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1 Introduction: Automatic Control Systems
Recent trends in the development of modern civilization have been in the direction of greater control. With the advent of the steam engine, and the material improvements brought about by the industrial revolution, man has had available greater quantities of power for his use. To use this power effectively, he has been required to learn how to control and how to regulate it. As part of the control process, certain standards have to be established. The performance of the equipment is compared to these standards, and, according to the difference, appropriate action is taken to bring about a closer correspondence between the desired objectives and the actual performance. The need for good control is present in many phases of our existence. We will limit ourselves in this class to a study of problems pertaining to the field of engineering applications of automatic control. The problem is to determine the desired objectives, and the best ways of producing those objectives. open-loop system
Figure 1 is a block diagram showing an open- loop control system. The input is sometimes called the reference, while the output is sometimes called the controlled variable. Disturbances can be present in the system.
Reference input, R
Controlled variable, C G
Disturbance input, D Figure 1. Open-Loop Control System.
Elements of a Simple Feedback Control System feedback control system
A feedback control system is a mechanism (or a set of systems and subsystems) that detects a discrepancy, and corrects it. Figure 2 is a block-diagram of a simplified feedback control system. The difference between the reference input and some function of the controlled variable is used to supply an actuating error signal to the control elements and the controlled system. The actuating error signal endeavors to reduce to zero the difference between the reference input and the controlled variable.
Reference Actuating input + error -
Control elements
Feedback, function of control variable
Controlled system
Controlled variable
Feedback element
Figure 2. Elements of a Simplified Feedback System.
disturbance function
In addition to the principal variables shown above, we may have a disturbance function. The disturbance function represents an unwanted inp ut to the system that causes the controller variable to differ from the reference input.
Disturbance elements Reference Actuating input + error -
Control elements
Feedback, function of control variable
Controlled system
Controlled variable
Feedback element
Figure 3. Elements of a Simplified Feedback System with disturbance.
Feedback versus Open-Loop Some advantages of feedback control over control without feedback are that lower tolerances and greater time delays can be permitted for the control elements. Also, feedback control systems lower the system’s sensitivity to disturbances and provide the ability to correct for these disturbances.
Examples of Control Systems For example, the basic elements of a ship’s steering control system are shown in figure 4. The desired angular heading for the ship is used to provide the reference input signal. The actual angular heading of the ship is the controlled variable. A signal proportional to the actual ship’s heading provides the feedback that is compared to the reference input signal. The error signal, which is proportional to the angular difference between the desired and the actual ship heading, is used to actuate the steering control motor. The steering motor, hydraulic or electric, positions the ship’s rudder that causes the ship to turn to the desired heading.
actual ship heading desired ship heading position error signal steering control motor ship rudder
Figure 4. Ship Steering System. Figure 5 shows a gun-positioning control system in which an electrical signal proportional to the desired gun position is the reference input. The controlled variable is a position proportional to the actual gun position. In this sort of system the control endeavors to maintain the control variable position equal at all times to the desired input position. Hydraulic or electric motors generally are the controlled system used to provide the main power for moving the gun carriage.
Figure 5. Gun Positioning System. Figure 6 illustrates a mill motor speed control system in which it is desired to operate the second drive motor at the same speed as the main drive motor. Frequently, a DC voltage proportional to the main DC motor speed is the reference input whereas the speed of the second DC motor is the controlled variable output. A DC tachometer used to indicate the output speed provides the feedback signal.
Figure 6. Mill Motor Speed Control.
Figure 7 shows a simple form of temperature control for use as part of a process control system. The reference input is a signal proportional to the desired process temperature. The actual temperature of the processed material is an indirectly controlled quantity. A signal proportional to the process temperature is fed back and compared with the signal proportional to the desired temperature. The difference signal operates the regulator that positions the throttle valve that controls the amount of steam flow to heat the material controlled in the process. The addition or withdrawal of material from the process can be considered to be a disturbing function that acts to alter the value of the temperature being controlled. Pneumatic control of the throttling valve is frequently employed.
Figure 7. Temperature Regulator on Process Control.
These examples are merely illustrative and are chosen to indicate a few of the many different types of systems for which control is used. They also indicate the high degree of similarity in form among these seemingly different controls.
General Comments on Feedback Control Systems a. The objective of a feedback control system is for the output to track the reference. This is equivalent to saying that the steady-state error must equal zero (after oscillations, output – input = 0). b. Assume systems are linear. We can then represent each block with a transfer function, T(s). c. e(t ) = r(t) – c(t) In a stable system (that converges), e(t) → 0. d. We must find a suitable transfer function for each block. This is called modeling, and can be really difficult. e. We can describe the system in the time domain via differential equations or state variables.
Feedback Control System Design and Analysis In order for the practicing engineer to arrive at a feedback control system design that best meets the requirements of a particular application, it is desirable that a gene ral design procedure be available. The design should be reliable in performance, economical in cost and operation, capable of ready manufacture, light, durable, and easily serviced. a) b) c) d) e)
Understand the problem requirements. Model the system. Find the overall transfer function. Evaluate the performance of your system (e.g. look at step response). If the performance is not suitable, then change a system parameter (e.g. amplifier gain) or add another subsystem. f) Check the design experimentally if possible.
Stability versus Performance The basic principle of feedback control (or closed- loop operation) tends to make for accurate control, as the control system endeavors continually to correct any error that exists. However, this corrective action can give rise to a dangerous condition of unstable operation when used with control elements having a large amount of amplification and significant delays in their time response.
An unstable control system is one that is no longer effective in maintaining the controlled variable very nearly equal to the desired value. Instead, large oscillations or erratic behavior of the controlled variable may take place, rendering the control useless. The requirements of stability and accuracy are mutually incompatible.
The Mathematical Basis of Stability The principal means for determining the stability of linear control systems are: 1. Locating, by analytical or graphical means, the actual position in the complex plane of each of the roots of the characteristic equation of the system. 2. Applying the Routh-Hurwitz stability criterion to the coefficients of the system’s characteristic equation. 3. Applying Nyquist’s criterion to a graphical plot of the open- loop response of the system as a function of frequency for a sinusoidal driving function. The labor involved in locating the exact position of the roots of the characteristic equation or in calculating their values is such as to limit the use of this method. The Routh criterion involves the use of a brief, simple algebraic process and permits the ready determination of the system stability. However, the graphical data necessary for applying the Nyquist criterion provide quantitative information on the degree of accuracy of the system, the degree of system stability, as well as the system stability itself. Hence, it is the Nyquist criterion in one or more of its modified forms that is used most extensively to determine system stability.
Classes of Systems Linear Systems vs. Nonlinear Systems linear
A system is linear if it satisfies the superposition principle and the homogeneity property (a linear combination of inputs gives the same linear combination of outputs). For example, if for a given system, the input x(t) corresponds to the output y(t), x1 (t) corresponds to y1 (t), and x2 (t) corresponds to y2 (t), the system is linear if: x1 (t) + x2 (t) corresponds to y1 (t) + y2 (t) k.x(t) corresponds to k.y(t)
(superposition) (homogeneity)
An affine system is a system that is “almost” linear – for example, y = a.u + b, which does not satisfy the superposition because of the constant factor b. nonlinear
A nonlinear system is any system that does not satisfy both the superposition principle and the property of homogeneity.
Dynamic vs. Static Systems static
dynamic
A static system is a system for which the output depends only on the present input (no memory). Algebraic relations can be used to describe a static system. A dynamic system is a system for which the output depends on past inputs as well as present inputs. Dynamic systems are usually described by differential and/or difference equations.
Recent trends in the development of modern civilization have been in the direction of greater control. With the advent of the steam engine, and the material improvements brought about by the industrial revolution, man has had available greater quantities of power for his use. To use this power effectively, he has been required to learn how to control and how to regulate it. As part of the control process, certain standards have to be established. The performance of the equipment is compared to these standards, and, according to the difference, appropriate action is taken to bring about a closer correspondence between the desired objectives and the actual performance. The need for good control is present in many phases of our existence. We will limit ourselves in this class to a study of problems pertaining to the field of engineering applications of automatic control. The problem is to determine the desired objectives, and the best ways of producing those objectives. open-loop system
Figure 1 is a block diagram showing an open- loop control system. The input is sometimes called the reference, while the output is sometimes called the controlled variable. Disturbances can be present in the system.
Reference input, R
Controlled variable, C G
Disturbance input, D Figure 1. Open-Loop Control System.
Elements of a Simple Feedback Control System feedback control system
A feedback control system is a mechanism (or a set of systems and subsystems) that detects a discrepancy, and corrects it. Figure 2 is a block-diagram of a simplified feedback control system. The difference between the reference input and some function of the controlled variable is used to supply an actuating error signal to the control elements and the controlled system. The actuating error signal endeavors to reduce to zero the difference between the reference input and the controlled variable.
Reference Actuating input + error -
Control elements
Feedback, function of control variable
Controlled system
Controlled variable
Feedback element
Figure 2. Elements of a Simplified Feedback System.
disturbance function
In addition to the principal variables shown above, we may have a disturbance function. The disturbance function represents an unwanted inp ut to the system that causes the controller variable to differ from the reference input.
Disturbance elements Reference Actuating input + error -
Control elements
Feedback, function of control variable
Controlled system
Controlled variable
Feedback element
Figure 3. Elements of a Simplified Feedback System with disturbance.
Feedback versus Open-Loop Some advantages of feedback control over control without feedback are that lower tolerances and greater time delays can be permitted for the control elements. Also, feedback control systems lower the system’s sensitivity to disturbances and provide the ability to correct for these disturbances.
Examples of Control Systems For example, the basic elements of a ship’s steering control system are shown in figure 4. The desired angular heading for the ship is used to provide the reference input signal. The actual angular heading of the ship is the controlled variable. A signal proportional to the actual ship’s heading provides the feedback that is compared to the reference input signal. The error signal, which is proportional to the angular difference between the desired and the actual ship heading, is used to actuate the steering control motor. The steering motor, hydraulic or electric, positions the ship’s rudder that causes the ship to turn to the desired heading.
actual ship heading desired ship heading position error signal steering control motor ship rudder
Figure 4. Ship Steering System. Figure 5 shows a gun-positioning control system in which an electrical signal proportional to the desired gun position is the reference input. The controlled variable is a position proportional to the actual gun position. In this sort of system the control endeavors to maintain the control variable position equal at all times to the desired input position. Hydraulic or electric motors generally are the controlled system used to provide the main power for moving the gun carriage.
Figure 5. Gun Positioning System. Figure 6 illustrates a mill motor speed control system in which it is desired to operate the second drive motor at the same speed as the main drive motor. Frequently, a DC voltage proportional to the main DC motor speed is the reference input whereas the speed of the second DC motor is the controlled variable output. A DC tachometer used to indicate the output speed provides the feedback signal.
Figure 6. Mill Motor Speed Control.
Figure 7 shows a simple form of temperature control for use as part of a process control system. The reference input is a signal proportional to the desired process temperature. The actual temperature of the processed material is an indirectly controlled quantity. A signal proportional to the process temperature is fed back and compared with the signal proportional to the desired temperature. The difference signal operates the regulator that positions the throttle valve that controls the amount of steam flow to heat the material controlled in the process. The addition or withdrawal of material from the process can be considered to be a disturbing function that acts to alter the value of the temperature being controlled. Pneumatic control of the throttling valve is frequently employed.
Figure 7. Temperature Regulator on Process Control.
These examples are merely illustrative and are chosen to indicate a few of the many different types of systems for which control is used. They also indicate the high degree of similarity in form among these seemingly different controls.
General Comments on Feedback Control Systems a. The objective of a feedback control system is for the output to track the reference. This is equivalent to saying that the steady-state error must equal zero (after oscillations, output – input = 0). b. Assume systems are linear. We can then represent each block with a transfer function, T(s). c. e(t ) = r(t) – c(t) In a stable system (that converges), e(t) → 0. d. We must find a suitable transfer function for each block. This is called modeling, and can be really difficult. e. We can describe the system in the time domain via differential equations or state variables.
Feedback Control System Design and Analysis In order for the practicing engineer to arrive at a feedback control system design that best meets the requirements of a particular application, it is desirable that a gene ral design procedure be available. The design should be reliable in performance, economical in cost and operation, capable of ready manufacture, light, durable, and easily serviced. a) b) c) d) e)
Understand the problem requirements. Model the system. Find the overall transfer function. Evaluate the performance of your system (e.g. look at step response). If the performance is not suitable, then change a system parameter (e.g. amplifier gain) or add another subsystem. f) Check the design experimentally if possible.
Stability versus Performance The basic principle of feedback control (or closed- loop operation) tends to make for accurate control, as the control system endeavors continually to correct any error that exists. However, this corrective action can give rise to a dangerous condition of unstable operation when used with control elements having a large amount of amplification and significant delays in their time response.
An unstable control system is one that is no longer effective in maintaining the controlled variable very nearly equal to the desired value. Instead, large oscillations or erratic behavior of the controlled variable may take place, rendering the control useless. The requirements of stability and accuracy are mutually incompatible.
The Mathematical Basis of Stability The principal means for determining the stability of linear control systems are: 1. Locating, by analytical or graphical means, the actual position in the complex plane of each of the roots of the characteristic equation of the system. 2. Applying the Routh-Hurwitz stability criterion to the coefficients of the system’s characteristic equation. 3. Applying Nyquist’s criterion to a graphical plot of the open- loop response of the system as a function of frequency for a sinusoidal driving function. The labor involved in locating the exact position of the roots of the characteristic equation or in calculating their values is such as to limit the use of this method. The Routh criterion involves the use of a brief, simple algebraic process and permits the ready determination of the system stability. However, the graphical data necessary for applying the Nyquist criterion provide quantitative information on the degree of accuracy of the system, the degree of system stability, as well as the system stability itself. Hence, it is the Nyquist criterion in one or more of its modified forms that is used most extensively to determine system stability.
Classes of Systems Linear Systems vs. Nonlinear Systems linear
A system is linear if it satisfies the superposition principle and the homogeneity property (a linear combination of inputs gives the same linear combination of outputs). For example, if for a given system, the input x(t) corresponds to the output y(t), x1 (t) corresponds to y1 (t), and x2 (t) corresponds to y2 (t), the system is linear if: x1 (t) + x2 (t) corresponds to y1 (t) + y2 (t) k.x(t) corresponds to k.y(t)
(superposition) (homogeneity)
An affine system is a system that is “almost” linear – for example, y = a.u + b, which does not satisfy the superposition because of the constant factor b. nonlinear
A nonlinear system is any system that does not satisfy both the superposition principle and the property of homogeneity.
Dynamic vs. Static Systems static
dynamic
A static system is a system for which the output depends only on the present input (no memory). Algebraic relations can be used to describe a static system. A dynamic system is a system for which the output depends on past inputs as well as present inputs. Dynamic systems are usually described by differential and/or difference equations.
