Optimal control of two-commercial aircraft dynamic system during approach

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system during approach

Fulgence Nahayo, Salah Khardi, Mounir Haddou

To cite this version:

Fulgence Nahayo, Salah Khardi, Mounir Haddou. Optimal control of two-commercial aircraft dynamic system during approach. General Mathematics Notes, 2011, 3 (2), pp. 27-49. �hal-00987121�

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www.i-csrs.org

Available free online at http://www.geman.in

Optimal control of two-commercial aircraft dynamic system during approach. The noise

levels minimization

F. Nahayo^1,2, S. Khardi² and M. Haddou³

1 University Claude Bernard of Lyon 1 (France) and University of Burundi (Burundi), E-Mail: [email protected]

2 The French Institute of Science and Technology for Transport, Development and Networks - Transport and Environment Laboratory, Lyon - France, E-Mail: [email protected]

3 University of Orl´eans. CNRS-MAPMO, France, E-Mail: [email protected] Received: December 2010, Accepted ... 2011

Abstract

This paper aims to reduce noise levels of two-aircraft landing simultaneously on approach. Constraints related to stability, performance and ﬂight safety are taken into account. The problem of optimal control is described and solved by a Sequential Quadratic Programming numerical method ’SQP’ when globalized by the trust region method. By using a merit function, a sequential quadratic pro- gramming method associated with global trust regions bypasses the non-convex problem. This method used a nonlinear interior point trust region optimization solver under AMPL. Among several possible solutions, it is shown that there is an optimal trajectory leading to a reduction of noise levels on approach.

Keywords: TRSQP algorithm, optimal control problem, aircraft noise lev- els, AMPL programming, KNITRO.

2000 MSC No: Use appropriate MSC Nos.

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1 Introduction

The aim of this work is the development of a theoretical model of noise optimization while maintaining a reliable evolution of the flight procedures of two commercial aircraft on approach. These aircraft are supposed to land successively on one runway without conflict [37]. It is all about the evolution of flight dynamics and minimization of noise for two similar commercial aircraft landing taking into account the energy constraint. This model is a non-linear and non-convex optimal control. It is governed by a system of ordinary non-linear differential equations [12]. The 3-D movement of the two planes is described by a system depending on ordinary non-linear differential equations with mixed constraints. The function to be minimized is the integral describing the overall level of noise emitted by the two aircraft on approach and collected on the ground. We take into account constraints related to joint stability, performance and flight safety.

The problem of optimal control is described and solved by a Trust Region Sequential Quadratic Programming method ’TRSQP’[3, 5, 10, 11, 30]. By using a merit function, a sequential quadratic programming method associated with global trust regions bypasses the non-convex problem. This method is established by following a tangent quadratic problem obtained from the opti- mality conditions of Karush-Khun-Tucker applied to the problem considering the objective function as the merit function.

The TRSQP methods are suggested as an option by a Nonlinear Interior point Trust Region Optimization solver ’KNITRO’ [33] under A Mathemat- ical Programming Modeling Language ’AMPL’[17, 27]. The global convergence properties are analyzed under diﬀerent assumptions on the approximate Hessian. Additional assumptions on the feasibility perturbation technique are used to prove quadratic convergence to points satisfying second-order suﬃcient conditions.

Details of the two-aircraft ﬂight dynamic, the noise levels, the constraints, the mathematical model of the two-aircraft acoustic optimal control problem and the trust region sequential quadratic programming method processing are presented in section 2, 3 and 4 while the numerical experiments are presented in the last section.

2 Mathematical Modelization

The motion of each aircraft Ai, i:= 1,2 is three dimensional analyzed with 3 frames: the landmark (O,−→X₁,−→Y ₁,−→Z₁), the aircraft frame (G_i,−→X_Gi,−→Y _Gi,−→Z_Gi) and the aerodynamic one (Gi,−→

Xai,−→ Y ai,−→

Zai) wherei:= 1,2 [6]. The transition between these three frames is shown easily [7]. In general, the equations of

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motion of each aircraft are summarized as:

−→F_ext_i − ^dm_dtⁱ−→V_a_i = ^mⁱ^d

−

→V _ai −M→_ext dt

Gi = _dt^d[I_G_i−→Ω_i]

d−→ X^o

dt = ^d−→ X¹

dt +−→Ω₁₀× −→X

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The index i = 1, 2 reﬂects the aircraft. In the system above, −→

Fext_i represents the external forces acting on the aircraft, m_i(t) the mass of the aircraft, v_i the airspeed of aircraft, −→

Mext_Gi the moments of each aircraft, J(Gi, Ai) the inertia matrix,^d−→X^o

dt is the derivative with respect to time of the vector X in vehicle-carried normal Earth frame R_O, ^d−→X¹

dt is the derivative with respect to time of the vector X in frame R₁, Ω_i the angular rotation of the aircraft and Ω10 is the angular velocity of the frame R1 relative to the frame RO. After transformations and simpliﬁcations, the system (1) becomes:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

V˙_a_i = _m¹

i[−m_igsinγ_a_i− ¹₂ρSV_a²

iC_D

+(cosα_a_icosβ_a_i+sinβ_a_i +sinα_a_icosβ_a_i)F_x_i − ^dm_dtⁱu_i−m_iΔAⁱ_u] β˙_a_i = _m¹

iV_ai[m_igcosγ_a_isinμ_a_i+ ¹₂ρSV_a²_iC_y_i

+[−cosα_a_isinβ_a_i +cosβ_a_i −sinα_a_isinβ_a_i]F_y_i − ^dm_dtⁱv_i−m_iΔAⁱ_v]

˙

α_a_i = _m ¹

iV_aicosβ_ai[m_igcosγ_a_icosμ_a_i− ¹₂ρSV_a²_iC_L_i+ [−sinα_a_i +cosα_a_i]F_z_i

−^dm_dtⁱw_i−m_iΔAⁱ_w]

˙

p_i = _AC^C₋_E₂{r_iq_i(B−C)−Ep_iq_i +¹₂ρSlV_a²_iC_l_i +₂

j=1Fj[y^b_M_i_jcosβm_ijsinαm_ij−z_M^b _i_jsinβm_ij]} +_AC^E₋_E₂{p_iq_i(A−B)−Er_iq_i+¹₂ρSlV_a²

iC_ni +₂

j=1F_j[x^b_M

ijsinβ_m_i_j −y_M^b

ijcosβ_m_i_jcosα_m_i_j]}

˙

q_i = _B¹{−r_ip_i(A−C)−E(p²_i −r_i²) +¹₂ρSlV_a²_iC_mi +2

j=1F_j[z_M^b _i_jcosβ_m_i_jcosα_m_i_j−x^b_M_i_jcosβ_m_i_jsinα_m_i_j]}

˙

ri = _AC^E₋_E₂{riqi(B−C) +Epiqi+ ¹₂ρSlV_a²_iCl_i

+₂

j=1F_j[y^b_M

ijcosβ_m_i_jsinα_m_i_j−z_M^b

ijsinβ_m_i_j]} +_AC^A₋_E₂{p_iq_i(A−B)−Er_iq_i+¹₂ρSlV_a²

iC_ni +₂

j=1F_j[x^b_M

ijsinβ_m_i_j −y_M^b

ijcosβ_m_i_jcosα_m_i_j]} X˙_G_i =V_a_icosγ_a_icosχ_a_i +u_w

Y˙_G_i =V_a_icosγ_a_isinχ_a_i +v_w Z˙_G_i =−V_a_isinγ_a_i +w_w

φ˙_i =p_i+q_isinφ_itanθ_i+r_icosφ_itanθ_i θ˙_i =q_icosφ_i−r_isinφ_i

ψ˙_i = ^sinφ_cosθⁱ

iq_i+^cosφ_cosθⁱ

ir_i

˙

mi =−2.01×10⁻⁵ ^(Φ⁻^μ⁻

K ηc)√

√ Θ

5η_n(1+η_tfλ)

G+0.2M²^ηd

ηtfλ−(1−λ)M 1

2ρSV_ai²CD

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where j means the engine index, the expressionsA =I_xx, B =I_yy, C =I_zz, E = I_xz are the inertia moments of the aircraft,ρis the air density, S is the aircraft reference area, l is the aircraft reference length, g is the acceleration due to gravity, C_D =C_D0+kC_L² is the drag coefficient, C_yi =C_yββ+C_yp^pl_V +C_yr^rl_V + C_{Y δ}_lδ_l+C_{Y δ}_nδ_nis the lateral forces coefficient, C_Li =C_Lα(α_a−α_a0)+C_Lδ_mδ_m+ C_LMM+C_Lq^q_V^bâ^l is the lift coefficient, C_li =C_lββ+C_lp^pl_V +C_lr_V^rl+C_lδ_lδ_l+C_lδ_nδ_n is the rolling moment coefficient, C_mi =C_m0+C_mα(α−α₀) +C_mδ_mδ_m is the pitching moment coefficient,C_ni =C_nββ+C_np^pl_V +C_nr^rl_V +C_nδ_lδ_l+C_nδ_nδ_nis the yawing moment coefficient, (x^b_{M ij}, x^b_{M ij}, x^b_{M ij}) is the position of the engine in the body frame, F = (F_xi, F_yi, F_zi) is the propulsive force, V_ai = (u_i, v_i, w_i) is the aerodynamic speed, (ΔAⁱ_u,ΔAⁱ_v,ΔAⁱ_w) is the complementary acceleration, (u_w, v_w, w_w) is the wind velocity, β_mij is the yaw setting of the engine and α_mij is the pitch setting of the engine. The mass change is reflected in the aircraft fuel consumption as described by E. Torenbeek [36] where the specific consumption is

C_SR_i = 2.01×10⁻⁵ (Φ−μ−^K_η_cⁱ)√ Θ

5η_n(1 +η_tf_iλ)

G_i+ 0.2M_i²_η^η^di

tfiλ−(1−λ)M_i with the generator function G:

G_i = (Φ− ^K_η_cⁱ)(1− ^1.01

η_i^ν−1^ν (K_i+μ_i)(1−_Φ_ηcηt^Ki )

) K_i =μ_i(c^ν−1^ν −1)

μ_i = 1 + ^ν⁻₂¹M_i²

The Nomenclature of engine performance variables are given by G the gas generator power function, G0 the gas generator power function (static, sea level), K_i the temperature function of compression process, M_i the flight Mach number, T4 the turbine Entry total Temperature, T0 the ambient temperature at sea level, T the flight temperature, while the nomenclature of engines yields is η_c = 0.85 the isentropic compressor efficiency, η_d_i = 1−1.3(^0.05

Re¹5)²(^0.5_M

i)²_D^L, the isentropic fan intake duct eﬃciency, L the duct length, D the inlet diameter, Re the reynolds number at the entrance of the nozzle, η_f_i = 0.86 − 3.13 × 10⁻²M_i the isentropic fan eﬃciency, η_i = ^1+η^di^γ−1² ^Mⁱ²

1+^γ−1₂ M_i² the gas Gen- erator intake stagnation pressure ratio, η_n = 0.97 the isentropic efficiency of expansion process in nozzle, η_t = 0.88 the isentropic turbine efficiency η_tf_i = η_tη_f_i, _c the overall pressure ratio (compressor), ν the ratio of specific heats ν = 1.4, λ the bypass ratio, μ_i the ratio of stagnation to static temperature of ambient air, Φ the nondimensional turbine entry temperature Φ = ^T_T⁴ and Θ the relative ambient temperature Θ = _T^T

0. The expressions α_ai(t), β_ai(t), θ_i(t), ψ_i(t), φ_i(t), V_a_i(t), X_G_i(t), Y_G_i(t), Z_G_i(t), p_i(t), q_i(t), r_i(t), m_i(t)

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are respectively the attack angle, the aerodynamic sideslip angle, the inclination angle, the cup, the roll angle, the airspeed, the position vectors, the roll velocity of the aircraft relative to the earth, the pitch velocity of the aircraft relative to the earth, the yaw velocity of the aircraft relative to the earth and the aircraft mass.

Transforming the system (2) in state function, one has:

dy_i(t)

dt =f_i(y_i(t), u_i(t)), i= 1,2 (3)

where the state vector is:

y_i(t) : [t₀, t_f]−→R¹³

y_i(t) = (α_ai(t), β_ai(t), θ_ai(t), ψ_ai(t), φ_i(t), V_a_i(t), X_G_i(t), Y_G_i(t), Z_G_i(t), p_i(t), q_i(t), r_i(t), m_i(t))

(4) The control vector is

u_i(t) : [t₀, t_f]−→R⁴

t−→u_i(t) = (δ_l_i(t), δ_m_i_(t), δ_n_i(t), δ_x_i(t)) (5)

where the expressions δ_l_i(t), δ_m_i_(t), δ_n_i(t), δ_x_i(t) are respectively the roll control, the pitch control, the yaw control and the thrust one. The dynamics relationship can be written as:

˙

y_i(t) =f_i(y_i(t), u_i(t), t),∀t ∈[0, T], y_i(0) =y_i0 (6)

The angles γ_a_i(t), χ_a_i(t), μ_a_i(t) corresponding respectively to the aerodynamic climb angle (air-path inclination angle), the aerodynamic azimuth (air-path track angle) and the air-path bank angle (aerodynamic bank angle) are not taken as state in this model.

To simplify the model, the atmosphere standards conditions are considered. The engine angles, the complementary acceleration and the aerodynamic sideslip angle are negligible because the wind is constant and there is no engine failure. With some complex mathematical transformations, the dynamic

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system (2) becomes:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

V˙_a_i = _m¹

i[−m_igsinγ_a_i −¹₂ρSV_a²

iC_D + (cosα_a_i+sinα_a_i)F_x_i −m˙_iu_i]

˙

α_a_i = _m ¹

iV_aicosβ_ai[m_igcosγ_a_icosμ_a_i− ¹₂ρSV_a²

iC_L_i +[cosα_a_i−sinα_a_i]F_z_i −m˙_iw_i]

˙

p_i = _AC^C₋_E₂{r_iq_i(B −C)−Ep_iq_i+¹₂ρSlV_a²_iC_l_i} +_AC^E₋_E₂{p_iq_i(A−B)−Er_iq_i+ ¹₂ρSlV_a²_iC_ni

˙

q_i = _B¹{−r_ip_i(A−C)−E(p²_i −r_i²) + ¹₂ρS_ilV_a²

iC_mi},

˙

r_i = _AC^E₋_E₂{r_iq_i(B −C) +Ep_iq_i+¹₂ρSlV_a²

iC_l_i +_AC^A₋_E₂{p_iq_i(A−B)−Er_iq_i+ ¹₂ρSlV_a²_iC_ni} X˙_G_i =V_a_icosγ_a_icosχ_a_i

Y˙_G_i =V_a_icosγ_a_isinχ_a_i Z˙_G_i =−V_a_isinγ_a_i

φ˙i =pi+qisinφitanθi+ricosφitanθi

θ˙i =qicosφi−risinφi

ψ˙_i = ^sinφ_cosθⁱ

iq_i+ ^cosφ_cosθⁱ

ir_i

˙

m_i =−2.01×10⁻⁵ ^(Φ⁻^μ⁻

K ηc)√

√ Θ

5ηn(1+η_tfλ

G+0.2M²^ηd

ηtfλ−(1−λ)M 1

2ρSV_ai²C_D

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By the combination of this system with the aircraft control, one has the two- aircraft dynamic ﬂight model as shown in (6). The state vector is

y_i(t) : [t₀, t_f]−→R¹²

y_i(t) = (α_ai(t), θ_ai(t), ψ_ai(t), φ_ai(t), V_a_i(t), X_G_i(t), Y_G_i(t), Z_G_i(t), p_i(t), q_i(t), r_i(t), m_i(t))

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This will be added to the cost function and constraint function for the aircraft optimal control problem as shown in the following paragraphs.

The objective function model. In this paper, the considered cost function is the Sound Exposure Level ’SEL’[1, 22, 28]:

SEL= 10log 1 t_o

t

10^0.1L^A1,dt^(t)dt

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whereto is the time reference taken equal to 1s andt the noise event interval.

[t₁₀, t_1f] and [t₂₀, t_2f] are the respective approach intervals for the ﬁrst and the

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second aircraft, the objective function is calculated as:

SEL1 = 10log 1

to

_t₂₀

t₁₀ 10^0.1L^A1,dt^(t)dt

, t∈[t10, t20] SEL₁₂ =SEL₁₁⊕SEL₂₁

= 10 log[_t¹

o

t_1f

t₂₀ 10^0.1L^A1,dt^(t)dt+ _t¹

o

t_1f

t₂₀ 10^0.1L^A2,dt^(t)dt], t∈[t₂₀, t_1f] SEL₂ = 10 log

1 to

_t_2f

t_1f 10^0.1L^A2,dt^(t)dt

, t∈[t_1f, t_2f] SEL_G = ^(t²⁰⁻^t¹⁰^)SEL¹^⊕(^t1f−t₂₀)^SEL12⊕(^t2f−t_1f)^SEL2

t_2f−t₁₀

= 10 log{_t_2f₋¹_t₁₀[(t₂₀−t₁₀)t₂₀

t₁₀ 10^0.1L^A1^(t)dt +(t_1f −t₂₀)t_1f

t₂₀ 10^0.1L^A1^(t)dt+ (t_1f −t₂₀)t_1f

t₂₀ 10^0.1L^A2^(t)dt +(t_2f −t_1f)t_2f

t_1f 10^0.1L^A2^(t)dt,]}, t∈[t₁₀, t_2f]

(10) where SEL_G is the cumulated two-aircraft noise and the operator ⊕ means the acoustic adding. Expressions L_A1(t), L_A2(t) are equivalent and reﬂect the aircraft jet noise given by the formula [1, 20]:

L_A1(t) = 141 + 10 log ρ1

ρ w

+ 10 log Ve

c 7.5

+ 10 logs₁ +3 log

2s₁ πd²₁ + 0.5

+ 5 logτ₁ τ₂

+10 log

⎡

⎢⎢

⎢⎣

1−v₂ v₁

me

+ 1.2

1 + s₂v₂² s₁v₁²

4

1 + s₂

s₁ 3

⎤

⎥⎥

⎥⎦

−20 logR+ ΔV + 10 log ρ

ρ_ISA 2

c c_ISA

4

where v₁ is the jet speed at the entrance of the nozzle, v₂ the jet speed at the nozzle exit, τ1 the inlet temperature of the nozzle, τ2 the temperature at the nozzle exit, ρ the density of air, ρ₁ the atmospheric density at the entrance of the nozzle, ρ_ISA the atmospheric density at ground, s₁ the entrance area of the nozzle hydraulic engine, s2 the emitting surface of the nozzle hydraulic engine, d₁ the inlet diameter of the nozzle hydraulic engine, V_e = v₁[1− (V /v₁) cos(α_p)]^2/3 the eﬀective speed (α_p is the angle between the axis of the motor and the axis of the aircraft), R the source observer distance, w the exponent variable deﬁned by: w = 3(V_e/c)^3.5

0.6 + (V_e/c)^3.5 − 1, c the sound velocity (m/s), m the exhibiting variable depending on the type of aircraft: me = 1.1

s₂ s₁; s₂

s₁ < 29.7, me = 6.0; s₂

s₁ ≥ 29.7, the term ΔV =−15log(C_D(M_c, θ))−10log(1−M cosθ), means the Doppler convection

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when C_D(M_c, θ) = [(1 +M_ccosθ)² + 0.04M_c²], M the aircraft Mac Number, M_c the convection Mac Number: M_c = 0.62(v₁−V cos(α_p))c, θ is the Beam angle.

Formula above leads to the objective function J_G12(y(t), u(t), i = 1,2) =

tg(y(t), u(t), t, i= 1,2)dt.

Constraints. The considered constraints concern aircraft flight speeds and altitudes, flight angles and control positions, energy constraint, aircraft separation, flight velocities of aircraft relative to the earth and the aircraft mass:

1. The vertical separation given by Z_G₁₂ = Z_G₂ −Z_G₁ where Z_G₁, Z_G₂ are respectively the altitude of the ﬁrst and the second aircraft and ZG₁₂ the altitude separation.

2. The horizontal separationX_G₁₂ =X_G₁−X_G₂ [13, 14, 38] whereX_G₁, X_G₂ are horizontal positions of the ﬁrst and the second aircraft and their separation distance.

3. The aircraft speed V_a_i must be bounded as follows 1.3V_s ≤ V_a_i ≤ V_if whereVsis the stall speed,Vif is the maximum speed andVio = 1.3Vsthe minimum speed of the aircraftA_i [15, 36], the roll velocity of the aircraft relative to the earth p_i ∈ [p_i0, p_if], the pitch velocity of the aircraft relative to the earth qi ∈ [qi0, qif] and the yaw velocity of the aircraft relative to the earth r_i ∈[r_i0, r_if] .

4. On the approach, the ICAO standards and aircraft manufacturers re- quire ﬂight angle evolution as follows: attack angle α_a_i ∈ [α_io, α_if], the inclination angle θ_i ∈[θ_i0, θ_if] and the roll angle φ_i ∈[φ_io, φ_if].

5. The aircraft control δ(t) = (δ_l_i(t), δ_m_i_(t), δ_n_i(t), δ_x_i(t)) keeps still between the position δli0 and δlif for the roll control, δmi0 and δmif for the pitch control, δ_ni0 and δ_nif for the yaw control andδ_xi0 andδ_xif for the thrust.

6. The mass m_i of the aircraft A_i is variable: m_i0 < m_i < m_if, i = 1,2.

This constraint results in energy consumption of the aircraft [8, 24].

On the whole, the constraints come together under the relationship:

k_1i(y_i(t), u_i(t))≤0

k_2i(y_i(t), u_i(t))≥0 (11)

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where

k(t) :R¹²×R⁴ −→R¹⁶,

(y_i(t), u_i(t))−→k_i(y_i(t), u_i(t))

k_1i(y_i(t), u_i(t)) = (α_i(t)−α_if, θ_i(t)−θ_if, ψ_i(t)−ψ_if, φ_i(t)−φ_if,

V_a_i(t)−V_aif, X_G_i(t)−X_Gif, Y_G_i(t)−Y_Gif, Z_G_i(t)−Z_Gif, p_i(t)−p_if, q_i(t)−q_if, r_i(t)−r_if, δ_l_i(t)−δ_l_i_f, δ_m_i(t)−δ_mif, δ_n_i(t)−δ_nif, δ_x_i(t)−δ_xif, m_i(t)−m_if)

k_2i(y_i(t), u_i(t)) = (α_i(t)−α_i0, θ_i(t)−θ_i0, ψ_i(t)−ψ_i0, φ_i(t)−φ_i0,

V_a_i(t)−V_ai0, X_G_i(t)−X_Gi0, Y_G_i(t)−Y_Gi0, Z_G_i(t)−Z_Gi0, p_i(t)−p_i0, q_i(t)−q_i0, r_i(t)−r_i0, δ_l_i(t)−δ_l_i₀, δ_m_i(t)−δ_mi0, δ_n_i(t)−δ_ni0, δ_x_i(t)−δ_xi0, m_i(t)−m_i0).

The following values reﬂect the digital applications considered for the two- aircraft [1, 7, 8, 36].

Table of limit digital values for the two-aircraft in approach phase

Constraint denomination maximum value minimum value

The Aircraft speed V_a1f=V_a2f= 200m/s V_a10=V_a20= 73.45m/s The A1 Aircraft altitude Z_G1f = 35×10² m Z_G10= 0m

The A2 Aircraft altitude Z_G2f = 41×10² m Z_G20= 0m

The aircraft roll control δ_l1f =δ_l2f = 0.0174 δ_l10=δ_l20=−0.0174 The pitch control δ_m1f=δ_m2f= 0.087 δ_m10=δ_m20= 0 The yaw control δ_n1f =δ_n2f = 0.314 δ_n10=δ_n20=−0.035 The thrust control δ_x1f =δ_x2f= 0.6 δ_x10=δ_x20= 0.2 The attack angle α_a1f=α_a2f = 12^◦ α_a10=α_a20= 2^◦ The inclination angle θ_a1f =θ_a2f = 7^◦ θ_a10=θ_a20=−7^◦ The air-path inclination angle γ_a1f =γ_a2f = 0^◦ γ_a10=γ_a20=−5^◦ The aerodynamic bank angle μ_a1f =μ_a2f = 3^◦ μ_a10=μ_a20=−2^◦ The air-path azimuth angle χ_a1f =χ_a2f = 5^◦ χ_a10=χ_a20=−5^◦ The roll angle φ_a1f =φ_a2f = 1^◦ φ_a10=φ_a20=−1^◦

The cup ψ_a1f=ψ_a2f= 3^◦ ψ_a10=ψ_a20=−3^◦

The limits of time t_1f = 600s,t_2f = 645s t₁₀= 0s,t₂₀= 45s The mass of the A1 Aircraft m₁₀1.1×10⁵ kg, m_1f 1.09055×10⁵ kg, The mass of the A2 Aircraft m₂₀1.10071×10⁵kg m_2f 1.09126×10⁵ kg The A300 inertia moments [8] A= 5.555×10⁶ kg m² B= 9.72×10⁶ kg m²

C= 14.51×10⁶kg m² E=−3.3×10⁴kg m² The Aircraft vertical separation Z₁₂= 2×10³f t6×10²m

The Aircraft longitudinal separation X_G₁₂= 5N M9×10³ m The Aircraft roll velocity relative to the

earth

p_1f=p_2f= 1^◦s⁻¹ p₁₀=p₂₀=−1^◦s⁻¹

The Aircraft pitch velocity relative to the earth

q_1f =q_2f = 3.6^◦s⁻¹ q₁₀=q₂₀= 3^◦s⁻¹

The Aircraft yaw velocity relative to the earth

r_1f =r_2f = 12^◦s⁻¹ r₁₀=r₂₀=−12^◦s⁻¹

Table 1.

The two-aircraft acoustic optimal control problem. The combination of the aircraft dynamic equation (3) and (7), the aircraft objective function from equations (10) and the the aircraft ﬂight constraints (11), the two-aircraft