Adaptive Disturbance Rejection Control for Automatic Carrier Landing

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Jul 5, 2016 - Automatic Carrier Landing System. Xin Wang, Xin Chen, and Liyan Wen. College of Automation Engineering, Nanjing University of AeronauticsΒ ...

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 system’s response to turbulence in carrier landing in [2]. A finite horizon technique was introduced to maintain a constant flight-path angle under the worst case conditions during carrier landing in [3]. An improvement in carrier landing performance was made by the incorporation of the direct lift control using 𝐻∞ outputfeedback synthesis [4]. As pointed out in [5], a fuzzy logic based carrier landing system was designed and the results

indicated that fuzzy logic could yield significant benefits for aircraft outer loop control. For the lateral-directional aircraft dynamics in carrier landing, a linear fractional transformation gain-scheduled controller was presented in [6]. The dynamic inversion technique was used in unmanned combat aerial vehicle on an aircraft carrier in [7]. In the absence of wind and sea state turbulence, the controller performed well. After adding wind and sea state turbulence, the controller performance was degraded. Adaptive control has become one of the most popular designs for failures and disturbances compensation. An output tracking model reference adaptive control (MRAC) scheme was developed for single-input/single-output systems in [8]. The related technical issues including design conditions, plant-model matching conditions, controller structures, adaptive laws, and stability analysis are presented in detail, with extensions to adaptive disturbance rejection. In [9], a combined direct and indirect MRAC statefeedback architecture was developed for MIMO dynamical systems with matched uncertainties and the methodology was extended to systems with a baseline controller. To solve the disturbance rejection problems, adaptive feedforward [10, 11], feedback control methods [12], terminal sliding-mode control method [13–15], 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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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

βˆ’π‘‡0 sin 𝛼0 βˆ’ 𝐷𝛼0

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 π‘šπ‘‰Μ‡ = 𝑇 cos 𝛼 βˆ’ 𝐷 βˆ’ π‘šπ‘” sin 𝛾

(1)

π‘šπ‘‰π›ΎΜ‡ = 𝑇 sin 𝛼 + 𝐿 βˆ’ π‘šπ‘” cos 𝛾. The kinematic equation is πœƒΜ‡ = π‘ž.

(2)

The moment equation is π‘žΜ‡ =

𝑀 . 𝐼𝑦

(3)

The navigation equation is β„ŽΜ‡ = 𝑉 sin 𝛾.

(4)

The identical equation is 𝛾 = πœƒ βˆ’ 𝛼.

(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

𝑇𝑉0 cos 𝛼0 βˆ’ 𝐷𝑉0

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 = 𝐴 1 π‘₯1 + 𝐡1 𝑒

(6)

𝑦1 = 𝐢1 π‘₯1 ,

where π‘₯1 = [V, 𝛼, π‘ž, πœƒ, β„Ž]𝑇 , 𝑒1 = [𝛿𝑒 , 𝛿𝑑 ]𝑇 , and 𝑦1 = [V, 𝛼, π‘ž, πœƒ, β„Ž]𝑇 are the system state vector, input vector, and output vector, respectively, and the matrices are

+ 𝑔 cos 𝛾0

0

βˆ’π‘” cos 𝛾0

0

𝑉0 cos 𝛾0

βˆ’

π·β„Ž0

[ π‘š π‘š π‘š ] [ ] [ ] [ 𝐿 𝑉0 βˆ’π‘‡0 cos 𝛼0 βˆ’ 𝐿 𝛼0 + π‘šπ‘” sin 𝛾0 𝐿 β„Ž0 ] βˆ’π‘” sin 𝛾0 [ ] [ ] βˆ’ 1 βˆ’ [ π‘šπ‘‰0 π‘šπ‘‰0 𝑉0 π‘šπ‘‰0 ] [ ] ], 𝐴1 = [ [ 𝑀V ] 𝑀 + 𝑇 cos 𝛼 𝑀 𝐿 𝑀 𝑀 𝐿 𝑀 𝑀 𝑔 sin 𝛾0 𝑔 sin 𝛾0 π‘ž0 𝛼0 𝛼0 0 0 𝛼̇ 0 𝛼̇ 0 𝛼̇ 0 [ 0 βˆ’ 𝛼̇ 0 ( 𝑉0 ) βˆ’ ( βˆ’ ) + βˆ’ ( ) 0 ] [ ] [ 𝐼𝑦 ] 𝐼𝑦 π‘šπ‘‰0 𝐼𝑦 𝐼𝑦 π‘šπ‘‰0 𝑉0 𝐼𝑦 𝐼𝑦 𝐼𝑦 𝑉0 [ ] [ ] [ 0 0 1 0 0 ] [ ] [

sin 𝛾0

βˆ’π‘‰0 cos 𝛾0

0

]

Mathematical Problems in Engineering

3

𝑇𝛿 cos 𝛼0 𝐷𝛿 βˆ’ 𝑑 βˆ’ 𝑒 [ ] π‘š π‘š [ ] [ ] 𝐿 𝛿𝑒 𝑇𝛿𝑑 sin 𝛼0 [ ] βˆ’ βˆ’ [ ] [ ] π‘šπ‘‰0 π‘šπ‘‰0 [ ] 𝐡1 = [ 𝑀 , 𝑀𝛼̇ 0 𝑇𝛿𝑑 sin 𝛼0 ] [ 𝛿𝑒 𝑀𝛼̇ 0 𝐿 𝛿𝑒 ] βˆ’ ( ) βˆ’ ( )] [ [ 𝐼𝑦 ] 𝐼𝑦 π‘šπ‘‰0 𝐼𝑦 π‘šπ‘‰0 [ ] [ ] 0 0 [ ] [ 1 [ [0 [ [ 𝐢1 = [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 [19–21], and the components are presented as 𝑒π‘₯ 𝑉𝑀 0 𝑋 ∈ [βˆ’βˆž, βˆ’1750) { { { { { { 0.04 { { (𝑋 + 1750) 𝑋 ∈ [βˆ’1750, βˆ’1710) { { { βˆ’1710 + 1750 { { { { 0.02 { { (𝑋 + 1870) 𝑋 ∈ [βˆ’1710, βˆ’1630) { { { βˆ’1630 + 1710 { { { { βˆ’0.004 { { { (𝑋 + 430) 𝑋 ∈ [βˆ’1630, βˆ’1550) { { βˆ’1550 + 1630 { { { { = { βˆ’0.028 (𝑋 + 1130) 𝑋 ∈ [βˆ’1550, βˆ’1340) { βˆ’1340 + 1550 { { { { { 0.01 { { (𝑋 + 1844) 𝑋 ∈ [βˆ’1340, βˆ’1160) { { { βˆ’1160 + 1340 { { { { 0.15 βˆ’ 0.038 { { { (𝑋 + 1418) 𝑋 ∈ [βˆ’1160, βˆ’400) { { βˆ’400 + 1160 { { { { { βˆ’0.15 { { 𝑋 ∈ [βˆ’400, βˆ’250) (𝑋 + 250) { { 150 { { { 𝑋 ∈ [βˆ’250, 0) {0

𝑒𝑦 𝑉𝑀 0 𝑋 ∈ [βˆ’βˆž, βˆ’2650) { { { { { { { βˆ’0.076 { { (𝑋 + 2650) 𝑋 ∈ [βˆ’2650, βˆ’2400) { { { βˆ’2400 + 2650 { { { { { { βˆ’0.088 + 0.076 { { (𝑋 + 3667) 𝑋 ∈ [βˆ’2400, βˆ’2200) { { βˆ’2200 + 2400 { { { { { { { βˆ’0.08 + 0.088 𝑋 ∈ [βˆ’2200, βˆ’1970) (𝑋 βˆ’ 330) ={ { βˆ’1970 + 2200 { { { { { { 0.02 + 0.08 { { 𝑋 ∈ [βˆ’1970, βˆ’750) (𝑋 + 994) { { { βˆ’750 + 1970 { { { { { { βˆ’0.01 βˆ’ 0.02 { { 𝑋 ∈ [βˆ’750, βˆ’250) (𝑋 + 417) { { { βˆ’250 + 750 { { { { { 𝑋 ∈ [βˆ’250, 0) {βˆ’0.01 𝑋 = βˆ’π‘‰ cos 𝛾 (𝑇0 βˆ’ 𝑑) βˆ’ 286,

(8) where 𝑋 is the distance between the aircraft and the ship center of pitch, negative after of ship, 𝑇0 is the total landing time, and 𝑑 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 𝑒π‘₯ and 𝑒𝑦 . Because the flight speed is far larger than the wind speed, we can get 𝑉𝑖 β‰ˆ 𝑉 + 𝑒π‘₯ 𝛼𝑖 β‰ˆ 𝛼 +

𝑒𝑦 𝑉

,

(9)

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Mathematical Problems in Engineering

where 𝑉𝑖 and 𝛼𝑖 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 = 𝐴 1 π‘₯1 + 𝐡1 𝑒1 + 𝐡1𝑑 𝑑 (𝑑)

𝐡1𝑑

𝑦1 = 𝐢1 π‘₯1 , (10) where 𝐴 1 , 𝐡1 , 𝐢1 , π‘₯1 , and 𝑒1 are defined in (6). The disturbance is 𝑑(𝑑) = [𝑒π‘₯ , 𝑒𝑦 ] and the matrix 𝐡1𝑑 is

𝑇 sin 𝛼0 βˆ’ 𝐷𝛼0 𝑇𝑉0 cos 𝛼0 βˆ’ 𝐷𝑉0 βˆ’ cos 𝛾0 [ ] π‘š π‘šπ‘‰0 [ ] [ ] [ ] 𝑇𝑉0 sin 𝛼0 + 𝐿 𝑉0 𝑇 sin 𝛼0 + 𝐿 𝛼0 [ ] [ ] 2 [ ] π‘šπ‘‰ π‘šπ‘‰ 0 [ ] 0 ]. =[ [𝑀 𝑀V0 𝑀𝛼̇ 0 𝐿 𝛼0 + 𝑇0 sin 𝛼0 𝑀𝛼0 ] [ 𝛼̇ 0 𝑇𝑉0 sin 𝛼0 + 𝐿 𝑉0 ] [ ] ( )βˆ’ ( )βˆ’ 2 [ 𝐼𝑦 π‘šπ‘‰0 𝐼𝑦 𝐼𝑦 𝐼𝑦 𝑉0 ] π‘šπ‘‰0 [ ] [ ] [ ] 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‡ (𝑑) = 𝐴π‘₯ (𝑑) + 𝐡𝑒 (𝑑) + 𝐡𝑑 𝑑 (𝑑) 𝑦 (𝑑) = 𝐢π‘₯ (𝑑) ,

(12)

where 𝐴 ∈ 𝑅𝑛×𝑛 , 𝐡 ∈ 𝑅𝑛×𝑀, 𝐡𝑑 ∈ 𝑅𝑀×𝑝 , and 𝐢 ∈ 𝑅𝑀×𝑛 are constant and unknown; π‘₯(𝑑) ∈ 𝑅𝑛 , 𝑒(𝑑) ∈ 𝑅𝑀, and 𝑦(𝑑) ∈ 𝑅𝑀 are the system state vector, input vector, and output vector, respectively; 𝑑(𝑑) = [𝑑1 (𝑑), . . . , 𝑑𝑝 (𝑑)]𝑇 ∈ 𝑅𝑝 is the disturbance vector. The disturbance signal 𝑑(𝑑) is unmatched with the control input 𝑒(𝑑), in the sense that 𝐡 and 𝐡𝑑 are not linearly dependent, 𝐡𝑑 =ΜΈ 𝐡𝛼 for any matrix 𝛼 ∈ 𝑅𝑀×𝑝 . The control objective is to design an adaptive statefeedback control signal 𝑒(𝑑) for (12), to make closed-loop signal boundedness and the output 𝑦(𝑑) track a chosen reference signal π‘¦π‘š (𝑑) generated from a reference model: π‘¦π‘š (𝑑) = π‘Šπ‘š (𝑠) [π‘Ÿ] (𝑑) ∈ 𝑅𝑀,

(13)

𝑀×𝑀

where π‘Šπ‘š (𝑠) ∈ 𝑅 is a stable transfer function matrix and π‘Ÿ(𝑑) ∈ 𝑅𝑀 is an external reference input signal for defining a desired π‘¦π‘š (𝑑). Note that, in this paper, we use the notation 𝑦(𝑑) = 𝐺(𝑠)[𝑒](𝑑) to represent the output 𝑦(𝑑) of a system whose transfer matrix is 𝐺(𝑠) and input is 𝑒(𝑑), a convenient notation for adaptive control systems. 3.2. Preliminaries and Assumptions Lemma 1 (see [8]). For any 𝑀×𝑀 strictly proper and full rank rational transfer matrix 𝐺(𝑠), there exists a lower triangular

(11)

polynomial matrix πœ‰π‘š (𝑠), defined as the left interactor matrix of 𝐺(𝑠), of the form 𝑑1 (𝑠) [ β„Žπ‘š (𝑠) [ 21 [ πœ‰π‘š (𝑠) = [ . [ . [ .

0

0

0

𝑑2 (𝑠)

0

0

.. .

.. .

.. .

] ] ] ], ] ]

(14)

π‘š π‘š [β„Žπ‘€1 (𝑠) β„Žπ‘€2 (𝑠) β‹… β‹… β‹… 𝑑𝑀 (𝑠)]

where β„Žπ‘–π‘—π‘š (𝑠), 𝑗 = 1, 2, . . . , 𝑀 βˆ’ 1, 𝑖 = 2, . . . , 𝑀, are some polynomials and 𝑑𝑖 (𝑠) are any chosen monic stable polynomials such that the high-frequency gain matrix of 𝐺(𝑠) defined as 𝐾𝑝 = limπ‘ β†’βˆž πœ‰π‘š (𝑠) 𝐺(𝑠). Lemma 2 (see [24]). Every 𝑀 Γ— 𝑀 real matrix 𝐾𝑝 with nonzero leading principal minors Ξ” 1 , Ξ” 2 , . . . , Ξ” 𝑀 can be uniquely factored as 𝐾𝑝 = πΏπ·π‘ˆ,

(15)

where 𝐿 ∈ 𝑅𝑀×𝑀 is unity lower triangular, π‘ˆ ∈ 𝑅𝑀×𝑀 is unity upper triangular, and 𝐷 = diag {𝑑1 , 𝑑2 , . . . , π‘‘π‘š } = diag {Ξ” 1 , Ξ” 2 Ξ”βˆ’1 , . . . , Ξ” π‘š Ξ”βˆ’1 π‘šβˆ’1 } .

(16)

Assumption 3. All zeros of 𝐺0 (𝑠) = 𝐢(𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡 are stabilizable and detectable. Assumption 4. 𝐺0 (𝑠) is strictly proper with full rank and has a known modified interactor matrix πœ‰π‘š (𝑠) such that 𝐾𝑝 = limπ‘ β†’βˆž πœ‰π‘š (𝑠)𝐺0 (𝑠) is finite and nonsingular (so that π‘Šπ‘š (𝑠) = βˆ’1 πœ‰π‘š (𝑠) can be chosen as the transfer matrix for the reference model system).

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5 (2) Break turn (1) Upwind leg

800 ft, 250 kt, level turn break at 45∼60 deg bank speed brake extended throttle idle

800 ft altitude, 300∼360 kt, autopilot

(5) Final approach

Line up with the deck centerline βˆ’3.5 deg flight 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

20∼30 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 𝐾1βˆ—π‘‡ ∈ 𝑅𝑀×𝑛 and 𝐾2βˆ—π‘‡ ∈ 𝑅𝑀×𝑀 are for the plant-model output matching, and 𝐾3βˆ— (𝑑) ∈ 𝑅𝑀 is used to cancel the effect of the disturbance 𝑑(𝑑).

𝛾 = βˆ’3.5∘

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

r

Air wake

Lemma 8 (see [25]). The matrix 𝐾𝑑 = limπ‘ β†’βˆž πœ‰π‘š (𝑠)𝐺𝑑 (𝑠) is finite if 𝑍0βˆ’1 (𝑠)𝑍𝑑 (𝑠) 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 𝐾𝑝 are nonzero, and their signs are known. From plant (12), the control and disturbance transfer functions are obtained as 𝐺0 (𝑠) = 𝐢(𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡 and 𝐺𝑑 (𝑠) = 𝐢(𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡𝑑 and are expressed in their left coprime polynomial matrix decompositions: 𝐺0 (𝑠) = π‘ƒπ‘™βˆ’1 (𝑠)𝑍0 (𝑠) and 𝐺𝑑 (𝑠) = π‘ƒπ‘™βˆ’1 (𝑠)𝑍𝑑 (𝑠), where 𝑃𝑙 (𝑠), 𝑍0 (𝑠) ∈ 𝑅𝑀×𝑀 and 𝑍𝑑 (𝑠) ∈ 𝑅𝑀×𝑝 are some polynomial matrices. Assumption 6. The transfer matrix 𝑍0βˆ’1 (𝑠)𝑍𝑑 (𝑠) 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 𝑒(𝑑) and the disturbance input 𝑑(𝑑) to the output 𝑦(𝑑) 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‘’βˆ— (𝑑) = 𝐾1βˆ—π‘‡ π‘₯ (𝑑) + 𝐾2βˆ— π‘Ÿ (𝑑) + 𝐾3βˆ— (𝑑) ,

(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‘š (𝑑). From plant (12), the input-output form is obtained as 𝑦 (𝑑) = 𝐺0 (𝑠) [𝑒] (𝑑) + 𝑦 (𝑑) ,

(18)

βˆ’1

with 𝑦(𝑑) = 𝐺0 (𝑠)[𝑑](𝑑) = 𝐢(𝑠𝐼 βˆ’ 𝐴) 𝐡𝑑 [𝑑](𝑑). Operate the interactor matrix (a polynomial matrix) Μ‡ = 𝐴π‘₯(𝑑) + 𝐡𝑒(𝑑) + 𝐡𝑑 𝑑(𝑑), 𝑦(𝑑) = πœ‰π‘š (𝑠) on plant (12), π‘₯(𝑑) 𝐢π‘₯(𝑑), to reach an expression of πœ‰π‘š (𝑠)[π‘¦π‘š ](𝑑) in a possible form: πœ‰π‘š (𝑠) [𝑦] (𝑑) = βˆ’πΎ0 π‘₯ (𝑑) + 𝐾𝑝 𝑒 (𝑑) + 𝐾𝑝1 𝑒̇ + β‹… β‹… β‹… + 𝐾𝑝𝑙0 𝑒(𝑙0 ) (𝑑) + 𝐾𝑑 𝑑 (𝑑) + 𝐾𝑑1 𝑑̇ (𝑑)

(19)

+ β‹… β‹… β‹… + 𝐾𝑑𝑙1 𝑑(𝑙1) (𝑑) , with some constant matrices 𝐾0 ∈ 𝑅𝑀×𝑛 , 𝐾𝑝 ∈ 𝑅𝑀×𝑀, 𝐾𝑝𝑗 ∈ 𝑅𝑀×𝑀, 𝑗 = 1, 2, . . . , 𝑙0 , 𝐾𝑑 ∈ 𝑅𝑀×𝑝 , and 𝐾𝑑𝑖 ∈ 𝑅𝑀×𝑝 , 𝑖 = 1, 2, . . . , 𝑙1 , for some integers 𝑙0 , 𝑙1 β‰₯ 0. From (12) and (18), we have π‘₯ (𝑠) = (𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡𝑒 (𝑠) + (𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡𝑑 𝑑 (𝑠) .

(20)

6

Mathematical Problems in Engineering

Expressing (19) in 𝑠 domain and using (20), we have πœ‰π‘š (𝑠) 𝑦 (𝑠) = βˆ’πΎ0 (𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡𝑒 (𝑠) + 𝐾𝑝 𝑒 (𝑠) + 𝐾𝑝1 𝑠𝑒 (𝑠) + β‹… β‹… β‹… + 𝐾𝑝𝑙0 𝑠𝑙0 𝑒 (𝑠) βˆ’ 𝐾0 (𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡𝑑 𝑑 (𝑠) + 𝐾𝑑 𝑑 (𝑠)

(21)

+ 𝐾𝑑1 𝑠𝑑 (𝑠) + β‹… β‹… β‹… + 𝐾𝑑𝑙1 𝑠𝑙1 𝑑 (𝑠) . From Assumption 4, 𝐾𝑝 = limπ‘ β†’βˆž πœ‰π‘š (𝑠)𝐺0 (𝑠) is finite and nonsingular and from Assumption 6, 𝐾𝑝𝑗 = 0, 𝑗 = 1, . . . , 𝑙0 , 𝐾𝑃 = 𝐾𝑝 , and 𝐾𝑑𝑗 = 0, 𝑗 = 1, . . . , 𝑙0 , 𝐾𝑑 = 𝐾𝑑 . Hence, we have πœ‰π‘š (𝑠) [𝑦] (𝑑) = βˆ’πΎ0 π‘₯ (𝑑) + 𝐾𝑝 𝑒 (𝑑) + 𝐾𝑑 𝑑 (𝑑) .

(23)

βˆ’1

𝑦 (𝑑) = 𝐢 (𝑠𝐼 βˆ’ 𝐴 βˆ’ 𝐡𝐾1βˆ—π‘‡ ) 𝐡𝐾2βˆ— [π‘Ÿ] (𝑑) βˆ’1 βˆ’1

𝑇

πœ‡π‘—βˆ— = [𝑑𝑗0 , 𝑑𝑗1 , . . . , π‘‘π‘—π‘žπ‘— ] ∈ π‘…π‘žπ‘— +1 , 𝑇

𝑓𝑗 (𝑑) = [1, 𝑓𝑗1 (𝑑) , . . . , π‘“π‘—π‘žπ‘— (𝑑)] ∈ π‘…π‘žπ‘— +1 , 𝑗 = 1, 2, . . . , 𝑝.

𝑑 (𝑑) = π‘βˆ—π‘‡ 𝑓 (𝑑) ,

π‘βˆ—π‘‡

πœ‡1𝑇 0𝑇(π‘ž2 +1) 0𝑇(π‘ž3 +1) β‹… β‹… β‹… 0𝑇(π‘žπ‘ +1) [ 𝑇 ] [0(π‘ž +1) πœ‡2βˆ—π‘‡ 0𝑇(π‘ž +1) β‹… β‹… β‹… 0𝑇(π‘ž +1) ] [ 1 ] 3 𝑝 π‘Γ—π‘ž ] =[ .. .. .. .. ] ∈ 𝑅 , [ .. [ . ] . . . . ] (28) [ 𝑇 𝑇 𝑇 βˆ—π‘‡ [0(π‘ž1 +1) 0(π‘ž2 +1) 0(π‘ž3 +1) β‹… β‹… β‹… πœ‡π‘ ] 𝑇

(24)

𝑓 (𝑑) = [𝑓1𝑇 (𝑑) 𝑓2𝑇 (𝑑) β‹… β‹… β‹… 𝑓𝑝𝑇 (𝑑)] ∈ π‘…π‘ž π‘ž = π‘ž1 + π‘ž2 + β‹… β‹… β‹… + π‘žπ‘ + 𝑝.

+ 𝐢 (𝑠𝐼 βˆ’ 𝐴 βˆ’ 𝐡𝐾1βˆ—π‘‡ ) 𝐡𝑑 𝑑 (𝑠) = π‘Šπ‘š (𝑠) [π‘Ÿ] (𝑑) = π‘¦π‘š (𝑑) . Remark 10. From (24), we can conclude that the plant-model matching conditions are 𝐢 (𝑠𝐼 βˆ’ 𝐴 βˆ’ =

βˆ’1 𝐡𝐾1βˆ—π‘‡ )

(27)

Hence, the disturbance 𝑑(𝑑) is expressed as

where 𝐾1βˆ—π‘‡ = πΎπ‘βˆ’1 𝐾0 , 𝐾2βˆ— = πΎπ‘βˆ’1 , and 𝐾3βˆ— (𝑑) = 𝐾3𝑑 𝑑(𝑑) with 𝐾3𝑑 (𝑑) = βˆ’πΎπ‘βˆ’1 𝐾𝑑 , which leads to the output matching: πœ‰π‘š (𝑠)[𝑦](𝑑) = π‘Ÿ(𝑑). We applied (23) to plant (12); we have

+ 𝐢 (𝑠𝐼 βˆ’ 𝐴 βˆ’ 𝐡𝐾1βˆ—π‘‡ ) 𝐡 [𝐾3βˆ— ] (𝑑)

The parameter matrix and the disturbance signal components are

(22)

From (17), the control law can be designed as 𝑒 (𝑑) = π‘’βˆ— (𝑑) = 𝐾1βˆ—π‘‡ π‘₯ (𝑑) + 𝐾2βˆ— π‘Ÿ (𝑑) + 𝐾3βˆ— (𝑑) ,

Remark 11. For model (26), if π‘žπ‘— = 0, then 𝑑𝑗 (𝑑) = 𝑑𝑗0 , 𝑗 = 1, 2, . . . , 𝑝, representing a constant disturbance signal. If we choose π‘žπ‘— = 1 and 𝑓𝑗1 (𝑑) = sin(𝑀𝑗1 𝑑), then 𝑑𝑗 (𝑑) = 𝑑𝑗0 + 𝑑𝑗1 sin(𝑀𝑗1 𝑑), 𝑗 = 1, 2, . . . , 𝑝, 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 π‘“π‘—π‘˜ (𝑑) in (26).

𝐡𝐾2βˆ—

βˆ— βˆ— βˆ— βˆ— βˆ— = [π‘˜3𝑑1 , π‘˜3𝑑2 , . . . , π‘˜3𝑑𝑝 ], π‘˜3𝑑𝑗 ∈ 𝑅𝑀, 𝑗 = With 𝐾3𝑑 1, 2, . . . , 𝑝, the disturbance rejection term 𝐾3βˆ— (𝑑) is parameterized as βˆ— βˆ— 𝐾3βˆ— (𝑑) = 𝐾3𝑑 𝑑 (𝑑) = 𝐾3𝑑 π‘βˆ—π‘‡ 𝑓 (𝑑) = Ξ¦βˆ—3 𝑓 (𝑑) ,

(29)

where the parameter matrix is

π‘Šπ‘š (𝑠) π‘Šπ‘š (𝑠) 𝐾2βˆ—βˆ’1 𝐾3βˆ—

(𝑠)

(25)

βˆ’1

βˆ— βˆ— βˆ— Ξ¦βˆ—3 = [πœ™31 , πœ™32 , . . . , πœ™3𝑝 ] ∈ π‘…π‘€Γ—π‘ž ,

+ 𝐢 (𝑠𝐼 βˆ’ 𝐴 βˆ’ 𝐡𝐾1βˆ—π‘‡ ) 𝐡𝑑 𝑑 (𝑠) = 0.

π‘ž = π‘ž1 + π‘ž2 + β‹… β‹… β‹… + π‘žπ‘ + 𝑝, (30)

For the plant-model matching condition (25), there exist constant matrices 𝐾1βˆ—π‘‡ and 𝐾2βˆ— and 𝐾3βˆ— (𝑑), if and only if the relative degree condition in Assumption 6 is satisfied. 3.4. Parameterizations of the Term 𝐾3βˆ— (𝑑). For the disturbance vector 𝑑(𝑑) ∈ 𝑅𝑝 , each element 𝑑𝑗 (𝑑) in (12) can be expressed as π‘žπ‘—

𝑑𝑗 (𝑑) = 𝑑𝑗0 + βˆ‘ π‘‘π‘—π‘˜ π‘“π‘—π‘˜ (𝑑) = πœ‡π‘—βˆ—π‘‡ 𝑓𝑗 (𝑑) , π‘˜=1

(26)

𝑗 = 1, 2, . . . , 𝑝, with some unknown constants 𝑑𝑗0 , π‘‘π‘—π‘˜ and some known bounded continuous signals π‘“π‘—π‘˜ , 𝑗 = 1, 2, . . . , 𝑝, π‘˜ = 1, 2, . . . , π‘žπ‘— .

βˆ— βˆ— = π‘˜3𝑑𝑗 πœ‡π‘—βˆ—π‘‡ ∈ 𝑅𝑀×(π‘žπ‘— +1) , 𝑗 = 1, 2, . . . , 𝑝. πœ™3𝑗

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‡ (𝑑) = (𝐴 + 𝐡𝐾1βˆ—π‘‡ ) π‘₯ (𝑑) + 𝐡𝐾2βˆ— π‘Ÿ (𝑑) + 𝐡𝐾3βˆ— (𝑑) + 𝐡𝑑 𝑑 (𝑑) + 𝐡 [𝑒 (𝑑) βˆ’ 𝐾1βˆ—π‘‡ π‘₯ (𝑑) βˆ’ 𝐾2βˆ— π‘Ÿ (𝑑) βˆ’ 𝐾3βˆ— (𝑑)] 𝑦 (𝑑) = 𝐢π‘₯ (𝑑) .

(31)

Mathematical Problems in Engineering

7

In view of (24), (25), and (31), the output tracking error equation is 𝑒 (𝑑) = 𝑦 (𝑑) βˆ’ π‘¦π‘š (𝑑) = π‘Šπ‘š (𝑠) πΎπ‘βˆ— [𝑒 βˆ’ 𝐾1βˆ—π‘‡ π‘₯ βˆ’ 𝐾2βˆ— π‘Ÿ βˆ’ 𝐾3βˆ— ] (𝑑) + 𝑓𝑝 (𝑑) ,

(32)

πΎπ‘βˆ— = 𝐾2βˆ—βˆ’1 , βˆ—π‘‡

where 𝑓𝑝 (𝑑) = 𝐢𝑒(𝐴+𝐡𝐾1 )𝑑 π‘₯(0) converges to zero exponenβˆ’1 tially fast due to the stability of 𝐴 + 𝐡𝐾1βˆ—π‘‡ and π‘Šπ‘š (𝑠) = πœ‰π‘š (𝑠). Hence, we have πœ‰π‘š (𝑠) [𝑒] (𝑑) = 𝐾𝑝 [𝑒 (𝑑) βˆ’ 𝐾1βˆ—π‘‡ π‘₯ (𝑑) βˆ’ 𝐾2βˆ— π‘Ÿ (𝑑) βˆ’ 𝐾3βˆ— (𝑑)] .

(33)

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

Μƒ = Ξ¦(𝑑) βˆ’ Ξ¦βˆ— and Φ𝑇(𝑑) = where the parameter error is Ξ¦(𝑑) 𝑇 [Ξ¦0 (𝑑), Ξ¦1 (𝑑)] is the estimate of unknown parameter matrix 𝑇 𝑇 𝑇 Ξ¦βˆ—π‘‡ = [Ξ¦βˆ—0 , Ξ¦βˆ—π‘‡ 1 ], πœ”(𝑑) = [𝑒 (𝑑), πœ” (𝑑)] , and πœ”(𝑑) = 𝑇 𝑇 𝑇 𝑇 βˆ— βˆ’1 [π‘₯ (𝑑), π‘Ÿ (𝑑), 𝑓 (𝑑)] , where Θ0 = (𝐿 βˆ’ 𝐼) is introduced to parameterize the unknown matrix 𝐿, which has a special form: 0 0 β‹…β‹…β‹… 0 [ βˆ— ] [ πœƒ21 0 β‹… β‹… β‹… 0] [ ] [ πœƒβˆ— πœƒβˆ— ] β‹… β‹… β‹… 0 [ 31 ] 32 [ ] βˆ— 𝑀×𝑀 . (41) Θ0 = [ . ]βˆˆπ‘… .. .. [ . ] . . 0] [ . [ ] [πœƒβˆ— 0 0] [ π‘€βˆ’11 β‹… β‹… β‹… ] βˆ— βˆ— [ πœƒπ‘€1 β‹… β‹… β‹… πœƒπ‘€π‘€βˆ’1 0] For such a matrix Ξ˜βˆ—0 , the parameter vectors are defined as βˆ— ∈ 𝑅, πœƒ2βˆ— = πœƒ21 𝑇

βˆ— βˆ— , πœƒ32 ] ∈ 𝑅2 , πœƒ3βˆ— = [πœƒ31

βˆ’1

𝐿 πœ‰π‘š (𝑠) [𝑒] (𝑑) = π·π‘ˆ (𝑒 (𝑑) βˆ’ 𝐾1βˆ—π‘‡ π‘₯ (𝑑) βˆ’ 𝐾2βˆ— π‘Ÿ (𝑑) βˆ’ 𝐾3βˆ— (𝑑)) .

.. .

(34)

βˆ— βˆ— βˆ— πœƒπ‘€βˆ’1 = [πœƒπ‘€βˆ’11 , . . . , πœƒπ‘€βˆ’1π‘€βˆ’2 ] ∈ π‘…π‘€βˆ’2 ,

We have the following equation: π‘ˆπ‘’ (𝑑) = 𝑒 (𝑑) βˆ’ (𝐼 βˆ’ π‘ˆ) 𝑒 (𝑑) .

πΏβˆ’1 πœ‰π‘š (𝑠) [𝑒] (𝑑) = 𝐷 [𝑒 (𝑑) βˆ’ (𝐼 βˆ’ π‘ˆ) 𝑒 (𝑑) βˆ’

βˆ’

𝐾2βˆ— π‘Ÿ (𝑑)

βˆ’

𝐾3βˆ—

(𝑑)]] .

βˆ— βˆ— βˆ— πœƒπ‘€ = [πœƒπ‘€1 , . . . , πœƒπ‘€π‘€βˆ’2 ] ∈ π‘…π‘€βˆ’1 ,

(35)

and their estimates are πœƒ2 (𝑑) = πœƒ21 (𝑑) ∈ 𝑅,

With (34) and (35), we have

π‘ˆ [𝐾1βˆ—π‘‡ π‘₯ (𝑑)

𝑇

πœƒ3 (𝑑) = [πœƒ31 (𝑑) , πœƒ32 (𝑑)] ∈ 𝑅2 ,

(36)

.. .

The new equation is πΏβˆ’1 πœ‰π‘š (𝑠) [𝑒] (𝑑) = 𝐷 [𝑒 (𝑑) βˆ’ Ξ¦βˆ—0 𝑒 (𝑑) βˆ’ Ξ¦βˆ—π‘‡ 1 πœ” (𝑑)] ,

(37)

βˆ—π‘‡ βˆ— βˆ— 𝑇 𝑇 where Ξ¦βˆ—π‘‡ 1 = [π‘ˆπΎ1 , π‘ˆπΎ2 , π‘ˆπΎ3𝑓 ] and πœ”(𝑑) = [π‘₯ (𝑑), π‘Ÿ (𝑑), 𝑓𝑇 (𝑑)]𝑇 . This new equation has a new controller structure:

𝑒 (𝑑) = Ξ¦0 𝑒 (𝑑) + Φ𝑇1 πœ” (𝑑) ,

(38)

where Ξ¦0 and Φ𝑇1 are the estimates of Ξ¦βˆ—0 and Ξ¦βˆ—π‘‡ 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¦βˆ—0 = (𝐼 βˆ’ π‘ˆ): 0 πœ™12 πœ™13 β‹… β‹… β‹…

πœ™1𝑀

] [ [0 0 πœ™23 β‹… β‹… β‹… πœ™2𝑀 ] ] [ ] [. . . . ] ∈ 𝑅𝑀×𝑀. [ .. .. Ξ¦0 = [ .. .. ] ] [ ] [0 0 β‹… β‹… β‹… 0 πœ™ [ π‘€βˆ’1𝑀 ] [0 0

β‹…β‹…β‹… 0

0

πœ‰π‘š (𝑠) [𝑒] (𝑑) +

πœƒπ‘€βˆ’1 (𝑑) = [πœƒπ‘€βˆ’11 (𝑑) , . . . , πœƒπ‘€βˆ’1π‘€βˆ’2 (𝑑)] ∈ π‘…π‘€βˆ’2 , πœƒπ‘€ (𝑑) = [πœƒπ‘€1 (𝑑) , . . . , πœƒπ‘€π‘€βˆ’2 (𝑑)] ∈ π‘…π‘€βˆ’1 . We introduce a filter β„Ž(𝑠) = 1/𝑓(𝑠), where 𝑓(𝑠) is chosen as a stable and monic polynomial whose degree is equal to the maximum degree of the modified interactor πœ‰π‘š (𝑠). Operating both sides of (40) by β„Ž(𝑠)𝐼𝑀 leads to πœ‰π‘š (𝑠) β„Ž (𝑠) [𝑒] (𝑑) + Ξ˜βˆ—0 πœ‰π‘š (𝑠) β„Ž (𝑠) [𝑒] (𝑑) (44)

Μƒ 𝑇 πœ”] (𝑑) . = π·β„Ž (𝑠) [Ξ¦ We define 𝑇

(45)

𝑇

(39)

πœ‚π‘– (𝑑) = [𝑒1 (𝑑) , . . . , π‘’π‘–βˆ’1 (𝑑)] ∈ π‘…π‘–βˆ’1 , 𝑖 = 2, 3, . . . , 𝑀.

(46)

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

]

Μƒ 𝑇 (𝑑) πœ” (𝑑) , (𝑠) [𝑒] (𝑑) = 𝐷Φ

(43)

𝑒 (𝑑) = πœ‰π‘š (𝑠) β„Ž (𝑠) [𝑒] (𝑑) = [𝑒1 (𝑑) , . . . , 𝑒𝑀 (𝑑)] ,

βˆ—π‘‡ 𝑒 (𝑑) + [0, πœƒ2βˆ—π‘‡ πœ‚2 (𝑑) , πœƒ3βˆ—π‘‡ πœ‚3 (𝑑) , . . . , πœƒπ‘€ πœ‚π‘€ (𝑑)]

From (37) and (38), we obtain a new error model: Ξ˜βˆ—0 πœ‰π‘š

(42)

(40)

𝑇

Μƒ πœ”] (𝑑) . = π·β„Ž (𝑠) [Ξ¦

𝑇

(47)

8

Mathematical Problems in Engineering

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

𝑇 πœ‚π‘€ (𝑑)] πœ€ (𝑑) = 𝑒 (𝑑) + [0, πœƒ2𝑇 πœ‚2 (𝑑) , πœƒ3𝑇 πœ‚3 (𝑑) , . . . , πœƒπ‘€

+ Ξ¨ (𝑑) πœ‰ (𝑑) ∈ 𝑅𝑀, where Ξ¨(𝑑) ∈ 𝑅

𝑀×𝑀

(48)

βˆ—

is the estimate of Ψ = 𝐷 and

𝜁 (𝑑) = β„Ž (𝑠) [πœ”] (𝑑) ,

𝑖 = 1, 2, . . . , 𝑀,

πœ‰ (𝑑) = Φ𝑇 (𝑑) 𝜁 (𝑑) βˆ’ β„Ž (𝑠) [Ξ¦π‘‡πœ”] (𝑑) ,

(49)

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

(50)

Μƒ (𝑑) πœ‰ (𝑑) , +Ξ¨

3.6. Adaptive Laws. Based on the error model (50), the adaptive laws are chosen as Ξ“πœƒπ‘– πœ€π‘– (𝑑) πœ‚π‘– (𝑑) π‘š2 (𝑑)

,

𝑖 = 2, . . . , 𝑀,

+ 𝐾2 (𝑑) π‘Ÿ (𝑑) + Ξ¦3 (𝑑) 𝑓 (𝑑) ,

(51)

πœ”1 (𝑑) =

π‘Ž (𝑠) [𝑒] (𝑑) , Ξ› (𝑠)

πœ”2 (𝑑) =

π‘Ž (𝑠) [𝑦] (𝑑) , Ξ› (𝑠)

πœ”3 (𝑑) =

𝑏 (𝑠) [𝑓] (𝑑) , Ξ› (𝑠)

𝑇

Ξ“πœ€ (𝑑) πœ‰ (𝑑) Ξ¨Μ‡ (𝑑) = βˆ’ , π‘š2 (𝑑) where Ξ“πœƒπ‘– = Ξ“πœƒπ‘– 𝑇 > 0, 𝑖 = 2, 3, . . . , 𝑀, and Ξ“ = Γ𝑇 > 0 are adaptive gains; 𝐷 is defined in (15); πœ€(𝑑) = [πœ€1 (𝑑), πœ€2 (𝑑), . . . , πœ€π‘€(𝑑)]𝑇 is calculated from (50); and 𝑀

π‘š2 (𝑑) = 1 + πœπ‘‡ (𝑑) 𝜁 (𝑑) + πœ‰π‘‡ (𝑑) πœ‰ (𝑑) + βˆ‘πœ‚π‘–π‘‡ πœ‚π‘– (𝑑) .

(52)

𝑖=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) πœƒπ‘– (𝑑) ∈ 𝐿∞ , 𝑖 = 2, 3, . . . , 𝑀, Ξ¦(𝑑) ∈ 𝐿∞ , Ξ¨(𝑑) ∈ 𝐿∞ , and πœ€(𝑑)/π‘š(𝑑) ∈ 𝐿2 ∩ 𝐿∞ ; Μ‡ ∈ 𝐿2 ∩ 𝐿∞ , and (ii) πœƒΜ‡ 𝑖 (𝑑) ∈ 𝐿2 ∩ 𝐿∞ , 𝑖 = 2, 3, . . . , 𝑀, Ξ¦(𝑑) 2 ∞ Μ‡ ∈𝐿 ∩𝐿 . Ξ¨(𝑑) 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 3–6,

(54)

where Φ𝑇1(𝑑), Φ𝑇2 (𝑑), Φ𝑇3𝑑 (𝑑), 𝐾2 (𝑑), and Ξ¦3 (𝑑) are adaptive estimates of the corresponding nominal controller parameters and

𝑇

π·πœ€ (𝑑) 𝜁 (𝑑) 𝑇 Ξ¦Μ‡ (𝑑) = βˆ’ , π‘š2 (𝑑)

(53)

where 𝐿 ∈ 𝑅𝑛×𝑀 is a gain matrix such that 𝐴 βˆ’ 𝐿𝐢 is stable, which is possible, and (𝐴 𝐢) is assumed to be detectable. Hence, we have 𝑒 (𝑑) = Φ𝑇1 (𝑑) πœ”1 (𝑑) + Φ𝑇2 (𝑑) πœ”2 (𝑑) + Φ𝑇3𝑑 (𝑑) πœ”3 (𝑑)

Μƒ = Ξ¨(𝑑) βˆ’ Ξ¨βˆ— and πœƒΜƒπ‘– = πœƒπ‘– βˆ’ πœƒβˆ— are the parameter where Ξ¨(𝑑) 𝑖 errors. This error model is choice for update laws.

πœƒΜ‡ 𝑖 (𝑑) = βˆ’

Proof (outline). The proof of this stability theorem can be established through using a unified framework. Because the control input 𝑒(𝑑) described in (38) depends on the state π‘₯(𝑑), it first needs to be expressed by using the system output 𝑦(𝑑) through establishing the state observer of the plant: Μ‚Μ‡ (𝑑) = (𝐴 βˆ’ 𝐿𝐢) π‘₯ Μ‚Μ‡ (𝑑) + 𝐡𝑒 (𝑑) + 𝐡𝑑 𝑑 (𝑑) + 𝐿𝑦 (𝑑) , π‘₯

𝑖 = 1, 2, . . . , 𝑀.

𝑇 𝑇 𝑇 ̃𝑇 πœ€ (𝑑) = [0, πœƒΜƒ2 πœ‚2 (𝑑) , πœƒΜƒ3 πœ‚3 (𝑑) , . . . , πœƒΜƒπ‘€πœ‚π‘€ (𝑑)] + 𝐷Φ

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‘‘β†’βˆž 𝑒(𝑑) = 0 with 𝑒(𝑑) = 𝑦(𝑑) βˆ’ π‘¦π‘š (𝑑).

(55)

= with π‘Ž(𝑠) = [𝐼𝑀 𝑠𝐼𝑀 β‹… β‹… β‹… π‘ π‘›βˆ’1 𝐼𝑀]𝑇 , 𝑏(𝑠) 𝑇 π‘›βˆ’1 [πΌπ‘ž π‘ πΌπ‘ž β‹… β‹… β‹… 𝑠 πΌπ‘ž ] , and Ξ›(𝑠) being a chosen monic stable polynomial of degree 𝑛, which has the same eigenvalues with 𝐴 βˆ’ 𝐿𝐢. Then, introducing the fictitious filters for the plant 𝑦(𝑑) = 𝐢(𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡𝑒(𝑑) + 𝐢(𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡𝑑 𝑑(𝑑) and using series transformations, the control input described as in (54) is transformed into the form 𝑒 (𝑑) = 𝐺11 (𝑠, β‹…) [𝑦] (𝑑) + 𝐺12 (𝑠, β‹…) [π‘Ÿ] (𝑑) + 𝐺13 (𝑠, β‹…) [𝑓] (𝑑) + 𝐺14 (𝑠, β‹…) [𝑓𝑝 ] (𝑑) ,

(56)

where 𝑦(𝑑) = β„Ž(𝑠)[𝑦](𝑑) (β„Ž(𝑠) is given below (43)) and 𝐺11 (𝑠, β‹…), 𝐺12 (𝑠, β‹…), 𝐺13 (𝑠, β‹…), and 𝐺14 (𝑠, β‹…) are proper stable operators with finite gains. Furthermore, a filtered version of the output signal 𝑦(𝑑) is expressed in a feedback framework: 𝑑

σ΅„© σ΅„©σ΅„© βˆ’π›Ό (π‘‘βˆ’πœ) π‘₯1 (𝜏) 󡄩󡄩𝑦 (𝑑)σ΅„©σ΅„©σ΅„© ≀ π‘₯0 + 𝛽1 ∫ 𝑒 1 0 𝜏

βˆ’π›Ό2 (πœβˆ’πœ”)

β‹… (∫ 𝑒 0

σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑦 (πœ”)σ΅„©σ΅„©σ΅„© π‘‘πœ”) π‘‘πœ,

(57)

Mathematical Problems in Engineering

9

Μ‡ for some 𝛽1 , 𝛼1 , 𝛼2 > 0, and π‘₯1 (𝑑) = β€–Ξ¦(𝑑)β€– + β€–πœ€(𝑑)β€–π‘š(𝑑) ∈ 2 ∞ 𝐿 ∩ 𝐿 . Applying the small gain lemma to (57), we conclude that 𝑦(𝑑) ∈ 𝐿∞ , and so 𝑦(𝑑), 𝑒(𝑑) ∈ 𝐿∞ . Thus, the signals satisfy πœ”(𝑑), 𝜁(𝑑), πœ‰(𝑑), π‘š(𝑑), πœ€ ∈ 𝐿∞ . Furthermore, Μ‡ Μ‡ ∈ 𝐿2 (Lemma 12) are satisfied, and in πœƒπ‘– , πœ€(𝑑)/π‘š(𝑑), Ξ¦(𝑑), Ξ¨(𝑑) turn πœ‰(𝑑) and 𝑒(𝑑) = 𝑦(𝑑) βˆ’ π‘¦π‘š (𝑑), such that 𝑒(𝑑) = 𝑦(𝑑) βˆ’ π‘¦π‘š (𝑑) converges to zero. βˆ’0.03858 18.984 [ [ βˆ’0.001028 βˆ’0.63253 [ [ 𝐴 = [ 7.8601𝐸 βˆ’ 5 βˆ’0.75905 [ [ 0 0 [ [ βˆ’0.04362 βˆ’249.76

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

βˆ’32.139

1

0.005612

1.3233𝐸 βˆ’ 4

] 3.7553𝐸 βˆ’ 6 ] ] βˆ’0.5183 βˆ’0.0079341 βˆ’3.0808𝐸 βˆ’ 7 ] ], ] ] 1 0 0 ] 0 249.76 0 ]

10.1 0 [ ] [βˆ’1.5446𝐸 βˆ’ 4 ] 0 [ ] [ 0.024656 βˆ’0.01077] 𝐡=[ ], [ ] [ ] 0 0 [ ] 0 0 [ ] 𝐢=[

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

The turbulence disturbances are described in [22, 23]; we can get 𝐡𝑑 = [𝐡𝑑1 , 𝐡𝑑2 ] , 𝑇

𝐡𝑑1 = [βˆ’0.0386 0.001 7.8601𝐸 βˆ’ 5 0 0] ,

(59)

𝑇

𝐡𝑑2 = [βˆ’0.05262 0.00251 βˆ’0.00304 0 0] . 4.2. Adaptive Control Design. For the aircraft system, the transfer function, 𝐺0 (𝑠) = 𝐢(𝑠𝐼 βˆ’ 𝐴)βˆ’1 𝐡, has stable zeros, 𝑠1 = βˆ’4.507, 𝑠2 = βˆ’0.91, and 𝑠3 = βˆ’0.5685, and is strictly proper and full rank. The interactor matrix is chosen as πœ‰π‘š (𝑠) = diag {𝑠 + 1 (𝑠 + 1)2 } .

(60)

The high-frequency matrix is 𝐾𝑝 = lim πœ‰π‘š (𝑠) 𝐺0 (𝑠) = [ π‘ β†’βˆž

2.2673

8.8455

βˆ’0.0902 βˆ’0.3521

],

(61)

and it is is finite and nonsingular and the matrix 𝐾𝑑 = lim πœ‰π‘š (𝑠) 𝐺𝑑 (𝑠) = [ π‘ β†’βˆž

(58)

βˆ’0.0386 βˆ’0.0526 βˆ’0.2481 βˆ’0.6246

]

(62)

is finite. From the specified left coprime polynomial matrix decompositions, 𝐺0 (𝑠) = π‘ƒπ‘™βˆ’1 (𝑠)𝑍0 (𝑠) and 𝐺𝑑 (𝑠) = π‘ƒπ‘™βˆ’1 (𝑠)𝑍𝑑 (𝑠), we can obtain lim π‘βˆ’1 π‘ β†’βˆž 0

(𝑠) 𝑍𝑑 (𝑠) = [

βˆ’0.017 βˆ’0.0064 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 Ξ“πœƒ = 100, 𝐷 = diag{2 2}, and Ξ“ = diag{10 10}. 4.3. Simulation Results. For this simulations study, the initial state is chosen as π‘₯0 (𝑑) = [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 βˆ’10

βˆ’20

βˆ’10

βˆ’20

βˆ’40

βˆ’40 30 40 Versus time (s)

70

0

βˆ’30

20

60

10

βˆ’30

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.

βˆ’50

10

50

60

70

βˆ’50

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.

𝐷𝛿𝑒 :

Aerodynamic drag derivative with respect to elevator deflection angle 𝑇, 𝐷, 𝐿: Engine thrust, aerodynamic drag force, and aerodynamic lift force 𝛼, πœƒ, 𝛾: Angle of attack, pitch, and flight-path slope β„Ž: Altitude of aircraft 𝑔: Gravitational acceleration π‘š: Mass of aircraft 𝑉: Airspeed of aircraft π‘ž: Pith rate

70

Mathematical Problems in Engineering

References

Adaptive control with disturbance

6 Throttle (unit)

11

4 2 0 βˆ’2

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.

𝐼𝑦 : 𝑀: 𝛿𝑒 , 𝛿𝑑 :

Moment of inertia in pitch The pitch moment Elevator deflection bias angle and engine throttle angle Aerodynamic pitch moment and drag 𝑀𝑉0 , 𝐷𝑉0 : derivative with respect to airspeed 𝑉0 𝑇𝑉0 , 𝐿 𝑉0 : Thrust and aerodynamic lift derivative with respect to airspeed 𝑉0 𝑀𝛿𝑒 , 𝐿 𝛿𝑒 : Aerodynamic pitch moment and lift derivative with respect to 𝛿𝑒 𝑉𝑀 , 𝑒π‘₯ , 𝑒𝑦 : Turbulence velocity and body axis components of 𝑉𝑀 𝑉0 , 𝛼0 , 𝛼̇ 0 , π‘ž0 , β„Ž0 : The trim value of aircraft state Aerodynamic pitch moment and drag 𝑀𝑉0 , 𝐷𝑉0 : derivative with respect to airspeed 𝑉0 𝑇𝑉0 , 𝐿 𝑉0 : Thrust and aerodynamic lift derivatives with respect to airspeed 𝑉0 𝑀𝛿𝑒 , 𝐿 𝛿𝑒 : Aerodynamic pitch moment and lift derivative with respect to elevator Aerodynamic drag derivative with 𝐷𝛼0 , π·β„Ž0 : respect to 𝛼0 and β„Ž0 𝐷𝛿𝑒 : Aerodynamic drag due to 𝛿𝑒 𝑀𝛼0 , 𝑀𝛼̇ 0 : Aerodynamic pitch moment with respect to 𝛼0 and 𝛼̇ 0 π‘€π‘ž0 : Aerodynamic pitch moment with respect to π‘ž0 𝑇0 : Benchmark aerodynamic thrust at the airspeed 𝑉0 𝑇𝛿𝑑 : Aerodynamic thrust derivatives with respect to the throttle Aerodynamic lift derivative with respect 𝐿 𝛼0 , 𝐿 β„Ž0 : to angle of attack and height.

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

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12 [17] X.-X. Yin, Y.-G. Lin, W. Li, H.-W. Liu, and Y.-J. Gu, β€œAdaptive sliding mode back-stepping pitch angle control of a variabledisplacement pump controlled pitch system for wind turbines,” ISA Transactions, vol. 58, pp. 629–634, 2015. [18] N. T. Nguyen, β€œOptimal control modification for robust adaptive control with large adaptive gain,” Systems & Control Letters, vol. 61, no. 4, pp. 485–494, 2012. [19] J.-Z. Geng, H.-L. Yao, and H. Zhang, β€œStudies on effect of air wake on slope and landing property of carrier aircraft,” Journal of System Simulation, vol. 21, no. 18, pp. 5940–5943, 2009. [20] D. D. Boskovic and J. Redding, β€œAn autonomous carrier landing system for unmanned aerial vehicles,” in Proceedings of the AIAA Conference on Guidance, Navigation, and Control, AIAA 2009-6264, Chicago, Ill, USA, August 2009. [21] β€œDepartment of defense handbook flying qualities of piloted aircraft. MIL-STD-1797A,” 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, β€œMultivariable adaptive control using high-frequency gain matrix factorization,” IEEE Transactions on Automatic Control, vol. 49, no. 7, pp. 1152–1157, 2004. [25] L. Wen, G. Tao, and H. Yang, β€œAdaptive turbulence compensation for multivariable nonlinear aircraft models,” in Proceedings of the American Control Conference (ACC ’15), pp. 5557–5562, 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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