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Apago PDF Enhancer This page intentionally left blank Apago PDF Enhancer E1IFC 10/27/ Page 1 Antenna Azimuth Position Control System. SOLUTION MANUAL Apago PDF Enhancer Solutions to Problems Student companion website Founded in , John Wiley & Sons, Inc. has been a. Control Systems Engineering, Sixth Edition by Norman S. Nise Advanced Electronic Communications Systems Wayne Tomasi Sixth Edition Kerry_Patterson,_Joseph_Grenny,_Ron_McMillan,_Al_(zlibraryexau2g3p_onion )lesforgesdessalles.info
Sustained yield harvesting principles ensure that the number of trees cut each year does not exceed the amount of new growth. Typically, the steady-state error decreases with an increase in gain and increases with a decrease in gain. Inherent in Determine a physical system and specifications from the requirements. Chapters2and3 cover modeling of open-loop systems, using frequency response techniques and state- space techniques, respectively. Again, the system position will change with a disturbance, such as an additional force, and the system will not detect or correct for the disturbance. Control is passed to the aerosurfaces as the orbiter descends into the atmosphere.
Robots designed by control system principles can compensate for human disabilities. Control systems are also useful in remote or dangerous locations. For example, a remote-controlled robot arm can be used to pick up material in a radioactive environment.
Control systems can also be used to provide convenience by changing the form of the input. For example, in a temperature control system, the input is a position on a thermostat. The output is heat. Thus, a convenient position input yields a desired thermal output. Another advantage of a control system is the ability to compensate for disturbances. Early elevators were controlled by hand ropes or an elevator operator. One of two modern Duo-liftelevatorsmakesitsway up the Grande Arche in Paris.
Two elevators are driven by one motor, with each car acting as a counterbalance to the other. Today, elevators are fully auto- matic, using control systems to regulate position and velocity.
The system must be able to yield the correct output even withadisturbance. Forexample,consideranantenna systemthat points inacommanded direction. Conse- quently, the system itself must measure the amount that the disturbance has repositioned the antenna and then return the antenna to the position commanded by the input.
Numerous biological control systems were built into the earliest inhabitants of our planet. Let us now look at a brief history of human-designed control systems. Awater clock invented by Ktesibios operated by having water trickle into a measuring container at a constant rate.
The level of water in the measuring container could be used to tell time. For water to trickle at a constant rate, the supply tank had to be kept at a constant level. Soon after Ktesibios, the idea of liquid-level control was applied to an oil lamp by Philon of Byzantium. The lamp consisted of two oil containers configured vertically.
The lower pan was open at the top and was the fuel supply for the flame. The closed upper bowl was the fuel reservoir for the pan below.
The containers were interconnected by two capillary tubes and another tube, called a vertical riser, which was inserted into the oil in the lower pan just below the surface. As the oil burned, the base of the vertical riser was exposed to air, which forced oil in the reservoir above to flow through the capillary tubes and into the pan.
The transfer of fuel from the upper reservoir to the pan stopped when the previous oil level in the pan was reestablished, thus blocking the air from entering the vertical riser. Hence, the system kept the liquid level in the lower container constant. The concept was further elaborated on by weighting the valve top. If the upward pressure from the boiler exceeded the weight, steam was released, and the pressure decreased.
Ifitdidnotexceedtheweight,thevalvedidnotopen,andthepressureinsidethe boiler increased. Thus, the weight on the valve top set the internal pressure of the boiler. Also in the seventeenth century, Cornelis Drebbel in Holland invented a purely mechanical temperature control system for hatching eggs.
The device used a vial of alcohol and mercury with a floater inserted in it. The floater was connected to a damper that controlled a flame. A portion of the vial was inserted into the incubator to sense the heat generated by the fire. As the heat increased, the alcohol and mercury expanded, raising the floater, closing the damper, and reducing the flame. Lower temperature caused the float to descend, opening the damper and increasing the flame.
Speed Control In , speed control was applied to a windmill by Edmund Lee. Increasing winds pitched the blades farther back, so that less area was available. As the wind 1 See Bennett and Mayr for definitive works on the history of control systems. William Cubitt improved on the idea in by dividing the windmill sail into movable louvers.
Also in the eighteenth century, James Watt invented the flyball speed governor to control the speed of steam engines. In this device, two spinning flyballs rise as rotational speed increases.
A steam valve connected to the flyball mechanism closes with the ascending flyballs and opens with the descending flyballs, thus regulating the speed. Stability, Stabilization, and Steering Control systems theory as we know it today began to crystallize in the latter half of the nineteenth century. In , James Clerk Maxwell published the stability criterion for a third-order system based on the coefficients of the differential equation.
In , Edward John Routh, using a suggestion from William Kingdon Clifford that was ignored earlier by Maxwell, was able to extend the stability criterion to fifth-order systems. This paper contains what is now known as the Routh-Hurwitz criterion for stability, which we will study in Chapter 6. A student of P. Chebyshev at the University of St. Petersburg in Russia, Lyapunov extended the work of Routh to nonlinear systems in his doctoral thesis, entitled The General Problem of Stability of Motion.
During the second half of the s, the development of control systems focused on the steering and stabilizing of ships. Other efforts were made to stabilize platforms for guns as well as to stabilize entire ships, using pendulums to sense the motion.
Twentieth-Century Developments It was not until the early s that automatic steering of ships was achieved. In , the Sperry Gyroscope Company installed an automatic steering system that used the elements of compensation and adaptive control to improve performance.
However, much of the general theory used today to improve the performance of automatic control systems is attributed to Nicholas Minorsky, a Russian born in It was his theoretical development applied to the automatic steering of ships that led to what we call today proportional-plus-integral-plus-derivative PID , or three-mode, con- trollers, which we will study in Chapters 9 and In the late s and early s, H.
Bode and H. Nyquist at Bell Telephone Laboratories developed the analysis of feedback amplifiers. These contributions evolved into sinusoidal frequency analysis and design techniques currently used for feedback control system, and are presented in Chapters 10 and In , Walter R.
Evans, working in the aircraft industry, developed a graphical technique to plot the roots of a characteristic equation of a feedback system whose parameters changed over a particular range of values.
This technique, now known as the root locus, takes its place with the work of Bode and Nyquist in forming the foundation of linear control systems analysis and design theory. We will study root locus in Chapters 8, 9, and Contemporary Applications Today, control systems find widespread application in the guidance, navigation, and control of missiles and spacecraft, as well as planes and ships at sea.
For example, 1. The rudder commands, in turn, result in a rudder angle that steers the ship. We find control systems throughout the process control industry, regulating liquid levels in tanks, chemical concentrations in vats, as well as the thickness of fabricated material. For example, consider a thickness control system for a steel plate finishing mill. Steel enters the finishing mill and passes through rollers. In the finishing mill, X-rays measure the actual thickness and compare it to the desired thickness.
Any difference is adjusted by a screw-down position control that changes the roll gap at the rollers through which the steel passes. This change in roll gap regulates the thickness. Modern developments have seen widespread use of the digital computer as part of control systems. For example, computers in control systems are for industrial robots, spacecraft, and the process control industry.
It is hard to visualize a modern control system that does not use a digital computer. The space shuttle contains numerous control systems operated by an onboard computer on a time-shared basis. In space, the flight control system gimbals rotates the orbital maneuvering system OMS engines into a position that provides thrust in the commanded direction to steer the spacecraft. For example, the elevons require a control system to ensure that their position is indeed that which was commanded, since disturbances such as wind could rotate the elevons away from the commanded position.
Similarly, in space, the gimbaling of the orbital maneu- vering engines requires a similar control system to ensure that the rotating engine can accomplish its function with speed and accuracy. Control systems are also used to control and stabilize the vehicle during its descent from orbit. Numerous small jets that compose the reaction control system RCS are used initially in the exoatmo- sphere, where the aerosurfaces are ineffective. Control is passed to the aerosurfaces as the orbiter descends into the atmosphere.
Inside the shuttle, numerous control systems are required for power and life support. For example, the orbiter has three fuel-cell power plants that convert hydrogen and oxygen reactants into electricity and water for use by the crew.
The fuel cells involve the use of control systems to regulate temperature and pressure. The reactant tanks are kept at constant pressure as the quantity of reactant diminishes. Sensors in the tanks send signals to the control systems to turn heaters on or off to keep the tank pressure constant Rockwell Interna- tional, Control systems are not limited to science and industry.
For example, a home heating system is a simple control system consisting of a thermostat containing a bimetallic material that expands or contracts with changing temperature. This expansion or contraction moves a vial of mercury that acts as a switch, turning the heater on or off. The amount of expansion or contraction required to move the mercury switch is determined by the temperature setting.
For example, in an optical disk recording system microscopic pits representing the information are burned into the disc by a laser during the recording process. During playback, a reflected laser beam focused on the pits changes intensity Figure 1.
The light intensity changes are converted to an electrical signal and processed as sound or picture. A control system keeps the laser beam positioned on the pits, which are cut as concentric circles. There are countless other examples of control systems, from the everyday to the extraordinary. As you begin your study of control systems engineering, you will become more aware of the wide variety of applications. We can consider these configurations to be the internal architecture of the total system shown in Figure 1.
It starts with a subsystem called an input transducer, which converts the form of the input to that used by the controller. The controller drives a process or a plant. The input is sometimes called the reference, while the output can be called the controlled variable.
Other signals, such as disturbances, are shown added to the controller and process outputs via summing junctions, which yield the algebraic sum of their input signals using associated signs.
For example, the plant can be a furnace or air conditioning system, where the output variable is temperature. The controller in a heating system consists of fuel valves and the electrical system that operates the valves. For example, if the controller is an electronic amplifier and Disturbance 1 is noise, then any additive amplifier noise at the first summing junction will also drive the process, corrupting the output with the effect of the noise.
The system cannot correct for these disturbances, either. Open-loop systems, then, do not correct for disturbances and are simply commanded by the input.
For example, toasters are open-loop systems, as anyone with burnt toast can attest. The controlled variable output of a toaster is the color of the toast. The device is designed with the assumption that the toast will be darker the longer it is subjected to heat. The toaster does not measure the color of the toast; it does not correct for the fact that the toast is rye, white, or sourdough, nor does it correct for the fact that toast comes in different thicknesses.
Other examples of open-loop systems are mechanical systems consisting of a mass, spring, and damper with a constant force positioning the mass. The greater the force, the greater the displacement. Again, the system position will change with a disturbance, such as an additional force, and the system will not detect or correct for the disturbance. If the professor adds a fourth chapter—a disturbance—you are an open-loop system if you do not detect the disturbance and add study time to that previously calculated.
The result of this oversight would be a lower grade than you expected. Closed-Loop Feedback Control Systems The disadvantages of open-loop systems, namely sensitivity to disturbances and inability to correct for these disturbances, may be overcome in closed-loop systems.
The generic architecture of a closed-loop system is shown in Figure 1. The input transducer converts the form of the input to the form used by the controller. An output transducer, or sensor, measures the output response and converts it into the form used by the controller.
For example, if the controller uses electrical signals to operate the valves of a temperature control system, the input position and the output temperature are converted to electrical signals. The input position can be converted to a voltage by a potentiometer, a variable resistor, and the output temperature can be converted to a voltage by a thermistor, a device whose electrical resistance changes with temperature.
The first summing junction algebraically adds the signal from the input to the signal from the output, which arrives via the feedback path, the return path from the output to the summing junction. In Figure 1. The result is generally called the actuating signal. Under this condition, the actuating signal is called the error. The closed-loop system compensates for disturbances by measuring the output response, feeding that measurement back through a feedback path, and comparing that response to the input at the summing junction.
If there is any difference between the two responses, the system drives the plant, via the actuating signal, to make a correction. Closed-loop systems, then, have the obvious advantage of greater accuracy than open-loop systems. They are less sensitive to noise, disturbances, and changes in the environment.
Transient response and steady-state error can be controlled more conveniently and with greater flexibility in closed-loop systems, often by a simple adjustment of gain amplification in the loop and sometimes by redesigning the controller. We refer to the redesign as compensating the system and to the resulting hardware as a compensator.
On the other hand, closed-loop systems are more complex and expensive than open-loop systems. A standard, open-loop toaster serves as an example: It is simple and inexpensive.
A closed-loop toaster oven is more complex and more expensive since it has to measure both color through light reflectivity and humidity inside the toaster oven. Thus, the control systems engineer must consider the trade-off between the simplicity and low cost of an open-loop system and the accuracy and higher cost of a closed-loop system.
In summary, systems that perform the previously described measurement and correction are called closed-loop, or feedback control, systems. Systems that do not have this property of measurement and correction are called open-loop systems.
Computer-Controlled Systems In many modern systems, the controller or compensator is a digital computer. The advantage of using a computer is that many loops can be controlled or compensated by the same computer through time sharing. Furthermore, any adjustments of the 1. The computer can also perform supervi- sory functions, such as scheduling many required applications. For example, the space shuttle main engine SSME controller, which contains two digital computers, alone controls numerous engine functions.
It monitors engine sensors that provide pressures, temperatures, flow rates, turbopump speed, valve positions, and engine servo valve actuator positions.
The controller further provides closed-loop control of thrust and propellant mixture ratio, sensor excitation, valve actuators, spark igniters, as well as other functions Rockwell International, We now expand upon the topic of performance and place it in perspective as we define our analysis and design objectives.
For example, we evaluate its transient response and steady-state error to determine if they meet the desired specifications. A control system is dynamic: It responds to an input by undergoing a transient response before reaching a steady-state response that generally resembles the input.
We havealreadyidentified these two responses and cited a position control system an elevator as an example. In this section, we discuss three major objectives of systems analysis and design: We also address some other design concerns, such as cost and the sensitivity of system performance to changes in parameters.
Transient Response Transient response is important. In the case of an elevator, a slow transient response makes passengers impatient, whereas an excessively rapid response makes them uncomfortable. If the elevator oscillates about the arrival floor for more than a second, a disconcerting feeling can result. Transient response is also important for structural reasons: Too fast a transient response could cause perma- nent physical damage.
In this book, we establish quantitative definitions for transient response. We then analyze the system for its existing transient response. Finally, we adjust parameters or design components to yield a desired transient response—our first analysis and design objective.
As we have seen, this response resembles the input and is usually what remains after the transients have decayed to zero.
For example, this response may be an elevator stopped near the fourth floor or the head of a disk drive finally stopped at the correct track. We are concerned about the accuracy of the steady-state response. An antenna tracking a satellite must keep the satellite well within its beamwidth in order not to lose track. Stability Discussion of transient response and steady-state error is moot if the system does not have stability.
In order to explain stability, we start from the fact that the total response of a system is the sum of the natural response and the forced response. When you studied linear differential equations, you probably referred to these responses as the homogeneousandtheparticularsolutions,respectively.
Naturalresponsedescribesthe way the system dissipates or acquires energy. The form or nature of this response is dependent only on the system, not the input. On the other hand, the form or nature of the forced response is dependent on the input.
In some systems, however, the natural response grows without bound rather than diminish to zero or oscillate. Eventually, the natural response is so much greater than the forced response that the system is no longer controlled. This condition, called instability, could lead to self-destruction of the physical device if limit stops are not part of the design.
For example, the elevator would crash through the floor or exit through the ceiling; an aircraft would go into an uncontrollable roll; or an antenna commanded to point to a target would rotate, line up with the target, but then begin to oscillate about the target with growing oscillations and increasing velocity until the motor or amplifiers reached their output limits or until the antenna was damaged structurally.
A time plot of an unstable system would show a transient response that grows without bound and without any evidence of a steady-state response. Control systems must be designed to be stable. That is, their natural response must decay to zero as time approaches infinity, or oscillate. In many systems the transient response you see on a time response plot can be directly related to the natural response.
Thus, if the natural response decays to zero as time approaches infinity, the transient response will also die out, leaving only the forced response.
If the system is stable, the proper transient response and steady-state error character- istics can be designed. Stability is our third analysis and design objective.
If you look at Figure 1. The transient response is the sum of the natural and forced responses, while the natural response is large. If we plotted the natural response by itself, we would get a curve that is different from the transient portion of Figure 1. The steady-state response of Figure 1.
Thus, the transient and steady-state responses are what you actually see on the plot; the natural and forced responses are the underlying mathematical components of those responses.
However, other important considerations must be taken into account. For example, factors affecting hardware selection, such as motor sizing to fulfill power requirements and choice of sensors for accuracy, must be considered early in the design. Finances are another consideration. Control system designers cannot create designs without considering their economic impact. Such considerations as budget allocations and competitive pricing must guide the engineer.
For example, if your product is one of a kind, you may be able to create a design that uses more expensive components without appreciably increasing total cost. However, if your design will be used for many copies, slight increases in cost per copy can translate into many more dollars for your company to propose during contract bidding and to outlay before sales. Another consideration is robust design. System parameters considered con- stant during the design for transient response, steady-state errors, and stability change over time when the actual system is built.
Thus, the performance of the system also changes over time and will not be consistent with your design. Un- fortunately, the relationship between parameter changes and their effect on per- formance is not linear. Thus, the engineer wants to create a robust design so that the system will not be sensitive to parameter changes. We discuss the concept of system sensitivity to parameter changes in Chapters 7 and 8.
This concept, then, can be used to test a design for robustness. In this section we will look at an example of a feedback control system.
The system introduced here will be used in subsequent chapters as a running case study to demonstrate the objectives of those chapters. A colored background like this will identify the case study section at the end of each chapter. Section 1. Antenna Azimuth: An Introduction to Position Control Systems A position control system converts a position input command to a position output response. Position control systems find widespread applications in antennas, robot arms, and computer disk drives.
The radio telescope antenna in Figure 1. In this section, we will look in detail at an antenna azimuth position control system that could be used to position a radio telescope antenna. We will see how the system works and how we can effect changes in its performance.
The discussion here will be on a qualitativelevel, with the objective of getting an intuitive feeling for the systems with which we will be dealing.
An antenna azimuth position control system is shown in Figure 1. The functions are shown above the blocks, and the required hardware is indicated inside the blocks.
Parts of Figure 1. A radio antenna is an example of a system with position controls. Let us look at Figure 1. The input command is an angular displace- ment. The potentiometer converts the angular displacement into a voltage. The signal and power amplifiers boost the difference between the input and output voltages. This amplified actuating signal drives the plant. The system normally operates to drive the error to zero.
When the input and output match, the error will be zero, and the motor will not turn.
Thus, the motor is driven only whentheoutputandtheinputdonotmatch. Thegreaterthedifferencebetweentheinput and the output, the larger the motor input voltage, and the faster the motor will turn. If we increase the gain of the signal amplifier, will there be an increase in the steady-state value of the output?
If the gain is increased, then for a given actuating signal, the motor will be driven harder. However, the motor will still stop when the actuating signal reaches zero, that is, when the output matches the input. The difference in the response, however, will be in the transients. Since the motor is driven harder, it turns faster toward its final position. Also, because of the increased speed, increased momentum could cause the motor to overshoot the final value and be forced by the system to return to the commanded position.
Thus, the possibility exists for a transient response that consists of damped oscillations that is, a sinusoidal response whose amplitude diminishes with time about the steady-state value if the gain is high.
The responses for low gain and high gain are shown in Figure 1. Let us now direct our attention to the steady-state position to see how closely the output matches the input after the transients disappear. We define steady-state error as the difference between the input and the output after the transients have effectively disappeared. The definition holds equally well for step, ramp, and other types of inputs.
Typically, the steady-state error decreases with an increase in gain and increases with a decrease in gain. In some systems, the steady-state error will not be zero; for these systems, a simple gain adjustment to regulate the transient response is either not effective or leads to a trade-off between the desired transient response and the desired steady-state accuracy. To solve this problem, a controller with a dynamic response, such as an electrical filter, is used along with an amplifier. With this type of controller, it is possible to design both the required transient response and the required steady-state accuracy without the trade-off required by a simple setting of gain.
However, the controller is now more complex. The filter in this case is called a compensator. Many systems also use dynamic elements in the feedback path along with the output transducer to improve system performance. Gain adjust- ments can affect performance and sometimes lead to trade-offs between the performance criteria. Compensators can often be designed to achieve performance specifications without the need for trade-offs.
Now that we have stated our objectives and some of the methods available to meet those objectives, we describe the orderly progression that leads us to the final system design. The antenna azimuth position control system discussed in the last section is representative of control systems that must be analyzed and designed.
Inherent in Determine a physical system and specifications from the requirements. Transform the physical system into a schematic.
Use the schematic to obtain a block diagram, signal-flow diagram, or state-space representation. If multiple blocks, reduce the block diagram to a single block or closed-loop system.
Analyze, design, and test to see that requirements and specifications are met. Draw a functional block diagram. For example, if testing Step 6 shows that requirements have not been met, the system must be redesigned and retested.
Sometimes requirements are conflicting and the design cannot be attained. In these cases, the requirements have to be respecified and the design process repeated. Let us now elaborate on each block of Figure 1. Step 1: Transform Requirements Into a Physical System We begin by transforming the requirements into a physical system. For example, in the antenna azimuth position control system, the requirements would state the desire to position the antenna from a remote location and describe such features as weight and physical dimensions.
Using the requirements, design specifications, such as desired transient response and steady-state accuracy, are determined. Perhaps an overall concept, such as Figure 1. Step 2: It indicates functions such as input transducer and controller, as well as possible hardware descriptions such as amplifiers and motors.
At this point the designer may produce a detailed layout of the system, such as that shown in Figure 1. Step 3: Create a Schematic As we have seen, position control systems consist of electrical, mechanical, and electromechanical components.
After producing the description of a physical system, the control systems engineer transforms the physical system into a schematic diagram. The control system designer can begin with the physical description, as contained in Figure 1.
The engineer must make approxi- mations about the system and neglect certain phenomena, or else the schematic will be unwieldy, making it difficult to extract a useful mathematical model during the next phase of the analysis and design sequence. The designer starts with a simple schematic representation and, at subsequent phases of the analysis and design sequence, checks the assumptions made about the physical system through analysis and computer simulation.
If the schematic is too simple and does not adequately account for observed behavior, the control systems engineer adds phenomena to the schematic that were previously assumed negligible. A schematic diagram for the antenna azimuth position control system is shown in Figure 1. When we draw the potentiometers, we make our first simplifying assumption by neglecting their friction or inertia. These mechanical characteristics yield a dynamic, rather than an instantaneous, response in the output voltage.
We assume that these mechanical effects are negligible and that the voltage across a potenti- ometer changes instantaneously as the potentiometer shaft turns.
A differential amplifier and a power amplifier are used as the controller to yield gain and power amplification, respectively, to drive the motor. Again, we assume that the dynamics of the amplifiers are rapid compared to the response time of the motor; thus, we model them as a pure gain, K.
A dc motor and equivalent load produce the output angular displacement. Both inductance and resistance are part of the armature circuit. In showing 16 Chapter 1 Introduction The designer makes further assumptions about the load. The load consists of a rotating mass and bearing friction. These decisions are not easy; however, as you acquire more design experience, you will gain the insight required for this difficult task. Step 4: One such model is the linear, time-invariant differential equation, Eq.
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