Autonomous Execution of Aircraft Supermaneuvers with Switching Nonlinear Backstepping Control

Majid Moghadam∗ Controls and Avionics Research Group, Aerospace Research Center, Istanbul Technical University, Istanbul, 34469, Turkey N. Kemal Ure† and Gokhan Inalhan‡ Department of Aeronautical Engineering Istanbul Technical University, Istanbul, 34469, Turkey

Control system design for agile maneuvering aircraft poses several challenges, such as underactuated nonlinear dynamics, input saturation limits and loss of control effectiveness in high angle of attack regions of the flight envelope. Previous work in the field developed a variety of switched nonlinear control methods to enable precise tracking of agile maneuver profiles. However, demonstrating autonomous execution of challenging agile maneuvers, such as the ones that require simultaneous tracking of both translational and attitude state variables is still an open problem. In this study, we develop switched nonlinear backstepping control laws tailored towards the execution of such agile maneuvers. In particular, we demonstrate the applicability of our design on a high fidelity F-16 model and two supermaneuvers: Pugachev’s Cobra and Rolling Circle. In addition, we present a numerical study for analysis of stability of these controllers by introducing the notion of Region of Recoverability (ROR). ROR plots outline the subsets of the state space where the switching of controllers is feasible, which helps the designer to assess the stability of switched control system.

Nomenclature

VT Airspeed, ft/s X,¯ Y,¯ Z¯ Body force components, lb α Angle of attack, rad L,¯ M,¯ N¯ Angular moments, lb.ft β Side slip angle, rad g Gravitational acceleration, ft/s2 q0, q1, q2, q3 Quaternion components ci Inertia parameters φ, θ, ψ Euler roll, pitch, and yaw angles, rad q¯ Dynamic pressure, lb/ft2 p, q, r Roll, pitch, and yaw rates, rad/s S Wing area, ft2 γ Flight path angle, rad B Wing span, ft χ Course angle, rad c¯ Mean aerodynamic chord, ft Np Inertial north position, ft C? Aerodynamic coefficients Ep Inertial east position, ft δa, δe, δr Control surface deflections, deg h Height, ft δlef Leading-edge flap deflection, rad 2 D, L, Y Drag, lift, and side forces, lb hE Angular momentum of engine, slug.ft /s T Thrust force, lb Tw/b Body to wind frame rotation matrix 2 2 g1, g2, g3 Gravity components, m/s Ps Static pressure, lb/ft m Mass, slug ref Reference input ∗Graduate Research Assistant, Department of Aeronautical and Astronautical Engineering, [email protected] †Assistant Professor, Department of Aeronautical and Astronautical Engineering, [email protected]. ‡Professor, Department of Aeronautical and Astronautical Engineering, [email protected].

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American Institute of Aeronautics and Astronautics I. Introduction

Agile maneuvers are often executed by piloted fighter aircraft for gaining the advantage in combat and/or evading threats. Such maneuvers are also commonly used in aerobatics shows. These maneuvers usually feature i) high angle of attack, ii) high body angular rates and iii) simultaneous tracking of attitude and translational states, such as rolling the aircraft while keeping the altitude constant. In the last two decades, design of feedback control systems that can track such maneuvers in an autonomous manner attracted a lot of interest in control and aerospace communities. This trend is partially correlated with the increasing number of threats to unmanned combat aerial vehicles (UCAVs). In order to evade these threats, UCAVs need to perform agile maneuvers, in addition to the classical autonomous control modes such as waypoint tracking.1 Another potential source of motivation is to perform automated aerobatics shows.2 Design of such control systems is challenging due to a number of reasons, such as highly nonlinear dynamics of the aircraft and envelope saturation limits. The main objective of this work is to push the state of the art in nonlinear flight control systems design to enable execution of challenging agile maneuvers.

A. Previous Work Similar to most nonlinear systems, stabilizing and controlling nonlinear six-degree-of-freedom (6DOF) air- craft dynamics to perform supermaneuvers initiated with design and implementation of linear control strate- gies.3 However, in most aggressive maneuvers, aircraft states deviate significantly from the equilibrium point around which the dynamics are linearized, which leads to loss of performance and even instability in some cases. Hence, nonlinear control strategies such as nonlinear dynamic inversion (NDI),4 sliding mode control (SMC),5 gain scheduling,6 and backstepping method (BS)7 have gained substantial interest in control system design for tracking agile maneuvers.

The simplicity of the design methodology and implementation made NDI and SMC attractive choices for researchers.8–12 Separating nonlinear dynamics into inner and outer loop dynamics and designing controller for each loop have been investigated by Snell.9 However, the inability of this method in stabilizing the aircraft while tracking translational variables that arises due to non-minimum phase (NMP) characteristic of the equations of motion4 motivated researchers to look for alternative solutions. Fiorentini and Serrani13 used a suitable redefinition of the internal dynamics and presented an adaptive flight path angle trajectory tracking control method. A number of early studies10, 14, 15 neglected NMP characteristics of the system on controller design and studied the effects of NMP for specific types of aircraft. On the other hand, SMC has the additional advantage of being robust to uncertainties in the system model. In addition, it is pos- sible to design a dynamic compensator to slightly overcome the NMP effect on controlling the aircraft.16 Chattering of the control inputs because of the switching nature of the design methodology in SMC is one of the disadvantages of this method, which can be addressed by using higher order sliding mode (HOSM) techniques.17, 18 However, because of the NMP dynamics, none of the aforementioned control techniques could demonstrate consistently superior tracking performance for challenging agile maneuvers that require simultaneous tracking of attitude and translational variables. The issues with NMP dynamics popularized alternative approaches such as BS. Backstepping is a Lyapunov based design method that gained a lot of attention in last decade. In this method, each state can be stabilized using either control input or another state as a virtual control that manipulates the state trajectory. A global stabilizing controller in each step is designed using Lyapunov’s direct method.4 Compared to NDI and SMC, backstepping offers a more flex- ible way of dealing with nonlinearities and internal dynamics of highly coupled nonlinear dynamics. More importantly, the control design does not necessarily suffer from NMP dynamics. Aircraft flight path angle control using BS method is studied by.19–21 In these works, aerodynamic coefficients are usually modeled as data lookup tables with considerable uncertainties. Adaptive and incremental backstepping methods have been used widely in the literature to make the controller robust to model uncertainties.22–27 A comparison between NDI and BS control in stabilizing the flight path angle of the aircraft is also investigated.28 However, it should be noted that due to underactuated nature of the aircraft dynamics, even with the BS method, most of the supermaneuvers cannot be tracked with a single controller, hence a switching strategy is needed. It is seen that, theoretically or practically, different types of agile maneuvers like Herbst,3 rapid turning,27 hover flight,29 and many kinds of maneuvers have been performed/analyzed in the previous work. However, a large portion of these maneuvers does not challenge the coupled nonlinear dynamics of the aircraft. For example, assume an aircraft that flies with a roll angle of 90◦ and tends to hold the altitude and course

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American Institute of Aeronautics and Astronautics angle at a constant value. In this orientation, most of the basic intuition regarding flight dynamics no longer hold true. For instance, lift force generated by the elevator deflection does not result in acceleration in longitudinal plane, instead, it produces an acceleration in lateral plane. Vice versa, side force pushes the aircraft in vertical plane. Designing nonlinear controllers for performing such maneuvers is not trivial, as the state-input relation cannot be decoupled into independent equations. The proposed controller should stabilize the aircraft while performing the desired aggressive maneuver. In addition, it is also a point of interest to execute more than one maneuver in a row to be able to perform a complete predefined aerobatic scenario. Frazzoli et al.30 used discrete maneuvers for motion planning of agile vehicles. Ure and Inalhan1 presented a multi-modal flight control framework and flight path planning to enable the vehicle to perform agile maneuvers. Attitude transition modes were used to recover the UAV’s orientation while passing between maneuvers.

B. Contributions In this work, we present a switched BS based approach for tracking agile maneuvers that demand simulta- neous tracking of translational and attitude variables. Our contributions can be listed as follows:

• We show that by designing a suitable switching strategy it is possible to execute highly challenging agile maneuvers that require simultaneous tracking of both attitude (such as roll and pitch angle) and trans- lational (such as altitude) variables. We validate our approach via simulations of two supermaneuvers: Pugachev’s cobra and Rolling Circle. Pugachev’s Cobra involves a large angle nose up motion while keeping the altitude as constant as possible and Rolling Circle involves a constant altitude coordinated turn where the aircraft keeps rolling around its roll axis. Simulations are performed with a high fidelity 6DOF nonlinear F-16 Model. To the best of our knowledge, this is the first time demonstration of a control system that enables autonomous execution of these two maneuvers. • In order to assess the stability and feasibility of the switching conditions, we present a graphical approach named Region of Recoverability (ROR). These regions are computed via simulations and they basically present the stability region of each local controller, which enables the designer to decide on the state-dependent switching rules for switching between controllers. Since the inappropriate switching rules can destabilize the system, this proposed graphical tool provides an important aid to the designer to overcome this issue. We present the application of this method in the analysis of switching conditions for Pugachev’s Cobra maneuver.

II. Aircraft Model

The nonlinear equations of motions of the six-degree-of-freedom model of the aircraft can be written as23, 31 1 V˙ = (−D + T cos α cos β + mg ) (1) T m 1 1 α˙ = q − (p cos α + r sin α) tan β + (−L − T sin α + mg3) (2) mVT cos β ˙ 1 β = p sin α − r cos α + (Y − T cos α sin α + mg2) (3) mVT

      q˙0 0 −p −q −r q0   1     q˙1 p 0 r −q q1   =     (4) q˙2 2 q −r 0 p  q2 q˙3 r q −p 0 q3

p˙ = (c1r + c2p)q + c3L¯ + c4(N¯ + hEq) (5) 2 2 q˙ = c5pr − c6(p − r ) + c7(M¯ − hEr) (6)

r˙ = (c8p − c2r)q + c4L¯ + c9(N¯ + hEq) (7)

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American Institute of Aeronautics and Astronautics 1 g γ˙ = [L cos φ − Y sin φ + T (cos α sin β sin φ + sin α cos φ)] − cos γ (8) mVT VT 1 χ˙ = [L sin φ + Y cos φ + T (sin α sin φ − cos α sin β cos φ)] (9) mVT cos γ

N˙ p = VT cos χ cos γ (10)

E˙ p = VT sin χ cos γ (11) ˙ h = VT sin γ (12) In which aerodynamic forces (D, L, Y ) are defined in wind frame. Aerodynamic forces normally are calculated and given as data lookup tables in body-fixed frame (X,¯ Y,¯ Z¯), so they need to be transformed to wind frame D = −X¯ cos α cos β − Y¯ sin β − Z¯ sin α cos β (13) Y = −X¯ cos α sin β + Y¯ cos β − Z¯ sin α sin β (14) L = X¯ sin α − Z¯ cos α (15) where forces and moments are related to non-dimensional aerodynamic coefficients as

X¯ =qSC ¯ X,T (16)

Y¯ =qSC ¯ Y,T (17)

Z¯ =qSC ¯ Z,T (18)

L¯ =qSBC ¯ L,T (19)

M¯ =qS ¯ cC¯ M,T (20)

N¯ =qSBC ¯ N,T (21) The gravity components are defined as26     g1 2(q1q3 − q0q2)g     g2 = Tw/b  2(q2q3 + q0q1)g  (22) 2 2 2 2 g3 (q0 − q1 − q2 + q3)g

III. F-16 Model Description

In this study the nonlinear high-fidelity model of the F-16 aircraft32 is used. The aerodynamic data tables which are taken from NASA report33 are valid for large ranges of angle of attack (−20◦ ≤ α ≤ 90◦) and side slip angle (−30◦ ≤ β ≤ 30◦).

A. Engine and Control Surfaces Engine and actuator models are borrowed from Stevens.34 Control surfaces are modeled as a first-order transfer function with saturation limits, rate limits, and time constants defined in table 1. The engine is modeled by first-order lag and a throttle gearing and a lookup data table for thrust as a function of operating power level, altitude, and Mach number. This model is taken from Stevens34 and described in more details in Sonneveldt and Lars31

B. Leading Edge Flap The model is also equipped with the leading-edge flap which enables the F-16 aircraft to fly at higher angle of attacks. The corresponding transfer function is 2s + 7.25 q¯ δlef = 1.38 α − 9.05 + 1.45. (23) s + 7.25 Ps where, Ps denotes the static pressure. The pilot, and accordingly the controller, does not have control on δlef directly. Deviation on this control surface results in changes in aerodynamic coefficients values.

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American Institute of Aeronautics and Astronautics Table 1: F-16 control surfaces.

Control surface Saturation limit Rate limit Time constant Elevator ± 25◦ 60 (deg/s) 0.0495 s Aileron ± 21.5◦ 80 (deg/s) 0.0495 s Rudder ± 30◦ 120 (deg/s) 0.0495 s

IV. Maneuver Descriptions

A. Pugachev’s Cobra For both military and aerobatic performances, Pugachev’s Cobra maneuver has been a point of interest among pilots. According to the fact that it challenges both aircraft maneuverability capacity at high angle of attacks and low speeds, which indicates the stall condition, and the pilot’s skill. Pugachev’s Cobra maneuver can be decomposed into three sub-maneuvers: level flight, cobra maneuver, and VT − θ transition. In level flight, aircraft flies in a trimmed condition with constant airspeed and altitude. This flight can be characterized by defining zero translational accelerations in every direction and zero angular rates.34 In Cobra maneuver aircraft increases the pitch angle and reaches to the vertical orientation or slightly beyond/before that while holding the flight path angle close to zero. This obviously results in considerable loss of the airspeed and increase in angle of attack. Therefore, VT − θ transition sub-maneuver is used to take the aircraft back to level flight by decreasing pitch angle and recovering the speed. Throughout the maneuver, aircraft gains altitude slightly and loses the speed considerably. A visualization of the execution can be found in Fig. 1.

Figure 1: Pugachev’s Cobra maneuver visualization

B. Rolling Circle Possibly one of the most challenging aerobatic maneuvers for both pilots and controllers is the Rolling Circle (Turning-Circle) maneuver. In its final form, the pilot manipulates all of the controls in a periodic manner. Intuitively, as shown in Fig. 2, the aircraft follows a circular trajectory in the lateral plane while rolling with a constant roll rate. Obviously, this maneuver challenges the pilot skills and controller ability in dealing with the non-minimum phase characteristic of the aircraft. It even becomes more challenging when the maneuver is executed at high roll rate. Manipulating RC remote controls should be done in a very fast rhythmic manner. Based on the authors’ knowledge, there is no autonomous controller in the literature to perform this supermaneuver. We are aiming to execute the Rolling Circle maneuver fully autonomously using switching BS method.

V. Backstepping Flight Control Design

As the EOM of the aircraft in section II imply, engine thrust and angular moments can be used as control inputs to stabilize the system. Therefore control input vector is given by

h iT U = T L¯ M¯ N¯ (24)

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American Institute of Aeronautics and Astronautics Figure 2: Rolling circle maneuver visualization.

In every step of the controller design, a specific number of states are selected and the stabilizing control law is calculated. Considering only equations of these states results in a nonlinear system with the same number of states as the number of control inputs. Fortunately, control inputs are mostly affine in aircraft nonlinear state equations. However, this does not hold for flight path angle dynamics, which is going to be discussed in section VI.B. Assume a nonlinear system in canonical form

X˙ = F(X,U) + G(X,U)U (25)

where X ∈ Rn, U ∈ Rn, F : Rn × Rn → Rn, G : Rn × Rn → Rn×n. In the backstepping control approach20, 22, 23 tracking error coordinates and its time derivative for constant reference are defined as

Z = Xref − X (26)

Z˙ = −X˙ (27) Now use the positive Lyapunov’s function 1 V = ZT Z (28) 2 to determine a stabilizing control input. To this aim let’s calculate its derivative

V˙ = ZT Z˙ (29) = ZT (−F(X,U) − G(X,U)U) (30)

choosing the stabilizing control law as

−1 Udes = G(X,U) (CZ − F(X,U)) (31)

gives V˙ = −ZT Z ≤ 0 (32) for C a positive diagonal control gain matrix. This guarantees the global stability under the invertibility of G(X,U).4 This approach calculates the desired thrust and angular moments in Eq. 24. However, finding eligible throttle and control deflections to achieve the calculated control effort is a matter. One modern approach is to define aerodynamic coefficients as adaptive linear functions of states and control deflections. This enables us to invert the functions and calculate the desired deflections. This approach is known as adaptive backstepping method.22, 23 Another approach is to design a control allocator that numerically maps the control demands into the actuator settings.20 Practically, aerodynamic coefficients are available

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American Institute of Aeronautics and Astronautics as data lookup tables consisting of huge amount of data. Therefore, optimization techniques are used to implement the control allocation. In this work, a simple algorithm to invert system model and find the desired throttle and control deflections is implemented. More details on this algorithm are given in Appendix. It should be mentioned that throughout this paper it’s assumed that full state feedback is available to the controller.

VI. Backstepping Controller Design for Pugachev’s Cobra Maneuver

Longitudinal model of the aircraft is considered to simulate Pugachev’s Cobra maneuver. Neglecting lateral directional dynamics is acceptable as the maneuver requires aircraft not to move on lateral plane. Performing Pugachev’s Cobra maneuver using full model (longitudinal and lateral) requires designing addi- tional controllers that keep roll rate and yaw rate at zero while performing the maneuver. These controllers will be discussed in section VIII. The longitudinal equation of motions of the aircraft can be written as

1 V˙ = (−D + T cos α − mg sin γ) (33) T m 1 α˙ = q + (−L − T sin α + mg cos γ) (34) mVT 1 γ˙ = (L + T sin α − mg cos γ) (35) mVT θ˙ = q (36) 1 q˙ = M¯ (37) Iy

A. Speed Controller As the Eq. 33 suggests, speed dynamics are in the canonical form given in Eq. 25. Thus, tracking error state and positive Lyapunov control function are defined as

zv = VT,ref − VT (38) 1 V = z2 (39) v 2 v Following the similar procedure in Section V nonlinear stabilizing speed controller can be calculated as

−1 Tdes = gv (cvzv − fv) (40)

in which cv is a positive controller parameter and fv and gv can be derived from Eq. 33 1 f = (−D − mg sin γ) (41) v m cos α g = (42) v m π note that gv becomes singular when α = k 2 with k a non-zero integer. Fortunately, neither the F-16 flight envelope nor the intended scenario include these values for α. Therefore, according to Lyapunov theorem for local stability (see e.g. Theorem 3.2 in Slotine and Li4), the calculated control law in Eq. 40 makes the closed loop system asymptotically stable.

B. Steady-Level Flight According to the fact that in steady-level flight, aircraft is in equilibrium point so there exist linear controllers in the literature34, 35 to stabilize the system. However, these controllers are valid for the scenarios in which states are not expected to deviate much from the nominal point, which is not the case in supermaneuvers like Pugachev’s Cobra. Feedback linearization, on the other hand, has become a point of interest in recent years.8–11 However, according to non-minimum phase characteristic of this method, especially near the stall

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American Institute of Aeronautics and Astronautics angle,19 NDI may fail to stabilize the system while performing agile maneuvers. Harkegard and Torkel19, 20 designed a flight path angle controller using BS method via taking advantage of inherent zero crossing property of lift force with respect to angle of attack but as the designed controller required real-time trim value for angle of attack, Gavilan et al.21 proposed an adaptive BS controller which stabilizes the system and follows the desired trajectory for flight path angle without any knowledge of aerodynamic coefficients and trim angle of attack. In this work, uncertainties in aerodynamic coefficients are not considered so the adaptive controller discussed in Gavinal et al.21 is going to be redefined for the system in Eqs. 33 - 37. Regarding the lift coefficient properties in F-16 aircraft this assumption can be made

Assumption 1. The lift coefficient CL is only a function of α. The reference xB body axis is chosen such that CL(0) = 0. This is feasible when xB is parallel to the aircraft zero-lift line. A typical plot for lift in Eq. 15 with respect to angle of attack is given in Fig. 3. This shows that the property αL(α) > 0 holds for all α ∈ R. Assuming constant reference input for γ and cos γ ≈ cos γref in steady state mode, the Eq. 35 becomes

Figure 3: Relation between lift force and angle of attack in F-16 aircraft

1 γ˙ = (L + T sin α − mg cos γref ) (43) mVT =∆ Φ(α) (44)

Call α0 the trim angle of attack that makes Φ(α0) = 0. Therefore, under the assumption 1, the property (α − α0)Φ(α) > 0 holds for all α in flight envelope. 2 Step 1. Let us define the error vector zγ ∈ R as " #" # zγ,1 γ − γref zγ = (45) zγ,2 θ − γref − α0

its time derivative can be calculated as " # η(zγ,2 − zγ,1) z˙γ = (46) q

where the scalar function η is defined as ∆ η(x − α0) = Φ(x) (47)

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American Institute of Aeronautics and Astronautics for every x ∈ R. Note that scalar function η(x) satisfies xη(x) ≥ 0 for all x ∈ R. Now let us define the first positive Lyapunov control function and its time derivative as

1 2 Vγ,1 = z (48) 2 γ1

V˙γ,1 = zγ,1η(zγ,2 − zγ,1) (49) (50)

choosing zγ,2,des = −cγ,1zγ,1 makes V˙γ negative definite for cγ,1 > −1. Taking zγ,2 as virtual input, the required pitch rate is going to be calculated in the next step. Step 2. Defining error dynamics as

z˜γ,2 = zγ,2 − zγ,2,des (51) results in

z˙γ,1 = η(˜zγ,2 − (1 + cγ,1)zγ,1) (52)

z˜˙γ,2 = q + cγ,1η(˜zγ,2 − (1 + cγ,1)zγ,1) (53) The positive Lyapunov function can be defined as36

Z z˜γ,2−(1+cγ,1)zγ,1 Vγ,2 = cγ,2Vγ,1 + η(s)ds (54) 0

for cγ,2 > 0, and the derivative:

V˙γ,2 = cγ,1zγ,1η + (−η + q)η (55) 2 = −η + (cγ,1zγ,1 + q)η (56)

where η = η(˜zγ,2 − (1 + cγ,1)zγ,1). The virtual control input that makes V˙γ,2 negative definite can be calculated as qdes = −cγ,3zγ,1 (57)

for cγ,3 > 0 as the tunable parameter. This controller forms the outer loop part of the flight path angle controller which makes the origin for error dynamics zγ,1 =z ˜γ,2 globally stable. Step 3. The inner loop pitch rate controller can be found by defining error state as

zq = qref − q (58) and corresponding positive Lyapunov function: 1 V = z2 (59) q 2 q Therefore by assuming constant reference input

V˙q = zqz˙q (60) 1 = zq(− M¯ ) (61) Iy 2 = −zq ≤ 0 (62) is achievable if M¯ des = cqIyzq (63)

for cq > 0. The required elevator deflection that results in the desired pitching moment is discussed in the appendix. Assuming constant reference inputs in the design procedure simplifies the controller structures. Otherwise, derivative of reference inputs appear in control laws which requires command filters.22, 37, 38 Therefore, for the sake of simplicity, we assume fixed reference inputs. However, this may not be a right assumption for virtual inputs (e.g. qref in Eq. 58), even though, when the actual references are fixed. But, in the simulation results, it will be shown that this simplification does not influence the controller performance significantly. Controller C1 comprises speed and flight path angle control laws given in Eqs. 40 and 72 respectively.

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American Institute of Aeronautics and Astronautics C. Cobra Maneuver (Controller C2) As discussed in section IV while executing Cobra maneuver aircraft increases the angle of attack while holding the altitude at a constant value. Motivated by the actual pilots’ procedure, and the longitudinal EOM, angle of attack (Eq. 34) can be controlled via pitch rate as a virtual command input. Alternatively, Cobra maneuver can be executed using pitch angle controller instead of angle of attack controller, because θ = α + γ always hold. In addition, as the engine thrust is affine in Eq. 35 so it can be used to hold the T flight path angle at the desired value during the maneuver. Therefore if we choose xθ,γ = [ θ γ ] and T Uθ,γ = [ MT¯ ] as the state and control input vectors, the corresponding EOM in Eqs. 34 and 35 appear in canonical form. By defining tracking error state vector as " # " # θ θ + q zθ,γ = − (64) γ γ ref the stabilizing Lyapunov based nonlinear controller C2 using Eq. 31 can be calculated as

−1 Uθ,γ,des = Gθ,γ (Cθ,γ zθ,γ − fθ,γ ) (65)

where Cθ,γ is a positive definite diagonal control gain matrix and " # 1 0 Iy Gθ,γ = (66) 0 sin α mVT " # q fθ,γ = (67) 1 (L − mg cos γ) mVT

Obviously, Gθ,γ is singular where α = 0 which, in fact, is in the flight envelope. Actually, while in level flight, α0 is a positive number, therefore, switching from C1 to C2 should not cause any problem. Because final state value just before switching, which is, in fact, the initial point for the next controller, is in the region of attraction (ROA) of C2, therefore, the non-singularity condition will not be violated.

D. VT − α Transition Mode (Controller C3) According to the fact that, we aim to perform continuous aggressive maneuvers by switching between different stabilizing controllers, therefore, it’s possible that α = 0 become an initial condition for a controller, including 1 C2. In this case using a VT − α transition mode which can deviate the state vector from the singular initial condition of the next controller can be a solution to this problem. VT − α controller (C3) can be calculated in a double loop structure. Because T and q are affine in VT and α dynamics given in Eqs. 33 and 34. Thus, T T if we choose xv,α = [ VT α ] and Uv,α = [ T¯ q¯ ] as the state and control input vectors, the nonlinear stabilizing outer loop controller can be found as

−1 Uv,α,des = Gv,α(Cv,αzv,α − fv,α) (68)

where Cv,α > 0 is diagonal gain matrix and " # VT,ref − VT zv,α = (69) αref − α

" cos α # m 0 Gv,α = (70) 0 1

" 1 # m (−D − mg sin γ) fv,α = (71) 1 (−L − T sin α + mg cos γ) mVT The control law becomes singular where α = 90◦ which is not in the flight envelope. The inner loop q stabilizing controller can be borrowed from Step 3 in section VI.B:

M¯ des = cqIy(qref − q) (72)

where cq > 0 is tunable controller parameter.

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American Institute of Aeronautics and Astronautics E. VT − θ Transition Mode (Controller C4) At the final stage of the Pugachev maneuver where pitch angle is around 90◦ and the aircraft lost considerable airspeed, switching to a transition mode which recovers the attitude and airspeed of the aircraft seems more logical than switching directly to the steady-level flight controller. This will be discussed in section VII in T more details. Obviously, [ VT θ ] dynamics are in canonical form in Eqs. 33 and 36, therefore

−1 Uv,θ,des = Gv,θ(Cv,θzv,θ − fv,θ) (73)

stabilizes the error dynamics for Cv,θ > 0 a diagonal gain matrix and " # VT,ref − VT zv,θ = (74) θref − θ − q " # cos α 0 G = m (75) v,θ 0 1 Iy " 1 # m (−D − mg sin γ) fv,θ = (76) q

F. Maneuver Execution Using Switching Control The closed-loop switching control system scheme is shown in Fig. 4. Starting with a steady-level flight (C1) and switching to Cobra maneuver controller (C2) at t = 10s then to C4 transition mode at t = 20s and finally to C1 at t = 50s, Pugachev’s Cobra maneuver is executed in MATLAB Simulink. The 3D trajectory, position and state trajectories are plotted in Fig. 5. The fact that aircraft loses considerable airspeed while gaining pitch angle is trivial. Note that the proposed controller family were able to hold the flight path angle close to zero while rotating the aircraft nose to the vertical position. Fig. 5.c shows that, although an small increase in height is happened, it is recovered successfully. Now let us perform a maneuver in which after finishing Cobra maneuver we switch directly to C1 without entering the transition mode C4. The system response is presented in Fig. 6. Obviously, the aircraft gained more altitude than the previous scenario. In addition, recovery of VT , α and θ lasted considerably longer than the scenario in which VT − θ transition mode was used.

Figure 4: Switching BS control system scheme for Pugachev’s Cobra maneuver

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American Institute of Aeronautics and Astronautics (a) 3D trajectory

state trajectory reference input controller switch 1000

ft/s T 500 V 0 100

° 50

0 50

° 0

-50 100

° 50

0 50 /s

° 0 q -50 0 10 20 30 40 50 60 time (s)

(b) State trajectories

104 3.8

3.7

Height (ft) 3.6 0 0.5 1 1.5 2 2.5 3 North (ft) 104 (c) Longitudinal position trajectory

Figure 5: Pugachev’s Cobra maneuver using switching BS method between C1, C2 and C4 controllers

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American Institute of Aeronautics and Astronautics (a) 3D trajectory

state trajectory reference input controller switch 1000

ft/s T 500 V 0 100

° 50

0 50

° 0

-50 100

° 50

0 50 /s

° 0 q -50 0 10 20 30 40 50 60 time (s)

(b) State trajectories

104 3.8

3.7

Height (ft) 3.6 0 0.5 1 1.5 2 2.5 North (ft) 104 (c) Longitudinal position trajectory

Figure 6: Pugachev’s Cobra maneuver using switching BS method between C1 and C2 controllers

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American Institute of Aeronautics and Astronautics VII. Lyapunov Based Numerical Recoverability Analysis and Comparison Between Controllers

In this section, we are going to analyze and compare the controllers’ recoverability after finishing the Cobra Maneuver. In the previous section, we have shown that while executing a typical Pugachev’s Cobra maneuver, in order to get back to steady-level flight, taking advantage of VT −θ transition mode to recover the attitude and airspeed seems more logical than directly switching to VT − γ controller. In this section, for the sake of justification, we aim to present a more comprehensive and in-depth analyzation on the recoverability and stability of these controllers. The term ”recoverability” of the aircraft from an abnormal orientation, like vertical nose-up position, can be defined in different ways. Each stable state trajectory settles to its final value in different settling time. In this section, we will introduce a numerical Lyapunov based recoverability analysis to compare the controllers and prove the need to perform switching between controllers. When the Cobra maneuver finishes, it’s desired to recover the airspeed and flight path angle in the shortest time, in order to achieve the steady-level flight. Obviously recovering time and quality is dependent on the last state vector value before switching to the intended controller. This value, in fact, is the initial point for the next controller. Starting from different initial points will result in a different stabilizing procedure, time and quality. Therefore, it seems wise to compare the controllers in recovering the aircraft starting from a specific initial point. The characteristics that are going to be compared are Lyapunov recovery time, accumulated T elevator saturation time, and the aircraft instability. Assume state vector X = [ x1 x2 ... xn ] is going to be stabilized by the controller, thus, let us define the state tracking error vector as

Z = X − Xref (77)

and positive Lyapunov function: 1 V = ZT QZ (78) 2 then it’s time derivative can be calculated: V˙ = ZT QZ˙ (79) where Q is a positive definite matrix. It’s well-known that V˙ < 0 indicates the stability of the system. Here we aim to perform a real-time calculation of V and V˙ . It’s possible to set a reliable Q matrix to normalize V and V˙ with respect to maximum acceptable tracking error and maximum acceptable stat derivative. This is feasible by setting  2  2 0 ... 0 z1,max  2   0 2 ... 0   z2,max  Q =   (80)  :: ... 0   2  0 0 ... 2 zn,max

where zi,max is the maximum acceptable error value for xi in steady state mode. A controller is assumed recoverable if both of these conditions hold for a specific period of time (Tmax)

V < 1 (81)

V˙ < V˙threshold (82) where   x˙ 1,max

h 1 1 1 i x˙ 2,max  V˙threshold = 2 ...   (83) z1,max z2,max zn,max    ...  x˙ n,max Now let us consider both C1 and C4 in stabilizing the aircraft for initial flight path angles in the range of ◦ ◦ ◦ ◦ ◦ ◦ −90 ≤ γ ≤ 50 and initial angle of attacks for −25 ≤ α ≤ 55 . Selecting zγ,max = 10 andγ ˙ max = 1 and Tmax = 5s, Lyapunov based γ recovering time for both controllers can be compared in Fig. 7.a. It can be seen that the transition mode controller beats the level flight controller especially in high initial flight path angles. Replying the same procedure for different initial conditions in θ − α plane results in Fig. 7.b. Both

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American Institute of Aeronautics and Astronautics 20 20

15 15

10 (s) s T s 10 T 5

0 5 50

0 100 50 0 0 ° -50 -50 -100 -50 (deg) 0 50 °

(a) Different initial γ values (b) Different initial θ − α values

Figure 7: Lyapunov based recovery time for VT − γ (red) and VT − θ (blue) controllers for different initial conditions controllers recovers the system on θ = α line on less than a few seconds. However, as the initial condition deviates from this line, the domination of the VT − θ controller becomes more evident. It’s worth nothing to compare two controllers without considering the actuator saturation. Fig. 8 reveals the accumulated elevator saturation time for both controllers while stabilizing the aircraft starting from different initial conditions in α−θ plane. This is a very important comparison between the controllers. VT −θ transition controller successes to stabilize the aircraft for most of the initial conditions without saturating the elevator. In contrast with VT − γ controller in which considerable amount of elevator saturation occurs, especially in the initial high angle of attacks and high pitch angles, which is the exact case in switching from Cobra maneuver to steady-level flight. This is an important result from the safety point of view. Because long-term actuator saturation may produce abnormal and unexpected moments which may result in instability of the actual aircraft.

30

20

10 saturation (s) e 0 20 0 20 40 -20 -20 0 ° °

Figure 8: Accumulated elevator saturation time for VT − γ (red) and VT − θ (blue) controllers for different initial α and θ values

Let us step further, and separate the scenarios as recoverable and unrecoverable ones. In this study, a

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American Institute of Aeronautics and Astronautics scenario is called recoverable if it holds all of the following conditions for s specific period of time (Tmax):   Tγ < Tγ,threshold   T < T  v v,threshold  V > 0 T (84) ◦  |α| < 170   |γ| < 100◦   |q| < 180◦/s in which flight path angle and speed recovery time, Tγ and Tv, are the time in which aircraft enters to conditions defined in Eq. 81 and Eq. 82 and stays still for Tmax seconds. Note that maximum tracking 2 error and maximum acceleration for VT is chosen as zv,max = 50 (ft/s) and V˙T,max = 10 (ft/s ). Finally by selecting Tγ,threshold = 5s and Tv,threshold = 30s we are all set to plot the Lyapunov based region of recoverability (ROR) of both controllers and make a comparison between them. Conditions in Eq. 84 are far simpler to comprehend than they appear. They, intuitively, indicate that a scenario is called recoverable if the aircraft attitude is in an acceptable form and aircraft’s airspeed and flight path angle recovers in less than 30 and 5 seconds, respectively. Following these definitions, ROR for both controllers are plotted in Fig. 9.

recoverable unrecoverable recoverable unrecoverable 150 150

100 100

50 50 ° 0 ° 0

-50 -50

-100 -100

-150 -150 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 ° °

(a) VT − γ controller (b) VT − θ controller

Figure 9: Lyapunov based ROR

It is evident that ROR of VT − θ controller is bigger than VT − γ controller. In addition, ROR of VT − γ controller does not include the region of execution of the Pugachev’s Cobra maneuver. This can be seen in Fig. 10 in which α − γ trajectory of a typical Pugachev maneuver is plotted. This fact also shows that it’s possible to switch between controllers with various RORs to stabilize the aircraft and be able to perform aggressive maneuvers which cannot be executed using only one control law. Therefore, it may be possible to take the boundaries of RORs as the switching surfaces to stabilize the aircraft while state trajectories move over α − θ plane. As discussed in Eq. 78 it is possible to define a single Lyapunov function that includes both VT and γ, or even all states. Then correspondingly, define a Tthreshold as the recoverability time threshold similar to Eq. 84 and plot the Lyapunov based ROR same as Fig. 9. However, in this final stage of the recoverability analysis, we want to plot ROR of the controllers for different values of Tthreshold. Taking advantage of this approach, three-dimensional ROR for both controllers can be seen in Fig. 11. Accordingly, it can be claimed that VT − θ transition controller is more reliable than VT − γ controller in recovering the aircraft from abnormal conditions, e.g. final stage of Cobra maneuver.

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American Institute of Aeronautics and Astronautics recoverable unrecoverable recoverable unrecoverable 150 150

100 100

50 50 Pugachev's

° Pugachev's ° 0 Cobra 0 trajectory Cobra region of execution

-50 -50

-100 -100

-150 -150 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 ° °

(a) VT − γ controller (b) VT − θ controller

Figure 10: Lyapunov based ROR and Pugachev’s Cobra maneuver trajectory

recoverable unrecoverable recoverable unrecoverable 10 10

8 8 (s) (s) 6 6

4 4 threshold threshold T T 2 2

0 0 100 50 100 0 0 0 ° ° 0 50 -100 -50 ° -100 -50 °

(a) VT − γ controller (b) VT − θ controller

Figure 11: Lyapunov based ROR for different values for recovery threshold time (Tthreshold)

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American Institute of Aeronautics and Astronautics VIII. Backstepping Controller Design for Rolling Circle Maneuver

According to the characterizations of Rolling Circle maneuver discussed in section IV.B, in order to execute this maneuver we should consider the full model of the aircraft given in section II. In contrast with the Pugachev’s Cobra maneuver, Rolling Circle can be performed using only one controller. But we will see that according to the under-actuated characterization of the system and actuator constraints, it may not be possible to stabilize all states while performing the maneuver. It will be shown that a state-based switching can stabilize the system while executing the maneuver. According to the EOM of the aircraft, engine thrust (T ) and rolling moment (L¯) are affine in Eqs. 1 and 5, thus, constant speed and roll rate throughout the maneuver can be performed using a decoupled VT − p controller. On the other hand, circular position trajectory on the lateral plane can be executed by holding the altitude (γ = 0) and rate of change of course angle (χ ˙ = constant) at constant values. While maneuvering ˙ in constant airspeed, this can be done by holding θ = θtrim and ψ = constant. This is feasible using a BS controller which takes the advantage of pitching and yawing moments as controller inputs to produce the rhythmic pitch and yaw rates while the aircraft rolls. This is not as trivial as it looks. Assume the condition where roll angle is close to zero. Thus, γ (or θ) can be kept at zero (at θtrim) by producing proper lift force (or M¯ ). Similarly, χ (or ψ) can be stabilized using side force (or ψ). Now, consider the situation where roll angle is close to 90◦. As the terms L cos φ in Eq. 8 and the term Y cos φ in Eq. 9 disappears, the controller loses its controllability in stabilizing the system and following the reference inputs. It will be shown that using a proper state-based switching law, the controller is capable of dealing with this uncontrollability nature of the aircraft dynamics.

A. VT − p Controller Let us firstly define the tracking error coordinates: " # " # VT VT zv,p = − (85) p p ref and control input vector: " # T Uv,p = (86) L¯

from Eqs. 1 and 5 it can be seen that VT and p dynamics are in canonical form described in Eq. 25. Therefore, nonlinear stabilizing controller can be calculated as

−1 Uv,p,des = Gv,p(zv,p − fv,p) (87)

where

" cos α cos β # m 0 Gv,p = (88) 0 c3

" 1 # m (−D + mg1) fv,p = (89) (c1r + c2p)q + c4(N¯ + hEq)

note that, the ROA of the controller is located in the flight envelope (α, β 6= 90◦). In addition, neither ◦ ◦ α = 90 nor β = 90 may ever happen in Rolling Circle maneuver. A procedure to how to pass from Tdes and L¯des to throttle, δth, and aileron, δa deflections is presented in control allocation block in the appendix.

B. θ − ψ Controller Because of the coupling dynamics between pitch and yaw angles, two decoupled controllers may never stabilize the system. Remember that aircraft constantly rolls, therefore, stabilizing θ via lift force and controlling ψ

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American Institute of Aeronautics and Astronautics using side force will fail to execute the maneuver. Equations of motions for pitch and yaw angles can be written as " # " #" # θ˙ cos φ − sin φ q = (90) ˙ sin φ cos φ ψ cos θ cos θ r " # q = Gθ,ψ (91) r

which is in the canonical form given in Eq. 25. Therefore, the stabilizing nonlinear controller can be found as −1 Uθ,ψ,des = Gθ,ψCθ,ψzθ,ψ (92)

where Cθ,ψ > 0 is a diagonal control gain matrix and " # " # θ θ zθ,ψ = − (93) ψ ψ ref " # q Uθ,ψ = (94) r (95)

Note that, θ = 90◦ is a singular condition for this controller which is not in flight envelope and may never happen while executing Rolling Circle maneuver. Apart from that, Gθ,ψ is non-singular for every values of φ and α. It’s worth mentioning that, in contrast with controllers in previous sections, off-diagonal elements of Gθ,ψ are not necessarily zero. This indicates that θ and ψ controllers are not decoupled. Thus, pitch and yaw rates are constantly being used to control θ and ψ interchangeably. Intuitively, this seems logical as mentioned in first part of this section. The inner loop controller for q and r can be found as

−1 Uq,r,des = Gq,r(Cq,rzq,r − fq,r) (96)

in which Cq,r > 0 is a diagonal gain matrix and zq,r is the error vector. Uq,r, Gq,r and fq,r can be derived from Eqs. 4 and 7.

C. Maneuver Execution Using Switching Control According to characteristics of the Rolling Circle maneuver discussed in section IV, it seems feasible to achieve the constant airspeed and roll rate using VT − p controller. The reference input to this controller can be chosen to be " # " # V 800 (ft/s) T = (97) p 10 (deg/s) ref Now, in order to follow a circular lateral trajectory we have two options available:

Option 1 : Using a fixed θ − ψ controller One option is to use θ −ψ controller to hold the altitude and course angle rate at a constant value by holding pitch angle close to trimmed vale and rate of change of yaw angle at a constant value. Therefore reference inputs can be selected as " # " # θ 3.1519 (deg) = (98) ψ 10t (deg) ref the executed maneuver visualization together with the state trajectories are plotted in Fig. 12 and Fig. 13. It seems that the controller was successful in executing the maneuver. However, the aircraft lost considerable altitude during the maneuver. This is according to the fact that the reference input to pitch angle is a fixed value while trimmed pitch value changes constantly. Also note that, by taking a look at actuator deflections, the extensive skills that are required to perform the maneuver by actual pilot becomes evident. Now if it’s

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American Institute of Aeronautics and Astronautics required to perform the maneuver with fast roll rates, it may be impossible for a human to manipulate the control stickers fast enough to perform the maneuver successfully, but the presented controllers are able to execute such fast aggressive maneuvers autonomously.

Figure 12: Performing Rolling Circle maneuver using VT − p and θ − ψ controllers

10 200

0 ° 0 ° -200 -10 0 20 40 60 80 -20 10

0 20 40 60 80 ° 0 400 -10 0 20 40 60 80

° 200 200 ° 0 0 -200 0 20 40 60 80 0 20 40 60 80 Time (s) Time (s) (a) Flight path angle and course angle (b) Euler angles

20 900 /s ° 0 ft/s T 800 p V -20 700 0 20 40 60 80 0 20 40 60 80 10 10 ° /s

° 0 0 q -10 -10 0 20 40 60 80 0 20 40 60 80 5 20 ° /s

° 0 0 r -5 -20 0 20 40 60 80 0 20 40 60 80 Time (s) Time (s) (c) Angular rates (d) Airspeed, angle of attack and side slip angle

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American Institute of Aeronautics and Astronautics 10500 10000

T 5000 10000 0 0 20 40 60 80 0

9500 ° e -2 -4 0 20 40 60 80 9000 5 Height (ft) ° a 0 -5 8500 0 20 40 60 80 50 ° r 0 8000 -50 0 20 40 60 80 0 20 40 60 80 Times (s) Time (s) (e) Height (f) Engine thrust and control surfaces

20000

15000

10000 East (ft)

5000

0 -1 -0.5 0 0.5 1 1.5 North (ft) 104 (g) North-east position

Figure 13: Trajectories while performing Rolling Circle maneuver using VT − p and θ − ψ controllers.

Option 2 : Using switching control It’s well-known that moment produced by rudder is far smaller than the elevator. This, in fact, justifies the altitude loss in option 1. For instance, when φ = 90◦ rudder is used to stabilize θ and elevator is used for ψ. The produced yawing moment by rudder is not sufficient to stabilize the aircraft on this orientation so rudder saturates and aircraft loses altitude accordingly. This problem may be tackled by performing the maneuver with high roll rates. However, according to rate limits and time constants in actuators, they may not be able to handle such fast maneuver. The basic idea behind this option is to accelerate roll rate whenever rudder is the dominant actuator for pitch stability. This can be implemented by defining two controllers C1 and C2. Both controllers stabilize the error dynamics of VT , p, θ and ψ. The only difference between them is the ◦/s ◦/s roll rate reference value. pref = 10 for C1 and pref = 25 for C2. As the aircraft constantly rolls, so switching occurs in a rhythmic manner. As the switching time depends on the φ value, so it can be called state-based switching law and can be described as follows: ( 20◦/s − 135◦ ≤ φ ≤ −45◦ or 45◦ ≤ φ ≤ 135◦ pref = (99) 25◦/s otherwise

Fig. 14 shows a graphical explanation of switching between C1 and C2 as the aircraft rolls. The maneuver is simulated in MATLAB and the responses are plotted in Fig. 15 and Fig. 16. In comparison with the controller in option 1, this switching controller was able to deal with the altitude loss and rudder saturations up to an acceptable degree.

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American Institute of Aeronautics and Astronautics �1 �1 �2 �2 �1 �1 �2 �2 �1

Figure 14: Switching law for performing Rolling Circle maneuver

Figure 15: Performing Rolling Circle maneuver using switching BS controllers

10 200

0 ° 0 ° -200 -10 0 20 40 60 80 -20 10

0 20 40 60 80 ° 0 400 -10 0 20 40 60 80

° 200 200 ° 0 0 -200 0 20 40 60 80 0 20 40 60 80 Time (s) Time (s) (a) Flight path angle and course angle (b) Euler angles

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American Institute of Aeronautics and Astronautics 40 900 /s T

° 20 800 V p 0 700 0 20 40 60 80 0 20 40 60 80 10 10 ° /s

° 0 0 q -10 -10 0 20 40 60 80 0 20 40 60 80 10 20 ° /s

° 0 0 r -10 -20 0 20 40 60 80 0 20 40 60 80 Time (s) Time (s) (c) Angular rates (d) Airspeed, angle of attack and side slip angle

10400 10000

T 5000 10200 0 0 20 40 60 80 0

10000 ° e -2 -4 9800 0 20 40 60 80 5 ° a Height (ft) 9600 0 -5 0 20 40 60 80 9400 50 ° r 0 9200 -50 0 20 40 60 80 0 20 40 60 80 Times (s) Time (s) (e) North position vs height (f) Engine thrust and control surfaces

20000

15000

10000 East (ft)

5000

0 -1 -0.5 0 0.5 1 1.5 North (ft) 104 (g) North-east position

Figure 16: Trajectories while performing Rolling Circle maneuver using switching BS controllers

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American Institute of Aeronautics and Astronautics IX. Conclusion and Future Works

In this study, we have studied the capabilities of switching control and Lyapunov based backstepping nonlinear control method in order to execute supermaneuvers. Lyapunov’s direct method is used to design a globally stabilizing and reference follower controller in every iteration of backstepping method. Because of the small number of research on supermaneuvers that push the aircraft limits and challenge the underactuated features of the vehicle, we have chosen to execute Pugachev’s Cobra and Rolling Circle supermaneuvers using switching nonlinear backstepping method. Using a numerical Lyapunov based recoverability analysis of the controllers, it has been shown that it’s practical to calculate the boundaries of the ROR of a controller family and use these borders as the switching surfaces. In the proposed simulation for Pugachev maneuver, a time- based switching was used to bring the proper controller into the loop to perform the maneuver, in addition, to make the states stay in the ROR. Finally, instead of time-based switching, a state-based switching law was used to stabilize the aircraft while performing the Rolling Circle maneuver. In other words, it has been shown that, it is possible to stabilize the aircraft using a proper switching between a family of controllers with different RORs. As for the future work, theoretical analysis of the proposed switching methods instead of simulation results can be presented. This may open the way for defining a more general form of the switching laws for a family of maneuvers.

Appendix: Conversion From Desired Moments to Control Surface Deflections (Control Allocation)

Through the paper, engine thrust and aerodynamic moments are considered as control inputs. In order to control the aircraft, it’s desired to achieve the control effort required in control law using a proper control 31 deflections (δth, δa, δe, δr). Engine model is fully explained in Sonneveldt. Therefore, it’s possible to inverse the engine model and produce the desired thrust by calculating the required throttle. This may not be feasible for actual aircraft but it works fine with the simulation. An overview of the total aerodynamic moment coefficients is given in table 2. These coefficients are used to sum the various aerodynamic contributions to the given force or moment. It can be immediately seen that the desired moments calculated in controllers, e,g, Eq. 31, can be transferred to the desired moment coefficient values using Eqs. 19 - 21: 1 C = L¯ (100) L,T,des qSB¯ 1 C = M¯ (101) M,T,des qS¯ c¯ 1 C = N¯ (102) N,T,des qSB¯ Now as aileron and rudder deflections are affine in these moment coefficients, as seen in table 2, the same inverting method can be used to calculated the reference δa,des and δr,des values.

Table 2: F-16 moment coefficients

Rolling-moment coefficient CL,T = f(α, β) + g(α, β)δa + h(α, β)δr

Pitching-moment coefficient CM,T = f(α, β, δe)

Yawing-moment coefficient CN,T = f(α, β) + g(α, β)δa + h(α, β)δr

Calculating the desired elevator deflection from the reference pitching moment is slightly more compli- cated than lateral moments. The reason for this is that δe is not affine in CM,T . In MATLAB simulations we used a simple data lookup table inverting algorithm to do so. According to the fact that, at any iteration α and β angles are available from state estimator, it’s possible to form a new one-dimensional lookup table consisting of CM,T and δe for a fixed α and β values. Finally, using 1-D data interpolation, δe,des can be found.

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American Institute of Aeronautics and Astronautics Acknowledgments

This work is supported by Scientific and Technological Research Council of Turkey (Turkish: TUB¨ ITAK˙ ) under the grant agreement 315M341.

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