Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
Note 1
Introduction
1. Introduction
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada
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Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
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 in refineries, and so on.
A system is generally made up of components which could be mechanical, electrical,
hydraulic, and/or pneumatic. The function of the control system is to regulate the output
of the system in accordance with 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 is measured by using
transducers and, fed back to the controller in order to correct deviations from the input
signal. This is referred to as feedback and is an essential element of closed-loop control
systems. When feedback is not used, we have an open-loop control 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 a power element within the system such as an electrical motor,
hydraulic valve/ram) that acts upon the quantity to be controlled.
Modern control is increasingly based on having an electrical signal as input and having
transducers that produce electrical signals as a measure of the output. These are then fed
into the controller which acts upon the system being controlled. Computers are
increasingly used in control applications.
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
the manipulated variable to the system to correct or limit deviation of the measured value
from a desired value.
Plants:
A plant may be a piece of equipment, perhaps just a set of machine parts functioning
together, the purpose of which is to perform a particular operation. In this book, we shall
call any physical object to be controlled (such as a mechanical device, a heating furnace, a
chemical reactor, or a spacecraft) a plant.
Processes:
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada
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Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
Defined as natural, progressively continuing operation or development marked by a series
of gradual changes that succeed one another in a relatively fixed way and lead toward a
particular result or end; or an artifical or voluntary, progressively continuing operation
that consists of a series of controlled actions or movements systematically directed toward
a particular result or end. In this book we shall call any operation to be controlled a
process. Examples are chemical, economic, and biological processes.
Systems:
A system is a combination of components that act together and perform a certain
objective. A system is 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, be interpreted 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 a disturbance is generated within the system, it is called internal, while an external
disturbance is generated outside the system and is an input.
Feedback Control systems:
A system that maintains a prescribed relationship between the output and the reference
input by comparing 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 temperature and comparing it with the reference temperature
(desired temperature), the thermostat turns the heating or cooling equipment on or off in
such a way as to ensure that the room temperature remains at a comfortable level
regardless of outside conditions.
3. Example
An example of a closed-loop system is a temperature-control system in a room. For this
system we wish 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, we must measure this variable. We call the system for
the measurement of a variable a sensor. In this system, the sensor is a thermistor.
Thermistor is a device which has a resistance that varies with temperature. By measuring
this resistance, we obtain a measure of the temperature.
Plant
Desired
Temperature
Warm
Air
Heating
Room
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada
Sub-System
Compensator
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Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
We defined the plant of a control system as that part of the system to be controlled. It is
assumed in this 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 is the actual temperature of the living area. The plant
is represented as shown in the above figure. In this example, the output of each of the
systems is connected to the input of the other, to form the closed loop. In most closedloop control systems, it is necessary to connect a third system into the loop to obtain
satisfactory characteristics for the total system. This additional system is called a
compensator or 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 the Desired
Temperature. If we want to change 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 is called 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 is not zero, in
a properly designed system the error signal causes a response in the plant such that the
magnitude 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.
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Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
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 openloop control systems. In other words, in an open-loop control system the output is neither
measured nor fed back for comparison with the input. One practical example is a
washing machine. Soaking, washing, and rinsing 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 each reference 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 perform the desired task. Open-loop control can be
used, in practice, only if the relationship between the input and output is known and if
there are neither internal nor external disturbances. Clearly, such systems are not
feedback control systems. Note that any control system that operates on a time basis is
open loop. For instance, traffic control by means of signals operated on a time basis is
another example of open-loop control.
Closed-loop control systems:
Feedback control systems are often referred to as closed-loop control systems. In
practice, the terms feedback control and closed-loop control are used interchangeably. In
a closed-loop control system the 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/or integrals), 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 order to reduce
system error.
Closed-loop versus open-loop control systems
An advantage of the closedloop control system is the fact that the use of feedback makes
the system response relatively insensitive to external disturbances and internal variations
in system parameters. It is thus possible to use relatively inaccurate and inexpensive
components to obtain the accurate control of a given plant, whereas doing so is
impossible in the open-loop case. From the point of view of stability, 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 may
tend to overcorrect errors that can cause oscillations of constant or changing amplitude.
5. The Control Problems
We may state the control problem as follows. A physical system or process is to be
accurately controlled through closed-loop, or feedback, operation. An output variable,
called the response, is adjusted as required by the error signal. This error signal is the
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada
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Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
difference between the system response, as measured by a sensor, and the reference
signal, which represents the desired system response.
Generally a controller, or compensator, is required to filter the error signal in order that
certain control criteria, or specifications, be satisfied. These criteria may involve, but not
be limited to:
1.
2.
3.
4.
Disturbance rejection
Steady-state errors
Transient response characteristics
Sensitivity to parameter changes in the plant
Solving in control problem generally involves
Choosing sensors to measure the plant output
1.
Choosing actuators to drive the plant
2.
Developing the plant, actuator, and sensor equations (models)
3.
Designing the controller based on the developed models and the control
criteria
4.
Evaluating the design analytically, by simulation, and finally, by testing the
physical system
5.
If the physical tests are unsatisfactory, iterating these steps
The relationship of mathematical analysis and design to physical-system design
procedures is depicted below.
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