Modelling and Control of Advanced Mechatronic System

THE UNIVERSITY OF MANCHESTER

DOCTORAL THESIS

Modelling and Control of

Advanced Mechatronic System

Author:

Xiaomo YAN

Supervisor:

Dr. Alexandru STANCU

Prof. Hong WANG

A thesis submitted in fulfillment of the requirements

for the degree of Doctor of Philosophy

in the Faculty of Science and Engineering

School of Electrical and Electronic Engineering

The University of Manchester

2017

http://www.manchester.ac.uk/

https://uk.linkedin.com/in/yan-xiaomo-17aba857

http://www.manchester.ac.uk/research/hong.wang/

http://www.manchester.ac.uk/research/hong.wang/

http://www.eee.manchester.ac.uk/

http://www.manchester.ac.uk/

2

Contents

List of Figures 5

List of Abbreviations 7

Abstract 8

Declaration of Authorship 9

Copyright Statement 10

Acknowledgements 12

1 Introduction 13

1.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . 18

1.2 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . . 20

2 Control Background and Preliminaries 21

2.1 Control Background . . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 Mechatronic Systems . . . . . . . . . . . . . . . . . . 21

2.1.2 Sliding Mode Control . . . . . . . . . . . . . . . . . 23

2.1.3 Fault Tolerant Control . . . . . . . . . . . . . . . . . 25

2.1.4 System Optimisation . . . . . . . . . . . . . . . . . . 28

2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.1 Continuous Nonlinear System . . . . . . . . . . . . 31

2.2.2 Discrete Time Nonlinear System . . . . . . . . . . . 32

3

2.3 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Continuous Time Systems . . . . . . . . . . . . . . . 32

2.3.2 Discrete Time Systems . . . . . . . . . . . . . . . . . 35

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Mechatronic System Modelling 37

3.1 Frame and Rotation . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.1 Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.2 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Forward Kinematic of 3-DOF Manipulator . . . . . . . . . 41

3.3 Inverse Kinematic . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Geometric solution for Horizontal Joint . . . . . . . 46

3.3.2 Geometric solution for Vertical Joints . . . . . . . . 47

3.4 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 Feedback Linearisation for Manipulator Modelling 49

3.4.2 Ackerman Controller . . . . . . . . . . . . . . . . . . 50

3.4.3 System State Observer Design . . . . . . . . . . . . 52

3.5 3-DOF Manipulator Visualization Programming . . . . . . 55

3.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 56

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Fault Matrix based Fault Tolerant Control 60

4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Terminal Sliding Mode Controller Design . . . . . . . . . . 62

4.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Disturbance Observer . . . . . . . . . . . . . . . . . 65

4.3.2 Fault Matrix Observer . . . . . . . . . . . . . . . . . 66

4.4 Fault Tolerant Controller Design . . . . . . . . . . . . . . . 69

4.5 Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . 71

4

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Set-point Re-planning 78

5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 78

5.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . 78

5.1.2 Terminal Sliding Mode . . . . . . . . . . . . . . . . . 80

5.2 Set-point Re-planning . . . . . . . . . . . . . . . . . . . . . 85

5.2.1 System Model with Disturbance . . . . . . . . . . . 86

5.2.2 Performance Optimisation . . . . . . . . . . . . . . . 88

5.2.3 Singular Problem . . . . . . . . . . . . . . . . . . . . 89

5.3 Case-study on Mobile Robot . . . . . . . . . . . . . . . . . . 93

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Conclusion and Future Work 104

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . 106

6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

References 109

0Word Count: 19617

5

List of Figures

1.1 Unimate Robot c©Unimation. . . . . . . . . . . . . . . . . . 13

1.2 IRB-6700 Manipulator c©ABB. . . . . . . . . . . . . . . . . . 14

1.3 Closed-loop System with Chattering in the output. . . . . . 16

1.4 Closed-loop System affected by Uncertainties. . . . . . . . 16

1.5 System Performance Deteriorating. . . . . . . . . . . . . . . 17

2.1 Function of Relay. . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Function of Saturation. . . . . . . . . . . . . . . . . . . . . . 30

2.3 Fuction of Dead-zone. . . . . . . . . . . . . . . . . . . . . . 30

2.4 Discrete Nonlinear System. . . . . . . . . . . . . . . . . . . 33

3.1 The Basic Frames. . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Point P in Frame A. . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Coordinate Rotation. . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Frames Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 Inertia Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Geometric analysis for 3-DOF manipulator. . . . . . . . . . 46

3.9 Linear Close-loop Subsystem. . . . . . . . . . . . . . . . . . 50

3.10 Full-Order Observer . . . . . . . . . . . . . . . . . . . . . . 53

3.11 Manipulator Simulation. . . . . . . . . . . . . . . . . . . . . 55

4.1 Fault Matrix Observer. . . . . . . . . . . . . . . . . . . . . . 67

4.2 System State (x1) Estimation. . . . . . . . . . . . . . . . . . 73

4.3 System State (x2) Estimation. . . . . . . . . . . . . . . . . . 74

6

4.4 Fault Matrix Estimation. . . . . . . . . . . . . . . . . . . . . 75

4.5 Disturbance Estimation. . . . . . . . . . . . . . . . . . . . . 76

5.1 Block Diagram of Set-point Replanning . . . . . . . . . . . 85

5.2 Holonomic Mobile Robot c©Festo Didactic. . . . . . . . . . 94

5.3 Block Diagram of Robotino Control. . . . . . . . . . . . . . 95

5.4 Position Error in x Direction . . . . . . . . . . . . . . . . . . 96

5.5 Position Error in y Direction. . . . . . . . . . . . . . . . . . 96

5.6 Velocity in x Direction. . . . . . . . . . . . . . . . . . . . . . 97

5.7 Velocity in y Direction . . . . . . . . . . . . . . . . . . . . . 98

5.8 Sliding Manifold on x Direction (sx). . . . . . . . . . . . . . 99

5.9 Sliding Manifold on y Direction (sy). . . . . . . . . . . . . . 99

5.10 Optimal system output error (ex). . . . . . . . . . . . . . . . 100

5.11 Optimal system output error (ey). . . . . . . . . . . . . . . . 101

5.12 System Performance Index. . . . . . . . . . . . . . . . . . . 102

7

List of Abbreviations

DOF Degree Of Freedom

SMC Sliding Mode Control

TSMC Terminal Sliding Mode Control

DTSMC Discrete Terminal Sliding Mode Control

MIMO Multi-Input Multi-Output

FTC Fault Tolerant Control

DC Direct Current

PI Proportional Integral

HJB Hamilton Jacobi Bellman

LQR Linear Quadratic Regulator

8

Abstractby Xiaomo YAN

Control of mechatronic systems remain an open problem in control the-

ory despite the research work worldwide in the last decade. Uncertain-

ties in mechatronic systems, which includes faults, and disturbance, will

often cause undesired behaviours, affecting the systems performances,

may lead to the system failure, or even causing safety issues.

Control reconfiguration is an active approach in control systems field.

However, controller reconfiguration involves changes in its parameters

and structure. System stability might not be able to be guaranteed during

the parameters tuning, which might cause more damage to system stabil-

ity, sometimes may cause safety issues. Due the on-line reconfiguration

has a scope, during which the system stability can not be guaranteed.

This leads that the systems must be turned off during reconfiguration

process. In many industrial areas, such as metallurgy, forging, and man-

ufacturing, shutting down the streamline leads to significant levels of lost

productivity and unacceptable economic losses.

As alternative to control reconfiguration approach, in this thesis two

methods are proposed to deal with faults and disturbances. The first

method is the fault matrix observer and the second one is the set-point

re-planning. The idea of both methods is to compensate the faults and

disturbances which affect the system performances without changing the

controller structure or controller parameters.

9

Declaration of Authorship

I, Xiaomo YAN, declare that this thesis titled, “Modelling and Control

of Advanced Mechatronic System” and the work presented in it are my

own. I confirm that:

• This work was done wholly or mainly while in candidature for a

research degree at this University.

• Where any part of this thesis has previously been submitted for a

degree or any other qualification at this University or any other in-

stitution, this has been clearly stated.

• Where I have consulted the published work of others, this is always

clearly attributed.

• Where I have quoted from the work of others, the source is always

given. With the exception of such quotations, this thesis is entirely

my own work.

• I have acknowledged all main sources of help.

• Where the thesis is based on work done by myself jointly with oth-

ers, I have made clear exactly what was done by others and what I

have contributed myself.

Signed:

Date: 13, Jun,2017

10

Copyright Statement

• The author of this thesis (including any appendices and/or sched-

ules to this thesis) owns certain copyright or related rights in it (the

’Copyright’) and s/he has given The University of Manchester cer-

tain rights to use such Copyright, including for administrative pur-

poses.

• Copies of this thesis, either in full or in extracts and whether in hard

or electronic copy, may be made only in accordance with the Copy-

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• The ownership of certain Copyright, patents, designs, trade marks

and other intellectual property (the ’Intellectual Property’) and any

reproductions of copyright works in the thesis, for example graphs

and tables (’Reproductions’), which may be described in this the-

sis, may not be owned by the author and may be owned by third

parties. Such Intellectual Property and Reproductions cannot and

must not be made available for use without the prior written per-

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11

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versity Library, The University Library’s regulations (see http://

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AcknowledgementsI would like to express my deepest thanks and sincere gratitude to my su-

pervisor, Prof. Hong Wang , not only for his thoughtful supervision, sin-

cere advice and guidance, but also for his unlimited assistance, valuable

consultation, helpful comments and critical remarks while reviewing the

thesis.

My deep thanks to my supervisor Dr. Alexandru Stancu for his great

support, valuable discussions and helpful comments while reviewing the

thesis.

I would also like to thank Control System Centre at the University

of Manchester for the excellent environment provided for postgraduate

students.

My special and sincere thanks to my beloved family. A special feel-

ing of gratitude to my loving parents, Mingyan Jiang and Wei Yan whose

words of encouragement and push for tenacity ring in my ears.

Last, but not least, I would like to dedicate this thesis to my wife

Ziwei for her love.

13

Chapter 1

Introduction

Robotic manipulators are mechatronic systems mimicking the func-

tionality of a human arm. The first robot manipulator was developed in

1947 by Argonne National Laboratory (Goertz 64). Because of its bionic

structure, researches noticed that the robotic manipulator should have a

wider use in industry. The first robotic manipulator patent was filed by

George Devol in 1954, which implies that the value of robotic manipula-

tor has been recognized by industrial sectors.

FIGURE 1.1: Unimate Robot c©Unimation.

Two years later, in 1956, a robotic company called Unimation is founded

by Devol and Joseph F. Engelberger to develop the industrial robot. As

Chapter 1. Introduction 14

the quick growing of industrial robotic field, ABB Robotics launched IRB

6 as the first commercial micro-processor controlled robot. With growing

awareness of possible capability of the manipulator, researchers start to

develop new control strategies and innovative manipulator structures for

special needs.

FIGURE 1.2: IRB-6700 Manipulator c©ABB.

As robotic manipulator has human like structure and adaptation to

the high-risk working environment, such as high radiation, extreme tem-

perature, it has become an attractive area in both industrial and research

areas.

At the beginning, manipulators, such as GE Master-Slave Manipula-

tor, are controlled manually, so the efficiency and quality are still rely on

the experience of operators. With the repeatability growing expectation

in industry, the manual operation becomes an obstacle in improving the

efficiency of manipulator.


Manipulator control is an important research in nonlinear system

control. In the trajectory control, the speed control is used to guarantee

that the end-effector will track set trajectory. As the dynamics of manipu-

lator has strong non-linearities, and system states are highly coupled, the

controller must have a good dynamic response and be adapt at stabiliz-

ing the nonlinear terms.

Proportional–integral–derivative (PID) controller is one of the famous

control strategies used in manipulators control due to its easy tuning

structure [1–3]. However, the easy parameter tuning brings some disad-

vantages. Because the tuning of PID is based on the system output, which

means the inner relationships of system states are ignored during the tun-

ing The neglect of the inner relationships of system states always cause

a energy waste. For instance, when the speed of system responses is in-

creased by the proportional coefficient, overshoots are introduced at the

same time. Then derivative coefficient has to be increased to reduce the

overshoot, which means derivative term goes against the proportional

term in the same PID controller.

Another control method widely used in industry is variable struc-

ture control (VSC). VSC is a discontinuous control method, achieving the

stability by switching between different controller structures. There are

two types of VSC, bang-bang control and sliding mode control. The con-

trol law used in bang-bang control is relay, which stabilizes the system by

thresholds of tracking errors. However, bang-bang control can only sta-

bilize the system states in the range of thresholds. Different with bang-

bang control, the states related structures used in sliding mode control

(SMC) can make the tracking a time variant reference. By choosing dif-

ferent structures of nonlinear sliding manifolds, the SMC is adaptable to

rapidly-change system states. As the sliding mode control is based on


the system dynamics, it is easier to analysis system stability of SMC than

that of PID.

FIGURE 1.3: Closed-loop System with Chattering in theoutput.

Figure 1.3 is the block diagram of the closed-loop system. Sliding

mode controller (SMC) is used as the control strategy. The convergence

speed is always described qualitative, such as ’fast converge’ and ’slow

converge’. In this thesis an analytical solution of converge time is pro-

posed, such that the convergence speed can be described more precisely.

Because of the imperfections in real systems in reality SMC always presents

the chartering phenomenon.

FIGURE 1.4: Closed-loop System affected by Uncertainties.

Uncertainties, which includes faults and disturbance, always deteri-

orate the system performance. Controller reconfiguration is always used


to improve the system performance. However, controller reconfigura-

tion requires to turn off the system, otherwise, the system’s stability can

not be guaranteed. Under the consideration of safety, the assembly lines

with mechatronic systems have to be turned off during the maintenance

period, which will cause a big lost to the factories. As the mechatronic

system has become an attractive area in industrial processes, most of the

mechatronic systems are off-the-shelf with embedded controllers which

makes controller structure hard to be changed.

FIGURE 1.5: System Performance Deteriorating.

This thesis proposes two solutions to improve the system perfor-

mance affected by uncertainties. The first solution is fault matrix based

FTC which focuses on the system performance damaged by the cou-

pled actuator faults and disturbances. The second solution is set-point

re-planning focusing on improving the system performance affected by

general uncertainties.

In fault matrix based FTC, a coefficient matrix is added on the con-

troller to compensate actuator faults. As faults always cause the changes

in the system behaviours, there is a connection between faults and the

system states. However, when both disturbances and faults exist in the

system, the disturbance will affect the relationship between system states


and actuator faults. If actuator fault can be isolated from disturbance, ac-

tuator faults can be compensated without changing controller structure.

In most instances, different kinds of uncertainties exist in the system

and are coupled to each other. To optimise system performance, opti-

mal control methods are always applied on the system. However, the

conventional system optimisation is also based on controller reconfigu-

ration. It will be convenient if we can find a method that optimises the

system performance, which is damaged by uncertainties, without chang-

ing controller.

1.1 Research Questions

• How can be computed analytically the convergence time for the

reaching phase in case of SMC?

• How to isolate, estimate and eliminate the actuator faults in the

nonlinear system when it is coupled with disturbance without chang-

ing controller?

• How to control the systems, even with reduced performances, in

presence of faults or any other types of disturbances without mod-

ifying the controller structure or parameters?

1.2 Thesis Objectives

This section gives a summary of the thesis objectives which are listed

as follows:


• Investigating the effectiveness of nonlinear control design for con-

tinuous nonlinear system, as well as for discrete nonlinear system

using a special nonlinear structure.

• Designing a new fault tolerant control strategy for continuous time

kinematic systems subject to faults and disturbances, using fault

matrix to link actuator fault to control signal.

• Introducing a novel discrete time system optimisation strategy to

mechatronic systems subject to uncertainties without changing the

original closed-loop.


1.3 Structure of Thesis

The rest of the thesis is organised as follows:

Chapter 2 contains a state of the art in areas related to this thesis.

In addition, some important stability theorems are introduced to help to

understand the methods proposed in this thesis.

Chapter 3 presents the modelling of n-DOF manipulator based on

Newton-Euler method. Then, parameters from ABB IRB 120 are used in

the Matlab based 3-DOF manipulator visualization program design. To

make the visualization program practical, both full-order observer and

reduced-order observer are implemented.

Chapter 4 introduces a fault matrix based FTC strategy. A terminal

slide mode control strategy is chosen to guarantee the system stability. In

addition, a fault matrix observer is designed to observe the relationships

between control signals and actuator faults.

Chapter 5 presents a novel operational control method. HJB equa-

tion is used to optimise the system performance. Due to the Cascade

control structure used in the design, multi-rate method is implemented

to reduce the affect from the delay between the outer loop and the inner-

loop.

Chapter 6 presents the conclusion and suggestions for future work.

21

Chapter 2

Control Background and

Preliminaries

2.1 Control Background

2.1.1 Mechatronic Systems

There are two types of techniques in finding manipulator dynamics,

numerical and symbolical. However, before any of the techniques can be

used, the problem of complexity need to be solved [4–14]. In 1972, Paul

[15] presented a numerical solution of finding configuration-dependent

6-DOF manipulator based on force balance. Targets are recognized by the

information given by vision subsystem, and then the strategy subsystem

guarantees that the arm is capable of grabbing the target cube. In 1977,

a general symbolical 3-DOF dynamics is given by using Euler-Lagrange

formula [16]. It is clear to see that the torques of the motors are related

to angles, angular velocities and other constant parameters of the manip-

ulator. They also considered the offset of links to make their dynamics

more adaptable. But it is also clear to see that the symbolic technique is

only available in small DOF structure for the reason of complexity.

Chapter 2. Control Background and Preliminaries 22

Indeed, it may not necessary to have all the details of manipulator

in control strategy design. But it turns out that Coriolis and centripetal

can be ignored to simplify the dynamics not only in low speed but also

high speed situations [17]. The dynamics left is still not ’brief’ enough to

be used in design, and people have never stop finding a more efficient

expression. As we can see that the most complex part in manipulator

dynamics is brought in by the inertia tensor, the dynamics will be sim-

plified greatly if they can be decoupled. Youcef-Touml and Asada [18]

attempted to solve this decoupling question in 1985. It turns out that

those components can only be decoupled when the revolute joints have a

specific structure, which is all joint axes must be orthogonal to each other.

In 1984, Tarn et al. [19] figure out a method called "External Linearisa-

tion, which remains more system features than Jacobian method. More-

over, a necessary and sufficient condition of decoupling is also given in

this paper. Kosuge and Furuta used Jacobian matrix to deduce the linear

mode. However, the disadvantage of Jacobian matrix is that the system

dynamics is linearised around a nominal point, which makes their dy-

namics only suitable for those slow-change processes [20]. Later in 1991,

Yurkovich et al. [21] applied feedback linearisation onto the manipulator

dynamics. By using this method, all the nonlinear components are elim-

inated by the first part of the controller, then the rest part of controller

only need to focus on maintaining the stability of a second order system

that represents the fundamental physical relationship.

Additionally, for a better understanding of the behaviours of manip-

ulator, a clear physical insight of manipulator is crucial [22, 23]. As a

result of the complex dynamics, the computation related manipulators

is always slow. It is essential to find a more efficient way to deal with

manipulator dynamics in the computer program. In 1981, Hewit and


Burdess [24] brought a fast control method by observing torque.

2.1.2 Sliding Mode Control

Sliding mode control (SMC) is known for its outstanding robust-

ness[25–34]. The SMC design can be considered as a mathematical ex-

planation of one basic reaction from actual operators. The sliding mode

controller is discontinuous, which can jump from one continuous surface

to another one. Utkin [35] proposed an idea of variable structure system

(VSS) control design. Furthermore, he also proved the existence of sliding

mode surface. Basing on Utkin’s method, Young [36] presented a prac-

tical example of SMC on 2-DOF manipulator. To apply the SMC control

strategy into practical processes, discrete time SMC control stability has

been proved based on a pair of selected state feedback control signal [37].

The amplitude of control has to be as small as possible, which is contrary

to the continuous system [38].

In 1996, Mnif et al. applied holonomic constraints to the manipulator

dynamics in SMC design. However, as problems were shown in these

methods [39–41], chattering phenomenon has long been a notable issue

in the SMC design. Those high-frequency components, which caused by

the ideal switch, will not only affect the system response but also damage

the switches of plants. To eliminate the chattering phenomenon, smooth

functions, such as saturation function, are usually used in the SMC de-

sign [42]. When the sliding manifold moves within the scope of the cho-

sen linear portion, the amplifier from saturation function slows the re-

sponse. As a good controller strategy, the convergence of the controller

need to be enhanced. Terminal sliding mode control (TSMC) had been


introduced in the early 90s, which guarantees the finite convergence time

[43–45].

In 1997, a new terminal sliding manifold had been proposed by Man

and Yu to guarantee the finite convergence for a type of linear MIMO

kinematic system [46]. The convergence time has been proved in the pa-

per, but the part that pulls the system states back to manifold remains

to be discontinuous. Young et al. [47] presented a practical SMC guide,

in which, boundary layer control was mentioned. Boundary layer con-

trol focuses on reducing the effect of chattering by replacing switching

function with the designed parasitic dynamics. But the boundary layer

control is doing a trade-off between chattering reduction and disturbance

rejection, which means it is necessary to find a balance between steady

state error and chattering. As the nonlinear sliding manifold introduced,

a singular problem has to be considered.

In 2002, a non-singular TSMC manifold has been proposed with an

additional limitation on the manifold parameter selection [48]. In the

conventional SMC design, a fast response relies on a large control gain,

which is another incentive of the chattering issue. Instead of using a lin-

ear function, nonlinear sliding manifolds give a more ’flexible’ response

[49–53]. To solve the chattering problem, Levant [54] proposed a high-

order SMC method, which is capable of guaranteeing the finite-time con-

vergence. The high-order SMC can reduce chattering while remaining

the robustness. The boundary of uncertainties and system relative de-

gree are needed. TSMC can only guarantee the system states converge to

the set-point within finite time, Polyakov [55] pointed out a certain type

of function, which guarantees a fast reach time with both large and small

variances. The consideration of their function is based on both fast con-

vergences on the small scalar and others. To approach the requirement,


they use the differences between different system states powers. Then on

account of the similar idea, the pull-back part is based on the norm of the

manifold.

2.1.3 Fault Tolerant Control

As the control theory has been approved by industry, the quality of

industrial products relies on control strategies more than ever [56–65].

Sometimes robustness is used to guarantee that controllers are capable of

working under harsh situations. However, there are demands for high

system performance, the robustness is always limited, which means the

inherent controller may not be able to guarantee the performance any

more when extra faults occur. Faults occur for many reasons, such as

ageing, temperature changing, and humidity increasing. Aiming for bet-

ter reliability and safety, the importance of fault tolerant control has been

increasingly recognized by more and more people [66–70]. By using the

fault tolerant control strategy, the system can continue working with a

reduced level performance rather than fail completely.

Fault tolerant control becomes challenging not only because of the

increasing the complexity of systems, but also the states coupling with

disturbance. Increasing the efficiency of the control strategy, accurate sys-

tem states value are needed. But in many practical industrial cases, states

required are not only the system outputs but also those hidden inside the

equipment. To achieve all the parameters needed, it is necessary to find

a way to gather those internal states. The observer is now a common

method used in control strategy design. The observer is mostly designed

by the computation based on knowledge of original system structure [71–


80]. By finding the mathematical relationships between observable sys-

tem outputs and states, those internal states’ value can be calculated.

There are two basic linear system state observer methods, full-order

and reduced order state observers [81]. The full-order observer is based

on the system output values and knowledge of system model. But some-

times, the system outputs are parts of the system states. It means there

might be another way to design the observer only for the states not in-

cluded in the output, which is named reduced-order observer. As most

of the practical industrial processes are nonlinear systems; it is neces-

sary to find methods that can estimate system states from nonlinear sys-

tems. The main idea of nonlinear observer designs [82], such as extended

Kalman filter, linearise the system around equilibrium points, then by us-

ing the constraints of systems, such as Lipschitz condition, to find out the

boundary of the Lyapunov candidate.

The internal relationships among sensor fault, actuator fault and dis-

turbance cannot be ignored during the strategy design. If those uncer-

tainties can be decoupled, it is possible to design an observer to estimate

the fault. To eliminate the disturbance, either disturbance observer or in-

crease the robustness of the controller can be used. For instance, Capisani

et al. [83] proposed two sliding mode control based fault observer when

either sensor fault or actuator fault occurs in the system. The basic idea

of fault matrix is representing the actuator faults basing on its relation-

ship with control inputs. Different from disturbance, faults always relate

to the system states [84]. Instead of analysing the linear system stability,

boundary of Jacobian matrix can also be used in tracking error analysis

to make sure zero is an equilibrium point of the observer [85].

However, the observer based strategy is not the only way to design


fault tolerant control strategy. Marx et al. [86] applied a H∞ filter to Youla

parametrized controller to minimize the responses of exogenous inputs.

In 2015, an adaptive fault tolerant control strategy was proposed by Liu

and his co-workers [87], the system they focus on is Markovian jump

system with constant parameters in each period. Both actuator fault and

disturbance have been considered.

In 2016, Defoort et al. [88] proposed an adaptive super-twisting ac-

tuator fault observer. However, super-twisting only works for second-

order system, which makes his method difficult to extend to higher order

system. The observer designed focus on estimate the real value of system

states, which means the state observer proposed only guarantee distur-

bance will not make system unstable. Actuator fault is approximated by

integrating the sign of estimation error, which reduce the chattering. But

due to the constant coefficient used in the fault estimation, the proposed

fault observer can only compensate the actuator fault roughly.

In 2017, Zhang et al. [89] proposed an adaptive fault-tolerant con-

trol strategy. Actuator fault is expressed by using control signal and bias

signals. The un-modelled nonlinear part in system is described by fuzzy

hyperbolic model, which transfer the non-linear system in to linear func-

tional. A fuzzy system state observer is designed based on the system

linear functional. However, disturbance is compensated by its boundary,

which means the proposed method can only improve the damage comes

from actuator fault, and guarantee the disturbance will not make the sys-

tem unstable.

Li et al. [90] proposed an adaptive fault tolerant control strategy in

2017. The proposed method focus on the SISO nonlinear system. A fuzzy


system states observer is designed. The un-modelled nonlinear part is ex-

pressed by fuzzy logic model. The actuator fault is expressed by control

signal. However, in their design, disturbance has not been taken into con-

sideration, which means their method cannot isolate actuator fault from

disturbance.

2.1.4 System Optimisation

The motive of optimal control theory is trying to find a result from

any possible solutions based on dynamic programming and maximum

principle [91–100]. To achieve the optimal value, feedback regulation is

always used in the design [101]. In 1961, Desoer et al. [102] presented a

force optimal tracking strategy, which not only guarantees the minimum

sample periods but also proves its finite time convergence. In the same

year, Smith [103] found a time-optimal method for high-order system,

which has a constraint of constant control signal value. His method is

based on optimising the switch time, but with the constant control signal

he used, chattering cannot be ignored. Instead of optimising the switch

time, a HJB based obstacle avoidance algorithm is proposed following

pseudo return function in 1997 [104]. In 2000, LQR method had been

applied on self-turning relay system [105]. To guarantee the fast conver-

gence, an auxiliary output had also been designed. However, conven-

tional optimal controls have focused on improving system performance

by optimising controller parameters, which is hard to . In 2014, T. Chai, S.

Qin, and H. Wang [106] proposed the idea of optimal operational control

for complex industrial processes. However, only the possible research

directions are given, methods to realize the ideas still remain to be open


questions. The main idea of the optimal operational control is to opti-

mise the damaged system performance on-line [107–110]. For the shake

of safety, the original control closed-loop is considered as inaccessible.

A. Wang, P. Zhou and H. Wang [111] proposed a set-point reselection

method for stochastic system. The reselected set-point is used to reshape

the system performance curve. The trough of performance curve has

been widen, so that the system performance is easier to converge the op-

timal point. However, only a brief mathematical description has been

given to generalize the idea. The authors also indicated that the conver-

gence of the set-point reselection remains to be an open question.

2.2 Preliminaries

Most of the practical industrial systems are nonlinear. There are sev-

eral typical nonlinear functions, such as relay, saturation, and dead-zone.

Compare with linear system, the existing of non-linearities makes the sys-

tem hard to be analysed.

FIGURE 2.1: Function of Relay.


FIGURE 2.2: Function of Saturation.

FIGURE 2.3: Fuction of Dead-zone.


2.2.1 Continuous Nonlinear System

The continuous nonlinear state equation is

x = f(t, x, u), (2.1)

where x ∈ Rn denotes the system state vector, t represents that the system

is time varying or non-autonomous, x is the derivative of system states,

f ∈ Rn is the nonlinear relationships vector relating to time and system

states. u is controller to be designed. Equation (2.1) is short for a finite

number of coupled differential equations

Definition 1. [112] A function f : Rn × Rm × R is Lipschitz with a Lipschitz

constant γ if for any vectors x, x ∈ Dx ⊂ Rn, and u ∈ Du ⊂ Rm and t ∈ It ⊂

R, with Dx, Du being the regions of interest and It being an time interval,

||f(x, u, t)− f(x, u, t) ≤ γ||x− x||, (2.2)

with γ > 0

x1 = f1(t, x1, . . . , xn, u1, . . . , um),

x2 = f2(t, x1, . . . , xn, u1, . . . , um),

...

xn = fn(t, x1, . . . , xn, u1, . . . , um).

(2.3)

From Equation (2.3), the controllers do not need to have the same dimen-

sions with system states, which is normally less than the dimensions of

system states. When f is time-invariant, the system equation is

x = f(x, u), (2.4)


Equation (2.4) is an overview of the nonlinear system, which includes all

components in f . However, it is necessary to find an expression which

can demonstrate more details of the system.

x = Ax+ f(x) + g(x)u, (2.5)

where Ax represents the linear part related to system states, f(x) denotes

the system inner structure, g(x) is the relationship between the controller

and system inputs.

2.2.2 Discrete Time Nonlinear System

As shown in Figure 2.4, the discrete time nonlinear system can be

considered as a sampled continuous nonlinear system with sampling

time T . The discrete time nonlinear state equation is

xk+1 = f(xk, uk), (2.6)

where xk represents the kth sample of system state, f is nonlinear rela-

tionships relating to system states xk and control input uk.

2.3 Stability Theory

2.3.1 Continuous Time Systems

A key area of control engineering is stability theory. Controllers are

designed to guarantee the stability of system. One of the foundations of


FIGURE 2.4: Discrete Nonlinear System.

the equilibrium point is Lyapunov function. System described in Equa-

tion (2.4) with a state feedback controller can be rewritten as

x = f(x), (2.7)

Definition 2. The equilibrium point x = 0 in Equation (2.7) is

• stable if, for each ε > 0, there is δ = δ(ε) > 0 such that

||x(0)|| < δ ⇒ ||x(t)|| < ε,∀t ≥ 0. (2.8)

• unstable if it is not stable.

• asymptotically stable if it is stable and δ can be chosen such that

||x(0)|| < δ ⇒ limt→∞

x(t) = 0. (2.9)

Theorem 2.3.1 (Lyapunov Stability Theorem). Let x = 0 be an equilibrium

point for (2.7) and D ⊂ Rn be a domain containing x = 0. Let V :D → R be a


continuously differentiable function such that

V (0) = 0 and V (x) > 0 in D − {0}

V ≤ 0 in D.

Then, x = 0 is stable. Moreover, if

V (x) < 0 in D − {0}.

then x = 0 is asymptotically stable.

Theorem 2.3.2 (LaSalle’s Invariance Principle Theorem). For autonomous

systems, let D ⊂ Rn be a domain containing the equilibrium point of origin. If

there exists a continuously differentiable, positive definite function V : D → R

such that

V =∂V

∂x= −W (x) ≤ 0 (2.10)

in D. Let

S = {x ∈ D|V (x) = 0}, (2.11)

and suppose that no solution can stay identically in S, other than the origin.

Then, the origin is asymptotically stable.

In addition, if D = Rn and V is radially unbounded, the origin is globally

asymptotically stable.

Remark 1. The candidate chosen in this thesis is the square of system state,

which enables the candidate to be an expression of energy consumption.

Remark 2. Because the Lyapunov candidate is also the sum of the square of

system states, it is obvious that the candidate will greater than 0 when x 6= 0.

To make sure the system is stable, the L2 distance of system states need to be

bounded, which means the feasible domain of the system states will be a round

area. So the derivation of the candidate is less than or equal to 0. If the derivative

of the candidate is less than 0, the Lyapunov candidate is describing a system

whose states are stable in a smaller round than the one picked initially. As the


feasible domain goes smaller and smaller, system states will go to the original

point of the area x = 0, which is asymptotically stable.

2.3.2 Discrete Time Systems

Nonlinear time varying discrete system can be written as

xk+1 = f(xk), (2.12)

where x ∈ Rn, and f ∈ Rn.

Definition 3. Consider the system (2.12), V changes to ∆V of the form

∆Vk+1 = V (xk+1)− V (xk). (2.13)

Corollary 1 ([37]). The equilibrium point 0 is locally stable in the sense of

Lyapunov if there exists a positive definite Lyapunov candidate V (xk) such that

∆Vk+1 ≤ 0, (2.14)

for k ≥ k0 and all x such that ||x|| < r for some r > 0 in a deleted neighbourhood

of 0.

2.4 Summary

In this chapter, a state-of-the-art on the recent methods in mecha-

tronic systems, sliding mode control, fault tolerant control, and system

optimisation is provides. There are highlighted the last achievements in

each domain together with the open research topics which are addressed

in this thesis. Because one of the most important problems in control


community is the stability analysis, some fundamental definitions and

theorems are provided at the end of this chapter. These concepts will be

used in the rest of the thesis for prove the proposed theorems.

37

Chapter 3

Mechatronic System Modelling

As mechatronic system modelling involves mechanical engineering,

electrical engineering, and materials engineering, the dynamics of the

mechatronic system is extremely complex. It is necessary to understand

the principles used in the mechatronic system before the design of con-

troller.

Robotic manipulators are widely used in industrial processes nowa-

days. The rigid n-DOF is driven by n motors, which are located at the

joints. Motors generate the necessary torques in movements. There are

two conventional methods of deriving the dynamics of robotic manipu-

lator, Newton-Euler formulation and Lagrangian formulation. These two

approaches are based on the two different but coexistent balances in all

systems.

Newton-Euler formulation builds on the force balance. Instead of

considering a complex mechatronic system, the process can be simplified

by moving the end point into the target area. With this simplification,

an involute process turns into a combination of rotations and transport-

ing, which perfectly fits into Newton-Euler method. Additionally, as the

kinematic system is practical, energy balance should also be satisfied dur-

ing the processes, which means Lagrangian method can also be used to

Chapter 3. Mechatronic System Modelling 38

achieve the dynamics. Because of the consistent result between Newton-

Euler method and Lagrangian method, the Newton-Euler method is used

in the thesis.

3.1 Frame and Rotation

Before starting achieving the dynamics of manipulator, it is neces-

sary to considering two questions, how to describe the similar functions

of each joint and how to map those functions into the same frame. In this

part, definitions of frames and rotation matrices are given.

3.1.1 Frame

There are four basic frames used in the derivative of kinematic sys-

tem.

FIGURE 3.1: The Basic Frames.

• Base Frame (B): Base frame normally affixes to the nonmoving part,

such as ground, to indicate reference point.


• Station Frame (S): Station frame is a task relevant frame, which in-

dicates the working area of the kinematic system.

• Wrist Frame (W): Wrist frame represents the coordinate on the last

joint of kinematic system.

• Tool Frame (T): Tool frame affixes to the end effector.

Figure 3.2 represents point P in a frame. The position of P can be ex-

pressed as

FIGURE 3.2: Point P in Frame A.

AP =

Px

Py

Pz

, (3.1)

where Px, Py and Px represent the positions of P along axes XA, YA and

ZA respectively.

3.1.2 Rotation

As shown in Figure 3.1 and Figure 3.2, the description of a point

not only relate to its position but also relate to the frame category. For

example, targets are always described by the position relates to station


frame SP ; meanwhile, end effector is described by the position relates

to tool frame TP . It is necessary to find a way to transfer positions that

relate to different frames into one frame.

In Figure 3.3, frame 1 shares the same origin point with frame 0 with

a rotation angle θ along direction Z. The position of point P in frame 1 is

known as 1P , which relates to frame 1.

1P =

1Px

1Py

1Pz

(3.2)

Basing on the trigonometric function, the relationship between the posi-

tion of P in frame 1 and in frame 0 can be written as,

0P = R

1Px

1Py

1Pz

. (3.3)

where R =

cos θ − sin θ 0

sin θ cos θ 0

0 0 1

is the rotation matrix along direction Z.

In the practical kinematic system, rotation always comes with shift-

ing. As shown in Figure 3.4, the distance between the origins of frame 1

and frame 2 is L. Frame 3 shares the same origin point with frame 2 with

an angle θ along the Z direction. The shifting of the origin points is

1P2,3,org =

Lx

Ly

Lz

. (3.4)


FIGURE 3.3: Coordinate Rotation.

When the position of P in frame 3 is known, the position of P in frame 1

can be,

1P

1

=

1 0 0 Lx

0 1 0 Ly

0 0 1 Lz

0 0 0 1

×

cos θ − sin θ 0 0

sin θ cos θ 0 0

0 0 1 0

0 0 0 1

×

Px

Py

Pz

1

. (3.5)

3.2 Forward Kinematic of 3-DOF Manipulator

Since motors’ density distribution is hard to be achieved, the mass of

motors is considered as a mass point, which means the inertia tensor can

be ignored. Transformation operator (i+1i T ) is used to describe the trans-

position between the relative frames. As the transformations between


FIGURE 3.4: Frames Transfer.

frames are invertible, we have i+1i T =i

i+1 T−1. The position matrix (iP ) of

a point in the ith frame can be transferred into the (i+ 1)th frame as,

i+1P =i+1i T iP. (3.6)

The method used here to derivate the manipulator dynamics is Newton-

Euler’s dynamics formulation [113]. Based on Newton’s second law, the

change of movement is related to its acceleration. As the constant length

every link has, changes of accelerations are only related to the angular

velocities.

The force of each joint has two components, the force needed to track

the trajectory and the force used to eliminate effects from other links. In

this part, the expression of the angular velocity of the (i+1)th link, which

is effected by the ith link, is given. The angular velocity of the (i + 1)th

link is

i+1ωi+1 =i+1i Riωi + θi+1

i+1Zi+1, (3.7)


where i+1ωi+1 is the angular velocity of (i+ 1)th link. i+1i R is the rotation

matrix from the ith link to the (i+ 1)th link. iωi is the angular velocity of

the ith link related to the ith frame. θi+1 is the increment of the (i+1)th an-

gle. i+1Zi+1 =

0

0

1

is the axis that the (i+ 1)th joint rotates along. i+1i Riωi

represents the shift caused by the rotation of the ith link. θi+1i+1Zi+1 is

the angular velocity needed for the (i + 1)th link. Then it is easy to have

the angular acceleration of the (i+ 1)th joint.

i+1ωi+1 =i+1i R iωi +i+1

i R iωi × θi+1i+1Zi+1 + θi+1

i+1Zi+1, (3.8)

where θi+1 is the angular acceleration needed to change θi+1. The pur-

pose of the outer interactions is to have the (i + 1)th link dynamics, use

θi+1i+1Zi+1 to represent the change of the frame.

i+1vi+1 = i+1i R[iωi × iPi+1 + iωi × (iωi × iPi+1) + ivi], (3.9)

where iPi+1 is the target position from the (i + 1)th frame into the ith

frame.

i+1vCi+1=i+1ωi+1 × i+1PCi+1

+ i+1ωi+1 × (i+1ωi+1 × i+1PCi+1)

+ i+1vi+1.

(3.10)

According to Newton’s Law, the force needed for reaching the designed

speed is

i+1Fi+1 = mi+1i+1vCi+1

. (3.11)


The force exerted on the ith joint from its neighbour is

ifi = ii+1R

i+1fi+1 + iFi. (3.12)

When talk about the rotation of rigid body, inertia tensor needs to be

FIGURE 3.5: Inertia Tensor.

considered for an accurate modelling. The rotation of rigid body can be

considered as the rotation of a combination of mass points, which is

I =

∫P × vdm, (3.13)

where P is the position vector of mass points. v is linear velocity. Iner-

tia tensor is used to describe the relationships between angular velocity,

torque and angular acceleration, which can be expressed as

I =

∫P × (ω × P )dm. (3.14)


Then according to Euler’s Law, the torque needed for the even mass dis-

tribution rigid body is

i+1Ni+1 = Ci+1Ii+1i+1ωi+1 + i+1ωi+1 × Ci+1Ii+1

i+1ωi+1. (3.15)

The torque exerted on the ith link from its neighbor is

ini = iNi + ii+1R

i+1ni+1 + iPCi × iFi + iPi+1 × ii+1R

i+1fi+1. (3.16)

Motors in each joint only supply the torque along the joint. The torque

from motor is

τi = inTiiZi. (3.17)

The torque of the ith link is used to drive the links above it.

Because every manipulator has its unique mass distribution, there is

no need for us to use a complicated distribution when trying to find gen-

eral dynamics. In this chapter, assume,

1) The base frame is ground.

2) The mass of each link is considered as a point at the end of each link.

3) Inertia tensor is ignored in the dynamics.

4) End-effector has no effect on the dynamics.

3.3 Inverse Kinematic

The inverse kinematic is a method to find out the angular positions

for each joint based on the position of the end-effector. There are two

basic solutions in the inverse kinematics, algebraic and geometric solu-

tions. The algebraic solution is based on the inverse transfer matrices;


the geometric solution is based on the geometric properties of the system

structure.

3.3.1 Geometric solution for Horizontal Joint

FIGURE 3.6: Geometric analysis for 3-DOF manipulator.

As mentioned above, the 3-DOF manipulator can be divided into

two groups. Joint 1 is the first group, and joint 2 and 3 are the second

group. From Figure 3.6, the set point of joint 1 is:

θ1 = ± arccos(Px√

P 2x + P 2

z

). (3.18)

From the expression of θ1, there is always a pair of solutions for θ1, and

that means when the manipulator passes the different half plant, the

value of θ1 might be changed.


3.3.2 Geometric solution for Vertical Joints

After finding out the set-point of joint 1, the problems now turns out

to be solving the inverse kinematics of 2-DOF manipulator. As shown in

FIGURE 3.7: Geometric analysis Horizontal Joints.

Figure 3.7,

m =√P 2z + P 2

x − l1 (3.19)

n =√P 2y +m2 (3.20)

By using the cosine rule:

ϕ = arccos(l22 + n2 − l23

2l2n) (3.21)

Then

θ2 = arccosPyn− ϕ (3.22)

θ3 = arcsin(Py − l2 × sin θ2

l3)− θ2 (3.23)


The inverse kinematic above is only working on the upper half plant (y ≥

0), when working in the other half, let

θ2 = arccosPyn

+ ϕ (3.24)

θ3 = arcsin(Py − l2 × sin θ2

l3) (3.25)

As shown in Figure 3.8, the multiple solutions exist not only in the

angles but also in the pose choosing. The pose problems may be solved

FIGURE 3.8: Geometric analysis Vertical Joints.

by trajectory optimisation. In some cases, the physical structure of the

manipulator may also constrain the other possible poses.


3.4 Controller Design

The dynamics of n-DOF manipulator is

θ = M(θ)−1(τ − V (θ, θ)−G(θ)). (3.26)

In the simulation, an alternative control signal ui is used to represent the

coupled input torque. After having the proper control signal design, in-

verse kinematic is used to decouple the torques. Let

x = θ, (3.27)

where [x1 x2 x3]T = [θ1 θ2 θ3]T and [x4 x5 x6]T = [θ1 θ2 θ3]T .

3.4.1 Feedback Linearisation for Manipulator Modelling

From Equation (3.26), the manipulator dynamics has strong non-

linearity with coupled inputs, which can be represented by a general

nonlinear expression,

x = f(x) + g(x)τ, (3.28)

where x = θ, f(x) = M(θ)−1(−V (θ, θ) − G(θ)), g(x) = M(θ)−1. Then an

alternative control signal vector u is used to take place of system input,

x = f(x) + u, (3.29)

where

u =

[0 0 0 u1 u2 u3

]T= g(x)τ. (3.30)


Then if the alternative input u is picked in a certain way, it is possible to

transfer the nonlinear dynamics into linear. Pick

ui = −fi(x) + vi, (3.31)

where vi represents the control signal needed to be designed. Then the

nonlinear system turns into linear system,

x =

[x4 x5 x6 v1 v2 v3

]T(3.32)

The system shown above can be divided into three simple linear systems,

each of which represents one of the basic physical definitions. However,

basing on the expression of ui, it is not hard to notice these three subsys-

tems are not independent.

FIGURE 3.9: Linear Close-loop Subsystem.

3.4.2 Ackerman Controller

In this system, Ackerman method is chosen as the method in con-

troller design. Before designing the controller, the controllability of the


system needs to be checked.

Qc =

[B AB

]=

0 1

1 0

(3.33)

rank[Qc] = 2 (3.34)

The controllable matrix is full rank, so the linear system is controllable.

To make the system stable, move the poles to −1 ± j, the characteristic

polynomials are

(λ+ 1− j)(λ+ 1 + j) = λ2 − 2λ+ 2 (3.35)

λ(A) = A2 + 2A+ 2I (3.36)

K = [0 1]Q−1λ(A) (3.37)

K =

[2 2

](3.38)

The controller is

ui = −fi(x)−Kx (3.39)

From the simulation result, the system has been settled and the static

error goes to zero.


3.4.3 System State Observer Design

Observability

In the practical industrial processes, some of the states cannot be re-

flected directly by the output; moreover, some are even hard to be de-

tected by other methods. But in the controller these states are needed,

so it is necessary to design an observer to observe these states for the

controller designing.

Before design the observers, the system must be observable. But in

some special cases, if the unobservable parts are asymptotically stable, it

is still possible to design the observer for the system. For the system

x = Ax+Bu

y = Cx

(3.40)

If the system is observable, the system must satisfies the following con-

straints

V =

C

CA

...

CAn−1

(3.41)

rank(V ) = n (3.42)

For the subsystems in the dynamics

A =

0 1

0 0

C =

[1 0

](3.43)


rank(V ) = 2 (3.44)

Then the system is observable.

Full-Order Observer Design

The full-order observer is a restructuring system basing on the sys-

tem dynamics. As shown in Figure 5.1, a feedback L(y − Cx) is brought

in to cancel the perturbation of the system dynamics. The observer can

FIGURE 3.10: Full-Order Observer.

be represented as:

˙x = (A− LC)x+ Ly +Bu

x(0) = x0

y = Cx

(3.45)


For the manipulator dynamics,

A =

0 1

0 0

B =

0

1

C =

[1 0

](3.46)

The dual coefficient matrix is:

A = AT , B = CT (3.47)

Design the poles of the observer located at λ∗1,2 = −5

λi(A− BK) = λ∗i (3.48)

The state feedback matrix K is

K =

[10 25

](3.49)

Let

L = KT (3.50)

Then

A− LC =

−10 1

−25 0

(3.51)

The full-order observer is

˙x =

−10 1

−25 0

x+

0

1

u+

10

25

x1 (3.52)


3.5 3-DOF Manipulator Visualization Program-

ming

To illustrate the dynamics, a functional configuration based simu-

lator result will be shown in this section. The manipulator simulation

used in this chapter is capable of showing torques and system response

on-line. A 3D working space is created to represent the practical station

frame. A 3-DOF manipulator model is created with one horizontal link

and two vertical links. Links of the manipulator are represented by solid

lines. The lengths of these segments are based on the practical parame-

ters. The mass distribution is assumed as a point at the end of each link.

The parameters used in this example is based on the ABB IRB 120 Indus-

trial Robot[114].

−0.8−0.6

−0.4−0.2

00.2

0.40.6

0.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y(m

)

z(m)x(m)

FIGURE 3.11: Manipulator Simulation.

Figure 3.11 is the interface of the simulator. A 3-DOF manipulator


model is displayed to demonstrate the pose of manipulator on-line. The

trajectory of the manipulator is marked by dashed line.

3.6 Simulation Results

Full-order observer has been applied on the manipulator simulation;

the system states used in control strategy design is based on the estima-

tions. The initial positions of the manipulator are placed at

θ01 = 20◦

θ02 = 30◦

θ03 = −20◦

(3.53)

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Time(s)

Angular Position of Joint 1 & Estimation (Degree)

x1 Estimation

x1

FIGURE 3.12: Position Response x1 based on Full-OrderObserver.

Figures 3.12, 3.13, and 3.14 are the angular positions responses of

manipulator with full-order observer. The dashed lines represent the es-

timation of angular positions, and the solid lines are the system states

values. Although the initial value of the observer states and system states


0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Time(s)

Angular Position of Joint 2 & Estimation (Degree)

x2 Estimation

x2


0 20 40 60 80 100−120

−100

−80

−60

−40

−20

0

Time(s)

An

gu

lar

Po

sitio

n o

f Jo

int

3 &

Est

ima

tion

(D

eg

ree

)

x3 Estimation

x3


are different, the states observer need few seconds to converge to the tar-

get value. The difference between also makes the gain in the observer

loop higher than that in the control loop, which introduces the oscillation

at the beginning of the process.

Figures 3.15, 3.16, and 3.17 are the angular velocity responses of the

system with full-order observer. The dashed lines are the estimations of

angular velocities, and the solid lines are the system angular velocities.


0 20 40 60 80 100−60

−40

−20

0

20

40

60

80

100

120

Time(s)

An

gu

lar

Ve

loci

ty o

f Jo

int

2 &

Est

ima

tion

(D

eg

ree

/se

c)

x4 Estimation

x4

FIGURE 3.15: Velocity Response x4 based on Full-OrderObserver.

0 20 40 60 80 100−100

−50

0

50

100

150

200

Time(s)

An

gu

lar

Ve

loci

ty o

f Jo

int

2 &

Est

ima

tion

(D

eg

ree

/se

c)

x5 Estimation

x5

FIGURE 3.16: Velocity Response x5 based on Full-OrderObserver.

The full-order observer estimate all the states of system, which means the

full-order can get more information from the system. As shown in the

figures, the estimations from full-order observer converge to real target

values faster than those from the reduced-order observer. However, this

fast convergence also brings oscillation at the beginning of the process.


0 20 40 60 80 100−120

−100

−80

−60

−40

−20

0

20

40

Time(s)

An

gu

lar

Ve

loci

ty o

f Jo

int

3 &

Est

ima

tion

(D

eg

ree

/se

c)

x6 Estimation

x6

FIGURE 3.17: Velocity Response x6 based on Full-OrderObserver

3.7 Summary

In this chapter, a 3-DOF manipulator simulator has been programmed.

Frames are defined since it confers greater conceptual clarity on the anal-

ysis. The dynamics of the 3-DOF manipulator is obtained by using the

Newton-Euler method. Due to the strong non-linearity that kinematic

system has, feedback linearisation is applied to simplify the dynamics.

As the torque inputs are coupled, torques of each joint are obtained by

using the inverse kinematic method. Basing on the practical situations,

system state observer is designed to observe the system state. The algo-

rithms proposed in Chapter 4 are tested in simulation using the simulator

developed in this chapter.

60

Chapter 4

Fault Matrix based Fault Tolerant

Control

With the increasing ageing and environmental damages, such as ex-

treme temperature, humidity, acidic, dynamics may not be accurate in

few months. Inaccurate dynamics may also be caused by actuator faults

or disturbance. When actuator fault is coupled with disturbance, the cur-

rent fault tolerant strategy may not work well anymore. so it is necessary

to find a method that can not only estimate actuator fault but also isolate

the actuator fault from disturbance.

4.1 Problem Formulation

The mechatronic system dynamics can be written as,

x = Ax+ F (x) + EgΘu+ E∆d(t)

y = Cx

(4.1)

where x =

[x1 x2

]Trepresents the system state vector. x1 ∈ Rn repre-

sents the position vector in mechatronic system, and x2 ∈ Rn is the speed

Chapter 4. Fault Matrix based Fault Tolerant Control 61

vector. A =

0 I

0 0

is the coefficient of linear part. F (x) =

[0 f1

]Tis the

nonlinear part of the system. E =

[0 Es

]Tis the new coefficient matrix

with Es = I .

F ∈ Rn is the vector representing both the linear and the nonlinear

parts in the system which satisfies Lipschitz condition, |f(a) − f(b)| ≤

δf |a − b|. E ∈ Rn×n is a coefficient matrix. g ∈ Rm×m represents the non-

linearities between control input and system, which also satisfies |g(a)−

g(b)| ≤ δg|a−b|. Θ ∈ Rm×m is fault matrix with a prior knowledge . u ∈ Rm

is the control input vector. y is the system output. ∆d represents the un-

modelled parts in the system, and is bounded by |∆d| < δd. To make the

system more practical, let us assume that system input is bounded, which

means |Θu| < δu.

The disturbance is generated by a linear exogenous system

ζ = Aoζ

∆d = Dζ

(4.2)

where ζ ∈ Rn×n, ∆ ∈ Rn. The system can be written in a form suitable for

control strategy design. We need to transfer the system into error model.

As a pre-phase work for that, let us define

e1 = x1 − xd

e2 = e1

(4.3)


then the system can be re-written as

e = Ae+ F + EgΘu+ E∆d(t)− Exd

y = C(e+ [xd xd]T )

(4.4)

If the strategy described in this paper is designed properly, both e1 and

e2 should converge to zero at the end, which means the system state x1

achieves the reference vector xd, and x2 guarantees that x1 stays with the

reference.

4.2 Terminal Sliding Mode Controller Design

In this section, a terminal sliding mode controller is proposed for

non-faulty system. Fault matrix is assumed to be constant Θ = I , where

I is an identity matrix.

Lemma 1 ([115]). If there exists a continuous radially unbounded function V :

Rn → R+ ∪ {0} such that 1) V (x) = 0 ⇒ x ∈ M ;2) any solution x(t) of

x = g(t, x), x(0) = x0 satisfies the inequality DV (x(t)) ≤ −(αV p(x(t)) +

βV q(x(t)))k for some α, β, q, p, k > 0, pk < 1, qk > 1, then the set M ⊂ Rn is

globally fixed-time attractive for the system x = g(t, x), x(0) = x0. and

T (x0) ≤ 1

αk(1− pk)+

1

βk(qk − 1),∀x0 ∈ Rn.

Based on the Lemma above, the sliding mode manifold can be ex-

pressed as

S =

e1,1 + α1,1e

γ1,11,1 + α2,1e

γ2,12,1

...

e1,n + α1,neγ1,n1,n + α2,ne

γ2,n2,n

, (4.5)


where ei,j represents the jth element in ei. To make the sliding manifold

applicable to Lemma 1, pick αi > 0, βi > 0. Integers γ1 and γ2 satisfy

γ1 = 2n1−12n1−1

∈ (2,∞), and γ2 = 2n2−12m2−1

∈ (1, 2).

Remark 3. The manifold can transfer to e2 = −( 1α2e1 + α1

α2eγ11 )

1γ2 , which con-

tains two components e1γ21 and e

γ1γ21 . As γ1 ∈ (2,∞) and γ2 ∈ (1, 2), γ1

γ2∈ (1,∞)

and 1γ2∈ (0, 1), it means e

1γ21 can guarantee the fast convergence when |e1| < 1,

and eγ1γ21 guarantees the fast convergence when |e1| ≥ 1.

Dynamics on Manifold for Non-faulty System

The control law requires s = 0 [116],

si =e1,n + α1,nd

dt(eγ1,n1,n ) + α2,n

d

dt(eγ2,n2,n )

=e2,i + α1,iγ1,ieγ1,i−11,i e2,i + α2,iγ2,ie

γ2,i−12,i (fi

+ Es,igiui − xd,i)

=e2,i + α1,iγ1,ieγ1,i−11,i e2,i + α2,iγ2,ie

γ2,i−12,i f

+ α2,iγ2,ieγ2,i−12,i Es,igiui − α2,iγ2,ie

γ2,i−12,i xd,i

=0,

(4.6)

where i = 1, 2, ..., n. From Equation (4.6), the control efforts can be ex-

pressed as,

u∗i =− (α2,iγ2,iEs,igi)−1(e

2−γ2,i2,i + α1,iγ1,ie

γ1,i−11,i e

2−γ2,i2,i

+ α2,iγ2,if − α2,iγ2,ixd,i),

(4.7)


Dynamics outside Manifold for Non-faulty System

Similar to the method described in [25], when the initial condition is

not on the manifold, it is necessary to design an additional part of con-

troller to pull the state towards manifold. Similar to the sliding manifold,

this additional part is structured as

u∗∗i = −(α2,iγ2,ieγ2,i−12,i Es,igi)

−1(β1,isξ1,ii + β2,is

ξ2,ii ), (4.8)

where β1,i and β2,i are positive constants, ξ1,i = 2n2−12m2−1

, ξ1,i = 2p2−12q2−1

,m2, n2, p2,

and q2 ∈ Z+. ξ1,i ∈ (0, 1) and ξ2,i > 1

Singular Problem

Due to the non-linearities of the sliding manifold, singular problem

should be taken into consideration. During the design of the dynamics

outside manifold, e1−γ2,i2,i appears in the controller. In addition, the range

of γ2,i ∈ (1, 2), which means e2,i = 0 is the singular point of the controller.

The singular problem is solved by introducing sinusoidal function as

µi =

sin(π

2

eγ2,i−1

2,i

τ) if e

γ2,i−12,i < τ

1 otherwise

(4.9)

By introducing the sinusoidal function designed above into the controller,

ui =− (α2,iγ2,iEs,igi)−1(e

2−γ2,i2,i + α1,iγ1,ie

γ1,i−11,i e

2−γ2,i2,i

+ α2,iγ2,if − α2,iγ2,ixd,i)−

µi(α2,iγ2,ieγ2,i−12,i Es,igi)

−1(β1,isξ1,ii + β2,is

ξ2,ii ),

(4.10)


To check the stability of the sliding mode controller, choose the Lyapunov

candidate in the form as,

V =1

2s2i (4.11)

where si is the ith manifold. The derivative of the candidate is

V =sisi

=si(e2,i + α1,iγ1,ieγ1,i−11,i e2,i + α2,iγ2,ie

γ2,i−12,i (fi

+ Es,igiui))

=− µi(β1,isξ1,i+1i + β2,is

ξ2,i+1i )

<0.

(4.12)

Based on the expression of the sliding manifold, V = 0 only when e = 0.

Due to the singular problem, the system convergence time can be roughly

estimated by duplicating the conclusion of Lemma 1.

4.3 Observer Design

Uncertainties can be caused by many different reasons, such as ag-

ing, the unmeasured components, working environment changing. In

the controller design, robustness is always in the list of the considera-

tions.

4.3.1 Disturbance Observer

The uncertainties in nonlinear systems always increase the difficulty

level of controller design. In this paper, uncertainties bring chattering

into the sliding mode controller, but if the values of uncertainties can be


estimated, the chattering is easy to deal with. In this section, a distur-

bance observer is used to achieve the dynamics of uncertainties. Moti-

vated by [117], the disturbance observer is suggested as

σ = (Ao − l(e)ED)σ + Aop(e)− l(e)(EDp(e)

+f(e) + Eg(e)Θu− Exd)

ζ = σ + p(e)

∆d = Dζ

, (4.13)

where l(e) is the gain function, p(e) is the nonlinear function to be de-

signed.

The nonlinear variable p(e) is picked as

p(e) = KLr−1f (Ce+ [xd xd]

T ) (4.14)

where r is the relative degree. The observer gain l(e) can be determined

by

l(e) = K∂Lr−1

f (Ce+ [xd xd]T )

∂e(4.15)

4.3.2 Fault Matrix Observer

In this section, an observer-based fault detection strategy is proposed

for the uncertain nonlinear system with both actuator and sensor faults.

Due to the existent of sensor fault, system states cannot be used in the

strategy design. As a result of actuator fault, fault matrix will no longer

be the same as its prior knowledge, which means

Θ = Constant 6= ΘH ,


FIGURE 4.1: Fault Matrix Observer.

where ΘH represents the value of healthy fault matrix.

As shown in Figure 4.1, the observer structure used in this section

focus on the accuracy of fault estimation.

The dynamics of the observer is expressed as

z = Ae+ F (e) + EgΘu+ LCe

−Exd

e = z + Ev

(4.16)

where z =

[z1 z2

]T∈ R2n represents the internal states of the unknown

input observer. e =

[e1 e2

]T∈ R2n is the vector of system error esti-

mation. By adding v on z, the fault matrix will not be effected by the

uncertainties.

Assumption 1. There exists a positive definite P that satisfies

PE = CT (4.17)


To make sure that the estimation will approach the real value of fault

matrix, stability is necessary to be checked. Choose the Lyapunov candi-

date as

V =1

2eTP e+

1

2tr(ΘT Θ) (4.18)

where e = e− e and Θ = Θ− Θ

The derivative of candidate

V =eTP ˙e+ tr(ΘT ˙Θ)

=eTP (Ae+ EgΘu− EgΘu− LCe

+ E∆d − Ev + F − F ) +m∑i=1

θi˙θi

≤− eT (PLC − PA− 1

2P TP − 1

2δ2fI

− 1

2PEETP − 1

2δ2uδ

2gI)e+ eTPET (∆d − v)

+m∑i=1

(ui

m∑j=1

(Ce)j gji + ˙θi)θi

(4.19)

From the expression above, the updating law of Θ is

˙Θ =

u1

m∑j=1

((Ce)j gj1) . . . 0

0. . . 0

0 . . . umm∑j=1

((Ce)j gjm)

(4.20)

The estimation of the disturbance is represented by ∆d. By choosing

v = ∆d (4.21)


the derivative of candidate becomes

V ≤− eT (PLC − PA− 1

2P TP − 1

2δ2fI

− 1

2PEETP − 1

2δ2uδ

2gI)e

(4.22)

If there exists a P that guarantees that (PLC − PA − 12P TP − 1

2δ2fI −

12PEETP

− 12δ2uδ

2gI) ≺ 0, then the fault matrix estimation will be achieved asymp-

totically.

4.4 Fault Tolerant Controller Design

Uncertainties always bring chattering into sliding mode controller.

Those chattering may be caused by the delay of the system and inaccu-

rate system model. To make the uncertainties become an available signal,

disturbance observer is applied here to estimate the value of the uncer-

tainties. By applying multi-layer structure, the inner loop focus on the

stability of system. Disturbance observer is added on outer loop to mon-

itor uncertainties and feedback to the inner loop so that the uncertainties

can be eliminated. But if there is any error between disturbance and its

estimation, then chattering will not be totally eliminated.

Combining Equations (4.10), (4.21), and (4.20), the chattering-reduced

TSMC expression is

u∗i =− (α2,iγ2,iEs,igi)−1(e

2−γ2,i2,i + α1,iγ1,ie

γ1,i−11,i e

2−γ2,i2,i

+ α2,iγ2,if)− (α2,iγ2,ieγ2,i−12,i Es,igi)

−1µi(β1,isξ1,ii

+ β2,isξ2,ii )− (gi)

−1∆d,i + (Es,igi)−1xd,i.

(4.23)


Based on the results above, both the fault matrix and disturbance

can be achieved, it is possible to design a fault tolerant control strategy to

improve the performance when faults occur. To eliminate the effect from

fault matrix, design the control signal as

u = Θ−1u∗ (4.24)

where Θ is the estimation of fault matrix.

The fault tolerant controller is

ui =− (α2,iγ2,iEs,igiθi)−1(e

2−γ2,i2,i + α1,iγ1,ie

γ1,i−11,i e

2−γ2,i2,i

+ α2,iγ2,if)− (α2,iγ2,ieγ2,i−12,i Es,igiθi)

−1µi(β1,isξ1,ii

+ β2,isξ2,ii )− (giθi)

−1∆d,i + (Es,igiθi)−1xd,i.

(4.25)

To check the stability of the system, choose the Lyapunov candidate as

Vi =1

2s2i . (4.26)

The derivative of the candidate is

Vi =sisi

=si[e2,i + α1,iγ1,ieγ1,i−11,i e2,i + α2,iγ2,ie

γ2,i−12,i (fi

+ Es,igiθiui − xd,i + Es∆d)]


γ2,i−12,i (fi

+ Es,igiθi

θiu∗i − xd,i + Es∆d)]

. (4.27)

As proved above, when the fault happens, Θ → Θ which means Θ

Θ→ I .

The candidate derivative changes into


Vi =sisi


γ2,i−12,i (fi

+ Es,igiu∗i − xd,i + Es∆d)]

=− µi(β1,isξ1,i+1i + β2,is

ξ2,i+1i )

≤0.

(4.28)

From the proof above, the effect from the fault matrix can be eliminated

by introducing of the FD strategy, and the system remained stable.

4.5 Simulation Result

In order to illustrate the theoretical method proposed in this chapter,

the simulator developed will be used in Chapter 3. The system dynamics

can be written as,

x = Ax+ Eg(x)Θu+ E∆d, (4.29)

where x ∈ R2 is the system state vector, E is the conscience representing

the coupling of disturbance and system input. Θ is fault matrix.u and ∆d

are the system input and disturbance. The system parameters are picked

as

A =

0 1

1 0

E =

0

1

g = 0.5 sin(x1) + 1. (4.30)

When no fault occurs, the fault matrix remains a constant value, let us

assume

ΘH = I. (4.31)


The disturbance in the system is generated by the function

ζ = Aoζ

∆d = Dζ

, (4.32)

where Ao =

0 −0.1

0.1 0

and D =

[0.5 0.5

], which introduces a cosine

disturbance into the system.

The sliding manifold can be written as

s = e1 + 0.2e23111 + 10e

752 , (4.33)

where e = x − [xd xd]. To satisfy the condition described in Equation

(4.12), the controller parameters are picked as

β1

β2

=

0.7

11

ξ1

ξ2

=

137

1115

. (4.34)

To solve the singular problem, µ is picked as

µi =

sin(π

2

e252

τ) if e

252 < 0.01

1 otherwise

. (4.35)

Based on the system dynamics, the disturbance observer gain K =

[5 0.5]. Then the nonlinear variable is determined by

l =

0 5

0 0.5

. (4.36)


In the FTC strategy design, let’s pick

P =

1 2

2 3

L =

2

0.3

. (4.37)

the eigenvalues of PLC − PA − 12PP − 2I − 1

2PEETP are −3.3133 and

−0.2867. The update law of Θ is designed as

˙θ = [0.5 sin(x1) + 1]u(2x1 + 3x2). (4.38)

0 10 20 30 40 50 60 70 80−1

−0.5

0

0.5

1

1.5

2

2.5

Time(s)

Sys

tem

Sta

te

x1 &

Est

ima

tion

(m

)

x1

x1 Estimation

r

FIGURE 4.2: System State (x1) Estimation.

Figure 4.2 is the position response(x1) and its estimation. The thick

dashed curve are the position error estimation, solid curve is the position

feedback. The thin dashed line is the reference signal. Due to the different

initial values system and estimator, the estimation error which is feed

back to observer is big, that makes the estimation increase rapidly. As


the both system dynamics and observer are in continuous time domain,

it needs a period to decrease the feed-back gain after estimation converge

to the system state, which causes the oscillation at the beginning of this

process. When reference changes to 2, there is an error between x1 and

its estimation, which can be caused by the estimation from x2.

0 10 20 30 40 50 60 70 80−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Ve

loci

ty x

2 &

Esi

tma

tion

(m

/s)

x2

x2 Estimation

FIGURE 4.3: System State (x2) Estimation.

Figure 4.3 is the velocity response(x2) of the system, which can be

considered as the changeing rate of x1. The dashed curve is the velocity

estimation, and solid curve is the system velocity As the initial value of

estimator is closer to the reference, the speed of system is faster than that

of estimator. When the estimation of x1 converge to its target state, the

estimation of x2 still above zero, which makes the estimation of x1 keep

increasing When the reference changes to 2, there is a depression in x2,

which is caused by the trigonometric function used in singular problem.


However, the observer is based on the system output and estimation er-

ror, which makes the estimation of x2 smooth.

0 10 20 30 40 50 60 70 80

0.8

1

1.2

1.4

1.6

1.8

2

Time(s)

Fault Matrix & Estimation

ΘΘ Estimation

FIGURE 4.4: Fault Matrix Estimation.

As shown in Figure 4.4, there is no actuator fault existing in the sys-

tem at the beginning of this simulation. Dashed curve is the estimation

from fault observer, and the solid curve is the value of the fault matrix

in the system. The initial values are placed differently from fault initial

matrix value, which cause the oscillation at the beginning. However, as

those target signals working on the system remain the same, x2 has an

overshoot at the beginning. After a short period, as the observer is get-

ting a better estimation, x2 comes back to normal.

In the second half of the process, the fault is set to the value 1.2; the

reference signal value is 2. At the beginning of the second half process,

fault matrix is underestimated for a short time, which makes Θ has an


overshoot. When Θ has overshoot, has overshoot, the system has inade-

quate control, which is shown in the curve of x2.

0 10 20 30 40 50 60 70 80−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time(s)

Dis

turb

an

ce &

Est

ima

tion

DisturbanceEstimation

FIGURE 4.5: Disturbance Estimation.

Figure 4.5 is the result from disturbance observer. Solid curve is the

disturbance generated by exogenous system, and dashed curve is the es-

timation from disturbance observer. As the big initial error from distur-

bance observer, there is an overshoot at the beginning. After the observer

states catch the system states, the disturbance can be observed accurately.

When reference changes to 2, there is only a small overshoot in the dis-

turbance estimation.


4.6 Summary

In this chapter, a fault matrix based fault tolerant control strategy

is proposed. System considered in this chapter is nonlinear and satis-

fies Lipschitz condition. To guarantee the stability of non-faulty system,

terminal sliding mode is used. Different with the conventional sliding

mode control, fixed-time sliding manifold can guarantee system states

converge to zero within fixed-time boundary.

As the existing singular problem, sine function is introduced to guar-

antee the stability of the system. However, as the system states do not en-

force the sliding manifold strictly, the time boundary that calculated may

not guarantee the system converge time any more But as the trigonomet-

ric function only works within a small range, it is possible to add a small

constant in the time boundary to cover the execution time of the trigono-

metric function.

Both disturbance and actuator fault exist in the system. Fault matrix

is used to isolate the actuator fault from disturbance. Moreover, the fault

matrix can also be used to find the relationship between actuator fault

and control signal, which can be used as the coefficient of the controller

to compensate the fault in the system. Disturbance are observed by dis-

turbance observer, the value of disturbance estimation is used to improve

the performance of fault observer.

78

Chapter 5

Set-point Re-planning

This chapter proposed a novel set-point re-planning method, which

optimise the deteriorating system performance without changing orig-

inal control loop. As a result of the long maintenance duration of the

manipulator, Robotino mobile robot is used as the experiment platform

for this method.

5.1 Problem Formulation

As most of the industrial processes are controlled by computers, the

signals used in the controller design are discrete.

5.1.1 System Model

The system dynamics considered here can be expressed as,

x1,k+1 = x1,k + hx2,k,

x2,k+1 = x2,k + h(fk + gkuk),

(5.1)

Chapter 5. Set-point Re-planning 79

where h is the system sampling time, xi,k = xi(kh), xk =

[x1,k x2,k

]T∈

R2n is the system states vector, fk ∈ Rn represents the nonlinear function

in the system, uk ∈ Rn is the control input vector. gk ∈ Rn is an invertible

matrix that typifies the nonlinear relationships between the control input

uk and system states xk.

In this section, the set-point vector is denoted by r∗ ∈ Rn, and the

system is considered in fault free situation and not affected by distur-

bances. However, the case of systems affected by faults and/or distur-

bances is addressed in section 5.2.1. The errors between system states

and set-point vector can be expressed as,

e1,k = x1,k − r∗k, (5.2)

where r∗k is the set-point vector at time instant k, e1,k ∈ Rn is the vector of

position error at time instant k. Based on relation in Equation (5.2), the

position error at time instant k + 1 can be expressed as

e1,k+1 = x1,k+1 − r∗k+1. (5.3)

Using Equation (5.1), (5.2), and (5.3), the error between two consecu-

tive steps become

e1,k+1 − e1,k = hx2,k − (r∗k+1 − r∗k). (5.4)

In mechatronic systems, and in particular for robotic manipulators, the

set-point for the position of each joint is known and it is given by the

user. There is not any set-point for speeds or accelerations available. In


order to overcome this issue, in this thesis is proposed to be used

e2,k = x2,k −1

h(r∗k+1 − r∗k), (5.5)

where e2 ∈ Rn is the vector of velocity error at time instant k. Based on

Equation (5.1), Equation (5.3) and Equation (5.6),

e1,k+1 = e1,k − h(x2,k −1

h(r∗k+1 − r∗k)), (5.6)

the dynamics for the errors in discrete time become

e1,k+1 =e1,k + he2,k,

e2,k+1 =e2,k + h(fk + gkuk) +1

h(−r∗k+2

+ 2r∗k+1 − r∗k),

(5.7)

In the next paragraph, a SMC for the errors in Equation (5.7) is pro-

posed. The role of the controller is to make the errors to converge at zero

in finite time. This control problem is similar with a trajectory tracking

strategy.

5.1.2 Terminal Sliding Mode

The control method proposed to make the errors from Equation (5.7)

to converge to zero in finite time is terminal sliding mode control. Based

on the contribution made by Man et al. [44], the non-singular sliding


manifold is picked as,

sk =

s1

s2

...

sn

=

e2,1,k + β1eq1p11,1,k

e2,2,k + β2eq2p21,2,k

...

e2,n,k + βneqnpn

1,n,k

, (5.8)

where β ∈ Rn is positive defined. e1,j,k and e2,j,k are the jth component at

time instant k. pi and qi are positive odd integers satisfying qipi∈ (1,∞),

where pi and qi are design parameters. The sliding manifold is a con-

straint to the system states to make the system states to converge to the

origin point following a certain trajectory.

Proposition 1. It is necessary to have qipi∈ (1,∞) so that the sliding manifold

structure choose in (5.8) is a terminal attractor.

Proof. This proof shows x = 0 is an attractor of the sliding manifold.

As discrete time and continuous time are two alternative frame, which

means

e1 = e2 ⇔ e1,k+1 − e1,k. (5.9)

The sliding manifold chosen in Equation (5.8) can be transfer to continu-

ous time frame, which is,

s = e2 + βeqp

1 . (5.10)

When system states are on the manifold,

s = 0. (5.11)


e1 from Equation (5.7) can be expressed in continuous time as,

de1

dt= −βe

qp

1 , (5.12)

then

dt = − 1

βe− qp

1 de1. (5.13)

The convergence time is

t = −e1→0∫e1(0)

1

βe− qp

1 de1

= − 1

β

1

1− qp

e1(0)1− qp

(5.14)

e1(0) is the initial value of e1. The convergence time only meaningful

when t > 0, so qp> 1.

Observation 1. When system states move along the manifold,

si,k+1 = si,k = 0. (5.15)

Then by combining Equation (5.8) and Equation (5.15), the equiva-

lent control law that keeps states on the manifold (5.8) is given by

u∗i,k =− g−1i,k fi,k + g−1

i,k

1

h[βie

qipi1,i,k − βi(e1,i,k

+ he2,i,k)qipi ] + g−1

i,k

1

h2(r∗i,k+2 − 2r∗i,k+1

+ r∗i,k),

(5.16)

where ui,k is the ith control signal at time instant k. However, if the sys-

tem’s initial values are not placed on the manifold (sk = 0), controller


shown in Equation (5.16) cannot transport system states to the sliding

manifold.

To make sure the convergence of the system with initial values out-

side sliding manifold, an additional term need to be added. Based on

[115], the new controller is written as,

ui,k =− g−1i,k fi,k + g−1

i,k

1

h[βie



i,k

1

h2(r∗i,k+2 − 2r∗i,k+1

+ r∗i,k)− g−1i,k

1

h(α1,is

n1,im1,i

i,k + α2,is

n2,im2,i

i,k ),

(5.17)

where m1,i,m2,i, n1,i, n2,i are positive odd integers which satisfy n1,i

m1,i∈

(0, 1) and n2,i

m2,i∈ (1,∞). Use the controller described in Equation (5.17)

taking the place of uk in Equation (5.7),

e1,i,k+1 =e1,i,k + he2,i,k,

e2,i,k+1 =e2,i,k + βieqipi1,i,k − βi(e1,i,k + he2,i,k)

qipi

− (α1,is

n1,im1,i

i,k + α2,is

n2,im2,i

i,k ).

(5.18)

To prove the stability of system with the TSMC, the convergence of

the system states has to be checked.

Lemma 2. For the error dynamics shown in Equation (5.7), if the controller is

designed as


i,k

1

h[βie



i,k

1

h2(r∗i,k+2 − 2r∗i,k+1

+ r∗i,k)− g−1i,k

1

h(α1,is

n1,im1,i

i,k + α2,is

n2,im2,i

i,k ),

(5.19)

where m1,i,m2,i, n1,i, n2,i are positive odd integers which satisfy n1,i

m1,i∈ (0, 1)

and n2,i

m2,i∈ (1,∞). Then the system state will converge to zero within finite

time.


Proof. By using the sliding manifold described in Equation (5.8), system

states are expected to stay on the manifold (si = 0) after reaching,

si,k+1 = si,k = 0. (5.20)

However, the initial values are not always placed on the sliding man-

ifold. To make the sliding manifold an attractor, it is necessary to prove

that the system states move toward the manifold when they are outside

the sliding manifold (si 6= 0), which means

|si,k+1| < |si,k|. (5.21)

If the convergence of the difference for the surfaces can be proved,

then the stability of the system is guaranteed. The difference between

si,k+1 and si,k is

si,k+1 − si,k

=e2,i,k+1 + βieqipi1,i,k+1 − e2,i,k − βie

qipi1,i,k

=e2,i,k + βieqipi1,i,k − βi(e1,i,k + he2,i,k)

qipi

− (α1,is

n1,im1,i

i,k + α2,is

n2,im2,i

i,k ) + βi(e1,i,k

+ he2,i,k)qipi − e2,i,k − βie

qipi1,i,k

=− (α1,is

n1,im1,i

i,k + α2,is

n2,im2,i

i,k ).

(5.22)

As described above,m1,m2, n1 and n2 are positive odd integer, which

guarantees that the numerator of nimi

+ 1 is even. αj,i is positively defined.

As the convergence of Equation (5.22) is proved in [55], it is obvious to


see |si,k+1| < |si,k|, if |si,k| 6= 0. Then

e2,i,k = −βieq1p11,i,k. (5.23)

The system is globally asymptotically stable. Based on Remark 4, e2,i,k is

a terminal attractor.

However, the same for other practical systems, mechatronic system

is vulnerable to uncertainties. The following method focus on improving

the system performance deteriorated by uncertainties.

5.2 Set-point Re-planning

When uncertainties appear in the system, system performance will

be deteriorated. In this section, a new method of set-point re-planning

is given to control the system even with reduced performances in the

presence of disturbances.

FIGURE 5.1: Block Diagram of Set-point Replanning

Figure 5.1 is the block diagram of set-point replanning. The con-

troller is designed based on the accurate system dynamics. An additive


set-point ∆r is designed to optimise the system performance in presence

of disturbances.

5.2.1 System Model with Disturbance

The system (5.1) can be written in presence of disturbances as in the

following x1,k+1 = x1,k + hx2,k + d1,

x2,k+1 = x2,k + h(fk + gkuk) + d2,

(5.24)

where d1 ∈ Rn and d2 ∈ Rn are the additive uncertainties in the system,

such as any unconsidered signal in the system dynamics. To improve the

system performance, an additional term ∆rζ is used to compensate the

effects of uncertainties. The new reference signal is

rk = ∆rζ + r∗k, (5.25)

where r∗k is the original reference vector. ∆r ∈ Rn is the new compen-

satory set-point to be designed. rk ∈ Rn is the new set-point that used in

the system. (•)ζ shows a different sampling time with the system states

because is computed by the outer loop (see Figure 5.1). As e is used to

represent the error between system states x and the original set-point r∗,

it is necessary to define another error vector for positions that represents

the deference between x and the new set-point r. To avoid confusion in

the notations, the new error vector will be epsilon. Define

ε1,k = x1,k − rk

ε2,k = x2,k − 1h(rk+1 − rk).

, (5.26)


where ε =

[ε1 ε2

]T∈ R2n is the vector representing the error between

system state x and new set-point r. ε1 ∈ Rn and ε2 ∈ Rn. h is the sampling

time. Based on the new vectors defined above, the error dynamics is

ε1,k+1 =ε1,k + hε2,k + d1,

ε2,k+1 =ε2,k + h(fk + gkuk) +1

h(−rk+2

+ 2rk+1 − rk) + d2,

, (5.27)

u is used to represent the controller based on the new error vector. h

is the length of system states sampling time. Similar to the definition

in Equation (5.17), the controller ui,k basing on the new error vector ε is

expressed as,


i,k

1

h[βiε

qipi1,i,k − βi(ε1,i,k

+ hε2,i,k)qipi ] + g−1

i,k

1

h2(r∗i,k+2 − 2r∗i,k+1

+ r∗i,k)− g−1i,k

1

h(α1,is

n1,im1,i

i,k + α2,is

n2,im2,i

i,k ),

(5.28)

where si,k = ε2,i,k + βiεqipi1,i,k is the sliding manifold related to the new

system state vector ε. fi,k = fi(εk) and εj,i,k = εj,i(kh). Combine the

system dynamics described in Equation (5.27) and controller ui,k together,

ε1,k+1 =ε1,k + hε2,k + d1

ε2,k+1 =ε2,k + βεqp

1,k − β(ε1,k + hε2,k)qp + d2

+1

h(−∆rk+2 + ∆2rk+1 −∆rk).

(5.29)

Equation (5.29) is the system dynamics basing on the time-varying set-

point. A novel set-point re-planning method is proposed to improve the

performance of the system expressed in Equation (5.29).


5.2.2 Performance Optimisation

To eliminate the effect of uncertainties, a set-point re-planning method

is proposed. Choose the performance index as [118]

J = eT1,Ne1,N +N−1∑ζ=0

eT1,ζe1,ζ , (5.30)

whereN is the number of sampling points. By choosing this performance

index, system states are expected to stay as close as possible to the set-

point r∗. The outer loop is designed based on this performance index.

Define Hamilton equation [118],

Hζ = eT1,ζe1,ζ + λTζ+1fζ (5.31)

where fζ is the system dynamics, λζ is Hamiltonian. Hamilton equation

is used in the design of outer loop. Based on the chosen performance

index, the Hamiltonian becomes

Hζ =eT1,ζe1,ζ + λTζ+1(ε1,ζ+hε2,ζ+d1

ε2,ζ+h(fζ+gζ uζ)+d2)

=n∑i=1

{ε21,i,ζ + 2ε1,i,ζ∆ri,ζ + ∆2ri,ζ

+ λ1,i,ζ+1(ε1,i,ζ + hε2,i,ζ + d1)

+ λ2,i,ζ+1[ε2,i,ζ + βiεqipi1,i,ζ − βi(ε1,i,ζ

+ hε2,i,ζ)qipi − (α1,is

n1,im1,i

i,ζ + α2,is

n2,im2,i

i,ζ )

− 1

h(∆ri,ζ+2 − 2∆ri,ζ+1 + ∆ri,ζ) + d2]}.

(5.32)

where the λ1 ∈ Rn and λ2 ∈ Rn are the costates, and e1 = ε1 + ∆r.

Remark 4. Because of the sampling time of set-point re-planning loop is bigger

than that of control loop, system state ((•)k) will be sampled multiple times


during one sampling period of re-planned set-point ((•)ζ). In other words, the

re-planned set-point remains the same during each sampling time considering in

Hamilton equation (∆ri,ζ = ∆ri,k+1 = .. 6= ∆ri,ζ+1).

The basic idea of Hamilton equation is to find the minimum value

by using the partial derivative related to the performance index.

λ1,i,ζ =∂Hζ

∂ε1,i,ζ

=λ2,i,ζ+1[βiqipiεqipi−1

1,i,ζ − βiqipi

(ε1,i,ζ

+ hε2,i,ζ)qipi−1 − (α1,i

n1,i

m1,i

s

n1,im1,i−1

i,ζ

+ α2,in2,i

m2,i

s

n2,im2,i−1

i,ζ )βiqipiεqipi−1

1,i,ζ ]

+ 2ε1,i,ζ + 2∆ri,ζ + λ1,i,ζ+1,

λ2,i,ζ =∂Hζ

∂ε2,i,ζ

=λ1,i,ζ+1h+ λ2,i,ζ+1[1− βihqipi

(ε1,i,ζ


n1

m1

s

n1,im1,i−1

i,ζ

+ α2,in2,i

m2,i

s

n2,im2,i−1

i,ζ )].

(5.33)

As Equation (5.33) has strong non-linearities, singular problem need

to be checked.

5.2.3 Singular Problem

As the component related to sn1,im1,i−1

i,ζ appears in the costate dynamics,

singular problem has to be considered. Meanwhile, n1

m1∈ (0, 1), the sin-

gular problem has to be taken into consideration when ε1 = 0 or when

the system reaches the manifold. To solve the singular problem, a new


coefficient is introduced [119],

µi,ζ =

sin(π

2

|s1−

n1,im1,i

i,ζ |τ

) if |s1−

n1,im1,i

i,ζ | ≤ τ

1 otherwise

, (5.34)

where τ is a small scalar that represents the execution area of the sinu-

soidal function. After introducing the new coefficient, the costates dy-

namics change into

λ1,i,ζ =∂Hζ

∂ε1,i,ζ

=λ2,i,ζ+1[βiqipiεqipi−1

1,i,ζ − βiqipi

(ε1,i,ζ


n1,i

m1,i

µi,ζ s

n1,im1,i−1

i,ζ

+ α2,in2,i

m2,i

s

n2,im2,i−1


1,i,ζ ]

+ 2ε1,i,ζ + 2∆ri,ζ + λ1,i,ζ+1,

λ2,i,ζ =∂Hζ

∂ε2,i,ζ

=λ2,i,ζ+1[1− βihqipi

(ε1,i,ζ + hε2,i,ζ)qipi−1

− (α1,in1,i

m1,i

µi,ζ s

n1,im1,i−1

i,ζ + α2,in2,i

m2,i

s

n2,im2,i−1

i,ζ )]

+ λ1,i,ζ+1h.

(5.35)

where µi,ζ is added in front of sn1,im1,i−1

i,ζ to solve the singular problem.

To make sure that the re-planned set-point have the extremum, ∆ri has

to satisfies the stationary condition, which is

0 =∂Hζ

∂∆ri,ζ

= 2ε1,i,ζ + 2∆ri,ζ −1

hλ2,i,ζ+1.

(5.36)


The re-planned set-point is

∆ri,ζ = −ε1,i,ζ +1

2hλ2,i,ζ+1, (5.37)

where ∆ri,ζ is the additive set-point on the ith set-point component.

As an additional optimal closed-loop is added, the stability of the

new closed-loop system has to be checked.

Theorem 5.2.1. Suppose that the system (5.7) with the controller shown in

Equation (5.28) is affected by uncertainties denoted by d1 and d2 . Then under

the re-planned set-point ∆rζ , the performance index described as Equation (5.30)

will be optimised, moreover, the system states will track the original set-point r∗.

Proof. Since the set-point update law (Equation (5.37)) is introduced, the

new set-point becomes rζ = r∗ζ + ∆rζ , where ∆rζ has been derived by

Equation (5.37). The feedback errors used in the controller design change

from e to ε. If the convergence of ε to e can be proved, then the system

states will track r∗.

By combining Equation (5.35) and Equation (5.37) together, the costates

relationships become,

λ1,i,ζ+1 =λ1,i,ζ − Aλ2,i,ζ+1,

λ2,i,ζ+1 =λ2,i,ζ − λ1,i,ζ+1h+Bλ2,i,ζ+1,

(5.38)

whereA =

1

h− βi

qipiεqipi−1

1,i,ζ + βiqipi


+ (α1,in1,i

m1,i

µi,ζ s

n1,im1,i−1

i,ζ

+ α2,in2,i

m2,i

s

n2,im2,i−1


1,i,ζ

(5.39)


andB =βih

qipi


+ (α1,in1,i

m1,i

µi,ζ s

n1,im1,i−1

i,ζ + α2,in2,i

m2,i

sn2m2−1

i,ζ ).

(5.40)

If the system has equilibrium points, then system states will remain

at one of them eventually. Assume the system described in Equation

(5.44) has equilibrium points, based on the properties of equilibrium points,

λi,ζ = λi,ζ+1. (5.41)

Combining the relationship between λi,ζ and λi,ζ+1 with Equation (5.44),

0 = Aλ2,i,ζ+1,

0 = −λ1,i,ζ+1h+Bλ2,i,ζ+1.

(5.42)

WhenAλ2,i,ζ+1 = 0, eitherA or λ2,i,ζ+1 should be zero. AssumeA = 0,

1

h=βi

qipiεqipi−1

1,i,ζ − βiqipi

(ε1,i,ζ + hε2,i,ζ)qipi−1 − (α1,i

n1,i

m1,i

µi,ζ s

n1,im1,i−1

i,ζ

+ α2,in2,i

m2,i

s

n2,im2,i−1


1,i,ζ

(5.43)

If |εi,ζ | = 0, equation becomes 1h

= 0, which is impossible. So it is neces-

sary to guarantee |εi,ζ | 6= 0 to compensate 1h

. However, as the convergence

of system states is guaranteed, which shows that |εi,ζ | → 0. As |εi,ζ | → 0,

A does not converge to zero.


Based on Equation (5.44), if let λ2,ζ+1 = 0,

λ1,i,ζ+1 =λ1,i,ζ ,

λ2,i,ζ =λ1,i,ζ+1h,

(5.44)

As the λ2,i,ζ = λ2,i,ζ+1, it is clear that the system will stay at λi,ζ+1 = 0.

When λ1 = λ2 = 0, combine the system dynamics, which is shown

in Equation (5.26), and re-planned set-point in Equation (5.37),

0 =x1,i,ζ − r∗i,ζ ,

0 =x2,i,ζ −1

h(r∗ζ+1 − r∗ζ),

(5.45)

then it is obvious that x→ r∗.

The theorem 5.2.1 proves the set-point replanning not only optimise

the system performance in presence of disturbance, but also guarantee

the system states keep tracking the original set-point r∗ in presence of

disturbances.

5.3 Case-study on Mobile Robot

With increasing interest in industrial automation, mobile robot has

been playing an important role in transportation, rescue, etc.. Mobile

robot always works in complex situations in which uncertainties are in-

evitable. Most of those applications rely on the accurate robot locations

feedback; however, the original controller may not able to guarantee that

accuracy when uncertainties occur.

Mobile robot is used in the experiment. The programming environ-

ment is Ubuntu Linux. Based on the special distribution of three wheels,


FIGURE 5.2: Holonomic Mobile Robot c©Festo Didactic.

TABLE 5.1: Robotino DC Motor Parameters.

DC motor(GR 42x25)Rated Voltage 24VDCRated Speed 3600RPMRated Torque 3.8NcmRated Current 0.9AStarting Torque 20NcmStarting Current 4ANo Load Speed 4200RPMMotor Weight 390g

Robotino is capable of moving omni-directionally. Wheels are driven by

DC motors, whose data specification is shown in Table I.

Figure 5.3 is the block diagram of the basic close-loop design of Robotino.

The trajectory planning agency generates the set-points based on the re-

quirements. The control signals are generated from an external com-

puter. The control signals are passed to Robotino through the inherent

Robotino’s WLAN-Link. The embedded system in Robotino transfers the

velocity setting values into torques. To guarantee the primary stability of

the mobile robot, Robotino has a PI controller inside.


Basing on the structure of sliding manifold shown in Equation (5.8), the

FIGURE 5.3: Block Diagram of Robotino Control.

sliding manifolds used here is

s = v + αpγ, (5.46)

where v represents the velocity os mobile robot, p is the position feedback

from encoder. In this practical test, choose γ = 119

, and α = 2.

The performance index used here is

J = eT1,Ne1,N +N−1∑k=0

eT1,ke1,k. (5.47)


0 50 100 150 200 250 300−1

−0.8

−0.6

−0.4

−0.2

0

Samples(0.03s/sample)

Po

sitio

n E

rro

r e

x(m

)

Sampled Position ErrorPosistion Error

FIGURE 5.4: Position Error in x Direction

0 50 100 150 200 250 300−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3


Po

sitio

n E

rro

r e

y(m

)

Sampled Position ErrorPosistion Error

FIGURE 5.5: Position Error in y Direction.

Figure 5.4 and Figure 5.5 are the position errors in x and y directions.


Dashed curves are the resampled position errors which are used in the

controller design, and solid curves are the position feedbacks. Uncertain-

ties have been added to the controller inputs. Controller’s inputs affected

by uncertainties are represented by solid curves. The figures also show

that when system works without the optimal loop, the robot’s stability is

still guaranteed. According to the hardware structure and programming

logic, the optimised control signal may not able to follow the system if

their sampling frequencies remain the same. Multi-rate sampling method

is used in the experiment. Dotted curves represent the re-sampled sig-

nals. The new sampling frequency is one tenth of the system’s clock fre-

quency.

0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2


Linear Velocity vx(m/s)

Sampled System Velocity System Velocity

FIGURE 5.6: Velocity in x Direction.


0 50 100 150 200 250 300−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Lin

ea

r V

elo

city

vy(m

/s)

Sampled System Velocity System Velocity

FIGURE 5.7: Velocity in y Direction

Velocities of the mobile in each direction are shown in Figure 5.6 and

5.7. Dashed curves are the resampled velocities going to be used in con-

troller design, and solid curves are the velocity feedbacks. Same as posi-

tions, the re-sampled velocities are shown in dotted curves. At the begin-

ning of process, as the big errors between robot position and set-point,

controller contributes a high gain, which rise the speed rapidly. As the

position errors decrease, the speed decreases smoothly. As the existing

uncertainties, position errors can not converge to zero, and speeds keep

chattering at the end of process.


0 50 100 150 200 250 300−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2


Slid

ing

Ma

nifo

ld

sx

sx based on Reselected Setpoint

sx

FIGURE 5.8: Sliding Manifold on x Direction (sx).

0 50 100 150 200 250 300−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4


Slid

ing

Ma

nifo

ld s

y

sy based on Reselected Setpoint

sy

FIGURE 5.9: Sliding Manifold on y Direction (sy).

Figures 5.8 and 5.9 show the differences of sliding manifolds based


on both r∗ and r. Solid curves are the sliding manifold with original

set-points, and dashed curves are from the system with re-planned set-

point. In Figure 5.8, the re-planned set-points increase the feedback error

at the beginning which offers a faster speed response. In Figure 5.9, the

re-planned set-point not only gives a faster response but also introduces

an overshoot in the sliding manifold in y direction. The difference in

the responses of sliding manifolds in x and y direction is caused by the

mechanical wearing and sensor ageing

0 50 100 150 200 250 300−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0


Po

sitio

n E

rro

r e

x(m

)

Optimized Position Error Position Error

FIGURE 5.10: Optimal system output error (ex).

Figure 5.10 and Figure 5.11 are the comparisons of optimised and

original position errors. Solid curves are original position feedbacks from

encoders, and dashed curves are the optimised position feedbacks. and

As the larger position errors that the re-planned set-point give, the mobile

robot goes to the target point faster than before. When steady state errors

exist in the system, ∆r also compensates the steady state errors. The


optimisation, the mobile robot not only has a faster response but also an

improved accuracy.

0 50 100 150 200 250 300−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2


Po

sitio

n E

rro

rey (

m)

230 240 250

−2

0

2

4

6

x 10−3

Optimized Position Error Position Error

FIGURE 5.11: Optimal system output error (ey).


0 50 100 150 200 250 3000

2

4

6

8

10

12

14

16

18

20


System Performance Index J(m2)

Optimized System Performance Index

System Performance Index

FIGURE 5.12: System Performance Index.

Figure 5.12 is the performance index of the mobile robot. Solid curve

is the optimised system performance index, and dashed curve are the

original system performance index As the faster response and smaller

system steady state errors, the accumulation of system states with re-

planned set-point is better than that with original set-point.

5.4 Summary

In this chapter, an operational optimal control strategy is designed

for the mechatronic system. Terminal sliding mode control is used to

make the system converge time adjustable.

Different with the conventional optimal control strategy, the strat-

egy proposed optimises the deteriorating system performance without


change the controller parameters. The multi-rate method is used to make

sure that the system states can follow the re-planned set-point.

Hamilton-Jacobin-Bellman(HJB) equation is applied on the outer loop

as the optimal method. However, as the complex structure of the origi-

nal closed loop, the singular problem exists. A trigonometric function is

used to solve the singular problem. The convergence and stability of the

system with the optimal loop is proved in Theorem 5.2.1.

104

Chapter 6

Conclusion and Future Work

6.1 Conclusion

In the first part of this thesis, a 3-DOF manipulator dynamics is built

based on the Newton-Euler method. Newton-Euler method is used to

achieved the dynamics. To make the dynamics easier to be tested, a visu-

alized program is designed. The control method used in the simulator is

state-feedback control. To make the simulator more practical, both full-

order and reduced-order observer have been applied to it.

In the second part, a fault matrix based fault tolerant control strategy

is proposed. A terminal sliding mode manifold is used in the controller

design to guarantee the system converge time. However, during the con-

troller design, (eγ2−12 )−1 appears. As γ2 ∈ (1, 2), singular problem has to

be considered. To solve the singular problem, a mathematical approxi-

mation, sin(x)x

= 1 when x → 0, is used. To eliminate the impact of dis-

turbance, a disturbance observer has been applied in the strategy. When

actuator fault exist in the system, fault matrix is used to represent the ac-

tuator fault based on its relationship with the control signal. Because of

the existence of sensor fault, system states cannot be used directly in the

fault tolerant control strategy design.

Chapter 6. Conclusion and Future Work 105

As both actuator fault and control signal are related to system state,

fault matrix is used to represent the relationship between them. To esti-

mate the fault matrix, a Luenberger-like fault observer is designed, which

means output error will be used in the design. By using the bounds of

system non-linearities and a time domain disturbance observer, actuator

fault is isolated. Different with the typical target system of observer, fault

matrix is coupled in the middle of other nonlinear parts (EgΘu). Inequal-

ity features are used to uncoupled the fault matrix. A simulation result

has been given in the last section to illustrate the method.

In the third part, a novel set-point re-planning method is proposed.

Robotino mobile robot is chosen as real platform. Similar to FTC, a termi-

nal sliding mode controller is applied on the system to guarantee system

output tracking the desired trajectory. Different with the sliding manifold

used in FTC design, a simpler sliding manifold is used to give controller

better applicability. To make the convergence time controllable, was cho-

sen qipi> 1. The reaching part is still based on the similar structure as

in the fault observer design. As the exponential function takes the place

of sign function, system states converge to set-point smoothly. However,

the robustness of sliding mode control is decreased.

When uncertainties exist in the system, system performance will be

damaged. In conventional optimal control, controller parameters need

to be changed. As the increasing acceptance of industrial automation,

mechatronic systems are provided with embedded controller, which makes

controllers inaccessible. Instead of changing controller parameters, set-

point re-planning method is proposed. Although an additional set-point

(∆r) is added to the system, system states are still expected to tracking the

original set-point (r∗). As the new set-point in the system is (r = ∆r+r∗),

system states error has to be update. Because the target trajectory is r∗,


system performance chosen here is based on the original error (e = x−r∗).

Hamilton-Jacobi-Bellman equation is used to achieve the update law. The

optimal loop is considered as an outer loop in the control scheme. If the

original control close-loop is considered as a black box, the stability of

optimal loop has to be proved, which is shown in Theorem 5.2.1.

6.2 Thesis Contributions

The contributions of this thesis are:

• Designed a terminal sliding mode controller. Exponential structure

guarantees the global fast convergence. The closed-loop system is

proved to be asymptotically stable. The singular problem in con-

troller design is solved by using mathematical approximation.

• Proposed a fault matrix based fault tolerant control strategy. Actu-

ator faults in the system are represented by using the relationships

between controller and actuator faults. Because disturbance is re-

lated to time, disturbance isolated by using a time relating distur-

bance observer.

• Proposed a set-point re-planning optimal method. Instead of chang-

ing controller parameters, set-point(rk = ∆rζ + r∗k) is used to opti-

mise the system performance. As the re-planned set-point (∆rζ) is

used to compensate the uncertainties, system states is still expected

to track the original set-point(r∗k). The convergence of the system

state is proved.

• Several papers related to the thesis have been published


• X. Yan, B. Tian and H. Wang, "An adaptive observer-based

fault detection and diagnosis for nonlinear systems with sen-

sor and actuator faults," International Conference on Advanced

Mechatronic Systems (ICAMechS), Beijing, China, 2015, pp. 491-

496.

• X. Yan, Z. Zuo, L. Yin, A. Wang and H. Wang, "Chattering-

free sliding mode control for MIMO nonlinear manipulator

systems based on adaptive neural networks," The 54th IEEE

Conference on Decision and Control (CDC), Osaka, Japan, 2015,

pp. 6300-6305.

• B. Tian, Z. Wang, X. Yan, H. Wang, "Fault-tolerant control scheme

for manipulator with actuator and sensor failure," International

Journal of Advanced Mechatronic Systems 2015, 6(4), pp. 166-173.

• B. Tian, Z. Zuo, X. Yan, H. Wang. "A fixed-time output feed-

back control scheme for double integrator systems," Automat-

ica, 2017, 80, pp. 17-24.


6.3 Future Work

In the thesis, the set-point re-planning strategy is designed for the

system with certain parameters, however, as loads of many mechatronic

systems, such as those in sorting, medical care, and transporting, are not

constants. It may necessary to let the set-point re-planning strategy more

adaptable to the time-varying loads.

With the increasing interests in machine learning, it is possible to

apply machine learning to fault tolerant control strategy. By using the

neural network, some of the prior knowledge about uncertainties, such

as boundary, may not be necessary any more However, as the possible

training time that neural network needs, it may not be an on-line method.

Most of the practical processes involve random signals, it is possible

to extend the set-point re-planning methods to stochastic system. As the

distributions of the stochastic system are ’static’, it is easy to extend the

set-point re-planning to the stochastic system. However, as the stochastic

noise exists in the system, Kalman filter may be used in the design to

have a better estimation of system states.

109

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