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Elements of Feedback Control
3.1 OBJECTIVES AND INTRODUCTION
Objectives
1. Know the definition of the following terms: input, output, feedback, error, open loop, and closed loop.
2. Understand the principle of closed-loop control.
3. Understand how the following processes are related to the closed-loop method of control: position feedback, rate feedback, and acceleration feedback.
4. Understand the principle of damping and its effect upon system operation.
5. Be able to explain the advantages of closed-loop control in a weapon system.
6. Be able to model simple first-order systems mathematically.
Introduction
The elements of feedback control theory may be applied to a wide range of physical systems. However, in engineering this definition is usually applied only to those systems whose major function is to dynamically or actively command, direct, or regulate themselves or other systems. We will further restrict our discussion to weapons control systems that encompass the series of measurements and computations, beginning with target detection and ending with target interception.
3.2 CONTROL SYSTEM TERMINOLOGY
To discuss control systems, we must first define several key terms.
- Input. Stimulus or excitation applied to a control system from an external source, usually in order to produce a specified response from the system.
- Output. The actual response obtained from the system.
- Feedback. That portion of the output of a system that is returned to modify the input and thus serve as a performance monitor for the system.
- Error. The difference between the input stimulus and the output response. Specifically, it is the difference between the input and the feedback.
A very simple example of a feedback control system is the thermostat. The input is the temperature that is initially set
into the device. Comparison is then made between the input and the temperature of the outside world. If the two are different, an error results and an output is produced that activates a heating or cooling device. The comparator within the thermostat continually samples the ambient temperature, i.e., the feedback, until the error is zero; the output then turns off the heating or cooling device. Figure 3-1 is a block diagram of a simple feedback control system.
Other examples are:
(1) Aircraft rudder control system
(2) Gun or missile director
(3) Missile guidance system
(4) Laser-guided projectiles
(5) Automatic pilot
3.3 CLOSED AND OPEN-LOOP SYSTEMS
Feedback control systems employed in weapons systems are classified as closed-loop control systems. A closed-loop system is one in which the control action is dependent on the output of the system. It can be seen from figure 3-1 and the previous description of the thermostat that these represent examples of closed-loop control systems. Open-loop systems are independent of the output.
3.3.1 Characteristics of Closed-Loop Systems
The basic elements of a feedback control system are shown in figure 3-1. The system measures the output and compares the measurement with the desired value of the output as prescribed by the input. It uses the error (i.e., the difference between the actual output and desired output) to change the actual output and to bring it into closer correspondence with the desired value.
Since arbitrary disturbances and unwanted fluctuations can occur at various points in the system, a feedback control system must be able to reject or filter out these fluctuations and perform its task with prescribed accuracies, while producing as faithful a representation of the desired output as feasible. This function of filtering and smoothing is achieved by various electrical and mechanical components, gyroscopic devices, accelerometers, etc., and by using different types of feedback. Posi- tion feedback is that type of feedback employed in a system in which the output is either a linear distance or an angular displacement, and a portion of the output is returned or fed back to the input. Position feedback is essential in weapons control systems and is used to make the output exactly follow the input. For example: if, in a missile-launcher control system, the position feedback were lost, the system response to an input signal to turn clockwise 10o would be a continuous turning in the clockwise direction, rather than a matchup of the launcher position with the input order.
Motion smoothing by means of feedback is accomplished by the use of rate and acceleration feedback. In the case of rate (velocity) feedback, a portion of the output displacement is differentiated and returned so as to restrict the velocity of the output. Acceleration feedback is accomplished by differentiating a portion of the output velocity, which when fed back serves as an additional restriction on the system output. The result of both rate and acceleration feedback is to aid the system in achieving changes in position without overshoot and oscillation.
The most important features that negative feedback imparts to a control system are:
(1) Increased accuracy--An increase in the system's ability to reproduce faithfully in the output that which is dictated by an input.
(2) Reduced sensitivity to disturbance--When fluctuations in the relationship of system output to input caused by changes within the system are reduced. The values of system components change constantly throughout their lifetime, but by using the self-cor-recting aspect of feedback, the effects of these changes can be minimized.
(3) Smoothing and filtering--When the undesired effects of noise and distortion within the system are reduced.
(4) Increased bandwidth--When the bandwidth of any system is defined as that range of frequencies or changes to the input to which the system will respond satisfactorily.
3.3.2 Block Diagrams
Because of the complexity of most control systems, a shorthand pictorial representation of the relationship between input and output was developed. This representation is commonly called the block diagram. Control systems are made up of various combinations of the following basic blocks.
Element. The simplest representation of system components. It is a labeled block whose transfer function (G) is the output divided by the input.
Summing Point. A device to add or subtract the value of two or more signals.
Splitting Point. A point where the entering variable is to be transmitted identically to two points in the diagram. It is sometimes referred to as a "take off point."
Control or Feed Forward Elements (G). Those components directly between the controlled output and the referenced input.
Reference variable or Input (r). An external signal applied to a control system to produce the desired output.
Feedback (b). A signal determined by the output, as modified by the feedback elements, used in comparison with the input signal.
Controlled Output (c). The variable (temperature, position, velocity, shaft angle, etc.) that the system seeks to guide or regulate.
Error Signal (e). The algebraic sum of the reference input and the feedback.
Feedback Elements (H). Those components required to establish the desired feedback signal by sensing the controlled output.
Figure 3-3 is a block diagram of a simple feedback control system using the components described above.
In the simplified approach taken, the blocks are filled with values representative of component values. The output (c) can be expressed as the product of the error (e) and the control element (G).
c = eG (3-1)
Error is also the combination of the input (r) and the feedback (b).
e = r - b (3-2)
But feedback is the product of the output and of the feedback element (H).
b = cH (3-3)
Hence, by substituting equation (3-3) into equation (3-2)
e = r - cH
and from equation (3-1)
e = c/G
c/G = r - cH (3-4)
c = Gr - cGH
c + cGH = Gr
c = G r
1 +GH (3-5)
It has then been shown that figure 3-3 can be reduced to an equivalent simplified block diagram,
G , shown below.
1 + GH
c = rG
1 + GH (3-6)
In contrast to the closed loop-system, an open-loop system does not monitor its own output, i.e., it contains no feedback loop. A simple open-loop system is strictly an input through a control element. In this case:
c = rG
The open-loop system does not have the ability to compensate for input fluctuations or control element degradation.
3.3.3 Motor Speed Control System
If the speed of a motor is to be controlled, one method is to use a tachometer that senses the speed of the motor, produces an output voltage proportional to motor speed, and then subtracts that output voltage from the input voltage. This system can be drawn in block diagram form as shown in figure 3-5. In this example
r = input voltage to the speed control system
G = motor characteristic of 1,000 rpm per volt of input
c = steady state motor speed in rpm
H = the tachometer characteristic of 1 volt per 250 rpm motor speed
Example. This example assumes that the input signal does not change over the response time of the system. Neglecting transient responses, the steady state motor speed can be determined as follows:
r = 10 volts
c = (e)(1000) rpm
e = c volts
1000
b = c volts
250
e = r - b
= 10 - c volts
250
Equating the two expressions for e and solving for c as in equation (3-4)
c = 10 - c volts
1000 250
c + 4c = 10,000 rpm
c = 2,000 rpm
Finally the error voltage may be found
e = c = 2 volts
1000
or by using the simplified equivalent form developed earlier as equation (3-6):
c = r G = 10V 1000 rpm/V = 2000 rpm
1 + GH 1 + 1000 rpm 1V
V 250 RPM
e = c = 2000 rpm = 2 volts
G 1000 rpm
V
3.4 RESPONSE IN FEEDBACK CONTROL SYSTEMS
In weaponry, feedback control systems are used for various purposes and must meet certain performance requirements. These requirements not only affect such things as speed of response and accuracy, but also the manner in which the system responds in carrying out its control function. All systems contain certain errors. The problem is to keep them within allowable limits.
3.4.1 Damping
Weapons system driving devices must be capable of developing suf-ficient torque and power to position a load in a minimum rise time. In a system, a motor and its connected load have sufficient inertia to drive the load past the point of the desired position as govern-ed by the input signal. This overshooting results in an opposite error signal reversing the direction of rotation of the motor and the load. The motor again attempts to correct the error and again overshoots the desired point, with each reversal requiring less correction until the system is in equilibrium with the input stimu-lus. The time required for the oscillations to die down to the desired level is often referred to as settling time. The magnitude of settling time is greatly affected by the degrees of viscous friction in the system (commonly referred to as damping). As the degree of viscous friction or damping increases, the tendency to overshoot is diminished, until finally no overshoot occurs. As damping is further increased, the settling time of the system begins to increase again.
Consider the system depicted in figure 3-6. A mass is attached to a rigid surface by means of a spring and a dashpot and is free to move left and right on a frictionless slide. A free body diagram of the forces is drawn in figure 3-7.
Newton's laws of motion state that any finite resultant of external forces applied to a body must result in the acceleration of that body, i.e.:
F = Ma
Therefore, the forces are added, with the frame of reference carefully noted to determine the proper signs, and are set equal to the product of mass and acceleration.
F(t) - Fspring - Fdashpot = Ma (3-7)
The force exerted by a spring is proportional to the difference between its rest length and its instantaneous length. The proportionality constant is called the spring constant and is usually designated by the letter K, with the units of Newtons per meter (N/m).
Fspring = Kx
The force exerted by the dashpot is referred to as damping and is proportional to the relative velocity of the two mechanical parts. The proportionality constant is referred to as the damping constant and is usually designated by the letter B, with the units of Newtons per meter per second N- sec.
m
Fdashpot = Bv
Noting that velocity is the first derivative of displacement with respect to time and that acceleration is the second derivative of displacement with respect to time, equation (3-7) becomes
F(t) - Kx - Bdx = Md2x dt dt2 (3-8)
rearranging
Md2x + Bdx + Kx = F(t)
dt2 dt
or
d2x + B dx + Kx = F(t) (3-9) dt2 M dt M M
This equation is called a second-order linear differential equation with constant coefficients.
Using the auxiliary equation method of solving a linear differential equation, the auxiliary equation
of (3-9) is:
s2 + Bs + K = 0
M M (3-10)
and has two roots
-B + B2 - 4K
M M M
s = (3-11) 2
and the general solution of equations (3-9) is of the form
x(t) = C1es1t + C2es2t (3-12)
where s1 and s2 are the roots determined in equation (3-10) and C1 and C2 are coefficients that can be determined by evaluating the initial conditions.
It is convenient to express B in terms of a damping coefficient as follows:
B = 2 MK
or
= B
2 MK
Then equation (3-10) can be written in the form:
0 = s2 + 2ns + n2 (3-13)
where
n = K and is the natural frequency of the system.
M
and
= B
M(2n)
= B
2 MK
For the particular value of B such that = 1 the system is critically damped. The roots of equation (3-10) are real and equal (s1 = s2), and the response of the system to a step input is of the form
x(t) = A(1 - C1te-s1t - C2e-s2t) (3-14)
The specific response is shown in figure 3-8a.
For large values of B (>1), the system is overdamped. The roots of equation (3-10) are real and unequal, and the response of the system to a step input is of the form
x(t) = A(1 - C1te-s1t - C2e-s2t)
Since one of the roots is larger than in the case of critical damp-ing, the response will take more time to reach its final value. An example of an overdamped system response is shown in figure 3-8b.
For small values of B such that <1 the system is underdamped. The roots of equation (3-10) are complex conjugates, and the general solution is of the form
x(t) = A[1 - e-t sin(t + )]
where is the imaginary part of the complex roots and is B , the real portion of s. 2M
The system oscillates at the frequency . For small values of , is very nearly the same as n in equation (3-13).
Examples of critically damped, overdamped, and underdamped system response are depicted in figures 3-8a-c respectively.
3.4.2 System Damping
The previous illustrations are characteristic of the types of motion found in most weapons tracking systems. In the case where the system is underdamped, in that the oscillations of overshoot are allowed to continue for a relatively long period of time, the system responds rapidly to an input order, but has relative difficulty in settling down to the desired position dictated by that input. Rapid initial response is a desirable characteristic in weapon control tracking systems if the system is to keep up with high-speed targets. However, the long settling time is an undesirable trait in a dynamic tracking system because during the settling time, the target will have moved, thus initiating a change in the system input prior to the system's responding adequately to the previous stimulus. It should be easy to extrapolate this condition over time to the point where the system can no longer follow the target and the track is lost.
Some of the more common methods of achieving damping are the employment of friction (viscous damping), feeding back electrical signals that are 180o out of phase with the input, or returning a DC voltage that is of opposite polarity to that of a driving voltage.
When the damping in a system becomes too great, the system will not overshoot, but its initial response time will become ex-cessive. This condition is known as overdamped. It is generally an undesirable condition in weapons systems because of the rela-tively slow initial response time associated with it.
When a system responds relatively quickly with no overshoot, the system is critically damped. In actual practice, systems are designed to be slightly underdamped, but approaching the critically damped condition. This accomplishes the functions of minimizing the system response time while at the same time minimizing over-shoot. It is called realistic damping. Figure 3-9 is a graphical representation of the relationship among the different conditions of damping.
Example. To illustrate the basic concepts of feedback control, consider the simple mechanical accelerometer that is employed in various configurations in missile guidance systems and inertial navigation systems. In this example, the accelerometer is employed in a guided-missile direction-control system. It is de-sired to hold the missile on a straight-line course. Lateral ac-celerations must be measured and compensation made by causing the steering-control surfaces to be actuated to produce a counter ac-celeration, thus assisting the overall guidance system in maintain-ing a steady course. This example depicts the system for left and right steering only; however, the up/down steering control is identical.
The accelerometer consists of a spring, a mass, and damping fluid all contained within a sealed case, with the entire assembly mounted in the missile.
The position, x, of the mass with respect to the case, and thus with respect to the potentiometer, is a function of the ac-celeration of the case. As the mass is moved by the results of lateral acceleration, motor drive voltage is picked off by the wiper arm of the potentiometer. The system is calibrated so that when no acceleration exists, the wiper arm is positioned in the center of the potentiometer and no drive voltage is fed to the motor.
As lateral accelerations of the missile occur, the mass is moved by the resulting force in a manner so as to pick off a voltage to steer the missile in a direction opposite to that of the input accelerations. As the missile is steered, an acceleration opposite to that initially encountered tends to move the mass in the opposite direction. The motion of the mass is opposed by the damping action of the fluid and the spring. To achieve a relatively rapid response and a minimum of overshoot, the choice of the viscosity of the fluid and the strength of the spring is critical.
3.5 SUMMARY
This chapter has presented a broad overview of the basic concepts of feedback control systems. These concepts are employed over a wide range of applications, including automatic target tracking systems, missile and torpedo homing systems, and gun and missile-launcher positioning systems.
A feedback control system consists of an input, an output, a controlled element, and feedback. System response should be as rapid as possible with minimum overshoot. To accomplish this, some means of damping is employed. If damping is weak, then the system is underdamped. This condition is characterized by rapid initial response with a long settling time. If damping is too strong or the system is overdamped, then the system response time is excessively long with no overshoot. The critically damped condition occurs when damping is weak enough to permit a relatively rapid initial response and yet strong enough to prevent overshoot. Control systems employed in modern weaponry are designed to be slightly underdamped but approaching the critically damped case. The result of this compromise in design is the achievement of rapid initial response with a minimum of overshoot and oscillation.
3.6 REFERENCES/BIBLIOGRAPHY
Bureau of Naval Personnel. Aviation Fire Control Technician 3 2. NAVPERS 10387-A. Washington, D.C.: GPO, 1971.
Commander, Naval Ordnance Systems Command. Weapons Systems Fundamentals. NAVORD OP 3000, vols. 2 & 3, 1st Rev. Washington, D.C.: GPO, 1971.
H.Q., U.S. Army Material Command: Servomechanisms. Pamphlet 706-136, Sec. I. Washington, D.C.: GPO, 1965.
Spiegel, Murry R. Applied Differential Equations. Englewood Cliffs, N.J.: Prentice-Hall Inc., 1967.
Weapons and Systems Engineering Dept. Weapons Systems Engineering. Vol. 2, Annapolis, Md.: USNA, 1983.
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