- For the system of Figure P10.5, do the following:a. Plot the Bode magnitude and phase plots.b. Assuming a second-order approximation, estimate the transient response of the system if K = 2.c. Use
- The unity feedback system of Figure P7.1,whereis to be designed to meet the following specifications: steady-state error for a unit step input = 0.1; damping ratio = 0.5; natural frequency = √10.
- The open-loop frequency response shown in Figure P10.8 was experimentally obtained from a unity feedback system. Estimate the percent overshoot and steady-state error of the closed-loop system. 60 40
- For the system shown in Figure P7.25, find the sensitivity of the steady state error for changes in K1 and in K2, when K1 = 100 and K2 = 0.1. Assume step inputs for both the input and the
- In Figure P7.13, P(s) = 5/s ; H(s) = 1anda. Find F(s) and G(s).b. Find the value of K that will result in zero steady-state error for a unit step input. C(s) R(s) (s+ 1)(s+ 3)(s² + 2s + 20) 200K
- a. Show that the sensitivity to plant changes in the system of Figure P7.13 iswhere L(s) = G(s)P(s)H(s) andb. Show that ST:P(s) = T(s)H(s)/F(s) = 1 for all values of s. C(s) _ F(s) L(s) T(s) = R(s)
- Given the system shown in Figure P7.24, find the sensitivity of the steady-state error to parameter a. Assume a step input. Plot the sensitivity as a function of parameter a. R(s) + K Cls) s(s + 2)(s
- Given the system shown in Figure P7.23, do the following:a. Derive the expression for the error, E(s) = R(s) - C(s), in terms of R(s) and D(s).b. Derive the steady-state error, e (∞), if R(s) and
- Derive Eq. (7.69) in the text.Data From Eq. (7.69): G1(s)G2(s) R(s) 1+ G(s)G2(s)H(s)] e(00) = lim sE(s) = lim s 8-0 G2(s) D(s) + G(s)G2(s)H(s)
- A dynamic voltage restorer (DVR) is a device that is connected in series to a power supply. It continuously monitors the voltage delivered to the load, and compensates voltage sags by applying the
- For the system shown in Figure P7.21, use MATLAB to find the following for K = 10, and K = 106:Figure P7.21:a. The system typeb. Kp, Kv, and Kac. The steady-state error for inputs of 30u(t), 30tu(t),
- For the system shown in Figure P7.20,a. What is the system type?b. What is the appropriate static error constant?c. What is the value of the appropriate static error constant?d. What is the
- Given the system shown in Figure P7.19, find the following:a. The system typeb. The value of K to yield 0.1% error in the steady state. R(s) + (s+ 1) (s+2) C(s) K FIGURE P7.19
- For each system shown in Figure P7.18, find the appropriate static error constant as well as the steady-state error, r(∞) - c(∞), for unit step, ramp, and parabolic inputs. R(s) s+ 4 C(s) 5 (s+
- For each system shown in Figure P7.17, find the following:a. The system typeb. The appropriate static error constantc. The input waveform to yield a constant errord. The steady-state error for a unit
- Derive Eq. (7.72) in the text, which is the final value of the actuating signal for non unity feedback systems.Data From Eq. (7.72): sR(s)G(s) eal (0) = lim- %3D -01+ G2(s)H1(s)
- In Figure P7.16, let G(s) = 5 and P(s) = 7/s + 2.a. Calculate the steady-state error due to a command input R(s) = 3/s with D(s) = 0.b. Verify the result of Part a using Simulink.c. Calculate the
- Design the values of K1 and K2 in the system of Figure P7.15 to meet the following specifications: Steady-state error component due to a unit step disturbance is -0.00001; steady-state error
- Find the total steady-state error due to a unit step input and a unit step disturbance in the system of Figure P7.14. D(s) R(s) + 1 100 C(s) s+5 s+2 FIGURE P7.14
- The transfer function from elevator deflection to altitude change in a Tower Trainer 60 Unmanned Aerial Vehicle isAn autopilot is built around the aircraft as shown in Figure P7.13, with F(s) = H(s)
- The system of Figure P7.12 is to have the following specifications: Kv = 20; ζ = 0.7. Find the values of K1 and Kf required for the specifications of the system to be met. O(s) + A(s) K1 s(s +
- For the system shown in Figure P7.11, use MATLAB to find the following:a. The system typeb. Kp, Kv, and Kac. The steady-state error for inputs of 100u(t), 100tu(t), and 100t2u(t) R(s) (s+9) 6(s +
- Repeat Problem33 for the system shown in Figure P7.10.Data From Problem 33:Given the system in Figure P7.9, find the following:a. The closed-loop transfer functionb. The system typec. The
- Given the system in Figure P7.9, find the following:a. The closed-loop transfer functionb. The system typec. The steady-state error for an input of 5u(t)d. The steady-state error for an input of
- Given the unity feedback control system of Figure P7.1, wherefind the following:a. K and a to yield Kv = 1000 and a 20% overshootb. K and a to yield a 1% error in the steady state and a 10%
- Given the unity feedback control system of Figure P7.1, wherefind the values of n, K, and a in order to meet specifications of 12% overshoot and Kv = 110. K G(s) = (v s)us
- The unity feedback system of Figure P7.1, whereis to be designed to meet the following requirements: The steady-state position error for a unit ramp input equals 1/10; the closed-loop poles will be
- The unity feedback system of Figure P7.1 has a transfer functionand is to follow a ramp input, r(t) = tu(t), so that the steady-state output position differs from the input position by 0.01 of the
- A second-order, unity feedback system is to follow a ramp input with the following specifications: the steady-state output position shall differ from the input position by 0.01 of the input velocity;
- For the unity feedback system of Figure P7.1, wherefind the minimum possible steady-state position error if a unit ramp is applied. What places the constraint upon the error? K G(s) %3! s(s + 4)(s
- Given the unity feedback system of Figure P7.1, wherefind the value of K to yield a steady-state error of 8%. K(s + 6) G(s) (s+ 2)(s + 10s + 29)
- For the unity feedback system of Figure P7.1, wherefind the value of K to yield a steady-state error of 0.4 for a ramp input of 27tu(t). K(s+ 13)(s+ 19) G(s) s(s + 6)(s +9)(s+22)
- For the unity feedback system of Figure P7.1, wherea. Find the system type.b. What error can be expected for an input of 12u(t)?c. What error can be expected for an input of 12tu(t)? K (s? + 6s + 6)
- The unity feedback system of Figure P7.1, whereis to have 1/6000 error between an input of 10tu(t) and the output in the steady state.a. Find K and n to meet the specification.b. What are Kp, Kv, and
- Given the system of Figure P7.8, design the value of K so that for an input of 100tu(t), there will be a 0.01 error in the steady state. R(s) K C(s) s(s + 2) 5s K FIGURE P7.8
- Given the unity feedback system of Figure P7.1, wherefind the value of Ka so that a ramp input of slope 30 will yield an error of 0.005 in the steady state when compared to the output. K(s + a) s(s +
- For the system shown in Figure P7.7,a. What value of K will yield a steady-state error in position of 0.01 for an input of (1/10)t?b. What is the Kv for the value of K found in Part a?c. What is the
- The steady-state error is defined to be the difference in position between input and output as time approaches infinity. Let us define a steady-state velocity error, which is the difference in
- Name two methods for calculating the steady-state error for systems represented in state space.
- What are the restrictions on the feedforward transfer function G2(s) in the system of Figure P7.6 to obtain zero steady-state error for step inputs if:a. G1(s) is a Type 0 transfer function;b. G1(s)
- For the unity feedback system of Figure P7.1, wherefind the value of K to yield a static error constant of 10,000. K(s + 2)(s+ 4)(s + 6) (s +5)(s + 7) G(s)
- For the unity feedback system of Figure P6.3 withdetermine the range of K for stability. K(s + 3)(s + 5) G(s) : (s- 2)(s- 4) %3D
- In the system of Figure P6.3, letFind the range of K for closed-loop stability when:a. a < 0; b < 0b. a < 0; b > 0c. a > 0; b < 0d. a > 0; b > 0 K(s - a) G(s) = %3D (9- s)s
- For the unity feedback system of Figure P6.3 withdetermine the range of K to ensure stability. K(s + 6) G(s): s(s + 1)(s+ 4) %3D
- Determine if the unity feedback system of Figure P6.3 with K(s? + 1) G(s) = %3D (s + 1)(s +2)
- Given the unity feedback system of Figure P6.3 withtell how many closed-loop poles are located in the right half-plane, in the left half-plane, and on the jω-axis. 8 G(s) s(s6 – 255 - s4 + 2s3 +
- Using the Routh-Hurwitz criterion and the unity feedback system of Figure P6.3 withtell whether or not the closed-loop system is stable. 1 G(s) = 2s4 + 5s3 + s? + 2s
- Given the unity feedback system of Figure P6.3 withtell how many poles of the closed-loop transfer function lie in the right half-plane, in the left half-plane, and on the jω-axis. 84 G(s) = s(s +
- In the system of Figure P6.3, letFind the range of K for closed-loop stability. K(s + 1) G(s) = s(s – 2)(s+ 3)
- Consider the unity feedback system of Figure P6.3 witha. Using the Routh-Hurwitz criterion, find the region of the s-plane where the poles of the closed-loop system are located.b. Use MATLAB to
- Figure P5.34 shows the diagram of an inverting operational amplifier.a. Assuming an ideal operational amplifier, use a similar procedure to the one outlined in Problem 52 to find the system
- Given the block diagram of the active suspension system shown in Figure P5.36, (Lin, 1997)a. Find the transfer function from a road disturbance r to the error signal e.b. Use the transfer function in
- Motion control, which includes position or force control, is used in robotics and machining. Force control requires the designer to consider two phases: contact and noncontact motions. Figure
- In Figure P7.16, the plant, P(s) = 48,500/s2 + 2.89s; represents the dynamics of a robotic manipulator joint. The system’s output, C(s), is the joint’s angular position (Low, 2005). The
- Letin Figure P8.3.a. Plot the root locus.b. Write an expression for the closed-loop transfer function at the point where the three closed-loop poles meet. Ks+ 3 G(s) = (9 + s);s
- Using the Nyquist criterion, find the range of K for stability for each of the systems in Figure P10.4. R(s) + K C(s) (s + 2) (s + 4)(s + 6) System 1 R(s) + K(s2 – 4s + 13) C(s) (s + 2)(s + 4)
- Use the results of Problem 17 to estimate the percent overshoot if the gain term in the numerator of the forward path of each part of the problem is respectively changed as follows:a. From 10 to 30b.
- Define phase margin.
- Write a program in MATLAB that will do the following:a. Allow a value of gain, K, to be entered from the keyboardb. Display the closed-loop magnitude and phase frequency response plots of a unity
- Name two different frequency response characteristics that can be used to determine a system’s transient response.
- Use MATLAB’s LTI Viewer with the Nichols plot to find the gain margin, phase margin, zero dB frequency, and 180° frequency for a unity feedback system with the forward-path transfer function 5(s+
- Name three different methods of finding the closed-loop frequency response from the open-loop transfer function.
- Write a program in MATLAB that will do the following:a. Make a Nichols plot of an open-loop transfer functionb. Allow the user to read the Nichols plot display and enter the value of Mpc. Make
- Briefly explain how to find the static error constant from the Bode magnitude plot.
- Using Bode plots, estimate the transient response of the systems in Figure P10.6. R(S) + C(s) 100(s + 2) S(s + 1)(s + 4) System 1 R(s) + E(s) 50(s + 3)(s + 5) C(s) s(s + 2)(s +4)(s + 6) System 2
- Describe the change in the open-loop frequency response magnitude plot if time delay is added to the plant.
- If the phase response of a pure time delay were plotted on a linear phase versus linear frequency plot, what would be the shape of the curve?
- Write a program in MATLAB that will use an open-loop transfer function, G(s), to do the following: a. Make a Bode plotb. Use frequency response methods to estimate the percent overshoot,
- When successively extracting component transfer functions from experimental frequency response data, how do you know when you are finished?
- The Bode plots for a plant, G(s), used in a unity feedback system are shown in Figure P10.7. Do the following:a. Find the gainmargin, phasemargin, zero dBfrequency, 180° frequency, and the
- Consider the system in Figure P10.9.a. Find the phase margin if the system is stable for time delays of 0, 0.1, 0.2, 0.5, and 1 second.b. Find the gain margin if the system is stable for each of the
- The open-loop dynamics from dc voltage armature to angular position of a robotic manipulator joint is given by P(s) = 48500 / s2 + 2:89s (Low, 2005).a. Draw by hand a Bode plot using asymptotic
- Fruit flies’ flight dynamics are interesting to study because they provide a proof-of-concept framework and inspiration for the invention of man-made machines. In an experiment (Roth, 2012), flies
- As discussed in Section 10.12, the Nyquist stability criterion can be applied to systems with pure time delay without the need for rational approximations as required in Problems 8.72 and 9.59. You
- Design the value of gain, K, for a gain margin of 10 dB in the unity feedback system of Figure P11.1 ifa.b.c. K G(s) (s + 4)(s + 10)(s + 15)
- Why is the phase margin increased above that desired when designing a lag compensator?
- The unity feedback system of Figure P11.1 withis operating with 15% overshoot. Using frequency response techniques, design a compensator to yield Kv = 50 with the phase-margin frequency and phase
- The unity feedback system shown in Figure P11.1 withis operating with 15% overshoot. Using frequency response methods, design a compensator to yield a fivefold improvement in steady-state error
- Design a lag compensator so that the system of Figure P11.1 whereoperates with a 45° phase margin and a static error constant of 100.
- When designing a lag-lead network, what difference is there in the design of the lag portion as compared to a separate lag compensator?
- For the system of Problem 6, do the following:a. Use frequency response methods to find the gain, K, required to yield about 15% overshoot. Make any required second-order approximations.b. Use
- Based upon your answer to Question 7, explain why lead networks do not cause instability.
- From the Bode diagram viewpoint, briefly explain how a lead network increases the speed of the transient response.
- Compare the following for uncompensated and lag-compensated systems designed to yield the same transient response: low-frequency gain, phase-margin frequency, gain curve value around the phase-margin
- Given the unity feedback system of Figure P11.1 withdo the following:a. Use frequency response methods to design a lag compensator to yield Kv = 1000 and 15% overshoot for the step response. Make any
- Why is a correction factor added to the phase margin required to meet the transient response?
- Design a PI controller for the system of Figure 11.2 that will yield zero steady-state error for a ramp input and a 9.48% overshoot for a step input. Motor Desired Power and Shaft Shaft position
- Write a MATLAB program that will design a PI controller assuming a second-order approximation as follows:a. Allow the user to input from the keyboard the desired percent overshootb. Design a PI
- Design a compensator for the unity feedback system of Figure P11.1 withto yield a Kv = 4 and a phase margin of 45°.
- Consider the unity feedback system of Figure P11.1 with The uncompensated system has about 55% overshoot and a peak time of 0.5 second when Kv = 10. Do the following:a. Use frequency response
- The unity feedback system of Figure P11.1 with is operating with 20% overshoot. [Section: 11.4]a. Find the settling time.b. Find Kp.c. Find the phase margin and the phase-margin frequency.d.
- Repeat the design of Example 11.3 in the text using a PD controller. Example 11.3 Design Lead Compensation Design D PROBLEM: Given the system of Figure 11.2, design a lead compensator to yield a 20%
- Repeat Problem 13 using a PD compensator. Data from Problem 13Consider the unity feedback system of Figure P11.1 with The uncompensated system has about 55% overshoot and a peak time of
- Write a MATLAB program that will design a lead compensator assuming second-order approximations as follows:a. Allow the user to input from the keyboard the desired percent overshoot, peak time, and
- Repeat Problem 17 for a PD controller.Data from Problem 17Write a MATLAB program that will design a lead compensator assuming second-order approximations as follows:a. Allow the user to input from
- Use frequency response methods to design a lag-lead compensator for a unity feedback system whereand the following specifications are to be met: percent overshoot = 10%, settling time = 0.2 second,
- Write a MATLAB program that will design a lag-lead compensator assuming second-order approximations as follows:a. Allow the user to input from the keyboard the desired percent overshoot, settling
- Under what conditions can inspection of the signal-flow graph of a system yield immediate determination of observability?
- Design an observer for the plant represented in cascade form. Transform the plant to observer canonical form for the design. Then transform the design back to cascade form. The characteristic
- An inverted pendulum mounted on a motor-driven cart was presented in Problem 30, Chapter 3. Its state space model was linearized (Prasad, 2012) around a stationary point, (x0 = 0), corresponding to