Download - Nonlinear Fault-Tolerant Guidance and Control for … Nonlinear Fault-Tolerant Guidance and Control for Damaged Aircraft Gong Xin Xu [email protected] Master of Applied Science Graduate

Nonlinear Fault-Tolerant Guidance and Control forDamaged Aircraft

by

Gong Xin Xu

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

Copyright c© 2011 by Gong Xin Xu

Abstract

Nonlinear Fault-Tolerant Guidance and Control for Damaged Aircraft

Gong Xin Xu

[email protected]

Master of Applied Science

Graduate Department of Aerospace Science and Engineering

University of Toronto

2011

This research work presents a fault-tolerant flight guidance and control framework to

deal with damaged aircraft. Damaged scenarios include the loss of thrust, actuator mal-

function and airframe damage. The developed framework objective is to ensure that

damaged aircraft can be stabilized and controlled at all times. The guidance system

is responsible for providing the airspeed, vertical and horizontal flight path angle com-

mands while considering aircraft dynamics. The control system, designed by the non-

linear state-dependent Riccati equation (SDRE) control method, is used to track the

guidance commands and to stabilize the damaged aircraft. The versatility of SDRE al-

lows it to passively adapt to the aircraft parameter variations due to damage. A novel

nonlinear adaptive control law is proposed to improve the controller performance. The

new control law demonstrated improved tracking ability. The framework is implemented

on the nonlinear Boeing 747 and NASA Generic Transport Model (GTM) to investigate

the simulation results.

ii

Acknowledgements

I would like to express my deepest gratitude to my thesis supervisor Professor Hugh Liu

for giving me the opportunity to work on the fault-tolerant flight control topic, and for

his continuous guidance and support throughout the research. His encouragement and

advice led me to the right path and are greatly appreciated.

I would also like to thank the other members of my research committee, Professor

Peter Grant and Professor Christopher Dameren for their valuable feedback and com-

ments.

My heartfelt appreciation also goes to the friends and colleagues at the Flight Systems

and Control (FSC) group at UTIAS, Chen, Connie, Difu, Everett, Jason, Keith, Sohrab

and others. They made my life at FSC an enjoyable and memorable experience.

I would also like to extend my deepest gratitude to my family for their unconditional

love and support.

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Contents

1 Introduction 1

1.1 Aircraft Flight Control System Design . . . . . . . . . . . . . . . . . . . 1

1.2 Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Research Objectives & Contribution . . . . . . . . . . . . . . . . . . . . 7

1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Aircraft Dynamics 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Aircraft Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Nonlinear Equations of Motion . . . . . . . . . . . . . . . . . . . . 11

2.3 Nonlinear Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Boeing 747-100/200 . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 NASA GTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Damaged Aircraft Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Trim Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Fault-tolerant Flight Guidance and Control Problem 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

iv

3.4 Aircraft Guidance Law Design . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Guidance Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 State-Dependent Riccati Equation Control Method 33

4.1 SDRE Control Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Stability and Optimality Analysis . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Optimality Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 The Art and Capabilities of SDRE . . . . . . . . . . . . . . . . . . . . . 40

4.4 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4.1 Loss-of-thrust example - UAV example . . . . . . . . . . . . . . . 46

4.4.2 Damaged Aircraft - B747 . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Adaptive State-Dependent Riccati Equation Control Method 62

5.1 Adaptive Control Method . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Stability Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3.1 Baseline Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.2 Adaptive Law Design . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Conclusions and Future Work 78

A Derivations 80

B State-dependent coefficients 83

v

Bibliography 87

List of Tables

2.1 Failure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Trimmed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Trimmed Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Straight and Level Flight Trim Results . . . . . . . . . . . . . . . . . . . 46

4.2 Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1 GTM Trim Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

List of Figures

1.1 Military and Civil Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Mechanical and FBW Systems . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Aircraft Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Boeing 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 NASA GTM Simulink Environment . . . . . . . . . . . . . . . . . . . . . 17

2.4 GTM Damage Case Example . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Trim Routine Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 B747 Trimmed Airspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 B747 Trimmed Roll Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.8 B747 Trimmed Angle of Attack . . . . . . . . . . . . . . . . . . . . . . . 23

2.9 B747 Trimmed Pitch Angle . . . . . . . . . . . . . . . . . . . . . . . . . 23

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2.10 B747 Trimmed Sideslip Angle . . . . . . . . . . . . . . . . . . . . . . . . 23

2.11 B747 Trimmed Yaw Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.12 B747 Trimmed Roll Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.13 B747 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.14 B747 Trimmed Pitch Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.15 B747 Lateral Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.16 B747 Trimmed Yaw Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.17 B747 Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Proposed Guidance and Control Framework . . . . . . . . . . . . . . . . 27

3.2 Guidance Law Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 SDRE Design Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Undamaged UAV: Airspeed Time History . . . . . . . . . . . . . . . . . 49

4.3 Undamaged UAV: Flight Path Angle Time History . . . . . . . . . . . . 49

4.4 Undamaged UAV: Angle of Attack Time History . . . . . . . . . . . . . . 49

4.5 Undamaged UAV: Pitch Rate Time History . . . . . . . . . . . . . . . . 49

4.6 Undamaged UAV: Throttle Control Time History . . . . . . . . . . . . . 49

4.7 Undamaged UAV: Elevator Time History . . . . . . . . . . . . . . . . . . 49

4.8 Undamaged UAV: Tracking Distance Error Time History . . . . . . . . . 50

4.9 Undamaged UAV: Altitude Time History . . . . . . . . . . . . . . . . . . 50

4.10 Damaged UAV: Airspeed Time History . . . . . . . . . . . . . . . . . . . 51

4.11 Damaged UAV: Flight Path Angle Time History . . . . . . . . . . . . . . 51

4.12 Damaged UAV: Angle of Attack Time History . . . . . . . . . . . . . . . 51

4.13 Damaged UAV: Pitch Rate Time History . . . . . . . . . . . . . . . . . . 51

4.14 Damaged UAV: Throttle Control Time History . . . . . . . . . . . . . . 52

4.15 Damaged UAV: Elevator Time History . . . . . . . . . . . . . . . . . . . 52

4.16 Damaged UAV: Pitch Angle Time History . . . . . . . . . . . . . . . . . 52

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4.17 Damaged UAV: Altitude Time History . . . . . . . . . . . . . . . . . . . 52

4.18 Loss-of-thrust Case II Simulation Results . . . . . . . . . . . . . . . . . . 53

4.19 B747 Actuator Damage: Roll Rate Time History . . . . . . . . . . . . . . 55

4.20 B747 Actuator Damage: Pitch Rate Time History . . . . . . . . . . . . . 55

4.21 B747 Actuator Damage: Yaw Rate Time History . . . . . . . . . . . . . 56

4.22 B747 Actuator Damage: Elevator Time History . . . . . . . . . . . . . . 56

4.23 B747 Actuator Damage: Aileron Time History . . . . . . . . . . . . . . . 57

4.24 B747 Actuator Damage: Rudder Time History . . . . . . . . . . . . . . . 57

4.25 B747: Flight Path Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.26 Damaged B747: Flight Path Angle Time History . . . . . . . . . . . . . 61

4.27 Damaged B747: Elevator Time History . . . . . . . . . . . . . . . . . . . 61

4.28 Damaged B747: Airspeed Time History . . . . . . . . . . . . . . . . . . . 61

4.29 Damaged B747: Throttle Control Time History . . . . . . . . . . . . . . 61

4.30 Damaged B747: Trajectory Time History . . . . . . . . . . . . . . . . . . 61

5.1 Adaptive framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 GTM Trim: angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3 GTM Trim: flight path angle . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 GTM stabilizing: angle of attack . . . . . . . . . . . . . . . . . . . . . . 75

5.5 GTM stabilizing: pitch rate . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.6 GTM stabilizing: elevator deflection . . . . . . . . . . . . . . . . . . . . . 75

5.7 GTM tracking: angle of attack . . . . . . . . . . . . . . . . . . . . . . . . 75

5.8 GTM tracking: elevator deflection . . . . . . . . . . . . . . . . . . . . . . 76

5.9 GTM tracking: elevator deflection comparison perfect vs damage . . . . . 76

5.10 GTM tracking: angle of attack comparison . . . . . . . . . . . . . . . . . 77

5.11 GTM tracking: elevator deflection comparison . . . . . . . . . . . . . . . 77

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Chapter 1

Introduction

1.1 Aircraft Flight Control System Design

The aircraft flight control system is a vital component, among other critical aircraft

systems, to ensure flight performance and safety. It was successfully introduced by the

Wright brothers in 1902. The original design features the three-axis control, with coupled

roll and yaw control to alleviate the adverse yaw effects [43]. Such a design paved

the foundation of the modern aircraft flight control system, which has flourished with

revolutionary changes.

By the 1950s, analog flight control computers emerged to allow artificial modification

of the aircraft handling qualities in addition to the basic autopilot stabilization tasks [16].

The Canadian Avro CF-105 Arrow interceptor (Fig. 1.1a) equipped with an analog flight

control computer demonstrated impressive performance capabilities. Subsequently, dig-

ital fly-by-wire (FBW) technology was introduced to replace the analog flight control

computers. In 1972, the technology was flown by an F-8 Crusader (Fig. 1.1b) in flight

experiments conducted by NASA. In the civil aviation field, Airbus A320 was the first

commercial airliner utilized the FBW control system on main control surfaces in 1987.

1

Chapter 1. Introduction 2

(a) Avro CF-105 Arrow (b) F-8 (c) Airbus A320

Fig. 1.1: Military and civil aircraft.Sources: (a) DND http://www.airforce.forces.gc.ca/v2/equip/resrc/images/hst/l-g/arrow4.jpg;

(b) NASA http://www.dfrc.nasa.gov/gallery/photo/F-8DFBW/Small/EC77-6988.jpg; (c) airliners.net

Concurrently, the conventional mechanical flight control system (Fig. 1.2a), as still

seen in small aircraft nowadays, also gradually evolved to the mechanical-hydraulic sys-

tems, which include a hydraulic system to generate actuators forces to move the control

surfaces. Although the hydro-mechanical system makes pilots flying the aircraft less de-

manding and allows large forces on the control surfaces, it adds additional complexity

and weight to the already highly complex mechanical system. The FBW flight control

system (Fig. 1.2b) became the solution to replace the previous two systems. Because the

digital computers are used to receive and send signals, it allows for easier control im-

plementation, thus better handling qualities. The digital FBW technology improved the

flight reliability, maneuverability as well as safety while providing drastic cost reduction.

(a) Hawk aircraft mechanical system (b) Dgital FBW flight control system

Fig. 1.2: Flight control systems [18]


1.2 Research Motivation

The advent of digital FBW system brings a new chapter to the flight control system

design. The computer-based system allows better handling, greater aircraft maneuver-

ability and agility, weight reduction and so on. These benefits come with the most

stringent safety requirement. For example, an FBW system must have the same level of

safety and integrity as the simple mechanical system. It means that the probability of

a failure occurring in the FBW system which would result in catastrophic consequences

to the aircraft must be less than 1 in 109 per flight hour [14]. The intense focus on the

safety level and the need to improve the system integrity require the FBW aircraft to

be fitted with a back up system. This system has the ability to generate fault-tolerant

control commands that take advantages of the system redundancy in terms of controls,

sensors, and computing. The control effector redundancy provides a unique opportunity

for the back up control system to reconfigure itself to mitigate and compensate for the

failure of the aircraft with the objective of increasing survivability.

In addition, recent civil aviation safety data show that about 16% of the accidents

that happened in between 1992 and 2007 belongs to the category of Loss of Control In-

flight (LOS-I), which is caused by pilot error, technical malfunctions, or unusual upsets

due to external disturbances [16]. For example, in the late 1970s an American Airlines

DC-10 crashed in Chicago (Flight 191, May 25, 1979) due to engine seperation. The pilot

only had 15s to react before the plane crashed. The subsequent investigations showed

the accident could have been avoided [36]. More recently, an El Al cargo Flight 1862

(October 4, 1992) fatal crash was also demonstated in simulation to be avoidable [31].

These catastrophic crashes underscore the need to have intelligent, fault-tolerant flight

control (FTFC) systems. NASA commenced the Integrated Resilient Aircraft Control

(IRAC) project of NASA Aviation Safety Program (AvSP) to investigate and research

advanced flight controls that can be implemented to ensure safety in the presence of

unforeseen, adverse conditions. Similarly, European GARTEUR Flight Mechanics Action


Group FM-AG(16) on fault-tolerant flight control has also focused on the topic.

In addition to the FBW system, the FTFC system is only activated when the fault is

positively detected and diagnosed on the aircraft. The FTFC system is specifically de-

signed to stabilize and control the aircraft without exacerbating the situation into a more

serious situation while ensuring post-damage performance with acceptable degradation

so that the survivability and safety are greatly enhanced.

1.3 Literature Review

The fault-tolerant flight control system design are reviewed in this section. In general,

fault-tolerant control systems can be categorized into passive and active systems. Passive

controllers are designed based on a pre-determined set of requirements. They are fixed

and are robust against a class of presumed faults [17]. Controllers need neither a fault

detection, isolation and diagnosis scheme nor controller reconfiguration, but only limited

fault-tolerant capabilities are achievable. On the other hand, as the name suggests, the

active controllers respond to the system malfunction by actively reconfiguring control laws

to ensure stability and performance requirements are met. Sometimes, certain degrees of

performance degradation have to be accepted due to severe damage. The overall design

objectives of fault-tolerant controllers are to meet the system transient and steady-state

performance requirements not only under normal operating conditions, but also in fault

situations. To achieve these objectives, a large number of controllers have been designed

and implemented on both linear and nonlinear systems. The rest of this section presents

some common control methods designed for fault-tolerant flight control, covering both

the passive and active systems.

Marcos and Balas [32] presented a quasi-linear parameter varying (LPV) model for

LPV control synthesis that guarantees stability and robustness of the closed loop sys-

tem. LPV was chosen to model the damaged aircraft because it is represented by its

varying parameter description. Three different linearization approaches were used in the


paper, namely Jacobian linearization, state transformation and function substitution.

Shin [49] presented an LPV model based on fault parameters. The fault parameters were

also scheduling parameters in this case and were estimated on-line by using a two-stage

adaptive Kalman filter. Aligned with the work of Shin, Shin and Gregory [48] presented

another LPV formulation that involved function substitution. Unlike the previous work,

aerodynamic coefficients were fitted from the plots and the uncertainties of which were

included in the model.

Maciejowski and Jones [31] applied Model Predictive Control (MPC) to the El Al

flight 1862 model based on the assumption that a perfect fault detection and isolation

model was available. Kale [26] and Miksch [35] also explored MPC as the control system.

However, the main disadvantage in their formulations [26,31,35] was that the cost func-

tion has two different horizons, predictive and control horizons. By doing so, however

the closed loop stability was not guaranteed and additional complexity is introduced. To

overcome the stability problem, Almeida and Leissling [2] presented a new formulation

using MPC in which fault-tolerant MPC with infinite prediction horizon approach was

studied. Since MPC is applied over a long horizon, it would be only possible to implement

such a method to dynamically slow systems, and hence not applicable to fault-tolerant

flight control systems. Very recently, Fabio [15] provided a possible alternative to solve

the problem by reducing the horizon to a shorter one.

The Sliding mode control (SMC) method was also studied extensively by [3, 4, 47].

In the first paper [3], Alwi et al made an assumption that there was no FDI available.

Unlike the MPC approach [31], the presented SMC did not require exact knowledge of

the post-damage aircraft. With the popularity of control allocation in the field of FTFC

systems, Shin et al [47] used the idea of control allocation, but an adaptive sliding mode

control method was implemented. Most of the research related to SMC are lack of a

detailed analysis in stability. As a result, in the paper by Alwi et al [4], the control

allocation and SMC were combined and a rigorous design procedure was presented.


Adaptive control is another popular method being widely used in the flight control

system [28, 40, 46]. A direct adaptive reconfigurable flight control method was proposed

by Kim et al [28] to deal with the nuisance of having a system identification process in the

indirect adaptive control. System identification was often used to accompany the indirect

adaptive control to handle the model mismatch and parameter uncertainty issues. The

timescale separation principle was applied in a model following scheme to control both

inner loop state and the outer loop state of the flight system simultaneously. Adaptive

control is often used in conjunction with a neural control scheme, particularly in nonlinear

dynamic systems. In the paper by Napolitano [40], an integrated sensor and actuator

failure detection, identification and accommodation within an FTFC system was studied.

Such an integration is able to detect failures and pass the information to the controller

with the goal of minimizing the false alarm rate and incorrect failure identification.

Lombaerts et al [30] presented the use of nonlinear dynamic inversion (NDI) technique

in which a real-time identified physical model of the damaged aircraft was included to

avoid NDIs sensitivity to modeling error. An Iterated Extended Kalman Filter (IEKF)

was used to estimate the aircraft states. The usage of which can potentially increase

the computational cost and lead to real-time implementation difficulties. DI and MPC

were combined together as a control method in [25] to tackle the FTFC problem. The

combination is intuitive since the DI provides a linearized model that MPC can work on.

As a result, a reconfigurable, nonlinear controller was designed.

The above literature review covers some of the prominent control methods used in

the field of fault-tolerant flight control system. However, few of the existing methods

provide a systematic and efficient design approach to deal with the problem. Necessary

features such as the flexibility and versatility, which these control methods lack, make the

design process difficult, especially when transferring from one model to another one. It is

also important to recognize the inherent nonlinear nature of aircraft dynamics. Nonlinear

control methods may be better to handle the fault-tolerant tasks. As a result, a nonlinear


control method that possesses the systematic and efficient design approach is proposed

in this research to deal with damaged aircraft.

Fault-tolerant flight control systems are often complemented by a robust guidance

system to achieve safe landing objective. For example, Menon et al. [33] implemented a

robust guidance algorithm for impaired aircraft based on a point mass nonlinear aircraft

model. The guidance algorithm was formulated with the finite interval differential game.

The guidance commands then were inverse transformed into the roll, pitch and yaw

attitude commands. Chawla et al [6] studied a partial integrated guidance and control

system based on the nonlinear dynamic inversion to perform obstacle avoidance of UAVs.

The collision cone concept was used in the derivation to transform the problem into a

sequential target interception problem. The guidance algorithm was then derived under

the frame of a collision cone. Other guidance algorithms based on optimization methods,

such as mixed integer linear programming or model predictive control techniques are not

suitable in our applications due to their heavy requirement of computational resources.

In this thesis, a robust, feedback based guidance algorithm is implemented for dam-

aged aircraft. The guidance algorithm takes into consideration damaged aircraft dynam-

ics to adjust its commands in a feedback fashion. It also needs little modification to

the existing control system architectures, unlike the above mentioned ones which require

guidance command transformations.

1.4 Research Objectives & Contribution

This research focuses on the design of nonlinear fault-tolerant flight control laws to en-

sure flight safety in the presence of adverse conditions. Additionally, it investigates the

integration of guidance laws and control laws in the context of damaged aircraft to guar-

antee fast response, and safe landing. The benefits of the proposed framework over the

existing ones are also explored.

Although there have been many control laws designed for the purpose of mitigat-


ing faults and recovering the flight performance, few have demonstrated the ability to

integrate with guidance laws to ensure safe landing. In addition, most of the exiting

controllers, as mentioned in the previous section, are categorized as linear controllers,

which require extensive gain scheduling to cover a wide flight envelope. This work aims

to provide a sophisticated, yet designer-friendly solution to achieve the objective of safe

landing of the damaged aircraft while taking the advantage of nonlinear control.

The benefits of integration are critical in the damaged case. The smooth integration

can support the post-damage planning, guidance, and control in a unified manner, which

not only saves precious time, but also increases the survivability and safety.

This work contributes to the research and development of the fault-tolerant flight

control system mainly in the following areas:

• To develop a nonlinear fault-tolerant flight control system to handle damaged air-

craft;

• To implement the guidance and control laws to expedite post-damage recovery and

ensure safe landing;

• To investigate and verify the proposed framework performance by comparing with

the existing control method.

1.5 Thesis Organization

The thesis is organized in the following manner. Chapter 2 presents both healthy and

damaged aircraft dynamics and modeling. The fault-tolerant flight guidance and control

problem is introduced in Chapter 3, which covers the problem formulation and design

framework. Chapter 4 focuses on the aircraft guidance law design. In Chapter 5, non-

linear fault-tolerant control methods are discussed. A novel nonlinear adaptive control

law is derived in Chapter 6. Simulation studies are performed in that chapter to demon-

strate its promising results. Finally, the concluding remarks and possible future works

are offered in Chapter 7.

Chapter 2

Aircraft Dynamics

This chapter presents the aircraft dynamics. In Section 2.2, the general nonlinear equa-

tions of motion (EOM) are derived and formulated. Section 2.3 introduces the nonlinear

aerodynamic coefficients that are included in the models. Section 2.4 deals with the

damaged aircraft modeling and Section 2.5 presents the optimization based trim routine

used to seek the steady state level flight condition.

2.1 Introduction

In this chapter, the general six degrees of freedom (6DoF) nonlinear EOM are introduced.

Two aircraft models, Boeing 747-100/200 and NASA Generic Transport Model (GTM),

are used throughout the thesis as simulation test beds. Both models are covered in details

in Section 2.3. The Damaged aircraft modeling is studied in Section 2.4. Several damage

scenarios as well as their possible outcomes are reviewed in that section.

2.2 Aircraft Modeling

Before diving into the derivation of the equations of motion, it is important to establish

the frames of reference. Throughout the thesis, the following right-handed and orthogonal

reference frames are used: the earth-fixed inertial reference frame, FE; the vehicle carried

local earth reference frame, FO whose origin is fixed at the centre of gravity of the vehicle

9

Chapter 2. Aircraft Dynamics 10

and is assumed to have the same orientation as FE; the wind-axes reference frame, FW ,

obtained by three successive rotations of horizontal flight path angle χ, vertical flight

path angle γ, and bank angle µ from FO; the stability-axes frame, FS, obtained from a

rotation from FW by a rotation of −β; the body-fixed frame, FB, obtained by rotations

of yaw angle ψ, pitch angle θ, and roll angle φ from FO. These frames are shown in Fig.

2.1.

Fig. 2.1: Aircraft Reference Frames

Transformation from one frame to another is done by using the rotational matrices.

For example, rotational matrices for FB to FS and FB to FW are defined in Eq.(2.1) and

Eq.(2.2), respectively.

FSB =

cosα 0 sinα

0 1 0

−sinα 0 cosα

(2.1)

FWB =

cosα · cosβ sinβ sinα · sinβ

−cosα · sinβ cosβ −sinα · sinβ

−sinα 0 cosα

(2.2)

In order to derive the equations of motion, a number of assumptions must be made:

• The aircraft is a rigid body;

• The earth is flat and non-rotating;

• The aircraft mass properties are constant, any mass variation is negligible;

• The aircraft has a plan of symmetry, which is the XBZB plane. It implies that

moment of inertia Iyz and Ixy are equal to zero. This assumption is valid for


undamaged aircraft. When aircraft suffer from asymmetric damage, the assumption

does not apply any more;

2.2.1 Nonlinear Equations of Motion

The aircraft equations of motion can be derived from Newton’s Second Law. Mathemat-

ically, Newton’s Second Law can be expressed as the following in the inertial frame

F =d

dt(mVt)|E (2.3)

M =d

dt(H)|E (2.4)

where F is the sum of all external forces; m is the aircraft mass; M represents the sum of

all external moments about the centre of the mass; H is the angular momentum about

the centre of mass.

The above equations can be written in the body-fixed frame, FB as

F =d

dt(mVt)|B + ω ×mVt (2.5)

M =d

dt(H)|B + ω ×H (2.6)

where ω is the total angular velocity of the aircraft with respect to the Earth. The vector

terms in Eq.(2.5) and Eq.(2.6) can be expressed as

Vt = ui + vj + wk (2.7)

ω = pi + qj + rk (2.8)

H = Iω (2.9)

where I is defined as

I =

Ix 0 −Ixz

0 Iy 0

−Ixz 0 Iz

(2.10)


Substituting Eq.(2.7)-Eq.(2.9) into Eq.(2.5) and Eq.(2.6) and expanding terms yields,

Fx = m(u+ qw − rv) (2.11)

Fy = m(v + ru− pw) (2.12)

Fz = m(w + pv − qu) (2.13)

Mx = pIx − rIxz + qr(Iz − Iy)− pqIxz (2.14)

My = qIy + pq(Ix − Iz) + (p2 − r2)Ixz (2.15)

Mz = rIz − pIxz + pq(Iy − Ix) + qrIxz (2.16)

where the external forces are the aerodynamic forces, thrust forces and gravity forces and

the external moments include the aerodynamic moments and the engine moments.

Fx = qSCxb + FTx −mgsinθ (2.17)

Fy = qSCyb + FTy +mgcosθsinφ (2.18)

Fz = qSCzb + FTz +mgcosθcosφ (2.19)

Mx = qSbClb +Mengx (2.20)

My = qScCmb +Mengy (2.21)

Mz = qSbCnb +Mengz (2.22)

where q = 12ρV 2

t is the dynamic pressure. The equations presented above are collected

together and rearranged into a set of twelve first order, aircraft equations of motion.

Force equations:

u = rv − qw −mgsinθ +1

m(qSCxb + FTx) (2.23)

v = −ru+ pw + gsinφcosθ +1

m(qSCyb + FTy) (2.24)

w = qu− pv + gcosφcosθ +1

m(qSCyb + FTy) (2.25)


Kinematic equations:φ

θ

ψ

=

1 sinφ · tanθ cosφ · tanθ

0 cosφ −sinφ

0 sinφcosθ

cosφcosθ

p

q

r

(2.26)

Moment equations:p

q

r

=

Ixx 0 −Ixz

0 Iyy 0

−Ixz 0 Izz

−1

Mx + (Iyy − Izz)qr + Ixzpq

My + (Izz − Ixx)pr + Ixz(r2 − p2)

Mz + (Ixx − Iyy)pq − Ixzqr

(2.27)

Navigation equations:xe

ye

he

=

cosθ · cosψ sinφ · sinθ · cosψ − cosφ · sinψ cosφ · sinθ · cosψ + sinφ · sinψ

cosθ · sinψ sinφ · sinθ · sinψ + cosφ · cosψ cosφ · sinθ · sinψ − sinφ · cosψ

sinθ −sinφ · cosθ −cosφ · cosθ

u

v

w

(2.28)

where (u, v, w) are the velocity components; (φ, θ, ψ) are the Euler angles, roll, pitch and

yaw angle; (p, q, r) are the roll, pitch and yaw rate; (xe, ye, he) are the inertial positions.

For the fault-tolerant flight control design, it is more sensible to introduce the air-

speed, angle of attack and slideslip angle as state variables to replace u, v, w in the force

equations. The main reasons are: first of all, Some of aerodynamic derivatives obtained

from wind tunnel or flight tests, are tabulated based on α, β. As a result, it is easier

to use these variables as state instead of converting from other variables. Thus, greater

accuracy may be preserved. Secondly, when aircraft suffer from abnormalities in flight,

their behavior can be difficult to predict. For instance, the upper limit of the pitch rate

q may reach as high as 0.2rad/s, similar to the case of agile aircraft and aircraft can fly

at a high airspeed (e.g. Vt = 60m/s). It means the term qu in eq.(2.25) may become as

large as 12g’s. However, in reality the upper limit of the normal acceleration can be only

a few g’s. Hence greater accelerations are introduced into equations because of the high

rotation rates which the body-axes experience. This means much less favorable computer


scaling and hence much poorer solution accuracy for a given computer precision if the

simulation is based on u, v and w instead of Vt, α, and β [45]. The following equations are

used to replace the force equations. Their derivations are included in the Appendix A.

α =1

mVtcosβ(−Fxsinα + Fzcosα +mVt(−pcosαsinβ + qcosβ − rsinαsinβ)) (2.29)

β =1

mVt(−Fxcosαsinβ + Fycosβ − Fzsinαsinβ −mVt(−psinα + rcosα)) (2.30)

Vt =1

m(Fxcosαcosβ + Fysinβ + Fzcosβsinα) (2.31)

Thus, the 6DOF state vector is

x =

[Vt α β φ θ ψ p q r xe ye he

]T(2.32)

2.3 Nonlinear Aerodynamic Coefficients

2.3.1 Boeing 747-100/200

As mentioned earlier, both B747 and GTM models are considered in the thesis. In this

section, the B747 nonlinear aerodynamic coefficients are presented. The NASA GTM

ones are briefly reviewed later in this section.

The Boeing 747-100/200 (Fig. 2.2) is an inter-continental wide-body transport with

four turbofan jet engines designed to operate from international airports. It exhibits a

wide array of characteristics (leading and trailing edge flaps, spoilers, variety of control

surfaces, four fan jet engines...) which make it the perfect representative for any of the

commercial airplanes flying today. The physical properties and aerodynamic data used

in this thesis are obtained from NASA technical reports [22,23]. The aerodynamic coef-

ficients are based on a number of stability derivatives, which are defined in the stability

frame of reference. Since the EOM are in FB, the aerodynamic coefficients must be in FB


as well. Thus, the following relationships are employed to accomplish the transformation.

CXb = −CDcosα + CLsinα (2.33)

CZb = −CDsinα− CLcosα (2.34)

Cmb = Cm (2.35)

CY b = CY (2.36)

Clb = Clcosα− Cnsinα (2.37)

Cnb = Clsinα + Cncosα (2.38)

Fig. 2.2: Boeing 747, Source: Airliners.net

The complete expressions of the coefficients can be found in the technical reports.

However, in order to facilitate the fault-tolerant flight controller investigation and de-

sign, the model complexity will be reduced by eliminating some stability derivatives

that contribute little to the overall aerodynamic coefficients. The following simplified

aerodynamic coefficient equations are used:

CL = CLbasic +dCLdq

qsc

2Vt+ (

dCLdδEI

δEI +dCLdδEO

δEO) (2.39)

CD = KCDbasic + (1−K)CDmach + ∆CDsideslip (2.40)

CY =dCYdβ

β +dCYdp

psb

2Vt+ ∆CYrudders (2.41)

Cl =dCldβ

β +dCldp

psb

2Vt+dCldr

rsb

2Vt+ ∆Clinbd ailerons

+ ∆Clrudders (2.42)

Cm = Cmbasic +dCm0.25

dq

qsc

2Vt+ (

dCm0.25

dδEIδEI +

dCm0.25

dδEOδEO) (2.43)

Cn =dCndβ

β +dCndp

psb

2Vt+dCndr

rsb

2Vt+ ∆Cninbd ailerons

+ ∆Cnrudders (2.44)


In the cases of lift and pitching moment coefficients, the contributing factors include

the basic lift and pitching moment coefficients, the dynamic stability derivatives dCLdq

and dCmdq

as well as the contributions from the both inboard and outboard elevators,

respectively. It is also assumed in this case that the centre of gravity coincides with the

aerodynamic centre at the quarter chord location. The drag coefficient, CD, is mainly

dictated by the basic drag coefficient, drag coefficient due to Mach number as well as the

sideslip angle. K is an aircraft specific constant. CY is determined from the contribution

of β, p, and rudders. Similarly, Cl and Cn depend on β, p, r, inboard ailerons, and

rudders.

The stability derivatives in Eq.(2.39)- Eq.(2.44) are then put into the look-up tables

(LUT) in Matlab for easy access during the simulation. Thus, the nonlinear B747 model

is obtained using the reduced aerodynamic coefficients and the previous derived nonlinear

EOM.

2.3.2 NASA GTM

The NASA GTM model is provided in a Simulink package (Fig. 2.3). It includes the

comprehensive aircraft information in terms of aerodynamic look-up tables as well as

Simulink blocks. The difference between the previous mentioned EOM and the one

implemented in the GTM model is that the aerodynamic forces are calculated based on

the center of pressure (CP) instead of the aerodynamic center (ac). It is believed that

because of the extensive wind tunnel data, such practices becomes possible. As a result,

the moment equation requires modification to accommodate the change. In addition, the

CG location is no longer assumed at the ac as in the case of B747. The moment equation

implemented in the model is the following:

M = Maero + Meng + (CP−CG)× Faero (2.45)


Fig. 2.3: NASA GTM Simulink Environment

2.4 Damaged Aircraft Modeling

Aircraft damage can range from single component failure/malfunction to severe airframe

and engine damage. Different failure situations pose different levels of severity and threat

to the flight safety. In the case of sensor failure, the original system could be recovered as

long as the correct information is available elsewhere, either from physically redundant

sensors or from observers or estimators based on analytical redundancy. Actuator failures

are more involved than the sensor case. After the actuator failure occurs, if the original

performance is still desired, the remaindering actuators have to operate beyond their

design capabilities. This means actuator saturation and further system performance

degradation. Thus, in the case of actuator failures the system should accept graceful

degradation in performance. The airframe structural damage can be the most difficult

to deal with. Not only does it compromise the aircraft integrity, but also alters the

aircraft original flight envelope. Thus, great effort must be put into the structural damage

scenarios. Additionally, it is important to have a close representation of the fault when


designing the fault-tolerant control. Table 2.1 lists a number of common fault scenarios

and their respective effects.

In this work, the primary focus is on actuator failures and airframe structural damage.

In the case of the B747, loss of actuator effectiveness will be considered. Let u be the

actuator vector of the control design,

u =

[u1 u2 · · · ui

]T(2.46)

where i = 1 · · · s (max number of actuators). Let Λ be the control effective matrix to

model the actuator faults. Λ is a diagonal matrix with positive elements.

Λ =

Λ1 0 . . . 0

0 Λ2 . . . 0

......

. . ....

0 0 . . . Λi

(2.47)

The actuator fault model is

u = Λu (2.48)

where 0 < Λi ≤ 1. When Λi = 1, it means no faults occurred in the ith actuator. If

Λi < 1, it implies the faults has impaired the ith actuator’s function.


Table 2.1: Failure modes [30]

Failure Mode Effect

Control loss on actuators Surface stuck at last position

Structural loss on control surfaceControl effectiveness reduced

minor change in aerodynamics

Engine(s) out Asymmetric thrust, increased drag due to β

Severe structural damage

Large change in possible operating region

significant change in aerodynamics,

mass and moments of inertia

Fig. 2.4: GTM Damage Case [20], Note the included model support string in the grids.

The NASA GTM model includes six damage models, ranging from rudder off to

left horizontal stabilizer off. Each damage scenario provides a unique design challenge.

However, this work concentrates on the damage scenario six (Fig. 2.4), which is the

loss of entire left horizontal stabilizer and the left elevator. The damage reduces the

longitudinal stability as well as the pitch control power due to the elimination of the left

elevator. The control asymmetry generates an undesired rolling moment which needs

to be compensated for with some roll control. In addition, the aircraft is no longer

symmetric along the fuselage centerline, which means off-diagonal inertias are non-zero

and the CG location is shifted forward, down and to the right of the fuselage centerline.

The GTM simulink model is based on a 5.5% scaled down aircraft model. For the sake of

simlicity, the simulink model is used instead of the scaled up aircraft model. The damage


involves the following variables change:

∆W = −0.59lbs (2.49)

∆x = −0.553in (2.50)

∆y = +0.088in (2.51)

∆z = −0.032in (2.52)

∆Ixx = −0.00918sl − ft2 (2.53)

∆Iyy = −0.27315sl − ft2 (2.54)

∆Izz = −0.28049sl − ft2 (2.55)

∆Ixz = −0.01559sl − ft2 (2.56)

∆Ixy = +0.04370sl − ft2 (2.57)

∆Iyz = +0.00265sl − ft2 (2.58)

where ∆W is the change in the aircplane weight; ∆x, ∆y and ∆z are the C.G. shift; The

rest describes the change in the moment of inertia.

2.5 Trim Analysis

Aircraft trim analysis is an important procedure to evaluate the aircraft behavior. As a

part of the analysis, the trim routine is used to find an equilibrium point of the aircraft

under a given set of constraints. Generally, an aircraft in-flight can be trimmed in

several conditions: steady-state level flight, steady state climbing/descending or constant

turning. The steady-state level flight is particularly interesting in this case. The steady-

state flight condition means that the time derivatives of the state variables are zero. A

steady-state point is often used as an initial point of a simulation. Thus, it is important

to find a set of control inputs and state values corresponded to an equilibrium point

of the system. As a result, the objective of the trim routine is to solve the aircraft

nonlinear equations of motion which are first order differential equations, to obtain state

and control vectors that ensure the time derivatives of state variables are zero.


A simplex optimization problem is formulated based on the cost function Eq.(2.59)

adopted from [50], aircraft dynamics and the steady-state level flight constraint, Eq.(2.60)

to obtain trimmed state and control vectors.

J = V 2t + 100(α2 + β2) + 10(p2 + q2 + r2) (2.59)

γ = 0 (2.60)

where γ = 0 means the flight path angle must be zero during the steady-state level flight.

Fig. 2.5: Trim routine procedure

The objective of the optimization is to minimize the cost function by varying the

control input variables. In the case of B747, the state vector is

x =

[Vt α β φ θ ψ p q r xe ye he

]Tand the control vector is

u =

[δth δe δa δr

]T, where δth is the thrust setting from 0-1; δe is the elevator deflec-

tion; δa is the aileron deflection; δr is the rudder deflection. Fig.2.5 illustrates the trim

routine procedure.

The numerical trim analysis is performed on the Boeing 747 model. The steady-state

flight condition is based on the aircraft altitude he and airspeed Vt, which are specified at

the beginning of the trim routine. In this case, he = 3000m and Vt = 150m/s are specified

for the aircraft descent and approach scenario. Table 2.2 and 2.3 list the trimmed state

and control vectors. Fig. 2.6- 2.17 show the 30 seconds open-loop time history.


Table 2.2: Trimmed States

State Value Derivative State Value Derivative

Vt 150m/s -5.31e-035 m/s2 p 0 rad/s 4.58e-018 rad/s2

α 9.36e-002 rad 0 rad/s q 0rad/s 0 rad/s2

β 0rad 1.79e-019 rad/s r 0rad/s 0 rad/s2

φ 0rad 0 rad/s xe 0m 150 m/s

θ 9.36e-002 rad 0 rad/s ye 0m -2.96e-016 m/s

ψ 0 rad 0 rad/s ze 3000m 0 m/s

Table 2.3: Trimmed Controls

Control Value Control Value

δth 1.90e-001 δe -8.54e-001deg

δa 9.36e-002deg δr 0deg

0 5 10 15 20 25 30149

149.5

150

150.5

151

Time (s)

Vt (

m/s

)

Fig. 2.6: airspeed

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

Time (s)

φ (d

eg)

Fig. 2.7: roll angle


0 5 10 15 20 25 305.2

5.25

5.3

5.35

5.4

5.45

5.5

Time (s)

α (d

eg)

Fig. 2.8: Angle of attack

0 5 10 15 20 25 305.2

5.25

5.3

5.35

5.4

5.45

5.5

Time (s)

θ (d

eg)

Fig. 2.9: pitch angle

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

Time (s)

β (d

eg)

Fig. 2.10: Sideslip angle

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

Time (s)

ψ (

deg)

Fig. 2.11: yaw angle

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

Time (s)

p (d

eg/s

)

Fig. 2.12: Roll rate

0 5 10 15 20 25 300

1000

2000

3000

4000

5000

Time (s)

x e (m

)

Fig. 2.13: xe

0 5 10 15 20 25 30−2

0

2

4

6

8

10x 10

−3

Time (s)

q (d

eg/s

)

Fig. 2.14: pitch rate

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

Time (s)

y e (m

)

Fig. 2.15: ye


0 5 10 15 20 25 30−1

−0.5

0

0.5

1

Time (s)

r (d

eg/s

)

Fig. 2.16: Yaw rate

0 5 10 15 20 25 302999

2999.5

3000

3000.5

3001

Time (s)

h e (m

)

Fig. 2.17: Altitude

2.6 Summary

This section covered the fundamental aircraft dynamics. The nonlinear equations of

motion were derived in this chapter. Two aircraft models were established to be used

as test beds for the fault-tolerant control design in the later sections. Furthermore, the

damaged aircraft dynamics were also presented. The trim routine was implemented to

find the steady state conditions for the simulation scenarios.

Chapter 3

Fault-tolerant Flight Guidance and

Control Problem

3.1 Introduction

In this chapter, the fault-tolerant flight guidance and control problem is identified and

formulated and the guidance law design is derived. The objective is to provide a feasible

framework that is capable of handling aircraft that have suffered from damage so that

stabilization and safe landing are achieved. The framework comprises the guidance and

control loops. Each loop has its own design approach and objective, but overall acts in

an integrated, continuous fashion.

3.2 Problem Formulation

In this section, a detailed description of the fault-tolerant flight guidance and control

problem is introduced. The proposed framework that provides a feasible and viable

solution is also presented in the section. When the aircraft encounters damage in flight,

the conventional control design may not be adequate and robust enough to handle the

situation. Eventually, the aircraft may become uncontrollable and unstable. Most of the

time, human pilot intervention is required to prevent the situation from deteriorating to

25

Chapter 3. Fault-tolerant Flight Guidance and Control Problem 26

the worst. However, human error is becoming a major contributing factor to aviation

accidents. Thus, it is desired to implement the advanced control system that is capable

of actively providing intelligent and effective actuator control in such situations as well

as to backup the conventional flight control in normal flight conditions.

In addition, damaged aircraft can behave drastically different from the original aircraft

specifications. The flight envelope can be altered as well. Since the aircraft dynamics

are intrinsically nonlinear, linear control methods sometimes are not adequate to handle

the complex dynamics. The nonlinear robust controller on the other hand is able to

tolerant significant aircraft parameter variation. For sudden, large scale behavior changes,

nonlinear controller is far superior than the linear controller which may not be able to

control the plant at all. The nonlinear controller does not require extensive and time-

consuming gain scheduling due to the large number of design points. The unpredictable

nature of damaged scenarios can also increase the level of complexity of gain scheduling in

the linear controller. Thus, the nonlinear controller is more suitable in the fault-tolerant

flight control.

Furthermore, the ultimate goal for any damaged aircraft is to land safely. A robust

guidance law is a pre-requisite for the control system. The robust guidance law should

take the altered aerodynamics and performance change into consideration when gener-

ating guidance commands. In addition, from a practical point of view, it is beneficial to

have the robust guidance design fitted into the existing flight control design system so

that little system modification is necessary.

The proposed fault-tolerant flight guidance and control framework addresses the issues

mentioned above and provides a sophisticated system to deal with damaged aircraft.

Fig. 3.1 depicts the proposed system.

The control part of the system adopts the conventional three loop flight control system

design with a separate speed controller. Alternatively, an integrated speed controller

can be augmented with the inner loop. Since the guidance loop generates the airspeed


tracking command, it is more sensible to have a separate speed control loop to track the

signal. In addition, the inner loop has fast states, such as the roll rate, pitch rate, and

yaw rate, and the airspeed dynamics is not as fast as the inner loop state; as a result,

the tracking performance may not be as accurate as expected. The inner loop consists of

the [ p; q ; r ] T state vector. The output is the control vector [δe ; δa ; δr]T . The inner

loop is responsible for stabilizing the aircraft as well as tracking the commands from the

outer loop. The outer loop is designed with [α ;β; µ ] T , where µ is the bank angle. The

flight path angle loop is responsible for keeping track of guidance commands, which are

[γ ; χ ] T , the vertical flight path angle and the heading angle. The guidance loop also

provides the speed command which feeds directly to the speed controller, whose output

is the throttle setting δth. The input to the guidance loop is based on the trajectory

information defined in the trajectory loop. State feedback is required for all of the loops.

Fig. 3.1: Proposed fault-tolerant flight guidance and control system

The control system design is based on the nonlinear state-dependent Riccati equation

(SDRE) control method. In each loop, the SDRE controller is designed to track the

commands generated by the previous loop. State feedback is required in each loop to

provide the information for SDRE. The guidance commands are generated by the zero

effort miss concept guidance law [41] that transforms the traditional trajectory track-

ing problem into the aircraft-target intercepting problem. The integration between the

guidance and control systems is done in a harmonious fashion. Little modification is

required to accommodate the guidance commands into the traditional three loop control

architecture.


3.3 Remarks

The proposed fault-tolerant flight guidance and control framework is the backbone of the

work. Not only does it provide the guideline to the guidance and control systems design,

but also illustrates a sophisticated and advanced system. In the following chapters, the

actual system design work are carried out in details. The flight simulation results are

also included.

3.4 Aircraft Guidance Law Design

In this section, the aircraft guidance law design is presented. The guidance law is used in

the framework to provide the flight path angle, the heading angle as well as the airspeed

commands. Additionally, the guidance law can be readily fit into the existing control

system architecture so that little modification is required during the system integration.

In the following sections, the detailed design is based on the work of No et al. [41].

The guidance system is based on the concept of zero effort miss, which is a common

notion in the missile guidance community and has been used in a number of proportional

navigation guidance laws [56]. Essentially, the aircraft under the guidance law commands

tries to intercept the reference trajectory on which an ideal imaginary aircraft flies. The

guidance law navigates the aircraft to follow the imaginary aircraft as close as possible.

By doing so, the trajectory tracking objective is achieved with minimum error. Thus, the

traditional trajectory tracking problem is reformulated into an aircraft-target intercept

problem.

3.5 Guidance Law Design

The core of the guidance law is the zero effort miss concept. Since the guidance law is

based on the aircraft-target interception problem, the zero effort vector is defined in such

a scenario. Let there be an ideal, imaginary target aircraft flying on the trajectory. The


real aircraft tries to intercept the imaginary target. Denote (d,v) and (d∗,v∗) as the

position and velocity vectors for the aircraft and target, respectively. These vectors can

be expanded into the components in the reference inertia frame (ex, ey, ez) as

d = dxex + dyey + dzez (3.1)

v = vxex + vyey + vzez (3.2)

d∗ = d∗xex + d∗yey + d∗zez (3.3)

v∗ = v∗xex + v∗yey + v∗zez (3.4)

Assume both the aircraft and target maintain their speed and direction, the distance

vector between them as shown in Fig. 3.2, at some time in future tf can be expressed as

dtgo = (d∗ − d) + (v∗ − v)tgo (3.5)

= Txex + Tyey + Tzez (3.6)

where tgo is the time-to-go until the future time tf ,

tgo = tf − t (3.7)

(Tx, Ty, Tz) are the components of the zero effort miss vector is the fixed frame.

Tx = d∗x − dx + (v∗x − vx)tgo (3.8)

Ty = d∗y − dy + (v∗y − vy)tgo (3.9)

Tz = d∗z − dz + (v∗z − vz)tgo (3.10)

The vector in Eq.(3.5) is often referred to as the zero effort miss vector.

Fig. 3.2: Zero effort miss vector


The actual guidance commands are derived through the use of a Lyapunov-like func-

tion.

V =1

2dtgo · dtgo (3.11)

=1

2(T 2

x + T 2y + T 2

z ) (3.12)

Taking time derivative of Eq.(3.12),

dV

dt= Tx(v

∗x − vx)tgo + Ty(v

∗y − vy)tgo + Tz(v

∗z − vz)tgo (3.13)

The velocity vector can then be expressed in terms of the flight path angle γ, and the

heading angle χ.

v = vcosγcosχex + vcosγsinχey − vsinγez (3.14)

Assuming the coordinate turn is achieved, so the sideslip angle β = 0, then the

heading angle can be approximated by the yaw angle ψ,

χ ≈ ψ (3.15)

Substituting the above two equations into the Eq.(3.13),

dV

dt= Tx(v

∗x − vcosψcosγ + vψsinψcosγ + vγcosψsinγ)tgo

+ Ty(v∗y − vsinψcosγ − vψcosψcosγ + vγsinψsinγ)tgo

+ Tz(v∗z + vsinγ + vγcosγ)tgo (3.16)

Following No et al [41], Eq.(3.13) can be transformed from the fixed frame (ex, ey, ez)

to a control frame (ev, eψ, eγ)which includes the airspeed, the flight angle and the head-

ing angle, where ev is the unit direction vector along the velocity; eγ is a unit vector

perpendicular to ev and is positive in the direction of the increasing longitudinal flight

path angle; eψ is along the direction of the increasing yaw angle and follows the right

hand rule. As a result,

dV

dt= (v∗v − v)Tvtgo + (v∗ψ − vψcosγ)Tψtgo + (v∗γ + vγ)Tγtgo (3.17)


where (v∗v , v∗ψ, v

∗γ) denote the target acceleration vector in the control frame. (Tv, Tγ, Tψ)

are the components in the control frame,

Tv = Txcosψcosγ + Tysinψcosγ − Tzsinγ (3.18)

Tψ = −Txsinψ + Tycosψ (3.19)

Tγ = Txcosψsinγ + Tysinψsinγ + Tzcosγ (3.20)

To ensure the Lyapunov stability theorem can be employed, Eq.(3.13) must be nega-

tive definiteness, which is achieved by

dV

dt= −2NV (3.21)

where N is a positive constant. One of the advantages of the guidance law is that the

aircraft dynamics are taken into consideration. However, the entire dynamics are too

complex to include. First order approximations are used to describe the control channels

for the airspeed v, the flight path angle γ, and the heading angle ψ. Speed control loop:

v =1

τv(vc − v) (3.22)

Flight path angle loop:

γ =1

τγ(γc − γ) (3.23)

Heading angle loop:

ψ =1

τψ(ψc − ψ) (3.24)

where τv, τγ and τψ are the time constants of each control loop. vc, γc and ψc are the

input commands to the control loops. Finally, Eq.(3.21) becomes,

dV

dt= (v∗v − v)MTvtgo + (v∗ψ − vψcosγ)Tψtgo + (v∗γ + vγ)Tψtgo (3.25)

= −2NV (3.26)

= −NT 2v −NT 2

ψ −NT 2γ (3.27)


As suggested by No et al. [41], the natural selection of guidance commands to satisfy

Eq.(3.27) appear to be,

vc = v +N

tgoτvTv + τvv

∗v (3.28)

ψc = ψ +N

tgo

τψvcosγ

Tψ +τψ

vcosγv∗ψ (3.29)

γc = γ − N

tgo

τγvTγ −

τγvv∗γ (3.30)

The set of guidance commands provide the airspeed, the flight path angle and the heading

angle to intercept the target. By enforcing a small miss distance error, in other words

keeping zero effort miss vector dtgo small, for short tgo, the aircraft follows the imaginary

target and stays on the desired trajectory with a small error.

As expected, the guidance laws Eq.(3.28), (3.29), and (3.30) are feedback based com-

mands. For the impaired aircraft case, the ideal aircraft must consider the impaired

aircraft performance degradation. For example,

v∗x = v · cosγ (3.31)

v∗z = v · sinγ (3.32)

where v and γ are the feedback values of the impaired aircraft.

3.6 Summary

In this chapter, The fault-tolerant flight guidance and control problem was formulated.

The guidance law design was introduced. It is based on the zero effort miss concept that

has been used in a number of proportional guidance law designs. The design transformed

the traditional guidance law into an aircraft-target interception problem. By intercepting

the target aircraft, the real aircraft stays on the desired trajectory with a small error.

The guidance law has several advantageous features. The guidance commands are based

on the feedback as well as the aircraft dynamics. The design parameters are similar to

the control gains, requiring proper tuning.

Chapter 4

State-Dependent Riccati Equation

Control Method

In this chapter, the control method implemented in the fault-tolerant flight guidance and

control framework is discussed in detail. The state-dependent Riccati equation (SDRE)

control method is reviewed first. The background mathematical preliminaries, control

problem formulation and design technique are also presented. The SDRE is a unique

control method among nonlinear control methods. It embraces the advantages of linear

controller design techniques while applying to nonlinear system dynamics. In the end,

simulation results are included and discussed.

The nonlinear controller design is intrinsically more difficult than the linear controller

design. It requires rigorous and sophisticated mathematical background to ensure proper

formulation and analysis are performed. Despite these difficult obstacles, research on the

topic of nonlinear control method has flourished and made noticeable advances in recent

years [24, 27]. However, there are still challenging questions awaiting to be answered in

the field. The lack of connection between the theoretic work and the practical implemen-

tation prevents many modern nonlinear control methods from being applied. In addition,

stability, performance and robustness continued to be the issues that nonlinear control

methods struggle to address satisfactorily.

33

Chapter 4. State-Dependent Riccati Equation Control Method 34

The SDRE control method appearers to be a very practical nonlinear control method

for the systematic design of nonlinear controllers. It has become very popular within the

control community over the last decade, providing an extremely effective algorithm for

synthesizing nonlinear feedback controls by allowing nonlinearities in the system states,

while additionally offering great design flexibility through design metrics [9]. The control

method was originally introduced by Pearson [44] in the 1970s and later refined by Wernli

and Cook [52]. In recent years, Cloutier, D’Souza and Mracek [11, 12, 38] independently

studied the control method. The SDRE method provides a straightforward and efficient

computational algorithm to solve difficult nonlinear problems, which are often compli-

cated by nonaffine-in-control, control, or state constraints. The backbone of the method

is state parameterization. It allows the nonlinear dynamics expressed by differential equa-

tions to be parametrized into the product of a matrix-valued function and the state vector

while preserving the original system nonlinearities. In the end, a linear-like structure is

obtained in state space form. The coefficients are state-dependent and non-unique. The

control method has been successfully implemented in a variety of practical applications

across disciplines. Specifically, Mracek and Cloutier [37] applied the SDRE method to a

full envelope missile longitudinal autopilot. Cimen [8] proposed an approximate SDRE

nonlinear tracking method which was used to design a supertanker’s autopilot. Gao [21]

implemented the SDRE control method in a re-entry tracking problem for a reusable

launch vehicle (RLV). Bogdanov [5] flight tested the SDRE controller on board a small

unmanned helicopter. Flight tests were flown to evaluate the accuracy of tracking under

SDRE control. These works demonstrate that the SDRE control method is a capable

nonlinear control method and has great potential in the practical implementations.

The control method solves an algebraic Riccati equation (ARE) to construct the sub-

optimal control law. The interesting fact is that because of the state-dependent nature

of the coefficients, the ARE is solved at each step with varying coefficients. It means the

feedback control gain varies at each step as well. This is certainly a desirable feature


of SDRE in the fault-tolerant flight control design. The control law can actively modify

itself in response to the aircraft parameter changes. In addition, extra design freedom is

available through the non-uniqueness of state-dependent coefficients.

In the following sections, the SDRE nonlinear control method is first reviewed in

Sec. 4.1 with the control problem formulation. The section also covers the state-dependent

coefficient parameterization or extended linearization. The SDRE stability and optimal-

ity analysis are offered in Sec. 4.2. The SDRE design techniques are presented in Sec. 4.3.

Simulation studies are followed in Sec. 4.4.

4.1 SDRE Control Method

Consider the general autonomous, affine-in-control, nonlinear system dynamics in the

form of,

x(t) = f(x) + B(x)u(t) x(0) = x0 (4.1)

where state vector x ∈ <n and control vector u ∈ <m; f : <n 7→ <n and B : <n 7→ <n×m

with B 6= 0, ∀x.

The nonlinear regulator problem is formed as the following. Minimize the infinite-

horizon performance index,

J =1

2

∫ ∞0

(xTQ(x)x + uTR(x)u)dt (4.2)

with respect to the state vector x and the control vector u subject to the nonlinear

system dynamics Eq.(4.1). The state and control weighting matrices Q(x), R(x) are

state-dependent, such that Q(x) is positive semi-definite and R(x) is positive definite

for all x. Additionally, Q(x) can be expressed as Q(x) = C(x)TC(x).

In order to proceed with the SDRE control law, the state-dependent coefficients

(SDCs) must be introduced. SDCs are obtained through a procedure known as ex-

tended linearization [19], apparent linearization [52], or SDC parameterization [12]. It


is a procedure to bring the nonlinear dynamics into a linear-like structure expressed by

SDCs in addition to the state and control vectors. It is important to assume,

Assumption 1. f(x) is continuously differentiable with respect to x for all x.

Assumption 2. Without the loss of generality, the origin x = 0 is an equilibrium point

of the system with u = 0. It implies f(0) = 0 and B(0) 6= 0.

so that the existence of a global SDC parameterization of f(x) is guaranteed [51]. As a

result, the nonlinear differential equations, Eq.(4.1) can be expressed as,

x = A(x)x + B(x)u(t) x(0) = x0 (4.3)

f(x) = A(x)x (4.4)

where A(x) and B(x) are the state-dependent coefficients. The following definitions are

associated with the SDCs.

Definition 1. A(x) is a controllable parameterization of the nonlinear system if the pair

A(x),B(x) is controllable for all x

Definition 2. A(x) is a stabilizable parameterization of the nonlinear system if the pair

A(x),B(x) is stabilizable for all x

Definition 3. A(x) is Hurwitz if all the eigenvalues of A(x) are in the open left plane

(negative real parts) for all x

In addition to the assumptions mentioned above, the following assumption must also

be met,

Assumption 3. A(·), B(·), Q(·), and R(·) are C1(<n) matrix-valued functions

Assumption 4. The pair A(x), B(x) and A(x),Q1/2(x) are pointwise stabilizable

and detectable SDC parameterizations of the nonlinear system 4.1 for all x, respectively.


The SDRE control design is similar to the Linear Quadratic Regulator (LQR) control

method. In the case of SDRE, the state-dependent Riccati equation is solved at each

step to construct the control law. The state feedback controller shares the similar form

with LQR.

u(x) = −R−1(x)BT(x)P(x)x (4.5)

where P(x) is the unique, symmetric, positive definite solution to the state-dependent

Riccati equation,

P(x)A(x) + AT(x)P(x)−P(x)B(x)R−1(x)BT(x)P(x) + Q(x) = 0 (4.6)

The closed loop dynamics become:x = [A(x)−B(x)R−1(x)BT(x)P(x)]x (4.7)

The nonlinear state feedback gain is,K(x) = R−1(x)BT(x)P(x) (4.8)

Fig. 4.1: SDRE design flowchart

Clearly, the control gain is dependent on the state vector x. It also varies every

time the SDRE is solved. The direct benefits of SDRE method is its simplicity and

effectiveness. There is no attempt to solve the Hamilton-Jacobi-Bellman equation. When

the coefficients and weighting matrices are constant, the SDRE problem becomes the

well-known LQR problem.


Fig. 4.1 shows the systematic procedures to construct the nonlinear state feedback

gain with the SDRE control method.

4.2 Stability and Optimality Analysis

4.2.1 Stability Analysis

Stability is an important issue for any controller. Nonlinear system stability is well

defined by the Lyapunov stability theories [27]. Although global asymptotic stability

of the closed-loop system is highly desirable, such a property is difficult to achieve and

prove. In the case of SDRE control method, global asymptotic stability can only be

proved in two special cases. In the first case, the closed-loop coefficient matrix ACL(x)

is assumed to possess a special structure. The second case involved the single system

state, n = 1. Despite the difficulty to prove the global asymptotic stability property, the

local asymptotic stability is well proven [10] and presented here.

Theorem 1. [39] Consider the nonlinear multivaribale system Eq.(4.1) with feedback

control Eq.(4.5) applied where x ∈ <n (n > 1) and P(x) is the unique, symmetric,

positive-definite, pointwise-stabilizing solution of the SDRE Eq.(4.6). Then, under As-

sumptions 3 and 4, the SDRE method produces a closed-loop solution which is locall

asymptotically stable

Proof. Using SDRE control, the closed-loop solution becomes x = ACL(x)x, where

ACL(x) is the closed-loop SDC matrix. From Riccati equation theory, ACL(x) is guar-

anteed to be stable at every point x. Under the smoothness assumptions of Assumption

4, P(x) is continuously differentiable and hence so is ACL(x). Applying the Mean Value

Theorem to ACL(x) gives,

ACL(x) = ACL(0) +∂ACL(z)

∂xx (4.9)

where ∂ACL(z)∂x

generates a tensor, and the vector z is that point on the line segment


joining the origin 0 and x. As a result,

x = ACL(x)x (4.10)

x = (ACL(0) +∂ACL(z)

∂xx)x (4.11)

x = ACL(0)x + xT∂ACL(z)

∂xx (4.12)

which gives,

x = ACL(0)x + ψ(x, z) ‖x‖ (4.13)

where ψ(x, z) = 1‖x‖x

T ∂ACL(z)∂x

x, such that lim‖x‖→0 ψ(x, z) = 0. Hence, in a neighbor-

hood about the origin, the linear term which has a constant stable coefficient matrix

ACL(0) dominates the higher-order terms, yielding the local asymptotic stability.

In the case of global asymptotic stability property, the following two Theorems apply,

Theorem 2. [12] If the closed-loop coefficient matrix ACL(x) is symmetric for all

x, then under the conditions given by the Assumptions 4 and 5, the SDRE closed-loop

solution is global asymptotically stable.

Theorem 3. In the scalar case (n=1), the SDRE closed-loop solution is globally asymp-

totically stable.

The proofs of the above Theorem can be found in the Cloutier et al. [12].

4.2.2 Optimality Analysis

The SDRE control method is often regarded as a sub-optimal nonlinear control method.

In this subsection, the optimality property of the SDRE method is addressed.

Assumption 5. A(x), B(x), P(x), Q(x) and R(x) along with their gradients ∂A(x)∂x

,

∂B(x)∂x

, ∂P(x)∂x

, ∂Q(x)∂x

, ∂R(x)∂x

are bounded in a neighborhood Ω about the origin.


Theorem 4. [39] In the general multivariable case (n > 1), the SDRE nonlinear feed-

back solution and its associated state and costate trajectories satisfy the first necessary

condition for optimality (∂H∂u

= 0) of the nonlinear optimal regulator problem Eq.(4.1)

and Eq.(4.2), where H is the Hamiltonian of the system. Additionally, if Assumption 5

holds, under asymptotic stability, as the state x is driven to zero, the second necessary

condition for optimality λ = −∂H∂x

is asymptotically satisfied at a quadratic rate.

The proof is available in the work of Mracek et al. [39]. Theorem 4 shows the sub-

optimality property of the SDRE control method. As the Theorem states, the second

condition for optimality is only satisfied asymptotically. However, similar to the stability

property, SDRE global optimality property is possible under special circumstances [7],

such as in the scalar case.

4.3 The Art and Capabilities of SDRE

The SDRE controller design provides an effective and systematic way to construct the

nonlinear controller. It also offers great design flexibility to the designer via state-

dependent matrices. In this section, some of the important SDRE design techniques

and their capabilities are covered. In order to implement the SDRE method, the sys-

tem must conform to the SDRE design requirements. In this case, the system must be

affinity-in-control and f(0) = 0. In this section, the techniques are also introduced to

deal with non-conforming nonlinear systems so that the SDRE control method can still

be applied to such systems. These concepts and techniques are implemented in the sim-

ulation examples to demonstrate the design approach, controller performance as well as

their promising results.

Design Freedom

One of the main advantages of the SDRE method is the design freedom, which is via

the state-dependent coefficients to capture system nonlinearity. The parameterization is


not unique in the multivariable case (n > 1) as mentioned earlier. The designer has the

choice to decide which parameterization fits best to the overall system design. Further-

more, additional design freedom can be introduced in the parameterization process to

enhance the system performance and stability. The SDC parameterization A(x) itself

can be factored into A(x, α), where α is the vector of free design parameters. The in-

troduction of α is not common to traditional methods and is unique in the SDRE. To

satisfy the conditions of ARE, the pair (A(x, α),B(x)) must be pointwise stabilizable.

The free design parameters are beneficial to the controller performance. Because of the

non-unique nature, it can be used to avoid singularities or loss of controllability. They

can also be used together with the state-dependent weighting matrices to enhance the

system flexibility as well as effect trade offs between performance, optimality, stability,

robustness, and disturbance rejection, thus offering a more robust nonlinear control law.

The following example demonstrates the extra degree of freedom concept. Consider the

nonlinear system,

x3 = x1x2 (4.14)

The state is x =

[x1 x2 x3

]T. There are three distinct parameterizations for x3

dynamics.

x3 =

[x2 0 0

]︸︷︷︸

A1(x)

x1

x2

x3

x3 =

[0 x1 0

]︸︷︷︸

A2(x)

x1

x2

x3

x3 =

[αx2 (1− α)x1 0

]︸︷︷︸

A3(x)

x1

x2

x3

(4.15)

Evidently, neither A1(x) nor A2(x) parameterization captures the fact that x3 dy-

namics depend on both x1 and x2. Thus, a better parametrization, A3(x, α), with the

extra degree of freedom, α, is preferred.

Integral Servomechanism

In order to perform command tracking, the SDRE controller can be implemented

as an integral servomechanism as demonstrated in [13]. A number of modifications to

the existing regulator problem is required. First, the state vector x is decomposed as


xT =

[xTR xT

N

], where xR is the vector including states to track the reference command

rc and xN is the vector including the rest state. The state vector x is then augmented

with the integral state of xR, xI,

x =

[xI xR xN

]T(4.16)

The augmented system is given by

˙x = A(x, α) + B(x)u (4.17)

where

A(x, α) =

0 I... 0

0 A(x, α)

B(x) =

0

B(x)

(4.18)

and the SDRE integral servo controller is given by

u = −R−1(x)BT(x)P(x)

xI −

∫rcdt

xR − rc

xN

(4.19)

In order for the SDRE to have a solution, the pointwise detectability condition must

be satisfied. This is accomplished by penalizing the integral states with the corresponding

non-zero diagonal elements of Q(x).

Non-conforming systems

The above mentioned techniques are only applicable to the conforming systems,

namely affinity-in-control, f(x) continuously differentiable and f(0) = 0. In order to

extend the SDRE method to general nonlinear systems, the non-conforming part of the

system must be converted. Then, the systematic design procedures are performed to con-

struct the control law. In this segment, a couple of techniques are presented to handle

non-conforming systems so that the newly converted system meets the conditions and

requirements set out in the previous sections.

The presence of state-independent terms: if the system possess state-independent

terms, sometimes called biased terms, b(t), which can be time varying, the assumption


f(0) = 0 is violated. Thus, the parameterization procedure can not be performed. There

are a number of ways to overcome the obstacle, b(t). Two pertinent approaches are briefly

covered here. In the aircraft flight control problem, any component of the velocity vector

Vt can be zero, but the speed of the airplane will not go to zero. In this case, the bias

term can be handled by multiplying and dividing by the squared of the magnitude of the

velocity vector as

b(t) =b(t)Vt

V2t

Vt (4.20)

In addition, a stable state z can be augmented to the system to solve the non-

conforming problem. For example

z(t) = −λz(t) (4.21)

with λ > 0. The biased term can be factored as

b(t) =b(t)

zz (4.22)

The second approach will be used more than the first approach given its simplicity

and flexibility. The first approach requires the velocity as a state of the system. It

poses a constraint on the system design. For some flight control systems, the velocity

is controlled by a separate speed controller. This implies the airspeed state is detached

from the main flight control design framework.

The presence of state-dependent terms which exclude the origin: Sinusoidal

functions in the aircraft nonlinear dynamic equations are examples of the state-dependent

terms which do not go to zero as the state goes to zero, e.g. cos(x). These terms like the

state-independent terms impede the direct parameterization of f(x) into A(x, α) due to

the violation of f(0) = 0 condition. To overcome this hurdle, the term must be shifted so

that it goes to zero as the state goes to zero while retaining the state dependency. This


is done by adding and subtracting a bias to the term. For example,

cosα = [cosα− 1] + 1 (4.23)

= [cosα− 1

α]α + 1 (4.24)

The function (cosα− 1) goes through zero as α approaches zero. The term can then

be factored as (cosα−1)α

α. In this instance, the biased term, 1, is created. The previously

mentioned technique can be used to deal with the additional term.

cosα = [(cosα− 1)

α]α +

1

zz (4.25)

This shifting procedure can be used for any state-dependent term which does not go

through the origin.

Non-affine in control: The nonlinear aircraft dynamics and nonlinear aerodynamic

coefficients dictate the fact that flight control design will have to deal with the non-

affinity in control. This means the control input can not be separated from the dynamics.

Dealing with the non-affinity in control problem is a difficult design task. Unlike other

nonlinear control methods, the SDRE method has superior capability to convert the

system dynamics into a pseudo-in control form that is actually affinity-in control. Then,

the systematic design approach can be implemented on the new system. Consider a

nonlinear in the control system represented by

x = f(x) + g(x,u) (4.26)

The nonlinear in control problem can be dealt with by introducing integral control

u = Cu + Du (4.27)

In its simplest form, C = 0 and D = I. The augmented system then isx

u

=

f(x) + g(x,u)

Cu

+

0

D

u (4.28)


which is affine in control. The above mentioned techniques can also be implemented in

the case that the system does not conform with the required structure. Direct parame-

terization can then be performed.

Numerical SDC: It is worth mentioning that in addition to analytical parameterization

to obtain the state-dependent coefficients, the numerical approach is also available to

perform the task. The idea is based on the perturbation of the state control vectors. The

approach is fully described in Menon et al.’s work [34].

4.4 Simulation Studies

This section focuses on the simulation studies of the guidance and SDRE flight system

framework. The framework is tested with two different damaged aircraft scenarios. The

SDRE method is implemented as the flight control law while the zero effort miss vec-

tor provides the guidance commands. In Chapter 2, Table 2.1 a list of damage cases

investigated in the work is given. In this section, only the loss-of-thrust and the eleva-

tor ineffectiveness cases are considered. The complete loss-of-thrust in flight forces the

aircraft to become a glider in the air. Any error in the calculation of the distance, and

maneuver to the nearest approachable runway can be catastrophic since the allowable

error margin is very small. In this example, the SDRE controller actively reconfigures

itself to accommodate the guidance commands and the aircraft impaired dynamics.

In the case of actuator damage, a complete flight control system with the speed

controller is designed and simulated. The flight control system consists of three loops:

the inner loop, the outer loop and the flight path angle loop as described in the previous

chapter. Similar to the loss-of-thrust example, the SDRE control law is also able to deal

with the aircraft behaviors change while maintaining the command tracking.


4.4.1 Loss-of-thrust example - UAV example

The loss-of-thrust example is based on the longitudinal dynamics of an electric powered

UAV [6]. The longitudinal state vector includes

[Vt α θ q xe he

]T. The control

input is

[δth δe

]T. The previous developed trim routine is first applied to the model to

obtain the steady-state flight condition as listed in Table 4.1.

Table 4.1: Straight and level flight trim results

States Value Derivative States Value Derivative

Vt 20 m/s 0 m/s2 q 0 rad/s 0 rad/s2

α 4.93e-2 rad 0 rad/s xe 0 m 20 m/s

θ 4.93e-2 rad 0 rad/s he 50 m 0 m/s

In the loss-of-thrust example, the damaged aircraft is commanded to track a reference

trajectory. The flight path angle command issued by the guidance law guides the aircraft

to track the reference glide slope during both the decent and flare phases.

We will formulate the control design problem as the following. The state and control

are:

x =

[ev α q θ edγ z εdγ

], u =

[δth δe

](4.29)

where e =

[ev ed

]Tis the tracking error.

ev = vc − Vt (4.30)

edγ = dγc − dγ (4.31)

The reference command dγc is zero. To further ensure dγ is exact zero, i.e. no steady-

state error, integral servomechanism control is employed in this case, εdγ . dγ is defined

as the perpendicular distance from the glide path [50].


The control objective is to regulate dγ, the off-glide-path-distance, to zero so that

the aircraft will remain on the glide path at all time. In the case of damage, this is

accomplished in the absence of the thrust control.

The component of velocity perpendicular to the glide path is given by:

dγ = Vtsin(γ − γc) = Vt(θ − α− γc) (4.32)

when (γ− γc) is small and γ = θ−α. To successfully track the trajectory, dγ = edγ must

be zero, so that γ = γc.

Applying the variable substitutions, one pair of SDC, stabilizable in the entire domain

of interest, is obtained. As mentioned earlier, SDC matrices in the multivariable case

are not unique. By using different approaches, a number of SDC variations are possible.

SDC detailed expressions are included in the Appendix B.

A(x) =

a11 a12 0 a14 0 a16 0

a21 a22 1 a24 0 a26 0

a31 0 0 0 0 a36 0

0 0 1 0 0 a46 0

a51 a52 0 a54 0 a56 0

0 0 0 0 0 −λ 0

0 0 0 0 −1 0 0

, B(x) =

b11 b12

b21 b22

b31 b32

0 0

0 0

0 0

0 0

(4.33)

Undamaged case

The proposed framework is first applied to the undamaged aircraft and compared with

the dynamic inversion (DI) control method. The goal is to track the glide path angle,

γc = −4 deg and land the aircraft while maintaining the airspeed. In the case of SDRE,

γ is indirectly controlled by dγ, whereas in the DI case it is tracked directly in the outer

loop.

The design parameters included in the framework are the proportionality constant,


N , and time-to-go, tgo, from the guidance law, and state and control weightings, Q and

R, from the SDRE as outlined in Table 4.2. A relatively large N needs to be used if one

wants the flight vehicle to follow the reference quickly. A relatively small tgo should be

employed if the precision is a more important measure of tracking performance [42].

Table 4.2: Design parameters

[H] N tgo Q R λ

32 1 diag1e5; 0.1; 0.1; 0.1; 5e5; 0; 5e5 diag1e5; 50 100

Given the fact that Vt and γ tracking are of interest, little weighting is placed on α,

q, and θ. In addition, the augmented slow state z does not require any penalty.

Figs. 4.2 - 4.9 show simulation results. Although the graphs of α, q, and θ are not

shown here for brevity, they are well behaved. Both the SDRE and DI are able to track

the reference trajectory relative well. However, the SDRE’s initial response is slower.

This may be corrected by further tuning the proportional guidance constant N in the

guidance law. In the case of DI, it is worth noting that near the end of the simulation

steady-state error exists. The small sudden variations in δth and δe are caused by the

commands from the guidance laws. The guidance commands adapt to the reference

trajectory change when the target aircraft is in transition from the decent to the flare

phase.


0 10 20 30 40 50 6019.6

19.7

19.8

19.9

20

20.1

20.2

20.3

Time (s)

Vt (

m/s

)

Vt

Vtc

Fig. 4.2: Airspeed

The dashed red line indicates the guidance command; the

solid blue line is the aircraft response.

0 10 20 30 40 50 60−5

−4

−3

−2

−1

0

1

Time (s)

γ (d

eg)

γγ

c

Fig. 4.3: Flight path angle

The dashed red line indicates the guidance command; the

solid blue line is the aircraft response.

0 10 20 30 40 50 602.7

2.75

2.8

2.85

2.9

2.95

Time (s)

α(de

g)


0 10 20 30 40 50 60−1.5

−1

−0.5

0

0.5

1

Time (s)

q (d

eg/s

)

Fig. 4.5: Pitch rate

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

Time (s)

δ th

Fig. 4.6: Throttle control

0 10 20 30 40 50 60−8

−6

−4

−2

0

2

4

6

Time (s)

δ e (

deg)

Fig. 4.7: Elevator deflection angle


0 10 20 30 40 50 60−0.5

0

0.5

1

Time (s)

Dis

tanc

e er

ror

(m)

Fig. 4.8: Tracking distance error

0200

400600

8001000

1200

−1

0

10

20

40

60

xe (m)

ye (m)

he (m)

he

Ref. traj.

Fig. 4.9: Altitude

The solid red line is the reference trajectory; the solid blue

line is the aircraft response.

Damaged case I

The SDRE control with the zero effort miss concept guidance law is implemented in

the case of loss-of-thrust. The aircraft losses its thrust force in flight. Essentially the

aircraft becomes a glider. Obviously in this case, the aircraft can no longer maintain its

speed. The guidance commands are based on an imaginary damaged aircraft flying on

the trajectory. This is different from the previous example where an ideal target flies

on the path. To ensure that the aircraft does not enter the stall region prematurely, a

steeper glide slope is required. It is expected such an adjustment are incorporated in the

flight management computer.

The control design is challenging in this case due to the small elevator upper limit,

δe ≤ 5deg. The elevator does not have a lot movement initially to respond to γcmd. A

few seconds of control saturation is observed in this case. In addition, when the aircraft

enters the flare phase, it is expected to pitch up and prepare for landing despite the

absence of the thrust control.

Fig. 4.10-4.17 show simulation results with the SDRE controller. Fig. 4.10 shows

the time history of the airspeed. As it is expected, the airspeed gradually decreases.

Its rate of decrease depends greatly on the aircraft rate of descent, which is reflected


by the flight path angle, shown in Fig. 4.11. The α and q plots concur with γ plot.

Fig. 4.14 shows the aircraft sudden loss-of-thrust. Fig. 4.17 illustrates the elevator time

history. Overall, SDRE performance is acceptable in the following aspects: first of all,

from the state point of view, at the end of the path, the aircraft speed is higher which

is desirable. It means the aircraft is not in the region of stall. Secondly, α, q, and θ are

more behaved in the SDRE case. For example, α at the end of simulation is about 8deg.

Furthermore, the SDRE control input does not saturate in the flare phase. Any control

input saturation poses a threat to the aircraft controllability. When the aircraft already

suffers from adverse conditions, the control saturations may well push the aircraft over

the limit causing catastrophic failures.

0 5 10 15 20 25 30 35 40 4514

15

16

17

18

19

20

Time (s)

Vt (

m/s

)

Fig. 4.10: Airspeed

0 5 10 15 20 25 30 35 40 45−5

−4

−3

−2

−1

0

1

Time (s)

γ (d

eg)


0 5 10 15 20 25 30 35 40 452

3

4

5

6

7

8

Time (s)

α (d

eg)


0 5 10 15 20 25 30 35 40 45−1.5

−1

−0.5

0

0.5

1

Time (s)

q (d

eg/s

)



0 5 10 15 20 25 30 35 40 450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (s)

δ th

Fig. 4.14: Throttle control.

The UAV throttle setting gradually drops to zero in 10

seconds.

0 5 10 15 20 25 30 35 40 45−20

−15

−10

−5

0

5

Time (s)

δ e (

deg)

Fig. 4.15: Elevator deflection angle.

Note the brief elevator saturation at the beginning of the

simulation.

0 5 10 15 20 25 30 35 40 45−1

0

1

2

3

4

5

6

7

Time (s)

θ (d

eg)

Fig. 4.16: Pitch angle

0200

400600

800

−1

0

10

20

40

60

xe (m)

ye (m)

he (m)

Fig. 4.17: UAV trajectory

Damaged case II

The SDRE control method is also compared with the existing nonlinear dynamic inversion

(NDI) control method to evaluate its performance [53, 54]. The comparison shows the

SDRE performs better than the NDI control law due to its reconfigurable ability. In this

investigation, the aircraft is subject to multiple adverse conditions in addition to the loss-

of-thrust. The control effectiveness constant Λ = 0.8 and 20% aerodynamic coefficient

uncertainties are introduced. The current research work is more focused on the proposed

control and guidance system design than the previous work, ensuring feasibility.

Fig. 4.18 illustrates the simulation comparison. The SDRE controller is sufficiently

robust against the imposed uncertainties. On the other hand, the uncertainties in the

coefficients lead to erroneous inversion in the DI case. Consequently, the DI controller fails


to land the aircraft. However, the DI’s sensitivity to modeling errors can be circumvented

if online model estimation is included, resulting in the expected performance similar to

SDRE performance is expected. The DI remedy is beyond the scope of this research.

0 5 10 15 20 25 30 35 40 4510

15

20

25Speed (m/s)

Time (s)

Vt (

m/s

)

0 5 10 15 20 25 30 35 40 45−5

0

5Flight path angle (deg)

Time (s)

γ (d

eg)

0 5 10 15 20 25 30 35 40 452

4

6

8α (deg)

Time (s)

α (d

eg)

0 5 10 15 20 25 30 35 40 45−2

−1

0

1Q (deg/s)

Time (s)

Q (

deg/

s)

0 5 10 15 20 25 30 35 40 45−5

0

5

10θ (deg)

Time (s)

θ (d

eg)

0 100 200 300 400 500 600 700−20

0

20

40

60Flight trajectory

xe (m)

h e (m

)

0 5 10 15 20 25 30 35 40 450

0.2

0.4Throttle setting

Time (s)

δ th

0 5 10 15 20 25 30 35 40 45−20

−10

0

10Elevator deflection (deg)

Time (s)

δ e (de

g)

SDREDI

SDREDI

SDREDI

SDREDIγcmd

SDREDI

SDREDI

SDREDI

Fig. 4.18: Loss-of-thrust II simulation results

4.4.2 Damaged Aircraft - B747

In this section, the B747 aircraft actuator damage case is studied. The simulation model

is developed based on the method mentioned in Chapter 2. The proposed design frame-

work 3.1 is fully implemented in this example. The framework includes a number of

feedback loops. The control design starts with the trim routine to find the straight and

level flight condition. The aforementioned optimization based trim routine is used to find

such a condition. The remainder of this section is dedicated to describe the control law


design.

Inner loop

The inner loop includes the state vector

[p q r

]T. To design the tracking controller,

the slow state z and integral servomechanism are augmented to the state vector to deal

with the bias terms and ensure zero steady-state error, respectively. This is also a chal-

lenging problem because the controls are highly nonlinear as they are embedded in the

aerodynamic coefficients as products of sines and cosines. The paremeterized dynamics

become,

p

q

r

p

q

r

z

=

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 a11 0 a13 a14

0 0 0 a21 0 a23 a24

0 0 0 a31 0 a33 a34

0 0 0 0 0 0 a44

∫p∫q∫r

p

q

r

z

+

0 0 0

0 0 0

0 0 0

0 b12 b13

b21 0 0

0 b32 b33

0 0 0

δe

δa

δr

(4.34)

The SDC elements are included in the Appendix B. The state and control weighting

matrices chosen are:

Q = diag(1e3; 1e3; 1e5; 0; 0; 0) R = diag(0.01; 0.01; 0.01) (4.35)

The selection of state and control weighting matrices is similar to the approach used in

the LQR control method. Since zero steady-state error is important in the tracking case,

the weighting on the integral states are nonzero. They can enhance the rise time and

reduce the overshoot. The control weightings also dictate the overall system response.

The inner loop stabilization is an important aspect of the fault-tolerant flight control

problem. Immediately after the damage, the aircraft must be stabilized before anything

else can be considered. Fortunately, the SDRE based control approach handles the sta-

bilization and tracking commands very well. The simulation example shows the tracking


of the inner state commands after the actuators damage are imposed to the system. Ele-

vators, ailerons and rudders suffer from 50%, 20% and 20% damage, respectively after 5

seconds. A doublet tracking command is generated for the roll, pitch and yaw states to

track at different times. Figs. 4.19 - 4.21 show the inner state tracking response, while

Figs. 4.22-4.24 illustrate the control surfaces deflection.

0 10 20 30 40 50−10

−8

−6

−4

−2

0

2

4

6

Time (s)

p (d

eg/s

)

pcmd

pp

damage

Fig. 4.19: Roll rate

The solid red line indicates the doublet command signal;

the dashed blue line is the SDRE controlled response;

the dashed black line represents the SDRE controlled damaged aircraft response

0 10 20 30 40 50−8

−6

−4

−2

0

2

4

6

Time (s)

q (d

eg/s

)

qcmd

qq

damage






0 10 20 30 40 50−8

−6

−4

−2

0

2

4

6

8

Time (s)

r (d

eg/s

)

rcmd

rrdamage

Fig. 4.21: Yaw rate




From the simulation results, the coupling between the roll and yaw rate is quite

evident. The tracking performance is done well with and without the actuators damage.

In cases of pdamage and rdamage, the tracking performance is slightly worse than that of

the perfect cases, p and r. The pitch rate tracking remains the same. The SDRE’s

adaptation ability is indirectly demonstrated here. The state-dependent feedback gain

contributes to the success of the post-damage command tracking.

0 10 20 30 40 50−15

−10

−5

0

5

10

15

20

Time (s)

δ e (

deg)

δe

δ edamage

Fig. 4.22: Elevator deflectionThe solid blue line indicated the healthy aircraft elevator deflection;

the dashed black line represents the damaged aircraft elevator deflection


0 10 20 30 40 50−20

−15

−10

−5

0

5

10

15

20

Time (s)

δ a (

deg)

δ a

δ a

damage

Fig. 4.23: Aileron deflection

The solid blue line indicated the healthy aircraft elevator deflection;


0 10 20 30 40 50−20

−10

0

10

20

30

Time (s)

δ r (

deg)

δ r

δ rdamage

Fig. 4.24: Rudder deflection

The solid blue line indicated the healthy aircraft elevator deflection;


The control surfaces plots show consistent surface deflections between the perfect and

damage cases. Although in the aileron case, the control saturation is experienced, it is

only for a short duration.

Outer loop

The outer loop design involves the state

[α β µ

]Tand control vector

[p q r

]T. µ

is the bank angle, used in lieu of the roll angle. The bank angle is expressed as the


following [1]. The derivation is included in Appendix A.

µ =pcosα + rsinα

cosβ+

1

mVt[qSCDsinβcosµtanγ + qSCY tanγcosµcosβ

+ qSCL(tanβ + tanγsinµ) + 4T (sinαtanγsinµ

+ sinαtanβ − cosαtanγcosµsinβ)]− gcosγcosµtanβ

Vt(4.36)

To ensure a coordinated turn takes place, the sideslip β angle command is zero. Similar

to the inner loop design, the integral tracking technique is used to eliminate the steady

state error. The SDC structure looks like,

α

β

µ

α

β

µ

z

=

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 a11 0 0 a14

0 0 0 a21 a22 a23 a24

0 0 0 a31 a32 a33 a34

0 0 0 0 0 0 a44

∫α∫β∫µ

α

β

µ

z

+

0 0 0

0 0 0

0 0 0

b11 b12 b13

b21 0 b23

b31 0 b33

0 0 0

p

q

r

(4.37)



Q = diag(1; 1; 1; 0; 0; 0) R = diag(100; 0.1; 0.01) (4.38)

Flight path loop

The flight path loop serves as a vital link between the guidance system and the control

system. It takes the commands from the guidance law and transforms them to the signals

for control loop to track. The loop consists of the state vector

[γ χ

]Tand the control

vector

[α µ

]T. The sideslip command is always zero to ensure coordinated turns. The


flight path loop possesses the following SDC structure,

γ

χ

γ

χ

z

=

0 0 1 0 0

0 0 0 1 0

0 0 a11 0 a13

0 0 0 0 a23

0 0 0 0 a33

∫γ∫χ

γ

χ

z

+

0 0

0 0

b11 0

0 b22

0 0

αµ

(4.39)



Q = diag(100; 100; 0; 0) R = diag(1e8; 1e3) (4.40)

The flight path angle loop tracking performance is shown in the Fig. 4.25. Although

the system is a little sluggish to track the flight path angle, the performance is acceptable

based on the fact the doublet flight path angle command is followed.

0 5 10 15 20 25 30 35 40−6

−4

−2

0

2

4

6

Time (s)

γ (d

eg)

γγ

cmd


Speed controller

Similar to the above designs, the SDRE controller is also implemented to control the

airspeed. However, unlike the previous state vectors, the airspeed does not need to be

regulated with high accuracy. In the interest of saving computational resources, the

integral servomechnisim technique is not used in the speed controller. Nevertheless, the

resulting airspeed tracking performance is still within the acceptable ranges. Essentially,


this loop is the scalar case SDRE design. The state is Vt and the control is δth. The

parameterized equation is,

Vt = [−qSmVt

(CDcosα− CLsinα)cosαcosβ − g

Vtsinθcosαcosβ

+qS

mVtCY sinβ +

g

Vtcosθsinφsinβ − qS

mVt(CDsinα + CLcosα)cosβsinα

+g

Vtcosθcosφcosβsinα]Vt + (

4T

mcosαcosβ − 0.0436

4T

mcosβsinα)δth (4.41)

where T is the single engine thrust force. The state and control weighting numbers chosen

for the speed controller are:

Q = 0.1 R = 0.01 (4.42)

The control law becomes,

δth = Kth(Vt)(Vt − Vtref ) + δth0 (4.43)

where Kth(Vt) is the state-dependent control gain; Vtref is the airspeed command and δth0

is the trimmed throttle setting.

Simulation example

A landing trajectory tracking problem simulated with actuators damage is presented

here. In this problem, the aircraft descends from 3000m to the runway following a pre-

defined trajectory, based on the guidance algorithm generating the tracking commands

for the control system. During the approach, in addition to the actuator damage, the

aircraft also suffers a thrust loss of 20%. The damages are included at the begining of the

simulation. Fig. 4.26 shows the aircraft flight path angle history during the appraoch. It

is expected to follow a -4 deg. glide slope. About 250 seconds, the aircraft is transitioning

from the glide slope to flare phase. The airspeed is maintained during the appraoch as

shown by Fig. 4.28. Under the guidance and control framework, the aircraft is able

to complete the trajectory and approach the runway without incident, as illustrate in

Fig. 4.30.


0 50 100 150 200 250 300−5

−4

−3

−2

−1

0

1

Time (s)

γ (d

eg)


0 50 100 150 200 250 300−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

δ e (

deg)

Fig. 4.27: Elevator deflection

0 50 100 150 200 250 300149.94

149.96

149.98

150

150.02

150.04

Time (s)

Vt (

m/s

)

Fig. 4.28: Airspeed

0 50 100 150 200 250 3000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (s)

δ th

Fig. 4.29: Throttle setting

0 1 2 3 4 5 6

x 104

−1

0

10

1000

2000

3000

4000

xe (m)

ye (m)

h e (m

)

Fig. 4.30: Trajectory history

4.5 Summary

In this chapter, the nonlinear state-dependent Riccati equation (SDRE) control method

was presented. The art and capabilities of the design techniques were also reviewed in

details. The fault-tolerant control system design was presented as well. The aircraft

damage scenarios, loss-of-thrust and actuator damage, were investigated with the imple-

mentation of the proposed fault-tolerant flight guidance and control system. Simulation

results were included to confirm the framework’s capbility to handle aircraft damage

cases.

Chapter 5

Adaptive State-Dependent Riccati

Equation Control Method

In this chapter, a novel nonlinear control method is proposed and implemented in the

fault-tolerant framework to evaluate its performance. The new control method is based on

the nonlinear state-dependent Riccati equation control method as the baseline controller,

but is also augmented with the model reference adaptive control method.

The research work in adaptive control started as early as the 1950’s. It was mainly

targeted at autopilot design for high performance aircraft. However, the tragic test flight

of X-15 diminished the interest in adaptive control from the community at that time. In

recent years, there has been increasing development of a coherent theory and practical

application of adaptive control, which has tremendous potential for the flight control com-

munity. Adaptive control deals with complex systems that have unpredictable parameter

derivations and uncertainties. Unlike robust control, which has the advantage of dealing

with disturbances, fast varying parameters, and unmodeled dynamics, adaptive control

is superior for handling uncertainties in constants or slowly varying parameters. When

the aircraft suffers from damage, the initial dynamics change can be sudden and drastic.

However the subsequent dynamics change will be slow once the aircraft is stabilized by

the control system. The aerodynamic coefficients variations are the main factors to the

62

Chapter 5. Adaptive State-Dependent Riccati Equation Control Method63

changes. In order to maintain reasonable performance and command tracking of a system

in the presence of uncertainty and possibly unknown variations in plant parameters, the

adaptive augmentation of a robust baseline controller should be considered.

Traditionally, the adaptive control law is augmented with the classic linear LQR

control law. The model reference adaptive control (MRAC) is used as the adaptive

algorithm in the augmentation. Such an arrangement has showed great results in dealing

with a system in the presence of uncertainty and unknown variations in plant parameters

[29]. The MRAC is based on the plant with a known structure, but the parameters are

unknown. The included reference model specifies the desired system response to the

command signals. The augmented controller with the adaptive algorithm adjusts itself

in response to the parameters variations and thus provides tracking. Fig. 5.1 shows the

basic MRAC framework.

Fig. 5.1: Adaptive framework

Many fault-tolerant flight control systems are designed based on the adaptive aug-

mentation with a baseline robust controller [29,55,57]. Lavretsky et al. [29] compared the

performance results between two adaptive laws augmented with the LQR baseline con-

trol law. The objective was to improve the transient characteristics. While both MRAC

and CMRAC adaptive laws were able to recover the tracking performance, the CMARC

did so with more desirable transient response. The simulation work was conducted on

the NASA Generic Transport Model (GTM). Furthermore, Dan et al [55] implemented

the adaptive augmented control law to deal with the actuator faults on an F-16 fighter

plane. The focus was on the online estimation of an eventual fault and the addition of

the adaptive augmented linear matrix inequality (LMI) method to reduce the fault effect


on the system without the need for a fault detection and isolation mechanism.

Although the linear control method is most often the choice when augmenting with

the adaptive law, the augmented system still needs gain scheduling to accommodate the

entire flight envelop. This procedure further complicates the already complex design

process. On the other hand, the advantages of nonlinear control method is that it offers

much more flexibility than its counterpart when dealing with augmentation. For example,

Lombaerts et al. [30] proposed a sophisticated flight control law based on nonlinear

dynamic inversion. It is well known that the dynamic inversion method is prone to

erroneous inversion due to model mismatch and uncertainties in the parameters. To

overcome the problem, a two-step algorithm was developed to estimate the parameters

online and adapt the parameter variations. The algorithm was implemented on the

SIMONA Research Simulator [30].

The proposed adaptive augmented control is based on the SDRE control method

and MRAC. The proposed framework has many advantages over the existing linear one.

First of all, the baseline SDRE method is a very effective and systematic control method.

Unlike many other nonlinear methods, it is based on a systematic design process. In other

words, the method can be readily implemented to different kinds of nonlinear systems.

Secondly, as demonstrated in the previous chapter, the SDRE method is able to deal

with complicated and difficult control problems involved with non-affine in control, state

and control constraints. These features are especially important in the context of fault-

tolerant flight control. Additionally, being a robust nonlinear control method itself, it has

the advantage of actively updating the feedback gain at each step to accommodate the

parameter changes. In a way, it also adapts itself to the varying environment. Finally,

using the SDRE as the baseline controller avoids the complex gain scheduling process in

the flight control system.

In the next section, the augmented control method is formally introduced. The math-

ematical formulation and problem definition are presented. Subsequently, the stability


analysis is included for the proposed controller to show its viability. Finally, the novel con-

trol method is implemented on the NASA GTM simulink model to test its performance.

Comparisons are also made in Section 5.3 to investigate the controller’s performance.

5.1 Adaptive Control Method

In this section, the novel adaptive state-dependent Riccai equation control method is

introduced. It is based on the nonlinear state-dependent Riccati equation control method

and the model reference adaptive control law. The augmentation aims to provide superior

tracking performance and transient response than its baseline controller.

Consider the general autonomous, affine-in-control, nonlinear system dynamics in the

form of,

x = f(x) + B(x)u(t) x(0) = x0 (5.1)

where the state vector x ∈ <n, and the control vectoru ∈ <m. Assume the dynamics

satisfy the SDRE conditions stated earlier, so the system can be expressed with state

dependent coefficients.

xp = Ap(xp)xp + Bp(xp)u (5.2)

where f = Ap(xp)xp. The control goal is to ensure bounded tracking. The system

controlled output,

y(x) = Cp(xp)xp ∈ <m (5.3)

tracks any bounded and possibly time-varying command signal r(t) ∈ <m with bounded

errors in the presence of the system uncertainties. Let

e(t) = y(t)− r(t) (5.4)

denote the system output tracking error. The system uncertainties can be reflected in

the state dependent coefficients. In the case of damage, A(x) and B(x) vary in terms of


their individual elements. As a result, new state-dependent matrices can be defined as,

A(x) = Ap(xp)−∆A(x) (5.5)

B(x) = Bp(xp)−∆B(x) (5.6)

Consider that

x = xp −∆x (5.7)

Taking the derivative of Eq.5.7 and Substitute Eq.(5.5) and (5.6) into Eq.(5.2),

x = A(x)x + B(x)u + [∆A(x)x + ∆B(x)u] (5.8)

where the δx contribution is small enough to be neglected when multiplied by state-

dependent matrix A(x). The time derivative of δx is assumed to zero. Eq.(5.8) represents

the expanded system dynamics including the uncertainties. It is also reasonable to assume

that the last two terms of Eq.5.8 are much less than the first two terms. As a result,

x = A(x)x+ B(x) (5.9)

Model reference adaptive control is based on the assumption of model matching. It as-

sumes that given a reference Hurwitz matrix Aref , there must exist, a possibly unknown,

gain matrix Kx such that,

Aref (x) = A(x) + B(x)KTx (5.10)

Substitute Eq.(5.10) into Eq.(5.8) and rearranging the terms with the relationships of

Eq.(5.5) and Eq.(5.6),

x = Aref (x)x + [−(B(x)KTx x + B(x)u] (5.11)

As a result,

u(x) = KTx (x)x (5.12)

where Kx(x) ∈ <n×m is the adaptive gain matrix, whose dynamics will be defined later.


The reference model dynamics are,

xref = Aref (xref )xref (5.13)

Let the system tracking error be,

e(x) = x− xref (5.14)

Note the tracking error is also state dependent. Its value varies at the each step. The

tracking error dynamics can be obtained by subtracting the reference state dynamics

from the system dynamics as,

e(x) = Aref (x)e(x) + B(x)(KTx (x)−KT

x (x))x (5.15)

The objective is to design state-dependent feedback gain Kx such that the track-

ing error goes to zero. The direct model reference adaptive control law and Lyapunov

arguments can be used as shown in the next section to derive the control law,

˙Kx = −Γxxe(x)TPref (x)B(x) (5.16)

where Pref (x) = P Tref (x) > 0 is the unique symmetric, state-dependent, positive definite

solution to the state-dependent algebraic Lyapunov equation,

ATref (x)Pref (x) + Pref (x)Aref (x) = −Qref (x) (5.17)

where Qref (x) = QTref (x) > 0 is similar to the state-dependent weighting matrix Q(x)

in the ARE. Using the state-dependent feedback gain in Eq.(5.16) and the relation in

Eq.(5.17), one can solve the above stated tracking problem with asymptotically stable

closed-loop dynamics. The in-depth stability analysis is presented in the next section.

Similar to the standard direct model reference adaptive control laws, the solution is valid

for any symmetric positive definite rates of adaptation Γx. As later demonstrated in the

example, the adaptation rate cannot be arbitrarily large, or unwanted oscillation and

unnecessary computational burden will occur.


5.2 Stability Studies

In this section, the stability analysis of the proposed state-dependent adaptive control

method is offered. It is important to note that the baseline controller of this novel

augmented control method is the nonlinear SDRE method. As proven in the previous

chapter, the SDRE control method is locally asymptotically stable and is only glob-

ally stable in special cases. Since the underlying controller is only locally stable, the

augmented control method is proved to be locally stable in this section. However, the

necessary global asymptotic stable condition will be given at the end of the analysis.

5.2.1 Stability Analysis

To prove the augmented control method is locally stable, a suitable Lyapunov function

candidate is chosen,

V (e,∆Kx) = eT(x)Pref (x)e(x) + trace(∆KTx Γ−1x ∆KxΓ) (5.18)

where Γx = ΓTx > 0 denotes the rate of adaptation. PTref (x) = Pref (x) > 0 is the state

dependent solution to the state dependent algebraic Lyapunov equation Eq.(5.17). The

stability analysis is mainly based on the following theorem,

Theorem 5. [27] Let x = 0 be an equilibrium point for Eq.(5.1) and D ⊂ <n be a

domain containing x = 0. Let V : [0,∞) × D 7→ < be a continuously differentiable

function such that

W1(x) ≤ V (t, x) ≤ W2(x) (5.19)

∂V

∂t+∂V

∂xf(t, x) ≤ −W3(x) (5.20)

∀t ≥ 0, ∀x ∈ D where W1(x), W2(x), and W3(x) are continuous positive definite functions

on D. Then, x = 0 is uniformly asymptotically stable.


Taking the time derivative of the Lyapunov function evaluated along the error dy-

namics trajectory gives,

V (x) =∂V

∂t+∂V

∂xf(t, x) (5.21)

=∂V

∂xf(t, x) (5.22)

where ∂V∂t

= 0 Based on Theorem 5, the following new theorem is established for the

proposed control method.

Theorem 6. Given a Lyapunov function candidate, if the function satisfies the conditions

in Theorem 5, then the closed loop system under the state-dependent adaptive feedback

control is locally asymptotically stable.

Proof. The derivative of the Lyapunov function becomes,

V (x) = e(x)Pref (x)e(x)+eT(x)Pref (x)e(x)+eT(x)Pref (x)e(x)+2trace(∆KTx Γ−1 ˙Kx)

(5.23)

Substitute the error dynamics as well as the Riccati differential equation into this,

P(x) = −AT(x)P(x)−P(x)A(x) + P(x)B(x)BT(x)P(x)−Q(x) (5.24)

However, the Lyapunov inequality is more appropriate in this case,

P(x) + AT(x)P(x) + P(x)A(x) + Q(x) ≤ 0 (5.25)

So the derivative of the Lyapunov function becomes,

V (x) ≤ [Aref (x)e(x) + B(x)(KTx (x)−KT

x (x))x]TPref (x)e(x) (5.26)

+ eT(x)[−ATref (x)Pref (x)−Pref (x)Aref (x)−Qref (x)]e(x) (5.27)

+ 2trace(∆KTx Γ−1 ˙Kx) (5.28)

Rearranging terms,

V (x) ≤ −eT(x)Qrefe(x) + 2eT(x)Pref (x)B(x)∆KTx x (5.29)

+ 2trace(∆KTx Γ−1 ˙Kx) (5.30)


Define the training error signal as

e(x) = BT(x)Pref (x)e(x) (5.31)

and applying the trace identity the Lyapunov function derivative becomes,

V (x) ≤ −eT(x)Qrefe(x) + 2trace(∆KTx xeT(x)) + 2trace(∆KT

x Γ−1 ˙Kx) (5.32)

and collecting and rearranging terms

V (x) ≤ −eT(x)Qrefe(x) + 2trace[∆KTx (Γ−1Kx + xeT)] (5.33)

and choosing the control gain to be,

˙Kx = −ΓxeT(x)Pref (x)B(x) (5.34)

implies that

V ≤ 0 (5.35)

Eq.(5.35) confirms the Lyapunov function satisfies the second condition, Eq.(5.20), in

Theorem 5. In other words, W3(x) = 0 in our case. The first condition can be satisfied

automatically given the fact that both W1(x) and W2(x) can be set to V (x) at each

step to ensure the condition. As a result, Theorem 5 is satisfied. However, one must

remember that the augmented controller relies on the baseline SDRE controller to provide

the Aref (x). As proved earlier in the Chapter, SDRE is locally stable. Thus, Aref (x) can

only be a local Hurwitz matrix. Thus, the augmented controller is locally asymptotically

stable.

As mentioned earlier, the global stable property of the augmented adaptive control

method is possible under special circumstances dictated by the SDRE control method.

The necessary condition to guarantee the global stable property is given here.

Corollary 1. The SDRE and MRAC augmented control method is globally stable if the

following conditions are met:


• SDRE controller is proven to be globally asymptotically stable

• Theorem 5 is satisfied

Proof. When the SDRE controller is globally asymptotically stable, it implies thatAref (x)

is always a Hurwitz matrix. As a result, the Theorem 6 derivation result V ≤ 0 is valid

in the global sense given the conditions of Theorem 5 are satisfied.

5.3 Simulation Studies

In this section, the proposed state-dependent adaptive control method is implemented

on the NASA GTM model to study its performance with airframe damage. The method

is also compared with the regular SDRE control method to investigate the performance

enhancement.

The damage scenario is provided by the GTM simulink model as damage case six,

see Sec. 2.4. The entire left stabilizer suffers damage, and this results the elimination of

the left elevator. A more detailed damage case description was presented in Chapter 2.

The nonlinear controller design for the GTM is particularly a difficult task. The

aircraft nonlinear dynamics equations are slightly different from the traditional flight

dynamics due to the offset between the center of pressure and center of mass. Secondly,

the nonlinear aerodynamic coefficients are complex. They require translation from the

simulink model to the analytic forms to fit the control design. Furthermore, similar

to the B747 design case, the system is also non-affine in control due to the nonlinear

characteristics of the aerodynamic coefficients.

α =1

mVtcosβ(−Fxsinα + Fzcos(α) +mVt(−pcosαsinβ + qcosβ − rsinαsinβ)) (5.36)

q = c5pr − c6(p2 − r2) + c7My (5.37)

To demonstrate the novel control method, the NASA GTM model short period dynamics

are considered. The following equations are the same ones identified in Chapter 2. The


forces and moment equations are,

Fx = qS(CXbasic + CXdynamic + CXcontrol) + Fxcontrol + Fxgravity (5.38)

Fz = qS(CXbasic + CXdynamic + CXcontrol) + Fzcontrol + Fzgravity (5.39)

My = qSc(Cmbasic + Cmdynamic) + rz qS(CXbasic + CXdynamic)

− rxqS(CZbasic + CZdynamic) +Myengine + qScCmcontrol + rz qSCXcontrol

− rxqSCZcontrol (5.40)

where rx = cpx − cgx and rz = cpz − cgz are the cp and cg offsets in x and z axes.

Similar to the B747 longitudinal case, Fx, Fy, and My represent the aerodynamic forces

and moments. In the forces case, the contributions include the aerodynamic coefficients,

(such as CXbaic , CXdynamic , CXcontrol), control terms (such as Fxcontrol), and the gravity

force terms (such as Fxgravity). These terms can all be expressed by a series of stability

derivatives that are included in the simulink model as look-up tables. In the case of the

moment, because of the difference between the cp and the cg, the aerodynamic forces

also contribute in the equation.

5.3.1 Baseline Control

The baseline control design is presented in this section. The first step is to parameterize

the above dynamics into the state-dependent coefficients form. The original dynamics

include bias terms. It means that a slow state z needs to be introduced to deal with the

issue.

x =

0 a12 a13

0 0 a23

0 0 a31

α

q

z

+

b11

b12

0

δe (5.41)

The SDC elements are included in Appendix B. The NASA GTM model is trimmed

to 3 deg. angle of attack, straight and level flight. In this example, the angle of attack

is selected to be the tracking signal. The integral servomechanism is formulated to


ensure zero steady state error in the tracking results. The SDRE tracking controller is

designed as outlined in the previous chapter. As a result, the augmented state-dependent

coefficients become

A =

0 1 0 0

0 0 a12 a13

0 0 0 a23

0 0 0 a31

B

0

b11

b22

0

(5.42)

The baseline nonlinear SDRE controller is designed with the assumption that there is

no fault or uncertainties in the system. The output of the system is the desired response

meeting all necessary control specifications. Thus, the output is used as the reference

signal for the adaptive control law to construct the augmented control law in dealing

with the damaged aircraft. The state-dependent feedback control gain is formulated as

ubl = −Kx(xref )Txref (5.43)

5.3.2 Adaptive Law Design

The Model reference adaptive control (MRAC) law is augmented with the SDRE con-

troller aiming to provide better performance results. The baseline controller is applied

to the perfect system to obtain the desired reference output which is used to formulate

the tracking error,

e = x− xref (5.44)

The tracking error is then used to derive the adaptive law as showed in the previous

section. The adaptive control law is

˙Kx = −Γxxe(x)TPref (x)B(x) (5.45)

Thus, the augmented controller provides the control action as the sum of the baseline


SDRE controller and the MRAC augmented one.

δe = ubl + uad (5.46)

It is worth mentioning that the initial conditions of the adaptive control law can be

set arbitrarily. In our case, it is set at the origin so that the adaptive law is added for

compensate the baseline controller.

5.3.3 Simulations

In this section, the SDRE+MRAC control method is implemented on the NASA GTM

simulink model to investigate the stability and tracking performance. The tracking results

are also compared with the baseline SDRE alone controller. The numerical example starts

with the trimmed flight condition, which is obtained using the trim function included

in the simulink model. In this case, the targeted condition includes the angle of attack

α and the flight path angle γ. The final numerical results are tabulated in Table 5.1

and plotted in Fig. 5.2- 5.3. The evaluated damage case is number six. It involves the

loss of the entire left stabilizer and hence the complete impairment of the left elevator.

The damage has direct impact on the aircraft longitudinal stability as well as the pitch

control power due to the left elevator elimination. The aircraft parameter changes are

provided in Eq.(2.67) - Eq.(2.76).

Table 5.1: GTM trim results

State Value State Value

α 3 deg γ 0

δe 2.66 deg TAS 93 knots


0 5 10 15 20 25 303

3

3

3

3

3

3

3

3

Time (s)

α (d

eg)


0 5 10 15 20 25 30−6

−4

−2

0

2

4

6

8x 10

−10

Time (s)

γ (d

eg)


0 5 10 15 20 25 30

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

Time (s)

α (d

eg)

SDRESDRE+MRAC


0 5 10 15 20 25 30−5

0

5

10

15

Time (s)q

(deg

/s)

SDRESDRE+MRAC


0 5 10 15 20 25 30−2

−1

0

1

2

3

Time (s)

δ e (

deg)

SDRESDRE+MRAC


0 5 10 15 20 25 301

2

3

4

5

6

Time (s)

α (d

eg)

α ref

α dam

α perfect


Stabilization

The stabilizing scenario is conducted by introducing the damage case in the middle

of normal flight, and to investigate the controller’s ability to stabilize the aircraft back

to the original state with the impaired aircraft dynamics. In this case, the damage is

introduced at 10 seconds. The SDRE+MRAC controller as well as the baseline SDRE

controller are implemented to study the performance. Figs. 5.4 - 5.6 show the results of

both SDRE and SDRE+MRAC controllers.


It is reasonable to conclude that both controllers are able to stabilize the aircraft and

maintain the initial angle of attack after the damage occurrence. Although steady state

error is observed in both case such results are expected due to the extensive left stabilizer

damage. From the figures, it is also clear that the performance of both controllers are

very similar making it difficult to decide which one is superior.

AOA tracking

The angle of attack tracking case is more involved than the stabilizing case. In this

case, a doublet tracking command is provided to the control system. First of all, the

reference tracking performance is obtained from the perfect system using the SDRE

controller. Unlike the stabilizing case, the damage is introduced at the beginning of the

simulation and the tracking command starts seconds later to give the system enough

time to settle down. Fig. 5.7 illustrates the comparison between the baseline SDRE

tracking controller for the damaged aircraft and the reference signal. As it is expected

the damaged aircraft tracking performance is worse than that of the the reference case

due to the extensive damage to the elevator. However, the baseline SDRE controller

is still able to manage the tracking. The result is acceptable. Fig. 5.8 illustrates the

elevator deflection that is required.

0 5 10 15 20 25 300

1

2

3

4

5

6

Time (s)

δ e (

deg)


0 5 10 15 20 25 30−4

−2

0

2

4

6

Time (s)

δ e (

deg)

δ e

dam

δe

perfect


Fig. 5.9 shows the comparison between the healthy and impaired aircraft elevator

deflection. As expected the damaged aircraft elevator performance degrades due to the


extensive damage on the surface. However, the control deflection is still well within the

travel limit.

Finally, Fig. 5.10 illustrates the simulation results for the SDRE+MRAC, baseline

SDRE, reference signal and command signal. It is clear that the SDRE+MRAC tracking

performance is superior to the baseline SDRE. From this figure, it is observed that the

augmented adaptive controlled signal tracks very close to the reference signal. While the

baseline SDRE controller performance is acceptable, the augmented control performance

does not have the drastic change that the baseline has between 10 and 15 seconds.

Fig. 5.11 provides the elevator deflection in all cases.

0 5 10 15 20 25 301

2

3

4

5

6

Time (s)

α (d

eg)

cmdrefSDRESDRE+MRAC


0 5 10 15 20 25 30−6

−4

−2

0

2

4

6

8

Time (s)

δ e (

deg)

δ e

ref

δ e

SDRE

δ e

SDRE+MRAC


5.4 Summary

In this chapter, a novel nonlinear augmented adaptive control law was successfully de-

veloped. The augmented control method utilizes the nonlinear state-dependent Riccati

equation control method as the baseline control law and augments it with a model refer-

ence adaptive control algorithm. Stability analysis confirms the local stability property of

the new control method. It is only locally stable due to the fact that the baseline control

law is locally asymptotic stable and global stability is only available in special cases. Sim-

ulation studies were also included to confirm the superiority of the novel control method

over the baseline SDRE control law in dealing with damaged aircraft tracking scenarios.

Chapter 6

Conclusions and Future Work

The objective of this thesis was to investigate and develop a nonlinear fault-tolerant flight

control design to deal with aircraft that suffer from the various degrees of damage, such as

loss of thrust, actuator malfunction and/or airframe damage. A design framework was

developed to accommodate the damaged aircraft while achieving close to the original

system performance.

The proposed design framework includes a guidance law and a control law to deal

with damaged aircraft. The guidance law was based on the zero-effort miss concept

to provide the flight path commands to the control system while taking into account

the impaired aircraft dynamics. The SDRE control method was used to design the

fault-tolerant flight control laws. The nonlinear SDRE control method demonstrated its

passive adaptive features and robust performance in the simulation studies. In order

to ensure the aircraft parameter variations due to damage were properly considered, a

novel nonlinear adaptive control law based on SDRE was proposed in this research. The

method demonstrated improved tracking performance and the potential to be considered

as an adaptive fault-tolerant flight control method. The proposed framework was applied

to nonlinear B747 and NASA GTM aircraft models to demonstrate its capability.

Future research will focus on the modeling and simulation of more sophisticated and

78

Chapter 6. Conclusions and Future Work 79

realistic aircraft damage scenarios. The simulation model fidelity is an important issue.

In order to obtain realistic simulation results, the aircraft simulation model should include

more relevant stability derivatives. This issue can be dealt with using two approaches.

The first one requires access to flight test data. Using actual flight test data to evaluate

the control method is the desired approach for future work. The second approach is to

analytically derive modified equations of motion that include damage effects so that the

complex coupling can be taken into account as part of the design process. In addition,

while the fault-tolerant flight control system provides promising results in terms of dealing

with in-flight aircraft damage, the current research work is only a preliminary, early

stage of work, which limits its actual application to the aircraft control system design

and application. Applying the fault-tolerant flight control systems to deal with damaged

aircraft will compliment the existing aircraft control system, alleviate the pilots workload

during this critical time and increase flying safety. Thus, working towards the verification

and validation of the fault-tolerant flight control is a desirable goal.

Appendix A

Derivations

In this section, the derivations of Vt, α, and β are provided.

Airspeed dynamics

It is known that the velocity components are equal tou

v

w

= Vt

cosα · cosβ

sinβ

sinα · cosβ

(A.1)

where,

Vt =√u2 + v2 + w2 (A.2)

α = arctan(w

u) (A.3)

β = arctan(v√

u2 + w2) (A.4)

Taking the time derivative of eq.(A.2),

Vt =uu+ vv + ww

Vt(A.5)

Substituting eq.(A.1) into eq.(A.5),

Vt = ucosαcosβ + vsinβ + wsinαcosβ (A.6)

Substituting u, v, and w from the force equations, and canceling out the roll, pitch and yaw rate terms

Vt =1

m(Fxcosαcosβ + Fysinβ + Fzcosβsinα) (A.7)

80

Appendix A. Derivations 81

Angle of attack dynamics

The similar approach can be used to obtain the α and β. Taking the time derivative of eq.(A.3),

α =uw − uwu2 + w2

(A.8)

u2 + w2 can be replaced by V 2t − v2 = V 2

t (1− sin2β) = V 2t cos

2β

Substituting eq.(A.1) into eq.(A.8),

α =wcosα− usinα

Vtcosβ(A.9)

Substituting u and w from the force equations as well as eq.(A.1),

α =1

mVtcosβ(−Fxsinα+ Fzcosα+mVt(−pcosαsinβ + qcosβ − rsinαsinβ)) (A.10)

Sideslip dynamics

Differentiating eq.(A.4) with respect to time yields,

β =v(u2 + v2)− v(uu+ ww)

V 2t

√u2 + w2

(A.11)

Based on eq.(A.1), the following terms can be replaced by,

u2 + w2 = V 2t cos

2β (A.12)

uv = V 2t sinβcosβcosα (A.13)

vw = V 2t sinβcosβsinα (A.14)

With the new replacements,

β =1

Vt(−ucosαsinβ + vcosβ − wsinαsinβ) (A.15)

Substituting u and w as well as eq.(A.1), after canceling terms, the new equation becomes,

β =1

mVt(−Fxcosαsinβ + Fycosβ − Fzsinαsinβ −mVt(−psinα+ rcosα)) (A.16)

Bank angle dynamics

The development of the bank angle dynamics starts from the wind axis force equations, which are similar

to the body axis equations,

vw =1

mFw − Ωwvw (A.17)

Appendix A. Derivations 82

The external aerodynamic forces, engine forces, and gravitational forces are

FAw =

−CD

−CY

−CL

Tw =

4Tcosβcosα

−4Tsinβcosα

4Tsinα

Ww =

−mgsinγ

mgsinµcosγ

mgcosµcosγ

(A.18)

The linear and angular velocity components in the wind axis are

vw =

Vt

0

0

ωw =

pw

qw

rw

Ωwvw =

0

Vtrw

−Vtqw

(A.19)

The following equations then can be readily obtained,

pw = pcosαcosβ + sinβ(q − α) + rsinαcosβ (A.20)

α = q − secβ(qw + pcosαsinβ + rsinαsinβ) (A.21)

qw =1

mVt(qSCL −mgcosµcosγ − 4Tsinα) (A.22)

rw =1

mVt(−qSCY −mgsinµcosγ − 4Tsinβcosα) (A.23)

The wind-axis kinematic equations then can be obtainedµ

γ

χ

=

1 sinµtanγ cosµtanγ

0 cosµ −sinµ

0 sinµsecγ cosµsecγ

pw

qw

rw

(A.24)

As a result, the bank angle dynamics, µ, is

mu = pw + (qwsinµ+ rwcosµ)tanγ (A.25)

With the appropriate substitutions,

µ =pcosα+ rsinα

cosβ+

1

mVt[qSCDsinβcosµtanγ + qSCY tanγcosµcosβ

+ qSCL(tanβ + tanγsinµ) + 4T (sinαtanγsinµ

+ sinαtanβ − cosαtanγcosµsinβ)]− gcosγcosµtanβ

Vt(A.26)

Appendix B

State-dependent coefficients

In the Appendix, the state-dependent coefficients of each example are presented.

Loss-of-thrust example SDC

a11 = −ρS

2m(CX1cosα+ CZ1sinα)Vt −

ρS

mVt0 (CX1cosα+ CZ1sinα)−

ρS

2mVtδe0 (CXδecosα+ CZδesinα)

−ρS

mVt0δe0 (CXδecosα+ CZδesinα) (B.1)

a12 = g · cosα0 · cosθ ·sinα

α+ g · sinα0 · cosθ ·

cosα− 1

α+Tmax

m· cosα0 ·

cosα− 1

α· δth0

−Tmax

m· sinα0 ·

sinα

α· δth0

(B.2)

a14 = −g · cosθ0 · cosα ·sinθ

θ− g · sinθ0 · cosα ·

cosθ − 1

θ(B.3)

a16 = −ρSV 2

t0

2mz(CX1

cosα+ CZ1sinα) +

g · sinα0 · cosθz

−g · sinθ0 · cosα

z−ρSV 2

t0δe0

2mz(CXδecosα+ CZδe sinα)

+Tmax · cosα0 · δth0

mz(B.4)

83

Appendix B. State-dependent coefficients 84

a21 =ρS

2m(CX1

sinα− CZ1cosα+ CXδeδe0sinα− CZδeδe0cosα) (B.5)

a22 =ρVt0SCX1

2mcosα0 ·

sinα

α+ρVt0Ssinα0CX1

2m

cosα− 1

α+ g ·

sinθ

Vt· cosα0 ·

sinα

α

−ρVt0SCZ1

cosα0

2m·cosα− 1

α+ρVt0Ssinα0CZ1

2m·sinα

α+ρVt0Scosα0CXδeδe0

2m

sinα

α

+ρVt0Ssinα0CXδe δe0

2m

cosα− 1

α−Tmaxδth0

cosα0

mVt

sinα

α−Tmaxδth0

sinα0

mVt

cosα− 1

α

−ρVt0Scosα0CZδeδe0

2m

cosα− 1

α+ρVt0Ssinα0CZδeδe0

2m

sinα

α(B.6)

a24 = g ·cosθ0cosα

Vt·cosθ − 1

θ− g ·

sinθ0cosα

Vt

sinθ

θ(B.7)

a26 =ρVt0Ssinα0CX1

2mz+g · sinα0 · cosα

Vtzsinθ +

g · cosθ0 · cosαVtz

+ρVt0Ssinα0CXδeδe0

2mz−Tmaxsinα0δth0

mVtz

−ρVt0SCZδeδe0

2mz+Q0

z−ρVt0SCZ1

cosα0

2mz(B.8)

a31 =c7

2ρVtScCm1 + c7Vt0ScCm1 +

c7

2ρVtScCmδeδe0 + c7ρVt0ScCmδe δe0 (B.9)

a36 =c7

2zρV 2t0Sc(Cm1 + Cmδe δe0 )−

c7 · dT · Tmax · δth0

z(B.10)

a46 =Q0

z(B.11)

a51 = θ + θ0 − α0 − γc (B.12)

a52 = −Vt − Vt0 (B.13)

a54 = Vt0 (B.14)

a56 =Vt0θ0 − Vt0α0 − Vt0γc

z(B.15)

CX1 = CX0 + CXα (α)α+ CXQ (α)Q (B.16)

CZ1 = CZ0 + CZα (α)α+ CZQ (α)Q (B.17)

Cm1 = Cm0 + Cmα (α)α+ CmQ (α)Q (B.18)

(B.19)

b11 =Tmaxcosα

m(B.20)

b12 = −ρSV 2

t

2m(CXδecosα+ CZδesinα) (B.21)

b21 = −Tmaxsinα

mVt(B.22)

b22 =ρSVt

2m(CXδe sinα+ CZδe cosα) (B.23)

b31 = −c7 · dT · Tmax (B.24)

b32 =c7

2ρV 2t ScCmde (B.25)


B747 example inner loop SDC

a11 = c2q (B.26)

a13 = c1q (B.27)

a14 =1

z[c3qSbCl1cosα− c3qSbCn1sinα+ c4qSbCl1sinα+ c4qSbCn1cosα] (B.28)

a21 = −c6p+ c5r (B.29)

a23 = c6r (B.30)

a24 =1

zc7qScCm1 (B.31)

a31 = c8q (B.32)

a33 = −c2q (B.33)

a34 =1

z[c4qSbCl1cosα− c4qSbCn1sinα+ c9qSbCl1sinα+ c9qSbCn1cosα] (B.34)

a44 = −1e4 (B.35)

b12 = c3qSbcosαClδa + c4qSbsinαClδa (B.36)

b13 = c3qSbcosαClδr − c3qSCnδrsinα+ c4qSbClδrsinα+ c4qSbCnδrcosα (B.37)

b21 = c7qScCmδe (B.38)

b32 = c4qSbClδacosα+ c9qSbClδasinα (B.39)

b33 = c4qSbClδrcosα− c4qSbCnδrsinα+ c9qSbClδrsinα+ c9qSbCnδrcosα (B.40)

Cl1 =dCl

dββ +

dCl

dp

psb

2Vt+dCl

dr

rsb

2Vt(B.41)

Cm1 = Cmbasic +dCm0.25

dq

qsc

2Vt(B.42)

Cn1 =dCn

dββ +

dCn

dp

psb

2Vt+dCn

dr

rsb

2Vt(B.43)

B747 Outer loop SDC

a11 =g

Vtsinθ

sinα

α(B.44)

a14 = −qS

mVtzCL +

1

mVtz(−4 · 0.0436Tδthcosα− 4Tδthsinα) (B.45)

b11 = −cosαsinβ (B.46)

b12 = cosβ (B.47)

b13 = −sinαsinβ (B.48)


a21 =g

Vtcosθcosφsinβ

sinα

α(B.49)

a22 =qS

mVtCD

sinβ

β+

g

V tsinθcosα

sinβ

β(B.50)

a23 =g

Vtcosθcosβ

sinφ

φ(B.51)

a24 =qSCY cosβ

mVtz−

4Tδthcosαsinβ

mVtz+

4 · 0.0436Tδthsinαsinβ

mVtz(B.52)

b21 = sinα (B.53)

b23 = −cosα (B.54)

a31 =4Tδth

mVttanβ

sinα

α(B.55)

a32 =qS

mVtCDcosµtanγ

sinβ

β+

qS

mVtcosβCL

sinβ

β−

4Tδth

mVtcosαtanγcosµ

sinβ

β−gcosγcosµ

Vtcosβ

sinβ

β(B.56)

a33 =qS

mVtCLtanγ

sinµ

µ+

4Tδth

mVtsinαtanγ

simµ

µ(B.57)

a34 =qS

mVtCY tanγcosµcosβ (B.58)

b31 =cosα

cosβ(B.59)

a44 = −1e3 (B.60)

B747 flight path loop SDC

a11 = −g

Vt

cosγ − 1

γ(B.61)

a13 = −qS

mVtz(CDsinβsinµ+ CY sinµcosβ − CLcosµ) (B.62)

b1 =4Tδth

mVtcosµ

sinα

α+

4Tδth

mVtsinµsinβ

cosα− 1

α(B.63)

a22 =1

mVtcosγz(qSCDsinβ + qSCY cosβ − 4Tδthsinβcosα) (B.64)

b22 =1

mVtcosγ(qSCDsinβ + qSCY cosβ − 4Tδthsinβcosα)

cosµ− 1

µ+

1

mVtcosγ(qSCL + 4Tδthsinα)

sinµ

µ(B.65)

NASA GTM short period mode SDC

a12 = 1 (B.66)

a13 = (−qSCXbasic − qSCXdynamic − FXengine − FXgravity )sinα

mVtz

+ (qSCZbasic + qSCZdynamic + FZengine + FZgravity )cosα

mVtz(B.67)

a23 =c7

z[qSc(Cmbasic + Cmdynamic ) + rz qS(CXbasic + CXdynamic )− rxqS(CZbasic + CZdynamic ) +Myengine ] (B.68)

b11 =−qSmVt

sinαCXδeδe

+qS

mVtcosα

CZδeδe

(B.69)

b22 = c7qS(cCmδeδe

+ rzCXδeδe

− rxCZδeδe

) (B.70)

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