design and build of a thruster controlled model of a large...

School of Mechanical Engineering

Level IV Design Project 2005

Design and Build of a Thruster Controlled Model of a Large Flexible

Space Station

Final Report

Danielle Moreau

Pamela Woods

Supervisor: Frank Wornle

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Executive Summary

This project documents the research and design of all components required to provide active

vibration suppression of a laboratory built test structure by applying predictive control. A

controller has been designed to provide active vibration suppression for the first bending

mode in the test structure using air jet thrusters as actuators.

A test rig was designed to simulate the flexible elements of a large flexible space structure.

The various devices involved with structural excitation and vibration control have been

investigated. These include shakers, jet thrusters and accelerometers.

Predictive control was selected as the control method to implement and an ARX model is

used in system identification. The controller was applied to a simulated sixth order system

and results showed that the controller is effective in suppressing vibrations in the plant.

The controller was then implemented on the microcontroller and applied to the test rig. Non-

linearities in the system caused an inaccurately identified system. This led to an incorrectly

calculated control output applied to the test rig.

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Disclaimer

Danielle Moreau and Pamela Woods declare the authentication of this report, that all of its

contents are purely their own.

Danielle Moreau

………………………………. ………………

(signature) (date)

Pamela Woods

………………………………. ………………

(signature) (date)

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Acknowledgements

Danielle Moreau and Pamela Woods wish to thank a number of people who have contributed

to this project and without whose contribution, results to date would not have been possible.

Danielle and Pamela would like to thank:

Andrei Koutousov

Ji (Sam) Lu

The lab technicians in the Adelaide University Workshop

Derek Franklin

George Osborne who has provided extensive technical support

And especially Frank Wornle

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Table of Contents

Executive Summary ................................................................................................................... i

Disclaimer ................................................................................................................................. ii

Acknowledgements .................................................................................................................. iii

1 Introduction....................................................................................................................... 1

1.1 Aim .......................................................................................................................... 1

1.2 Background Information .......................................................................................... 1

1.3 Literature Review..................................................................................................... 1

1.4 Project Specification ................................................................................................ 4

2 Model Concept Solutions and Alternatives ...................................................................... 6

2.1 Initial Model Concept .............................................................................................. 6

2.1.1 Research Structure of the Politecnico di Milano............................................. 6

2.1.2 Variations of the Previous Research Structure................................................ 7

2.1.3 Initial Concept Analysis ................................................................................ 13

2.2 Test Rig Mechanical Design .................................................................................. 14

2.2.1 Sprinkler Concept.......................................................................................... 15

2.2.2 Mechanical Design Alternatives of Test Rig Elements................................. 16

2.2.2.1 Truss Arms................................................................................................ 16

2.2.2.2 Rotational Coupling Device...................................................................... 18

2.2.2.3 Coupling Housing .................................................................................... 19

2.2.2.3.1 Housing Concept 1 ............................................................................. 19

2.2.2.3.2 Housing Concept 2 ............................................................................. 20

2.2.2.3.3 Final Housing Design ......................................................................... 22

3 Excitation and Structural Vibration ................................................................................ 26

3.1 Technical Elements of Excitation and Vibration ................................................... 27

3.1.1 Method of Forced Excitation: Shakers .......................................................... 27

3.1.2 Measuring Vibration: Accelerometers .......................................................... 28

3.1.3 Air Jet Thrusters ............................................................................................ 28

4 Control Design................................................................................................................ 30

4.1 Selection of Control Method.................................................................................. 30

4.1.1 Sliding Mode Control.................................................................................... 31

4.1.2 Predictive Control ......................................................................................... 31

4.2 Selected Control Method ...................................................................................... 32

4.3 Predictive Control Mathematical Formulation....................................................... 33

4.4 Computational Procedure....................................................................................... 38

4.5 Predictive Controller Hand Calculations ............................................................... 39

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4.6 System Identification of a 3-DOF System ............................................................. 47

4.6.1 Results for System Identification Performed Without Noise in the System . 49

4.6.2 Results for System Identification with a Small Amount of Noise in the

System ........................................................................................................... 54

4.6.3 Results for System Identification with a Large Amount of Noise in the

System ........................................................................................................... 58

4.6.4 Summary of System Identification Results ................................................... 62

4.7 Performing System Identification Off-line ............................................................ 65

4.8 Controller Results for the Three Degree of Freedom System................................ 66

4.8.1 Results for Deadbeat Control ........................................................................ 66

4.8.2 Results for the Controller With p = 6, q = 50................................................ 69

5 Experimental Setup..........................................................Error! Bookmark not defined.

5.1 Initial Excitation of Strips .......................................Error! Bookmark not defined.

5.2 Method of Forced Vibration ...................................Error! Bookmark not defined.

5.3 Control Process .......................................................Error! Bookmark not defined.

5.4 Software ..................................................................Error! Bookmark not defined.

5.4.1 RTMC9S12 – Target and SIMULINK...........Error! Bookmark not defined.

5.4.2 Graphical User Interface (GUI)......................Error! Bookmark not defined.

5.4.3 Target Program and its Subsystems ...............Error! Bookmark not defined.

5.4.3.1 Excitation Mode.........................................Error! Bookmark not defined.

5.4.3.2 Off Mode....................................................Error! Bookmark not defined.

5.4.3.3 Control Mode .............................................Error! Bookmark not defined.

5.4.3.4 System Identification .................................Error! Bookmark not defined.

5.4.3.5 Accelerometers ..........................................Error! Bookmark not defined.

5.5 Hardware.................................................................Error! Bookmark not defined.

5.5.1 Power Supply .................................................Error! Bookmark not defined.

5.5.2 Test Rig Connections to Microcontroller .......Error! Bookmark not defined.

6. Results and Discussion ....................................................Error! Bookmark not defined.

6.1 System Identification Results..................................Error! Bookmark not defined.

6.1.2 Excitation of A Single Mode..........................Error! Bookmark not defined.

6.1.3 Poor I/O Data .................................................Error! Bookmark not defined.

6.1.4 System Non-Linearities..................................Error! Bookmark not defined.

7 Achievement of Project Aims and Future Work..............Error! Bookmark not defined.

7.1 Future Work ............................................................Error! Bookmark not defined.

8 Costing of the Fabrication and Materials.........................Error! Bookmark not defined.

9 Conclusion .......................................................................Error! Bookmark not defined.

10 References........................................................................Error! Bookmark not defined.

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Appendix A Solid Edge Draft Workshop Drawings ......Error! Bookmark not defined.

Appendix B ANSYS Mode Shape Simulations and Session Editors..Error! Bookmark

not defined.

Appendix C Predictive Controller Flowcharts and MATLAB M-FileError! Bookmark

not defined.

Appendix D System Identification Results For a 3 DOF System. Error! Bookmark not

defined.

Appendix E Circuit Diagram .........................................Error! Bookmark not defined.

Appendix F Software MATLAB M-Files and Simulink Block Diagrams.............Error!

Bookmark not defined.

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1 Introduction

1.1 Aim

This project investigates active vibration control in a large, flexible space structure. The aim

of the project is therefore to provide active vibration suppression in a laboratory built test

structure by applying predictive control. A controller will be designed to provide active

vibration suppression in the test structure. Air jet thrusters will be installed on the test

structure and employed by the controller to suppress the structural vibrations.

1.2 Background Information

Today, the active control of space structures is an extensive and topical field of research

within aerospace communities around the world (Casella, 1996). The mechanical

characteristics of such structures are so distinct that a new research class has been developed,

namely Large Space Structures, to cater for their individual topology (Allen, 1999). Large

Space Structures are categorised by their large number of vibration modes, which cause

unacceptable disturbance levels within the structure without active vibration control (Allen,

1999). Active vibration control of large space structures suppresses structural vibrations and

is used therefore to optimize the performance of the structure and of other structural pointing

instruments such as antennas and solar panels and is also used to reduce structural weight for

economical launch conditions (Allen, 1999).

1.3 Literature Review

A number of previous research projects have focused on applying active vibration control to a

large experimental laboratory structure. The most fundamental of these projects include

investigations by NASA into active vibration control of large space structures (McGinley,

1997) and a number of research projects conducted at the Dipartimento di Ingegneria

Aerospaziale - Politecnico di Milano, Italy, over the last two decades (Casella, 1995).

Previous projects have investigated active vibration control of a laboratory truss structure by

employing a variety of control laws and actuators.

In the previous research projects conducted at the Politecnico di Milano, the laboratory rig

constructed for active vibration control is a modular truss. The modular truss has a total

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length of 19 m, 54 bays, a mass of 75 kg and is built from commercially available polyvinyl

chloride elements as shown in figure 1 (Allen, 1999). The truss was suspended by three pairs

of soft springs to ensure that the rigid and elastic modes of vibration are decoupled.

Figure 1: Research Structure at the Politecnico di Milano (Allen, 1999)

The design of the truss structure ensures that the bending vibration modes occur only in the

vertical or horizontal direction so that control can be applied only to the vibration modes in

the horizontal direction to simplify the control problem (Casella, 1996).

Previous research involved investigating a range of control systems that utilise Air Jet

Thruster (AJT) actuators, Piezoelectric Active Members (PAM) or a combination of both

elements. Both of these elements provide active vibration control for the first vibration

modes, with the AJT actuators being efficient in suppressing large vibrations in a structure

(Casella, 1996). The rigid modes of vibration are ignored as those on the model truss would

be completely different from those in the space station structure in a space environment

(Casella, 1996). Previous research has seen a number of non-linear control techniques being

implemented and these include Sliding Mode Control, Variable Structure Control and Multi

Pulse Width Modulation Control. Previous researchers have discovered that the space

environment requires a simple and reliable control system and, as such, bang-bang techniques

are an effective solution (Allen, 1999). Air jet thrusters are on/off devices employing bang-

bang techniques as these actuators operate discontinuously in 2 states. Air jet thrusters

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produce a constant force allowing implementation of simple digital switching logic (Dozio,

2002).

Previous research into vibration control of large flexible space structures has produced

satisfactory experimental results. In the research project ‘Nonlinear Controllers for Vibration

Suppression in a Large Flexible Space Structure’ where a controller employing air jet thruster

actuators was implemented, the experiment was conducted as follows. Initially, a mechanical

excitation system was used to excite the structure into a steady oscillation. The excitation

device was disconnected from the structure so that control could begin. The controller then

counteracted the oscillatory motion until its conclusion (Casella, 1996). The controller does

this by setting the real time jet thruster task to 1 or 0, at appropriate time instances so that the

jet thrusters provide a force to counteract vibrations (Dozio, 2002). Results for employing air

jet thrusters as actuators have seen vibration suppression to occur below a time of two

seconds for the first four modes of vibration as shown in figure 2 (Casella, 1996). With no

control applied, vibrations would decay naturally in a time of the order of hundreds of

seconds. Thus vibration suppression in a time of two seconds is very satisfactory.

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Figure 2: Velocity and Control Graphs Demonstrating Control Time to Steady State (Casella,

1996)

The different control techniques employed by various projects have all produced satisfactory

methods of suppressing vibrations in the laboratory truss structure. The research suggests

though, that such control techniques should be applied to a more complicated truss structure

to ensure validity (Vander Velde, 1983).

This project is being undertaken to replicate the results obtained by previous research with the

intention of using the test rig constructed in this project for undergraduate demonstrations. To

date, there is a lack of demonstration equipment in this area of research for undergraduate

students in the Adelaide University Mechanical Engineering School.

1.4 Project Specification

The project specification is to provide vibration control to a large flexible space structure. A

control system that utilizes constant force air jet thrusters as actuators will be investigated to

suppress vibrations in a laboratory structure. A number of control techniques are to be

investigated for active vibration control of the structure before selection of the most suitable

control technique. Initially, the controller will be developed in a MATLAB M-file and

SIMULINK model using predictive control before being implemented on a microcontroller.

A model of a large, flexible space station will be constructed to test the predictive control

theory. The test rig will consist of flexible linear or truss elements and will contain air jet

thrusters and a device to generate forced excitation. The structure will be purposely excited to

its first few modes of vibration and then the performance of the controller acting on the rig

will be determined by its ability to suppress the structural vibrations with the air jet thrusters.

The aim of this project is to investigate active vibration control of a large, flexible space

structure and to implement a control system that will damp vibrations in a test rig. In order to

achieve this aim, the following project stages are to be completed:

1. Test Rig Design

A test rig simulating the flexible properties of space station structure is to be designed and

constructed.

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2. ANSYS Simulation

To determine the feasibility of the test rig design and its response to excitation, a modal

analysis of the truss arms is conducted on ANSYS.

3. Controller Design

A controller is developed in MATLAB using predictive control, before being implemented on

a small microcontroller.

4. Controller Implementation on the Working Model

The test will rig will be excited and the controller will be used to suppress the structural

vibrations using the air jet thrusters.

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2 Model Concept Solutions and Alternatives

The first project objective is the design and construction of a test rig simulating the flexible

properties of a space station. The following section documents the initial test rig design

concept, the different test rig alternatives investigated and the final design concept that has

been selected for construction.

2.1 Initial Model Concept

The initial test rig concept is derived from the structure used in successful projects at the

Politecnico di Milano. The initial test rig design comprises of a truss structure suspended at a

number of locations along its axis. A number of different truss configurations have been

investigated and simulated on ANSYS to determine their response to excitation. This

simulation was performed to determine structure feasibility and suitability for construction

and control.

Firstly, the configuration of the truss structure developed at the Politecnico di Milano was

investigated. Subsequent simulations focused on a number of variations of this structure.

2.1.1 Research Structure of the Politecnico di Milano

Previous research projects at the Politecnico di Milano have led to the design of an

experimental space structure for the investigation of active vibration control in this project.

The experimental structure is a modular beam-like truss, consisting of cubic bays with one

diagonal on each cubic face. This minimizes torsional bending moments due to gravity.

Furthermore, it leads to the decoupling of the bending modes in the horizontal or vertical

plane (Bernelli-Zazzera, 2000). See table 1 for the properties of the truss structure.

Table 1: Truss Structure Properties and Property Value (Dozio, 2002)

Truss Structure Property Property Value

Mass 75 kg

Length 19 m

Number of cubic bays 54

Number of nodes 220

Construction material Commercial PVC elements

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The structure, as shown in figure 3, is made from PVC tubes that are connected with steel bolt

assemblies to produce the cubes.

Figure 3: Previous Research Structure (Casella, 2002)

The structure is suspended from the ceiling using three pairs of steel springs with stiffness of

104 N/m. The springs are located at positions of 15%, 50% and 85% of the total truss

extension. This ensures an acceptable decoupling of the rigid and elastic bending modes

(Allen, 1999). The rigid bending modes of vibration can thus be ignored. This is logical, as

those in the laboratory structure are completely different to those found in a real space station

in space (Casella, 1996). Another advantage of this suspension design is that the control

system can be used to suppress vibrations in the horizontal plane only (Allen, 1999).

2.1.2 Variations of the Previous Research Structure

Three different truss configurations based on the truss built in previous projects have been

considered for this project. The three variations of the truss built in previous projects were

analysed using ANSYS to find their response to excitation. It is not feasible to construct a

truss that is the same length as that constructed at the Politecnico di Milano. This is due to the

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budget and time constraints on this project. In order to reduce the length, all models

considered are scaled versions of the truss described above but with slightly different

formation of the individual bays.

The Milan structure has a length of 19 m with 54 cubic bays. Therefore, the length of each

element is 19/54 = 0.352 m.

It is proposed that the truss structure to be built should be a length of approximately 3 m.

Therefore, it must be scaled by a factor of 3/19 = 0.1579.

To produce a model with a length of 3 m, it is therefore required that each element must be a

length of 0.16x0.352 ≈ 5.6 cm.

For an element length of 5.6 cm, the total truss length would be 5.6x54 = 302.4 cm =3.024 m.

A number of different truss configurations were modelled on ANSYS to find the model with a

wide range of vibration modes. The different truss models investigated are a cubic truss, a

triangular truss and a cubic truss with diagonals. Each of these models is described in the

following sections.

Cubic Truss

A single bay, the truss length and a section of the cubic truss are shown in figure 4.

0,0,0 0.056,0,0

0.056,0.056,0

0.056,0.056,0.056

0,0,0.056

0,0.056,0.056

0,0.056,0

z

x y

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Figure 4: Single Bay, Truss Length and Section in Cubic Formation

The ANSYS results of modelling a single bay of the cubic truss formation are shown in table

2 for the first five bending modes. Plots of the undeformed and deformed nodal solution for

the first five bending modes are shown in appendix B

Table 2: ANSYS Results Showing the Frequency of Each of the First Five Bending Modes for a

Single Bay in Cubic Truss Formation

Modal Set Frequency (Hz)

1 14.112

2 36.451

3 50.401

4 73.030

5 106.95

Triangular Truss

A single bay, the truss length and a section of the triangular truss are shown in figure 5.

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Figure 5: Single bay, Truss Length and Section in Triangular Formation

The ANSYS results of modelling a single bay of the triangular truss formation are shown in

table 3 for the first five bending modes. Plots of the undeformed and deformed nodal solution

for the first five bending modes are shown in Appendix B.

0,0,0 0.056,0,0

0.056,0.056,0

0.056,0.056,0.056

0,0,0.056

0,0.056,0.056

z

x y

0.028

0,0.056,0

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Single Bay in Triangular Truss Formation


1 18.665

2 90.523

3 119.11

4 132.10

5 205.31

Cubic Truss with diagonals

A single bay, the truss length and a section of the cubic truss with diagonals are shown in

figure 6.

0,0,0 0.056,0,0

0.056,0.056,0

0.056,0.056,0.056

0,0,0.056

0,0.056,0.056

0,0.056,0

z

x

y

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Figure 6: Single Bay, Truss Length and Section in Cubic Formation with Diagonals

The results of modelling a single bay of the cubic truss formation with diagonals are shown in

table 4 for the first five bending modes. Plots of the undeformed and deformed nodal solution

for the first five bending modes are shown in Appendix B.


Single Bay in Cubic Truss Formation With Diagonals


1 20.704

2 40.910

3 42.209

4 42.803

5 43.314

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2.1.3 Initial Concept Analysis

The analysis of the proposed models revealed a lack of flexibility in terms of changing their

structural configuration. The design is highly complicated and thus restrictive if altering the

structural configuration is desired. In order to first apply control to a simple structure and to

then determine the effectiveness of the control method by testing it on a more complicated

structure, the test rig design is to allow a change in the truss configuration.

The test rig design is then reformulated to consist of two arms mounted on a central fixed

pivot. This design allows for a more realistic simulation of the space environment as the truss

arms are fixed only at one end, allowing free motion at the other end. The final rig is designed

in a modular way. Starting with a single arm, additional segments can be attached. This

allows the complexity of the system to be increased gradually and therefore the truss arms

have a variety of modified structural configurations. Control of a single strip formation as

shown in figure 7 will first be performed to ensure the controller adequately suppresses

structural vibrations in this simple truss formation. The controller will then be applied to a

more complicated structure such as the four arm truss formation as shown in figure 8. Finally,

the length of the arms will be extended as shown in figure 9 by attaching additional arms

through use of connecting blocks. The design of this model and a number of concept

alternatives are discussed in the following sections.

Figure 7: Test rig in Single Strip Formation

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Figure 8: Test Rig in 4 Strip Formation

Figure 9: Test Rig in 4 Strip Formation with Extended Length

2.2 Test Rig Mechanical Design

The test rig consists of two arms mounted on a central pivot that is capable of rotation around

a vertical axis. The thrusters are mounted onto the arms to allow jet propulsion, the concept

utilized by a rotational garden sprinkler. The rotational coupling is mounted on a large base so

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that the housing and arms are capable of uninhibited motion. A number of different concepts

for the design of the test rig have been investigated as discussed in the following sections.

2.2.1 Sprinkler Concept

The mechanical design of the test rig is based on the concept of a rotational garden sprinkler.

The principle of the rotational sprinkler can be seen in figure 10. Figure 10 shows that water

flows into the sprinkler in the direction of the arrow A, causing the sprinkler arms to revolve,

that is to be jet-propelled, in the direction of the arrow B as described by Newton’s third law.

This concept of using jet propulsion to rotate the arms of the sprinkler has been applied to the

test rig. In the test rig design, air flows through a tube into the air jet thrusters, causing the test

rig arms to revolve in the opposite direction.

Figure 10: Principle of Jet Propulsion in a Revolving Sprinkler

The final test rig design incorporates the jet propulsion concept utilized by the rotational

sprinkler.

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2.2.2 Mechanical Design Alternatives of Test Rig Elements

The primary mechanical elements of the final test rig design are the truss arms, the rotational

coupling device, the rotational housing and the connecting blocks.

2.2.2.1 Truss Arms

Functional Specification

The truss arms are to simulate the flexibility of a large space structure and provide a medium

of vibration suitable to demonstrate the ability of the controller to suppress structural

vibrations. For this purpose, the arms are required to be long, flexible elements that can be set

into vibration easily however they must be of sufficient strength to carry the load of the air jet

thrusters and of other measuring and excitation devices.

Materials Considered

A number of materials have been considered for construction of the truss arms and these

include carbon fibre rods, Delrin, PVC welding rods and plastic sheets as well as the use of

commercially available construction sets. Properties of each of the materials are presented in

table 5. The table includes the mechanical properties, modulus of elasticity and density, as

well as the most readily available form of the material.

The table includes several types of Carbon Fibre of various fibre diameters as their properties

differ greatly with changing type. Table 5 shows that carbon fibre has a higher value of

modulus of elasticity, compared to Delrin, Plastic Sheets and Poly-Vinyl Chloride Welding

rods. Thus, carbon fibre is the least flexible material.

Material formation is an important factor in the selection of the truss arm material. Sheet or

strip elements have the advantage of giving the structure a high stiffness in the vertical

direction while allowing transverse vibrations in the horizontal direction. In the presence of

gravity, sagging of the truss arms is possible. Sagging should be minimized as it can

permanently deform the arms and is not suitable for simulating non-gravity conditions.

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Creating the truss arms from strip elements as opposed to cylindrical beam shaped elements

will reduce sagging in the test rig arms.

Table 5: Material Type and Associated Properties

Material Modulus of Elasticity

(N/m2)

Density (kg/ m3) Readily

Available Form

Carbon Fibre (Cityplastics, 2005)

Carbon, PAN HM

Fibre Diameter: 8 [µm]

380 x 109 (N/m2) 1.60-1.70 x 103 Rods of Diameter

3-10mm

Carbon, PAN HT

Fibre Diameter: 9 [µm]

230 x 109 (N/m2) 1.60-1.70 x 103 Rods of Diameter

3-10mm

Carbon, pitch GP

Fibre Diameter:

10-13 [µm]

27-35 x 109 (N/m2) 1.60-1.70 x 103 Rods of Diameter

3-10mm

Carbon, pitch HP

Fibre Diameter

9-18 [µm]

150-480 x 109 (N/m2) 1.80-2.15 x 103 Rods of Diameter

3-10mm

Delrin (Dupont, 2005)

Delrin, average value 3.1 x 109 5.10 x 102

Printed Circuit Board Sheets (Matweb, 2005)

Printed circuit board

sheets, average value

1.0 x 109 1.40 x 103 Sheets of thickness

1.25 mm

Poly-Vinyl Chloride Welding (Matweb, 2002)

Rigid PVC, average

value

2.8 x 108 1.16 - 1.45 x 103 Rods of Diameter

4 mm

Plasticised PVC, average

value

0.3 to 1.9 x 108 1.15-1.40 x 103 Rods of Diameter

4 mm

Construction Sets

Another alternative is to construct the test rig arms from plastic construction sets that are

readily available from toy and model shops. These sets have the tubular elements and

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connecting pieces that would make assembly of a truss structure very simple. A variety of

these construction sets include Meccano, Lego and K Nex systems.

After investigation of a number of construction sets including Meccano, Lego and K Nex, the

only possible solution was found to be K Nex systems as they contain truss structural

elements and connection pieces of a suitable size. But the most appropriate set containing 96

elements is priced at $100 and with 4 sets required to produce the test rig arms, a total of $400

would be necessary. Thus this option is not economically viable.

Based on the above design criteria, printed circuit board plastic sheets are selected as the

construction material of the truss arms. Plastic sheets are the only material that readily

supports the test rig arm construction from strip elements. The sheet formation allows the

strips to be directly cut from the sheets. The printed circuit board sheets are available at

minimal cost from the workshop which is a major advantage over such materials as carbon

fibre rods that are 14 $/m.

2.2.2.2 Rotational Coupling Device


The primary function of the rotational coupling device is to generate an uninhibited rotating

motion of the truss arms.

The technology in the sprinkler can be replicated to produce a rotational coupling device for

the test rig. An alternative is to incorporate sprinkler coupling, with the sprinkler head

removed, directly into the test model to simplify the design and to minimise labour. To

determine the sealing capabilities of the sprinkler, the pressure is increased to three bar

through the sprinkler’s main pipe. Although the seal withstands the applied pressure

adequately, the friction force between the rotating and stationary parts is too large for smooth

rotation, resulting in a broken, jarred sliding motion of the sprinkler. Although this motion is

adequate for sprinkler usage, this coupling device is not sufficient for the test rig application.

Therefore a modified coupling device based on the sprinkler technology is required.

The final coupling device consists of a bearing and a seal at either end of a central bar. The

coupling enables air to be transferred without leakage from a regulated air supply through the

central bar of the housing and exit without leakage through tubing that feeds into the air jet

thrusters.

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The electrical cables that supply signals to the actuators and sensors are fastened to the

housing containing the rotational coupling device. The cables are sufficiently flexible to

enable signals to be coupled from the H-Bridge circuitry to the flexible arms. To prevent the

wires from becoming tangled, the coupling is limited to 360 degree rotation by a stopper.

2.2.2.3 Coupling Housing


The primary function of the coupling housing is to house the rotational coupling device, the

air supply tubes and electrical wiring and to support the truss arms. Thus, the coupling

housing needs to be of sufficient strength to support the load of the truss arms and have

adequate internal space for electrical wiring and the air supply tubing. Several housing

concepts and a description of the final housing design are discussed in the following sections.

2.2.2.3.1 Housing Concept 1

The first housing concept describes the truss arms directly inserted into the main housing

where they are held in place with small grub screws. The housing contains a number of inserts

for the strips on two opposite faces and therefore the test rig consists of truss arms extended

on either side of the main housing. On the housing face, inserts are placed horizontally and

vertically, enabling a number of different possible truss configurations. The truss arms can

therefore consist of a single vertical or horizontal strip, a triangular truss formation when

three strips are inserted as shown in figure 11 or a cubic truss formation when four strips are

inserted.

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Figure 11: Housing Concept 1: Housing Face with Strip Inserts in Triangular Formation

This housing concept however, is inadequate. As the strips are held in place on only one edge,

excitation of the strips causes them to pivot on the grub screws resulting in an irregular

bending motion. In order to achieve uniform bending, the strips require clamping on both

their top and underside edge so that they act as cantilever beams.

2.2.2.3.2 Housing Concept 2

To overcome the design problem experienced in the first housing concept, the strips are to be

clamped and then inserted into the housing. Blocks are to be inserted into the main housing to

allow clamping of both sides of the strip. These blocks have initially been designed as an E

shape that allows two strip inserts per block as shown in figure 12.

The E block is fastened to the main housing by a screw located at its central arm. To clamp

the strip adequately, vice technology is replicated. Based on a vice, a small block has been

inserted above the strip and screwed in place so that the strip is clamped on both edges.

Final Report ___________________________________________________________________________________


Figure 12: Housing Concept 2: E Blocks with the Strip Inserted and Vice Technology

An E block is fastened to 2 opposite housing faces. To increase the length of the truss arms,

two E blocks are placed back to back at the end of one strip and held together with a screw as

shown in figure 13. This allows the arms to be extended by another strip length once control

of a single arm is achieved. Rotating the E block 90 degrees about the fixed screw at the main

housing allows the two strips to be aligned parallel to the vertical axis producing another

configuration for control to be applied.

Figure 13: Two E Blocks Back to Back to Extend Truss Arm Length

Final Report ___________________________________________________________________________________


To increase the flexibility of the truss arms, the E blocks concept is extended. Instead of

blocks that hold inserts for two strips, the blocks hold only one strip as shown in figure 14, a

maximum of four of these individual blocks can be held to a housing face. These blocks

contain a single strip insert and lips that hold screws for housing placement. A central screw

acts as a rotational pivot to rotate the blocks to any angle to change the orientation of the strip.

Figure 14: Single Strip Block Design with Lip

This design is inadequate in that the screws do not allow rotation of the block and hence

rotation of the strip, without the removal of each individual screw. This is a laborious task to

perform each time the truss formation requires alteration. A new design for mounting the

block is therefore required in order to allow the beam to be rotated with ease. This is achieved

in the final housing design concept.

2.2.2.3.3 Final Housing Design

The final housing design also consists of a thin rectangular rod rather than a large, heavy

block and thus requires less force and momentum to rotate.

The final design of the main housing in single strip formation is shown in figure 15. To

minimize the required size of the housing, printed circuit board plates can be attached either

side of the main housing to allow up to four truss arms to be attached.

Final Report ___________________________________________________________________________________


Figure 15: Final Housing Design in Single Strip Formation

C Blocks

C blocks are the used to connect the strips to the main housing. The C Blocks are an extension

of the E Block design used in housing design concepts 1 and 2. A single strip is held in the C

block using a wedge as shown in figure 16.

Figure 16: Final Housing Concept: C blocks with Vice Technology

Final Report ___________________________________________________________________________________


The wedge block is clamped to the C block using a grub screw. A single screw positioned in

the centre of the block in used in conjunction with a spring washer and top hat bush bearing.

To hold the block securely to the housing while allowing the blocks to be rotated relative to

the housing by hand.

To increase the length of the truss arms, two C blocks are placed back-to-back separated by a

plate made from the strip material. The plate is positioned between C blocks to provide a

smooth surface for each block to rotate against. The C-blocks are held to the plate by a single

central screw, and rotation is possible via a spring washer and top hat bush nut in one C

block, and a single top hat bush nut in the second C block. Rotation is possible because the

top hat bush nut is threaded on the inside but has a smooth outer surface, which is in contact

with the smooth inside of the drilled hole.

Slide Along Blocks

The shakers, accelerometers and thrusters are attached to the truss via the top face of slide

along blocks. These blocks sit onto the strips and can be repositioned. They can be fastened to

the strip with a grub screw. This design feature allows the position of the accelerometers,

shakers and thrusters to be altered. This is advantageous as it allows the shakers to be placed

at different locations to excite the structure into its resonance frequency depending on the

mode of the truss formation and the truss configuration itself. It also allows the

accelerometers to be placed at locations where they can easily detect the vibration of the strip

and for the thrusters to be located where they are capable of maximum control.

For single strip formation, the accelerometers, shakers and thrusters are attached to a C block

at one end of the strip, as shown in figure 17 When the truss structure becomes more

complicated and the position of the sensors and actuators needs to be altered, the sliding

blocks will be used to attach the jet thrusters and accelerometers to the strips.

Final Report ___________________________________________________________________________________


Figure 17: Air Jet Thruster, Measurement and Excitation Devices Attached to the Truss Arm

The final test rig design is shown in figure 18.

Figure 18: Final Test Rig Design

Final Report ___________________________________________________________________________________


3 Excitation and Structural Vibration

Forced excitation must be applied to the structure to test the rig response to the controller in

suppressing the structural vibrations. Vibrations are possible in any system with mass and

stiffness (Vibrations, 2004). Vibrations are the result of energy being transferred between

kinetic and potential energies in a system. Most real systems have many modes of oscillation,

all of which are associated with a natural resonance frequency (Vibrations, 2004). Resonances

are inherent properties of the system and therefore depend on the systems material properties

and boundary conditions. Each mode is defined by a natural or resonance frequency and a

mode shape (Schwarz, 2005).

In this project, individual beam elements are first controlled before increasing the complexity

of the truss formations. Beams are capable of transverse, longitudinal and torsional vibrations

(Vibrations, 2004) as depicted in figure 19. Transverse vibrations of beams are those

vibrations that are perpendicular to the length of the beam. Longitudinal beam vibrations are

deflections of the beams in the longitudinal direction or along its length where as torsional

vibrations of beams are caused by vibrations in an angular direction about the beam centre

axis in a plane perpendicular to the cross section of the shaft (Vibrations, 2004). In this

project, transverse vibrations of beams are investigated.

Figure 19: Transverse, longitudinal and torsional beam vibrations

Final Report ___________________________________________________________________________________


3.1 Technical Elements of Excitation and Vibration

The rig is to be excited and the subsequent vibration measured and then controlled. This

requires a number of technical elements including shakers, accelerometers and air jet

thrusters. The performance and properties of each of these elements are discussed below.

3.1.1 Method of Forced Excitation: Shakers

Excitation of structures can be transient, random (continuous, burst or periodic) or sinusoidal.

Impact testing is the most common form of forced excitation but in situations where the

material is delicate or requires a force of a specific duration, shakers are an excitation solution

(Peeters, 2000). Excitation types determine the method and are described as follows:

� Shakers are an inexpensive excitation method

� If a structure has low frequency modes, those below 1 Hz, excitation with a shaker is

the most suitable option. Ambient or drop excitation sources are not a solution for this

situation

� If continuous monitoring is required, ambient excitation should be used. If

intermittent monitoring is required, impact testing is a solution (Peeters, 2005)

For exciting the test rig arms, a shaking device is the most suitable option. The beam has low

frequencies of vibration and requires a force with a specified duration to excite the structure.

Shakers are used to produce a wide range of controllable vibrations in systems (BKSV, 2005).

They allow the possible simultaneous excitation of a number of points along the rig arms.

Shakers can excite the system either mechanically (electromechanical shaker) or hydraulically

(electrohydraulic shaker) (Tongue, 1996). The most common shaker is the electromechanical

shaker and that is the device to be used in this project. Electromechanical shakers vibrate due

to an alternating current. To excite the system, shakers are capable of producing a time

varying force which they apply onto the structure causing it to vibrate (Tongue, 1996). As the

electrical input signal into the shaker is controllable, a variety of input waveforms are possible

including random and sinusoidal (Tongue, 1996). Shakers are commonly attached to a

structure using a stinger which is a long slender rod that imparts the excitation force onto the

structure along the axis of the stinger (Schwarz, 2005).

Final Report ___________________________________________________________________________________


Disadvantages of shakers include that securing the shaker to the system is difficult (Schwarz,

2005) and shakers have their own mass and stiffness that could possibly affect the response of

the structure (Tongue, 1996). This is an important consideration when designing the structure

and when locating the shaker on the structure. Shakers have many advantages over other

excitation devices such as a variety of excitations are possible at a range of force levels and

with a high degree of repeatability (Tongue, 1996).

The shakers are placed onto aluminium blocks so that their position remains unfixed. This

allows the shakers to be placed at different locations to excite the structure into its resonance

frequency depending on the mode of the truss formation and the truss configuration itself.

3.1.2 Measuring Vibration: Accelerometers

The accelerometers selected to detect arm vibrations are the ADXL202JQC dual axis

accelerometers from Analog Devices. Each accelerometer is mounted on a small circuit board

that is attached to the aluminium blocks. Accelerometers measure the acceleration at a single

point and in both the horizontal and vertical direction (Schwarz, 2005) and as such the

accelerometers need to be positioned in such a way as to measure the transverse accelerations

of the test rig arms. For this reason, the accelerometers mounted on the aluminium blocks can

be repositioned to points of maximum vibration. For control of the first mode of vibration in a

single arm arrangement, the accelerometers are located at the free end of the strip to measure

the point of maximum acceleration.

3.1.3 Air Jet Thrusters

Previous projects have implemented air jet thrusters, active members or a combination of both

as the actuators. Each device has its own advantages. Air jet thrusters are efficient in

controlling large structural vibrations as they produce a force that is aligned with the

displacement of the structure caused by the vibrations (Casella, 1996). Piezoelectric actuators

expend only renewable electrical energy but have disadvantages including an inefficiency

compared to jet thrusters. This is because the piezoelectric actuators require high forces due to

the production of a bending moment over the diameter of the structure (Casella, 1996).

In this project air jet thrusters are implemented as the controller actuators as stated in the

project specification. Many space control systems utilise jet thrusters for altitude or shape

control (Dozio, 2002). Air jet actuators are on/off devices in that they operate discontinuously

Final Report ___________________________________________________________________________________


in two states. Air jet thrusters produce a constant force that is very simple to implement using

digital switching control (Dozio, 2002).

Each actuator consists of an electrovalve that opens to let air flow through and closes to

prevent air passing through a nozzle. When opened, the air force produced has the same

direction as the nozzle axis (Dozio, 2002). In previous research the on/off action of the

actuators is controlled by relays driven by the outputs of a PC. To prevent discharges when

the relay is opened, each electrovalve is coupled with a diode (Dozio, 2002).

The actuators are attached to aluminium blocks so that the position of the thrusters can be

altered. For control of the first mode of vibration in a single test rig arm, a shaker, an

accelerometer and an air jet thruster are located on a single block positioned at the free end of

the single strip to simplify the control problem as shown in figure 17.

Final Report ___________________________________________________________________________________


4 Control Design

A controller is designed to suppress the vibrations in the rig by controlling its arm movement

using air jet thrusters as actuators. A predictive control algorithm has been selected for this.

The required computational procedure to generate the control input using predictive control is

subsequently formulated. This computational procedure is then implemented in MATLAB

and its ability to suppress vibrations in a simulated system is demonstrated.

4.1 Selection of Control Method

Advancements in modern aerospace structures have seen a minimisation in their structural

weight and the use of materials with a low rigidity. Consequently, such structures are

extremely flexible and sensitive to excitation of the low range frequency modes (Bianchi,

2005). Active control methods can be used to damp these low range structural vibrations,

obtaining better performance than can be achieved by passively altering the structure

(Bianchi, 2005).

Selection of the most suitable active control method for this application is based on a number

of factors including controller robustness, the practicality of the design and the adaptivity of

the controller to a number of structural configurations and time varying system parameters

(Bianchi, 2005).

The controller design must meet the required robustness in order to function in the presence

of a number of model uncertainties (Bianchi, 2005). A large flexible space structure has an

infinite number of vibration modes which can be excited at different times (Bianchi, 2005).

Due to the high computational load caused by such a large number of vibration modes, the

system must be approximated by a reduced-order model. Approximating the system may lead

to control laws which can go unstable. Therefore controller robustness is an important design

factor (Bianchi, 2005). Depending on the type and location of the actuators and sensors,

different control strategies have varying performance and robustness of the closed loop

behaviour (Ehmann, 2005).

The practicality of the control method includes not only satisfying the practical and technical

limitations of the system hardware and software, but also the control engineering and

programming abilities of the project group members.

Final Report ___________________________________________________________________________________


Lastly, the need for the control method to be adaptive is due to both the unknown system

parameters and a changing structural configuration. The system plant is often unknown and

hence on-line identification is required through an adaptive control approach (Bianchi, 2005).

Also construction in space, thermal distortion and reorientation of subsystems requires system

parameters to be adjusted before the control commands are calculated, generating the need for

an adaptive control approach (Bianchi, 2005).

Previous research has seen a number of active control techniques applied to large space

structures and these include sliding mode control or variable structure control and predictive

control. These techniques have both been successful in suppressing structural vibrations in a

laboratory built test structure and hence the control method employed here will be a

previously tested control method to ensure successful use in this application.

4.1.1 Sliding Mode Control

Sliding Mode Control (SMC) consists of a control law that switches with infinite speed to

drive the system on a specified state trajectory, called the sliding surface. This control type is

fundamentally non-linear and is a type of pulse modulated control system (Allen, 1999). The

system can be shaped using the sliding surface and defining the relative importance of the

each mode according to the available control power. Experimental results using Sliding Mode

Control were found to be very satisfactory when compared with those obtained in previous

studies using different control methods on the same structure (Allen, 1999).

The simple on/off control action required when using air jet thrusters as actuators is preserved

using sliding mode control (Allen, 1999). The resulting control action is robust in the

presence of system uncertainties and external disturbances (Allen, 1999).

4.1.2 Predictive Control

Predictive control is an adaptive control technique in which the predicted system response at a

certain number of time steps into the future is used to determine the present control action

(Phan, 1998). Predictive control can be formulated in state-space form as well as using input

output model (Phan, 1998).

Final Report ___________________________________________________________________________________


4.2 Selected Control Method

The selected control method is predictive control. The motivation for selecting predictive

control, other than possessing the required robustness and adaptability, is that the

computational procedure is clearly outlined in ‘Predictive Controllers for Feedback

Stabilization’ (Phan, 1998).

The important features of predictive control are illustrated in figure 20.

Figure 20: Predictive control block diagram (Kvaternik, 2000)

The system has control inputs u, measured outputs y and is subject to disturbances d and

measurement noise. The two steps involved in the predictive control strategy are

1. Identification of the system

2. Use of the identified model to produce a controller (McGinley, 1997)

Performing system identification makes the predictive control method adaptive. The model

used in system identification is the Autoregressive Moving Average with eXogenous input

(ARMAX or for short ARX). This model is then used for design of the predictive control law.

Predictive control is a robust control method as system identification is performed in the

presence of disturbances which, as such do not have to be modelled separately. The control

approach is therefore both feedforward and feedback (Kvaternik, 2000).

Output (y) Input (u)

uid = random input

Plant (System)

System Identification

Predictive Control (Feedforward/Feedback)

Disturbances (d)

Final Report ___________________________________________________________________________________


The ARX model states that the current output at time step k is a combination of previous input

and output measurements. The relationship between past input and output measurements are

described in the ARX model of the following form

)()2()1()()2()1()( 2121 pkukukupkykykyky pp −++−+−+−++−+−= βββααα ……

(Phan, 1998)

This ARX model has order p and coefficients αi and βi that can be identified from the input

and the output data. These parameters are then used to determine the predictive control

algorithm. The coefficients are calculated during system identification. When the ARX model

is used to describe an observer, parameters αi and βi are termed the Markov parameters (Phan,

1998). The characteristics of this observer change with model order. With no noise present in

the system, the ARX model of minimum order relates to a deadbeat observer. In the presence

of noise, the ARX model of larger order relates to an observer with minimised prediction

error (Phan, 1998).

4.3 Predictive Control Mathematical Formulation

The predictive control method is explained in ‘Predictive Controllers for Feedback

Stabilization’ (Phan, 1998). The mathematical formulation to design the predictive controller

from system input-output data is outlined below.

The first step in computing the predictive control input is system identification. The ARX

model used in system identification has the form

)()2()1()()2()1()( 2121 pkukukupkykykyky pp −++−+−+−++−+−= βββααα ……

where p is the selected ARX model order.

For an nth order SISO system the coefficients αi and βi are scalars and for an nth order MIMO

system with m outputs and r inputs, each αi is an m x m matrix and each βi is an m x r matrix.

The coefficients of the ARX model can be determined from the input and output data. For a

set of l data points, the ARX model at each time step is described by

Y=PV

where P is a (1 x 2p) matrix of ARX coefficients

[ ]pp αβαβ ,,,,P 11 …=

(1)

(2)

(3)

Final Report ___________________________________________________________________________________


and Y and V, which are (1 x (l-p)) and ((2p-2) x ( l-p)) matrices respectively, which are

comprised of past input and output data

[ ])(,),1(),(Y lypypy …+=

−

−

−−

−−

=

)()1()0(

)()1()0(

)1()()1(

)1()()1(

V

plyyy

pluuu

lypypy

lupupu

⋯

⋯

⋮⋮⋮⋮

⋯

⋯

Therefore the ARX parameters can be computed from

P=YV+

Where + denotes the pseudoinverse of a non square matrix.

In order for the controller to take dynamic output feedback form, the ARX model Y=PV takes

the form

)1()1()1()( 111 −+−+−= kzkukyky βα

where

)()()()1(

)3()3()3()2(

)2()2()2()1(

1

3332

2221

pkzpkupkypkz

kzkukykz

kzkukykz

pppp −+−+−=+−

−+−+−=−

−+−+−=−

− βα

βα

βα

⋮

Shifting the time indices gives

)()()()1(

)()()()1(

)()()()1(

)()()()1(

1

3332

2221

111

kzkukykz

kzkukykz

kzkukykz

kzkukyky

pppp ++=+

++=+

++=+

++=+

− βα

βα

βα

βα

⋮

This process puts the system in observable canonical form

(4)

(5)

(6)

(7)

(8)

(9)

(10)

Final Report ___________________________________________________________________________________


)()(

)()()1(

kzCky

kuBkzAkz

p

pp

=

+=+

where

000

000

00

00

,

)(

)(

)(

)(

)( 3

2

1

1

2

1

=

=

−

⋯

⋮⋱⋮⋮⋮

⋯

⋯

⋯

⋮

p

p

p

I

I

A

kz

kz

kz

ky

kz

α

α

α

α

=

= 000 ,

3

2

1

⋯

⋮

ICB p

p

p

β

β

β

β

From the state equations, the state of the system at each successive time step can be written as

z(k+1) = Apz(k) + Bpu(k)

[ ]

[ ]

−+

++=+−+

+

+=+++=+

−+−

)(

)1(

)(

,,,)()1(

)1(

)(,)()1()()()2(

1

22

rnku

ku

ku

BBABAkzArnkz

ku

kuBBAkzAkuBkuBAkzAkz

pppp

rn

p

rn

p

pppppppp

⋮⋯

⋮

The state )1( +−+ rnkz is driven to zero and the record of previous inputs needed to bring

state z(k) to zero is the solution of

(11)

(12)

Final Report ___________________________________________________________________________________


[ ]

[ ] )(

)(

)1(

)(

,,,

)(

)1(

)(

,,,)(0

1

1

kzA

rnku

ku

ku

BBABA

rnku

ku

ku

BBABAkzA

rn

ppppp

rn

p

pppp

rn

p

rn

p

+−−

−+−

−=

−+

+

−+

++=

⋮⋯

⋮⋯

The state at k is therefore a linear combination of inputs u(k) to u(k+n-r). For a single-input

systems, the state z(k) can be brought to zero in n steps corresponding to a deadbeat

controller. Multi-input systems allow the state to be brought to zero in a faster time. The

deadbeat controller in state-feedback form is

)()( kzGku D=

Where the deadbeat gain GD places the eigenvalues of Ap+BpGD at the origin. The deadbeat

solution requires excessive control effort and therefore it is required that the state of the

system is brought to zero within q steps, where q>n-r+1 is the prediction horizon. Substituting

q into (12) gives

[ ]

−+

−++=+ −

)1(

)2(

)(

,,,)()(1

qku

qku

ku

BBABAkzAqkz pppp

q

p

q

p

⋮⋯

The minimum control energy solution which does not saturate the actuators is obtained from

setting )( qkz + to zero. Therefore

−+

−+

)1(

)2(

)(

qku

qku

ku

⋮= [ ] )(,,,

1kzABBABA

q

ppppp

q

p

+−− ⋯

This solution motivates a q-step predictive controller based on ARX representation of model

order p. The predictive control law is of the form

)()( ),( kGzuku qp ==

(12)

(13)

(14)

(15)

(16)

Final Report ___________________________________________________________________________________


where G is the first r-row partition of - [ ] q

ppppp

q

p ABBABA+−

,,,1⋯ . Therefore, for a SISO

system, to find u(k), G is the 1st row of - [ ] q

ppppp

q

p ABBABA+−

,,,1⋯ .

In the state space domain, the system is governed by

)()()1( kzGBAkz pp +=+

G is divided into p partitions to convert (16) to dynamic output feedback form

[ ] ,,,, 321 pggggG …=

where each gi is r x m and therefore the predictive controller becomes

kzgkzgkzgkygku ppqp ()()()()( 123121),( −++++= … )

⋮

… )1()1()2(

)2()1()1()1()1()1()(

3

3222221

+−++−++−

+−+−+−=−+−+−=

pkupkyku

kykukykzkukykz

pp βαβ

αβαβα

)1()1()(1 +−++−=− pkupkykz ppp βα

Finally, the controller takes the compact dynamic output feedback form

)()2()1(

)()2()1()(

21

21),(

pkuHkuHkuH

pkyGkyGkyGku

p

pqp

−++−+−

+−++−+−=

…

…

where the controller matrices have the form

[ ]

[ ] pp

p

p

p

p

gGggggG

ggggG

α

α

α

α

α

α

α

α

α

13

2

1

13212

3

2

1

3211

,,,,,,

,,,,,

=

=

=

− …

⋮

…

⋮

…

(17)

(18)

(19)

Final Report ___________________________________________________________________________________


[ ]

[ ] pp

p

p

p

p

gHggggH

ggggH

β

β

β

β

β

β

β

β

β

13

2

1

13212

3

2

1

3211

,,,,,,

,,,,,

=

=

=

− …

⋮

…

⋮

…

The dimension of gi is r x m, αi is m x m and βi is m x r; therefore, the dimension of each Gi is

r x m and each Hi is r x r.

4.4 Computational Procedure

The computational procedure is the process required to determine the predictive control input

for each time step from input and output measurements. The following steps, shown in flow

chart format in Appendix C, have been implemented in a MATLAB M-file to generate the

controller output at each time step based on the mathematical formulation in section 4.3.

1. Specify the controller parameters:

p = ARX model order

q = prediction horizon

l = number of data points

2. Perform system identification using SYS_ID_FUNC:

2.1 Form data matrices Y and V and determine the coefficients of the ARX

model in matrix P using equations (2) to (6)

2.2 Place the system in observable canonical form as shown in (10)

3. Perform predictive control algorithm using PREDICTIVE_FUNC:

3.1 Compute the gain matrix G from equation (16)

(20)

Final Report ___________________________________________________________________________________


3.2 Compute the controller gain matrices Gi and Hi from equation (20)

3.3 Evaluate the control law given by equation (19)

The complete M-file is included in Appendix C.

4.5 Predictive Controller Hand Calculations

To demonstrate the computational procedure in which the predictive control input is

calculated, hand calculations are performed for the first three time steps for the system shown

in figure 21.

Figure 21: Mass/spring/damper system (Cazzolato, 2003)

Defining the system

The system is defined by system matrices

[ ] 0D 0 1 1

0

B

1 0

==

=

−−= C

mm

d

m

kA

k= m = d =1

Defining a continuous-time system

SYSc = SS(A, B, C, D)

where sample rate and interval are

fs = 100

Ts = 1/fs

Final Report ___________________________________________________________________________________


Computing the discrete-time model (zoh)

SYSd = c2d(SYSc, Ts)

The system matrices therefore become

[ ] 0D 0 1 00995.0

4.893x10B

0.99 00995.0

.009950 1 -5

==

=

−= CA

with a sampling time of 0.01s

System Parameters

Selecting the system parameters to be

l = 3, set of data points

q = 2, the prediction horizon

p = 2, the order of the ARX model

r = 1, the number of inputs

Input Data

Producing a random binary input of column length 10

u_new = idinput(11, 'rbs', [0 1], [])

−

−

−

−

=

1

1

1

1

1

1

1

1

1

1

1

_ newu

The initial state of the system is defined to be

Final Report ___________________________________________________________________________________


=0

00x

Simulated Response

The discrete simulated system response to the random input is generated

[y, x] = dlsim(SYSd.a, SYSd.b, SYSd.c, SYSd.d, u_new, x0)

=

=

0.0373 0.0026

0.0478 0.0022

0.0382 0.0017

0.0286 0.0014

0.0389 0.0011

0.0293 0.0007

0.0195 0.0005

0.0097 0.0003

0.0198 0.0002

0.0100 0.0000

0 0

x

0026.0

0022.0

0017.0

0014.0

0011.0

0007.0

0005.0

0003.0

0002.0

0

0

y

System Identification performed off-line

System identification, in which the coefficients of the ARX model are found, is performed

off-line in order to reduce the computational time during control. Data matrices Y and V are

generated by retrieving values from matrix y of simulated system outputs and from controller

output matrix u_new.

Producing data matrix Y from past input data gives

[ ] [ ]-3-3 0.3458x10 0.1987x10 y(3) y(2) ==Y

Note: The first element of y is y(0), therefore y(p) = y(2 ) is the 3rd element of matrix y

Producing data matrix V from past input and output data gives

Final Report ___________________________________________________________________________________


−

=

=

0 0

1 1

.00020 0

1 1

)1( )0(

)1( )0(

)2( )1(

)2( )1(

yy

uu

yy

uu

V

The pseudo inverse of V is found to be

=+

0 0.5 0.0001 0.5-

0 0.5 0 0.5 V

Determining matrix P of ARX coefficients from P = YV+ gives

[ ]

]8.6170x10 100.2723x 3.9303x10 100736.0[

0 0.5 0.0001 0.5-

0 0.5 0 0.5 0.3458x10 0.1987x10

9-3 8-3

3-3-

−−−=

•=

x

P

Therefore the ARX coefficients are

β1 = 3100736.0 −− x

α 1 = -83.9303x10 ≈ 0

β 2 = 3100.2723x −

α 2 = -98.6170x10 ≈ 0

System in Observable Canonical Form

Using the ARX coefficients, the system is placed into observable canonical form

=

=

= 1

01

0

1

22

1

p

pp

C

BAβ

β

α

α

=

−=

=

−

−

01

102723.0

100736.0

0 8.6170x10

1 3.9303x10

3

3

9-

-8

p

pp

C

x

xBA

Computing Gain Matrix G

Final Report ___________________________________________________________________________________


Computing matrix [ ]21 , ggG = , which is the first row of [ ] 2, pppp ABBA

+−

−=

=

−•

=

−

−−

−

−

−

313-

33

13-

3

3

3

9-

-8

102723.0 6.3407x10-

100736.0102723.0 ,

6.3407x10-

102723.0

102723.0

100736.0

0 8.6170x10

1 3.9303x10

x

xxBBA

x

x

xBA

ppp

pp

Taking the pseudoinverse of matrix [ApBp,Bp]

[ ]

−=− +

36-

33

103.6731 8.5546x10

100.9928 106731.3,

x

xxBBA ppp

=

•

==

−

7- 8-

-77

9-

-8

9-

-82

0.0862x10 3.9303x10

10 x 0.3930 100862.0

0 8.6170x10

1 3.9303x10

0 8.6170x10

1 3.9303x10 xAAA ppp

Computing [ ] 2, pppp ABBA

+−

[ ]

××

××=

××

××•

××

××−=−

−+

3-12-

3-3-

7- 8-

-77

36-

332

100.0317- 101.3177-

100.1529- 10 0.0317-

100.0862 103.9303

10 0.3930 100862.0

103.6731 108.5546

100.9928 106731.3, pppp ABBA

Therefore, G, which is the first row of [ ] 2, pppp ABBA

+− , is found to be

[ ]33 101529.0 100317.0 −− −−= xxG

g1 = -0.317 x 10-4

g2 = -0.153 x 10-3

Computing Controller Gain Matrices

Computing the controller gain matrices G1, G2 and H1, H2 gives

Final Report ___________________________________________________________________________________


[ ]

[ ]734-

2

7

3

3

33

1

119-3-

2

11

9-

-8

33

1

100862.0102723.0 10-0.317

103930.0102723.0

100736.0101529.0 100317.0

100273.08.6170x10 10-0.1529

102562.0108.6170

103.9303101529.0 100317.0

−−

−

−

−−−

−

−−−

×−=×××=

×−=

×

×−•×−×−=

×−=××=

×−=

×

×•×−×−=

H

H

G

G

Computing Predictive Control Input

Therefore, the predictive controller input is

8

7

71111

7

71111

2121

107.4

1100862.0

1103930.00022.0100273.00026.0102562.0

)9(100862.0

)10(103930.0)9(100273.0)10(102562.0)11(

)2()1()2()1()(

−

−

−−−

−

−−−

×≈

−××−

−××−××−××−=

××−

××−××−××−=

−+−+−+−=

u

uyyu

kuHkuHkyGkyGku

Applying the Predictive Control Input to the System

Applying control to system using

)()(

)()()1(

kxCky

kuBkxAkx

p

pp

=

+=+

where the initial state

x0 = x(end, :)’

0373.010325.3

0373.001)1(

10325.3

0373.0107.4

102723.0

100736.0

0.0373

0.0026

0 8.6170x10

1 3.9303x10)1(

0.0373

0.0026

11

11

8

3

3

9-

8-

0

=

•

=

=•

−+

•

=

=

−

−−

−

−

xy

xx

x

xx

x

Final Report ___________________________________________________________________________________


This new output value y and the new control input value u_new become the most recent

entries of the initial simulated matrix y and the initial random control matrix u_new

respectively. The oldest and hence the first entry in matrix y and u_new are removed so that

they remain a constant size.

Therefore, matrices u_new and y become

=

×

−

−

−

−

=

− 0.0373

0.0026

0.0022

0.0017

0.0014

0.0011

0.0007

0.0005

0.0003

0.0002

0

107.4

1

1

1

1

1

1

1

1

1

1

_

8

ynewu

Calculating the Predictive Control Input and Corresponding System Output for the Second

Time Step

As system identification is performed offline such that αi and βi and matrices Ap, Bp and Cp are

the constant, the control matrix G and hence control gain matrices G1, G2, H1, and H2 also

remain constant. Therefore, all that remains to be recalculated in the next time step is the new

control input and the corresponding system output.

The new predictive control input is

9

7

871111

7

71111

2121

1068

1100862.0

107.4103930.00026.0100273.00373.0102562.0

)10(100862.0

)11(103930.0)10(100273.0)11(102562.0

)10()11()10()11()12(

-x.

u

uyy

uHuHyGyGu

≈

−××−

×××−××−××−=

××−

××−××−××−=

+++=

−

−−−−

−

−−−

The resulting controlled output is

Final Report ___________________________________________________________________________________


)()(

)()()1(

kxCky

kuBkxAkx

p

pp

=

+=+

The new input control value u and output value y become the most recent values in matrices

of previous input data u_new and output data y. The oldest or first element of these matrices

is then removed to ensure their size remains constant. Therefore, the input and output

matrices become

=

×

−

−

−

−

=

−

−

99

8

104986.1

0.0373

0.0026

0.0022

0.0017

0.0014

0.0011

0.0007

0.0005

0.0003

0.0002

1068

107.4

1

1

1

1

1

1

1

1

1

_

x

y

x.

newu

-

Calculating the Predictive Control Input and Corresponding System Output for the Third

Time Step

The next control action can then be calculated from

14-

87

9711911

7

71111

2121

-1.0926x10

107.4100862.0

106.8103930.00373.0100273.0104986.1102562.0

)11(100862.0

)12(103930.0)11(100273.0)12(102562.0)13(

)2()1()2()1()(

≈

×××−

××−××−××−=

××−

××−××−××−=

−+−+−+−=

−−

−−−−−

−

−−−

xx

u

uyyu

kuHkuHkyGkyGku

The resulting controlled output is

Final Report ___________________________________________________________________________________


)()(

)()()1(

kxCky

kuBkxAkx

p

pp

=

+=+

Once again, the new input value u and output value y become the most recent entires in the

matrix of past input data u_new and output data y. This process of calculating the new

predictive input and corresponding output is continued for the defined number of time steps.

4.6 System Identification of a 3-DOF System

System identification, using the ARX model, is performed on a 3-DOF system in MATLAB.

The aim of this system identification is to determine the effectiveness of the ARX model

when modelling a real system and its ability to faithfully reproduce the system in the presence

of noise.

System identification is performed by altering the ARX model order p and for the following

three cases

• without noise in the system

• with a small amount of noise of the order of 1/100 of the size of the input added to the

system

• with noise increased to 1/10 the size of the input added to the system

9

10-

9-

10-

9-

9

3

3

119-

8-

10498.13.1907x10

1.4986x1001)1(

3.1907x10

1.4986x10106.8

102723.0

100736.0

10325.3

0373.0

0 8.6170x10

1 3.9303x10)1(

−

−

−

−

−

=

•

=

=•

−+

•

=

xy

xx

x

xx

10

18

10

18

10

14

3

3

10-

9-

9-

8-

1019.3

10938.9

10190.301)1(

10938.9

10190.3100926.1102723.0

100736.0

3.1907x10

1.4986x10

0 8.6170x10

1 3.9303x10)1(

−

−

−

−

−

−

−

−

=

•

=

=−•

−+

•

=

x

x

xy

x

xxx

xx

Final Report ___________________________________________________________________________________


The three degree of freedom flexible system used for system identification is

ukxxcxM =++ ɺɺɺ

=

−

−+−

−+

+

−

−+−

−+

+

3

2

1

3

2

1

33

3322

221

3

2

1

33

3322

221

3

2

1

3

2

1

0

0

0

0

00

00

00

u

u

u

x

x

x

kk

kkkk

kkk

x

x

x

cc

cccc

ccc

x

x

x

m

m

m

ɺ

ɺ

ɺ

ɺɺ

ɺɺ

ɺɺ

mNsccc

mNkkk

kgmmm

/1.0

/1000

1

321

321

321

===

===

===

ui = control force applied to each mass

The positions of the three masses are yi = xi, i=1,2,3.

In order to model the three degree of freedom system, substitution of state variables of is

performed as follows

)3(

3323

)2(

23212

)1(

1211

...1.01.0

...1.02.01.0

...1.02.0

uxxx

uxxxx

uxxx

=+−

=−+−

=−+

ɺɺɺɺ

ɺɺɺɺɺ

ɺɺɺɺ

)()( 12

1

11

1

12

211

11

kzCzuMkxxCuMxz

zxz

xz

−−=−−==

==

=

−−ɺɺɺɺ

ɺɺ

)()( 34

1

22

1

24

423

23

kzCzuMkxxCuMxz

zxz

xz

−−=−−==

==

=

−−ɺɺɺɺ

ɺɺ

)()( 56

1

33

1

36

635

35

kzCzuMkxxCuMxz

zxz

xz

−−=−−==

==

=

−−ɺɺɺɺ

ɺɺ

Resulting in state space equations:

DuCxy

BuAxx

+=

+=ɺ

In matrix form, for the SISO case, where the position of mass three is the output

Final Report ___________________________________________________________________________________


=

−−

−−

−−

=

0

0

0

0

1

0

1.010001.0100000

100000

1.010002.020001.01000

001000

001.010002.02000

000010

BA

0010000 =

= DC

System identification is performed by firstly generating a random binary excitation input. The

resultant plant output generated by the random input is produced using the lsim command.

The digitised input and output time histories at l = 100 data points are used to form the data

matrices Y and V. The matrix of ARX coefficients P is then determined by solving the

multiplication of data matrix Y and the pseudoinverse of data matrix V.

The normalised difference between the ARX model and the plant output y is determined and

the results are presented graphically for the three cases. Modelling the plant and the ARX

model against separate input data is a better indication of the validity of the model as it

suggests that the model will be valid irrespective of the input data selection. Hence this is a

good indication that close predictions will be obtained. Therefore the plant and the ARX

model are compared using an impulse response and a random input signal different to that

used to generate the ARX model.

4.6.1 Results for System Identification Performed Without Noise in the System

System identification is firstly performed with an ARX model order of p = 4. With this model

order, the sum of squared errors between the plant output and the ARX model is 1.5 x 10-6.

Figure 22 displays the output of the ARX model and the plant respectively, when subjected to

random input data. Figure 23 shows the impulse responses of ARX model and plant. Figure

24 illustrates their response to a random noise signal.

Final Report ___________________________________________________________________________________


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3

4x 10

-3

time [s]

model (dotted), plant (solid)

Response to random noise signal, matching initial states

ARX

y

Figure 22: Output of the ARX Model and the Plant

Figure 23: Impulse Responses of ARX Model and Plant

Final Report ___________________________________________________________________________________


Figure 24: ARX Model and Plant Response to a Random Noise Signal

The plant is sixth order with three modes of vibration. Each of the modes is second order. The

characteristics of the plant are shown in table 6.

Table 6: Plant Characteristics

Eigenvalue Damping Frequency (rad/s)

-9.90 x 10-3 + 14.1i 7.04 x 10-4 14.1

-9.90 x 10-3 – 14.1i 7.04 x 10-4 14.1

-7.77 x 10-3 + 39.4i 1.97 x 10-3 39.4

-7.77 x 10-3 – 39.4i 1.97 x 10-3 39.4

-1.62 x 10-3 + 57.0i 2.85 x 10-3 57.0

-1.62 x 10-3 – 57.0i 2.85 x 10-3 57.0

From the characteristics of the system, three spectral peaks corresponding to the three modes

of vibration are expected to be located at the following frequencies

14.1 rad/s ≈ 2.24 Hz

39.4 rad/s ≈ 6.27 Hz

57.0 rad/s ≈ 9.07 Hz

Final Report ___________________________________________________________________________________


The power spectral density of the plant output signal y, in figure 25, displays three peaks

corresponding to the three modes of vibration occurring at the frequencies as

expected.

0 5 10 15 20 25 30 35 40 45 5010

-8

10-7

10-6

10-5

10-4

10-3

Pxx - X Power Spectral Density

Frequency

Figure 25: Plant Power Spectral Density

As the plant has three modes of vibration and each mode is second order, an ARX model

order of p < 6 prevents all of the modes from being captured. For an ARX model order of p =

4 as used in figures 22 to 24, the first two modes are captured and but the third mode is not.

This results in a high variance between the plant output and the ARX model. If p < 6 then not

all of the modes of the three degree of freedom system are captured, resulting in a high

variance between the plant output and the ARX model. Therefore, only an ARX model order

value of p ≥ 6 is valid for the sixth order system.

Using an ARX model order of p = 6 results in a variance of 2.28 x 10-22 between the ARX

model and plant, with the results shown graphically in figures 26 to 28.

Final Report ___________________________________________________________________________________


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



Final Report ___________________________________________________________________________________



In the absence of noise, using an ARX model of order p = 6 corresponds to a deadbeat

observer. Graphical results for changing the model order from p = 4 to p = 100 are

documented in Appendix D. It can be seen from results in Appendix D that as the model order

is increased, the variance between the ARX model and the plant decreases. In the presence of

no noise, a model order of p = 6 is adequate to reproduce the plant faithfully.

4.6.2 Results for System Identification with a Small Amount of Noise in the System

In order to determine the robustness of the ARX model in the presence of a noise and to

examine its performance with different input signals, a random binary noise signal of size 1 x

10-6 is added to the plant output. The ARX model is then generated from the plant response

with added noise.

To make the identification of the ARX model robust with respect to noise, a higher order

ARX model is necessary. An ARX model order of p = 6 produces a variance of 6.90 x 10-6

between the plant and the ARX model in the presence of a small noise signal, with results

shown graphically in figures 29 to 31. This is a much larger variance than was achieved for

the same model order with no noise in the system.

Final Report ___________________________________________________________________________________


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-3

time [s]



ARX

y



Final Report ___________________________________________________________________________________



With noise added to the system, a minimum ARX model order of p = 10, resulting in a

squared error of 2.94 x 10-9 between the plant and the ARX model, is required to reproduce

the plant faithfully, as shown in figures 32 to 34.

Final Report ___________________________________________________________________________________


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

3

4x 10

-3

time [s]



ARX

y



Final Report ___________________________________________________________________________________



Results for changing the model order from p = 4 until p = 100 are documented in Appendix

D. Graphical results in Appendix D demonstrate that as the model order is increased, the error

also decreases.

4.6.3 Results for System Identification with a Large Amount of Noise in the System

To examine the performance of the ARX model in the presence of a larger noise signal, the

size of the random noise signal added to the plant is increased to 1 x 10-4. Therefore, the noise

signal is 1/10th the size of the input signal. The ARX model is then generated from the plant

with the increased noise signal added.

A higher model order than was previously adequate, is required to make the identification of

the ARX model robust with respect to the large noise signal added to the system. For a model

order of p = 6, the variance between the plant and ARX model with a large of amount noise in

the system, is found to be 3.35 x 10-6. System identification results are shown graphically in

figures 35 to 37 for an ARX model order of p = 6 and with a large amount of noise in the

system. This is a larger variance than was achieved for a model order of p = 6 in the previous

two cases.

Final Report ___________________________________________________________________________________


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6x 10

-3

time [s]



ARX

y



Final Report ___________________________________________________________________________________



To reproduce this signal faithfully, the required ARX model order is p = 20. For an ARX

model order of p = 20, the variance between the signals produced by the plant and the ARX

model is 1.24 x 10-8 with graphical results shown in figures 38 to 40.

Final Report ___________________________________________________________________________________


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y


0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4

5x 10

-4

time [s]


Impulse response


Final Report ___________________________________________________________________________________


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

-3

time [s]


Response to random input noise signal


Results for changing the model order from p = 4 until p = 100 are documented in Appendix

D. The graphical results demonstrate that as the model order is increased, the variance also

decreases when a large noise signal is added to the system.

4.6.4 Summary of System Identification Results

The variance between the ARX model and the plant for the three cases and for increasing

values of ARX model order p are documented in table 7.

Final Report ___________________________________________________________________________________


Table 7: Variances for the three cases between the plant and the ARX model of increasing model

order p values

p Variance

(without noise)

Variance

(with noise)

Variance

(with increased noise)

4 1.50 x 10-6

2.72 x 10-6 6.67 x 10-6

6 2.27 x 10-22

6.90 x 10-6 3.35 x 10-6

10 1.68 x 10-25

2.94 x 10-9 3.03 x 10-6

16 6.11 x 10-27

9.23 x 10-13 2.03 x 10-7

20 3.25 x 10-29 2.05 x 10-12

1.24 x 10-8

30 2.23 x 10-31 2.62 x 10-14

3.55 x 10-10

50 1.89 x 10-31

2.63 x 10-14 4.03 x 10-10

80 1.95 x 10-31 2.31 x 10-14

2.39 x 10-10

100 1.76 x 10-31 3.10 x 10-14

4.00 x 10-10

Figure 41 shows the results in table 7 represented graphically. Figure 41 demonstrates that as

noise is added to the system and then increased in amplitude to 1 x 10-6, the variance between

the plant and the ARX model for a particular ARX model order p also increases. The figure

shows that above the optimum ARX model order of p = 25, the variance remains almost

constant for increasing model order. Increasing the ARX model order to above a value of 25

does not decrease the variance between the plant and the ARX model to the extent that the

added computation time is warranted.

Final Report ___________________________________________________________________________________


Figure 41: Log of Variance Versus ARX Model Order p

Log of Variance for Model Order p

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

020

40

60

80

100

120

p

Variance

Log of Variance Without noise

Log of Variance With a Small Noise Added

Log of Variance With a Large Noise Signal Added

Final Report ___________________________________________________________________________________


4.7 Performing System Identification Off-line

An important consideration is whether the system should be re-identified and the control law

matrices re-calculated each sampling interval. System identification and re-calculation of the

control law requires considerable computational effort and hence it would be desirable to

minimise the number of times system identification has be performed. System identification

would need to occur each period for accurate vibration suppression, if the coefficients of the

plant vary considerably in from period to period.

By monitoring consecutive values of the ARX coefficients for a model order of p = 25 and for

a prediction horizon of q = 20, it can be determined whether system identification should be

performed off-line or in real time. Graphical results of comparing the coefficients of the ARX

model for a model order of 25 and a prediction horizon of 20 is shown in figures 42 and 43.

0 5 10 15 20 25-1

-0.5

0

0.5

1

1.5

2x 10

-3 Beta Variance

Beta number

Beta Value

Figure 42: Beta Values for p = 25 and q = 20

Final Report ___________________________________________________________________________________


Figure 43: Alpha Values for p = 25 and q = 20

Figures 42 and 43 demonstrate that there is a minimal variance between the ARX coefficients

calculated for the first 20 time steps. Therefore, in order to reduce the computational time,

system identification will be performed off-line.

4.8 Controller Results for the Three Degree of Freedom System

As system identification demonstrated the ARX model faithfully reproduces the plant, the

final step to ensure that the controller suppresses structural vibrations is to apply the

controller to the sixth order system discussed in section 4.6. A deadbeat controller with p = q

= 6 and a controller with an ARX model order of p = 6 and prediction horizon of q = 50 are

applied to the three degree of freedom system.

4.8.1 Results for Deadbeat Control

The deadbeat controller takes the form

u(k) = GDz(k)

Final Report ___________________________________________________________________________________


where the deadbeat gain GD places the eigenvalues of the closed loop system Ap+BpGD at the

origin of the complex plane (Phan, 1998). The deadbeat controller brings the system to zero in

the fastest possible time. The smallest order ARX model corresponds to p = pmin =6. Using p=

6 and q = qmin =6 corresponds a deadbeat controller.

The deadbeat controller is applied to the simulated plant response generated from a sinusoidal

input signal. The corresponding input and output histories are shown in figure 44. The first 10

time steps are the uncontrolled plant response, after which the controller is switched on and

the controlled plant response is shown in blue.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3Closed Loop Response

Time Step

Output (red = uncontrolled, blue = controlled)

Final Report ___________________________________________________________________________________


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06Control Input

Time Step

Input u (red = controller off, blue = controller on)

Figure 44: Deadbeat Controller Input and Output Control Histories

Figure 44 demonstrates that vibration suppression occurs in approximately 4 time steps. This

is the fastest time to drive the system to zero.

The deadbeat controller places the eigenvalues of the closed loop system Ap+BpGD at the

origin of the complex plane. The eigenvalues of the closed loop system are found to be

0.6232 x 10-3 + 0.36 x 10-3 i

0.6232 x 10-3 - 0.36 x 10-3 i

-0.0003 x 10-3 + 0.7195 x 10-3 i

-0.0003 x 10-3 - 0.7195 x 10-3 i

-0.6230 x 10-3 + 0.3595 x 10-3 i

-0.6230 x 10-3 - 0.3595 x 10-3 i

which are located approximately at the origin of the complex plane as shown in figure 45.

Controller Output

Final Report ___________________________________________________________________________________


-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real Part

Imaginary Part

Eigenvalues of (Ap+Bp+G)z(k)

Figure 45: Eigenvalues of Ap+BpGD

Therefore the deadbeat controller successfully suppresses structural vibrations in the plant in

minimal time.

4.8.2 Results for the Controller With p = 6, q = 50

The prediction horizon is now increased to determine the time to suppress structural

vibrations with a larger prediction horizon. In this case, the controller is computed with p = 6

and q = 50 and then applied to the plant generated from the same sinusoidal input as used in

section 4.7.1. The corresponding input and output control histories are shown in figure 46.

Final Report ___________________________________________________________________________________


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Figure 46: Input and Output Control Histories for a Controller with p = 50 and q =6

Closed Loop Response

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time Step

Output Plant (red = uncontrolled, blue = controlled)

Closed Loop Response

Controller Input

Time Step

Input (red = uncontrolled, blue = controlled)

Controller Output

Final Report ___________________________________________________________________________________


In this case, vibration suppression successfully occurs in approximately 14 time steps. This is

as compared to suppression occurring in approximately 4 times steps when using a deadbeat

controller. While it takes longer for vibration suppression to occur, the magnitude of the

control gains does not appear to be reduced by increasing the prediction horizon. This is

unexpected as increasing the prediction horizon should decrease the control energy required.

Final Report ___________________________________________________________________________________


5 Experimental Setup

The final project objective is to implement the controller on the test rig. The experimental

setup used to achieve this aim is outlined below including the control process and

implementation through software and hardware.

5.1 Initial Excitation of Strips

Before the controller is applied, the test rig arms must be excited. In order to demonstrate the

full effect of the controller, it is necessary to cause maximum vibrations in the strips. This

requires that the strips be excited at their resonant frequency. The resonant frequency is found

from accelerometer measurements of the strips in vibration. The period of the accelerometer

output is used to estimate the resonant frequency. In single strip form, the period is 700 ms

and thus the resonance frequency is 1.42 Hz as shown in figure 47.

Figure 47: Accelerometer Data of Strip in Vibration

5.2 Method of Forced Vibration

Final Report ___________________________________________________________________________________


It has been found through experimentation that the shakers are unsuitable for exciting the test

rig arm at its natural frequency. The shakers have a switching frequency that is too high to

excite the strips at resonance. Instead, the air jet thrusters are used to excite the test rig arms.

5.3 Control Process

The process required to control the first mode of vibration in the test rig when in single strip

formation is shown in figure 49. This is performed at a sampling frequency of 10 Hz.

Figure 48: Steps Required to Control the First Mode of Vibration in the Test Rig

Step 1: Excitation of test rig arms

Pulses are sent to the air jet thrusters to excite the arms at their

resonant frequency

Step 2: Excitation is stopped

Air jet thrusters are switched off

Step 3: System identification

Requires the sensor and actuator I/O data and is performed off-line.

Step 4: Excitation of the test rig arms

Using the same pulse as generated in step 1

Step 5: Apply the Controller

Final Report ___________________________________________________________________________________


5.4 Software

To implement the controller on the test rig, the process described in section 5.3 is

implemented in a SIMULINK block diagram together with I/O blocks. A build process can

then be started using the RTMC9S12 –Target, which generates C-code from the block

diagram and then compiles and downloads the code onto the microcontroller.

5.4.1 RTMC9S12 – Target and SIMULINK

In order to implement the developed algorithms in real-time, a host-target software program,

RTMC9S12 –Target, is used to transform a SIMULINK block diagram in MATLAB 6.5.1

into C-code, which can then interface with the microcontroller (Wornle, 2005). Using

Mathworks’ Real-Time Workshop and the Embedded Coder, a SIMULINK block diagram

can be turned into C-code and then compiled and downloaded into the Flash ROM of the

Dragon-12 microcontroller (Wornle, 2005).

SIMULINK is an environment in which the physical system and developed algorithms can be

modeled as a block diagram. For real time applications, I/O blocks are connected to the

sensors and actuators, which in this case are accelerometers and air jet thrusters. Signal data is

saved and displayed using the MATLAB host PC and the parameters can be changed from the

host PC while the target is running in real-time.

5.4.2 Graphical User Interface (GUI)

A GUI is a pictorial interface to a program. It makes programs user friendly by providing

intuitive controls such as sliders, push buttons and displays as shown in figure 49. When a

user clicks on a button, the MATLAB code that implements the function of the button is

executed. The GUI has displays for the control and accelerometer data and a button that starts

each of the following modes when pressed:

1. Excitation

2. Stop vibrations

3. System identification

4. Controller on

Final Report ___________________________________________________________________________________


Each of the modes is an enabled subsystem on the SIMULINK target program, with the

exception of system identification, which is implemented in an M-file. Pressing a button

determines which subsystem is enabled, either off mode, excitation mode or control mode.

The M-file detailing call back functions of the GUI and the target SIMULINK block diagram

are shown in Appendix F.

Figure 49: The GUI

5.4.3 Target Program and its Subsystems

The target program has an enabled subsystem for each mode and converts the accelerometer

data from analogue to digital.

5.4.3.1 Excitation Mode

The test rig arms are excited using the air jet thrusters. A square wave is generated at twice

the resonant frequency and is sent to the pulse pins of the air jet thrusters. The sign of the

pulse is switched at twice the frequency to switch the air jet thrusters in alternate directions.

The excitation mode SIMULINK block diagram is shown in Appendix F.

Final Report ___________________________________________________________________________________


5.4.3.2 Off Mode

To turn the thrusters off after excitation mode, a zero signal is sent to the actuators pulse and

sign pins. The off mode SIMULINK block diagram is shown in Appendix F.

5.4.3.3 Control Mode

The control mode generates the control input to be applied at each time step. Calculation of

the control input requires sensor and actuator I/O data and the gain matrices G and H

calculated in system identification. The magnitude and sign of the control input is sent to the

air jet thrusters through the pulse and sign pins respectively. The control mode SIMULINK

block diagram is shown in Appendix F.

5.4.3.4 System Identification

The predictive control algorithm implemented in real-time requires an identified system. As

the coefficients of the ARX model do not vary greatly over a given time period as discussed

in section 4.7, system identification is performed off-line and directly in a MATLAB M-file.

System identification is performed to determine the coefficients of the ARX model and hence

the gain matrices, G and H used to calculate the control input. The actuator and sensor I/O

data is used in system identification. The system identification M-file is shown in Appendix

F.

5.4.3.5 Accelerometers

The accelerometers measure the acceleration of the test rig arms in both the x and y direction.

The accelerometers are an analogue device and hence require analogue to digital conversion

achieved by using a 10-bit analogue to digital converter in the target program. This output

data is used in system identification and in calculating the new control input.

Final Report ___________________________________________________________________________________


5.5 Hardware

The shaker, air jet thrusters and the accelerometers require connections to the microcontroller

and to a power supply.

5.5.1 Power Supply

The shakers, air jet thrusters and accelerometers all require a power supply. The shakers and

air jet thrusters require power from an external voltage source. The required voltage for the

shakers is 5 V and 20 V is required for the air jet thrusters. The accelerometers are powered

directly by the on board microcontroller power supply.

5.5.2 Test Rig Connections to Microcontroller

The microcontroller used is the Wytec Dragon-12 Board of type MC9S12DP256 as shown in

figure 50.

Figure 50: Wytec Dragon-12 Board Type: MC9S12DP256 EVB (Wornle, 2005)

Final Report ___________________________________________________________________________________


The two air jet thrusters require a PWM and direction connection to the microcontroller as

shown in figure 51. The air jet thrusters are connected to the microcontroller PWM port -

PORT H to pins 0 to 3 to generate the PWM and direction signals.

Figure 51: Air Jet Thruster and Shaker Connections

The two accelerometers measure acceleration in both the x and y directions. The

accelerometers monitor the vibrations and convert them into an analogue electrical signal.

The accelerometers are therefore an analogue device and hence require analogue to digital

conversion. To perform analogue to digital conversion, each accelerometer is connected to

two pins of the ATD bank 0 to convert each of the measurements in the x and y direction to

digital.

A circuit diagram displaying all power and microcontroller connections is shown in appendix

G.

Power and ground connection to external power supply

PWM and direction connection to microcontroller

Air jet thruster H - bridge

Shaker H - bridge Accelerometer

Final Report ___________________________________________________________________________________


6. Results and Discussion

To determine the ability of the controller to suppress structural vibrations, the test rig is

initially excited and then the controller is applied. Currently however, when control mode is

activated, an incorrect thruster output is generated. This is likely to be due to incorrectly

identifying the system.

To determine whether the error was in calculating the control input in the SIMULINK target

program, the subsystem is tested using dummy input variables. When the dummy inputs are

used, the control input is correctly calculated in the SIMULINK control mode subsystem.

This indicates that the inputs to the control algorithm generated through system identification

are incorrect.

The test rig plant is estimated using the ARX model and this is used in the real-time algorithm

for determining the control input signal. Therefore, to determine why the controller does not

suppress vibrations, the system identification results using I/O data are examined.

6.1 System Identification Results

System identification is performed using the experimental I/O data and a sampling frequency

of 10Hz. An I/O data set of l = 100 samples is used to generate the ARX model. The accuracy

of the model is expressed through the variance value. For an ARX model order of p = 2, the

variance between the plant and the ARX model is 0.030 and results are shown graphically in

figure 52.

Final Report ___________________________________________________________________________________


Figure 52: ARX Model and Plant Output for a Model Order of p = 2

Despite the variance being of the order of 10-2, the ARX model is not a good fit, as shown in

figure 52. An ARX model order of p = 2 should be adequate to replicate the plant as it is a

simple system when excited to its first mode of vibration. However, a model order of p = 6 is

tested to determine if this improves the model fit. Changing the ARX model order to p = 6

produces a variance of 0.032. Changing the ARX model order to p = 6 has little effect on

improving the model fit as shown in figure 53.

Final Report ___________________________________________________________________________________


Figure 53: ARX Model and Plant Output for a Model Order of p = 6

Contributions to a poor model fit possibly include:

1. Excitation of a single mode

2. Poor I/O data

3. The presence of non-linearities in the system

6.1.2 Excitation of a Single Mode

Rather than exciting all of the modes, the test rig is excited at its resonant frequency.

Excitation of a single mode could explain why the ARX model is a poor fit. This is because

the other modes of vibration are not used in generating the ARX model. To test if this is the

case, system identification is performed on a first order linear plant with excitation of only a

single mode. The input is a square pulse generated at the resonant frequency of the plant.

The first order plant:

1.0

42.12

=

=

ζ

πωn

Final Report ___________________________________________________________________________________


0 01 1

0

2

10

2=

=

=

−−= DCBA

nn ζωω

An ARX model order of p = 6 is used as this should adequately capture the 1st order plant.

The variance between the plant and the ARX model is 3.93 x 10-30. Figure 54 displays the

ARX model and the plant when subjected to the square pulse input. Figure 55 shows the

impulse responses of the ARX model and the plant. Figure 56 illustrates their response to a

random noise signal.

Figure 54: ARX model and Plant When Subjected to Square Pulse Input

Final Report ___________________________________________________________________________________


Figure 55: Impulse Responses of the ARX Model and the Plant

Figure 56: Response to a Random Noise Signal

Final Report ___________________________________________________________________________________


The small variance demonstrates that the ARX model replicates the first order plant

accurately when it is excited at a single frequency. To ensure the validity of this result, the

ARX model is generated for a third order plant excited at its natural frequency. The input is a

square pulse generated at the resonant frequency of the plant.

The third order plant:

0 010101

1

0

1

0

1

0

)12.11(6 )12.11( 0 0 0 0

1 0 0 0 0 0

0 0 )72.3(4 )72.3( 0 0

0 0 1 0 0 0

0 0 0 0 2

0 0 0 0 1 0

1.0

42.12

2

2

2

=

=

=

−−

−−

−

=

=

=

DCB

A

n

nn

nn

n

ωζω

ωζω

ζωω

ζ

πω

An ARX model order of p = 6, produces a variance of 4.68 x 10-27 between the plant and the

ARX model with results shown graphically in figures 57 to 59.

Final Report ___________________________________________________________________________________


Figure 57: ARX model and Plant When Subjected to Square Pulse Input

Figure 58: Impulse Responses of the ARX Model and the Plant

Final Report ___________________________________________________________________________________


Figure 59: Response to a Random Noise Signal

System identification of the first and third order systems excited at their natural frequencies

produces a good model fit. This demonstrates that excitation of a single frequency should not

be the cause of the inaccurate ARX model produced when using the test rig I/O data.

6.1.3 Poor I/O Data

The poor model fit is possibly due to incorrect I/O data. That is, the I/O data is not

representative of the physical system response. MATLAB’s System Identification Toolbox,

IDENT, is used to determine whether the ARX model is a good fit to the I/O data. The first

half of the I/O data is used as the working data set and the second half of the data set is used

as validation data. The different ARX models with their corresponding fits to the output data

are

ARX210: na = 2, nb =1 and nk = 0, produces a fit of 90.19 %

ARX220: na = 2, nb =2 and nk = 0, produces a fit of 91.99 %

ARX660: na = 6, nb = 6 and nk = 0, produces a fit of 92.81 %

Final Report ___________________________________________________________________________________


The ARX models generated by IDENT have a low variance between the plant and the ARX

model. Therefore, the I/O data is sufficient in generating an accurate ARX model. The results

are shown graphically in figure 60.

Figure 60: ARX Models Generated Using IDENT Compared to Output Data

The ARX model of order p = 6 generated through the system identification M-file and the

ARX model generated by IDENT, ARX660 are now compared. The model coefficients

obtained by the two methods are compared in table 8.

Table 8: Comparison of ARX Coefficients

ARX Model A

Coefficients

IDENT ARX660

Model A Coefficients

ARX Model B

Coefficients

IDENT ARX660

Model B Coefficients

1 1 0 -0.00920

-0.411 -0.0619 0.00160 3.15 x 10-5

0.0139 0.0422 -0.00440 -0.00387

0.389 0.212 0.000300 0.00266

0.219 0.145 -0.00120 -0.00456

-0.0182 -0.00116 0.00510 0.00480

-0.176 -0.493

The ARX model has coefficients comparable to those of the ARX660 model. The difference

between the coefficients is not large enough to suggest that the poor model fit is caused by

incorrect I/O data.

ARX660 ARX210 ARX220 Output

Final Report ___________________________________________________________________________________


6.1.4 System Non-Linearities

It is highly likely that the poor model fit is due to non-linearities in the system. The controller

is designed assuming a linear system. This is done because it greatly simplifies the analysis of

the system. However, all of the systems in the real world have some non-linear components,

thus the model will always be an approximation of the system behaviour. If the non-linearities

in the plant are pronounced, the linear ARX model cannot reproduce the plant faithfully. It

rather replicates the linear elements of the system.

A system is linear near equilibrium. The system is furthest from its equilibrium when it is

excited at its resonant frequency and hence is largely non-linear. The test rig must be excited

at its resonant frequency due to limitations of the sensors and actuators. Amplitudes of

acceleration must be large enough to be measured by the accelerometers and be of

comparable magnitude to the force of the air jet thrusters to be controlled.

In order to determine if non-linearities in the system have caused the poor model fit, system

identification is performed using the third order plant with falsified non-linear output

measurements y produced by adding random data. The variance between the plant and the

ARX model is 0.0752 with results shown in figure 61.

Figure 61: ARX Model and Plant Using Falsified Output Measurements

Final Report ___________________________________________________________________________________


This suggests that system non-linearities are a cause of the poor model fit. The system is only

linear when it is excited to small vibrations, thus accurate system identification of the plant is

possible at small vibrations. As discussed previously, this is not a viable option due to

limitations on the actuators and sensors.

A system is dynamically stable when it responds to perturbations in a proportionate way. In

unstable systems, perturbations are of out of proportion to the size of the system. The

departure of the linear ARX model from the non – linear system may lead to instabilities in

the ARX model.

The poles of the ARX model are

0.6354 + 0.7778i 0.6354 - 0.7778i 0.6080 -0.3797 + 0.5882i -0.3797 - 0.5882i -0.7033 A pole on the unit circle indicates instability. The pole at –0.7 is close to the unit circle and

hence indicates possible instability of the ARX model.

When the ARX model is generated using the test rig I/O data with a set of 100 data points, the

instabilities are evident, as shown in figure 62. It can be seen that the ARX model tracks the

plant locally, but after approximately 15 seconds, the ARX model becomes unstable. If it is

stable, it is expected that the ARX model will go to zero regardless of the initial conditions.

The simulation however, shows that the ARX model approaches infinite amplitude.

Physically, this is impossible. The test rig has physical limitations such as saturation that

prevent it from becoming unstable. Saturation is not present in the ARX model however, so it

becomes unstable when excited at its natural frequency.

Final Report ___________________________________________________________________________________


Figure 62: Instabilities of the ARX Model

This indicates that the reason for the controller inadequately suppressing structural vibrations

is due to the non-linearities in the plant. System identification is performed using a linear

ARX model on a system that is highly non-linear when excited at its resonant frequency. This

leads to the ARX model being a poor fit. With an inaccurately modelled system, the controller

will be unable to suppress the structural vibrations.

Final Report ___________________________________________________________________________________


7 Achievement of Project Aims and Future Work

This project is being investigated as a direct result of successful completion of previous

projects conducted by NASA and at the Politecnico di Milano. However, the likelihood of

producing a working model with active vibration control that is comparable to that of

previous projects at the Politecnico di Milano, while desirable, is not highly realistic as the

resources available and time span spent on the project were far more extensive than are

utilised here. In this project, the level of success achieved in applying active vibration control

to the structure have been dependent on the design of the test rig itself and in the details and

application of the control system.

The aim of this project has been to investigate active vibration control of a large, flexible

space structure and to implement a control system that will damp vibrations in a test rig. In

order to achieve this aim, the following project stages have been completed:

1. Test Rig Design

A test rig simulating the flexible properties of space station structure has been designed and

constructed.

2. ANSYS Simulation

To determine the feasibility of the test rig design and its response to excitation, a modal

analysis of the truss arms has been conducted on ANSYS.

3. Controller Design

A controller has been developed in MATLAB using predictive control, before being

implemented on a microcontroller.

While having been performed, the final project goal of controller implementation on the test

rig still requires future work to ensure that the controller adequately suppresses vibrations

7.1 Future Work

In the previous sections, project aims and conclusions based on results achieved so far were

discussed. From the discussion, a number of points to be addressed in the future can be

identified. The future work is summarised below.

Final Report ___________________________________________________________________________________


1. Before any future experiments can be carried out, the controller must first

successfully suppress vibrations in the test rig. This requires that the system identification

technique be corrected. Either a non-linear model needs to be used to identify the system or

the test rig needs to be excited to smaller, measurable amplitudes of vibration that can be

controlled by the force of the air jet thrusters.

2. Once control of the test rig in single strip formation has been achieved, the controller

should be applied to a more complicated structure to ensure validity. This includes application

to the test rig in four arm formation and in extended arm formation as shown in figures 8 and

9 in section 2.1.3. Another test rig configuration is suggested by the rectangular truss structure

used at the Politecnico di Milano. The test rig could be modified to a rectangular truss

structure by attaching cross arms to the strips in four arm formation.

3. Once successful implementation of the controller has been achieved, the controller

should be applied to suppress the first few vibration modes. Successful control of the different

modes will require altering the location of the accelerometers and the air jet thrusters.

4. The accelerometers could be replaced by position transducers. Presently, using

acceleration measurements as the output data means that the acceleration would be driven to

zero by the controller. In order to drive the position to zero, acceleration needs to be

integrated twice. Using position transducers would eliminate the noise introduced by the

double integrations.

A further extension goal is to apply different control methods to the test rig. This would allow

direct comparisons to the vibration suppression obtained using Predictive Control. In

particular, the effectiveness of various control methods to adapt to different structural

configurations can be explored. During system identification, the system is modelled using an

ARX model. Future extension goals may also include experimentation with different model

types.

Final Report ___________________________________________________________________________________


8 Costing of the Fabrication and Materials

Material costing was kept to budget due to the great majority of the materials being sourced

from workshop.

The C blocks and the slide along blocks have been constructed using workshop resources. A

total of 20 C blocks and accompanying top hat bush nuts and wedges and the 16 slide along

blocks have been produced. The materials to manufacture these elements include:

� Plastic sheets to produce the strips for the test rig arms

� Fasteners including:

� Grub screws M4 x 6 x 36

� Aluminium to produce:

� C blocks x 20

� Wedges x 20

� Top hat bush nuts x 20

� slide along blocks x 16

The following complete elements were also constructed using workshop resources:

� Rotational coupling device

� Housing block

� Base plate

� Shakers

� Air Jet thrusters

Items that have been purchased with the project budget are detailed on table 9.

Final Report ___________________________________________________________________________________


Table 9: Price Table Detailing Items and Cost

Item Price

Carbon Fibre Rod $ 12.00

Accelerometers x 5 $ 21.35 x 5 = $106.75

Pan recessed head screws M3 x 16 1 bag at $ 1.40

Plain Washers M6 1 bag at $ 1.20

Spring Washers M6 1 bag at $ 2.60

Total $ 123.95

Final Report ___________________________________________________________________________________


9 Conclusion

This project investigates active vibration control in a large, flexible space structure. A test rig

has been designed to simulate the flexible conditions of a space structure and a controller has

been designed to provide active vibration suppression in the test structure using the ARX

model and predictive control. Air jet thrusters have been installed on the test structure. The air

jet thrusters are used to excite the test rig arms to its natural resonance and are employed by

the controller to suppress the structural vibrations.

Initial simulation results of a three degree of freedom system suggested that the controller

accurately suppresses structural vibrations. When the controller was applied to the test rig

however, non-linearities in the system prevented an accurate controller output.

Even though the controller was not successfully implemented as intended, valuable

experience was gained from the process of learning how to use the RTMC9S12 – Target and

the Wytec Dragon-12 Board. With this knowledge, and the test results from the work

completed this year, development of a working prototype that suppresses structural vibrations

in the test rig is highly feasible.

Final Report ___________________________________________________________________________________


10 References

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Space Station, Control Engineering Practice No 8 (2000) pp.861-871, Dipartmento di

Informatica e Sistemistica, Pavia, Italy

BKSV, 2005: <http://www.bksv.com/2796.asp>, Denmark

Bernelli-Zazzera F, Dozio L, Mantegazza P, 2000: Multi-Pulse-Width-Modulated Control of a

Large Flexible Structure, Journal of Chinese Society of Mechanical Engineers, Vol. 21, No.1,

pp.77-85 (2000), China

Casella F, 2002: Modeling and Control for Vibration Suppression in a Large Flexible

Structure with Jet Thrusters and Piezoactuators, IEEE Transactions on Control Systems

Technology, Vol 10. No 4, July 2002 pp 589-598, Dipartimento di Electronica e

Informazione, Politecnico di Milano, Italy

Casella F, Locatelli A, Schiavoni N, 1996: Nonlinear Controllers For Vibration Suppression

in a Large Flexible Structure, Control Engineering Practice, Vol.4, No.6, pp.791-806,

Dipartimento di Electronica e Informazione, Politecnico di Milano, Italy

Cityplastics, 2005:<http://www.cityplastics.com.au/pvc.html>

CNCI, 2005: <http://cnci.org.za/inf/leaflets_html/fibre.html>

Dimarogonas A, 1996: Vibration for Engineers, 2nd Ed, Prentice Hall, New Jersey, USA

Final Report ___________________________________________________________________________________


Dozio L, Bernelli Zazzera F, Mantegazza P, 2002: Experimental Analysis of the Multi-Pulse-

Width Modulation Control of a large Space Structure Using On/Off Jet Thruster, Active 2002

July 15-17, Politecnico di Milano, Italy

Dupont Plastics, 2005: <http://plastics.dupont.com/NASApp/myplastics/Mediator?id=0>

Ebianchi, L. Dozio, G.L. Ghiringhelli, P. Mantegazza, 2005: Complex Control Systems,

Applications of DIAPM-RTAI at DIAPM, Dipartmento di Ingegneria Aerospaziale del

Politecnico, Milano, Italy

Ehmann C, Nordmann R, 2005: Comparison of Control Strategies for Active Vibration

Control of Flexible Structure, Mechatronics and Machine Acoustics, Darmstadt University of

Technology, Darmstadt, Germany

Flowbiz, 2005: <http://www.flowbiz.com/VA/nonmetal.htm>

Friedland, Bernard 1986: Control System Design: An Introduction to State-Space Methods

McGraw-Hill

Kvaternik R, Bennett R and Juang J, 2000: Exploratory Studies in Generalized Predictive

Control for Active Aeroelastic Control of Tiltrotor Aircraft

Matweb, 2005: <http://www.matweb.com/search/SpecificMaterial.asp?bassnum=O5600>

McGinley W and Shen J, 1997: Dynamic Analysis of a Two Member Manipulator Arm:

Progress Report, USA

<http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19970017807_1997027682.pdf>

Final Report ___________________________________________________________________________________


Minh Q. Phan and Jer-Nan Juang, 1998: Predictive Controllers for Feedback Stabilization,

Journal of Guidance, Control, and Dynamics. Vol 21, No. 4, (1998)

MSE, 2005: <http://www.mse.mtu.edu/~drjohn/my4150/class14/class14.html>

Peeters B, Maeck J, De Roeck, G, 2000: Vibration-based Damage detection in Civil

Engineering: Excitation sources and Temperature Effects, Institute of Physics Publishing

Schwarz B, Richardson M, 2005: Experimental Modal Analysis, Vibrant Technology INC,

<http://www.vibetech.com/papers/paper28.pdf>, California

Smart Materials and Structures,

<http://ej.iop.org/links/q06/zGd83HQQn1A6,nX5Ndvt0w/sm1314.pdf>, Belgium

Tongue B, 1996: Principles of Vibration, Oxford University Press, New York, USA

Vander Velde W, He J 1983: Design of Space Station Control Systems Using On-Off

Thrusters, J. Guidance Vol 6, No.1, Jan-Feb 1983 pp53-60, America

Wornle, F. 2005: RTMC9S12-Target: A Simulink Target for Real-Time Control Using

Freescale MC9S12DP256B/C Microcontrollers, Australia

<http://www.mecheng.adelaide.edu.au/robotics/WWW_Devs/Dragon12/rtmc9S12Target/rtm

c9S12-Target.pdf>

2004: ‘MECH ENG 3012 Dynamics and Control II (Automatic Control II) Lecture Notes’,

prepared by Ben Cazzolato, School of Mechanical Engineering The University of Adelaide

Final Report ___________________________________________________________________________________


2004: ‘MECH ENG 3012 Vibrations + MECH ENG 3028 Dynamics and Control Lecture

Notes’, prepared by Anthony Zander, School of Mechanical Engineering The University of

Adelaide

Final Report ___________________________________________________________________________________

___________________________________________________________________________________Design and Build of a Thruster Controlled Model of a Large Flexible Space Station -100-

10. Appendices

Final Report ___________________________________________________________________________________


Appendices

Appendix A Solid Edge Draft Workshop Drawings

Appendix B ANSYS Mode Shape Simulations and Session Editors for:

1. Cubic Truss

2. Triangular Truss

3. Cubic Truss with Diagonals

Appendix C Predictive Controller MATLAB M-File With 3 DOF Plant

Appendix D System Identification Results For a 3 DOF System

Appendix E Circuit Diagram

Appendix F Software MATLAB M-Files and SIMULINK Block Diagrams

Final Report ___________________________________________________________________________________


Appendix A Solid Edge Draft Workshop Drawings

Final Report ___________________________________________________________________________________


Figure A1: C Block - Back to Back Type

Final Report ___________________________________________________________________________________


Figure A2: C Block - Main Housing Type

Final Report ___________________________________________________________________________________


Figure A3: Sliding Block

Final Report ___________________________________________________________________________________


Figure A4: Top Hat Bush Type 1

Final Report ___________________________________________________________________________________


Figure A5: Top Hat Bush Type 2

Final Report ___________________________________________________________________________________


Figure A6: Wedge

Final Report

___________________________________________________________________________________


Appendix B ANSYS Mode Shape Simulations and Session Editors

ANSYS Simulations of a Single Bay of the Cubic Truss for Modes 1 to 5

Figure B1: ANSYS Simulation of a Single Bay of the Cubic Truss for Mode 1

Final Report

___________________________________________________________________________________




Final Report

___________________________________________________________________________________


ANSYS Simulations of a Single Bay of the Triangular Truss for Modes 1 to 5

Figure B6: ANSYS Simulation of a Single Bay of the Triangular Truss for Mode 1


Final Report

___________________________________________________________________________________




Final Report

___________________________________________________________________________________


ANSYS Simulations of a Single Bay of the Cubic Truss with Diagonals for Modes 1 to 5

Figure B11: ANSYS Simulation of Single Bay of Cubic Truss with Diagonals for Mode 1


Final Report

___________________________________________________________________________________



Figure B14: ANSYS Simulation of a Single Bay of Cubic Truss with Diagonals for Mode 4

Final Report

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Figure B15: ANSYS Simulation of a Single Bay of Cubic Truss with Diagonals for Mode 5

Final Report

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Session Editor for ANSYS Simulations of a Single Bay of the Cubic Truss for Modes 1 to 5

/BATCH

/COM,ANSYS RELEASE 7.0SP11UP20030909 10:59:29 04/14/2005

RESUME,'rectangular_1','db','.'

/COM,ANSYS RELEASE 7.0SP11UP20030909 10:59:29 04/14/2005

/PREP7

!*

!*

R,1,1.96e-005,1,0.1,1,0,1,

!*

R,1,1.96e-005,1,0.1,1,0,1,

!*

/UI,MESH,OFF

FINISH

/SOL

!*

ANTYPE,2

/STATUS,SOLU

SOLVE

!*

FINISH

/PREP7

!*

R,1,1.96e-005,1,0.1,1,0,1,

!*

R,1,1.96e-005,1,0.1,1,1,1,

!*

R,1,1.96e-005,1,0.1,1,1,1,

!*

!*

R,1,1.96e-005,1,0.1,1,1,1,

!*

FINISH

/SOL

!*

ANTYPE,2

Final Report

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/STATUS,SOLU

SOLVE

FINISH

/PREP7

!*

ET,2,BEAM4

!*

KEYOPT,2,2,0

KEYOPT,2,6,0

KEYOPT,2,7,0

KEYOPT,2,9,0

KEYOPT,2,10,0

!*

!*

R,2,0.0000196, , , , , ,

RMORE, , , , , , ,

!*

R,2,1.96e-005,0,0,0,0,0,

RMORE,0,0,0,0,0,0,

!*

FINISH

/SOL

FINISH

/PREP7

!*

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDE,EX,1

MPDE,EY,1

MPDE,EZ,1

MPDE,NUXY,1

MPDE,NUYZ,1

MPDE,NUXZ,1

MPDE,PRXY,1

MPDE,PRYZ,1

MPDE,PRXZ,1

MPDE,GXY,1

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MPDE,GYZ,1

MPDE,GXZ,1

MPDATA,EX,1,,1E+009

MPDATA,PRXY,1,,0.3

FINISH

/SOL

!*

ANTYPE,2

/STATUS,SOLU

/STATUS,SOLU

SOLVE

FINISH

/PREP7

!*

R,2,1.96e-005,1,1,1,1,1,

RMORE,1,1,1,1,1,1,

!*

R,2,1.96e-005,1,1,1,1,1,

RMORE,1,1,1,1,1,1,

!*

FINISH

/SOL

!*

ANTYPE,2

/STATUS,SOLU

SOLVE

FINISH

! /EXIT,NOSAV

Session Editor for ANSYS Simulations of a Single Bay of the Triangular Truss for Modes 1 to 5

NOPR ! Suppress printing of UNDO process

/PMACRO ! Echo following commands to log

RESUME, single_tri,db,F:\FINALY~1\ANSYS\

/PREP7

!*

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ET,1,BEAM4

!*

!*

R,1,0.5E-05,0.104E-10,0.417E-12,0.005,0.001, ,

RMORE, , , , , , ,

!*

!*

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDATA,EX,1,,1000000000

MPDATA,PRXY,1,,0.3

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDATA,DENS,1,,1410

FLST,5,9,4,ORDE,5

FITEM,5,157

FITEM,5,-162

FITEM,5,315

FITEM,5,-316

FITEM,5,321

CM,_Y,LINE

LSEL, , , ,P51X

CM,_Y1,LINE

CMSEL,S,_Y

!*

!*

CMSEL,S,_Y1

LATT,1,1,1, , , ,

CMSEL,S,_Y

CMDELE,_Y

CMDELE,_Y1

!*

FLST,2,9,4,ORDE,5

FITEM,2,157

FITEM,2,-162

FITEM,2,315

FITEM,2,-316

Final Report

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FITEM,2,321

LMESH,P51X

/UI,MESH,OFF

FINISH

/SOL

FLST,2,2,3,ORDE,2

FITEM,2,109

FITEM,2,-110

!*

/GO

DK,P51X, , , ,0,ALL, , , , , ,

!*

ANTYPE,2

MSAVE,0

!*

MODOPT,LANB,5

EQSLV,SPAR

MXPAND,5, , ,1

LUMPM,0

PSTRES,0

!*

MODOPT,LANB,5,0,1000000000, ,OFF

/STATUS,SOLU

SOLVE

FINISH

/POST1

SET,FIRST

SET,LIST

SET,FIRST

PLDISP,1

SET,NEXT

PLDISP,1

SET,NEXT

PLDISP,1

SET,NEXT

FCCH

!*

Final Report

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PLDISP,1

SET,NEXT

PLDISP,1

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

SET,PREVIOUS

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

SET,NEXT

SET,PREVIOUS

SET,PREVIOUS

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

SET,PREVIOUS

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

SET,NEXT

SET,PREVIOUS

SET,PREVIOUS

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

)/GOP ! Resume printing after UNDO process

Session Editor for ANSYS Simulations of a Single Bay of the Cubic Truss with Diagonals for

Modes 1 to 5

Final Report

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/NOPR ! Suppress printing of UNDO process


RESUME, single_diagonals,db,F:\FINALY~1\ANSYS\

/PREP7

L, 4, 8

!*

ET,1,BEAM4

!*

!*

!*

R,1,0.5E-05,0.104E-10,0.417E-12,0.005,0.001, ,

RMORE, , , , , , ,

!*

!*

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDATA,EX,1,,100000000

MPDATA,PRXY,1,,0.3

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDATA,DENS,1,,1410

FLST,5,18,4,ORDE,2

FITEM,5,1

FITEM,5,-18

CM,_Y,LINE

LSEL, , , ,P51X

CM,_Y1,LINE

CMSEL,S,_Y

!*

!*

CMSEL,S,_Y1

LATT,1,1,1, , , ,

CMSEL,S,_Y

CMDELE,_Y

CMDELE,_Y1

!*

FLST,2,18,4,ORDE,2

Final Report

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FITEM,2,1

FITEM,2,-18

LMESH,P51X

FINISH

/SOL

FLST,2,2,3,ORDE,2

FITEM,2,4

FITEM,2,8

!*

/GO

DK,P51X, , , ,0,ALL, , , , , ,

!*

ANTYPE,2

MSAVE,0

!*

MODOPT,LANB,5

EQSLV,SPAR

MXPAND,5, , ,1

LUMPM,0

PSTRES,0

!*


/STATUS,SOLU

SOLVE

FINISH

/POST1

SET,LIST

SET,FIRST

PLDISP,1

SET,NEXT

PLDISP,1

SET,NEXT

PLDISP,1

SET,NEXT

PLDISP,1

SET,PREVIOUS

SET,NEXT

Final Report

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SET,NEXT

PLDISP,1

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

SET,PREVIOUS

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

SET,PREVIOUS

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

SET,PREVIOUS

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

SET,PREVIOUS

/EFACE,1

AVPRIN,0, ,

!*

PLNSOL,U,X,1,1

)/GOP ! Resume printing after UNDO process

Final Report

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Session Editor for ANSYS Simulation of Strips

/NOPR ! Suppress printing of UNDO process


FINISH ! Make sure we are at BEGIN level

/CLEAR,NOSTART ! Clear model since no SAVE found

! WE SUGGEST YOU REMOVE THIS LINE AND THE FOLLOWING STARTUP LINES

/input,menust,tmp,'',,,,,,,,,,,,,,,,1

/GRA,POWER

/GST,ON

/PLO,INFO,3

/GRO,CURL,ON

/CPLANE,1

/REPLOT,RESIZE

WPSTYLE,,,,,,,,0

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L, 1, 2

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SECDATA,0.02,0.00125,0,0,0,0,0,0,0,0

/UI,BEAM,OFF

/REPLOT

LPLOT

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ET,1,BEAM4

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R,1,0.25E-04,0.822E-09,0.326E-11,0.02,0.00125, ,

RMORE, , , , , , ,

!*

R,1,2.5e-005,8.22e-010,3.26e-012,0.02,0.00125,0,

RMORE,0,0,0,0,0,0,

!*

!*

MPTEMP,,,,,,,,

MPTEMP,1,0

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MPDATA,EX,1,,1000000000

MPDATA,PRXY,1,,0.3

MPTEMP,,,,,,,,

MPTEMP,1,0

MPDATA,DENS,1,,1410

/UI,MESH,OFF

/UI,MESH,OFF

FINISH

/SOL

!*

ANTYPE,2

!*

MSAVE,0

!*

MODOPT,LANB,5

EQSLV,SPAR

MXPAND,5, , ,1

LUMPM,0

PSTRES,0

!*


KPLOT

FINISH

/PREP7

SMRT,6

/UI,MESH,OFF

/UI,MESH,OFF

/UI,MESH,OFF

SMRT,OFF

SMRT,6

/UI,MESH,OFF

/REPLOT

LPLOT

FINISH

/SOL

/ZOOM,1,RECT,-0.677553,0.0659231 ,-0.720783805904 ,0.0393264059479

/REPLOT

Final Report

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/AUTO,1

/REP,FAST

FINISH

/PREP7

LMESH, 1

FINISH

/SOL

FLST,2,1,3,ORDE,1

FITEM,2,1

!*

/GO

DK,P51X, , , ,0,ALL, , , , , ,

FINISH

/POST1

FINISH

Final Report

___________________________________________________________________________________


Appendix C Predictive Controller Flowcharts and MATLAB M-File

MAIN:

Figure C1: Predictive Controller Flowchart of MAIN Program

End

Y

Generate random input and apply to the system to

be controlled (plant)

Measure system output

Store i/p-o/p’s values in u_stored and y_stored

Loop i times?

Store current i/p-o/p pair

Discard oldest i/p-o/p pair

predictive_func

Calculate controller output and apply

to the plant

N

sys_id_func

Perform off line system identification

Measure system output

User to define: q (the prediction horizon)

p (order of the ARX model

l (the number of data points)

START

Final Report

___________________________________________________________________________________


SYS_ID_FUNC:

Y(i,j)=u_stored(h,1)

Increment h and j by 1

j=(l-p+1)?

N

Y

V_plus=pinv(v)

P=Y*V_plus

Set j=2

Set i=1

Return Ap,Bp,Cp,

alpha

Set h=p+1

Set j=1

j=l-p+1?

Set i=1

Set j=1

Is i even? N Y

H=p-2+j-count+1

V(i,j)=u_stored(h,1)

Increment j by 1

H=p-2+j-count+1

V(i,j)=y_stored(h,1)

Increment j by 1

j=(l-p+1)?

alpha(i,1)=P(1,j)

Bp(i,1)=P(1,j-1)

Increment i and j

i=p?

Iden=eye(p,p-1)

Ap=cat(2,alpha,iden)

Cp=eye(1,p)

N

Y

Y Y

N N

START

Final Report

___________________________________________________________________________________


Figure C2: Predictive Controller Flowchart of SYS_ID_FUNC

PREDICTIVE_FUNC:

1

Ap(q-i)

*Bp;

G_row_orig=cat(2,G_row_orig,G_ro

w_new)

N

Y

G_row_orig=Ap(q-1)

*Bp

Set i=2

i=q?

G_row=-1*pinv(G_row_orig)*(Apq)

G=G_row(1,:)

G_new=G(r,(1:p))*alpha((1:p),1)

H_new=G(r,(1:p))*Bp((1:p),1)

Set j=2

i=p?

Y

alpha_new=alpha((i:p),1)

G_next=G(r,(1:p-i+1))*alpha_new;

G_new=cat(2,G_new,G_next);

beta=Bp((i:p),1)

H_next=G(r,(1:p-i+1))*beta

H_new=cat(2,H_new,H_next)

Increment i by 1

Set u_new=0

Set i=1

u_new=u_new+G_new(r,i)*y_stored(end-i+1,1)+H_new(r,i)*u_stored(end-i+1,1)

Increment i by 1

i=p?

N

N

Y

START

Return u_new

Final Report

___________________________________________________________________________________


Figure C3: Predictive Controller Flowchart of PREDICTIVE_FUNC

Final Report

___________________________________________________________________________________


MATLAB M-File of System Identification and Control of Simulated 3-DOF System

function[u_stored, y_stored] = main() clear all % example system (see paper) m = 1; % m1 = m2 = m3 = 'm' = 1 kg c = 0.1; % c1 = c2 = c3 = 'c' = 0.1 N*s/m k = 1000; % k1 = k2 = k3 = 'k' = 1000 N/m % MM = [m 0 0; 0 m 0; 0 0 m]; % CC = [2*c -c 0; -c 2*c -c; 0 -c 2*c]; % KK = [2*k -k 0; -k 2*k -k; 0 -k 2*k]; % % A = [zeros(size(MM)), eye(size(MM)); -MM\KK, -MM\CC]; % B = [0; 0; 0; 1; 0; 0]; % input: force applied to first mass % C = [0 0 1 0 0 0]; % output: displacement (X1) of third mass (-> x3) % D = 0; A = [0 1 0 0 0 0; -2*k/m -2*c/m k/m c/m 0 0; 0 0 0 1 0 0; k/m c/m -2*k/m -2*c/m k/m c/m; ... 0 0 0 0 0 1; 0 0 k/m c/m -2*k/m -2*c/m]; B = [0;1;0;0;0;0]; % u1 C = [0 0 0 0 1000 0]; % x3 D = 0; % define continuous-time system equation of the plant SYSc = SS(A, B, C, D); % sample rate / sample interval fs = 100; Ts = 1/fs; % compute discrete-time system (zoh) SYSd = c2d(SYSc, Ts); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % STEP 1.1 SELECT VALUE OF P GOVERNING THE SPEED OF THE OBSERVER %%%%% REQUIRE USER INPUT % l is the number of data points % q is the prediction horizon % p is the order of the ARX model q = 6; p = 6; l = 10; r = 1;

Final Report

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% INITIAL INPUTS u's and OUTPUT's y's %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u_new = idinput(51, 'sine', [0 1], []); % random input column vector, length=100 %u_new=ones(101,1) % initial state x0 = zeros(size(SYSd.b)); [y, x] = dlsim(SYSd.a, SYSd.b, SYSd.c, SYSd.d, u_new, x0); % generates actual system response to random input first time y_plot=y %%%%%%%%%%%%%%%%%%%%% INITIALISATION FOR PLOTTING %%%%%%%%%%%% % initial state x0 = x(end, :)'; t2 = 0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u_stored=[]; y_stored=[]; % off-line sys-ID u_stored = [u_stored(2:end); u_new]; y_stored = [y_stored(2:end); y]; [Ap, Bp, Cp, alpha] = sys_id_func(q, p, l, r, u_stored, y_stored); % initial state x0 = zeros(size(Bp)); % u_stored = zeros(size(u_stored)); % y_stored = zeros(size(u_stored)); u_new = u_new(end); y = y(end); for m=1:40 % store current i/p-o/p pair (u_new, y) u_stored = [u_stored(2:end); u_new]; y_stored = [y_stored(2:end); y]; u_new = predictive_func(q, p, l, r, Ap, Bp, Cp, alpha, y_stored, u_stored) % function call for predictive control algorithm %u_new = 0; %%%%%%%%%%%%%%%% PLOTTING %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % t_sim = [0:Ts/10:Ts]; u_sim = [u_new];

Final Report

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%[y_sim, x] = dlsim(SYSd.a, SYSd.b, SYSd.c, SYSd.d, u_sim,x0); % MANUAL CALCULATION OF CONTROLLER OUTPUT DUE TO NEW % PREDICTIVE INPUT x=Ap*x0+Bp*u_sim; y_sim=Cp*x; % y is 1x1 y = y_sim(end,:); t2 = t2+Ts; x0 = x; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % u_new and y are now looped back into the start of program % and placed on top of u_stored and y_stored in preparation % for next loop end % ends main for loop % plot results tt = Ts*[0:length(y_stored)-1]; figure plot(tt(1:10), y_stored(1:10), 'r.-') hold on plot(tt(10:end), y_stored(10:end), '.-') figure b=tt(1:10) a=zeros(length(b)-1) plot(tt(1:9),a,'r.-') hold on plot(tt(9:end),[0;u_stored(10:end)],'.-') end % ends main function %________________________FUNCTION DEFINITIONS_________________ %_____________________________________________________________ % STORE_FUNC STORES U_NEW's AND Y's IN COLUMN VECTORS U_STORED and Y_STORED function a = store_func(in1, in2) % in1=in1(1:end-1); % keeps as circular array % in2=in2(1:end-1); % each new y found will be stored in start of y_stored, % while oldest y is deleted in keeping with "circular" column array a=[in2; in1]; % stores in column vector end

Final Report

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%____________________________________________________________ % SYS_ID_FUNC PERFORMS SYSTEM IDENTIFICATION TO PRODUCE MATRICES A,B,C function [Ap, Bp, Cp, alpha] = sys_id_func(q, p, l, r, u_stored, y_stored) % STEP 1.2 FORM DATA MATRIX Y % EQN 10 % Y is row vector containing values of column vector [y(p),y(p+1)...y(l)] Y=[]; h=p+1; % Note y(p) is actually the p+1 element of y vector for j=1:(l-p+1) % Refers to column number in Y Y(1,j) = y_stored(h,1); % Forms Y using value from input vector y h=h+1; % Increments to select value from next column of y end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% STEP 1.3 FORM DATA MATRIX V i=0; j=0; count=0; V=[]; for i=1:2*p % Refers to row number in V R=rem(i,2); % Divides row number by 2 to test whether even or odd row. if R==1 % Odd rows contain inputs for j=1:(l-p+1) % Refers to column number in V h=p-2+j-count+1; %decrements starting p-1 to zero V(i,j)=u_stored(h,1); end else % Even rows contain outputs for j=1:(l-p+1) %column number in V h=p-2+j-count+1;

% note u(p-1) is actually the pth element of % u. in this case row 2,4,6,8 V(i,j)=y_stored(h,1); end count=count+1; end end

Final Report

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% STEP 1.4 FIND MATRIX P V_plus=pinv(V); % Finds pseudo inverse of V P=Y*V_plus; % Finds matrix P %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % STEP 2. PUT SYSTEM IN OBSERVABLE CANONICAL FORM Ap=[]; Bp=[]; % Observable canonical matrices alpha=[]; % Matrix used in A % Finding alpha and B i=0; j=2; for i=1:p % Number of rows of alpha alpha(i,1)=P(1,j); Bp(i,1)=P(1,(j-1)); i=i+1; j=j+2; end iden=eye(p,(p-1)); % Identity matrix is part of matrix A Ap=cat(2,alpha,iden); % Finds matrix A Bp; % Finds matrix C Cp=eye(1,p); end %_____________________________________________________________ % PREDICTIVE_FUNC function u_new = predictive_func(q, p, l, r, Ap, Bp, Cp, alpha, y_stored, u_stored) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % STEP 3. FINDING MATRIX G AND CONTROLLER GAIN MATRICES Gi, Hi G_row_orig=Ap^(q-1)*Bp;% this is the first column for i=2:q % Subsequent columns G_row_new=Ap^(q-i)*Bp; G_row_orig=cat(2,G_row_orig,G_row_new); % columns are placed in concatenated matrix end G_row=-1*pinv(G_row_orig)*(Ap^q); % -[Ap^q-1*Bp..,ApBp,Bp]+*Ap^q (takes pseudo inverse)

Final Report

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G=G_row(1,:); % First r-row partition of -[Ap^q-1*Bp..,ApBp,Bp]+*Ap^q G_new=[]; % G's stored in a concatenated matrix, ie. [G1 G2.. Gp] H_new=[]; % H's stored in a concatenated matrix, ie. [H1 H2.. Hp] G_new=G(r,(1:p))*alpha((1:p),1); % Left-most matrix G1 H_new=G(r,(1:p))*Bp((1:p),1); % Left-most matrix H1 for i=2:p % Calculates G2, G3...Gp alpha_new=alpha((i:p),1); % smaller and smaller alpha matrices G_next=G(r,(1:p-i+1))*alpha_new; % calculates G2,G3...Gp G_new=cat(2,G_new,G_next); % stores in concatenated matrix beta=Bp((i:p),1); % beta matrices decreasing in size H_next=G(r,(1:p-i+1))*beta; % calculates H2,H3...Hp H_new=cat(2,H_new,H_next); % stores in concatenated matrix end eigz=eig(Ap+Bp*G) plot(eigz,'o','MarkerSize',6) %zplane(eigz) % STEP 4. COMPACT DYNAMIC OUTPUT FEEDBACK FORM % G_new = 1e5*[-0.992, 3.88, -6.51, 5.79, -2.71, 0.53]; % H_new = [-5.51 -17.09 -31.8 -22.92 -3.93 -0.065]; u_new=0; for i=1:p % loops for each time step u_new=u_new+G_new(r,i)*y_stored(end-i+1,1)+H_new(r,i)*u_stored(end-i+1,1); % compact dynamic output feedback controller form % u_new is 1x1 end u_new; end

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Appendix D System Identification Results For a 3 DOF System

System Identification Results For No Noise in the System

ARX Model Order p = 4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3

4x 10

-3

time [s]



ARX

y

Figure D1: Response to Random Input Signal, Matching Initial States, For No Noise in the

System with ARX Model Order p = 4

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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response

Figure D2: Impulse Response For No Noise in the System with ARX Model Order p = 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-3

-2

-1

0

1

2

3x 10

-3

time [s]



Figure D3: Response to Random Input Signal For No Noise in the System with ARX Model

Order p = 4

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time [s]




Order p = 6

ARX Model Order p =10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6x 10

-3

time [s]



ARX

y



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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

2

3

4x 10

-3

time [s]




Order p = 10

Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time [s]




Order p = 16


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-3

time [s]



ARX

y



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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-3

time [s]




Order p = 20

Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time [s]




Order p = 50


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

time [s]




Order p = 80

Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3x 10

-3

time [s]




Order p = 100

Final Report

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System Identification Performed With a Small Amount of Noise in the System

ARX Model Order p= 4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-3

time [s]



ARX

y

Figure D25: Response to Random Input Signal, Matching Initial States, with Small Amount of

Noise in the System with ARX Model Order p = 4

0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response

Figure D26: Impulse Response For Small Amount of Noise in the System with ARX Model

Order p = 4

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-3

time [s]



ARX

y

Figure D27: Response to Random Input Signal For Small Amount of Noise in the System with



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-3

time [s]



ARX

y



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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4

5x 10

-4

time [s]


Impulse response


Order p = 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time [s]





Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

-4

-3

-2

-1

0

1

2

3

4x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 10

Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

4x 10

-3

time [s]






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



Final Report

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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5x 10

-3

time [s]





Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y

igure D36: Response to Random Input Signal, Matching Initial States, with Small Amount of


0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 20

Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time [s]






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6x 10

-3

time [s]



ARX

y



Final Report

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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time [s]





Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 80

Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

2

3

4x 10

-3

time [s]






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4

-3

-2

-1

0

1

2

3

4x 10

-3

time [s]



ARX

y



Final Report

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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3x 10

-3

time [s]





Final Report

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System Identification Performed With A Large Noise Signal in the System


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y

Figure D48: Response to Random Input Signal, Matching Initial States, with Large Amount of


0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response

Figure D49: Impulse Response For Large Amount of Noise in the System with ARX Model

Order p = 4

Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time [s]



Figure D50: Response to Random Input Signal For Large Amount of Noise in the System with


ARX model Order p = 6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6x 10

-3

time [s]



ARX

y



Final Report

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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-3

-2

-1

0

1

2

3

4x 10

-3

time [s]



Figure D53: Response to Random Input Signal For Large Amount of Noise in the System with


Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4

5x 10

-4

time [s]


Impulse response


Order p = 10

Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

time [s]






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



Final Report

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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4

5x 10

-4

time [s]


Impulse response


Order p = 16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-3

-2

-1

0

1

2

3

4x 10

-3

time [s]





Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4

5x 10

-4

time [s]


Impulse response


Order p = 20

Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

-3

time [s]






0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

2

3x 10

-3

time [s]





Final Report

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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4

5x 10

-4

time [s]


Impulse response


Order p = 50

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

time [s]



ARX

y



Final Report

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-8

-6

-4

-2

0

2

4

6

8x 10

-3

time [s]



ARX

y



0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 80

Final Report

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3x 10

-3

time [s]






0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-6

-4

-2

0

2

4

6x 10

-3

time [s]



ARX

y



Final Report

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0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4x 10

-4

time [s]


Impulse response


Order p = 100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-3

-2

-1

0

1

2

3

4x 10

-3

time [s]





Final Report

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Appendix E Circuit Diagram

Final Report

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Appendix F Software MATLAB M-Files and Simulink Block Diagrams

GUI Callback Functions M-File GUI Callback Functions M-File function varargout = test2(varargin) gui_Singleton = 1; gui_State = struct('gui_Name', mfilename, ... 'gui_Singleton', gui_Singleton, ... 'gui_OpeningFcn', @test2_OpeningFcn, ... 'gui_OutputFcn', @test2_OutputFcn, ... 'gui_LayoutFcn', [] , ... 'gui_Callback', []); if nargin & isstr(varargin{1}) gui_State.gui_Callback = str2func(varargin{1}); end if nargout [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:}); else gui_mainfcn(gui_State, varargin{:}); end % End initialization code % --- Executes just before test2 is made visible. function test2_OpeningFcn(hObject, eventdata, handles, varargin) % This function has no output args, see OutputFcn. % hObject handle to figure % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % varargin command line arguments to test2 (see VARARGIN) % Choose default command line output for test2 handles.output = hObject; % Update handles structure guidata(hObject, handles); % UIWAIT makes test2 wait for user response (see UIRESUME) % uiwait(handles.figure1); % --- Outputs from this function are returned to the command line. function varargout = test2_OutputFcn(hObject, eventdata, handles) % varargout cell array for returning output args (see VARARGOUT); % hObject handle to figure % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % Get default command line output from handles structure varargout{1} = handles.output; % --- Executes during object creation, after setting all properties. function slider1_CreateFcn(hObject, eventdata, handles) % hObject handle to slider1 (see GCBO)

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% eventdata reserved - to be defined in a future version of MATLAB % handles empty - handles not created until after all CreateFcns called % Hint: slider controls usually have a light gray background, change % 'usewhitebg' to 0 to use default. See ISPC and COMPUTER. usewhitebg = 1; if usewhitebg set(hObject,'BackgroundColor',[.9 .9 .9]); else set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor')); end % --- Executes during object creation, after setting all properties. function edit1_CreateFcn(hObject, eventdata, handles) % hObject handle to edit1 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles empty - handles not created until after all CreateFcns called % Hint: edit controls usually have a white background on Windows. % See ISPC and COMPUTER. if ispc set(hObject,'BackgroundColor','white'); else set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor')); end % --- Executes on slider movement. function slider1_Callback(hObject, eventdata, handles) % hObject handle to slider1 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % Hints: get(hObject,'Value') returns position of slider % get(hObject,'Min') and get(hObject,'Max') to determine range of slider myValue = get(hObject, 'Value'); set(handles.edit1, 'String', num2str(myValue)); freePortSend(1, 115200, 1, 1, 0, myValue) % set 'f' to 'myValue' function edit1_Callback(hObject, eventdata, handles) % hObject handle to edit1 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % Hints: get(hObject,'String') returns contents of edit1 as text % str2double(get(hObject,'String')) returns contents of edit1 as a double myValue = str2num(get(hObject,'String')); if(~isempty(myValue))

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myMax = get(handles.slider1, 'Max'); myMin = get(handles.slider1, 'Min'); if(myValue < myMin) myValue = myMin; elseif(myValue > myMax) myValue = myMax; end set(handles.slider1, 'Value', myValue); set(hObject, 'String', num2str(myValue)); freePortSend(1, 115200, 1, 1, 0, myValue) % set 'f' to 'myValue' end % --- Executes on button press in pushbutton1. function pushbutton1_Callback(hObject, eventdata, handles) % hObject handle to pushbutton1 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) set(handles.edit1, 'String', '2.84') set(handles.slider1, 'Value', 2.84) freePortSend(1, 115200, 1, 1, 0, 2.84) % set 'f' to '2.84' % --- Executes on button press in pushbutton2. function pushbutton2_Callback(hObject, eventdata, handles) % hObject handle to pushbutton2 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) myString = get(hObject, 'String'); f = get(handles.slider1, 'Value'); if(strcmp(myString, 'Excitation off')) set(hObject, 'String', 'Excitation on') set(handles.pushbutton3, 'String', 'Controller off') freePortSend(1, 115200, 1, 1, 0, f) % set 'f' to '0' disp('Switching to sysID mode') freePortSend(1, 115200, 0, 1, 0, 2) % switch to 'excitation' mode else set(hObject, 'String', 'Excitation off') disp('Switching to off mode') freePortSend(1, 115200, 1, 1, 0, f) % set 'f' to '0' freePortSend(1, 115200, 0, 1, 0, 1) % switch to 'off' mode end % --- Executes on button press in pushbutton3. function pushbutton3_Callback(hObject, eventdata, handles) % hObject handle to pushbutton3 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) myString = get(hObject, 'String'); if(strcmp(myString, 'Controller off')) set(hObject, 'String', 'Controller on') set(handles.pushbutton2, 'String', 'Excitation off')

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disp('Switching to controller mode') freePortSend(1, 115200, 1, 1, 0, 0) % set 'f' to '0' freePortSend(1, 115200, 2, 6, 0, H) % download parameter vector freePortSend(1, 115200, 3, 6, 0, G) % download parameter vector freePortSend(1, 115200, 0, 1, 0, 3) % switch to 'control' mode else set(hObject, 'String', 'Controller off') disp('Switching to off mode') freePortSend(1, 115200, 0, 1, 0, 1) % switch to 'off' mode end % --- Executes on button press in pushbutton4. function pushbutton4_Callback(hObject, eventdata, handles) % hObject handle to pushbutton4 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) global accel1 accel2 pulse; set(handles.pushbutton2, 'String', 'Excitation off') set(handles.pushbutton3, 'String', 'Controller off') freePortSend(1, 115200, 0, 1, 0, 1) % switch to 'off' mode freePortSend(1, 115200, 2, 1, 0, 1) % turn model 'off' % display data axes(handles.axes1) % control pulses plot(pulse.time, pulse.signals.values, 'r.-'); grid kk = axis; axis([kk(1) kk(2) -0.5 1.5]); axes(handles.axes2) % accelerometer 1 plot(accel1.time, accel1.signals.values, 'r.-'); grid %kk = axis; %axis([0 30 kk(3) kk(4)]); axes(handles.axes3) % accelerometer 2 plot(accel1.time, accel1.signals.values, 'r.-'); grid %kk = axis; %axis([0 30 kk(3) kk(4)]); % call sys-ID program sys_ID_alone; % --- Executes on button press in radiobutton2. function radiobutton2_Callback(hObject, eventdata, handles) % hObject handle to radiobutton2 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) % Hint: get(hObject,'Value') returns toggle state of radiobutton1

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myValue = get(hObject, 'Value'); if(myValue == 1) set(hObject, 'String', 'Controller on') else set(hObject, 'String', 'Controller off') end % --- Executes on figure close... function myClosereq(hObject, eventdata, handles) % hObject handle to pushbutton1 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) global run; run = 0; disp('goodbye...') closereq System Identification M-file %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %l is the number of data points %q is the prediction horizon %p is the order of the ARX model clear all close all % % paper model % A=[0 1 0 0 0 0; -2000 -0.2 1000 0.1 0 0; 0 0 0 1 0 0; 1000 0.1 -2000 ... % -0.2 1000 0.1; 0 0 0 0 0 1; 0 0 1000 0.1 -1000 -0.1]; % B=[0; 1; 0; 0; 0; 0] % C=[0 0 0 0 1 0]; % % Output is location of mass 3, corresponding the state x5 % D=0; % first order model, fres = 1.42 Hz, zeta = 0.1 wn = 2*pi*1.42; z = 0.1; A=[0 1; -wn^2 -2*z*wn]; B=[0; 1] C=[1 0]; % Output is location of mass 3, corresponding the state x5 D=0; % 3rd order model % A1=[0 1; -(1.2*wn)^2 -2*z*(1.2*wn)]; % B1=[0; 1] % C1=[1 0]; % A2=[0 1; -(3.72*wn)^2 -2*(z*2)*(3.72*wn)]; % B2=[0; 1] % C2=[1 0]; % A3=[0 1; -(11.12*wn)^2 -2*(z*3)*(11.12*wn)]; % B3=[0; 1] % C3=[1 0]; % % A = [A1 zeros(2,4); zeros(2,2) A2 zeros(2,2); zeros(2,4) A3]; % B = [B1;B2;B3]; % C = [C1 C2 C3]; % output y = y1 + y2 + y3 % D = 0;

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SYS = SS(A,B,C,D) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % STEP 1.1 SELECT VALUE OF P GOVERNING THE SPEED OF THE OBSERVER % l is the number of data points % q is the prediction horizon % p is the order of the ARX model q=6; p=6; l=100; r=1; u_stored=[]; % u_stored will be an array of 100 values y_stored=[]; u_stored=[]; %u_new=idinput(101,'PRBS',[0 1],[0 5]); % column vector, input length=100 %fs=100; % sampling frequency k = 1; while(k<=100) u_new(k,1) = 1; u_new(k+1,1) = -1; k = k+1; end; u_new(end) = -u_new(end-1); fs = 1.42*2*10; u_stored = k t = 1/fs*[0:length(u_new)-1]; y_stored=lsim(SYS,u_new,t); y_plant_v=y_stored % for i=1:101 % R=rem(i,2); % % if R==1 % u_stored(i) = U(i); % % else % u_stored(i) = -U(i); % % end % end % u_stored=u_stored'; % % % for i=1:100 % R=rem(i,2); % Divides row number by 2 to test whether even or odd row. % % if R==1 % u_new(i) = UN(i); % % else % u_new(i) = UN(i); % % end % end % u_new=u_new';

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%av_accel = mean(accel1.signals.values); % y_stored = accel1.signals.values((end-100):end)-av_accel; % y_plant_v = accel1.signals.values((end-99):end)-av_accel; % PREDICTIVE CONTROL %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Y is row vector containing values of column vector [y(p),y(p+1)...y(l)] Y=[]; h=p+1; % Note y(p) is actually the p+1 element of y vector for j=1:(l-p+1) % Refers to column number in Y Y(1,j)=y_stored(h,1); % Forms Y using value from input vector y h=h+1; % Increments to select value from next column of y end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i=0; j=0; count=0; V=[]; for i=1:2*p % Refers to row number in V R=rem(i,2); % Divides row number by 2 to test whether even or odd row. if R==1 % Odd rows contain inputs for j=1:(l-p+1) % Refers to column number in V h=p-2+j-count+1; %decrements starting p-1 to zero V(i,j)=u_stored(h,1); end else % Even rows contain outputs for j=1:(l-p+1) %column number in V h=p-2+j-count+1; % note u(p-1) is actually the pth element of u. in this case row 2,4,6,8 V(i,j)=y_stored(h,1); end count=count+1; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V_plus=pinv(V); % Finds psuedo inverse of V P=Y*V_plus; % Finds matrix P % STEP 2. PUT SYSTEM IN OBSERVABLE CANONICAL FORM % EQNS 16 and 17

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A=[]; B=[]; % Observable canonical matrices alpha=[]; % Matrix used in A % Finding alpha and B i=0; j=2; for i=1:p % Number of rows of alpha alpha(i,1)=P(1,j); B(i,1)=P(1,(j-1)); i=i+1; j=j+2; end % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % y_arx % filter with coefficients of:a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... % + b(nb+1)*x(n-nb)- a(2)*y(n-1) - ... - a(na+1)*y(n-na) arx.b=[]; arx.b(1,1)=0; arx.a(1,1)=1; for i=1:p arx.b(1,(i+1))=B(i,1); arx.a(1,(i+1))=-alpha(i,1); end arx.b; arx.a; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Variance y_v = filter(arx.b, arx.a, u_new); % arx model output % plant output = ACCELEROMETER VALUES vari = 1/length(y_v) * sum((y_v - y_plant_v).^2); %finds variance - the normalised difference between arx model output & %plant output disp(['Variance: ' num2str(vari)]) % display results figure plot(t, y_v, 'r.-', t, y_plant_v, 'b') xlabel('time [s]') ylabel('model (dotted), plant (solid)') legend('ARX','y') title('Response to random noise signal, matching initial states')

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Figure F1:Target Program

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Figure F2: Off Mode

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Figure F3: Excitation Mode

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Figure F4: Control Mode

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Figure F5:Buffer

design and build of a thruster controlled model of a large...

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