tesis en ingles

Robust Control for Offshore Steel JacketPlatforms under Wave-Induced Forces

Submitted by

Dongsheng Han, Bachelor of Science

A Thesis Submitted for the DegreeMaster of Informatics

(Masters By Research)

School of Computing Sciences

Faculty of Business and Informatics

Central Queensland University

July 2007 (Firstly Submitted)

January 2008 (Finally Submitted)

Approved for Submission by Principal Supervisor

Professor Qing-Long Han

Declaration

The work contained in this thesis has not been previously submitted either in whole

or in part for a degree at Central Queensland University or any other tertiary in-

stitution. To the best of my knowledge and belief, the material presented in this

thesis is original except where due reference is made in text.

Dongsheng Han

20 July, 2007

Abstract

This thesis is concerned with robust control of an offshore steel jacket platform

subject to nonlinear wave-induced forces. Since time delay and uncertainty are

inevitably encountered for an offshore structure and their existence may induce in-

stability, oscillation and poor performance, it is very significant to study on how the

delay and uncertainty affect the offshore structure. In this thesis, a memory robust

control strategy is, for the first time, proposed to reduce the internal oscillations

of the offshore structure under wave-induced forces, so as to ensure the safety and

comfort of the offshore structure.

Firstly, when the system’s states are adopted as feedback, memory state feedback

controllers are introduced for the offshore structure. By using Lyapunov-Krasovskii

stability theory, some delay-dependent stability criteria have been established, based

on which, and by combining with some linearization techniques, memory state feed-

back controllers are designed to control the offshore structure. The simulation re-

sults show that such controllers can effectively reduce the internal oscillations of the

offshore structure subject to nonlinear wave-induced forces and uncertainties. On

the other hand, a new Lyapunov-Krasovskii functional is introduced to derive a less

conservative delay-dependent stability criterion. When this criterion is applied to

the offshore structure, an improved memory state feedback controller with a small

gain is obtained to control the system more effectively, which is sufficiently shown

by the simulation.

Secondly, when the system’s outputs are adopted as feedback, memory dynamic

output feedback controllers are considered for the offshore structure. By employing

a projection theorem and a cone complementary linearization approach, memory

dynamic output feedback controllers are derived by solving some nonlinear mini-

mization problem subject to some linear matrix inequalities. The simulation results

show that the internal oscillations of the offshore structure subject to nonlinear

wave-induced forces are well attenuated.

Finally, robust H∞ control is fully investigated for the offshore structure. By em-

ploying Lyapunov-Krasovskii stability theory, some delay-dependent bounded real

lemmas have been obtained, under which, via a memory state feedback controller or

a dynamic output feedback controller, the resulting closed-loop system is not only

asymptotically stable but also with a prescribed disturbance attenuation level. The

simulation results illustrate the validity of the proposed method.

Acknowledgements

I am deeply indebted to my Principal Supervisor, Professor Qing-Long Han, from

the Central Queensland University (CQU) whose help, sincere suggestions and en-

couragement helped me in all the time of research for and writing of this thesis. His

insight and enthusiasm for research have enabled me to accomplish this work and

are truly appreciated.

It is a pleasure to thank my Associate Supervisor, Dr. Xiefu Jiang, who made

many efforts on my research at CQU. A special thank goes to my another Associate

Supervisor, Dr. Xian-Ming Zhang, whose patient and meticulous guidance and

invaluable suggestions are indispensable to the completion of this thesis.

I would like to take this opportunity to thank the Faculty of Business and Infor-

matics for giving me permission to commence this thesis in the first instance, to do

the necessary research work and to use faculty’s source.

Last, but not least, I thank my parents, who have been a constant source of

support and love.

Table of Contents

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Offshore Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Robust Control . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Significance of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Outline and Contribution of the Thesis . . . . . . . . . . . . . . . . . 11

Chapter 2: Modeling of an Offshore structure . . . . . . . . . . . . . . 15

Chapter 3: State Feedback Control . . . . . . . . . . . . . . . . . . . . . 19

3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Nominal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20


3.2.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.4 An Improved Delay-Dependent Stabilization Criterion . . . . . 38

3.3 Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 A Norm-Bounded Uncertainty . . . . . . . . . . . . . . . . . . 44


3.3.3 A Polytopic Uncertainty . . . . . . . . . . . . . . . . . . . . . 57


3.4 State Feedback H∞ Control . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.1 An H∞ Control for Nominal Systems . . . . . . . . . . . . . . 70


3.4.3 An Improved Controller Design Scheme . . . . . . . . . . . . . 81


3.4.5 An H∞ Control for Uncertain Systems . . . . . . . . . . . . . 86

3.4.6 A Norm-Bounded Uncertainty . . . . . . . . . . . . . . . . . . 86


3.4.8 A Polytopic Uncertainty . . . . . . . . . . . . . . . . . . . . . 99


3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Chapter 4: Dynamic Output Feedback Control . . . . . . . . . . . . . .111

4.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2 Nominal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113


4.2.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 115


4.3 Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125


4.4 Output Feedback H∞ Control . . . . . . . . . . . . . . . . . . . . . . 131


4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Chapter 5: Conclusion and Future Work . . . . . . . . . . . . . . . . . .143

Appendix A: Lemmas Referred . . . . . . . . . . . . . . . . . . . . . . . .147

Appendix B: Useful Theories . . . . . . . . . . . . . . . . . . . . . . . . .149

B.1 Stability of Time-Delay Systems . . . . . . . . . . . . . . . . . . . . 149

B.1.1 Stability Concept: . . . . . . . . . . . . . . . . . . . . . . . . . 149

B.1.2 Lyapunov-Krasovskii Stability Theorem: . . . . . . . . . . . . 150

B.2 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 150

B.3 The LMI Toolbox of Matlab . . . . . . . . . . . . . . . . . . . . . . 151

B.4 An H∞ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Appendix C: Simulation Diagrams . . . . . . . . . . . . . . . . . . . . .155

C.1 The Simulation Diagram of the Controller Design from Proposition 3 155






C.7 The Simulation Diagram of the Output Feedback Control . . . . . . . 162

C.8 The Simulation Diagram of the Robust Output Feedback Control . . 162

C.9 The Simulation Diagram of the Output Feedback H∞ Control . . . . 164

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167

List of Tables

3.1 Comparison results about minimum of γ for various h. . . . . . . . . 86

3.2 The achieved minimum values of γ for various h . . . . . . . . . . . . 99

List of Figures

1.1 Offshore drill platforms . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Offshore steel jacket platform . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Steel jacket structure with an AMD . . . . . . . . . . . . . . . . . . . 15

3.1 The displacement of the first floor with no control and ω = 1.8 . . . . 28

3.2 The displacement of the second floor with no control and ω = 1.8 . . 28

3.3 The displacement of the third floor with no control and ω = 1.8 . . . 29

3.4 The displacement of the first floor with no control and ω = 0.5773 . . 29

3.5 The displacement of the second floor with no control and ω = 0.5773 30

3.6 The displacement of the third floor with no control and ω = 0.5773 . 30

3.7 The displacement of the first floor via the controller (3.3) with (3.20)

for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.8 The displacement of the second floor via the controller (3.3) with

(3.20) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.9 The displacement of the third floor via the controller (3.3) with (3.20)

for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


(3.20) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


(3.21) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35


(3.21) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


(3.28) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42


(3.28) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.25 The displacement of the first floor when no control is used to the

system with norm-bounded uncertainties and ω = 1.8 . . . . . . . . . 46

3.26 The displacement of the second floor when no control is used to the


3.27 The displacement of the third floor when no control is used to the



system with norm-bounded uncertainties and ω = 0.5773 . . . . . . . 48






for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49


(3.36) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


(3.36) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52


(3.37) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


(3.37) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


system with polytopic uncertainties and ω = 1.8 . . . . . . . . . . . . 61






system with polytopic uncertainties and ω = 0.5773 . . . . . . . . . . 63






for ω = 1.8 (polytopic uncertainties) . . . . . . . . . . . . . . . . . . . 64


(3.52) for ω = 1.8 (polytopic uncertainties) . . . . . . . . . . . . . . . 65




for ω = 0.5773 (polytopic uncertainties) . . . . . . . . . . . . . . . . . 66


(3.52) for ω = 0.5773 (polytopic uncertainties) . . . . . . . . . . . . . 66




system under disturbance and ω = 1.8 . . . . . . . . . . . . . . . . . 74






system under disturbance and ω = 0.5773 . . . . . . . . . . . . . . . . 75






for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


(3.65) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.63 The displacement of the third floor via the controller (3.55) with

(3.65) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


(3.65) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


(3.65) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


(3.68) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


(3.68) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84


(3.68) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 84


(3.68) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.73 The displacement of the first floor of the uncontrolled system with

norm-bounded uncertainties under disturbance and ω = 1.8 . . . . . . 88

3.74 The displacement of the second floor of the uncontrolled system with


3.75 The displacement of the third floor of the uncontrolled system with



norm-bounded uncertainties under disturbance and ω = 0.5773 . . . . 90






for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


(3.74) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


(3.74) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


(3.74) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


(3.74) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


(3.75) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


(3.75) for ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


(3.75) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


(3.75) for ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


polytopic type uncertainties under disturbance and ω = 1.8 . . . . . . 101






polytopic type uncertainties under disturbance and ω = 0.5773 . . . . 102











3.100The displacement of the first floor via the controller (3.55) with (3.80)


3.101The displacement of the second floor via the controller (3.55) with


3.102The displacement of the third floor via the controller (3.55) with


4.1 The displacement of the first floor when the output feedback con-

troller is used and ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . . . 120

4.2 The displacement of the second floor when the output feedback con-


4.3 The displacement of the third floor when the output feedback con-


4.4 The displacement of the first floor when the output feedback con-

troller is used and ω = 0.5773 . . . . . . . . . . . . . . . . . . . . . . 122

4.5 The displacement of the second floor when the output feedback con-


4.6 The displacement of the third floor when the output feedback con-


4.7 The displacement of the first floor when the robust output feedback

controller is used and ω = 1.8 . . . . . . . . . . . . . . . . . . . . . . 127

4.8 The displacement of the second floor when the robust output feedback


4.9 The displacement of the third floor when the robust output feedback


4.10 The displacement of the first floor when the robust output feedback

controller is used and ω = 0.5773 . . . . . . . . . . . . . . . . . . . . 128

4.11 The displacement of the second floor when the robust output feedback


4.12 The displacement of the third floor when the robust output feedback


4.13 The displacement of the first floor when the robust H∞ output feed-

back controller is used and ω = 1.8 . . . . . . . . . . . . . . . . . . . 136

4.14 The displacement of the second floor when the robust H∞ output

feedback controller is used and ω = 1.8 . . . . . . . . . . . . . . . . . 136

4.15 The displacement of the third floor when the robust H∞ output feed-

back controller is used and ω = 1.8 . . . . . . . . . . . . . . . . . . . 137

4.16 The displacement of the first floor when the robust H∞ output feed-

back controller is used and ω = 0.5773 . . . . . . . . . . . . . . . . . 137

4.17 The displacement of the second floor when the robust H∞ output

feedback controller is used and ω = 0.5773 . . . . . . . . . . . . . . . 138

4.18 The displacement of the third floor when the robust H∞ output feed-

back controller is used and ω = 0.5773 . . . . . . . . . . . . . . . . . 138

C.1 The simulation diagram of the Controller Design from Proposition 3 . 155

C.2 The simulation diagram of the Controller Design from Proposition 7 . 157

C.3 The simulation diagram of the Controller Design from Proposition 12 159

C.4 The simulation diagram of the Controller Design from Proposition 15 160

C.5 The simulation diagram of the output feedback control . . . . . . . . 163

C.6 The simulation diagram of the robust output feedback control . . . . 163

C.7 The simulation diagram of the output feedback H∞ control . . . . . . 164

List of Symbols

Rn n-dimension real space

Rm×n set of all real m by n matrices

L2[0, +∞) space of square integrable functions on [0, +∞)

AT (resp. xT ) transpose of matrix A (resp. vector x)

‖x‖ Euclidean norm of vector x

‖A‖ norm of matrix A

A ≤ B B − A is a symmetric positive semi-definite matrix

P > 0 P is a symmetric positive definite matrix

P < 0 P is a symmetric negative definite matrix

I identity matrix of appropriate dimensions

∗ denotes a symmetric term in a symmetric, i.e.[P W∗ Q

]=

[P W

W T Q

]

diag... denotes a block-diagonal matrix

Chapter 1

Introduction

1.1 Offshore Structures

The history of structural designs can be roughly divided into three eras: classical,

modern, and post-modern. Classical civil structural designs deal only with static

loads. The modern designs add specifications on the dynamic response. Today, in

the post-modern era, major civil infrastructures must be designed to satisfy both

static and dynamic requirements in the presence of a specified class of environmental

loads. In practice, many civil structures, for example, vibratory platforms, bridges,

tall buildings, etc., are subjected to different types of environmental loading such as

strong winds, earthquake motions, etc., which induce severe vibration. Moreover,

excessive displacement and velocity of the structures endanger the safety. Since the

theory of structural vibration control was built, much attention has been paid on

this topic in the past decades. As a well-known application, the offshore structure

has been widely investigated recently, see, for example, [32, 34, 35, 37, 54, 56, 61].

Offshore structures are a class of typical vibrating structures under external

loads, which have been widely applied as operating station for the offshore exploita-

tion, floating breakwater, offshore fish-farming platform and combination of the

entertainment facilities. They have evolved from very stiff and relatively shallow-

water structures in 1940s to very flexible deep-water ones in recent years. Shortly

after the turn of this century, more than 6,500 offshore oil and gas installations

2 CHAPTER 1. INTRODUCTION

Figure 1.1: Offshore drill platforms

have been set up around the world, in some 53 countries [72]. There are a number

of different types of permanent offshore platforms used for a particular range, see

Figure 1.1 [74]. The most common type is the steel tubular jacket type structure

which is one kind of fixed platform. The steel structures are built on pile ground-

work composed of welded steel pipes. These pipes with 1 to 2 meters diameter are

fixed into the seabed and the depth up to 100 meters [73]. A drawing of steel jacket

structure is shown in Figure 1.2 [71].

Located in hostile marine environments, offshore structures are commonly equipp-

ed with a helicopter pad, drilling derrick, cranes, offices and accommodations. They

are typically subjected to severe loads due to not only gravity and operating loads

but also winds, waves and also currents. And all these loads may induce the exces-

sive vibration of the structures. Therefore, the risk of failure in these structures is

not only higher than other structures, but also the possibility of local or major dam-

age and considerable human discomfort due to vibrations are more likely. To ensure

safety, the horizontal displacement of the structures needs to be limited. Moreover,

for the comfort of people who work on the platforms, acceleration also needs to be

1.1. OFFSHORE STRUCTURES 3

Figure 1.2: Offshore steel jacket platform

restricted.

Vibration control of offshore structures can be achieved by increasing the cross-

sectional area of individual elements and/or adding bracing members to the frames,

so as to shift the natural frequencies away from the resonating of frequencies. How-

ever, it will multiply cost for meeting excessive construction materials. An alter-

native approach is to implement a passive or active damping method to regulate

the motion of structures. The passive approach requires the detailed understanding

of structural dynamics and materials properties. In [53], Patil and Jangid (2005)

studied the response of offshore steel jacket platforms installed with energy dissipa-

tion devices such as viscoelastic, viscous and friction dampers. In [44], Lee et al.

(2006) presented a typical tension-leg type of floating platform incorporated with

the tuned liquid column damper (TLCD) device. Although passive control devices

can mitigate the vibration of offshore structures without requiring external energy,

the performance may be limited by the environment and choices of materials. On

the other hand, an active control mechanism can be effective over a wide frequency

range with desired reduction in the dynamic response. It requires sensors and ac-


tuators connected through a feedback control. In [71], Zribi et al. (2004) presented

nonlinear and robust control schemes for offshore steel jacket platforms by using

Lyapunov theory and an optimal control approach. In [45], Li et al. (2003) devel-

oped an H2 control algorithm for controlling the lateral vibration of a jacket-type

offshore platform by using an active mass damper (AMD). In [48], Ma et al. (2006)

studied the structural vibration control of the same platform as the one in [45], and

presented a feedback and feedforward optimal control (FFOC) law which was shown

more efficient than a classical state feedback optimal control (SFOC) in reducing the

displacement of the same platform. Based on the stochastic dynamic programming

principle and stochastic averaging method for quasi-Hamiltonian systems, Luo and

Zhu (2006) [47] proposed a nonlinear stochastic optimal control (NSOC) strategy

for offshore platforms under wave loading.

In a word, the offshore steel jacket platforms subject to wave forces have been

extensively investigated in recent years, and some nice fruits have been obtained on

this issue. However, all the studies aforementioned are under such an assumption

that the system model is exactly known, also, the delay effects have never been

considered in feedback channels. In fact, the delay and uncertainty are inevitably

encountered for a practical system in control process and signal processing fields

due to actuator speed limit of mechanical systems, transportation time and so on.

Their existences usually result in instability and degrade the performances of the

system under consideration. Therefore, it is of much more practical significance to

investigate how the delay and uncertainty affect the offshore steel jacket platforms,

which motivates this thesis.

1.2. TIME-DELAY SYSTEMS 5

1.2 Time-Delay Systems

1.2.1 Stability Analysis

Consider the following system

x(t) = A0x(t) + Bx(t− h)

x(t) = φ(t), t ∈ [−h, 0](1.1)

where x(t) ∈ Rn denotes the system state, h > 0 is a constant time delay and φ(t) is

an initial condition, which guarantees the existence and uniqueness of the solution

of (1.1).

As is well known, during the past decades, the study on stability for system (1.1)

has been attracted much attention, see, for example, [8, 10, 11, 12, 21, 22, 23, 25,

26, 27, 31, 40, 46, 59, 60, 62, 69]. Two types of stability conditions have been formu-

lated. One is delay-independent, while the other is delay-dependent. As the names

imply, the former doesn’t contain any delay information, while the later depends

on information of a delay. As expected, delay-dependent stability criteria are less

conservative than delay-independent ones in applications. The achieved maximum

admissible upper bound (MAUB) of a delay then becomes a main performance index

to measure the conservatism of a delay-dependent stability condition. The larger

MAUB, the less conservatism.

In the existing literature, there are two kinds of Lyapunov-Krasovskii function-

als, i.e. complete Lyapunov-Krasovskii functionals and simple Lyapunov-Krasovskii

functionals, for estimating the maximum time-delay bound the system can tolerate

and still retain stability. Using the complete Lyapunov-Krasovskii functionals as

those in [17, 18, 19, 25, 26], one can obtain the maximum time-delay bound which

is very close to the analytical limit. Employing the simple Lyapunov-Krasovskii

functionals usually yields conservative results. However, the results based on the

simple Lyapunov-Krasovskii functionals can be easily applied to controller synthe-

sis and filter design. Hence, it is still an attractive topic for finding some simple


Lyapunov-Krasovskii functionals, by which one can have less conservative results.

For comparison, we use a numerical example for system (1.1) with

A =

( −2 00 −0.9

), B =

( −1 0−1 −1

).

The analytical limit for stability for the numerical example is calculated to be

hanalytical = 6.17258. In the existing literature, in order to derive a delay-dependent

stability criterion, one transforms system (1.1) into a system with a distributed

delay, i.e.

x(t) = (A + B)x(t)−B

∫ t

t−h

[Ax(ξ) + Bx(ξ − h)]dξ. (1.2)

Choose a Lyapunov-Krasovskii function

V (t, xt) = xT (t)Px(t), P = P T > 0, (1.3)

and apply Razumikhin Theorem to obtain hmax = 0.9041. As pointed out by Gu

et al. (2003) [19] (Example 5.3), for this example, the stability of system (1.1) is

equivalent to that of system (1.2). The conservatism of the result is due to the

application of the Razumikhin Theorem. For some systems (1.1) with different

system’s matrices, the model transformation (1.2) may induce additional dynamics.

To reduce the conservatism, instead of transforming system (1.1) into (1.2), one

transforms it into

x(t) = (A + B)x(t)−B

∫ t

t−h

x(ξ)dξ. (1.4)

Then choosing a Lyapunov-Krasovskii functional

V (t, xt) = xT (t)Px(t) +

∫ t

t−h

xT (ξ)Qx(ξ)dξ

+

∫ 0

−h

∫ t

t+θ

xT (ξ)BT RBx(ξ)dξdθ, (1.5)

where P = P T > 0, Q = QT > 0, R = RT > 0, and using the bounding technique

for some cross term yield hmax = 4.3588 [52]. This result was also derived by


decomposing delayed term matrix as B = B1 + B2 in [22]. In [8], the author

introduced a descriptor transformation

x(t) =y(t), (1.6)

y(t) =(A + B)x(t)−B

∫ t

t−h

y(ξ)dξ. (1.7)

A Lyapunov-Krasovskii functional was chosen as

V (t, xt)=

(x(t)y(t)

)T (I 00 0

)(P1 0P2 P3

)(x(t)y(t)

)

+

∫ t

t−r

xT (ξ)Qx(ξ)dξ +

∫ 0

−r

∫ t

t+θ

yT (ξ)ATd RAdy(ξ)dξdθ, (1.8)

where P1 = P T1 > 0, Q = QT > 0 and R = RT > 0. Use this model transformation

and bounding technique for cross terms [52] to obtain hmax = 4.4721 in [9]. In [28],

the authors introduce some slack variables (free-weighting matrices) to derive the

same result.

In [27], the author avoided using model transformation on system (1.1) and

proposed the following Lyapunov-Krasovskii functional


∫ t

t−h

xT (ξ)Qx(ξ)dξ

+

∫ t

t−h

(h− t + ξ)xT (ξ)(hR)x(ξ)dξ, (1.9)

which is equivalent to

V (t, xt)=xT (t)Px(t) +

∫ t

t−r

xT (ξ)Qx(ξ)dξ +

∫ 0

−r

∫ t

t+θ

xT (ξ)(rR)x(ξ)dξdθ (1.10)

where P = P T > 0, Q = QT > 0 and R = RT > 0. Instead of using the bounding

technique for some cross term, the author used the following bounding

−∫ t

t−h

xT (ξ)(hR)x(ξ)dξ ≤(

x(t)x(t− h)

)T ( −R RR −R

)(x(t)

x(t− h)

)(1.11)

to derive the maximum allowed delay bound as hmax = 4.4721. Compared with

the above mentioned results, the most advantage of the result in [27] is that the


stability condition which was formulated in an LMI form, was very simple and

easily applied to controller design, and did only include the Lyapunov-Krasovskii

functional matrices variables P,Q and R, which means that no additional matrix

variable was involved. From the computation point of view, it is clear to see that

testing the result in [27] is less time-consuming than some existing results in the

literature. However, the result hmax = 4.4721 is not close enough to the analytical

limit hanalytical = 6.17258 and work needs to be done to arrive at a value much closer

to the analytical limit. Therefore, the natural question is: How can one improve the

result by using simple Lyapunov-Krasovskii functionals? Answer to this question

will significantly enhance the stability analysis and controller synthesis of time-delay

systems. It seems that using the existing simple Lyapunov-Krasovskii functionals

can not realize the outcome even if one introduces more additional matrices variables

apart from Lyapunov-Krasovskii functional matrices variables. One way to solve the

problem is to choose a new Lyapunov-Krasovskii functional. For this purpose, we

propose the following new simple Lyapunov-Krasovskii functional


∫ 0

−h2

∫ t

t+s

xT (θ)Rx(θ)dθds

+

∫ t

t−h2

[x(s)

x(s− h2)

]T [Q1 Q2

QT2 Q3

] [x(s)

x(s− h2)

]ds (1.12)

where xt is defined as xt = x(t + θ),∀θ ∈ [−h, 0] and P = P T > 0, R = RT > 0,[Q1 Q2

QT2 Q3

]=

[Q1 Q2

QT2 Q3

]T

> 0. Based on (1.12), one can have a delay-dependent

stability criterion. Applying this new criterion, one obtains the maximum allowed

delay bound as hmax = 5.7175, which significantly improves the result hmax = 4.4721

in the above mentioned references. One can clearly see that we have made a very

significant step towards the analytical limit for stability of the system.

1.2.2 Robust Control

The robust control of a linear system with time delay is also an important problem

both in theory and in practice, much effort has been done on this topic, see, for


example, [5, 7, 13, 16, 24, 29, 36, 38, 39, 41, 42, 55, 57, 58, 64, 66, 67, 68] etc. For

simplicity, we consider the following system

x(t) = A0(t)x(t) + A1(t)x(t− h) + B(t)u(t)

x(t) = φ(t), t ∈ [−h, 0](1.13)

where u ∈ Rm is the control input; the coefficient matrices A0(t), A1(t), B(t) are

uncertain, which possibly belong to one of the following uncertainties.

Norm-bounded uncertainty

[A0(t) A1(t) B(t)] = [A0 A1 B] + DF (t)[E0 E1 Eb] (1.14)

where A0, A1, B,E0, E1, Eb are known real matrices with appropriate dimen-

sions and F (t) is a unknown time-varying matrix satisfying F T (t)F (t) ≤ I.

Polytopic uncertainty

[A0(t) A1(t) B(t)] ∈

q∑i=1

αi[Ai0 Ai

1 Bi];

q∑i=1

αi = 1, αi ≥ 0

(1.15)

where Ai0, A

i1, B

i, (i = 1, 2, · · · , q) are constant real matrices.

The basic robust control problem is to seek a suitable controller such that the

resulting closed-loop system of (1.13) is asymptotically stable for all uncertainties

satisfying (1.14) or (1.15). To this aim, two control schemes are well used. One is

to design a state feedback controller as

u(t) = K1x(t) + K2x(t− h) (1.16)

where K1, K2 are called controller gains, which are to be determined. The other one

is to design a dynamic output feedback controller to stabilize the original system

(1.13). Assume the output of the system (1.13) is

y(t) = C0x(t) + C1x(t− h) (1.17)


where C0 and C1 are constant real matrices with appropriate dimensions. The

dynamic output feedback controller is of form

xc(t) = Acxc(t) + Bcy(t)

u(t) = Ccxc(t) + Dcy(t)(1.18)

where Ac, Bc, Cc, Dc are controller parameters to be determined.

In the recent years, the robust control for system (1.13) has been widely inves-

tigated based on linear matrix inequality technique and a lot of controller design

methods have been proposed. To mention it, the parameter tuning approach is ad-

dressed to design a state feedback controller in [9, 67]. Since the parameter tuning

approach needs to restrict some matrix variables to some special structures, this

method usually leads to much conservative results. To achieve much better results,

Moon et al proposed an iterative algorithm to work out the controller parameters in

[49], which, recently, has been gained much attention in control synthesis, one can

see [16, 66] etc. On the other hand, dynamic output feedback control has also been

extensively studied [2, 29, 30, 37]. A projection approach has been a main tool to

obtain an output feedback controller, see, for example, [2, 30] and reference therein.

1.3 Significance of This Thesis

• Since humankind stepped into the 21st century, the need of energy has been

augmenting sharply. Therefore, accompanied by increasingly exhausted ter-

restrial energy, more and more attention has been devoted into offshore energy

drilling. Meanwhile, offshore structures provide the necessary possibility for

such an effort. Nevertheless, comparing with terrestrial exploring platforms,

complex ocean environment brings much larger challenge to the safety of the

offshore structures. Specifically, dynamic loading, such as earthquake, wind,

wave, deep water current and so on, can induce large motion of offshore plat-

forms, further potentially endanger the safety and comfort of the platforms.

Hence, it is a meaningful research topic on how to control an offshore structure.

1.4. OUTLINE AND CONTRIBUTION OF THE THESIS 11

• On the other hand, time delays are unavoidable for practical systems, and

their existences usually lead to instability or performance degradation of a

system. It is of great significance to study the effect of time delays on an

offshore structure both in practice and in theory.

1.4 Outline and Contribution of the Thesis

In this thesis, the robust stability and control problem is investigated for offshore

steel jacket platforms subject to nonlinear wave-induced forces. In order to reduce

the internal oscillations of the platforms subject to irregular wave forces, two types of

control schemes are proposed, i.e. state feedback control and dynamic output feed-

back control. Based on Lyapunov-Krasovskii stability theory, some delay-dependent

stabilization criteria are obtained in terms of linear matrix inequalities. By employ-

ing an iterative algorithm, the controllers are then easily obtained thanks to the

Matlab LMI Toolbox.

The outline of the thesis is given as follows:

In Chapter 2, a typical offshore steel jacket platform is described. After analyzing

its structure and parameter characteristics, a mathematical model is built, based

on which, two control strategies are introduced to reduce the internal oscillations in

the next sections.

In Chapter 3, a memory state feedback control is considered for the offshore steel

jacket platform with uncertainty. Firstly, a delay-dependent stability condition is

obtained for a nominal closed-loop system, based on which, the controller is then

designed by employing linearization approaches. The simulation results show that

internal oscillations are effectively reduced when this controller is applied to the

offshore steel jacket platform. Secondly, the above derivatives are successfully ex-

tended to the system with either norm-bounded or polytopic uncertainties. Finally,

an H∞ control stabilization is also developed for this offshore platform, and some


delay-dependent criteria are obtained in terms of LMIs, which guarantee the con-

sidered system is asymptotically stable with a prescribed disturbance attenuation

level γ via a state feedback controller.

In Chapter 4, a memory dynamic output feedback control is studied for the

offshore steel jacket platform with uncertainties. A sufficient condition for the ex-

istence of such a controller is first established for a nominal system. In virtue of a

projection theorem and a cone complementary linearization algorithm, a dynamic

output feedback controller is then designed, which can be easily solved by using the

Matlab LMI Toolbox. Simulation results illustrate that the dynamic output feed-

back controller can more effectively reduce the internal oscillations of the offshore

platform subject to wave forces when the system outputs are adopted as feedback.

Then, these results are extended to suit for the uncertain system. Similar to the pre-

vious chapter, the robust H∞ control is also discussed for the system via a dynamic

output feedback controller.

In Chapter 5, this thesis is summarized and some future research topics on

offshore steel jacket platforms are also proposed.

The contributions of the thesis can be summarized in the following

(1) In practice, we study, for the first time, the effect of time delays on the offshore

structure system. Two types of controllers with time delays are proposed to

reduce the internal oscillations of the offshore structure platforms. When the

system states are adopted as feedback, a memory state feedback controller

is designed; while, a memory dynamic output feedback controller is achieved

when the system outputs are considered as feedback;

(2) Two types of uncertainties, namely norm-bounded uncertainty and polytopic

uncertainty, are considered for offshore structure platforms and some sufficient

conditions for robust control are obtained;

1.4. OUTLINE AND CONTRIBUTION OF THE THESIS 13

(3) An H∞ control problem is investigated for the offshore structure platforms,

such that not only the internal oscillations can be reduced via above two types

of controller, but also the system is of a prescribed disturbance attenuation

level; and

(4) In theory, in order to reduce the conservatism of delay-dependent stability

conditions in the literature, a new Lyapunov-Krasovskii functional is first in-

troduced.

Chapter 2

Modeling of an Offshore structure

Because nonlinear self-excited hydrodynamic force is the most dominant external

persistence, we consider an offshore steel jacket platform under wave-induced forces,

see Figure 2.1 [1]. An active tuned mass damper (AMD) mechanism mounted on

top is connected to a hydraulic servo mechanism. The operation of the hydraulic

servo is driven by active control forces regulated by a designed controller.

Figure 2.1: Steel jacket structure with an AMD

The natural frequencies and mode shapes of the undamped free vibration can

be calculated by one of the available structural dynamic softwares. For simplicity

of presentation, we will only consider the first two modes of vibration because these

two modes are the most dominant and hence the most important for the design of

16 CHAPTER 2. MODELING OF AN OFFSHORE STRUCTURE

the control. The equation of motion of the first two modes of vibrations with the

coupled AMD are

z1 = −2ξ1ω1z1 − ω21z1 − ΦT

1 (Fa − FAMD) + ft1 + ft2,

z2 = −2ξ2ω2z2 − ω22z2 − ΦT

2 (Fa − FAMD) + ft3 + ft4,

y = −2ξT ωT (y − U8)− ω2T (y − U8) + 1

mTu,

(2.1)

where z1 and z2 are the generalized coordinates of vibration modes 1 and 2, re-

spectively; y is the horizontal displacement of the AMD; ω1 and ω2 are the natural

frequencies of the first two modes of vibrations; ξ1 and ξ2 are the damping ratio in

the first two modes of vibrations; Φ1 and Φ2 are the first and second mode shapes

vectors, respectively; FAMD is the passive control force vector due to the AMD; ξT

is the damping ratio of the AMD; CT , mT and KT are the damping, the mass and

the stiffness of the AMD, respectively; ωT =√

KT /mT is the natural frequency of

the AMD; U8 is the horizontal displacement of joint 8 in the offshore structure; u is

the control action of the system; and ft1, ft2, ft3, ft4 are the nonlinear self-excited

hydrodynamic force terms [71].

The horizontal displacement of joint 8 in the offshore structure is

U8 = φ1z1 + φ2z2, (2.2)

where φ1 and φ2 are the contributions of the first two mode shapes. Noting that

ΦT1 Fa = φ1u , ΦT

2 Fa = φ2u, (2.3)

the control forces due to the AMD can explicitly be written as

ΦT

1 FAMD = φ1[KT (y − U8) + CT (y − U8)],

ΦT2 FAMD = φ2[KT (y − U8) + CT (y − U8)]

(2.4)

The nonlinear self-excited hydrodynamic force terms can be modeled by using

Morison equation.

Using Equations (2.1)-(2.4), the equations of motion of a steel jacket platform

subjected to nonlinear self-excited hydrodynamic forces can be written as

17

z1 =− 2ξ1ω1z1 − ω21z1 − φ1KT (φ1z1 + φ2z2) + φ1KT y

− φ1CT (φ1z1 + φ2z2) + φ1CT y − φ1u + ft1 + ft2,

z2 =− 2ξ2ω2z2 − ω22z2 − φ2KT (φ1z1 + φ2z2) + φ2KT y

− φ2CT (φ1z1 + φ2z2) + φ2CT y − φ2u + ft3 + ft4,

y =− 2ξT ωT y + 2ξT ωT (φ1z1 + φ2z2)− ω2T y + ω2

T (φ1z1 + φ2z2)

+1

mT

u.

(2.5)

Let x1(t) = z1(t), x2(t) = z1(t), x3(t) = z2(t), x4(t) = z2(t), x5(t) = y(t), and

x6(t) = y(t). The system (2.5) can be written as

x(t) = Ax(t) + Bu(t) + Fg(x, t) (2.6)

where

x(t) =[

x1(t) x2(t) x3(t) x4(t) x5(t) x6(t)]T

, (2.7)

B =[

0 −φ1 0 −φ2 0 1mT

]T, (2.8)

F =

[0 1 0 0 0 00 0 0 1 0 0

]T

, (2.9)

g(x, t) =

[ft1 + ft2

ft3 + ft4

], (2.10)

and

A =

0 1 0 0 0 0A21 A22 −KT φ1φ2 −CT φ1φ2 φ1KT φ1CT

0 0 0 1 0 0−KT φ1φ2 −CT φ1φ2 A43 A44 φ2KT φ2CT

0 0 0 0 0 1ω2

T φ1 2ξT ωT φ1 ω2T φ2 2ξT ωT φ2 −ω2

T −2ξT ωT

(2.11)

with A21 = −ω21 − KT φ2

1, A22 = −2ξ1ω1 − CT φ21, A43 = −ω2

2 − KT φ22, and A44 =

−2ξ2ω2 − CT φ22.

Remark 1. The nonlinear term g(x, t) in (2.6) is uniformly bounded and it can be

assumed that the nonlinear term g(x, t) satisfies the following cone-bounding con-

straint:

‖g(x, t)‖ ≤ µ ‖x(t)‖ , (2.12)

18 CHAPTER 2. MODELING OF AN OFFSHORE STRUCTURE

where µ is a positive scalar.

For the purpose of simulation, the same system’s parameters setting from Abdel-

Rohman [1] is used for simulation studies. The data for the waves are H = 12.19m,

h = 76.2m, λ = 182.88m. The structure consists of cylindrical steel tube members.

The density of steel is 7730.7kg/m3; the density of water, ρw = 1025.6kg/m3. The

weight of the concrete deck carried by the steel members is 6672.3 × 103N, and

Uow = 0.122m/ sec . The dimensions of the structural elements are given in Figure

1. The project areas, volumes, and masses of each member in the structure can

be found in [1]. Using these data, the wave force parameters at each joint can be

calculated [1].

The natural frequencies of the first two modes of vibrations are ω1 = 1.818rps

and ω2 = 10.8717rps. The structural damping in each mode is considered to be

0.005, hence ζ1 = ζ2 = 0.005. The contribution of the first two modes shapes are

φ1 = −0.003445 and φ2 = 0.003463.

The TMD parameters are chosen to tune with the first mode such that ωT =

1.8rps, ζT = 0.15 and KT = 1551.5, CT = 256.

Using the values of the parameters of the system, we find that

A =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819

0 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 0 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

;

B =[

0 0.003445 0 −0.00344628 0 0.00213]T

.

In the following chapters, two kinds of control approaches, i.e. state feedback

control and dynamic output feedback control, will be proposed for the above offshore

steel jacket platform.

Chapter 3

State Feedback Control

For practical systems, time delay and uncertainty are unavoidable due to modeling

errors and data transmission, which usually result in instability and degrade per-

formance of the corresponding systems. In this chapter, for an offshore steel jacket

platform, in order to effectively reduce the internal system oscillations, a memory

state feedback controller will be introduced. Based on Lyapunov-Krasovskii func-

tional stability theory and some linearization methods, a memory state feedback

controller will be designed, under which, via numerical simulation, the considered

system can be effectively controlled, that is, the amplitudes of the internal oscilla-

tions are greatly decreased.

In addition, robust and H∞ control via a state feedback control will also be

investigated for the system with uncertainties, either norm-bounded or polytopic.

3.1 Problem Formulation

Consider an offshore steel jacket platform of a general form

x(t) = A(t)x(t) + B(t)u(t) + Fg(x, t)

x(t) = φ(t), t ≤ 0(3.1)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control, g(x, t) ∈ Rp is the

nonlinear self-excited hydrodynamic force vector, which is uniformly bounded and

20 CHAPTER 3. STATE FEEDBACK CONTROL

satisfies the following cone-bounding constraint:

‖g(x, t)‖ ≤ µ ‖x(t)‖ , (3.2)

where µ is a positive scalar; A(t) and B(t) are system matrices, which may be

time-varying, but are known to belong to a certain compact set Ω, that is,

(A(t), B(t)) ∈ Rn×(n+m) ⊂ Ω

and φ(t) denotes an initial condition. In this chapter, suppose that the states of

system (3.1) are adopted as feedback, in this situation, we introduce a state feedback

controller

u(t) = K1x(t) + K2x(t− h), (3.3)

where K1 and K2 are constant matrices, which are to be determined, delay h is

assumed to be constant satisfying h > 0.

Remark 2. Clearly, the controller (3.3) is dependent on both the current and the

past states, which is called a memory state feedback controller when K2 6= 0, while

a memoryless one when K2 = 0.

The main aim of this chapter is to design a controller of form (3.3) such that

the resulting closed-loop system by (3.1) and (3.3) is asymptotically stable. In the

sequel, we first consider the controller design for the nominal system of (3.1), and

then these results are extended to suit for system (3.1) with either norm-bounded

or polytopic uncertainty. Finally, a robust H∞ control problem of the system is

considered.

3.2 Nominal Systems

In this section, we focus on the stabilization of the nominal system of (3.1). In this

case, we suppose A(t) ≡ A and B(t) ≡ B for all t ≥ 0, where A and B are constant

3.2. NOMINAL SYSTEMS 21

real matrices. The resulting closed-loop system of (3.1) with (3.3) is given as

x(t) = (A + BK1)x(t) + BK2x(t− h) + Fg(x, t),

x(t) = φ(t), t ∈ [−h, 0],(3.4)


Based on an integral inequality established recently and the Lyapunov-Krasovskii

stability theory, a delay-dependent stability criterion is first derived, which is stated

as follows.

Proposition 1. For some given scalars µ > 0 and h > 0, system (3.4) is asymp-

totically stable if there exist matrices P > 0, Q > 0, R > 0 and M1, M2, Z1, Z2, Z3

of appropriate dimensions such that

Ξ :=

Ξ11 Ξ12 PF h(A + BK1)T

∗ Ξ22 0 hKT2 BT

∗ ∗ −I hF T

∗ ∗ ∗ −hR−1

< 0, (3.5)

Ψ :=

R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0, (3.6)

where

Ξ11 := P (A + BK1) + (A + BK1)T P

+ Q + µ2I + MT1 + M1 + hZ1,

Ξ12 := PBK2 −MT1 + M2 + hZ2,

Ξ22 := −Q−MT2 −M2 + hZ3.

Proof. Choose a Lyapunov-Krasovskii functional candidate as

V (xt) = xT (t)Px(t) +

∫ t

t−h

xT (s)Qx(s)ds

+

∫ 0

−h

dθ

∫ t

t+θ

xT (s)Rx(s)ds, (3.7)

where xt = x(t + α), α ∈ [−h, 0] and P > 0, Q > 0, R > 0 are to be determined.

Taking the derivative of V (xt) with respect to time t along the trajectory of system


(3.4) yields

V (xt) =2xT (t)Px(t) + xT (t)Qx(t)

− xT (t− h)Qx(t− h)

+ hxT (t)Rx(t)−∫ t

t−h

xT (s)Rx(s)ds (3.8)

According to Lemma 3 (see Appendix A), for any Mi, Zj (i = 1, 2, j = 1, 2, 3) with

appropriate dimensions, if (3.6) holds, then

−∫ t

t−h

xT (s)Rx(s)ds ≤[

x(t)x(t−h)

]T [ρ11 ρ12

∗ ρ22

] [x(t)

x(t−h)

](3.9)

where

ρ11 := MT1 + M1 + hZ1

ρ12 := −MT1 + M2 + hZ2

ρ22 := −MT2 −M2 + hZ3

In addition, noting from (3.2) that

0 ≤ µ2xT (t)x(t)− gT (x, t)g(x, t). (3.10)

Introducing a new vector

η(t) = [xT (t) xT (t− h) gT (x, t)]T

and substituting (3.9) and (3.10) into (3.8) gives

V (t, xt) ≤ ηT (t)Φ + hΓT RΓη(t)

where

Φ :=

Ξ11 Ξ12 PF∗ Ξ22 0∗ ∗ −I

,

Γ := [A + BK1 BK2 F ]


with Ξ11, Ξ12 and Ξ22 are defined in (3.5). Clearly, if the matrix inequalities (3.5)

and (3.6) hold, then using the Schur complement yields Φ+hΓT RΓ < 0. Therefore,

there exists a sufficient small number δ > 0 such that V (t, xt) ≤ −δxT (t)x(t) < 0

for x(t) 6= 0, which guarantees the asymptotical stability of the closed-loop system

(3.4) by Lyapunov-Krasovskii stability theorem. This completes the proof.

Proposition 1 provides a delay-dependent stability condition for system (3.4).

Clearly, this condition is nonlinear due to nonlinear terms, such as PBK1 etc. Below,

in order to get controller gains K1 and K2, we will propose two schemes to linearize

the matrix inequalities (3.5) and (3.6).

3.2.2 Controller Design

In this section, we are concerned with the controller design based on Proposition 1.

To this aim, an equivalent version of Proposition 1 is first presented in the following.

Proposition 2. For some given positive scalars µ > 0 and h > 0, system (3.4) is

asymptotically stable if there exist matrices X > 0, Q > 0, R > 0 and Y1, Y2, M1, M2,

Z1, Z2, Z3 of appropriate dimensions such that

Ξ :=

Ξ11 Ξ12 F hXAT + hY T1 BT µX

∗ Ξ22 0 hY T2 BT 0

∗ ∗ −I hF T 0∗ ∗ ∗ −hR 0∗ ∗ ∗ ∗ −I

< 0 (3.11)

Ψ :=

XR−1X M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0 (3.12)

where

Ξ11 := XAT + AX + BY1 + Y T1 BT + Q + MT

1 + M1 + hZ1,

Ξ12 := BY2 − MT1 + M2 + hZ2,

Ξ22 := −Q− MT2 − M2 + hZ3.

Moreover, the controller gains are given by K1 = Y1X−1 and K2 = Y2X

−1.


Proof. Now, we will prove the matrix inequalities (3.5) and (3.6) are equivalent

to (3.11) and (3.12), respectively. In fact, noting that P > 0 in (3.5), then P is

invertible, define

T1 := diagP−1, P−1, I, I

T2 := diagP−1, P−1, P−1

and set X = P−1, Y1 = K1P−1, Y2 = K2P

−1, R = R−1 and

Q M1 M2

∗ Z1 Z2

∗ ∗ Z3

:=

X 0 0∗ X 0∗ ∗ X

Q M1 M2

∗ Z1 Z2

∗ ∗ Z3

X 0 0∗ X 0∗ ∗ X

then we have

T T1 ΞT1 =

Ξ11 + µ2X2 Ξ12 F hXAT + hY T1 BT

∗ Ξ22 0 hY T2 BT

∗ ∗ −I hF T

∗ ∗ ∗ −hR

T T2 ΨT2 = Ψ

where Ξ and Ψ are defined in (3.5) and (3.6), respectively. Thus, by the Schur

complement, one obtains

Ξ < 0 ⇐⇒ Ξ < 0

Ψ ≥ 0 ⇐⇒ Ψ ≥ 0

which completes the proof.

We are now in a position to design the desired controller. From Proposition 2,

matrix inequality (3.11) is linear on matrix variables, while (3.12) not. However,

only one nonlinear term, i.e. XR−1X, is included in (3.12). This means that only

if this term is linearized, then (3.12) is converted into an LMI. By employing the

Matlab LMI Toolbox, the controller can be easily solved out based on the LMIs. In

the following, two methods will be introduced to linearize this nonlinear term.


According to the fact that

XR−1X ≥ 2X − R, (3.13)

we have

XR−1X M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥

2X − R M1 M2

∗ Z1 Z2

∗ ∗ Z3

This leads to a controller design approach based on the solutions of two LMIs, which

is stated in the following.

Proposition 3. For some given positive scalars µ > 0 and h > 0, system (3.4)

with K1 = Y1X−1 and K2 = Y2X

−1 is asymptotically stable if there exist matrices

X > 0, Q > 0, R > 0 and Y1, Y2, M1, M2, Z1, Z2, Z3 of appropriate dimensions such

that (3.11) and

2X − R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0 (3.14)

Although Proposition 3 provides an approach to design a controller, the obtained

results are usually conservative since it requires 2X − R > 0, which is a restriction

on variables X and R. To remove this constraint, we can introduce a new matrix

variable S > 0 such that

XR−1X ≥ S (3.15)

which is equivalent to

[R−1 X−1

X−1 S−1

]≥ 0 (3.16)

Following the above line yields the second method to design the suitable controller

of form (3.3), which is shown in the following proposition.

Proposition 4. For some given positive scalars µ > 0 and h > 0, system (3.4)

with K1 = Y1X−1 and K2 = Y2X

−1 is asymptotically stable if there exist matrices


X > 0, Q > 0, R > 0, S > 0,X > 0, R > 0,S > 0 and Y1, Y2, M1, M2, Z1, Z2, Z3 of

appropriate dimensions such that (3.11) and

S M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0 (3.17)

XX = I, RR = I, SS = I. (3.18)

Clearly, Proposition 4 is a non-convex feasibility problem due to the equality

constraints in (3.18). So it cannot be directly solved by using Matlab LMI Toolbox.

However, employing the cone complementary approach proposed in [4], we can con-

vert the non-convex feasibility problem into a nonlinear minimization one subject

to LMIs.

A Nonlinear Minimization Problem

Minimize Tr(XX + RR+ SS)

Subject to (3.11), (3.17) and[R XX S

]≥ 0,

[X II X

]≥ 0,

[R II R

]≥ 0,

[S II S

]≥ 0. (3.19)

Similar to [4], an iterative algorithm can be proposed to solve the above nonlinear

minimization problem, which is stated below.

Algorithm 2.1: Maximize h for a given scalar µ > 0.

Step 1 Choose a sufficiently small initial value hini such that (3.11),(3.17)

and (3.19) are feasible. Set hso = hini.

Step 2 Find a feasible set

(X0, Q0, R0, S0,X 0, R0,S0, Y 0i , M0

i , Z0j (i = 1, 2, j = 1, 2, 3)) satisfy-

ing (3.11),(3.17) and (3.19). Set l = 0.

Step 3 Solve the following LMI problem for the variables (X,X , R, R, S,S)

Minimize Tr(X lX + X lX + RlR+ RlR + SlS + S lS)Subject to (3.11), (3.17), (3.19)

Set X l+1 = X,X l+1 = X , Rl+1 = R, Rl+1 = R, Sl+1 = S,S l+1 = S.


Step 4 If matrix inequality (3.12) is satisfied, then set hso = hini and in-

crease γini to some extent and go back to Step 2. If the condition

(3.12) is not satisfied within a specified number of iterations, then

exit, otherwise, set l = l + 1 and go to Step 3.

Remark 3. The proposed algorithm provides a procedure to obtain a suboptimal

upper bound h of delay size. It is worth noting that the inequality (3.12) is used as

the stopping criterion in the above algorithm since it is numerically very difficult to

exactly obtain the minimum value, 36, of Tr(X lX +X lX + RlR+RlR+SlS+S lS).

Remark 4. Based on Proposition 1, two controller design schemes are proposed, i.e.

the Controller Design from Proposition 3 and the Controller Design from Proposition

4. In general, the former scheme can tolerate a larger upper bound h than the later

one. Moreover, when these two schemes are applied to the offshore structure in

(2.6), even for the same admissible bound h, the obtained controller gain by the

later is smaller than that by the former, which is shown in the next section.

3.2.3 Simulation Results

In the previous section, a memory state feedback controller is introduced to control

an offshore structure platform and two controller design schemes are proposed. Now,

we aim to show the effectiveness of the proposed methods via numerical simulations.

For simulation to the offshore structure platform (2.6), two cases of wave fre-

quency are taken into consideration, ω = 1.8 rps and ω = 0.5773 rps. We set

µ = 0.1. In the sequel, we compare the state curves under controller (3.3) with

h = 0.1 by two schemes with those by no control.

The responses of the offshore structure with no control are shown in Figures 3.1

- 3.6. The uncontrolled lateral displacements of the three floors of jacket for ω =

1.8 and ω = 0.5773 are shown in Figures 3.1-3.3 and Figures 3.3-3.6, respectively.

Clearly, for ω = 1.8, the oscillation amplitudes of the three floors range from -2.25


0 10 20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5First floor − No Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.1: The displacement of the first floor with no control and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5Second floor − No Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.2: The displacement of the second floor with no control and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−3

−2

−1

0

1

2

3Third floor − No Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.3: The displacement of the third floor with no control and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First floor − No Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.4: The displacement of the first floor with no control and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Second floor − No Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.5: The displacement of the second floor with no control and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Third floor − No Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.6: The displacement of the third floor with no control and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4First floor − State Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.7: The displacement of the first floor via the controller (3.3) with (3.20)for ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Second floor − State Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.8: The displacement of the second floor via the controller (3.3) with (3.20)for ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Third floor − State Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.9: The displacement of the third floor via the controller (3.3) with (3.20)for ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5First floor − State Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Second floor − State Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Third floor − State Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4First floor − State Feedback Control (Nonlinear Minimization)

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Second floor − State Feedback Control (Nonlinear Minimization)

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Third floor − State Feedback Control (Nonlinear Minimization)

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5First floor − State Feedback Control (Nonlinear Minimization)

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Second floor − State Feedback Control (Nonlinear Minimization)

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Third floor − State Feedback Control (Nonlinear Minimization)

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



ft to 2.25 ft, from -2.5 ft to 2.5 ft, and from -2.6 ft to 2.6 ft, respectively. The

responses for ω = 0.5773 oscillate within the range of 1.5 ft, 1.6 ft and 1.7 ft peak

to peak, respectively (see Figures 3.2 - 3.4).

However, under the controller (3.3), the amplitudes of displacements for both

ω = 1.8 and ω = 0.5773 can be effectively controlled. In fact, for the Controller

Design from Proposition 3, the controller gains K1 and K2 in (3.3) are given by

Ks1 :

K1 = [−0.3168 −0.0612 4.8474 0.3767 −0.1314 −0.3290]× 104

K2 = [0.0359 0.0061 −1.3969 0.0348 0.1165 0.0597]× 104

(3.20)

The responses of the offshore structure under controller Ks1 are also given in Figures

3.7 - 3.12. Compared with Figures 3.1 - 3.6, clearly, the displacements of the three

floors are greatly decreased. When the system reaches a steady state, the amplitudes

are just 14%, 16% and 16% of those with no control for ω = 1.8, respectively. For

ω = 0.5773, the amplitudes of displacement are also decreased, which can be seen

from Figures 3.10 - 3.12. (See Appendix C.1 for the simulation diagram)

Next, we will show the effectiveness of the Controller Design from Proposition

4. The obtained controller gains by this scheme are

Ks2 :

K1 = [−0.9575 −0.6353 0.1208 1.7778 1.1862 −1.4151]× 103

K2 = [0.0012 −0.0022 0.0348 0.0008 0.0003 0.0021]

(3.21)

Obviously, compared with the Controller Design from Proposition 3, the gains of

Ks2 are much smaller than those of Ks1, which means that the Controller Design

from Proposition 4 requires less energy than the Controller Design from Proposition

3 to control the offshore structure. On the other hand, Figures 3.13 - 3.18 plot the

response curves of offshore structure by Ks2, from which it is not difficult to see

that these amplitudes are little smaller than those in Figures 3.7 - 3.12, respectively.

From the view of both energy consumption and the decrease of internal oscillation,

the Controller Design from Proposition 4 is more effective than the Controller De-

sign from Proposition 3. (The simulation diagram of the Controller Design from


Proposition 4 is similar to that of the Controller Design from Proposition 3. It’s

omitted.)

3.2.4 An Improved Delay-Dependent Stabilization Crite-rion

In the previous section, Proposition 1 provides a delay-dependent stability condition

for system (3.4) by using a widely employed Lyapunov-Krasovskii functional (LKF)

(3.7). As pointed out in [46], it seems impossible to reduce the conservatism of the

obtained delay-dependent stability conditions by using this LKF. In this section,

we introduce a new LKF, by which a less conservative delay-dependent stability

criterion is obtained. Interestingly, based on this new stability condition, the result-

ing controller can control the offshore structure more effectively than the Controller

Design from Proposition 3.

The new LKF candidate is given as


∫ 0

−h2

ds

∫ t

t+s

xT (θ)Rx(θ)dθ

+

∫ t

t−h2

[x(s)

x(s− h2)

]T [Q1 Q2

QT2 Q3

] [x(s)

x(s− h2)

]ds (3.22)

where P, Q1, Q2, Q3, R ∈ Rn, P > 0, Q =

[Q1 Q2

QT2 Q3

]> 0, R > 0 to be deter-

mined.

Similar to Proposition 1, the following proposition can be easily obtained, the

proof is omitted.

Proposition 5. For some given positive scalars µ and h, system (3.4) is asymptot-

ically stable if there exist matrices P > 0, Q =

[Q1 Q2

QT2 Q3

]> 0, R > 0 and Mi, Zj

(i = 1, 2, j = 1, 2, 3) of appropriate dimensions such that

Ω11 Ω12 PBK2 PF h2(A + BK1)

T

∗ Ω22 −Q2 0 0∗ ∗ −Q3 0 h

2KT

2 BT

∗ ∗ ∗ −I h2F T

∗ ∗ ∗ ∗ −h2R−1

< 0, (3.23)


R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0, (3.24)

where

Ω11 := P (A + BK1) + (A + BK1)T P + Q1 + µ2I + MT

1 + M1 +h

2Z1,

Ω12 := Q2 −MT1 + M2 +

h

2Z2,

Ω22 := Q3 −Q1 −MT2 −M2 +

h

2Z3.

In order to show the less conservatism of Proposition 5, we take a well known

example from [22].

Example 1. Consider a linear system with a time delay

x(t) =

[−2 00 −0.9

]x(t) +

[−1 0−1 −1

]x(t− h) (3.25)

References [10, 28, 62, 65, 69] calculated the maximum admissible upper bound

(MAUB) of delay h, and the obtained results were all the same, i.e. 4.4721. However,

Proposition 5 can achieve a much larger MAUB, i.e. 5.7175, which sufficiently shows

the less conservatism of the new delay-dependent stability criterion by employing

the new LKF (3.22).

Similarly, we can also design a state feedback controller based on Proposition 5.

For simplicity, we just give the same scheme as Proposition 3 and the other design

scheme like Proposition 4 is omitted.

Proposition 6. For some given positive scalars µ and h, system (3.4) with K1 =

Y1X−1 and K2 = Y2X

−1 is asymptotically stable if there exist matrices X > 0, Q =[Q1 Q2

QT2 Q3

]> 0, R > 0 and Y1, Y2, M1, M2, Z1, Z2, Z3 of appropriate dimensions such

that

Ω11 Ω12 BY2 F h2(XAT + Y T

1 BT ) µX

∗ Ω22 −Q2 0 0 0∗ ∗ −Q3 0 h

2Y T

2 BT 0∗ ∗ ∗ −I h

2F T 0

∗ ∗ ∗ ∗ −h2R 0

∗ ∗ ∗ ∗ ∗ −I

< 0, (3.26)


2X − R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0. (3.27)

where

Ω11 := AX + BY1 + XAT + Y T1 BT + Q1 + MT

1 + M1 +h

2Z1,

Ω12 := Q2 − MT1 + M2 +

h

2Z2,

Ω22 := Q3 − Q1 − MT2 − M2 +

h

2Z3.

Proposition 6 provides another design scheme based on a new delay-dependent

stability condition. It is interesting that, for the offshore structure, the obtained

controller (3.3) by the Controller Design from Proposition 6 is more effective than

Ks1 by the Controller Design from Proposition 3. To show this, using the same

setting as the Controller Design from Proposition 3, that is, h = 0.1 and µ = 0.1,

the controller gains are given by

Ks3 :

K1 = [−1.3364 −0.3302 3.3623 1.5286 0.7710 −1.1746]× 103

K2 = [5.1730 1.0975 −75.1301 −1.4988 2.7732 6.7567]

(3.28)

The responses of the offshore structure via controller (3.3) with Ks3 are shown in

Figures 3.19 - 3.24. (The simulation diagram of the Controller Design from Propo-

sition 6 is similar to that of the Controller Design from Proposition 3. It’s omitted.)

Comparing with Figures 3.7 - 3.12 by the Controller Design from Proposition 3, we

find that

À the oscillation amplitudes of the jacket platform for ω = 0.5773 are almost the

same as those by the Controller Design from Proposition 3;

Á the oscillation amplitudes for ω = 0.5773 are little worse than those by the

Controller Design from Proposition 3, but control results are also good; and

Â the controller gain is greatly reduced;


0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

First floor − Improved State Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Second floor − Improved State Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Third floor − Improved State Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

First floor − Improved State Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Second floor − Improved State Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Third floor − Improved State Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



which are enough to illustrate the validity and superiority of the Controller Design

from Proposition 6.

3.3 Uncertain Systems

In the previous section, we discussed the controller design for a nominal offshore

structure system and three schemes have been derived to reduce the internal oscilla-

tions. However, when the system matrices are subject to uncertainties, the obtained

results fail to this situation. As is well known, depending on the uncertainty type, we

can have different results. In this section, we will consider two types of uncertainties,

one is a norm-bounded uncertainty and the other polytopic uncertainty.

3.3.1 A Norm-Bounded Uncertainty

For the norm-bounded uncertainty, it is assumed that

[A(t) B(t)] = [A B] + LG(t)[Ea Eb] (3.29)

where A,B,L, Ea and Eb are constant matrices with appropriate dimensions, and

G(t) is a time-varying real matrix with Lebesgue-measurable elements satisfying

GT (t)G(t) ≤ I, ∀t. (3.30)

The closed-loop system of (3.1) with (3.29) connecting (3.3) is then given by

x(t) = [(A + BK1) + LG(t)(Ea + EbK1)]x(t)

+ [BK2 + LG(t)EbK2]x(t− h) + Fg(x, t),

x(t) = φ(t), t ∈ [−h, 0],

(3.31)

Based on Lemma 2 (see Appendix A), we have

Proposition 7. For some given positive scalars µ and h, system (3.31) with (3.30)

is robustly stable if there exist matrices X > 0, Q > 0, R > 0, and Y1, Y2, M1, M2,

3.3. UNCERTAIN SYSTEMS 45

Z1, Z2, Z3 of appropriate dimension such that (3.14) and

Ξ11 Ξ12 F hXAT + hY T1 BT µX λL (EbY1 + EaX)T

∗ Ξ22 0 hY T2 BT 0 0 Y T

2 ETb

∗ ∗ −I hF T 0 0 0∗ ∗ ∗ −hR 0 λhL 0∗ ∗ ∗ ∗ −I 0 0∗ ∗ ∗ ∗ ∗ −λI 0∗ ∗ ∗ ∗ ∗ ∗ −λI

< 0, (3.32)

where Ξ11, Ξ12 and Ξ22 are defined in (3.11). Moreover, the controller gains are

given by K1 = Y1X−1 and K2 = Y2X

−1.

Proof. According to Proposition 3, if (3.11) and (3.14) holds, system (3.4) is asymp-

totically stable. Replacing A and B in (3.11) with A + LG(t)Ea and B + LG(t)Eb,

respectively, we find that (3.11) for system (3.31) with (3.30) is equivalent to the

following

Ξ + H1G(t)N1 + HT1 GT (t)NT

1 < 0 (3.33)

where

H1 = [LT 0 0 hLT 0]T

N1 = [EbY1 + EaX EbY2 0 0 0]

and Ξ is defined in (3.11). By Lemma 2 (see Appendix A), (3.33) holds for any G(t)

satisfying (3.30) if and only if there exists a positive number λ > 0 such that

Ξ + λH1HT1 + λ−1NT

1 N1 < 0. (3.34)

Applying the Schur complement to (3.34) yields (3.32). This completes the proof.

Similarly, the following result is straightforward from Proposition 6.

Proposition 8. For some given positive scalars µ and h, system (3.31) with (3.30)

is robustly stable if there exist matrices X > 0, Q1 > 0, Q2 > 0, Q3 > 0, R1 > 0,


R2 > 0 and Y1, Y2, M1, M2, Z1, Z2, Z3 of appropriate dimensions such that (3.27)

and

Ω11 Ω12 BY2 F ρ15 µX λL ρ18

∗ Ω22 −Q2 0 0 0 0 0∗ ∗ −Q3 0 h

2Y T

2 BT 0 0 Y T2 ET

b

∗ ∗ ∗ −I h2F T 0 0 0

∗ ∗ ∗ ∗ −h2R 0 h

2λL 0

∗ ∗ ∗ ∗ ∗ −I 0 0∗ ∗ ∗ ∗ ∗ ∗ −λI 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI

< 0, (3.35)

where

ρ15 =h

2(XAT + Y T

1 BT ),

ρ18 = (EbY1 + EaX)T .

and Ω11, Ω12, Ω22 are defined in (3.26). Moreover, the controller gains are given by

K1 = Y1X−1 and K2 = Y2X

−1.


0 10 20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5First floor − Uncontrolled System with Uncertainty

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.25: The displacement of the first floor when no control is used to the systemwith norm-bounded uncertainties and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−3

−2

−1

0

1

2

3Second floor − Uncontrolled System with Uncertainty

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.26: The displacement of the second floor when no control is used to thesystem with norm-bounded uncertainties and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−3

−2

−1

0

1

2

3Third floor − Uncontrolled System with Uncertainty

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.27: The displacement of the third floor when no control is used to thesystem with norm-bounded uncertainties and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First floor − Uncontrolled System with Uncertainty

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.28: The displacement of the first floor when no control is used to the systemwith norm-bounded uncertainties and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Second floor − Uncontrolled System with Uncertainty

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.29: The displacement of the second floor when no control is used to thesystem with norm-bounded uncertainties and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Third floor − Uncontrolled System with Uncertainty

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.30: The displacement of the third floor when no control is used to thesystem with norm-bounded uncertainties and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4First floor − Robust State Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Second floor − Robust State Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Third floor − Robust State Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First floor − Robust State Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Second floor − Robust State Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Third floor − Robust State Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4First floor − Improved Robust State Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Second floor − Improved Robust State Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Third floor − Improved Robust State Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



0 10 20 30 40 50 60 70 80 90 100−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First floor − Improved Robust State Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Second floor − Improved Robust State Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Third floor − Improved Robust State Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


Now, for the offshore structure system, suppose the system matrices are subject

to norm-bounded uncertainties. Then we will show, via numerical simulation, the

effectiveness of the proposed methods - Propositions 7 and 8. The system matrices

are assumed to be of the form (3.29) with

L = [0 0 0.001 0 0 0.01]T ,

Ea = [1 0 0 0 0 0],

Eb = 0.

The wave frequency ω is also separated to two cases: 1.8 rps and 0.5773 rps, and

h = 0.1, µ = 0.1.

Figures 3.25 - 3.30 plot the states of the uncertain system without control. It’s

easy to see that the oscillation amplitudes are different from those of the system

with no uncertainty, see Figures 3.1 - 3.6, especially for the responses for ω = 1.8

(see Figures 3.25 - 3.27). That is, the uncertainty brings much influence to internal

oscillations of the offshore structures, so it is of much significance in practice to take

the uncertainty into account.


By the Controller Design from Proposition 7 deriving from Propositions 7, the

controller gains can be obtained as

Ks4 :

K1 = [−0.2555 −0.1584 1.6036 0.1897 0.1050 −0.1216]× 104

K2 = [0.3045 0.0542 −1.4487 0.0115 0.0394 0.1280]× 103

(3.36)

Figures 3.31 - 3.36 show all the states of the uncertain system by the Controller

Design from Proposition 7. (The simulation diagram is shown in Appendix C.2.)

Comparing with Figures 3.25 - 3.30, we find that the responses of the platform are

controlled in reasonable scopes instead of devastatingly drastic oscillations. In fact,

it is clear from Figures 3.31 - 3.33 that the amplitudes of the platform for ω = 1.8

are about only 14% of those shown in Figures 3.25 - 3.27. For ω = 0.5773, from

Figures 3.34 - 3.36, the internal oscillations induced by wave and the uncertainty are

also reduced, the amplitudes of the displacement of three floors are decreased from

1.5 ft (in Figure 3.28) to 1.2 ft (in Figure 3.34), from 1.6 ft (in Figure 3.29) to 1.3

ft (in Figure 3.35) and from 1.8 ft (in Figure 3.30) to 1.4 ft (in Figure 3.36), peak

to peak, respectively. These figures attest the effectiveness of the design control

technique in reducing the uncontrolled responses.

On the other hand, the Controller Design from Proposition 8 deriving from

Propositions 8 yields the controller gains as

Ks5 :

K1 = [−1.6666 −0.4307 5.8123 2.0646 0.6845 −1.4603]× 103

K2 = [89.7419 5.3661 −703.9537 −11.9643 40.3774 83.3679]

(3.37)

The state responses of the system under study via a memory controller by the

Controller Design from Proposition 8 are shown in Figures 3.37 -3.40, from which, we

can see that the internal oscillations are almost the same as those by the Controller

Design from Proposition 7. However, it is clear that the gain Ks5 is much smaller

than Ks4, which means the Controller Design from Proposition 8 is more efficient

and economical than the Controller Design from Proposition 7 in the sense of energy


consumption. (The simulation diagram of the Controller Design from Proposition

8 is similar to that of the Controller Design from Proposition 7. It’s omitted.)

3.3.3 A Polytopic Uncertainty

For the polytopic uncertainty, (A(t), B(t)) in (3.1) is assumed to be bounded by a

given convex bounded polyhedral domain Σ, i.e.

(A(t), B(t)) ∈ Σ :=

q∑

j=1

ξj(Aj, Bj),

q∑j=1

ξj = 1, ξj ≥ 0

(3.38)

The closed-loop system of (3.1) with (3.38) under controller (3.3) becomes

x(t) = [A(t) + B(t)K1]x(t) + B(t)K2x(t− h) + Fg(x, t),

x(t) = φ(t), t ∈ [−h, 0],(3.39)

Following the ideas in [50, 51], we choose a parameter-dependent Lyapunov-

Krasovskii functional candidate as

V (xt) =

q∑j=1

[xT (t)ξjPjx(t) +

∫ t

t−h

xT (s)ξjQjx(s)ds

+

∫ 0

−h

dθ

∫ t

t+θ

xT (s)ξjRjx(s)ds]

4=

q∑j=1

Vj(xt). (3.40)

where xt = x(t + α), α ∈ [−h, 0] and Pj > 0, Qj > 0, Rj > 0 are to be determined.

Then we have

Proposition 9. The system (3.1) with (3.38) is robustly stable if there exist matrices

Pj > 0, Qj > 0, Rj > 0, M1j,M2j, Z1j, Z2j, Z3j and N of appropriate dimensions,

such that

Υj :=

Υ11j Υ12j Υ13j NT F∗ Υ22j NT BjK2 NT F∗ ∗ Υ33j 0∗ ∗ ∗ −I

< 0 (3.41)

and

Rj M1j M2j

∗ Z1j Z2j

∗ ∗ Z3j

≥ 0, (3.42)


hold for j = 1, 2, · · · , q, where

Υ11j = Qj + µ2I + MT1j + M1j + hZ1j + NT (Aj+BjK1) + (Aj+BjK1)

T N,

Υ12j = Pj −NT + (Aj + BjK1)T N,

Υ22j = hRj −NT −N,

Υ13j =−MT1j + M2j + hZ2j + NT BjK2,

Υ33j =−MT2j −M2j + hZ3j.

Proof. Taking the derivative of V (xt) with respect to time t along the trajectory of

system (3.39) yields

V (xt) =

q∑j=1

Vj(xt) (3.43)

where

Vj(xt) = 2xT (t)ξjPjx(t) + xT (t)ξjQjx(t)− xT (t− h)ξjQjx(t− h)

+ hxT (t)ξjRjx(t)−∫ t

t−h

xT (s)ξjRjx(s)ds (3.44)

By Lemma 3 (see Appendix A), for any matrices M1j,M2j and Z1j, Z2j, Z3j, if (3.42)

is true, then so is the following

−∫ t

t−h

xT (s)ξjRjx(s)ds

≤ ξj

[x(t)

x(t− h)

]T [MT

1j + M1j −MT1j + M2j

∗ −MT2j −M2j

] [x(t)

x(t− h)

]

+ hξj

[x(t)

x(t− h)

]T [Z1j Z2j

∗ Z3j

] [x(t)

x(t− h)

], (3.45)

Similar to [28], for any N with appropriate dimensions, it is clear from (3.39) that

the following is true

0 = 2xT (t)NT [−x(t) + (A(t) + B(t)K1)x(t) + B(t)K2x(t− h) + Fg(x, t)]

Noting that (3.38), we have

0 = 2

q∑j=1

ξjxT (t)NT [−x(t) + (Aj + BjK1)x(t) + BjK2x(t− h) + Fg(x, t)] (3.46)


In addition, from (3.2)

0 ≤ µ2xT (t)x(t)− gT (x, t)g(x, t) (3.47)

Substituting (3.44), (3.45), (3.46), (3.47) into (3.43) gives

V (xt) ≤q∑

j=1

ξjψT (t)Υjψ(t)

where Υj is defined in (3.41) and

ψT (t) = [ xT (t) xT (t) xT (t− h) gT (x, t) ]

Thus, if (3.41) and (3.42) hold for j = 1, 2, · · · , q, then there exists a δ > 0 such

that V (xt) ≤ −δ ‖x(t)‖2 < 0 for x(t) 6= 0, which guarantees that system (3.39) is

robustly stable for polytopic uncertainty (3.38). This completes the proof.

The following proposition provides a controller design method for system (3.39)

with polytopic uncertainty (3.38).

Proposition 10. The system (3.39) with (3.38) is robustly stable if there exist

matrices Pj > 0, Qj > 0, Rj > 0, Mij, Zkj, N and Yi of appropriate dimensions,

where i = 1, 2, j = 1, 2, ...q, k = 1, 2, 3, such that the following LMIs hold for

j = 1, 2, ...q :

Υ(j) :=

Υ11 Υ12 Υ13 F µNT

∗ Υ22 BjY2 F 0

∗ ∗ Υ33 0 0∗ ∗ ∗ −I 0∗ ∗ ∗ ∗ −I

< 0, (3.48)

Rj M1j M2j

∗ Z1j Z2j

∗ ∗ Z3j

≥ 0, (3.49)

where

Υ11 = Qj + MT1j + M1j + hZ1j


+AjN + BjY1 + (AjN + BjY1)T ,

Υ12 = Pj − N + (AjN + BjY1)T ,

Υ22 = hRj − N − NT ,

Υ13 = −MT1j + M2j + hZ2j + BjY2,

Υ33 = −MT2j − M2j + hZ3j.


−1.

Proof. From Proposition 9, if (3.41) and (3.42) hold, system (3.39) with (3.38) is

robustly stable. In addition, from (3.41), noting that N is invertible. Pre- and

post-multiplying both sides of matrix inequality (3.41) by

diag(

NT)−1

,(NT

)−1,

(NT

)−1, I

,

and its transpose, respectively, yields

Υ(j) =

Υ11 Υ12 Υ13 F∗ Υ22 BjK2N

−1 F∗ ∗ Υ33 0∗ ∗ ∗ −I

< 0 (3.50)

where

Υ11 =(NT

)−1QjN

−11 + µ2

(NT

)−1N−1 +

(NT

)−1MT

1jN−1

+(NT

)−1M1jN

−1 + h(NT

)−1Z1jN

−1

+ (Aj + BjK1)N−1 +

(NT

)−1(Aj + BjK1)

T ,

Υ12 =(NT

)−1PjN

−1 −N−1 +(NT

)−1(Aj + BjK1)

T ,

Υ22 = h(NT

)−1RjN

−1 −N−1 − (NT

)−1,

Υ13 = − (NT

)−1MT

1jN−1 +

(NT

)−1M2jN

−1

+ h(NT

)−1Z2jN

−1 + BjK2N−1,

Υ33 = − (NT

)−1MT

2jN−1 − (

NT)−1

M2jN−1 + h

(NT

)−1Z3jN

−1.

Meanwhile, pre- and post-multiplying both sides of matrix inequality (3.42) by

diag(

NT)−1

,(NT

)−1,

(NT

)−1

,


and its transpose, respectively, yields

(NT

)−1RjN

−1(NT

)−1M1jN

−1(NT

)−1M2jN

−1

∗ (NT

)−1Z1jN

−1(NT

)−1Z2jN

−1

∗ ∗ (NT

)−1Z3jN

−1

≥ 0.

Let

(NT

)−1QjN

−1 = Qj,(NT

)−1PjN

−1 = Pj,(NT

)−1RjN

−1 = Rj,(NT

1

)−1Z1jN

−1 = Z1j,(NT

)−1Z2jN

−1 = Z2j,(NT

)−1Z3jN

−1 = Z3j,(NT

)−1M1jN

−1 = M1j,(NT

)−1M2jN

−1 = M2j,N−1 = N , K1N

−1 = Y1,K2N

−1 = Y2.

Applying the Schur complement to (3.50), we obtain (3.49) and (3.48). This com-

pletes the proof.


0 10 20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0

1

2

3

4First floor − Uncontrolled System with Polytopic Type Uncertainty

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.43: The displacement of the first floor when no control is used to the systemwith polytopic uncertainties and ω = 1.8

Let’s consider the closed-loop system (3.39) with

A(t) = A + ∆A, B(t) = B, (3.51)


0 10 20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0

1

2

3

4Second floor − Uncontrolled System with Polytopic Type Uncertainty

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.44: The displacement of the second floor when no control is used to thesystem with polytopic uncertainties and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0

1

2

3

4Third floor − Uncontrolled System with Polytopic Type Uncertainty

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.45: The displacement of the third floor when no control is used to thesystem with polytopic uncertainties and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5First floor − Uncontrolled System with Polytopic Type Uncertainty

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.46: The displacement of the first floor when no control is used to the systemwith polytopic uncertainties and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5Second floor − Uncontrolled System with Polytopic Type Uncertainty

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.47: The displacement of the second floor when no control is used to thesystem with polytopic uncertainties and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5Third floor − Uncontrolled System with Polytopic Type Uncertainty

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.48: The displacement of the third floor when no control is used to thesystem with polytopic uncertainties and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4First floor − Robust State Feedback Control (Polytopic Type Uncertainties)

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.49: The displacement of the first floor via the controller (3.3) with (3.52)for ω = 1.8 (polytopic uncertainties)


0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4Second floor − Robust State Feedback Control (Polytopic Type Uncertainties)

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.50: The displacement of the second floor via the controller (3.3) with (3.52)for ω = 1.8 (polytopic uncertainties)

0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5Third floor − Robust State Feedback Control (Polytopic Type Uncertainties)

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.51: The displacement of the third floor via the controller (3.3) with (3.52)for ω = 1.8 (polytopic uncertainties)


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6


Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Second floor − Robust State Feedback Control (Polytopic Type Uncertainties)

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.53: The displacement of the second floor via the controller (3.3) with (3.52)for ω = 0.5773 (polytopic uncertainties)


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6


Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


where A,B defined in (2.6) and

∆A =

0 0 0 0 0 00 0 0 0 0 0δ1 0 0 0 0 00 0 0 0 0 00 δ2 0 0 0 00 0 0 0 0 0

with

δ1 ∈[ −0.09, 0.09

], δ2 ∈

[ −0.15, 0.15].

Clearly, (3.39) with (3.51) is a system with polytopic uncertainty, with four

vertices as (Aj, B) (j = 1, 2, 3, 4), where

A1 =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819−0.09 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 −0.15 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

;


A2 =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819−0.09 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 0.15 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

;

A3 =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819

0.09 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 −0.15 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

;

A4 =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819

0.09 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 0.15 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

;

As previously presented, two cases of the wave frequency are taken into consid-

eration as 1.8 rps and 0.5773 rps, and the value of µ is 0.1. In the following, we

compare the simulation result when u(t) = 0 with that via memory controller (3.3)

with h = 0.1.

Figures 3.43 - 3.48 show the states of the system with polytopic type uncertainty

when no control is added. For ω = 1.8, the displacements of the three floors are

between −4 ft and 4 ft (see Figures 3.43 - 3.45). Figures 3.46 - 3.48 show that the

amplitudes of the responses are about -2.6 ft peak to peak. Recalling Figures 3.1 -

3.6, we can see how the uncertainty badly degrades the performance of the offshore

structure system. However, the memory state feedback controller (3.3) based on

Proposition 10 can be used to effectively control this uncertain system. To see this,

the controller gains are given as

Ks6 :

K1 = [−0.2266 −0.0659 1.1847 0.2477 0.0463 −0.1982]× 104

K2 = [−18.5225 −12.7426 169.8950 16.7815 −6.8169 −19.5826]

(3.52)

3.4. STATE FEEDBACK H∞ CONTROL 69

The state responses of the controlled system are demonstrated in Figures 3.49 -

3.54. They are the displacements of the three floors for ω = 1.8 and ω = 0.5773 in

sequence. (The simulation diagram of the Controller Design from Proposition 10 is

similar to that of the Controller Design from Proposition 8 except the definitions

of fcn. See Appendix C.3 for their definitions.) From Figures 3.49 - 3.51, we find

that responses of these floors, for ω = 1.8, oscillate between -0.3 ft and 0.35 ft,

between -3.2 ft and 3.9 ft, and between -0.35 ft and 0.41 ft, respectively, which are

merely about 10% of the corresponding responses of the uncontrolled system (see

Figures 3.43 - 3.45). The amplitudes of the displacement for ω = 0.5773 shown

in Figures 3.52 - 3.54 are reduced to 1.2 ft, 1.3 ft and 1.4 ft peak to peak, which

obtain 40% decrease compared with Figures 3.46 - 3.48. Under the actions of the

designed controller, the maximum amplitudes of the three floors are reduce into an

acceptable scope successfully.

3.4 State Feedback H∞ Control

In this section, we focus on an H∞ control problem for the offshore steel jacket

platform. The system can be described as follows:

x(t) = A(t)x(t) + B(t)u(t) + Fg(x, t) + Bwω(x, t),

z(t) = Czx(t),

x(t) = φ(t), t ≤ 0

(3.53)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control, z(t) ∈ Rr is the con-

trolled output, w(t) ∈ Rl is the external disturbance, which belongs to L2[0, +∞),

g(x, t) ∈ Rp is the nonlinear self-excited hydrodynamic force vector, which is uni-

formly bounded and satisfies the following cone-bounding constraint:

‖g(x, t)‖ ≤ µ ‖x(t)‖ , (3.54)

where µ is a positive scalar; φ(t) denotes an initial condition; F, Bw, and Cz are

constant real matrices with appropriate dimensions, A(t) and B(t) are system ma-


trices, which may be time-varying, but are known to belong to a certain compact

set Ω, that is,

(A(t), B(t)) ∈ Rn×(n+m) ⊂ Ω

The goal of the section is to develop an H∞ controller

u(t) = K1x(t) + K2x(t− h), (3.55)

where K1 and K2 are constant matrices, which are to be determined, delay h is

assumed to be constant satisfying h > 0, such that

(i) the following closed-loop system

x(t) = (A(t) + B(t)K1)x(t) + B(t)K2x(t− h)

+ Fg(x, t) + Bwω(t),

z(t) = Czx(t),

x(t) = φ(t), t ≤ 0

(3.56)

with ω(t) = 0 is asymptotically stable; and

(ii) under the condition φ(t) = 0, the H∞ performance

‖z(t)‖2 ≤ γ ‖ω(t)‖2

of the closed-loop system (3.56) is guaranteed for all nonzero ω(t) ∈ L2[0, +∞) and

a prescribed γ > 0.

In what follows, first, we will consider an H∞ controller design problem for the

nominal case of system (3.56). Then, the robust H∞ state feedback control issue of

system (3.56) with uncertainty will be investigated.

3.4.1 An H∞ Control for Nominal Systems

In this subsection, we aim at designing an H∞ controller for the nominal system. In

this case, let A(t) ≡ A and B(t) ≡ B for all t ≥ 0. Then, the resulting closed-loop

system can be described as


x(t) = (A + BK1)x(t) + BK2x(t− h)


z(t) = Czx(t),

x(t) = φ(t), t ≤ 0

(3.57)

Choose a Lyapunov-Krasovskii functional candidate as


∫ t

t−h

xT (s)Qx(s)ds +

∫ 0

−h

dθ

∫ t

t+θ

xT (s)Rx(s)ds,

where P, Q, R ∈ Rn×n, P > 0, Q > 0 and R > 0.

Proposition 11. For some given positive scalars µ, γ and h, system (3.57) is

asymptotically stable and satisfies ‖z(t)‖2 ≤ γ ‖ω(t)‖2 for all nonzero ω(t) ∈ L2[0, +∞),

if there exist matrices P > 0, Q > 0, R > 0 and M1, M2, Z1, Z2, Z3 of appropriate

dimensions such that

z :=

z11 z12 PF PBw h(A + BK1)T

∗ z22 0 0 hKT2 BT

∗ ∗ −I 0 hF T

∗ ∗ ∗ −γ2I hBTw

∗ ∗ ∗ ∗ −hR−1

< 0, (3.58)

R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0, (3.59)

where

z11 = P (A + BK1) + (A + BK1)T P + Q

+µ2I + MT1 + M1 + hZ1 + CT

z Cz,

z12 = PBK2 −MT1 + M2 + hZ2,

z22 = −(1− d)Q−MT2 −M2 + hZ3.

Proof. Firstly, we consider the asymptotic stability of system (3.57) with ω(t) = 0.

When ω(t) = 0, system (3.57) is degraded to system (3.4). According to Proposition

1, if the linear matrix inequalities (3.5) and (3.6) hold, system (3.4) is asymptotically

stable. Therefore, if (3.5) and (3.6) hold, system (3.57) with ω(t) = 0 is asymptot-

ically stable. Clearly, (3.6) is the same as (3.59), and (3.58) implies (3.5) by the


Schur complement. Then, if (3.58) and (3.59) hold, system (3.57) with ω(t) = 0

is asymptotically stable. So, in the following, we only need to prove the H∞ per-

formance ‖z(t)‖2 ≤ γ ‖ω(t)‖2 is guaranteed for all nonzero ω(t) ∈ L2[0, +∞) under

zero initial condition. For this, taking the derivative of V (xt) with respect to time

t along the trajectory of system (3.57) yields

V (xt) =2xT (t)Px(t) + xT (t)Qx(t)− xT (t− h)Qx(t− h)

+ hxT (t)Rx(t)−∫ t

t−h

xT (s)Rx(s)ds (3.60)

Introducing a new vector

ψT (t) =[

xT (t) xT (t− h) gT (x, t) ωT (t)].

by using Lemma 3 (see Appendix A) and noting that (3.54), after simple manipu-

lation, we have

V (xt) + zT (t)z(t)− γ2ωT (t)ω(t) ≤ ψT (t)[Θ + hΓT RΓ

]ψ(t) (3.61)

where

Θ :=

z11 z12 PF PBw

∗ z22 0 0∗ ∗ −I 0∗ ∗ ∗ −γ2I

,

Γ :=[

A + BK1 BK2 F Bw

]

with z11, z12, z22 being defined in (3.58).

Obviously, if (3.58) is feasible, then applying the Schur complement gives Θ +

hΓT RΓ < 0, which leads to

V (xt) + zT (t)z(t)− γ2ωT (t)ω(t) ≤ 0. (3.62)

Integrating both sides of (3.62) from 0 to ∞ yields

∫ ∞

0

[zT (t)z(t)− γ2ωT (t)ω(t)

]dt ≤ V (xt)|t=0 − V (xt)|t=∞.


Under zero initial condition φ(t) = 0, one has V (xt)|t=0 = 0, thus

∫ ∞

0

[zT (t)z(t)− γ2ωT (t)ω(t)

]dt ≤ 0,

which means ‖z(t)‖2 ≤ γ ‖ω(t)‖2 . This completes the proof.

Proposition 11 provides a bounded real lemma for system (3.57), which can

ensure that the system is not only asymptotical stability but also of a prescribed

disturbance attenuation level γ. In order to solve out the controller gains K1 and

K2, similar to proof procedure of Propositions 2 and 3, we have


asymptotically stable and satisfies ‖z(t)‖2 ≤ γ ‖ω(t)‖2 for all nonzero ω(t) ∈ L2[0,∞),

if there exist matrices X > 0, Q > 0, R > 0 and Y, M1, M2, Z1, Z2, Z3 of appro-

priate dimensions such that LMIs

z11 z12 F Bw h(XAT + Y T1 BT ) µX XCT

z

∗ z22 0 0 hY T2 BT 0 0

∗ ∗ −I 0 hF T 0 0∗ ∗ ∗ −γ2I hBT

w 0 0∗ ∗ ∗ ∗ −hR 0 0∗ ∗ ∗ ∗ ∗ −I 0∗ ∗ ∗ ∗ ∗ ∗ −I

< 0 (3.63)

2X − R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0. (3.64)

where

z11 = XAT + Y T1 BT + AX + BY1 + Q + MT

1 + M1 + hZ1,

z12 = BY2 − MT1 + M2 + hZ2,

z22 = −Q− MT2 − M2 + hZ3.


−1.


0 10 20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5First floor − Uncontrolled System with Disturbance

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.55: The displacement of the first floor when no control is used to the systemunder disturbance and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−3

−2

−1

0

1

2

3Second floor − Uncontrolled System with Disturbance

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.56: The displacement of the second floor when no control is used to thesystem under disturbance and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−3

−2

−1

0

1

2

3Third floor − Uncontrolled System with Disturbance

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.57: The displacement of the third floor when no control is used to thesystem under disturbance and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1First floor − Uncontrolled System with Disturbance

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.58: The displacement of the first floor when no control is used to the systemunder disturbance and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Second floor − Uncontrolled System with Disturbance

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.59: The displacement of the second floor when no control is used to thesystem under disturbance and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Third floor − Uncontrolled System with Disturbance

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.60: The displacement of the third floor when no control is used to thesystem under disturbance and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

First floor − Hinf


Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Second floor − Hinf


Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.62: The displacement of the second floor via the controller (3.55) with(3.65) for ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Third floor − Hinf


Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

First floor − Hinf


Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Second floor − Hinf


Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Third floor − Hinf


Time(sec)

Thi

rd F

loor

Res

pons

e(ft)




Now, we will show the validity of the proposed method for the offshore structure sys-

tem when there exists an external disturbance. For simulation, the wave frequency

is still set to be 1.8 or 0.5773 rps. Let µ = 0.1, γ = 3 and

Bw =[

1 0 0.1 0 0 0]T

,

Cz =[

0 0.5 0 0 0 0]T

.

When u(t) = 0, i.e. no control is added to the offshore platform, in this case,

Figures 3.55 - 3.60 depict the response of all the states of the system. The displace-

ments for ω = 1.8 are drawn in Figures 3.55 - 3.57, respectively, and in Figures 3.57

- 3.58 for ω = 0.5773. It’s easy to see that the external disturbance significantly

influences the amplitudes of the system. However, under memory controller (3.55)

with h = 0.1, the system can be effectively controlled. In fact, by Proposition 12,

the controller gains are given by

Ks7 :

K1 = [−0.3770 −0.2423 4.6490 0.4048 0.0828 −0.1934]× 104

K2 = [273.8290 −30.0347 −812.6438 21.0510 15.0692 42.9539]× 103

(3.65)

The state responses of the controlled system are shown in Figures 3.61 - 3.66.

(The simulation diagram is described in Appendix C.4.) It is clear from Figures

3.61 - 3.63 that the amplitudes of displacement for ω = 1.8 are reduced to 0.4

ft, 0.45 ft and 0.5 ft peak to peak, which are only about 10% of the uncontrolled

responses of the system. From Figures 3.64 - 3.66, we find that the responses for for

ω = 0.5773 oscillate between -0.35 ft and 0.35 ft, between -0.38 ft and 0.4 ft, and

between -0.4 ft and 0.42 ft, respectively, which are decreased about 45%. Therefore,

the simulation results have shown that the obtained controller has well reduced the

internal oscillations of the offshore structure subject to the external disturbance.


3.4.3 An Improved Controller Design Scheme

As shown in the previous section, by employing the new LKF (3.22), we can obtain a

less conservative delay-dependent criterion. It is interesting that, based on this new

condition, the resulting controller can act on the offshore structure more efficiently.

In what follows, we also extend this LFK (3.22) to construct an H∞ controller for

the offshore structures.

By choosing the LFK (3.22), an improved delay-dependent stability criteria can

be formulated as follows.



if there exist matrices X > 0,

[Q1 Q2

QT2 Q3

]> 0, R > 0 and Y1, Y2, Mi, Zj (i = 1, 2,

j = 1, 2, 3) of appropriate dimensions such that

ℵ11 ℵ12 BY2 F Bwh2(XAT + Y T

1 BT ) µX XCTz

∗ ℵ22 −Q2 0 0 0 0 0∗ ∗ −Q3 0 0 h

2Y T

2 BT 0 0∗ ∗ ∗ −I 0 h

2F T 0 0

∗ ∗ ∗ ∗ −γ2I h2BT

w 0 0∗ ∗ ∗ ∗ ∗ −h

2R 0 0

∗ ∗ ∗ ∗ ∗ ∗ −I 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

< 0, (3.66)

XR−1X M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0, (3.67)

where

ℵ11 :=AX + BY1 + XAT + Y T1 BT + Q1 + MT

1 + M1 +h

2Z1,

ℵ12 :=Q2 − MT1 + M2 +

h

2Z2,

ℵ22 :=Q3 − Q1 − MT2 − M2 +

h

2Z3.

Proof. Since the proof is similar to that of Proposition 11, it is thus omitted.

Noting that

XR−1X ≥ 2X − R,


then a sufficient condition, which is in the form of LMIs, can be obtained in the

following.




[Q1 Q2

QT2 Q3

]> 0, R > 0 and Y1, Y2, Mi, Zj (i = 1, 2,

j = 1, 2, 3) of appropriate dimensions such that (3.66) and

2X − R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0.

Moreover, the controller gains are given by K1 = Y1X−1, and K2 = Y2X

−1.


0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

First floor − Improved Hinf


Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


Here, we compare the obtained performances of the offshore structure by two

control schemes: the Controller Design from Proposition 12 and the Controller

Design from Proposition 14. Under the same parameter values as the Controller

Design from Proposition 12, the achieved controller gains by the Controller Design


0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Second floor − Improved Hinf


Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Third floor − Improved Hinf


Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

First floor − Improved Hinf


Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Second floor − Improved Hinf


Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Third floor − Improved Hinf


Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


from Proposition 14 are given by

Ks8 :

K1 = [−0.8389 −0.3904 5.7361 0.2219 0.0080 −0.3967]× 104

K2 = [0.6931 0.0413 −2.8987 0.0654 0.0795 0.2371]× 103

(3.68)

The simulation results of the offshore structure by the Controller Design from

Proposition 14 (Ks8) are shown in Figures 3.67 - 3.72. Comparing Figures 3.67 -

3.72 with Figures 3.61 - 3.66, we find internal oscillations of the three floors by the

Controller Design from Proposition 14 are almost the same as the Controller Design

from Proposition 12. (The simulation diagram is similar to that of the Controller

Design from Proposition 12. So it’s omitted.)

However, we calculate the achieved minimum H∞ performance values γ by both

the Controller Design from Proposition 12 and the Controller Design from Proposi-

tion 14, and the obtained values γ for various h are listed in Table 3.1.

From the table, the Controller Design from Proposition 14 can achieve much less

minimum γ than the Controller Design from Proposition 12. Therefore, the Con-

troller Design from Proposition 14 not only reduces internal oscillations effectively,


Table 3.1: Comparison results about minimum of γ for various h.

h 0.1 0.2 0.3 0.4 0.5 0.6Proposition 12 0.56023 0.61417 0.66017 1.00800 - -Proposition 14 0.53918 0.56018 0.58746 0.61414 0.63837 0.66016

but also ensures much less H∞ performance value. That is, the Controller Design

from Proposition 14, indeed, is an improvement over the Controller Design from

Proposition 12.

3.4.5 An H∞ Control for Uncertain Systems

In this subsection, we consider a robust H∞ control problem. For two types of

uncertainties, namely norm-bounded uncertainty and polytopic uncertainty, robust

H∞ controllers are designed, which can not only ensure the asymptotical stability

and guarantee a prescribed H∞ level as well for the offshore structures.

3.4.6 A Norm-Bounded Uncertainty

For the norm-bounded uncertainty, it is assumed that

[A(t) B(t)] = [A B] + LG(t)[Ea Eb]

where the meanings of A,B,L, Ea, Eb and G(t) are the same as those in (3.29). The

closed-loop system is then given by

x(t) =[A+BK1+LG(t)(Ea+EbK1)]x(t)+[BK2+LG(t)EbK2]x(t− h)


z(t) =Czx(t),

x(t) =φ(t), t ∈ [−h, 0]

(3.69)

Based on Lemma 2 (see Appendix A) and following the proofs of Propositions

12 and 14, the results below are not difficult to obtain. The proofs str omitted.




if there exist λ > 0 and matrices X > 0, Q > 0, R > 0 and Y, M1, M2, Z1, Z2, Z3

of appropriate dimensions such that

z11 z12 F Bw σ1 µX XCTz λL σ2

∗ z22 0 0 h2Y T

2 BT 0 0 0 Y T2 ET

b

∗ ∗ −I 0 h2F T 0 0 0 0

∗ ∗ ∗ −γ2I h2BT

w 0 0 0 0∗ ∗ ∗ ∗ −h

2R 0 0 h

2λL 0

∗ ∗ ∗ ∗ ∗ −I 0 0 0∗ ∗ ∗ ∗ ∗ ∗ −I 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI

< 0, (3.70)

2X − R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0, (3.71)

where

σ1 := h(XAT + Y T1 BT ),

σ2 := (EbY1 + EaX)T ,

and z11, z12, z22 are defined in (3.63). Furthermore, the controller gains are given

by K1 = Y1X−1 and K2 = Y2X

−1.




[Q1 Q2

QT2 Q3

]> 0, R > 0 and Y1, Y2, Mi, Zj (i = 1, 2,

j = 1, 2, 3) of appropriate dimensions such that

ℵ11 ℵ12 BY2 F Bw σ3 µX XCTz λL σ4

∗ ℵ22 −Q2 0 0 0 0 0 0 0∗ ∗ −Q3 0 0 h

2Y T

2 BT 0 0 0 Y T2 ET

b

∗ ∗ ∗ −I 0 h2F T 0 0 0 0

∗ ∗ ∗ ∗ −γ2I h2BT

w 0 0 0 0∗ ∗ ∗ ∗ ∗ −h

2R 0 0 h

2λL 0

∗ ∗ ∗ ∗ ∗ ∗ −I 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI

< 0 (3.72)


2X − R M1 M2

∗ Z1 Z2

∗ ∗ Z3

≥ 0. (3.73)

where

σ3 := h(XAT + Y T1 BT ),

σ4 := (EbY1 + EaX)T ,

and ℵ11, ℵ12, ℵ22 are defined in (3.66). Moreover, the controller gains are given by

K1 = Y1X−1, and K2 = Y2X

−1.


0 10 20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5First floor − Uncontrolled Uncertain System under Disturbance

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.73: The displacement of the first floor of the uncontrolled system withnorm-bounded uncertainties under disturbance and ω = 1.8

Now, we show the validity of Propositions 15 and 16. Consider system (3.57)

with the parameters: µ = 0.1, γ = 3, h = 0.1 and

Bw =[

1 0 0.1 0 0 0]T

,

Cz =[

0 0.5 0 0 0 0],

L =[

0 0 0.001 0 0 0.01]T

,


0 10 20 30 40 50 60 70 80 90 100−3

−2

−1

0

1

2

3Second floor − Uncontrolled Uncertain System under Disturbance

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.74: The displacement of the second floor of the uncontrolled system withnorm-bounded uncertainties under disturbance and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−3

−2

−1

0

1

2

3Third floor − Uncontrolled Uncertain System under Disturbance

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.75: The displacement of the third floor of the uncontrolled system withnorm-bounded uncertainties under disturbance and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1First floor − Uncontrolled Uncertain System under Disturbance

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.76: The displacement of the first floor of the uncontrolled system withnorm-bounded uncertainties under disturbance and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Second floor − Uncontrolled Uncertain System under Disturbance

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.77: The displacement of the second floor of the uncontrolled system withnorm-bounded uncertainties under disturbance and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5Third floor − Uncontrolled Uncertain System under Disturbance

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.78: The displacement of the third floor of the uncontrolled system withnorm-bounded uncertainties under disturbance and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

First floor − Robust Hinf


Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Second floor − Robust Hinf


Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Third floor − Robust Hinf


Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8



Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8



Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8



Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


0 10 20 30 40 50 60 70 80 90 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

First floor − Improved Robust Hinf


Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Second floor − Improved Robust Hinf


Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Third floor − Improved Robust Hinf


Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

First floor − Improved Robust Hinf


Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Second floor − Improved Robust Hinf


Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Third floor − Improved Robust Hinf


Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


Ea =[

1 0 0 0 0 0];

The other parameter values can be found in Chapter 2.

When u(t) = 0, Figures 3.73 - 3.78 depict the responses of all the states of

the uncertain system for different wave frequencies. Comparing with Figures 3.1 -

3.6, which are for the nominal system, we find that the responses of the uncertain

system become more drastic. For example, the amplitudes of the displacement for

ω = 0.5773 has increased by nearly 13% (see Figure 3.73 - 3.75). So, it’s necessary

to take measures to cripple such passive effects and constrain the oscillations of the

states.

However, based on Propositions 15 and 16, the obtained controllers can effec-

tively control the uncertain system, which are shown in the following.

By solving LMIs (3.70) and (3.71), the Controller Design from Proposition 15

gives the controller gains as

Ks9 :

K1 = [−0.1941 −0.1424 1.9398 0.3026 0.1145 −0.0999]× 104

K2 = [0.2640 0.0649 −2.0823 −0.0230 0.0326 0.1235]× 103

(3.74)


The simulation diagram is shown in Appendix C.5. And the state responses of

system (3.57) under the Controller Design from Proposition 15 are shown in Figures

3.79 - 3.84. Comparing Figure 3.79 with Figure 3.73, we find that the curve, which

indicates the displacement of the first floor under control, oscillates in a range about

from -0.25 ft to 0.3 ft, not in the other range from -2.4 ft to 2.4 ft any longer when

no control is applied. Similarly sharp decrease occurs to the displacements of the

second and third floors (see Figures 3.80 and 3.81). Comparing with Figures 3.76

- 3.78, their oscillation amplitudes in Figures 3.82 - 3.84 are reduced from 1.7 ft to

1.2 ft, from 1.8 ft to 1.4 ft and from 1.9 ft to 1.5 ft peak to peak, respectively, which

obtain nearly 30% reduction. These results show that the servo under the Controller

Design from Proposition 15 motivates the AMD to vibrate in some scope, so that

the internal oscillations of the offshore structure can be restrained successfully.

On the other hand, by LMIs (3.72) and (3.73), the Controller Design from Propo-

sition 16 gives the controller gains as

Ks10 :

K1 = [−0.9159 −0.3360 6.4051 0.2540 0.0617 −0.3493]× 104

K2 = [0.8114 0.0912 −2.8218 0.0571 0.0593 0.2468]× 103

(3.75)

The state responses under the Controller Design from Proposition 16 are shown in

Figures 3.85 - 3.90. Comparing Figures 3.85 - 3.90 with Figures 3.79 - 3.84, we

find the internal oscillations of the offshore structure under the Controller Design

from Proposition 16 are slightly weaker than those under the Controller Design

from Proposition 15. Taking the displacement of the first floor for ω = 1.8 as an

example, the amplitude is 0.35 ft peak to peak under the Controller Design from

Proposition 16, but 0.55 ft under the Controller Design from Proposition 15. (The

simulation diagram of the Controller Design from Proposition 16 is similar to that

of the Controller Design from Proposition 15, so it’s omitted.)

As a byproduct, we compare the achieved H∞ performance by the Controller

Design from Proposition 15 with that by the Controller Design from Proposition 16.


For different values of h, the obtained minimum H∞ levels, γ, both by the Controller

Design from Proposition 15 and by the Controller Design from Proposition 16, are

listed in Table 3.2, from which, the Controller Design from Proposition 16 can

achieve better H∞ performance than the Controller Design from Proposition 15.

Table 3.2: The achieved minimum values of γ for various h

h 0.1 0.2 0.3 0.4 0.5 0.6Proposition 15 0.56067 0.61502 0.66156 1.01410 - -Proposition 16 0.53937 0.56061 0.58807 0.61505 0.63947 0.66150

3.4.8 A Polytopic Uncertainty

For the polytopic uncertainty, (A(t), B(t)) in (3.56) is assumed to be bounded by

a given convex bounded polyhedral domain Σ, i.e.

(A(t), B(t)) ∈ Σ :=

q∑

j=1

ξj(Aj, Bj),

q∑j=1

ξj = 1, ξj ≥ 0

(3.76)

System (3.53) with (3.76) under controller (3.55) becomes

x(t) = [A(t) + B(t)K1]x(t) + B(t)K2x(t− h) + Fg(x, t) + Bwω(t),

z(t) = Czx(t),

x(t) = φ(t), t ∈ [−h, 0]

(3.77)

Based on a parameter Lyapunov-Krasovskii functional approach, we have the

following conclusion. The proof is similar to that in Propositions 9 and 10.

Proposition 17. For some given positive scalars µ, γ and h, system (3.77) with

polytopic uncertainty (3.38) is asymptotically stable and satisfies ‖z(t)‖2 ≤ γ ‖ω(t)‖2

for all nonzero ω(t) ∈ L2[0,∞), if there exist matrices Pj > 0, Qj > 0, Rj > 0,


M1j, M2j, Z1j, Z2j, Z3j, N and Y1, Y2 of appropriate dimensions such that

Υ11 Υ12 Υ13 F Bw µNT NT CTz

∗ Υ22 BjY2 F Bw 0 0

∗ ∗ Υ33 0 0 0 0∗ ∗ ∗ −I 0 0 0∗ ∗ ∗ ∗ −γ2I 0 0∗ ∗ ∗ ∗ ∗ −I 0∗ ∗ ∗ ∗ ∗ ∗ −I

< 0 (3.78)

Rj M1j M2j

∗ Z1j Z2j

∗ ∗ Z3j

≥ 0 (3.79)

hold for j = 1, 2, · · · , q, where

Υ11 =Qj + MT1j + M1j + hZ1j + AjN + BjY1 + (AjN + BjY1)

T ,

Υ12 =Pj − N + ε(AjN + BjY1)T ,

Υ22 =hRj − εN − εNT ,

Υ13 =− MT1j + M2j + hZ2j + BjY2,

Υ33 =− MT2j − M2j + hZ3j.


−1.


Consider the closed-loop system (3.77) with the following parameter matrices

A(t) = A + ∆A, B(t) = B,

where A,B defined in (2.6) and

∆A =

0 0 0 0 0 00 0 0 0 0 0δ1 0 0 0 0 00 0 0 0 0 00 δ2 0 0 0 00 0 0 0 0 0

with

δ1 ∈[ −0.09, 0.09

], δ2 ∈

[ −0.15, 0.15].


0 10 20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0

1

2

3

4First floor − Uncontrolled Polytopic Type Uncertain System under Disturbace

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.91: The displacement of the first floor of the uncontrolled system withpolytopic type uncertainties under disturbance and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0

1

2

3

4Second floor − Uncontrolled Polytopic Type Uncertain System under Disturbace

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.92: The displacement of the second floor of the uncontrolled system withpolytopic type uncertainties under disturbance and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−4

−3

−2

−1

0

1

2

3

4Third floor − Uncontrolled Polytopic Type Uncertain System under Disturbace

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.93: The displacement of the third floor of the uncontrolled system withpolytopic type uncertainties under disturbance and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1First floor − Uncontrolled Polytopic Type Uncertain System under Disturbace

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 3.94: The displacement of the first floor of the uncontrolled system withpolytopic type uncertainties under disturbance and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Second floor − Uncontrolled Polytopic Type Uncertain System under Disturbace

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.95: The displacement of the second floor of the uncontrolled system withpolytopic type uncertainties under disturbance and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5Third floor − Uncontrolled Polytopic Type Uncertain System under Disturbace

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 3.96: The displacement of the third floor of the uncontrolled system withpolytopic type uncertainties under disturbance and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25


State Feedback Control (Polytopic Type)

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)


0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25



Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.98: The displacement of the second floor via the controller (3.55) with(3.80) for ω = 1.8 (polytopic uncertainties)


0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25



Time(sec)

Thi

rd F

loor

Res

pons

e(ft)


0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5



Time(sec)

Firs

t Flo

or R

espo

nse(

ft)



0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5



Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 3.101: The displacement of the second floor via the controller (3.55) with(3.80) for ω = 0.5773 (polytopic uncertainties)

0 10 20 30 40 50 60 70 80 90 100

−0.6

−0.4

−0.2

0

0.2

0.4

0.6



Time(sec)

Thi

rd F

loor

Res

pons

e(ft)



Clearly, (3.77) with (3.51) is a system with polytopic uncertainty, with four vertices

as (Aj, Bj) (j = 1, 2, 3, 4), where Bj = B and

A1 =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819−0.09 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 −0.15 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

;

A2 =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819−0.09 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 0.15 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

;

A3 =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819

0.09 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 −0.15 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

;

A4 =

0 1 0 0 0 0−3.3235 −0.0212 0.0184 0.0030 −5.3449 −0.8819

0.09 0 0 1 0 00.0184 0.0030 −118.1385 −0.1117 5.3468 0.8822

0 0.15 0 0 0 1−0.0114 −0.0019 0.0114 0.0019 −3.3051 −0.5454

.

Let h = 0.1, γ = 3, µ = 0.1, and

Bw =[

1 0 0.1 0 0 0]T

,

Cz =[

0 0.5 0 0 0 0].

When u(t) = 0, that is, no control is added into the system, Figures 3.91 -

3.96 demonstrate the responses of all the states of the system with polytopic type

uncertainty. Comparing with Figures 3.1 - 3.6, we can see the duplex influences

of ploytopic type uncertainty and disturbance to the system. For example, the

amplitude of the displacement of the first floor for ω = 1.8 has increased by nearly

40% (see Figure 3.91); and even the displacement of the first floor for ω = 0.5773


has also increased by about 12% (see Figure 3.94). Therefore, it’s necessary to

take measures to weaken the effect brought by uncertainty and disturbance, and to

reduce the internal oscillations induced by wave forces. For this aim, a robust H∞

state feedback controller based on Proposition 17 is proposed. The gains are given

by

Ks11 :

K1 = [−7.1284 −0.9221 8.8308 −0.1684 −0.6949 −2.7038]× 104

K2 = [−247.1548 −79.4067 534.3352 −6.2850 −29.7022 −116.5531]

(3.80)

The state responses of the system with polytopic uncertainties by controller

(3.55) with Ks11 are shown in Figures 3.97 - 3.102. (The simulation diagram of the

Controller Design from Proposition 17 is similar to that of the Controller Design

from Proposition 15 except the definitions of fcn. See Appendix C.6 for details.)

Compared with Figures 3.91 - 3.96, the oscillation amplitudes of the three floors

for ω = 1.8 and ω = 0.5773 decrease sharply. In fact, from Figures 3.91 - 3.93,

for ω = 1.8, the amplitudes of the three floors are only 6% of those in Figures

3.97 - 3.99). Also, from Figures 3.100 - 3.102, it’s easy to see that the oscillation

amplitudes of the displacements reduce from 1.7 ft to 1 ft, from 1.9 ft to 1.1 ft and

from 2 ft to 1.2 ft peak to peak, respectively. All these have illustrated that the

controller (3.3) with Ks11 can reduce the internal oscillations and ensure the safety

and comfort of the offshore structure with polytopic uncertainty.

3.5 Conclusion

In order to effectively reduce the internal system oscillations, when the system states

are adopted as feedback, a memory state feedback approach is, for the first time,

proposed in this chapter for an offshore steel jacket platform. Based on Lyapunov-

Krasovskii stability theory, some delay-dependent robust stability criteria for the

closed-loop system have been obtained, where two classes of uncertainties, namely,

norm-bounded uncertainty and polytopic uncertainty, are taken into account. Then,

3.5. CONCLUSION 109

these criteria are used to design the desired controllers incorporating with some ma-

trix techniques. Simulation results sufficiently show the effectiveness of the proposed

memory controllers on the offshore structures. On the other hand, by introducing

a new Lyapunov-Krasovskii functional, an improved delay-dependent stability con-

dition is obtained, which is less conservative than those in the literature. Finally,

robust H∞ control for the offshore system has also been investigated, and an H∞

controller has been designed, which guarantees the system is asymptotically stable

with a prescribed H∞ performance.

Chapter 4

Dynamic Output FeedbackControl

In Chapter 3, the state feedback control for the offshore steel jacket platform was

considered when the system states are adopted as feedback. However, if the system’s

outputs are taken as feedback, we can design a dynamic output feedback controller

to reduce the internal systems oscillations, which motivates the study in this chap-

ter. Based on Lyapunov-Krasovskii stability theory and a project theorem, some

sufficient conditions for the existence of dynamic output feedback controllers will be

obtained, which are in terms of the solutions of a set of linear matrix inequalities.

Moreover, a robust H∞ performance for the system with norm-bounded uncertainty

will also be considered. The simulation results will be given to illustrate the validity

of the proposed method.

4.1 System Description

Let’s reconsider the offshore steel jacket platform with uncertainties as follows:

x(t) = (A + ∆A)x(t) + Bu(t) + Fg(x, t)

y(t) = Cx(t)

x(t) = φ(t), t ≤ 0

(4.1)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control, y(t) ∈ Rq is the

measurement output, g(x, t) ∈ Rp is the nonlinear self-excited hydrodynamic force

112 CHAPTER 4. DYNAMIC OUTPUT FEEDBACK CONTROL

vector, which is uniformly bounded and satisfies the following cone-bounding con-

straint:

‖g(x, t)‖ ≤ µ ‖x(t)‖ , (4.2)

where µ is a positive scalar; A, B, C, and F are constant real matrices with appro-

priate dimensions, ∆A represents norm-bounded uncertainty of the following form

4A(t) := LG(t)Ea, (4.3)

where G(t) is an uncertainty matrix bounded by I, i.e.

GT (t)G(t) ≤ I, ∀t, (4.4)

Assume that the outputs of system (4.1) are adopted as feedback, under which

we will design a dynamic output feedback controller as

ξ(t) = AKξ(t) + BKy(t− h)

u(t) = CKξ(t) + DKy(t− h)(4.5)

where ξ(t) ∈ Rk is the state of the controller, AK , BK , CK and DK are matrices to

be determined and h is a constant time-delay, such that the closed-system combined

by (4.1) and (4.5) is asymptotically stable.

Introducing an augmented vector

ϕ(t) := [xT (t) ξT (t)]T

then the closed-loop system combined by (4.1) and (4.5) is given by

ϕ(t) = (A0 + ∆A0)ϕ(t) + A1Eϕ(t− h) + ET Fg(x, t) (4.6)

where

A0 :=

[A BCK

0 AK

], ∆A0 :=

[∆A 00 0

], A1 :=

[BDKCBKC

], E :=

[In×n

0n×k

]T

. (4.7)

Let

K :=

[DK CK

BK AK

]


and

A00 :=

[A 00 0

], B00 :=

[B 00 I

], C00 :=

[C0

], D00 =

[0 00 I

]. (4.8)

Then, the matrices A0, A1, ∆A0 can be restated as

A0 = A00 + B00KD00,

A1 = B00KC00,

∆A0 = ET LG(t)EaE. (4.9)

In the following, firstly, we will discuss the dynamic output feedback control

for the nominal system of (4.1), then the obtained results will be extended to suit

for the uncertain system (4.1). Finally, the H∞ control issue via dynamic output

feedback controller will be also investigated.

4.2 Nominal Systems

In this section, we will focus on the nominal system of (4.1), in this case, assume

∆A ≡ 0. The resulting closed-loop system of (4.6) is given by

ϕ(t) = A0ϕ(t) + A1Eϕ(t− h) + ET Fg(x, t) (4.10)


Proposition 18. For given positive scalars µ and h, system (4.10) is asymptotically

stable if there exist real matrices P > 0, Q > 0 and R > 0 of appropriate dimensions

such that

Θ :=

Θ11 PA1 + ET R PET F hAT0 ET R

∗ −Q−R 0 hAT1 ET R

∗ ∗ −I hF T R∗ ∗ ∗ −R

< 0 (4.11)

where

Θ11 := PA0 + AT0 P + ET (Q−R + µ2I)E.

Proof. Choose a Lyapunov functional candidate as

V (ϕt) = ϕT (t)Pϕ(t) +

∫ t

t−h

ϕT (s)ET QEϕ(s)ds


+ h

∫ 0

−h

ds

∫ t

t+s

ϕT (θ)ET REϕ(θ)dθ, (4.12)

where ϕt = x(t+α), α ∈ [−h, 0] and P > 0, Q > 0, R > 0 to be determined. Taking

the derivative of V (t, ϕt) with respect to t along the trajectory of (4.10) yields

V (ϕt) = 2ϕT (t)Pϕ(t)

+ ϕT (t)ET QEϕ(t)− ϕT (t− h)ET QEϕ(t− h)

+ h2ϕT (t)ET REϕ(t)− h

∫ t

t−h

ϕT (θ)ET REϕ(θ)dθ. (4.13)

By Lemma 1 (see Appendix A), one obtains

−h

∫ t

t−h

ϕT (θ)ET REϕ(θ)dθ

≤[

ϕ(t)Eϕ(t− h)

]T [ −ET RE ET R∗ −R

] [ϕ(t)

Eϕ(t− h)

]. (4.14)

Noting (4.2), we have

µ2ϕT (t)ET Eϕ(t)− gT (x, t)g(x, t) ≥ 0. (4.15)

Let

ηT (t) =[

ϕT (t) ϕT (t− h)ET gT (x, t)]. (4.16)

Combining (4.13) with (4.14) and (4.15), yields

V (t, xt) ≤ ηT (t)Φ + h2ΓT RΓ

η(t)

where

Φ :=

Θ11 PA1 + ET R PET F∗ −Q−R 0∗ ∗ −I

Γ :=[EA0 EA1 F

]

with Θ11 being defined in (4.11). If (4.11) is feasible, then Φ + h2ΓT RΓ < 0 by the

Schur complement. So there exists a scalar δ > 0 such that V (ϕt) ≤ −δ ‖ϕ(t)‖2 < 0

for ϕ(t) 6= 0, which guarantees the closed-loop system (4.10) is asymptotically stable.

This completes the proof.


Proposition 18 provides a delay-dependent stability criterion for the nominal

system (4.10), from which, however, the controller parameters cannot be directly

obtained. In order to solve out the controller parameters, we will follow the lines in

[30], and a parameterized controller design is presented in terms of a set of LMIs,

which is stated in the following.

4.2.2 Controller Design

To solve out the controller parameters, by substituting (4.9) into (4.11), after simple

algebraic manipulation, (4.11) can be rewritten as

Ω + ΣΠKΛT + ΛKT ΠT ΣT < 0 (4.17)

where

Ω :=

Ω11 ET R PET F hAT00E

T R∗ −Q−R 0 0∗ ∗ −I hF T R∗ ∗ ∗ −R

Σ := diagP, I, I, hR

Π :=[

BT00 0 0 BT

00ET

]T

Λ :=[

D00 C00 0 0]T

with

Ω11 := PA00 + AT00P + ET (Q−R + µ2I)E

From Lemma 4, inequality (4.17) is solvable for some K if and only if

Π⊥Σ−1ΩΣ−1(Π⊥)T < 0 (4.18)

Λ⊥Ω(Λ⊥)T < 0 (4.19)

where Π⊥ and Λ⊥ denote the orthogonal complements of Π and Λ, respectively. Let

W1 and [W T2 W T

3 ] be the orthogonal complements of CT and [BT BT ]T , respectively.

Then, Π⊥ and Λ⊥ can be chosen as

Π⊥ =

W T2 0 0 0 W T

3

0 0 I 0 00 0 0 I 0

T


Λ⊥ =

I 0 0 0 00 0 W T

1 0 00 0 0 I 00 0 0 0 I

T

. (4.20)

In order to simplify (4.18) and (4.19), P and P−1 can be partitioned as

P =

[Y NNT 4

], P−1 =

[X M

MT 4]

, (4.21)

where X, Y ∈ Rn×n, M , N ∈ Rn×k, 4 means irrelevant matrix, and positive definite

matrices X, Y satisfy [X II Y

]≥ 0. (4.22)

Substituting (4.8) and (4.21) into (4.18) and (4.19), then (4.18) is equivalent to

z :=

z11 W T

2 XR (W T2 + W T

3 )F∗ −Q−R 0∗ ∗ −I

< 0 (4.23)

where

z11 := W T2 [AX + XAT + X(Q−R + µ2I)X]W2

+ W T2 XAT W3 + W T

3 AXW2 − h−2W T3 R−1W3

and (4.19) is equivalent to

Ξ :=

Ξ11 RW1 Y F hAT R∗ W T

1 (−Q−R)W1 0 0∗ ∗ −I hF T R∗ ∗ ∗ −R

< 0 (4.24)

where

Ξ11 := Y A + AT Y + Q−R + µ2I

In summary, based on Proposition 18 and above analysis, an existence sufficient

condition of parameterized controllers of form (4.5) is obtained, which is stated as

Proposition 19. Let W1 and [W T2 W T

3 ]T be the orthogonal complements of CT and

[BT BT ]T , respectively. For given scalars µ > 0 and h > 0, the dynamic output


feedback control problem for system (4.10) is solvable if there exist real matrices

X > 0, Y > 0, Q > 0 and R > 0 of appropriate dimensions such that (4.22), (4.23)

and (4.24) are feasible.

Noting that Proposition 19 does not present the computation of the controller

itself, but the existence of parameterized controllers. Moreover, the obtained matrix

inequalities are not linear due to such terms as XQX inz11. In order to compute the

desired controller, first, we convert the non-convex feasibility problem of Proposition

19 into a nonlinear minimization problem subject to LMIs by employing a cone

complementary linearization algorithm in [4]. For this aim,

Pre- and post-multiplying both sides of z in (4.23) by T T := diagI, X, I, Iand its transpose, respectively, yields

T TzT =

z11 W T

2 XRX (W T2 + W T

3 )F∗ −XQX −XRX 0∗ ∗ −I

< 0 (4.25)

Introducing new matrix variables S > 0 and Z > 0 such that XRX ≥ S and

XQX ≥ Z, then since

[ −W T2 XRXW2 W T

2 XRX∗ −XRX

]

=

[W T

2

I

] [ −XRX XRXXRX −XRX

] [W2

I

]

=

[W T

2

I

] [I −I0 I

] [0 00 −XRX

] [I 0−I I

] [W2

I

]

≤[

W T2

I

] [I −I0 I

] [0 00 −S

] [I 0−I I

] [W2

I

]

=

[ −W T2 SW2 W T

2 S∗ −S

].

Therefore, if XRX ≥ S > 0 and XQX ≥ Z > 0, then (4.25) is implied byz11 W T

2 S (W T2 + W T

3 )F∗ −Z − S 0∗ ∗ −I

< 0 (4.26)

where

z11 :=W T2 [AX + XAT + X(Q + µ2I)X − S]W2


+ W T2 XAT W3 + W T

3 AXW2 − h−2W T3 R−1W3

By the Schur complement, (4.26) is equivalent to

z11 W T2 S (W T

2 + W T3 )F W T

2 X µW T2 X

∗ −Z − S 0 0 0∗ ∗ −I 0 0∗ ∗ ∗ −Q−1 0∗ ∗ ∗ ∗ −I

< 0 (4.27)

where

z11 :=W T2 [AX + XAT − S]W2 + W T

2 XAT W3

+ W T3 AXW2 − h−2W T

3 R−1W3

On the other hand, applying the Schur complement gives

XRX ≥ S ⇐⇒[

R X−1

X−1 S−1

]≥ 0 (4.28)

XQX ≥ Z ⇐⇒[

Q X−1

X−1 Z−1

]≥ 0 (4.29)

Let

R = R−1, X = X−1, S = S−1, Z = Z−1, Q = Q−1.

then a new sufficient condition for the existence of the dynamic output feedback

controller is derived as follows.

Proposition 20. Let W1 and [W T2 W T

3 ] be the orthogonal complements of CT and

[BT BT ]T , respectively. For given scalars µ > 0 and h > 0, the dynamic output

feedback control problem for system (4.10) is solvable if there exist real matrices

X > 0, Y > 0, Q > 0, R > 0, R > 0, X > 0, S > 0, S > 0, Z > 0, Z > 0 and

Q > 0 of appropriate dimensions such that (4.22), (4.24) and

z11 W T2 S (W T

2 + W T3 )F W T

2 X µW T2 X

∗ −Z − S 0 0 0∗ ∗ −I 0 0∗ ∗ ∗ −Q 0∗ ∗ ∗ ∗ −I

< 0 (4.30)


[R XX S

]≥ 0 (4.31)

[Q XX Z

]≥ 0 (4.32)

RR = I, XX = I, SS = I, ZZ = I, QQ = I (4.33)

where

z11 := W T2 [AX + XAT − S]W2 + W T

2 XAT W3 + W T3 AXW2 − h−2W T

3 RW3.

Clearly, matrix inequalities (4.22),(4.24), (4.30)-(4.32) are all linear on matrix

variables, while (4.33) are a set of equality constraints. Fortunately, these con-

straints can be converted into a nonlinear minimization problem subject to LMIs by

employing the Cone Complementary Problem (CCP) proposed in [4]. The key idea

of the CCP can be stated as: if the LMI

[S II S

]≥ 0

is feasible on the n × n matrix variables S > 0 and S > 0, then Tr(SS) ≥ n, and

Tr(SS) = n if and only if SS = I. According to the above CCP lines, the non-

convex feasibility problem formulated by (4.22),(4.24), (4.30)-(4.32) and (4.33) can

be considered as a minimization problem involving LMI conditions as

Nonlinear Minimization Problem Subject to LMIs

Minimize Tr(XX + RR + QQ + ZZ + SS)Subject to (4.22), (4.24), (4.30)-(4.32), and[

R II R

]≥ 0,

[X II X

]≥ 0,

[Q II Q

]≥ 0,

[Z II Z

]≥ 0,

[S II S

]≥ 0.

(4.34)

The iterative algorithm proposed in Algorithm 2.1 can be modified to solve the

above nonlinear minimization problem.

Now, we can present the computation of the desired controller. First, compute

some solutions by solving the above nonlinear minimization problem (4.34); second,


compute two full-column rank matrices M,N ∈ Rn×k such that

MNT = I −XY. (4.35)

Then, the unique solution P can be thus calculated from the following equation

[Y INT 0

]= P

[I X−1

0 MT

]. (4.36)

Finally, the controller parameter K =

[DK CK

BK AK

]can be easily obtained by solv-

ing LMI (4.11) with known P .


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First floor − Output Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 4.1: The displacement of the first floor when the output feedback controlleris used and ω = 1.8

In Chapter 3, we discussed the control problem of the offshore structure system.

When the system states are adopted as feedback, the system can be effectively con-

trolled by a memory state feedback controller. The internal oscillations are greatly

reduced to a suitable level. However, when the system outputs are considered as

feedback, we can exploit Proposition 20 to seek a dynamic output feedback con-

troller to reduce the internal oscillations of the system. Let h = 0.02 and µ = 0.1,


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Second floor − Output Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 4.2: The displacement of the second floor when the output feedback controlleris used and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Third floor − Output Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 4.3: The displacement of the third floor when the output feedback controlleris used and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First floor − Output Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 4.4: The displacement of the first floor when the output feedback controlleris used and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Second floor − Output Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 4.5: The displacement of the second floor when the output feedback controlleris used and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Third floor − Output Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 4.6: The displacement of the third floor when the output feedback controlleris used and ω = 0.5773

solving the nonlinear minimization problem (4.34) yields

X =

0.9739 −0.2793 0.0131 −0.3556 −0.1356 −0.4666−0.2793 6.3799 0.4450 0.5945 0.3318 2.13220.0131 0.4450 0.3279 −0.1595 0.0184 0.2631−0.3556 0.5945 −0.1595 37.1101 −0.2603 0.3984−0.1356 0.3318 0.0184 −0.2603 1.2820 −0.3217−0.4666 2.1322 0.2631 0.3984 −0.3217 2.0653

Y =

0.0002 −0.0000 −0.0000 0.0000 0.0159 0.0142−0.0000 0.0000 0.0000 −0.0000 −0.0056 −0.0013−0.0000 0.0000 0.0003 −0.0000 −0.0011 −0.00520.0000 −0.0000 −0.0000 0.0000 0.0002 0.00020.0159 −0.0056 −0.0011 0.0002 7.6689 0.87610.0142 −0.0013 −0.0052 0.0002 0.8761 2.6154

× 105

R =

34.3152 −2.1196 −2.4177 0.3390 7.6227 1.11032.1196 −10.8303 −3.4722 −0.4904 −1.7991 −3.4295−2.4177 −3.4722 19.9992 0.2347 −2.3994 −7.04510.3390 −0.4904 0.2347 0.3808 0.2910 −0.67807.6227 −1.7991 −2.3994 0.2910 43.4562 0.98181.1103 −3.4295 −7.0451 −0.6780 0.9818 26.8163


Q =

1.0733 −0.0616 −0.4189 −0.0096 0.3589 0.4931−0.0616 0.2663 −0.1686 −0.0206 −0.0827 −0.0228−0.4189 −0.1686 3.7791 0.0488 −0.2449 −0.6789−0.0096 −0.0206 0.0488 0.0278 −0.0049 −0.04170.3589 −0.0827 −0.2449 −0.0049 0.7836 0.36730.4931 −0.0228 −0.6789 −0.0417 0.3673 1.1980

Then, from (4.35) we obtain

M =

−0.1846 −0.0274 −0.4305 −0.2927 −0.3011 −0.77690.7583 −0.0636 −0.2661 −0.1514 0.5419 −0.18340.0862 −0.0212 −0.5942 −0.4188 −0.3304 0.59540.0320 −0.2172 0.5690 −0.7923 −0.0158 −0.01060.3395 0.8567 0.1906 −0.0770 −0.3235 −0.06210.5170 −0.4622 0.1753 0.2870 −0.6338 −0.0662

N =

−0.0464 0 0 0 0 00.0063 0.0054 0 0 0 00.0144 −0.0061 0.0001 0 0 0−0.0008 0.0001 −0.0000 0.0001 0 0−6.4494 −8.6021 0.0039 0.0203 −0.0456 0−7.5054 2.6504 −0.0100 −0.0043 0.0027 −0.0098

× 105

Finally, a desired output feedback controller is thus obtained as (4.5) with

AK =

−25.9976 2.3183 20.2489 −22.8328 −4.1473 4.973612.8569 −2.0863 −8.3747 9.4323 1.7165 −2.0621−34.1903 20.8718 −22.8694 35.8562 −4.5279 5.5525−34.4200 16.2495 −9.9625 16.9789 −5.1585 6.50804.4404 7.5349 −24.6161 34.1852 0.6353 0.252963.9004 −23.7410 −3.1931 −11.2750 7.7576 −17.4902

BK =

−0.0002 −0.0227 0.0014−0.0055 0.0097 −0.0019−1.0327 −1.1987 −0.0571−0.6641 −0.9038 −0.01820.2092 −0.5066 0.1348−6.4840 2.1548 −1.6073

CK =[ −4.0071 0.5979 3.1251 −3.5246 −0.6401 0.7672

]× 104

DK =[ −11.4847 −33.9770 −0.2274

].

See Appendix C.7 for the simulation diagram. Under the obtained output feedback

controller, the simulation results of the state response of the system are given in

Figures 4.1 - 4.6, from which, it is clear that the internal oscillations are effectively

controlled. For example, the amplitude of vibration of the third floor for ω = 1.8


has decreased by about 73% (compare Figure 4.3 with Figure 3.3). Figures 4.4 - 4.6

have plotted the states for ω = 0.5773. In a word, under the operation of design

control scheme, the responses of the platform have reduced greatly as compared

with the uncontrolled case.

4.3 Uncertain Systems

Now, we consider system (4.1) with norm-bounded uncertainty (4.3).

Based on Proposition 20, replacing A in (4.24) and (4.30) with (A + 4A), we

find that (4.24) and (4.30) for (4.1) are, respectively, equivalent to the following

conditions:

Ξ + Γ1G(t)Γ2 + ΓT2 GT (t)ΓT

1 < 0 (4.37)

and

z+ Γ3G(t)Γ4 + ΓT4 GT (t)ΓT

3 < 0. (4.38)

where

Γ1 : =[

LT Y 0 0 hLT R]T

,

Γ2 : =[

Ea 0 0 0],

Γ3 : =[

LT (W2 + W3) 0 0 0]T

,

Γ4 : =[

EaXW2 0 0 0].

Using Lemma 2 (see Appendix A), (4.37) and (4.38) hold for any G(t) satisfying

(4.4) if and only if there exist positive scalers λ > 0 and ε > 0 such that

Ξ + λΓ1ΓT1 + λ−1Γ2

T Γ2 < 0 (4.39)

and

z+ εΓ3ΓT3 + ε−1ΓT

4 Γ4 < 0. (4.40)


which, by the Schur complement, can be represented as

Ξ11 RW1 Y F hAT R Y L λETa

∗ W T1 (−Q−R)W1 0 0 0 0

∗ ∗ −I hF T R 0 0∗ ∗ ∗ −R hRL 0∗ ∗ ∗ ∗ −λI 0∗ ∗ ∗ ∗ ∗ −λI

< 0, (4.41)

z11 W T2 S α13 W T

2 X µW T2 X α16 W T

2 XETa

∗ −Z − S 0 0 0 0 0∗ ∗ −I 0 0 0 0∗ ∗ ∗ −Q 0 0 0∗ ∗ ∗ ∗ −I 0 0∗ ∗ ∗ ∗ ∗ −εI 0∗ ∗ ∗ ∗ ∗ ∗ −εI

< 0 (4.42)

where α13 = (W T2 + W T

3 )F, α16 = ε(W T2 + W T

3 )L, and Ξ11, z11 are defined in (4.24)

and (4.30), respectively.

Therefore, nonlinear minimization problem (4.34) can be modified to design a

robust controller for system (4.1), which is stated as

Minimize Tr(XX + RR + QQ + ZZ + SS)Subject to (4.22), (4.31), (4.32), (4.41), (4.42), and[

R II R

]≥ 0,

[X II X

]≥ 0,

[Q II Q

]≥ 0,

[Z II Z

]≥ 0,

[S II S

]≥ 0.

(4.43)


For comparison, we set h = 0.02, µ = 0.1 and

L =[

0 0 0.001 0 0 0.01]T

,

Ea =[

1 0 0 0 0 0].

Comparing Figures 3.1 - 3.6 with Figures 3.25 - 3.30, we find that the states of the

system with uncertainty become more oscillatory than that of the system with no

uncertainty, that is, uncertainty indeed degrades the system performance. In the

following, assume that the outputs are adopted as feedback, we will design a robust


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First floor − Robust Output Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 4.7: The displacement of the first floor when the robust output feedbackcontroller is used and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Second floor − Robust Output Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 4.8: The displacement of the second floor when the robust output feedbackcontroller is used and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Third floor − Robust Output Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 4.9: The displacement of the third floor when the robust output feedbackcontroller is used and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8First floor − Robust Output Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 4.10: The displacement of the first floor when the robust output feedbackcontroller is used and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Second floor − Robust Output Feedback Control

Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 4.11: The displacement of the second floor when the robust output feedbackcontroller is used and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Third floor − Robust Output Feedback Control

Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 4.12: The displacement of the third floor when the robust output feedbackcontroller is used and ω = 0.5773


dynamic output feedback controller to reduce the oscillations. Solve the nonlinear

minimization problem (4.43), and also from (4.35), we have

X =

0.9819 −0.2382 0.0238 −0.3716 −0.1554 −0.4042−0.2382 6.5575 0.4362 1.4347 0.3511 2.15160.0238 0.4362 0.3181 −0.1429 0.0246 0.2561−0.3716 1.4347 −0.1429 35.2285 −0.2669 0.8984−0.1554 0.3511 0.0246 −0.2669 1.1753 −0.2491−0.4042 2.1516 0.2561 0.8984 −0.2491 1.9786

Y =

0.0002 −0.0000 −0.0000 0.0000 0.0071 0.0065−0.0000 0.0000 0.0000 −0.0000 −0.0026 −0.0005−0.0000 0.0000 0.0004 −0.0000 −0.0002 −0.00260.0000 −0.0000 −0.0000 0.0000 0.0001 0.00010.0071 −0.0026 −0.0002 0.0001 4.0692 0.26700.0065 −0.0005 −0.0026 0.0001 0.2670 1.3051

× 106

Q =

1.0335 −0.0693 −0.4557 −0.0160 0.3563 0.4699−0.0693 0.2377 −0.1407 −0.0244 −0.0960 −0.0379−0.4557 −0.1407 4.0913 0.0655 −0.2813 −0.7205−0.0160 −0.0244 0.0655 0.0388 −0.0088 −0.06090.3563 −0.0960 −0.2813 −0.0088 0.8355 0.37450.4699 −0.0379 −0.7205 −0.0609 0.3745 1.2169

R =

72.1236 −4.2052 −6.7008 0.7737 12.4381 1.1549−4.2052 22.4673 −4.9218 −1.5549 −2.8950 −7.4955−6.7008 −4.9218 40.1781 0.4039 −6.5749 −11.35530.7737 −1.5549 0.4039 1.4329 0.5770 −2.073212.4381 −2.8950 −6.5749 0.5770 87.3544 1.23881.1549 −7.4955 −11.3553 −2.0732 1.2388 53.6202

M =

−0.1665 −0.0607 −0.3505 −0.3426 −0.2964 −0.80030.7573 −0.0185 −0.2809 −0.0707 0.5477 −0.20570.0852 −0.0124 −0.5787 −0.5200 −0.2718 0.55990.1822 −0.2983 0.6476 −0.6769 −0.0181 −0.00250.3097 0.8618 0.2054 −0.0910 −0.3277 −0.05940.5123 −0.4051 0.0432 0.3752 −0.6562 −0.0124

N =

−0.0215 0 0 0 0 00.0025 0.0025 0 0 0 00.0073 −0.0032 0.0002 0 0 0−0.0004 0.0001 −0.0001 0.0001 0 0−2.6908 −4.5215 0.0007 0.0157 −0.0532 0−3.7994 1.3809 −0.0068 −0.0084 0.0032 −0.0117

× 106

4.4. OUTPUT FEEDBACK H∞ CONTROL 131

Then, the dynamic output feedback controller is given as

AK =

−47.6189 −2.6575 50.7682 −46.8550 −8.5812 11.055821.1351 −0.0906 −19.8367 18.2366 3.4111 −4.3565−60.4355 20.3676 −0.3215 3.6266 −7.3594 9.7559−91.0058 11.1410 52.5115 −48.5904 −14.3507 18.62284.2809 9.5237 −34.0269 38.1094 −1.1880 −1.6500

135.8845 −14.4186 −85.6415 65.9191 22.4684 −37.1408

,

BK =

0.0245 0.0163 0.0106−0.0271 −0.0003 −0.0060−1.3555 −1.5639 0.0810−1.1833 −1.4937 0.07430.5561 −0.9351 0.0981−12.6807 3.7234 −2.3869

,

CK =[ −7.4312 −0.1929 7.9140 −7.3066 −1.3380 1.7214

]× 104,

DK =[ −13.7838 30.8852 7.8082

]

By the dynamic output feedback controller, the displacement responses of the

three floors for ω = 1.8 are depicted in Figures 4.7 - 4.9. (See the simulation diagram

in Appendix C.8.) It’s easy to see that the amplitudes of displacement reduce by

about 70% (compare with Figures 3.25 and 3.27). Moreover, as shown in Figures

4.10 - 4.12, the amplitudes of vibration for ω = 0.5773 are 15% cut off (compare

with Figures 3.28 and 3.30). Finally, we find that the controller can process the

internal oscillations of offshore steel jacket platform positively.

4.4 Output Feedback H∞ Control

In this section, we consider an H∞ performance of system (4.1). The system is

restated as

x(t) = (A + ∆A)x(t) + Bu(t) + Fg(x, t) + Bwω(x, t)

y(t) = Cx(t)

z(t) = Czx(t)

x(t) = φ(t), t ≤ 0

(4.44)

where z(t) ∈ Rt is the controlled output, w(t) ∈ Rl is the external disturbance,

which belongs to L2[0, +∞), Bw, and Cz are constant real matrices with appropriate


dimensions. Introducing a augmented vector

ϕ(t) := [xT (t) ξT (t)]T

then the closed-loop system combined by (4.44) and (4.5) is given by

ϕ(t) = (A0 + ∆A0)ϕ(t) + A1Eϕ(t− h)

+ ET Fg(x, t) + ET Bwω(x, t)

z(t) = CzEϕ(t)

(4.45)

where ∆A0, A0, A1, E are defined in (4.7).

The aim below is to seek an H∞ output feedback controller (4.5) such that

(1) The closed-loop system (4.45) is asymptotically stable for any admissible un-

certainty satisfying (4.3); and

(2) Under zero initial condition, ‖z(t)‖2 ≤ γ ‖ω(t)‖2 is guaranteed for all nonzero

ω(t) ∈ L2[0, +∞), where γ > 0 is a prescribed scalar.

For this purpose, firstly, we discuss the H∞ control for the normal case of system

(4.45), and the obtained results are then extended to the uncertain case. Suppose

∆A = 0, then the normal system can be expressed as

ϕ(t) = A0ϕ(t) + A1Eϕ(t− h(t)) + ET Fg(x, t)

+ ET Bwω(t),

z(t) = CzEϕ(t).

(4.46)

Now, choose a Lyapunov-Krasovskii functional candidate as

V (xt) = ϕT (t)Pϕ(t) +

∫ t

t−h

ϕT (s)ET QEϕ(s)ds

+ h

∫ 0

−h

ds

∫ t

t+s

ϕT (θ)ET REϕ(θ)dθ (4.47)

where P > 0, Q > 0, R > 0, then we have




if there exist real matrices P > 0, Q > 0 and R > 0 of appropriate dimensions such

that

Ψ =

Ψ11 PA1 + ET R PET D PET Bw hAT0 ET

∗ −Q−R 0 0 hAT1 ET

∗ ∗ −I 0 hF T

∗ ∗ ∗ −γ2I hBTw

∗ ∗ ∗ ∗ −R−1

< 0 (4.48)

where

Ψ11 = AT0 P + PA0 + ET (Q−R)E + µ2ET E + ET CT

z CzE.

Proof. From Proposition 18, if (4.11) holds, (4.10) is asymptotically stable. By the

Schur complement, (4.48) implies (4.11). Therefore, system (4.46) is asymptotically

stable if (4.48) holds. In the sequel, we only prove that ‖z(t)‖2 ≤ γ ‖ω(t)‖2 is

guaranteed for all nonzero ω(t) ∈ L2[0,∞) if (4.48) is feasible. To see this, taking

the time derivative of V (xt) in (4.47) along the trajectory of (4.46) yields

V (xt) = 2ϕT (t)Pϕ(t) + ϕT (t)ET QEϕ(t)− ϕT (t− h)ET QEϕ(t− h)

+ h2ϕT (t)ET REϕ(t)− h

∫ t

t−h

ϕT (θ)ET REϕ(θ)dθ. (4.49)

Define

ηT (t) =[

ϕT (t) ϕT (t− h)ET gT (x, t) ωT (t)].

By applying Lemma 1 (see Appendix A), we have

V (xt)− γ2ωT (t)ω(t) + zT (t)z(t) ≤ ηT (t)[Ψ + h2ΓT RΓ]η(t), (4.50)

where

Ψ :=

Ψ11 PA1 + ET R PET D PET Bw

∗ −Q−R 0 0∗ ∗ −I 0∗ ∗ ∗ −γ2I

Γ := [EA0 EA1 F Bw]

If matrix inequality (4.48) holds, then Ψ + h2ΓT RΓ < 0 by the Schur complement,

thus

V (xt)− γ2ωT (t)ω(t) + zT (t)z(t) < 0. (4.51)


Integrating both sides of (4.51) from 0 to ∞ yields

∫ ∞

0

[zT (s)z(s)− γ2ωT (s)ω(s)]ds < V (xt)|t=0 − V (xt)|t=∞. (4.52)

Under the zero initial condition, we have

∫ ∞

0

[zT (s)z(s)− γ2ωT (s)ω(s)]ds < 0. (4.53)

That is, ‖z(t)‖2 ≤ γ ‖ω(t)‖2 . This completes the proof.

For the uncertain system (4.45), based on Proposition 21, we also have



if there exist λ > 0, real matrices P > 0, Q > 0, and R > 0 of appropriate dimen-

sions such that

Ψ11 PA1 + ET R PET D PET Bw hAT0 ET PET L λET ET

a

∗ −Q−R 0 0 hAT1 ET 0 0

∗ ∗ −I 0 hF T 0 0∗ ∗ ∗ −γ2I hBT

w 0 0∗ ∗ ∗ ∗ −R−1 L 0∗ ∗ ∗ ∗ ∗ −λI 0∗ ∗ ∗ ∗ ∗ ∗ −λI

< 0 (4.54)

where Ψ11 is defined in (21).

Similar to Proposition 20, the H∞ controller design based on Propositions 21

and 22 can be converted into nonlinear minimization problems. For instance, robust

H∞ output feedback controller can be obtained by solving the following nonlinear

minimization problem as

Minimize Tr(XX + RR + QQ + ZZ + SS)Subject to (4.22), (4.31), (4.32), Σ1 < 0, Σ2 < 0, and[

R II R

]≥ 0,

[X II X

]≥ 0,

[Q II Q

]≥ 0,

[Z II Z

]≥ 0,

[S II S

]≥ 0, ε > 0, λ > 0.

(4.55)


where

Σ1 :=

ρ1 RW1 Y F Y Bw hAT R Y L λETa

∗ W T1 (−Q−R)W1 0 0 0 0 0

∗ ∗ −I 0 hF T R 0 0∗ ∗ ∗ −γ2I hBT

wR 0 0∗ ∗ ∗ ∗ −R L 0∗ ∗ ∗ ∗ ∗ −λI 0∗ ∗ ∗ ∗ ∗ ∗ −λI

Σ2 :=

ρ2 W T2 S ρ3 ρ4 ρ5 W T

2 XETa W T

2 X µW T2 X W T

2 CZ

∗ −Z − S 0 0 0 0 0 0 0∗ ∗ −I 0 0 0 0 0 0∗ ∗ ∗ −γ2I 0 0 0 0 0∗ ∗ ∗ ∗ −εI 0 0 0 0∗ ∗ ∗ ∗ ∗ −εI 0 0 0∗ ∗ ∗ ∗ ∗ ∗ −Q 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

with

ρ1 := Y A + AT Y + Q−R + µ2I,

ρ2 := W T2 [AX + XAT − S]W2 + W T

2 XAT W3 + W T3 AXW2 − h−2W T

3 RW3,

ρ3 := (W T2 + W T

3 )F,

ρ4 := (W T2 + W T

3 )Bw,

ρ5 := ε(W T2 + W T

3 )L.

The controller parameters can be obtained similar to that proposed in the pre-

vious section and thus it is omitted.


To show the validity of the proposed approach, let h = 0.02, µ = 0.1, γ = 2, and

Bw =[

1 0 0.1 0 0 0]T

,

Cz =[

0 0.5 0 0 0 0],

L =[

0 0 0.001 0 0 0.01]T

,

Ea =[

1 0 0 0 0 0].


0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5


Output Feedback Control

Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 4.13: The displacement of the first floor when the robust H∞ output feedbackcontroller is used and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5



Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 4.14: The displacement of the second floor when the robust H∞ outputfeedback controller is used and ω = 1.8


0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5



Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 4.15: The displacement of the third floor when the robust H∞ output feed-back controller is used and ω = 1.8

0 10 20 30 40 50 60 70 80 90 100

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5



Time(sec)

Firs

t Flo

or R

espo

nse(

ft)

Figure 4.16: The displacement of the first floor when the robust H∞ output feedbackcontroller is used and ω = 0.5773


0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8



Time(sec)

Sec

ond

Flo

or R

espo

nse(

ft)

Figure 4.17: The displacement of the second floor when the robust H∞ outputfeedback controller is used and ω = 0.5773

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8



Time(sec)

Thi

rd F

loor

Res

pons

e(ft)

Figure 4.18: The displacement of the third floor when the robust H∞ output feed-back controller is used and ω = 0.5773


Figures 3.73 - 3.78 have showed the responses of all the states of the uncertain system

under the external disturbance (no control). In order to weaken the passive effects of

the uncertainty and the external disturbance, and constrain the internal oscillations

inducd by the wave forces, we design a robust H∞ output feedback controller under

the assumption that the outputs are adopted as feedback. By solving the nonlinear

minimization problem (4.55), and we obtain the following values:

X =

0.8920 −0.2880 0.0132 −0.3571 −0.0960 −0.6045−0.2880 6.4149 0.4542 0.0103 0.4285 2.35980.0132 0.4542 0.3101 −0.1612 0.0227 0.2586−0.3571 0.0103 −0.1612 35.1609 −0.2893 0.1598−0.0960 0.4285 0.0227 −0.2893 1.3604 −0.3514−0.6045 2.3598 0.2586 0.1598 −0.3514 2.3298

Y =

0.0001 −0.0000 −0.0000 0.0000 0.0049 0.0047−0.0000 0.0000 −0.0000 −0.0000 −0.0019 −0.0001−0.0000 −0.0000 0.0004 −0.0000 −0.0002 −0.00220.0000 −0.0000 −0.0000 0.0000 0.0001 0.00010.0049 −0.0019 −0.0002 0.0001 3.1825 0.22320.0047 −0.0001 −0.0022 0.0001 0.2232 1.0495

× 105

Q =

1.4201 −0.1067 −0.6063 −0.0118 0.5036 0.8241−0.1067 0.2436 −0.1599 −0.0211 −0.1548 −0.1000−0.6063 −0.1599 4.2173 0.0508 −0.2548 −0.7412−0.0118 −0.0211 0.0508 0.0348 0.0003 −0.04300.5036 −0.1548 −0.2548 0.0003 0.8050 0.44810.8241 −0.1000 −0.7412 −0.0430 0.4481 1.3537

R =

95.5147 −5.7053 −10.1872 0.7876 20.6250 5.8309−5.7053 23.5080 −5.2679 −1.3370 −4.2067 −7.4061−10.1872 −5.2679 39.7708 0.2020 −5.9512 −11.01280.7876 −1.3370 0.2020 1.3087 0.7261 −1.557920.6250 −4.2067 −5.9512 0.7261 88.8340 0.90665.8309 −7.4061 −11.0128 −1.5579 0.9066 52.7852

M =

−0.1896 0.0095 −0.3789 −0.2279 0.1773 0.85850.7648 −0.0119 −0.2059 −0.1664 −0.5674 0.15120.0770 −0.0151 −0.6787 −0.4336 0.3522 −0.4702−0.0382 −0.1938 0.5269 −0.8266 0.0054 0.00580.2963 0.8609 0.2071 −0.0809 0.3445 0.05470.5329 −0.4700 0.1814 0.2062 0.6354 0.1264


N =

−0.0171 0 0 0 0 00.0014 0.0025 0 0 0 00.0067 −0.0029 0.0001 0 0 0−0.0004 0.0001 −0.0001 0.0001 0 0−2.4977 −4.0859 0.0044 0.0075 0.0225 0−3.3454 1.2286 −0.0016 −0.0065 −0.0032 0.0048

× 105

and the H∞ controller is given by

AK =

−55.1086 4.5678 31.6434 −39.9572 8.6959 −10.695925.0105 −2.9557 −13.0193 16.3788 −3.6237 4.4169−71.1069 25.0372 −6.3232 18.0134 10.9393 −13.0617−61.1169 13.9139 13.1385 −14.4612 9.7647 −11.756029.4671 −11.3510 13.1744 −30.4525 −9.7332 3.7473−176.8769 8.5685 96.8820 −84.4714 37.1686 −62.9122

BK =

0.0697 0.0159 0.0205−0.0337 −0.0023 −0.0082−2.3202 −1.9839 −0.1448−1.3075 −1.2734 −0.08151.9676 1.3790 0.473444.3579 −7.3372 9.3754

CK =[ −9.4697 1.0410 5.4371 −6.8718 1.4942 −1.8343

]× 104

DK =[

25.1348 37.7031 15.4304]

See the simulation diagram in Appendix C.9. Figures 4.13 - 4.18 show the state

responses of the controlled system. From Figures 4.13 - 4.15, for ω = 1.8, the

displacements have decreased by about 80% (See Figures 3.73 - 3.75). Compared

Figures 4.16 - 4.18 with Figures 3.76 - 3.78, the amplitudes of vibration for ω =

0.5773 achieve about 35% decrease. All these have shown that the proposed H∞

controller can effectively control the offshore structure system.

4.5 Conclusion

The design problem of dynamic output feedback control for the offshore steel jacket

platform under wave excitations has been studied. Based on Lyapunov-Krasovskii

stability theory and a project theorem, some sufficient conditions for the existence

of dynamic output feedback controllers has been obtained in terms of a set of linear

4.5. CONCLUSION 141

matrix inequalities. Furthermore, robust H∞ performance for the system with norm-

bounded uncertainties has also been investigated. Finally, the simulation results has

indicated that the proposed control schemes are capable of significantly reducing the

internal oscillations of the offshore steel jacket platform.

Chapter 5

Conclusion and Future Work

Conclusion

The control problem of offshore steel jacket platforms has been studied in this

thesis. In order to ensure the safety and comfort of the offshore structure, a mem-

ory control strategy has been proposed for the first time to reduce the internal

oscillations.

In Chapter 3, when the system states are adopted as feedback, we have designed

a memory state feedback controller. By using Lyapunov-Krasovskii functional the-

ory, some delay-dependent stability criteria have been first established, based on

which, and by combining with some linearization techniques, two controller design

approaches have then been given, which only depend on the solutions of a set of

LMIs. The simulation results have shown that such a controller can effectively reduce

the internal oscillations of the offshore structure subject to nonlinear wave-induced

forces.

In Chapter 4, when the system outputs are considered as feedback, a memory

dynamic output feedback controller has been addressed. By employing a projection

theorem and a cone complementary linearization approach, a memory dynamic out-

put feedback controller has been derived by solving a nonlinear minimization subject

to a set of LMIs. The obtained controller has been applied to an offshore structure

subject to nonlinear wave-induced forces, and the internal oscillations have been

well attenuated.

144 CHAPTER 5. CONCLUSION AND FUTURE WORK

In addition, an H∞ control for the offshore structure has been investigated for

the first time. By employing Lyapunov-Krasovskii theory, some delay-dependent

bounded real lemmas have been obtained, under which, via a memory H∞ controller,

the resulting closed-loop system is not only asymptotically stable but also with a

prescribed disturbance attenuation level. The simulation results have shown the

validity of the proposed method.

On the other hand, we proposed a new Lyapunov-Krasovskii functional to study

the stability for systems with time delay. With its help, a new delay-dependent

stability criterion has been obtained, which has been shown to be less conservative

than those in the literature. Furthermore, based on this criterion, a memory state

feedback controller for the offshore structure has also been designed. It has been

found that the controller, which is of less gain, can effectively ensure safety and

comfort of the offshore structure.

Future work

Accompanied by the sharply increase of energy need, offshore structures are de-

veloping into deeper and deeper ocean, which have to meet steeply growing menaces,

such as wave, wind, current, earthquake, etc. How these adverse factors effect the

offshore deep-water platforms needs to be further investigated in the future.

In this thesis, memory controller design is proposed to reduce the internal os-

cillations of an offshore structure. However, offshore structures are only a small

piece of a cake in structural vibration control area. How to extend this line to other

civil structures to constrain server vibration induced by environmental loads is a

challenging topic in our future work.

Finally, as we know, network based control has emerged as a topic of significant

interest in the control community in the recent years due to its more advantages,

such as low cost, high reliability and simple installation, etc. How to introduce the

network to an offshore structure and how to more effectively control this structure

145

are the main aim to study in the future.

Appendix A

Lemmas Referred

Lemma 1. [27] For any constant matrix W ∈ Rn×n,W = W T > 0, scalar γ > 0,

and vector function x : [−γ, 0] → Rn such that the following integration is well

defined, then

−γ

∫ 0

−γ

xT (t + ξ)Wx(t + ξ)dξ (A.1)

≤ (xT (t) xT (t− γ)

) ( −W WW −W

)(x(t)

x(t− γ)

). (A.2)

Lemma 2. [63]Given matrices Q = QT , H, E and R = RT > 0 of appropriate

dimensions,

Q + HFE + ET F T HT < 0

for all F satisfying F T F ≤ R, if and only if there exists some λ > 0 such that

Q + λHHT + λ−1ET RE < 0.

Lemma 3. [69]Let x(t) ∈ Rn be a vector-valued function with first-order continuous-

derivative entries. Then, the following integral inequality holds for any matrices X,

M1, M2 ∈ Rn×n, and Z ∈ R2n×2n, and a scalar function h := h(t) ≥ 0:

−∫ t

t−h

xT (s)Xx(s)ds ≤ ξT (t)Υξ(t) + hξT (t)Zξ(t)

where

Υ :=

[MT

1 + M1 −MT1 + M2

∗ −MT2 −M2

],

148 APPENDIX A. LEMMAS REFERRED

ξ(t) :=

[x(t)

x(t− h)

],

[X Y∗ Z

]≥ 0

with

Y :=[

M1 M2

].

Lemma 4. [14] Consider a symmetric matrix Ξ ∈ Rn×n and two matrices Π and Γ

with column dimension n. Then there exists a matrix Θ of compatible dimensions

such that

Ξ + ΠT ΘΓ + ΓT ΘT Π < 0

if and only if (Π⊥)T ΞΠ⊥ < 0 and (Γ⊥)T ΞΓ⊥ < 0, where Π⊥ and Θ⊥ denote the

orthogonal complements of Π and Θ, respectively.

Appendix B

Useful Theories

B.1 Stability of Time-Delay Systems

This part is borrowed from [19]. A time-delay system can be described by a retarded

functional differential equation as:

x(t) = f(t, xt), (B.1)

with

x(t) = φ(t), t ∈ [t0 − r, t0

], (B.2)

where x(t) ∈ Rn, f : R×C→ Rn and φ(·) is the initial condition function of (B.1).

x(θ) = x(t+θ), −r ≤ θ ≤ 0. (B.1) indicates that the derivative of the state variables

x at time t depends on t and x(ξ) for t− r ≤ ξ ≤ t.

B.1.1 Stability Concept:

For the system described by (B.1), the trivial solution x(t) = 0 is said to be stable

if for any t0 ∈ R and any ε > 0, there exists a δ = δ(t0, ε) > 0 such that ‖xt0‖c < δ

implies ‖x(t)‖ < ε for t ≥ t0.

• It is said to be asymptotically stable if it is stable, and for any t0 ∈ R and

any ε > 0, there exists a δa = δa(t0, ε) > 0 such that ‖xt0‖c < δa implies

limt→∞

x(t) = 0.

150 APPENDIX B. USEFUL THEORIES

• It is said to be uniformly stable if it is stable and δ = δ(t0, ε) can be chosen

independently of t0.

• It is uniformly asymptotically stable if it is uniformly stable and there exists

a δa > 0 such that for any η > 0, there exists a T = T (δa, η), such that

‖xt0‖c < δ implies ‖x(t)‖ < η for t ≥ t0 + T and t0 ∈ R.

• It is globally (uniformly) asymptotically stable if it is (uniformly) asymptoti-

cally stable and δa can be an arbitrarily large, finite number.

B.1.2 Lyapunov-Krasovskii Stability Theorem:

Suppose f : R × C → Rn in (B.1) maps (bounded sets in C) into a bounded sets

in Rn, and that u, v, w are continuous nondecreasing functions, where additionally

u(s) and v(s) are positive for s > 0, and u(0) = v(0) = 0. If there exists a continuous

differentiable functional V : R× C→ Rn such that

u(‖φ(0)‖) ≤ V (t, φ) ≤ v (‖φ(0)‖c) ,

and

V (t, φ) ≤ −w(‖φ(0)‖),

then the trivial solution of (B.1) is uniformly stable. If w(s) > 0 for s > 0, then it

is uniformly asymptotically stable. If, in addition, lims→∞

u(s) = ∞, then it is globally

uniformly asymptotically stable.

B.2 Linear Matrix Inequalities

This part is borrowed from [3]. A strict linear matrix inequality (LMI) has the

general form of

F (x) , F0 +m∑

i=1

xiFi > 0, (B.3)

where x =(

x1 x2 · · · xm

) ∈ Rm is a vector consisting of m variable, and the

symmetric matrices Fi = F Ti ∈ Rn, i = 0, 1, . . . , m are m+1 given constant matrices.

B.3. THE LMI TOOLBOX OF MATLAB 151

An LMI may also be nonstrict, where ” > ” is replaced by ” ≥ ”. Notice that the

variables appear linearly on the left hand side of the inequality. The basic LMI

problem is to find whether or not there exists an x ∈ Rm such that (B.3) is satisfied.

An LMI is an affine inequality constraint on the design variables. Specifications

such as regional pole placement, robust stability, LQG performance, or RMS gain

attenuation can be expressed as LMIs.

In many practical problems, the parameters may appear nonlinearly in their most

natural form. So some important techniques can be applied to transform them into

an LMI form. The popular one is as follows.

Schur Complement: For matrices A, B, C, the inequality

[A BBT C

]> 0

is equivalent to the following two inequalities

A > 0,

C −BT A−1B > 0.

B.3 The LMI Toolbox of Matlab

This part is borrowed from [15]. Commercial software of the LMI control toolbox is

a set of tools for use with Matlab. It provides state-of-the-art optimization routines

to solve linear matrix inequalities. It also includes specialized tools for LMI-based

analysis and design of control systems.

In the past few years, LMI solvers have emerged as powerful tools to solve convex

optimization problems that arise in many analysis and design applications, such as

control, identification, filtering, and structural design. While the toolbox emphasis

is on control, its LMI capabilities make it useful in any area where LMI techniques

are applicable.

The toolbox provides a fully integrated general-purpose environment for specify-


ing and solving LMI problems. It offers a powerful and user-friendly environment for

developing LMI-based applications. The toolbox uses a structured representation

of LMI constraints that boosts efficiency and minimizes memory requirements.

The commands or functions which are often used in the research include:

• Setlimis: initialize the LMI description.

• Lmivar: define a new matrix variable.

• Lmiterm: specify the term content of an LMI.

• Feasp: compute a solution to the feasibility problem.

• Mincx: compute a solution to the optimization problem.

• Gevp: generalize eigenvalue minimization under LMI constraints.

Example 2. Find a symmetric matrix P satisfying the LMIs

AT1 P + PA1 < 0,

AT2 P + PA2 < 0,

AT3 P + PA3 < 0,

P > I. (B.4)

where

A1 =

[ −1 21 −3

], A2 =

[ −0.8 1.51.3 −2.7

], A3 =

[ −1.4 0.90.7 −2

].

The Matlab program can be written as follows to solve this question.

setlmis([])

P=lmivar(1,[2 1])

lmiterm([1 1 1 P],1,A1,’s’) % LMI #1

lmiterm([2 1 1 P],1,A2,’s’) % LMI #2

B.4. AN H∞ CONTROL 153

lmiterm([3 1 1 P],1,A3,’s’) % LMI #3

lmiterm([-4 1 1 P],1,1) % LMI #4: P

lmiterm([4 1 1 0],1) % LMI #4: I

lmis=getlmis

[tmin, xfeas]=feasp (lmis)

Though running this program, if it can be found that a tmin < 0, the LMIs (B.4)

are solvable. The answer is P =

[270.8 126.4126.4 155.1

].

B.4 An H∞ Control

H∞-norm:

‖G‖∞ , supω

σ(G(jω))

sup : the least upper bound

σ : the greatest singular value

H∞ Control Problem

This part is borrowed from [70]. Consider a system described by the block

diagram

The plant G and controller K are assumed to be real-rational and proper. It

will be assumed that state space models of G and K are available and that their


realizations are assumed to be stabilizable and detectable. z, w, u, v are controlled

output, external disturbance, output vector, control vector, respectively.

Optimal Control: find all admissible controllers K(s) such that the close-loop

transfer function Twz(s) from the external disturbance w to controlled output z

satisfies that ‖Twz(s)‖∞ is minimized.

However, in practice it is often not necessary and sometimes even undesirable

to design an optimal controller, and it is usually much cheaper to obtain controllers

that are very close in the norm sense to the optimal ones, which will be called

suboptimal controllers.

Suboptimal H∞ Control: Given γ > 0, find all admissible controllers K(s),

if there are any, such that ‖Twz(s)‖∞ < γ.

Appendix C

Simulation Diagrams

C.1 The Simulation Diagram of the Controller

Design from Proposition 3

Figure C.1: The simulation diagram of the Controller Design from Proposition 3

In Figure C.1, Embedded MATLAB Function fct, dct is defined as follows:

function ftt = fct(u)

t=u(1);

x=u(2:7);

w=1.8;

if t==0

156 APPENDIX C. SIMULATION DIAGRAMS

ali=0;

a2i=0;

else

ali=x(2);

a2i=x(4);

end c2=1.694*cos(-w*t)-(-0.30419e-2*ali-0.44904e-2*a2i);

c3=4.25*cos(-w*t)+0.24-(-0.32902e-2*ali-0.15346e-2*a2i);

c4=11.67*cos(-w*t)+0.4-(-0.3445e-2*ali+0.34628e-2*a2i);

c6=1.694*cos(1.5708-w*t)-(-0.30419e-2*ali-0.44904e-2*a2i);

c7=4.25*cos(1.5708-w*t)+0.24-(-0.32902e-2*ali-0.15346e-2*a2i);

c8=11.67*cos(1.5708-w*t)+0.4-(-0.3445e-2*ali+0.34628e-2*a2i);

f2=808.071*abs(c2)*c2+6799.929*sin(-w*t);

f3=1247.282*abs(c3)*c3+26899.393*sin(-w*t);

f4=439.214*abs(c4)*c4+27017.722*sin(-w*t);

f6=808.071*abs(c6)*c6+6799.929*sin(1.5708-w*t);

f7=1247.282*abs(c7)*c7+26899.393*sin(1.5708-w*t);

f8=439.214*abs(c8)*c8+27017.722*sin(1.5708-w*t);

ft1=-0.30419e-2*f2-0.329e-2*f3-0.3445e-2*f4;

ft2=-0.30419e-2*f6-0.329e-2*f7-0.3445e-2*f8;

ft3=-0.44904e-2*f2-0.15346e-2*f3+0.34628e-2*f4;

ft4=-0.44904e-2*f6-0.15346e-2*f7+0.34628e-2*f8;

ftt=[0;ft1+ft2;0;ft3+ft4;0;0];% wave frequency

%end fct

function dtt = dct(u)

x=u(1:6); dt1=-0.30419e-2*x(1)-0.44904e-2*x(3);

dt2=-0.32902e-2*x(1)-0.15346e-2*x(3);

dt3=-0.3445e-2*x(1)+0.34628e-2*x(3); dtt=[dt1;dt2;dt3];

C.2. THE SIMULATION DIAGRAM OF THE CONTROLLER DESIGN FROMPROPOSITION 7 157

%end dct

Note: The definition of function fct, dct are also used in the later simulations

throughout this thesis.




In Figure C.2, Embedded MATLAB Function fcn is defined as follows:

function y = fcn(u,v,Gt)

A=[ 0 1 0 0 0 0

-3.3235 -0.0212 0.0184 0.0030 -5.3449 -0.8819

0 0 0 1 0 0

0.0184 0.0030 -118.1385 -0.1117 5.3468 0.8822

0 0 0 0 0 1

-0.0114 -0.0019 0.0114 0.0019 -3.3051 -0.5454 ];

B=[0;0.003445; 0;-0.00344628;0;0.00213];


L=[0;0;0.001;0;0;0.01];

Ea=[1,0,0,0,0,0];

A2=L*Gt*Ea; %the norm-bounded uncertainties

A3=(A+A2);

K1 =1.0e+004 *[-0.2555 -0.1584 1.6036 0.1897 0.1050 -0.1216];

K2 =1.0e+003 *[0.3045 0.0542 -1.4487 0.0115 0.0394 0.1280]; %controller gains

y=(A3+B*K1)*u+B*K2*v;

G(t) is used to output a uniformly distributed random signal ranging between

-10 and 10.

Embedded MATLAB Functions fct, dct are the same as those described in Ap-

pendix C.1.



See Figure C.2. Embedded MATLAB Function fcn in the Controller Design from

Proposition 10 is defined as follows:


A=[ 0 1 0 0 0 0

-3.3235 -0.0212 0.0184 0.0030 -5.3449 -0.8819

0 0 0 1 0 0

0.0184 0.0030 -118.1385 -0.1117 5.3468 0.8822

0 0 0 0 0 1

-0.0114 -0.0019 0.0114 0.0019 -3.3051 -0.5454 ];

B=[0;0.003445; 0;-0.00344628;0;0.00213];

L = [0;0;3;0;0;0];

Ea=[1,0,0,0,0,0];

L1 = [0;0;0;0;5;0];


Ea1=[0,1,0,0,0,0];

A2=L*Gt*Ea;

A3=L1*Gt*Ea1; %the polytopic uncertainties

K1 =1.0e+004 *[-0.2266 -0.0659 1.1847 0.2477 0.0463 -0.1982];

K2 =[-18.5225 -12.7426 169.8950 16.7815 -6.8169 -19.5826]; % h=0.1,

mu=0.1;contrller gains

y=(A+A2+A3+B*K1)*u+B*K2*v; %end fcn

G(t) is used to output a uniformly distributed random signal ranging between

-0.03 and 0.03.


pendix C.1.




See Figure C.3, where w(t) is used to output a uniformly distributed random

signal ranging between -100 and 100, which is treated as the external disturbance.



pendix C.1.




The simulation diagram is depicted in Figure C.4, where Embedded MATLAB

Function fcn is defined as follows:


A=[ 0 1 0 0 0 0

-3.3235 -0.0212 0.0184 0.0030 -5.3449 -0.8819

0 0 0 1 0 0

0.0184 0.0030 -118.1385 -0.1117 5.3468 0.8822

0 0 0 0 0 1

-0.0114 -0.0019 0.0114 0.0019 -3.3051 -0.5454 ];

B=[0;0.003445; 0;-0.00344628;0;0.00213];

L=[0;0;0.001;0;0;0.01];


Ea=[1,0,0,0,0,0];

A2=L*Gt*Ea; %the norm-bounded uncertainties

A3=(A+A2);

K1 =1.0e+004 *[-0.1941 -0.1424 1.9398 0.3026 0.1145 -0.0999];

K2 =1.0e+003 *[0.2640 0.0649 -2.0823 -0.0230 0.0326 0.1235]; % h=0.1,


y=(A3+B*K1)*u+B*K2*v; %end fcn

w(t) is used to output a uniformly distributed random signal ranging between

-100 and 100, which is treated as the external disturbance; G(t) is used to output a

uniformly distributed random signal ranging between -10 and 10.


pendix C.1.



See Figure C.4, where Embedded MATLAB Function fcn in the Controller Design

from Proposition 17 is described as follows:


A=[ 0 1 0 0 0 0

-3.3235 -0.0212 0.0184 0.0030 -5.3449 -0.8819

0 0 0 1 0 0

0.0184 0.0030 -118.1385 -0.1117 5.3468 0.8822

0 0 0 0 0 1

-0.0114 -0.0019 0.0114 0.0019 -3.3051 -0.5454 ];

B=[0;0.003445; 0;-0.00344628;0;0.00213];

L = [0;0;3;0;0;0];

Ea=[1,0,0,0,0,0];


L1 = [0;0;0;0;5;0];

Ea1=[0,1,0,0,0,0];

A2=L*Gt*Ea;

A3=L1*Gt*Ea1; %the polytopic uncertainties

K1 =1.0e+004 *[-7.1284 -0.9221 8.8308 -0.1684 -0.6949 -2.7038];

K2 =[-247.1548 -79.4067 534.3352 -6.2850 -29.7022 -116.5531]; % h=0.1,


y=(A+A2+A3+B*K1)*u+B*K2*v; %end fcn

w(t) is used to output a uniformly distributed random signal ranging between

-100 and 100, which is treated as the external disturbance; G(t) is used to output a

uniformly distributed random signal ranging between -0.03 and 0.03.


pendix C.1.

C.7 The Simulation Diagram of the Output Feed-

back Control

The simulation diagram is depicted in Figure C.5.

C.8 The Simulation Diagram of the Robust Out-

put Feedback Control

The simulation diagram is depicted in Figure C.6, where Embedded MATLAB Func-

tion fcn is defined as follows:

function y = fcn(u,Ft)

A=[ 0 1 0 0 0 0

-3.3235 -0.0212 0.0184 0.0030 -5.3449 -0.8819

0 0 0 1 0 0

C.8. THE SIMULATION DIAGRAM OF THE ROBUST OUTPUTFEEDBACK CONTROL 163

Figure C.5: The simulation diagram of the output feedback control

Figure C.6: The simulation diagram of the robust output feedback control


0.0184 0.0030 -118.1385 -0.1117 5.3468 0.8822

0 0 0 0 0 1

-0.0114 -0.0019 0.0114 0.0019 -3.3051 -0.5454 ];

L=[0;0;0.001;0;0;0.01];

Ea=[1,0,0,0,0,0];

A2=L*Ft*Ea; %the norm-bounded uncertainties

y=(A+A2)*u;%end fcn

F(t) is used to output a uniformly distributed random signal ranging between -10

and 10.


pendix C.1.

C.9 The Simulation Diagram of the Output Feed-

back H∞ Control

Figure C.7: The simulation diagram of the output feedback H∞ control

The simulation diagram is shown in Figure C.7, where Embedded MATLAB

Functions fct, dct are the same as those described in Appendix C.1, and Embedded

C.9. THE SIMULATION DIAGRAM OF THE OUTPUT FEEDBACK H∞CONTROL 165

MATLAB Functions fcn is the same as that described in Appendix C.8. And w(t)

is used to output a uniformly distributed random signal ranging between -100 and

100. F(t) is used to output a uniformly distributed random signal ranging between

-10 and 10.

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[73] http://www.esru.strath.ac.uk/EandE/Web sites/98-9/offshore/steel.htm (Ac-

cess Date: March 16, 2006 2:33 p.m.).

REFERENCES 175

[74] http://www.naturalgas.org/naturalgas/extraction offshore.htm (Access Date:

2006 1:33 p.m.).

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