Adaptive Disturbance Rejection Control for Automatic Carrier Landing System

An adaptive disturbance rejection algorithm is proposed for carrier landing system in the final-approach.The carrier-based aircraft dynamics and the linearized longitudinal model under turbulence conditions in the final-approach are analyzed. A stable adaptive control scheme is developed based on LDU decomposition of the high-frequency gain matrix, which ensures closed-loop stability and asymptotic output tracking. Finally, simulation studies of a linearized longitudinal-directional dynamics model are conducted to demonstrate the performance of the adaptive scheme.


Introduction
The automatic carrier landing system requires that the aircraft arrives at the touchdown point in a proper sink speed and a small margin error for position.The key requirements of this problem are that the aircraft must remain within tight bounds on a three-dimensional flight path while approaching the ship and then touch down in a relatively small area with acceptable sink rate, angular attitudes, and speed.Further, this must be accomplished with limited control authority for varying conditions of wind turbulence and ship air wake.
During the past decades, research on the improvement of the automatic carrier landing system had received much attention.A vertical rate and vertical acceleration reference were used in the control law to reduce the turbulence effects and deck motion in [1].A noise rejection filter was added to the control algorithm to decrease the sensitivity to noise and an optimization of the control gains was then performed to prevent degradation of the system's response to turbulence in carrier landing in [2].A finite horizon technique was introduced to maintain a constant flight-path angle under the worst case conditions during carrier landing in [3].An improvement in carrier landing performance was made by the incorporation of the direct lift control using  ∞ outputfeedback synthesis [4].As pointed out in [5], a fuzzy logic based carrier landing system was designed and the results indicated that fuzzy logic could yield significant benefits for aircraft outer loop control.For the lateral-directional aircraft dynamics in carrier landing, a linear fractional transformation gain-scheduled controller was presented in [6].The dynamic inversion technique was used in unmanned combat aerial vehicle on an aircraft carrier in [7].In the absence of wind and sea state turbulence, the controller performed well.After adding wind and sea state turbulence, the controller performance was degraded.
Adaptive control has become one of the most popular designs for failures and disturbances compensation.An output tracking model reference adaptive control (MRAC) scheme was developed for single-input/single-output systems in [8].The related technical issues including design conditions, plant-model matching conditions, controller structures, adaptive laws, and stability analysis are presented in detail, with extensions to adaptive disturbance rejection.In [9], a combined direct and indirect MRAC statefeedback architecture was developed for MIMO dynamical systems with matched uncertainties and the methodology was extended to systems with a baseline controller.To solve the disturbance rejection problems, adaptive feedforward [10,11], feedback control methods [12], terminal sliding-mode control method [13][14][15], and back-stepping control designs [16,17] were proposed.In [18], an extension of biobjective optimal control modification for unmatched uncertain systems was proposed.However, the existing methods are mainly for the matched disturbance rejection, while there exists certain difficulty of achieving tracking performance, especially for the unmatched disturbances.
In this paper, an adaptive control scheme is proposed to handle wind during carrier landing.The main contributions of this paper are described as follows: (1) With unmatched disturbance, the aircraft models in air-wake turbulence conditions during the carrier landing are analyzed.The longitudinal linearized model of a carrier-based aircraft dynamics is constructed on the final-approach.
(2) Adaptive LDU decomposition-based state-feedback controller is designed to relax design conditions, including adaptive laws and stability analysis.
( The rest of this paper is organized as follows.In Section 2, we present the aircraft model with disturbances during the carrier landing phase.In Section 3, we propose adaptive designs to solve the aircraft disturbance compensation.We illustrate an application of the proposed adaptive design to aircraft wind disturbance rejection control.In Sections 4 and 5 some simulation results and conclusions are discussed.

Longitudinal Model of Carrier-Based Aircraft on Final-Approach Dynamics in the Air Wake
The overall carrier landing task for a fixed-wing aircraft is shown in Figure 1.The final-approach leg is typically entered from the last turn until the touchdown on the carrier deck, as illustrated in Figure 2. The turbulence is the major source of glide path and touchdown errors.In this phase, the longitudinal reference flight state is chosen as a steady rectilinear flight in air wake at a constant velocity, with constant angle of attack and a flight-path angle.The flaps and the gear are totally lowered, and two control means are employed to control the flight-path vector in the vertical plane: elevator and engine thrust.In this section, the longitudinal linear aircraft model and carrier air wake are described.

Nonlinear Aircraft Longitudinal Equations in the Calm Air.
Both of the bank and sideslip angles are zero; the decoupling longitudinal of the nonlinear equations is described in the calm circumstance.The longitudinal aircraft dynamics equations are presented as follows.
The force equations are The kinematic equation is The moment equation is The navigation equation is The identical equation is

Longitudinal Linearized Model of Aircraft in the Calm
Air.The linear model of the longitudinal flight dynamics is constructed based on the small-perturbation equation.The linearized longitudinal flight dynamics is described as where  1 = [V, , , , ℎ]  ,  1 = [  ,   ]  , and  1 = [V, , , , ℎ]  are the system state vector, input vector, and output vector, respectively, and the matrices are

Longitudinal Linearized Model of Aircraft in the Carrier
Air Wake.The steady component of the carrier air wake is taken into account to provide some disturbances, as a basis of our simulations.

Turbulence Description.
The steady component of the carrier air wake is taken into account to the simulation.The superstructure and deck/hull features of an aircraft carrier are known to generate turbulent airflow behind the carrier.This region of turbulent air has become known as the burble and it is often encountered by pilots immediately after an aircraft carrier.This turbulent region of air has adverse effects on landing aircraft and can cause pilots to bolter, missing the arresting wires and requiring another landing attempt.The burble components are determined from look-up tables scheduled on the aircraft distance behind the ship in [19][20][21], and the components are presented as where  is the distance between the aircraft and the ship center of pitch, negative after of ship,  0 is the total landing time, and  is the present time.

Longitudinal Linearized Model of Carrier-Based Aircraft in
Air-Wake Disturbance.The linear model of the aircraft under the air-wake disturbance is addressed in [22,23].The airspeed and angle of attack are susceptible to   and   .
Because the flight speed is far larger than the wind speed, we can get where   and   are the airspeed and angle of attack affected by disturbances.From ( 6) and ( 9), the linearized longitudinal dynamics of aircraft under turbulence conditions can be modeled as where  1 ,  1 ,  1 ,  1 , and  1 are defined in (6).The disturbance is () = [  ,   ] and the matrix  1 is

Adaptive Disturbance Rejection Design
In this section, to solve turbulence compensation problem, an adaptive disturbance rejection design is developed for multivariable systems with unmatched input disturbances.

Problem Formulation.
Consider the linear time-invariant system as where  ∈  × ,  ∈  × ,   ∈  × , and  ∈  × are constant and unknown; () ∈   , () ∈   , and () ∈   are the system state vector, input vector, and output vector, respectively; () = [ 1 (), . . .,   ()]  ∈   is the disturbance vector.The disturbance signal () is unmatched with the control input (), in the sense that  and   are not linearly dependent,   ̸ =  for any matrix  ∈  × .The control objective is to design an adaptive statefeedback control signal () for ( 12), to make closed-loop signal boundedness and the output () track a chosen reference signal   () generated from a reference model: where   () ∈  × is a stable transfer function matrix and () ∈   is an external reference input signal for defining a desired   ().Note that, in this paper, we use the notation () = ()[]() to represent the output () of a system whose transfer matrix is () and input is (), a convenient notation for adaptive control systems.
Remark 7. Assumption 3 is for output matching and internal signal stability.Assumption 4 is for choosing the reference system model for adaptive control.Assumption 5 is for designing adaptive parameter update laws.Assumption 6 is the relative degree condition from the control input () and the disturbance input () to the output () for the design of a derivative-free disturbance rejection scheme.

Nominal Disturbance Rejection Design.
With the knowledge of the plant and disturbance parameters, the nominal state-feedback controller is where the nominal parameters  *  1 ∈  × and  *  2 ∈  × are for the plant-model output matching, and  * 3 () ∈   is used to cancel the effect of the disturbance ().
Lemma 8 (see [25]).The matrix Based on Lemma 8, the existence of a nominal controller ( 17) is established a follows.
Theorem 9.For plant (12) with the unmatched disturbances, under Assumptions 3 and 6, there exists a state-feedback control law, to make the boundedness of all closed-loop signals, disturbance rejection, and output tracking of the reference   ().

Stability Analysis.
To analyze the closed-loop system stability, we first establish some desired properties of the adaptive parameter update laws mentioned above.

Simulation Study
4.1.Aircraft Model in Turbulence.The proposed multivariable adaptive disturbance rejection scheme is applied to a carrier landing system using LDU based decomposition.The aircraft longitudinal model defined in planted ( 6) is derived in [26].
The system parameter matrices are described as The turbulence disturbances are described in [22,23]; we can get The high-frequency matrix is which means that the relative degree condition in Assumption 6 can be ensured.The related gain parameters in adaptive laws (51) are chosen as Γ  = 100,  = diag{2 2}, and Γ = diag{10 10}.

Simulation Results.
For this simulations study, the initial state is chosen as  0 () = [240 0 0 0 295], and the initial controller parameters are set as 70% of their true values.As shown in Figures 3 and 4, the LDU based adaptive controller can ensure that the aircraft system output signal tracks the reference height tightly.Figures 5 and 6 show tracking performances of the automatic carrier landing system where the adaptive controller is used under the final-approach leg. Figure 7 shows the surface deflections and power control, when the aircraft receives a time varying turbulence.From the simulations, the automatic carrier landing system with the proposed adaptive controller is well performed in the turbulence.This indicates the disturbance adaptive controller can be used in carrier air-wake in the final-approach air condition.

Conclusions
In this paper, a multivariable disturbance rejection scheme is presented to solve the wind turbulence problem.The statefeedback output tracking MRAC scheme is designed based on the LDU decomposition of the high-frequency gain matrix.The aircraft carrier landing system under aircraft carrier air wake is analyzed.The proposed LDU decomposition-based disturbance rejection techniques are used to solve a typical carrier landing aircraft turbulence compensation problem.Finally, simulation results have been presented to show that MRAC-based disturbance rejection scheme is an effective method of the carrier landing system with the disturbances.  : Moment of inertia in pitch : The pitch moment   ,   : Elevator deflection bias angle and engine throttle angle   0 ,   0 : Aerodynamic pitch moment and drag derivative with respect to airspeed  0   0 ,   0 : Thrust and aerodynamic lift derivative with respect to airspeed  0    ,    : Aerodynamic pitch moment and lift derivative with respect to     ,   ,   : Turbulence velocity and body axis components of    0 ,  0 , α 0 ,  0 , ℎ 0 : The trim value of aircraft state   0 ,   0 : Aerodynamic pitch moment and drag derivative with respect to airspeed  0   0 ,   0 : Thrust and aerodynamic lift derivatives with respect to airspeed  0    ,    : Aerodynamic pitch moment and lift derivative with respect to elevator   0 ,  ℎ 0 : Aerodynamic drag derivative with respect to  0 and ℎ 0    : Aerodynamic drag due to     0 ,  α 0 : Aerodynamic pitch moment with respect to  0 and α 0   0 : Aerodynamic pitch moment with respect to  0  0 : Benchmark aerodynamic thrust at the airspeed  0    : Aerodynamic thrust derivatives with respect to the throttle   0 ,  ℎ 0 : Aerodynamic lift derivative with respect to angle of attack and height.

Figure 1 :
Figure 1: Procedures of carrier landing for the aircraft.

Figure 3 :Figure 4 :
Figure 3: Final landing phase altitude for the aircraft.