System Modelling · 2007. 11. 7. · system, braking system, . . . Tutorial 1 - 6 Antenna Positioning Control System Original system: the antenna with electric motor drive systems.

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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