Consider the cascaded subsystems shown in Figure P5.29. If

Reduce the block diagram shown in Figure P5.1 to a single transfer function, T(s) = C(s)/R(s) Use the following methods: (a) Block diagram reduction [Section: 5.2] (b) MATLAB

Find the closed-loop transfer function, T(s) = C(s)/R(s) for the system shown in Figure P5.2, using block diagram reduction. [Section: 5.2]

Find the equivalent transfer function, T(s) = C(s)/R(s) , for the system shown in Figure P5.3. [Section: 5.2]

Reduce the system shown in Figure P5.4 to a single transfer function, T(s) = C(s)/R(s). [Section: 5.2]

Find the transfer function, T(s) = C(s)/R(s), for the system shown in Figure P5.5. Use the following methods: (a) Block diagram reduction [Section: 5.2] (b) MATLAB. Use the following transfer functions: \(\begin{aligned} & G_1(s)=1 /(s+7), G_2(s)=1 /\left(s^2+2 s+3\right) \\ & G_3(s)=1 /(s+4), G_4(s)=1 / s \\ & G_5(s)=5 /(s+7), G_6(s)=1 /\left(s^2+5 s+10\right) \\ & G_7(s)=3 /(s+2), G_8(s)=1 /(s+6) \end{aligned}\) Hint: Use the append and connect commands in MATLAB’s Control System Toolbox.

Reduce the block diagram shown in Figure P5.6 to a single block, T(s) = C(s)/R(s). [Section: 5.2]

Find the unity feedback system that is equivalent to the system shown in Figure P5.7. [Section: 5.2]

Given the block diagram of a system shown in Figure P5.8, find the transfer function \(G(s)=\theta_{22}(s) / \theta_{11}(s)\). [Section: 5.2]

Reduce the block diagram shown in Figure P5.9 to a single transfer function, T(s) = C(s)/R(s). [Section: 5.2]

Reduce the block diagram shown in Figure P5.10 to a single block representing the transfer function, T(s) = C(s)/R(s) . [Section: 5.2]

For the system shown in Figure P5.11, find the percent overshoot, settling time, and peak time for a step input if the system’s response is underdamped. (Is it? Why?) [Section: 5.3]

For the system shown in Figure P5.12, find the output, c(t), if the input, r(t), is a unit step. [Section: 5.3]

For the system shown in Figure P5.13, find the poles of the closed-loop transfer function, T(s) = C(s)/R(s). [Section: 5.3]

For the system of Figure P5.14, find the value of K that yields 10% overshoot for a step input. [Section: 5.3]

For the system shown in Figure P5.15, find K and \(\alpha\) to yield a settling time of 0.12 second and a 20% overshoot. [Section: 5.3]

For the system of Figure P5.16, find the values of \(K_1\) and \(K_2\) to yield a peak time of 1 second and a settling time of 2 seconds for the closed-loop system’s step response. [Section: 5.3]

Find the following for the system shown in Figure P5.17: [Section: 5.3] (a) The equivalent single block that represents the transfer function, T(s) = C(s)/R(s). (b) The damping ratio, natural frequency, percent overshoot, settling time, peak time, rise time, and damped frequency of oscillation.

For the system shown in Figure P5.18, find \(\zeta, \omega_n\), percent overshoot, peak time, rise time, and settling time. [Section: 5.3]

A motor and generator are set up to drive a load as shown in Figure P5.19. If the generator output voltage is \(e_g(t)=K_f i_f(t)\), where \(i_f(t)\) is the generator's field current, find the transfer function \(G(s)=\theta_o(s) / E_i(s)\). For the generator, \(K_f=2 \Omega\). For the motor, \(K_t=2 \mathrm{~N}-\mathrm{m} / \mathrm{A}\), and \(K_b=2 \mathrm{~V}-s / \mathrm{rad}\).

Find \(G(s)=E_0(s) / T(s)\) for the system shown in Figure P5.20.

Find the transfer function \(G(s)=E_o(s) / T(s)\) for the system shown in Figure P5.21.

Label signals and draw a signal-flow graph for each of the block diagrams shown in the following problems: [Section: 5.4] (a) Problem 1 (b) Problem 3 (c) Problem 5

Draw a signal-flow graph for each of the following state equations: [Section: 5.6] (a) \(\dot{\mathbf{x}}=\left[\begin{array}{rrr}0 & 1 & 0 \\ 0 & 0 & 1 \\ -2 & -4 & -6\end{array}\right] \mathbf{x}+\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right] r\) \(\begin{aligned} y & =\left[\begin{array}{lll} 1 & 1 & 0 \end{array}\right] \mathbf{x} \\\) (b) \(\dot{\mathbf{x}} & =\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & -3 & 1 \\ -3 & -4 & -5 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right] r \\ y & =\left[\begin{array}{lll} 1 & 2 & 0 \end{array}\right] \mathbf{x} \end{aligned}\) (c) \(\begin{aligned} \dot{\mathbf{x}} & =\left[\begin{array}{rrr} 7 & 1 & 0 \\ -3 & 2 & -1 \\ -1 & 0 & 2 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right] r \\ y &, =\left[\begin{array}{lll} 1 & 3 & 2 \end{array}\right] \mathbf{x} \end{aligned}\)

Given the system below, draw a signal-flow graph and represent the system in state space in the following forms: [Section: 5.7] (a) Phase-variable form (b) Cascade form \(G(s)=\frac{200}{(s+10)(s+20)(s+30)}\)

Repeat Problem 24 for \(G(s)=\frac{20}{s(s-2)(s+5)(s+8)}\) [Section: 5.7]

Using Mason’s rule, find the transfer function, T(s) = C(s)/R(s), for the system represented in Figure P5.22. [Section: 5.5]

Using Mason’s rule, find the transfer function, T(s) = C(s)/R(s), for the system represented by Figure P5.23. [Section: 5.5]

Use Mason’s rule to find the transfer function of Figure 5.13 in the text. [Section: 5.5]

Use block diagram reduction to find the transfer function of Figure 5.21 in the text, and compare your answer with that obtained by Mason’s rule. [Section: 5.5]

Represent the following systems in state space in Jordan canonical form. Draw the signal-flow graphs. [Section: 5.7] (a) \(G(s)=\frac{(s+1)(s+2)}{(s+3)^2(s+4)}\) (b) \(G(s)=\frac{(s+2)}{(s+5)^2(s+7)^2}\) (c) \(G(s)=\frac{(s+4)}{(s+2)^2(s+5)(s+6)}\)

Represent the systems below in state space in phase-variable form. Draw the signal-flow graphs. [Section: 5.7] (a) \(G(s)=\frac{s+3}{s^2+2 s+7}\) (b) \(G(s)=\frac{s^2+2 s+6}{s^3+5 s^2+2 s+1}\) (c) \(G(s)=\frac{s^3+2 s^2+7 s+1}{s^4+3 s^3+5 s^2+6 s+4}\)

Repeat Problem 31 and represent each system in controller canonical and observer canonical forms. [Section: 5.7]

Represent the feedback control systems shown in Figure P5.24 in state space. When possible, represent the open-loop transfer functions separately incascadeandcompletethe feedback loops with the signal path from output to input. Draw your signal-flow graph to be in one-to-one correspondence to the block diagrams (as close as possible). [Section: 5.7]

You are given the system shown in Figure P5.25. [Section: 5.7] (a) Represent the system in state space in phase-variable form. (b) Represent the system in state space in any other form besides phase-variable.

Repeat Problem 34 for the system shown in Figure P5.26. [Section: 5.7]

Use MATLAB to solve Problem 35.

Represent the system shown in Figure P5.27 in state space where \(x_1(t), x_3(t)\), and \(x_4(t)\), as shown, are among the state variables, c(t) is the output, and \(x_2(t)\) is internal to \(X_1(s) / X_3(s)\). [Section: 5.7]

Consider the rotational mechanical system shown in Figure P5.28. (a) Represent the system as a signal-flow graph. (b) Represent the system in state space if the output is \(\theta_2(t)\).

Given a unity feedback system with the forward-path transfer function \(G(s)=\frac{8}{s(s+8)(s+10)}\) use MATLAB to represent the closed-loop system in state space in (a) phase-variable form; (b) parallel form.

Consider the cascaded subsystems shown in Figure P5.29. If \(G_1(s)\) is represented in state space as \(\begin{aligned} \dot{\mathbf{x}}_1 & =\mathbf{A}_1 \mathbf{x}_1+\mathbf{B}_1 r \\ y_1 & =\mathbf{C}_1 \mathbf{x}_1 \end{aligned}\) and \(G_2(s)\) is represented in state space as \(\begin{aligned} & \dot{\mathbf{x}}_2=\mathbf{A}_{\mathbf{2}} \mathbf{x}_2+\mathbf{B}_2 y_1 \\ & y_2=\mathbf{C}_2 \mathbf{x}_2 \end{aligned}\) show that the entire system can be represented in state space as

Consider the parallel subsystems shown in Figure P5.30. If \(G_1(s)\) is represented in state space as \(\begin{aligned} \dot{\mathbf{x}}_{\mathbf{1}} & =\mathbf{A}_{\mathbf{1}} \mathbf{x}_{\mathbf{1}}+\mathbf{B}_{\mathbf{1}} r \\ y_1 & =\mathbf{C}_{\mathbf{1}} \mathbf{x}_{\mathbf{1}} \end{aligned}\) and \(G_2(s)\) is represented in state space as \(\begin{aligned} & \dot{\mathbf{x}}_2=\mathbf{A}_{\mathbf{2}} \mathbf{x}_2+\mathbf{B}_2 r \\ & y_2=\mathbf{C}_{\mathbf{2}} \mathbf{x}_2 \end{aligned}\) show that the entire system can be represented in state space as \(\begin{aligned} {\left[\begin{array}{c} \dot{\mathbf{x}}_1 \\ \cdots \\ \dot{\mathbf{x}}_2 \end{array}\right] } & =\left[\begin{array}{ccc} \mathbf{A}_1 & \vdots & \mathbf{0} \\ \cdots & \vdots & \cdots \\ \mathbf{0} & \vdots & \mathbf{A}_2 \end{array}\right]\left[\begin{array}{c} \mathbf{x}_1 \\ \cdots \\ \mathbf{x}_2 \end{array}\right]+\left[\begin{array}{c} \mathbf{B}_1 \\ \cdots \\ \mathbf{B}_2 \end{array}\right] r \\ y & =\left[\begin{array}{lll} \mathbf{C}_1 & \vdots & \mathbf{C}_2 \end{array}\right]\left[\begin{array}{c} \mathbf{x}_1 \\ \cdots \\ \mathbf{x}_2 \end{array}\right] \end{aligned}\)

Consider the subsystems shown in Figure P5.31 and connected to form a feedback system. If G(s) is represented in state space as \(\begin{aligned} \dot{\mathbf{x}}_1 & =\mathbf{A}_1 \mathbf{x}_1+\mathbf{B}_1 e \\ y & =\mathbf{C}_1 \mathbf{x}_1 \end{aligned}\) and \(\mathrm{H}_2(s)\) is represented in state space as \(\begin{aligned} & \dot{\mathbf{x}}_2=\mathbf{A}_2 \mathbf{x}_2+\mathbf{B}_2 y \\ & \rho=\mathbf{C}_2 \mathbf{x}_2 \end{aligned}\) show that the closed-loop system can be represented in state space as \(\begin{aligned} & {\left[\begin{array}{c} \dot{\mathbf{x}}_1 \\ \cdots \\ \dot{\mathbf{x}}_2 \end{array}\right]=\left[\begin{array}{cc} \mathbf{A}_1: & -\mathbf{B}_1 \mathbf{C}_2 \\ \cdots & \cdots \\ \mathbf{B}_2 \mathbf{C}_1: & \mathbf{A}_2 \end{array}\right]\left[\begin{array}{c} \mathbf{x}_1 \\ \cdots \\ \mathbf{x}_2 \end{array}\right]+\left[\begin{array}{c} \mathbf{B}_1 \\ \cdots \\ \mathbf{0} \end{array}\right] r} \\ & y=\left[\mathbf{C}_1 \vdots 0\right]\left[\begin{array}{c} \mathbf{x}_1 \\ \cdots \\ \mathbf{x}_2 \end{array}\right] \\ & \end{aligned}\)

Given the system represented in state space as follows: [Section: 5.8] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} -1 & -7 & 6 \\ -8 & 4 & 8 \\ 4 & 7 & -8 \end{array}\right] \mathbf{x}+\left[\begin{array}{r} -5 \\ -7 \\ 5 \end{array}\right] r \\ & y=\left[\begin{array}{lll} -9 & -9 & -8 \end{array}\right] \mathbf{x} \end{aligned}\) convert the system to one where the new state vector, z, is \(\mathbf{z}=\left[\begin{array}{rrr} -4 & 9 & -3 \\ 0 & -4 & 7 \\ -1 & -4 & -9 \end{array}\right] \mathbf{x}\)

Repeat Problem 43 for the following system: [Section: 5.8] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} -5 & 1 & 1 \\ 9 & -9 & -9 \\ -9 & -1 & 8 \end{array}\right] \mathbf{x}+\left[\begin{array}{r} 9 \\ -4 \\ 0 \end{array}\right] r \\ & y=\left[\begin{array}{lll} -2 & -4 & 1 \end{array}\right] \mathbf{x} \end{aligned}\) and the following state-vector transformation: \(\mathbf{z}=\left[\begin{array}{rrr} 5 & -4 & 9 \\ 6 & -7 & 6 \\ 6 & -5 & -3 \end{array}\right] \mathbf{x}\)

Diagonalize the following system: [Section: 5.8] \(\begin{aligned} & \dot{\mathbf{x}}=\left[\begin{array}{rrr} -5 & -5 & 4 \\ 2 & 0 & -2 \\ 0 & -2 & -1 \end{array}\right] \mathbf{x}+\left[\begin{array}{r} -1 \\ 2 \\ -2 \end{array}\right] r \\ & y=\left[\begin{array}{lll} -1 & 1 & 2 \end{array}\right] \mathbf{x} \end{aligned}\)

Repeat Problem 45 for the following system: [Section: 5.8] \(\begin{aligned} \dot{\mathbf{x}} & =\left[\begin{array}{ccc} -10 & -3 & 7 \\ 18.25 & 6.25 & -11.75 \\ -7.25 & -2.25 & 5.75 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 1 \\ 3 \\ 2 \end{array}\right] r \\ y & =\left[\begin{array}{lll} 1 & -2 & 4 \end{array}\right] \mathbf{x} \end{aligned}\)

Diagonalize the system in Problem 46 using MATLAB.

Find the closed-loop transfer function of the Unmanned Free-Swimming Submersible vehicle’s pitch control system shown on the back endpapers (Johnson, 1980).

Repeat Problem 48 using MATLAB.

Use Simulink to plot the effects of nonlinearities upon the closed-loop step response of the antenna azimuth position control system shown on the front endpapers, Configuration 1. In particular, consider individually each of the following nonlinearities: saturation (\(\pm 5\) volts), backlash (deadband width 0.15), deadzone (-2 to +2), as well as the linear response. Assume the preamplifier gain is 100 and the step input is 2 radians.

Problem 8 in Chapter 1 describes a high-speed proportional solenoid valve. A subsystem of the valve is the solenoid coil shown in Figure P5.32. Current through the coil, L, generates a magnetic field that produces a force to operate the valve. Figure P5.32 can be represented as a block diagram (Vaughan, 1996). (a) Derive a block diagram of a feedback system that represents the coil circuit, where the applied voltage, \(v_g(t)\), is the input, the coil voltage, \(v_L(t)\), is the error voltage, and the current, i(t), is the output. (b) For the block diagram found in Part a, find the Laplace transform of the output current, I(s). (c) Solve the circuit of Figure P5.32 for I(s), and compare to your result in Part b.

Figure P5.33 shows a noninverting operational amplifier. Assuming the operational amplifier is ideal, (a) Verify that the system can be described by the following two equations: \(\begin{aligned} & v_o=A\left(v_i-v_o\right) \\ & v_1=\frac{R_i}{R_i+R_f} v_o \end{aligned}\) (b) Check that these equations can be described by the block diagram of Figure P5.33(b). (c) Use Mason's rule to obtain the closed-loop system transfer function \(\frac{V_o(s)}{V_i(s)}\). (d) Show that when \(A \rightarrow \infty, \frac{V_o(s)}{V_i(s)}=1+\frac{R_f}{R_i}\).

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 equations. b. Draw a corresponding block diagram and obtain the transfer function \(\frac{V_o(s)}{V_i(s)}\). c. Show that when \(A \rightarrow \infty, \frac{V_o(s)}{V_i(s)}=-\frac{R_f}{R_i}\).

Figure P5.35(a) shows an n-channel enhancement-mode MOSFET source follower circuit. Figure P5.35(b) shows its small-signal equivalent (where \(R_i=R_1 \| R_2\) ) (Neamen, 2001). a. Verify that the equations governing this circuit are \(\frac{v_{i n}}{v_i}=\frac{R_i}{R_i+R_s} ; \quad v_{g s}=v_{i n}-v_o ; \quad v_o=g_m\left(R_s \| r_o\right) v_{g s}\) b. Draw a block diagram showing the relations between the equations. c. Use the block diagram in Part b to find \(\frac{V_o(s)}{V_i(s)}\).

A car active suspension system adds an active hydraulic actuator in parallel with the passive damper and spring to create a dynamic impedance that responds to road variations. The block diagram of Figure P5.36 depicts such an actuator with closed-loop control. In the figure, \(K_t\) is the spring constant of the tire, \(M_{U S}\) is the wheel mass, r is the road disturbance, \(x_1\) is the vertical car displacement, \(x_3\) is the wheel vertical displacement, \(\omega_0^2=\frac{K_t}{M_{U S}}\) is the natural frequency of the unsprung system and \(\varepsilon\) is a filtering parameter to be judiciously chosen (Lin, 1997). Find the two transfer functions of interest: (a) \(\frac{X_3(s)}{R(s)}\) (b) \(\frac{X_1(s)}{R(s)}\)

The basic unit of skeletal and cardiac muscle cells is a sarcomere, which is what gives such cells a striated (parallel line) appearance. For example, one bicep cell has about \(10^5\) sarcomeres. In turn, sarcomeres are composed of protein complexes. Feedback mechanisms play an important role in sarcomeres and thus muscle contraction. Namely, Fenn’s law says that the energy liberated during muscle contraction depends on the initial conditions and the load encountered. The following linearized model describing sarcomere contraction has been developed for cardiac muscle: \(\begin{aligned} {\left[\begin{array}{c} \dot{A} \\ \dot{T} \\ \dot{U} \\ \dot{S L} \end{array}\right] } & =\left[\begin{array}{cccc} -100.2 & -20.7 & -30.7 & 200.3 \\ 40 & -20.22 & 49.95 & 526.1 \\ 0 & 10.22 & -59.95 & -526.1 \\ 0 & 0 & 0 & 0 \end{array}\right]\left[\begin{array}{c} A \\ T \\ U \\ S L \end{array}\right]+\left[\begin{array}{c} 208 \\ -208 \\ -108.8 \\ -1 \end{array}\right] u(t) \\ y & =\left[\begin{array}{llll} 0 & 1570 & 1570 & 59400 \end{array}\right]\left[\begin{array}{c} A \\ T \\ U \\ S L \end{array}\right]-6240 u(t) \end{aligned}\) where A= density of regulatory units with bound calcium and adjacent weak cross bridges \((\mu \mathrm{M})\) T= density of regulatory units with bound calcium and adjacent strong cross bridges (M) U= density of regulatory units without bound calcium and adjacent strong cross bridges (M) SL= sarcomere length (m) The system's input is u(t)= the shortening muscle velocity in meters/second and the output is y(t)= muscle force output in Newtons (Yaniv, 2006). Do the following: (a) Use MATLAB to obtain the transfer function \(\frac{Y(s)}{U(s)}\). (b) Use MAtLAB to obtain a partial- fraction expansion for \(\frac{Y(s)}{U(s)}\) (c) Draw a signal-flow diagram of the system in parallel form. (d) Use the diagram of Part c to express the system in state-variable form with decoupled equations.

An electric ventricular assist device (EVAD) has been designed to help patients with diminished but still functional heart pumping action to work in parallel with the natural heart. The device consists of a brushless dc electric motor that actuates on a pusher plate. The plate movements help the ejection of blood in systole and sac filling in diastole. System dynamics during systolic mode have been found to be: \(\left[\begin{array}{c} \dot{x} \\ \dot{v} \\ \dot{P}_{a o} \end{array}\right]=\left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & -68.3 & -7.2 \\ 0 & 3.2 & -0.7 \end{array}\right]\left[\begin{array}{c} x \\ v \\ P_{a o} \end{array}\right]+\left[\begin{array}{c} 0 \\ 425.4 \\ 0 \end{array}\right] e_m\) The state variables in this model are x, the pusher plate position, v, the pusher plate velocity, and \(P_{a o}\), the aortic blood pressure. The input to the system is \(e_m\), the motor voltage (Tasch, 1990). (a) Use MATLAB to find a similarity transformation to diagonalize the system. (b) Use MATLAB and the obtained similarity transformation of Part a to obtain a diagonalized expression for the system.

In an experiment to measure and identify postural arm reflexes, subjects hold in their hands a linear hydraulic manipulator. A load cell is attached to the actuator handle to measure resulting forces. At the application of a force, subjects try to maintain a fixed posture. Figure P5.37 shows a block diagram for the combined arm-environment system. In the diagram, \(H_r(s)\) represents the reflexive length and velocity feedback dynamics; \(H_{a c t}(s)\) the activation dynamics, \(H_i(s)\) the intrinsic act dynamics; \(H_h(s)\) the hand dynamics; \(H_e(s)\) the environmental dynamics; \(X_a(s)\) the position of the arm; \(X_h(s)\) the measured position of the hand; \(F_h(s)\) the measured interaction force applied by the hand; \(F_{\text {int }}(s)\) the intrinsic force; \(F_{r e f}(s)\) the reflexive force; A(s) the reflexive activation; and D(s) the external force perturbation (de Vlugt, 2002). (a) Obtain a signal-flow diagram from the block diagram. (b) Find \(\frac{F_h(s)}{D(s)}\).

Use LabVIEW’s Control Design and Simulation Module to obtain the controller and the observer canonical forms for: \(G(s)=\frac{s^2+7 s+2}{s^3+9 s^2+26 s+24}\)

A virtual reality simulator with haptic (sense of touch) feedback was developed to simulate the control of a submarine driven through a joystick input. Operator haptic feedback is provided through joystick position constraints and simulator movement (Karkoub, 2010). Figure P5.38 shows the block diagram of the haptic feedback system in which the input \(u_h\) is the force exerted by the muscle of the human arm; and the outputs are \(y_s\), the position of the simulator, and \(y_j\), the position of the joystick. (a) Find the transfer function \(\frac{Y_s(s)}{U_h(s)}\). (b) Find the transfer function \(\frac{Y_j(s)}{U_h(s)}\).

Some medical procedures require the insertion of a needle under a patient’s skin using CT scan monitoring guidance for precision. CT scans emit radiation, posing some cumulative risks for medical personnel. To avoid this problem, a remote control robot has been developed (Piccin, 2009). The robot controls the needle in position and angle in the constraint space of a CT scan machine and also provides the physician with force feedback commensurate with the insertion opposition encountered by the type of tissue in which the needle is inserted. The robot has other features that give the operator the similar sensations and maneuverability as if the needle was inserted directly. Figure P5.39 shows the block diagram of the force insertion mechanism, where \(F_h\) is the input force and \(X_h\) is the output displacement. Summing junction inputs are positive unless indicated with a negative sign. By way of explanation, Z= impedance; G= transfer function; \(C_i=\) communication channel transfer functions; F= force; X= position. Subscripts h and m refer to the master manipulator. Subscripts s and e refer to the slave manipulator. (a) Assuming \(Z_h=0, C_1=C_s, C_2=1+C_6\) and \(C_4= -C_m\) use Mason's Rule to show that the transfer function from the operators force input \(F_h\) to needle displacement \(X_h\) is given by \(Y(s)=\frac{X_h(s)}{F_h(s)}=\frac{Z_m^{-1} C_2\left(1+G_s C_s\right)}{1+G_s C_s+Z_m^{-1}\left(c_m+C_2 Z_e G_s C_s\right)}\) (b) Now with \(Z_h \neq 0\) show that \(\frac{X_h(s)}{F_h(s)}=\frac{Y(s)}{1+Y(s) Z_h}\)

A hybrid solar cell and diesel power distribution system has been proposed and tested (Lee, 2007). The system has been shown to have a very good uninterruptible power supply as well as line voltage regulation capabilities. Figure P5.40 shows a signal flow diagram of the system. The output, \(V_{\text {Load }}\), is the voltage across the load. The two inputs are \(I_{C f}\), the reference current, and \(I_{D i s t}\), the disturbance representing current changes in the supply. (a) Refer to Figure P5.40 and find the transfer function \(\frac{V_{\text {Load }}(s)}{I_{C f}(s)}\). (b) Find the transfer function \(\frac{V_{\text {Load }}(s)}{I_{\text {Dist }}(s)}\).

Continuous casting in steel production is essentially a solidification process by which molten steel is solidified into a steel slab after passing through a mold, as shown in Figure P5.41(a). Final product dimensions depend mainly on the casting speed \(V_p\) (in \(\mathrm{m} / \mathrm{min}\) ), and on the stopper position X (in %) that controls the flow of molten material into the mold (Kong, 1993). A simplified model of a casting system is shown in Figure P5.41 (b) (Kong, 1993) and (Graebe, 1995). In the model, \(H_m=\) mold level (in mm); \(H_t=\) assumed constant height of molten steel in the tundish; \(D_z=\) mold thickness = depth of nozzle immerged into molten steel; and \(W_t=\) weight of molten steel in the tundish. For a specific setting let \(A_m=0.5\) and \(G_x(s)=\frac{0.63}{s+0.926}\) Also assume that the valve positioning loop may be modeled by the following second-order transfer function: \(G_V(s)=\frac{X(s)}{Y_C(s)}=\frac{100}{s^2+10 s+100}\) and the controller is modeled by the following transfer function: \(G_C(s)=\frac{1.6\left(s^2+1.25 s+0.25\right)}{s}\) The sensitivity of the mold level sensor is \(\beta=0.5\) and the initial values of the system variables at t=0- are: R(0-)=0 ; \(Y_C(0-)=X(0-)=41.2\) ; \(\Delta H_m(0-)=0\); \(H_m(0-)=-75 ; \Delta V_p(0-)=0\); and \(V_p(0-)=0\). Do the following: (a) Assuming \(v_p(t)v is constant \(\left[\Delta v_p=0\right]\), find the closed-loop transfer function \(T(s)=\Delta H_m(s) / R(s)\). (b) For r(t)=5 u(t), \(v_p(t)=0.97 u(t)\), and \(H_m(0-)=-75 \mathrm{~mm}\), use Simulink to simulate the system. Record the time and mold level (in array format) by connecting them to Workspace sinks, each of which should carry the respective variable name. After the simulation ends, utilize MATLAB plot commands to obtain and edit the graph of \(h_m(t)\) from t=0 to 80 seconds.

A simplified second-order transfer function model for bicycle dynamics is given by \(\frac{\varphi(s)}{\delta(s)}=\frac{a V}{b h} \frac{\left(s+\frac{V}{a}\right)}{\left(s^2-\frac{g}{h}\right)}\) The input is \(\delta(s)\), the steering angle, and the output is \(\varphi(s)\), the tilt angle (between the floor and the bicycle longitudinal plane). In the model parameter a is the horizontal distance from the center of the back wheel to the bicycle center of mass; b is the horizontal distance between the centers of both wheels; h is the vertical distance from the center of mass to the floor; V is the rear wheel velocity (assumed constant); and g is the gravity constant. It is also assumed that the rider remains at a fixed position with respect to the bicycle so that the steer axis is vertical and that all angle deviations are small (Åstrom, 2005). (a) Obtain a state-space representation for the bicycle model in phase-variable form. (b) Find system eigenvalues and eigenvectors. (c) Find an appropriate similarity transformation matrix to diagonalize the system and obtain the state-space system's diagonal representation.

It is shown in Figure 5.6(c) that when negative feedback is used, the overall transfer function for the system of Figure 5.6(b) is \(\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s) H(s)}\) Develop the block diagram of an alternative feedback system that will result in the same closed-loop transfer function, C(s)/R(s), with G(s) unchanged and unmoved. In addition, your new block diagram must have unity gain in the feedback path. You can add input transducers and/or controllers in the main forward path as required.

The purpose of an Automatic Voltage Regulator is to maintain constant the voltage generated in an electrical power system, despite load and line variations, in an electrical power distribution system (Gozde, 2011). Figure P5.42 shows the block diagram of such a system. Assuming \(K_a=10, T_a=0.1\), \(K_e=1, T_e=0.4, K_g=1, T_g=1, K_s=1, T_s=0.001\), and the controller, \(G_{P I D}(s)=1.6+\frac{0.4}{s}+0.3 s\), find the closed-loop transfer function, \(T(s)=\frac{\Delta V_t(s)}{\Delta V_{r e f}(s)}\), of the system, expressing it as a rational function.

A drive system with an elastically coupled load was presented in Problem 71, Chapter 4. The mechanical part of this drive (Thomsen, 2011) was reduced to a two-inertia model. Using slightly different parameters, the following transfer function results: \(G(s)=\frac{\Omega_L(s)}{T(s)}=\frac{25\left(s^2+1.2 s+12500\right)}{s\left(s^2+5.6 s+62000\right)}\) Here, \(T(s)=T_{e m}(s)-T_L(s)\), where \(T_{e m}(s)=\) the electromagnetic torque developed by the motor, \(T_L(s)=\) the load torque, and \(\Omega_L(s)=\) the load speed. The drive is shown in Figure P5.43 as the controlled unit in a feedback control loop, where \(\Omega_r(s)=\) the desired (reference) speed. The controller transfer function is \(G_C(s)=K_p+\frac{K_I}{s}=4+\frac{0.5}{s}\) and provides an output voltage =0-5.0 volts. The motor and its power amplifier have a gain, \(K_M=10 \mathrm{~N}\)-m/volt. (a) Find the minor-loop transfer function, \(D(s)=\frac{\Omega_L(s)}{T_{e m}(s)}\) analytically or using MATLAB. (b) Given that at t=0, the load speed \(\omega_L(t)=0 \mathrm{rad} / \mathrm{sec}\) and a step reference input \(\omega_r(t)=260 u(t), \mathrm{rad} / \mathrm{sec}\), is applied, use MATLAB (or any other program) to find and plot \(\omega_L(t)\). Mark on the graph all of the important characteristics, such as percent overshoot, peak time, rise time, settling time, and final steady-state value.

Integrated circuits are manufactured through a lithographic process on a semiconductor wafer. In Lithography, similarly to chemical photography, a semiconductor wafer is covered with a photosensitive emulsion and then selectively exposed to light to form the electronic components. Due to miniaturization, this process is to be performed with nanometer accuracy and at the highest possible speed. Sophisticated apparatus and methods have been developed for this purpose. Figure P5.44 shows the block diagram of a scanner dedicated to this purpose (Butler, 2011). Use Mason’s Rule to find: (a) The transfer function \(\frac{X_{s s}(s)}{R(s)}\). (b) The transfer function \(\frac{X_{l s}(s)}{R(s)}\).

A boost converter is a dc-to-dc switched power supply in which the voltage output is larger than the voltage input. A block diagram for a peak current mode controlled converter (Chen, 2013) is shown in Figure P5.45. Find the transfer function \(\frac{\hat{v}_o(s)}{\hat{v}_i(s)}\).

In Problem 64 of Chapter 2, a three-phase ac/dc converter that supplies dc to a battery charging system (Graovac, 2001) was introduced. Each phase had an ac filter represented by the equivalent circuit of Figure P2.38. You were asked to show that the following equation gives the s-domain relationship between the inductor current, \(I_{a c F}(s)\), and two active sources: a current source, \(I_{a c R}(s)\), representing a phase of the ac/dc converter, and the supply phase voltage, \(V_a(s)\) : \(\begin{aligned} I_{a c F}(s)= & I_{a c F 1}(s)+I_{a c F 2}(s)=\frac{1+R C s}{L C s^2+R C s+1} I_{a c R}(s) \\ & +\frac{C s}{L C s^2+R C s+1} V_a(s) \end{aligned}\) (a) Derive an s-domain equation for \(V_c(s)\). (b) Given that \(R=1 \Omega, L=1 \mathrm{mH}\), and \(C=20 \mu \mathrm{F}, i_{a c R}(t)=10 u(t)\) amps, \(v_a(t)=20 t u(t)\) volts, \({ }^{10}\) and assuming zero initial conditions, use Simulink to model this system and plot the inductor current, \(i_{a c F}(t)\), and the capacitor voltage, \(v_c(t)\), over a period from 0 to \(15 \mathrm{~ms}\).

The motor and load shown in Figure P5.46(a) are used as part of the unity feedback system shown in Figure P5.46(b). Find the value of the coefficient of viscous damping, \(D_L\), that must be used in order to yield a closed-loop transient response having a 20% overshoot.

Assume that the motor whose transfer function is shown in Figure P5.47(a) is used as the forward path of a closed-loop, unity feedback system. (a) Calculate the percent overshoot and settling time that could be expected. (b) You want to improve the response found in Part a. Since the motor and the motor constants cannot be changed, an amplifier and a tachometer (voltage generator) are inserted into the loop, as shown in Figure P5.47. Find the values of \(K_1\) and \(K_2\) to yield a 20% overshoot and a settling time of 0.25 second.

The system shown in Figure P5.48 will have its transient response altered by adding a tachometer. Design K and \(K_2\) in the system to yield a damping ratio of 0.69. The natural frequency of the system before the addition of the tachometer is \(10 \mathrm{rad} / \mathrm{s}\).

The Mechanical system shown in Figure P5.49(a)is used as part of the unity feedback system shown in Figure P5.49(b). Find the values of M and D to yield 20% overshoot and 2 seconds settling time.

Assume ideal operational amplifiers in the circuit of Figure P5.50. (a) Show that the leftmost operational amplifier works as a subtracting amplifier. Namely, \(v_1=v_o-v_{i n}\). (b) Draw a block diagram of the system, with the subtracting amplifier represented with a summing junction, and the circuit of the rightmost operational amplifier with a transfer function in the forward path. Keep R as a variable. (c) Obtain the system's closed-loop transfer function. (d) For a unit step input, obtain the value of R that will result in a settling time \(T_s=1 \mathrm{msec}\). (e) Using the value of R calculated in Part d, make a sketch of the resulting unit step response.

Given the rotational system shown in Figure P4.23 (Problem 78 of Chapter 4), do the following: a. Using the transfer function you derived for that system, \(G(s)=\Theta_1(s) / T(s)\), where \(\Theta_1(s)\) is the angular displacement of the first shaft, find the value of \(n=N_1 / N_2\) that yields a settling time of 10 seconds for a step input in torque. b. If this rotational system is the controlled unit, G(s), in the feedback system of Figure P5.51, find the values of \(\zeta, \omega_n\), % O.S., and \(T_s\) for a controller gain \(K=4 \mathrm{~N}-\mathrm{m} / \mathrm{rad}\) and r(t)=u(t) radians.

A process is simulated by the second-order passive circuit, shown in Figure P5.52, where the feedback amplifier, controller, and final control element are represented by op-amp circuits. (a) Denoting the input and output as \(R(s)=V_i(s)\) and \(C(s)=V_o(s)\), with R(s)-C(s)=E(s), and noting that the feedback amplifier has a unity gain, draw a block diagram for this feedback control system, where \(G_C(s), G_F(s)\), and \(G_P(s)\) are the transfer functions of the controller, final control element, and the process, respectively. (b) Find the value of \(R_P\) that makes the circuit representing the process critically damped. (c) Noting that the proportional controller is simply an amplifier, \(G_C(s)=K_P\), find the value of its gain \(K_P\) that results in dominant closed-loop poles with a damping ratio, \(\zeta=0.5\), and a settling time, \(T_s=4 \mathrm{~ms}\). Verify that the other pole is nondominant. What would be the appropriate value of the controller potentiometer, \(R_F\), given that its tolerance is \(\pm 10 \%\) ?

Control of HIV/AIDS. Given the HIV system of Problem 84 in Chapter 4 and repeated here for convenience (Craig, 2004): \(\begin{aligned} {\left[\begin{array}{c} \dot{T} \\ \dot{T}^* \\ \dot{v} \end{array}\right]=} & {\left[\begin{array}{ccc} -0.04167 & 0 & -0.0058 \\ 0.0217 & -0.24 & 0.0058 \\ 0 & 100 & -2.4 \end{array}\right]\left[\begin{array}{l} T \\ T^* \\ v \end{array}\right] } \\ + & {\left[\begin{array}{c} 5.2 \\ -5.2 \\ 0 \end{array}\right] u_1 } \\ y= & {\left[\begin{array}{lll} 0 & 0 & 1 \end{array}\right]\left[\begin{array}{c} T \\ T^* \\ v \end{array}\right] } \end{aligned}\) Express the system in the following forms: (a) Phase-variable form (b) Controller canonical form (c) Observer canonical form Finally, (d) Use MATLAB to obtain the system’s diagonalized representation.

Hybrid vehicle. Figure P5.53 shows the block diagram of a possible cascade control scheme for an HEV driven by a dc motor (Preitl, 2007). Let the speed controller \(G_{S C}(s)=100+\frac{40}{s}\), the torque controller and power amp \(K_A G_{T C}(s)=10+\frac{6}{s}\), the current sensor sensitivity \(K_{C S}=0.5\), the speed sensor sensitivity \(K_{S S}=0.0433\). Also, following the development in previous chapters \(\frac{1}{R_a}=1 ; \eta_{\text {tot }} K_t=1.8\); \(k_b=2 ; \quad D=k_f=0.1 ; \frac{1}{J_{\text {tot }}}=\frac{1}{7.226} ; \frac{r}{i_{\text {tot }}}=0.0615\) ; and \(\rho C_w A v_0 \frac{r}{i_{\text {tot }}}=0.6154\) (a) Substitute these values in the block diagram, and find the transfer function, \(T(s)=V(s) / R_v(s)\), using block diagram reduction rules. [Hint: Start by moving the last \(\frac{r}{i_{\text {tot }}}\) block to the right past the pickoff point.] (b) Develop a Simulink model for the original system in Figure P5.53. Set the reference signal input, \(r_V(t)=4 u(t)\), as a step input with a zero initial value, a step time =0 seconds, and a final value of 4 volts. Use X-Y graphs to display (over the period from 0 to 8 seconds) the response of the following variables to the step input: (1) change in car speed (m/s), (2) car acceleration \(\left(\mathrm{m} / \mathrm{s}^2\right)\), and (3) motor armature current (A). To record the time and the above three variables (in array format), connect them to four Workspace sinks, each of which carries the respective variable name. After the simulation ends, utilize MATLAB plot commands to obtain and edit the three graphs of interest.

Parabolic trough collector. Effective controller design for parabolic trough collector setups is an active area of research. One of the techniques used for controller design (Camacho, 2012) is Internal Model Control (IMC). Although complete details of IMC will not be presented here, Figure P5.54(a) shows a block diagram for the IMC setup. Use of IMC assumes a very good knowledge of the plant dynamics. In Figure P5.54(a), the actual plant is \(P(s) . \tilde{P}(s)\) is a software model that mimics the plant functions. G(s) is the controller to be designed. It is also assumed that all blocks represent linear time-invariant systems and thus the superposition theorem applies to the system. (a) Use superposition (by assuming D(s)=0 ) and Mason's gain formula to find the transfer function \(\frac{C(s)}{R(s)}\) from command input to system output. (b) Use superposition (by assuming R(s)=0 ) and Mason's gain formula to find the transfer function \(\frac{C(s)}{D(s)}\) from disturbance input to system output. (c) Use the results of Parts a and b to find the combined output C(s) due to both system inputs. (d) Show that the system of Figure P5.54(a) has the same transfer function as the system in Figure P5.54(b) when \(G_C(s)=\frac{G(s)}{1-G(s) \tilde{P}(s)}\).