Aerospace Systems Modelling & Control Revision PDF

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Thomas Palmer 29 January 2020

Aerospace Systems Modelling & Control 5ENT1062 Lecture 1: Introduction What is a System? A set of things working together as a mechanism. A system is a combination of sub systems A system may be a subsystem of a larger system A system has inputs and outputs A system is dynamic. How can we control a System? Identify the outputs we want to control Identify the input that effects the output Adjust the input so that we obtain the desired output. Actuators - Electric motors, Linear Actuators, Hydraulic, Pneumatic Sensors - Strain gauges, Pressure sensors, Gyroscopic sensors, Accelerometers, Thermistors

Modelling Air Speed of an Aircraft:

Pilots Desired Airspeed

Pilot Interface

Engine

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Aircraft

Actual Air Speed

Propulsion System

Gear Box

Air Frame

Propellor

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Control of Air Speed of an Aircraft:

Pilot Interface

Propellor

Engine

Gear Box

Air Frame

Systems Modelling: Develop mathematical models for each sub system (differential equations) Join mathematical models of subsystems together to give overall systems model Use subsystems model to predict dynamic and steady state performance. Computer Simulation, Calculus.

Mathematical Modelling Dynamic Systems: Find a set of differential equations tat describe the relationship between the inputs and outputs No model will be perfect Keep the equations as simple as possible. Use a linear time - invariant model whenever possible

Continuous Linear Time - Invariant Systems:

Linear implies that the equation only contains scaled derivative variables. Time-Invariant implies that the scaling factors do not vary with time. Modelling Air Speed of an Aircraft:

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Where: m = mass c = drag

Lecture 2: 1st & 2nd Order Systems First Order Systems:

Example 1: What is the gain?

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Example 2: What is the time constant?

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Solution of ODE’s using Laplace Transforms: 1. TRANSFORMATION - Transform using Laplace tables 2. MANIPULATION - Y(s) = ? 3. INVERSE TRANSFORMATION - Partial fractions to make Y(s) the sum of standard Laplace transforms. Hence find Y(t) from Laplace transform tables.

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Transfer Functions:

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Transfer Function of a Standard 1st Order System:

Step Response of a Standard 1st Order System:

Example: Modelling Cabin Temperature Newton’s Law of Cooling - The rate of change of temperature is proportional to the difference in temperature between atmospheric temperature and the object.

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Standard Form for a 2nd Order Differential Equation:

Step Response of Underdamped 2nd Order System:

Under-Damped Systems:

Example - Landing Gear Suspension System:

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Standard Form for a 2nd Order Transfer Function:

Landing Gear Suspension System:

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Over-Damped Systems:

Example:

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Critically Damped Systems:

Summary:

Lecture 3: Block Manipulation & Unity Feedback Systems Block Manipulation - The idea is to use a set of rules to gradually reduce a complex system consisting of a number of subsystems down to a single block that contains the transfer function for the whole system.

Blocks in Series:

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Proof:

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Example for Two Water Tanks in Series:

Closed Loop:

Flow Rate Control:

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Blocks in Parallel: Moving a block round a pick off point:

Interlocking Loops:

Steady State Output: This is the value of the output after everything has settled down following a change of input.

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Steady State Output from a Transfer Function:

Unity Feedback & Series Controllers: Unity feedback allows us to feedback the output to compare to the input. If there is an error a controller may be used to charge the input to the system we wish to control called plant.

Proportional Control: Plant input is proportional to the error between the desired output and actual output. Gives one degree of freedom. Implemented mechanically using a gearbox. Implemented electrically using an amplifier.

Example - Satellite Tracking Antenna:

Steady State Error: This is the difference between the input value and the output value after everything has settled down following a change of input.

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Steady State Error of Unity Feedback Systems:

In the example…

Classification of Unity Feedback Systems: We can classify unity feedback systems by their order and type number. The order of a system is given by the order of the open loop characteristic equation. The type number for a system is given by the number of roots of the open loop characteristic equation that are zero.

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Position Error Constant: If we assume the output of a unity feedback system is position then the reference signal must be the desired position. If the input is a unit step, the steady state error will be a position error. Velocity Error Constant: If the input is a ramp function the steady state output will be a constant velocity assuming the system is stable.

Acceleration Error Constant: If the input is a parabolic function the steady state output will be a constant acceleration assuming the system is stable.

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Lecture 4: Introduction to Sensors - Temperature, Pressure & Selection Basic Block Diagram:

Sensor Parameters: Capacity, Temperature, Angular Rate, Current, Force, Position, Displacement, Acceleration, Pressure, Flow Rate, RPM, Torque, Voltage

Example: Environmental Cooling Systems (ECS) Avionics and Passenger Low Air Low Alarms - Temperature and flow sensors Cooling Effects Detectors (CED) - Integrated temperature and flow sensors Bleed Air - High temperature, mass flow and pressure sensors Cabin Temperature - Multiple point temperature sensor Cabin Crew Ventilation - Temperature and flow sensors

Transducers: Transducers are devices that convert energy from one form to another. This energy type may be: Electrical, Mechanical, Electromagnetic, Chemical, Acoustic, Thermal or Photoelectric. They are widely used in electrical measurements of instruments or devices.

Sensor Characteristics: Sensitivity - The minimum input of the measured parameter that will create a detectable output change. For example, the change in output voltage for a given change in input parameter. Accuracy - The maximum difference between the indicated and actual reading. For example, if a sensor reads a force of 10N with a +/- 1% accuracy, the force could be between 9.9N and 10.1N. Resolution - The smallest incremental change of the input parameter that a sensor can detect. For example, can a temperature sensor detect a 0.5 degree change or only a 2 degree change.

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Linearity - In a linear sensor, the input has a linear relationship with the output signal. It is a desirable feature in most sensors. The conversion from the sensor output (voltage) to a calculated quantity becomes more complex for a non linear relationship. Range - This is the maximum or minimum of the applied parameter the sensor can measure. For example, sensors reading an angular rotation may only rotate 200 degrees and therefore it will not read angular rotation greater than 200 degrees.

Response Time - This is the time required for a sensor output to change from its previous state to a final settles value within a specified tolerance. Dynamic Response - The frequency range in which the sensor can operate. Typically sensors will have an upper and lower limit. Environmental - Sensors have some operational limits such as temperature, humidity, oil pressures. For example, sensors may work in relative humidity from 10% to 80%. Calibration - Many sensors will need calibration to set the relationship between the input and output. For example, temperature sensors may beed to be adjusted so that the measured temperature is the same as the actual temperature. This may need to be performed frequently. Cost - Generally more precision costs more. Some sensors have a small cost but the final conditioning equipment costs are significant.

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Precision - Refers to the degree of reproducibility of a measurement. If exactly the same value was measured a number of times, an ideal sensor would output the same value every time.

Examples of Temperature Sensors: These sensors provide temperature feedback to the system controller to make decisions. Decisions that it can make include Over-Temperature shutdown or turn on/off cooling fan. Thermistor, RTD, Thermocouple, LM35

Thermocouple Temperature Sensors - These measure the temperature by correlating the voltage difference between the junction of two different metal alloys. The differential voltage is measured and compared with the known voltage / temperature difference to determine the measured temperature. Two thin metal wires welded together form a junction. Almost any type of metal can be used. There are preferred metals used for their predictable output voltages and ability to withstand large temperature gradients. The junction between two metals generate a voltage that is a function of temperature. This is known as the thermoelectric or seebeck effect. Types of Thermocouples: There are several types of thermocouples which are categorised based on sensitivity and range. B, S, T and R type thermocouples are less sensitive but have the advantage of a higher range, They are also known as platinum type thermocouples. They can measure large temperatures and are usually used in furnaces or combustion chambers. J and K type thermocouples have high sensitivity but a limited range.

Considerations of Thermocouples: Connection Problems - Measurement errors can be caused by unintentional thermocouple junctions. It is important to use the correct type of thermocouple extensions wired if needed.

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Example for Two Water Tanks in Series:

Closed Loop:

Flow Rate Control:

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Blocks in Parallel: Moving a block round a pick off point:

Interlocking Loops:

Steady State Output: This is the value of the output after everything has settled down following a change of input.

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Steady State Output from a Transfer Function:

Unity Feedback & Series Controllers: Unity feedback allows us to feedback the output to compare to the input. If there is an error a controller may be used to charge the input to the system we wish to control called plant.

Proportional Control: Plant input is proportional to the error between the desired output and actual output. Gives one degree of freedom. Implemented mechanically using a gearbox. Implemented electrically using an amplifier.

Example - Satellite Tracking Antenna:

Steady State Error: This is the difference between the input value and the output value after everything has

settled down following a change of input.

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Steady State Error of Unity Feedback Systems:

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In the example…

Classification of Unity Feedback Systems: We can classify unity feedback systems by their order and type number. The order of a system is given by the order of the open loop characteristic equation. The type number for a system is given by the number of roots of the open loop characteristic equation that are zero.

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Position Error Constant: If we assume the output of a unity feedback system is position then the reference signal must be the desired position. If the input is a unit step, the steady state error will be a position error. Velocity Error Constant: If the input is a ramp function the steady state output will be a constant velocity assuming the system is stable.

Acceleration Error Constant: If the input is a parabolic function the

steady state output will be a constant acceleration assuming the system is stable.

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Lecture 4: Introduction to Sensors - Temperature, Pressure & Selection Basic Block Diagram:

Sensor Parameters: Capacity, Temperature, Angular Rate, Current, Force, Position, Displacement, Acceleration, Pressure, Flow Rate, RPM, Torque, Voltage

Example: Environmental Cooling Systems (ECS) Avionics and Passenger Low Air Low Alarms - Temperature and flow sensors Cooling Effects Detectors (CED) - Integrated temperature and flow sensors Bleed Air - High temperature, mass flow and pressure sensors Cabin Temperature - Multiple point temperature sensor Cabin Crew Ventilation - Temperature and flow sensors

Transducers: Transducers are devices that convert energy from one form to another. This energy type may be: Electrical, Mechanical, Electromagnetic, Chemical, Acoustic, Thermal or Photoelectric. They are widely used in electrical measurements of instruments or devices.

Sensor Characteristics: Sensitivity - The minimum input of the measured parameter that will create a detectable output change. For example, the change in output voltage for a given change in input parameter. Accuracy - The maximum difference between the indicated and actual reading. For example, if a sensor reads a force of 10N with a +/- 1% accuracy, the force could be between 9.9N and 10.1N. Resolution - The smallest incremental change of the input parameter that a sensor can detect. For example, can a temperature sensor detect a 0.5 degree change or only a 2 degree change.

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Linearity - In a linear sensor, the input has a linear relationship with the output signal. It is a desirable feature in most sensors. The conversion from the sensor output (voltage) to a calculated quantity becomes more complex for a non linear relationship. Range - This is the maximum or minimum of the applied parameter the sensor can measure. For example, sensors reading an angular rotation may only rotate 200 degrees and therefore it will not read angular rotation greater than 200 degrees.

Response Time - This is the time required for a sensor output to change from its previous state to a final settles value within a specified tolerance. Dynamic Response - The frequency range in which the sensor can operate. Typically sensors will have an upper and lower limit. Environmental - Sensors have some operational limits such as temperature, humidity, oil pressures. For example, sensors may work in relative humidity from 10% to 80%. Calibration - Many sensors will need calibration to set the relationship between the input and output. For example, temperature sensors may beed to be adjusted so that the measured temperature is the same as the actual temperature. This may need to be performed frequently. Cost - Generally more precision costs more. Some sensors have a small cost but the final conditioning equipment costs are significant.

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Precision - Refers to the degree of reproducibility of a measurement. If exactly the same value was measured a number of times, an ideal sensor would output the same value every time.

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Examples of Temperature Sensors: These sensors provide temperature feedback to the system controller to make decisions. Decisions that it can make include Over-Temperature shutdown or turn on/off cooling fan. Thermistor, RTD, Thermocouple, LM35

Thermocouple Temperature Sensors - These measure the temperature by correlating the voltage difference between the junction of two different metal alloys. The differential voltage is measured and compared with the known voltage / temperature difference to determine the measured temperature. Two thin metal wires welded together form a junction. Almost any type of metal can be used. There are preferred metals used for their predictable output voltages and ability to withstand large temperature gradients. The junction between two metals generate a voltage that is a function of temperature. This is known as the thermoelectric or seebeck effect. Types of Thermocouples: There are several types of thermocouples which are categorised based on sensitivity and range. B, S, T and R type thermocouples are less sensitive but have the advantage of a higher range, They are also known as platinum type thermocouples. They can measure large temperatures and are usually used in furnaces or combustion chambers. J and K type thermocouples have high sensitivity but a limited range.

Considerations of Thermocouples: Connection Problems - Measurement errors can be caused by unintentional thermocouple junctions. It is important to use the correct type of thermocouple extensions wired if needed.

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Lead Resistance - Thermocouple wires that are thin have a high resistance. They can experience signal noise or errors side to the input impedance. For this reason, the leads should be kept short. Noise - One of the biggest issues with thermocouples. The output of thermocouples are very prone to signal noise, interference can be minimised by making the cable a twisted couple. Thermal Shunting - Mass of thermocouples, mounting location, self heating/cooling effects of the thermocouple can all contribute to inaccuracies when measuring the temperature. De-Calibration - Oxidisation and corrosion can affect the calibration and function of the thermocouple over time. Sensitivity & Range - Thermocouples have the greatest range of the different temperature sensors however the voltage produced per degree increase and decrease frequently at the junction. Time Response - The time it takes for the thermocouple to reach steady state (thermal equilibrium with the surroundings) differs based on the wire thickness. Thicker wires can take 20s with small gauge wires taking 20ms on average. Signal Conditioning - The output voltage from the thermocouple is very small (around 50mV). Therefore the output signal needs to be amplified considerably to be used in practical applications.

Advantages:

Robust / resistance shock and vibration Wide temperature range Simple to manufacture Requires no excitation power No self-heating Can be miniaturised Disadvantages: Produces relatively low output signal Output signal is non-linear Requires a sensitive measuring device Requires a stable measuring device Low signal level - susceptible to noise

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RTD Temperature Sensors: Resistance Temperature Sensor (RTD). Measures temperature by correlating resistance of metallic elements with the temperature. Composed of certain metallic elements, whose change in resistance is a function of temperature. Small excitation current is passed over the element, resulting in voltage which is proportional to the resistance of the element. Voltage is then converted to calibrated temperature. Change in temperature is detected by the change in resistance of the wire.

Effectively utilised for temperature measurement by using in bridge circuits. A Three/Four wire bridge co...