Flight Level Control Problem

The flight level tracking that plays an important role in autopilot systems has received considerable attentions from many researchers (Lygeros, 2003; Lygeros et al., 1999; Cook, 1997; Tomlin et al., 1996; Etkin and Redi, 1996). A commercial aircraft‘s cruising altitude is typically assigned a flight level by air traffic control (ATC). To ensure aircraft separation, each aircraft has its own flight level separated by a few hundred feet; however, changes in flight level do happen occasionally and must be cleared by ATC. At all other times, the aircraft crew must ensure that they remain within the allowed bounds of their assigned level.

At the same time, they must also maintain limits on factors such as speed, flight path angle, and acceleration imposed by limitations of airframe and engine and passenger comfort requirements or to avoid dangerous situations such as aerodynamic stall. In this paper, the flight level tracking problem is formulated into an optimal control problem. For safety reasons, the speed of the aircraft and the flight path angle must be kept within a safe “aerodynamic envelope” (Tomlin et al., 1996) that can be translated into the dynamic constraints of the optimal control problem. A flight level tracking problem and a minimum time problem are outlined in the following sections and then solved using the proposed solver.

6.1.1 Aircraft Model

Much ATC research (e.g., Cook, 1997; Etkin and Redi, 1996) has applied a point mass model to describe aircraft motion, considering only aircraft movement in a lateral direction. In Figure 6.1, three coordinate frames are used to describe aircraft motion: Xg-Yg denotes the ground frame; Xb-Yb,the body frame; and Xw-Yw, the wind frame. In addition,

θ

,

γ

, and

α

denote the rotation angle between the frames; V∈ \ represents the speed of the aircraft, which is aligned with the positive Xw direction; and h is the aircraft’s altitude.

The equations of the motion can be derived from the force balance relationships:

where T is the thrust exerted by the engine, D is the aerodynamic drag, and L is the aerodynamic lift. By applying basic aerodynamics, the lift (L) and drag (D) can be approximated by

where CL, CD, and c are dimension-less lift and drag coefficients, s is the wing surface area and

ρ

is the air density.

According to the admissible optimal control formulation described in Section 3.4, the air model can be formulated by a three-state model with a state variable vector x(t) = [x1, x2, x3]^T

= [V,

γ

, h]^T and a control input vector u(t) = [u1, u2]^T = [T,

θ ]

^T. By approximating

α

with a small angle, the equations of the motion (system equations) can be written as

1 2

This model, proposed by Lygeros et al. (1999) and adopted here, extends the three dimensions of an aerodynamic envelope protection problem. Taking into the consideration of safety conditions, the aircraft speed and flight path angle are bounded in a rectangular limitation called a “safe aerodynamic envelop.” Following Tomlin et al. (1996), Lygeros (2003) proposed a simplified aerodynamic envelope that is adopted in this paper and translated into the following dynamic constraints:

min 1 max

Based on the NLP formulation described in Section 2.2, these constraints can be treated as dynamic constraints and rewritten as follows:

1 1 min 2 1 max

To illustrate the capabilities of the proposed method, the flight level tracking problem and the minimum time problem have been chosen.

Case I: Flight level tracking problem

This tracking problem is to find the controls that will maintain the system state x(t) as close as possible to the desired state r(t) in the interval [t0, tf]. The performance index for the tracking problem can be written as

0

2

0 ^tf

( ) ( )

( )_t

J = ∫

t

^x t − ^r t

^Q

^dt

^(6.6)

where Q(t) is a real symmetric n × n matrix that is positive semi-definite for all

t ∈⎣ ⎡ t

₀

, t

_f

⎤ ⎦

^. The flight level tracking problem involves keeping the aircraft as near as possible to the desired level and aircraft speed. Therefore, the performance index can be represented as

( ) ( ) ( )

where x1d is the desired aircraft speed, x2d is desired flight path angle and x3d is the assigned altitude.

Case II: Minimum time problem

The minimum time problem is to transfer a system from an arbitrary initial state x(t0) = x0

to a specified target set St in minimum time. The performance index for the minimum time

problem can be written as

0 0 0

tf

f t

J = − = t t ∫ d ^t

^(6.8)

where tf is the first instant of time when x(t) and St intersect. In some emergencies, the aircraft crew is asked to change their level as soon as possible.

6.1.2 Numerical examples

The following parameters, outlined here for case I, are used in both cases:

a

L = 65.3 Kg/m,

a

D = 3.18 Kg/m,

m = 160×10

³ Kg,

g = 9.81 m/s

²,

θ

min = -20°, γmin = -20,

c = 6, θ

max = 25°, γmax = 25,

T

min = 60×10³ N,

T

min = 120×10³ N,

V

min = 92 m/s,

V

max = 170 m/s,

h

min = -150 m,

h

max = 150 m The initial values of the state variables are

x

0 = [100, 20, -120]^T (6.9)

and the purpose of this problem is to find a suitable control for maintaining the flight level and keeping the aircraft altitude at the assigned level. Thus the desired states are set with following values

r(t) = [150, 0, 0]

^T. (6.10)

In addition to the dynamic constraints proposed in Eq. (6.5), the control inputs are also limited within the following bounds:

min 1 max

Substituting these parameters into Eqs. (6.3) and (6.7), the flight tracking problem is solved by the OCP solver. The numerical results are shown in Figure 6.2. As shown in Figure 6.2(a), all states meet the constraints, and the flight level and aircraft speed return to the

desired states. Table 6.1 shows the user subroutines for this case. Obviously, the OCP solver provides an easily usable tool for solving dynamic optimization problems.

Case II: Minimum time problem

In this problem, the aircraft crew is asked to increase their altitude in minimum time. The initial and final altitude are h0 = 0 m and hf

= 500 m, respectively. All constraints imposed on

case I are also imposed on this case. The initial state x0 = [100, 0, 0]^T, the final time, tf, obtained by using the AOCP, is 73.98 seconds, and the final altitude is 499.928 m. The control histories shown in Figure 6.3(a) and (b) give the state trajectories, which, as the figure illustrates, all fall within the safe “aerodynamic envelope” (i.e., meet the dynamic constraints).

在文檔中 動態系統最佳化設計之計算方案與其應用 (頁 123-127)