The purpose of the control electronics is to control the wing sweep angle, the pitch of
the stabilator, and the motion of the active control surfaces. However, since the
implementation of the active control surfaces is still in inchoate stages and the stabilator is
missing, only the control electronics as applied to the wing sweep angle is discussed in detail.
An overview of the previous analog control system, from 1978, is followed by design and
testing of the current analog system. AWT's progress on control electronics will conclude
with an overview of the implementation scheme of the digital control system. However, first
we will digress and review the necessary background information to fully understand the
control electronics.
the stabilator, and the motion of the active control surfaces. However, since the
implementation of the active control surfaces is still in inchoate stages and the stabilator is
missing, only the control electronics as applied to the wing sweep angle is discussed in detail.
An overview of the previous analog control system, from 1978, is followed by design and
testing of the current analog system. AWT's progress on control electronics will conclude
with an overview of the implementation scheme of the digital control system. However, first
we will digress and review the necessary background information to fully understand the
control electronics.
6.1 Node Voltage Analysis
Before analyzing the amplifying circuits of control electronics, it is necessary to take
an aside to refresh our knowledge of node voltage analysis. Node voltage analysis is
important to our project in understanding the amplifying circuit analysis in the following
sections. Node voltage analysis is based on Kirchhoff's current law, KCL. KCL may be
considered the conservation of mass applied to a circuit. KCL states that all the current
going into and out of a node must be accounted for. A typical node in a circuit is illustrated
below in figure 23
an aside to refresh our knowledge of node voltage analysis. Node voltage analysis is
important to our project in understanding the amplifying circuit analysis in the following
sections. Node voltage analysis is based on Kirchhoff's current law, KCL. KCL may be
considered the conservation of mass applied to a circuit. KCL states that all the current
going into and out of a node must be accounted for. A typical node in a circuit is illustrated
below in figure 23
Applying KCL to the node in figure 23 results in the following equation
[6.1.1]
Equation 6.1.1 means that the current going into a node is equal to the current coming out
of the node.
of the node.
Using node voltage analysis, the currents from the KCL equation are written in terms
of the voltages at the node. The simple circuit in figure 24 will be used to illustrate node
voltage analysis.
of the voltages at the node. The simple circuit in figure 24 will be used to illustrate node
voltage analysis.
Consider the three nodal points, a, b, and r. The voltage across the batteries on the left and
right are given by v1 and v2, respectively. The reference node is r; notice that the
reference node can encompass the entire bottom half of the circuit. The node voltage at
node a refers to the voltage difference between a and r:
right are given by v1 and v2, respectively. The reference node is r; notice that the
reference node can encompass the entire bottom half of the circuit. The node voltage at
node a refers to the voltage difference between a and r:
[6.1.2]
Likewise, the node voltage at node b refers to the voltage difference between b and r:
[6.1.3]
Typically the reference node, r, is grounded, i.e. voltage equals zero. Therefore, the node
voltage at node a and b would be the true voltage and not just the voltage change from the
reference.
voltage at node a and b would be the true voltage and not just the voltage change from the
reference.
As mentioned before, the purpose of the node voltage technique is to describe the
currents in terms of the node voltages. In figure 24the current from a to b, iab, is the
voltage at a minus the voltage at b divided by the resistance between a and b
currents in terms of the node voltages. In figure 24the current from a to b, iab, is the
voltage at a minus the voltage at b divided by the resistance between a and b
[6.1.4]
The statement above concerning the current, iab, is the main purpose of the node voltage
technique; it serves as a mnemonic device to express the current. However, the statement
only applies to nodes with resistors between them. The node voltage technique is favorable
in analyzing electronic circuits, such as those containing op amps, because many of the
components are always grounded. Thus each component can have the same reference
node.
technique; it serves as a mnemonic device to express the current. However, the statement
only applies to nodes with resistors between them. The node voltage technique is favorable
in analyzing electronic circuits, such as those containing op amps, because many of the
components are always grounded. Thus each component can have the same reference
node.
6.2 Operational Amplifier Analysis
An operational amplifier, or op amp for short, is the most important component in an
analog control system. Op amps may be used to do calculations such as integration,
summation, and even logarithmic calculations. For the purposes of this project, the op amp
is the 'brains' of the control system. The op amp compares the reference value with the
actual value and outputs the necessary value to control the process. This project relies
specifically on two specific op amp techniques: the summing amplifier and the inverting
amplifier.
analog control system. Op amps may be used to do calculations such as integration,
summation, and even logarithmic calculations. For the purposes of this project, the op amp
is the 'brains' of the control system. The op amp compares the reference value with the
actual value and outputs the necessary value to control the process. This project relies
specifically on two specific op amp techniques: the summing amplifier and the inverting
amplifier.
There exists a possible source of confusion when referring to an amplifying circuit and
an op amp. An op amp is a component of a summing amplifier or an inverting amplifier.
Consequently, referring the gain of the op amp differs from the gain of the entire amplifying
circuit; the same applies to the current, voltage, etc.
an op amp. An op amp is a component of a summing amplifier or an inverting amplifier.
Consequently, referring the gain of the op amp differs from the gain of the entire amplifying
circuit; the same applies to the current, voltage, etc.
6.3 Inverting Amplifier
The schematic of an inverting amplifier is shown below in figure 25The inverting
amplifier circuit is composed of an op amp; two resistors, Ri, and RF; and an input voltage
signal, v1. Analysis of the inverting amplifier is based on the assumptions that the op amp is
operating in its linear amplifying region and that the gain of the op amp is very high [17]. The
assumption that the op amp is operating in its 'linear amplifying region' is expresses by the
input-output characteristics as shown in figure 26
amplifier circuit is composed of an op amp; two resistors, Ri, and RF; and an input voltage
signal, v1. Analysis of the inverting amplifier is based on the assumptions that the op amp is
operating in its linear amplifying region and that the gain of the op amp is very high [17]. The
assumption that the op amp is operating in its 'linear amplifying region' is expresses by the
input-output characteristics as shown in figure 26
Figure 26 shows that the op amp is governed by the following equation
[6.3.1]
Equation 6.3.1 states that the output voltage, vout, of the op amp is equal to the difference
between the voltage at the input terminals, v+ - v-, times some gain, A. The assumption
that the gain of the amplifier is very high leads to the following equation:
between the voltage at the input terminals, v+ - v-, times some gain, A. The assumption
that the gain of the amplifier is very high leads to the following equation:
[6.3.2]
For instance, if v-out < 15 volt and A = 106, then v+ - v- < 15/106 = 15 mV. Therefore, the
voltages at the input terminals, v+ and v-, will be equal within 15 mV or less. Furthermore,
for the schematic in figure 25, the positive input terminal of the op amp is grounded;
consequently, v+ = 0 and v- 0. Thus, the current resulting from vin may be written using
the node voltage analysis technique from section 6.1
voltages at the input terminals, v+ and v-, will be equal within 15 mV or less. Furthermore,
for the schematic in figure 25, the positive input terminal of the op amp is grounded;
consequently, v+ = 0 and v- 0. Thus, the current resulting from vin may be written using
the node voltage analysis technique from section 6.1
[6.3.3]
The inner resistance, Rinner, between the input terminals of the op amp is illustrated
in figure 27. Typically, the resistance between the input terminals has a large value of
resistance; this results in negligible current into the positive and negative terminals of the
op amp
in figure 27. Typically, the resistance between the input terminals has a large value of
resistance; this results in negligible current into the positive and negative terminals of the
op amp
[6.3.4]
Negligible current into the negative input terminal of the op amp in figure 25 implies that the
current that flows through Ri is the same current that flows through RF. Therefore, using
iin, the output voltage of the inverting amplifier circuit may be calculated
current that flows through Ri is the same current that flows through RF. Therefore, using
iin, the output voltage of the inverting amplifier circuit may be calculated
[6.3.5]
Rearranging equation 6.3.5, the voltage gain of the entire inverting amplifier circuit may be
expressed as follows
expressed as follows
[6.3.6]
Equation 6.3.6 shows that an inverting amplifier may be used to change the polarity of an
incoming voltage signal; this property of the inverting amplifier's gain is useful in conjunction
with a summing amplifier.
incoming voltage signal; this property of the inverting amplifier's gain is useful in conjunction
with a summing amplifier.
6.4 Summing Amplifier
A summing amplifier is a modification of an inverting amplifier as shown in figure 28.
Notice that the summing amplifier is the inverting amplifier, from figure 25, plus an additional
input voltage signal. The summing amplifier circuit consists of three types of components:
three resistors, R1, R2, and RF; two input voltage signals, v1 and v2; and an op amp.
Notice that the summing amplifier is the inverting amplifier, from figure 25, plus an additional
input voltage signal. The summing amplifier circuit consists of three types of components:
three resistors, R1, R2, and RF; two input voltage signals, v1 and v2; and an op amp.
Using the same assumptions from the inverting amplifier section, the op amp gain is large
and is operating in its linear range; analysis of the summing amplifier is similar to the
inverting amplifier. The difference between them is the calculation of the current coming
from the two voltage inputs. The input current from the voltage source, iin, is calculated
using the node voltage technique
and is operating in its linear range; analysis of the summing amplifier is similar to the
inverting amplifier. The difference between them is the calculation of the current coming
from the two voltage inputs. The input current from the voltage source, iin, is calculated
using the node voltage technique
[6.4.1]
Recall from section 6.3, since the positive input terminal of op amp is grounded, v+ = 0; the
input voltage of the negative terminal of the op amp is approximately zero, v- 0.
input voltage of the negative terminal of the op amp is approximately zero, v- 0.
The output voltage of the summing amplifier, vout, draws the input current, iin, through RF;
the output voltage may be determined as follows
the output voltage may be determined as follows
[6.4.2]
Equation 6.4.2 shows the output voltage as the negative of the weighted sum of the input
voltages. The negative polarity of the output voltage of the summing amplifier is a problem
that may be addressed by adding an inverting amplifier. An example is shown in figure 29.
voltages. The negative polarity of the output voltage of the summing amplifier is a problem
that may be addressed by adding an inverting amplifier. An example is shown in figure 29.
For simplicity, the resistance values throughout the schematic in figure 29 are all equal. The
output voltage of the summing amplifier is calculated from equation 6.4.2
output voltage of the summing amplifier is calculated from equation 6.4.2
[6.4.3]
The output voltage of the summing amplifier is now the input voltage source of the inverting
amplifier. The final output voltage of the summing amplifier/inverting amplifier system is
determined by substituting the output voltage of the summing amplifier for the input voltage
of the inverting amplifier in equation 6.3.6
amplifier. The final output voltage of the summing amplifier/inverting amplifier system is
determined by substituting the output voltage of the summing amplifier for the input voltage
of the inverting amplifier in equation 6.3.6
[6.4.4]
6.5 Block Diagrams of Closed Loop Feedback Systems
A review of the block diagram representation of a control system is provided in this
section. A standard closed loop feedback system is presented in block diagram form as
shown in figure 30. The purpose of the control system is to control the output variable, Y.
Ideally, the output variable is supposed to mirror the input variable, R. Each block in the
diagram represents a major component of the control system. The chain of occurrences in
a feedback system begins with the input. The input is the reference value that output
should equal. The actual value of the output is obtained by the sensor and compared to the
input. The difference between the input value and the output value is the error. The error is
input into the controller, GC. The purpose of the controller is to reduce the error. The signal
from the controller is then input to the actuator, GA. An actuator is the device that
converts the signal from the controller to a mechanical force. The force applied by the
actuator changes the output according to some governing equations of motion inherent to
the process dynamics block, Gp. The output, Y, is again measured by the sensor and
compared to the input to determine the error. Hence the name closed loop feedback
system.
section. A standard closed loop feedback system is presented in block diagram form as
shown in figure 30. The purpose of the control system is to control the output variable, Y.
Ideally, the output variable is supposed to mirror the input variable, R. Each block in the
diagram represents a major component of the control system. The chain of occurrences in
a feedback system begins with the input. The input is the reference value that output
should equal. The actual value of the output is obtained by the sensor and compared to the
input. The difference between the input value and the output value is the error. The error is
input into the controller, GC. The purpose of the controller is to reduce the error. The signal
from the controller is then input to the actuator, GA. An actuator is the device that
converts the signal from the controller to a mechanical force. The force applied by the
actuator changes the output according to some governing equations of motion inherent to
the process dynamics block, Gp. The output, Y, is again measured by the sensor and
compared to the input to determine the error. Hence the name closed loop feedback
system.
In the block diagram representation, each block represents a mathematical model of
the equations of motion governing a particular part of the system. Typically, the equations
of motion are derived in the form of a differential equation expressed as a function of time.
However, in the block diagram representation, the equations of motion are transformed
from the time domain to the s-domain with a LaPlace transform. The purpose of the
transformation is that a differential equation in the time-domain may be reduced to a
polynomial in the s-domain. For instance, consider the equations of motion governing the
process dynamics block. Assume that the relationship between the signal from the actuator,
MV(s), and the output of the system, Y(s), may be expressed by the differential equation as
follows
the equations of motion governing a particular part of the system. Typically, the equations
of motion are derived in the form of a differential equation expressed as a function of time.
However, in the block diagram representation, the equations of motion are transformed
from the time domain to the s-domain with a LaPlace transform. The purpose of the
transformation is that a differential equation in the time-domain may be reduced to a
polynomial in the s-domain. For instance, consider the equations of motion governing the
process dynamics block. Assume that the relationship between the signal from the actuator,
MV(s), and the output of the system, Y(s), may be expressed by the differential equation as
follows
[6.5.1]
A, B, C, Q, and R are arbitrary coefficients. By convention, a variable expressed in lower
case is representative of the time domain; a variable expressed in upper case symbolizes
the s-domain. The LaPlace transformation of equation 6.5.1 is shown in the following steps.
A table of LaPlace transformations is given in Appendix B.
case is representative of the time domain; a variable expressed in upper case symbolizes
the s-domain. The LaPlace transformation of equation 6.5.1 is shown in the following steps.
A table of LaPlace transformations is given in Appendix B.
[6.5.2]
The transfer function may be considered the ratio of the output to the input of the block
in a block diagram. Assuming all initial conditions are zero, the transfer function is given by
the following equation
in a block diagram. Assuming all initial conditions are zero, the transfer function is given by
the following equation
[6.5.3]
A general representation of a transfer function is presented below
[6.5.4]
In general the transfer function, G, is a quotient of two polynomial functions, p(s) and q(s).
As mentioned before, the LaPlace transform allows the differential equation in the time-
domain to be represented as a polynomial in the s-domain. The polynomial expression of the
differential equation is written in the blocks of the block diagram. The output of the block is
always the transfer function of the block times whatever is input to the block
As mentioned before, the LaPlace transform allows the differential equation in the time-
domain to be represented as a polynomial in the s-domain. The polynomial expression of the
differential equation is written in the blocks of the block diagram. The output of the block is
always the transfer function of the block times whatever is input to the block
6.6 Bolding's Analog System
This section is provided as a review of the original analog circuitry used with the wing-
stabilator model. The previous analog control system is shown in figure 31. In this context,
analog means that information is encoded into an electrical signal that it is proportional to
the quantity being represented [17]. For example, in this particular project, measurements
of the wing sweep angle or the pitch of the stabilator are taken with Direct Current
Differential Transformers (DCDT). A DCDT is a transducer calibrated such that a measured
wing sweep angle or stabilator pitch angle outputs a corresponding voltage.
stabilator model. The previous analog control system is shown in figure 31. In this context,
analog means that information is encoded into an electrical signal that it is proportional to
the quantity being represented [17]. For example, in this particular project, measurements
of the wing sweep angle or the pitch of the stabilator are taken with Direct Current
Differential Transformers (DCDT). A DCDT is a transducer calibrated such that a measured
wing sweep angle or stabilator pitch angle outputs a corresponding voltage.
The top panel of the analog system, shown in figure 31 above, is the signal monitor; it
is used in monitoring the incoming accelerometer signals. The bottom element is the power
supply. The power supply's main function was to provide energy to move the stabilator;
however, when a change in the wing sweep was desired, power was diverted from the
stabilator to the wing. The remaining three elements are the control electronics. The control
electronics are composed entirely of electronics modules. The advantage of the modular
construction is that it allows the quick replacement of defective circuits and testing on a
modular basis. [1] The function of the control electronics belongs to two subsystems: The
Configuration Control System and the Control Law Implementation System.
is used in monitoring the incoming accelerometer signals. The bottom element is the power
supply. The power supply's main function was to provide energy to move the stabilator;
however, when a change in the wing sweep was desired, power was diverted from the
stabilator to the wing. The remaining three elements are the control electronics. The control
electronics are composed entirely of electronics modules. The advantage of the modular
construction is that it allows the quick replacement of defective circuits and testing on a
modular basis. [1] The function of the control electronics belongs to two subsystems: The
Configuration Control System and the Control Law Implementation System.
6.6.1 Configuration Control System
The function of the configuration control system is to control the wing sweep position. The
entirety of the configuration control system is composed of three major electronic
components: the position control module, the Servo-selector, and the Servoamplifier. The
components are shown in figure 32.
entirety of the configuration control system is composed of three major electronic
components: the position control module, the Servo-selector, and the Servoamplifier. The
components are shown in figure 32.
Figure 33 shows a schematic of the configuration control system.
・Notice the position control section in figure 33 above. The purpose of the position control
module is two fold:
module is two fold:
・Amplify the signal from the wing sweep position sensor to a powerful enough signal for the
wing sweep meter. The wing sweep meter is calibrated to physically display the current
angle of the wing sweep.
wing sweep meter. The wing sweep meter is calibrated to physically display the current
angle of the wing sweep.
・Provide a wing sweep command in the form of a variable voltage.
The function of the wing sweep meter amplifier is trivial but necessary. The signal coming
from the wing sweep sensor is too weak to drive the wing sweep meter; the amplifier adds
energy to the signal to drive the wing sweep meter. The wing sweep meter is shown below in
figure 34.
from the wing sweep sensor is too weak to drive the wing sweep meter; the amplifier adds
energy to the signal to drive the wing sweep meter. The wing sweep meter is shown below in
figure 34.
A power supply in conjunction with a series of resistors, as shown in figure 35,
provides a variable voltage to set off a series of reactions within the circuitry to drive the
wing sweep position. The output voltage of the circuit in figure 35 is calibrated to represent
a particular wing sweep angle. For example, an output voltage of 5 volts might represent the
fully swept forward wing on the display of the wing sweep meter, while an output voltage of -
5 volts would represent a fully aft wing on the wing sweep meter display.
provides a variable voltage to set off a series of reactions within the circuitry to drive the
wing sweep position. The output voltage of the circuit in figure 35 is calibrated to represent
a particular wing sweep angle. For example, an output voltage of 5 volts might represent the
fully swept forward wing on the display of the wing sweep meter, while an output voltage of -
5 volts would represent a fully aft wing on the wing sweep meter display.
Turning the knob on the position control module varies the voltage output which initiates a
chain of occurrences that lead to actually moving the wing; the next step in the chain of
occurrences begins with the servo-selector module.
chain of occurrences that lead to actually moving the wing; the next step in the chain of
occurrences begins with the servo-selector module.
In the servo-selector module, the variable voltage from the position control module
along with the feedback from the position sensor from the wing is input into two wing-
stabilator/norm-emergency switches. The switches, which are manually operated, are shown
on figure 36.
along with the feedback from the position sensor from the wing is input into two wing-
stabilator/norm-emergency switches. The switches, which are manually operated, are shown
on figure 36.
The purpose of the switches is to allow the operator to determine the "state" of operation.
For example, the wing-stabilator switch basically determines whether power is given to the
wing or the stabilator, but the norm-emergency switch is primarily for the wing. In "normal"
mode, depending on the position of the wing-stabilator switch, the wing sweep can be
controlled with the wing sweep control knob on the position control module or the stabilator
can be controlled with the control laws implemented in the servo-amplifier module, explained
in the next section. However, when in "emergency" mode, the wing is swept fully forward so
as to stave off the onset of flutter and preserve the model.
For example, the wing-stabilator switch basically determines whether power is given to the
wing or the stabilator, but the norm-emergency switch is primarily for the wing. In "normal"
mode, depending on the position of the wing-stabilator switch, the wing sweep can be
controlled with the wing sweep control knob on the position control module or the stabilator
can be controlled with the control laws implemented in the servo-amplifier module, explained
in the next section. However, when in "emergency" mode, the wing is swept fully forward so
as to stave off the onset of flutter and preserve the model.
In addition to the wing-stabilator/norm-emergency switches, the servo-selector
module is home for another power supply for the Wing Sweep DCDT, a solenoid valve, and a
voltage to current amplifier in the servoamplifier module. The solenoid value is the medium
by which power is given to either the wing or the stabilator. The wing-stabilator/norm-
emergency switches control whether or not power is given to the solenoid valve. The
solenoid valve in turn determines whether hydraulic power is given to the wing or the
stabilator. A DCDT amplifier also resides in the servo-selector module. Similar to the wing
sweep meter amplifier in the position control module, the DCDT amplifier buttresses the
feedback signal from the DCDT position sensor on the wing to a signal usable by the wing-
stabilator/norm-emergency circuitry. The next step in the configuration control system
is the servoamplifier module
module is home for another power supply for the Wing Sweep DCDT, a solenoid valve, and a
voltage to current amplifier in the servoamplifier module. The solenoid value is the medium
by which power is given to either the wing or the stabilator. The wing-stabilator/norm-
emergency switches control whether or not power is given to the solenoid valve. The
solenoid valve in turn determines whether hydraulic power is given to the wing or the
stabilator. A DCDT amplifier also resides in the servo-selector module. Similar to the wing
sweep meter amplifier in the position control module, the DCDT amplifier buttresses the
feedback signal from the DCDT position sensor on the wing to a signal usable by the wing-
stabilator/norm-emergency circuitry. The next step in the configuration control system
is the servoamplifier module
The servoamplifier module is similar to the summing amplifier/inverting amplifier schematic
shown in figure 29. The additional diodes, capacitors, and transistors serve as protections
devices in the circuit. A schematic of the servoamplifier is shown in figure 37 below.
shown in figure 29. The additional diodes, capacitors, and transistors serve as protections
devices in the circuit. A schematic of the servoamplifier is shown in figure 37 below.
Details involved in understanding the circuitry of the servoamplifier module are not
necessary because AWT found no use for the servoamplifier electronics. For simplicity, the
servoamplifier module of the Configuration Control System is considered a "black box"
proportional controller. A basic closed loop feedback block diagram control system is
presented in figure 38.
necessary because AWT found no use for the servoamplifier electronics. For simplicity, the
servoamplifier module of the Configuration Control System is considered a "black box"
proportional controller. A basic closed loop feedback block diagram control system is
presented in figure 38.
The servoamplifier module is highlighted in figure 38. Wing sweep reference angle information
and the actual wing sweep position information are both input to the Servoamplifier module.
The servoamplifier module is designed to interpret magnitude of the incoming voltage as
representative of the actual wing sweep angle. The error, the difference between the
desired reference value and the actual value, is then computed and input to the controller
section of the Servoamplifier module. The controller then outputs the necessary controls
actions, in the form of voltage, to the servo valve and actuators to move the wing. The
means by which the error is computed and the controller performs the necessary control
action is governed by the transfer functions of the "black box" circuitry of the
Servoamplifier module.
and the actual wing sweep position information are both input to the Servoamplifier module.
The servoamplifier module is designed to interpret magnitude of the incoming voltage as
representative of the actual wing sweep angle. The error, the difference between the
desired reference value and the actual value, is then computed and input to the controller
section of the Servoamplifier module. The controller then outputs the necessary controls
actions, in the form of voltage, to the servo valve and actuators to move the wing. The
means by which the error is computed and the controller performs the necessary control
action is governed by the transfer functions of the "black box" circuitry of the
Servoamplifier module.
6.6.2 Control Law Implementation System
The purpose of the Control Law Implementation System is to process the data from
the acceleration sensors on the stabilator and wing and perform the necessary control
actions to vary the pitch of the stabilator in order to suppress flutter. Control laws are
used to determine the control actions necessary to suppress flutter given position and
acceleration data from the sensors. A control law is a theoretical model of the process. For
example, if the aerodynamic forces change the pitch of the stabilator, the control law
attempts to predict the motion of the wing-stabilator system. Using the prediction of
motion, control actions can be taken to suppress undesired motions of the system.
the acceleration sensors on the stabilator and wing and perform the necessary control
actions to vary the pitch of the stabilator in order to suppress flutter. Control laws are
used to determine the control actions necessary to suppress flutter given position and
acceleration data from the sensors. A control law is a theoretical model of the process. For
example, if the aerodynamic forces change the pitch of the stabilator, the control law
attempts to predict the motion of the wing-stabilator system. Using the prediction of
motion, control actions can be taken to suppress undesired motions of the system.
Implementation of the Control Law System is the most academically challenging
aspect of this project. Bolding's work on control law implementation using analog control
systems is very difficult to understand. Before trying to follow the control law
implementation scheme of Bolding, AWT recommends that any future groups begin by
learning some of the basics in advanced control law theory.
aspect of this project. Bolding's work on control law implementation using analog control
systems is very difficult to understand. Before trying to follow the control law
implementation scheme of Bolding, AWT recommends that any future groups begin by
learning some of the basics in advanced control law theory.
6.7 Current Analog Control System
The analog control system presently in use is shown below in figures 38 and 40. The
Educational Servo model es151, shown in figure 39, is a pre-fabricated circuit made by
Feedback Co. An electrical circuit diagram of the entire circuit is given in Appendix A.
Educational Servo model es151, shown in figure 39, is a pre-fabricated circuit made by
Feedback Co. An electrical circuit diagram of the entire circuit is given in Appendix A.
The Educational Servo model ES 151b, shown in figure 40, is the motor designed for use in
conjunction with model ES 151. The circuitry of the Educational Servo may be made to
control either the position or rotational speed of the model.
conjunction with model ES 151. The circuitry of the Educational Servo may be made to
control either the position or rotational speed of the model.
The Educational Servo System, shown in figures 39 and 40, is presented in terms of a
block diagram in figure 41.
block diagram in figure 41.
A walkthrough of the diagram begins with the input angle. The circular dial on the left
of figure 39 is used to set a specific input angle. A potentiometer is calibrated to output a
particular voltage for a given angle. Notice that the block diagram in figure 41 is a positive
feedback loop instead of a negative feedback loop from figure Z. The positive feedback and
the block with the "-1" are used to represent the actions of the summing amplifier in the
circuit. The summing amplifier outputs the negative of the sum of the input voltage and the
output voltage. The transfer functions of the summing amplifier are derived in section 6.4.
The output of the summing amplifier is considered the error. The error is the difference
between the input angle and the displayed angle on the ES 151b. Notice that the error is
still the difference of the input and the output voltage. As mentioned before the summing
amplifier outputs the negative of the sum of two voltages. Feeding back the negative of the
sum implicitly produces a negative feedback.
of figure 39 is used to set a specific input angle. A potentiometer is calibrated to output a
particular voltage for a given angle. Notice that the block diagram in figure 41 is a positive
feedback loop instead of a negative feedback loop from figure Z. The positive feedback and
the block with the "-1" are used to represent the actions of the summing amplifier in the
circuit. The summing amplifier outputs the negative of the sum of the input voltage and the
output voltage. The transfer functions of the summing amplifier are derived in section 6.4.
The output of the summing amplifier is considered the error. The error is the difference
between the input angle and the displayed angle on the ES 151b. Notice that the error is
still the difference of the input and the output voltage. As mentioned before the summing
amplifier outputs the negative of the sum of two voltages. Feeding back the negative of the
sum implicitly produces a negative feedback.
The error is then input to the controller. The job of the controller is to output a
voltage signal that reduces the error. The implementation of four major types of analog
process controllers using analog circuitry is discussed in Appendix D. The four types of
controllers discussed are a proportional, proportional-derivative, proportional-integrating,
and a proportional-integrating-derivative. The name of the controller refers to the form of
its transfer function in the s-plane. Furthermore, in Appendix D, the differential equation of
the analog circuitry has been derived and converted to the s-plane to show that the circuit
provides the desired transfer function. The goal of Appendix D is to show the multiple types
of transfer functions available for Gc in figure 41.
voltage signal that reduces the error. The implementation of four major types of analog
process controllers using analog circuitry is discussed in Appendix D. The four types of
controllers discussed are a proportional, proportional-derivative, proportional-integrating,
and a proportional-integrating-derivative. The name of the controller refers to the form of
its transfer function in the s-plane. Furthermore, in Appendix D, the differential equation of
the analog circuitry has been derived and converted to the s-plane to show that the circuit
provides the desired transfer function. The goal of Appendix D is to show the multiple types
of transfer functions available for Gc in figure 41.
The process dynamic block is a model of everything that occurs beginning with an
input signal from the controller to the voltage signal fed back to the summing amplifier. In
general, obtaining a mathematical model of the process dynamics is the most time
consuming and costly component a controls project. Fortunately, one of the advisors to
AWT, Dr. Senent, hinted that the model of the Educational Servo from the power amplifying
stages to the motor out was a second order system. The process of finding the process of
the model and validating the model are discussed in sections 6.8 and 6.9.
input signal from the controller to the voltage signal fed back to the summing amplifier. In
general, obtaining a mathematical model of the process dynamics is the most time
consuming and costly component a controls project. Fortunately, one of the advisors to
AWT, Dr. Senent, hinted that the model of the Educational Servo from the power amplifying
stages to the motor out was a second order system. The process of finding the process of
the model and validating the model are discussed in sections 6.8 and 6.9.
6.8 Modeling the Process Dynamics
Dr. Senent suspected that the process dynamics block in figure 41 would be of the form as
follows.
follows.
[6.8.1]
To determine if the suspicion is correct, the response of the motor to a known input signal
must be observed; this lead to AWT's first hands-on experiment. The first step in
conducting the experiment was to isolate the transfer function block of the process
dynamics as shown in figure 42.
must be observed; this lead to AWT's first hands-on experiment. The first step in
conducting the experiment was to isolate the transfer function block of the process
dynamics as shown in figure 42.
Isolating the process dynamics block on the Educational servo is presented in figure 44.
On the face of the Educational Servo system exists female plugs labeled 'A' through 'R'. The
process dynamics of the position control was isolated by inputting a known voltage into the
female plug labeled 'T' and measuring the response at the plug labeled 'R'. A digital
oscilloscope was needed to take the measurement of the response of the output. To the
delight of AWT, the output response of the process dynamics was observed to be similar to
the graph in figure 43. The response is shown in figure 45. This meant that the time
constant, t, and the gain of the process dynamics block, K, could be measured off of the
graph and then a mathematical model of the process dynamics may be determined.
process dynamics of the position control was isolated by inputting a known voltage into the
female plug labeled 'T' and measuring the response at the plug labeled 'R'. A digital
oscilloscope was needed to take the measurement of the response of the output. To the
delight of AWT, the output response of the process dynamics was observed to be similar to
the graph in figure 43. The response is shown in figure 45. This meant that the time
constant, t, and the gain of the process dynamics block, K, could be measured off of the
graph and then a mathematical model of the process dynamics may be determined.
Before data was taken, an analysis of how to get the coefficients off of the display of
the oscilloscope was conducted. According to the table of Laplace Transforms in Appendix
B an input step voltage of Vin would be of the following form in the s-domain
the oscilloscope was conducted. According to the table of Laplace Transforms in Appendix
B an input step voltage of Vin would be of the following form in the s-domain
[6.8.2]
Therefore the output of the process dynamics block of the form in equation 6.8.1 would be
as follows
as follows
[6.8.3]
The technique of partial fraction expansion is need to determine the coefficients, a, b, and c.
A review of partial fractions is given in Haykin [18]. The coefficients a, b, and c were found
to be
, -
, and 
, respectively. Therefore, the output of the process
dynamics block is as follows
A review of partial fractions is given in Haykin [18]. The coefficients a, b, and c were found
to be
, - , and , respectively. Therefore, the output of the process dynamics block is as follows
[6.8.4]
The inverse Laplace transform was then used to convert the output of the process to the
time domain. The inverse Laplace transform was taken from tables in Appendix B. The
result was simplified as follows
time domain. The inverse Laplace transform was taken from tables in Appendix B. The
result was simplified as follows
[6.8.5]
Equation 6.8.5 may be manipulated to determine the coefficients 
and t. Take the limit
of y(t) as t goes to infinity
and t. Take the limit of y(t) as t goes to infinity
[6.8.6]
With the equation for the output written in this form, we can draw a fictitious line as seen in
figure 43. The slope of the line is . The slope, m, can be read off of a digital oscilloscope and
with the value of the input voltage, the gain of the process dynamics, K, can be determined
as follows
figure 43. The slope of the line is . The slope, m, can be read off of a digital oscilloscope and
with the value of the input voltage, the gain of the process dynamics, K, can be determined
as follows
[6.8.7]
The time constant, t, is the x-intercept of the fictitious line. However, it is important to note
that the form of the input is a step function at t =0. Therefore, the time constant
measured off of the oscilloscope is measured from the instant in time when the step voltage
is applied. In other words, on the oscilloscope, t =0 is defined as the point when the step
voltage is applied.
that the form of the input is a step function at t =0. Therefore, the time constant
measured off of the oscilloscope is measured from the instant in time when the step voltage
is applied. In other words, on the oscilloscope, t =0 is defined as the point when the step
voltage is applied.
AWT conducted an experiment to determine the constants in equation 6.8.1. Ten
arbitrary voltages were applied to the so-called process dynamics block of the Educational
Servo. From the ten input voltages, ten measurements were taken of x-intercepts and the
slope of the response curve. The data is given in appendix C. The average value of the
process dynamics gain, K, and the time constant, t, were calculated. Anomalous data was
removed and attributed to bad measurements off of the oscilloscope. AWT's mathematical
model of the process dynamics of the position control function of the Educational Servo is
presented as follows
arbitrary voltages were applied to the so-called process dynamics block of the Educational
Servo. From the ten input voltages, ten measurements were taken of x-intercepts and the
slope of the response curve. The data is given in appendix C. The average value of the
process dynamics gain, K, and the time constant, t, were calculated. Anomalous data was
removed and attributed to bad measurements off of the oscilloscope. AWT's mathematical
model of the process dynamics of the position control function of the Educational Servo is
presented as follows
[6.8.8]
The units of K are 1/s and the unit of t is sec.
6.9 Mathematical Model Validation
In theory, a mathematical model of the process dynamics of the system allows AWT
to predict the response of the position control system of the Educational Servo. For
instance, on the Educational Servo, an input of a reference angle of 80o corresponds to
inputting 8 volts into the system. The block diagram representation of the positional control
system of the Educational Servo may then be used to solve for the response of the system
to 8 volt input; the Matlab affiliated program, Simulink, was specifically designed for this.
With Simulink, the block diagram can be entered as shown in figure 46
to predict the response of the position control system of the Educational Servo. For
instance, on the Educational Servo, an input of a reference angle of 80o corresponds to
inputting 8 volts into the system. The block diagram representation of the positional control
system of the Educational Servo may then be used to solve for the response of the system
to 8 volt input; the Matlab affiliated program, Simulink, was specifically designed for this.
With Simulink, the block diagram can be entered as shown in figure 46
The transfer function of the process dynamics is input directly as seen in equation 6.8.8.
For simplicity, let the controller be a proportional controller with C= 1.0498; a proportional
controller is reviewed in Appendix D. With Simulink, the response of the output of the output
angle and/or the output voltage to the potentiometer can be simulated for any given input.
AWT chose to simulate the output voltage that feeds the output potentiometer versus the
input voltage that comes from the input potentiometer. On figure 46, this is the response at
point b to an input at point a. This simulation was chosen because the voltage at point a
and b can be directly displayed on a digital oscilloscope for comparison. The result of the
Simulink simulation is presented in figure 47.
For simplicity, let the controller be a proportional controller with C= 1.0498; a proportional
controller is reviewed in Appendix D. With Simulink, the response of the output of the output
angle and/or the output voltage to the potentiometer can be simulated for any given input.
AWT chose to simulate the output voltage that feeds the output potentiometer versus the
input voltage that comes from the input potentiometer. On figure 46, this is the response at
point b to an input at point a. This simulation was chosen because the voltage at point a
and b can be directly displayed on a digital oscilloscope for comparison. The result of the
Simulink simulation is presented in figure 47.
Before AWT can claim that the simulation exactly represents the real-life process, the
simulation must be compared with the oscilloscope display of the response of the output
voltage to an input voltage of 8 volts. The same parameters that were in the simulation
must be mirrored in the experiment. The setup of the experiment on the face of the
Educational Servo is shown in figure 48 below.
simulation must be compared with the oscilloscope display of the response of the output
voltage to an input voltage of 8 volts. The same parameters that were in the simulation
must be mirrored in the experiment. The setup of the experiment on the face of the
Educational Servo is shown in figure 48 below.
On figure 48, there is an arrow pointing to a variable resistor that controls the gain of the
proportional controller. According to equation D.2 in Appendix D, setting the resistance to 4.
98 KW provides a proportional control of 1.0498. Setting the input angle to 80o provides an
input voltage of 8 volts. The response of the output voltage to the input 8 volts is shown in
figure 49. Remember that the response of the output voltage to the input voltage is
representative of the response of the output angle to the input angle.
proportional controller. According to equation D.2 in Appendix D, setting the resistance to 4.
98 KW provides a proportional control of 1.0498. Setting the input angle to 80o provides an
input voltage of 8 volts. The response of the output voltage to the input 8 volts is shown in
figure 49. Remember that the response of the output voltage to the input voltage is
representative of the response of the output angle to the input angle.
A comparison of the actual response displayed on the oscilloscope to the simulated
response with Simulink shows that the mathematical model of the position control system of
the Educational Servo is reasonable. The difference between the settling times of the two
responses is around 0.5 sec and the difference between the maximum overshoot is 1 volt.
response with Simulink shows that the mathematical model of the position control system of
the Educational Servo is reasonable. The difference between the settling times of the two
responses is around 0.5 sec and the difference between the maximum overshoot is 1 volt.
AWT has demonstrated the angular position control of the Educational Servo. Future
groups can follow a similar process and formulate a mathematical model of the rotation
control of the Educational Servo. Furthermore, future groups can design an implementation
scheme utilizing the angular position control and rotation control characteristics in the wing-
stabilator model. AWT suggests that the position control feature of the Educational Servo
may be used to control the wing sweep; while the rotational control feature could be used in
conjunction with a crank-shaft to oscillate a flap. However, depending on the
implementation scheme of future groups, the mathematical model of the educational servo
developed by AWT must be modified.
groups can follow a similar process and formulate a mathematical model of the rotation
control of the Educational Servo. Furthermore, future groups can design an implementation
scheme utilizing the angular position control and rotation control characteristics in the wing-
stabilator model. AWT suggests that the position control feature of the Educational Servo
may be used to control the wing sweep; while the rotational control feature could be used in
conjunction with a crank-shaft to oscillate a flap. However, depending on the
implementation scheme of future groups, the mathematical model of the educational servo
developed by AWT must be modified.
6.10 Digital Control System
The fundamental difference between a digital control system and an analog control
system is that a digital system utilizes digital signals and a digital computer to control a
process [19]. The primary difference is illustrated below in figure 50.
system is that a digital system utilizes digital signals and a digital computer to control a
process [19]. The primary difference is illustrated below in figure 50.
Notice that the digital system shown above is a modification of the controller section of the
analog system in figure 25. Instead of designing an amplifying circuit, as seen in figure 23, to
provide the transfer function necessary to control the process, a computer is used. The
benefit of the digital controller is that an extensive knowledge of electronic circuits is not
required; however one must be proficient in Control Theory and programming.
analog system in figure 25. Instead of designing an amplifying circuit, as seen in figure 23, to
provide the transfer function necessary to control the process, a computer is used. The
benefit of the digital controller is that an extensive knowledge of electronic circuits is not
required; however one must be proficient in Control Theory and programming.
6.10.1 Digital-to-Analog and Analog-to-Digital Converters
The signals input into a computer and output signals of the computer are in digital
form. An analog-to-digital converter must be used to change the analog voltage signal to a
digital signal for the computer to use. Furthermore, when the computer has computed and
output the necessary control actions, a digital-to-analog converter must be used to
convert the digital signal of the computer to an analog voltage signal to control the process.
form. An analog-to-digital converter must be used to change the analog voltage signal to a
digital signal for the computer to use. Furthermore, when the computer has computed and
output the necessary control actions, a digital-to-analog converter must be used to
convert the digital signal of the computer to an analog voltage signal to control the process.
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