Automatic Control

 

 

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, 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

 

CONTENTS   CONTENTS Introduction  ……………………………………………………………………… Introduction ………………………………… …………………………………………. ……. 4 Principles of automatic control…………………………… control………………………………………………………… …………………………….. .. 5 Role of feedback …………………………………………………… ………………………………………………………………………... …………………... 10 Assignments (Homework #1) …………………………………………………… …………………………………………………………... ……... 13  EDUCATIONAL OBJECTIVE:  principles understanding the feedback and feedforward control contro l  

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)…………………………………… #2)…………………………………….… .… 36  EDUCATIONAL OBJECTIVE: ability to obtain a mathematical description of a dynamic system in the appropriate form  

….. 38 Chapter 2. Mathematical description of control systems ……………………… systems ………………………….. Typical dynamic blocks ....................................................……………………............… ................................................. ...……………………............… 38 Block diagrams ................................................................…………............………….… ................................... .............................…………............………….… 57 Signal-flow graphs .......................................................................………………………. ............................................... ........................………………………. 66 Assignments (Homework #4) ........................................................……………………... ................ ........................................……………………... 77 Example Test ........................ 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  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 ....................................................... ...................................................................………………………1 ............………………………10 03 Assignments (Homework #5) ................................................... ........................................................……………………...1 .....……………………...11 13  EDUCATIONAL OBJECTIVE: ability to utilize existing analytical and numerical techniques and software tools developed in control engineering 

 ........………………………115 Chapter 4. Analysis of continuous-time continuous-tim e control systems systems ........………………………115   2

 

Stability analysis ............................................................................……………………...1 ................................................. ...........................……………………...11 15 Relative stability ............................................................................……………………...1 ................................................ ............................……………………...12 29 Analysis of system statics ..............................................................…………………….. .............................................. ................……………………....138 Analysis of system dynamics ..........................................................……………………. .............................................. ............…………………….15 152 2 Assignments (Homework #6, Homework #7, and an d Homework #8) ..…………………....16 ..…………………....161 1 Example Test .............. Test ........................................................................... ....................................................................………………….... .......…………………....164 164  EDUCATIONAL OBJECTIVE: ability to assess properties of an existing control system   Chapter 5. Design of continuous-time control systems ...........…………………….. systems ...........……………………..166 166

Design considerations and problem definition ................................………...….………166 S-domain design ...........................................................................……………………...170 ...........................................................................……………………...170 S-domain design. Pole placement .................................................…………...………... .............................. ...................…………...………... 187 Frequency-domain design . . ……………………………………………………………211 Assignments (Homework #9 and Homework #10) ........................…………………….229 ...... ..................…………………….229  ................................................................................……………………. ........................................................……………………. 233 Example Test Test ........................  EDUCATIONAL OBJECTIVE: ability to design a control system compliant with design specifications 

Chapter 6. Introduction to digital control  control ......................…………………………... 234 Discrete-time representation of continuous signals ......... ........................…………………… ...............…………………… 235 Discrete-time domain description of dynamic dyn amic 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

TOTAL

100 points

  3

 

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 m machines, achines, devices and processes requires very short response time, large amount of complex, repetitious analytical analytic al and mechanical operations, and low tolerance to errors that are well beyond human human abilities. Automation became the only alternative for continuing the technical progress. While design of a particular automatic control co ntrol 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 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 matrix theory for control systems 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 engineering offers its servic services es and general methods to electrical, mechanical, mecha nical, chemical, chemica l, aerospace and power engineering, as well as m metallurgy, etallurgy, biology, material science, etc. Control theory provides a ffoundation oundation 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 Ztransforms, optimization optimization techniques and applied statistics. On this foundation methods and models of control are formulated. Computer application skills allow control engineers to utilize m modern odern software tools implementing control engineering techniques and a nd facilitating numerical simulation, analysis and design of control systems. systems.

Microprocessor background is required required for the

  4

 

implementation of control laws through a microprocessor and interfacing microprocessors with system components. Knowledge of circuits and electronics is crucial for understanding hardware implementation implementat ion 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.  automated.  PRINCIPLES OF AUTOMATIC CONTROL Automation implies development of a technical system capable of self-control. self-control. Any deviation from the required status in such a system must result in the generation of control con trol efforts,  partially or completely eliminating this deviation. It is important that the control efforts are generated without any participation of a human h uman operator, who is responsible only for the definition of the "required status". An automatically controlled system is expected to maintain its its actual status 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 mechanism operates according to the following principle: - an error , i.e., a discrepancy between b etween 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 ov erall 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 is typically responsib responsible le for the m major ajor physical transformation transformation that constitutes the process to to be controlled. The feedback path also performs physical ttransformations, ransformations, 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 v oltage and frequency control system of an ind industrial ustrial 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 line (4). The speed of the turbine (and the frequency of the generated voltage) is controlled by the valve in the   5

 

Disturbance CONTROL

FORWARD PATH: The physical process to be a utomatically co ntrolled  ntrolled 

Actual status of the process

EFFORT

 _  FEEDBACK PATH: Definition of control e fforts fforts

ERROR 

+

Desired status of the process

 

Figure 1 - Closed-loop control system.

steam line (5). The voltage of the generator is controlled 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 servo-motor and gearbox that allow for manipulation of the position of the valve in the ssteam team 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  decreas  decreases es. The voltage voltage signal V1 generated by the voltmeter (12) becomes lower than the reference signal R 1, 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. 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   6

 

4

5

11

2

3

12

1

L R

10

C

6  _  V1

9

8

7

16

14 E2

V2  _ 

E1

13

+

+

R 1

R 2

 

Figure 2 - Control systems of an industrial power generator resulting in the reduction of the EMF and the controlled voltage. Successful selection selection of the function implemented in the control module (14), assures that the change in the field current promptly  balances the effect of changing chang ing loads and other disturbing factors. It could be seen tha thatt 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 an d compared with the velocity reference signal R 2. 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 th thee 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 a ctual 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 exa mple presents the principle of operation of p particular articular control systems, the following general definitions are needed to discuss d iscuss a generic control system. 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.   7

 

Controlled variable represents quantitatively the actual operation, o peration, status, or performance of the controlled process. Control system is a combination of components performing 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 qua ntitative characterization of the actual operation, status o orr 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 ccontrol ontrol efforts transforming the error signal into the control signal, in accordance with the control law. The following formula formula presents an example of a control law:

( ) = K e(t ) + K

U t

1

2

∫

  de(t ) ( ) +K 

e t dt

3

dt 

 

where e(t) and U(t) are the error and control signals. Appropriate selection of a control law, both its configuration and parameters (K 1, K 2, K 3), is crucial for the operation of the entire system, and constitutes one of the central issues in control engineering. engine ering. 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.   8

 

The schematics of Fig. 2, exhibit two control systems: the velocity (frequency) control co ntrol and the voltage control. The controlled plant of the first system system is the transformation of the energy of compressed steam into the energy of rotating 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 the voltmeter in the power line. The reference signals are special 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 tto o the required value, the small voltages 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. 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. formula. Feedback control in market economy is economy is the mechanism behind the "supply/demand" formula. A manufacturing process process could be viewed as the the controlled plant. The capital investments investments constitute the actuation signal. The output is represented by the am amount ount 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 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 destruction of this mechanism mechanism in so-called "socialist" sys systems tems have resulted in the complete deterioration of economies.) harmony mechanism in nature. It could be said Feedback control in a biological system is system is a harmony that the deer population within a particular geographic region is regulated by a biological control system. The death and birth processes 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, w which hich depends   9

 

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 (control efforts) affecting the death and birth processes. The structure described described represents one of the mechanisms mechanisms responsible for balance in nature. Any attempt to disconnect disconnect the closed chain of processes may result in undesirable und esirable 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 no t on the basis of the phenomena responsible for the occurrence of the error. 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 proportional voltage signal, 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 ffurnace. urnace. Any deviation of the actual indoor temperature from the required one will be detected and eliminated 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. 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 expected error. 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 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 before it 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   10

 

the outdoor temperature. The following is the list of advantages and drawbacks of both control technique techniques. s. FEEDBACK: Advantage:: a general-purpose error correction mechanism reacts to the error itself, not to the Advantage factors causing it Drawback: the Drawback:  the error correction mechanism is activated by occurring errors, which results in the increased response time. Note that Note  that the closed-loop of mechanical/electrical/i mechanica l/electrical/information nformation transformations is formed formed only in a feedback control system. FEEDFORWARD: Advantage: it reacts to particular disturbances before they cause system errors, which results in Advantage: a very short response time, plan t only from specific disturbances, acting as a specialDrawback: it "protects" the controlled plant Drawback:  purpose error correction mechanism. Note that the sequence of mechanical/electrical/information transformations taking place on a Note that feedforward control system system does not form a closed loop. Feedforward control is also known as openloop control. loop  control. It could be concluded, therefore, that feedback normally presents the most practical technique for the development of a control system. system. However, in a situation when the controlled 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 for a complete modification of the dynamic properties of a controlled plant, and a nd for the reduction of system sensitivity to external signals and “internal” syst system em  parameters. For example, exa mple, 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 fl flight ight dynamics of a gui guided ded missile, as fuel is being spent, is practically negligible only due to the special feedback mechanisms mechanisms.. At the same time, introduction of feedback may result in system instability, i.e., development   11

 

of a potentially self-destructive system. 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. 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), (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 sys system, tem, the human operator 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 - the system configuration and all system components are known. It is required to evaluate system behavior and performance characteristics. components, specifically the controlled plant, are known. The system Design - major - major system components,  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.

  12

 

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 ssystem ystem 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 pumping water from the lake or to the lake using a reversible pump. Suggest a control procedure for m maintaining aintaining the required required water level a)  using the feedback principle  b)  using the feedforward principle Provide schematics and explain advantages and drawbacks of both systems.

  13