System Modelling · 2007. 11. 7. · system, braking system, . . . Tutorial 1 - 6 Antenna...
Transcript of System Modelling · 2007. 11. 7. · system, braking system, . . . Tutorial 1 - 6 Antenna...
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The University of New BrunswickDepartment of Electrical and Computer Engineering
Fredericton, NB, E3B 5A3 Canada
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System Modelling System Modelling
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U(s) G(s)
H(s)
Y(s) Systems
System
Dis
turb
ance
s
Control inputs
System outputs
Engineering systemsBiological systems
Information systems
envi
ronm
ent
subsystem
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• The system’s boundary depends upon the defined objective function of the system.
• The system’s function is expressed in terms of measured output variables.
• The system’s operation is manipulated through the control input variables.
• The system’s operation is also affected in an uncontrolled manner through the disturbance input variables.
System Variables
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Car and Driver Example
Objective function: to control the direction and speed of the car.
Outputs: actual direction and speed of the car
Control inputs: road markings and speed signs
Disturbances: road surface and grade, wind, obstacles.
Possible subsystems: the car alone, power steering system, braking system, . . .
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Antenna Positioning Control System
Original system: the antenna withelectric motor drive systems.Control objective: to point theantenna in a desired reference direction.Control inputs: drive motor voltages.Outputs: the elevation and azimuth of the antenna.Disturbances: wind, rain, snow.
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. Original system: the antenna with electric motor drive systems.
Control objective: to point the antenna in a desired reference direction.
Control inputs: drive motor voltages.
Outputs: the elevation and azimuth of the antenna.
Disturbances: wind, rain, snow.
Antenna Positioning Control System
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Antenna Control SystemFunctional Block Diagram
Physical VariablesInformation Variables
AntennaMotorPoweramp
Diff.amp
Ref.input
Anglesensor
volts volts
volts
+_
power torque Angularposition
Antenna SystemWind force
Feedback Path
Error
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Control System Components
System or process (to be controlled)
Actuators (converts the control signal to a power signal)
Sensors (provides measurement of the system output)
Reference input (represents the desired output)
Error detection (forms the control error)
Controller (operates on the control error to form the control signal, sometimes called compensators)
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Feedback System Characteristics
LoadKl
MotorKm
AmpKa
Speed sensorKs
Referencespeed
u+
_
Disturbancetorque
ωo
Open loop system
Feedback Path
ωr–
+
Td
Tm
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Control System Design Objectives
Primary Objectives:
1. Dynamic stability
2. Accuracy
3. Speed of response
Addition Considerations:
4. Robustness (insensitivity to parameter variation)
5. Cost of control
6. System reliability
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Control System Design Steps
Define the control system objectives.
Identify the system boundaries.
define the input, output and disturbance variables
Determine a mathematical model for the components and subsystems.
Combine the subsystems to form a model for the whole system.
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Control System Design Steps
Apply analysis and design techniques to determine the control system structure and parameter values of the control components, to meet the design objectives.
Test the control design on a computer simulation of the system.
Implement and test the design on the actual process or plant.
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System ModelingSystem Modeling
Purpose of models in control systems:Purpose of models in control systems:1. The mathematical model of a system is 1. The mathematical model of a system is
the basis for all control system analysis the basis for all control system analysis and design methods.and design methods.
2. A detailed model allows some 2. A detailed model allows some verification of the performance of the verification of the performance of the control system through control system through simulationsimulation, before , before it is implemented on the actual system.it is implemented on the actual system.
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Types of ModelsTypes of Models
Physical ModelsPhysical Models
scale modelsscale models
analogue modelsanalogue models
Mathematical ModelsMathematical Models
analytically basedanalytically based
experimentally basedexperimentally based
A model for a given system depends upon:
• defined system boundaries
• objective of the study
• level of approximationrequired
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Types of ModelsTypes of Models
A design model will often have many assumptions and simplifications made to allow the use of analytical methods (we will normally require linear, time-invariantmodels).
For verification studies, all model details are included and the model equations are then solved numerically, i.e. computer simulation.
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The Modeling ProcessThe Modeling Process1. Define the purpose or objective of the model1. Define the purpose or objective of the model..
Identify system boundaries, functional blocks, Identify system boundaries, functional blocks, interconnecting variables, inputs and outputs. interconnecting variables, inputs and outputs. Construct a functional block diagram.Construct a functional block diagram.
2. Determine the model for each component or 2. Determine the model for each component or subsystem.subsystem.
Apply known physical laws when possible, Apply known physical laws when possible, otherwise use experimental data to identify inputotherwise use experimental data to identify input--output relationships.output relationships.
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The Modeling ProcessThe Modeling Process
3. Integrate the subsystem models into an overall 3. Integrate the subsystem models into an overall system model.system model.
Combine equations, eliminate variables, check Combine equations, eliminate variables, check for sufficient equations to solve the system.for sufficient equations to solve the system.
4. Verify the model validity and accuracy.4. Verify the model validity and accuracy.
Implement a simulation of the model equations Implement a simulation of the model equations and compare with experimental data for the and compare with experimental data for the same conditions.same conditions.
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The Modeling ProcessThe Modeling Process
5. Make simplifications to create an approximate 5. Make simplifications to create an approximate model suitable for control design.model suitable for control design.
Linearization of model equations
Reduce the order of the model by eliminating unimportant dynamics
Use lumped parameter approximations for distributed parameter system.
trade-offModel Complexity Model Accuracy
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Electrical Component ModelsElectrical Component Modelsii
++vv__
ii++vv__
ii ++vv__
voltage/current voltage/charge
Inductance v = L di/dt v = L dq2/dt2
Resistance v = R i v = R dq/dt
Capacitance v = 1/C ∫ i dt v = 1/C q
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Mechanical Translation ModelsMechanical Translation Models
MM
xx
ffxx
BB
ff
ff xx
force/velocity force/position
Mass f = M dv/dt f = M dx2/dt2
Viscousfriction
f = B v f = B dx/dt
Spring f = k ∫ v dt f = k x
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Mechanical Rotational ModelsMechanical Rotational Models
T, T, θθ
JJ
BB
T, T, θθ
T, T, θθ
ss
torque/velocity torque/position
Inertia T = J dω/dt T = J dθ2/dt2
Viscousfriction
T = B ω T = B dθ/dt
Stiffness T = s ∫ ω dt T = s θ
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Transformation ModelsTransformation Models
Transformer
Lever
Gears
ii11
vv11
ii22
vv22
NN22NN11
TT1 1 , , θθ11
TT2 2 , , θθ22
NN11
NN22
ff11 , x, x11
ff22 , x, x22
LL11
LL22
v1 N1=
v2 N2
i1 N2=
i2 N1
f1 L2=
f2 L1
x1 L1=
x2 L2
T1 N1=
T2 N2
θ1 N2=
θ2 N1
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Classes of Model EquationsClasses of Model Equations
•• Continuous differential equations.Continuous differential equations.
•• Discrete difference equations.Discrete difference equations.
•• Algebraic equationsAlgebraic equations
Within each class there are subclasses of equations. The following shows the subclasses for differential equations.
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Differential EquationsDifferential Equations
Partial OrdinarPartial Ordinaryy
Linear NonlinearLinear Nonlinear
Time invariant Time varyingTime invariant Time varying
LTI - Linear, time invariant, ordinary differential equations are required for control analysis and design.
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ApproximationsApproximations
partial diff. eqs. partial diff. eqs. ordinary diff. eqs.ordinary diff. eqs.
lumped parameterlumped parameter
nonlinear eqs. nonlinear eqs. linear eqs.linear eqs.LinearizationLinearization
time varying time varying time invarianttime invariantsequence of modelssequence of models