In Example 10.7.4 the steady-state error for a unit-ramp

Sketch the root locus plot of \(3 s^2+12 s+k=0\) for \(k \geq 0\). What is the smallest possible dominant time constant, and what value of k gives this time constant?

Sketch the root locus plot of \(3 s^2+c s+12=0\) for \(c \geq 0\). What is the smallest possible dominant time constant, and what value of c gives this time constant? What is the value of \(\omega_n\) if \(\zeta<1\)?

Sketch 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 \(L_a I s^2+\left(R_a I+c L_a\right) s+c R_a+K_b K_T=0\) Use the following parameter values: \(\begin{array}{clrl}K_b=K_T & =0.1 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} & I & =12 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^2 \\R_a & =2 \Omega & L_a & =3 \times 10^{-3} \mathrm{H}\end{array}\)

Sketch the root locus plot of \(m s^2+12 s+10=0\) for \(m \geq 2\). What is the smallest possible dominant time constant, and what value of m gives this time constant?

In the following equations, identify the root locus plotting parameter K and its range in terms of the parameter p, where \(p \geq 0\). a. \(6s^2 + 8s + 3p = 0\) b. \(3s^2 + (6 + p)s + 5 + 2p = 0\) c. \(4s^3 + 4ps^2 + 2s + p = 0\)

Consider a unity feedback system with the plant \(G_p(s)\) and the controller \(G_c(s)\). PID control action is applied to the plant \(G_p(s)=\frac{s+10}{(s+1)(s+2)}\) The PID controller has the transfer function \(G_c(s)=K_P\left(1+\frac{1}{T_l s}+T_D s\right)\) Use the values \(T_I=0.2\) and \(T_D=0.5\). a. Identify the open-loop poles and zeros. b. Identify the root locus parameter K in terms of \(K_P\). c. Identify the closed-loop poles and zeros for the case \(K_P=10\).

In parts (a) through (f), sketch the root locus plot for the given characteristic equation for \(K \geq 0\). a. s(s + 5) + K = 0 b. s(s + 7)(s + 9) + K = 0 c. \(s^2 + 3s + 5 + K(s + 3) = 0\) d. s(s + 4) + K(s + 5) = 0 e. \(s(s^2 + 3s + 5) + K = 0\) f. s(s + 3)(s + 7) + K(s + 4) = 0

PID control action is applied to the plant \(G_p(s)=\frac{s+10}{(s+2)(s+5)}\) The PID controller has the transfer function \(G_c(s)=K_P\left(1+\frac{1}{T_l s}+T_D s\right)\) Use the values \(T_I=0.2\) and \(T_D=0.5\). Plot the root locus with the proportional gain \(K_P\) as the parameter.

Consider the following equation where the parameter p is nonnegative. \(4s^3 + (25 + 5p)s^2 + (16 + 30p)s + 40p = 0\) a. Put the equation in standard root locus form and define a suitable root locus parameter K in terms of the parameter p. b. Obtain the poles and zeros, and sketch the root locus plot.

In the following equation, \(K \geq 0\). \(s^2(s+9)+K(s+1)=0\) a. Obtain the root locus plot. b. Obtain the value of K at the breakaway point, and obtain the third root for this value of K. c. What is the smallest possible dominant time constant for this equation?

Consider the following equation where the parameter K is nonnegative. \((2s + 5)(2s^2 + 14s + 49) + K s(2s + 1)(2s + 3) = 0\) a. Determine the poles and zeros, and sketch the root locus plot. b. Use the plot to set the value of K required to give a dominant time constant of \(\tau = 0.5\). Obtain the three roots corresponding to this value of K.

In the following equations, identify the root locus plotting parameter K and its range in terms of the parameter p, where \(p \geq 0\). a. \(9s^3 + 6s^2 ? 5ps + 2 = 0\) b. \(4s^3 ? ps^2 + 2s + 7 = 0\) c. \(s^2 + (3 ? p)s + 4 + 4p = 0\)

In parts (a) through (f), Obtain the root locus plot for \(K \leq 0\) for the given characteristic equation. a. s(s + 5) + K = 0 b. \(s^2 + 3s + 3 + K(s + 3) = 0\) c. \(s(s^2 + 3s + 3) + K = 0\) d. s(s + 5)(s + 7) + K = 0 e. s(s + 3) + K(s + 4) = 0 f. s(s + 6) + K(s ? 4) = 0

The plant transfer function for a particular process is \(G_p(s)=\frac{8-s}{s^2+2 s+3}\) We wish to investigate the use of proportional control action with this plant. a. Obtain the root locus and determine the range of values of the proportional gain \(K_P\) for which the system is stable. b. Determine the value of \(K_P\) required to give a time constant of \(\tau=2 / 3\). c. Plot the unit step response of the plant for \(K_P=1\). A process containing a negative sign in the numerator of its transfer function is called a "reverse reaction" process. What is the effect of the negative sign in the numerator of \(G_p(s)\)?

The plant transfer function for a particular process is \(G_p(s)=\frac{26+s-2 s^2}{s(s+2)(s+3)}\) We wish to investigate the use of proportional control action with this plant. a. Obtain the root locus and determine the range of values of the proportional gain \(K_P\) for which the system is stable. b. Determine the value of \(K_P\) required to give \(\zeta=1\). c. Plot the unit step response of the plant. What is the effect of the negative sign in the numerator of \(G_p(s)\) ?

Control of the attitude \(\theta\) of a missile by controlling the fin angle \(\phi\), as shown in Figure P11.16, involves controlling an inherently unstable plant. Consider the specific plant transfer function \(G_p(s)=\frac{\Theta(s)}{\Phi(s)}=\frac{1}{s^2-5}\) a. Determine the PD control gains so that the steady-state error for a step command is zero, the closed-loop damping ratio is 0.707, and the dominant closed-loop time constant is 0.1. b. Use the root locus to evaluate the performance of the resulting controller in light of the specifications \(\zeta=0.707\) and \(\tau=0.1\) if the plant transfer function \(G_p(s)\) has an uncertainty \(\Delta\) due to fuel consumption, where \(G_p(s)=\frac{1}{s^2-5-\Delta}, \quad 0 \leq \Delta \leq 1\)

The use of a motor to control the rotational displacement of an inertia I is shown in Figure P11.17. The open-loop transfer function of the plant for a specific application is \(G_p(s)=\frac{6}{s(2 s+2)(3 s+24)}\) a. Use the root locus plot to show that it is not possible with proportional control to achieve a dominant time constant of less than 2.07 sec for this plant. b. Use PD control to improve the response, so that the dominant time constant is 0.5 sec or less and the damping ratio is 0.707 or greater. To do this, first select a suitable value for \(T_D\), then plot the locus with \(K_P\) as the variable.

Proportional control action applied to the heat flow rate \(q_i\) can be used to control the temperature of the oven shown in Figure P11.18. Consider the specific plant \(G_p(s)=\frac{T_1(s)}{Q_i(s)}=\frac{s+10}{s^2+5 s+6}\) Use the root locus plot to obtain the smallest damping ratio this system can have. Obtain the value of the proportional gain \(K_P\) required to minimize the dominant time constant with \(\zeta=0.707\), and determine this time constant.

Proportional control action applied to the flow rate \(q_{m i}\) can be used to control the liquid height, as shown in Figure P11.19. Consider the specific plant \(G_p(s)=\frac{H_2(s)}{Q_{m i}(s)}=\frac{5(s+4)}{(s+3)(s+p)}\) The proportional gain is \(K_P=2\). The likely value of p is p = 7, but it is known that p might be greater than 7 . Use the root locus plot to investigate the roots of the system for \(p \geq 7\).

Proportional control action applied to the flow rate \(q_{mi}\) can be used to control the liquid height of the system shown in Figure P11.20. Consider the specific plant \(G_p(s)=\frac{H_2(s)}{Q_{m i}(s)}=\frac{1}{s^2+3 s+2}\) Use the root locus plot to design a PI controller for this system to minimize the dominant time constant, with a damping ratio of \(\zeta= 0.707\).

Design a PID controller applied to the motor torque T to control the robot arm angle \(\theta\) shown in Figure P11.21. Consider the specific plant \(G_p(s)=\frac{\Theta(s)}{T(s)}=\frac{4}{3 s^2+3}\) The dominant closed-loop roots must have \(\zeta=0.5\) and a time constant of 1 .

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 PID controller 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 \(\phi\) and \(\dot{\phi}\) can be measured.

Use of a motor to control the position of a certain load having inertia, damping, and elasticity gives the following plant transfer function. See Figure P11.23. \(G_p(s)=\frac{\Theta(s)}{V(s)}=\frac{0.5}{\left(s^2+s+1\right)(s+0.5)}\) a. Use the ultimate cycle method to compute the controller gains for P, PI, and PID control. b. Plot and compare the unit-step responses for the three designs obtained in (a). If the PID response is unsatisfactory, tune the gains to improve the performance.

Figure P11.24 shows an electrohydraulic position control system whose plant transfer function for a specific system is \(G_p(s)=\frac{Y(s)}{F(s)}=\frac{5}{2 s^3+10 s^2+2 s+4}\) a. Use the ultimate cycle method to design P, PI, and PID controllers. b. Plot and compare the unit-step responses for the three designs obtained in (a). If the PID response is unsatisfactory, tune the gains to improve the performance.

A certain plant has the transfer function \(G_p(s)=\frac{4 p}{\left(s^2+4 \zeta s+4\right)(s+p)}\) where the nominal values of \(\zeta\) and p are \(\zeta=0.5\) and p = 1. a. Use Ziegler-Nichols tuning to compute the PID gains. Obtain the resulting closed-loop characteristic roots. b. Use the root locus to determine the effect of a variation in the parameter p over the range \(0.4 \leq \zeta \leq 0.6\). c. Use the root locus to determine the effect of a variation in the parameter p over the range \(0.5 \leq p \leq 1.5\).

The plant transfer function of the system in Figure P11.26 for a specific case is \(G_p(s)=\frac{8}{(2 s+2)(s+2)(4 s+12)}\) a. Use the ultimate cycle method to compute the PID gains. b. Plot the unit-step response. If the response is unsatisfactory, use the root locus plot to explain the result, and try to improve the response by tuning the gains.

Consider the PI-control system shown in Figure P11.27 where I = 5 and c = 0. It is desired to obtain a closed-loop system having \(\zeta = 1\) and \(\tau = 0.1\). Let \(m_{\text{max}} = 20\) and \(r_{\text{max}} = 2\). Obtain \(K_P\) and \(K_I\) .

Consider the PI-control system shown in Figure P11.27 where I = 10 and c = 20. It is desired to obtain a closed-loop system having \(\zeta= 1\) and \(\tau = 0.1\). a. Obtain the required values of \(K_P\) and \(K_I\) , neglecting any saturation of the control elements. b. Let \(m_{\text{max}} = r_{\text{max}} = 1\). Obtain \(K_P\) and \(K_I\) . c. Compare the unit-step response of the two designs.

Consider the PI-control system shown in Figure P11.27 where I = 7 and c = 5. It is desired to obtain a closed-loop system having \(\zeta = 1\) and \(\tau = 0.2\). Let \(m_{\text{max}} = 20\) and \(r_{\text{max}} = 5\). Obtain \(K_P\) and \(K_I\) .

a. Design a PI and an I controller with internal feedback for the plant \(G_p(s) = 1/4s\). See Figure P11.30. We are given that \(m_{\text{max}} = 6\) and \(r_{\text{max}} = 3\). Set \(\zeta = 1\). b. Evaluate the unit-step response of each design. c. Evaluate the unit-ramp response of each design.

Compare the performance of the critically damped controllers shown in Figure P11.30 with the plant \(G_p(s) = 1/I s\) having the following inputs: a. A unit-ramp disturbance b. A sinusoidal disturbance c. A sinusoidal command input

A certain field-controlled de motor with load has the following parameter values. \(\begin{array}{rlrl}L & =2 \times 10^{-3} \mathrm{H} & R & =0.6 \Omega \\K_T & =0.04 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} & & c=0 \\I & =6 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^2 & &\end{array}\) 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 \(\zeta = 0.707\).

In Figure P11.33 the input u is an acceleration provided by the control system and applied in the horizontal direction to the lower end of the rod. The horizontal displacement of the lower end is y. The linearized form of Newton's law for small angles gives \(m L \ddot{\theta}=m g \theta-m u\) a. Put this model into state variable form by letting \(x_1=\theta\) and \(x_2=\dot{\theta}\). b. Construct a state variable feedback controller by letting \(u=k_1 x_1+k_2 x_2\). Over what ranges of values of \(k_1\) and \(k_2\) will the controller stabilize the system? What does this formulation imply about the displacement y ?

Figure P.11.34 illustrates an active vibration control scheme for a two-mass system. An electrohydraulic actuator between the two masses provides a force that acts on both and is under feedback control. The system model is \(\begin{aligned}& m_1 \ddot{x}_1=k_1\left(y-x_1\right)-k_2\left(x_1-x_2\right)-c\left(\dot{x}_1-\dot{x}_2\right)-f \\& m_2 \ddot{x}_2=k_2\left(x_1-x_2\right)+c\left(\dot{x}_1-\dot{x}_2\right)+f\end{aligned}\) The given parameter values are \(m_1=50 \ \mathrm{~kg}, m_2=250 \ \mathrm{~kg}, k_1=1.5 \times 10^5 \ \mathrm{~N} / \mathrm{m}, k_2=1.2 \times 10^4 \ \mathrm{~N} / \mathrm{m}\), and \(c=100 \ \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}\). a. Put the model into state variable form. b. Assume that we can measure all four state variables, and use P action with state-variable feedback. In the original passive system, \(k_2=1.6 \times 10^4 \mathrm{~N} / \mathrm{m}\) and \(c=98 \mathrm{~N} \cdot \mathrm{m} / \mathrm{s}\), which resulted in characteristic roots at \(s=-1.397 \pm 69.94 j, s=-0.168 \pm 7.779 j\). Compute the values of the feedback gains so that the closed-loop roots will be near those of the passive system.

Figure P11.35a is the circuit diagram of a speed-control system in which the dc motor voltage \(v_a\) 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 \(v_a\) is varied by changing the generator input voltage \(v_f\). The voltage \(v_a\) is related to the generator field current \(i_f\) by \(v_a=K_f i_f\). Figure P11.35b is a diagram of a feedback system for controlling the speed by measuring it with a tachometer and varying the voltage \(v_f\). Use the following values in SI units. \(\begin{aligned}& L_f=0.2 \quad R_f=2 \quad K_t=1 \\& L_a=0.2 \quad R_a=1 \quad K_b=K_T=0.5 \\& K_f=50 \quad I=10 \quad c=20 \\&\end{aligned}\) a. Develop a state variable model of the plant that includes the generator, the motor, and the load. Include the load torque \(T_L\) as a disturbance. b. 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.

The following equations are the model of the roll dynamics of a missile ([Bryson, 1975]). See Figure P11.36. \(\begin{aligned}\dot{\delta} & =u \\\dot{\omega} & =-\frac{1}{\tau} \omega+\frac{b}{\tau} \delta \\\dot{\phi} & =\omega\end{aligned}\) where \(\quad \delta=\) aileron deflection b = aileron effectiveness constant u = command signal to the aileron actuator \(\phi=\) roll angle,\(\omega=\) roll rate Using the specific values \(b=10 \mathrm{~s}^{-1}\) and \(\tau=1 \mathrm{~s}\), and assuming that the state variables \(\delta, \omega\), and \(\phi\) can be measured, develop a linear state-feedback controller to keep \(\phi\) near 0 . The dominant roots should be \(s=-10 \pm 10 j\), and the third root should be s = -20.

Many winding applications in the paper, wire, and plastic industries require a control system to maintain proper tension. Figure P11.37 shows such a system winding paper onto a roll. The paper tension must be held constant to prevent internal stresses that will damage the paper. The pinch rollers are driven at a speed required to produce a paper speed \(v_p\) at the rollers. The paper speed as it approaches the roll is \(v_r\). The paper tension changes as the radius of the roll changes or as the speed of the pinch rollers change. The paper has an elastic constant k so that the rate of change of tension is \(\frac{d T}{d t}=k\left(v_r-v_p\right)\) For a paper thickness d, the rate of change of the roll radius is \(\frac{d R}{d t}=\frac{d}{2} W\) The inertia of the windup roll is \(I=\rho \pi W R^4 / 2\), where \(\rho\) 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. a. Assuming that the paper thickness is small enough so that \(\dot{R} \approx 0\) for a short time, develop a state-variable model with the motor voltage e and the paper speed \(v_p\) as the inputs. b. Modify the model developed in part (a) to account for R and I being functions of time.

An electro-hydraulic positioning system is shown in Figure P11.38. Use the following values. \(\begin{aligned}K_a & =10 \mathrm{~V} / \mathrm{A} \quad K_1=10^{-2} \mathrm{in} . / \mathrm{V} \\K_2 & =3 \times 10^5 \mathrm{sec}^{-3} \quad K_3=20 \mathrm{~V} / \mathrm{in} . \\\zeta & =0.8 \quad \omega_n=100 \mathrm{rad} / \mathrm{sec} \quad \tau=0.01 \mathrm{sec}\end{aligned}\) a. Develop a state-variable model of the plant with the controller current \(i_c\) as the input and the displacement y as the output. b. Assuming that proportional control is used so that \(G_c(s)=K_P\), develop a state model of the system with \(y_r\) 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 \(K_P\).

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 state variable feedback controller 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 \(\phi, \dot{\phi}, x\), and \(\dot{x}\) can be measured.

The 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, PI, and PID control.

A 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 Ziegler-Nichols process reaction method to compute the PID gains.

Use MATLAB to obtain the root locus plot of \(5s^2 + cs + 45 = 0\) for \(c \geq 0\).

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

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

Use MATLAB to obtain the root locus plot of the system shown in Figure P11.45 in terms of the variable \(k_2 \geq 0\). Use the values m = 2, c = 8, and \(k_1 = 26\). What is the value of \(k_2\) required to give \(\zeta = 0.707\)?

Use MATLAB to obtain the root locus plot of the system shown in Figure P11.46 in terms of the variable \(c_2 \geq 0\). Use the values \(m = 2, c_1 = 8\), and k = 26. What is the smallest possible dominant time constant and the associated value of \(c_2\)?

Use MATLAB to obtain the root locus plot of \(s^3+13 s^2+52 s+60+K=0\) for \(K \geq 0\). Is it possible for any dominant roots of this equation to have a damping ratio in the range \(0.5 \leq \zeta \leq 0.707\) and an undamped natural frequency in the range \(3 \leq \omega_n \leq 5\)?

(a) Use MATLAB to obtain the root locus plot of \(2 s^3+12 s^2+16 s+K=0\) for \(K \geq 0\). (b) Obtain the value of K required to give a dominant root pair having \(\zeta=0.707\). (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 \(K /\left(2 s^3+12 s^2+16 s+K\right)\).

Use 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 \(L_a I s^2+\left(R_a I+c L_a\right) s+c R_a+K_b K_T=0\) Use the following parameter values: \(\begin{array}{rlrl}K_b=K_T & =0.1 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} & I & =4 \times10^{-5} \mathrm{~kg} \cdot \mathrm{m}^2 \\R_a & =2 \Omega & L_a & =3 \times 10^{-3} \mathrm{H}\end{array}\)

Consider the two-mass model shown in Figure P11.50. Use the following numerical values: \(m_1=m_2=1, k_1=1, k_2=4\), and \(c_2=8\). a. Use MATLAB to obtain the root locus plot in terms of the parameter \(c_1\). b. Use the root locus plot to determine the value of \(c_1\) required to give a dominant root pair having a damping ratio of \(\zeta=0.707\). c. Use the root locus plot to determine the value of \(c_1\) 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.

In parts (a) through (f), use MATLAB to obtain the root locus plot for the given characteristic equation for \(K \geq 0\). a. s(s + 5) + K = 0 b. s(s + 7)(s + 9) + K = 0 c. \(s^2 + 3s + 5 + K(s + 3) = 0\) d. s(s + 4) + K(s + 5) = 0 e. \(s(s^2 + 3s + 5) + K = 0\) f. s(s + 3)(s + 7) + K(s + 4) = 0

In parts (a) through (f), use MATLAB to obtain the root locus plot for \(K \leq 0\) for the given characteristic equation. a. s(s + 5) + K = 0 b. \(s^2 + 3s + 3 + K(s + 3) = 0\) c. \(s(s^2 + 3s + 3) + K = 0\) d. s(s + 5)(s + 7) + K = 0 e. s(s + 3) + K(s + 4) = 0 f. s(s + 6) + K(s ? 4) = 0

Consider the equation \(s^3 + 10s^2 + 24s + K = 0\) a. Use MATLAB to obtain the value of K required to give dominant roots with \(\zeta= 0.707\). 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 \(\tau= ?\). Obtain the three roots corresponding to this value of K.

Consider the equation \(s^3 + 9s^2 + (8 + K)s + 2K = 0\) 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 \(\pm 10 \%\) about the value found in part (a).

Consider the equation \(s^3+9 s^2+(8+K) s+2 K=0\) Use the sgrid function to determine if it is possible to obtain a dominant root having a damping ratio in the range \(0.5 \leq \zeta \leq 0.707\). If so, use MATLAB to obtain the value of K required to give the largest possible value of \(\zeta\) in the range \(0.5 \leq \zeta \leq 0.707\).

Consider the equation \(s^3+10 s^2+24 s+K=0\) Use the sgrid function to determine if it is possible to obtain a dominant root having a damping ratio in the range \(0.5 \leq \zeta \leq 0.707\), and an undamped natural frequency in the range \(2 \leq \omega_n \leq 3\). If so, use MATLAB to obtain the value of K required to give the largest possible value of \(\zeta \omega_n\) in the ranges stated.

In Example 10.7.4 the steady-state error for a unit-ramp disturbance is \(1 / K_I\). 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 \(K_I\). Using the values given for \(K_P\) and \(K_D\) in that example, obtain a root locus plot with \(K_I\) as the variable. Discuss what happens to the damping ratio and time constant of the dominant root as \(K_I\) is increased.

In Example 10.8.3 the steady-state error for a unit-ramp command is \(-4 / K_I\). 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 \(K_I\). Using the values given for \(K_P\) and \(K_D\) in that example, obtain a root locus plot with \(K_I\) as the variable. Discuss what happens to the damping ratio and time constant of the dominant root as \(K_I\) is increased.

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. Suppose the actuator saturation limits are \(\pm 5\). 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 an appropriate value for \(K_A\) and to plot the output response and the actuator output versus time.

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 \(\zeta=0.707\). Suppose there is a rate limiter of \(\pm 0.1\) 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.

A certain dc motor has the following parameter values: \(\begin{aligned}L & =2 \times 10^{-3} \mathrm{H} \quad R=0.6 \Omega \\K_T & =0.04 \mathrm{~N} \cdot \mathrm{m} / \mathrm{A} \quad c=0 \\I & =6 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^2\end{aligned}\) 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 \(\zeta = 0.707\). 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. c. Suppose the motor current is limited to \(\pm 2 \ \mathrm{A}\). 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.

Consider the liquid-level controller designed in Example 10.10.1, whose Simulink diagram is shown in Figure 10.10.1. Modify the model to include a Rate Limiter block to limit the rate of \(q_1\), in front of the Saturation block. The limits on the rate should be \(\pm 20\). Use this model to obtain plots of the response of the height \(h_2\) to a unit-step command and a unit-step disturbance. Compare these responses to those found in Example 10.10.1.