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Aerospace Systems

, Volume 1, Issue 1, pp 63–71 | Cite as

Automatic landing system design via multivariable model reference adaptive control

  • Yawen Wang
  • Qifu Li
  • Bei LuEmail author
Original Paper
  • 292 Downloads

Abstract

The landing of a civil transport aircraft is one of the most critical phases due to parametric uncertainties and strong crosswind conditions. In this paper, separate controllers are designed for longitudinal and lateral-directional channels for the landing phase, which is divided into the final approach, flare, and decrab. A multivariable model reference adaptive control scheme is implemented with state feedback for output tracking. The safety and flight performance of the autolanding control system are demonstrated through Monte Carlo simulations of a nonlinear civil transport aircraft model.

Keywords

Civil transport aircraft Autolanding design Adaptive control Output tracking Monte Carlo simulations 

List of symbols

\( D_{\text{LG}} \)

Distance to threshold

\( H \)

Altitude

\( H_{\text{LG}} \)

Landing gear height

\( n = [n_{x} ,n_{y} ,n_{z} ] \)

Load factors

\( ss_{\text{LG}} \)

Landing gear sideslip angle

\( u = [\delta_{\text{th}} ,\delta_{\text{e}} ,\delta_{\text{a}} ,\delta_{\text{r}} ] \)

Control inputs (thrust, elevator, aileron, and rudder)

\( V = [u,v,w] \)

Translational speeds

\( V_{a} \)

True airspeed

\( V_{c} \)

Calibrated airspeed

\( V_{g} \)

Ground speed

\( V_{Z} \)

Vertical airspeed

\( V_{{Z_{{\rm {LG}}} }} \)

Landing gear vertical speed

\( w = [w_{x} ,w_{y} ,w_{z} ] \)

Wind speeds

\( X = [x,y,z] \)

Position of the center of gravity of the aircraft

\( Y_{\text{LG}} \)

Deviation from runway axis

\( \alpha \)

Angle of attack

\( \beta \)

Aerodynamic sideslip angle

\( \Delta = [\Delta_{\text{loc}} ,\Delta_{\text{gld}} ] \)

ILS noises (localizer noise and glide noise)

\( \Delta y \)

Localizer deviation

\( \Delta z \)

Glide deviation

\( \chi \)

Flight path azimuth angle

\( \varOmega = [p,q,r] \)

Angular rates

\( \varPhi = [\phi ,\theta ,\psi ] \)

Attitude angles

1 Introduction

Today for a civil transport aircraft, most of the flight segments can be carried out by the autopilot, which guarantees the safety and stability of flights. However, the final approach and flare segments are still challenging tasks as high airworthiness standards have to be met in poor visibility conditions. In the aeronautical industry, the Certification Specifications for All Weather Operations (CS-AWO) [1] taken by European Aviation Safety Agency (EASA) and the Advisory Circular (AC) 120-28D [2] taken by the Federal Aviation Administration (FAA) provide acceptable means for obtaining approval of operations in Category III (CAT III) landing. With the help of CAT III instrument landing systems (ILS) that are now widely used in airports, automatic landing control laws have been proposed and are achieved by classical control techniques [3]. However, those automatic landing control laws are far from being robust to system uncertainties (e.g., mass, position of center of gravity (CG), etc.) or environmental phenomena (e.g., crosswinds, temperature variations, etc.). For this reason, ONERA, the French Aerospace Lab, and the AIRBUS Company have proposed a challenge aiming at inspiring new control systems that are able to improve the robustness. The challenge involves an ILS approach and flare from 1000 ft (304.8 m) above runway until touchdown [4].

Over the past two decades, there have been numerous research results of automatic landing control design for dealing with the civil transport aircraft landing problem. For instance, the linear-quadratic-gaussian with loop-transfer recovery method was proposed by Ghalia and Alouani [5] to design an automatic landing controller for a typical commercial aircraft encountering a wind shear. A robust automatic landing controller was designed in Alpert [6] using a fixed-order \( H_{\infty } \) control method and was successfully applied to vertical speed tracking with the ground effect in consideration. An adaptive sliding mode control technique was used by Bouadi et al. [7] for flight path tracking, and the controller provided fault tolerance against aerodynamic parameter uncertainties, external disturbances, and modeling inaccuracies. Other control methods based on soft computing, such as fuzzy logic and neuro networks, can be found in Raj and Tattikota [8], Pashikar and Sundararajan et al. [9].

Model reference adaptive control (MRAC) is a fundamental adaptive control method, which is based on the state equation design and does not depend on the mathematical model of the controlled object. It has strong adaptive ability and robustness to time-varying system parameters and external disturbances. The multivariable MRAC design has been employed in many applications, such as flight control [10], fault-tolerant control with parametric and structural uncertainties caused by damages [11].

The objective of this paper is to design an autopilot system with the integration of the MRAC method and classical control techniques to enable a correct landing despite parametric variations. The paper is organized as follows. In Sect. 2, the aircraft model and automatic landing segments are described. Then, the main theorem of multivariable MRAC method is introduced in Sect. 3. Based on this theorem, the detailed automatic landing controller design process is given in Sect. 4 and the control performance is validated through Monte Carlo simulations in Sect. 5. Finally, conclusions are drawn in Sect. 6.

2 Civilian aircraft landing control problem

The Civilian Aircraft Landing Challenge is a benchmark proposed by ONERA and AIRBUS. In this section, the aircraft model provided for the benchmark is first briefly described, followed by a description of the automatic landing segments.

2.1 The aircraft model

The benchmark aircraft model is a standard six-degrees-of-freedom 12th-order rigid-body aircraft model, representing a large, twin-engined, transport aircraft. During the landing of the aircraft, we need to consider both the longitudinal and lateral motions of the aircraft. Figure 1 shows the input and output information of the nonlinear aircraft simulation model, which has a total of 9 inputs (4 control inputs, 3 wind inputs, and 2 ILS noise inputs) and 40 outputs (16 states, 5 non-measured outputs (available for visualization only), and 19 measured outputs (available for feedback)) [12]. Descriptions of the notations are listed in Nomenclature.
Fig. 1

Nonlinear aircraft simulation model

Four actuators are available and are approximated by magnitude and rate-limited first-order filters, whose characteristics are summarized in Table 1.
Table 1

Characteristics of engines and actuators

Parameter

Time-constant

Lower-bound

Upper-bound

Rate-limit

Engines (EPR)

2 s

0.95

1.6

0.1

Elevators

0.07 s

− 25 deg

25 deg

20 deg/s

Ailerons

0.06 s

− 55 deg

55 deg

60 deg/s

Rudder

0.2 s

− 30 deg

30 deg

30 deg/s

A more detailed description of this model can be found in Biannic and Boada-Bauxell [3]

2.2 Description of the landing phase

As illustrated in Fig. 2, the landing phase for a civil transport aircraft is divided into three phases [12, 13].
Fig. 2

The landing phase for a civil transport aircraft

  • Final approach phase In this phase, the aircraft needs to track the glide-slope and localizer path command signals generated by the ILS. Meanwhile, the calibrated airspeed is kept constant and the sideslip angle is close to zero.

  • Flare phase The flare controller replaces the final approach longitudinal controller when the height of the airplane landing gear \( H_{\text{LG}} \) is less than 15 m. In this phase, the throttle is set to idle, so the calibrated airspeed is no longer to be controlled and only the elevator is effective to reduce the vertical speed \( V_{z} \).

  • Alignment phase (or decrab mode) This phase replaces the final approach lateral outer control loop when the height of the airplane landing gear is less than 9 m. The objective of this phase is to minimize lateral effort at touchdown to meet airworthiness standards.

2.3 Overview of the autoland architecture

As briefly discussed above, the landing phase is divided into three segments, and for each, controllers need to be designed separately for different landing requirements. Both longitudinal and lateral control systems consist of two nested loops: inner and outer control loops. The above control systems consisting of glide, localizer, flare and alignment subsystems are assembled as shown in Fig. 3 to define the automatic landing control system.
Fig. 3

The autoland architecture

The longitudinal and lateral outer loops (lon outer loop and lat outer loop) generate the commanded signals \( (V_{cc} ,V_{zc} ) \) for the longitudinal axis and \( (n_{yc} ,\phi_{c} ) \) for the lateral axis, respectively, to inner loops (Lon Inner Loop and Lat Inner Loop). Along the longitudinal axis, when the airplane landing gear height \( (H_{\text{LG}} ) \) is less than 15 m above runway, the desired vertical speed signal \( V_{zc} \) is modified by Flare controller to ensure a soft landing and the same architecture is used along the lateral axis. When the height of the landing gear is less than 9 m, the desired lateral load factor \( n_{yc} \) and roll angle \( \phi_{c} \) signals are determined by Decrab controller.

3 Model reference adaptive control

In this section, the theoretical background of a multivariable MRAC method based on the state feedback and output tracking is introduced and more details can be found in Guo et al. [14].

3.1 Problem statement

Consider an M-input and M-output linear time-invariant system described by
$$ \Delta \dot{x}(t) = A\Delta x(t) + B\Delta u(t),\quad \Delta y(t) = C\Delta x(t) $$
(1)
with \( A \in R^{n \times n} \), \( B \in R^{n \times M} \) and \( C \in R^{M \times n} \) being state, control and output matrices, and \( \Delta x(t) \in R^{n} \), \( \Delta u(t) \in R^{M} \) and \( \Delta y(t) \in R^{M} \) being the system state, input and output vector signals.
The control objective is to design an adaptive state feedback control law \( \Delta u(t) \) to reduce the effect of uncertainties inherent in \( A \), \( B \) and \( C \) matrices and to make the output \( \Delta y(t) \) track a desired signal \( \Delta y_{m} (t) \) generated by a reference model. The mathematical expressions of the reference model are given by
$$ \begin{aligned} \Delta y_{m} (t) & = W_{m} (s)[\Delta r](t) \\ W_{m} (s) & = {\text{diag}}\left\{ {\frac{1}{{d_{1} (s)}}, \ldots ,\frac{1}{{d_{M} (s)}}} \right\} \\ d_{i} (s) & = s^{{l_{i} }} + a_{i1} s^{{l_{i} - 1}} + \cdots + a_{{il_{i} }}. \\ \end{aligned} $$
(2)

To simplify the representation, the symbols of Laplace transformation and inverse Laplace transformation in the first expression are omitted without causing misunderstanding. \( \Delta r(t) \in R^{M} \) is a bounded external reference input signal and \( W_{m} (s) \) is the reference model transfer function matrix. \( d_{i} (s) \) is a monic stable polynomial of degree \( l_{i} \) with \( l_{i} = n - m \), where n and m are the degrees of denominator and numerator polynomials of system transfer function \( G(s) = C(sI - A)^{ - 1} B \).

The structure of the MRAC is illustrated in Fig. 4, where the ‘Reference model’ block generates the expected reference signal \( \Delta y_{m} (t) \) and the ‘Adaptive law’ block updates the control gain calculated by the ‘State feedback controller’ block, so that the tracking error asymptotically converges to zero, i.e., \( \lim_{t \to \infty } (\Delta y(t) - \Delta y_{m} (t)) = 0 \).
Fig. 4

Adaptive control system structure

3.2 State feedback control

For the linearized system (1), a fixed matching controller is given by:
$$ \Delta u(t) = K_{1}^{*T} \Delta x(t) + K_{2}^{*} \Delta r(t), $$
(3)
where \( K_{1}^{*} \in R^{n \times M} \) and \( K_{2}^{*} \in R^{M \times M} \) are the nominal controller parameters [15], to make
$$ G_{c} (s) = C(sI - A - BK_{1}^{{*{\text{T}}}} )^{ - 1} BK_{2}^{*} = W_{m} (s). $$
(4)
The controller parameters are
$$ K_{1}^{*T} = K_{p}^{ - 1} K_{0} , \, K_{2}^{*} = K_{p}^{ - 1}, $$
(5)
where \( K_{p} \) is a finite and nonsingular high frequency gain matrix and is defined as [16]
$$ K_{p} = \mathop {\lim }\limits_{s \to \infty } \xi_{m} \left( s \right)G\left( s \right) = \left[ {\begin{array}{*{20}c} {c_{1} A^{{l_{1} - 1}} B} \\ {c_{2} A^{{l_{2} - 1}} B} \\ \vdots \\ {c_{M} A^{{l_{M} - 1}} B} \\ \end{array} } \right] $$
(6)
where \( \xi_{m}^{{}} (s) \in R^{M \times M} \) is a lower triangular polynomial matrix defined as a modified left interactor matrix of the transfer matrix \( G(s), \) having the form of
$$ \xi_{m}^{{}} (s){ = }W_{m}^{ - 1} (s) $$
(7)
and
$$ \begin{aligned} K_{0} & = [K_{01} ; \cdots ;K_{0M} ] \\ K_{0i} & = - c_{i} A^{li} - a_{i1} c_{i} A^{li - 1} - \cdots - a_{{il_{i} }} c_{i} \\ \end{aligned} $$
(8)
with \( c_{i} \) \( (i = 1, \cdots ,M) \) being the ith row of matrix C, and \( a_{ij} \) being the coefficient of the element \( s^{{l_{i} - j}} \) in the polynominal \( d_{i} (s) \) (Goodwin and Salgado [17]).
A MRAC control law is designed as:
$$ \Delta u(t) = K_{1}^{\rm T} (t)\Delta x(t) + K_{2} (t)\Delta r(t), $$
(9)
where \( K_{1}^{\rm T} (t) \) and \( K_{2} (t) \) are the estimates of \( K_{1}^{*T} \) and \( K_{2}^{*} \), which are updated by some stable adaptive laws.
Substituting the control law (9) into the plant model (1), we have
$$ \left\{ {\begin{array}{l} {\Delta \dot{x}(t) = (A + BK_{1}^{*T} )\Delta x(t) + BK_{2}^{*} \Delta r(t)} \hfill \\ \qquad \qquad\,\, { + B[K_{1}^{\rm T} - K_{1}^{*T} ]\Delta x(t) + B[K_{2} - K_{2}^{*} ]\Delta r(t)} \hfill \\ {\Delta y(t) = C\Delta x(t)} \hfill \\ \end{array} } \right. $$
(10)
and the output tracking error is
$$ \Delta e(t) = \Delta y(t) - \Delta y_{m} (t) = W_{m} (s)K_{p}^{{}} [\tilde{\varTheta }^{\rm T} \omega ](t), $$
(11)
where \( \varTheta^{\rm T} = [K_{1}^{\rm T} ,K_{2}^{{}} ] \), \( \varTheta^{*T} = [K_{1}^{*T} ,K_{2}^{*} ] \), \( \omega (t) = [\Delta x^{\rm T} (t),\Delta r^{\rm T} (t)]^{\rm T} \), and \( \tilde{\varTheta }(t) = \varTheta (t) - \varTheta^{*} \).

3.3 Adaptive law

From (7) and (11), we obtain
$$ \xi_{m} (s)[\Delta e](t) = K_{P} \tilde{\varTheta }^{\rm T} (t)\omega (t). $$
(12)
The LDS decomposition can be used to deal with the uncertainty in the high frequency gain matrix \( K_{p} \) [18],
$$ K_{p} = L_{s} D_{s} S, $$
(13)
where \( S \) is an \( M \times M \) symmetric positive definite matrix, \( L_{s} \) is an \( M \times M \) unit lower triangular matrix, and \( D_{S} \) is an \( M \times M \) diagonal matrix.
Substituting (13) into (12) gives
$$ L_{s}^{ - 1} \xi_{m} (s)[\Delta e](t) = D_{s} S\tilde{\varTheta }^{\rm T} (t)\omega (t). $$
(14)
Parameterize the unknown matrix \( L_{s} \) by introducing \( \varTheta_{0}^{*} \):
$$ \varTheta_{0}^{*} = L_{s}^{ - 1} - I = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {\quad i \le j \le M} \hfill \\ {\theta_{ij}^{*} ,} \hfill & {\quad j < i < M} \hfill \\ \end{array} } \right. . $$
(15)
Then substituting \( L_{s} \) into \( \varTheta_{0}^{*} \) yields
$$ \begin{aligned} & \xi_{m} (s)[\Delta e](t) + \varTheta_{0}^{*} \xi_{m} (s)[\Delta e](t) \\ & \quad = D_{s} S\tilde{\varTheta }^{\rm T} (t)\omega . \\ \end{aligned} $$
(16)
Introduce a filter \( h(s) = 1/f(s) \) and \( f(s) \) is a stable and monic polynomial whose degree equals to the degree of \( \xi_{m} (s) \). Operating both sides of (16) by \( h(s)I_{M} \) leads to:
$$ \begin{aligned} & \bar{e} + [0,\theta_{2}^{*T} \eta_{2} (t),\theta_{3}^{*T} \eta_{3} (t), \ldots ,\theta_{M}^{*T} \eta_{M} (t)]^{\rm T} \\ & \quad = D_{s} Sh(s)[\tilde{\varTheta }^{\rm T} (t)]\omega (t), \\ \end{aligned} $$
(17)
where
$$ \begin{aligned} \bar{e}(t) & = \xi_{m} (s)h(s)[\Delta e](t) = [\bar{e}_{1} (t), \cdots, \bar{e}_{M} (t)]^{\rm T} \\ \eta_{i} (t) & = [\bar{e}_{1} (t),\bar{e}_{2} (t), \cdots, \bar{e}_{i - 1} (t)]^{\rm T} \in R^{i - 1} \\ \theta_{i}^{*} & = [\theta_{i1}^{*} ,\theta_{i2}^{*} , \cdots, \theta_{ii - 1}^{*} ]^{\rm T} , \, i = 2,3, \ldots ,M. \\ \end{aligned} $$
(18)
Based on this parameterized error equation, introduce the estimation error signal:
$$ \begin{aligned} \varepsilon (t) & = [0,\theta_{2}^{\rm T} \eta_{2} (t), \ldots ,\theta_{M}^{\rm T} \eta_{M} (t)]^{\rm T} \\ & \quad + \psi (t)\xi (t) + \bar{e}(t), \\ \end{aligned} $$
(19)
where \( \theta_{i} (t), \, i = 2,3, \ldots ,M \) are the estimates of \( \theta_{i}^{*} \), and \( \psi (t) \) is the estimate of \( \psi^{*} = D_{s} S \), and
$$ \begin{aligned} \xi (t) & = \varTheta^{\rm T} (t)\zeta (t) - h(s)[\varTheta^{\rm T} \omega ](t) \\ \zeta (t) & = h(s)[\omega ](t). \\ \end{aligned} $$
(20)
From (17)–(20), derive that
$$ \begin{aligned} \varepsilon (t) & = [0,\tilde{\theta }_{2}^{\rm T} \eta_{2} (t),\tilde{\theta }_{3}^{\rm T} \eta_{3} (t), \ldots ,\tilde{\theta }_{M}^{\rm T} \eta_{M} (t)]^{\rm T} \\ & \quad + \tilde{\psi }(t)\xi (t) + D_{s} S\tilde{\varTheta }^{\rm T} (t)\zeta (t), \\ \end{aligned} $$
(21)
where \( \tilde{\theta }_{i} (t) = \theta_{i} (t) - \theta_{i}^{*} ,i = 2,3, \ldots ,M \), \( \tilde{\psi }(t) = \psi (t) - \psi^{*} \) are the related parameter errors. Suggest the following laws:
$$ \begin{aligned} \dot{\varTheta }^{\rm T} (t) & = \frac{{ - D_{s} \varepsilon (t)\zeta^{\rm T} (t)}}{{m^{2} (t)}} \\ \dot{\varPsi }(t) & = \frac{{ - \varGamma \varepsilon (t)\xi^{\rm T} (t)}}{{m^{2} (t)}} \\ \dot{\theta }_{i} (t) & = - \frac{{\varGamma_{\theta i} \varepsilon_{i} (t)\eta_{i} (t)}}{{m^{2} (t)}},\quad i = 2,3, \ldots ,M, \\ \end{aligned} $$
(22)
where the signal \( \varepsilon (t) = [\varepsilon_{1} (t),\varepsilon_{2} (t), \cdots \varepsilon_{M} (t)]^{\rm T} \) is computed from (19), \( \varGamma_{\theta i} = \varGamma_{\theta i}^{\rm T} > 0 \), \( i = 2,3, \ldots ,M, \) and \( \varGamma = \varGamma^{\rm T} > 0 \) are adaption gain matrices and \( m(t) = \sqrt {1 + \zeta^{\rm T} (t)\zeta (t) + \xi^{\rm T} (t)\xi (t) + \sum\limits_{i = 2}^{M} {\eta_{i}^{\rm T} (t)\eta_{i} (t)} } \) is a standard normalization signal.

Guo et al. [19] proved through stability analysis that the above MRAC scheme with state feedback control law updated by the adaptive laws can guarantee the stability of the closed-loop system. Meanwhile, closed-loop signals can asymptotically track reference signals.

4 Automatic landing control design

In this section, the MRAC method is integrated with classical proportional-integral-derivative (PID) control to design an automatic landing control system. According to the control structure introduced in Sect. 2.3, the landing phase is divided into three segments, the final approach, flare, and decrab. The task of outer control loops is to provide the corresponding inner loops with appropriate reference signals to address the specific control objectives. On the other hand, the inner loops must track those signals despite perturbations and uncertainties. Note that the state signals are \( u,v,w,p,q,r,\phi ,\theta ,\psi \), all of which can be measured to apply the MRAC algorithm.

4.1 Longitudinal control architecture

During the final approach, the trajectory of the aircraft must track the ILS beam as close as possible. This can be achieved by minimizing the glide deviation \( (\Delta z) \). Moreover, the nominal longitudinal slope trajectory is \( \gamma_{gld} = - 3 \, \deg \) and the calibrated airspeed \( (V_{c} ) \) needs to be kept at \( 66{\text{ m/s}} \). Therefore, in this phase, the outer loop is to generate the reference vertical speed by feeding back the glide deviation \( (\Delta z) \) as \( \dot{\Delta }z \approx V_{z} \) and the inner loop is a vertical speed \( (V_{Z} ) \) and calibrated airspeed \( (V_{c} ) \) tracker.

Longitudinal outer control

An elementary proportional controller can be used to keep the aircraft on the nominal glide path, while the commanded calibrated airspeed \( (V_{c} ) \) variations \( V_{{c_{c} }} = \Delta V_{{c_{c} }} \) are simply set to 0:
$$ \begin{aligned} V_{{c_{c} }} & = 0 \\ V_{{z_{c} }} & = k_{{p_{z} }} \cdot \Delta_{z}. \\ \end{aligned} $$
(23)

Longitudinal inner control

The linearized longitudinal flight dynamic is described as
$$ \begin{aligned} \dot{x}_{\text{lon}} & = A_{\text{lon}} x_{\text{lon}} + B_{\text{lon}} u_{\text{lon}} \\ y_{\text{lon}} & = C_{\text{lon}} x_{\text{lon}}, \\ \end{aligned} $$
(24)
where \( x_{\text{lon}} = \left[ {u,w,q,\theta } \right]^{\rm T} ,\;u_{\text{lon}} = \left[ {\delta_{th} ,\delta_{e} } \right]^{\rm T} ,\;y_{\text{lon}} = \left[ {V_{c} ,V_{z} } \right]^{\rm T} \) are the system longitudinal state vector, input vector, and output vector, respectively.
The reference system transfer matrix can be chosen as:
$$ W_{m} \left( s \right){ = }\left[ {\begin{array}{*{20}c} {\frac{1}{{s + p_{1} }}} & 0 \\ 0 & {\frac{1}{{\left( {s + p_{2} } \right)\left( {s + p_{3} } \right)}}} \\ \end{array} } \right] $$
(25)
where \( p_{1} = - 1, \, p_{2} = p_{3} = - 0.5 \).
For the adaptive laws, we choose:
$$ \begin{aligned} D_{s} & = {\text{diag}}\{ 0.0005,0.0005\} , \\ \varGamma_{{_{{\theta_{2} }} }} & = 10, \, \\ \varGamma & = {\text{diag}}\{ 1000,1000\} . \\ \end{aligned} $$
(26)

4.2 Lateral-directional control architecture

The aircraft is held level \( (\phi = 0) \) and the sideslip angle \( (\beta ) \) is controlled to be zero with the coordinated use of the ailerons and rudder during landing. During the final approach, the aircraft minimizes the localizer deviation \( (\Delta y) \) to track the ILS localizer beam and the bank angle \( \phi_{c} \) reference is provided by feeding back the localizer deviation \( (\Delta y) \). The reference for lateral load factor \( n_{y} \) which replaces the sideslip angle \( (\beta ) \) is zero.

Lateral-directional outer control

The lateral deviation \( (\Delta y) \) is controlled via a proportional-derivative controller and the reference lateral load factor \( n_{y} \) is simply set to zero in this phase:
$$ \begin{aligned} n_{{y_{c} }} & = 0 \\ \phi_{c} & = k_{{p_{y} }} \cdot \Delta_{y} + k_{{d_{y} }} \cdot \dot{\Delta }_{y}. \\ \end{aligned} $$
(27)

Lateral-directional inner control

The linearized lateral flight dynamic is described as:
$$ \begin{aligned} \dot{x}_{{\rm lat}} &= A_{{\rm lat}} x_{{\rm lat}} + B_{{\rm lat}} u_{{\rm lat}} \hfill \\ y_{{\rm lat}} &= C_{{\rm lat}} x_{{\rm lat}}, \hfill \\ \end{aligned} $$
(28)
where \( x_{{\rm lat}} = \left[ {v,p,r,\phi ,\psi } \right]^{\rm T} , \, u_{{\rm lat}} = \left[ {\delta_{a} ,\delta_{r} } \right]^{\rm T} , \, y_{{\rm lat}} = \left[ {n_{y} ,\phi } \right]^{\rm T} \) are the system lateral state vector, input vector and output vector, respectively.
The reference system transfer matrix can be chosen as:
$$ W_{m} \left( s \right){ = }\left[ {\begin{array}{*{20}c} {\frac{1}{{s + p_{1} }}} & 0 \\ 0 & {\frac{1}{{\left( {s + p_{2} } \right)\left( {s + p_{3} } \right)}}} \\ \end{array} } \right] $$
(29)
where \( p_{1} = - 0.5, \, p_{2} = p_{3} = - 1 \).
For the adaptive laws, we choose
$$ \begin{aligned} D_{s} & = {\text{diag}}\{ 0.0006,0.0006\} , \\ \varGamma_{{_{{\theta_{2} }} }} & = 10, \, \\ \varGamma & = {\text{diag}}\{ 5000,5000\} . \\ \end{aligned} $$
(30)

4.3 Flare

The objective of the flare phase which is activated at the height of 15 m is to provide a vertical speed control so that the aircraft hits the runway with a lower vertical speed. Moreover, the throttle is set back to idle, so the only remaining effective control input is the elevator. In this paper, the following structure is used:
$$ \delta_{e} = K_{p} (V_{zc} - V_{z} ), $$
(31)
where the reference signal equals to \( - 0.75{\text{ m/s}} \).

4.4 Decrab

The objective of the align phase which starts at the height of 9 m is to make sure that the plane coincides with the runway centerline at touchdown. Thus, the lateral outer loop is simply changed as follows: the reference lateral load factor \( n_{y} \) is determined based on the feedback heading angle error \( \psi \),
$$ n_{{y_{c} }} = k_{\text{align}} \cdot \Delta \psi $$
(32)
and the reference \( \phi \) is set to zero to keep the plane wings level.

5 Simulation results

In this section, we apply the automated landing control law to the nonlinear transport aircraft model to assess its effectiveness. Two sets of simulations are conducted in MATLAB/Simulink, one under a nominal condition and the other with parametric variations. The preliminary simulation results verified that the control system designed in this paper can accomplish the landing task accurately and maintain the closed-loop stability and robustness when the aircraft encounters some uncertainties (e.g., mass, CG position, and temperature variations).

5.1 Nominal condition

In a nominal condition (mass = 150 t and \( x_{\text{cg}} = 21\% \)), the process of landing is given in Figs. 5, 6, and 7. In these figures, the blue lines are the closed-loop responses using the above MRAC method and the red lines are the closed-loop responses using a PID controller given in Biannic and Boada-Bauxell [4]. Overall, the controlled performance for both methods is acceptable. The simulation results demonstrate that the responses of the aircraft during landing are close to the expectation. During the final approach, the trajectory of the aircraft keeps close to the ILS beam and the calibrated airspeed is basically maintained at \( 66{\text{ m/s}} \). The aircraft is held level \( (\phi = 0) \) and the aerodynamic sideslip angle \( (\beta ) \) is controlled to zero after 50 s.
Fig. 5

The longitudinal variables

Fig. 6

The lateral variables

Fig. 7

The aircraft trajectory

5.2 Monte Carlo assessment

The assessment is carried out through Monte Carlo analysis using 1000 simulations. Each landing configuration is obtained by dispersing a set of three parameters and Table 2 lists the distribution characteristics for those parameters.
Table 2

Distributions for parameters affecting the landing performances

Parameter

Distribution

Min

Max

Aircraft mass

Uniform

60 t

180 t

Center of gravity

Uniform

15%

41%

Temperature

Uniform

− 69 °C

+ 40 °C

For each case, according to the CS-AWO airworthiness criteria, we need to evaluate the four parameters listed in Table 3.
Table 3

Risks evaluation

Parameter

Average

Limit

Longitudinal touchdown earlier than a point on the runway 60 m (200 ft) from the threshold. (HTP60)

\( 10^{ - 6} \)

\( 10^{ - 5} \)

Lateral touchdown with the outboard landing gear greater than 21 m (70 ft) from the runway centreline, assuming a 45 m (150 ft) runway. (YTP)

\( 10^{ - 6} \)

\( 10^{ - 5} \)

Bank angle such that wing tip touches ground before wheels. (PHI)

\( 10^{ - 8} \)

\( 10^{ - 7} \)

Lateral velocity or slip angle for structural limit load. (SSTP)

\( 10^{ - 6} \)

\( 10^{ - 5} \)

In this Monte Carlo evaluation, the Gaussian distribution of the longitudinal and lateral requirements are evaluated. The plots of Figs. 8 and 9 are histograms of the variables in Table 3 based on 1000 landings for MRAC and PID controllers, respectively. Figure 10 gives the resulting cumulative distribution plots. To satisfy the airworthiness requirements, the graphs should stay out of the yellow and red shaded areas. It can be seen that for MRAC controller all the average longitudinal and lateral requirements are fulfilled, but for PID controller the requirements do not meet the airworthiness standards.
Fig. 8

The histogram for MRAC controller

Fig. 9

The histogram for PID controller

Fig. 10

The resulting cumulative distribution

The mean values and standard deviations of airworthiness-evaluated parameters are listed in Table 4, showing that the performance of the control system is statistically good for MRAC controller.
Table 4

Distributions for parameters affecting the landing performances

Parameter

Mean (MRAC)

Variance (MRAC)

Mean (PID)

Variance (PID)

HTP60 (m)

7.1968

2.1164

6.5022

6.5022

YTP (m)

− 0.0096

0.0013

− 0.2225

− 0.2225

PHI (deg)

− 6.6281e−04

2.9268e−08

0.1191

0.1191

SSTP (deg)

0.0016

6.9623e−07

0.1242

0.1242

6 Conclusion

This paper presents an automated landing system, in which the landing phase is divided into the final approach, flare, and decrab phases. Separate controllers are designed for longitudinal and lateral-directional channels by integrating the PID control method and state feedback output tracking multivariable MRAC scheme. The performance is demonstrated through Monte Carlo nonlinear simulations with uncertainties taken into account. In addition, the proposed control scheme is compared with the traditional PID control method. The Monte Carlo assessment results show that the MRAC method is better than the traditional PID method in the presence of uncertainties.

Notes

Acknowledgments

This research was sponsored by the National Natural Science Foundation of China under the Grant #61473186.

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Copyright information

© Shanghai Jiao Tong University 2018

Authors and Affiliations

  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiChina

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