L₁ adaptive control of a generic

fighter aircraft

A N D R E A S M Y L E U S

Master of Science Thesis Stockholm, Sweden 2013

L₁ adaptive control of a generic fighter aircraft

A N D R E A S M Y L E U S

Master’s Thesis in Systems Engineering (30 ECTS credits) Master Programme in Aerospace Engineering (120 credits)

Royal Institute of Technology year 2013 Supervisor at SAAB Aeronautics, Linköping

was Daniel Simon and Peter Rosander Supervisor at KTH was Per Enqvist Examiner was Per Enqvist TRITA-MAT-E 2013:12 ISRN-KTH/MAT/E--13/12--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

This master’s thesis was performed at the section of Flight Control Systems at SAABAeronautics in Linkoping as a part of my Master of Science in Aerospace Engineeringat KTH, Stockholm. This report examines the use of L1 adaptive control to stabilizethe inner longitudinal and lateral loops of a generic fighter aircraft, in the event offailure of the system that measures current speed and altitude.

The philosophy of the L1 adaptive controller is to decouple the adaptation fromthe control loop by using a state-predictor based adaptation scheme, still only com-pensating for the uncertainties within the bandwidth of the control channel by the useof low-pass filters.

The main goal of the project was to investigate in the tuning of the L1 adaptivecontroller with respect to the nonlinear uncertainties related to the failure, and witha limited sampling rate of 60 Hz. The desired closed-loop dynamics for the state-predictor was designed by linearising the aircraft dynamics in a point in the middle ofthe flight envelope and placing the poles of the system with respect to flying qualities.The modified piecewise constant adaptation law was chosen as adaptation law, whichachieves faster adaptation by increasing the sampling rate, yielding better performanceat a given sample rate compared to the piecewise constant adaptation law [1]. All thestates were transformed to discrete time in order to be implementable digitally.

Results have shown that augmenting a state-feedback controller with a L1 adaptivecontroller increases robustness in the whole flying envelope, with good flying qualities.Problems were discovered in the low speed regions of the envelope, where the L1 adap-tive controller did not provide the desired performance. A switching scheme betweentwo L1 adaptive controllers was examined. The switch between the controllers wasdone by knowing when the landing gear was up or down. The second state-predictorwas designed with linearised dynamics in landing speed and altitude. The switchingscheme was flown in a simulator with a nonlinear generic fighter aircraft model withgood results.

1

Sammanfattning

Detta examensarbete utfordes vid sektionen for styrsystem pa SAAB Aeronautics iLinkoping som en del av min Master of Science i Aerospace Engineering vid KTH,Stockholm. Rapporten undersoker anvandningen av L1 adaptiv reglering for att sta-bilisera de inre longitudinella och laterala looparna i ett generisk stridsflygplan, ihandelse av ett fel pa det system som mater aktuell hastighet och hojd.

Filosofin med L1 adaptiv reglering ar att frankoppla anpassning fran reglerkretsenmed en prediktorbaserad anpassningslag, men kompenserar bara for osakerheterernainom bandbredden for styrkanalen med hjalp av lagpassfilter.

Det huvudsakliga malet med projektet var att undersoka finjusteringen av en L1

adaptiv regulator med avseende pa de olinjara osakerheterna i handelse av ett fel,och med en begransad samplingsfrekvens pa 60 Hz. Den onskade dynamiken for detslutna systemet i prediktorn designades genom att linjarisera flygplanets dynamik ien punkt i mitten av flygenveloppen och placera polerna for systemet med avseendepa flygegenskaper. Den modifierade styckvis konstanta anpassninglagen valdes somanpassninglag. Den uppnar snabbare anpassning genom att oka samplingshastighetenoch ger battre prestanda vid en given samplingsfrekvens jamfort med den styckviskonstant anpassninglagen [1]. Alla tillstand transformerades till diskret tid for attkunna implementeras digitalt.

Resultaten har visat att genom att utoka en vanlig tillstandsaterkoppling med enL1 adaptiv regulator okar robustheten i hela flygenveloppen med goda flygegenskaper.Problem upptacktes i de laga hastighetsregionerna, dar den L1 adaptiva regulatorninte ger onskad prestanda. En vaxlingsmetodik mellan tva L1 adaptiva regulatorer un-dersoktes. Vaxlingen mellan regulatorerna gjordes genom att veta nar landningsstalletar uppe eller nere. Den andra tillstandsprediktorn utformades med linjariserad dy-namik i landningshastighet och hojd. Vaxlingsmetodiken flogs i en simulator med enolinjar generisk flygplansmodell med goda resultat.

2

Acknowledgements

First of all I would like to express my gratitude towards Sam Nicander for making this the-sis possible. I would also like to thank my supervisors Daniel Simon and Peter Rosanderwho have provided invaluable help and support throughout the project. Anders Petterssonalso deserve my gratitude for helping me with valuable insights about L1 adaptive control.Finally I would like to thank all the SAAB employees that have contributed to my projectand given me positive impressions of SAAB.

Andreas MyleusLinkoping, April 8, 2013

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Contents

1 Introduction 71.1 Historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Flight mechanics 82.1 The VEGAS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The rigid-body equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Aerodynamic forces and moments . . . . . . . . . . . . . . . . . . . . 102.3 Linearised equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Short-period mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Lateral mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Flight control 143.1 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Inner/outer loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Gain scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.3 Nonlinear control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Adaptive flight control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 L1 adaptive control 174.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Modified piecewise constant adaptation law . . . . . . . . . . . . . . . . . . 20

4.3.1 L1 adaptive control architecture. . . . . . . . . . . . . . . . . . . . . 214.3.2 Definitions and sufficient condition for stability . . . . . . . . . . . . 224.3.3 Definitions of the performance bounds . . . . . . . . . . . . . . . . . 234.3.4 Analysis of the L1 adaptive controller . . . . . . . . . . . . . . . . . . 244.3.5 Discrete time implementation . . . . . . . . . . . . . . . . . . . . . . 26

4.4 Gradient descent adaptive law . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5 L1 adaptive controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.5.1 Flying qualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.5.2 Sampling frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5.3 Reference system design . . . . . . . . . . . . . . . . . . . . . . . . . 294.5.4 Model errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.5.5 Low-pass filter bandwidth design . . . . . . . . . . . . . . . . . . . . 324.5.6 Control surface dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 334.5.7 Multiple reference systems . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Results 355.1 Linear models with baseline controller . . . . . . . . . . . . . . . . . . . . . . 355.2 Linear models with baseline and L1 controller . . . . . . . . . . . . . . . . . 37

5.2.1 High-speed controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.2 Low-speed controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 VEGAS nonlinear model with baseline and L1 controller . . . . . . . . . . . 425.3.1 High-speed controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3.2 Low-speed controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 StyrSim simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4

6 Discussion 476.1 Linear models simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 VEGAS model simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3 StyrSim simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7 Conclusions 487.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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Nomenclature

vB = vehicle velocity vector with vector elements u, v and wwB = vehicle angular velocity vector with vector elements p, q and rFB = forces acting on the aircraft, with elements X, Y and ZMB = moments acting on the aircraft, with elements L, M and NIB = mass inertia matrixφ, θ ψ = roll, pitch and yaw euler anglesg = gravitational constantρ = air density

V∞ = length of velocity vector, V∞ =√u2 + v2 + w2

α = angle of attackβ = angle of side-slipqd = ρV 2

∞/2 = dynamic pressureS = aerodynamic reference areac = mean cordb = wing spanm = massIx = mass inertia around the x-axisCxy = non-dimensional aerodynamic coefficient = dCx

dy

δc, δe, δa, δr = control surface deflections, canard, elevator, aileron, rudderx(t), u(t), y(t) = state, control input and output vectorAm, Bm, Cm = state-space matrices of the desired closed-loop dynamicsAmd , Bmd , Cmd = discrete time state-space matrices of the desired closed-loop dynamicsω0, ω = nominal system input gain matrix, system input gain matrixf(·), f1(·), f2(·) = unknown nonlinear function, matched, unmatched componentBum = constant matrix, BT

mBum = 0 and the rank of [Bm, Bum] = nBumd = constant matrix, BT

mdBumd = 0 and the rank of [Bmd , Bumd ] = n

B,Bd = [Bm, Bum], [Bmd , Bumd ]M(s) = desired closed loop-systemHm(s) = transfer function from u(s) to y(s)r(t) = reference signal vectorKg = feed-forward constant gain matrixIn = identity matrix with size n× nx(t), ˙x(t), x(t) = predicted state vector of desired system response, d

dtof x, x(t)− x(t)

σ1(t), σ2(t) = matched, unmatched adaptive estimates

ω(t), θ1(t), θ2(t) = adaptive estimatesΓ = adaptive gainC(s) = low-pass filterKxy = feedback matrix; x = p, ry; y = H/L, see subscriptsunom, uad(t) = nominal state-feedback, L1 adaptive control signalζ, ωn/fn = damping ratio, undamped natural frequency (rad/s)/(Hz)τ = time constant for roll responseSubscriptsp, ry = pitch, roll-yawL,H = low-speed controller, high-speed controllerm,um = matched, unmatchedid = identifiedtrue = trueid = identified

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1 Introduction

The dynamics of an aircraft vary considerably in different parts of the flight envelope. Todeal with this, gain scheduling of multiple linear controllers, has traditionally been used.Typically this requires that the current speed and altitude of the aircraft is measurable. Ifthe systems which measure speed and altitude fail, a backup system with a controller thatcan adjust to that situation is required. In recent years, various methods for adaptive controlhave grown increasingly popular since they can automatically adapt to the varying dynamicsof the aircraft.

The task I was given was to investigate in the L1 adaptive method with respect toaircraft control and loss of airspeed and altitude data. In other words, it will have to beable to compensate for the lack of air data information (such as dynamic pressure), but also,e.g., an error in mass and mismodelling of non-linear aerodynamic coefficients. Anotherrequirement is that it would also need to compensate for changing control surface authorityin the envelope.

For practical reasons it would also need to be implementable digitally, with discrete-timemodels, control law and adaptation law with a sampling frequency of 60 Hz.

1.1 Historical perspective

Research in adaptive control was initiated in the 1950s by the design of autopilots for high-performance aircraft. It was found that aircraft which operated over a wide range of speedsand altitudes could not be controlled with just one ordinary linear feedback in the wholeflight envelope. Therefore more advanced controllers were needed to cope with differentflight conditions. Gain scheduling was found to be a suitable strategy for flight control. Theinterest in adaptive control diminished partly because of the problems in dealing with thetechniques of the time. In [2], this period of time is called the brave era because “there wasa very short path from idea to flight test with very little analysis in between”. The tragicflight test accident of the X-15-3 [3] is an example of this and gave adaptive control a badreputation.

In the 1960s there was more done which contributed to adaptive control. Topics suchas state-space, stability theory, stochastic control theory, dynamic programming and systemidentification increased the understanding of adaptive processes [4].

Adaptive control was reborn again in the 1970s and early 1980s when proofs of stabilityof adaptive systems appeared, where the merge between robust control and system identifi-cation had a particular role [4]. Two architectures of adaptive control emerged, direct andindirect methods. For the direct methods the control parameters are estimated directly, whilefor the indirect methods the system parameters are estimated, and the control parametersare obtained from a design procedure from these estimates.

The late 1980s and early 1990s gave new insights about the robustness of adaptive con-trollers together with investigations of nonlinear systems. The primary reason for introduc-ing adaptive control was to obtain techniques that could adapt to changes in the systemdynamics and disturbances.

In the beginning of the century adaptive control gained more interest and the first resultson L1 adaptive control [5] was presented in 2006, with aerospace applications in mind. Sincethen flight test have been performed [6], showing the potential for the theory.

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2 Flight mechanics

Before cruising into the world of flight control we need to be comfortable with the dynamicsof the system that is going to be controlled. The goal of this section is to introduce thedifferential equations of motion in the Body-axes frame of an aircraft.

2.1 The VEGAS model

The aircraft model used in this project is called VEGAS and consists of a nonlinear aerodatamodel, rigid body dynamics, servo dynamics, engine model, atmospheric model, and so on.The aerodynamic derivatives used in VEGAS are part of a generic fighter jet model known asADMIRE (AeroDATA Model In Research Environment). This model was jointly developedby SAAB, the Royal Institute of Technology and the Swedish Defence Research Agency [7].The ADMIRE model is open for anyone to use and similar to the Gripen fighter aircraft, asit includes the same types of control surfaces and has similar characteristics, such as unstablepitch dynamics in the subsonic region. This makes it suitable for a feasibility study like this.

The full VEGAS model has 9 states, 13 inputs and 28 outputs, which is excessive for thepurpose of this project. To capture the relevant dynamics for the problem we define

x = [β, α, p, q, r]T (2.1)

u = [δe, δa, δr]T (2.2)

Figure 1: Body- and Wind-axes coordinate system of a Gripen fighter aircraft with controlsurfaces, rotational rates, angles and non-dimensional aerodynamic forces and moments.[Courtesy of SAAB].

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2.2 The rigid-body equations

The general rigid-body equations are described by Euler’s general equations of unsteadymotion, derived by considering force and moment equilibriums around the aircraft’s centerof gravity. Assuming a Body-axes (see Figure 1) coordinate system we can start by defining

vB = [u, v, w]T (2.3)

wB = [p, q, r]T (2.4)

FB = [X, Y, Z]T (2.5)

MB = [L,M,N ]T (2.6)

Newton’s second law can be expressed in the following vector equations, assuming mass isconstant

FB = m ˙vB +m(wB × vB) (2.7)

MB = IB ˙wB + wB × IBwB (2.8)

IB =

Ix −Ixy −Ixz−Ixy Iy −Iyz−Ixz −Iyz Iz

(2.9)

This can be written in component form in the following way by assuming Ixy = Iyz = 0 (Ixzis the symmetry plane)

X = m(u+ qw − rv + g sin θ) (2.10)

Y = m(v + ru− pw − g cos θ sinφ) (2.11)

Z = m(w + pv − qu− g cos θ cosφ) (2.12)

L = Ixp− Ixz r + qr(Iz − Iy)− Ixzpq (2.13)

M = Iy q + rp(Ix − Iz) + Ixz(p2 − r2) (2.14)

N = −Ixzp+ Iz r + pq(Iy − Ix) + Ixzpq (2.15)

where the g is the gravity vector and the relationship between the angular velocities and theEuler angular rates is the following, see also Figure 2,φθ

ψ

=

1 sinφ tan θ cosφ tan θ0 cosφ − sinφ0 sinφ sec θ cosφ sec θ

pqr

(2.16)

The force and moment components acting on the aircraft are composed of aerodynamic andpropulsive contributions. In control theory you usually want to describe the equations ofmotion on the following form

x(t) = f(t, x(t), u(t)) (2.17)

From (2.7) and (2.8) we can express the following velocity and angular velocity derivatives

˙vB = FB/m− wB × vB (2.18)

˙wB = I−1B (MB − wB × IBwB) (2.19)

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Figure 2: Basic aerodynamic and flight mechanic angles, as well as body rotational ratesp, q and r. Subscript B denotes Body-axes, compare with Figure 1. Subscript E denotesEarth-axes.

which results in the following equations

u = rv − qw − g sin θ +X/m (2.20)

v = −ru+ pw + g cos θ sin θ + Y/m (2.21)

w = qu− pv + g cos θ cosφ+ Z/m (2.22)pqr

=

Ix 0 −Ixz0 Iy 0−Ixz 0 Iz

−1 L+ q (Ixz p− Iz q) + Iy q rM − p (Ixz p− Iz q)− r (Ix p− Ixz r)

N + q (Ix p− Ixz r)− Iy p q

(2.23)

These nonlinear equations of motion are used in VEGAS, but could also be used as a modelwhen testing the controller in, e.g., MATLAB.

2.2.1 Aerodynamic forces and moments

The aerodynamics forces (FB) and moments (MB) can be written as a product betweenthe dynamic pressure qd = ρV 2

∞/2, a reference area S and/or a reference length lref and an

aerodynamic dimensionless coefficient Cxy

(= ∂Cx

∂y

). The reference area is usually the wing

area and the reference length, the mean aerodynamic cord, (c), or the wing span, (b). Thedimensionless coefficients are nonlinear functions of mainly the speed, body rates, attitude,control surface deflections and aeroelastic effects. These are close to impossible to computeanalytically and are usually found by wind-tunnel tests.

2.3 Linearised equations of motion

The nonlinear model, with the relevant states, needs to be linearised in order to be used in theL1 adaptive control framework. The equations can be linearised using the small-disturbancetheory [8]. When applying the theory we assume that the motion of the aircraft consists ofsmall deviations around a steady flight condition, so called trimmed flight. The theory cannot be applied to all problems, e.g. stalled flight, but in most cases the theory is sufficientlyaccurate. Trimmed flight condition means that the aircraft is flying with a constant attitudeand altitude by using appropriate constant control surface deflections. The relevant states

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and control inputs are

x = [v, w, p, q, r]T (2.24)

u = [δc, δe, δa, δr]T (2.25)

By using a 50 /50 combination of δc and δe, so that the actual input δe is scaled 100 % onboth canards and elevators. In other words, we use the same amount of control effort onboth the canards and the elevators to control the pitch of the aircraft. Therefore, we canrewrite the input vector as

u = [δe, δa, δr]T (2.26)

All the variables in the equations of motion are replaced by a reference value plus a smalldisturbance, e.g.,

X = X0 + ∆X (2.27)

and for convenience, the flight condition is assumed to be symmetric, propulsive forcesconstant and the x-axis aligned along the direction of the aircraft’s velocity vector. Considerthe X-force equation (2.10) with the small-disturbance theory

X0 + ∆X −mg sin(θ0 + ∆θ) (2.28)

= m

[d

dt(u0 + ∆u) + (q0 + ∆q)(w0 + ∆w)− (r0 + ∆r)(v0 + ∆v)

](2.29)

If we neglect the products of the disturbance and the previous assumptions hold, the equationbecomes

X0 + ∆X −mg sin(θ0 + ∆θ) = m∆u (2.30)

This equation can be rewritten to the following, by using a common trigonometric relation-ship,

X0 + ∆X −mg(sin θ0 + ∆θ cos θ0) = m∆u (2.31)

Further, if all the disturbance quantities are set to 0, we have the following trimmed flightcondition

X0 −mg sin θ0 = 0 (2.32)

This reduces the X-force equation to

∆X −mg∆θ cos θ0 = m∆u (2.33)

∆X is the change in aerodynamic and propulsive force in the x-direction and can be expressedas a Taylor series in terms of disturbance variables. Typical disturbance variables can beu,w, δe, which results in

∆X =∂X

∂u∆u+

∂X

∂w∆w +

∂X

∂δe∆δe (2.34)

where ∂X∂u

= Xu and so on, are called the stability derivatives.

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2.3.1 Short-period mode

Instead of looking at the full model with all the states it is possible to decouple the longitu-dinal short-period motion. Elevator control input (δe) mainly affect the z-axis speed (w) andthe pitch rate (q). An approximation of the short-period mode of motion can be obtainedby assuming ∆u = 0 and neglecting the X-force equation [8][

∆w∆q

]=

[Zw u0

Mw +MwZw Mq +Mwu0

] [∆w∆q

]+

[Zδe

Mδe +MwZδe

]∆δe (2.35)

It is the movement of the air around the aircraft that creates the aerodynamic forces and mo-ments. Therefore it is more suitable to use the Wind-axes reference system when consideringthe equations for controlling the aircraft. The Wind-axes system is the Body-axes referencesystem aligned with the movement of the aircraft relative to the air, see V∞ in Figure 1.Because flight control is only concerned with short term objectives, we can transform to theWind-axes using the short-period approximation in [8]

tanα = w/u0 → ∆α ≈ ∆w

u0

(2.36)

where the definition of Mα = u0Mw, Za = u0Zw and Mα = u0Mw [8]. These relationshipsgives the following[

∆α∆q

]=

[ Zau0

1

Mα +MαZαu0

Mq +Mα

] [∆α∆q

]+

[Zδe

Mδe + Mα

u0Zδe

]∆δe (2.37)

where

Zw =−(CLα + 2CD0)qdS

mu0

Zδe =CZδeqdS

m

Mw =CmaqdSc

u0IyMq =

CmqqdSc2

2u0Iy

Mδe =CmδeqdSc

IyMw =

CmαqdSc2

2u20Iy

The linearisation from VEGAS, with a trimmed flight condition inMach = 0.6 andAltitude =5000 m, resulted in the following[

∆α∆q

]=

[−1.16 0.9264.32 −1.26

] [∆α∆q

]+

[0.36728.4

]∆δe (2.38)

2.3.2 Lateral mode

It is not possible to decouple pure yawing motion from roll motion, e.g. an aileron controlinput affects both the roll rate (p) and the yaw rate (r). These coupling effects need to betaken into consideration in a joint lateral mode model.

Consider the nonlinear lateral equations of motion from Section 2.2, by rearranging theterms and considering the small-disturbance theory we obtain [8]

∆v∆p∆r

=

Yv Yp −(u0 − Yr)Lv + Ixz

IxNv Lp + Ixz

IxNp Lr + Ixz

IxNr

Nv + IxzIxLv Np + Ixz

IxLp Nr + Ixz

IxLr

∆v∆p∆r

(2.39)

+

0 YδrLδa + Ixz

IxNδa Lδr + Ixz

IxNδr

Nδa + IxzIxLδa Nδr + Ixz

IxLδr

[∆δa∆δr

](2.40)

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where

Xy =Xy

1− (I2xz/(IxIz))

(2.41)

X = [L,N ]T (2.42)

y = [v, p, r, δa, δr]T (2.43)

By once again transforming to the Wind-axes using the approximation in [8]

tan β = v/u0 → ∆β ≈ ∆v

u0

(2.44)

and assuming that Ixz = 0, results in∆β∆p∆r

=

Yβ/u0 Yp/u0 −(1− Yr/u0)Lβ Lp LrNβ Np Nr

∆β∆p∆r

+

0 Yδr/u0

Lδa LδrNδa Nδr

[∆δa∆δr

](2.45)

where

Yβ =CyβqdS

mYp =

CypqdSb

2mu0

Yr =CyrqdSb

2mu0

Yδr =Cyδr qdS

m

Lβ =ClβqdSb

IxLp =

ClpqdSb2

2Ixu0

Lr =ClrqdSb

2

2Ixu0

Lδr =Clδr qdSb

Ix

Lδa =ClδaqdSb

IxNβ =

CnβqdSb

IzNp =

CnpqdSb2

2Izu0

Nr =CnrqdSb

2

2Izu0

Nβ =CnβqdSb

IzNp =

CnpqdSb2

2Izu0

Nr =CnrqdSb

2

2Izu0

Nδr =Cnδr qdSb

Iz

Nδa =CnδaqdSb

Iz

The linearisation from VEGAS, with a trimmed flight condition inMach = 0.6 andAltitude =5000 m, resulted in the following∆β

∆p∆r

=

−0.247 0.048 −0.944−25.1 −2.23 0.4844.48 −0.127 −0.456

∆β∆p∆r

+

−0.051 0.06856.3 5.585.31 −5.67

[∆δa∆δr

](2.46)

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3 Flight control

The following section describes the newly developed L1 adaptive control strategy that hasbeen used in this study. But before introducing the strategy it is good to have a basicknowledge about automatic control, current methods for flight control and the principles ofadaptive control.

3.1 Feedback control

The concept of feedback control is to control the dynamic behaviour of a system. This isusually done by measuring the states that are important for the dynamics of the system andcomparing with a preferred reference input which includes information about the desiredoutput of the system. The error is fed to the controller which determines the current controlsignal. The control signal sets the actuators, which affect the system, and the same processrepeats itself. This is typically implemented in a computer. Typical states that are measuredfor an aircraft include velocity, attitude and angular rates. Actuators on an aircraft aretypically the different control surfaces, such as the ailerons and elevators. To control asystem in the best possible way, a good understanding of the system is needed. Some controlmethods, such as PID-control does not use a model in the controller structure. The methodstill requires some knowledge of the system in order to manually tune the controller. Othermethods which are model-based, such as LQG/LQR or MPC requires a linear mathematicalmodel of the system when designing the controller. The mathematical model is usuallyexpressed by differential equations that describe the relationship between the states and theinput in the following way

x(t) = f(t, x(t), u(t)) (3.1)

One difficulty lies in identifying f(·), but usually it is enough if the model is a good approx-imation of the real system to achieve similar results in simulations. Section 2 concerns themodelling of an aircraft.

3.1.1 Inner/outer loop

An aircraft feedback control system is commonly divided into an inner and an outer loop.The inner loop usually controls the actuators to achieve, e.g., a desired angle of attack, whilethe outer loop specifies the angle of attack based on, e.g., desired altitude. The outer loopis effectively an autopilot and can be overridden by the pilot when using the control stick.

It is also common to decouple the longitudinal and lateral modes. This decoupling isdescribed further in Section 2, but in simple words we design one controller for the desiredpitch rate and one for the desired yaw angle and roll rate. The focus of the control design inthis project is to implement the L1 in these inner loops where the nonlinear effects are themost prominent.

Like stated above I have chosen to let the controller set a desired pitch rate q. This willeffectively trim the aircraft to the current flight condition when the stick is zeroed.

14

3.1.2 Gain scheduling

Gain scheduling is perhaps one of the most popular approaches to nonlinear control design.With the gain scheduling strategy, the dynamics of the aircraft is linearised at given operatingpoints in the envelope, assuming that the dynamics in a small region close to the operatingpoints is close to linear. Nonlinear aerodynamic coefficients can also be linearised, e.g.,schedule α over different linearised Cmα . The controller design can then be done with classiclinear control methods such as PID-control, pole-placement or LQR/LQG. The same is donefor possible combinations of e.g. speed, altitude and configuration [9]. To succeed it isimportant to find measurable variables that correlate well with changes in the dynamics.The control parameters are then altered between depending on auxiliary measurements ofthese states. Gain scheduling can be regarded as a mapping from process parameters tocontrol parameters [4]. It can be implemented as a function or table look-up. See Figure 3for an illustration of gain scheduling.

Controller System

Gain scheduler

referencestate

control signal

feedback

Auxiliary measurments

Figure 3: Controller with gain scheduling

An advantage of gain scheduling is that linear stability analysis and control theory canbe used for nonlinear systems to give hints about the behaviour. Local stability does nothowever guarantee that the overall system is stable with respect to, e.g., time. An obviousdisadvantage is that it takes a long time to develop and analyse all the controllers. The factthat all controllers are computed off-line means that there is no feedback to compensate foran incorrect schedule [4].

3.1.3 Nonlinear control

Another approach to flight control is to use nonlinear control methods such as sliding-modecontrol, backstepping or feedback linearisation [4]. These methods are usually based onnonlinear models of the system and the goal is to make one controller valid for the wholeregime. Designing such nonlinear models are often difficult and may not be able to accountfor all the varying dynamics.

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3.2 Adaptive flight control

“An adaptive controller is a controller with adjustable parameters and a mechanism for ad-justing the parameters.” [4]

As stated before, an essential part of control design is to have knowledge of the systemto be controlled. However, it is impossible to develop models which includes all the dynam-ics of the system.

The idea behind adaptive control is to design the controller for a single flight condition,and adapt to the changes of the system on-line. The changes are formulated as nonlinearparameter variations to a linear system. In this application, control surface deflectionsmainly produce aerodynamic moments which results in angular velocity, thus body ratesp, q and r correspond to the matched uncertainties. An error in mass, mismodelling ofaerodynamic coefficients or lack of air data information (such as dynamic pressure) howeverproduce unmatched model errors.

Figure 4 shows a general structure of an adaptive control system. It uses the controlsignal together with the output of the system to generate a compensation in the controllerin order to adapt to the changes of the system.

Controller System

Adaptive Law

referencestate

control signal

feedback

Figure 4: Scheme of an adaptive control system.

As mentioned before, there are two types of architectures of adaptive control, direct andindirect methods. In the direct methods the control parameters are estimated directly,while in the indirect methods the system parameters are estimated on-line, and the controlparameters are obtained from a design procedure.

MRAC (Model Reference Adaptive Control) is an example of an adaptive control strategy,which can be implemented in either direct or indirect form. L1 was born from a reformulationof MRAC [5] with the addition of a low-pass filter in the feedback loop. The benefit of thiswas that one could design the controller not to let high frequency adaptation content to thecontrol signal.

16

4 L1 adaptive control

Even though current control schemes obviously work they have their limitations. Theselimitations include the lack of robustness to changing system dynamics or unforeseen eventssuch as actuator failure. An adaptive scheme could, e.g., relieve a baseline controller, wherein the nominal case it would behave as usual, but in an unforeseen event the adaptivecontroller would extend the robustness. However, up until recently adaptive schemes havenot been able to prove robustness a priori and have relied on Monte-Carlo simulations todetermine the best adaptation rate for various flight conditions [10]. Fast adaptation oftenled to high frequencies in control signals and increased sensitivity to time delays.

The philosophy of the L1 adaptive controller is to decouple the adaptation and thecontrol loops by using a predictor-based fast adaptation scheme, still only compensating forthe uncertainties within the bandwidth of the control channel. The estimation rate is onlylimited by the available CPU power, while the robustness is limited by the available controlbandwidth.

There are two types of L1 fast adaptation schemes: the gradient descent adaptive law,see Section 4.4 and [6], which achieves fast adaptation by increasing the adaptive gain, andthe piecewise constant adaptive law, see Section 4.3 and [6][1], which achieves the same goalby increasing the sampling frequency.

By increasing the adaptation rate, the input and output of the system can be renderedarbitrarily close to the corresponding signals of the reference system that defines the desiredclosed-loop response, in both transient and steady state. The architecture does not requirepersistence of excitation, gain scheduling, control reconfiguration or high-gain feedback con-troller [10].

The architecture I have chosen for my controller design is a modified piecewise constantpresented in [1], where main difference from the from the controller in [11] is an additionalterm to the adaptive law. The controller in [11] has successfully been used in NASAs GTM(AirSTAR) [12] and the Boeing X-48B [13].

In general you can say that the L1 adaptive controller based on a linear identified system onthe form

x(t) = Aidx(t) +Bidu(t) (4.1)

is designed to control a general unknown nonlinear system on the form, see (2.17),

x(t) = fx(t, x(t)) + fu(t, x(t), u(t)) (4.2)

The error between the systems above is identified on-line and corrected for by the controller.

4.1 Norms

Some of the norms that will be used later are introduced below.

‖A‖∞ = max1≤i≤m

m∑i=1

|aij|,max of absolute row sum

‖A‖2 =√λmax(A ∗ A) = σmax(A)

‖G(s)‖L1 =

∫ ∞0

|g(τ)|dτ , if G(s) is exponentially stable and proper

‖x‖L∞ = max(|xi|)

17

4.2 Problem formulation

Consider the following MIMO system with nonlinear uncertainties:

x(t) = Amx(t) +Bmωu(t) + f(t, x(t)) (4.3)

y(t) = Cmx(t) (4.4)

where x(t) ∈ Rn is the measured system state vector, u(t) ∈ Rm is the control signal (m ≤ n),y(t) ∈ Rm is the regulated output.

Am ∈ Rn×n is a known Hurwitz (eig(Am) < 0) matrix specifying the desired closed-loopdynamics, Bm ∈ Rn×m is a full-rank known constant matrix, Cm ∈ Rm×n is a known full-rank constant matrix, (Am, Bm) is controllable and (Am, Cm) is observable. ω ∈ Rm×m isthe uncertain system input gain matrix, f : Rn×R→ Rn is an unknown nonlinear function.The initial states are assumed to be inside an arbitrarily large know set; ‖x0‖∞ ≤ ρ0 < ∞with known ρ0 > 0.

Also consider the true system formulation in (4.2),

x(t) = fx(t, x(t)) + fu(t, x(t), u(t)) (4.5)

assuming that fu(·) is linear and time-varying with respect to u(t), according to

fu(t, x(t)) = Btrue(t, x(t))u(t) (4.6)

where Btrue ∈ Rn×m is the true unknown time-varying input matrix. Substituting (4.6) in(4.2) results in

x(t) = fx(t, x(t)) +Btrue(t, x(t))u(t) (4.7)

Also define,

f∆(t, x(t)) = fx(t, x(t))− Aidx(t) (4.8)

B∆(t, x(t)) = Btrue(t, x(t))−Bidω (4.9)

where Aid and Bid are the identified linear system matrices, fx : R × Rn → Rn and f∆ :R × Rn → Rn are unknown nonlinear functions, B∆ ∈ Rn×m and ω ∈ Rm×m are unknownmatrices. We can see that ω is redundant, a change in ω would only change B∆. Substituting(4.8) and (4.9) to (4.7) gives the following identified system with model errors.

x(t) = Aidx(t) +Bidωu(t) + f∆(t, x(t)) +B∆(t, x(t))u(t) (4.10)

Introducing a control signal with a nominal linear state-feedback part and an adaptive part,where F (s) is the actuator dynamics,

u(s) = F (s)unom(s) + F (s)uad(s) (4.11)

and by considering assumptions and lemmas in [6] we can write

u(t) = ωunom(t) + σunom(t) + ωuad(t) + σuad(t) (4.12)

where σu(t) , σunom(t) + σuad(t), which results in

x(t) = Aidx(t) +Bid(ωunom(t) + ωuad(t) + σu(t)) + f∆(t, x(t))

+ B∆(t, x(t))(ωunom(t) + ωuad(t) + σu(t))

18

Inserting unom = −Kxyx(t) gives

x(t) = Aidx(t) +Bid(−ωKxyx(t) + ωuad(t) + σu(t)) + f∆(t, x(t))

+ B∆(t, x(t))(−ωKxyx(t) + ωuad(t) + σu(t)) (4.13)

= (Aid −BidKxy︸ ︷︷ ︸Am

)x(t) +BidKxyx(t) +Bid(ωuad(t) + σu(t))−BidKxyx(t)ωx(t) + f∆(t, x(t))

+ B∆(t, x(t))(−ωKxyx(t) + ωuad(t) + σu(t))

= Amx(t) +Bid(ωuad(t) + σu(t)) +BidKxy(I− ω)x(t) + f∆(t, x(t))

+ B∆(t, x(t))(−ωKxyx(t) + ωuad(t) + σu(t))

By defining

f(t, x(t)) = Bid(Kxy(I− ω)x(t) + σu(t)) + f∆(t, x(t)) (4.14)

+ B∆(t, x(t))(−ωKxyx(t) + ωuad(t) + σu(t))

we can write

x(t) = Amx(t) +

Bm︷︸︸︷Bid ωuad(t) + f(t, x(t)) (4.15)

y(t) = Cmx(t) (4.16)

which is on the same form as the original problem formulation (4.3).

The system (4.15) can be written as

x(t) = Amx(t) +Bm(ωuad(t) + f1(t, x(t)) +Bumf2(t, x(t)) (4.17)

y(t) = Cmx(t) (4.18)

where Bum ∈ Rn×(n−m) is a constant matrix such that BTmBum = 0, and the rank of

[Bm, Bum] = n, while f1 : Rn × R → Rm and f2 : Rn × R → Rn−m are unknown non-linear functions that satisfy [

f1(t, x(t))f2(t, x(t))

]= B−1f(t, x(t)) (4.19)

B ,[Bm, Bum

](4.20)

Here f1 represents the matched component of the unknown nonlinearities and f2 representsthe unmatched uncertainties.

The system uncertainties are assumed to satisfy the following assumptions:Assumption 3.2.1.1 (Boundedness of fi(t, 0)).There exist Bi > 0 such that

‖fi(t, 0)‖∞ ≤ Bi0 (4.21)

holds for all t ≥ 0, and for i = 1, 2.

Assumption 3.2.1.2 (Semiglobal Lipschitz condition)For any δ > 0, there exist a positive K1δ , K2δ , such that

‖fi(t, x1)− fi(t, x2)‖∞ ≤ Kiδ‖x1 − x2‖∞, i = 1, 2 (4.22)

19

hold for all ‖x1‖∞, ‖x2‖∞ ≤ δ, uniformly in t.

Assumption 3.2.1.3For any δ > 0, there exist a positive Lδ such that∣∣∣∣∣∣∣∣∂f(t, x)

∂t

∣∣∣∣∣∣∣∣∞≤ Lδ (4.23)

for all ‖x‖∞ ≤ δ, uniformly in t.

Assumption 3.2.1.4 (Partial knowledge of uncertain system input gain).The system input gain matrix ω is assumed to be an unknown (non-singular) strictly row-diagonally dominant matrix with sgn(ωii) known. Also, we assume that there exists a knowncompact convex set Ω such that ω ∈ Ω ⊂ Rm×m, and that a nominal system input gain ω0 ∈ Ωis known.

4.3 Modified piecewise constant adaptation law

The control objective is to design a full-state feedback L1 adaptive controller which ensuresthat y(t) tracks the output response of a desired system M(s) defined as

M(s) , Hm(s)Kg (4.24)

Hm(s) = Cm(sIn − Am)−1Bm (4.25)

Kg = Hm(0) = −(CmA−1m Bm)−1 (4.26)

where Kg is a feed-forward constant gain matrix, to a given bounded reference signal r(t)both in transient and steady-state. It ensures in steady state, to couple one reference signalto one output signal by unity gain. Hm(s) is the transfer function from u(s) to y(s)

y(s) = Hm(s)u(s) = Cm(sIn − Am)−1Bmu(s) (4.27)

This control architecture produces a mismatch between the desired system and the actualplant. The estimated signals, which are based on the prediction error, are used in thecontroller to compensate for the uncertainties, but only within the bandwidth of the filters.This estimation scheme is a modified piecewise constant adaptive law, first introduced in[1]. The main difference from the controller in [11] is the adaptive law. The function h(t)remembers the influence of the uncertainties from the previous step to improve the nextstep. Figure 5 visualizes the control architecture and the equations are listed below in thissection.

20

4.3.1 L1 adaptive control architecture.

Plant

referencexcontrol signal

State predictor

-+

Adaptation Lawσ

x_hat

x_tilde

σ1+

inv(Hm(s))Hum(s)

Kg

σ2

+- C(s)

Figure 5: Schematics of the L1 adaptive controller.

State-predictorBy parametrizing the nonlinear functions f1(t, x(t)) and f2(t, x(t)) in (4.17) we obtain thefollowing state-predictor

˙x(t) = Amx(t) +Bm(ω0uad(t) + σ1(t)) +Bumσ2(t) (4.28)

y(t) = Cmx(t) (4.29)

where ω0 is the nominal system input gain, σ1(t) ∈ Rm and σ2(t) ∈ R(n−m) are the adaptivematched and unmatched estimates. For each element in the input u there is a matched σ andunmatched signals are added so that the total number of σ are equal to the number of states.

Adaptation laws

h(t) = h(iTs) (4.30)

h((i)Ts) = −x(iTs) + h((i− 1)Ts) (4.31)

σ(t) =

[σ1(t)σ2(t)

]=

[σ1(iTs)σ2(iTs)

], t ∈ [iTs, (i+ 1)Ts) (4.32)[

σ1(iTs)σ2(iTs)

]= −Φ(Ts)

−1eAmTsx(iTs) + Φ(Ts)−1h(iTs), i = 0, 1, 2, ... (4.33)

Φ(Ts) = A−1m (eAmTs − In)B (4.34)

x(t) , x(t)− x(t)→ x(iTs) , x(iTs)− x(iTs) (4.35)

The time argument iTs effectuates zero order sample and hold at sampling time intervalsTs using index i, hence the name “piecewise constant”. The adaptive law [1] is derived byintegrating the error dynamics

˙x(t) = Amx(t) +B(σ(t)− σ(t)) (4.36)

where

σ(t) =

[(ω − ω0)u(t) + η1(t)

η2(t)

], ηi(t) = fi(t, x(t)) (4.37)

from time iTs to iTs + t, yielding in

x(iTs + t) = eAmTsx(iTs) +

∫ t

0

eAm(t−ξ)Bdξσ(iTs)−∫ t

0

eAm(t−ξ)Bσ(iTs + ξ)dξ (4.38)

21

Thus, at the sample time we have

x((i+ 1)Ts) = eAmTsx(iTs) + Φ(Ts)σ(iTs)−∫ Ts

0

eAm(t−ξ)Bσ(iTs + ξ)dξ (4.39)

Inserting the adaptation law (4.33) in to (4.39) results in

x((i+ 1)Ts) = eAmTsx(iTs)

+ Φ(Ts)(−Φ(Ts)−1eAmTsx(iTs) + Φ(Ts)

−1h(iTs))

−∫ Ts

0

eAm(t−ξ)Bσ(iTs + ξ)dξ (4.40)

which is also

x((i+ 1)Ts) = h(iTs)−∫ Ts

0

eAm(t−ξ)Bσ(iTs + ξ)dξ (4.41)

From the recursion of the adaptive law (4.31) we have that

x((i+ 1)Ts) = h(iTs)− h((i+ 1)Ts) (4.42)

which yields in

h(iTs) =

∫ Ts

0

eAm(t−ξ)Bσ((i− 1)Ts + ξ)dξ (4.43)

The proof is omitted in this text, but can be found in [1].

Control lawTo compensate for the uncertainties and track the reference signal r(t) with zero steady-stateerror the control law in the frequency domain is given by

u(s) = C(s)(Kgr(s)− σ1(s)−H−1m (s)Hum(s)σ2(s))−Kxyx(s) (4.44)

The L1 adaptive controller is defined by combining adaptive control law (4.44), the predictor(4.28), the adaptation rule (4.33) subject to the L1-norm condition in (4.51).

4.3.2 Definitions and sufficient condition for stability

Introduce the following transfer matrices

Hxm(s) , (sIn − Am)−1Bm (4.45)

Hxum(s) , (sIn − Am)−1Bum (4.46)

Hm(s) , CmHxm(s) = Cm(sIn − Am)−1Bm (4.47)

Hum(s) , CmHxum(s) = Cm(sIn − Am)−1Bum (4.48)

and let xin(t) be the signal of the Laplace transform xin(s) , (sIn − Am)−1x0 and ρin ,‖(sIn − Am)−1‖L1ρ0. Since Am is Hurwitz and x0 is bounded we have that ‖xin‖L∞ ≤ ρin.Where ‖x0‖L∞ ≤ ρ0 as previously defined.

The design of the controller involves the matrix K ∈ Rm×m, which is a feedback gain matrix,and D(s), which is a m×m strictly proper transfer matrix. The following filter

C(s) , ωKD(s)(Im + ωKD(s))−1 (4.49)

22

has to have static gain C(0) = Im, for all ω ∈ Ω. The choice of D(s) needs also to ensurethat C(s)H−1

m (s) is a proper stable transfer matrix. A simple choice of D(s) is 1sIm which

givesC(s) = ωK(sIm + ωK)−1 (4.50)

For the proofs of stability and performance bounds, the choice of K and D(s) also needs toensure that, for a given ρ0, there exists a ρr > ρin such that the following L1-norm conditionholds:

‖Gm(s)‖L1 + ‖Gum(s)‖L1`0 <ρr − ‖Hxm(s)C(s)Kg‖L1‖r‖L∞ − ρin

K1ρrρr +B0

(4.51)

where

Gm(s) , Hxm(s)(Im − C(s)) (4.52)

Gum(s) , (In −Hxm(s)C(s)H−1m (s)C)Hxum(s) (4.53)

while

`0 ,K2ρr

K1ρr

, B0 , max

B10,

B20

`0

where B10, B20 were defined in (4.21). Further, for an arbitrary constant γx > 0, let ρx =ρr + γx and

γx ,‖Hxm(s)C(s)H−1

m (s)C‖L1

1− ‖Gm(s)‖L1K1ρr − ‖Gum(s)‖L1K2ρr

γ0 + β (4.54)

where γ0 and β are arbitrarily small positive constants, such that γx ≤ γx. Also let ρu =ρur + γu, where

ρur , ‖ω−1C(s)‖L1(K1ρr +B10) (4.55)

+ ‖ω−1C(s)H−1m (s)Hum(s)‖L1(K2ρr +B20) (4.56)

+ ‖ω−1C(s)Kg‖L1‖r‖L∞ (4.57)

γu , (‖ω−1C(s)‖L1K1ρr + ‖ω−1C(s)H−1m (s)Hum(s)‖L1K2ρr )γx (4.58)

+ ‖ω−1C(s)H−1m (s)C‖L1 γ0 (4.59)

4.3.3 Definitions of the performance bounds

Introduce Ts > 0 and let it be sample time of the adaptation. This sample period canbe directly related to the computation performance of the CPU in which the controller isimplemented, and also the capabilities of the sensors that sample the states of the system.

For i = 1, 2, let B0 = maxiBi0, Kδ = max

iKiδ, for any δ > 0. Also let

α1(t) , ‖eAmt‖∞ (4.60)

α2(t) ,∫ t

0

‖eAm(t−ξ)BΦ−1(Ts)eAmTs‖∞dξ (4.61)

α3(t) ,∫ t

0

‖eAm(t−ξ)B‖∞dξ (4.62)

where Φ(Ts) was defined in (4.34) and B in (4.20).

Define the following bounds, for i = 1, 2, 3,

αi(Ts) = maxt∈[0,Ts]

αi(t) (4.63)

23

Also define

dx , ‖Am‖∞ρx + maxω∈Ω‖Bm(ω − ω0)‖∞ρu + ‖B‖∞(Kρxρx +B0) (4.64)

ρσ , maxω∈Ω‖(ω − ω0)‖∞ρu + (Kρxρx +B0) (4.65)

du , ‖s(Im +KD(s)ω0)−1KD(s)‖L1(‖Kg‖L1‖r‖∞ (4.66)

+ (1 + ‖H−1m (s)Hum(s)‖L1)(2α1(Ts) + 1)ρσ) (4.67)

dσ , maxω∈Ω‖(ω − ω0)‖∞du + (Kρxdx + Lρx) (4.68)

Finally let,

ς(Ts) = α3(Ts)dσTs (4.69)

γ0(Ts) , (α1(Ts) + α2(Ts))α3(Ts)dσTs + 2α3(Ts)dσTs (4.70)

Lemma 1: The following limit holds according to [1]:

limTs→0

γ0(Ts) = 0 (4.71)

4.3.4 Analysis of the L1 adaptive controller

Closed-loop reference systemTo analyse the stability and performance of the L1 adaptive controller, first assume that ωand f(x(t), t) are known, which results in the following non-adaptive controller:

xref (t) = Amxref (t) +Bm(ωuref (t) + f1(xref (t), t)) (4.72)

+ Bumf2(xref (t), t) (4.73)

uref (s) = ω−1C(s)(Kgr(s)− η1ref (s)−H−1m (s)Hm(s)η2ref ) (4.74)

yref (t) = Cxref (t) (4.75)

where ηiref (t) , fi(xref (t), t), for i = 1, 2.

Lemma 2: According to [1], the following holds for the closed-loop system in (4.72)-(4.75),if ‖x0‖∞ ≤ ρ0 and subject to the L1-norm in (4.51):

‖xref‖L∞ ≤ ρr, and ‖uref‖L∞ ≤ ρur (4.76)

Proof. Proof is omitted here, but the author of [1] refers that it is similar to the proof forLemma 2 in [11].

Prediction Error SignalSubtracting (4.17) from (4.105) gives the following prediction error dynamics

˙x(t) = Amx(t) +Bmη1 +Bumη2(t) (4.77)

where

η1(t) = σ1 − ((ω − ω0)uad(t) + η1(t)) (4.78)

η2(t) = σ2 − η2(t)) (4.79)

ηi(t) = fi(x(t), t), for i = 1, 2 (4.80)

24

If Ts is chosen that the following is satisfied

γ0(Ts) < γ0 (4.81)

then the prediction error x(t) can be reduced both in transient and steady-state by reducingTs. The error in the prediction signal plays an important role because it holds information ofthe uncertainties in the system. The estimates that are extracted from the prediction errorare used in the controller to compensate for the uncertainty effects.

Lemma 3: According to [1], let Ts satisfy (4.81) and if

‖xτ‖L∞ ≤ ρx, and ‖uτ‖L∞ ≤ ρu (4.82)

for some τ > 0, then the following holds for the prediction dynamics (4.77)

‖xτ‖L∞ ≤ γ0 (4.83)

Proof. The proof is omitted here, but can be found in [1].The bound on γ0 for the prediction error dynamics decreases as Ts decreases. The result ofthe proof is that the bound γ0 decays faster than in the method in [11]. This is due to themodified adaptation law, which results in multiplication of Ts to γ0. This methods is thusproven to need less computational power that the method introduced in [11].

Transient and Steady-State Performance

Theorem 1: According to [1], let Ts be chosen to satisfy (4.81). Given the closed-loopsystem with the L1 adaptive controller, subject to the L1-norm condition (4.51), and theclosed-loop reference system in (4.72)-(4.75), if ‖x0‖∞ < ρ0, then

‖x‖L∞ ≤ ρx (4.84)

‖u‖L∞ ≤ ρu (4.85)

‖x‖L∞ ≤ γ0 (4.86)

‖xref − x‖L∞ ≤ γx (4.87)

‖uref − u‖L∞ ≤ γu (4.88)

‖yref − y‖L∞ ≤ ‖C‖∞γx (4.89)

Proof. Proof is omitted here, but the author of [1] refers that it is similar to the proof in[6].

25

4.3.5 Discrete time implementation

In order to be implementable as a discrete time controller all the states in the model haveto be in discrete time. This is achieved by discretizing the state-predictor.

Consider the continuous state-predictor dynamics in (4.28)

˙x(t) = Amx(t) +Bm(ω0uad(t) + σ1(t)) +Bumσ2(t) (4.90)

y(t) = Cmx(t) (4.91)

This can be written as, assuming zero-order hold for the input u(t) and ˆσi(t),

x((i+ 1)Ts) = Amdx(iTs) +Bmd(ω0uad(iTs) + σ1(iTs)) +Bumdσ2(iTs) (4.92)

y(iTs) = Cmdx(iTs) (4.93)

where

Amd = eAmTs (4.94)

Bmd = A−1m (eAmTs − In)Bm (4.95)

Bumd = N (BTmd

) (4.96)

rank([Bmd , Bumd ]) = n (4.97)

Bd = [Bmd , Bumd ] (4.98)

Cmd = Cm (4.99)

If we once again consider the error dynamics, but this time in discrete time

x((i+ 1)Ts) = Amdx(iTs) +Bd(σ(iTs)− σ(iTs)) (4.100)

To once again cancel out all the terms except the one with σ(iTs) we need to formulate thefollowing adaptation law

σ(iTs) = −B−1d Amdx(iTs) +B−1

d h(iTs) (4.101)

Inserting (4.101) in to (4.100)

x((i+ 1)Ts) = Amdx(iTs) +Bd((−B−1d Amdx(iTs) +B−1

d h(iTs))− σ(iTs)) (4.102)

once again yields in

x((i+ 1)Ts) = h(iTs)−Bdσ(iTs) (4.103)

whereh(iTs) = Bdσ((i− 1)Ts) (4.104)

by considering the recursion in (4.31). This formulation of the adaptation law has not beenproven in theory, but has worked in simulations.

26

4.4 Gradient descent adaptive law

In this section, the L1 adaptive control strategy with the gradient descent adaptive law ispresented. This method achieves fast adaptation by increasing the adaptive gain (Γ). Whencombining the adaptation law defined in (4.107)-(4.111), where Γ is chosen as a large value,with a limited sampling rate, the result is most likely that the adaptive estimates bouncesbetween the projection bounds. This makes the system quickly unstable [14]. The structureis presented as orientation.

State-predictorBy parametrizing the nonlinear functions f1(t, x(t)) and f2(t, x(t)) in (4.17) we obtain thefollowing state-predictor

˙x(t) = Amx(t) +Bm(ωuad(t) + θ1(t)‖xt‖∞ + σ1(t)) +Bum(θ2(t)‖xt‖∞ + σ2)(4.105)

y(t) = Cmx(t) (4.106)

where ω ∈ Rm×m, θ1 ∈ Rm, σ1 ∈ Rm, θ2 ∈ Rn−m, σ2 ∈ Rn−m are the adaptive estimates.

Adaptation lawThe adaptive estimates are updated according to the following adaptation law

ω(t) = ΓProj(ω(t),−(xT (t)PBm)TuT (t)) (4.107)

θ1(t) = ΓProj(θ1(t),−(xT (t)PBm)T‖xt‖∞) (4.108)

σ1(t) = ΓProj(σ1(t),−(xT (t)PBm)T ) (4.109)

θ2(t) = ΓProj(θ2(t),−(xT (t)PBum)T‖xt‖∞)) (4.110)

σ2(t) = ΓProj(σ2(t),−(xT (t)PBum)T ) (4.111)

where x(t) , x(t) − x(t), Γ ∈ R+ is the adaption gain, P = P T > 0 is the solution to thealgebraic Lyapunov equation Am

TP + PAm = −Q, for some arbitrary Q = QT > 0. Theprojection operation ensures that ω(t) ∈ Ω, ‖θi(t)‖∞ ≤ θbi , |σi(t)| ≤ σbi , where Ω, θbi , σbi aredefined in [6].

Control lawTo compensate for the uncertainties and track the reference signal r(t) with zero steady-stateerror the following control law is given

u(s) = −C(s)η(s) (4.112)

and η is the Laplace transform of the signal

η(t) , ωuad(t) + η1(t) + η2m − rg(t) (4.113)

where rg(s) , Kgr(s), η2m , H−1m (s)Humη2, ηi , θi(t)‖x(t)‖∞ + σi(t), i = 1, 2

The lemmas and proofs mainly omitted in this text, but can be found in [6].

27

4.5 L1 adaptive controller design

As mentioned before, the L1 adaptive control with the modified piecewise constant adap-tation law [1] has been used in this study. The controller structure has been implementedin Simulink, and then later code-generated to C-code in order to merge with the nonlinearVEGAS model. The following sections describe the process of tuning the controller to getthe desired qualities.

4.5.1 Flying qualities

Before designing any flight control system the engineer needs to know what degree of controlis required for the pilot to consider the aircraft safe and flyable. The flying qualities expectedby the pilot depend mostly on the type of aircraft and the flight phase of the mission.

Flying qualities are largely related to the dynamic and control characteristics of theaircraft. If we for example consider the short-period motion as a second-order linear transferfunction

G(s) =ω2n

s2 + 2ζωns+ ω2n

(4.114)

The question that needs to be answered is what values should ζ (damping ratio) and ωn (un-damped natural frequency) have to satisfy the pilot. Researchers have studied this problemusing simulators and flight test aircraft, and a common rating system is the Cooper-Harperscale [8]. Flying qualities research provides engineers with information on how to assess theflying qualities early in the design. Figure 6 shows the levels of short-period flying qualitiesrelated to the damping ratio and the undamped natural frequency. This kind of figure isusually referred to as a thumbprint plot and it has been used as a guideline for placing thepoles of the longitudinal desired closed-loop system.

Further guidelines from [8] and [15] were used in order to place the poles to achieve goodflying qualities, see Section 4.5.3. They included requirements for dutch roll damping ratioand natural frequency and also the speed dependant time-constant of the roll response.

Figure 6: Short-period flying qualities. [8]

28

4.5.2 Sampling frequency

The piecewise adaptive law states that a higher sampling frequency results in better perfor-mance, see Lemma 3. In this project a fixed sampling frequency of 60 Hz is chosen, whichis the maximum execution rate of the VEGAS model.

4.5.3 Reference system design

When designing the reference system for the state-predictor in (4.92) the method [16] ofplacing the poles of the nominal linearised model to get the desired response is used. Com-pared to using a generic reference system to specify the response, this method takes thecross-coupling effects in the B-matrix into consideration. With this method you can specifythe desired response for the system in terms of damping ratio and undamped natural fre-quency characteristics. Consider the following continuous linearised short-period dynamicsin the trimmed state of Mach = 0.6 and Altitude = 5000 m.[

∆α∆q

]=

[−1.16 0.9264.32 −1.26

]︸ ︷︷ ︸

ApH

[∆α∆q

]+

[0.36728.4

]︸ ︷︷ ︸BpH

δe

By using the following command in MATLAB:

KpH = place(ApH , BpH , PpH ) (4.115)

where

PpH =[−ζpHωnpH + iωnpH

√1− ζ2

pH,−ζpHωnpH − iωnpH

√1− ζ2

pH

](4.116)

with damping ζpH = 0.9, and fnpH = 0.5 Hz → ωnpH = (0.5× 2π) rad/s. This gives

KpH = (0.323, 0.110) (4.117)

The desired closed-loop dynamics thus become[∆α∆q

]= (ApH −BpHKpH )

[∆α∆q

]+BpHδe (4.118)

[∆α∆q

]=

[−1.27 0.885−4.84 −4.38

]︸ ︷︷ ︸

AmpH

[∆α∆q

]+

[0.36728.4

]︸ ︷︷ ︸BmpH

δe

The same methodology is adopted for the lateral dynamics (2.46), but here we have to place3 poles. Where τry corresponds to the time constant for the roll response,

PryH =[−1/τryH ,−ζryHωnryH + iωnryH

√1− ζ2

ryH,−ζryHωnryH − iωnryH

√1− ζ2

ryH

](4.119)

with damping ζryH = 0.9, τryH = 3 s and fnryH = 0.5 Hz→ ωnryH = (0.5× 2π) rad/s.

29

4.5.4 Model errors

The uncertainties considered in the text are

f∆(t, x(t)) = A∆(t)x(t) (4.120)

B∆(t, x(t)) = B∆(t) (4.121)

which are variations of the system dynamics as the speed and altitude changes. The changesin the dynamics are assumed not to change with respect to the current states. Further weknow that F (0) = 1, which gives ω = 1, thus f(t, x(t)) from (4.15) results in

f(t, x(t)) = (Bid +B∆(t))σu(t) + A∆(t)x(t) +B∆(t)(−Kxyx(t) + uad(t)) (4.122)

To be able to formulate the L1-norm we need to calculate K1ρr and K2ρr , consider assumption3.2.1.2 and rewrite to the following

‖fi(t, x1)− fi(t, x2)‖∞‖x1 − x2‖∞

≤ Kiδ , i = 1, 2 (4.123)

Inserting f(·) into the inequality yields in

‖(((Bid +B∆(t))σu(t) + uad(t))− ((Bid +B∆(t))σu(t) + uad(t)))

‖x1 − x2‖∞(4.124)

+((A∆(t)x1 −B∆(t)Kxyx1)− (A∆(t)x2 −B∆(t)Kxyx2))‖∞‖x1 − x2‖∞

=

where the first term is equal to zero, resulting in

‖((A∆(t)x1 −B∆(t)Kxyx1)− (A∆(t)x2 −B∆(t)Kxyx2))‖∞‖x1 − x2‖∞

=

‖(A∆(t)(x1 − x2)−B∆(t)Kxy(x1 − x2))‖∞‖x1 − x2‖∞

(‖A+B‖≤‖A‖+‖B‖)≤

‖(A∆(t)(x1 − x2))‖∞‖x1 − x2‖∞

+‖(−B∆(t)Kxy(x1 − x2))‖∞

‖x1 − x2‖∞

which gives us the following, with respect to (4.20),

maxA∆(t),(x1−x2)6=0

(‖I+B

−1A∆‖∞)

+ maxB∆(t),(x1−x2)6=0

(‖I+B

−1B∆Kxy‖∞)≤ K1ρr (4.125)

maxA∆(t),(x1−x2)6=0

(‖I−B−1A∆‖∞

)+ max

B∆(t),(x1−x2)6=0

(‖I−B−1B∆Kxy‖∞

)≤ K2ρr (4.126)

where

I+ =

1 0 00 1 00 0 0

I− =

0 0 00 0 00 0 1

(4.127)

if we have, e.g., 2 matched states and 1 unmatched state. For the longitudinal model, weget K1ρr = 3.54 and K1ρr = 0.069.

30

To get a feel for the model errors associated with the loss of speed and altitude information Ihave included Figure 7. They present parameter variations of linearised models for differentmach speeds and altitudes with respect to a nominal linearised model in Mach = 0.6 andAltitude = 5000 m.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

||A∆||2/||A||

2

Mach

Altitude = 5000 m

(a) ‖A∆‖2

‖A‖2as a function of Mach

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

||B∆||2/||B||

2

Mach

Altitude = 5000 m

(b) ‖B∆‖2

‖B‖2as a function of Mach

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

||A∆||2/||A||

2

Altitude (m)

Mach = 0.6

(c) ‖A∆‖2

‖A‖2as a function of Altitude

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.2

0.4

0.6

0.8

1

1.2

1.4

||B∆||2/||B||

2

Altitude (m)

Mach = 0.6

(d) ‖B∆‖2

‖B‖2as a function of Altitude

Figure 7: Different norms as a function of Mach and Altitude

Further, if f(t, x(t)) is globally Lipschitz with uniform a Lipschitz constant K, then (4.51)

‖Gm(s)‖L1 + ‖Gum(s)‖L1`0 < limρr→∞

ρr − ‖Hxm(s)C(s)Kg‖L1‖r‖L∞ − ρinK1ρrρr +B0

→ (4.128)

‖Gm(s)‖L1 + ‖Gum(s)‖L1`0 <1

K1ρr

→ (4.129)

‖Gm(s)‖L1K1ρr + ‖Gum(s)‖L1K2ρr < 1 (4.130)

which gives us the L1-norm criteria to relate to.

31

4.5.5 Low-pass filter bandwidth design

Specifying K and D(s) to get the desired filter C(s) as in (4.144), while satisfying theL1-norm (4.51) is an open problem. We know from theory that the filters have to be BIBO-stable with static gain C(0) = 1 and not let frequencies higher that the effective controlbandwidth through [6]. A large K improves load disturbance attenuation but more moremeasurement noise is injected, causing large actuator demands. This noise will be dampedby the actuator and plant dynamics, but can cause actuator wear and undesired excitationof the plant dynamics [17]. One simple choice of D(s) is 1

s, which was chosen for this

application.The system will follow the desired system dynamics better as ‖Gm(s)‖L1 and ‖Gum(s)‖L1

are decreased, see (4.51). It is important to notice that the stability and performanceguarantees are only valid if the nonlinear functions fi(t, x(t)) fit in to the assumptions of(4.3). Reducing the L1-norms will make robustness to deviations that do not fit into (4.3)smaller [18]. The assumptions of the nonlinear functions do not consider for example time-delays in the system.

The low-pass filters designed for the L1 controller are first-order, discrete time filterswith a sampling period of 1/60 s. They are based on the s-domain filters, but discretizedas below. A design was chosen to have separate filters for the matched and unmatchedestimates. For the unmatched estimate a cascaded filter was used, inspired by [18][19]. Thisgives the following control law:

σ2pre(z) = Cum1(z)σ2(z) (4.131)

u(z) = Kgr(z)−Kxyx(z)− Cm(z)σ1(z) (4.132)

− Cum2(z)Hm(z)−1Hum(z)σ2pre(z) (4.133)

Short-period filtersFor the given choice of D(s) = 1

s, consider Figure 8 with ‖Gm(s)‖L1K1ρr and ‖Gum(s)‖L1K2ρr

as a function of the bandwidth of the matched and unmatched filters for the longitudinalmodel. The derivation of K1ρr and K2ρr can be found in Section 4.5.4.

0 50 100 150 200 250 300 3500

5

10

15

20

25

ω (rad/s)

||G

m(s)||

L1

K1

||Gum

(s)||L

1

K2

||Gm

(s)||L

1

K1+||G

um(s)||

L1

K2

Figure 8: L1-norms as a function of low-pass filter bandwidths

32

We can see from Figure 8 that increasing the bandwidth of the filters decreases the norms.However if we chose bandwidths of the filters that are lower than the available controlbandwidth of 20 rad/s, the L1-norm is not met.

To at least ensure that we let frequencies above the closed-loop reference system responsebandwidth, we construct the following filters with inspiration from [18].

Cm(s) =5ωnp

s+ 5ωnp→ Cm(z) =

(1− e−5ωnpTs)z

z − e−5ωnpTs(4.134)

Cum1(s) =ωnp

s+ ωnp→ Cum1(z) =

(1− e−ωnpTs)zz − e−ωnpTs

(4.135)

Cum2(s) =1.2ωnp

s+ 1.2ωnp→ Cum2(z) =

(1− e−1.2ωnpTs)z

z − e−1.2ωnpTs(4.136)

(4.137)

Lateral filtersThe same problem with the L1-norm exists for the lateral model, so lateral filters have alsobeen chosen with inspiration from [18]. Below is a list of the relevant filters.

Cm1(s) =1.2ωnry

s+ 1.2ωnry→ Cm1(z) =

(1− e−1.2ωnryTs)z

z − e−1.2ωnryTs(4.138)

Cm2(s) =

1τryωnry

s+ 1τryωnry

→ Cm2(z) =(1− e−

1τry

Ts)z

z − e−1τry

Ts(4.139)

Cum1(s) =ωnry

s+ ωnry→ Cum1(z) =

(1− e−ωnryTs)zz − e−ωnryTs

(4.140)

Cum2(s) =1.8ωnry

s+ 1.8ωnry→ Cum2(z) =

(1− e−1.8ωnryTs)z

z − e−1.8ωnryTs(4.141)

4.5.6 Control surface dynamics

Control surface dynamics is modelled with a first order low-pass filter with a bandwidthof 20 rad/s (≈ 3 Hz). This is implemented in Simulink by adding a discrete time transferfunction block to the input signal going to the state-predictor. The model from commandedsurface deflection to actual deflection becomes

δ(s) = F (s)δcmd(s) =20

s+ 20δcmd(s)→ (4.142)

δ(z) = F (z)δcmd(z) =0.2835

z − 0.7165δcmd(z) (4.143)

Rate limit and saturation blocks are also added to the control signal before the state-predictor, see Figure 5, to create an overall nonlinear model. Time-delays to the actuatorshave not been considers. The extension of considering actuator dynamics does not changethe architecture [14], however the low-pass filters are now defined as

C(s) , F (s)ωKD(s)(Im + F (s)ωKD(s))−1 (4.144)

where K and D(s) need to ensure that C(s) is a strictly proper and stable transfer functionwith static gain C(0) = 1 for all F ∈ F∆. For the proofs of stability, this filter needs to

33

ensure the L1-norm (4.51) holds.

Table 1 lists the limits and constraints of the VEGAS model.Min Max

δe, δa, δr −30 30

δe, δa, δr −50/s 50/s

Table 1: Constraints on the VEGAS model.

4.5.7 Multiple reference systems

The simulations have shown that one controller can not cope with all the uncertainties in theflying envelope, especially in low speed flying (landing). An idea was developed to increasethe number of controllers, one for low speed, low altitude (landing) and one for high speed,high altitude (normal flying). Switching between these two was done by knowing when thelanding gear was down or up. The benefit of having an extra controller, and state-predictor,is that one can specify and have control over the system response more accurately. Thestate-predictor would have linearised system dynamics in the point of the envelope usedfor landing (Mach = 0.2 and Altitude = 1000 m). The desired short-period poles for thelow-speed controller were placed with the same method as the high-speed controller, withdamping ζpL = 0.9, and fnpL = 0.5. For the lateral low-speed controller, the poles wereplaced to with damping ζryL = 0.9, τryL = 2 s and fnryL = 0.5.

The active controller feeds its control signal to both the actuators and the inactive con-troller. The switch is done by knowing when the landing gear is up or down. The state-feedback is also switched.

In other words, when approaching the landing strip and lowering the gear, the low-speedcontroller will be switched to, and the high-speed will run in the background, fed by thecontrol signal (uadL) that the low-speed control produces. Figure 9 visualizes the switchingstrategy.

reference

L1 Low-speed

L1 High-speed

Landing Gear Up/Down

u_ad_L

u_ad_HSystem

u_tot

Landing Gear Down

-K_high+

+ -K_low

x

x

Landing Gear Up

Landing Gear Up/Down

Landing Gear Up/Down

u

u

Figure 9: Switching strategy between two L1 controllers.

34

5 Results

The results are presented here and are later discussed in Section 6. In Section 5.1 I will showsimulations with a baseline controller designed to give nominal performance and handlingqualities in Mach = 0.6 and Altitude = 5000 m, without any model errors. The models Ihave used in the simulations are linearised models from VEGAS at different mach speedsand altitudes. Figures of q-/p- and β-step responses are presented.

Section 5.2 contains simulations with a baseline controller designed to give nominal per-formance and handling qualities at Mach = 0.6 and Altitude = 5000 m, augmented witha L1 adaptive controller designed in the same point in the envelope. It also contains alow-speed controller designed in the same way but in the envelope point Mach = 0.2 andAltitude = 1000 m. Figures of q-/p- and β-step responses are presented for both the high-speed and the low-speed controllers. The models I have used here are also linearised modelsfrom VEGAS at different mach speeds and altitudes.

In Section 5.3 the same controllers presented in Section 5.2 are simulated, but withthe nonlinear VEGAS model, see Section 2.1. Figures of q-/p- and β-step responses arepresented for the high-speed controller. Figures of the effect of added mass and a change ofx-axis center of gravity for the low-speed controller with a q-step is presented.

In Section 5.4 I will finally present a recorded flight in a simulator with the same con-trollers as in Section 5.2 and 5.3. The simulator also runs the VEGAS model. This finalsection exists to illustrate how the adaptive estimates σi vary during real reference inputsfrom a pilot.

5.1 Linear models with baseline controller

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4α/αss

[deg

]

Time (s)0 1 2 3 4 5

0

0.5

1

1.5

2

2.5q

[deg

/s]

Time (s)

0 1 2 3 4 5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time (s)

[deg

]

δe

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

Figure 10: Baseline q-steps at 5000 m for different mach speeds.

35

0 1 2 3 4 5−0.1

0

0.1β

[deg

]

Time (s)0 1 2 3 4 5

0

0.5

1

1.5p

[deg

/s]

Time (s)

0 1 2 3 4 50

0.05

0.1r

[deg

/s]

Time (s)0 1 2 3 4 5

−0.02

0

0.02

0.04

δa

[deg

]

Time (s)

0 1 2 3 4 50

0.05

0.1

Time (s)

[deg

]

δr

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

Figure 11: Baseline p-steps at 5000 m for different mach speeds.

0 1 2 3 4 50

0.5

1

1.5β

[deg

]

Time (s)0 1 2 3 4 5

−5

0

5p

[deg

/s]

Time (s)

0 1 2 3 4 5−2

−1

0r

[deg

/s]

Time (s)0 1 2 3 4 5

0

1

2

δa

[deg

]

Time (s)

0 1 2 3 4 5−0.5

0

0.5

Time (s)

[deg

]

δr

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

Figure 12: Baseline β-steps at 5000 m for different mach speeds.

36

5.2 Linear models with baseline and L1 controller

5.2.1 High-speed controller

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4α/αss

[deg

]

Time (s)0 1 2 3 4 5

0

0.5

1

1.5

2

2.5q

[deg

/s]

Time (s)

0 1 2 3 4 5−1

−0.5

0

0.5

1

σ1 (matched)

Time (s)0 1 2 3 4 5

−4

−3

−2

−1

0

1

2x 10

−4 σ2 (unmatched)

Time (s)

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

0 1 2 3 4 5−0.5

0

0.5

1

1.5

uad

[deg

]

Time (s)0 1 2 3 4 5

−0.8

−0.6

−0.4

−0.2

0

unom

[deg

]

Time (s)

0 1 2 3 4 5−1

−0.5

0

0.5

1

Time (s)

[deg

]

δe = u

tot

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

Figure 13: Baseline + L1 q-steps at 5000m for different mach speeds.

37

0 2 4−0.05

0

0.05

0.1

0.15β

[deg

]

Time (s)0 2 4

0

0.2

0.4

0.6

0.8

1

1.2

1.4p

[deg

/s]

Time (s)0 2 4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35r

[deg

/s]

Time (s)

0 2 4−0.2

−0.15

−0.1

−0.05

0

0.05

σ11

(matched)

Time (s)0 2 4

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (s)

σ12

(matched)

0 2 4−2

0

2

4

6

8x 10

−5σ2 (unmatched)

Time (s)

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

0 2 40

0.05

0.1

0.15

0.2

0.25

uad

(δa)

[deg

]

Time (s)0 2 4

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

unom

(δa)

[deg

]

Time (s)0 2 4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

utot

(δa)

[deg

]

Time (s)

0 2 4−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

uad

(δr)

[deg

]

Time (s)0 2 4

−0.05

0

0.05

0.1

0.15

Time (s)

[deg

]

unom

(δr)

0 2 4−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

utot

(δr)

[deg

]

Time (s)

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

Figure 14: Baseline + L1 p-steps at 5000 m for different mach speeds.

38

0 2 40

0.2

0.4

0.6

0.8

1

1.2

1.4β

[deg

]

Time (s)0 2 4

−3

−2

−1

0

1

2p

[deg

/s]

Time (s)0 2 4

−2

−1.5

−1

−0.5

0

0.5r

[deg

/s]

Time (s)

0 2 4−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

σ11

(matched)

Time (s)0 2 4

−2

−1

0

1

2

Time (s)

σ12

(matched)

0 2 4−2

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4σ2 (unmatched)

Time (s)

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

0 2 4−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

uad

(δa)

[deg

]

Time (s)0 2 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

unom

(δa)

[deg

]

Time (s)0 2 4

−0.5

0

0.5

1

1.5

utot

(δa)

[deg

]

Time (s)

0 2 40.5

1

1.5

2

2.5

3

3.5

uad

(δr)

[deg

]

Time (s)0 2 4

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Time (s)

[deg

]

unom

(δr)

0 2 40

0.5

1

1.5

2

2.5

3

utot

(δr)

[deg

]

Time (s)

Mach=0.2Mach=0.4Mach=0.6Mach=0.8Mach=1.0

Figure 15: Baseline + L1 β-steps at 5000 m for different mach speeds.

39

5.2.2 Low-speed controller

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4α/αss

[deg

]

Time (s)0 1 2 3 4 5

0

0.5

1

1.5

2

2.5q

[deg

/s]

Time (s)

0 1 2 3 4 5−2

−1

0

1

2

3

4

5

σ1 (matched)

Time (s)0 1 2 3 4 5

−1

0

1

2

3

4

5x 10

−4 σ2 (unmatched)

Time (s)

Mach=0.2Mach=0.3Mach=0.4

Figure 16: Baseline + L1 q-steps at 1000 m for different mach speeds.

0 2 4−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02β

[deg

]

Time (s)0 2 4

0

0.2

0.4

0.6

0.8

1

1.2

1.4p

[deg

/s]

Time (s)0 2 4

0

0.05

0.1

0.15

0.2

0.25r

[deg

/s]

Time (s)

0 2 4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

σ11

(matched)

Time (s)0 2 4

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time (s)

σ12

(matched)

0 2 4−15

−10

−5

0

5x 10

−5σ2 (unmatched)

Time (s)

Mach=0.2Mach=0.3Mach=0.4

Figure 17: Baseline + L1 p-steps at 1000 m for different mach speeds.

40

0 2 40

0.2

0.4

0.6

0.8

1

1.2

1.4β

[deg

]

Time (s)0 2 4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2p

[deg

/s]

Time (s)0 2 4

−2

−1.5

−1

−0.5

0r

[deg

/s]

Time (s)

0 2 4−2

−1.5

−1

−0.5

0

0.5

1

1.5

σ11

(matched)

Time (s)0 2 4

−15

−10

−5

0

5

10

15

20

Time (s)

σ12

(matched)

0 2 4−2

−1.5

−1

−0.5

0

0.5

1x 10

−4σ2 (unmatched)

Time (s)

Mach=0.2Mach=0.3Mach=0.4

Figure 18: Baseline + L1 β-steps at 1000 m for different mach speeds.

41

5.3 VEGAS nonlinear model with baseline and L1 controller

5.3.1 High-speed controller

0 1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

α/αs

s [d

eg]

Time (s)0 1 2 3 4 5

−5

0

5

10

15

20

q [d

eg/s

]

Time (s)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

θ/θs

s [d

eg]

Time (s)0 1 2 3 4 5

0

1

2

3

4

5

n z [m/s

2 ]

Time (s)

Mach=0.30Mach=0.40Mach=0.60Mach=0.80Mach=1.00

Figure 19: VEGAS Baseline + L1 q-steps (r(t) = 5 rad/s (normalized)) at 5000 m fordifferent mach speeds.

0 1 2 3 4 5−20

−15

−10

−5

0

5

10

15

20

β [d

eg]

Time (s)

0 1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

p/ps

s [d

eg/s

]

Time (s)

Mach=0.30Mach=0.40Mach=0.60Mach=0.80Mach=1.00

Figure 20: VEGAS baseline + L1 p-steps (r(t) = 100 rad/s (normalized)) 5000 m for differentmach speeds.

42

0 1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

β/βs

s [d

eg]

Time (s)0 1 2 3 4 5

−20

−15

−10

−5

0

5

10

15

20

p [d

eg/s

]

Time (s)

Mach=0.30Mach=0.40Mach=0.60Mach=0.80Mach=1.00

Figure 21: VEGAS Baseline + L1 β-steps (r(t) = 5 deg (normalized)) 5000 m.

43

5.3.2 Low-speed controller

0 1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

α/αs

s [d

eg]

Time (s)0 1 2 3 4 5

−1

0

1

2

3

4

q [d

eg/s

]

Time (s)

0 1 2 3 4 5

0.4

0.5

0.6

0.7

0.8

0.9

1

θ/θs

s [d

eg]

Time (s)0 1 2 3 4 5

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

n z [m/s

2 ]

Time (s)

Mach=0.20Mach=0.30Mach=0.40

Figure 22: VEGAS baseline + L1 q-steps at 1000 m for different mach speeds (Mass = 9100kg).

0 1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

α/αs

s [d

eg]

Time (s)0 1 2 3 4 5

−1

0

1

2

3

4

q [d

eg/s

]

Time (s)

0 1 2 3 4 50.4

0.5

0.6

0.7

0.8

0.9

1

θ/θs

s [d

eg]

Time (s)0 1 2 3 4 5

0.9

1

1.1

1.2

1.3

n z [m/s

2 ]

Time (s)

Mach=0.20Mach=0.30Mach=0.40

Figure 23: VEGAS Baseline + L1 q-steps 1000 m for different mach speeds (Mass = 14000kg and xcg = +0.1 m).

44

0 1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

α/αs

s [d

eg]

Time (s)0 1 2 3 4 5

−1

0

1

2

3

4

q [d

eg/s

]

Time (s)

0 1 2 3 4 50.4

0.5

0.6

0.7

0.8

0.9

1

θ/θs

s [d

eg]

Time (s)0 1 2 3 4 5

0.9

1

1.1

1.2

1.3

n z [m/s

2 ]

Time (s)

Mach=0.20Mach=0.30Mach=0.40

Figure 24: VEGAS baseline + L1 q-steps at 1000 m for different mach speeds (Mass = 14000kg and xcg = −0.1 m).

45

5.4 StyrSim simulations

Figure 25: Flight path of simulator run.

0 100 200 300

0

10

20

α

[deg

]

Time (s)0 100 200 300

02040

q

[deg

/s]

Time (s)

0 100 200 300

−20

0

20

σ1 (matched high)

Time (s)0 100 200 300

−505

10x 10

−3 σ2 (unmatched high)

Time (s)

0 100 200 300−20

020406080

σ1 (matched low)

Time (s)0 100 200 300

−0.010

0.010.02

σ2 (unmatched low)

0 100 200 300−1

0

1

2Landing gear out (1/0)

Time (s)0 100 200 300

0.20.40.60.8

11.2

Mach

Time (s)

Figure 26: Short-period controller states and adaptive estimates captured during simulatorrun.

46

6 Discussion

6.1 Linear models simulations

In Figure 10, 11 and 12 step responses in q, p and β is presented for the baseline high-speedcontroller with static gain of one for the nominal model. The dotted lines can be seen asguidelines for the step responses, the values should be within the lines to be regarded tohave good flying qualities, see Section 4.5.1. All three figures illustrate that for the machand altitude where the controller is designed for, gives good flying qualities. But when thespeed is lower or higher (the model changes), the step responses stabilize the system, butwith poor or inadequate flying qualities.

When comparing Figure 10 and Figure 13, one can see that the addition of the L1 adaptivecontrol signal has increased robustness, the controller is able so give good or acceptable flyingqualities in almost all speeds and altitudes.

One can also see in Figure 13 that the controller does not give good flying qualities inthe low speed region of Mach = 0.2 (Altitude = 5000 m). The same behaviour can alsobe seen in Figure 14 and 15 for low speed with the lateral controller. In the simulator, theeffects of poor controller performance also prominent in the low speeds, especially for thelateral mode. Distinct dutch roll oscillations were noticed when approaching the low speedregions. In practice one might not fly at those extreme low speeds at that altitude.

At lower altitudes, the pitch and lateral controller were able to stabilize better at lowerspeeds, but not good enough. This was the motivation to formulate a strategy to place asecond controller at low speed, low altitude (landing), see section 4.5.7. In Figure 16, 17and 18 step responses in q, p and β is presented for the low-speed controller. Oscillations inthe step responses can be seen for Mach = 0.4 in all figures, this is because of the nonlinearactuator model that has been implemented. If the actuator model is removed, the oscillationsdisappear with the implication that you hit the rate limits of the actuators.

6.2 VEGAS model simulations

The high-speed and low-speed short-period and lateral controllers were migrated to thenonlinear VEGAS model, where the results of step responses of the high-speed controllercan be found in Figure 19, 20 and 21. Only the step responses with adequate flying qualitiesare shown. At 5000 m, the high-speed controller manages to adapt to the uncertainties quitewell down to Mach = 0.3 with a desired (reference) q of 5 rad/s.

The step responses for the short-period mode are similar to the linear model simulations,see Figure 10. This points out that the linear models for the short-period mode are quiteeffective to capture the dynamics. As for the p- and β-step responses, one can see that thelinear models differ a great deal from the nonlinear model. An explanation to this is quitelikely that we are not rolling around the velocity vector, in other words, we have some αwhen rolling. When flying in the simulator, the oscillations seen in, e.g., Figure 20 are notvery prominent.

Overall, the high-speed controller has a hard time to cope with the uncertainties relatedto the low speeds (Mach ≤ 0.3), the speeds very much critical for landing. Figure 22 showsthe q-steps for different mach speeds for the low-speed controller that give good handlingqualities.

Figure 23 shows the same q-steps with an addition of 4900 kg and moving the x-axis centerof gravity forward 0.1 m. The controller has problems in controlling at designed speed of

47

Mach = 0.2, much likely because of the shorter leverage arm around the aerodynamic centerfor the actuators to work with.

Figure 24 illustrates the same q-steps with an addition of 4900 kg and moving the x-axiscenter of gravity backward 0.1 m. The controller is able to cope with this disturbance.

6.3 StyrSim simulation

A simulator is a powerful tool for the designer to evaluate a control system. I have used sucha tool to get direct feedback on how the aircraft behaves when acting as a pilot. Naturalreference inputs are easier to create with a joystick instead of manually specifying them ona desktop simulator. Effects from tuning are easier to grasp as a pilot than looking at plotsof step responses. The visual feedback from the projected landscape and dials in the cockpitis invaluable.

The flight path from one of the recorded flights made in the simulator can be seen inFigure 25. A take-off was made, a switch between the controllers, some loops, rolls andhigh-g turns. Then an approach was done together with a switch back to the low-speedcontroller. After a successful landing, the recording was stopped.

In Figure 26 a recording of α, q, σi (for both the low-speed and high-speed controllers),Mach and the landing gear switch is presented.

What is interesting here is how the adaptive estimates vary, the uncertainties related tothe problem are relatively slow and high frequency content in this flight is not noticeable.This is most likely because of the slow varying dynamics related to this problem, speed andaltitude vary slowly when flying. The signals are also relatively restricted in the range ofmagnitude. Turbulence rejection has not been a focus during test flights, but has been casu-ally tested with relative success. Time-delays or measurement noise has not been regarded.Neither has issues related to flow separation at high α, which could introduce high frequentuncertainties, been addressed.

7 Conclusions

This interesting case study has opened up the world of adaptive control for me, where L1

adaptive control has been a good first approach to understand the basics of adapting touncertainties. The results are promising for this application, even with a limited adaptationrate of 60 Hz, which might open new doors in exploring the benefits of adaptive control. Onecontroller, with the modified piecewise constant adaptation law, with one reference systemis not able to cope with all the uncertainties related to the loss of speed and altitude data.It increases robustness, but not for the whole flight envelope.

Because of the problems at low speeds, a switching strategy was formulated between twoL1 adaptive controllers, without a theoretical foundation for this approach. It was howeverflown in a simulator with relative success. The high-speed controllers in both the longitudinaland lateral modes are effective down to around Mach = 0.3 if Altitude ≤ 5000 m, and aboutMach = 0.4 at Altitude > 5000 m. The low-speed controllers, at least for the longitudinalcase, can cope with speeds of about Mach = 0.15− 0.4 at Altitude < 3000 m.

A pitch-unstable high-performance fighter aircraft is a complex system to master. Ihave learnt that when you think you have it figured out, cross-coupling effects, constraints,hardware limitations, outer world etc., come and bite you when you least expect it. Thishas made me more hungry to understand the complex motion a high-performance fighteraircraft paints in the sky, and the control that is related to it.

48

The tuning of the filters has been an open problem throughout the project, even if thecontrol channel bandwidth limits the design. Trail and error methods have been adoptedwith inspiration from other peoples work to get a good balance between performance androbustness. The state-predictor was transformed to discrete time with a reformulation inthe adaptive law. This structure has not been proven in theory, but has given satisfactoryresults in simulations.

Problems were most prominent with the lateral controller, the L1 adaptive controller hadbig problems in stabilizing the inner loops by itself. The addition of the baseline controllerwas very much needed to achieve good performance. Overall, there were more issues relatedto the lateral MIMO controller than the longitudinal SISO.

The baseline controller managed to provide similar performance in the design point of theflight envelope as the augmented controller, but in all other cases, the augmented controllerincreased performance. Tuning was also done to reduce the bandwidth of the closed-loopdesired system in the state-predictor to calm the desired system response down to be ableto account for the lower speeds, without success. Overall the L1 adaptive controller withthe modified piecewise constant adaptive law can be considered to increase robustness andperformance inner loops in this application, with reservation for the lateral inner loops inlow speeds. More work still needs to be done in order to integrate it in to a complex flightcontrol system.

7.1 Future work

This work is one small step on the way of hopefully adopting adaptive control in real flyingapplications in the future. Further developments of the controller could include the additionof feedback linearisation and more studies on noise attenuation (turbulence, measurementnoise etc.). Another addition to this work could be to look at some sort of parameter varyingstate-predictor, full nonlinear state-predictor or schedule multiple linear state-predictors.The motivation for this could be relieve the L1 adaptive controller with the addition of amore complete model. Another factor is that the desired closed-loop response might vary inthe flight envelope, at take-off you want to specify one desired response and when flying atsupersonic speeds you want another.

Time-delays between sensors and actuators and other structures for D(s) are factorsthat could be investigated in. The addition of better models for the actuator dynamics withrespect to the theory could also be considered in the future. Theoretical proof of the discretetime state-predictor and the reformulation of the adaptive law should be evaluated.

An natural step forward from this project would be to implement the control algorithmsin a technology demonstrator of some sort, like the NASAs GTM (AirSTAR) [12]. Thiscould give invaluable insights to the problems that can arise when including the hardwarefactor.

49

References

[1] Li, Z. and Hovakimyan, N., “L1 Adaptive Controller for MIMO Systems with Un-matched Uncertainties using Modified Piecewise Constant Adaptation Law,” 51st IEEEConference on Decision and Control , dec 2012, pp. 7303–7308.

[2] Astrom, K. J., “Adaptive control around 1960,” 34th IEEE Conference Decision andControl , New Orleans, LA, 1995, pp. 2784–2789.

[3] Dydek, Z. T., Annaswamy, A. M., and Lavretsky, E., “Adaptive Control and the NasaX-15-3 Flight Revisited,” IEEE Control Systems Magazine, jun 2010.

[4] Astrom, K. J. and Wittenmark, B., Adaptive Control , Dover Publications, Inc., Mineola,2nd ed., 2008, ISBN-10: 0-486-46278-1.

[5] Cao, C. and Hovakimyan, N., “Design and analysis of a novel L1 adaptive controller,part I,” American Control Conferencel , Minneapolis, Minnesota, USA, jun 2006, pp.3397–3402.

[6] Hovakimyan, N. and Cao, C., L1 Adaptive Control Theory - Guaranteed Robustnesswith Fast Adaptation, Society for Industrial and Applied Mathematics, 1st ed., 2010,ISBN-13: 978-0-898717-04-4.

[7] “ADMIRE Model,” http://www.foi.se/admire, Accessed 2012-11-15.

[8] Nelson, R. C., Flight Stability and Automatic Control , McGraw-Hill Book Co, 2nd ed.,1998, ISBN-10: 0-07-115838-3.

[9] Etkin, B. and Reid, L. D., Dynamics of Flight - Stability and Control , John Wiley andSons, Inc., 3rd ed., 1996, ISBN-10: 0-471-03418-5.

[10] Hovakimyan, N., Cao, C., Kharisov, E., Xargai, E., and Gregory, I. M., “L1 AdaptiveControl for Safety-Critical Systems,” IEEE Control Systems Magazine, oct 2010.

[11] Xargay, E., Hovakimyan, N., and Cao, C., “L1 Adaptive Controller for Multi-InputMulti-Output Systems in the Presence of Nonlinear Unmatched Uncertainties,” 2010American Control Conference, jun-jul 2010, pp. 874–879.

[12] Xargay, E., Hovakimyan, N., Dobrokhodov, V., Statnikov, R. B., Kaminer, I., Cao, C.,and Gregory, I. M., “Flight test of L1 adaptive controller on the NASA AirSTAR flighttest vehicle,” AIAA Guidance, Navigation and Control Conf., AIAA-2010-8015 , 2010.

[13] Leman, T. J., L1 adaptive control augmentation system for the X-48B aircraft , Master’sthesis, University of Illinois at Urbana-Champaign, 2009.

[14] Holhjem, Ø. H., L1 Adaptive Control of the Inner Control Loops of an F-16 Aircraft ,Master’s thesis, Norwegian University of Science and Technology, jun 2012.

[15] Gunnarsson, K. S., “Kravdokument for styrlagsdesign,” Tech. rep., SAAB Aerosystems,2001.

[16] Griffin, B. J., Burken, J. J., and Xargay, E., “L1 Adaptive Control AugmentationSystem with Application to the X-29 Lateral/Directional Dynamics: A Multi-InputMulti-Output Approach,” 2010 AIAA Guidance, Navigation, and Control Conference,aug 2010.

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[17] Pettersson, A., Astrom, K., Robertsson, A., and Johansson, R., “Analysis of linear L1adaptive control architectures for aerospace applications,” Decision and Control (CDC),2012 IEEE 51st Annual Conference on, dec 2012, pp. 1136–1141.

[18] Pettersson, A., Astrom, K. J., Robertsson, A., and Johansson, R., “Augmenting L1adaptive control of piecewise constant type to a fighter aircraft. Performance and ro-bustness evaluation for rapid maneuvering,” 2012 AIAA Guidance, Navigation, andControl Conference, Minneapolis, Minnesota, aug 2012.

[19] Gregory, I. M., Xargay, E., Cao, C., and Hovakimyan, N., “L1 Adaptive Flight ControlSystem Systematic Design and Verification and Validation of Control Metrics,” AIAA-2010-7773 , 2010.

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