Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 7345056, 12 pages http://dx.doi.org/10.1155/2016/7345056

Research Article Adaptive Disturbance Rejection Control for Automatic Carrier Landing System Xin Wang, Xin Chen, and Liyan Wen College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China Correspondence should be addressed to Xin Chen; [email protected] Received 27 March 2016; Revised 25 June 2016; Accepted 5 July 2016 Academic Editor: Jean J. Loiseau Copyright Β© 2016 Xin Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.

1. 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 systemβs 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 π»β 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 [13β15], and back-stepping control designs [16, 17] were proposed. In [18], an extension of biobjective optimal control modification for unmatched uncertain

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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. (3) The proposed LDU decomposition-based disturbance rejection techniques are used to solve a typical carrier landing aircraft turbulence compensation problem. Extensive simulation results are obtained through a longitudinal aircraft dynamic model during aircraft landing. 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.

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

βπ0 sin πΌ0 β π·πΌ0

2.1. 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 ππΜ = π cos πΌ β π· β ππ sin πΎ

(1)

πππΎΜ = π sin πΌ + πΏ β ππ cos πΎ. The kinematic equation is πΜ = π.

(2)

The moment equation is πΜ =

π . πΌπ¦

(3)

The navigation equation is βΜ = π sin πΎ.

(4)

The identical equation is πΎ = π β πΌ.

(5)

2.2. 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

2. Longitudinal Model of Carrier-Based Aircraft on Final-Approach Dynamics in the Air Wake

ππ0 cos πΌ0 β π·π0

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.

π₯Μ 1 = π΄ 1 π₯1 + π΅1 π’

(6)

π¦1 = πΆ1 π₯1 ,

where π₯1 = [V, πΌ, π, π, β]π , π’1 = [πΏπ , πΏπ‘ ]π , and π¦1 = [V, πΌ, π, π, β]π are the system state vector, input vector, and output vector, respectively, and the matrices are

+ π cos πΎ0

0

βπ cos πΎ0

0

π0 cos πΎ0

β

π·β0

[ π π π ] [ ] [ ] [ πΏ π0 βπ0 cos πΌ0 β πΏ πΌ0 + ππ sin πΎ0 πΏ β0 ] βπ sin πΎ0 [ ] [ ] β 1 β [ ππ0 ππ0 π0 ππ0 ] [ ] ], π΄1 = [ [ πV ] π + π cos πΌ π πΏ π π πΏ π π π sin πΎ0 π sin πΎ0 π0 πΌ0 πΌ0 0 0 πΌΜ 0 πΌΜ 0 πΌΜ 0 [ 0 β πΌΜ 0 ( π0 ) β ( β ) + β ( ) 0 ] [ ] [ πΌπ¦ ] πΌπ¦ ππ0 πΌπ¦ πΌπ¦ ππ0 π0 πΌπ¦ πΌπ¦ πΌπ¦ π0 [ ] [ ] [ 0 0 1 0 0 ] [ ] [

sin πΎ0

βπ0 cos πΎ0

0

]

Mathematical Problems in Engineering

3

ππΏ cos πΌ0 π·πΏ β π‘ β π [ ] π π [ ] [ ] πΏ πΏπ ππΏπ‘ sin πΌ0 [ ] β β [ ] [ ] ππ0 ππ0 [ ] π΅1 = [ π , ππΌΜ 0 ππΏπ‘ sin πΌ0 ] [ πΏπ ππΌΜ 0 πΏ πΏπ ] β ( ) β ( )] [ [ πΌπ¦ ] πΌπ¦ ππ0 πΌπ¦ ππ0 [ ] [ ] 0 0 [ ] [ 1 [ [0 [ [ πΆ1 = [0 [ [0 [ [0

0

0

]

0 0 0 0 ] 1 0 0 0] ] 0 1 0 0] ]. ] 0 0 1 0] ] 0 0 0 1] (7)

2.3. 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. 2.3.1. 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 [19β21], and the components are presented as π’π₯ ππ€ 0 π β [ββ, β1750) { { { { { { 0.04 { { (π + 1750) π β [β1750, β1710) { { { β1710 + 1750 { { { { 0.02 { { (π + 1870) π β [β1710, β1630) { { { β1630 + 1710 { { { { β0.004 { { { (π + 430) π β [β1630, β1550) { { β1550 + 1630 { { { { = { β0.028 (π + 1130) π β [β1550, β1340) { β1340 + 1550 { { { { { 0.01 { { (π + 1844) π β [β1340, β1160) { { { β1160 + 1340 { { { { 0.15 β 0.038 { { { (π + 1418) π β [β1160, β400) { { β400 + 1160 { { { { { β0.15 { { π β [β400, β250) (π + 250) { { 150 { { { π β [β250, 0) {0

π’π¦ ππ€ 0 π β [ββ, β2650) { { { { { { { β0.076 { { (π + 2650) π β [β2650, β2400) { { { β2400 + 2650 { { { { { { β0.088 + 0.076 { { (π + 3667) π β [β2400, β2200) { { β2200 + 2400 { { { { { { { β0.08 + 0.088 π β [β2200, β1970) (π β 330) ={ { β1970 + 2200 { { { { { { 0.02 + 0.08 { { π β [β1970, β750) (π + 994) { { { β750 + 1970 { { { { { { β0.01 β 0.02 { { π β [β750, β250) (π + 417) { { { β250 + 750 { { { { { π β [β250, 0) {β0.01 π = βπ cos πΎ (π0 β π‘) β 286,

(8) where π is the distance between the aircraft and the ship center of pitch, negative after of ship, π0 is the total landing time, and π‘ is the present time. 2.3.2. 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 π’π₯ and π’π¦ . Because the flight speed is far larger than the wind speed, we can get ππ β π + π’π₯ πΌπ β πΌ +

π’π¦ π

,

(9)

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Mathematical Problems in Engineering

where ππ and πΌπ 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 π₯Μ 1 = π΄ 1 π₯1 + π΅1 π’1 + π΅1π π (π‘)

π΅1π

π¦1 = πΆ1 π₯1 , (10) where π΄ 1 , π΅1 , πΆ1 , π₯1 , and π’1 are defined in (6). The disturbance is π(π‘) = [π’π₯ , π’π¦ ] and the matrix π΅1π is

π sin πΌ0 β π·πΌ0 ππ0 cos πΌ0 β π·π0 β cos πΎ0 [ ] π ππ0 [ ] [ ] [ ] ππ0 sin πΌ0 + πΏ π0 π sin πΌ0 + πΏ πΌ0 [ ] [ ] 2 [ ] ππ ππ 0 [ ] 0 ]. =[ [π πV0 ππΌΜ 0 πΏ πΌ0 + π0 sin πΌ0 ππΌ0 ] [ πΌΜ 0 ππ0 sin πΌ0 + πΏ π0 ] [ ] ( )β ( )β 2 [ πΌπ¦ ππ0 πΌπ¦ πΌπ¦ πΌπ¦ π0 ] ππ0 [ ] [ ] [ ] 0 0 [ ] 0 0 [ ]

3. 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. 3.1. Problem Formulation. Consider the linear time-invariant system as π₯Μ (π‘) = π΄π₯ (π‘) + π΅π’ (π‘) + π΅π π (π‘) π¦ (π‘) = πΆπ₯ (π‘) ,

(12)

where π΄ β π πΓπ , π΅ β π πΓπ, π΅π β π πΓπ , and πΆ β π πΓπ are constant and unknown; π₯(π‘) β π π , π’(π‘) β π π, and π¦(π‘) β π π are the system state vector, input vector, and output vector, respectively; π(π‘) = [π1 (π‘), . . . , ππ (π‘)]π β π π is the disturbance vector. The disturbance signal π(π‘) is unmatched with the control input π’(π‘), in the sense that π΅ and π΅π are not linearly dependent, π΅π =ΜΈ π΅πΌ for any matrix πΌ β π πΓπ . The control objective is to design an adaptive statefeedback control signal π’(π‘) for (12), to make closed-loop signal boundedness and the output π¦(π‘) track a chosen reference signal π¦π (π‘) generated from a reference model: π¦π (π‘) = ππ (π ) [π] (π‘) β π π,

(13)

πΓπ

where ππ (π ) β π is a stable transfer function matrix and π(π‘) β π π is an external reference input signal for defining a desired π¦π (π‘). Note that, in this paper, we use the notation π¦(π‘) = πΊ(π )[π’](π‘) to represent the output π¦(π‘) of a system whose transfer matrix is πΊ(π ) and input is π’(π‘), a convenient notation for adaptive control systems. 3.2. Preliminaries and Assumptions Lemma 1 (see [8]). For any πΓπ strictly proper and full rank rational transfer matrix πΊ(π ), there exists a lower triangular

(11)

polynomial matrix ππ (π ), defined as the left interactor matrix of πΊ(π ), of the form π1 (π ) [ βπ (π ) [ 21 [ ππ (π ) = [ . [ . [ .

0

0

0

π2 (π )

0

0

.. .

.. .

.. .

] ] ] ], ] ]

(14)

π π [βπ1 (π ) βπ2 (π ) β β β ππ (π )]

where βπππ (π ), π = 1, 2, . . . , π β 1, π = 2, . . . , π, are some polynomials and ππ (π ) are any chosen monic stable polynomials such that the high-frequency gain matrix of πΊ(π ) defined as πΎπ = limπ ββ ππ (π ) πΊ(π ). Lemma 2 (see [24]). Every π Γ π real matrix πΎπ with nonzero leading principal minors Ξ 1 , Ξ 2 , . . . , Ξ π can be uniquely factored as πΎπ = πΏπ·π,

(15)

where πΏ β π πΓπ is unity lower triangular, π β π πΓπ is unity upper triangular, and π· = diag {π1 , π2 , . . . , ππ } = diag {Ξ 1 , Ξ 2 Ξβ1 , . . . , Ξ π Ξβ1 πβ1 } .

(16)

Assumption 3. All zeros of πΊ0 (π ) = πΆ(π πΌ β π΄)β1 π΅ are stabilizable and detectable. Assumption 4. πΊ0 (π ) is strictly proper with full rank and has a known modified interactor matrix ππ (π ) such that πΎπ = limπ ββ ππ (π )πΊ0 (π ) is finite and nonsingular (so that ππ (π ) = β1 ππ (π ) can be chosen as the transfer matrix for the reference model system).

Mathematical Problems in Engineering

5 (2) Break turn (1) Upwind leg

800 ft, 250 kt, level turn break at 45βΌ60 deg bank speed brake extended throttle idle

800 ft altitude, 300βΌ360 kt, autopilot

(5) Final approach

Line up with the deck centerline β3.5 deg ο¬ight path-angle 0.75 nmile behind the carrier 45 deg position 295 ft altitude

Landing gear down below 250 kt

90 deg position 370 ft altitude (4) Base leg 0.75 to 1 mile

(3) Downwind leg

20βΌ30 deg bank angle descending to 450 ft

Landing checklist descending to 600 ft

Figure 1: Procedures of carrier landing for the aircraft.

Final approach 295 ft

where the nominal parameters πΎ1βπ β π πΓπ and πΎ2βπ β π πΓπ are for the plant-model output matching, and πΎ3β (π‘) β π π is used to cancel the effect of the disturbance π(π‘).

πΎ = β3.5β

rela Glidesl tive o to th pe e ca r r ie

r

Air wake

Lemma 8 (see [25]). The matrix πΎπ = limπ ββ ππ (π )πΊπ (π ) is finite if π0β1 (π )ππ (π ) is proper.

0.75 nmile

Based on Lemma 8, the existence of a nominal controller (17) is established a follows.

Figure 2: Final carrier landing phase.

Assumption 5. The leading principal minors of the highfrequency gain matrix πΎπ are nonzero, and their signs are known. From plant (12), the control and disturbance transfer functions are obtained as πΊ0 (π ) = πΆ(π πΌ β π΄)β1 π΅ and πΊπ (π ) = πΆ(π πΌ β π΄)β1 π΅π and are expressed in their left coprime polynomial matrix decompositions: πΊ0 (π ) = ππβ1 (π )π0 (π ) and πΊπ (π ) = ππβ1 (π )ππ (π ), where ππ (π ), π0 (π ) β π πΓπ and ππ (π ) β π πΓπ are some polynomial matrices. Assumption 6. The transfer matrix π0β1 (π )ππ (π ) is proper. 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 π’(π‘) and the disturbance input π(π‘) to the output π¦(π‘) for the design of a derivative-free disturbance rejection scheme. 3.3. Nominal Disturbance Rejection Design. With the knowledge of the plant and disturbance parameters, the nominal state-feedback controller is π’β (π‘) = πΎ1βπ π₯ (π‘) + πΎ2β π (π‘) + πΎ3β (π‘) ,

(17)

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 π¦π (π‘). From plant (12), the input-output form is obtained as π¦ (π‘) = πΊ0 (π ) [π’] (π‘) + π¦ (π‘) ,

(18)

β1

with π¦(π‘) = πΊ0 (π )[π](π‘) = πΆ(π πΌ β π΄) π΅π [π](π‘). Operate the interactor matrix (a polynomial matrix) Μ = π΄π₯(π‘) + π΅π’(π‘) + π΅π π(π‘), π¦(π‘) = ππ (π ) on plant (12), π₯(π‘) πΆπ₯(π‘), to reach an expression of ππ (π )[π¦π ](π‘) in a possible form: ππ (π ) [π¦] (π‘) = βπΎ0 π₯ (π‘) + πΎπ π’ (π‘) + πΎπ1 π’Μ + β β β + πΎππ0 π’(π0 ) (π‘) + πΎπ π (π‘) + πΎπ1 πΜ (π‘)

(19)

+ β β β + πΎππ1 π(π1) (π‘) , with some constant matrices πΎ0 β π πΓπ , πΎπ β π πΓπ, πΎππ β π πΓπ, π = 1, 2, . . . , π0 , πΎπ β π πΓπ , and πΎππ β π πΓπ , π = 1, 2, . . . , π1 , for some integers π0 , π1 β₯ 0. From (12) and (18), we have π₯ (π ) = (π πΌ β π΄)β1 π΅π’ (π ) + (π πΌ β π΄)β1 π΅π π (π ) .

(20)

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Expressing (19) in π domain and using (20), we have ππ (π ) π¦ (π ) = βπΎ0 (π πΌ β π΄)β1 π΅π’ (π ) + πΎπ π’ (π ) + πΎπ1 π π’ (π ) + β β β + πΎππ0 π π0 π’ (π ) β πΎ0 (π πΌ β π΄)β1 π΅π π (π ) + πΎπ π (π )

(21)

+ πΎπ1 π π (π ) + β β β + πΎππ1 π π1 π (π ) . From Assumption 4, πΎπ = limπ ββ ππ (π )πΊ0 (π ) is finite and nonsingular and from Assumption 6, πΎππ = 0, π = 1, . . . , π0 , πΎπ = πΎπ , and πΎππ = 0, π = 1, . . . , π0 , πΎπ = πΎπ . Hence, we have ππ (π ) [π¦] (π‘) = βπΎ0 π₯ (π‘) + πΎπ π’ (π‘) + πΎπ π (π‘) .

(23)

β1

π¦ (π‘) = πΆ (π πΌ β π΄ β π΅πΎ1βπ ) π΅πΎ2β [π] (π‘) β1 β1

π

ππβ = [ππ0 , ππ1 , . . . , ππππ ] β π ππ +1 , π

ππ (π‘) = [1, ππ1 (π‘) , . . . , ππππ (π‘)] β π ππ +1 , π = 1, 2, . . . , π.

π (π‘) = πβπ π (π‘) ,

πβπ

π1π 0π(π2 +1) 0π(π3 +1) β β β 0π(ππ +1) [ π ] [0(π +1) π2βπ 0π(π +1) β β β 0π(π +1) ] [ 1 ] 3 π πΓπ ] =[ .. .. .. .. ] β π , [ .. [ . ] . . . . ] (28) [ π π π βπ [0(π1 +1) 0(π2 +1) 0(π3 +1) β β β ππ ] π

(24)

π (π‘) = [π1π (π‘) π2π (π‘) β β β πππ (π‘)] β π π π = π1 + π2 + β β β + ππ + π.

+ πΆ (π πΌ β π΄ β π΅πΎ1βπ ) π΅π π (π ) = ππ (π ) [π] (π‘) = π¦π (π‘) . Remark 10. From (24), we can conclude that the plant-model matching conditions are πΆ (π πΌ β π΄ β =

β1 π΅πΎ1βπ )

(27)

Hence, the disturbance π(π‘) is expressed as

where πΎ1βπ = πΎπβ1 πΎ0 , πΎ2β = πΎπβ1 , and πΎ3β (π‘) = πΎ3π π(π‘) with πΎ3π (π‘) = βπΎπβ1 πΎπ , which leads to the output matching: ππ (π )[π¦](π‘) = π(π‘). We applied (23) to plant (12); we have

+ πΆ (π πΌ β π΄ β π΅πΎ1βπ ) π΅ [πΎ3β ] (π‘)

The parameter matrix and the disturbance signal components are

(22)

From (17), the control law can be designed as π’ (π‘) = π’β (π‘) = πΎ1βπ π₯ (π‘) + πΎ2β π (π‘) + πΎ3β (π‘) ,

Remark 11. For model (26), if ππ = 0, then ππ (π‘) = ππ0 , π = 1, 2, . . . , π, representing a constant disturbance signal. If we choose ππ = 1 and ππ1 (π‘) = sin(π€π1 π‘), then ππ (π‘) = ππ0 + ππ1 sin(π€π1 π‘), π = 1, 2, . . . , π, representing time-variant period disturbances. A large class of practical disturbances in control applications can be approximated by a proper selection of this basis function πππ (π‘) in (26).

π΅πΎ2β

β β β β β = [π3π1 , π3π2 , . . . , π3ππ ], π3ππ β π π, π = With πΎ3π 1, 2, . . . , π, the disturbance rejection term πΎ3β (π‘) is parameterized as β β πΎ3β (π‘) = πΎ3π π (π‘) = πΎ3π πβπ π (π‘) = Ξ¦β3 π (π‘) ,

(29)

where the parameter matrix is

ππ (π ) ππ (π ) πΎ2ββ1 πΎ3β

(π )

(25)

β1

β β β Ξ¦β3 = [π31 , π32 , . . . , π3π ] β π πΓπ ,

+ πΆ (π πΌ β π΄ β π΅πΎ1βπ ) π΅π π (π ) = 0.

π = π1 + π2 + β β β + ππ + π, (30)

For the plant-model matching condition (25), there exist constant matrices πΎ1βπ and πΎ2β and πΎ3β (π‘), if and only if the relative degree condition in Assumption 6 is satisfied. 3.4. Parameterizations of the Term πΎ3β (π‘). For the disturbance vector π(π‘) β π π , each element ππ (π‘) in (12) can be expressed as ππ

ππ (π‘) = ππ0 + β πππ πππ (π‘) = ππβπ ππ (π‘) , π=1

(26)

π = 1, 2, . . . , π, with some unknown constants ππ0 , πππ and some known bounded continuous signals πππ , π = 1, 2, . . . , π, π = 1, 2, . . . , ππ .

β β = π3ππ ππβπ β π πΓ(ππ +1) , π = 1, 2, . . . , π. π3π

Next, the adaptive disturbance rejection design will be studied for the plant with uncertainties from the plant and unmatched disturbances. 3.5. Error Equation. Applying (23) to plant (12), the closedloop system becomes π₯Μ (π‘) = (π΄ + π΅πΎ1βπ ) π₯ (π‘) + π΅πΎ2β π (π‘) + π΅πΎ3β (π‘) + π΅π π (π‘) + π΅ [π’ (π‘) β πΎ1βπ π₯ (π‘) β πΎ2β π (π‘) β πΎ3β (π‘)] π¦ (π‘) = πΆπ₯ (π‘) .

(31)

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7

In view of (24), (25), and (31), the output tracking error equation is π (π‘) = π¦ (π‘) β π¦π (π‘) = ππ (π ) πΎπβ [π’ β πΎ1βπ π₯ β πΎ2β π β πΎ3β ] (π‘) + ππ (π‘) ,

(32)

πΎπβ = πΎ2ββ1 , βπ

where ππ (π‘) = πΆπ(π΄+π΅πΎ1 )π‘ π₯(0) converges to zero exponenβ1 tially fast due to the stability of π΄ + π΅πΎ1βπ and ππ (π ) = ππ (π ). Hence, we have ππ (π ) [π] (π‘) = πΎπ [π’ (π‘) β πΎ1βπ π₯ (π‘) β πΎ2β π (π‘) β πΎ3β (π‘)] .

(33)

To deal with the uncertainty of the high-frequency gain matrix πΎπ , the LDU decomposition of πΎπ is used in (33), so that we have

Μ = Ξ¦(π‘) β Ξ¦β and Ξ¦π(π‘) = where the parameter error is Ξ¦(π‘) π [Ξ¦0 (π‘), Ξ¦1 (π‘)] is the estimate of unknown parameter matrix π π π Ξ¦βπ = [Ξ¦β0 , Ξ¦βπ 1 ], π(π‘) = [π’ (π‘), π (π‘)] , and π(π‘) = π π π π β β1 [π₯ (π‘), π (π‘), π (π‘)] , where Ξ0 = (πΏ β πΌ) is introduced to parameterize the unknown matrix πΏ, which has a special form: 0 0 β β β 0 [ β ] [ π21 0 β β β 0] [ ] [ πβ πβ ] β β β 0 [ 31 ] 32 [ ] β πΓπ . (41) Ξ0 = [ . ]βπ .. .. [ . ] . . 0] [ . [ ] [πβ 0 0] [ πβ11 β β β ] β β [ ππ1 β β β πππβ1 0] For such a matrix Ξβ0 , the parameter vectors are defined as β β π , π2β = π21 π

β β , π32 ] β π 2 , π3β = [π31

β1

πΏ ππ (π ) [π] (π‘) = π·π (π’ (π‘) β πΎ1βπ π₯ (π‘) β πΎ2β π (π‘) β πΎ3β (π‘)) .

.. .

(34)

β β β ππβ1 = [ππβ11 , . . . , ππβ1πβ2 ] β π πβ2 ,

We have the following equation: ππ’ (π‘) = π’ (π‘) β (πΌ β π) π’ (π‘) .

πΏβ1 ππ (π ) [π] (π‘) = π· [π’ (π‘) β (πΌ β π) π’ (π‘) β

β

πΎ2β π (π‘)

β

πΎ3β

(π‘)]] .

β β β ππ = [ππ1 , . . . , πππβ2 ] β π πβ1 ,

(35)

and their estimates are π2 (π‘) = π21 (π‘) β π ,

With (34) and (35), we have

π [πΎ1βπ π₯ (π‘)

π

π3 (π‘) = [π31 (π‘) , π32 (π‘)] β π 2 ,

(36)

.. .

The new equation is πΏβ1 ππ (π ) [π] (π‘) = π· [π’ (π‘) β Ξ¦β0 π’ (π‘) β Ξ¦βπ 1 π (π‘)] ,

(37)

βπ β β π π where Ξ¦βπ 1 = [ππΎ1 , ππΎ2 , ππΎ3π ] and π(π‘) = [π₯ (π‘), π (π‘), ππ (π‘)]π . This new equation has a new controller structure:

π’ (π‘) = Ξ¦0 π’ (π‘) + Ξ¦π1 π (π‘) ,

(38)

where Ξ¦0 and Ξ¦π1 are the estimates of Ξ¦β0 and Ξ¦βπ 1 and Ξ¦0 is upper triangular with zero diagonal elements (only its nonzero elements are estimated). The matrix Ξ¦0 has the same strictly form as that of Ξ¦β0 = (πΌ β π): 0 π12 π13 β β β

π1π

] [ [0 0 π23 β β β π2π ] ] [ ] [. . . . ] β π πΓπ. [ .. .. Ξ¦0 = [ .. .. ] ] [ ] [0 0 β β β 0 π [ πβ1π ] [0 0

β β β 0

0

ππ (π ) [π] (π‘) +

ππβ1 (π‘) = [ππβ11 (π‘) , . . . , ππβ1πβ2 (π‘)] β π πβ2 , ππ (π‘) = [ππ1 (π‘) , . . . , πππβ2 (π‘)] β π πβ1 . We introduce a filter β(π ) = 1/π(π ), where π(π ) is chosen as a stable and monic polynomial whose degree is equal to the maximum degree of the modified interactor ππ (π ). Operating both sides of (40) by β(π )πΌπ leads to ππ (π ) β (π ) [π] (π‘) + Ξβ0 ππ (π ) β (π ) [π] (π‘) (44)

Μ π π] (π‘) . = π·β (π ) [Ξ¦ We define π

(45)

π

(39)

ππ (π‘) = [π1 (π‘) , . . . , ππβ1 (π‘)] β π πβ1 , π = 2, 3, . . . , π.

(46)

From (45) and (46) in (44), we obtained

]

Μ π (π‘) π (π‘) , (π ) [π] (π‘) = π·Ξ¦

(43)

π (π‘) = ππ (π ) β (π ) [π] (π‘) = [π1 (π‘) , . . . , ππ (π‘)] ,

βπ π (π‘) + [0, π2βπ π2 (π‘) , π3βπ π3 (π‘) , . . . , ππ ππ (π‘)]

From (37) and (38), we obtain a new error model: Ξβ0 ππ

(42)

(40)

π

Μ π] (π‘) . = π·β (π ) [Ξ¦

π

(47)

8

Mathematical Problems in Engineering

Based on the parameterized error equation (47), an estimation error signal is introduced: π

π ππ (π‘)] π (π‘) = π (π‘) + [0, π2π π2 (π‘) , π3π π3 (π‘) , . . . , ππ

+ Ξ¨ (π‘) π (π‘) β π π, where Ξ¨(π‘) β π

πΓπ

(48)

β

is the estimate of Ξ¨ = π· and

π (π‘) = β (π ) [π] (π‘) ,

π = 1, 2, . . . , π,

π (π‘) = Ξ¦π (π‘) π (π‘) β β (π ) [Ξ¦ππ] (π‘) ,

(49)

It then follows from (44) (47) and (48) that π

(50)

Μ (π‘) π (π‘) , +Ξ¨

3.6. Adaptive Laws. Based on the error model (50), the adaptive laws are chosen as Ξππ ππ (π‘) ππ (π‘) π2 (π‘)

,

π = 2, . . . , π,

+ πΎ2 (π‘) π (π‘) + Ξ¦3 (π‘) π (π‘) ,

(51)

π1 (π‘) =

π (π ) [π’] (π‘) , Ξ (π )

π2 (π‘) =

π (π ) [π¦] (π‘) , Ξ (π )

π3 (π‘) =

π (π ) [π] (π‘) , Ξ (π )

π

Ξπ (π‘) π (π‘) Ξ¨Μ (π‘) = β , π2 (π‘) where Ξππ = Ξππ π > 0, π = 2, 3, . . . , π, and Ξ = Ξπ > 0 are adaptive gains; π· is defined in (15); π(π‘) = [π1 (π‘), π2 (π‘), . . . , ππ(π‘)]π is calculated from (50); and π

π2 (π‘) = 1 + ππ (π‘) π (π‘) + ππ (π‘) π (π‘) + βπππ ππ (π‘) .

(52)

π=2

3.7. Stability Analysis. To analyze the closed-loop system stability, we first establish some desired properties of the adaptive parameter update laws mentioned above. Lemma 12 (see [8]). The adaptive laws ensure that (i) ππ (π‘) β πΏβ , π = 2, 3, . . . , π, Ξ¦(π‘) β πΏβ , Ξ¨(π‘) β πΏβ , and π(π‘)/π(π‘) β πΏ2 β© πΏβ ; Μ β πΏ2 β© πΏβ , and (ii) πΜ π (π‘) β πΏ2 β© πΏβ , π = 2, 3, . . . , π, Ξ¦(π‘) 2 β Μ βπΏ β©πΏ . Ξ¨(π‘) Based on Lemma 12, the following desired closed-loop system properties are established. Theorem 13. For plant (12) with uncertainties from the system parameters and disturbance (26) under Assumptions 3β6,

(54)

where Ξ¦π1(π‘), Ξ¦π2 (π‘), Ξ¦π3π (π‘), πΎ2 (π‘), and Ξ¦3 (π‘) are adaptive estimates of the corresponding nominal controller parameters and

π

π·π (π‘) π (π‘) π Ξ¦Μ (π‘) = β , π2 (π‘)

(53)

where πΏ β π πΓπ is a gain matrix such that π΄ β πΏπΆ is stable, which is possible, and (π΄ πΆ) is assumed to be detectable. Hence, we have π’ (π‘) = Ξ¦π1 (π‘) π1 (π‘) + Ξ¦π2 (π‘) π2 (π‘) + Ξ¦π3π (π‘) π3 (π‘)

Μ = Ξ¨(π‘) β Ξ¨β and πΜπ = ππ β πβ are the parameter where Ξ¨(π‘) π errors. This error model is choice for update laws.

πΜ π (π‘) = β

Proof (outline). The proof of this stability theorem can be established through using a unified framework. Because the control input π’(π‘) described in (38) depends on the state π₯(π‘), it first needs to be expressed by using the system output π¦(π‘) through establishing the state observer of the plant: ΜΜ (π‘) = (π΄ β πΏπΆ) π₯ ΜΜ (π‘) + π΅π’ (π‘) + π΅π π (π‘) + πΏπ¦ (π‘) , π₯

π = 1, 2, . . . , π.

π π π Μπ π (π‘) = [0, πΜ2 π2 (π‘) , πΜ3 π3 (π‘) , . . . , πΜπππ (π‘)] + π·Ξ¦

and the reference model (13), the LDU decomposition-based MRAC scheme with the adaptive controller (38) and adaptive parameter update laws (51) guarantees closed-loop system boundedness and asymptotic output tracking limπ‘ββ π(π‘) = 0 with π(π‘) = π¦(π‘) β π¦π (π‘).

(55)

= with π(π ) = [πΌπ π πΌπ β β β π πβ1 πΌπ]π , π(π ) π πβ1 [πΌπ π πΌπ β β β π πΌπ ] , and Ξ(π ) being a chosen monic stable polynomial of degree π, which has the same eigenvalues with π΄ β πΏπΆ. Then, introducing the fictitious filters for the plant π¦(π‘) = πΆ(π πΌ β π΄)β1 π΅π’(π‘) + πΆ(π πΌ β π΄)β1 π΅π π(π‘) and using series transformations, the control input described as in (54) is transformed into the form π’ (π‘) = πΊ11 (π , β ) [π¦] (π‘) + πΊ12 (π , β ) [π] (π‘) + πΊ13 (π , β ) [π] (π‘) + πΊ14 (π , β ) [ππ ] (π‘) ,

(56)

where π¦(π‘) = β(π )[π¦](π‘) (β(π ) is given below (43)) and πΊ11 (π , β ), πΊ12 (π , β ), πΊ13 (π , β ), and πΊ14 (π , β ) are proper stable operators with finite gains. Furthermore, a filtered version of the output signal π¦(π‘) is expressed in a feedback framework: π‘

σ΅© σ΅©σ΅© βπΌ (π‘βπ) π₯1 (π) σ΅©σ΅©π¦ (π‘)σ΅©σ΅©σ΅© β€ π₯0 + π½1 β« π 1 0 π

βπΌ2 (πβπ)

β (β« π 0

σ΅© σ΅©σ΅© σ΅©σ΅©π¦ (π)σ΅©σ΅©σ΅© ππ) ππ,

(57)

Mathematical Problems in Engineering

9

Μ for some π½1 , πΌ1 , πΌ2 > 0, and π₯1 (π‘) = βΞ¦(π‘)β + βπ(π‘)βπ(π‘) β 2 β πΏ β© πΏ . Applying the small gain lemma to (57), we conclude that π¦(π‘) β πΏβ , and so π¦(π‘), π’(π‘) β πΏβ . Thus, the signals satisfy π(π‘), π(π‘), π(π‘), π(π‘), π β πΏβ . Furthermore, Μ Μ β πΏ2 (Lemma 12) are satisfied, and in ππ , π(π‘)/π(π‘), Ξ¦(π‘), Ξ¨(π‘) turn π(π‘) and π(π‘) = π¦(π‘) β π¦π (π‘), such that π(π‘) = π¦(π‘) β π¦π (π‘) converges to zero. β0.03858 18.984 [ [ β0.001028 β0.63253 [ [ π΄ = [ 7.8601πΈ β 5 β0.75905 [ [ 0 0 [ [ β0.04362 β249.76

4. 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

0

β32.139

1

0.005612

1.3233πΈ β 4

] 3.7553πΈ β 6 ] ] β0.5183 β0.0079341 β3.0808πΈ β 7 ] ], ] ] 1 0 0 ] 0 249.76 0 ]

10.1 0 [ ] [β1.5446πΈ β 4 ] 0 [ ] [ 0.024656 β0.01077] π΅=[ ], [ ] [ ] 0 0 [ ] 0 0 [ ] πΆ=[

1 0 0 0 0 ]. 0 0 0 0 1

The turbulence disturbances are described in [22, 23]; we can get π΅π = [π΅π1 , π΅π2 ] , π

π΅π1 = [β0.0386 0.001 7.8601πΈ β 5 0 0] ,

(59)

π

π΅π2 = [β0.05262 0.00251 β0.00304 0 0] . 4.2. Adaptive Control Design. For the aircraft system, the transfer function, πΊ0 (π ) = πΆ(π πΌ β π΄)β1 π΅, has stable zeros, π 1 = β4.507, π 2 = β0.91, and π 3 = β0.5685, and is strictly proper and full rank. The interactor matrix is chosen as ππ (π ) = diag {π + 1 (π + 1)2 } .

(60)

The high-frequency matrix is πΎπ = lim ππ (π ) πΊ0 (π ) = [ π ββ

2.2673

8.8455

β0.0902 β0.3521

],

(61)

and it is is finite and nonsingular and the matrix πΎπ = lim ππ (π ) πΊπ (π ) = [ π ββ

(58)

β0.0386 β0.0526 β0.2481 β0.6246

]

(62)

is finite. From the specified left coprime polynomial matrix decompositions, πΊ0 (π ) = ππβ1 (π )π0 (π ) and πΊπ (π ) = ππβ1 (π )ππ (π ), we can obtain lim πβ1 π ββ 0

(π ) ππ (π ) = [

β0.017 β0.0064 2.7493 1.7739

],

(63)

which means that the relative degree condition in Assumption 6 can be ensured. The related gain parameters in adaptive laws (51) are chosen as Ξπ = 100, π· = diag{2 2}, and Ξ = diag{10 10}. 4.3. Simulation Results. For this simulations study, the initial state is chosen as π₯0 (π‘) = [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.

10

Mathematical Problems in Engineering Adaptive control without disturbance

350

350

300

300

250

250

200 150

200 150

100

100

50

50

0

0

10

20

30 40 Versus time (s)

50

Adaptive control with disturbace

400

Height (ft)

Height (ft)

400

60

0

70

0

Reference trajectory Adaptive control

40

30

30

20

20

Tracking error (ft)

Tracking error (ft)

40

10 0 β10

β20

β10

β20

β40

β40 30 40 Versus time (s)

70

0

β30

20

60

10

β30

10

50

Adaptive control with disturbance

50

0

30 40 Versus time (s)

Figure 5: Final landing phase altitude for the aircraft.

Adaptive control without disturbance

50

20

Reference trajectory Adaptive control

Figure 3: Final landing phase altitude for the aircraft.

β50

10

50

60

70

β50

10

20

30 40 Versus time (s)

50

60

Height reference tracking error

Height reference tracking error

Figure 4: Final landing phase tracking error for the aircraft.

0

Figure 6: Final landing phase tracking error for the aircraft.

5. Conclusions

Nomenclature

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.

π·πΏπ :

Aerodynamic drag derivative with respect to elevator deflection angle π, π·, πΏ: Engine thrust, aerodynamic drag force, and aerodynamic lift force πΌ, π, πΎ: Angle of attack, pitch, and flight-path slope β: Altitude of aircraft π: Gravitational acceleration π: Mass of aircraft π: Airspeed of aircraft π: Pith rate

70

Mathematical Problems in Engineering

References

Adaptive control with disturbance

6 Throttle (unit)

11

4 2 0 β2

0

10

20

0

10

20

30 40 Versus time (s)

50

60

70

50

60

70

Elevator (deg)

40 30 20 10 0

30

40

Versus time (s)

Figure 7: Final landing phase control signal.

πΌπ¦ : π: πΏπ , πΏπ‘ :

Moment of inertia in pitch The pitch moment Elevator deflection bias angle and engine throttle angle Aerodynamic pitch moment and drag ππ0 , π·π0 : derivative with respect to airspeed π0 ππ0 , πΏ π0 : Thrust and aerodynamic lift derivative with respect to airspeed π0 ππΏπ , πΏ πΏπ : Aerodynamic pitch moment and lift derivative with respect to πΏπ ππ€ , π’π₯ , π’π¦ : Turbulence velocity and body axis components of ππ€ π0 , πΌ0 , πΌΜ 0 , π0 , β0 : The trim value of aircraft state Aerodynamic pitch moment and drag ππ0 , π·π0 : derivative with respect to airspeed π0 ππ0 , πΏ π0 : Thrust and aerodynamic lift derivatives with respect to airspeed π0 ππΏπ , πΏ πΏπ : Aerodynamic pitch moment and lift derivative with respect to elevator Aerodynamic drag derivative with π·πΌ0 , π·β0 : respect to πΌ0 and β0 π·πΏπ : Aerodynamic drag due to πΏπ ππΌ0 , ππΌΜ 0 : Aerodynamic pitch moment with respect to πΌ0 and πΌΜ 0 ππ0 : Aerodynamic pitch moment with respect to π0 π0 : Benchmark aerodynamic thrust at the airspeed π0 ππΏπ‘ : Aerodynamic thrust derivatives with respect to the throttle Aerodynamic lift derivative with respect πΏ πΌ0 , πΏ β0 : to angle of attack and height.

Competing Interests The authors declare that they have no competing interests.

[1] J. M. Urnes, R. K. Hess, R. F. Moomaw, and R. W. Huff, βHdot automatic carrier landing system for approach control in turbulence,β Journal of Guidance, Control, and Dynamics, vol. 4, no. 2, pp. 177β183, 1981. [2] A. L. Prickett and C. J. Parkes, βFlight testing of the F/A-18E/F automatic carrier landing system,β in Proceedings of the IEEE Aerospace Conference, pp. 52593β52612, March 2001. [3] S. Yuan and Y. Yang, βDesign of automatic carrier landing system using π»β control,β in Proceedings of the 3rd World Congress on Intelligent Control and Automation, pp. 3449β3451, IEEE, Hefei, China, June 2000. [4] R. J. Niewoehner and I. I. Kaminer, βDesign of an autoland controller for an F-14 aircraft using π»β synthesis,β Journal of Guidance, Control, and Dynamics, vol. 19, no. 3, pp. 656β663, 1996. [5] M. Steinberg, βDevelopment and simulation of An F/A-18. Fuzzy logic automatic carrier landing system,β in Proceedings of the 2nd IEEE International Conference on Fuzzy Systems, pp. 797β802, April 1993. [6] I. Fialho, G. I. Balas, and J. Renfrow, βOn the design of LPV controllers for the F-14 aircraft lateral-directional axis during powered approach,β in Proceedings of the American Control Conference, pp. 123β127, June 1997. [7] N. A. Denison, Automated Carrier Landing of an Unmanned Combat Aerial Vehicle Using Dynamic Inversion, Department of Aeronautical and Astronautical Engineering Graduate School of Engineering and Management, Air Force Institute of Technology, Air Force University, 2007. [8] G. Tao, Adaptive Control Design and Analysis, John Wiley & Sons, Hoboken, NJ, USA, 2003. [9] E. Lavretsky, βCombined/composite model reference adaptive conrol,β in Proceedings of the AIAA Guidance, Navigation, and Control Conference, AIAA 2009-6065, Chicago, Ill, USA, August 2009. [10] H. I. Basturk and M. Krstic, βState derivative feedback for adaptive cancellation of unmatched disturbances in unknown strict-feedback LTI systems,β Automatica, vol. 50, no. 10, pp. 2539β2545, 2014. [11] H. I. Basturk and M. Krstic, βAdaptive sinusoidal disturbance cancellation for unknown LTI systems despite input delay,β Automatica, vol. 58, pp. 131β138, 2015. [12] X. Y. Guo and M. Bodson, βEquivalence between adaptive feedforward cancellation and disturbance rejection using the internal model principle,β International Journal of Adaptive Control and Signal Processing, vol. 24, no. 3, pp. 211β218, 2010. [13] X. Chen, βAdaptive sliding mode control for discrete-time multi-input multi-output systems,β Automatica, vol. 42, no. 3, pp. 427β435, 2006. [14] Z. Zhao, J. Zhang, L. Sun, and D. Zhang, βTerminal sliding mode control with adaptive law for uncertain nonlinear system,β Mathematical Problems in Engineering, vol. 2015, Article ID 519090, 7 pages, 2015. [15] Y. Guo, S.-M. Song, and X.-H. Li, βTerminal sliding mode control for attitude tracking of spacecraft based on rotation matrix,β Mathematical Problems in Engineering, vol. 2015, Article ID 187924, 9 pages, 2015. [16] Q. Zou, F. Wang, and Q. Zong, βNonlinear constrained adaptive backstepping tracking control for a hypersonic vehicle with uncertainty,β Mathematical Problems in Engineering, vol. 2015, Article ID 237453, 16 pages, 2015.

12 [17] X.-X. Yin, Y.-G. Lin, W. Li, H.-W. Liu, and Y.-J. Gu, βAdaptive sliding mode back-stepping pitch angle control of a variabledisplacement pump controlled pitch system for wind turbines,β ISA Transactions, vol. 58, pp. 629β634, 2015. [18] N. T. Nguyen, βOptimal control modification for robust adaptive control with large adaptive gain,β Systems & Control Letters, vol. 61, no. 4, pp. 485β494, 2012. [19] J.-Z. Geng, H.-L. Yao, and H. Zhang, βStudies on effect of air wake on slope and landing property of carrier aircraft,β Journal of System Simulation, vol. 21, no. 18, pp. 5940β5943, 2009. [20] D. D. Boskovic and J. Redding, βAn autonomous carrier landing system for unmanned aerial vehicles,β in Proceedings of the AIAA Conference on Guidance, Navigation, and Control, AIAA 2009-6264, Chicago, Ill, USA, August 2009. [21] βDepartment of defense handbook flying qualities of piloted aircraft. MIL-STD-1797A,β Department of Defense, USA, 1995. [22] B. Etkin, Dynamics of Atmospheric Flight, John Wiley & Sons, New York, NY, USA, 1972. [23] M. V. Cook, Flight Dynamics Principles, Butterworth-Heinemann, London, UK, 2007. [24] A. K. Imai, R. R. Costa, L. Hsu, G. Tao, and P. V. Kokotovic, βMultivariable adaptive control using high-frequency gain matrix factorization,β IEEE Transactions on Automatic Control, vol. 49, no. 7, pp. 1152β1157, 2004. [25] L. Wen, G. Tao, and H. Yang, βAdaptive turbulence compensation for multivariable nonlinear aircraft models,β in Proceedings of the American Control Conference (ACC β15), pp. 5557β5562, July 2015. [26] B. L. Stevens and F. L. Lewis, Aircraft Control and Simulation, Wiley-Interscience, New York, NY, USA, 2003.

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Research Article Adaptive Disturbance Rejection Control for Automatic Carrier Landing System Xin Wang, Xin Chen, and Liyan Wen College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China Correspondence should be addressed to Xin Chen; [email protected] Received 27 March 2016; Revised 25 June 2016; Accepted 5 July 2016 Academic Editor: Jean J. Loiseau Copyright Β© 2016 Xin Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.

1. 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 systemβs 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 π»β 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 [13β15], and back-stepping control designs [16, 17] were proposed. In [18], an extension of biobjective optimal control modification for unmatched uncertain

2

Mathematical Problems in Engineering

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. (3) The proposed LDU decomposition-based disturbance rejection techniques are used to solve a typical carrier landing aircraft turbulence compensation problem. Extensive simulation results are obtained through a longitudinal aircraft dynamic model during aircraft landing. 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.

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

βπ0 sin πΌ0 β π·πΌ0

2.1. 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 ππΜ = π cos πΌ β π· β ππ sin πΎ

(1)

πππΎΜ = π sin πΌ + πΏ β ππ cos πΎ. The kinematic equation is πΜ = π.

(2)

The moment equation is πΜ =

π . πΌπ¦

(3)

The navigation equation is βΜ = π sin πΎ.

(4)

The identical equation is πΎ = π β πΌ.

(5)

2.2. 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

2. Longitudinal Model of Carrier-Based Aircraft on Final-Approach Dynamics in the Air Wake

ππ0 cos πΌ0 β π·π0

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.

π₯Μ 1 = π΄ 1 π₯1 + π΅1 π’

(6)

π¦1 = πΆ1 π₯1 ,

where π₯1 = [V, πΌ, π, π, β]π , π’1 = [πΏπ , πΏπ‘ ]π , and π¦1 = [V, πΌ, π, π, β]π are the system state vector, input vector, and output vector, respectively, and the matrices are

+ π cos πΎ0

0

βπ cos πΎ0

0

π0 cos πΎ0

β

π·β0

[ π π π ] [ ] [ ] [ πΏ π0 βπ0 cos πΌ0 β πΏ πΌ0 + ππ sin πΎ0 πΏ β0 ] βπ sin πΎ0 [ ] [ ] β 1 β [ ππ0 ππ0 π0 ππ0 ] [ ] ], π΄1 = [ [ πV ] π + π cos πΌ π πΏ π π πΏ π π π sin πΎ0 π sin πΎ0 π0 πΌ0 πΌ0 0 0 πΌΜ 0 πΌΜ 0 πΌΜ 0 [ 0 β πΌΜ 0 ( π0 ) β ( β ) + β ( ) 0 ] [ ] [ πΌπ¦ ] πΌπ¦ ππ0 πΌπ¦ πΌπ¦ ππ0 π0 πΌπ¦ πΌπ¦ πΌπ¦ π0 [ ] [ ] [ 0 0 1 0 0 ] [ ] [

sin πΎ0

βπ0 cos πΎ0

0

]

Mathematical Problems in Engineering

3

ππΏ cos πΌ0 π·πΏ β π‘ β π [ ] π π [ ] [ ] πΏ πΏπ ππΏπ‘ sin πΌ0 [ ] β β [ ] [ ] ππ0 ππ0 [ ] π΅1 = [ π , ππΌΜ 0 ππΏπ‘ sin πΌ0 ] [ πΏπ ππΌΜ 0 πΏ πΏπ ] β ( ) β ( )] [ [ πΌπ¦ ] πΌπ¦ ππ0 πΌπ¦ ππ0 [ ] [ ] 0 0 [ ] [ 1 [ [0 [ [ πΆ1 = [0 [ [0 [ [0

0

0

]

0 0 0 0 ] 1 0 0 0] ] 0 1 0 0] ]. ] 0 0 1 0] ] 0 0 0 1] (7)

2.3. 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. 2.3.1. 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 [19β21], and the components are presented as π’π₯ ππ€ 0 π β [ββ, β1750) { { { { { { 0.04 { { (π + 1750) π β [β1750, β1710) { { { β1710 + 1750 { { { { 0.02 { { (π + 1870) π β [β1710, β1630) { { { β1630 + 1710 { { { { β0.004 { { { (π + 430) π β [β1630, β1550) { { β1550 + 1630 { { { { = { β0.028 (π + 1130) π β [β1550, β1340) { β1340 + 1550 { { { { { 0.01 { { (π + 1844) π β [β1340, β1160) { { { β1160 + 1340 { { { { 0.15 β 0.038 { { { (π + 1418) π β [β1160, β400) { { β400 + 1160 { { { { { β0.15 { { π β [β400, β250) (π + 250) { { 150 { { { π β [β250, 0) {0

π’π¦ ππ€ 0 π β [ββ, β2650) { { { { { { { β0.076 { { (π + 2650) π β [β2650, β2400) { { { β2400 + 2650 { { { { { { β0.088 + 0.076 { { (π + 3667) π β [β2400, β2200) { { β2200 + 2400 { { { { { { { β0.08 + 0.088 π β [β2200, β1970) (π β 330) ={ { β1970 + 2200 { { { { { { 0.02 + 0.08 { { π β [β1970, β750) (π + 994) { { { β750 + 1970 { { { { { { β0.01 β 0.02 { { π β [β750, β250) (π + 417) { { { β250 + 750 { { { { { π β [β250, 0) {β0.01 π = βπ cos πΎ (π0 β π‘) β 286,

(8) where π is the distance between the aircraft and the ship center of pitch, negative after of ship, π0 is the total landing time, and π‘ is the present time. 2.3.2. 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 π’π₯ and π’π¦ . Because the flight speed is far larger than the wind speed, we can get ππ β π + π’π₯ πΌπ β πΌ +

π’π¦ π

,

(9)

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Mathematical Problems in Engineering

where ππ and πΌπ 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 π₯Μ 1 = π΄ 1 π₯1 + π΅1 π’1 + π΅1π π (π‘)

π΅1π

π¦1 = πΆ1 π₯1 , (10) where π΄ 1 , π΅1 , πΆ1 , π₯1 , and π’1 are defined in (6). The disturbance is π(π‘) = [π’π₯ , π’π¦ ] and the matrix π΅1π is

π sin πΌ0 β π·πΌ0 ππ0 cos πΌ0 β π·π0 β cos πΎ0 [ ] π ππ0 [ ] [ ] [ ] ππ0 sin πΌ0 + πΏ π0 π sin πΌ0 + πΏ πΌ0 [ ] [ ] 2 [ ] ππ ππ 0 [ ] 0 ]. =[ [π πV0 ππΌΜ 0 πΏ πΌ0 + π0 sin πΌ0 ππΌ0 ] [ πΌΜ 0 ππ0 sin πΌ0 + πΏ π0 ] [ ] ( )β ( )β 2 [ πΌπ¦ ππ0 πΌπ¦ πΌπ¦ πΌπ¦ π0 ] ππ0 [ ] [ ] [ ] 0 0 [ ] 0 0 [ ]

3. 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. 3.1. Problem Formulation. Consider the linear time-invariant system as π₯Μ (π‘) = π΄π₯ (π‘) + π΅π’ (π‘) + π΅π π (π‘) π¦ (π‘) = πΆπ₯ (π‘) ,

(12)

where π΄ β π πΓπ , π΅ β π πΓπ, π΅π β π πΓπ , and πΆ β π πΓπ are constant and unknown; π₯(π‘) β π π , π’(π‘) β π π, and π¦(π‘) β π π are the system state vector, input vector, and output vector, respectively; π(π‘) = [π1 (π‘), . . . , ππ (π‘)]π β π π is the disturbance vector. The disturbance signal π(π‘) is unmatched with the control input π’(π‘), in the sense that π΅ and π΅π are not linearly dependent, π΅π =ΜΈ π΅πΌ for any matrix πΌ β π πΓπ . The control objective is to design an adaptive statefeedback control signal π’(π‘) for (12), to make closed-loop signal boundedness and the output π¦(π‘) track a chosen reference signal π¦π (π‘) generated from a reference model: π¦π (π‘) = ππ (π ) [π] (π‘) β π π,

(13)

πΓπ

where ππ (π ) β π is a stable transfer function matrix and π(π‘) β π π is an external reference input signal for defining a desired π¦π (π‘). Note that, in this paper, we use the notation π¦(π‘) = πΊ(π )[π’](π‘) to represent the output π¦(π‘) of a system whose transfer matrix is πΊ(π ) and input is π’(π‘), a convenient notation for adaptive control systems. 3.2. Preliminaries and Assumptions Lemma 1 (see [8]). For any πΓπ strictly proper and full rank rational transfer matrix πΊ(π ), there exists a lower triangular

(11)

polynomial matrix ππ (π ), defined as the left interactor matrix of πΊ(π ), of the form π1 (π ) [ βπ (π ) [ 21 [ ππ (π ) = [ . [ . [ .

0

0

0

π2 (π )

0

0

.. .

.. .

.. .

] ] ] ], ] ]

(14)

π π [βπ1 (π ) βπ2 (π ) β β β ππ (π )]

where βπππ (π ), π = 1, 2, . . . , π β 1, π = 2, . . . , π, are some polynomials and ππ (π ) are any chosen monic stable polynomials such that the high-frequency gain matrix of πΊ(π ) defined as πΎπ = limπ ββ ππ (π ) πΊ(π ). Lemma 2 (see [24]). Every π Γ π real matrix πΎπ with nonzero leading principal minors Ξ 1 , Ξ 2 , . . . , Ξ π can be uniquely factored as πΎπ = πΏπ·π,

(15)

where πΏ β π πΓπ is unity lower triangular, π β π πΓπ is unity upper triangular, and π· = diag {π1 , π2 , . . . , ππ } = diag {Ξ 1 , Ξ 2 Ξβ1 , . . . , Ξ π Ξβ1 πβ1 } .

(16)

Assumption 3. All zeros of πΊ0 (π ) = πΆ(π πΌ β π΄)β1 π΅ are stabilizable and detectable. Assumption 4. πΊ0 (π ) is strictly proper with full rank and has a known modified interactor matrix ππ (π ) such that πΎπ = limπ ββ ππ (π )πΊ0 (π ) is finite and nonsingular (so that ππ (π ) = β1 ππ (π ) can be chosen as the transfer matrix for the reference model system).

Mathematical Problems in Engineering

5 (2) Break turn (1) Upwind leg

800 ft, 250 kt, level turn break at 45βΌ60 deg bank speed brake extended throttle idle

800 ft altitude, 300βΌ360 kt, autopilot

(5) Final approach

Line up with the deck centerline β3.5 deg ο¬ight path-angle 0.75 nmile behind the carrier 45 deg position 295 ft altitude

Landing gear down below 250 kt

90 deg position 370 ft altitude (4) Base leg 0.75 to 1 mile

(3) Downwind leg

20βΌ30 deg bank angle descending to 450 ft

Landing checklist descending to 600 ft

Figure 1: Procedures of carrier landing for the aircraft.

Final approach 295 ft

where the nominal parameters πΎ1βπ β π πΓπ and πΎ2βπ β π πΓπ are for the plant-model output matching, and πΎ3β (π‘) β π π is used to cancel the effect of the disturbance π(π‘).

πΎ = β3.5β

rela Glidesl tive o to th pe e ca r r ie

r

Air wake

Lemma 8 (see [25]). The matrix πΎπ = limπ ββ ππ (π )πΊπ (π ) is finite if π0β1 (π )ππ (π ) is proper.

0.75 nmile

Based on Lemma 8, the existence of a nominal controller (17) is established a follows.

Figure 2: Final carrier landing phase.

Assumption 5. The leading principal minors of the highfrequency gain matrix πΎπ are nonzero, and their signs are known. From plant (12), the control and disturbance transfer functions are obtained as πΊ0 (π ) = πΆ(π πΌ β π΄)β1 π΅ and πΊπ (π ) = πΆ(π πΌ β π΄)β1 π΅π and are expressed in their left coprime polynomial matrix decompositions: πΊ0 (π ) = ππβ1 (π )π0 (π ) and πΊπ (π ) = ππβ1 (π )ππ (π ), where ππ (π ), π0 (π ) β π πΓπ and ππ (π ) β π πΓπ are some polynomial matrices. Assumption 6. The transfer matrix π0β1 (π )ππ (π ) is proper. 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 π’(π‘) and the disturbance input π(π‘) to the output π¦(π‘) for the design of a derivative-free disturbance rejection scheme. 3.3. Nominal Disturbance Rejection Design. With the knowledge of the plant and disturbance parameters, the nominal state-feedback controller is π’β (π‘) = πΎ1βπ π₯ (π‘) + πΎ2β π (π‘) + πΎ3β (π‘) ,

(17)

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 π¦π (π‘). From plant (12), the input-output form is obtained as π¦ (π‘) = πΊ0 (π ) [π’] (π‘) + π¦ (π‘) ,

(18)

β1

with π¦(π‘) = πΊ0 (π )[π](π‘) = πΆ(π πΌ β π΄) π΅π [π](π‘). Operate the interactor matrix (a polynomial matrix) Μ = π΄π₯(π‘) + π΅π’(π‘) + π΅π π(π‘), π¦(π‘) = ππ (π ) on plant (12), π₯(π‘) πΆπ₯(π‘), to reach an expression of ππ (π )[π¦π ](π‘) in a possible form: ππ (π ) [π¦] (π‘) = βπΎ0 π₯ (π‘) + πΎπ π’ (π‘) + πΎπ1 π’Μ + β β β + πΎππ0 π’(π0 ) (π‘) + πΎπ π (π‘) + πΎπ1 πΜ (π‘)

(19)

+ β β β + πΎππ1 π(π1) (π‘) , with some constant matrices πΎ0 β π πΓπ , πΎπ β π πΓπ, πΎππ β π πΓπ, π = 1, 2, . . . , π0 , πΎπ β π πΓπ , and πΎππ β π πΓπ , π = 1, 2, . . . , π1 , for some integers π0 , π1 β₯ 0. From (12) and (18), we have π₯ (π ) = (π πΌ β π΄)β1 π΅π’ (π ) + (π πΌ β π΄)β1 π΅π π (π ) .

(20)

6

Mathematical Problems in Engineering

Expressing (19) in π domain and using (20), we have ππ (π ) π¦ (π ) = βπΎ0 (π πΌ β π΄)β1 π΅π’ (π ) + πΎπ π’ (π ) + πΎπ1 π π’ (π ) + β β β + πΎππ0 π π0 π’ (π ) β πΎ0 (π πΌ β π΄)β1 π΅π π (π ) + πΎπ π (π )

(21)

+ πΎπ1 π π (π ) + β β β + πΎππ1 π π1 π (π ) . From Assumption 4, πΎπ = limπ ββ ππ (π )πΊ0 (π ) is finite and nonsingular and from Assumption 6, πΎππ = 0, π = 1, . . . , π0 , πΎπ = πΎπ , and πΎππ = 0, π = 1, . . . , π0 , πΎπ = πΎπ . Hence, we have ππ (π ) [π¦] (π‘) = βπΎ0 π₯ (π‘) + πΎπ π’ (π‘) + πΎπ π (π‘) .

(23)

β1

π¦ (π‘) = πΆ (π πΌ β π΄ β π΅πΎ1βπ ) π΅πΎ2β [π] (π‘) β1 β1

π

ππβ = [ππ0 , ππ1 , . . . , ππππ ] β π ππ +1 , π

ππ (π‘) = [1, ππ1 (π‘) , . . . , ππππ (π‘)] β π ππ +1 , π = 1, 2, . . . , π.

π (π‘) = πβπ π (π‘) ,

πβπ

π1π 0π(π2 +1) 0π(π3 +1) β β β 0π(ππ +1) [ π ] [0(π +1) π2βπ 0π(π +1) β β β 0π(π +1) ] [ 1 ] 3 π πΓπ ] =[ .. .. .. .. ] β π , [ .. [ . ] . . . . ] (28) [ π π π βπ [0(π1 +1) 0(π2 +1) 0(π3 +1) β β β ππ ] π

(24)

π (π‘) = [π1π (π‘) π2π (π‘) β β β πππ (π‘)] β π π π = π1 + π2 + β β β + ππ + π.

+ πΆ (π πΌ β π΄ β π΅πΎ1βπ ) π΅π π (π ) = ππ (π ) [π] (π‘) = π¦π (π‘) . Remark 10. From (24), we can conclude that the plant-model matching conditions are πΆ (π πΌ β π΄ β =

β1 π΅πΎ1βπ )

(27)

Hence, the disturbance π(π‘) is expressed as

where πΎ1βπ = πΎπβ1 πΎ0 , πΎ2β = πΎπβ1 , and πΎ3β (π‘) = πΎ3π π(π‘) with πΎ3π (π‘) = βπΎπβ1 πΎπ , which leads to the output matching: ππ (π )[π¦](π‘) = π(π‘). We applied (23) to plant (12); we have

+ πΆ (π πΌ β π΄ β π΅πΎ1βπ ) π΅ [πΎ3β ] (π‘)

The parameter matrix and the disturbance signal components are

(22)

From (17), the control law can be designed as π’ (π‘) = π’β (π‘) = πΎ1βπ π₯ (π‘) + πΎ2β π (π‘) + πΎ3β (π‘) ,

Remark 11. For model (26), if ππ = 0, then ππ (π‘) = ππ0 , π = 1, 2, . . . , π, representing a constant disturbance signal. If we choose ππ = 1 and ππ1 (π‘) = sin(π€π1 π‘), then ππ (π‘) = ππ0 + ππ1 sin(π€π1 π‘), π = 1, 2, . . . , π, representing time-variant period disturbances. A large class of practical disturbances in control applications can be approximated by a proper selection of this basis function πππ (π‘) in (26).

π΅πΎ2β

β β β β β = [π3π1 , π3π2 , . . . , π3ππ ], π3ππ β π π, π = With πΎ3π 1, 2, . . . , π, the disturbance rejection term πΎ3β (π‘) is parameterized as β β πΎ3β (π‘) = πΎ3π π (π‘) = πΎ3π πβπ π (π‘) = Ξ¦β3 π (π‘) ,

(29)

where the parameter matrix is

ππ (π ) ππ (π ) πΎ2ββ1 πΎ3β

(π )

(25)

β1

β β β Ξ¦β3 = [π31 , π32 , . . . , π3π ] β π πΓπ ,

+ πΆ (π πΌ β π΄ β π΅πΎ1βπ ) π΅π π (π ) = 0.

π = π1 + π2 + β β β + ππ + π, (30)

For the plant-model matching condition (25), there exist constant matrices πΎ1βπ and πΎ2β and πΎ3β (π‘), if and only if the relative degree condition in Assumption 6 is satisfied. 3.4. Parameterizations of the Term πΎ3β (π‘). For the disturbance vector π(π‘) β π π , each element ππ (π‘) in (12) can be expressed as ππ

ππ (π‘) = ππ0 + β πππ πππ (π‘) = ππβπ ππ (π‘) , π=1

(26)

π = 1, 2, . . . , π, with some unknown constants ππ0 , πππ and some known bounded continuous signals πππ , π = 1, 2, . . . , π, π = 1, 2, . . . , ππ .

β β = π3ππ ππβπ β π πΓ(ππ +1) , π = 1, 2, . . . , π. π3π

Next, the adaptive disturbance rejection design will be studied for the plant with uncertainties from the plant and unmatched disturbances. 3.5. Error Equation. Applying (23) to plant (12), the closedloop system becomes π₯Μ (π‘) = (π΄ + π΅πΎ1βπ ) π₯ (π‘) + π΅πΎ2β π (π‘) + π΅πΎ3β (π‘) + π΅π π (π‘) + π΅ [π’ (π‘) β πΎ1βπ π₯ (π‘) β πΎ2β π (π‘) β πΎ3β (π‘)] π¦ (π‘) = πΆπ₯ (π‘) .

(31)

Mathematical Problems in Engineering

7

In view of (24), (25), and (31), the output tracking error equation is π (π‘) = π¦ (π‘) β π¦π (π‘) = ππ (π ) πΎπβ [π’ β πΎ1βπ π₯ β πΎ2β π β πΎ3β ] (π‘) + ππ (π‘) ,

(32)

πΎπβ = πΎ2ββ1 , βπ

where ππ (π‘) = πΆπ(π΄+π΅πΎ1 )π‘ π₯(0) converges to zero exponenβ1 tially fast due to the stability of π΄ + π΅πΎ1βπ and ππ (π ) = ππ (π ). Hence, we have ππ (π ) [π] (π‘) = πΎπ [π’ (π‘) β πΎ1βπ π₯ (π‘) β πΎ2β π (π‘) β πΎ3β (π‘)] .

(33)

To deal with the uncertainty of the high-frequency gain matrix πΎπ , the LDU decomposition of πΎπ is used in (33), so that we have

Μ = Ξ¦(π‘) β Ξ¦β and Ξ¦π(π‘) = where the parameter error is Ξ¦(π‘) π [Ξ¦0 (π‘), Ξ¦1 (π‘)] is the estimate of unknown parameter matrix π π π Ξ¦βπ = [Ξ¦β0 , Ξ¦βπ 1 ], π(π‘) = [π’ (π‘), π (π‘)] , and π(π‘) = π π π π β β1 [π₯ (π‘), π (π‘), π (π‘)] , where Ξ0 = (πΏ β πΌ) is introduced to parameterize the unknown matrix πΏ, which has a special form: 0 0 β β β 0 [ β ] [ π21 0 β β β 0] [ ] [ πβ πβ ] β β β 0 [ 31 ] 32 [ ] β πΓπ . (41) Ξ0 = [ . ]βπ .. .. [ . ] . . 0] [ . [ ] [πβ 0 0] [ πβ11 β β β ] β β [ ππ1 β β β πππβ1 0] For such a matrix Ξβ0 , the parameter vectors are defined as β β π , π2β = π21 π

β β , π32 ] β π 2 , π3β = [π31

β1

πΏ ππ (π ) [π] (π‘) = π·π (π’ (π‘) β πΎ1βπ π₯ (π‘) β πΎ2β π (π‘) β πΎ3β (π‘)) .

.. .

(34)

β β β ππβ1 = [ππβ11 , . . . , ππβ1πβ2 ] β π πβ2 ,

We have the following equation: ππ’ (π‘) = π’ (π‘) β (πΌ β π) π’ (π‘) .

πΏβ1 ππ (π ) [π] (π‘) = π· [π’ (π‘) β (πΌ β π) π’ (π‘) β

β

πΎ2β π (π‘)

β

πΎ3β

(π‘)]] .

β β β ππ = [ππ1 , . . . , πππβ2 ] β π πβ1 ,

(35)

and their estimates are π2 (π‘) = π21 (π‘) β π ,

With (34) and (35), we have

π [πΎ1βπ π₯ (π‘)

π

π3 (π‘) = [π31 (π‘) , π32 (π‘)] β π 2 ,

(36)

.. .

The new equation is πΏβ1 ππ (π ) [π] (π‘) = π· [π’ (π‘) β Ξ¦β0 π’ (π‘) β Ξ¦βπ 1 π (π‘)] ,

(37)

βπ β β π π where Ξ¦βπ 1 = [ππΎ1 , ππΎ2 , ππΎ3π ] and π(π‘) = [π₯ (π‘), π (π‘), ππ (π‘)]π . This new equation has a new controller structure:

π’ (π‘) = Ξ¦0 π’ (π‘) + Ξ¦π1 π (π‘) ,

(38)

where Ξ¦0 and Ξ¦π1 are the estimates of Ξ¦β0 and Ξ¦βπ 1 and Ξ¦0 is upper triangular with zero diagonal elements (only its nonzero elements are estimated). The matrix Ξ¦0 has the same strictly form as that of Ξ¦β0 = (πΌ β π): 0 π12 π13 β β β

π1π

] [ [0 0 π23 β β β π2π ] ] [ ] [. . . . ] β π πΓπ. [ .. .. Ξ¦0 = [ .. .. ] ] [ ] [0 0 β β β 0 π [ πβ1π ] [0 0

β β β 0

0

ππ (π ) [π] (π‘) +

ππβ1 (π‘) = [ππβ11 (π‘) , . . . , ππβ1πβ2 (π‘)] β π πβ2 , ππ (π‘) = [ππ1 (π‘) , . . . , πππβ2 (π‘)] β π πβ1 . We introduce a filter β(π ) = 1/π(π ), where π(π ) is chosen as a stable and monic polynomial whose degree is equal to the maximum degree of the modified interactor ππ (π ). Operating both sides of (40) by β(π )πΌπ leads to ππ (π ) β (π ) [π] (π‘) + Ξβ0 ππ (π ) β (π ) [π] (π‘) (44)

Μ π π] (π‘) . = π·β (π ) [Ξ¦ We define π

(45)

π

(39)

ππ (π‘) = [π1 (π‘) , . . . , ππβ1 (π‘)] β π πβ1 , π = 2, 3, . . . , π.

(46)

From (45) and (46) in (44), we obtained

]

Μ π (π‘) π (π‘) , (π ) [π] (π‘) = π·Ξ¦

(43)

π (π‘) = ππ (π ) β (π ) [π] (π‘) = [π1 (π‘) , . . . , ππ (π‘)] ,

βπ π (π‘) + [0, π2βπ π2 (π‘) , π3βπ π3 (π‘) , . . . , ππ ππ (π‘)]

From (37) and (38), we obtain a new error model: Ξβ0 ππ

(42)

(40)

π

Μ π] (π‘) . = π·β (π ) [Ξ¦

π

(47)

8

Mathematical Problems in Engineering

Based on the parameterized error equation (47), an estimation error signal is introduced: π

π ππ (π‘)] π (π‘) = π (π‘) + [0, π2π π2 (π‘) , π3π π3 (π‘) , . . . , ππ

+ Ξ¨ (π‘) π (π‘) β π π, where Ξ¨(π‘) β π

πΓπ

(48)

β

is the estimate of Ξ¨ = π· and

π (π‘) = β (π ) [π] (π‘) ,

π = 1, 2, . . . , π,

π (π‘) = Ξ¦π (π‘) π (π‘) β β (π ) [Ξ¦ππ] (π‘) ,

(49)

It then follows from (44) (47) and (48) that π

(50)

Μ (π‘) π (π‘) , +Ξ¨

3.6. Adaptive Laws. Based on the error model (50), the adaptive laws are chosen as Ξππ ππ (π‘) ππ (π‘) π2 (π‘)

,

π = 2, . . . , π,

+ πΎ2 (π‘) π (π‘) + Ξ¦3 (π‘) π (π‘) ,

(51)

π1 (π‘) =

π (π ) [π’] (π‘) , Ξ (π )

π2 (π‘) =

π (π ) [π¦] (π‘) , Ξ (π )

π3 (π‘) =

π (π ) [π] (π‘) , Ξ (π )

π

Ξπ (π‘) π (π‘) Ξ¨Μ (π‘) = β , π2 (π‘) where Ξππ = Ξππ π > 0, π = 2, 3, . . . , π, and Ξ = Ξπ > 0 are adaptive gains; π· is defined in (15); π(π‘) = [π1 (π‘), π2 (π‘), . . . , ππ(π‘)]π is calculated from (50); and π

π2 (π‘) = 1 + ππ (π‘) π (π‘) + ππ (π‘) π (π‘) + βπππ ππ (π‘) .

(52)

π=2

3.7. Stability Analysis. To analyze the closed-loop system stability, we first establish some desired properties of the adaptive parameter update laws mentioned above. Lemma 12 (see [8]). The adaptive laws ensure that (i) ππ (π‘) β πΏβ , π = 2, 3, . . . , π, Ξ¦(π‘) β πΏβ , Ξ¨(π‘) β πΏβ , and π(π‘)/π(π‘) β πΏ2 β© πΏβ ; Μ β πΏ2 β© πΏβ , and (ii) πΜ π (π‘) β πΏ2 β© πΏβ , π = 2, 3, . . . , π, Ξ¦(π‘) 2 β Μ βπΏ β©πΏ . Ξ¨(π‘) Based on Lemma 12, the following desired closed-loop system properties are established. Theorem 13. For plant (12) with uncertainties from the system parameters and disturbance (26) under Assumptions 3β6,

(54)

where Ξ¦π1(π‘), Ξ¦π2 (π‘), Ξ¦π3π (π‘), πΎ2 (π‘), and Ξ¦3 (π‘) are adaptive estimates of the corresponding nominal controller parameters and

π

π·π (π‘) π (π‘) π Ξ¦Μ (π‘) = β , π2 (π‘)

(53)

where πΏ β π πΓπ is a gain matrix such that π΄ β πΏπΆ is stable, which is possible, and (π΄ πΆ) is assumed to be detectable. Hence, we have π’ (π‘) = Ξ¦π1 (π‘) π1 (π‘) + Ξ¦π2 (π‘) π2 (π‘) + Ξ¦π3π (π‘) π3 (π‘)

Μ = Ξ¨(π‘) β Ξ¨β and πΜπ = ππ β πβ are the parameter where Ξ¨(π‘) π errors. This error model is choice for update laws.

πΜ π (π‘) = β

Proof (outline). The proof of this stability theorem can be established through using a unified framework. Because the control input π’(π‘) described in (38) depends on the state π₯(π‘), it first needs to be expressed by using the system output π¦(π‘) through establishing the state observer of the plant: ΜΜ (π‘) = (π΄ β πΏπΆ) π₯ ΜΜ (π‘) + π΅π’ (π‘) + π΅π π (π‘) + πΏπ¦ (π‘) , π₯

π = 1, 2, . . . , π.

π π π Μπ π (π‘) = [0, πΜ2 π2 (π‘) , πΜ3 π3 (π‘) , . . . , πΜπππ (π‘)] + π·Ξ¦

and the reference model (13), the LDU decomposition-based MRAC scheme with the adaptive controller (38) and adaptive parameter update laws (51) guarantees closed-loop system boundedness and asymptotic output tracking limπ‘ββ π(π‘) = 0 with π(π‘) = π¦(π‘) β π¦π (π‘).

(55)

= with π(π ) = [πΌπ π πΌπ β β β π πβ1 πΌπ]π , π(π ) π πβ1 [πΌπ π πΌπ β β β π πΌπ ] , and Ξ(π ) being a chosen monic stable polynomial of degree π, which has the same eigenvalues with π΄ β πΏπΆ. Then, introducing the fictitious filters for the plant π¦(π‘) = πΆ(π πΌ β π΄)β1 π΅π’(π‘) + πΆ(π πΌ β π΄)β1 π΅π π(π‘) and using series transformations, the control input described as in (54) is transformed into the form π’ (π‘) = πΊ11 (π , β ) [π¦] (π‘) + πΊ12 (π , β ) [π] (π‘) + πΊ13 (π , β ) [π] (π‘) + πΊ14 (π , β ) [ππ ] (π‘) ,

(56)

where π¦(π‘) = β(π )[π¦](π‘) (β(π ) is given below (43)) and πΊ11 (π , β ), πΊ12 (π , β ), πΊ13 (π , β ), and πΊ14 (π , β ) are proper stable operators with finite gains. Furthermore, a filtered version of the output signal π¦(π‘) is expressed in a feedback framework: π‘

σ΅© σ΅©σ΅© βπΌ (π‘βπ) π₯1 (π) σ΅©σ΅©π¦ (π‘)σ΅©σ΅©σ΅© β€ π₯0 + π½1 β« π 1 0 π

βπΌ2 (πβπ)

β (β« π 0

σ΅© σ΅©σ΅© σ΅©σ΅©π¦ (π)σ΅©σ΅©σ΅© ππ) ππ,

(57)

Mathematical Problems in Engineering

9

Μ for some π½1 , πΌ1 , πΌ2 > 0, and π₯1 (π‘) = βΞ¦(π‘)β + βπ(π‘)βπ(π‘) β 2 β πΏ β© πΏ . Applying the small gain lemma to (57), we conclude that π¦(π‘) β πΏβ , and so π¦(π‘), π’(π‘) β πΏβ . Thus, the signals satisfy π(π‘), π(π‘), π(π‘), π(π‘), π β πΏβ . Furthermore, Μ Μ β πΏ2 (Lemma 12) are satisfied, and in ππ , π(π‘)/π(π‘), Ξ¦(π‘), Ξ¨(π‘) turn π(π‘) and π(π‘) = π¦(π‘) β π¦π (π‘), such that π(π‘) = π¦(π‘) β π¦π (π‘) converges to zero. β0.03858 18.984 [ [ β0.001028 β0.63253 [ [ π΄ = [ 7.8601πΈ β 5 β0.75905 [ [ 0 0 [ [ β0.04362 β249.76

4. 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

0

β32.139

1

0.005612

1.3233πΈ β 4

] 3.7553πΈ β 6 ] ] β0.5183 β0.0079341 β3.0808πΈ β 7 ] ], ] ] 1 0 0 ] 0 249.76 0 ]

10.1 0 [ ] [β1.5446πΈ β 4 ] 0 [ ] [ 0.024656 β0.01077] π΅=[ ], [ ] [ ] 0 0 [ ] 0 0 [ ] πΆ=[

1 0 0 0 0 ]. 0 0 0 0 1

The turbulence disturbances are described in [22, 23]; we can get π΅π = [π΅π1 , π΅π2 ] , π

π΅π1 = [β0.0386 0.001 7.8601πΈ β 5 0 0] ,

(59)

π

π΅π2 = [β0.05262 0.00251 β0.00304 0 0] . 4.2. Adaptive Control Design. For the aircraft system, the transfer function, πΊ0 (π ) = πΆ(π πΌ β π΄)β1 π΅, has stable zeros, π 1 = β4.507, π 2 = β0.91, and π 3 = β0.5685, and is strictly proper and full rank. The interactor matrix is chosen as ππ (π ) = diag {π + 1 (π + 1)2 } .

(60)

The high-frequency matrix is πΎπ = lim ππ (π ) πΊ0 (π ) = [ π ββ

2.2673

8.8455

β0.0902 β0.3521

],

(61)

and it is is finite and nonsingular and the matrix πΎπ = lim ππ (π ) πΊπ (π ) = [ π ββ

(58)

β0.0386 β0.0526 β0.2481 β0.6246

]

(62)

is finite. From the specified left coprime polynomial matrix decompositions, πΊ0 (π ) = ππβ1 (π )π0 (π ) and πΊπ (π ) = ππβ1 (π )ππ (π ), we can obtain lim πβ1 π ββ 0

(π ) ππ (π ) = [

β0.017 β0.0064 2.7493 1.7739

],

(63)

which means that the relative degree condition in Assumption 6 can be ensured. The related gain parameters in adaptive laws (51) are chosen as Ξπ = 100, π· = diag{2 2}, and Ξ = diag{10 10}. 4.3. Simulation Results. For this simulations study, the initial state is chosen as π₯0 (π‘) = [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.

10

Mathematical Problems in Engineering Adaptive control without disturbance

350

350

300

300

250

250

200 150

200 150

100

100

50

50

0

0

10

20

30 40 Versus time (s)

50

Adaptive control with disturbace

400

Height (ft)

Height (ft)

400

60

0

70

0

Reference trajectory Adaptive control

40

30

30

20

20

Tracking error (ft)

Tracking error (ft)

40

10 0 β10

β20

β10

β20

β40

β40 30 40 Versus time (s)

70

0

β30

20

60

10

β30

10

50

Adaptive control with disturbance

50

0

30 40 Versus time (s)

Figure 5: Final landing phase altitude for the aircraft.

Adaptive control without disturbance

50

20

Reference trajectory Adaptive control

Figure 3: Final landing phase altitude for the aircraft.

β50

10

50

60

70

β50

10

20

30 40 Versus time (s)

50

60

Height reference tracking error

Height reference tracking error

Figure 4: Final landing phase tracking error for the aircraft.

0

Figure 6: Final landing phase tracking error for the aircraft.

5. Conclusions

Nomenclature

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.

π·πΏπ :

Aerodynamic drag derivative with respect to elevator deflection angle π, π·, πΏ: Engine thrust, aerodynamic drag force, and aerodynamic lift force πΌ, π, πΎ: Angle of attack, pitch, and flight-path slope β: Altitude of aircraft π: Gravitational acceleration π: Mass of aircraft π: Airspeed of aircraft π: Pith rate

70

Mathematical Problems in Engineering

References

Adaptive control with disturbance

6 Throttle (unit)

11

4 2 0 β2

0

10

20

0

10

20

30 40 Versus time (s)

50

60

70

50

60

70

Elevator (deg)

40 30 20 10 0

30

40

Versus time (s)

Figure 7: Final landing phase control signal.

πΌπ¦ : π: πΏπ , πΏπ‘ :

Moment of inertia in pitch The pitch moment Elevator deflection bias angle and engine throttle angle Aerodynamic pitch moment and drag ππ0 , π·π0 : derivative with respect to airspeed π0 ππ0 , πΏ π0 : Thrust and aerodynamic lift derivative with respect to airspeed π0 ππΏπ , πΏ πΏπ : Aerodynamic pitch moment and lift derivative with respect to πΏπ ππ€ , π’π₯ , π’π¦ : Turbulence velocity and body axis components of ππ€ π0 , πΌ0 , πΌΜ 0 , π0 , β0 : The trim value of aircraft state Aerodynamic pitch moment and drag ππ0 , π·π0 : derivative with respect to airspeed π0 ππ0 , πΏ π0 : Thrust and aerodynamic lift derivatives with respect to airspeed π0 ππΏπ , πΏ πΏπ : Aerodynamic pitch moment and lift derivative with respect to elevator Aerodynamic drag derivative with π·πΌ0 , π·β0 : respect to πΌ0 and β0 π·πΏπ : Aerodynamic drag due to πΏπ ππΌ0 , ππΌΜ 0 : Aerodynamic pitch moment with respect to πΌ0 and πΌΜ 0 ππ0 : Aerodynamic pitch moment with respect to π0 π0 : Benchmark aerodynamic thrust at the airspeed π0 ππΏπ‘ : Aerodynamic thrust derivatives with respect to the throttle Aerodynamic lift derivative with respect πΏ πΌ0 , πΏ β0 : to angle of attack and height.

Competing Interests The authors declare that they have no competing interests.

[1] J. M. Urnes, R. K. Hess, R. F. Moomaw, and R. W. Huff, βHdot automatic carrier landing system for approach control in turbulence,β Journal of Guidance, Control, and Dynamics, vol. 4, no. 2, pp. 177β183, 1981. [2] A. L. Prickett and C. J. Parkes, βFlight testing of the F/A-18E/F automatic carrier landing system,β in Proceedings of the IEEE Aerospace Conference, pp. 52593β52612, March 2001. [3] S. Yuan and Y. Yang, βDesign of automatic carrier landing system using π»β control,β in Proceedings of the 3rd World Congress on Intelligent Control and Automation, pp. 3449β3451, IEEE, Hefei, China, June 2000. [4] R. J. Niewoehner and I. I. Kaminer, βDesign of an autoland controller for an F-14 aircraft using π»β synthesis,β Journal of Guidance, Control, and Dynamics, vol. 19, no. 3, pp. 656β663, 1996. [5] M. Steinberg, βDevelopment and simulation of An F/A-18. Fuzzy logic automatic carrier landing system,β in Proceedings of the 2nd IEEE International Conference on Fuzzy Systems, pp. 797β802, April 1993. [6] I. Fialho, G. I. Balas, and J. Renfrow, βOn the design of LPV controllers for the F-14 aircraft lateral-directional axis during powered approach,β in Proceedings of the American Control Conference, pp. 123β127, June 1997. [7] N. A. Denison, Automated Carrier Landing of an Unmanned Combat Aerial Vehicle Using Dynamic Inversion, Department of Aeronautical and Astronautical Engineering Graduate School of Engineering and Management, Air Force Institute of Technology, Air Force University, 2007. [8] G. Tao, Adaptive Control Design and Analysis, John Wiley & Sons, Hoboken, NJ, USA, 2003. [9] E. Lavretsky, βCombined/composite model reference adaptive conrol,β in Proceedings of the AIAA Guidance, Navigation, and Control Conference, AIAA 2009-6065, Chicago, Ill, USA, August 2009. [10] H. I. Basturk and M. Krstic, βState derivative feedback for adaptive cancellation of unmatched disturbances in unknown strict-feedback LTI systems,β Automatica, vol. 50, no. 10, pp. 2539β2545, 2014. [11] H. I. Basturk and M. Krstic, βAdaptive sinusoidal disturbance cancellation for unknown LTI systems despite input delay,β Automatica, vol. 58, pp. 131β138, 2015. [12] X. Y. Guo and M. Bodson, βEquivalence between adaptive feedforward cancellation and disturbance rejection using the internal model principle,β International Journal of Adaptive Control and Signal Processing, vol. 24, no. 3, pp. 211β218, 2010. [13] X. Chen, βAdaptive sliding mode control for discrete-time multi-input multi-output systems,β Automatica, vol. 42, no. 3, pp. 427β435, 2006. [14] Z. Zhao, J. Zhang, L. Sun, and D. Zhang, βTerminal sliding mode control with adaptive law for uncertain nonlinear system,β Mathematical Problems in Engineering, vol. 2015, Article ID 519090, 7 pages, 2015. [15] Y. Guo, S.-M. Song, and X.-H. Li, βTerminal sliding mode control for attitude tracking of spacecraft based on rotation matrix,β Mathematical Problems in Engineering, vol. 2015, Article ID 187924, 9 pages, 2015. [16] Q. Zou, F. Wang, and Q. Zong, βNonlinear constrained adaptive backstepping tracking control for a hypersonic vehicle with uncertainty,β Mathematical Problems in Engineering, vol. 2015, Article ID 237453, 16 pages, 2015.

12 [17] X.-X. Yin, Y.-G. Lin, W. Li, H.-W. Liu, and Y.-J. Gu, βAdaptive sliding mode back-stepping pitch angle control of a variabledisplacement pump controlled pitch system for wind turbines,β ISA Transactions, vol. 58, pp. 629β634, 2015. [18] N. T. Nguyen, βOptimal control modification for robust adaptive control with large adaptive gain,β Systems & Control Letters, vol. 61, no. 4, pp. 485β494, 2012. [19] J.-Z. Geng, H.-L. Yao, and H. Zhang, βStudies on effect of air wake on slope and landing property of carrier aircraft,β Journal of System Simulation, vol. 21, no. 18, pp. 5940β5943, 2009. [20] D. D. Boskovic and J. Redding, βAn autonomous carrier landing system for unmanned aerial vehicles,β in Proceedings of the AIAA Conference on Guidance, Navigation, and Control, AIAA 2009-6264, Chicago, Ill, USA, August 2009. [21] βDepartment of defense handbook flying qualities of piloted aircraft. MIL-STD-1797A,β Department of Defense, USA, 1995. [22] B. Etkin, Dynamics of Atmospheric Flight, John Wiley & Sons, New York, NY, USA, 1972. [23] M. V. Cook, Flight Dynamics Principles, Butterworth-Heinemann, London, UK, 2007. [24] A. K. Imai, R. R. Costa, L. Hsu, G. Tao, and P. V. Kokotovic, βMultivariable adaptive control using high-frequency gain matrix factorization,β IEEE Transactions on Automatic Control, vol. 49, no. 7, pp. 1152β1157, 2004. [25] L. Wen, G. Tao, and H. Yang, βAdaptive turbulence compensation for multivariable nonlinear aircraft models,β in Proceedings of the American Control Conference (ACC β15), pp. 5557β5562, July 2015. [26] B. L. Stevens and F. L. Lewis, Aircraft Control and Simulation, Wiley-Interscience, New York, NY, USA, 2003.

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