Control of Unmanned Aerial Vehicles using Non-linear Dynamic Inversion

Non-linear Dynamic Inversion

Master’s Thesis

Division of Automatic Control

Department of Electrical Engineering

Linköping University

Mia Karlsson

LiTH-ISY-EX-3284-2002

Linköping 2002

Non-linear Dynamic Inversion

Master’s Thesis

Division of Automatic Control

Department of Electrical Engineering

Linköping University

Mia Karlsson

LiTH-ISY-EX-3284-2002

Supervisors:

M.Sc. Jonas Lövgren, Saab AB.

Lic. Ola Härkegård, ISY, Linköping University.

Examiner:

Prof. Torkel Glad, ISY, Linköping University.

Institutionen för Systemteknik 581 83 LINKÖPING 2002-12-20 Språk Language Rapporttyp Report category ISBN Svenska/Swedish X Engelska/English Licentiatavhandling

X Examensarbete ISRN LITH-ISY-EX-3284-2002

C-uppsats

D-uppsats Serietitel och serienummer

Title of series, numbering

ISSN

Övrig rapport

URL för elektronisk version

http://www.ep.liu.se/exjobb/isy/2002/3284/

Titel Title

Design av styrlagar för obemannade farkoster med hjälp av exakt linjärisering Control of Unmanned Aerial Vehicles using Non-linear Dynamic Inversion

Författare

Author

Mia Karlsson

Sammanfattning

Abstract

This master's thesis deals with the control design method called Non-linear Dynamic Inversion (NDI) and how it can be applied to Unmanned Aerial Vehicles (UAVs). In this thesis, simulations are conducted using a model for the unmanned aerial vehicle SHARC (Swedish Highly

Advanced Research Configuration), which Saab AB is developing.

The idea with NDI is to cancel the non-linear dynamics and then the system can be controlled as a linear system. This design method needs much information about the system, or the output will not be as desired. Since it is impossible to know the exact mathematical model of a system, some kind of robust control theory is needed. In this thesis integral action is used.

A problem with NDI is that the mathematical model of a system is often very complex, which means that the controller also will be complex. Therefore, a controller that uses pure NDI is only discussed, and the simulations are instead based on approximations that use a cascaded NDI. Two such methods are investigated. One that uses much information from aerodata tables, and one that uses the derivatives of some measured outputs. Both methods generate satisfying results. The outputs from the second method are more oscillatory but the method is found to be more robust. If the signals are noisy, indications are that method one will be better.

Nyckelord

Keyword

Non-linear Dynamic Inversion, exact linearization, cascaded structure, integral action, flight con-trol, Unmanned Aerial Vehicles

This master’s thesis deals with the control design method called Non-linear Dynamic Inversion (NDI) and how it can be applied to Unmanned Aerial Vehicles (UAVs). In this thesis, simulations are conducted using a model for the unmanned aerial vehicle SHARC (Swedish Highly Advanced Research Configuration), which Saab AB is developing.

The idea with NDI is to cancel the non-linear dynamics and then the system can be controlled as a linear system. This design method needs much information about the system, or the output will not be as desired. Since it is impossible to know the exact mathematical model of a system, some kind of robust control theory is needed. In this thesis integral action is used.

A problem with NDI is that the mathematical model of a system is often very complex, which means that the controller also will be complex. Therefore, a controller that uses pure NDI is only discussed, and the simulations are instead based on approximations that use a cascaded NDI. Two such methods are investigated. One that uses much infor-mation from aerodata tables, and one that uses the derivatives of some measured out-puts. Both methods generate satisfying results. The outputs from the second method are more oscillatory but the method is found to be more robust. If the signals are noisy, indications are that method one will be better.

This thesis finalizes my Master of Science in Applied Physics and Electrical Engineer-ing at LinköpEngineer-ing university.

I have written this master’s thesis at Saab Aerospace, department Gripen Handling Qualities, from August to December 2002. It has been a wonderful time, thanks to the people working there. I would especially like to thank my supervisor Jonas Lövgren and my vice-supervisor Karin Ståhl-Gunnarsson for being helpful and always finding time for my questions.

Another person that I would like to thank is my supervisor at Linköping university, Ola Härkegård, with whom I have had many interesting discussions.

I am also very grateful to all of you who have read the drafts of this thesis.

I would also like to thank my family and friends, for their support and for giving me such a good time during my studies.

Finally I would like to thank Markus, making my life happy, and to whom I am also grateful for making me interested in this very exciting and fascinating world of engi-neering.

Linköping, December 2002

Symbol Unit Definition

α rad angle of attack

β rad sideslip angle

b m wing span

m aerodynamic mean chord

C_l - rolling moment coefficient

C_L - lifting force coefficient

C_m - pitching moment coefficient

C_n - yawing moment coefficient

C_N - normal force coefficient

C_T - tangential force coefficient

C_Y - side force coefficient

δ_a rad aileron deflection

δ_e rad elevator deflection

δ_r rad rudder deflection

φ rad roll angle

F_T N engine thrust force

γ rad flight path angle

g m/s2 acceleration due to gravity

I_x, I_y, I_z, I_xz, I_yz, I_xy kgm2 moment of inertia

L Nm rolling moment

L_lift N lift force

m kg aircraft mass M - Mach number M Nm pitching moment n_z - load factor N N normal force N Nm yawing moment

p rad/s roll rate

θ rad pitch angle

q rad/s pitch rate

q_a N/m2 dynamic pressure

ρ kg/m3 air density

r rad/s yaw rate

S m2 reference area

T N tangential force

u m/s longitudinal velocity

v m/s lateral velocity

V_T m/s true airspeed

ω_n rad/s natural frequency

(x_e, y_e, z_e) m earth-axes coordinate system

ψ rad yaw angle

Y N side force

1 Introduction . . . .. . . . .. . . .. . . . .. . . .. . . 1 1.1 Background .. . . . .. . . .. . . . .. . . .. . . 1 1.2 Objectives. . .. . . . .. . . .. . . . .. . . .. . . 1 1.3 SHARC . . . .. . . . .. . . .. . . . .. . . .. . . 1 1.4 Limitations. .. . . . .. . . .. . . . .. . . .. . . 2 1.5 Outline . . . . .. . . . .. . . .. . . . .. . . .. . . 2 2 Flight Mechanics. . .. . . . .. . . .. . . . .. . . .. . . 5 2.1 General Theory . .. . . .. . . . .. . . .. . . 5

2.2 Flight Mechanics Specific for SHARC . . . .. . . 9

2.2.1 Forces and Aerodynamic Coefficients.. . . 9

2.3 Control Surfaces. .. . . .. . . . .. . . .. . . 10

2.3.1 Servos. . . .. . . .. . . . .. . . .. . . 11

3 Non-linear Dynamic Inversion . . . . .. . . . .. . . .. . . 13

3.1 Pure Non-linear Dynamic Inversion . . . .. . . 13

3.2 Cascaded Non-linear Dynamic Inversion . . . .. . . 14

3.3 An Example, Pure NDI versus Cascaded NDI . . . 14

4 Non-linear Dynamic Inversion Applied to SHARC . .. . . 17

4.1 Pure Non-linear Dynamic Inversion . . . .. . . 17

4.2 Cascaded Non-linear Dynamic Inversion, Method One . . . 20

4.2.1 Longitudinal Mode. . .. . . . .. . . .. . . 20

4.2.2 Lateral Mode . . . .. . . . .. . . .. . . 21

4.3 Cascaded Non-linear Dynamic Inversion, Method Two. . . 25

4.3.1 Longitudinal Mode. . .. . . . .. . . .. . . 26

4.3.2 Lateral Mode . . . .. . . . .. . . .. . . 27

4.4 Control of the Load Factor. . .. . . . .. . . .. . . 31

4.4.1 Control of the Load Factor by the Angle of Attack. . . 31

4.4.2 Control of the Load Factor by the Elevator . . . 32

4.5 Robustness. .. . . . .. . . .. . . . .. . . .. . . 32

4.6 Poles of the Closed Loop System .. . . .. . . 34

4.6.1 General Theory. . . .. . . . .. . . .. . . 35

4.6.2 System without Integrators. . . .. . . 36

4.6.3 System with Integrators . . .. . . .. . . 38

5 Simulation . . . .. . . . .. . . .. . . . .. . . .. . . 39

5.1 Control Systems Requirements . . .. . . .. . . 39

5.2 Deciding the Parameters . . . .. . . . .. . . .. . . 40

5.3 Simulation Environment . . . .. . . . .. . . .. . . 42

Bibliography . . . .. . . . .. . . .. . . . .. . . .. . . 63

Appendix A. . . .. . . . .. . . .. . . . .. . . .. . . 65

This master’s thesis has been carried out at Saab Aerospace, department Gripen Handling Qualities. The task is to investigate the control design method Non-linear Dynamic Inversion (NDI) and apply it to the unmanned aerial vehicle SHARC (Swed-ish Highly Advanced Research Configuration).

1.1 Background

Non-linear Dynamic Inversion has been applied to flight control design during the two last decades, see [1]. The reason for investigating NDI is that Saab wants an easy trol method that is applicable for non-linear systems. When for example Gripen is con-trolled, LQ-design is used. LQ-design is a method, where different controllers have to be designed for different flight cases, see [2]. The hope is that NDI is an easier method to use and that the same parameters can be used in the entire flight envelope. The inten-tion is that the controller then easily can be adapted to other aircraft.

1.2 Objectives

The purpose is to design control laws for the SHARC in both longitudinal and lateral motion using the design method Non-linear Dynamic Inversion. In order for the SHARC to behave well, even if there are model errors in the design, robust control laws have to be implemented.

1.3 SHARC

Sweden has tested UAV systems since the end of the 1960’s. An example is the French UAV system UGGLAN that is used for tactical reconnaissance within the brigades.

The UAVs can be divided into different groups: High-Altitude Long Endurance (HALE), Medium-Altitude Long Endurance (MALE), Unmanned Reconnaissance Aerial Vehicles (URAV), Unmanned Combat Aerial Vehicles (UCAV), and Tactical Unmanned Aerial Vehicles (TUAV).

In Sweden, the work with UAVs began in 1998 within the framework of the National Aeronautical Research Programme (NFFP). Nine configurations were defined, and one of them was SHARC. At the same time a working team called IPT UAV was formed. It will suggest the direction for Swedish military activity on UAVs during 2001 to 2020.

Figure 1.1: The SHARC.

The SHARC is an UCAV that has a low signature feature, a range of 1000 km and it will have high surviveability and a low cost in serial production. It will also be autono-mous. It is 10 meters long, has a wingspan of 8 meters, and a take-off-weight of about 5000 kg.

1.4 Limitations

The controllers calculated in this master’s thesis will not be tested on a real aircraft, only on a model of the SHARC. On the other hand, model errors will be implemented to simulate the reality. Another limitation is that the model not includes noise. The lim-its and the dynamics for the control surfaces of SHARC, are not definitive defined. The dynamics of the control surfaces are simulated by servos, that are approximations of the system.

1.5 Outline

In Chapter 2, the general theory of flight mechanics is presented together with SHARC-specific theory, such as dependencies of aerodynamic coefficients, control surfaces, and servos.

In Chapter 4 the theory is applied to the SHARC. A controller that uses pure NDI is discussed and two controllers that use cascaded NDI are designed. Chapter 4 also deals with controllers that make the system more robust, and descriptions for the closed loop system are derived.

In Chapter 5 simulations are conducted and parameters are decided in order for the sys-tem to behave according the requirements.

In this chapter, flight mechanics needed in order to design controllers for the SHARC is presented.

2.1 General Theory

The motion of an aircraft can be divided into rotational and translational motion. The rotational motion is the rotation around the body axes (x_b, y_b, z_b) and the motion can be expressed in angular velocities (p, q, r) and moments (L, M, N), see Figures 2.1 and 2.2.

Figure 2.1: Definitions of moments, forces etc.

Ψ β x_e V_T x_b q, M y_b p, L χ

Figure 2.2: Definitions of moments, forces, velocities etc.

Translational motion is movement in the directions of the body axes (x_b, y_b, z_b) and the motion can be described by the velocities (u, v, w) and the forces (X, Y, Z), see Figures 2.1, 2.2, and 2.3. The coordinate system relative earth, (x_e, y_e, z_e) is also seen. In this thesis the forces X and Z are not used, bust instead the tangential force

and the normal force

Figure 2.3: Definitions of forces, coordinates, attitudes and velocities.

In Figures 2.1 and 2.3 the true airspeed V_T is seen. The definition of true airspeed is:

where is the velocity of centre of gravity relative earth and is the velocity of wind relative earth, both expressed in (x_b, y_b, z_b).

Another definition needed is the Mach number defined as:

where a is the speed of the sound. φ y_e w _z e z_b, Z, y_b, Y, v r, N T = –X N = –Z θ γ α V_T x_b, X, u x_e L_lift V_T = v–v_w v v_w M VT a ---=

There are some angles that describe the attitude of the aircraft, for example the euler angles (φ, θ, ψ), see Figures 2.1, 2.2, and 2.3, the angle of attackα, see Figure 2.3, and the sideslip angleβ,see Figure 2.1. The definitions forα andβ are:

In Figure 2.3, the lift force L_lift is seen and is defined as:

(2.1)

The lift force L_liftis written like this to get a non-dimensional expression, where C_Tand C_N are aerodynamic coefficients corresponding to the tangential and normal forces, and q_a is the dynamic pressure defined as:

Using the lift force L_lift, the load factor n_z can be defined as:

(2.2)

This is an approximation that is good enough for smallα. Normally n_z is defined as

The motion of an aircraft can also be separated into longitudinal and lateral motion. Longitudinal motion considers the forces N and T and the moment M. The equations used in this thesis to describe longitudinal motion are and , see (2.3) and (2.10). Lateral motion considers the force Y and the moments L and N. The equations used to describe the lateral motion in this thesis are , , and , see (2.4), (2.11), and (2.9).

In order to be able to control the SHARC in both longitudinal and lateral mode, equa-tions for , , , , and are needed. An expression for is found in [3]:

(2.3)

where F_Tis the engine thrust force.

An expression for is found in [3]:

(2.4) where α arc w u ----    tan = β arc v V_T ---    sin =

L_lift = q_aSC_L = q_aS(–C_Tsin( )α +C_Ncos( )α )

q_a 1 2 ---ρV_T2 = n_z Llift mg ---= n_z N mg ---= α˙ q˙ β˙ _{r˙ p˙} α˙ q˙ β˙ r˙ p˙ α˙ α˙ q (pcosα+rsinα) β 1 mV_Tcosβ --- –L_lift F_T α mg(cosαcosθcosφ+ sinαsinθ)

+ sin – ( ) + tan – = β˙ β˙ _p ( )α _r ( )α 1 mV_T ---(Y –F_Tcos( )α sin( )β +mg₃) + cos – sin =

(2.5)

and Y is the side force where

The equations for , ,and are flat-earth approximations that do not include rotation of the earth and its curvature. The equations are found in [4]:

(2.6) (2.7) (2.8)

Solving (2.6), (2.7), and (2.8) for , , and yields:

(2.9)

(2.10)

(2.11)

The moments can also be expressed in non-dimensional aerodynamic coefficients:

(2.12) (2.13) (2.14) g₃ = g(cos( )β cos( )θ sin( )φ +sin( )β cos( )α sin( )θ –sin( )α sin( )β cos( )θ cos( )φ )

Y = q_aS C_Y q˙ p˙ r˙ L = I_xp˙–I_zx(r˙+ pq)–(I_y–I_z)qr M = I_yq˙–I_zx(r2– p2)–(I_z–I_x)rp N = I_zr˙–I_zx(p˙ qr– )–(I_x–I_y)pq q˙ p˙ r˙ p˙ L I_zx N I– zxqr+(Ix–Iy)pq I_z ---+ pq     _I y–Iz ( )qr + + 1 Izx 2 I_xI_z ---–       I_x ---= q˙ M Izx r 2 p2 – ( ) (I_z–I_x)rp + + I_y ---= r˙ N I_zx L+Izxpq+(Iy–Iz)qr I_x ---–qr     _I x–Iy ( )pq + + 1 Izx 2 I_xI_z ---–       I_z ---= L = q_aSbC_l M = q_aS cC_m N = q_aSbC_n

2.2 Flight Mechanics Specific for SHARC

In this section, aerodynamic coefficients, control surfaces and servos specific for the SHARC is presented.

2.2.1 Forces and Aerodynamic Coefficients

The aerodynamic coefficients can be found in tables calculated using results from wind tunnel tests. For the SHARC, the dependencies of the aerodynamic coefficients are pre-sented below.

(2.15)

where

(2.16)

The expression in (2.16) is necessary later in this thesis, in order to be able to solve for the control surfaceδ_e. It is expressed by a Taylor expansion, where only the first deriv-ative is included to simplify the calculations. C_m0 is calculated as:

The derivative of the aerodynamic coefficient is approximated by:

(2.17)

whereδ_eis taken from the previous sample. The value of∆in (2.17) is set to some con-venient number to prevent the derivatives from becoming discontinuous. A good value for∆is 3, that is chosen after some tests have been performed. It would be preferred if the aerodata tables already had contained information about the derivatives.

Other needed aerodynamic coefficients are:

(2.18) (2.19) (2.20) (2.21)

Some of these expressions are approximated using Taylor expansion in the same way as for C_m. The new expressions are:

C_m = C_m(M, ,α β)+C_m( )α˙ +C_m( )q +C_m( )δ_e C_m( )δ_e C_{m0 δ} e d dC_m δ_e + ≈ C_m0 C_m( )δ_{e δ} e d dC_m δ_e – = δ_e d dC_m C_m(δ_e+∆)–C_m(δ_e–∆) 2∆ ---≈ C_l = C_l(M, ,α β)+C_l( )p +C_l( )r +C_l( )δ C_n = C_n(M, ,α β)+C_n( )r +C_n( )δ C_N = C_N(M, ,α β)+C_N( )α˙ +C_N( )δ_e C_T = C_T(M, ,α β)+C_T( )δ_a +C_T( )δ_r +C_T( )δ_e

(2.22) (2.23)

(2.24)

Using the aerodata tables, one has to be certain that the inputs to the tables are part of the tables. Otherwise the outputs from the tables are set to zero. A solution to the prob-lem is to limit those signals, such as the commanded control surface deflections.

2.3 Control Surfaces

The control surfaces that make it possible to control the SHARC are seen in Figure 2.4

.

They are the elevatorδ_e, the aileronδ_a, and the rudderδ_r.

Figure 2.4: The control surfaces for the SHARC.

The definitions of the control surfaces are

One has to decide which variables of the SHARC that should be controlled. Usually, for ordinary aircraft,αis controlled at low speed, and n_z at high speed in longitudinal mode. This is because pilots feel most comfortable if the variables are controlled in that

C_l C_l(M, ,α β) C_l( )p C_l( )r δ_a d d C_l( )δδ_{a a} δ_r d d C_l( )δδ_{r r} C_l0 + + + + + = C_n C_n(M, ,α β) C_n( )r δ_a d d C_n( )δδ_{a a} δ_r d d C_n( )δδ_{r r} C_n0 + + + + = C_N C_N(M, ,α β) C_N( )α˙ C_{N 0 δ} e d dC_N δ_e + + + = δ_a δ_e,δ_r δ_a δ_e δe left _δ e right + 2 ---= δ_a δa left _δ a right – 2 ---= δ_r δr left _δ r right – 2 ---=

way. Even for UAVs, it is natural to control these variables. The velocity when the angle of attack is no longer controlled, but instead the load factor n_z, is called Corner speed. At the altitude 1000 meters, this occurs at about Mach 0.44. To decide Corner speed, the commandedα is set to its maximum, 15 degrees, and the Mach number is increased until n_z reaches its maximum value, which for the SHARC is six.

In lateral mode the angular velocity p and the sideslip angleβare to be controlled. The aim is to keepβto zero, except at landing if the wind is coming from the side. In this thesis, the SHARC will roll around the body axis x_b. Other alternatives would be to roll around the velocity vector V_T, or around a vector that points in a direction where mis-siles are fired.

One must also decide which control surfaces that are supposed to control pitch, yaw and roll. One way is to control the moments and decide the angles of the control sur-faces by solving a linearly constrained quadratic programming problem, see [5] and [6]. One advantage with this approach is that if one control surface is damaged, then the other surfaces can be optimized to control the aircraft. The calculations are part of a so called control selector, see [7], that is needed if the control surfaces are redundant. This is however not the case in this thesis. The three angular accelerations that are of interest, are all controlled by three surfaces. The relations are found in tables that origin from wind tunnel tests.

There are three equations and three control surfaces, which means that all surfaces can be solved for. To simplify, letδ_e control pitch andδ_r andδ_a together control yaw and roll. Even ifδ_eaffects roll and yaw, as well asδ_randδ_aaffect pitch, the contribution is pretty small and can be approximated by zero.

2.3.1 Servos

In the model of the SHARC, servos for the control surfaces had to be added, see Figure 2.5. The same model of a servo was added for all control surfaces. The model is simple but is a good approximation of general servo dynamics.

Figure 2.5: The servo used for all control surfaces.

p˙ = f _p(δ_a, ,δ_e δ_r) q˙ = f_q(δ_a, ,δ_e δ_r) r˙ = f_r(δ_a, ,δ_e δ_r)

0.05s+11

In this thesis, two different ways to control the SHARC are investigated, pure Non-lin-ear Dynamic Inversion and cascaded Non-linNon-lin-ear Dynamic Inversion. The word “pure” is used to make a distinction between NDI and the approximated variant, which uses a cascaded NDI.

3.1 Pure Non-linear Dynamic Inversion

Non-linear Dynamic Inversion (NDI), also known as exact linearization, see [2], is a structured way to cancel the dynamics and then control the system as a linear system. The states are linearized by letting:

...

...

Then let

where matrix L is choosen to get proper poles, and solve for u. If the nthstate derivative includes a control signal, the system can be exactly linearized. If the control signal occurs before the last state derivative, the entire closed loop system is not linearized and problems will occur if some states are unstable.

z₁ = y z₂ = y˙ z_n = y(n–1) z˙₁ = z₂ z˙₂ = z₃ z˙_n = ψ(z u, ) z˙_n = l₀r–Lz = ψ(z u, )

3.2 Cascaded Non-linear Dynamic Inversion

The other method used is a cascaded NDI, also known as “NDI using time scale sepa-ration”. The system is divided into an inner and an outer loop. The dynamics in the inner loop has to be faster than the dynamics in the outer loop. For example, the output from the inner loop in longitudinal mode is the angular velocity q, which is faster than the angle of attackα, which is output from the outer loop, because . The control surfaces have direct effect on , , and , since the surfaces produce the correspond-ing moments, soδ_e has more effect on than on .

A cascaded NDI also cancels the dynamics of a system, but the cancellation is not exact. It relies on the approximation that the inner loop is so fast, that the time it takes to perform something in the inner loop is approximated by zero. The behavior of the closed loop system is decided by the demands of the derivatives in the inner and outer loops. An approximated description of the closed loop system can also be made and consequently the approximated poles can be calculated. The cascaded control structure for the pitch mode is demonstrated in Figure 3.1, whereη andξ are some arbitrary functions.

Figure 3.1: The cascaded control structure for the pitch mode.

3.3 An Example, Pure NDI versus Cascaded NDI

To see how these two methods differ, a simple example will be presented by studying the following system:

(3.1) (3.2) (3.3)

where f is some arbitrary function. Start with pure NDI and define the new states:

The derivatives are:

(3.4) and α˙ ≈q p˙ q˙ r˙ q˙ α˙ α’=... q’=... η α_d q_d δ_e q _α ξ x˙₁ = – f x( )₁ +x₂ x˙₂ = u y = x₁ z₁ = x₁ z₂ = x˙₁ = – f x( )₁ +x₂ z˙₁ = z₂ = – f x( )₁ +x₂ z˙₂ = – f˙ x( )₁ x˙₁+ x˙₂ = – f˙ x( )₁ (– f x( )₁ +x₂)+u = v

The signal v is designed for the system to get proper poles:

(3.5)

The system can then be expressed as:

Set (3.4) and (3.5) equal and solve for u expressed in the original states. The control signal u will generate the output y, with performances defined by the signal v.

(3.6)

Now the second method with the cascaded NDI, will be examined. With the first method the desired closed loop system was designed with the help of matrix L. With the second method the desired closed system is designed by the derivatives in (3.7) and (3.8) where x_1d and x_2d are the desired signals.

(3.7) (3.8)

Use (3.1) and (3.7), and solve for the x₂ that generates the desired :

(3.9)

Now an approximation is made. The inner loop is supposed to be much faster than the outer loop, which means that x₂ can be approximated with x_2d. Equation (3.9) can therefore be written as:

(3.10)

Use (3.2) and (3.8), and solve for the u that generates the wanted :

(3.11)

Insert (3.10) into (3.11) and solve for u:

(3.12)

The control signals for both methods now can be compared. Equation (3.6) contains , which is not included in (3.12). The control signal calculated using pure NDI is more complex, but the advantage is that the system is exactly linearized in contrast to the cascaded NDI where approximations are made.

y =

∫∫

∫∫

In this chapter some different control methods using Non-linear Dynamic Inversion (NDI), are applied to the SHARC. First, pure NDI is evaluated. Then two controllers that use a cascaded NDI are calculated. The cascaded NDI method uses approxima-tions that will help simplify the expressions. For example, the inner loops are assumed to be much faster than the outer loops, and the dynamics for the control surfaces are not included, because the surfaces are assumed to move instantly.

4.1 Pure Non-linear Dynamic Inversion

As mentioned before, NDI is an exact linearization and the equations are complex, which soon will be noticed.

Start with the pitch mode, where the angle of attackαis to be controlled. The servo for the control surface angle , see Figure 2.5, can be expressed by:

(4.1)

Create the following states:

where is found in (2.3). The derivatives of the states are: δ_e* δ˙_e δe * _δ e – 0.05 ---= z₁ = α z₂ = α˙ z₃ = α˙˙ α˙ z˙₁ = α˙ = z₂

(4.2)

In the equation for there are some derivatives needed, for example that includes the moment M, see (2.10). The moment M is expressed by the elevatorδ_e, but the parame-ter of inparame-terest is , see Figure 2.5. To be able to express , the state derivative is also needed together with (4.1). The expression for is huge and there are also many derivatives that must be realized, for example the derivatives of the aerodynamic coeffi-cients and . To avoid this huge expression, it may be better to eliminate the deriva-tives of those variables that vary slowly, for example V_Tand F_T. Next step is to make a closed loop system, let

(4.3)

Design v by a state feedback, where L is chosen to get proper poles:

(4.4)

Set (4.3) and (4.4) equal and solve for .

Continue with the lateral mode. Because of the servos for the control surfacesδ_a and

δ_r, three states are needed. The equations for the servos are:

(4.5)

(4.6) z˙₂ z₃= α˙˙ q˙ (p˙cos( )α – psin( )αα ˙ +r˙sin( )α +rcos( )αα ˙) ( )β

pcos( )α +rsin( )α ( ) β˙ β ( ) cos 2 ---+ tan       m v˙( _Tcos( )β –v_Tsin( )ββ ˙) m2v_T2cos( )β 2 --- –L F_T ( )α mg(cos( )α cos( )θ cos( )φ + sin( )α sin( )θ )

+ sin – ( ) 1 mv_Tcos( )β --- –L˙ F˙_T ( )α –F_Tcos( )αα ˙ mg ( )αα ˙ ( )θ cos( )φ α

( )(_–sin( )θθ ˙_cos( )φ _–cos( )θ _sin( )φφ ˙)_{+ cos}( )αα _˙sin( )θ _+sin( )α _cos( )θθ ˙) cos + cos sin – ( + sin – ( ) + – – = = z˙₃ = ˙˙˙α = … z˙₂ q˙ δ_e* _δ e * z˙₃ z˙₃ F˙T z˙₃ = v v = l₀α_d–Lz δ_e* δ˙_a δa * _δ a – 0.05 ---= δ˙_r δr * _δ r – 0.05 ---=

The decision was to control the sideslip angleβ, therefore the states are:

where is found in (2.4). The derivatives of the states are:

(4.7)

The state derivative is necessary because and are needed in the expression instead ofδ_aandδ_r. Equations (4.5) and (4.6) are used for that purpose. The expression for is not derived in this thesis. The expression is huge and contains many deriva-tives, which must be approximated by filters. Now the closed loop will be shaped. Design v by letting:

Let

and solve for . This expression contains , so more equations are needed. Another variable to be controlled is p, hence create the new states:

z₄ = β z₅ = β˙ z₆ = β˙˙ β˙ z˙₄ = β˙ = z₅ z˙₅ β˙˙ z₆ p˙sin( )α pcos( )αα ˙ r˙ ( )α r ( )αα ˙ 1 mV_T --- q_aS C˙_Y q˙_aS C_Y F˙_T ( )α ( )β F_T ( )αα ˙ ( )β α ( )_cos( )ββ ˙ cos + cos sin – ( ) – mg ( )ββ ˙ ( )θ ( )φ ( )β sin( )θθ ˙sin( )φ θ ( )_cos( )φφ ˙ cos + – (

) ( )ββ ˙ _{( )α ( )θ sin( )β sin( )αα ˙sin( )θ}

α ( )_cos( )θθ ˙ cos + – (

) (cos( )αα ˙sin( )β +sin( )α cos( )ββ ˙) ( )θ ( )φ

α

( )_sin( )β (_–sin( )θθ ˙_cosφ_{– cos}( )θ _sin( )φφ ˙) sin + cos cos – + sin cos cos + cos + sin cos sin – ( ) + sin cos – + ( ) v˙_T mv_T2 --- q_aS C_Y F_T ( )α ( )β mg ( )β ( )θ ( )φ β

( )cos( )α sin( )θ – sin( )α sin( )β cos( )θ cos( )φ

sin + sin cos cos ( ) + sin cos – ( ) – + sin + cos – + = = = z˙₆ = … z˙₆ δ_a* δ_r* z˙₆ v = l₀β_d–Lz z˙₆ = v δ_a* _δ r *

The derivatives of the states are:

The equation for is found in (2.9). The moments L and N in that equation contain the surfaces deflectionsδ_a andδ_r. To be able to express in terms of and , is also needed. Once again this is a huge expression that not will be calculated in this thesis. It also contains derivatives that have to be approximated in some way. Design the signal v as:

Let

and solve for . Now there are two equations where is solved for. Let these equa-tions be equal and solve for .

Now the controller for the system is complete, and if no approximations are made, the system behaves exactly as desired. Unfortunately, the expressions are too complex, and contains too many derivatives that are difficult to create. One simplification is to say that the dynamics of the servos are so fast, that the expressions for the servos are not needed. In that case z₃, z₆, and z₈will not be used and the expressions for the controller will be smaller.

4.2 Cascaded Non-linear Dynamic Inversion, Method One

Now, a method that uses a cascaded NDI is presented, see [8] and [9]. The control sig-nals in both longitudinal and lateral mode will be calculated.

4.2.1 Longitudinal Mode

At low speed the controlled variable is the angle of attackα and at high speed the con-trolled variable is exchanged for the load factor n_z.The equations used whenα is con-trolled are (2.3) and (2.9) and the equations used for the pitch moment are (2.13), (2.15), and (2.16). The desired and can be expressed as a constant times the differ-ence between the desired state and the measured state, which gives a first order system:

(4.8) (4.9)

These equations determine how fast the system will respond on desired commands.

z₇ = p z₈ = p˙ z˙₇ = p˙ z˙₈ = … p˙ δ_a* _δ r * z˙₈ v = l₀p_d–Lz z˙₈ = v δ_a* _δ a * δ_r* α˙ q˙ α˙ = k_α(α_d–α) q˙ = k_q(q_d–q)

Solving for q in (2.3) and exchanging for the desired one in (4.8), gives that with this

q, the desired is obtained and also the desiredα. Since the inner loop is faster than the outer loop, as described in Chapter 3.3, the approximation can be made, which yields:

(4.10)

Continue with the inner loop. Use (2.10) and exchange for the desired one in (4.9):

(4.11)

Ifδ_ein (4.11) is solved for, thatδ_ewill generate the desired . Insert (4.10) into (4.11) and solve forδ_e:

(4.12)

Equation (4.12) is used during simulation. If the inner loop is fast enough, will almost conduct like what is expressed in (4.8), and so also the angle of attackα.

4.2.2 Lateral Mode

Lateral mode consists of both yaw and roll mode. In lateral mode, the same variables are controlled in both low and high speed. The variables are the roll rate p and the side-slip angleβand they are controlled by the surfacesδ_a andδ_r. At landing, β is some-times commanded to a certain value, otherwise the desiredβ is normally zero. In roll mode, the angular velocity p is controlled. Use (2.9) together with (2.12) and (2.14) and exchange for k_p(p_d-p). Solving forδ_a means that with thisδ_a, the desired

and consequently the desired p, are obtained. α˙ α˙ q≈q_d q_d k_α(α_d–α) (pcosα+rsinα) β 1 mv_Tcosβ ---–F_Tsinα mg(cosαcosθcosφ+ sinαsinθ)–q_aS(–C_Tsin( )α +C_Ncos( )α )

+     – tan + = q˙ k_q(q_d–q) q_aS c C_m(M, ,α β) C_m( )α˙ C_m( )q C_m0 δ_e d dC_m δ_e + + + +       I_y ---I_zx(r2– p2)+(I_z–I_x)rp I_y ---+ = q˙ δ_{e δ} e d d C_m( )δ_e    –1 k_qI_y k_α(α_d–α) (pcosα+rsinα) ( )β 1 mV_Tcosβ

--- F_T α mg(cosαcosθcosφ+sinαsinθ) q_aS C_Tsin( )α +C_Ncos( )α – ( ) ⋅ – + sin – ( ) – q – tan +     _I zx r 2 p2 – ( )–(I_z–I_x)rp –     1 q_aS c ---–C_m(M, ,α β)–C_m( )α˙ –C_m( )q –C_m0 ⋅     ⋅ = α˙ p˙ p˙

(4.13)

In yaw mode, both an inner loop and an outer loop are needed. In the outer loop the sideslip angleβ is controlled, and in the inner loop the faster yaw rate r is controlled. Use the expression for in (2.4). Exchanging for k_β(β_d-β) and r for r_d and then solv-ing for r_d, means that with this r_d, the desired is obtained and thus the desiredβ.

(4.14)

Continue with the inner loop. Use the expression for in (2.11) together with (2.12) and (2.14), and exchange for k_r(r_d-r). Ifδ_a is solved for, that δ_a would generate the desired r. Insert (4.14) into that equation and solve forδ_a.

δ_a 1 q_aSb _δ a d dC_l δ_a ( ) Izx I_z ---δ_a d dC_n δ_a ( ) +       --- k_p(p_d– p) 1 Izx 2 I_zI_x ---–       I_x q_aSb C_l(M, ,α β) C_l( )p C_l( )r δ_r d d C_l( )δδ_{r r} C_l0 + + + +     _I y–Iz ( )qr – I_zx q_aSb C_n(M, ,α β) C_n( )r δ_r d d C_n( )δδ_{r r} C_n0 + + +     _I zxqr – +(I_x–I_y)pq I_z ---pq +           – –           = β˙ _β˙ β˙ r_d k_β(β_d–β)+ p – ( )α 1 mv_T ---(Y –F_Tcos( )α sin( )β +mg₃) + sin α ( ) cos ---= r˙ r˙

(4.15) δ_a 1 q_aSb δ_α d d C_nIzx I_x ---δ_α d dC_l       --- 1 I 2 zx I_xI_z ---I_z –       k_r p ( )α –k_β(β_d–β) 1 mV_T ---(q_aS C_Y–F_Tcos( )α sin( )β +mg₃) + sin α ( ) cos --- –r           ⋅ q_aSb C_n(M, ,α β) C_n( )r _δ r d d C_nδ_r C_n0 + + +     I_zx 1 I_x --- q_aSb C_l(M, ,α β) C_l( )p C_l( )r δ_r d d C_lδ_r C_l0 + + + +     I_zxpq (I_y–I_z)qr + +     _–qr     _I x–Iy ( )pq – – –           =

Let (4.13) be equal to (4.15) and solve forδ_r:

(4.16)

Equations (4.13) and (4.16) are implemented in the simulation environment. If the inner loop for the yaw mode is fast enough, the control signal generates a that almost will be as desired. The response of also follow of the expression for the desired . In roll mode there is no inner loop, so would be as desired if problems such as model errors and noise not existed.

δ_r 1 q_aSb --- dδr dC_l – Izx I_z ---δ_r d dC_n – δ_a d dC_l I_zx I_z --- _δ a d dC_n + --- dδr dC_n I_zx I_x ---δ_r d dC_l + δ_a d d C_n Izx I_x --- _δ a d dC_l + ---+            –1 1 δ_α d d C_n Izx I_x ---+ δ_α d dC_l       --- 1 I2zx I_xI_z ---–       I_zk_r p ( )α –k_β(β_ref –β) 1 mV_T ---(q_aS C_Y –F_Tcos( )α sin( )β +mg₃) + sin α ( ) cos ---r –           q_aSb C( _n(M, ,α β)+C_n( )r +C_n0) I_zx (qaSb C( l(M, ,α β)+Cl( )p +Cl( )r +Cl0)+Izxpq+(Iy–Iz)qr) I_z ---–qr     – I_x–I_y ( )pq – –           1 δ_a d dC_l I_zx I_z ---δ_a d dC_n +       --- k_p(p_d– p) 1 Izx 2 I_zI_x ---–       I_x q_aSb C_l(M, ,α β) C_l( )p C_l( )r C_l0 + + + ( )–(I_y–I_z)qr I_zx (qaSb C( n(M, ,α β)+Cn( )r +Cn0)–Izxqr+(Ix–Iy)pq) I_z ---+ pq     – –       – ⋅               ⋅ = δ_r β˙ β β˙ _p˙

4.3 Cascaded Non-linear Dynamic Inversion, Method Two

The expressions for the control signals earlier in Chapter 4.2 are big, therefore another method that brings less complicated equations would be preferable. In this section, a cascaded NDI that uses fewer aerodynamic coefficients is tested. Instead the deriva-tives of some measured outputs are used together with measured control surface deflec-tions. The problem is, that it is difficult to use derivatives of measured signals because the signals may be noisy. The derivatives are implemented as high pass filters with the break-frequency 20 rad/s. That is, for low frequencies it is a normal derivation, but for frequencies above 20 rad/s, the gain of the transfer function is constant. This filter is used to avoid high frequency signals from being amplified too much. The filter is shown in Figure 4.1 whereα is input and its derivative is output.

Figure 4.1: The filter for the derivation.

The control signals are calculated by substracting the desired d and sensored s equa-tions for , , , , and . These less complicated equations are then used in the same way as in the first method. That is in a cascaded control structure, where the inner loops are assumed to be much faster than the outer loops. This can be read about in [10], where also results from flight tests are presented.

The equations for the moments L, M, and N used for this controller, are the following:

(4.17) (4.18) α _s α´ s 20+1 p˙ q˙ r˙ α˙ β˙ L q_aSb C_l(M, ,α β) C_l( )p C_l( )r _δ a d d C_l ( )δ_{a δ} r d d C_l ( )δ_r C_l0 + + + + +     q_aSb Γ δ_a d d + (C_l)δ_a δ_r d d C_l ( )δ_r +     = = M q_aS c C_m(M, ,α β) C_m( )α˙ C_m( )q δ_e d d C_m ( )δ_e C_m0 + + + +     q_aS c Λ δ_e d d + (C_m)δ_e     = =

(4.19)

4.3.1 Longitudinal Mode

Start with the pitch mode and use (2.3). In the first equation below, there are desired signals d and in the second equation there are sensored signals s.

(4.20)

(4.21)

where

(4.22)

Subtract (4.21) from (4.20) and use (4.22):

(4.23)

Solve for q_d, that is the angular velocity needed to obtain the desired .

(4.24)

The angular velocity q_dis obtained from the inner loop where the elevatorδ_eis used to control , which also means that q is controlled. Use (2.10) and (4.18):

(4.25)

(4.26)

where

(4.27)

Subtract (4.26) from (4.25) and use (4.27).

N q_aS b C_n(M, ,α β) C_n( )r δ_a d d C_n ( )δ_{a δ} r d d C_n ( )δ_r C_n0 + + + +     q_aS b Ω _δ a d d C_n( )δδ_{a a δ} r d d C_n( )δδ_{r r} + +     = = α˙_d q_d (pcosα+rsinα) β 1 mv_Tcosβ ---(–L F_T α mg(cosαcosθcosφ+sinαsinθ) )

+ sin – + tan – = α˙_s q_s (pcosα+rsinα) β 1 mv_Tcosβ ---(–L F_T α mg(cosαcosθcosφ+sinαsinθ) )

+ sin – + tan – = α˙_d = k_α(α_d–α) k_α(α_d–α) α– ˙_s = q_d–q_s α˙ q_d = k_α(α_d–α) α– ˙_s+q_s q˙ q˙_d q_aS c Λ _δ e d d + (C_m)δ_ed     _I zx r 2 p2 – ( ) (I_z–I_x)rp + + I_y ---= q˙_s q_aS c Λ δ_e d d + (C_m)δ_es     _I zx r 2 p2 – ( ) (I_z–I_x)rp + + I_y ---= q˙_d = k_q(q_d–q)

(4.28)

Insert (4.24) into (4.28) and solve forδ_ed. That control signal will generate a that is pretty much the same as the desired one, if the inner loop is fast enough.

(4.29)

4.3.2 Lateral Mode

The aileronδ_a and the rudderδ_r are used to control . Use (2.9), (4.17), and (4.19):

(4.30)

(4.31)

(4.32)

Subtract (4.31) from (4.32), use (4.32), and solve forδ_ad:

I_y(k_q(q_d–q)–q˙_s) q_aS c _δ e d dC_m δ_ed–δ_es ( ) = α˙ δ_ed ((k_α(α_d–α) α– ˙_s)k_q–q˙_s) Iy q_aS c δ_e d dC_m ---+δ_es = p˙ p˙_d q_aSb Γ _δ r d dC_l δ_{rd δ} a d dC_l δ_ad + +       I_zx q_aSb Ω _δ r d dC_n δ_{rd δ} a d dC_n δ_ad + +       I_zxqr – +(I_x–I_y)pq       ₁ I_z ----+ pq       I_y–I_z ( )qr + +       ₁ 1 Izx 2 I_xI_z ---–       I_x ---= p˙_s q_aSb Γ _δ r d dC_l δ_{rs δ} a d dC_l δ_as + +       I_zx q_aSb Ω _δ r d dC_n δ_{rs δ} a d dC_n δ_as + +       I_zxqr – +(I_x–I_y)pq       ₁ I_z ---- + pq       I_y–I_z ( )qr + +       ₁ 1 Izx 2 I_xI_z ---–       I_x ---= p˙_d = k_p(p_d– p)

(4.33)

Continue with the yaw mode. Start with the outer loop whereβis to be controlled. Use (2.4) where g₃ is found in (2.5):

(4.34) (4.35) (4.36)

Subtract (4.35) from (4.34), use (4.36), and solve for r_d:

(4.37)

With the angular velocity r_d in (4.37), the desired , and also the desiredβ are obtained. The yaw rate is controlled by the surfacesδ_a andδ_r in the inner loop. Use (2.11), (4.17), and (4.19): (4.38) δ_ad 1 δ_a d dC_l I_zx I_z --- _δ a d d C_n + --- kp(pd– p)– p˙s q_aSb --- 1 Izx 2 I_xI_z ---–       I_x δ_{as δ} a d dC_l I_zx I_z ---δ_a d d C_n +       δ_rd–δ_rs ( ) _δ r d dC_l I_zx I_z ---δ_r d d C_n +       – +       = β˙_{d p} ( )α _r d ( )α 1 mv_T ---(Y –F_Tcos( )α sin( )β +mg₃) + cos – sin = β˙_{s p ( )α r} s ( )α 1 mv_T ---(Y –F_Tcos( )α sin( )β +mg₃) + cos – sin = β˙_{d k} β(βd–β) = r_d β ˙_{s k} β(βd–β) – α ( ) cos ---+r_s = β˙ r˙_d 1 1 Izx 2 I_xI_z ---–       I_z --- q_aS b Ω δ_a d d C_n ( )δ_{ad δ} r d d C_n ( )δ_rd + +     _I x–Iy ( )pq I_zx q_aSb Γ δ_a d d + (( )δC_{l ad}) _δ r d d C_l ( )δ_rd +     _I zxpq (Iy–Iz)qr + + I_x ---–qr                + +      =

(4.39)

(4.40)

Subtract (4.39) from (4.38) and use (4.38):

(4.41)

Insert (4.37) into (4.41) and solve forδ_ad:

(4.42) r˙_s 1 1 Izx 2 I_xI_z ---–       I_z --- q_aS b Ω _δ a d d C_n ( )δ_{as δ} r d d C_n ( )δ_rs + +     _I x–Iy ( )pq I_zx q_aSb Γ _δ a d d + (( )δC_{l as}) _δ r d d C_l ( )δ_rs +     _I zxpq (Iy–Iz)qr + + I_x ---–qr                + +      = r˙_d = k_r(r_d–r) k_r(r_d–r)–r˙_s ( ) 1 Izx 2 I_xI_z ---–       I_z q_aSb _δ a d d C_n(δ_ad–δ_as) δ_r d d C_n(δ_rd–δ_rs) I_zx I_x ---δ_a d d C_l(δ_ad –δ_as) δ_r d d C_l(δ_rd–δ_rs) +       + +   = δ_ad 1 δ_a d dC_n I_zx I_x ---δ_a d dC_l +       --- kβ(βd–β) β ˙_s – α ( ) cos ---(–k_r)–r˙_s       1 Izx 2 I_xI_z ---–       I_z q_aSb ---δ_a d dC_n I_zx I_x ---δ_a d dC_l +       δ_as (δ_rd–δ_rs) δ_r d dC_n I_zx I_x ---δ_r d dC_l +       – +               =

There are now two equations whereδ_adis solved for, (4.33) and (4.42). Set those equa-tions equal and solve forδ_rd:

(4.43)

Equations (4.29), (4.33), and (4.43) are used during simulation. As can be seen, these equations are less complicated than those generated by the design methods “Pure Non-linear Dynamic Inversion” and “Cascaded Non-Non-linear Dynamic Inversion Method One”. δ_rd 1 δ_r d dC_l – Izx I_z ---δ_r d dC_n – δ_a d d C_l Izx I_z --- _δ a d dC_n + --- dδr dC_n I_zx I_x ---δ_r d dC_l + δ_a d d C_n Izx I_x --- _δ a d dC_l + ---+             --- 1 δ_a d dC_n I_zx I_x ---δ_a d dC_l +       --- kβ(βd–β) β ˙_s – α ( ) cos ---k_r – ( )–r˙_s ⋅     1 I_zx2 I_xI_z ---–       I_z q_aSb ---δ_a d dC_n I_zx I_x --- _δ a d dC_l +       δ_{as δ} r d dC_n I_zx I_x --- _δ r d dC_l +       δ_rs + + ⋅               1 δ_a d dC_l I_zx I_x --- _δ a d dC_n +       --- (k_p(p_d– p)– p˙_s) 1 Izx 2 I_xI_z ---–       I_x q_aSb --- _δ a d dC_l I_zx I_z --- _δ a d dC_n +       δ_as δ_r d dC_l I_zx I_z ---δ_r d dC_n +       δ_rs + +               –                 =

4.4 Control of the Load Factor

At high speed, the load factor n_zis controlled. The control signals in this section can be used together with the earlier calculated control signals in longitudinal and lateral mode.

Preferably, there would be an expression for so the desired dynamics could be demanded in the same way as for the other derivatives, that is

One expression looks like:

(4.44)

One would prefer to have n_z in an outer loop and the pitch rate q in an inner loop. Therefore equation (4.44) must contain q in some way. An expression for is pre-sented below, see also (2.1), (2.20), and (2.21).

There are already equations for and but there may be problems with the other expressions. Therefore, less complicated alternatives will instead be investigated.

4.4.1 Control of the Load Factor by the Angle of Attack

One way to control n_zis to command a desired n_z, transform the demand into a desired

α, and then control α with the same controller as before. The task is now to find an expression that relates n_z toα, as:

An expression for n_z is presented below, see also (2.1).

(4.45)

Substituting n_zd andα_d for n_z andα, and solving forα_d yields:

(4.46)

In (4.46) the following equations are needed:

n˙_z n˙_zd k_n z(nzd–nz) = n˙_z L mg ---˙ ≈ qaS C˙L mg ---= C˙L C˙_L α d dC_L t d dα β d dC_L t d dβ α˙ d dC_L t d d_α ˙ +… ⋅ + ⋅ + ⋅ = α˙ β˙ n_zd = k⋅α_d+m n_z L mg ---≈ qaS CL mg --- qaS mg --- C_L0 α d dC_L α +     ⋅ ≈ = α_d mg q_aS ---n_zd–C_L0 α d dC_L ---≈

(4.47) (4.48)

where

and

Equation (4.46) is implemented in the simulation environment.

4.4.2 Control of the Load Factor by the Elevator

In this section a n_z-controller that do not use anα-controller is calculated. Use (2.1), (2.2), and (2.24). If the angle of attack is small, sin(α) can be approximated by zero.

Solving for the elevatorδ_e and substituting n_zd for n_z yields:

(4.49)

This controller can also be calculated without the approximation where sin(α) is set to zero. Equation (4.49) has been implemented in the simulation environment. Unfortu-nately the derivative in the denominator becomes zero after some simulation time. The problem is that the inputs to the tables, where the aerodynamic coefficients are calcu-lated, get values that are not part of the tables. The result is that some aerodynamic coefficients are set to zero, which leads to division by zero. The underlying problem is that it is difficult to control n_z orα without an inner loop that controls the pitch rate q.

4.5 Robustness

One problem when the SHARC is controlled, is that the model of the system is not per-fect and therefore a robust controller is needed. Analyzing the robustness of a system, the different coefficients can be changed to see how the system reacts. In this section

α d dC_L α d dC_T α ( )–C_T ( )α α d dC_N α ( )–C_Nsin( )α cos + cos sin – = C_L0 C_L α d dC_L α – = α d dC_N α d d C_N(M, ,α β) α d d C_N( )α˙ + α d d C_N( )q α d d C_N( )δ_e + + = α d dC_T α d d C_T(M, ,α β) α d d C_T( )δ_a + α d d C_T( )δ_r α d d C_T( )δ_e + + = C_N(M, ,α β)+C_N( )α˙ C_N( )q δ_e d d C_N( )δδ_{e e} C_{N 0} + + +     _{( )α q} aS cos = n_zmg δ_e n_zdmg q_aScos( )α ---–C_N(M, ,α β)–C_N( )α˙ –C_N( )q –C_{N 0} δ_e d d C_N( )δ_e ---=