Answer: Given a unity feedback system with the

Find analytical expressions for the magnitude and phase response for each G(s) below. [Section: 10.1] a. G(s) = \(\frac{1}{s(s+2)(s+4)}\) b. G(s) = \(\frac{(s+5)}{(s+2)(s+4)}\) c. G(s) = \(\frac{(s+3)(s+5)}{s(s+2)(s+4)}\)

For each function in Problem 1, make a plot of the log-magnitude and the phase, using log-frequency in rad/s as the ordinate. Do not use asymptotic approximations. [Section: 10.1]

For each function in Problem 1, make a polar plot of the frequency response. [Section: 10.1]

For each function in Problem 1, sketch the Bode asymptotic magnitude and asymptotic phase plots. Compare your results with your answers to Problem 1. [Section: 10.2]

Sketch the Nyquist diagram for each of the systems in Figure P10.1. [Section: 10.4]

Draw the polar plot from the separate magnitude and phase curves shown in Figure P10.2. [Section: 10.1]

Draw the separate magnitude and phase curves from the polar plot shown in Figure P10.3. [Section: 10.1]

Write a program in MATLAB that will do the following: a. Plot the Nyquist diagram of a system b. Display the real-axis crossing value and frequency Apply your program to a unity feedback system with G(s) = \(\frac{K(s+5)}{\left(s^2+6 s+100\right)\left(s^2+4 s+25\right)}\)

Using the Nyquist criterion, find out whether each system of Problem 5 is stable. [Section: 10.3]

Using the Nyquist criterion, find the range of K for stability for each of the systems in Figure P10.4. [Section: 10.3]

For each system of Problem 10, find the gain margin and phase margin if the value of K in each part of Problem 10 is [Section: 10.6] a. K = 1000 b. K = 100 c. K = 0:1

Write a program in MATLAB that will do the following: a. Allow a value of gain, K, to be entered from the keyboard b. Display the Bode plots of a system for the entered value of K c. Calculate and display the gain and phase margin for the entered value of K Test your program on a unity feedback system with G(s)= K/[s(s + 3)(s + 12)].

Use MATLAB's LTI Viewer to find the gain margin, phase margin, zero dB frequency, and (180^{\circ}\) frequency for a unity feedback system with G(s) = \(\frac{8000}{(s+6)(s+20)(s+35)}\) Use the following methods: a. The Nyquist diagram b. Bode plots

Derive Eq. (10.54), the closed-loop bandwidth in terms of \(\zeta\) and \(\omega_n\0 of a two-pole system. [Section: 10.8]

For each closed-loop system with the following performance characteristics, find the closed-loop bandwidth: [Section: 10.8] a. \(\zeta\) = 0.2, \(T_s\) = 3 seconds b. \(\zeta\) = 0.2, \(T_p\) = 3 seconds c. \(T_s\) = 4 seconds, \(T_p\) = 2 seconds d. \(\zeta\) = 0.3, \(T_r\) = 4 seconds.

Consider the unity feedback system of Figure 10.10. For each G(s) that follows, use the M and N circles to make a plot of the closed-loop frequency response: [Section: 10.9] a. G(s) = \(\frac{10}{s(s+1)(s+2)}\) b. G(s) = \(\frac{1000}{(s+3)(s+4)(s+5)(s+6)}\) c. G(s) = \(\frac{50(s+3)}{s(s+2)(s+4)}\)

Repeat Problem 16, using the Nichols chart in place of the M and N circles. [Section: 10.9]

Using the results of Problem 16, estimate the percent overshoot that can be expected in the step response for each system shown. [Section: 10.10]

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: [Section: 10.10] a. From 10 to 30 b. From 1000 to 2500 c. From 50 to 75

Write a program in MATLAB that will do the following: a. Allow a value of gain, K, to be entered from the keyboard b. Display the closed-loop magnitude and phase frequency response plots of a unity feedback system with an open loop transfer function, KG(s) c. Calculate and display the peak magnitude, frequency of the peak magnitude, and bandwidth for the closed-loop frequency response and the entered value of K Test your program on the system of Figure P10.5 for K = 50.

Use MATLAB's LTI Viewer with the Nichols plot to find the gain margin, phase margin, zero dB frequency, and \(180^{\circ}\) frequency for a unity feedback system with the forward-path transfer function G(s) = \(\frac{5(s+6)}{s\left(s^2+4 s+15\right)}\)

Write a program in MATLAB that will do the following: a. Make a Nichols plot of an open-loop transfer function b. Allow the user to read the Nichols plot display and enter the value of \(M_p\) c. Make closed-loop magnitude and phase plots d. Display the expected values of percent overshoot, settling time, and peak time e. Plot the closed-loop step response Test your program on a unity feedback system with the forward-path transfer function G(s) = \(\frac{5(s+6)}{s\left(s^2+4 s+15\right)}\) and explain any discrepancies.

Using Bode plots, estimate the transient response of the systems in Figure P10.6. [Section: 10.10]

For the system of Figure P10.5, do the following: [Section: 10.10] 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 MATLAB or any other program to check your assumptions by simulating the step response of the system.

Write a program in MATLAB that will use an open-loop transfer function, G(s), to do the following: a. Make a Bode plot b. Use frequency response methods to estimate the percent overshoot, settling time, and peak time c. Plot the closed-loop step response Test your program by comparing the results to those obtained for the systems of Problem 23.

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 gain margin, phase margin, zero dB frequency, \(180^{\circ}\) frequency, and the closed-loop bandwidth. b. Use your results in Part a to estimate the damping ratio, percent overshoot, settling time, and peak time.

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. [Sections: 10.10, 10.11]

Consider the system in Figure P10.9. [Section: 10.12] 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 time delays given in Part a. c. For what time delays mentioned in Part a is the system stable? d. For each time delay that makes the system unstable, how much reduction in gain is required for the system to be stable?

Given a unity feedback system with the forward-path transfer function G(s) = \(\frac{K}{(s+1)(s+3)(s+6)}\) and a delay of 0.5 second, find the range of gain, K, to yield stability. Use Bode plots and frequency response techniques. [Section: 10.12]

Given a unity feedback system with the forward-path transfer function G(s) = \(\frac{K}{s(s+1)(s+15)}\) and a delay of 0.2 second, make a second-order approximation and estimate the percent overshoot if K = 30. Use Bode plots and frequency response techniques. [Section: 10.12]

Use the MATLAB function pade(T,n) to model the delay in Problem 30. Obtain the unit step response and evaluate your second-order approximation in Problem 30.

For the Bode plots shown in Figure P10.10, determine the transfer function by hand or via MATLAB. [Section: 10.13]

Repeat Problem 32 for the Bode plots shown in Figure P10.11. [Section: 10.13]

An overhead crane consists of a horizontally moving trolley of mass \(m_T\) dragging a load of mass \(m_L\), which dangles from its bottom surface at the end of a rope of fixed length, L. The position of the trolley is controlled in the feedback configuration shown in Figure 10.20. Here, G(s) = KP(s), H = 1, and P(s) = \(\frac{X_T(s)}{F_T(s)}=\frac{1}{m_T} \frac{s^2+\omega_0^2}{s^2\left(s^2+a \omega_0^2\right)}\) The input is \(f_T(t)\), the input force applied to the trolley. The output is \(x_T(t)\), the trolley displacement. Also, \(\omega_0\) = \(\sqrt{\frac{g}{L}}\) and a = \(\left(m_L+m_T\right) / m_T\) (Marttinen, 1990) Make a qualitative Bode plot of the system assuming a > 1.

A room's temperature can be controlled by varying the radiator power. In a specific room, the transfer function from indoor radiator power, \(\dot{Q}\), to room temperature, T in \({ }^{\circ} \mathrm{C}\) is (Thomas, 2005) P(s) = \(\frac{T(s)}{\dot{Q}(s)}\) = \(\frac{\left(1 \times 10^{-6}\right) s^2+\left(1.314 \times 10^{-9}\right) s+\left(2.66 \times 10^{-13}\right)}{s^3+0.00163 s^2+\left(5.272 \times 10^{-7}\right) s+\left(3.538 \times 10^{-11}\right)}\) The system is controlled in the closed-loop configuration shown in Figure 10.20 with G(s) = KP(s), H = 1. a. Draw the corresponding Nyquist diagram for K = 1. b. Obtain the gain and phase margins. c. Find the range of K for the closed-loop stability. Compare your result with that of Problem 58, Chapter 6.

The open-loop dynamics from dc voltage armature to angular position of a robotic manipulator joint is given by P(s) = \(\frac{48500}{s^2+2.89 s}\) (Low, 2005). a. Draw by hand a Bode plot using asymptotic approximations for magnitude and phase. b. Use MATLAB to plot the exact Bode plot and compare with your sketch from Part a.

Problem 47, Chapter 8 discusses a magnetic levitation system with a plant transfer function P(s) = \(-\frac{1300}{s^2-860^2}\) (Galvão, 2003). Assume that the plant is in cascade with an M(s) and that the system will be controlled by the loop shown in Figure 10.20, where G(s) = M(s)P(s) and H = 1. For each M(s) that follows, draw the Nyquist diagram when K = 1, and find the range of closed-loop stability for K > 0. a. M(s) = -K b. M(s) = \(-\frac{K(s+200)}{s+1000}\) c. Compare your results with those obtained in Problem 47, Chapter 8.

A simple modified and linearized model for the transfer function of a certain bicycle from steer angle \((\delta)\) to roll angle \((\varphi)\) is given by (Åstrom, 2005) P(s) = \(\frac{\varphi(s)}{\delta(s)}\) = \(\frac{12(s+20)}{s^2+25}\) Assume the rider can be represented by a gain K, and that the closed-loop system is shown in Figure 10.20 with G(s) = KP(s) and H = 1.

The control of the radial pickup position of a digital versatile disk (DVD) was discussed in Problem 48, Chapter 9. There, the open-loop transfer function from coil input voltage to radial pickup position was given as (Bittanti, 2002) P(s) = \(\frac{0.63}{\left(1+\frac{0.36}{305.4} s+\frac{s^2}{305.4^2}\right)\left(1+\frac{0.04}{248.2} s+\frac{s^2}{248.2^2}\right)}\) Assume the plant is in cascade with a controller, M(s) = \(\frac{0.5(s+1.63)}{s(s+0.27)}\) and in the closed-loop configuration shown in Figure 10.20, where G(s) = M(s)P(s) and H = 1. Do the following: a. Draw the open-loop frequency response in a Nichols chart. b. Predict the system's response to a unit step input. Calculate the % OS, \(c_{\text {final }}\), and \(T_s\). c. Verify the results of Part b using MATLAB simulations.

A ship's roll can be stabilized with a control system. A voltage applied to the fins' actuators creates a roll torque that is applied to the ship. The ship, in response to the roll torque, yields a roll angle. Assuming the block diagram for the roll control system shown in Figure P10.12, determine the gain and phase margins for the system.

The linearized model of a particular network link working under TCP/IP and controlled using a random early detection (RED) algorithm can be described by Figure 10.20 where G(s) = M(s)P(s); H = 1, and (Hollot, 2001) M(s) = \(\frac{0.005 L}{s+0.005}\); P(s) = \(\frac{140625 e^{-0.1 s}}{(s+2.67)(s+10)}\) a. Plot the Nichols chart for L = 1. Is the system closed-loop stable? b. Find the range of L for closed-loop stability. c. Use the Nichols chart to predict %OS and \(T_s\) for L = 0.95. Make a hand sketch of the expected unit step response. unit step response simulation.

In the TCP/IP network link of Problem 41 , let L = 0.8, but assume that the amount of delay is an unknown variable. a. Plot the Nyquist diagram of the system for zero delay, and obtain the phase margin. b. Find the maximum delay allowed for closed-loop stability.

An experimental holographic media storage system uses a flexible photopolymer disk. During rotation, the disk tilts, making information retrieval difficult. A system that compensates for the tilt has been developed. For this, a laser beam is focused on the disk surface and disk variations are measured through reflection. A mirror is in turn adjusted to align with the disk and makes information retrieval possible. The system can be represented by a unity feedback system in which a controller with transfer function \(G_C(s)\) = \(\frac{78.575(s+436)^2}{(s+132)(s+8030)}\) and a plant P(s) = \(\frac{1.163 \times 10^8}{s^3+962.5 s^2+5.958 \times 10^5 s+1.16 \times 10^8}\) form an open loop transmission L(s) = G_c(s)P(s) (Kim, 2009). a. Use MATLAB to obtain the system’s Nyquist diagram. Find out if the system is stable. b. Find the system’s phase margin. c. Use the value of phase margin obtained in b. to calculate the expected system’s overshoot to a step input. d. Simulate the system’s response to a unit step input and verify the %OS calculated in c.

The design of cruise control systems in heavy vehicles, such as big rigs, is especially challenging due to the extreme variations in payload. Assume that the frequency response for the transfer function from fuel mass flow to vehicle speed is shown in Figure P10.13. This response includes the dynamics of the engine, the gear box, the propulsion shaft, the differential, the drive shafts, the chassis, the payload, and tire dynamics. Assume that the system is controlled in a closed-loop, unity-feedback loop using a proportional compensator (van der Zalm, 2008). a. Make a plot oftheNyquist diagram that corresponds to the Bode plot of Figure P10.13. b. Assuming there are no open-loop poles in the right half-plane, find out if the system is closed-loop stable when the proportional gain K = 1. c. Find the range of positive K for which the system is closed-loop stable.

Use LabVIEW with the Control Design and Simulation Module and MathScript RT Module to do the following: Modify the CDEx Nyquist Analysis.vi to obtain the range of K for stability using the Nyquist plot for any system you enter. In addition, design a LabVIEW VI that will accept as an input the polynomial numerator and polynomial denominator of an open-loop transfer function and obtain a Nyquist plot for a value of K = 10,000. Your VI will also display the following as generated from the Nyquist plot: (1) gain margin, (2) phase margin, (3) zero dB frequency, and (4) 180 degrees frequency. Use the system and results of Skill-Assessment Exercise 10.6 to test your VIs.

Use LabVIEW with the Control Design and Simulation Module, and MathScript RT Module to build a VI that will accept an open-loop transfer function, plot the Bode diagram, and plot the closed-loop step response. Your VI will also use the CD Parametric Time Response VI to display (1) rise time, (2) peak time, (3) settling time, (4) percent overshoot, (5) steady-state value, and (6) peak value. Use the system in SkillAssessment Exercise 10.9 to test your VI. Compare the results obtained from your VI with those obtained in Skill-Assessment Exercise 10.9.

The block diagram of a cascade system used to control water level in a steam generator of a nuclear power plant (Wang, 2009) was presentedinFigureP6.14.Inthatsystem, the level controller, \(G_{LC}\)(s), is the master controller and the feed-water flow controller, \(G_{FC}\)(s), is the slave controller. Consider that the inner feedback loop is replaced by its equivalent transfer function, \(G_{WX}\)(s). Using numerical values in (Wang, 2009) and (Bhambhani, 2008) the transfer functions with a 1 second pure delay are: \(G_{f_w}(s)\) = \(\frac{2 \cdot e^{-\tau s}}{s\left(T_1 s+1\right)}\) = \(\frac{2 \cdot e^{-s}}{s(25 s+1)}\) \(G_{W X}(s)\) = \(\frac{(4 s+1)}{3(3.333 s+1)}\) \(G_{L C}(s)\) = \(K_{P_{L C}}+K_{D_{L C}} s\) = 1.5(10 s+1) Use MATLAB or any other program to: a. Obtain Bode magnitude and phase plots for this system using a fifth-order Padé approximation (available in MATLAB). Note on these plots, if applicable, the gain and phase margins. b. Plot the response of the system, c(t), to a unit step input, r(t) = u(t). Note on the c(t) curve the rise time, Tr, the settling time, Ts, the final value of the output, and, if applicable, the percent overshoot, %OS, and mid peak time, Tp. c. Repeat the above two steps for a pure delay of 1.5 seconds.

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 are stimulated to follow, in flight, an oscillating vertical bar. Through frequency response measurements, the obtained open loop transfer function from stimulus stripe position to a voltage proportional to wingbeat amplitudes is G(s) = e^{-0.032 s} \(\frac{0.181 s^2+1.23 s+8.68}{s^3+20.6 s^2+277 s+1098}\) Assume a unit feedback system. a. Make a Bode plot of the open-loop transfer function. b. Find the gain and phase margins. c. Use a computer program to make a plot of the corresponding Nyquist diagram. d. Use the result in Part b to find the range of K for closed-loop stability if the open-loop transfer function becomes KG(s).

In order to self-balance a bicycle, its open-loop transfer function is found to be (Lam, 2011): G(s) = \(\frac{\theta(s)}{U(s)}\) = \(\frac{334019}{s^4+5126.16 s^3+2470.7 s^2+428419 s-34040}\) where \(\theta(s)\) is the angle of the bicycle with respect to the vertical, and U(s) is the voltage applied to the motor that drives a flywheel used to stabilize the bicycle. Note that the bicycle is open-loop unstable with one open-loop pole in the right half-plane. a. Draw the Nyquist diagram of the system. b. Find the system's gain and phase margins. c. Assuming a unit feedback system, find the range of K for closed-loop stability if the forward path transfer function is KG(s). d. Assuming a second-order approximation, what is the expected %OS if K = 0.141? e. Use a computer program to simulate your system for a unit-step response using the value of K in Part d.

Modify the MATLAB program you developed in Problem 10.20 to do the following: a. Display the closed-loop magnitude and phase frequency response plots for the drive system (Thomsen, 2011) presented in Problem 56, Chapter 8. Using the graph properties, specify the value of K in the Bode plot title. b. Calculate and display the closed-loop transfer function, T(s), the peak magnitude, frequency of the peak magnitude, and bandwidth for the closed-loop frequency response at the following two values of the proportional controller’s gain, K = \(K_P\) = 3.2 and 10.

A linear model of the \(\alpha\)-subsystem of a grid connected voltage-source converter (VSC) with a Y-Y transformer (Mahmood, 2012) was presented in Problem 69, Chapter 8. In Figure P8.20(b), \(G_C(s)\) = K and \(G_P(s)\) is given in a pole zero form (with a unity gain and slightly modified parameters) as follows: \(G_P(s)\) = \(\frac{V_a(s)}{M_a(s)}\) = \(\frac{(s+2200)}{(s+220)\left(s^2+120 s+16 \times 10^{6)}\right.}\) Use MATLAB and frequency response techniques to obtain the Bode plots for this system and find the following: a. The range of K for system stability b. The gain margin, phase margin, zero dB frequency, and \(180^{\circ}\) frequency, if K = \(5 \times 10^5\).

A new measurement-based technique to design fixed structure controllers for unknown SISO systems, which does not require system identification, has been proposed. The fourth-order transfer function shown below and modified to have a unity steady state gain is used as an example (Khadraoui, 2013). G(s) = \(\frac{0.1111\left(4 s^2+5 s+1\right)}{s^4+3.1 s^3+0.85 s^2+0.87 s+0.1111}\) The interested reader is referred to the reference to explore this new technique. In this problem and its companion design problem in Chapter 11, however, we take a standard approach as covered in Chapters 10 and 11. Assuming that a cascade connected proportional controller, GC(s) = K, is used, utilize MATLAB and frequency response techniques to obtain the Bode plots for this system and find: a. The range of K for system stability b. The gain margin, phase margin, zero dB frequency, and \(180^{\circ}\) frequency, if K = 0.3.

Control of HIV/AIDS. The linearized model for an HIV/AIDS patient treated with RTIs was obtained in Chapter 6 as (Craig, 2004); P(s) = \(\frac{Y(s)}{U_1(s)}=\frac{-520 s-10.3844}{s^3+2.6817 s^2+0.11 s+0.0126}\) a. Consider this plant in the feedback configuration in Figure 10.20 with G(s) = P(s) and H(s) = 1. Obtain the Nyquist diagram. Evaluate the system for closed loop stability. b. Consider this plant in the feedback configuration in Figure 10.20 with G(s) = -P(s) and H(s) = 1. Obtain the Nyquist diagram. Evaluate the system for closed-loop stability. Obtain the gain and phase margins.

Hybrid vehicle. In Problem 71, MATLAB Chapter 8 , we used MATLAB to plot the root locus for the speed control of an HEV rearranged as a unity-feedback system, as shown in Figure P7. 31 (Preit1, 2007). The plant and compensator were given by G(s) = \(\frac{K(s+0.6)}{(s+0.5858)(s+0.0163)}\) and we found that K = 0.78, resulted in a critically damped system. a. Use MATLAB or any other program to plot the following: i. The Bode magnitude and phase plots for that system, and ii. The response of the system, c(t), to a step input, r(t) = 4 u(t). Note on the c(t) curve the rise time, \(T_r\), and settling time, \(T_s\), as well as the final value of the output. b. Now add an integral gain to the controller, such that the plant and compensator transfer function becomes G(s) = \(\frac{K_1\left(s+Z_c\right)(s+0.6)}{s(s+0.5858)(s+0.0163)}\) where \(K_1\) = 0.78 and \(Z_C\) = \(\frac{K_2}{K_1}\) = 0.4. Use MATLAB or any other program to do the following: i. Plot the Bode magnitude and phase plots for this case. ii. Obtain the response of the system to a step input, r(t) = 4 u(t). Plot c(t) and note on it the rise time, \(T_r\), percent overshoot, sOS, peak time, \(T_p\), and settling time, \(T_s\). c. Does the response obtained in Parts a or b resemble a second-order overdamped, critically damped, or underdamped response? Explain.