Control of a Reusable Launch Vehicle448845/FULLTEXT02.pdfthe research and development of fully...

Control of a Reusable Launch Vehicle

JOHAN KNÖÖS

Examensarbete Stockholm, 2011

Abstract

This report examines different control design methods, linear as well as nonlinear,for a suborbital reusable launch vehicle. An investigation of the natural vehiclebehavior is made, after which a baseline linear controller is constructed to fulfillcertain handling quality criteria. Thereafter the nonlinear cascade control methodsblock backstepping and nonlinear dynamic inversion via time scale separation areexamined and used to construct two nonlinear controllers for the vehicle. Optimalcontrollers, in terms of three different criteria, are found using the genetic optimiza-tion algorithm differential evolution. The optimal controllers are compared, and itis found that nonlinear dynamic inversion via time scale separation performs betterthan block backstepping with respect to the cases investigated. The results sug-gest control design by global optimization is a viable and promising complement toclassical methods.

Keywords: Flight control systems, nonlinear control, global optimization

Acknowledgments

I am indebted to my supervisors Fredrik Berefelt and John W.C. Robinson at theSwedish Defence Research Agency, FOI, for their continual guidance and support.

I would also like to thank Professor Xiaoming Hu at the Division of Optimizationand Systems Theory at the Department of Mathematics at KTH Royal Institute ofTechnology for his advice and support.

Contents

Aerodynamic Nomenclature 6

1. Background 81.1. Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Vehicle Model 112.1. Actuator Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3. Vehicle Characteristics 143.1. Control Effectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2. Open Loop Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1. Pull Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2.2. Velocity Vector Roll . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4. Baseline Controller 214.1. Longitudinal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2. Lateral Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5. Nonlinear Control 345.1. Motivating Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2. Block Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2.1. Extension of the Model . . . . . . . . . . . . . . . . . . . . . . 375.3. Nonlinear Dynamic Inversion . . . . . . . . . . . . . . . . . . . . . . . 395.4. Application to Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.4.1. BBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.4.2. NDI-TSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6. Optimization 446.1. Differential Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Contents 5

6.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

A. Derivations and Proofs 56A.1. Dynamics of α, β, and ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.2. Invertibility of the Virtual Control Gain Matrix . . . . . . . . . 58

Aerodynamic Nomenclature

α angle of attack

β sideslip angle

δa aileron angle

δe elevator angle

δr rudder angle

J moment of inertia tensor

ωn undamped actuator frequency

ψ heading

τ actuator time constant

θ elevation

ϕ bank

δ control surface angles

ω rotational velocity

F aero aerodynamic force

Mb moment

u control surface inputs

v velocity

ζ actuator damping coefficient

g acceleration due to gravity

l rolling moment

m airframe mass

m pitching moment

n yawing moment

p roll rate

q pitch rate

r yaw rate

u velocity along body-fixed x-axis

V airspeed

Contents 7

v velocity along body-fixed y-axis

w velocity along body-fixed z-axis

xe x-position in Earth coordinates

ye y-position in Earth coordinates

ze z-position in Earth coordinates

CHAPTER 1

Background

Since the advent of the Space Age, the need for more inexpensive and reliable access tospace has been ever-increasing. The technological and economical barriers involved inlaunching vehicles into space have made space access prohibitive to many prospects,commercial and otherwise. In recent years, significant efforts have been made inthe research and development of fully reusable launch vehicles (RLVs) in order toprovide lower-cost and more reliable access to space (see [4]), not only in scientificcontexts such as transportation to and from the International Space Station, butalso for novel ideas such as space tourism. These initiatives have been undertakenby several organizations in various parts of the world, including NASA in the UnitedStates and the European Space Agency (ESA) and Virgin Galactic in Europe.

Figure 1.1.: Geometry of the ALPHA.

One project aimed at facilitating space access is FAST20XX (Future High-AltitudeHigh-Speed Transport 20XX), an ESA research program intended to evaluate and

1.1. Scope and Objectives 9

lay the groundwork for future high-altitude high-speed transportation. The initia-tive consists of two sub-projects, one concerning the implementation of the suborbitalaircraft ALPHA (Airplane Launched PHoenix Aircraft) of Fig. 1.1, whose primarymission is essentially to provide the means for space tourism, and the other witha hypersonic passenger aircraft, SpaceLiner. Both projects have many technologicalobstacles, including exceedingly high temperatures upon atmospheric reentry and ex-cessive accelerations (possibly unsuitable for humans). To tackle these complications,the projects have been divided into smaller tasks and distributed to 17 specialist com-panies and institutions in various countries across Europe. This study was conductedat one of these institutions: the Swedish Defence Research Agency, FOI.

1.1. Scope and Objectives

FOI’s task is to design control and stability augmentation systems for the manualsteering system of the ALPHA vehicle. Since the vehicle expends all available fuelduring its climb to the upper atmosphere, no engine power is available during itsdescent. At high altitudes, the low air pressure combined with the vehicle’s lack ofengine power makes the vehicle difficult to control and best suited for steering byautopilot. The design of the manual steering system is therefore restricted to lowaltitudes. Its role as a backup for a potentially malfunctioning autopilot provides notonly the means to control the vehicle during some unforeseeable disturbance or eventwith which the autopilot is unable to cope, but also as a psychological reassurancefor the pilot and the passengers.

In order to spare the ALPHA from unnecessarily high surface temperatures uponits descent through the upper atmosphere, its trajectory has been defined in such away as to, among other things, minimize thermal load. At high altitudes where theaircraft is flying at speeds of several Mach, deviations from this trajectory cannot beallowed due to these factors. When the aircraft has reached an altitude of ca. 13 km,it performs a braking maneuver to dispel kinetic energy. After this point, the vehicleflies more like a regular aircraft and can be treated as such. This study is thereforerestricted to the state trajectory defined between the altitudes of 12.5 km (the altitudereached immediately after the braking maneuver) and 0 km (see Fig. 1.2). During thisfinal descent, the nominal Mach number does not exceed 0.5. In order to allow formaneuvers that possibly cause a positive acceleration while still staying under thenumerically problematic transonic region, the maximum Mach number of interestthat is studied is set to 0.8.

The study begins with an investigation of the airframe’s behavior (Chapter 3), fol-lowed by the design of a baseline linear controller consisting of two decoupled sub-controllers: one for the aircraft’s longitudinal modes and one for its lateral modes(Chapter 4). Thereafter two nonlinear controllers are constructed using cascade con-trol methods (Chapter 5), specifically backstepping and nonlinear dynamic inversion

10 Chapter 1. Background

using time scale separation. The performance of these controllers is investigated usingglobal optimization tools (Chapter 6).

Time [s]

Altitude[km]

Ascent and Initial DescentFinal Descent

0 200 400 600 800 1000 12000

20

40

60

80

100

120

Figure 1.2.: The reference trajectory.

1.2. Notation

Vectors and matrices are written in bold font. Vectors are by default column vectors.The Euclidean norm of a vector v is written as ‖v‖. A variable preceded by a∆ signifies a deviation from the nominal value. A primed variable signifies a newvariable related to the unprimed variable of the same name. Rotational matrices areexpressed as T ab, where a and b are the source and destination reference frames,respectively. The derivative of a function f : Rm → Rn with respect to a vectorx ∈ Rm is denoted by the Jacobian matrix ∂f/∂x, which at row i and column jhas the derivative ∂fi/∂xj . The transpose of a matrix A is written AT. The openinterval between a and b is denoted by (a, b) and the closed, left-open, and right-openby [a, b], (a, b], and [a, b), respectively. The set R+ denotes the strictly positive realaxis. A diagonal square matrix A ∈ Rn×n with elements a1, a2, . . . , an along thediagonal is written diag(a1, a2, . . . , an). No notational distinction is made between afunction and its Laplace transform; the distinction is evident by the context in whichthe function appears.

CHAPTER 2

Vehicle Model

The model of the airframe forms the foundation of all flight simulations and controldesigns made in this study. To construct the model, two right-handed Cartesiancoordinate systems are utilized: the aircraft-fixed and the Earth-fixed coordinatesystems. The aircraft-fixed coordinate system has its origin at the aircraft center ofmass and its x-, y-, and z-axes pointing through the nose, through the starboardwing, and down, respectively. The Earth-fixed coordinate system has its x-, y-, andz-axes pointing north, east, and down, respectively. The origin is usually set at thestarting point of a flight, at ground level.

Using these coordinate systems, the following standard six-degree-of-freedom (hence-forth 6-DOF) model describing the translation and rotation of the airframe in itsown coordinates is used as a foundation for the model (the derivations of which canbe found in several standard references, e.g. [10, 15]):

v =1

mF aero + T eb

00g

− ω × v (2.1)

ω = J−1(Mb − ω × Jω), (2.2)

where v =[u v w

]Tis the vehicle’s velocity, m its mass, F aero ∈ R3 the aero-

dynamic force, g the magnitude of the acceleration due to gravity, ω =[p q r

]Tthe angular velocity, J ∈ R3×3 the moment of inertia tensor (here assumed positive

definite), and Mb =[l m n

]Tthe external moment1.

1The variable m can be assumed to refer to the aircraft mass unless explicitly referred to as pitchingmoment.

12 Chapter 2. Vehicle Model

To describe the vehicle’s orientation and position in the Earth-fixed coordinate sys-tem, the following six kinematic equations are needed, which can be found in [10]:ϕθ

ψ

=

1 sinϕ tan θ cosϕ tan θ0 cosϕ − sinϕ0 sinϕ/ cos θ cosϕ/ cos θ

ω (2.3)

xe

ye

ze

= T bev, (2.4)

where xe, ye, and ze are the positions in the Earth coordinate system. The angles ψ,θ, and ϕ are the Euler angles representing heading, elevation, and bank, respectively,which describe three successive transformations used to rotate the Earth frame tothe aircraft frame, namely rotations about (1) the Earth z-axis, (2) the y-axis of therotated coordinate system, and finally (3) the x-axis of the second rotated coordinatesystem. The order in which the rotations are performed is important since one tripleof angles does not give a unique orientation for all rotational orders. The order givenabove (heading, elevation, bank) is a convention used in the aircraft community.

The force F aero and the moment Mb in Eqs. (2.1) and (2.2) are functions of theaircraft state as well as its position and orientation. In this study, they are ap-proximated using aerodata—data collected from CFD2 calculations and wind tunnelmeasurements. These data are parameterized by, among other variables, Mach num-ber. The lowest Mach number measured is 0.2, which is used as a lower Mach numberlimit for designs and simulations in this study as well.

Additional variables that are used throughout this text are the airspeed V , angle ofattack α, and sideslip angle β, which in terms of the defined states can be expressedas

V = ‖v‖ (2.5)

α = arctanw

u(2.6)

β = arcsinv

V. (2.7)

Conversely, the following also holds:

v = V

cosα cosβsinβ

sinα cosβ

. (2.8)

The angles of attack and sideslip are shown visually in Figs. 2.1 and 2.2, respectively.Their domains are set to (−2, 24) and (−30, 30), respectively, in accordance withaerodata limits.

2Computational fluid dynamics.

2.1. Actuator Dynamics 13

v

α

x

Figure 2.1.: The angle of attack.

v

β

x

Figure 2.2.: The sideslip angle.

2.1. Actuator Dynamics

In order to create a more realistic model of the airframe, the 6-DOF system presentedabove is augmented to take the control surface actuator dynamics into account. Theseare modeled with first order dynamics as

τ δ = δ − u, (2.9)

where τ is the actuators’ time constant and δ and u the vectors of actuator positionsand control inputs, respectively. The time constant for the ALPHA’s actuators isunknown and is therefore approximated by taking that of the actuators of the similarNASA X-38 vehicle, which according to [7] have an undamped natural frequencyωn = 26 s−1 and damping ratio ξ = 0.707, i.e. τ = 1

ξωn≈ 54 ms.

CHAPTER 3

Vehicle Characteristics

For an appropriate control design to be made, the basic characteristics and behaviorof the aircraft need to be investigated. For example, an airframe may be naturallystable and thus not require a stability augmentation system, or it may be naturallyunstable and have insufficient control authority. This analysis is done by observinghow the airframe responds to control surface deflections as well as determining theconditions necessary for the vehicle to be in an equilibrium state.

3.1. Control Effectors

A control system is based on ability to manipulate the airframe state, which is doneusing its control effectors. From the start of its descent, the ALPHA has depletedits fuel and thus has no thrust available. The remaining means for control of theaircraft are its aerodynamic surfaces. These are given with their abbreviated namesand functions in Table 3.1 and shown with their positions in Fig. 3.1. Positive surfacedeflections are defined to follow the industry convention of using the right-hand rule(see [10, 15]) and are shown in Table 3.2. The limits of the actuators are given inTable 3.3 and their maximum allowable rate is conservatively set to rmax = 45 s−1.

The inboard as well as outboard surfaces on the wings can be used as both elevatorsand ailerons. To avoid potentially unnecessarily complicated control allocation logic,the inboard elevators EI are chosen for pitch control and the outboard ailerons AOfor roll control, i.e. EO and AI are not engaged throughout this design. This choiceis based on the fact that the outboard ailerons, despite having roughly the samesurface area as their inboard counterparts, are farther from the vehicle’s main axis

3.2. Open Loop Behavior 15

Control surface Name Primary control

Rudder RU YawBody flap BF Pitch (primarily for trimming)In- and outboard elevators EI & EO PitchIn- and outboard ailerons AI & AO RollSpeed brakes SB Velocity

Table 3.1.: Control surfaces and their functions.

Control surface Positive deflection

Rudder Trailing edge leftElevators and body flap Trailing edge downAilerons Starboard trailing edge down

Table 3.2.: Positive directions of control surface deflections.

and therefore can exert more moment using the same deflection. Unless otherwisespecified, “elevator” henceforth refers to EI and “aileron” to AO.

Speed brake

Rudder

Body flap

Inboard elevator/aileron

Outboard elevator/aileron

Figure 3.1.: The ALPHA and its control surfaces.

Using this knowledge of the control surfaces, one can begin to investigate the aircraftopen loop behavior.

3.2. Open Loop Behavior

The aircraft’s open loop behavior is its natural behavior without a control systemengaged. This behavior, or characteristics and limitations, of the airframe is in-

16 Chapter 3. Vehicle Characteristics

Control surface Lower limit [deg] Upper limit [deg]

RU −10 10SB 0 40AO −20 20AI −10 10BF −15 20EO −10 10EI −20 20

Table 3.3.: Control surface limits.

vestigated in the context of equilibrium states, i.e. states in which the aircraft is“trimmed” (which are also used as linearization points in the linear control design inChapter 4). It is first useful to make the observation that the most general steadystate motion (in terms of the body equations) an aircraft can make is helical. Thismotion can be decomposed into rotations about the lateral axis (pull ups) and aboutthe velocity vector (velocity vector rolls).

3.2.1. Pull Up

In flight with constant angle of attack and constant pitching rate the aircraft is said tobe performing a pull up. Contained in this class of maneuvers are not only upward anddownward rotations but also straight and level flight. The states are found by solving

the equation[α(x) q(x)

]T= 0, where x =

[vT ωT ϕ θ ψ xe ye ze

]Tis the

aircraft state.

3.2.2. Velocity Vector Roll

A velocity vector roll is achieved when the angles of attack and sideslip as wellas angular velocity are held constant. This is tantamount to a constant rotationof the airframe about its velocity vector, a movement integral to maneuvers such as

coordinated turns. The trim states are found by solving[α(x) β(x) ω(x)T

]T= 0,

where x is as defined above.

3.3. Results

Figures 3.2 and 3.3 show the longitudinal control surface deflections (body flap andelevator) required for steady state pull ups of different normal accelerations at dif-ferent Mach numbers. First, the body flap is used to attempt a steady state. If itsaturates, the elevator is engaged. The essential trend is that with higher dynamic

3.3. Results 17

pressure, less deflection is needed. In particular, the body flap is capable of keepingthe vehicle in steady state at high dynamic pressures. The more the height and/orspeed increases, however, the more the elevator needs to be engaged.

Mach

nz

Body Flap [deg]

Mach

nz

Elevator [deg]

-15 -10 -5 0

0.2 0.4 0.6

-15 -10 -5

0.2 0.4 0.60

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

Figure 3.2.: Body flap and elevator deflections needed for a steady state pull up at differentnormal acceleration factors nz = qV/g and Mach numbers. The Mach–nz region that is solidwhite in both graphs represents untrimmable states. Altitude: 0 km.

Figures 3.4 and 3.5 show the control surface deflections and angles of attack andsideslip for different roll rates at 0 km and 12.5 km, respectively. It is apparent inFig. 3.5 that the airframe cannot sustain a roll at low dynamic pressures (see theregion between Mach 0.2 and 0.4, for example). The limiting factor appears to bepredominantly the angle of attack boundaries (−2 < α < 24) at lower roll ratesand the elevator at higher roll rates. In other words, there is not enough lift tokeep the airframe in the roll. At high roll rates, one might expect the ailerons tosaturate. However, it is apparent from both figures that the limiting factor is therudder limitations.


Mach

nz

Body Flap [deg]

Mach

nz

Elevator [deg]

-10 -5 0 5

0.2 0.4 0.6

-15 -10 -5

0.2 0.4 0.60

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

Figure 3.3.: Body flap and elevator deflections needed for a steady state pull up at differentnormal acceleration factors nz = qV/g and Mach numbers. The Mach–nz region that is solidwhite in both graphs represents untrimmable states. Altitude: 12.5 km.

3.3. Results 19

Elevator [deg]

Mach

Rollrate

[deg/s]

Rudder [deg]

Mach

Rollrate

[deg/s]

Aileron [deg]

Mach

Rollrate

[deg/s]

Angle of Attack [deg]

Mach

Rollrate

[deg/s]

Sideslip Angle [deg]

Mach

Rollrate

[deg/s]

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

-0.5

0

0.5

0

100

24681012

0

100

-15

-10

-5

0

0

100

02468

0

100

-20246

0

100

Figure 3.4.: Control surface deflections as well as angles of attack and sideslip needed forvelocity vector rolls of varying roll rates and Mach numbers. Altitude: 0 km.


Elevator [deg]

Mach

Rollrate

[deg/s]

Rudder [deg]

Mach

Rollrate

[deg/s]

Aileron [deg]

Mach

Rollrate

[deg/s]

Angle of Attack [deg]

Mach

Rollrate

[deg/s]

Sideslip Angle [deg]

Mach

Rollrate

[deg/s]

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

-1

-0.5

0

0.5

0

100

10

20

0

100

-15

-10

-5

0

0

100

02468

0

100

-5051015

0

100

Figure 3.5.: Control surface deflections as well as angles of attack and sideslip needed forvelocity vector rolls of varying roll rates and Mach numbers. Altitude: 12.5 km.

CHAPTER 4

Baseline Controller

In order to have a means for evaluating the performance and response of the nonlinearcontroller, a baseline linear controller is designed for the family of general linearizedsystems

x =∂f(x,u)

∂x

∣∣∣∣x=x0u=u0

∆x+∂f(x,u)

∂u

∣∣∣∣x=x0u=u0

∆u, (4.1)

where x ∈ Rn is the aircraft state, u ∈ Rp the control input, and f : Rn × Rp → Rnits dynamics, of the original nonlinear dynamics

x = f(x,u) (4.2)

at several equilibrium points[xT

0 uT0

]Tin the flight envelope. The variables and

dynamics are defined in more detail below.

The airframe’s longitudinal and lateral modes are in the linear approximation as-sumed to be decoupled. The linear controller design can therefore be divided intotwo design tasks: pitch channel control (longitudinal) and roll channel control (lat-eral). Since the vehicle in question is not intended to make any extreme maneuvers,this approximation is valid for most of the vehicle’s flight and the controllers relyingon it can be expected to behave well.

The design procedure for each of the control loops of the two channels is as follows:

1. A loop structure is posited with gains left undefined.

2. Desired handling qualities (HQs) are constructed as closed loop pole boundariesin the complex plane. Because of its ubiquity, a subset of the HQ requirementsoutlined by the US Department of Defense in [2] is used (shown in Table 4.1).

22 Chapter 4. Baseline Controller

In terms of the classifications presented therein, the ALPHA can be categorizedas a class II airplane (“medium weight, low-to-medium maneuverability”), andthe flight phase of interest is consistent with what is classified as categoryB (“nonterminal Flight Phases that are normally accomplished using gradualmaneuvers and without precision tracking”).

3. For several points in the flight envelope (parameterized by altitude and Machnumber):

a) The system is linearized around a trim point (see Section 3.2)[xT

0 uT0

]T.

b) The HQ boundaries are mapped into gain parameter space by Theorem 1.Here, the polynomial p(s, q) defined in Eq. (4.3) is the denominator of therelevant transfer function and the parameter vector q is a vector containingthe loop gains. The variable α parameterizes the HQ boundaries in thecomplex plane. Equation (4.5) is solved for q to find the parameter spaceboundaries.

c) The region that fulfills the HQs is located and an arbitrary point inside itis chosen for the gains.

4. Gain scheduling: the controllers are “glued” together into a global controllerin the flight envelope by means of interpolation of loop gains using schedulingvariables, here altitude and Mach number.

Theorem 1. Let a : Rm → Rn+1 be a function of a parameter vector q ∈ Rm andp : C× Rm → R a polynomial family parameterized by q as

p(s, q) =[1 s . . . sn

]a(q). (4.3)

Then q∗ belongs to the set of roots

q : p(σ(α) + jω(α), q) = 0, α ∈ [α−, α+] (4.4)

if and only if [d0(α) d1(α) . . . dn(α)

0 d0(α) . . . dn−1(α)

]a(q∗) =

[00

](4.5)

for some α ∈ [α−, α+], where

d0(α) = 1 (4.6)

d1(α) = 2σ(α) (4.7)

di(α) = 2σ(α)di−1 −[σ2(α) + ω2(α)

]di−2, i = 2, 3, . . . , n. (4.8)

Proof. See [1].

4.1. Longitudinal Control 23

Flight mode HQ Minimum Maximum

Short period mode Damping 0.3 2CAP 0.085 3.6Im s 6= 0

Dutch roll mode Damping ξ 0.08Natural frequency ωn 0.4ξωn 0.1Im s 6= 0

Roll subsidence mode Time constant 0 1.4Im s = 0

Table 4.1.: Desired handling qualities.

4.1. Longitudinal Control

The task of this pitch channel control and stability augmentation system is to imple-ment the HQs for the aircraft’s longitudinal modes: the short period mode, charac-terized by a rapid oscillation of the angle attack but with nearly constant airspeedand altitude, and phugoid mode, a slower pitch, altitude, and airspeed oscillationwithout significant change of angle of attack. Phugoid mode, however, is of such along period that it can easily be controlled by the pilot. This discussion is thereforelimited to control of the higher frequency short period mode.

The desired HQs for short period mode are summarized in Table 4.1 and consist ofconditions on the damping (that the closed loop poles lie in a certain cone) and onthe control anticipation parameter (CAP). In addition, the mode is required to beoscillatory since the framework of mapping boundaries from the complex space isincompatible with damping ratios ξ > 1. The CAP (see [3, 10]) is the ratio of initialpitch acceleration to steady state normal acceleration. In other words, it is a measureof how much acceleration to anticipate from an initial one. Mathematically, it canbe written as

CAP =ω2

nsp

∂nz/∂α, (4.9)

where ωnsp is the undamped natural frequency of the short period mode and nz thenormal acceleration (i.e. the acceleration in the negative z-direction) in multiples ofg. This equation is solved for ωnsp for the minimum and maximum CAP, resultingin an annulus in the complex plane. Together, the HQs form the shape shown inFig. 4.1.

Relevant quantities for the short period mode are angle of attack, pitch rate, normalacceleration, and position of the elevator actuator. It is well-known that pilots aremore sensitive to pitch rate at low speeds and to acceleration at high speeds (see[18]). The authors of [18] therefore propose a combination of the two quantities.


Re

Im

ωn,min

ωn,max

ξmin

ξmax

00

Figure 4.1.: Longitudinal HQ boundaries. Inactive boundaries not shown.

This combination, called C∗, is given by the linear combination nz +Kq. In steadystate, i.e. in the equilibrium state with constant angle of attack, the relation

q =nzg

V(4.10)

holds. To find a suitable K, the two components of C∗ are set equal to each other:

nz = Kq = Knzg

VCO, (4.11)

where VCO is the velocity at which the two quantities should have equal influence.The authors of [18] choose VCO = 122 m s−1, resulting in K = 12.4.

Similar to the longitudinal control loops in [12] and [15], a control loop for C∗ isconstructed, shown in Fig. 4.2. Using the variables mentioned above, the transferfunction from elevator to nz can be expressed as Gnz(s) = Nnz(s)/D(s) and fromelevator to q as Gq(s) = Nq(s)/D(s) using the reduced linearized system

x = Alng∆x+Blng∆ue, (4.12)

where x =[α q nz δe

]T, ∆x =

[∆α ∆q ∆nz ∆δe

]T, and the state and con-

trol matrices Alng ∈ R4×4 and Blng ∈ R4, respectively, are Jacobian matrices deter-mined by numerical linearization. The transfer function from C∗cmd to C∗ can thus

4.2. Lateral Control 25

be derived from the control loop as

GC∗(s) =[Nnz(s) + 12.4Nq(s)]KI

sD(s) + [Nnz(s) + 12.4Nq(s)] (KI +KPs). (4.13)

Σ KI/s ΣC∗cmd

ALPHAEI

KP

Σ 12.4

−−

C∗

nzq

Figure 4.2.: Longitudinal control loop.

Theorem 1 is applied to the denominator of the transfer function in Eq. (4.13) tomap the boundaries to parameter space, after which acceptable gains KI and KP canbe chosen.

4.2. Lateral Control

In addition to the longitudinal controller, a (somewhat more advanced) lateral con-troller must be implemented. The controller has the following objectives:

1. To let the pilot initiate a coordinated turn1 simply by specifying the turn rate(the rate of change of the heading ψ). The control loop calculates the bankangle and control surface deflections.

2. To fulfill the HQ requirements for the lateral modes: Dutch roll mode, a coupledroll and yaw movement, roll subsidence mode, a mode of pure rolling, and spiralmode. The spiral mode is usually so slow that it is suppressed unconsciouslyby the pilot. However, it is required that its root stay real. For the Dutch roll,a damping ratio of at least 0.08, a natural frequency of 0.4, and a product ofthe two of at least 0.1 are required. The roll subsidence mode needs to be keptnon-oscillatory (i.e. one real eigenvalue) and have a time constant of at most1.4 s. The Dutch roll mode is kept oscillatory (as the short period mode is inthe longitudinal controller) by setting the maximum damping ratio ξmax = 1,i.e. it is at most critically damped.

To fulfill these objectives, the control loop in Fig. 4.3 is constructed with an innerloop similar to the roll-angle hold autopilot presented in [15]. The input is the desired

1A coordinated turn is, as defined in [15], a turn where there is no aerodynamic force along thelateral axis. Furthermore, the airspeed and angular velocity need to be constant for the aircraftto be in steady state.


Re

Im

(ξωn)min

ξn,min

ωn,min

Real roll root× Roll time constant

−1/1.4 00

Figure 4.3.: Lateral HQ boundaries. Inactive boundaries not shown.

ParameterLookup

Σ

KI/s

Σϕcmd

K1 Σ

ALPHA

K2

ϕ−

AO p

ψcmd

RUcmd

Figure 4.4.: Lateral control loop.

turn rate ψ. The bank angle ϕ and rudder position RU needed for turn coordinationfor the specified ψ are computed offline for a range of different turn rates at differentparts of the flight envelope. This computation also includes the necessary values forthe longitudinal states, which are passed to the integrated C∗ controller (not shownin Fig. 4.3).

Through this loop, the transfer function from ϕcmd to ϕ is

G(s) =K1Nϕ(s)(s+KI)

sD(s)− sK2Np(s) +K1Nϕ(s)(s+KI), (4.14)

where the transfer function from AO to ϕ is Nφ(s)/D(s) and from AO to p isNp(s)/D(s). These transfer functions are derived from the reduced linearized systemfor the lateral channel, which for the Dutch roll, roll subsidence, and spiral modes

4.3. Results 27

depends on the sideslip angle β, roll rate p, yaw rate r, roll angle ϕ, and aileronactuator position:

x = Alat∆x+Blat∆ua, (4.15)

where x =[β p r ϕ δa

]T, ∆x =

[∆β ∆p ∆r ∆ϕ ∆δa

]T, and the state

and control matrices Alat ∈ R5×5 and Blat ∈ R5, respectively, are Jacobian matricesdetermined by numerical linearization.

As in the longitudinal case, Theorem 1 is used to map the boundaries (the shapesof which are shown in Fig. 4.3) from the complex plane to parameter space to findsatisfactory K1 and K2. Before applying Theorem 1, however, KI is set to 1/50 (cho-sen by trial-and-error) to keep the parameter space two-dimensional for simplicity.Contrary to the longitudinal controller, the lateral controller needs to fulfill the HQsfor multiple modes. Therefore, the boundaries mapped into parameter space need tobe satisfied simultaneously. In other words, the region of intersection of all allowableregions needs to be found.

4.3. Results

Figure 4.5 shows the mapping of the denominator of the longitudinal controller’stransfer function in Eq. (4.13) to parameter space for three different Mach numbers atone altitude. Apparently, the two active conditions are maximum natural frequencyand minimum damping, the region S between which is that of acceptable values. Thisregion shrinks with increasing Mach number as it becomes more difficult to damposcillations and lower the natural frequency. For several altitudes and Mach numbers,a point inside the region is chosen for usage in interpolation in a gain scheduler, i.e. afunction parameterized by the flight envelope position (altitude and Mach number)

which gives a vector[KP KI

]Tas output.

Figures 4.6 and 4.7 show the responses to a unit step in commanded C∗ for twoaltitudes at different Mach numbers using the gain scheduled longitudinal C∗ controlloop. The vehicle exhibits a fast and well-damped response to the input under theregulation of the longitudinal control loop.

Proceeding in a similar manner as in the the longitudinal control loop implemen-tation, the denominator of the lateral controller’s transfer function in Eq. (4.14) ismapped to parameter space for three different Mach numbers at one altitude, shownin Figure 4.8. As in the longitudinal case, the region S shrinks as the Mach numberincreases. The active boundaries in this case, however, are the ones that ensure realroll subsidence and imaginary Dutch roll roots as well as a satisfactory time constant

of the roll subsidence mode. A gain scheduler with the vector of gains[K1 K2

]Tas

output is made by choosing a point inside S for several altitudes and Mach numbersand then interpolating between them.


replacemen

KP

KI

S

KP

KI

S

KP

KI

S

-3 -2 -1 0 1-3 -2 -1 0 1

-3 -2 -1 0 1

-10

-5

0

-10

-5

0

-10

-5

0

Figure 4.5.: Boundaries for the longitudinal parameters mapped by Theorem 1 at an altitudeof 3 km for, from left to right and top to bottom, Mach numbers 0.4, 0.6, and 0.8. The mapfor Mach 0.2 is missing since the aircraft is untrimmable with only the body flap at that point.Origins of boundaries as defined in Fig. 4.1.

Governed by this gain scheduled controller, two coordinated turns with different turnrates ψ at different parts of the flight envelope are made, the results of which areshown in Figures 4.9 and 4.10. In both cases the turn rates reach the commandedrates, and so do the bank angles ϕ (with a small steady state error, however). Theovershoots in ϕ are quite large, but the damping ratios are within the acceptablerange. With the exception of some excursions in the transient region when thevehicle is entering the turn, the aerodynamic side force Faero,y = mgny tends to zero,which, along with the condition of constant angular velocity and constant airspeed,makes the turn coordinated.

4.3. Results 29

C*

Time [s]

ResponseCommand

C*

Time [s]

C*

Time [s]0 5 10 15

0 5 10 15

0 5 10 15

0

0.5

1

0

0.5

1

0

0.5

1

Figure 4.6.: C∗ unit step responses at an altitude of 3 km at, in order from top to bottom,Mach 0.4, 0.6, and 0.8.


C*

Time [s]

ResponseCommand

C*

Time [s]

C*

Time [s]0 5 10 15

0 5 10 15

0 5 10 15

0

0.5

1

0

0.5

1

0

0.5

1

Figure 4.7.: C∗ unit step responses at an altitude of 9 km at, in order from top to bottom,Mach 0.6, 0.7, and 0.8.

4.3. Results 31

K1

K2

S

K1

K2

S

↓

K1

K2

S

ց

-60 -40 -20 0-60 -40 -20 0

-60 -40 -20 0

0

5

10

15

0

5

10

15

0

5

10

15

Figure 4.8.: Boundaries for the lateral parameters mapped by Theorem 1 at an altitude of3 km for, from left to right and top to bottom, Mach numbers 0.4, 0.6, and 0.8. The map forMach 0.2 is missing since the aircraft is untrimmable with only the body flap at that point.Origins of boundaries as defined in Fig. 4.3.


ψ[deg/s]

Time [s]

ResponseCommand

ϕ[deg]

Time [s]

ny

Time [s]0 5 10 15 20

0 5 10 15 20

0 5 10 15 20

-0.5

0

0.5

0

50

100

0

10

20

Figure 4.9.: Responses to a step in ψcmd to 4 s−1 at an altitude of 3 km at Mach 0.4. Thevariable ny is the lateral aerodynamic acceleration in terms of g: ny = 1

mgFaero,y.

4.3. Results 33

ψ[deg/s]

Time [s]

ResponseCommand

ϕ[deg]

Time [s]

ny

Time [s]0 5 10 15 20

0 5 10 15 20

0 5 10 15 20

-0.5

0

0.5

0

50

100

0

10

20

Figure 4.10.: Responses to a step in ψcmd to 5 s−1 at an altitude of 9 km at Mach 0.6.The variable ny is the lateral aerodynamic acceleration in terms of g: ny = 1

mgFaero,y.

CHAPTER 5

Nonlinear Control

The advantages of using nonlinear control methods over linear ones are manifold,in particular the class of controllers of systems in cascade form on which this studyis focused. This class of controls includes, for example, backstepping and nonlineardynamic inversion. In contrast to linear control methods, these cascade methodsare not based on linear approximations of the system’s dynamics and are thus builton a more solid foundation. Since the nonlinear system description is valid globally(in terms of the flight envelope), gain scheduling is not required. Furthermore, theimplementation of these controllers is straightforward.

This is not to say, however, that these controllers are devoid of difficulties. Forexample, the dynamics must be in strict-feedback form. Additionally, there is noclear-cut manner in which to control “output variables” as in the linear case.

Despite these difficulties, the benefits of this methodology outweigh its drawbacks interms of performance and effort required in implementation. This is true particularlyin the case of aircraft, whose motion is essentially governed by a chain of integrators:commanded control surface deflection rates are integrated into moments, momentsare integrated into rotational velocities, which finally are integrated into orientationalangles. This inherent property makes aircraft well-suited for cascade designs.

5.1. Motivating Idea

Consider the system

xi = fi(x1, x2, . . . , xi) + gi(x1, x2, . . . , xi)xi+1, i = 1, 2, . . . , n (5.1)

5.2. Block Backstepping 35

of order n, with input xn+1. The idea behind the recursive backstepping and non-linear dynamic inversion via time scale separation methods is to control xi by usingxi+1 as a virtual control, which is chosen in such a way as to steer xi to the desiredpoint. The first virtual control x2 is chosen in such a way as to steer x1 to the originwhile at the same time giving it certain desired dynamics fdes

1 (x1). Next, x3 is chosento steer x2 to its desired value. This process continues recursively for all remainingstates xi, i = 3, 4, . . . , n, and can be done using varying degrees of accuracy, as isevident below.

5.2. Block Backstepping

Block backstepping (BBS), which can be seen as the most general method in theclass of cascade-based controls since it relies on the fewest assumptions possible, isa Lyapunov-based1 design technique for deriving control laws for systems in strict-feedback form. An early presentation of the scalar form of the method can be foundin [9], and of the generalized vector form, or block form, applied in a flight setting in[14]. As in other cascade methods like nonlinear dynamic inversion, the method ofbackstepping essentially allows one to supplant the dynamics of systems with desiredones.

Following the terminology presented in [11, 17], a BBS controller is constructed andextended with input and output integrators. For simplicity, consider the system

x = f(x) + g(x)y (5.2)

y = h(x,y) + k(x,y)u, (5.3)

where x ∈ Rm, y ∈ Rn, u ∈ Rp, the functions f : Rm → Rm and h : Rm × Rn →Rn are smooth and vanish at the origin, and the functions g : Rm → Rm×n andk : Rm×Rn → Rn×p smooth and nonsingular. The objective is to stabilize the originof x and prescribe new dynamics fdes(x) (the conditions for which are given below)for the x-subsystem given by Eq. (5.2). This is done by choosing a virtual controllaw for the x-subsystem ydes(x) that satisfies2

f(x) + g(x)ydes(x) = fdes(x). (5.4)

and introducing an error variable y = y − ydes(x). This is needed since the state ycan deviate from the desired state ydes(x). The objective of assigning new dynamicsfor x amounts to stabilizing the origin of y, which in turn results in the state y

1Lyapunov theory is covered in several standard references, e.g. [8].2Here one could choose to, as in [6], only cancel the unstable dynamics in f to lessen control effector

movement. The trade-off is less freedom when postulating new dynamics. This study thereforecancels all dynamics regardless of the potential stability of terms therein.

36 Chapter 5. Nonlinear Control

“chasing” the desired state ydes(x). Define h(x, y) = h(x, y+ydes(x)) and k(x, y) =k(x, y + ydes(x)). Equations (5.2) and (5.3) can then be written as3

x = fdes(x) + g(x)y (5.5)

˙y = h(x, y) + k(x, y)u− ydes(x, y). (5.6)

The origins of x and y can, in the sense of Lyapunov, be globally asymptoticallystabilized by constructing a quadratic control Lyapunov function (clf)

V (x, y) =1

2xTQxx+

1

2yTQyy (5.7)

with positive definite weighting matrices Qx ∈ Rm×m and Qy ∈ Rn×n, and choosinga control law u such that the clf’s time derivative

V (x, y) = xTQxx+ yTQy ˙y

= xTQx

(fdes(x) + g(x)y

)+ yTQy

(h(x, y) + k(x, y)u− ydes(x, y)

) (5.8)

is negative definite along the solution trajectory. The control law needs to cancelpotentially indefinite terms in V and is additionally exploited to synthesize newerror dynamics hdes

y (x, y) for y. One possible control law is

u = k(x, y)−1(hdesy (x, y)− h(x, y)−Q−1

y g(x)TQxx+ ydes(x, y)), (5.9)

where it is assumed that the desired dynamics fdes(x) for the x-subsystem andhdesy (x, y) for the y-subsystem are smooth and chosen such that xTQxf

des(x) and

yTQyhdesy (x, y), respectively, are negative definite (e.g. fdes(x) = Ax, where A ∈

Rn×n is negative definite). Combining Eqs. (5.5), (5.6), and (5.9), one arrives at thesystem

x = fdes(x) + g(x)y (5.10)

˙y = hdesy (x, y)−Q−1

y g(x)TQxx, (5.11)

which gives x the desired dynamics when y = 0, i.e. when y has “caught up” withydes(x).

3The function ydes(x, y) is the derivative of ydes(x) along the solution trajectory, i.e.

d

dt

(ydes(x)

)=∂ydes(x)

∂xx =

∂ydes(x)

∂x(f(x) + g(x)y) =

∂ydes(x)

∂x

(fdes(x) + g(x)y

),

hence the y-dependence.

5.2. Block Backstepping 37

5.2.1. Extension of the Model

Extension of this framework to include, for example, higher derivative states or inte-grator states is straightforward and requires only recursive application of the methoddescribed above.

Adding Derivative States

Derivative states can be added to account for inputs that are integrated before en-tering the model, hence their name input integrators. To augment the model withan input integrator, one can proceed as follows. Note that the objective is still togive the x-subsystem new dynamics. Consider, for example, the dynamics

u = fD(x,y,u) + gD(x,y,u)v, (5.12)

where v ∈ Rq, fD : Rm × Rn × Rp → Rp vanishes at the origin, and gD : Rm ×Rn × Rp → Rp×q nonsingular. First, the error u = u − udes(x, y) is formed andthe functions fD(x, y, u) = fD(x, y + ydes(x), u + udes(x, y)) and gD(x, y, u) =gD(x, y+ydes(x), u+udes(x, y)) defined, where udes(x, y) is the virtual control lawfor the y-subsystem chosen to satisfy Eq. (5.9) in place of u. This error variableis then used in a positive definite function 1

2 uTQuu, where Qu ∈ Rp×p, which is

appended to the clf given in Eq. (5.7) to include the new dynamics:

VD(x, y, u) = V (x, y) +1

2uTQuu. (5.13)

As a result of including the extended dynamics, u cannot be controlled directly.Therefore, ˙y is first rewritten to explicitly contain the deviation:

˙y = h(x, y) + k(x, y)u− ydes(x, y)

= h(x, y) + k(x, y)(u− udes(x, y) + udes(x, y))− ydes(x, y)

= h(x, y) + k(x, y)u+ k(x, y)udes(x, y)− ydes(x, y),

(5.14)

leading to the clf time derivative (with terms already canceled by udes(x, y))4

VD(x, y, u) = xTQxx+ yTQy ˙y + uTQu ˙u

= xTQxfdes(x)

+ yTQy

(hdesy (x, y) + k(x, y)u

)+ uTQu

(fD(x, y, u) + gD(x, y, u)v − udes(x, y, u)

).

(5.15)

4The function udes(x, y, u) is the derivative of udes(x, y) along the solution trajectory. The u-dependence appears because of the dependence thereon by ˙y.


The fdes and hdesy terms are, as before, negative definite and need not be canceled.

New terms appear not only in ˙u, but also in ˙y. To fulfill the requirement of negativedefiniteness, the control law is chosen as

v = gD(x, y, u)−1(fdes

D,u(x, y, u)− fD(x, y, u)

−Q−1u k(x, y)TQyy + udes(x, y, u)

),

(5.16)

where fdesD,u(x, y, u) is the desired error dynamics for u, chosen such that the function

uTQufdesD,u(x, y, u) is negative definite. This results in the system

x = fdes(x) + g(x)y (5.17)

˙y = hdesy (x, y) + k(x, y)u−Q−1

y g(x)TQxx (5.18)

˙u = fdesD,u(x, y, u)−Q−1

u k(x, y)TQyy. (5.19)

Adding Integrator States

Similarly to adding derivative states, the model can be extended to include inte-grator states, or output integrators. The motivation for including them is not onlyto eliminate steady state errors, but also to synthesize (approximate) second orderdynamics. For simplicity, derivative states are excluded from this presentation. Asabove, the objective is still to prescribe new dynamics for the x-subsystem. Considerthe system

η = f I(η) + gI(η)x, (5.20)

where η ∈ Rr, f I : Rr → Rr vanishes at the origin, and gI : Rr → Rr×m nonsingular.Prescribing new dynamics for this equation would require completely relinquishingthe synthesized dynamics for the x-subsystem. The objective is therefore to merelystabilize the equation under the assumption that ηTQηf I(η) is negative definite fora positive definite weighting matrix Qη. First, the clf is augmented with a negativedefinite function of the new state:

VI(η,x, y) =1

2ηTQηη + V (x, y). (5.21)

Then, the time derivative is calculated5:

VI(η,x, y) = ηTQηη + xTQxx+ yTQy ˙y

= ηTQη(f I(η) + gI(η)x)

+ xTQx(fdes(x) + g(x)y)

+ yTQy

(h(x, y) + k(x, y)u− ydes(x, y)

).

(5.22)

5Note that it is allowed that all “lower” states also be functions of η, e.g. h = h(η,x,y).

5.3. Nonlinear Dynamic Inversion 39

This function is made negative definite along the solution trajectory by using theaugmented function

fdes′(η,x) = fdes(x)−Q−1x gI(η)TQηη (5.23)

in place of the previously defined fdes(x).

5.3. Nonlinear Dynamic Inversion

Nonlinear dynamic inversion via time-scale separation (NDI-TSS) is another tech-nique in the cascade family of controls, which, like BBS, allows one to modify asystem’s dynamics. It has been used in settings similar to this study, e.g. in [7]. Ithas been shown in [17] that NDI-TSS is very closely related to BBS and that thetwo can be regarded as belonging to the same class of control methods. Unlike BBS,however, the NDI-TSS control method is not formulated using Lyapunov theory, eventhough its stability has been proved therewith in [13]. This proof, and the controlmethod in general, are predicated on the assumption that the subsystems consideredare evolving on different time scales. In other words, fast variables need to be fastenough to be able to be considered as control inputs to slow variables.

In accordance with the terminology presented above, a system with input and outputintegrators is stabilized under the separation of time scales assumption by choosingthe virtual control law udes(x, y) and the control law v as

udes(x, y) = k(x, y)−1(hdesy (x, y)− h(x, y)

)(5.24)

v = gD(x, y, u)−1(fdes

D,u(x, y, u)− fD(x, y, u))

(5.25)

in place of the BBS control laws in Eqs. (5.9) and (5.16), respectively. The virtualcontrol law ydes(x) is as defined in Eq. (5.4) and the desired dynamics for the x-subsystem as in Eq. (5.23). With the exception of certain terms being absent, thecontrol laws look very similar to those of the BBS controller. They are in fact asimplified case of the BBS control laws. For each subsystem, the time derivativeof the desired signal as well as the inter-subsystem term that guarantees a negativedefinite clf are omitted. Comparing the closed loop system

x = fdes(x) + g(x)y (5.26)

˙y = hdesy (x, y) + k(x, y)u− ydes(x, y) (5.27)

˙u = fdesD,u(x, y, u)− udes(x, y, u) (5.28)

with that resulting from the BBS method given in Eqs. (5.17) to (5.19), one seesthat the omission of the terms in the control laws leads to the time derivatives of thevirtual controls remaining. These terms are small in comparison to the other terms


when there is time scale separation. As noted in [17], in the limit of complete timescale separation as well as when Qy = k1I, Qu = k2I, k1 →∞, k1/k2 → 0, the BBSand NDI-TSS methods formally converge.

In terms of stability, BBS is clearly superior since it is based on fewer assumptions.The cost for this guaranteed stability is two extra terms, one of which involves apotentially complicated differentiation.

5.4. Application to Aircraft

As previously stated, the aircraft model given in Chapter 2 lends itself to cascadecontrol due to its structure. One way of formulating an aircraft’s dynamics so thatit fits the structure required to utilize backstepping and nonlinear dynamic inversionis to choose the errors between the angles of attack and sideslip, bank angle, rota-tional velocity vector, the moment, and their respective commanded values as statevariables. The following fully nonlinear controllers6 are specializations of the lawspresented above.

Let

Ω =[α β ϕ

]T(5.29)

Ω = Ω−Ωcmd (5.30)

ω = ω − ωdes(η, Ω) (5.31)

Mb = Mb −Mdesb (η, Ω, ω), (5.32)

where Ωcmd is the vector of commanded angles, Mdesb (η, Ω, ω) the signal needed

to stabilize the origin of ω, and ωdes(η, Ω) the signal needed to stabilize the originof Ω. The controller functions in such a way as to drive the angles of attack andsideslip as well as bank angle to their commanded values using the vehicle’s rotationalvelocity, which in turn is driven to the appropriate value by exerting moments usingthe control surfaces. This is possible insofar as the actuators and their rates staywithin their bounds.

The orientational dynamics can be expressed as

Ω = f(Ω, δ) + g(Ω)ω, (5.33)

6They are also completely decoupled from the linear controller presented in Chapter 4.

5.4. Application to Aircraft 41

where

f(Ω, δ) =1

mV

secβ 00 10 0

[ − sinα 0 cosα− cosα sinβ cosβ − sinα sinβ

]F tot (5.34)

g(Ω) =

− cosα tanβ 1 − sinα tanβsinα 0 − cosα

1 sinϕ tan θ cosϕ tan θ

(5.35)

F tot = F tot(Ω, δ). (5.36)

For derivations of these equations, see Section A.1. The dependence of the forceF tot and the function f on the vector of actuator positions δ is due to deflections ofthe control surfaces not purely exerting moments on the vehicle, but also (relativelysmall) forces. This dependence makes a standard BBS design impossible and is inthis study neglected so that F tot = F tot(Ω) and f = f(Ω).

Let f(Ω) = f(Ω + Ωcmd) − Ωcmd7 and g(Ω) = g(Ω + Ωcmd). Then, differentiation

of Eq. (5.30) yields

˙Ω = f(Ω) + g(Ω)ω

= f(Ω) + g(Ω)ωdes(η, Ω) + g(Ω)ω.(5.37)

Using Eq. (2.2), the dynamics for the rotational velocity can be written as

ω = h(ω) + kMb, (5.38)

where h(ω) = −J−1(ω×Jω) and k = J−1 (J is assumed positive definite). Lettingh(ω) = h(ω + ωdes(η, Ω)) and differentiating Eq. (5.31), the error dynamics of therotational velocity is

˙ω = h(ω)− ωdes(η, Ω, ω) + kMb

= h(ω)− ωdes(η, Ω, ω) + kMdesb (η, Ω, ω) + kMb.

(5.39)

In order to account for model errors and synthesize approximate second order dynam-ics, a (virtual) integrator of the orientational error states is included in the controlleras a state η ∈ R3:

η = −Kηη + Ω, (5.40)

where Kη = diag(Kη,α,Kη,β,Kη,ϕ) has strictly positive diagonal elements. Drivingthis state to zero forces Ω to Ωcmd as t → ∞. The term −Kηη makes the Ω-integrator “leak” and is necessary for the positive definiteness of the clf, or, indirectly,to ensure stability of the origin of η.

7Here, the formulation is that of a tracking problem. As such, the system is nonautonomous (dueto the time dependence of Ωcmd). However, the Lyapunov stability arguments presented above,which are formulated for a setpoint problem, can be shown to be valid in a tracking setting aswell by a generalization of the autonomous case as shown in Theorems 4.8 and 4.9 in [8].


Since the actuators are modeled as first order systems, the moment exerted on thevehicle cannot be controlled directly. However, by controlling the actuator deriva-tives, one is essentially controlling the moment’s derivative. Therefore, the model isextended with the dynamics8

Mb = ν. (5.41)

Combining Eqs. (5.37) and (5.39) to (5.41) and differentiating Eq. (5.32), the com-plete system can be written as

η = −Kηη + Ω (5.42)

˙Ω = f(Ω) + g(Ω)ωdes(η, Ω) + g(Ω)ω (5.43)

˙ω = h(ω)− ωdes(η, Ω, ω) + kMdesb (η, Ω, ω) + kMb (5.44)

˙Mb = ν − Mdesb (η, Ω, ω,Mb). (5.45)

This system is in strict-feedback form and can therefore be controlled using BBS orNDI-TSS. To choose a control law, desired dynamics for the states Ω, ω, and Mb

are chosen as the linear functions

fdesΩ (η, Ω) = −KΩΩ− Qη

QΩ

η (5.46)

hdesω (ω) = −Kωω (5.47)

jdesMb

(Mb) = −KMbMb, (5.48)

respectively, where KΩ = diag(Kα,Kβ,Kϕ), Kω = diag(Kp,Kq,Kr), and KMb=

diag(Kl,Km,Kn) have strictly positive diagonal elements and Qη, QΩ ∈ R+. Then,

ωdes(η, Ω) is chosen in accordance with Eq. (5.4) as

ωdes(η, Ω) = g(Ω)−1(f

desΩ (η, Ω)− f(Ω)

), (5.49)

which is possible when g(Ω) is invertible, or, as can be seen in Section A.2, for all θsatisfying |θ| ≤ 45, a condition fulfilled for all maneuvers performed in this study.At this point in the construction of the controller, the BBS and NDI-TSS methodsdiverge.

5.4.1. BBS

From Eq. (5.49), the virtual control Mdesb (η, Ω, ω) can be chosen:

Mdesb (η, Ω, ω) = k−1

(h

desω (ω)− h(ω) + ωdes(η, Ω, ω)−

QΩ

Qωg(Ω)TΩ

), (5.50)

8Determining the signal to send to the actuators can be decoupled from the design and solved as aseparate problem, namely solving the equation τMb = ∂Mb

∂δ(δ − u) for δ, where Mb = Mb(δ)

and Eq. (2.9) has been used.

5.4. Application to Aircraft 43

where Qω ∈ R+. Finally, Mb is stabilized by choosing the control law

ν = jdesMb

(Mb) + Mdesb (η, Ω, ω,Mb)− Qω

QMb

kTω, (5.51)

where QMb∈ R+.

Combining the system given by Eqs. (5.42) to (5.45) with the desired dynamics givenby Eqs. (5.46) to (5.48) and the control laws given by Eqs. (5.49) to (5.51), the closedloop system becomes

η = −Kηη + Ω (5.52)

˙Ω = −KΩΩ− QηQΩ

η + g(Ω)ω (5.53)

˙ω = −Kωω −QΩ

Qωg(Ω)TΩ + kMb (5.54)

˙Mb = −KMbMb −

QωQMb

kTω. (5.55)

5.4.2. NDI-TSS

The NDI-TSS controller construction is completed in a similar manner, the differ-ence being that the time derivatives of the desired dynamics, ωdes(η, Ω, ω) and

Mdesb (η, Ω, ω,Mb), as well as extra terms to ensure a negative definite clf time

derivative are omitted so that the virtual control Mdesb (η, Ω, ω) and control input ν

become

Mdesb (η, Ω, ω) = k−1

(h

desω (ω)− h(ω)

)(5.56)

ν = jdesMb

(Mb), (5.57)

respectively, in place of Eqs. (5.50) and (5.51). This results in the closed loop sys-tem

η = −Kηη + Ω (5.58)

˙Ω = −KΩΩ + g(Ω)ω (5.59)

˙ω = −Kωω − ωdes(η, Ω, ω) + kMb (5.60)

˙Mb = −KMbMb − M

desb (η, Ω, ω,Mb). (5.61)

CHAPTER 6

Optimization

In nonlinear control, analytical tools such as Lyapunov theory are used extensively.The usage of these tools leads to control laws which guarantee stability, but whichoften are conservative. This, in turn, implies potentially excessive control loop gainsand unnecessary control action.

The field of mathematical optimization provides a means of analyzing and removingthe shortcomings of relying solely on analytical methods as well as making theiroutcomes more applicable to the real world. Many control problems can be posed asor augmented with the constrained nonlinear optimization problem

minθ

J(θ)

subject to h(θ) = 0,(6.1)

where J : X → R is the objective function (for engineering applications generally notconvex), X = θ ∈ Rn : h(θ) = 0 the feasible set, and h : Rn → Rm the m equalityconstraints (see [5] for an in-depth discussion of the subject).

In general, one is interested in the global minimum. Since the function J is potentiallynon-convex, global optimization methods need to be employed. In practice, these aremuch more complex than local optimizers. This, coupled with the fact that designsin aircraft control often depend on many parameters, implies programs which inthe past would have been essentially unsolvable due to their numerical complexity.However, with the exponential increase in computing power over the last few years,solving them is no longer unfeasible, even when they depend on many parameters.This leads to a new dimension in control design which can be used to solve myriadproblems.

6.1. Differential Evolution 45

6.1. Differential Evolution

In this study, the global optimization method used to solve some of these problemsis differential evolution (DE), a global unconstrained optimization method inventedby Storn and Price (see [16]) in the class of genetic optimization algorithms. Like allreal-world global optimization methods, DE is a heuristic one. As such, it provides noguarantee of finding the global extremum. However, it has been shown in [16] to bean effective and competitive algorithm for many classical optimization problems.

The objective is to locate the point of the global extremum of a function J : Rn → Rwithout constraints. Drawing on the principles of biological evolution, the algo-rithm attempts this by letting a fixed-size population of agents (candidate solutions)successively evolve into new populations by means of stochastic transformation. In-cidentally, having a population with constituents that can evaluate the function in-dependently of each other makes the algorithm scalable and well-suited for parallelcomputing. The objective function values given by the agents in a population arealways less than or equal to the objective function values given by their respective“parents”. There is currently no standard way of determining whether the objectivefunction value is “good enough”. Here, the algorithm is terminated when the num-ber of generations reaches a certain point. Analogous to the biological function ofgenetic recombination, DE also incorporates a process known as crossover to increasepopulation diversity. Algorithm 1 shows the steps of the method.

Algorithm 1. Unconstrained Differential Evolution

0. Create the initial generation with a population consisting of N ≥ 4 agentsx1,x2, . . . ,xN chosen randomly1 from Rn. Let I := 1, 2, . . . , N be the set ofindices. Define the termination criterion by defining the maximum number ofgenerations kmax ∈ N.

For each generation k = 1, 2, 3, . . .:

1. If k = kmax, terminate.

2. For each xi =[xi,1 xi,2 . . . xi,n

]T, i ∈ I:

a) Choose three mutually distinct indices u, v, w randomly from I \ i.

b) Mutation: generate a vector

x′ = xu + F (xv − xw), (6.2)

where F ∈ (0, 2] is an amplification factor.

1The agents in the population are not required to be unique. Therefore the population is not a set.

46 Chapter 6. Optimization

c) Crossover: generate a recombinant vector y =[y1 y2 . . . yn

]Twith

elements

yj =

x′j for j ∈ l ∈ 1, 2, . . . , n : rl ≤ kCO ∪ rxi,j otherwise,

(6.3)

j ∈ 1, 2, . . . , n, where r1, r2, . . . , rn are chosen randomly from [0, 1], kCO

is the “crossover” constant chosen from [0, 1] to facilitate diversity, andr is an integer chosen randomly from 1, 2, . . . , n to ensure that not allelements are taken from xi.

d) If J(y) < J(x), then set zi := y. Otherwise set zi := xi.

3. Set xi := zi ∀i ∈ I.

6.2. Application

Other studies, e.g. [14], have shown the efficacy of global optimization methods inshaping time domain responses. Here, the feasibility of the methodology presentedabove is illustrated with three design objectives concerned with inherent system prop-erties, formulated as separate minimizations of (a) the system’s L2 gain, (b) actuatorflutter, and (c) the maximum actuator rate. Each minimization is done for bothBBS and NDI-TSS at several points i in the flight envelope using the control lawsgiven in Section 5.4. The properties are optimized using the DE algorithm withcrossover constant kCO = 0.8, amplification factor F = 0.5, and population sizeN = 10. For the remainder of this section, a dot in a subscript implies the expressionis valid for both the BBS and NDI-TSS cases, e.g. f· = g· implies fBBS = gBBS andfNDI−TSS = gNDI−TSS (but not fBBS = gNDI−TSS).

In order for the simulations to be realistic, constraints need to be added. The nor-malized time domain response of a ∆αcmd = −2 pull up (i.e. a pitching down motionof 2) at point i in the flight envelope

yi,·(t) =αi,·(t)− α0

∆αcmd, (6.4)

where α0 is the initial (trimmed) value of the angle of attack, is required to lie withinthe bounds presented in Fig. 6.1. Furthermore, constraints are added to keep theelevator from saturating (see Table 3.3 for its limits) and its rate below rmax (seeSection 3.1). In order for the time derivative of the commanded value (which is usedby the controllers) to be finite, the step is replaced with a ramp of duration 0.5 s, sothat

αcmd(t) = α0 +

∆αcmd

t0.5 for 0 ≤ t < 0.5

∆αcmd for 0.5 ≤ t,(6.5)

6.2. Application 47

which, normalized, is

u(t) =αcmd(t)− α0

∆αcmd. (6.6)

The remaining commanded values βcmd and ϕcmd in the vector Ωcmd are set to theirtrimmed values. The constraints are formulated as equality constraints and collectedin the non-negative function

h·(θ·) =∑i

∫ T

0max

0,

δe,i,·(t)− δe,max

δe,min − δe,i,·(t)

δe,i,·(t)− rmax

yi,·(t)− f+(t)f−(t)− yi,·(t)

dt, (6.7)

which is zero on the feasible set and non-zero on the complement of the feasible set.Here θ· is the vector of parameters (defined below for BBS and NDI-TSS), T thesimulation time, δe,i,·(t) the elevator deflection starting at point i in the flight enve-lope, δe,max and δe,min the maximum and minimum elevator deflections, respectively,f+(t) and f−(t) the upper and lower boundaries shown in Fig. 6.1, respectively, andthe maximum is performed componentwise. The functions δe,i,·(t) and yi,·(t) havean implicit θ· dependence. Since the DE algorithm does not provide a built-in wayof minimizing a function subject to constraints, one can “encourage” it to keep thesearch within the feasible set X by penalizing the objective function when it leavesX by adding the penalty function

1

2h·(θ·)

Tσh·(θ·) (6.8)

to the objective function, where σ ∈ R5×5 is a positive definite diagonal matrix ofpenalty parameters chosen in such a way that the penalty function dominates overthe objective function on Rn \ X. Note that the domain of the objective functionnow needs to encompass all of Rn.

The vector of parameters (all defined in Section 5.4) that define the search space isgiven by

θBBS =[QΩ Qω QMb

Kα Kq Km

]T(6.9)

for BBS, i.e. the weights of the blocks in the clf and the magnitudes of the poles ofthe desired dynamics. NDI-TSS does not require the weights for the ω nor the Mb

blocks, so the parameter vector becomes

θNDI−TSS =[QΩ Kα Kq Km

]T. (6.10)

Both the BBS and NDI-TSS optimizations have Qη = 1.

Analogous to the H∞ norm for linear systems (see [19]), the L2 gain is a propertyworthy of consideration when choosing the parameters of a system since it gives


Time [s]

Normalized

step

respon

se

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 6.1.: Time domain response envelope as used in [7].

a measure of the sensitivity of an output to an input. A system with input u ∈Lm2 (0 ∪ R+)2 resulting in an output y ∈ Ln2 (0 ∪ R+) is said to have an L2 gainof at most γ ∈ R+ when, for all u and all finite T > 0,∫ T

0‖y(t)‖2 dt ≤ γ2

∫ T

0‖u(t)‖2 dt. (6.11)

Here, the output y(t) is replaced with the pitching moment m(t) caused by the inputu(t), which in turn is replaced with the normalized commanded angle of attack u(t),so that the gain gives a measure of how “violently” the controller performs maneuvers.Two simplifications are made for a rough numerical approximation of the gain:

1. Only a single input is considered.

2. T is set to 6 s, the maximum time for which the time domain response envelopeis defined (see Fig. 6.1). This limit is sufficient to capture the essential behaviorof the response.

The objective is to find the maximum of the gains with respect to the examinedpoints i of the flight envelope, i.e.

fγ,·(θ·) = maxi

(∫ T0 mi,·(t)

2 dt∫ T0 u(t)2 dt

)1/2

(6.12)

2For X ⊂ R, Ln2 (X) = f : X → Rn :

∫X‖f(t)‖2 dt <∞.

6.2. Application 49

is used as an unconstrained objective function, where mi,·(t) is the pitching moment(implicitly dependent on θ·) starting at point i. Essentially, one is minimizing thepitching moment the controller requires for the vehicle to reach the commanded angleof attack.

Another useful criterion to consider when choosing control loop gains is the movementof the actuators. For energy saving purposes as well as to minimize “wear and tear”on the actuators, one would like to minimize their movement. Here, the sum of theelevator actuator time derivatives’ L2 norms

fL2,·(θ·) =∑i

(∫ T

0|δe,i,·(t)|2 dt

)1/2

(6.13)

is used as an unconstrained objective function, where δe,i,·(t), implicitly dependenton θ·, is the elevator deflection starting at point i.

Finally, the maximum of the actuator rates—akin to their maximum power—is min-imized. This is done using the maximum of the uniform norm of the actuator rateswith respect to the examined points i of the flight envelope

f∞,·(θ·) = maxi,t|δe,i,·(t)| (6.14)

as the unconstrained objective function.

In conclusion, optimizations are performed for BBS and NDI-TSS for the followingthree parameters:

1. L2 gain from commanded angle of attack to pitching moment for comparingmoment exertion on the airfame using the objective function

Jγ,·(θ·) = fγ,·(θ·) +1

2h·(θ·)

Tσh·(θ·). (6.15)

2. Actuator flutter for comparing energy usage of the actuators using the objectivefunction

JL2,·(θ·) = fL2,·(θ·) +1

2h·(θ·)

Tσh·(θ·). (6.16)

3. Actuator rates for comparing maximum power of the actuators using the ob-jective function

J∞,·(θ·) = f∞,·(θ·) +1

2h·(θ·)

Tσh·(θ·). (6.17)


6.3. Results

The results of the three optimizations described above are shown in Tables 6.1 to 6.3.For each controller, several generations are listed with the function values given by theoptimal agents in each respective population, as well as the dispersion of the agentsnormalized by the bounds of the parameters. The dispersion is a useful measure toconsider when determining when to terminate the optimization. The less dispersedthe agents become, the more certain one can be about having found a local minimum(that may also is a global one).

All of the BBS optimizations are done with completely random initial populations (asis evident by the—in comparison to the final values—very large function values in theearly generations, caused by penalty terms), and the optimizer manages to find thefeasible set in each minimization. In the case of the NDI-TSS optimizations, however,the optimizer has difficulty locating the feasible set. Therefore, the optimizations arerestarted with one agent in the initial population in the feasible set. The remainingagents are, as in the case of the BBS optimizations, chosen randomly.

Generation Best function value Dispersion

BBS 1 5.422× 1010 6× 10−1

50 3.386× 102 1× 10−1

100 3.358× 102 3× 10−2

500 3.349× 102 2× 10−3

1000 3.349× 102 8× 10−4

5000 3.349× 102 9× 10−17

NDI-TSS 1 4.799× 102 6× 10−1

50 7.537× 101 2× 10−1

100 7.462× 101 3× 10−2

500 7.459× 101 2× 10−6

1000 7.459× 101 5× 10−7

5000 7.459× 101 6× 10−17

Table 6.1.: L2 gain minimization.

The results from the L2 gain optimizations are summarized in Table 6.1. The NDI-TSS controller reaches a smaller value than the BBS controller, suggesting it ingeneral uses less pitching moment to reach the values it is given as commands. Thedifferences in behavior of the two optimized controllers can be seen visually in Fig. 6.2.The optimal BBS controller is much faster than the optimal sluggish but economicalNDI-TSS controller. The trade-off for this speed is more elevator deflection, hencemore pitching moment exertion.

As in the L2 gain optimizations, the NDI-TSS controller is capable of a smaller L2

norm, in this case measuring the actuator energy usage, than the BBS controller. The

6.4. Conclusion 51


BBS 1 1.238× 107 7× 10−1

50 2.804× 101 3× 10−1

100 2.674× 101 3× 10−1

500 2.533× 101 3× 10−4

1000 2.513× 101 1× 10−6

5000 2.512× 101 6× 10−9

NDI-TSS 1 3.586× 101 6× 10−1

50 5.013 2× 10−1

100 4.848 7× 10−2

500 4.820 1× 10−2

1000 4.820 1× 10−3

5000 4.820 2× 10−17

Table 6.2.: L2 norm minimization.

results of the optimizations are shown in Table 6.2. The superiority of the optimalNDI-TSS controller is visually apparent in Fig. 6.3 by the movement of the elevator.It keeps the elevator fairly constant throughout the maneuver, whereas the optimalBBS controller outputs some relatively large deflections in the transient region. Justlike the controllers optimal in L2 gain, the BBS controller is fast and the NDI-TSScontroller sluggish.

Finally, the uniform norm optimizations, i.e. power minimizations, are summarizedin Table 6.3. As in the previous two cases, NDI-TSS is more conservative. Unlikethe other two optimizations, one can see from the response in Fig. 6.4 that theBBS controller is slower and has a larger overshoot. Also, the elevator movement issmoother, which is what one would expect when minimizing elevator power.

In all three cases, the respective optimal NDI-TSS controllers are “better” in thesense that the their optimizations reach lower objective function values than theiroptimal BBS counterparts.

6.4. Conclusion

As can be seen in the responses of the optimal controllers, the NDI-TSS controllersare very sluggish and seem to not even reach steady state. A fairer comparisonbetween BBS and NDI-TSS could be made if the time domain boundaries were madestricter and if the objective function also were a function of the deviation of theresponse from a desired one, such as a typical second order response. Furthermore,more extreme maneuvers could be investigated. These investigations may revealBBS to be the superior control method of the two. In addition to these changes, the



BBS 1 5.423× 107 6× 10−1

50 3.509× 101 1× 10−1

100 3.355× 101 2× 10−1

500 1.160× 101 2× 10−5

1000 1.157× 101 1× 10−6

5000 1.156× 101 1× 10−16

NDI-TSS 1 4.217× 101 6× 10−1

50 4.330 1× 10−1

100 4.305 2× 10−1

500 4.290 7× 10−4

1000 4.290 8× 10−10

5000 4.290 8× 10−10

Table 6.3.: Uniform norm minimization.

objective functions could be made more accurate. For example, the L2 gain can bebetter approximated by testing more inputs, and the norms examined by testing moreinitial conditions, i.e. performing maneuvers in more points of the flight envelope.

Global optimization using genetic algorithms proved a useful tool not only in choosingparameters for BBS and NDI-TSS controllers, but also in comparing them. Thismethod of design allows for the efficient choice of system parameters for a host ofobjectives, e.g. minimization of energy loss and actuator movement. These resultssuggest that global optimization in nonlinear control is a viable and promising designmethodology.

6.4. Conclusion 53

replacemen

Time [s]

Normalized

step

respon

se

BBSNDICommandBoundaries

EI

Time [s]0 1 2 3 4 5 6

0 1 2 3 4 5 6

-10

-5

0

0

0.2

0.4

0.6

0.8

1

1.2

Figure 6.2.: Time domain responses and elevator deflections using BBS and NDI-TSS con-trollers with parameters optimized for minimum moment exertion.


replacemen

Time [s]

Normalized

step

respon

se


EI

Time [s]0 1 2 3 4 5 6

0 1 2 3 4 5 6

-10

-5

0

0

0.2

0.4

0.6

0.8

1

1.2

Figure 6.3.: Time domain responses and elevator deflections using BBS and NDI-TSS con-trollers with parameters optimized for actuator energy conservation.

6.4. Conclusion 55

replacemen

Time [s]

Normalized

step

respon

se


EI

Time [s]0 1 2 3 4 5 6

0 1 2 3 4 5 6

-10

-5

0

0

0.2

0.4

0.6

0.8

1

1.2

Figure 6.4.: Time domain responses and elevator deflections using BBS and NDI-TSS con-trollers with parameters optimized for actuator power conservation.

APPENDIX A

Derivations and Proofs

A.1. Dynamics of α, β, and ϕ

Using Eqs. (2.1), (2.6), and (2.8) and

F tot = F aero + T eb

00mg

, (A.1)

the dynamics for the angle of attack α is derived as follows:

α =d

dtarctan

w

u

=1

1 + (wu )2

uw − wuu2

=1

V 2 − v2(wV cosα cosβ − uV sinα cosβ)

=1

V 2(1− sin2 β)(wV cosα cosβ − uV sinα cosβ)

=secβ

V(w cosα− u sinα)

=secβ

V

[− sinα 0 cosα

](F tot

m− ω × v

)

= secβ[− sinα 0 cosα

]F tot

mV−

q sinα cosβ − r sinβr cosα cosβ − p sinα cosβp sinβ − q cosα cosβ

=

secβ

mV

[− sinα 0 cosα

]F tot +

[− cosα tanβ 1 − sinα tanβ

]ω.

(A.2)

A.1. Dynamics of α, β, and ϕ 57

Similarly, using Eqs. (2.1), (2.7), and (2.8) and the above definition of F tot, thesideslip dynamics is:

β =d

dtarcsin

v

V

=1√

1− v2

V 2

V v − vVV 2

=1√

V 2 − v2

(v − v

VV)

=1√

V 2(1− sin2 β)

(v − v

V 2vTv

)=

secβ

V

([0 1 0

]− v

V 2vT)v

=secβ

V

([0 1 0

]− sinβ

[cosα cosβ sinβ sinα cosβ

]T)v

=1

V

[− cosα sinβ secβ − tanβ sinβ − sinα sinβ

]v

=1

V

[− cosα sinβ cosβ − sinα sinβ

](F tot

m− ω × v

)=

1

mV


]F tot

+[cosα sinβ − cosβ sinα sinβ

] q sinα cosβ − r sinβr cosα cosβ − p sinα cosβp sinβ − q cosα cosβ

=

1

mV


]F tot +

[sinα 0 − cosα

]ω.

(A.3)

The combination of Eqs. (A.2) and (A.3) along with Eq. (2.3) leads to the relationαβϕ

= f(α, β, ϕ, δ) + g(α, β, ϕ)ω, (A.4)

where

f(α, β, ϕ, δ) =1

mV

secβ 00 10 0

[ − sinα 0 cosα− cosα sinβ cosβ − sinα sinβ

]F tot (A.5)

g(α, β, ϕ) =

− cosα tanβ 1 − sinα tanβsinα 0 − cosα

1 sinϕ tan θ cosϕ tan θ

. (A.6)

58 Appendix A. Derivations and Proofs

A.2. Invertibility of the Virtual Control Gain Matrix

Proposition 1. The virtual control gain matrix g(Ω) given in Eq. (5.35) is invertiblefor all θ satisfying |θ| ≤ 45 and all α, β ∈ [−30, 30].

Proof. Assume g(Ω) is singular for some θ satisfying |θ| ≤ 45. Then its determinant

det g(Ω) = − cosα tanβ sinϕ tan θ cosα

− sinα(cosϕ tan θ + sinϕ tan θ sinα tanβ)− cosα

= − cos2 α tanβ sinϕ tan θ − sin2 α sinϕ tan θ tanβ

− sinα cosϕ tan θ − cosα

= (− tanβ sinϕ− sinα cosϕ) tan θ − cosα

(A.7)

is zero, so that

cot θ =tanβ sinϕ+ sinα cosϕ

− cosα. (A.8)

However, the domain of α and β is [−30, 30], so that

| cot θ| =∣∣∣∣[tanβ secα tanα

] [sinϕcosϕ

]∣∣∣∣≤∥∥∥∥[tanβ secα

tanα

]∥∥∥∥=

√(tanβ secα)2 + tan2 α

≤√

(tan 30 sec 30)2 + tan2 30

=

√√√√(√3

3

2√3

)2

+

(√3

3

)2

=√

7/3,

(A.9)

i.e.

|θ| ≥ arctan3√

7

7> 45. (A.10)

This is an obvious contradiction of the initial assumption and thus completes theproof.

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[17] Johan Thunberg and John W.C. Robinson. Block backstepping, NDI and relatedcascade designs for efficient development of nonlinear flight control laws. InProceedings of the AIAA Guidance, Navigation and Control Conference andExhibit, Honolulu, Hawaii, August 2008. AIAA paper AIAA 2008-6960.

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Examensarbete E357 i Optimeringslära och systemteori

September 2011

www.math.kth.se/optsyst/

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