Chapter 1 Introduction to Control System

Chapter 1Introduction to Feedback System and Control problem

Introduction to the control Problem Control Systems are playing an important role in development of modern civilization and technology. Every system we come across today, has some type of control engineering involved in it. For example , a home heating system; a refrigerator, an air conditioner, an automobile; all these are examples of control systems. Take any sector of industry, you will find control systems everywhere. Say the space technology and weapon systems, the robotics, the power plants all are the industrial sectors where you find control systems.Basic Structure and Terminology The system to be controlled is given in various names : The most common being a process or a plant or the controlled system itself. In process industries, particularly chemicals, petroleum, steam power etc there are applications where we require the control of temperature, pressure, liquid level in vessels, humidity, composition and the like. All these applications generally considered as process control applications.During the Second World War the need of automatic airplane pilots, the gun positioning systems, radar antenna control systems ,and the like, led to more scientific approaches in the control engineering field. To solve these problems theory of servomechanisms was developed .Basic Structure and Terminology( cont..)Note that the word servomechanism originated from the words servo means the slave or the servant and the mechanism. So it means a servo system is a system which is slave to the command and this was the requirement of the weaponry system during the World War II. Fig. 1.1 in the lecture note shows that the I/O configuration of a process ( or plant). The output of this particular process are the response variable. the variable or the attribute of the process you want to control and this controlled will be exercised by, what we refer to as, the manipulated variable. The manipulated variable is subject to control by the controller we are going to design. And the requirement of the control is that there is unavoidable and undesirable disturbance acting on the process. This point maybe very carefully noted. The disturbance is uncalled for and it is beyond control . As shown in fig 1.2 .

Basic Structure and Terminology( cont..)The disturbance may be originated outside the process environment or the disturbance may also be originated from within the process. For example, the parameters of the process are subject to change with time .The manipulated variable is the variable to which the process is going to react and the reaction of this process to the manipulated variable is going to make the response variable follow your commands. So it means, naturally speaking, the manipulated variable must be under the control of the controller. The controller is going to control the manipulated variable and the function of this controller is to make the response variable follow the system commands. Basic Structure and Terminology( cont..)Now in particular case, the requirement of this particular system or the requirement of this controller is to force the controlled variable follow the command in spite of random unknown disturbances acting on the system. In Fig. 1.3. in open-loop control system, the controller directly gets the information about the command signal from the user. This information which the controller gets from the user as a command signal is effectively utilized by this controller. The controller acts upon the signal to generate a manipulated variable whose function is to force the controlled variable follow the command. So, in this particular case the only information available to the controller is the command signal. Basic Structure and Terminology( cont..)The plant information this controller was intelligent about the plant information at certain point of time and a controller was designed to realize the objective of the controlled variable following the command signal. But in due course if the disturbance on to the plant changes because of the reaction of the environment or because of the changes within the plant what will happen, your controller is ignorant about those changes. It means the controller which was designed earlier for a specific information about the disturbance is no more effective in making the controlled variable follow the command. So it means there will an error between the controlled variable and the command signal because of the changes in the disturbance variable and hence this type of controlled structure which in the literature is referred to as open-loop control, the open-loop meaning that the loop has not been closed to give you the information about the controlled variable. Basic Structure and Terminology( cont..)In Fig 1.4 shows the closed-loop control wherein the controller is intelligent and it gets information about the current status of the controlled variable as well. So in this particular scheme if this random effect comes on the process in that particular case the controller becomes ineffective to track these particular disturbances or to take action so as to nullify the effect of this disturbance on the process. So the controller should be more intelligent. What is the intelligence required? The additional information the controller requires is the disturbance. It should get the information about the disturbance also so that it knows that in spite of that disturbance the manipulated variable is to be manipulated, is to be controlled so that this variable follows the command. Basic Structure and Terminology( cont..)There is sensor here, this particular sensor gets the information about the controlled variable and passes on this information to the controller. note that controller is not intelligent, it gets the information about the disturbances indirectly through the controlled variable, it does not get the information directly in this particular feedback structure. Now the function of the controller is the following: It compares the actual controlled variable, the information available from the sensor from the command signal and generates an error signal and that error signal it utilizes to generate a suitable control signal and that control signal manipulates the signal to the plant so as to reduce the error to zero. Basic Structure and Terminology( cont..)So this particular system is an error self-nulling process wherein we will find that the error between the command and the actual variable is reduced to zero because of the controller action. This particular system in the literature is referred to as the closed-loop system, the name is evident from the structure of the system or it is also called a feedback control system. the major source of the problem in this particular structure is the sensor itself. Recall that in the earlier structure on open-loop control the sensor was not there. This is the additional hardware introduced in this particular structure because of the requirement of making the controller more intelligent. Now this sensor when this piece of hardware is introduced in this particular loop it is going to introduce problems of noise. Basic Structure and Terminology( cont..)in the process of measurement it generates high frequency noise and this high frequency noise also gets injected into the loop. So the plant and the controller in addition to getting the useful signal about the command signal and the controlled variable gets this noise signal which is a high frequency signal and the plant will react to that signal as well and control may not be effective. This particular problem was not there in the open-loop structure. However, suitable noise filter can be installed in the loop so that this particular problem of noise can be taken care of. Now what is the requirement on the controller. The requirement on the controller to make this particular system robust. The word robust is used in the control literature to emphasize the need of control so as to make it insensitive to disturbances and parameter variations. one of the primary requirement of the feedback system is to make the controlled system a robust one. Example 1.1 Automobile Driving systemIn fig 1.5 shows the automobile driving system There are two output variables. In the previous example, recall, there was only one output variable one controlled variable. The two output variables in this system are (the heading and the speed). Now the plant in this particular system is the vehicle. What is the disturbance acting on the system? It is the wind force plus road traffic conditions . The two input command signals is the direction of the highway and the speed limits as given by traffic signals is the another command variables. Now in the case manipulated variables For this heading the manipulated variable should be steering position, this steering control and the manipulated variable here is the accelerator or the brake position to control the speed. Example 1.1 Automobile Driving system (cont)A system with multiple input and multiple output variables in the literature is referred to as an MIMO very standard name we will be using Multi-Input Multi-Output system or a multivariable system. In contrast to this system, the earlier example was an example of an SISO Single Input Single Output system. These terms are quite frequently used in the literature called a scalar system . in this particular case one manipulated variable here may affect both the controlled variables as far as this plant is concerned. This is called the coupling between the input variables and the output variables or the interaction between the input and the output variables. This interaction makes the control decide the task a very difficult task That is why the design of multivariable systems is quite difficult in nature because of the interaction .Example 1.1 Automobile Driving system (cont)We can decouple this system into two SISO systems for the purpose of design because the interaction is negligible. The actuator here is ( foot or hands). In this case error detector has to be identified there is no hardware error detector . The eyes of the driver acts as an error detectorThe control logic is in the brain of the driver and a suitable action is taken by the driver depending upon the control logic he has set in his brain so that the manipulated variable through the actuators are suitably controlled so as to make the controlled variables follow the command. When the command signals are changing with time and the purpose of the controller is to make the controlled variable follow the time varying commands then this control system is referred to as a tracking system or regulator system / command following system.Example 1.1 Automobile Driving system (cont)The driver is in the feedback loop, it is not complete hardware system. But the case of a bathroom toilet tank, there is no human operator in the loop. So, in these systems wherein there is no human operator in the loop it is called automatic control systems . the systems the feedback systems wherein the human operator in the loop, well, may be referred to as man-machine control systems. Our emphasis throughout the course is going to be on automatic control systems. The complete hardware will be described which is to be used inside the loop to realize the objective of control.

Example 1.3 Servomechanism for steering of antenna In fig 1.11 shows the servomechanism for steering of antenna . It is a typical radar application where the requirement is this; that the antenna is aligned to the target the target plane and it is required to follow the target. So it means, in this particular case the command which is given by the radar sensor the radar sensor detects the deviation between the antenna axis and the targets position and the error between the antenna axis and the targets position is the command signal for the servo system. Now this command signal is given to the servo mechanism which is required to steer the antenna and the problem is that the antenna should monitor or should follow the command so as to reduce the error to zero. Example 1.3 Servomechanism for steering of antenna (cont..) There are two degrees of freedom: The elevation angle around the horizontal axis, the azimuth angle beta which is around the vertical axis. So, in this particular case naturally this becomes a multivariable system as per the terminology given because it is a two input case. However, as was the case in the earlier example of automobile driving system fortunately in this particular case the interaction is small and we can neglect the interaction. if the coupling or the interaction can be neglected in that particular case from the point of view of design a multivariable system can be treated as a subset of single input single output systems .

Example 1.3 Servomechanism for steering of antenna (cont..)We can design a servo system for driving the antenna as far as the elevation angle is concerned and other servo system for driving the azimuth angle or for controlling the azimuth angle of the antenna. So now let us take a typical system which will be required for the azimuth control of the antenna assuming that interaction between the two subsystems is negligibly small and we can treat the system as a single input single output system. Here is a simple block diagram description for driving the azimuth angle of the antenna this is beta the controlled variable the azimuth angle. Beta r will become the command signal as given by the radar sensor. So it means this particular variable has to reach this block which is the error detector and hence a suitable sensor is required here for sensing the azimuth position of the antenna. Example 1.3 Servomechanism for steering of antenna (cont..)Here shaft angle encoder is used . The output of this device is a digital input which is in proportion to the analog signal given. So it means here it is a digital signal, the feedback signal which is proportional to the azimuth angle beta. so there is computer which accepts two signals: one the command signal; the command signal could also be or will also be in this particular case a digital command proportional to the required beta r and feedback signal is a digital signal proportional to the actual beta the actual angle. So this computer which is a controller in this particular case generates an error between the two beta r minus beta and this particular error is used to generate a suitable manipulated variable a suitable manipulated signal which drives the motor so as to force this beta to follow beta r. Example 1.3 Servomechanism for steering of antenna (cont..)Here D/A converter is used . Here as u it is a controlled signal this signal is converted into an analog signal because my actuating device is an analog device. here a power amplifier is used . So this becomes the signal power given to the power amplifier, this particular signal controls the power supply to the motor so that this particular motor which generates the torque is able to drive the angle beta. Note that between the motor shaft and this beta is the antenna used a gear train. Example 1.3 Servomechanism for steering of antenna (cont..)the antenna the torque requirement for the antenna is much larger than the torque produced by a typical motor. In this particular case a DC motor has been used, a DC armature controlled motor has been used. The torque produced by the motor is lesser than the requirement of the antenna so for the torque magnification a suitable gear system is designed so that this particular torque the motor torque is able to drive the antenna axis. So in this case where is the feedback loop. You will find that the feedback loop in this particular case is through this particular channel that is the output is fed back and it is compared with the reference signal. Example 1.3 Servomechanism for steering of antenna (cont..)In comparison to the earlier block diagrams you will find one change here and that change is the following that in addition to this feedback signal one additional feedback signal is being given and that signal is through the tachogenerator. This is the tachogenerator which is coupled to the motor shaft. So it means the output here is going to be a voltage which is proportional to the shaft velocity. So, in addition to the position you are feeding back the velocity also. This is your command signal beta r the command for the azimuth angle given by the radar system. This is your feedback signal beta coming from the shaft angle encoder in suitable physical form may be a digital number voltage or any other, depends upon the hardware we are going to use. These two are compared and therefore this becomes the error detector block .Example 1.3 Servomechanism for steering of antenna (cont..)After comparison I have a signal which is e cap the actuating error signal. This particular error signal compares is the actuating signal for this particular system. Now this signal e cap. now we have another signal coming from this and here is your tacho; a tachogenerator over here which gives you a signal which is proportional to the derivative of the controlled variable. Now this signal is the signal given to the power amplifier. The function of the power amplifier is to change the power level so as to meet the requirements of the motor.

Example 1.3 Servomechanism for steering of antenna (cont..)here motor plus load as the plant. So in this particular case what is load; load is the antenna, the antenna is being driven through a gear train and the gear is coupled to the motor shaft .for model based design the antenna, the gears and the motor shaft all will be taken up as the plant of the system and the typical parameters of the plant will become moment of inertia J and the viscous friction B. the earlier systems the mass, the damper and the spring were taken as the elements of physical model of a mechanical system. Similarly, for rotational systems the typical parameters of physical model are moment of inertia J and the friction B. What is a Control Systems ? (cont..)System An interconnection of elements and devices for a desired purpose.A Control System consists of subsystems and processes (or plants) assembled to control the outputs of a process.

What is a Control Systems ? (cont..)The Control system is that means by which any quantity of interest in a machine , mechanism or other Equipment is maintained or altered in accordance with a desired manner. Consider, for example, the deriving system of an Automobile .Speed of the Automobile is a function of the position of its accelerator. The desired speed can be maintained ( or a desired change in speed can be achieved) by controlling pressure on the accelerator Pedal.This Automobile deriving system ( accelerator, carburetor and Engine- Vehicle ) constitutes a control system. The Basic Control System

The Basic Control System (cont..)For the Automobile deriving system the input ( Command ) signal is the force on the accelerator Pedal which through linkages causes the carburetor valve to open( close) so as to increase or decrease fuel ( Liquid Form) flow to the engine bringing the engine-vehicle speed ( controlled Variable ) to the desired value.The diagrammatic representation of fig 1.1 is known as Block diagram representation where each block represents an Element, a plant, mechanism , Device etc. Each Block has an input and Output signal which are linked by a relationship characterizing the block. It may be noted that the signal flow through the block is Unidirectional.

Control System another viewA Control System is an arrangement of physical components connected/related in such a manner as to command, direct or regulate itself or another system.Typical Examples of Control Systems: Central Temperature ControlFluid Level maintenance systemsBattery Voltage ControlHuman has numerous control systems built in it.

Human like Control

How.. Control Systems in Robotics Perspective

Industry

.Everywhere

Advantages of a Control SystemPower amplification: Radar antenna positioned by the low-power rotation of a knob at the input, requires a large amount of power for its output rotation. Control system will produce the needed power amplification/power gain.Remote control :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.

Advantages of a Control System (cont..)Convenience of input form :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.Compensation for disturbances :Another advantage of a control system is the ability to compensate for disturbances. Typically, we control such variables as temperature in thermal systems, position and velocity in mechanical systems, and voltage, current, or frequency in electrical systems. The system must be able to yield the correct output even with a disturbance. For example, consider an antenna system that points in a commanded direction. If wind forces the antenna from its commanded position, or if noise enters internally, the system must be able to detect the disturbance and correct the antenna's position.

Response CharacteristicsConsider a control system for an elevator. When the fourth-floor button is pressed on the ground floor, the elevator rises to the fourth floor with a speed and floor-leveling accuracy designed for passenger comfort. The push of the fourth-floor button is an input that represents our desired output, shown as a step function in Figure 1.2. The performance of the elevator can be seen from the elevator response curve in the figure.

Response Characteristics (cont..)A control system provides an output or response for a given input or stimulus. For example, when the Fourth floor of an elevator is pushed on the ground floor, the elevator rises to the fourth floor with a speed and floor- leveling accuracy designed for passenger comfort . The push of the fourth floor button is the input and is represented by a step command. The input represents what we would like the output to be after the elevator has stopped; the elevator itself follows the displacement described by the curve marked elevator Response. Two factors make the output different from the input. First, compare the instantaneous change of the input against the gradual change of the output .Response Characteristics (cont..)The state changes through a path that is related to the physical device and the way it acquires or dissipates energy. Thus, the elevator undergoes a gradual change as it rises from the first floor to the fourth floor. We call this party of the response the transient response.After the transient response, a physical system approaches its steady- state response, which is its approximation to the commanded or desired response. For the elevator example, this response Occurs when the elevator reaches the fourth floor. The accuracy of the elevators leveling with the floor is a second factor that could make the output different from the input . We call this different steady-state errorClosed- Loop ControlLet us reconsider the automobile deriving system. The route, speed and acceleration of the automobile are determined and controlled by the driver by observing traffic and road conditions and by properly manipulating the accelerator, gear-lever, brakes and steering wheel, etc. Suppose the deriver wants to maintain a speed of 50KM per hour ( Desired output). He accelerates the automobile to this speed with the help of the accelerator and then maintains it by holding the accelerator steady. No error in the speed of the automobile occurs so long as there are no gradients or other disturbances along the road. Closed- Loop Control ( cont)The actual speed of the automobile is measured by the speedometer and indicated on its dial.The deriver reads the speed dial visually and compares the actual speed with the desired one mentally. If there is a deviation of speed from the desired speed, accordingly he takes the decision to increase or decrease the speed.The decision is executed by change in pressure of his foot ( through muscular power ) on the accelerator pedal.The fig. 1.2 follow a closed loop, i.e, information about the instantaneous state of the output is feedback to the input and is used to modify it in a such manner as to achieve the desired output. It is on account of this basic difference that the system of Fig 1.1 is called an Open-loop system, while fig 1.2 is called Closed- loop system.

Closed- Loop Control ( cont)

Manually Controlled systems Vs Automatic Control systemsA simple block diagram of an automobile steering mechanism is shown in fig. 1.3 (a). The deriver senses visually and by tactile means( body movement ) the error between the actual and desired directions of the automobile as in Fig. 1.3 (b). Additional information is available to the deriver from the feel (sensing) of the steering wheel through his hands, these information's constitute the feedback signals which are interpreted by the drivers brain, who then signals his hand to adjust the steering wheel accordingly. In fact unless human being(s) are left out in a control system study practically all control systems are a sort of closed-loop system .Systems of the type represented in Figs. 1.2 and 1.3 involve continuous manual control by a human operator .these are classified as Manually Controlled Systems .

Manually Controlled systems Vs Automatic Control systems(cont..)In many complex and fast-acting systems, the presence of human element in the control loop is undesirable because the system response may be too rapid for an operator to follow or the demand on operators skill may be unreasonably high. Furthermore, some of the systems. e.g ., missiles, are self-destructive and in such systems human element must be excluded. Even in the situations where manual control could be possible, an economic case can often be made out for reduction of human supervision.Thus in most situations the use of some equipment which performs the same intended function as continuously employed human operator is preferred . A system incorporating such an equipment is known as automatic control system.Manually Controlled systems Vs Automatic Control systems(cont..)In fact in most situations an automatic control system could be made to perform intended functions better than a human operator, and could further be made to perform such functions as would be impossible for a human operator .The general block diagram of an automatic control system which is characterized by a feedback loop, is shown in fig 1.4.An error detector compares a signal obtained through feedback elements, which is the function of the output response, with the reference input.Any difference between these two signals constitutes an error or actuating signal, which actuates the control elements .The control element in turn alter the conditions in the plant (controlled member ) in a such a manner as to reduce the original error.

Open-loop ControlAs stated already, any physical system which does not automatically correct for variation in its output, is called an open-loop system. Such a system may be represented by the block diagram of Fig. 1.6 . In these systems the output remains constant for a constant input signal provided the external conditions remain un altered. The output may be changed to any desired value by appropriately changing the input signal but variations in external conditions or internal parameters of the system may cause the output to vary from the desired value in un controlled fashion .

Open-Loop SystemsA generic open-loop system is shown in Figure 1.6(a). 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.The distinguishing characteristic of an open-loop system is that it cannot compensate for any disturbances that add to the controller's driving signal (Disturbance 1 in Figure 1.6(a)).

Open-Loop Systems (cont)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 output of an open-loop system is corrupted not only by signals that add to the controller's commands but also by disturbances at the output (Disturbance 2 in Figure 1.6(a)). 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.Open-Loop Systems (cont)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.

Closed-Loop (Feedback Control) SystemsThe disadvantages of open-loop systems, namely sensitivity to disturbances and inability to correct for these disturbances, may be overcome in closed-loop systems.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. 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 1.6(b), the output signal is subtracted from the input signal. The result is generally called the actuating signal. Closed-Loop (Feedback Control) Systems (cont..)However, in systems where both the input and output transducers have unity gain (that is, the transducer amplifies its input by 1), the actuating signal's value is equal to the actual difference between the input and the output. 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. If there is no difference, the system does not drive the plant, since the plant's response is already the desired response.Advantages of Closed Loop systemGreater accuracy than open-loop systemsTransient and steady-state responses can be controlled more easilyMore complex and expensive than open-loop systemsRequires monitoring the plant output

Missile Launching and Guidance system The missile launching and guidance system of fig 1.10 is a sophisticated example of military applications of feedback control. The target plane is sited by a rotating radar antenna which then locks in and continuously tracks the target.Depending upon the position and velocity of the plane as given by the radar output data , the launch computer calculates the firing angle in terms of a launch command signal, which when amplified through a power amplifier derives the launcher ( derive motor) . The launcher angular position is feedback to launch computer and the missile is triggered as soon as the error between the launch command signal and the missile firing angle becomes zero.54Missile Launching and Guidance system (cont)After being fired the missile enters the radar beam which is tracking the target.The control system contained within the missile now receives a guidance signal from the beam which automatically adjusts the control surface of the missile such that the missile rides along the beam, finally homing on to the target .it is important to note that the actual missile launching and guidance system is far more complex requiring control of guns bearing as well as elevation.

Automatic Aircraft landing systemThe automatic aircraft landing system is simplified form of Fig. 1.5 . The system consists of three basic parts : the aircraft, the radar unit and the controlling unit. The radar unit measures the approximate vertical and lateral positions of the aircraft which are then transmitted to the controlling unit .From these measurements, the controlling unit calculates appropriate pitch and bank commands. These commands are then transmitted to the aircraft autopilots which in turn cause the aircraft to respond.Assuming that the lateral control system and the vertical control system are independent .57Automatic Aircraft landing system( cont)In fig. 1.5 (b). The aircraft lateral position, y(t), is the lateral distance of the aircraft from the extended centerline of the landing area on the deck of the aircraft carrier. The control system attempts to force y(t) to zero. The radar unit measures y(kT) is the sampled value of y(t), with T= 0.05s and k= 0,1,2,3 .Two unwanted inputs called disturbances appear in to the system. These are (i) wind gust affecting the position of the aircraft and (ii) radar noise present in the measurement of the aircraft position .58

Analysis and Design ObjectivesAnalysis is the process by which a system's performance is determined. For example, we evaluate its transient response and steady-state error to determine if they meet the desired specifications.Design is the process by which a system's performance is created or changed. For example, if a system's transient response and steady-state error are analyzed and found not to meet the specifications, then we change parameters or add additional components to meet the 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 have already identified these two responses and cited a position control system (an elevator) as an example.Analysis and Design Objectives (cont..)The three major objectives of systems analysis and design: producing the desired transient response, reducing steady-state error, and achieving stability. 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 permanent physical damage.Analysis and Design Objectives (cont..)Steady-State Response : Another analysis and design goal focuses on the steady-state response. 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 elevator must be level enough with the floor for the passengers to exit, and a read/write head not positioned over the commanded track results in computer errors. An antenna tracking a satellite must keep the satellite well within its beam width in order not to lose track.Analysis and Design Objectives (cont..)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. Natural response describes the 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. Thus, for a linear system, we can write : Total Response = Natural Response + Forced ResponseAnalysis and Design Objectives (cont..)For a control system to be useful, the natural response must (1) eventually approach zero, thus leaving only the forced response, or (2) oscillate. 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.Analysis and Design Objectives (cont..)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 characteristics can be designed. Stability is our third analysis and design objective.The Design process of control system

Step-3: Draw Schematic

Step-4: Draw Block Diagram

Mathematical ModelsModel the system mathematically using physical laws. Kirchoffs Voltage Law - The sum of voltages around a closed path is zero. Kirchoffs Current Law - The sum of currents flowing from a node is zero. Newtons Laws - The sum of forces on a body is zero (considering mass times acceleration as a force).The sum of moments on a body is zero.The model describes the relationship between the input and the output of the dynamic system.

Step-5: Reduce the Block Diagram

Step-6: Analyze and Design

The input signal is the desired position of the antenna. Several common forms of input functions are used for test purposes

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