INTRODUCTION TO AUTOMATIC CONTROL CLASS NOTES FOR EE361 BY VICTOR A. SKORMIN, Ph.D. Professor, Watson School of Engineering, Electrical Engineering Department Binghamton University who is grateful to his former student, Dr. Michael Elmore, Senior Staff Systems

Engineer of Lockheed Martin-Owego for valuable suggestions and corrections

Some students tend to hibernate In my Control class

And some are chronically late And I have little chance

To teach about overshoot And settling time, as well, And how a right-sided root Drives everything to hell, The final value theorem,

The pole placement rule ... And this is true, I'm teaching them

But they are sick of school But life is cynical and tough And playing games with men

And you will stop your happy laugh One rainy day, and then

Will realize that SUNY is The least of all headaches

And real life's the toughest quiz Like Skormin never makes

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CONTENTS

Introduction ……………………………………………………………………………. 4

Principles of automatic control………………………………………………………….. 5 Role of feedback ………………………………………………………………………... 10 Assignments (Homework #1) …………………………………………………………... 13

EDUCATIONAL OBJECTIVE: understanding the feedback and feedforward control principles

Chapter 1. Mathematical description of dynamic systems ……………….……….. 14

Time-domain description ...............................................................…………………….. 16 S-domain description ....................................................................……………………... 18 Frequency-domain description ......................................................………………….…. 28 Assignments (Homework #2 and Homework #2)…………………………………….… 36

EDUCATIONAL OBJECTIVE: ability to obtain a mathematical description of a dynamic system in the appropriate form

Chapter 2. Mathematical description of control systems ………………………….. 38

Typical dynamic blocks ....................................................……………………............… 38 Block diagrams ................................................................…………............………….… 57 Signal-flow graphs .......................................................................………………………. 66 Assignments (Homework #4) ........................................................……………………... 77 Example Test ................................................................................……………………... 79

EDUCATIONAL OBJECTIVE: ability to utilize existing methods of describing a control system as a combination of particular modules

Chapter 3. Common control engineering techniques ...........……………………….. 80

Numerical simulation .....................................................................……………………... 80 Loop and closed-loop transfer function .........................................……………………... 84 Computation of system poles and zeros .........................................…………………….. 87 Frequency-domain techniques: Nyquist procedure .........................……………………. 88 Frequency-domain techniques: Bode plots .....................................…………………….. 92 Root locus techniques ...................................................................………………………103 Assignments (Homework #5) ........................................................……………………...113

EDUCATIONAL OBJECTIVE: ability to utilize existing analytical and numerical techniques and software tools developed in control engineering

Chapter 4. Analysis of continuous-time control systems ........………………………115

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Stability analysis ............................................................................……………………...115 Relative stability ............................................................................……………………...129 Analysis of system statics ..............................................................……………………...138 Analysis of system dynamics ..........................................................…………………….152 Assignments (Homework #6, Homework #7, and Homework #8) ..…………………....161 Example Test ..................................................................................…………………....164

EDUCATIONAL OBJECTIVE: ability to assess properties of an existing control system

Chapter 5. Design of continuous-time control systems ...........…………………….. 166

Design considerations and problem definition ................................………...….………166 S-domain design ...........................................................................……………………... 170 S-domain design. Pole placement .................................................…………...………... 187 Frequency-domain design . . ……………………………………………………………211 Assignments (Homework #9 and Homework #10) ........................…………………….229 Example Test ................................................................................……………………. 233

EDUCATIONAL OBJECTIVE: ability to design a control system compliant with design specifications

Chapter 6. Introduction to digital control ......................…………………………... 234

Discrete-time representation of continuous signals ........................…………………… 235 Discrete-time domain description of dynamic systems ...................…………………... 243 Analysis of discrete-time control systems ......................................…………………… 249 Z-domain design of control systems ..............................................……………………. 254 Assignments (Homework #11) …….….…………….....................…………………… 263 EDUCATIONAL OBJECTIVE: ability to apply Z-domain techniques for assessing

properties of the existing and design of new digital control systems

References ....................................................................................……………………... 264

ASSIGNMENTS & GRADING POLICY

1. Homework Assignments - 20 points 2. Test #1 - 20 points 3. Test #2 - 20 points 4. Test #3 - 20 points 5. Final Exam - 20 points T O T A L 100 points

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INTRODUCTION

Automatic control is a discipline that is approximately seventy years old. Developments in

various fields of engineering have resulted in very sophisticated machines, devices and

manufacturing processes. Successful operation of these machines, devices and processes requires

very short response time, large amount of complex, repetitious analytical and mechanical operations,

and low tolerance to errors that are well beyond human abilities. Automation became the only

alternative for continuing the technical progress.

While design of a particular automatic control system constitutes an electrical and/or mechanical

engineering problem, general control theory was formulated only by 1955. It was found that all

control systems operate according to the same principle known as the negative feedback. Linear

differential equations in combination with Laplace, Fourier and later, Z-transform techniques were

suggested as the main mathematical tool of the new theory. Stability issues were rediscovered and

successfully incorporated in the control theory. Special engineering-oriented methods of system

analysis and design were formulated.

Introduction of computers became the beginning of the new era in control engineering. First,

application of computers allowed for full-scale implementation of powerful mathematical tools

provided by numerical analysis and matrix theory for control systems analysis and design. Second,

computers allowed for numerical simulation of control and dynamic systems, providing the most

accurate and thorough analytical and design tools. Third, a computer became a part of a control

system, implementing in software the most sophisticated control laws.

Modern age controls became one of the most mathematical and computer-saturated fields of

engineering. Interdisciplinary by nature, control engineering offers its services and general methods

to electrical, mechanical, chemical, aerospace and power engineering, as well as metallurgy, biology,

material science, etc. Control theory provides a foundation for such new disciplines as cybernetics,

robotics, and bioengineering. Computer-based control engineering allows for the development of

new technologies utilizing physical phenomena that are inherently unstable.

A successful control engineer has a strong mathematical background, which includes the theory

of complex variables and functions, differential equations, matrix theory, Laplace-, Fourier- and Z-

transforms, optimization techniques and applied statistics. On this foundation methods and models

of control are formulated. Computer application skills allow control engineers to utilize modern

software tools implementing control engineering techniques and facilitating numerical simulation,

analysis and design of control systems. Microprocessor background is required for the

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implementation of control laws through a microprocessor and interfacing microprocessors with

system components. Knowledge of circuits and electronics is crucial for understanding hardware

implementation aspects of control systems. Finally, a general knowledge of physics, mechanics, and

engineering is needed for understanding the nature of the system to be automated.

PRINCIPLES OF AUTOMATIC CONTROL

Automation implies development of a technical system capable of self-control. Any

deviation from the required status in such a system must result in the generation of control efforts,

partially or completely eliminating this deviation. It is important that the control efforts are

generated without any participation of a human operator, who is responsible only for the definition

of the "required status". An automatically controlled system is expected to maintain its actual status

consistent with the required status in spite of various disturbing effects.

It was noted that the ability of a complex system to maintain its status without any "intelligent"

supervision is based on a so-called negative feedback mechanism. A negative feedback mechanism

operates according to the following principle:

- an error, i.e., a discrepancy between the actual status and the required status of the system, is

detected,

- a control effort, defined as a certain function of the error, is generated,

- the direction (sign) of the control effort is always defined such that the detected error will be

reduced or eliminated, i.e., the overall effect of the control effort on the system is expected to

be equal (or close) to the effect of disturbing factors, taken with a minus sign (negative

feedback).

It can be noted that a control system, implementing the negative feedback principle, has a

distinctive closed-loop chain of resources/energy/signal/information transformations, as shown in

Fig. 1. The forward path is typically responsible for the major physical transformation that

constitutes the process to be controlled. The feedback path also performs physical transformations,

for example transformation of electric signals, but the electric power of these signals is very low: the

signals are used as carriers of information.

The following is an example of a voltage and frequency control system of an industrial power

generator. Fig. 2 shows schematics of its automatic control systems. The power generating unit

consists of a power generator (1), a turbine (2), a steam generator (3), and the steam line (4). The

speed of the turbine (and the frequency of the generated voltage) is controlled by the valve in the

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steam line (5). The voltage of the generator is controlled by the current in the field (excitation)

winding (6). Module (7) represents devices that allow for manipulation of the excitation current

flowing through the field winding. Components (8), (9) and (10) are electronics block, servo-motor

and gearbox that allow for manipulation of the position of the valve in the steam line. The power

generator is connected to a varying electrical load (11) that consists of inductance, resistance, and

capacitance components that, acting together, affect voltage and speed of the generator (frequency).

It should be understood that without any control the speed of the turbine and voltage of the generator

would exhibit unacceptable fluctuations caused by a large number of external and internal factors,

such as the load of the generator, status of the steam generator, status of bearings, etc. It is difficult

to visualize a crew of human operators, manually controlling the steam valve (5) and the field

current circuitry (7); these functions are performed by an automatic system.

Let us assume that due to the increased load (i.e., decreased load resistance) the output voltage

of the generator decreases. The voltage signal V1 generated by the voltmeter (12) becomes lower

than the reference signal R1, defined by process operators. This results in the appropriate value and

polarity of the error signal E1 defined by the error detector (13) as the difference between the actual

voltage and the reference. As shown below, the error signal is transformed by special control

module (14) into an intermediate signal, which controls the field current through special circuitry (7)

thus affecting the magnetic flux and increasing the electromotive force (EMF) of the generator. This

action results in the increase of the controlled voltage until the error is eliminated. In the case when

the actual voltage is higher than the value of the reference signal, the error has the opposite sign, thus

FORWARD PATH:

The physica l p rocessto be automatica lly co ntro lled

FEED BACK PATH:Definition o f contro l e fforts

Actua l status o f the p rocess

Des ired status of the proc ess

ERRO R

+

_

CONTROLEFFORT

Disturbance FORWARD PATH:The physica l p rocess

to be automatica lly co ntro lled

FORWARD PATH:The physica l p rocess

to be automatica lly co ntro lled

FEED BACK PATH:Definition o f contro l e fforts

FEED BACK PATH:Definition o f contro l e fforts

Actua l status o f the p rocess

Des ired status of the proc ess

ERRO R

+

_

CONTROLEFFORT

Disturbance

Figure 1 - Closed-loop control system.

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resulting in the reduction of the EMF and the controlled voltage. Successful selection of the function

implemented in the control module (14), assures that the change in the field current promptly

balances the effect of changing loads and other disturbing factors. It could be seen that improper

selection of this function would result in the failure of the described system.

As shown in Fig. 2, the velocity of the turbine-generator assembly is being transformed into a

voltage signal V2 by a tachogenerator (15) and compared with the velocity reference signal R2. The

resultant error E2 represents the difference between the actual and desired velocities, and through

control module (16), power amplifier (8), servomotor (9), and gearbox (10) is used to increase or

decrease the opening of the valve (5) in the steam line, thus affecting the steam flow and the velocity

of the turbine. It should be emphasized that such a system must be well balanced, i.e., provide "as

much control effort as necessary" to maintain the actual velocity of the generator (and therefore the

frequency of the generated voltage) equal to the required one in spite of various external effects

(such as variation of the load of the generator or fluctuations in the operation of the steam generator).

While the above example presents the principle of operation of particular control systems, the

following general definitions are needed to discuss a generic control system.

To control - means to maintain a particular operation, status, or performance of a physical

process.

Controlled plant or controlled process - is the physical process, i.e., the combination of physical

transformations, which must be maintained according to a precisely defined operational regime.

LR

C

_

_

+

+

1

23

4 511

6

7

8

9

10

V1

12

R1E1

1314

16E2 R2

V2

LR

C

_

_

+

+

1

23

4 511

6

7

8

9

10

V1

12

R1E1

1314

16E2 R2

V2

Figure 2 - Control systems of an industrial power generator

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Controlled variable represents quantitatively the actual operation, status, or performance of the

controlled process.

Control system is a combination of components performing control functions. A control system

typically forms a closed-loop circuit with the controlled process in the forward path.

Actuation signal symbolizes the control efforts applied to the controlled plant in order to provide

the desired effects on its status or performance.

A controlled plant can be viewed as a system that has an actuation signal(s) in the input and the

controlled variable(s) in the output.

Transducer (sensor) is a technical device that transforms a controlled variable into an electrical

signal thus providing the quantitative characterization of the actual operation, status or performance

of the controlled process.

Reference is the signal that represents the desired operation, status, or performance of a

controlled process. The controlled variables (referred to above) are represented by particular low

power electric signals following some scale. The reference signals have the same order of

magnitude and power as the signals representing controlled variables, but are defined by the human

operators of the process.

Disturbance signals represent all external (and sometimes internal) factors that result in the

undesirable deviations of controlled variables from their required values.

Error signal is the difference between the actual and desired values of controlled variable, or

between the reference and transducer signals.

Controller is an analog or digital device that defines the control efforts transforming the error

signal into the control signal, in accordance with the control law. The following formula presents an

example of a control law:

( ) ( ) ( ) ( )U t K e t K e t dt Kde t

dt= + +∫1 2 3

where e(t) and U(t) are the error and control signals. Appropriate selection of a control law, both its

configuration and parameters (K1, K2, K3), is crucial for the operation of the entire system, and

constitutes one of the central issues in control engineering.

Servomechanism is an electric, hydraulic, or pneumatic device that performs power

amplification of the control signal, generating a control effort.

Actuator is the device, driven by the servomechanism, which directly affects the controlled

process by applying the actuation (controlled input) signal.

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The schematics of Fig. 2, exhibit two control systems: the velocity (frequency) control and the

voltage control. The controlled plant of the first system is the transformation of the energy of

compressed steam into the energy of rotating turbine-generator assembly. The controlled variables

of the system are, obviously, the velocity of the turbine and the line voltage. The opening of the

valve in the steam line and the field current are the actuation signals. The tachogenerator (a small dc

generator) serves as the transducer, representing the velocity by a proportional low power dc signal.

Another transducer is the voltmeter in the power line. The reference signals are special voltages

defined by the human operator via a potentiometers; it is expected that when the velocity of the

turbine is exactly equal to its required value, the reference signal is equal to the signal of the

transducer. Similarly, when the line voltage is exactly equal to the required value, the small voltages

signal from the voltmeter is equal to the reference signal. The difference between the feedback and

reference signals constitutes the velocity and the voltage errors. The controllers of this system can

be implemented as an analog or a digital computer programmed in accordance with the particular

control laws. The control law must be selected based on known (or assumed) differential equations

of all system components.

Negative feedback mechanisms can be easily detected in many biological, economical and

physical systems capable of maintaining equilibrium.

Feedback control in market economy is the mechanism behind the "supply/demand" formula.

A manufacturing process could be viewed as the controlled plant. The capital investments

constitute the actuation signal. The output is represented by the amount of the product, i.e., the

supply. The role of the reference signal is played by the demand. The discrepancy between the

supply and demand, the system error, plays the major role in the price definition. Sales generate the

profit that in accordance with some control law is transformed into capital investments in

manufacturing. This closed-loop mechanism provides the only mechanism capable of maintaining

the balance between the supply and demand, providing that the man-made control law is properly

defined. (All known cases of destruction of this mechanism in so-called "socialist" systems have

resulted in the complete deterioration of economies.)

Feedback control in a biological system is a harmony mechanism in nature. It could be said

that the deer population within a particular geographic region is regulated by a biological control

system. The death and birth processes in such a system constitute the controlled process. The size

of the population is, obviously, the controlled variable. This variable can be defined in terms of the

required amount of vegetation to be consumed. The available amount of vegetation, which depends

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on the soil productivity and weather conditions, serves as the reference signal. The system error,

defined as the difference between the available and required amount of vegetation, is one of the

major factors (control efforts) affecting the death and birth processes. The structure described

represents one of the mechanisms responsible for balance in nature. Any attempt to disconnect the

closed chain of processes may result in undesirable effects.

ROLE OF FEEDBACK

The examples clearly point at one very important property of the discussed control systems: the

control effort is being defined on the basis of the error signal, not on the basis of the phenomena

responsible for the occurrence of the error. This is a typical feature of all feedback systems, a

feedback system corrects the error without asking "why this error has occurred".

Assume, for example, that an automatic system to maintain the indoor temperature is being

developed. The indoor temperature is affected by the operation of the furnace (the control effort)

and by such disturbances as outdoor temperature, efficiency of the heater, by the doors and windows

that may or may not be completely closed, etc. The feedback approach implies that the indoor

temperature, represented by a proportional voltage signal, is being monitored. This voltage,

subtracted from the reference voltage, representing the desired indoor temperature, constitutes the

error signal. The error is converted into a control signal, and finally, in the fuel flow of a furnace.

Any deviation of the actual indoor temperature from the required one will be detected and eliminated

through the feedback mechanism, regardless what caused this deviation, fluctuation of the outdoor

temperature, opened window, or fluctuation of the efficiency of the furnace.

But there is an alternate approach, known as the feedforward control. It implies that the factors

leading to the occurrence of system errors are being monitored, and a control effort is being

generated and applied to the controlled process to compensate for the expected error. In the case of

a temperature control system, it is possible to monitor the outdoor temperature as the major factor

responsible for the fluctuations of the indoor temperature. Since it is known that the drop of the

outdoor temperature will eventually result in the drop of the indoor temperature, the fuel flow could

be increased immediately as the drop of the outdoor temperature has been detected. It is quite

important that the feedforward approach theoretically can completely eliminate the error: the drop of

the outdoor temperature can be compensated by the increase of the fuel flow before it will result in

the error! However, the described feedforward temperature control system would definitely fail to

offset the effects of opened windows, changing efficiency of the furnace, or any factor, other than

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the outdoor temperature.

The following is the list of advantages and drawbacks of both control techniques.

FEEDBACK:

Advantage: a general-purpose error correction mechanism reacts to the error itself, not to the

factors causing it

Drawback: the error correction mechanism is activated by occurring errors, which results in the

increased response time.

Note that the closed-loop of mechanical/electrical/information transformations is formed only in

a feedback control system.

FEEDFORWARD:

Advantage: it reacts to particular disturbances before they cause system errors, which results in

a very short response time,

Drawback: it "protects" the controlled plant only from specific disturbances, acting as a special-

purpose error correction mechanism.

Note that the sequence of mechanical/electrical/information transformations taking place on a

feedforward control system does not form a closed loop. Feedforward control is also known as open-

loop control.

It could be concluded, therefore, that feedback normally presents the most practical technique

for the development of a control system. However, in a situation when the controlled plant is

affected by a small number of dominant disturbances, which can be properly monitored, the

feedback could be supplemented by a feedforward mechanism, responsible for dominant

disturbances.

Generally speaking, the feedback mechanism presents an ideal tool for a control system

designer. It can be shown that it allows for a complete modification of the dynamic properties of a

controlled plant, and for the reduction of system sensitivity to external signals and “internal” system

parameters. For example, a small change in the computer code implementing the control law in

modules (14), (16) of the control system of Fig. 2 can result in the same effects on the system

properties as very expensive modifications of the power generator. Effects of the road conditions on

the speed of an automobile could be virtually eliminated by the activation of the cruise control

system, which implies velocity feedback. Effect of varying mass on the flight dynamics of a guided

missile, as fuel is being spent, is practically negligible only due to the special feedback mechanisms.

At the same time, introduction of feedback may result in system instability, i.e., development

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of a potentially self-destructive system. Poor knowledge of control principles can also lead to an

inefficient control system, which requires unnecessarily high control efforts.

It is important to realize that feedback control mechanisms are present in both automatic and

manually controlled systems. In a manually controlled system the human operator unavoidably

becomes the part of the control loop. He/she is responsible for monitoring the controlled variable

(for example speed of the vehicle using a speedometer), comparing the desired value of the

controlled variable with the actual value, detecting the discrepancy (error), implementing the control

law (that reflects mood, emotions, fatigue, etc.), and finally, generating the control effort (by

applying pressure at the brake or gas pedals). In an automatic system, the human operator “stays out

of the loop” and is responsible only for specifying the required value of the controlled variable,

while the feedback mechanism detects the position error and implements some control law to

generate the control effort.

There are two major types of problems in control engineering; the first is the assessment of

system properties and behavior, and the second deals with selection of appropriate feedback

mechanisms:

Analysis - the system configuration and all system components are known. It is required to

evaluate system behavior and performance characteristics.

Design - major system components, specifically the controlled plant, are known. The system

behavior and performance characteristics are specified. It is required to select the system

configuration and the control law to assure the desired system performance.

Both the analysis and the design problems can be solved only on the basis of the mathematical

description of the system, which consists of the mathematical descriptions of particular system

components and the overall system configuration.

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ASSIGNMENTS (HOMEWORK #1)

1.1. Identify major control system components of the voltage control system shown in Fig. 2.

1.2. Develop a schematic of the temperature control system, described above, that combines

feedback and feedforward mechanisms. Identify major control system components and their

functions.

1.3. Water level in a reservoir of a chemical plant can increase due to rains, and decrease due to

consumption and evaporation. It is controlled by pumping water from the lake or to the lake

using a reversible pump. Suggest a control procedure for maintaining the required water level

a) using the feedback principle

b) using the feedforward principle

Provide schematics and explain advantages and drawbacks of both systems.