EE Control Systems Sample Career Avenues

7/25/2019 EE Control Systems Sample Career Avenues

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Control Systems 2015

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Control

Systems


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IndexChapter: -1 Control systems : Introduction1.1 Introduction

1.2 Definitions and Terminology

1.3 Example of Control Systems

1.4 Closed-Loop and Open-Loop Control Systems

1.5 Closed-loop versus open-loop control systems

1.6 The Control Problems

1.7 Response Characteristics and System Configurations

1.8 Analysis and Design Objectives

1.9 The Design Process

Chapter: - 2 Laplace Transform and Transform Function

2.1 Dynamic Systems

2.2 Review of Complex Variables and Complex Functions

2.3 Review of Laplace Transform

2.4 Transfer Functions

Chapter: - 3 Reduction of Multiple Subsystems

3.1 Introduction

3.2 Block diagrams

3.3 Signal Flow Graphs

3.4 Masons rule

Chapter: -4Modeling a Control System

4.1 Analogous Systems

4.2 Electrical Systems

4.3 Mechanical Systems


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Chapter: - 5Time Response

5.1 Poles, Zeros, and System Response

5.2 First Order Systems

5.3 Second Order System: Introduction

5.4 The General Second Order System5.5 Type of a transfer function and steady-state errors

5.6 Sensitivity

Chapter: - 6 Control Action

6.1 Introduction

6.2 Proportional Control

6.3 Derivative Control

6.4 Integral Control

6.5 Proportional plus Integral plus Derivative Control

6.6 PID controller theory further explanation

Chapter: - 7 STABILITY

7.1 Introduction

7.2 Routh-Hurwitz criterion

7.3 Routh - Hurwitz criterion: special cases

7.4 Routh-Hurwitz criterion: additional examples

Chapter: - SYSTEM STABILITY

8.1 Review of System Stability and Some Concepts Related to Poles and Zero.8.2 Sketch the Polar plot of Frequency Response

8.3 Nyquist Criterion and Diagram

8.4 Cauchys Principle of Argument

8.5 Nyquist Criterion

Chapter:- Bode Plots and Gain Adjustments Compensation

9.1 Introduction

9.2 Plotting frequency response

9.3 Asymptotic Approximations: Bode Plots

9.4 Bode plots for Second-order9.5 Stability, Gain Margin, and Phase Margin via Bode Plots


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Chapter: - Root Locus Techniques

10.1 Introductions10.2 The root locus concept

10.3 Construction Rules

Chapter: - Compensation of Control Systems

11.1 LEAD COMPENSATOR

11.2 LEAD COMPENSATOR DESIGN USING ROOT LOCUS TECHNIQUES

11.3 BODE DESIGN OF LEAD COMPENSATOR

11.4 LAG COMPENSATOR DESIGN USING BODE PLOTS

11.5 BODE DESIGN OF LAG COMPENSATOR

11.6 LAG-LEAD COMPENSATOR DESIGN USING BODE PLOTS

11.7 Design procedure: Compensator Structure

Chapter:- State space representation of control system

12.1 Introduction

12.2 SOME OBSERVATIONS

10.3 THE GENERAL STATE-SPACE REPRESENTATION

12.4 APPLYING THE STATE-SPACE REPRESENTATION

12.5 CONVERTING A TRANSFER FUNCTION TO STATE SPACE

12.6 CONVERTING A STATE SPACE TO TRANSFER FUNCTION

12.6 CONTROLLABILITY AND OBSERVABILITY

12.7 METHODS INVOLVING EIGEN VALUES

12.8 FREQUENCY-DOMAIN SOLUTION OF THE STATE EQUATION

12.9 FINDING THE STATE TRANSITION MATRIX12.10 STATE EQUATION FROM TRANSFER FUNCTIONS


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CHAPTER 1

Control systems : Introduction

1.1 Introduction

Control systems are everywhere around us, they regulate temperature in homes, stabilize aircraft and

spacecraft, affect production of food by regulating the quality and purity of produce, are used inrefineries, and so on.

A system is generally made up of components which could be mechanical, electrical, hydraulic, and/orpneumatic. The function of the control system is to regulate the output of the system in accordancewith some input signal. The controlled output may be position, velocity, acceleration, temperature, etc.The output is required to track and follow the input signal as closely as possible. The output ismeasured by using transducers and, fed back to the controller in order to correct deviations from theinput signal. This is referred to as feedback and is an essential element of closed-loopcontrol systems.When feedback is not used, we have an open-loopcontrol system.

In closed-loop control systems, feedback is used for calculating the error which is the difference

between the input and output signals. The error signal is then filtered, amplified and applied to apower element within the system such as an electrical motor, hydraulic valve/ram) that acts upon thequantity to be controlled.

Modern control is increasingly based on having an electrical signal as input and having transducersthat produce electrical signals as a measure of the output. These are then fed into the controller whichacts upon the system being controlled. Computers are increasingly used in control applications.

1.2. Definitions and Terminology

Controlled Variable and Manipulated Variable:The controlled variable is the quantity or condition that is measured and controlled. The manipulated

variable is the quantity or condition that is varied by the controller so as to affect the value of the

controlled variable. Normally, the controlled variable is the output of the system.

Control:

Control means measuring the value of the controlled variable of the system and applying themanipulated variable to the system to correct or limit deviation of the measured value from a desiredvalue.

Plants:

A plant may be a piece of equipment, perhaps just a set of machine parts functioning together, thepurpose of which is to perform a particular operation. In this book, we shall call any physical object tobe controlled (such as a mechanical device, a heating furnace, a chemical reactor, or a spacecraft) aplant.

Processes:Defined as natural, progressively continuing operation or development marked by a series of gradualchanges that succeed one another in a relatively fixed way and lead toward a particular result or end;or an artificial or voluntary, progressively continuing operation that consists of a series of controlledactions or movements systematically directed toward a particular result or end. In this book we shallcall any operation to be controlled a process. Examples are chemical, economic, and biological

processes.


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Systems:A system is a combination of components that act together and perform a certain objective. A systemis not limited to physical ones. The concept of the system can be applied to abstract, dynamic

phenomena such as those encountered in economics. The word system should, therefore, beinterpreted to imply physical, biological, economic, and the like, systems.

Disturbances:

A disturbance is a signal that tends to adversely affect the value of the output of a system. If adisturbance is generated within the system, it is called internal, while an external disturbance isgenerated outside the system and is an input.

Feedback Control systems:

A system that maintains a prescribed relationship between the output and the reference input bycomparing them and using the difference as a means of control is called a feedback control system.An example would be a room-temperature control system. By measuring the actual room temperatureand comparing it with the reference temperature (desired temperature), the thermostat turns the heatingor cooling equipment on or off in such a way as to ensure that the room temperature remains at acomfortable level regardless of outside conditions.

1.3. Example of Control Systems

An example of a closed-loop system is a temperature-control system in a room. For this system wewish to maintain, automatically, the temperature of the room at a desired value. To control any

physical variable, which we usually call a signal,we must know the value of this variable, that is, wemust measure this variable. We call the system for the measurement of a variable a sensor. In thissystem, the sensor is a thermistor. Thermistor is a device which has a resistance that varies withtemperature. By measuring this resistance, we obtain a measure of the temperature.


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We defined theplant of a control system as that part of the system to be controlled. It is assumed inthis example that the temperature is increased by activating a gas furnace or a heating sub-system.Hence the plant input is the electrical signal that activates the furnace, and the plant output signal isthe actual temperature of the living area. The plant is represented as shown in the above figure. In thisexample, the output of each of the systems is connected to the input of the other, to form the closed

loop. In most closed-loop control systems, it is necessary to connect a third system into the loop toobtain satisfactory characteristics for the total system. This additional system is called a compensatoror a controller.

The usual form of a single loop closed-loop control system is as follows:

The system input is a reference signal; usually we want the system output to be equal to this input.In the temperature-control system, this input is the setting of theDesired Temperature. If we want tochange the temperature, we change the system input. The system output is measured by the sensor,and this measured value is compared with (subtracted from) the input. This difference signal iscalled the error signal, or simply the error. If the output is equal to the input, this difference is zero,and no signal reaches the plant. Hence the plant output remains at its current value. If the error isnot zero, in a properly designed system the error signal causes a response in the plant such that themagnitude of the error is reduced. The compensator is a filter for the error signal, since usually

satisfactory operation does not occur if the error signal is applied directly to the plant.

1.4. Closed-Loop and Open-Loop Control Systems

Open-loop control systems:Those systems in which the output has no effect on the control action are called open-loop controlsystems. In other words, in an open-loop control system the output is neither measured nor fed backfor comparison with the input. One practical example is a washing machine. Soaking, washing, andrinsing in the washer operate on a time basis. The machine does not measure the output signal, that is,the cleanliness of the clothes.

In any open-loop control system the output is not compared with the reference input. Thus, to eachreference input there, corresponds a fixed operating condition; as a result, the accuracy of the system

depends on calibration. In the presence of disturbances, an open-loop control system will not performthe desired task. Open-loop control can be used, in practice, only if the relationship between the inputand output is known and if there are neither internal nor external disturbances. Clearly, such systemsare not feedback control systems. Note that any control system that operates on a time basis is openloop. For instance, traffic control by means of signals operated on a time basis is another example ofopen-loop control.


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Closed-loop control systems:

Feedback control systems are often referred to as closed-loop control systems. In practice, the termsfeedback control and closed-loop control are used interchangeably. In a closed-loop control systemthe actuating error signal, which is the difference between the input signal and the feedback signal(which may be the output signal itself or a function of the output signal and its derivatives and/orintegrals), is fed to the controller so as to reduce the error and bring the output of the system to a

desired value. The term closed-loop control always implies the use of feedback control action in orderto reduce system error.

1.5. Closed-loop versus open-loop control systems

An advantage of the closed loop control system is the fact that the use of feedback makes the systemresponse relatively insensitive to external disturbances and internal variations in system parameters. Itis thus possible to use relatively inaccurate and inexpensive components to obtain the accurate controlof a given plant, whereas doing so is impossible in the open-loop case. From the point of view ofstability, the open-loop control system is easier to build because system stability is not a major

problem. On the other hand, stability is a major problem in the closed-loop control system, which maytend to overcorrect errors that can cause oscillations of constant or changing amplitude.

1.6. The Control Problems

We may state the control problem as follows. A physical system or process is to be accuratelycontrolled through closed-loop, or feedback, operation. An output variable, called the response, isadjusted as required by the error signal. This error signal is the difference between the systemresponse, as measured by a sensor, and the reference signal, which represents the desired systemresponse.

Generally a controller, or compensator, is required to filter the error signal in order that certain controlcriteria, or specifications, are satisfied. These criteria may involve, but not be limited to:1. Disturbance rejection

2. Steady-state errors3. Transient response characteristics4. Sensitivity to parameter changes in the plant

Solving in control problem generally involves1. Choosing sensors to measure the plant output2. Choosing actuators to drive the plant3. Developing the plant, actuator, and sensor equations (models)4. Designing the controller based on the developed models and the control criteria5. Evaluating the design analytically, by simulation, and finally, by testing the physical system6. If the physical tests are unsatisfactory, iterating these steps


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controller and process outputs via summing junctions.The open-loop system cannot correctfor these disturbances.Examples toasters, washing machine (washing process)

Figure 1.3

Closed-Loop (Feedback Control) Systems The disadvantages of open-loop systems may

be overcome in closed-loop system as shown in Figure 1.3. An output transducer/ sensor,

measures the output response and converts into the form used by controller. The closed-loop systems measured the output response 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.

Examples air conditioning, lift, washing machine (water level control)

Figure 1.4

Controller

Process

Or Plant+

-

Error

SummingJunction

Input

Transducer

OutputTransducer= sensor

SummingJunction

SummingJunction

+ +

+ +

Disturbances 1 Disturbances 2

Controller

ProcessOr Plant

InputTransducer

SummingJunction

SummingJunction

+ +

+ +

Disturbances 1 Disturbances 2

Input= ref


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The differences between open and closed-loop system are:

Closed-Loop System Open-Loop System

Have the feedback path. Does not have the feedback path.Greater accuracy. Not accurate.Less sensitive to noise, disturbances and

changes in the environment.

Sensitive to noise, disturbances and changes

in the environment.The system can compare the output responsewith the input and make a correction if there isany difference.

The system cannot correct the disturbances.

More complex and expensive. Simple and inexpensive.

1.8. Analysis and Design Objectives

Control systems are dynamic: they respond to an input by undergoing a transient response beforereaching steady-state response that generally resembles the input.The 3 major objectives are:

o Producing the desired transient responseo Reducing steady-state erroro Achieving stability

Transient Response

Important in control systemEx: In the case of an elevator, a slow transient makes passenger impatient, whereas an excessivelyrapid response makes them uncomfortable. Too fast a transient response could cause permanent

physical damage. Therefore, we have to analyze the system for its existing transient response. Then,adjust parameters or design components to yield a desired transient response.

Steady-State Response

This response resembles the input and is usually what remains after the transients have decayed tozero. We define steady-state errors quantitatively, analyze a systems steady-state error, and thendesign corrective action to reduce this error.

1.9. The Design Process

The design of a control system follows these steps:1. Determine a physical system and specifications from requirements.2. Draw a functional block diagram.3. Represent the physical system as a schematic.4. Use the schematic to obtain a mathematical model such as a block diagram.5. Reduce the block diagram.6. Analyze and design the system to meet specified requirements and specifications that

include stability, transient response and steady-state performance.


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This topic is for those students who dont have the knowledge of Laplace Transform and Transform

Function. If you are already comfortable with these topics you can skip this chapter.

CHAPTER 2

Laplace Transform and Transform Function

2.1. Dynamic Systems

This course is about controlling dynamic systems. What makes dynamic systems unique anddistinctive is that dynamic systems have a memory. Their present output, y(t), does not just depend onthe present input signal value u(t), but on past inputs u() for t.

The way this memory is described in mathematical form is through integral/differential equations. Inthis course, we will learn that even simple dynamic systems are modeled by derivative and integralterms.

The Laplace transform allows us to transform these integral/differential equations into simpler

algebraic equations and solving them. It is a method for solving differential equations andcorresponding initial and boundary value problems.

2.2. Review of Complex Variables and Complex Functions

Analysis of dynamic systems relies heavily on the theories associated with complex variables andcomplex functions. A complex number has a real part and an imaginary part, both of which areconstant. If the real part and/or imaginary part are variables, a complex number is called a complexvariable. In the Laplace transformation, we use the notation s as a complex variable; that is,s= +jwhere is the real part and is the imaginary part.

Complex function:

A complex function F(s), a function of s, has a real part and an imaginary part orF(s) = Fx+ jFy

where Fx is the real part and jFy is the imaginary part. F(s) can be represented graphically in thecomplex plane as follows:

I

Re

Fy

Fx

|F|

F(s)

)(sF-

Fy


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The magnitude of F(s) is denoted by |F| and is equal to22

yx FF . Its angle is obtained as)/(tan 1 xy FF

. The angle is measured counterclockwise from the positive real axis.

The complex conjugate of F(s) is defined as yxjFFF

. Another form in which complex conjugate

is denoted is*

F .Note that the magnitude of F(s), denoted by |F|, can also be defined as:22

yx FFFFF

2.3. Review of Laplace Transform

We can convert many common functions, such as sinusoidal functions, differential equations, andexponential functions, into algebraic functions of a complex variable s. Operations such asdifferentiation and integration can be replaced by algebraic operations in the complex plane. Thus, alinear differential equation can be transformed into an algebraic equation in a complex variable s. Ifthe algebraic equation in s is solved for the dependent variable, then the solution of the differentialequation (the inverse Laplace transform of the dependent variable) may be found by use of a Laplacetransform table or by use of the partial-fraction expansion technique.

An advantage of the Laplace transform method is that it allows the use of graphical techniques forpredicting the system performance without actually solving system differential equations. Anotheradvantage of the Laplace transform method is that, when we solve the differential equation, both thetransient component and steady-state component of the solution can be obtained simultaneously.

Let us define

f(t) = a function of time t such thatf (t) = 0 for t < 0

s = a complex variableL = an operational symbol indicating that the quantity that it prefixes is to be transformed by the

Laplace integral

0)( dtetf st

.F(s) = Laplace transform off (t)

Then the Laplace transform off (t) is given by

0

)()()]([ dtetfsFtfL st

Laplace transform of common functions have been tabulated [see the text book page 40] and are usedto transform the time domain functionsf(t)into their Laplace form directly from tables and, vice versa.

Important Properties of Laplace Transforms

Real differentiation theorem.

The Laplace transform of the derivative of a functionf (t) is given by

)0()(])(

[ fssFdt

tdfL

where f (0) is the initial value off (t) evaluated at t = 0.Similarly, we obtain the following relationship for the second derivative off(t) applies:


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)0()0()(])(

[ 22

2

sffsFsdt

tfdL

and for theth

n derivative:

)0()0(...)0()0()(])(

[)1()2(

21

nn

nnn

n

n

ffsfsfssFsdt

tfdL

In most of the problems that we will consider in this course, it is assumed that the initial values of f (t)

and its derivatives are equal to zero, then the Laplace transform of theth

n derivative off (t)is given by

)(sFsn .

Real-integration theorem.

Similarly to above, the Laplace transform of dttf )( exists and is given by

sf

s

sF

dttfL

)0()(

)(

1

for zero initial value,

s

sFdttfL

)()(

.

Final-value theorem.

The final-value theorem is extensively used throughout the course and relates the steady-state behavioroff (t) to the behavior of sF(s) in the neighborhood of s = 0. The final-value theorem may be stated asfollows:

If f (t) and df (t)/dt are Laplace transformable, if F(s) is the Laplace transform of f (t), and if)(lim tft exists, then

)(lim)(lim 0 ssFtf st

This formula allows you to calculate the final steady state value off(t)by using F(s)without having toconvert F(s)back into its time domain representationf(t).

Other important properties of Laplace transforms are tabulated in the text book page 41.

Inverse Laplace Transformation

The reverse process of finding the time functionf(t) from the Laplace transform F(s) is

called the inverse Laplace transformation. The notation for the inverse Laplace transformation is1

L ,and the inverse Laplace transform can be found from F(s)by the following inversion integral:

jc

jc

stdsesF

jtfsFL )(

2

1)()]([1

for t>0where cis a constant related to the stability and convergence of the integral.


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Evaluating the inversion integral appears complicated. In practice, we seldom usethis integral for finding f(t). Instead, we usually decompose the F(s) into a simpler form by using

partial fraction expansion and, then use a conversion table to obtain the time domain equivalent of thepartial fractions.

If a particular transform F(s) cannot be found in a table, then we expand it into partial fractions and

write F(s) in terms of simple functions of s for which the inverse Laplace transforms are alreadyknown.

Partial-fraction expansion method for finding inverse Laplace transforms:

For problems in control systems analysis, F(s), the Laplace transform of f (t), frequently occurs in theformF(s) =B(s)/A(s) where A(s) and B(s) are polynomials in s. In the expansion of F(s) = B(s)/A(s) into a

partial-fraction form, it is important that the highest power of s inA(s) be greater than the highestpower of s inB(s). If such is not the case, the numeratorB(s)must be divided by the denominatorA(s)in order to produce a polynomial in splus a remainder (a ratio of polynomials in s whose numerator isof lower degree than the denominator).

If F(s) is broken up into components,)(...)()( 1 sFsFsF n then its inverse Laplace transform is

obtained as:

)(...)()(...)()( 11

111 tftfsFLsFLsFL nn

The advantage of the partial-fraction expansion approach is that the individual terms of F(s), resultingfrom the expansion into partial-fraction form, are very simple functions of s; consequently, it is notnecessary to refer to a Laplace transform table if we memorize several simple Laplace transform pairs.It should be noted, however, that in applying the partial-fraction expansion technique in the search forthe inverse Laplace transform of F(s)=B(s)/A(s) the roots of the denominator polynomial A(s) must

be obtained in advance. That is, this method does not apply until the denominator polynomial hasbeen factored.

Partial-fraction expansion whenF(s) involves distinct poles only:

Consider F(s) written in the factored form

))...((

))...((

)(

)()(

1

1

n

m

psps

zszsK

sA

sBsF

for m< n

wherep andzare either real or complex quantities.

If F(s)involves distinct poles only, then it can be expanded into a sum of simple partial fractions asfollows:

n

n

ps

a

ps

a

sA

sBsF

...

)(

)()(

1

1


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where ka

(k= 1, 2, . . . , n) are constants. The coefficient ka

is called the residue at the pole at

kps . The value of ka can be found by multiplying both sides of the above Equation by )( kps

and letting kps

which gives:

kps

kksAsBpsa

)()(

since:

tp

k

k

k keaps

aL

1

thenf(t) is obtained astp

n

tp neaeatf ...)( 11 for all .0t

2.4. Transfer Functions

A mathematical model of a dynamic system is defined as a set of equations that represents thedynamics of the system accurately or, at least, fairly well. Note that a mathematical model is not

unique to a given system. A system may be represented in many different ways and, therefore, mayhave many mathematical models, depending on one's perspective. The degree of accuracy andcomplexity of models is very much dependent on the use and application that they are intended for.

The dynamics of many systems, whether they are mechanical, electrical, thermal, economic,biological, and so on, may be described in terms of differential equations. Such differential equationsmay be obtained by using physical laws governing a particular system, for example, Newton's laws formechanical systems and Kirchhoff's laws for electrical systems. We must always keep in mind thatderiving a reasonable mathematical model is the most important part of the entire analysis.

Mathematical models

Mathematical models may assume many different forms. We will consider be considering the

transient-response or frequency-response analysis of single-input-single-output, linear, time-invariantsystems. Here, a transfer function representation is more convenient than any other. Once amathematical model of a system is obtained, various analytical and computer tools can be used foranalysis and synthesis purposes.

Simplicity versus accuracy

It is possible to improve the accuracy of a mathematical model by increasing its complexity. In somecases, we include hundreds of equations to describe a complete system at a certain level of accuracy.If extreme accuracy is not needed, however, it is preferable to obtain only a reasonably simplifiedmodel. In fact, we are generally satisfied if we can obtain a mathematical model that is adequate forthe problem under consideration. It is important to note, however, that the results obtained from theanalysis are valid only to the extent that the model approximates a given dynamic system.

In general, in solving a new problem, we find it desirable first to build a simplified model so that wecan get a general feeling for the solution. A more complete mathematical model may then be built andused for a more complete analysis.


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Linear systems

A system is called linear if the principle of superposition applies. The principle of superposition statesthat the response produced by the simultaneous application of two different forcing functions is thesum of the two individual responses. Hence, for the linear system, the response to several inputs can

be calculated by treating one input at a time and adding the results. It is this principle that allows oneto build up complicated solutions to the linear differential equation from simple solutions.

Linear time-invariant systems and linear time-varying systems

A differential equation is linear if the coefficients are constants or functions only of the independentvariable. Dynamic systems that are composed of linear time-invariant lumped-parameter componentsmay be described by linear time-invariant (constant coefficient) differential equations. Such systemsare called linear time-invariant (orlinear constant-coefficient) systems. Systems that are represented

by differential equations whose coefficients are functions of time are called linear time-varyingsystems. An example of a time-varying control system is a spacecraft control system. (i.e. mass of aspacecraft changes due to fuel consumption.)

Nonlinear systems

A system is nonlinear if the principle of super-position does not apply. Thus, for a nonlinear systemthe response to two inputs cannot be calculated by treating one input at a time and adding the results.Although many physical relationships are often represented by linear equations, in most cases actualrelationships are not quite linear. In fact, a careful study of physical systems reveals that even so-called "linear systems" are really linear only in limited operating ranges. In practice, manyelectromechanical systems, hydraulic systems, pneumatic systems, and so on, involve nonlinearrelationships among the variables. For example, the output of a component may saturate for largeinput signals. There may be a dead space that affects small signals. (The dead space of a component isa small range of input variations to which the component is insensitive). Square-law nonlinearity mayoccur in some components. For instance, dampers used in physical systems may be linear for low-velocity operations but may become nonlinear at high velocities, and the damping force may become

proportional to the square of the operating velocity. Examples of characteristic curves for these

nonlinearities are shown in your text.

Transfer Functions

In control theory, functions called transfer functions are commonly used to characterize the input-output relationships of components or systems that can be described by linear, time-invariant,differential equation.

The transfer function of a linear, time-invariant, differential equation system is defined as the ratio ofthe Laplace transform of the output (response function) to the Laplace transform of the input (drivingfunction) under the assumption that all initial conditions are zero. Consider the linear time-invariantsystem defined by the following differential equation:

xbxbxbxbyayayayamm

mm

onn

nn

o

1

)1(

1

)(

1

)1(

1

)(

......

whereyis the output of the system andxis the input. The transfer function of this system is obtainedby taking the Laplace transforms of both sides of the above equation, under the assumption that allinitial conditions are zero.


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Transfer function = nnnn

o

mm

mm

o

asasasa

bsbsbsbsG

11

1

11

1

...

...)(

By using the concept of transfer function, it is possible to represent system dynamics by algebraicequations in s. If the highest power of s in the denominator of the transfer function is equal to n, the

system is called an nth-order system.

Comments on transfer function

The applicability of the concept of the transfer function is limited to linear, time-invariant, differentialequation systems. The transfer function approach, however, is extensively used in the analysis anddesign of such systems. In what follows, we shall list important comments concerning the transferfunction. (Note that in the list a system referred to is one described by a linear, time invariant,differential equation.)

1. The transfer function of a system is a mathematical model in that it is an operational method ofexpressing the differential equation that relates the output variable to the input variable.

2. The transfer function is a property of a system itself independent of the magnitude and nature

of the input or driving function.3. The transfer function includes the units necessary to relate the input to the output; however, it

does not provide any information concerning the physical structure of the system. (i.e. transferfunctions of many physically different systems can be identical.)

4. If the transfer function of a system is known, the output or response can be studied for variousforms of inputs with a view toward understanding the nature of the system.

5. If the transfer function of a system is unknown, it may be established experimentally byintroducing known inputs and studying the output of the system. Once established, a transferfunction gives a full description of the dynamic characteristics of the system, as distinct fromits physical description.

To derive the transfer function, we proceed according to the following steps.1. Write the differential equation for the system.2. Take the Laplace transform of the differential equation, assuming all initial conditions are zero.3. Take the ratio of the Laplace transforms of the output to the input. This ratio is the transfer

function.


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20

Objectives

After this c

3.1 Introdu

We have b

More com

response of

single trans

Then, we c

information

3.2 Block d

In the previ

presented.

variable to t

input variab

graphically

following fi

model and t

Many syste

few more

component

areer Aven

apter, you

educe a blo

rom input to

nalyze and

ction

en working

licated syst

a single tra

er function.

n apply th

about the e

iagrams

us section

y definitio

he Laplace

le, Y(s)be t

denoting th

gure. The b

he arrows r

s are com

lements su

parts of a bl

ues

Reduct

hould be ab

k diagram o

output

esign for tra

with indivi

ems are re

sfer functi

analytical

tire system

he concept

the transfe

ransform o

e output va

relationshi

ock represe

present sig

osed of mu

h as summ

ock diagra

X(s)

CH

ion of

le to:

multiple su

sient respon

dual subsys

resented b

n can be ca

techniques

.

f a transfer

r function i

the input v

riable, and

Y(s) = X(s

ts a transfe

als (e.g. ele

ltiple subsy

ing junctio

for a linear

F

G(

PTER

ultiple

systems to

se, a system

tems repres

the interc

culated, we

of the previ

function fo

the ratio o

riable. Let

(s) be the t

)U(s)is thr

r function c

ctrical volta

tems. Whe

and picko

, time-invar

igure 3.1

)

3

Subsys

single bloc

onsisting of

ented by a

onnection o

want to rep

ous chapter

a linear ti

the Laplace

X(s)be the

ransfer func

ugh a block

rrespondin

ge from a p

multiple s

f points m

ant system

Y(s)

Control S

tems

representing

multiple sub

lock with i

f many su

resent multi

s and obtai

e-invariant

transform

Laplace tra

ion. Onem

diagram, a

to a syste

sition sens

bsystems a

st be adde

re shown i

stems 20

the transfer

ystems

s input and

systems. Si

ple subsyst

transient r

system was

f the output

nsform of t

ethod of

shown in t

s mathem

r).

e interconn

to the bl

Figure 3.1.

15

function

output.

nce the

ms as a

esponse

e)

e

tical

ected, a

ck. All


20/26

21

We will no

transfer fun

reducing m

Cascade F

Figure 3.2(

transfer fun

Parallel Fo

Figure 3.3(

in Figure 3.

Feedback

The third t

simplified

areer Aven

examine

ction repres

re complic

rm

) shows an

ction.

rm

) shows an

3(b).

orm

opology is

odel. We o

ues

ome comm

entation for

ted systems

example of

example of

the feedba

btain the eq

n topologi

each of th

to a single

cascaded su

F

arallel sub

F

k form as

ivalent, or

s for interc

m. These

lock.

systems an

igure 2.2

ystems. Th

igure 3.3

shown in

losed-loop,

nnecting s

ommon top

d Figure 3.

equivalent

Figure 3.4(

transfer fu

Control S

bsystems a

ologies will

(b) shows t

transfer fun

) and Fig

ction show

stems 20

d derive th

form the

e equivale

ction, Ge(s)

re 3.4(b) s

in Figure 3

15

e single

asis for

t single

appears

hows a

.4(c).


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22

Moving Bl

Figure 3.5

past a sum

Figure 3.6

past a picko

areer Aven

cks to Cre

hows equi

ing junctio

hows equi

ff point.

ues

te Familia

alent block

.

alent block

Figure 3.

Forms

diagrams f

diagrams f

4 (a), (b) an

rmed when

rmed when

d (c)

transfer fu

transfer fu

Control S

ctions are

ctions are

stems 20

oved left

oved left

15

r right

r right


22/26

d

se

a

s

s

23

igure 3.5:

iagram alge

umming junquivalent fo

oving a bl

.to the left

umming jun

.to the righ

umming jun

Figure 3.

Block dia

algebra fo

pointsequivalen

for movin

a.to the l

pickoff po

b.to the r

pickoff po

areer Aven

lock

bra for

ctionsrms for

ck

ast a

ction;

t past a

ction

ram

r pickoff

forms

a block

ft past a

int;

ght past a

int

ues

Control S stems 2015


23/26

24

Ex 3.1 Red

Answer:

areer Aven

ce the bloc

ues

diagram s

own in Fig

F

re 3.7 to a

igure 3.7

ingle transf

Control S

er function.

stems 20

15


24/26

25

Ex 3.2 Red

Answer:

areer Aven

cethe bloc

ues

k diagram s own in Fig

F

ure 3.8 to a

igure 3.8

single trans

Control S

er function.

stems 20

15


25/26

26

Do-it-your

3.9.

Answer:

s

sT

2

)(4

areer Aven

elf: Find t

ss

s

2

12

3

Rule

Closed loop

system

Addition or

subtraction

Moving a

starting poin

Moving a

summing po

Cascaded

elements

ues

e equivalen

R(s)

R(s)

R(s)

Table 3-1

t

int

G1(s)

G2

R(s)

+

B(

R(s) +

G1(s

transfer fu

F

Original syst

+

+

G2(

Some rules

(s)

G(s)

-

)

G

-

ction, T(s)

igure 3.9

m

B(s)

C(s))

or block dia

C(s)

C(s)

C(s)(s)

C(s)G(s)

(s)

C(s)/R(s) f

Reduce

R(s)

ram reductio

R(s)

R(s)

R(s) G(

G(s

G(s

B(s)

R(s)

Control S

r the syste

d system

B

1(s)G2(s)

1(s)+G2(s)

C(s)s)

)

+

stems 20

shown in

C(s)

(s)

(s)

(b)

(s)

C(s)

15

igure


26/26


Exercise: Consider the transfer function of the system shown in Fig. 3-9a. The final transfer functionis shown in Fig. 3-9d. Note that the first reduction involves a parallel combination; the secondinvolves a cascade combination.

C(s)

C(s)

-+

C(s)R(s) +

-G1(s)

Fig.3-9 Obtaining transfer function by block diagram reduction

G3(s)

G2(s)

(a)

(b)

+

+

H1(s)

G4(s)

H2(s)

-+

R(s) +

-G1(s)

H2(s)

)(

)(

4

1

sG

sH

G2(s)+G3(s) G4(s)C(s)

H2(s)

R(s) +

-)]()()[(1

)()]()([)(

321

4325

sGsGsH

sGsGsGsG

G1(s)

(c)

(d)

R(s))()()(1

)()(251

51

sHsGsGsGsG

First reduction

Next reduction

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