 of a missile by controlling the fin angle cp. as shown in Figure P12.1, involves controlling an inherently unstable plant. Consider the specific plant transfer function

Gp(s) = ( (s) / ( (s) = 1 / 5s2 – 6

Design a PD compensator for this plant. The dominant roots of the closed loop system must have ( = 0.707 and (n = 0.5.

Q:Consider a plant whose open-loop transfer function is G(s) H (s)Consider a plant whose open-loop transfer function is

G(s) H (s) = 1 / s [(s + 2)2 + 9]

The complex poles near the origin give only slightly damped oscillations that are considered undesirable. Insert a gain Kc and a compensator Gc (s)in series to speed up the closed-loop response of the system. Consider the following for Gc(s):

a. The lead compensator

b. The lag compensator

c. The so-called reverse-action compensator

Gc (s) = 1 – T1s / T2s + 1

Obtain the root locus plots for the compensated system using each compensator. Use the compensator parameters. Determine which compensator gives the best response.

Q:A) The equations of motion of the inverted pendulum modela) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that ( is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use the linearized model developed in part (b) to design a series compensator to stabilize the pendulum angle near ( = 0. It is required that the 2% settling time be no greater that 4 s and that the response be nonoscillatory. This means that the dominant root should be real and no greater than -1. No restriction is placed on the motion of the base. Assume that only ( can be measured.

Q:A certain unity feedback system has the following open-loop systemA certain unity feedback system has the following open-loop system transfer function.

G(s) = 5K / s3 + 6s2 + 5s

Obtain the Bode plots and compute the phase and gain margins for

a. K = 2

b. K =20

c. Use the Bode plots to determine the upper limit on K for the system to be stable. Which is the limiting factor: the phase margin or the gain margin?

Q:Figure P12.13 shows a pneumatic positioning system, where the displacementFigure P12.13 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume that

Figure P12.13

The pressure p2 is constant, and consider the specific plant

Gp(s) = X (s)/P1 (s) = 1 / 100s2 + s

With the following series PD compensator is used to control the pressure.

Gc (s) = 2 + 19s

Obtain the Bode plots for this system, and determine the phase and gain margins.

Gp (s) = H2 (s) / Q1 (s) = 25 / 5s2 + 6s + 5

With the following series PI compensator is used to control it.

Gc (s) = 7s + 64 / s

Obtain the Bode plots for this system, and determine the phase and gain margins.

Figure P12.14

Q:Rolling motion of a ship can be reduced by usingRolling motion of a ship can be reduced by using feedback control to vary the angle of the stabilizer fins, much like ailerons are used to control aircraft roll. Figure PI2.15 is the block diagram of a roll control system in which the roll angle is measured and used with proportional control action. Determine the phase and gain margins of the system if (a) Kp = 1, (b) KP = 10, and (c) Kp = 100. Determine the stability properties for each case. If the system is unstable, what effect will this have on the ship roll?

Q:The following transfer functions are the forward transfer function G(s)The following transfer functions are the forward transfer function G(s) and the feedback transfer function H(s) for a system whose closed-loop transfer function is

G (s) / 1 + G (s) H (s)

For each case determine the system type number, the static position and velocity coefficients, and the steady-state errors for unit-step and unit-ramp inputs.

a. G (s) H (s) = 20/s

b. G (s) H (s) = 20 / 5s + 1

c. G (s) H (s) = 7 / s2

Q:Remote control of systems over great distance, such as requiredRemote control of systems over great distance, such as required with robot space probes, may involve relatively large time delays in sending commands and receiving data from the probe. Consider a specific system using proportional control, where the total dead time is D = D1 + D2 = 100 sec (Figure P12.18). The plant is

Gp(s) = 1 / 100s + 1

How large can the gain KP be without the system being unstable?

Q:Hot-air heating control systems for large buildings may involve significantHot-air heating control systems for large buildings may involve significant dead time if there is a large distance between the furnace and the room being

Heated (Figure P12.19). Proportional control applied to the specific plant

Gp(s) = 1 / 0.1s + 1

Has the gain KP = 10. The time units are minutes. How large can the dead time D between the controller and the plant be if the phase margin must be no less than 40o?

Gp (s) = X (s) / P1 (s) = K / s2 + 2s

With K = 4, the damping ratio is ( = 0.5, the natural frequency is (n = 2 rad/s, and the steady-state ramp error is 0.5.

Figure P12.2

a. Design an electrical compensator to obtain (n = 4 while keeping ( = 0.5. Obtain the compensator’s resistances if C= 1 Î¼F.

b. Suppose that with K= 4, the original system gives a satisfactory transient response, but the ramp error must be decreased to 0.05. Design a compensator to do this.

Gp (s) = 1 / s(s2 + 3s + 2)

So that the static velocity error coefficient will be Cu = 5/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 40o.

Figure P12.20

Q:The speed wi of the load is to be controlledThe speed wi of the load is to be controlled with the torque T acting through a fluid coupling (see Figure P12.5). Design a compensator for the specific plant

Gp (s) = Ω2 (s) / T (s) = 4 / s2 + 2s

So that the static velocity error coefficient will be Cu = 20/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:Design a compensator for the plant Gp (s) = 2 /Design a compensator for the plant

Gp (s) = 2 / s2 + 2s

So that the static velocity error coefficient will be Cu = 20/sec and the phase margin at least 45o.

Q:Figure P12.2 shows a pneumatic positioning system, where the displacementFigure P12.2 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume that the pressure p2 is constant, and consider the specific plant

Gp (s) = X (s) / P1 (s) = 1 / s2 + s

Design a compensator for the plant so that Cu = 20/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:The block diagram of a position control system is shownThe block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant

Gp (s) = 1 / s (s + 5) (s + 1)

That will give a static velocity error coefficient of Cu = 50/sec and closed loop roots with a damping ratio of ( = 0.5.

Q:The block diagram of a position control system is shownThe block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant

Gp (s) = 1 / s (s2 + 3s + 2)

So that the static velocity error coefficient will be Cu = 10/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.

Q:Consider a unity-feedback system having the open-loop transfer function G(s) =Consider a unity-feedback system having the open-loop transfer function

G(s) = (2n / s(s + 2 ( (n)

Derive the following expression for this system’s phase margin

Q:Automatic guided vehicles are used in factories and warehouses toAutomatic guided vehicles are used in factories and warehouses to transport materials. They require a guide path in the floor and a control system for sensing the guide path and adjusting the steering wheels. Figure P12.28 is a block diagram of such a control system. Obtain the transfer function Gc (s) so that the step response has an overshoot no greater than 20% with a 2%

Figure P12.28

Settling time of no more than 1 s, and a steady-state unit-ramp error of no more than 0.1 m.

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