Using Linear Fractional Transformations for Clearance of Flight Control Laws

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Using Linear Fractional

Transformations for Clearance Of Flight

Control Laws

Master’s thesis

performed in Automatic Control by

J¨orgen Hansson Reg nr: LiTH-ISY-EX-3420-2003

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Using Linear Fractional

Transformations for Clearance Of Flight

Control Laws

Master’s thesis

performed in Automatic Control, Dept. of Electrical Engineering

at Link¨opings universitet Performed for SAAB AB

by J¨orgen Hansson

Reg nr: LiTH-ISY-EX-3420-2003

Supervisor: M.Sc. Fredrik Karlsson SAAB AB

Lic. Karin St˚ahl Gunnarsson SAAB AB

Lic. Ragnar Wallin ISY, Link¨opings Univeristet Examiner: Doc. Anders Hansson

ISY, Link¨opings Universitet Link¨oping, 1st October 2003

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Avdelning, Institution Division, Department Datum Date Spr˚ak Language ¤ Svenska/Swedish ¤ Engelska/English ¤ Rapporttyp Report category ¤ Licentiatavhandling ¤ Examensarbete ¤ C-uppsats ¤ D-uppsats ¤ ¨Ovrig rapport ¤

URL f¨or elektronisk version

ISBN ISRN

Serietitel och serienummer

Title of series, numbering

ISSN Titel Title F¨orfattare Author Sammanfattning Abstract Nyckelord Keywords

Flight Control Systems are often designed in linearization points over a flight envelope and it must be proven to clearance authorities that the system works for different parameter variations and failures all over this envelope.

In this thesisµ-analysis is tried as a complement for linear analysis in the frequency plane. Using this method stability can be guaranteed for all static parameter combinations modelled and linear criteria such as phase and gain margins and most unstable eigenvalue can be included in the analysis. A way of including bounds on the parameter variations using parameter dependent Lyapunov functions is also tried.

To performµ-analysis the system must be described as a Linear Frac-tional Transformation (LFT). This is a way of reformulating a param-eter dependent system description as an interconnection of a nominal linear time invariant system and a structured parameter block.

A linear and a rational approximation of the system are used to make LFTs. These methods are compared. Four algorithms for calculation of the upper and lower bounds ofµ are evaluated. The methods are tried on VEGAS, a SAAB research aircraft model.

µ-analysis works quite well for linear clearance. The rational

approx-imation LFT gives best results and can be cleared for the criteria men-tioned above. A combination of the algorithms is used for best results. When the Lyapunov based method is used the size of the problem grows quite fast and, due to numerical problems, stability can only be guaranteed for a reduced model.

Division of Automatic Control, Dept. of Electrical Engineering

581 83 Link¨oping 1st October 2003

—

LITH-ISY-EX-3420-2003 —

http://www.ep.liu.se/exjobb/isy/2003/3420/

Using Linear Fractional Transformations for Clearance Of Flight Con-trol Laws

Klarering av Styrlagar för Flygplan med hjälp av Linjära Rationella Transformationer

J¨orgen Hansson

× ×

Linear Fractional Transformation (LFT), Structured Singular Value (SSV), Linear Matrix Inequality (LMI),µ-analysis, Lyapunov function, Flight Clearance, Stability Margin

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Abstract

Flight Control Systems are often designed in linearization points over a flight envelope and it must be proven to clearance authorities that the system works for different parameter variations and failures all over this envelope.

In this thesis µ-analysis is tried as a complement for linear analysis in the frequency plane. Using this method stability can be guaranteed for all static parameter combinations modelled and linear criteria such as phase and gain margins and most unstable eigenvalue can be in-cluded in the analysis. A way of including bounds on the parameter variations using parameter dependent Lyapunov functions is also tried. To perform µ-analysis the system must be described as a Linear Fractional Transformation (LFT). This is a way of reformulating a pa-rameter dependent system description as an interconnection of a nom-inal linear time invariant system and a structured parameter block.

A linear and a rational approximation of the system are used to make LFTs. These methods are compared. Four algorithms for calcu-lation of the upper and lower bounds of µ are evaluated. The methods are tried on VEGAS, a SAAB research aircraft model.

µ-analysis works quite well for linear clearance. The rational

ap-proximation LFT gives best results and can be cleared for the criteria mentioned above. A combination of the algorithms is used for best re-sults. When the Lyapunov based method is used the size of the problem grows quite fast and, due to numerical problems, stability can only be guaranteed for a reduced model.

Keywords: Linear Fractional Transformation (LFT), Structured Sin-gular Value (SSV), Linear Matrix Inequality (LMI), µ-analysis, Lyapunov function, Flight Clearance, Stability Margin

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Preface

This master’s thesis is the final step between me and my Master of Science Degree in Applied Physics and Electrical Engineering. It was performed at SAAB AB, Division of Gripen Aeronautical Engineering, Section of Flying/Handling Qualities, GDFF, in corporation with the Division of Automatic Control at Link¨opings Universitet. This work was done during the spring semester and summer of 2003. This thesis is part of the FMV FoT 25 research project.

Acknowledgments

I would like to thank Dr.Martin Hayes and Ph.D student Peter Iordanov at the University of Limerick for helping me out with the optimization procedure for lower bounds on µ. I would also like to thank Professor Anders Helmersson for big help with calculation of the frozen analysis and the FFM. A big thank you to my supervisor and examiner at the university Ragnar Wallin and Docent Anders Hansson for rewarding discussions and hints. Thanks also to my supervisors at SAAB, Fredrik Karlsson and Karin St˚ahl Gunnarsson for taking time for me in their busy working days. Thank you to all the people at section GDFF at SAAB for making my thesis time a pleasant one. My classmates Petter Frykman proofread the report and Anders Ekman opposed on the the-sis. Finally I thank my girlfriend Malena and my family for putting out with my wining and giving me support in general during my studies.

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List of Figures

2.1 Flight envelope. . . 9

2.2 Centre of gravity diagram.. . . 11

3.1 Inertial and aircraft fixed coordinate frames. . . 14

3.2 Angles and rates for roll (φ, p), pitch (θ, q), yaw (ψ, r) . . . . 14

3.3 Forces, moments and velocities. . . 15

3.4 Static stability. . . 16

3.5 Centre of gravity and moment. . . 17

3.6 Motion of short period mode. . . 18

3.7 Motion of phugiod mode. . . 19

4.1 Linear mapping.. . . 21

4.2 Feedback. . . 22

4.3 State-space system as LFT. . . 23

4.4 State-space system. . . 24

4.5 The star product. . . 24

4.6 1-dimensional Trends & Bands approximation. . . 26

4.7 Conservative approximation.. . . 27

4.8 1-dimensional rational approximation. . . 27

5.1 Interconnected system.. . . 31

5.2 System with ∆-feedback. . . 33

5.3 s-plane interpretation of SSV. . . 34

5.4 Uncertainty in frequency plane. . . 36

5.5 Complex uncertainty. . . 36

5.6 Gain and phase margins on Nyquist plot. . . 37

5.7 Exclusion regions in the Nichols and Nyquist plane. . . 38

5.8 Adding exclusion region. . . 39

5.9 Stability margin. . . 39

5.10 Eigenvalue specification. . . 40

6.1 Adding scaling matrices. . . 42

6.2 System augmented with frequency. . . 43 xiii

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6.3 LFT for dividing frequency. . . 44

6.4 Imaginary augmentation of real LFT description. . . 45

6.5 Offset into RHP. . . 46

7.1 Stability regions. . . 48

7.2 Allowed ∆ and ˙∆ intervals. . . 57

7.3 Models of envelope. . . 57

8.1 VEGAS in SYSTEMBUILD. . . 59

8.2 Control surfaces on VEGAS. . . 60

8.3 LFT controller used for analysis. . . 61

9.1 Comparison of µ-calculations. . . . 66

9.2 Upper µ with different grids. . . . 66

9.3 Peak zoomed, all algorithms used. . . 66

10.1 CMα as function of Mach number. . . 69

10.2 SIMULINKinterconnection. . . 71

10.3 Envelope covered by model. . . 72

10.4 Comparison of LFT:s and linearization point in nominal case. . . 72

10.5 Comparison of LFTs with adjusted compensation parameters. . 73

10.6 Trends & Bands LFT results. . . 74

10.7 Rational LFT results. . . 75

11.1 Stability of Vegas.. . . 77

11.2 Upper µ for exclusion regions . . . . 78

11.3 Nichols plot of nominal and worst case system. . . 79

11.4 Comparison of signals, nominal and worst case Mach 0.2. . . 80

11.5 Migration of short period poles. . . 80

11.6 µ for eigenvalue specifications. . . . 81

11.7 Worst case poles. . . 81

12.1 Difference in parameters. . . 84

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Part I

Introduction

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Chapter 1

Introduction

1.1 Background

The clearance of a flight control system is a very important and time-consuming process. Due to hard regulations on aircraft systems for safety reasons it must be proven that the systems will work. Since the aircraft dynamics and the aerodynamics are very nonlinear and the control system is linear and often designed in discrete points, verifi-cation of the control system is very important. To consider variations in aircraft dynamics and aerodynamic coefficients, a few variations in the parameters are investigated and then simulation of the nonlinear system is performed to make sure that the system will work and re-main stable for all possible variations. Since the number of parameters that can vary in a complete aircraft system is quite large, this is a very time consuming process and although simulations are used to find all strange and normal flight cases a full certainty of having covered all cases can never be given. The methods tried in this thesis covers all static parameter variations modeled and can be used for linear analysis to make sure the system is stable and fulfils requirements on stability and eigenvalues. The worst case parameter combinations found from this analysis can then be used to find out which configurations and flight cases it is important to look at in the nonlinear simulations.

1.2 Problem formulation

In the GARTEUR AG(11) research project some advanced techniques for the clearance of flight control laws have been developed and tried (Fielding et al. 2002). SAAB AB, who participated in this project, are interested in using some of these methods as a complement to parts

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4 Chapter 1. Introduction

of the current clearance and design process. The object of this thesis was to try some of these methods, based on a system described as a Linear Fractional Transformation. Most interesting for SAAB at this point was µ-analysis. The thesis had the following goals.

• Describe the steps necessary to perform the analysis.

• Try the methods on VEGAS, a SAAB research aircraft model. • A number of envelope points should be tried and uncertainties

should be included in the model.

• Information about properties between the envelope points should

be found if possible.

• A few clearance criteria should be included e.g. stability margin

and eigenvalues. The parameter combination giving the worst case should be found if possible.

• If time is left, try some other method also based on Linear

Frac-tional Transformations.

1.3 Method

This is what was done in this thesis.

µ-analysis

The main goal was to try µ-analysis. Most of the theory was obtained from Fielding et al. (2002) were this already had been done for flight clearance purposes. Other scientific articles related to this area were also studied.

• To perform µ-analysis the system must be described as a Linear

Fractional Transformation (LFT). Two numerical ways of gener-ating such LFT descriptions, one linear approximation and one ra-tional approximation, were tried and compared. These were used on data from a nonlinear simulation model VEGAS with Mach number, centre of gravity and pitch moment coefficient depend-ing on angle of attack as varydepend-ing parameters. These parameters were chosen after consultation with the supervisors at SAAB. A controller which already was a LFT was used to stabilize the air-craft.

• To calculate the structured singular value, µ, reliable algorithms

were necessary. Four different algorithms for calculating upper and lower bounds were tried and compared.

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1.4. Relation to previous work 5

• The stability margin and worst case unstable poles criteria were

included in the analysis.

Parameter dependent Lyapunov functions

This method differs from µ-analysis but it also demands a LFT descrip-tion of the system. Since it includes variadescrip-tions on the parameter rates in the analysis it was interesting for SAAB. Therefore this analysis method was applied to one of the LFT descriptions of VEGAS.

1.4 Relation to previous work

Here the differences between the scientific articles used for this thesis and the achieved results are presented. In Fielding et al. (2002) several ways of generating LFTs are suggested. None of them utilizes rational approximations which are used in this thesis. A rational function can generally approximate a surface with a lower order function compared with a polynomial. In Mannchen et al. (2002) the Trends & Bands ap-proach is suggested for generation of LFTs. The µ-analysis results are compared with the results from a LFT generated by a numerical min-max approach and a LFT based on symbolical expressions of a physical model. In this thesis the Trends & Bands method is compared with a LFT based on rational approximations. In Iordanov et al. (2003) a comparison of existing µ-calculation algorithms is done but the Finite Frequency Method is not included but utilized to improve the results. Here the Finite Frequency Method is evaluated together with three other algorithms. All of the µ-analysis clearance criteria in this thesis have earlier been tried on Trends & Bands LFTs based on the HIRM+ aircraft model with the RIDE controller in Fielding et al. (2002).

1.5 Tools

The software tools used in this thesis are

MATRIXX& SYSTEMBUILD where the nonlinear aircraft simulation model VEGAS is implemented. This environment was used for linearization and data collection from the nonlinear model. MATLAB & SIMULINK was used for all other calculations and

simu-lations. The toolboxes mainly used were

LFR Toolbox used to make LFT descriptions.

Optimization Toolbox used for min-max optimization and uti-lized by some of the µ-calculation algorithms.

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6 Chapter 1. Introduction

Control System Toolbox for handling of state-space descrip-tions systems and Nyquist and Nichols plots.

µ-analysis and Synthesis Toolbox where the mu command is

implemented.

1.6 Thesis Outline

The thesis is organized as follows:

Chapter 2 gives a short background and introduction to the flight clearance process.

Chapter 3 provides some basic concepts from flight mechanics neces-sary to understand the model later developed.

Chapter 4 presents the main theory for Linear Fractional Transfor-mations and how they can be generated and reduced.

Chapter 5 introduces the Structured Singular Value (SSV) and presents how flight clearance criteria such as phase and gain margins and most unstable eigenvalue can be included in the analysis. Chapter 6 gives an overview of the algorithms used to calculate the

SSV.

Chapter 7 contains a way to include bounds on the rates of parameter variations in the analysis.

Chapter 8 introduces the VEGAS model and the controller used for analysis.

Chapters 9–11 present the results. The µ-algorithms are compared, LFTs of VEGAS with uncertainties included are generated and compared. Clearance of VEGAS with controller is performed. Chapter 12 summarizes the thesis, draws conclusions and gives

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Part II

Theory

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Chapter 2

Flight Clearance

Modern fighter aircrafts are often aerodynamically unstable to achieve better performance. Thus they can’t be flown without a flight control system. To get the aircraft certified to fly, the flight control system must be approved by the clearance authority by proving it to be fully functional throughout the flight envelope in presence of different failures and parameter variations.

2.1 Gain Scheduled Controllers

Since the aircraft dynamics are very dependent on e.g. Mach number, angle of attack and altitude, gain scheduling is often used to control the system. The system is linearized over a flight envelope depending on those three parameters and a control law is designed in each discrete point. The overall control system then interpolates the different control laws based on the current parameter state.

α alt

M

Figure 2.1. Flight envelope.

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10 Chapter 2. Flight Clearance

Using this design method a few points must be considered:

• The control laws are designed for linearized systems but will they

work when applied to the real nonlinear system?

• The control laws are designed in discrete points of the envelope

and fulfil the specifications in these points but what happens be-tween these points?

• The dynamics also depends on various additional parameters such

as e.g. aerodynamic coefficients, mass and centre of gravity. Will the system work when these parameters vary?

To make sure that the control system works for all possible parameter variations, some of the possible combinations are tried out, often the design points and the limits of the variations. Then a lot of nonlinear simulation is performed to verify the functionality over the whole en-velope and the values which have not explicitly been tested. This is a very time consuming task since the envelope is big and there are a lot of possible parameter combinations.

2.2 Parameter dependence

There are many parameters affecting an aircraft and its Flight Control System (FCS) but generally they can be divided into four groups: Configuration dependent variabilities such as centre of gravity (c.g.)

and inertia. They are depending on e.g. the amount of fuel and the current stores.

Aerodynamic uncertainties depending on e.g. the current aircraft profile, the wind and so on.

Hardware dependent variabilities such as changes in the sensor or actuator dynamics.

Air data system dependent tolerances such as measurement er-rors in Mach number or angle of attack.

Often at least the probable interval variations of the parameters are known, e.g. a specification on longitudinal c.g. position can be seen in figure 2.2.

2.3 Clearance Process

Clearance of an aircraft and its control system is a tedious task with many steps. One example of the industrial clearance process steps could be (Fielding et al. 2002):

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2.3. Clearance Process 11 A/C m ass [kg ]

Longitudinal c.g. position [%m.a.g]

Figure 2.2. Centre of gravity diagram.

Generation of analysis model involves making a nonlinear simula-tion model with parameter uncertainties. This model is used to make linear (small perturbed, trimmed) models.

Familiarization with aircraft and controller to get a picture of the unaugumented aircraft dynamics. Plots of aerodynamic sta-bility and control derivatives are studied.

Trend studies on the effect of uncertainties to see how different parameters affect handling and stability.

Linear stability analysis where e.g. open-loop stability margins and eigenvalues are calculated for a narrow grid of the envelope and for different uncertainties.

Linear handling analysis where evaluations of different time and frequency domain criteria are studied and worst cases are found. Nonlinear analysis by off-line and manned simulation to eval-uate the flying characteristics with and without uncertainties and find handling and control problems.

Clearance report where the manoeuvre and flight envelope limita-tions based on the earlier analysis are visualized and derived. Improvement of clearance based on special investigations on

re-duced stability margins.

Utilizing µ-analysis could facilitate the linear analysis part of the clear-ance process. Using the methods described in this thesis open loop stability margins and eigenvalues can be analyzed in and between the points in the envelope for a linear approximation of the system. This approximation can be nonlinear in parameters. The worst case static parameter combination can be found and utilized to perform nonlinear analysis where needed.

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Chapter 3

Basic Flight Mechanics

This chapter contains basic flight mechanics necessary for understand-ing the model and the analysis results. The longitudinal linearized equa-tions of motion are derived from the nonlinear and some properties of the solutions to these linear equations are discussed. The short period approximation of the linearized equations is later used for construc-tion of an LFT descripconstruc-tion. For a more thorough survey on aircraft dynamics e.g. Etkin (1972) is recommended.

3.1 Coordinate frames

When describing an aircraft three coordinate frames are usually used (figure 3.1).Two fixed to the aircraft, one with the x axis fixed to the aircraft for describing the motion, SA, and one with the x axis fixed to

the velocity vector for describing the aerodynamic forces acting on the plane SW. The third system is an inertial system fixed to the earth’s

surface describing the aircrafts movement relative the earth SI. The

same vector can be described in two frames and v = xiˆxi+ yiyˆi+ ziˆzi=

= xbxˆb+ ybyˆb+ zbˆzb

if one coordinate frame rotates with angular velocityω relative another frame then the theorem of Coriolis (3.1) can be used to transform the motion from one frame to the other.

d dt ¯¯ ¯¯ i v = d dt ¯¯ ¯¯ b v +ω × v (3.1) 13

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14 Chapter 3. Basic Flight Mechanics ˆ xA ˆ yA ˆ zA ˆ xI ˆ yI ˆ zI SA SI

Figure 3.1. Inertial and aircraft fixed coordinate frames.

3.1.1 Euler angles

The angles used for describing SArelatively SI are called Euler angles,

ΦA, and defined in figure 3.2 together with respectively rate, ¯ωA.

ΦA=        φ θ ψ        ¯ωA= ˙ΦA=        p q r        φ p q α θ ψ _r

Figure 3.2. Angles and rates for roll (φ, p), pitch (θ, q), yaw (ψ, r) and angle of attack α.

3.2 Forces, moments and velocities

The directions of the forces, FA and moments, TA (3.2) acting on the plane described in the body fixed frame are defined in figure 3.3 where

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3.3. Moment of inertia 15

directions of the velocities ¯vA also can be seen.

FA=        X Y Z        TA=        L M N        ¯vA=        u v w        (3.2) X, u Y, v Z, w M N L

Figure 3.3. Forces, moments and velocities.

The longitudinal forces, FA,l, and moments, TA,l, depend on the aerodynamic coefficients Ciin (3.3). ¯q is the aerodynamic pressure and

S is the wing area.

FA,l = X Z = ¯ qSCX ¯ qSCZ Horisontal F orce V ertical F orce

TA,l = M = qSC¯ M P itching moment

(3.3)

The aerodynamic coefficients are dimensionless and dependent on a lot of parameters e.g. velocity, altitude and the aircrafts profile. They can not be described analytically but are estimated empirically from wind tunnel tests and flight tests.

3.3 Moment of inertia

An aircraft can usually be assumed to be symmetric in the xz-plane. This gives the inertia tensor in (3.4).

I =        Ix 0 −Ixz 0 Iy 0 −Ixz 0 Iz        (3.4)

3.4 Static Stability

When an aircraft is subject to a disturbance in α an aerodynamic mo-ment starts acting on it. If this momo-ment brings it back, the system is

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16 Chapter 3. Basic Flight Mechanics

said to be statically stable. Using the sign conventions in figure 3.4 a positive alpha disturbance should produce a negative pitch moment and vice versa if the system is statically stable. Modern fighter aircrafts without a controller are often longitudinal unstable for subsonic veloc-ities. This is the case for the Swedish aircraft Gripen and the VEGAS model used in this thesis. When the aircraft is in force and moment balance it is said to be trimmed. The static stability is dependent on e.g. the pitch moment coefficient and the centre of gravity.

M < 0

∆α > 0

Figure 3.4. Static stability

Pitch moment coefficient

The pitch moment coefficient CM describes how the pitch moment

act-ing on the aircraft due to aerodynamics changes in the envelope. It consists of several components (3.5).

CM = CM,base+ CMα. . . (3.5)

where CMα is the pitch moment coefficient depending on angle of

at-tack.

3.4.1 Centre of gravity

The centre of gravity also affects the static stability. Simply described the longitudinal moment depends on the moments caused by the wings and by the tail as in figure 3.5. For a disturbance in α the tail moment should force the plane back. This is the case when CMα < 0. If the

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3.5. Nonlinear Equations of Motion 17

CMα increases making the plane more unstable. The aerodynamic

cen-tre (a.c.), is a point on the aircraft where the moments do not depend on the angle of attack. Both the centre of gravity and the aerodynamic centre move around due to the dynamics.

xac

xcg

lt

Figure 3.5. Centre of gravity and moment.

3.5 Nonlinear Equations of Motion

The longitudinal nonlinear equations are derived from regular rigid body dynamics. FT is the thrust force from the engine.

X + FT+ mg sin θ = m( ˙u + qw− rv)

Z + mg cos φ cos θ = m( ˙w + pv− qu)

M = Iyq + rp(I˙ x− Iz) + Ixz(p2− r2)

˙

θ = q cos φ− r sin φ

Table 3.1. Nonlinear longitudinal equations of rigid body motion.

3.6 Linearization

When linearizing the nonlinear longitudinal equations of motion all lat-eral components such as r, p, v disappear. For small disturbances the aircrafts lateral movement is not affected by the longitudinal. Assum-ing trimmed flight condition i.e. force and moment balance and flyAssum-ing straight forward with constant velocity gives u = u0, w = 0 and v = 0.

Xu∆u + Xw∆w− mg = m∆ ˙u

Zu∆u + Zw∆w + Zδe∆δe+ Zδc∆δc+ mg = m∆ ˙w− mu0∆q

Mu∆u + Mw∆w + Mq∆q + Mδe∆δe+ Mδc∆δe = Iy∆ ˙q

∆ ˙θ = ∆q

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18 Chapter 3. Basic Flight Mechanics

where Xi= ∂X_∂i ¯¯_i=i₀, Zi= . . . and so on. Dividing (3.6) with m in force

equations, Iy in the moment equation and rearranging to a state-space

description gives     ∆ ˙u ∆ ˙w ∆ ˙q ∆ ˙θ     =     ¯ Xu X¯w 0 0 ¯ Zu Z¯w u0 0 ¯ Mu M¯w M¯q 0 0 0 1 0         ∆u ∆w ∆q ∆θ     + (3.7) +     0 0 ¯ Zδe Z¯δc ¯ Mδe M¯δc 0 0     · ∆δe ∆δc ¸

3.6.1 Phugoid and short period mode

When solving the linearized longitudinal equations of motion typically two complex conjugated solutions are achieved. These are the linear system modes. One corresponds to a fast heavily damped oscillation and the other to a slow lightly damped oscillation. By making some approximations these modes can be separated. When designing a lon-gitudinal controller these modes must be suppressed, especially the fast short period mode.

Short period mode

The short period mode is fast and heavily damped, typical values can be e.g. a frequency of ωsp= 3 rad/s and a damping coefficient of ξsp= 0.4.

This mode has the properties that

• ∆u is small compared to ∆w. • ∆w and ∆θ are in phase. • ∆α ≈ ∆θ

which describes a pitch oscillation as in figure 3.6 This motion is

ap-Figure 3.6. Motion of short period mode. proximated by (3.8) · ∆ ˙w ∆ ˙q ¸ = · Zw u0 Mw Mq ¸ · ∆w ∆q ¸ + · Zδe Zδc Mδe Mδc ¸ · ∆δe ∆δc ¸ (3.8)

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3.6. Linearization 19

Utilizing that ∆α≈ ∆w

u0 for small disturbances a state transformation

as in section 4.4.1 can be done on (3.8) changing ∆w to ∆α (3.9) · ∆ ˙α ∆ ˙q ¸ = · Zα 1 Mα Mq ¸ · ∆α ∆q ¸ + · Zδe Zδc Mδe Mδc ¸ · ∆δe ∆δc ¸ (3.9) Phugoid mode

The phugoid mode is lightly damped and slow, e.g. frequency ωp= 0.1

rad/s and damping ξp= 0.04.

• ∆w is small compared to ∆u and ∆w ≈ 0 so the velocity vector

and x vector are the same, i.e. the aircrafts nose is always in the direction of motion.

• ∆u is approximately 90◦ _{before ∆θ.}

which give the motion in figure 3.7.

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Chapter 4

Linear Fractional

Transfor-mation

In this chapter the Linear Fractional Transformation referred to as LFT is introduced. The big advantage of LFTs is that linear time-varying systems that depend rationally on parameters can be described as an interconnection of a nominal linear time invariant system and a structured parameter block. Analysis of e.g. stability can then be done using these LFTs. The interested can read more in e.g. Zhou et al. (1996).

4.1 Upper and Lower LFTs

Consider a linear mapping or system (figure 4.1) where M can be com-plex M :C 7→ C w1 w2 z1 z2 M11 M12 M21 M22

Figure 4.1. Linear mapping.

z1 = M11w1+ M12w2

z2 = M21w1+ M22w2 (4.1)

Now put another block called ∆∈ C in feedback on the upper ports of the system (figure 4.2)

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22 Chapter 4. Linear Fractional Transformation w2 z2 ∆ M11 M12 M21 M22 Figure 4.2. Feedback.

This gives the following transfer function from w2→ z2

z1 = M11w1+ M12w2

z2 = M21w1+ M22w2

w1 = ∆z1

z1 = M11∆z1+ M12w2⇔ z1= (I− M11∆)−1M12w2

z2 = M21∆z1+ M22w2⇔ z2= (M21∆(I− M11∆)−1M12+ M22)w2

This is called an upper linear fractional transformation. If the feedback is put on the two lower ports instead a similar transfer function can be derived.

Definition 4.1 The upper and lower linear fractional transformations

of M and ∆ are defined by

Fl(M, ∆l) , M11+ M12∆l(I− M22∆l)−1M21 (4.2)

Fu(M, ∆u) , M22+ M21∆u(I− M11∆u)−1M12 (4.3)

(I− M11∆u) and (I− M22∆l) must be invertible for respective LFT

to exist. M11 in (4.2) or M22 in (4.3) can be considered the nominal

system and the rest describes how the system vary with the parameters ∆.

4.1.1 State-space systems

A regular state-space system can be described using LFTs by consid-ering the integration 1

s as an artificial feedback (figure 4.3).

z = Aw + Bu (4.4)

y = Cw + Du (4.5)

w = 1

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4.1. Upper and Lower LFTs 23 u y 1 s A B C D

Figure 4.3. State-space system as LFT.

Using this approach a linear state-space system that depends nonlin-early on parameters can be described as an interconnection of a linear time invariant system and a parameter block.

Example 4.1

Consider a scalar state-space system depending nonlinearly on δ

˙x = δ 2_{+ 1} 1 + δ x + bu = x + δ µ δ− 1 1 + δ ¶ x + bu y = cx + du Taking w1= δ ³ δ−1 1+δ ´ x = δw2 gives z1 = w2 z2 = δx− x − w2 ˙x = x + w1+ bu w1 = δz1 w2 = δz2 Finally let δx = w3 ˙x = 1 1 0 0 b z1= 0 0 1 0 0 z2=−1 0 −1 1 0 z3= 1 0 0 0 0 y = c 0 0 0 d

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24 Chapter 4. Linear Fractional Transformation 1 s δI3x3 M u y

Figure 4.4. State-space system.

4.2 The Star Product

If two linear mappings such as (4.1) are interconnected as in figure 4.5 the star product is obtained. It is a generalization of LFTs.

y1 z1 z2 y2 u1 w1 w2 u2 Q11 Q12 Q21 Q22 M11 M12 M21 M22

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4.3. LFT generation 25 y1 = Q11u1+ Q12w1 z1 = Q21u1+ Q22w1 z2 = M11w2+ M12u2 y2 = M21w2+ M22u2 w1 = z2⇔ z1= (I− Q22M11)−1Q21u1+ (I− Q22M11)−1Q22M12u2 w2 = z1⇔ z2= (I− M11Q22)−1M11Q21u1+ (I− M11Q22)−1M12u2 y1 = (Q11+ Q12(I− M11Q22)−1M11Q21)u1+ Q12(I− M11Q22)−1M12u2 y2 = M21(I− Q22M11)−1Q21u1+ (Q11+ M21(I− Q22M11)−1Q22M12)u2

Definition 4.2 The star product Q ? M ∈ C is defined as

Q ? M = · Q11 Q12 Q21 Q22 ¸ ? · M11 M12 M21 M22 ¸ = · Fl(Q, M11) Q12(I− M11Q22)−1M12 M21(I− Q22M11)−1Q21 Fu(M, Q22) ¸ (4.7) (I− M11Q22) and (I− Q22M11) must be invertible. The star product

notation can be used for upper and lower LFTs which are special cases of the star product.

Fl(M, ∆l) = M ? ∆l (4.8)

Fu(M, ∆u) = ∆u? M (4.9)

4.3 LFT generation

There are many ways to make LFT descriptions of nonlinear systems, but generally they can be divided into two approaches. Making an analytical LFT description from the nonlinear symbolic equations or approximating functions from numerical data. In this thesis the latter approach is chosen since it normally can be done with less effort and the emphasis of this thesis is on the analysis methods. Two ways of making numerical approximations are tried. Both ways adapt functions to state space data.

4.3.1 Linear Parameter Varying Systems

State space data can be obtained by e.g. finding stationary points for a nonlinear aircraft model and then linearize the model around these points for different flight cases and parameter combinations. A system described by such functions is called a Linear Parameter Varying Sys-tem (LPV) since it is linear in states but the state coefficients varies

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26 Chapter 4. Linear Fractional Transformation

with the parameters as in (4.10). Note that one parameter could be

time. _· ˙x1 ˙x2 ¸ = · A11(δ) A12(δ) A21(δ) A22(δ) ¸ · x1 x2 ¸ (4.10) This LPV can always be reformulated as an LFT if Aii(δ) is rational

or polynomial by doing steps similar to example 4.1.

4.3.2 Trends & Bands approach

The Trends & Bands approach is a linear approximation suggested by Mannchen et al. (2002) and is used quite successfully in the GARTEUR project. The idea is to approximate the trend of the data with linear parameters and adding the largest deviation, if it is significant, as an additional parameter making a band around the function. By doing this the physical meaning of the function is preserved in the trend param-eters and the whole physical system is covered by the approximation. The one parameter case is seen in figure 4.6. The aij element in the

δerr

aij

aij = f (δk)

aij0+ aijkδk

δk

Figure 4.6. 1-dimensional Trends & Bands approximation. system matrix, A, depending on a parameter aij = f (δk) would get the

following approximation

aij = aij0+ aijkδk+ δerr (4.11)

If the element depends on two parameters the approximation is a plane and when the element is depending on more than two parameters the approximation is a multidimensional linear regression plane. To get the coefficients e.g. min-max optimization can be used. In min-max opti-mization the coefficients are chosen such that the maximal deviation is minimized. The advantage of this method is the simplicity and since all matrix elements have the same form in the approximation it can be automatically generated. However, this is a linear approximation, so if the true function has a narrow peak, as seen in figure 4.7, the

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4.3. LFT generation 27

approximation will cover a lot of systems besides the real and intro-duce conservatism. This way of making LFTs will later be referred to as T&B.

δerr

aij

δk

Figure 4.7. Conservative approximation.

4.3.3 Approximation with rational functions

To get a less conservative LFT description, rational functions can be adapted to data from linearization points (4.12). Rational functions are preferred before polynomials since more advanced surfaces can be approximated with lower order functions.

aij =

b0+ b1δk+ b1δ2k+ . . .

1 + a0δk+ . . .

+ δerr (4.12)

Using this approach the model will be more accurate compared to T&B since the approximated surface will follow the original function better as shown in figure 4.8.

δk

aij

δerr

Figure 4.8. 1-dimensional rational approximation.

This will make the compensation parameters, wich represents the er-ror, smaller and reduce the conservatism. More effort is needed to find the functions but if some physical knowledge of the system exists this effort can be reduced. The number of parameters in the ∆- block can

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28 Chapter 4. Linear Fractional Transformation

be quite large if a high order rational function is needed for a good approximation.

4.4 Preparation of approximation data

The data used to approximate functions can be normalized in several ways to achieve better numerical properties.

4.4.1 Normalizing states

If a state is transformed with a constant matrix T such that T x = ˆx

then T ˙x = ˙ˆx since T is constant and

T−1˙ˆx = AT−1x + Buˆ ⇔ ˙ˆx = T AT−1x + T Buˆ (4.13)

y = CT−1x + Duˆ (4.14)

The two systems · _˜ A B˜ ˜ C D˜ ¸ = · T AT−1 T B CT−1 D ¸ (4.15)

will have the same input-output properties. This can be used e.g. to scale the state-space matrices for better numerical properties or to change some of the states.

4.4.2 Normalizing parameter data

If the parameter data used in the estimation is normalized to the in-terval [−1, 1] the size of estimated coefficients can be compared. A co-efficient which is small compared to the other is probably insignificant and can be removed without making the approximation error larger. This is useful in the rational function approach.

4.5 Model reduction

The LFT model should have as low order as possible. Low order reduces numerical problems, time for calculation and makes it easier for the optimization algorithms. The methods below are used in this thesis for clever realization and model reduction.

4.5.1 Tree decomposition

A LFT will have the same number of parameters in the ∆-block as the number of parameters in the LPV if the reformulation is done in

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4.5. Model reduction 29

the simplest possible way, without any simplifications. This is called the order of the LFT. This order can be reduced for a polynomial LPV, such as the T&B, if the LFT is built using tree decomposition (Magni 2001). The idea is to factorize and sum decompose the polynomial matrix in turns, until no more transformations can be done. This is implemented in the Onera Toolbox, (Magni 2001), command symtreed.

Example 4.2

Consider a polynomial matrix of order six,

S(∆) =

·

δ1δ2+ δ3

δ1δ2δ3

¸

by first applying a sum decomposition · δ1δ2+ δ3 δ1δ2δ3 ¸ = · δ3 0 ¸ + · δ1δ2 δ1δ2δ3 ¸ then a factorization · δ3 0 ¸ + · δ1δ2 δ1δ2δ3 ¸ = · δ3 0 ¸ + · 1 δ3 ¸ δ1δ2

the order has been reduced from 6 to 4.

There are of course several ways of doing this by factorizing other pa-rameters and doing the decompositions and factorizations in another order. The best way is not known in advance so a number of combina-tions must be tried and the one giving the lowest order is used.

4.5.2 n-D decomposition

n-D decomposition is implemented in the Onera Toolbox command minlfr. By doing transformations and step by step eliminating the nonobservable and noncontrollable states in the transformed LFT the order can be reduced (Magni 2001).

4.5.3 µ-sensitivities

By using µ-sensitivities as described in section 5.2, parameters with small influence on the µ-value are found. These can be removed without making the model considerably more optimistic in the sense that it will produce only a slightly lower µ-value.

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Chapter 5

Structured Singular Values

5.1 Structured Singular Values

The Structured Singular Value (SSV) , µ, was first suggested by Doyle in 1982 as a way of analyzing systems with uncertainties. The SSV is applied to systems described by Linear Fractional Transformations and is the inverse of the largest parameter combination, in a 2-norm sense, which will make the system unstable. Using this analysis, called

µ-analysis, stability can be verified and parameter combinations which

give the worst performance can be found.

5.1.1 Small gain theorem

A conservative yet useful theorem is the small gain theorem. The the-orem is sufficient but not necessary, meaning that if the thethe-orem does not grant stability the system can still be stable.

+ + y1 y2 e₂ r1 r2 S1 S2

Figure 5.1. Interconnected system.

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32 Chapter 5. Structured Singular Values

Theorem 5.1 (Small gain theorem) Consider two stable systems in a

feedback interconnection as in figure 5.1. The closed loop system is sta-ble if the product of the gains is smaller than 1.

kS2kkS1k < 1 (5.1)

This can be interpreted as when the loop is circulated the signal should not be amplified as it internally would go towards infinity. Therefore the total gain should always be smaller than 1. The limit, i.e. when the total gain is 1, can algebraically be verified by (5.2)

det(I− S2S1) = 0 (5.2)

The theorem is conservative in at least two senses.

• The sign of the feedback doesn’t affect the theorem.

• If the norms are taken for a whole frequency range they could

take their maximum value at different frequencies.

5.1.2 Norms

To get a value on the gain of a system with multiple inputs and outputs vector and matrix norms can be used. The vector p-norm is

kukp= p v u u tXm i=1 |ui|p (5.3)

with two important special cases p = 2,∞

kuk2 = 2 v u u tXm i=1 |ui|2= u∗u (5.4) kuk∞ = ∞ v u u tXm i=1 |ui|∞= max i (|ui|) (5.5) (5.6) the matrix norm induced by the vector norm is

kAkp= sup x6=0 kAxkp kxkp (5.7) with p = 2,∞ kAk2 = p λmax(A∗A) = ¯σ(A) (5.8) kAk∞ = max 1≤i≤m n X j=1 |aij| (row sum) (5.9)

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5.1. Structured Singular Values 33

The 2-norm is also called the maximal singular value and used as a measurement of the largest gain of a matrix. If A is square this is the absolute value of the largest eigenvalue.

5.1.3 Structured Singular Value

Consider a LFT description as in figure 5.2 with a LTI system M (jω) interconnected with a structured parameter block ∆, which is nominally stable for ∆ = 0.

y u

∆

M11(s) M12(s)

M21(s) M22(s)

Figure 5.2. System with ∆-feedback.

Then according to the small gain theorem, this system is stable if ¯ σ(M11(s))¯σ(∆)≤ 1 ⇔ ¯σ(∆) ≤ 1 ¯ σ(M11(s)) (5.10) This is a stability test for an unstructured ∆ but it is conservative in this case since it doesn’t utilize the structure of ∆ which is diagonal or block diagonal.

∆(s) = diag{∆1(s), . . . , ∆N(s)}

¯

σ(∆i(s)) ≤ k (5.11)

Instead the largest singular value of ∆ i.e. the largest k that won’t turn the system unstable is desired. This is defined as

Definition 5.1 (Structured Singular Value) Given a matrix M ∈ C

and an uncertainty structure ∆∈ C, the structured singular value µ is µ∆(M ) = µ min ∆∈∆¯σ(∆) : det(I− M∆) = 0 ¶−1 (5.12)

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Real axis

Imaginary axis

Figure 5.3. s-plane interpretation of SSV.

The system must be nominally stable i.e. all nominal eigenvalues must be in the left half plane (LHP) for µ-analysis to be performed. Usually

µ is calculated along the imaginary axis and an s-plane interpretation of

the structured singular value can be seen in figure 5.3. A system that is nominally stable has all it’s eigenvalues in the LHP. As the parameters change, some of the eigenvalues move towards the imaginary axis and will eventually reach the axis. When they reach the axis the gain of the system is one (5.13).

k∆(s)k2kM11(s)k2= 1 (5.13)

If the smallest parameter combination making the poles reach the axis is larger than the allowed parameter values, the system is stable for all allowed parameter variations. The possible parameter (∆) intervals are usually normalized so that k∆k_∞ ≤ 1 implies that the system is stable for µ∆≤ 1. Since µ is the inverse of the largest singular value of

the smallest ∆ making the system unstable, it can also be seen as the largest allowed gain of M11. If ∆ is diagonal then ¯σ(∆) is the largest

absolute value of ∆.

5.1.4 Upper and lower bounds

Practically the exact µ-value is hard to find. Therefore upper and lower bounds are calculated. Finding the upper bound can be formulated as a convex optimization problem such that the true maximum for a scaled system can be found. The scaling makes the bound a little conservative and the bound is not the true µ-value. The lower bound problem is nonconvex why a local minimum are often found. If the global minimum is found it is the true µ-value.

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5.2. µ-sensitivities 35

5.2 µ-sensitivities

To find out which parameters of the LFT that have the most or least influence on the µ-value, µ-sensitivities can be used (Braatz & Morari 1991). This can be interesting for model reduction where the parameters that has the least influence on µ can be removed without making the model more optimistic. It can also be used to find the most important parameters to realize what part of the model to improve to make it more accurate. Definition 5.2 (µ-sensitivities) ∂µi≡ · lim ∆_αi→0+ µ(M (αi))− µ(M(αi− ∆αi)) ∆αi ¸ αi=1 (5.14) α = diag{α1Ir1, . . . , αnIrn} (5.15)

Where ri is the uncertainty block dimension.

M (α) = · αM11 √ αM12 √ αM21 M22 ¸ (5.16)

It can be interpreted as the change in the maximum µ-value when a parameter is reduced a little bit. If the change is significant, the parameter has a large influence on the µ-value and the system.

5.3 Modelling of Clearance Criteria

Clearance criteria given in the frequency domain can be modelled and included in the analysis. This makes µ-analysis quite useful for this kind of clearance. However, in the flight industry many criteria are given in the time domain, which makes it harder to include them in the

µ-analysis framework.

5.3.1 Uncertainty modelling

For most control designs and control analysis, a mathematical model of the real plant is used. This model will always differ from the real plant and will not behave exactly as the plant. An example can be seen in figure 5.4. These differences will be referred to as uncertainties and must be modelled to make the model more accurate.

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|G(jω)| nominal

real plant

ω

Figure 5.4. Uncertainty in frequency plane.

Real Uncertainties

To model a parameter that varies in an interval, real uncertainties are used. This models a parameter which is not known, but the probable interval is known or a parameter which is known to vary within an interval.

c ∈ [0.8, 1.6]

c ∈ {1.2 + (0.4)δ : δ ∈ R, |δ| ≤ 1}

Complex uncertainties

Complex uncertainties are used to model e.g. uncertain dynamics. A complex uncertainty is a region in the s-plane as in figure 5.5

c∈ {1.2 + (0.4)δ : δ ∈ C, |δ| ≤ 1} 0.8 1.6 0.4 Imagina ry a xis Real axis

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5.4. Phase and Gain margins 37

5.4 Phase and Gain margins

Linear stability criteria such as gain and phase margins can be in-cluded in the LFT-description and µ-analysis, making it easy to verify specifications on them. The Nyquist criterion says that if the Nyquist curve does not enclose -1 in the Nyquist plane, the system is stable. The distance and angle to this point, as seen in figure 5.6, is the gain respectively phase margin.

-1

1

Gm

ϕm

Figure 5.6. Gain and phase margins on Nyquist plot.

5.4.1 Nichols exclusion regions

Based on the Nyquist criterion, Nichols exclusion regions can be used to verify gain and phase margins. The critical point i.e. -1 in the Nyquist plane or−180◦ in the Nichols plane is surrounded with a region. The margins are fulfilled if the Nichols curve of the system doesn’t pass through this region. This region is not exact but based on experience. It can quite easily be approximated and included in the LFT description making it possible to use µ-analysis (Bates et al. 2001, Fielding et al. 2002). The simplest approach is to approximate the region with an ellipse as seen in figure 5.7 and describe this as a multiplicative uncer-tainty in the frequency region (5.19).

|L(jω)|2 dB G2 m +(∠L(jω) + 180 ◦₎2 P2 m = 1 (5.17)

This ellipse is described by (5.17) where∠ is the argument of L(jω). It can be mapped to a circle with centre−a and radius r in the Nyquist

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38 Chapter 5. Structured Singular Values −220 −210 −200 −190 −180 −170 −160 −150 −140 −8 −6 −4 −2 0 2 4 6 8

Open Loop Phase [deg]

Open Loop Gain [dB]

(a) Nichols plane

−2.5 −2 −1.5 −1 −0.5 0 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Re L(jω) Im L(j ω ) (b) Nyquist plane

Figure 5.7. Exclusion regions in the Nichols and Nyquist plane.

plane using (5.18) Gm = 20 log10(a + r) Pm = cos−1( a2− r2+ 1 2a ) (5.18) P (s) = P1(s)(−a + ∆) (5.19)

where P1(s) is the nominal model and ||∆||∞ < r, ∆ ∈ C . This can

also be written as

P (s) = aP1(s)(1 + WN∆) (5.20)

where||∆||_∞< 1 and WN =−r_a. This is added to all loops

simultane-ously or one loop at a time to check the stability. A multivariable system can be unstable for simultaneous perturbations although it passes the one loop at a time test. The loops can be broken on the sensor side or the actuator side. A block diagram description is seen in figure 5.8. Constants for the two usual margins shown in figure 5.7 are found in table 5.1.

Gain Phase a Wn

Nominal 6dB 36.87◦ 1.25 0.6 Perturbed 4.5dB 28.44◦ 1.14 0.47

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5.4. Phase and Gain margins 39

WN ∆N

a P1

K

Figure 5.8. Adding exclusion region.

5.4.2 Stability margin and worst case parameter

com-bination

The stability margin is used as a tool for comparing stability. The inner Nichols exclusion region is defined to have stability margin one and a stability margin less than one is hence considered insufficient. How much this region can be scaled before it touches the system is the stability margin ρ as in figure 5.9. The region is scaled in the analysis by scaling r which is the same as scaling WN in (5.20). To find the worst

ρ = 1

ρ = 1.75

Figure 5.9. Stability margin.

case parameter combination the region must be scaled and µ calculated iteratively until a peak value of one is reached. The scale factor on WN

when the µ value is one is the stability margin. Another way suggested by Kureemun et al. (2001) is to define ρ by (5.21).

ρ = 1

µ (5.21)

When the region just touches the systems Nichols curve a µ-value of one should be achieved. A µ smaller than one corresponds to a larger region and so on.

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5.5 Most unstable eigenvalue

Usually µ is calculated by doing a frequency grid i.e. calculating µ on the imaginary axis in the s-plane as described in section 5.1.3. But the grid can be done over some other line in the s-plane if the nominal poles all are inside the region surrounded by the frequency grid. The usual calculating algorithms will work. This line in the s-plane can be utilized for including specifications on the most unstable eigenvalues in the analysis.

5.5.1 Specifications

A specification on unstable eigenvalues can be seen in (5.22). It can be translated into five lines in the s-plane as in figure 5.10.

<{λ} = 0 , f or ω∈ Ω1 = |ω| ≥ 0.15rad/s <{λ} ≤ ln 2 20 , f or ω∈ Ω2 = 0≤ |ω| ≤ 0.15rad/s <{λ} ≤ ln 2 7 , f or ω∈ Ω3 = {0} (5.22)

By calculating µ on a grid consisting of those lines it can be verified if

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Re Im section 1 section 2 section 3 section 4 section 5 µ≤ 1 µ≥ 1

Figure 5.10. Eigenvalue specification.

any of the poles pass the lines (Mannchen et al. 2001). To find out the value of the most unstable pole the lines must be moved iteratively until

µ becomes one. The point of the line corresponding to the peak value is

the most unstable eigenvalue. By doing this with a lower µ-algorithm the parameter combination that generates this eigenvalue can be found. The worst case poles can also be found by using the Pole Placement Approach described in section 6.2.3 and moving a line into the RHP until a µ-value of one is found.

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Chapter 6

µ-calculations

There exist several approaches to calculate upper and lower bounds on

µ. Tight bounds are wanted to draw the right conclusions about the

system since if 1 is between the bounds not much can be said. In this thesis a few calculating algorithms are tried out and compared. Com-puting µ when the problem consists of real repeated uncertainties is a much more difficult task then for mixed or strictly imaginary uncertain-ties. In a physical model, the uncertainties are often real and repeated so such problems need to be solved. There are basically two types of algorithms.

Grid based algorithms which calculate a µ-value on each predefined point of a frequency axis grid.

Peak finding algorithms where frequency is included in the algo-rithm and a µ peak value and the corresponding frequency are found.

6.1 Upper

µ-calculations

All the upper bound algorithms recast the problem as a convex op-timization problem. They will always find the maximal value but the result can differ a bit from the true value since the system is scaled.

6.1.1 Scaling

To find an upper bound on µ the system is usually scaled. How to recast the problem when the parameters are imaginary is shown. If the parameters are real or both real and imaginary other scalings are introduced. The problem is scaled with a matrix D as in figure 6.1. This

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42 Chapter 6. µ-calculations

D should be positive definite, Hermitian i.e. D∗ = D, and commute with ∆ so that D∆ = ∆D. ∆ ∆ D D−1 D D−1 M M u u y y

Figure 6.1. Adding scaling matrices.

The new largest singular value is ¯σ(D∆D−1). A search for both a D matrix and the upper µ value should be done (6.1).

inf

D ¯σ(D∆D

−1₎ _(6.1)

This is a convex problem so a global solution can always be found if it exists. Even if an upper bound for the original problem is impossible to find, the new scaled problem may be solved if the scaling matrix D is found.

6.1.2 µ-Analysis and Synthesis Toolbox

The command mu, (Balas et al. 1993), solves the upper bound over a grid of frequency points. Static µ is calculated in each point. This algorithm can handle large problems with up to 100 uncertainties. It gives conservative bounds on the real µ and since the frequency axis is calculated on a grid it can miss narrow points. The bound found is

µ∆(M (jω)) = β > 0 if ¯ σ µ (I + G2_l)−14₍DlM D −1 r β − jGm)(I + G 2 r)− 1 4 ¶ ≤ 1 (6.2)

and there exists scaling matrices Gl, Gm, Gr, Dland Dr.

6.1.3 Finite Frequency Method

In the finite frequency method, (Helmersson 1995), frequency is added as an additional uncertainty to the LFT description. By doing this, a complete frequency interval is covered in one computation. The compu-tation can be done with any upper bound algorithm e.g. mu described above. When large frequency intervals are added the resulting upper

Using Linear Fractional Transformations for Clearance of Flight Control Laws

Using Linear Fractional

Transformations for Clearance Of Flight

Control Laws

Using Linear Fractional

Transformations for Clearance Of Flight

Control Laws

Abstract

Preface

Acknowledgments

Contents

I

Introduction

1

II

Theory

7

III

Results

63

List of Figures

Part I

Introduction

Chapter 1

Introduction

1.1

Background

1.2

Problem formulation

1.3

Method

1.4

Relation to previous work

1.5

Tools

1.6

Thesis Outline

Part II

Theory

Chapter 2

Flight Clearance

2.1

Gain Scheduled Controllers

2.2

Parameter dependence

2.3

Clearance Process

Chapter 3

Basic Flight Mechanics

3.1

Coordinate frames

3.1.1

Euler angles

3.2

Forces, moments and velocities

3.3

Moment of inertia

3.4

Static Stability

3.4.1

Centre of gravity

3.5

Nonlinear Equations of Motion

3.6

Linearization

3.6.1

Phugoid and short period mode

Chapter 4

Linear Fractional

Transfor-mation

4.1

Upper and Lower LFTs

4.1.1

State-space systems

4.2

The Star Product

4.3

LFT generation

4.3.1

Linear Parameter Varying Systems