Solve the following differential equations using classical

A variable resistor, called a potentiometer, is shown in Figure P1.1. The resistance is varied by moving a wiper arm along a fixed resistance. The resistance from A to C is fixed, but the resistance from B to C varies with the position of the wiper arm. If it takes 10 turns to move the wiper arm from A to C, draw a block diagram of the potentiometer showing the input variable, the output variable, and (inside the block) the gain, which is a constant and is the amount by which the input is multiplied to obtain the output. An animation PowerPoint presentation (PPT) demonstrating this system is available for instructors at www.wiley.com/college/nise. See Potentiometer. [Section 1.4: Introduction to a Case Study]

A temperature control system operates by sensing the difference between the thermostat setting and the actual temperature and then opening a fuel valve an amount proportional to this difference. Draw a functional closed loop block diagram similar to Figure 1.8(d) identifying the input and output transducers, the controller, and the plant. Further, identify the input and output signals of all subsystems previously described. [Section 1.4: Introduction to a Case Study]

An aircraft’s attitude varies in roll, pitch, and yaw as defined in Figure P1.2. Draw a functional block diagram for a closed-loop system that stabilizes the roll as follows: The system measures the actual roll angle with a gyro and compares the actual roll angle with the desired roll angle. The ailerons respond to the roll-angle error by undergoing angular deflection.The Aircraft Aircraft responds to this angular deflection, producing a roll angle rate. Identify the input and output transducers, the controller, and the plant. Further, identify the nature of each signal. [Section 1.4: Introduction to a Case Study]

We can build a control system that will automatically adjust amotorcycle’s radio volume as the noise generated by the motorcycle changes. The noise generated by the motorcycle increases with speed. As the noise increases, the system increases the volume of the radio. Assume that the amount of noise can be represented by a voltage generated by the speedometer cable, and the volume of the radio is controlled by a dc voltage (Hogan, 1988). If the dc voltage represents the desired volume disturbed by the motorcycle noise, draw the functional block diagram of the automatic volume control system, showing the input transducer, the volume control circuit, and the speed transducer as blocks. Also, show the following signals: the desired volume as an input, the actual volume as an output, and voltages representing speed, desired volume, and actual volume. An animation PowerPoint presentation (PPT) demonstrating this system is available for instructors at www.wiley.com/college/nise. See Motorcycle. [Section 1.4: Introduction to a Case Study]

A dynamometer is a device used to measure torque and speed and to vary the load on rotating devices. The dynamometer operates as follows to control the amount of torque:A hydraulic actuator attached the axle presses attire against a rotating flywheel. The greater the displacement of the actuator,the greater the force that is applied to the rotating flywheel. A strain gage load cell senses the force. The displacement of the actuator is controlled by an electrically operated valve whose displacement regulates fluid flowing into the actuator (D’Souza, 1988). Draw a functional block diagram of a closed-loop system that uses the described dynamometer to regulate the force against the tire during testing. Show all signals and systems. Include amplifiers that power the valve, the valve, the actuator and load, and the tire. [Section 1.4: Introduction to a Case Study]

During a medical operation an anesthesiologist controls the depth of unconsciousness by controlling the concentration of isoflurane in a vaporized mixture with oxygen and nitrous oxide. The depth of anesthesia is measured by the patient’s blood pressure. The anesthesiologist also regulates ventilation, fluid balance, and the administration of other drugs. In order to free the anesthesiologist to devote more time to the latter tasks, and in the interest of the patient’s safety, we wish to automate the depth of anesthesia by automating the control of isoflurane concentration.Draw a functional block diagram of the system showing pertinent signals and subsystems (Meier, 1992). [Section 1.4: Introduction to a Case Study]

The vertical position, x(t), of a grinding wheel is controlled by a closed-loop system. The input to the system is the desired depth of grind, and the output is the actual depth of grind. The difference between the desired depth and the actual depth drives the motor, resulting in a force applied to the work. This force results in a feed velocity for the grinding wheel (Jenkins, 1997). Draw a closed-loop functional block diagram for the grinding process, showing the input, output, force, and grinder feed rate. [Section 1.4: Introduction to a Case Study]

You are given a high-speed proportional solenoid valve. A voltage proportional to the desired position of the spool is applied to the coil. The resulting magnetic field produced by the current in the coil causes the armature to move. A push pin connected to the armature moves the spool. A linear voltage differential transformer (LVDT) that outputs a voltage proportional to displacement senses the spool’s position. This voltage can be used in a feedback path to implement closed-loop operation (Vaughan, 1996). Draw a functional block diagram of the valve, showing input and output positions, coil voltage, coil current, and spool force. [Section 1.4: Introduction to a Case Study]

The human eye has a biological control system that varies the pupil diameter to maintain constant light intensity to the retina. As the light intensity increases, the optical nerve sends a signal to the brain, which commands internal eye muscles to decrease the pupil’s eye diameter. When the light intensity decreases, the pupil diameter increases. a. Draw a functional block diagram of the light-pupil system indicating the input, output, and intermediate signals; the sensor; the controller; and the actuator. [Section 1.4: Introduction to a Case Study] b. Under normal conditions the incident light will be larger than the pupil. If the incident light is smaller than the diameter of the pupil, the feedback path is broken (Bechhoefer, 2005).Modify your block diagram from Part a. to show where the loop is broken.What will happen if the narrow beam of light varies in intensity, such as in a sinusoidal fashion? c. It has been found (Bechhoefer, 2005) that it takes the pupil about 300 milliseconds to react to a change in the incident light. If light shines off center to the retina, describe the response of the pupil with delay present and then without delay present.

A Segway \(^{\circledR 5}\) Personal Transporter (PT) (Figure P1.3) is a two-wheeled vehicle in which the human operator stands vertically on a platform. As the driver leans left, right, forward, or backward, a set of sensitive gyroscopic sensors sense the desired input. These signals are fed to a computer that amplifies them and commands motors to propel the vehicle in the desired direction. One very important feature of the PT is its safety: The system will maintain its vertical position within a specified angle despite road disturbances, such as uphills and downhills or even if the operator over-leans in any direction. Draw a functional block diagram of the PT system that keeps the system in a vertical position. Indicate the input and output signals, intermediate signals, and main subsystems. (http://segway.com)

In humans, hormone levels, alertness, and core body temperature are synchronized through a 24-hour circadian cycle. Daytime alertness is at its best when sleep/wake cycles are in sync with the circadian cycle. Thus alertness can be easily affected with a distributed work schedule, such as the one to which astronauts are subjected. It has been shown that the human circadian cycle can be delayed or advanced through light stimulus. To ensure optimal alertness, a system is designed to track astronauts’ circadian cycles and increase the quality of sleep during missions.Core body temperature can be used as an indicator of the circadian cycle. A computer model with optimum circadian body temperature variations can be compared to an astronaut’s body temperatures. Whenever a difference is detected, the astronaut is subjected to a light stimulus to advance or delay the astronaut’s circadian cycle (Mott, 2003). Draw a functional block diagram of the system. Indicate The Input and output signals, intermediate signals, and main subsystems.

Tactile feedback is an important component in the learning of motor skills such as dancing, sports, and physical rehabilitation. A suit with white dots recognized by a vision system to determine arm joint positions with millimetric precision was developed. This suit is worn by both teacher and student to provide position information. (Lieberman, 2007). If there is a difference between the teacher’s positions and that of the student, vibrational feedback is provided to the student through eight strategically placed vibrotactile actuators in the wrist and arm. This placement takes advantage of a sensory effect known as cutaneous rabbit that tricks the subject to feel uniformly spaced stimuli in places where the actuators are not present. These stimuli help the student adjust to correct the motion. In summary, the system consists of an instructor and a student having their movements followed by the vision system. Their movements are fed into a computer that finds the differences between their joint positions and provides proportional vibrational strength feedback the student. Draw a block diagram describing the system design.

Some skillful drivers can drive and balance a four wheeled vehicle on two wheels. To verify that a control system can also drive a car in this fashion, a prototype using an RC (remote-controlled) car is equipped with a feedback control system (Arndt, 2011). In a simplified system model,the roll angle at which the car balances was calculated a priori and found to be \(52.3^{\circ}\). This value was used as the desired input. The desired input is compared with the actual roll angle and the difference is fed to a controller that feeds a servo motor indicating the desired wheel steering angle that controls the vehicle’s roll angle on two wheels. The car’s actual roll angle is measured using a hinged linkage that rolls along the ground next to the vehicle and is connected to a potentiometer. Draw a block diagram indicating the system functions. Draw blocks for the system controller, the steering servo, and the car dynamics. Indicate in the diagram the following signals: the desired roll angle, the steering wheel angle, and the actual car roll angle.

Moored floating platforms are subject to external disturbances such as waves, wind, and currents that cause them to drift. There are certain applications, such as diving support, drilling pipe-laying, and tanking between ships in which precise positioning of moored platforms is very important (Muñoz-Mansilla, 2011). Figure P1.4 illustrates a tethered platform in which side thrusters are used for positioning. A control system is to be designed in which the objective is to minimize the drift, Y, and an angular deviation from the vertical axes, \(\phi\) (not shown). The disturbances acting on the system's outputs are the force, F, and the torque, M, caused by the external environment. In this problem, the plant will have one input, the force delivered by the thrusters \(\left(\mathrm{F}_u\right)\) and two outputs, Y and \(\phi\). Note also that this is a disturbance attenuation problem, so there is no command input. Draw a block diagram of the system indicating the disturbances F and M, the control signal \(\mathrm{F}_u\), and the outputs Y and \(\phi\). Your diagram should also have blocks for a controller, the one-input two-output plant, and a block indicating how the disturbances affect each of the outputs.

In the Case Study of Section 1.4, an antenna azimuth angle is controlled, and its corresponding block diagram is shown in Figure 1.8(d) in the text. There, the sensor used to measure the antenna’s azimuth angle is a potentiometer. a. Modify The block diagram of the sensor used to measure the antenna’s angle is an accelerometer. b. Modify The block diagram of the sensor used to measure the antenna’s angle is a gyroscope.

Figure P1.5 shows the topology of a photo-voltaic (PV) system that uses solar cells to supply electrical power to a residence with hybrid electric vehicle loads (Gurkavnak, 2009). The system consists of a PV array to collect the sun’s rays, a battery pack to store energy during the day, a dc/ac inverter to supply ac power to the load, and a bidirectional dc/dc converter to control the terminal voltage of the solar array according to a maximum power point tracking (MPPT) algorithm. In case of sufficient solar power (solar insolation), the dc/dc converter charges the battery and the solar array supplies power to the load through the dc/ac inverter.With Less or no solar energy (solar non-insolation), power is supplied from the battery to the load through the dc/dc converter and the dc/ac inverter. Thus, the dc/dc converter must be bidirectional to be able to charge and discharge the battery. With the MPPT controller providing the reference voltage, the converter operates as a step-up converter (boost) to discharge the battery if the battery is full or a step-down (buck) converter, which charges the battery if it is not full. In Figure P1.5, the Inverter is controlled by the Power Manager and Controller through the Current Controller. The Power Manager and Controller directs the Inverter to take power either from the battery, via the Bidirectional Converter, or the solar array, depending upon the time of day and the battery state of charge (SOC). Draw the following two functional block diagrams for this system: a. A diagram that illustrates the conversion of solar irradiation into energy stored in the battery. In that diagram, the input is the solar irradiance, r(t), and the output is the battery voltage, \(v_b(t)\). b. The main diagram, in which the input is the desired output voltage, \(v_r(t)\), and the output is the actual inverter output voltage, \(v_o(t)\). Both of these functional block diagrams should show their major components, including the PV array, MPPT controller, dc/dc converter, battery pack, dc/ac inverter, current controller, and the Power Manager and Controller. Show all signals, including intermediate voltages and currents, time of day, and the SOC of the battery.

Oil drilling rigs are used for drilling holes for identification of oil or natural gas sources and for extraction. An oil drilling system can be thought of as a drill inside a straw, which is placed inside a glass. The straw assembly represents the drill string, the drill surrounded by fluid, and the glass represents the volume, the annulus, around the drill string through which slurry and eventually oil will flow as the drilling progresses. Assume that we want to control the drill pressure output, \(P_d(t)\), with a reference voltage input, \(V_d(t)\). A control loop model (Zhao, 2007) consists of a drill-pressure controller, drill motor subsystem, pulley subsystem, and drill stick subsystem. The output signal of the latter, the drill pressure, \(P_d(t)\), is measured using a transducer, which transmits a negative feedback voltage signal, \(V_b(t)\), to the drill pressure controller. That signal is compared at the input of the controller to the reference voltage, \(V_r(t)\), Based on the error, \(e(t)=V_r(t)-V_b(t)\), the drill pressure controller sets the desired drill speed, \(\omega_d\), which is the input to the drill motor subsystem whose output is the actual drill speed, \(\omega_a\), which is the input to the pulley subsystem. The output of the pulley system drives the drill stick subsystem. The drill stick subsystem may be severely affected by environmental conditions, which may be represented as disturbances acting between the pulley and stick subsystems. Draw a functional block diagram for the above system, showing its major components as well as all signals.

Given the electric network shown in Figure P1.6. [Review] a. Write the differential equation for the network if v(t)=u(t), a unit step. b. Solve the differential equation for the current, i(t), if there is no initial energy in the network. c. Make a plot of your solution if R/L=1.

Repeat Problem 18 using the network shown in Figure P1.7. Assume \(R=1 \Omega, L=0.5 \mathrm{H}\), and 1/LC=16. [Review]

Solve the following differential equations using classical methods. Assume zero initial conditions. [Review] a. \(\frac{d x}{d t}+7 x=5 \cos 2 t\) b. \(\frac{d^2 x}{d t^2}+6 \frac{d x}{d t}+8 x=5 \sin 3 t\) c. \(\frac{d^2 x}{d t^2}+8 \frac{d x}{d t}+25 x=10 u(t)\)

Solve the following differential equations using classical methods and the given initial conditions: [Review] a. \(\begin{aligned} & \frac{d^2 x}{d t^2}+2 \frac{d x}{d t}+2 x=\sin 2 t \\ & x(0)=2 ; \frac{d x}{d t}(0)=-3 \end{aligned}\) b. \(\begin{aligned} & \frac{d^2 x}{d t^2}+2 \frac{d x}{d t}+x=5 e^{-2 t}+t \\ & x(0)=2 ; \frac{d x}{d t}(0)=1 \end{aligned}\) c. \(\frac{d^2 x}{d t^2}+4 x=t^2\) \(x(0)=1 ; \frac{d x}{d t}(0)=2\)

Control of HIV/AIDS. As of 2012, the number of people living worldwide with Human Immunodeficiency Virus/Acquired Immune Deficiency Syndrome (HIV/ AIDS) was estimated at 35 million, with 2.3 million new infections per year and 1.6 million deaths due to the disease (UNAIDS, 2013). Currently there is no known cure for the disease, and HIV cannot be completely eliminated in an infected individual. Drug combinations can be used to maintain the virus numbers at low levels, which helps prevent AIDS from developing. A common treatment for HIV is the administration of two types of drugs: reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). The amount in which each of these drugs is administered is varied according to the amount of HIV viruses in the body (Craig, 2004). Draw a block diagram of a feedback system designed to control the amount of HIV viruses in an infected person. The plant input variables are the amount of RTIs and PIs dispensed. Show blocks representing the controller, the system under control, and the transducers. Label the corresponding variables at the input and output of every block.

Hybrid vehicle. The use of hybrid cars is becoming increasingly popular. A hybrid electric vehicle (HEV) combines electric machine(s) with an internal combustion engine (ICE), making it possible (along with other fuel consumption–reducing measures, such as stopping the ICE at traffic lights) to use smaller and more efficient gasoline engines. Thus, the efficiency advantages of the electric drivetrain are obtained, while the energy needed to power the electric motor is stored in the onboard fuel tank and not in a large and heavy battery pack. There are various ways to arrange the flow of power in a hybrid car. In a serial HEV (Figure P1.8), the ICE is not connected to the drive shaft. It drives only the generator, which charges the batteries and/or supplies power to the electric motor(s) through an inverter or a converter. The HEVs sold today are primarily of the parallel or split-power variety. If the combustion engine can turn the drive wheels as well as the generator, then the vehicle is referred to as a parallel hybrid, because both an electric motor and the ICE can drive the vehicle. A parallel hybrid car (Figure P1.9) includes a relatively small battery pack (electrical storage) to put out extra power to the electric motor when fast acceleration is needed. See (Bosch, 5th ed., 2000), (Bosch, 7th ed., 2007), (Edelson, 2008), (Anderson, 2009) for more detailed information about HEV. As shown in Figure P1.10, split-power hybrid cars utilize a combination of series and parallel drives (Bosch, 5th ed., 2007). These cars use a planetary gear (3) as a split-power transmission to allow some of the ICE power to be applied mechanically to the drive. The other part is converted into electrical energy through the alternator (7) and the inverter (5) to feed the electric motor (downstream of the transmission) and/or to charge the high-voltage battery (6). Depending upon driving conditions, the ICE, the electric motor, or both propel the vehicle. Draw a functional block diagram for the cruise (speed) control system of: a. A serial hybrid vehicle, showing its major components, including the speed sensor, electronic control unit (ECU), inverter, electric motor, and vehicle dynamics; as well as all signals, including the desired vehicle speed, actual speed, control command (ECU output), controlled voltage (inverter output), the motive force (provided by the electric motor), and running resistive force; b. A parallel hybrid vehicle, showing its major components, which should include also a block that represents the accelerator, engine, and motor, as well as the signals (including accelerator displacement and combined engine/motor motive force); c. A split-power HEV, showing its major components and signals, including, in addition to those listed in Parts a and b, a block representing the planetary gear and its control, which, depending upon driving conditions, would allow the ICE, the electric motor, or both to propel the vehicle, that is, to provide the necessary total motive force.

Parabolic trough collector. A set of parabolic mirrors can be used to concentrate the sun’s rays to heat a fluid flowing in a pipe positioned at the mirrors’ focal points (Camacho, 2012). The heated fluid, such as oil, for example, is transported to a pressurized tank to be used to create steam to generate electricity or power an industrial process. Since the solar energy varies with time of day, time of year, cloudiness, humidity, etc., a control system has to be developed in order to maintain the fluid temperature constant. The temperature is mainly controlled by varying the amount of fluid flow through the pipes, but possibly also with a solar tracking mechanism that tilts the mirrors at appropriate angles. Assuming fixed mirror angles, draw the functional block diagram of a system to maintain the fluid temperature a constant. The desired and actual fluid temperature difference is fed to a controller followed by an amplifier and signal conditioning circuit that varies the speed of a fluid circulating pump. Label the blocks and links of your diagram, indicating all the inputs to the system, including external disturbances such as solar variations, cloudiness, humidity, etc.