I Introduction
One of the most difficult tasks in aircraft control is achieving a safe and comfortable landing. A review of accident statistics indicates that over 45 percent of all general aviation accidents occur during the approach and landing phases of a flight [1]. A closer look shows that the cause of over 90 percent of those cases was pilot related. Loss of control was also a major contributing factor in 33 percent of these cases. In many cases, the accidents have happened due to not landing within a targeted landing time [2]. Aircraft landing has fives phases: the base leg, the glide slope, the flareout, the touchdown, and the afterlanding roll. The glide slope and the flareout are the most crucial phases. In the glide slope phase the aircraft descends along a predefined straight line at a steady state descent angle; whereas, in the flareout phase the aircraft gradually raises its nose while landing. There are several performance requirements and constraints associated with aircraft landing. The aircraft needs to follow the desired trajectory and land smoothly within a targeted landing time. In addition, the aircraft landing controller should be robust to wind turbulence, measurement noise, and actuator failures.
Many researchers have applied various control techniques to design an automatic aircraft landing control system. Different control methods such as classical control [5, 19], optimal control [2, 15, 8, 21], adaptive control [14, 20], nonlinear control [18, 13], and intelligent control [9, 10, 6] have been used to solve the automatic aircraft landing problem. In [17], the flareout maneuver control was successfully implemented using the classical feedback control method. [5] implemented a PID controller combined with an evolutionary technique to tune control gains. Generally, classical controllers often lack optimality in achieving desired performances. [2] applied an optimization technique called the parametric expansion method to synthesize a linear timevarying aircraft landing controller. In [7, 10, 12, 15, 3] issues such as robustness to measurement noise, actuator faults, wind disturbances, and flight uncertainties were investigated and , LQR/LQG, mixed control techniques were employed to design optimal and robust aircraft landing controllers. Recently, concepts from adaptive control and intelligent control such as adaptive synthesis based on dynamic inversion theory [10, 16, 18]
[6], quantitative feedback theory [20], sliding mode control [14], structured singular value
synthesis, or fuzzy techniques [4] were used to design automatic aircraft landing controllers. In [13, 3, 14] nonlinear control techniques were applied to improve the control performance of the landing controller. Most of the aforementioned techniques are based on the asymptotic convergence of tracking errors. This does not guarantee a safe, smooth, and comfortable landing, which often requires the aircraft to accurately track a particular landing profile while landing within a given period of time and satisfying performance requirements and flight constraints.The contribution of this paper is, therefore, designing an optimal controller that guarantees the finitetime convergence of the landing trajectory tracking errors. For this purpose, we first design the desired landing path which satisfies performance requirements and flight constraints. The flareout maneuver phase is designed as an exponential path, which connects end point of glide slope phase and the touchdown point on the runway. Furthermore, the touchdown point is captured as a boundary condition. In addition, performance requirements and constraints on control input and output signals are determined. Subsequently, we formulate the tracking problem by designing a performance index which should be minimized in a targeted landing time. We then have converted the aircraft landing problem to a Linear Quadratic Tracking (LQT) problem where we minimize the designed cost function for a linear model of the aircraft respecting boundary conditions. Control gains are found by solving differential Riccati equations backward in time using the Dormand Prince algorithm. The simulation results are analyzed to show that the developed trajectory tracking system can handle a range of angle of attack and headwind disturbances.
In the remainder of this paper, in Section II, the aircraft landing problem is formulated. In Section III, the desired trajectory and performance index are designed by incorporating the performance requirements and flight constraints. An LQT controller is developed for aircraft landing by minimizing the designed performance index to guide the aircraft to follow the desired landing trajectory. In Section IV, the proposed flight controller is simulated and the results are discussed. Finally, Section V concludes this paper.
Ii Problem formulation
The aircraft landing process is described in Figure 1. The glide slope phase and flareout phase are the most critical steps. After the base leg, the aircraft should descend towards the runway centerline with the flight path angle between 2.5 to 3.5 deg. As the aircraft descends to a certain altitude above the ground, the flareout phase begins. In the flareout phase of landing, the pitch angle of the airplane must be gradually adjusted to have a value between 0 to 10 deg, and the aircraft should approach the touchdown point smoothly. The flareout phase is the most difficult part of the landing maneuver, as the runway is not in the pilot’s field of view and the touchdown point has to be accurately reached within a targeted time smoothly, following the landing trajectory. In this paper, we assume that the lateral and longitudinal dynamics of the aircraft are decoupled, and the lateral controller [11] takes care of the lateral deviations from the center line of the runway. The focus of this paper, therefore, is on the design of a longitudinal controller to track the desired landing trajectory.
The longitudinal motion of the aircraft is governed entirely by the elevator deflection angle . The longitudinal dynamics of an aircraft is given by the socalled short period equation of motion [2] of the aircraft:
(1) 
where is pitch rate, is short period gain, is path time constant, is short period resonant frequency, and is short period damping factor. Assuming the velocity of the aircraft to be constant, pitch angle rate and altitude are related [2] through vertical acceleration as:
(2) 
Combining (1) and (2), a fourthorder linear differential equation is found:
(3) 
Substituting
(4a)  
(4b) 
in (3), the new aircraft equation of motion becomes
(5) 
from which one can derive, the state space representation of the aircraft’s model as:
(6a)  
(6b) 
where , ,
,
, , , , ,
and C is the identity matrix.
Our aim is to design a controller for the aircraft to follow the landing trajectory with the minimum tracking error given as:
(7) 
where is the desired landing trajectory over the landing time interval , . Then, given an aircraft model in (6) and boundary conditions as and , we address the following problems for the aircraft landing:
Problem 1
Design a desired landing flight trajectory which captures performance requirements and flight constraints.
Problem 2
Design a performance index to capture performance requirements and constraints
Problem 3
Design a controller that tracks the desired trajectory and satisfies the given performance requirements and constraints.
Iii Landing trajectory tracking
Iiia Trajectory design
The aircraft landing trajectory is the desired aircraft landing path, which the aircraft follows over a specified period of time . Landing trajectory must ensure safe and comfortable landing. The desired landing path is expressed as a desired altitude which is a function of forward distance. Its important phases are the glide slope phase and flareout phase as shown in Figure 3. The glide slope phase is the first part of a landing path. It is modeled as a straight line path whose slope is given in terms of the flight path angle . The other part is the flareout path which is initiated when the aircraft reaches to a certain height above the landing surface. Flareout path is an exponential path with a near zero flight path angle at touchdown. In Figure 3, the desired landing path is shown in forward distance (X)altitude (h)plane.
In the glide slope phase the aircraft tracks a straight line path pointing toward the beginning of the runway. The desired altitude, , is computed based on a given flight path angle :
(8) 
As soon as the flareout phase starts, the desired altitude is computed as:
(9) 
where is a constant which defines the curvature of flareout maneuver path, depending on the distance between the origin of Xh plane and the touchdown point.
The objective of a good, stabilized final approach is to descend at an angle and airspeed that permits the airplane to reach the desired touchdown point with minimum floating; in essence, a semistalled condition [1]. To accomplish this, it is essential that both the descent angle and the airspeed be accurately controlled. Since on a normal approach the power setting is not fixed as in a poweroff approach, the power and pitch attitude are adjusted simultaneously as necessary to control the airspeed and the descent angle, or to attain the desired altitudes along the approach path. In this paper, we focus on the flareout phase to achieve a safe and comfortable landing. In (9), , and are unknowns. We can solve for these unknowns from , , and under the following constraints:

Slope continuity constraint: For a smooth phase change, the slope at the beginning of the flareout path should be equal to the slope of the glide slope path. Thus, the slope of glide slope path can be calculated as
(10) Similarly, the slope at the beginning of flareout path is
(11) Using (10) and (11), is computed as
(12)
Under the assumption of constant aircraft forward velocity
, at any time moment
, the forward distance on the flareout curve will be approximately(15) 
Substituting from (15) in (9), the landing trajectory in flareout phase is governed by
(16) 
where . By differentiating (16), the desired rate of descent in flareout phase is:
(17) 
The value of the pitch angle is crucial only during the last few sec of landing before the touchdown point. In flareout maneuver phase, the desired pitch angle, , is::
(18) 
Also, for a safe and comfortable landing, the pitch rate should be:
(19) 
IiiB Performance requirements
The following landing requirements and flight constraints are considered for the aircraft landing in this paper:

The landing path should be an exponential path such that it ensures a safe and comfortable landing.

The rate of descent should be nonzero in order to avoid overshooting. In addition, the rate of descent at touchdown should be a small negative value in order to prevent over stressing of landing gear. Requirements to avoid undesirable behaviors such as landing gear over stress, aircraft floating down runway and aircraft overshooting runway are captured by magnitude of rate of descent at touchdown point. Typically based on type of the aircraft, a value between 60 to 180 ft/min is considered as an ideal rate of descent at touchdown point.

The lower limit on the pitch angle should be in order to prevent the nose wheel of an aircraft from touching down first, and the upper limit on the pitch angle should be to prevent the tail gear from touching down first:

During landing, in the flareout maneuver phase, the angle of attack must remain below 80 percent of the stall value. The stall value is assumed to be [2]. Hence, the rate of change of angle attack is restricted to a value less than 20 percent of the stall value:

The longitudinal motion control is mainly executed by the elevator. To avoid saturation, the elevator is not permitted to operate against the mechanical stops during the landing process. Thus, the deflection of the elevator is restricted between and .
In addition, the aircraft should follow the designed landing trajectory and reduce the tracking error within the targeted landing time with minimum control effort . This can be captured by the following performance index:
(20) 
where , , , are all symmetric weighting parameters. The terms outside the integral ensure that the states are close to the desired values at the touchdown point, avoiding an early or a late landing. On the other hand, the terms inside the integral push the aircraft to track the desired landing trajectory with minimum control effort in the landing time .
IiiC Controller design based on linear quadratic tracking
Given the aircraft model in (6) and the boundary conditions and , our aim is to find an optimal control that guides the aircraft to follow the desired landing trajectory over the time interval , minimizing the performance index in (20), as described below:
(21) 
subject to
Defining the costate vector
, where and , and defining the Hamiltonian function , the constrained optimization in (21) will be converted to(22) 
whose solution can be found from , where is the variation of the functional . The optimal control function will be
(23a)  
(23b) 
in which and can be found by solving the following differential equations in a backward way with a given boundary conditions:
(24a)  
(24b)  
(24c)  
(24d) 
Iv Simulation
A simulation environment is set up by integrating the MATLAB and FlightGear simulation Software in which we simulated an aircraft landing from the altitude of 95 ft to the runway 10L of San Francisco airport. The information on the approach plate (see Table IV) of runway 10L of the airport is used to design a safe and comfortable landing trajectory for an aircraft with model parameters given in Table I.
Symbol  Parameter value  
1  0.95  
2  40  
3  1  
4  0.5 
(25) 
Table IV summarizes parameters extracted from the approach plate. Similarly, assuming this particular aircraft has a nominal initial altitude of 100 ft, the desired flight path angle of 3 deg, and the pitch angle of 0 deg, then using (12) to (15), we found the desired landing trajectory parameters , , , , as shown in Table IV.
The designed desired landing trajectory is plotted in Figure 4 in solid blue. The desired altitude and the desired altitude rate are exponential functions while the desired pitch angle and the desired pitch rate are constant functions. Both exponential functions and constant functions have derivatives of all orders. Therefore, the designed landing trajectory is a smooth landing trajectory.
Name  Parameter value  
1  34346  
2  1800  
3  3957 
Name  Parameter value  
1  0.00049  
2  256  
3  0.1385  
4  6.68  
5  1908 
Parameter Name  Parameter value  
1  0 sec  
2  20 sec  
3  95 ft  
4  14 ft/sec  
5  0.05 rad  
6  0 rad/sec  
7  0 ft  
8  0 ft/sec  
9  0 rad  
10  0 rad/sec 
Iva Case I: Simulation results for the initial condition given in Table Iv
For the given initial conditions in Table IV, the simulated aircraft landing trajectory is shown in Figure 4 in red. The controller gain is found solving (24d) in a backward way using the DormandPrince algorithm. To respect the Constrains C1C5 in Section IIIB, we have tuned the performance index parameters , , and to values in (25). Comparing the desired altitude , the desired altitude rate , and their respective simulation values in Figure 4, we can see that the aircraft has accurately tracked the desired landing trajectory in a desired landing time of 20 sec satisfying Constraints C1C5 in Section IIIB. This comparison is summarized in Table V.
IvB Case II: Simulation results for different initial conditions
Head wind and deviation of a desired flight path angle from a desired value of 3 deg introduces a deviation in the gliding distance and the pitch angle which in turn give rise to a deviation in the initial altitude of the flareout phase as shown in Figure 7.
However, the designed controller can handle these deviations in the initial altitude and pitch angle by adjusting control input in order to maintain the rate of descent and the desired approach airspeed, while avoiding the saturation region of the elevator actuator (Constraint C5) .
Acceptable deviations in the initial altitude and pitch angle around the desired landing trajectory is determined by varying them correlationally until the control limiting values are attained instantaneously. Accordingly, and are found to be 20 ft and 1 deg respectively, and the corresponding region of operation is shown in Figure 8 in green.
V conclusion
In this paper, a smooth trajectory tracking system was designed for a fixedwing aircraft landing. The aircraft landing problem was converted to a finitetime LQT problem. Information on the approach plate of the runway was employed to design a desired smooth landing trajectory. We considered multiple performance requirements and constraints to design an optimal controller which minimizes the trajectory tracking error. The designed controller gives an accurate tracking performance for the flareout phase of aircraft landing. The simulation result demonstrated the finitetime convergence of the trajectory tracking error. Moreover, the robustness of the designed controller was evaluated against different initial conditions due to the headwind disturbances. This work can be extended by employing a nonlinear model of an aircraft in the presence of turbulent crosswinds and system uncertainties.
Acknowledgment
This research is supported by Air Force Research Laboratory and OSD under agreement number FA87501520116.
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