## ABSTRACT

This research work is aimed at the development of an observer-based dynamic output feedback

controller for stabilization and tracking of nonlinear systems. The developed controller is

designed after the immersion-invariance and internal model principle (IMP) frameworks and targets

non-square systems such as rotational-translational actuator (RTAC), cart-driven inverted pendulum

(CIP) and quadrotor unmanned aerial vehicle (UAV). However non-square multiple-input

multiple-output (MIMO) systems such as the UAV represented the principal system of choice for

their structural properties. Non-square MIMO systems are systems that have more inputs than

outputs (over-actuated) or vice-versa (under-actuated) and reflect the structures of many real world

systems. The developed immersion invariance error feedback control law(IIEFCL) is used to solve

stabilization and robust tracking problems of non-square MIMO non-linear systems. The output

feedback internal model based observer is developed and tested with the RTAC, CIP and UAV

while the immersion invariance stabilizing controller is developed and tested on the RTAC system.

The output feedback controller showed good stability response on the selected models while the

immersion invariance method displayed a good transient phase stability and tracking results with

the addition of a robust state feedback feature to the underlying controller. The obtained settling

times for the output feedback stabilization results were 2.7s, 1.113s and 0.6435s respectively for the

three systems. The immersion-invariance control law acting as a robustifier to another controller

produced zero percent overshoot and tracking error. The results showed attainment of desired

stability and tracking and also quick convergence, disturbance rejection and handling of transient

oscillations such as finite time escape or transient instability phenomena, from which many nonlinear

systems do not recover after they occur. The IIEFCL was developed for the Quadrotor UAV

and the results obtained were compared with some other standard nonlinear controllers that have

been used in QUAV control. The metric for comparison was the integral of the squared control

input (ISCI) signal. Results obtained compared favourably with existing nonlinear control laws.

The IIEFCL showed the most improvement of 92.92% improvement over the backstepping conviii

trol law, it had a 72.92% improvement over the feedback linearization control law and the least

improvement was with respect to the sliding mode control law where only 66.225% improvement

was recorded. Simulations were made using Matlab/Simulink and embedded C++ tools.

ix

## TABLE OF CONTENTS

Title page i

Declaration ii

Certification iii

Dedication iv

Acknowledgments v

Abstract viii

Table of Contents x

List of Figures xvii

List of Tables xx

List of Acronyms and Symbols xxi

Chapter 1: INTRODUCTION 1

1.1 Background to the Study 1

1.2 Motivation 8

1.3 Significance of Study 10

x

1.4 Statement of the Problem 10

1.5 Aim and Objectives 12

1.6 Methodology 12

1.7 Thesis Outline 13

Chapter 2: LITERATURE REVIEW 15

2.1 Introduction 15

2.2 Review of Fundamental Concepts 15

2.2.1 Output regulation 15

2.2.2 Output feedback-based regulation 27

2.2.3 Stabilization using immersion and invariance 31

2.2.4 Observers for control systems 33

2.2.5 MIMO dynamic model 41

2.3 Review of Similar Works 44

Chapter 3: METHODOLOGY 53

3.1 Introduction 53

3.1.1 Development of the output feedback controller 54

3.1.2 RTAC Parameters 61

3.1.3 Output regulation for the CIP system 62

3.1.4 Regulator equations for the CIP 65

3.1.5 CIP Parameters 67

3.1.6 Output regulator for the QUAV 68

3.1.7 Quadrotor UAV Parameters 71

xi

3.2 Development of immersion-invariance stabilizing controller 71

3.2.1 Immersion invariance control of the RTAC or TORA system 72

3.3 Development of the Immersion Invariance Output Error Feedback Regulator for

Non-Square Systems 77

3.3.1 Feedback stabilization control design 77

3.3.2 Stabilizing controller by Lyapunov approach 81

3.3.3 Revisiting output error feedback regulator design for the UAV 83

3.3.4 Computation of generalized immersion relations 86

3.3.5 Output error feedback control law 88

3.3.6 Immersion-invariance controller for the quadrotor 89

3.3.7 Actuator model formulation 90

3.3.8 Augmented system model 93

3.3.9 Selection of target dynamic for the quadrotor 94

3.3.10 UAV state feedback controller design for immersion invariance 95

3.3.11 Proof of boundedness and stability of immersion invariance control law 98

3.3.12 Synthesis of immersion invariance output feedback observer-based control 100

3.3.13 Design of the error feedback observer 101

3.4 Validation of the Developed Immersion Invariance Output Feedback Control Law 104

Chapter 4: RESULTS AND DISCUSSION 107

4.1 Introduction 107

4.2 Output Regulation Experiments 107

4.2.1 RTAC Regulation Experiments 108

4.2.2 CIP Output Regulation 111

xii

4.2.3 Quadrotor UAV Output Regulation 113

4.2.4 Summary of Output Regulation Experiments 116

4.3 Immersion Invariance Stabilization Experiments 117

4.3.1 RTAC Immersion Invariance 117

4.3.2 RTAC immersion invariance stabilization controller 117

4.3.3 Linear immersion invariance Tracking control with observation 121

4.3.4 Robust RTAC stabilization and tracking experiments with disturbance injection

121

4.4 Immersion Invariance Error Feedback Control Results 125

4.4.1 QUAV unforced stabilization results 126

4.4.2 Exosystem profile 127

4.4.3 Immersion-invariance error feedback regulator results 128

4.4.4 IIEFCL state output results 132

4.4.5 Characterization of the control laws 139

4.4.6 Comparison of results from various control laws 150

Chapter 5: SUMMARY AND CONCLUSIONS 152

5.1 Summary 152

5.2 Conclusions 153

5.3 Limitation 155

5.4 Contribution to Knowledge 156

5.5 Recommendation for Future Work 157

References 165

xiii

Appendix A: Code listing for the RTAC Output Feedback Regulator Problem 167

A.0.1 RTAC parameters are defined here from Avis et al., 2013 167

A.0.2 Substitution of params 167

A.0.3 Definition of the exosystem matrix as either constant or variable 167

A.0.4 Computing controllability 167

A.0.5 Discussion 168

A.0.6 Computing observability 168

A.0.7 Computing observability 168

A.0.8 Construct a stabilizing gain Kx for A+BKx 168

A.0.9 Developed method for matching the poles 169

A.0.10 Computing the values in the gains L,L1,L2,G1,G2 169

A.0.11 Computing the gains L=L1 169

A.0.12 Computing the gains L1 and L2 for the matrix G2 170

A.0.13 Compute the L2 gain from developed method 170

A.0.14 Computing Kv 171

A.0.15 Compute the global K matrix Kavis=[Kxavis Kvavis] 171

A.0.16 Solve the sylvester equations from knowlwegde of Kx and Kv 172

A.0.17 Computing the other gain of the observer G1 172

A.0.18 Test for the controllability of Aobsv, Bobsv 172

A.0.19 Test for stability of G1 and G2 matrices 173

A.0.20 Check for Hurwitz stability of the closed loop system matrix below 173

A.0.21 Calling model here 173

xiv

## CHAPTER ONE

INTRODUCTION

1.1 Background to the Study

The design of stabilizing controllers is a major task and requirement for the control engineer. While

notions of stability vary from linear to nonlinear systems, the main ideas and goals are the same;

which is to force the evolution of the trajectories of a system towards an equilibrium point or an

equilibrium set. This is the same goal in either linear bounded-input bounded-output (BIBO) stability

or more involved nonlinear counterparts like the Lyapunov stability variants and structural

stability. Output regulation, which is the core theme of this proposal, requires that for any system

to be successfully controlled, it must first be established to be stabilizable (Byrnes et al., 1990).

This stability being enforced in the presence of constraints such as known or unknown disturbances

and uncertainties. The system requirement is that all trajectories of interest or system states remain

asymptotically close to or converges exactly to the desired equilibrium point and remains there

for all time (Isidori & Byrnes, 1990). In the absence of stability, the ultimate desire for affecting

the behavior of any system through regulation using suitable control signals is not possible; such

regulation being demanded in the presence of the stated constraints which can interact with any

real system to cause instability and deviation from correct behavior. Examples of such uncertainties

and disturbances include; un-modeled dynamics, high frequency parasitic components, large

inputs and other complex coupling phenomena in multiple-input multiple-output (MIMO) systems

(Kokotovic, 1985).

The search for a globally stabilizing and asymptotically tracking feedback controller has seen various

control schemes put forward for consideration (Andrieu & Praly 2009; Astolfi & Praly 2015).

Another name for such a robust stabilization and tracking control scheme is output regulation. Pioneering

output regulation control designs include: linear single-input single-output (SISO) and

1

MIMO systems by Francis and Wonham (1976) and the equivalent nonlinear SISO and MIMO

treatment by Byrnes and Isidori (1990). This has also seen the achievement of various results

ranging from local, semi-global to global. However, while stabilization in the local sense has been

considered for non-linear systems since the early 1990â€™s, it is the stabilization in a large domain

such as semi-global stabilization by Teel and Praly (1993), Battilotti (2001) and global stabilization

by Chen and Huang (2004), Peixoto et al., (2009) that are good prospects in the search for a

global stabilizing feedback compensator. More specifically global robust regulation concerns have

dominated the research in many of the literature on stabilization of nonlinear systems (Astolfi &

Praly, 2017).

Continuing research in output regulation over the years has seen a stream of new results added by

Huang (2004), Chen and Huang (2004), Serrani (2005), Isidori (2011), Khalil and Praly (2014),

Astolfi and Praly (2015). These works contributed to the development of workable designs. Specifically

considering the analysis of the internal structure of a MIMO system and studies of how this

structure was affected by feedback and output injection. The obtained results were used to solve

such problems as stabilization, tracking, system decoupling and fault isolation. In the case of

non-linear systems, most of the methods for the analysis of the internal structure have been conveniently

extended beyond that originally developed by Francis and Wonham (1976). According

to Isidori (2011), recent works showed that the study of problems of output feedback design for

MIMO nonlinear systems has come to an almost complete stall. Some reasons given for this difficulty

in tackling the non-linear MIMO problem include simple and complex interconnections

between various states and inputs and difficulty in transferring observer control design from linear

to nonlinear MIMO systems (Isidori, 2011; Wang et al., 2014).

Therefore regulation of the output of any system must be preceded by satisfiable stabilization properties

in the implemented closed-loop control architecture if regulation of the system output must

be achieved. Figure 1.1 depicts such a control architecture that is well known in control systems

theory (Hoagg & Bernstein, 2006). The variables yc(t), e(t), u(t) and y(t) represent the command

2

or reference input, error signal, control input and output respectively. Such potential instability justifies

the case for the presence of a robust regulator that maintains the stability of the system state

and regulates the system output towards the asymptotic tracking of reference signals and rejection

of unwanted or undesired disturbances. In order to effectively handle a wide range of disturbances

Figure 1.1: Standard Regulator Structure (Hoagg and Bernstein, 2006)

and system uncertainties, a different architecture is needed as given by the internal model-based

output feedback controller. The design depicted in Figure 1.2 captures the internal model principle

Figure 1.2: Standard Output Feedback Regulator Structure (Valmorbida and Galeani, 2013)

(IMP) philosophy, which requires the resulting controller to have asymptotic tracking and disturbance

rejection properties (Kokotovic & Arcak, 2001). The internal model F, steady state control

and state variables uss and xss, plant and observer are P and ^ P respectively. Observer estimates

3

^x and ^!, unstable states ~x, exogenous signal ! and output error e. This is made possible by embedding

a copy of the actual system in the control architecture. This stabilizing control ensures the

tracking of a given reference trajectory so that the output error is regulated to zero. The parts of the

robust observer-based regulator consist of the plant, the observer which acts as the internal model,

the steady state stabilizer and steady state generator ( and respectively) and finally the exosystem

supplying the external inputs to the system. This framework also assumes knowledge of the reference

signals/disturbances.

In developing an observer-based output feedback control law, specific design methods depend on

the kind of information available for feedback (Astolfi & Praly, 2017). In general this technique

for robust regulation of a dynamic systems output has been categorized by Serrani (2005) into state

and output error feedback. The output feedback approach to regulation of a systems response is

more practicable in tackling the robust stabilization and tracking question. Considering that the

control system designer is assured of having the entire output present for measurement as against

the possibility of having individual state for measurement. Output feedback stabilizing regulator

also has other advantages which include; ease of practical implementation, multi-functionality

within the single design e.g. a single controller design being able to serve as both stabilizer and

steady state regulator for multiple manipulated variables and applicability for disturbance rejection

(Astolfi & Praly, 2017; Chang et al., 2015).

Therefore, the developed observer-based output feedback controller is used to achieve robustness,

adaptive behavior and importantly, to ensure global asymptotic stability (GAS) of system states.

Based on the principle of internal model, the output feedback controller ensures simultaneous robust

stabilization and tracking of reference inputs or rejection of reference disturbances (Francis &

Wonham, 1976). However, it often achieves this in a local domain of the work space (under small

signal perturbation). Another scheme for stabilization is the immersion and invariance method,

known for yielding globally stabilizing and adaptive designs for controllers in the presence of

parametric uncertainties (Astolfi & Ortega, 2003). Immersion and invariance method does not

4

however have the robustness characteristics of the internal model output feedback design.

The robust output regulator with structure shown in Figure 1.2 is proposed for modification. This

modification is necessary because it is not possible to always know every profile or function of the

disturbance or uncertainty to be encountered. Therefore while output measurement by an observer

is still necessary, the observer-based output feedback nomenclature remains open to increased robustness

and adaptive design features. The development of such stabilizing regulators has seen

the use of different implementation architectures using general principles of internal model control

(IMC) such as model predictive control (MPC) framework, one example of which is the model predictive

regulator of Aguilar and Krener (2014), state and error observer framework by Carnevale

et al., (2012), strictly adaptive framework, immersion-invariance framework by Astolfi and Ortega

(2003), disturbance observer (Shim et al., 2015).

Extending the treatment to systems with uncertainties, Wiese et al., (2015) and Shim et al., (2015)

have also addressed the problem of adaptive control designs. The uncertainties being sources of instability

in either the transient or steady operating phase is affected by the observer gains. Whereas

high gain designs which are ideal for local or semi-global stabilization are beneficial for quick

convergence, high gains have a detrimental effect in the transient regime where large error occurs.

High gains also have a detrimental effect on the measurement error in steady state regime where

the error (usually small), is derived from model uncertainties. Switched gain observer design has

also been studied by Ahrens and Khalil (2008).

Therefore, having established the existence of different approaches and the several attempts made

to obtain a globally stabilizing output feedback controller (Andrieu & Praly, 2009), this work

proposes another approach for non-square MIMO non-linear systems, based on the identified developments.

The approach will make use of a hybridization of techniques from both internal

model principle-based output-feedback and immersion-invariance stabilization theory. The developed

control scheme shall tradeoff the strengths and weaknesses of each method. While the

5

output regulation framework requires the unrealistic knowledge of every disturbance and uncertainty,

such requirements are absent for Immersion-Invariance stabilization. Therefore the strong

stability results from Immersion-Invariance are combined with the strong tracking results from

output regulation to produce an improved controller.

This work has given an alternative and detailed development of an immersion and invariance

observer-based output feedback regulator that has characteristics of global stability. The design

builds on the premise that global asymptotic stabilization implies global asymptotic tracking that

assures semi-global stabilization in output feedback (Teel & Praly, 1993). The current synthesis

of a stabilizing observer-controller framework has brought together established techniques which

are principally from internal model principle and immersion-invariance design. The design specifically

addresses stabilization and tracking a non-square MIMO nonlinear systems. Figure 1.3

summarizes the flow of information in an output-regulated feedback controller that is based on the

IMP. This flowchart starts with the definition of the plant, exosystem and controller, formation of

an autonomous closed loop system, analysis of the resulting structure of the resulting autonomous

system, solution of the regulator equations for the system. A, B and C are the system transmission

matrix, input matrix and output matrix respectively. Wr is the controllability or reachability

matrix, Kx, Kv are the stabilizing state and disturbance feedback gains. u is the regulator control

variable. L1, L2, G1, G2 are observer gains.

6

Figure 1.3: Output Error Feedback Regulator Flowchart

Similarly, the flowchart for setting up an Immersion and Invariance stabilization (Astolfi and

Ortega, 2003) scheme is shown in Figure 1.4. x is the system state in original coordinates, is system

state after immersion, K is stabilizing feedback matrix, f(:) and g(:) are state and input vector

fields. (:) is an immersion map and (:) is the zero manifold while (:; ðŸ™‚ is the off-manifold

control law. Following the establishment of the output regulation scheme using observer output

feedback and immersion invariance, a closed loop architecture was implemented. By extending

the standard architecture for observer-based output regulation, this work proposed an immersion

7

Figure 1.4: Immersion Invariance Flowchart

and invariance output feedback architecture for robust and adaptive stabilization of system state

and asymptotic convergence of system output error to zero.

1.2 Motivation

Investigations by Wang et al., (2014) in which the dynamic output feedback in MIMO system is

achieved in the context of a high gain observer (HGO) forms the principal motivation for this work.

Astolfi and Praly (2015) have also looked at the extension of the output feedback problem by in-

8

vestigating the effects of small un-modeled discrepancies which could also cover external sources

arising from reference signals and disturbance inputs. However, practice shows that not all disturbance

inputs are of small magnitude. In the case of MIMO non-linear systems that are controlled

via output feedback, the problem of asymptotic stabilization, despite of its obvious relevance in

dynamic systems remains unresolved. Literature evidence shows that inquiry in this direction has

received little attention (Wang et al., 2014). This is moreso for non-square MIMO systems where

works addressing observer-based output feedback for non-square uncertain systems are also limited

(Isidori, 2011; Shim et al., 2015).

While output feedback presupposes the existence of some form of estimator or observer of state,

certain constraints can be identified with the availability of such measurement (Nazrulla & Khalil,

2008). Firstly, not all states are usually available for measurement (Kazantis & Kravaris, 1998;

Kaldmae & Kotta, 2013), making the full state information problem of output regulation more ideal

than practical for the task. Aside from the full state information being rarely available, design methods

based on the availability of partial state information are important (Wang et al., 2014), as well

as designs that consider the output error information (Astolfi & Praly, 2017). Such designs have

proved to be more practically feasible and have necessitated output feedback techniques which

although applied successfully in SISO type systems, have not seen a similar level of progress in

nonlinear MIMO systems (Khalil & Praly, 2014; Isidori, 2011). More recent works have utilized

immersion (Astolfi and Ortega, 2003), embedding the partial or reduced state output observer as

an internal model of the system while respecting certain structural conditions that guarantees system

analysis. Other works have utilized non-linear coordinate transformation. These have become

more practical approaches to solving the dynamic Output Regulation problem but mainly in square

systems. Considering these facts, the problem of achieving robust stabilization-tracking via output

feedback is still largely open, especially for non-square MIMO non-linear systems (Wang et al.,

2014).

The question therefore becomes, how to achieve GAS in the presence of disturbances and model

9

uncertainties of varying magnitudes using a control law that is robust and adaptive under all operating

conditions? Specifically taking into account the inability to know accurately all possible

uncertainties that might enter into a system. One answer to this question was the control law provided

by the developed immersion-invariance output feedback controller. The appeal for such a

control law follows the established fact that stabilization and tracking without robustness and inherent

adaptation leads to instability, reduced control authority, loss of control, increased energy

usage and poor transient performance. Therefore the developed scheme did not need to have full

information of the possible uncertainties or disturbances to achieve the control goals.

1.3 Significance of Study

The significance of this work is the extension of the existing treatment of MIMO output regulation

or stabilization-tracking analysis to specifically address non-square MIMO nonlinear systems. The

utilized approach firstly, uses output-regulated output feedback control as a basis. Secondly, the

current treatment is extended in the particular direction immersion and invariance setting that allows

for design of globally stabilizing control laws. Lastly, identifying and making use of relevant

design parameters from the two previous techniques. It focuses on the development of a practical

framework to be applied towards achieving robust stabilization and tracking irrespective of

intervening uncertainties. This new framework is an Immersion-Invariance output feedback strategy.

Therefore, elements of the system which might prevent the attainment of global asymptotic

stability (GAS) such as finite time escape phenomena like peaking are addressed.

1.4 Statement of the Problem

Section 1.2 presented globally stabilizing control laws for nonlinear and linear systems as an open

problem in the context of nonlinear, non-square systems utilizing output feedback controllers. One

of the difficulties in achieving such stabilizing control laws using output feedback regulation concept

remains the existence and uniqueness of solutions to the regulator equations which is not

assured for all systems. Because, while it can be checked that such a regulator exists (neces-

10

sity), obtaining the regulator (sufficiency) remains a difficult task. Moreso, for non-square systems

where analysis sometimes requires squaring the plant before output feedback is applied. Seeing

the difficulty in achieving stability using the analytical route, certain literature developed a constructive

approach using eigenvalue pole placement and gain matrix designs which when properly

done, renders the system internally stable. The attraction for such a method rests with the fact

that if subsets of the system were designed to be stable then the global system when assembled

together retained such stability. The identified difficulty could be described in a sense as structural.

This description has meaning in non-square systems with respect to size of inputs and outputs.

Also more fundamental was the size reduction in systems with reduced rank leading to nominally

uncontrollable and unobservable systems. It has been observed that some of these characteristics

that are influenced by the system structure were readily identifiable using differential geometric

techniques for analyzing such nonlinear systems.

System stability in the global context requires control laws that are robust for all possible and acceptable

inputs. Therefore, the continued demand for controlled systems to have good stability and

tracking performance cannot be overemphasized. Also, overcoming the potentially destabilizing

effect of disturbances and uncertainties is key to driving the error term in any controlled system to

zero. The problem can therefore be summarized as the regulation of any given systemâ€™s state to an

invariant zeroing manifold from which it is impossible to escape and at the lowest possible control

cost. The implemented solutions used both state and output feedback principles to ensure the continued

internal stability and output regulation of the selected systemsâ€™ states. This was achieved

in the presence of structural imbalance inherent to non-square systems and also in the presence of

potential destabilizing factors such as external disturbances and inputs that were captured in the

utilized exosystem structure.

11

1.5 Aim and Objectives

The aim of this work is: development and synthesis of a dynamic output feedback observer for

stabilization and tracking in non-square MIMO nonlinear systems.

The objectives of the study are therefore as follows:

1. Development of an internal model based output regulation framework for stabilization of

non-square MIMO nonlinear systems.

2. Development of an immersion and invariance stabilization framework for non-square MIMO

nonlinear systems.

3. Development of an immersion and invariance output feedback control law for stabilization

and tracking of non-square MIMO nonlinear systems.

4. Validation of the immersion-invariance output feedback theory and control law using standard

benchmark non-square MIMO systems (cart-driven inverted pendulum (CIP) , rotational/

translational proof-mass actuator (RTAC) and a fixed rotary wing quadrotor unmanned

aerial vehicle (QUAV) .

1.6 Methodology

The methodology for implementation of the stated objective items are:

1. Internalmodel principle(IMP)-based output regulation framework for stabilization and tracking

(a) Adoption of relevant mathematical background for output regulation

(b) Analysis of necessary and sufficient conditions for existence of solutions for output

regulated control

12

(c) Implementation of the IMP regulation framework with internal model-based observer

on the following non-square dynamic: Rotational Translational Actuator (RTAC), Cart

Driven Inverted Pendulum (CIP), Quadrotor Unmanned Aerial Vehicle (QUAV).

(d) Simulation experiments of the output regulated control law in Matlab/Simulink

2. Immersion and invariance (I&I) stabilization framework

(a) Adoption of relevant mathematical preliminaries for immersion and invariance stabilization

(b) Analysis of the necessary and sufficient conditions for existence of solutions for an

immersion and invariance stabilizing control law

(c) Implementation of the immersion and invariance framework with standard observer on

selected benchmark non-square dynamic

(d) Simulation experiments of the control law in Matlab/Simulink

3. Development of an immersion and invariance output feedback framework

(a) Parameter identification for use in developed output regulated output feedback scheme

(b) Implementation of the immersion and invariance observer output feedback framework

on the non-square MIMO QUAV model.

(c) Simulation in Matlab/Simulink of the developed control law.

4. Validation on selected non-square dynamic system and fine-tuning(using neural tools) where

needed of the selected parameters in the developed controller of methodology item (3) to

ensure the goal of GAS and perfect tracking

1.7 Thesis Outline

The organisation of the thesis is as follows; Chapter One has laid-out in brief the overall goals to

be achieved with the introduction, motive, significance of study, statement of the problem, thesis

13

aim and objectives and ends with a discussion of the method used in achieving the set goals.

Chapter Two has given a detailed review of pertinent concepts and literature. Chapter Three has

addressed the materials and methods applied in-so-far as this thesis report is concerned, Chapter

Four presented the results for discussion and Chapter Five concludes the body of the report. All

cited references have been included in the references section while relevant additional material

such as codes and certain parameter derivations have been placed in the appendix section.

14

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