Consider a unity-feedback system having the open-loop

Control of the attitude \(\theta\) of a missile by controlling the fin angle \(\phi\), as shown in Figure P12.1, involves controlling an inherently unstable plant. Consider the specific plant transfer function \(G_p(s)=\frac{\Theta(s)}{\Phi(s)}=\frac{1}{5 s^2-6}\) Design a PD compensator for this plant. The dominant roots of the closed loop system must have \(\zeta=0.707\) and \(\omega_n=0.5\).

Figure P12.2 shows a pneumatic positioning system, where the displacement x is controlled by varying the pneumatic pressure \(p_1\). Assume that the pressure \(p_2\) is constant, and consider the specific plant \(G_p(s)=\frac{X(s)}{P_1(s)}=\frac{K}{s^2+2 s}\) With K = 4, the damping ratio is \(\zeta=0.5\), the natural frequency is \(\omega_n=2 \ \mathrm{rad} / \mathrm{s}\), and the steady-state ramp error is 0.5. a. Design an electrical compensator to obtain \(\omega_n = 4\) while keeping \(\zeta = 0.5\). Obtain the compensator’s resistances if \(C = 1 \ \mu F\). 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.

It is desired to control the angular displacement \(\theta\) of a space vehicle by controlling the applied torque T supplied by thrusters (Figure P12.3). The plant model is \(G_p(s)=\frac{\Theta(s)}{T(s)}=\frac{5}{s^2}\) Design a compensator for this plant. The system must have a settling time no greater than 4 sec and a steady-state error of zero for a step command, and the dominant roots of the closed-loop system must have \(\zeta \geq 0.45\).

When proportional control is applied to the following plant using a gain of \(K_P=1\), the closed-loop roots are satisfactory, but the static velocity error coefficient must be increased to \(C_v=5 / \mathrm{sec}\). \(G_p(s)=\frac{1}{s^3+3 s^2+2 s}\) Design a compensator for this plant that keeps the closed-loop roots near their original locations but increases \(C_v\) to 5 .

The speed \(\omega_1\) of the load is to be controlled with the torque \(T_d\) acting through a fluid coupling (Figure P12.5). Design a compensator for the specific plant \(G_p(s)=\frac{\Omega_1(s)}{T_d(s)}=\frac{1}{s^2+s}\) The static velocity error coefficient must be \(C_v=10 / \mathrm{sec}\), the dominant roots of the closed-loop system must have \(\zeta=0.5\) and \(\omega_n=2\).

The block diagram of a speed control system is shown in Figure P12.6. For a particular system with proportional control, \(G_1(s) = K_P\), the open-loop transfer function is \(G(s) =\frac {K_P}{s(s + 2)}\) With \(K_P = 4\), the damping ratio is \(\zeta = 0.5\), the natural frequency is \(\omega_n = 2 \ \mathrm{rad/sec}\), and \(C_v = 2/\mathrm{sec}\). Design a compensator that will give a static velocity error coefficient of \(C_v = 20/\mathrm{sec}\), a phase margin of at least \(40^{\circ}\), and a gain margin of at least 6 dB.

The block diagram of a position control system is shown in Figure P12.7. For a particular system with proportional control, \(G_1(s)=K_P\), the open-loop transfer function is \(G(s)=\frac{2.5 K_P}{s(s+2)(0.25 s+2)}\) Design a compensator to give \(s=-2 \pm j 2 \sqrt{3}\) and \(C_v=80 / \mathrm{sec}\).

It is desired to control the angular displacement \(\theta\) of a space vehicle by controlling the applied torque T supplied by thrusters (Figure P12.8a). The plant model is \(G_p(s)=\frac{\Theta(s)}{T(s)}=\frac{10}{s^2}\) The feedback sensor that measures the displacement has a time constant of 0.1 sec (Figure P12.8b). Design a compensator so that the closed loop system will have a time constant of 1 sec and a damping ratio of 0.5.

The plant transfer function for the angular displacement \(\theta\) of an inertia I subjected to a control torque T is (see Figure P12.8a) \(G_p(s)=\frac{\Theta(s)}{T(s)}=\frac{1}{I s^2}\) Suppose that I = 5 and that the output of the controller is the torque T. Use the root locus to investigate whether or not a controller transfer function of the following form will give a settling time of no more than 2 sec and an overshoot of no more than 5%. \(G_c(s)=K \frac{s+a}{s+b}\) Obtain suitable values of K, a, and b. What happens if we try to "cancel" one of the poles at the origin by placing the zero very close to the origin?

Consider a plant whose open-loop transfer function is \(G(s) H(s)=\frac{1}{s\left[(s+2)^2+9\right]}\) The complex poles near the origin give only slightly damped oscillations that are considered undesirable. Insert a gain \(K_c\) and a compensator \(G_c(s)\) in series to speed up the closed-loop response of the system. Consider the following for \(G_c(s)\); a. The lead compensator b. The lag compensator c. The so-called reverse-action compensator \(G_c(s)=\frac{1-T_1 s}{T_2 s+1}\) Obtain the root locus plots for the compensated system using each compensator. Use the compensator gain \(K_c\) as the locus parameter. Without computing specific values for the compensator parameters, determine which compensator gives the best response.

a) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about \(\phi=0\), assuming that \(\dot{\phi}\) is very small. b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and \(g=9.81 \mathrm{~m} / \mathrm{s}^2\). c) Use the linearized model developed in part (b) to design a series compensator to stabilize the pendulum angle near \(\phi=0\). 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 \(\phi\) can be measured.

A certain unity feedback system has the following open-loop system transfer function. \(G(s) = \frac{5K}{s^3 + 6s^2 + 5s}\) 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?

Figure P12.13 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure \(p_1\). Assume that the pressure \(p_2\) is constant, and consider the specific plant \(G_p(s)=\frac{X(s)}{P_1(s)}=\frac{1}{100 s^2+s}\) with the following series PD compensator is used to control the pressure. \(G_c(s)=2+19 s\) Obtain the Bode plots for this system, and determine the phase and gain margins.

The height \(h_2\) in Figure P12.14 can be controlled by adjusting the flow rate \(q_1\). Consider the specific plant \(G_p(s)=\frac{H_2(s)}{Q_1(s)}=\frac{25}{5 s^2+6 s+5}\) with the following series PI compensator is used to control it. \(G_c(s)=\frac{7 s+64}{s}\) Obtain the Bode plots for this system, and determine the phase and gain margins.

Rolling 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 P12.15 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) \(K_P = 1\), (b) \(K_P = 10\), and (c) \(K_P = 100\). Determine the stability properties for each case. If the system is unstable, what effect will this have on the ship roll?

The open-loop transfer function of a certain unity feedback system is \(G(s) = K \frac{10s + 1}{2s^3 + 3s^2 + s}\) Compute the value of the gain K so that the phase margin will be \(60^{\circ}\).

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 \(\frac{G(s)}{1 + G(s)H(s)}\) 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. \(G(s)H(s) =\frac {20}{s}\) b. \(G(s)H(s) =\frac{20}{5s + 1}\) c. \(G(s)H(s) =\frac{7}{s^2}\)

Remote 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 \(D = D_1 + D_2 = 100 \ \mathrm{sec}\) (Figure P12.18). The plant is \(G_{p}(s) =\frac{1}{100s + 1}\) How large can the gain \(K_P\) be without the system being unstable?

Hot-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 \(G_{p} (s) =\frac{1}{0.1s + 1}\) has the gain \(K_P = 10\). 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 \(40^{\circ}\)?

The block diagram of a position control system is shown in Figure P12.20. Design a compensator for the particular plant \(G_{p} (s) =\frac{1}{s(s^2 + 3s + 2)\) so that the static velocity error coefficient will be \(C_v = 5/\mathrm{sec}\), the gain margin will be no less than 10 dB, and the phase margin no less than \(40^{\circ}\).

The speed \(\omega_2\) 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 \(G_{p} (s) =\frac{\Omega_2(s)}{T (s)} = \frac {4}{s^2 + 2s}\) so that the static velocity error coefficient will be \(C_v = 20/ \mathrm{sec}\), the gain margin will be no less than 10 dB, and the phase margin no less than \(50^{\circ}\).

Design a compensator for the plant \(G_p(s)=\frac{2}{s^2+2 s}\) so that the static velocity error coefficient will be \(C_v=20 / \mathrm{sec}\) and the phase margin at least \(45^{\circ}\).

Figure P12.2 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure \(p_1\). Assume that the pressure \(p_2\) is constant, and consider the specific plant \(G_p(s)=\frac{X(s)}{P_1(s)}=\frac{1}{s^2+s}\) Design a compensator for the plant so that \(C_v=20 / \mathrm{sec}\), the gain margin will be no less than 10 dB, and the phase margin no less than \(50^{\circ}\).

The block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant \(G_p(s)=\frac{5}{s(s+5)(s+1)}\) that will give a static velocity error coefficient of \(C_v=50 / \mathrm{sec}\) and closed loop roots with a damping ratio of \(\zeta=0.5\).

The block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant \(G_p(s)=\frac{1}{s\left(s^2+3 s+2\right)}\) so that the static velocity error coefficient will be \(C_v=10 / \mathrm{sec}\), the gain margin will be no less than 10 dB, and the phase margin no less than \(50^{\circ}\).

Consider a unity-feedback system having the open-loop transfer function \(G(s)=\frac{\omega_n^2}{s\left(s+2 \zeta \omega_n\right)}\) Derive the following expression for this system's phase margin. \(\mathrm{PM}=\tan ^{-1} \frac{2 \zeta}{\sqrt{-2 \zeta^2+\sqrt{1+4 \zeta^4}}}\)

Two independent feedback control systems are needed to position the pen of a flatbed x-y plotter. Consider only the control system for the x axis. An optical encoder provides feedback of the pen’s location. Assume that the dynamics of the motor and pen assembly are described by the following plant transfer function. \(G_{p} (s) =\frac{X(s)}{V(s)} =\frac{0.005}{s(0.1s + 1)(0.001s + 1)}\) where x is the pen location in meters and v is the motor voltage. Design a compensator to achieve zero steady-state error for a step input and a 2% settling time of no more than 1 s, with an overshoot of no more than 4%.

Automatic 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 \(G_c(s)\) so that the step response has an overshoot no greater than 20% with a 2% settling time of no more than 1 s, and a steady-state unit-ramp error of no more than 0.1 m.

With the PI gains set to \(K_P = 6\) and \(K_I = 50\) for the plant \(G_{p} (s) =\frac{1}{s + 4}\) the time constant is \(\tau = 0.2\) and the damping ratio is \(\zeta = 0.707\). a. Compute the gain and phase margins. b. Suppose there is dead time D = 0.1 between the controller and the plant. Compute the gain and phase margins, and plot the unit-step response. c. How large can the dead time be without the system becoming unstable?

In Example 12.1.4 a lag-lead compensator was designed by canceling the plant pole at s = ?3 with a compensator zero. Suppose the plant model is slightly inaccurate and the plant pole is really at s = ?3.2. Evaluate the resulting unit-step response and unit-ramp response in terms of the performance specifications.

Solve Problem 12.2 using MATLAB.

Solve Problem 12.4 using MATLAB.

Solve Problem 12.5 using MATLAB.

Solve Problem 12.6 using MATLAB.

Solve Problem 12.12 using MATLAB.

Solve Problem 12.13 using MATLAB.

Solve Problem 12.14 using MATLAB.

Solve Problem 12.15 using MATLAB.

Solve Problem 12.18 using MATLAB.

Solve Problem 12.28 using MATLAB.

Solve Problem 12.29 using MATLAB.

Refer to Example 12.1.1, in which a PID controller was designed to control the plant \(1/s^2\). Using the gain values calculated in that example, construct a Simulink model of the system using the PID controller block and including the disturbance torque shown in Figure 12.1.7. Use the model to obtain plots of the responses to a unit-step command and a unit-step disturbance.

Consider Problem 12.19. Use Simulink to determine how large the dead time can be before the system becomes unstable.