# Adaptive Disturbance Rejection Control for Automatic Carrier Landing System

Authors: Xin Wang; Xin Chen; Liyan Wen
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

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.


---

## Body

## 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∞ output-feedback 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 state-feedback 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 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 Section2, 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.


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

The overall carrier landing task for a fixed-wing aircraft is shown in Figure1. The final-approach leg is typically entered from the last turn until the touchdown on the carrier deck, as illustrated in Figure 2. The turbulence is the major source of glide path and touchdown errors. In this phase, the longitudinal reference flight state is chosen as a steady rectilinear flight in air wake at a constant velocity, with constant angle of attack and a flight-path angle. The flaps and the gear are totally lowered, and two control means are employed to control the flight-path vector in the vertical plane: elevator and engine thrust. In this section, the longitudinal linear aircraft model and carrier air wake are described.Figure 1
Procedures of carrier landing for the aircraft.Figure 2
Final carrier landing phase.


### 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(1)mV˙=Tcos⁡α-D-mgsin⁡γmVγ˙=Tsin⁡α+L-mgcos⁡γ.The kinematic equation is(2)θ˙=q.The moment equation is(3)q˙=MIy.The navigation equation is(4)h˙=Vsin⁡γ.The identical equation is(5)γ=θ-α.


### 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(6)x˙1=A1x1+B1uy1=C1x1,where x1=[v,α,q,θ,h]T, u1=[δe,δt]T, and y1=[v,α,q,θ,h]T are the system state vector, input vector, and output vector, respectively, and the matrices are(7)
A
1
=
T
V
0
cos
⁡
α
0
-
D
V
0
m
-
T
0
sin
⁡
α
0
-
D
α
0
m
+
g
cos
⁡
γ
0
0
-
g
cos
⁡
γ
0
-
D
h
0
m
-
L
V
0
m
V
0
-
T
0
cos
⁡
α
0
-
L
α
0
+
m
g
sin
⁡
γ
0
m
V
0
1
-
g
sin
⁡
γ
0
V
0
-
L
h
0
m
V
0
M
v
0
I
y
-
M
α
˙
0
I
y
L
V
0
m
V
0
M
α
0
I
y
-
M
α
˙
0
I
y
L
α
0
+
T
0
cos
⁡
α
0
m
V
0
-
g
sin
⁡
γ
0
V
0
M
q
0
I
y
+
M
α
˙
0
I
y
-
M
α
˙
0
I
y
g
sin
⁡
γ
0
V
0
0
0
0
1
0
0
sin
⁡
γ
0
-
V
0
cos
⁡
γ
0
0
V
0
cos
⁡
γ
0
0
,
B
1
=
-
D
δ
e
m
-
T
δ
t
cos
⁡
α
0
m
-
L
δ
e
m
V
0
-
T
δ
t
sin
⁡
α
0
m
V
0
M
δ
e
I
y
-
M
α
˙
0
I
y
L
δ
e
m
V
0
-
M
α
˙
0
I
y
T
δ
t
sin
⁡
α
0
m
V
0
0
0
0
0
,
C
1
=
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
.


### 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(8)
u
x
V
w
=
0
X
∈
-
∞
,
-
1750
0.04
-
1710
+
1750
X
+
1750
X
∈
-
1750
,
-
1710
0.02
-
1630
+
1710
X
+
1870
X
∈
-
1710
,
-
1630
-
0.004
-
1550
+
1630
X
+
430
X
∈
-
1630
,
-
1550
-
0.028
-
1340
+
1550
X
+
1130
X
∈
-
1550
,
-
1340
0.01
-
1160
+
1340
X
+
1844
X
∈
-
1340
,
-
1160
0.15
-
0.038
-
400
+
1160
X
+
1418
X
∈
-
1160
,
-
400
-
0.15
150
X
+
250
X
∈
-
400
,
-
250
0
X
∈
-
250,0
u
y
V
w
=
0
X
∈
-
∞
,
-
2650
-
0.076
-
2400
+
2650
X
+
2650
X
∈
-
2650
,
-
2400
-
0.088
+
0.076
-
2200
+
2400
X
+
3667
X
∈
-
2400
,
-
2200
-
0.08
+
0.088
-
1970
+
2200
X
-
330
X
∈
-
2200
,
-
1970
0.02
+
0.08
-
750
+
1970
X
+
994
X
∈
-
1970
,
-
750
-
0.01
-
0.02
-
250
+
750
X
+
417
X
∈
-
750
,
-
250
-
0.01
X
∈
-
250,0
X
=
-
V
cos
⁡
γ
T
0
-
t
-
286
,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 t 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 ux and uy. Because the flight speed is far larger than the wind speed, we can get(9)Vi≈V+uxαi≈α+uyV,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 (10)x˙1=A1x1+B1u1+B1ddty1=C1x1,where A1, B1, C1, x1, and u1 are defined in (6). The disturbance is dt=[ux,uy] and the matrix B1d is (11)
B
1
d
=
T
V
0
cos
⁡
α
0
-
D
V
0
m
cos
⁡
γ
0
-
T
sin
⁡
α
0
-
D
α
0
m
V
0
T
V
0
sin
⁡
α
0
+
L
V
0
m
V
0
T
sin
⁡
α
0
+
L
α
0
m
V
0
2
M
α
˙
0
I
y
T
V
0
sin
⁡
α
0
+
L
V
0
m
V
0
-
M
v
0
I
y
M
α
˙
0
I
y
L
α
0
+
T
0
sin
⁡
α
0
m
V
0
2
-
M
α
0
I
y
V
0
0
0
0
0
.


## 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(1)mV˙=Tcos⁡α-D-mgsin⁡γmVγ˙=Tsin⁡α+L-mgcos⁡γ.The kinematic equation is(2)θ˙=q.The moment equation is(3)q˙=MIy.The navigation equation is(4)h˙=Vsin⁡γ.The identical equation is(5)γ=θ-α.


## 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(6)x˙1=A1x1+B1uy1=C1x1,where x1=[v,α,q,θ,h]T, u1=[δe,δt]T, and y1=[v,α,q,θ,h]T are the system state vector, input vector, and output vector, respectively, and the matrices are(7)
A
1
=
T
V
0
cos
⁡
α
0
-
D
V
0
m
-
T
0
sin
⁡
α
0
-
D
α
0
m
+
g
cos
⁡
γ
0
0
-
g
cos
⁡
γ
0
-
D
h
0
m
-
L
V
0
m
V
0
-
T
0
cos
⁡
α
0
-
L
α
0
+
m
g
sin
⁡
γ
0
m
V
0
1
-
g
sin
⁡
γ
0
V
0
-
L
h
0
m
V
0
M
v
0
I
y
-
M
α
˙
0
I
y
L
V
0
m
V
0
M
α
0
I
y
-
M
α
˙
0
I
y
L
α
0
+
T
0
cos
⁡
α
0
m
V
0
-
g
sin
⁡
γ
0
V
0
M
q
0
I
y
+
M
α
˙
0
I
y
-
M
α
˙
0
I
y
g
sin
⁡
γ
0
V
0
0
0
0
1
0
0
sin
⁡
γ
0
-
V
0
cos
⁡
γ
0
0
V
0
cos
⁡
γ
0
0
,
B
1
=
-
D
δ
e
m
-
T
δ
t
cos
⁡
α
0
m
-
L
δ
e
m
V
0
-
T
δ
t
sin
⁡
α
0
m
V
0
M
δ
e
I
y
-
M
α
˙
0
I
y
L
δ
e
m
V
0
-
M
α
˙
0
I
y
T
δ
t
sin
⁡
α
0
m
V
0
0
0
0
0
,
C
1
=
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
.


## 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(8)
u
x
V
w
=
0
X
∈
-
∞
,
-
1750
0.04
-
1710
+
1750
X
+
1750
X
∈
-
1750
,
-
1710
0.02
-
1630
+
1710
X
+
1870
X
∈
-
1710
,
-
1630
-
0.004
-
1550
+
1630
X
+
430
X
∈
-
1630
,
-
1550
-
0.028
-
1340
+
1550
X
+
1130
X
∈
-
1550
,
-
1340
0.01
-
1160
+
1340
X
+
1844
X
∈
-
1340
,
-
1160
0.15
-
0.038
-
400
+
1160
X
+
1418
X
∈
-
1160
,
-
400
-
0.15
150
X
+
250
X
∈
-
400
,
-
250
0
X
∈
-
250,0
u
y
V
w
=
0
X
∈
-
∞
,
-
2650
-
0.076
-
2400
+
2650
X
+
2650
X
∈
-
2650
,
-
2400
-
0.088
+
0.076
-
2200
+
2400
X
+
3667
X
∈
-
2400
,
-
2200
-
0.08
+
0.088
-
1970
+
2200
X
-
330
X
∈
-
2200
,
-
1970
0.02
+
0.08
-
750
+
1970
X
+
994
X
∈
-
1970
,
-
750
-
0.01
-
0.02
-
250
+
750
X
+
417
X
∈
-
750
,
-
250
-
0.01
X
∈
-
250,0
X
=
-
V
cos
⁡
γ
T
0
-
t
-
286
,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 t 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 ux and uy. Because the flight speed is far larger than the wind speed, we can get(9)Vi≈V+uxαi≈α+uyV,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 (10)x˙1=A1x1+B1u1+B1ddty1=C1x1,where A1, B1, C1, x1, and u1 are defined in (6). The disturbance is dt=[ux,uy] and the matrix B1d is (11)
B
1
d
=
T
V
0
cos
⁡
α
0
-
D
V
0
m
cos
⁡
γ
0
-
T
sin
⁡
α
0
-
D
α
0
m
V
0
T
V
0
sin
⁡
α
0
+
L
V
0
m
V
0
T
sin
⁡
α
0
+
L
α
0
m
V
0
2
M
α
˙
0
I
y
T
V
0
sin
⁡
α
0
+
L
V
0
m
V
0
-
M
v
0
I
y
M
α
˙
0
I
y
L
α
0
+
T
0
sin
⁡
α
0
m
V
0
2
-
M
α
0
I
y
V
0
0
0
0
0
.


## 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(8)
u
x
V
w
=
0
X
∈
-
∞
,
-
1750
0.04
-
1710
+
1750
X
+
1750
X
∈
-
1750
,
-
1710
0.02
-
1630
+
1710
X
+
1870
X
∈
-
1710
,
-
1630
-
0.004
-
1550
+
1630
X
+
430
X
∈
-
1630
,
-
1550
-
0.028
-
1340
+
1550
X
+
1130
X
∈
-
1550
,
-
1340
0.01
-
1160
+
1340
X
+
1844
X
∈
-
1340
,
-
1160
0.15
-
0.038
-
400
+
1160
X
+
1418
X
∈
-
1160
,
-
400
-
0.15
150
X
+
250
X
∈
-
400
,
-
250
0
X
∈
-
250,0
u
y
V
w
=
0
X
∈
-
∞
,
-
2650
-
0.076
-
2400
+
2650
X
+
2650
X
∈
-
2650
,
-
2400
-
0.088
+
0.076
-
2200
+
2400
X
+
3667
X
∈
-
2400
,
-
2200
-
0.08
+
0.088
-
1970
+
2200
X
-
330
X
∈
-
2200
,
-
1970
0.02
+
0.08
-
750
+
1970
X
+
994
X
∈
-
1970
,
-
750
-
0.01
-
0.02
-
250
+
750
X
+
417
X
∈
-
750
,
-
250
-
0.01
X
∈
-
250,0
X
=
-
V
cos
⁡
γ
T
0
-
t
-
286
,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 t 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 ux and uy. Because the flight speed is far larger than the wind speed, we can get(9)Vi≈V+uxαi≈α+uyV,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 (10)x˙1=A1x1+B1u1+B1ddty1=C1x1,where A1, B1, C1, x1, and u1 are defined in (6). The disturbance is dt=[ux,uy] and the matrix B1d is (11)
B
1
d
=
T
V
0
cos
⁡
α
0
-
D
V
0
m
cos
⁡
γ
0
-
T
sin
⁡
α
0
-
D
α
0
m
V
0
T
V
0
sin
⁡
α
0
+
L
V
0
m
V
0
T
sin
⁡
α
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## 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(12)x˙t=Axt+But+Bddtyt=Cxt,where A∈Rn×n, B∈Rn×M, Bd∈RM×p, and C∈RM×n are constant and unknown; xt∈Rn, ut∈RM, and yt∈RM are the system state vector, input vector, and output vector, respectively; dt=d1t,…,dptT∈Rp is the disturbance vector. The disturbance signal dt is unmatched with the control input ut, in the sense that B and Bd are not linearly dependent, Bd≠Bα for any matrix α∈RM×p.The control objective is to design an adaptive state-feedback control signalut for (12), to make closed-loop signal boundedness and the output yt track a chosen reference signal ymt generated from a reference model: (13)ymt=Wmsrt∈RM,where Wms∈RM×M is a stable transfer function matrix and rt∈RM is an external reference input signal for defining a desired ymt. Note that, in this paper, we use the notation yt=Gsut to represent the output yt of a system whose transfer matrix is Gs and input is ut, a convenient notation for adaptive control systems.


### 3.2. Preliminaries and Assumptions

Lemma 1 (see [8]).
For anyM×M strictly proper and full rank rational transfer matrix Gs, there exists a lower triangular polynomial matrix ξms, defined as the left interactor matrix of Gs, of the form (14)ξms=d1s000h21msd2s00⋮⋮⋮⋮hM1mshM2ms⋯dMs,where hijm(s),j=1,2,…,M-1,i=2,…,M, are some polynomials and dis are any chosen monic stable polynomials such that the high-frequency gain matrix of Gs defined as Kp=lims→∞ξmsGs.Lemma 2 (see [24]).
EveryM×M real matrix Kp with nonzero leading principal minors Δ1,Δ2,…,ΔM can be uniquely factored as(15)Kp=LDU,where L∈RM×M is unity lower triangular, U∈RM×M is unity upper triangular, and (16)D=diag⁡d1,d2,…,dm=diag⁡Δ1,Δ2Δ-1,…,ΔmΔm-1-1.Assumption 3.
All zeros ofG0s=CsI-A-1B are stabilizable and detectable.Assumption 4.
G 0 s is strictly proper with full rank and has a known modified interactor matrix ξms such that Kp=lims→∞ξmsG0s is finite and nonsingular (so that Wm(s)=ξm-1s can be chosen as the transfer matrix for the reference model system).Assumption 5.
The leading principal minors of the high-frequency gain matrixKp are nonzero, and their signs are known.
From plant (12), the control and disturbance transfer functions are obtained as G0s=CsI-A-1B and Gds=CsI-A-1Bd and are expressed in their left coprime polynomial matrix decompositions: G0s=Pl-1sZ0s and Gds=Pl-1sZds, where Pls,Z0s∈RM×M and Zds∈RM×p are some polynomial matrices.Assumption 6.
The transfer matrixZ0-1sZds is proper.Remark 7.
Assumption3 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 ut and the disturbance input dt to the output yt 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(17)u∗t=K1∗Txt+K2∗rt+K3∗t,where the nominal parameters K1∗T∈RM×n and K2∗T∈RM×M are for the plant-model output matching, and K3∗t∈RM is used to cancel the effect of the disturbance dt.Lemma 8 (see [25]).
The matrixKd=lims→∞⁡ξmsGds is finite if Z0-1sZds is proper.Based on Lemma8, the existence of a nominal controller (17) is established a follows.Theorem 9.
For plant (12) with the unmatched disturbances, under Assumptions 3 and 6, there exists a state-feedback control law, to make the boundedness of all closed-loop signals, disturbance rejection, and output tracking of the reference ymt.
From plant (12), the input-output form is obtained as (18)yt=G0sut+y¯t,with y¯t=G0sdt=CsI-A-1Bddt.Operate the interactor matrix (a polynomial matrix)ξms on plant (12), x˙t=Axt+But+Bddt, yt=Cxt, to reach an expression of ξmsymt in a possible form: (19)ξmsyt=-K¯0xt+K¯put+K¯p1u˙+⋯+K¯pl0ul0t+K¯ddt+K¯d1d˙t+⋯+K¯dl1dl1t,with some constant matrices K¯0∈RM×n, K¯p∈RM×M, K¯pj∈RM×M, j=1,2,…,l0, K¯d∈RM×p, and K¯di∈RM×p, i=1,2,…,l1, for some integers l0,l1≥0. From (12) and (18), we have (20)xs=sI-A-1Bus+sI-A-1Bdds.Expressing (19) in s domain and using (20), we have (21)ξmsys=-K¯0sI-A-1Bus+K¯pus+K¯p1sus+⋯+K¯pl0sl0us-K¯0sI-A-1Bdds+K¯dds+K¯d1sds+⋯+K¯dl1sl1ds.From Assumption 4, Kp=lims→∞ξmsG0s is finite and nonsingular and from Assumption 6, Kpj=0, j=1,…,l0, K¯P=Kp, and Kdj=0, j=1,…,l0, K¯d=Kd. Hence, we have (22)ξmsyt=-K¯0xt+Kput+Kddt.From (17), the control law can be designed as (23)ut=u∗t=K1∗Txt+K2∗rt+K3∗t,where K1∗T=Kp-1K¯0, K2∗=Kp-1, and K3∗t=K3ddt with K3dt=-Kp-1Kd, which leads to the output matching: ξmsyt=rt. We applied (23) to plant (12); we have (24)yt=CsI-A-BK1∗T-1BK2∗rt+CsI-A-BK1∗T-1BK3∗t+CsI-A-BK1∗T-1Bdds=Wmsrt=ymt.Remark 10.
From (24), we can conclude that the plant-model matching conditions are (25)CsI-A-BK1∗T-1BK2∗=WmsWmsK2∗-1K3∗s+CsI-A-BK1∗T-1Bdds=0.For the plant-model matching condition (25), there exist constant matrices K1∗T and K2∗ and K3∗t, if and only if the relative degree condition in Assumption 6 is satisfied.


### 3.4. Parameterizations of the TermK3∗t

For the disturbance vectordt∈Rp, each element djt in (12) can be expressed as (26)djt=dj0+∑k=1qjdjkfjkt=μj∗Tfjt,j=1,2,…,p,with some unknown constants dj0, djk and some known bounded continuous signals fjk,j=1,2,…,p,k=1,2,…,qj.Remark 11.
For model (26), if qj=0, then djt=dj0,j=1,2,…,p, representing a constant disturbance signal. If we choose qj=1 and fj1t=sin⁡wj1t, then djt=dj0+dj1sin⁡wj1t,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 fjkt in (26).The parameter matrix and the disturbance signal components are(27)μj∗=dj0,dj1,…,djqjT∈Rqj+1,fjt=1,fj1t,…,fjqjtT∈Rqj+1,j=1,2,…,p.Hence, the disturbancedt is expressed as (28)dt=N∗Tft,N∗T=μ1T0q2+1T0q3+1T⋯0qp+1T0q1+1Tμ2∗T0q3+1T⋯0qp+1T⋮⋮⋮⋮⋮0q1+1T0q2+1T0q3+1T⋯μp∗T∈Rp×q,ft=f1Ttf2Tt⋯fpTtT∈Rqq=q1+q2+⋯+qp+p.WithK3d∗=k3d1∗,k3d2∗,…,k3dp∗, k3dj∗∈RM, j=1,2,…,p, the disturbance rejection term K3∗t is parameterized as (29)K3∗t=K3d∗dt=K3d∗N∗Tft=Φ3∗ft,where the parameter matrix is (30)Φ3∗=ϕ31∗,ϕ32∗,…,ϕ3p∗∈RM×q,q=q1+q2+⋯+qp+p,ϕ3j∗=k3dj∗μj∗T∈RM×qj+1,j=1,2,…,p.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 closed-loop system becomes (31)x˙t=A+BK1∗Txt+BK2∗rt+BK3∗t+Bddt+But-K1∗Txt-K2∗rt-K3∗tyt=Cxt.In view of (24), (25), and (31), the output tracking error equation is (32)et=yt-ymt=WmsKp∗u-K1∗Tx-K2∗r-K3∗t+fpt,Kp∗=K2∗-1,where fpt=CeA+BK1∗Ttx0 converges to zero exponentially fast due to the stability of A+BK1∗T and Wm(s)=ξm-1(s). Hence, we have (33)ξmset=Kput-K1∗Txt-K2∗rt-K3∗t.To deal with the uncertainty of the high-frequency gain matrixKp, the LDU decomposition of Kp is used in (33), so that we have (34)L-1ξmset=DUut-K1∗Txt-K2∗rt-K3∗t.We have the following equation: (35)Uut=ut-I-Uut.With (34) and (35), we have (36)L-1ξmset=Dut-I-Uut-UK1∗Txt-K2∗rt-K3∗t.The new equation is (37)L-1ξmset=Dut-Φ0∗ut-Φ1∗Tωt,where Φ1∗T=UK1∗T,UK2∗,UK3f∗ and ωt=xTt,rTt,fTtT. This new equation has a new controller structure: (38)ut=Φ0ut+Φ1Tωt,where Φ0 and Φ1T are the estimates of Φ0∗ and Φ1∗T and Φ0 is upper triangular with zero diagonal elements (only its nonzero elements are estimated). The matrix Φ0 has the same strictly form as that of Φ0∗=I-U: (39)Φ0=0ϕ12ϕ13⋯ϕ1M00ϕ23⋯ϕ2M⋮⋮⋮⋮00⋯0ϕM-1M00⋯00∈RM×M.From (37) and (38), we obtain a new error model: (40)ξmset+Θ0∗ξmset=DΦ~Ttω¯t,where the parameter error is Φ~(t)=Φ(t)-Φ∗ and ΦT(t)=[Φ0(t),Φ1T(t)] is the estimate of unknown parameter matrix Φ∗T=[Φ0∗,Φ1∗T], ω¯(t)=[uT(t),ωT(t)]T, and ω(t)=xT(t),rT(t),fT(t)T, where Θ0∗=L-1-I is introduced to parameterize the unknown matrix L, which has a special form:(41)Θ0∗=00⋯0θ21∗0⋯0θ31∗θ32∗⋯0⋮⋮⋮0θM-11∗⋯00θM1∗⋯θMM-1∗0∈RM×M.For such a matrix Θ0∗, the parameter vectors are defined as (42)θ2∗=θ21∗∈R,θ3∗=θ31∗,θ32∗T∈R2,⋮θM-1∗=θM-11∗,…,θM-1M-2∗∈RM-2,θM∗=θM1∗,…,θMM-2∗∈RM-1,and their estimates are (43)θ2t=θ21t∈R,θ3t=θ31t,θ32tT∈R2,⋮θM-1t=θM-11t,…,θM-1M-2t∈RM-2,θMt=θM1t,…,θMM-2t∈RM-1.We introduce a filter hs=1/fs, where fs is chosen as a stable and monic polynomial whose degree is equal to the maximum degree of the modified interactor ξms. Operating both sides of (40) by hsIM leads to (44)ξmshset+Θ0∗ξmshset=DhsΦ~Tω¯t.We define (45)e¯t=ξmshset=e¯1t,…,e¯MtT,(46)ηit=e¯1t,…,e¯i-1tT∈Ri-1,i=2,3,…,M.From (45) and (46) in (44), we obtained (47)e¯t+0,θ2∗Tη2t,θ3∗Tη3t,…,θM∗TηMtT=DhsΦ~Tω¯t.Based on the parameterized error equation (47), an estimation error signal is introduced: (48)εt=e¯t+0,θ2Tη2t,θ3Tη3t,…,θMTηMtT+Ψtξt∈RM,where Ψ(t)∈RM×M is the estimate of Ψ∗=D and (49)ζt=hsω¯t,i=1,2,…,M,ξt=ΦTtζt-hsΦTω¯t,i=1,2,…,M.It then follows from (44) (47) and (48) that(50)εt=0,θ~2Tη2t,θ~3Tη3t,…,θ~MTηMtT+DΦ~T+Ψ~tξt,where Ψ~(t)=Ψ(t)-Ψ∗ and θ~i=θi-θi∗ are the parameter errors. This error model is choice for update laws.


### 3.6. Adaptive Laws

Based on the error model (50), the adaptive laws are chosen as (51)θ˙it=-Γθiεitηitm2t,i=2,…,M,Φ˙Tt=-DεtζTtm2t,Ψ˙t=-ΓεtξTtm2t,where Γθi=ΓθiT>0,i=2,3,…,M, and Γ=ΓT>0 are adaptive gains; D is defined in (15); ε(t)=ε1(t),ε2(t),…,εM(t)T is calculated from (50); and (52)m2t=1+ζTtζt+ξTtξt+∑i=2MηiTηit.


### 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)
θi(t)∈L∞, i=2,3,…,M, Φ(t)∈L∞, Ψ(t)∈L∞, and ε(t)/m(t)∈L2∩L∞;(ii)
θ˙i(t)∈L2∩L∞, i=2,3,…,M, Φ˙(t)∈L2∩L∞, and Ψ˙(t)∈L2∩L∞.Based on Lemma12, 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, 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 limt→∞⁡e(t)=0 with e(t)=y(t)-ym(t).Proof (outline).
The proof of this stability theorem can be established through using a unified framework. Because the control inputut described in (38) depends on the state xt, it first needs to be expressed by using the system output yt through establishing the state observer of the plant: (53)x^˙t=A-LCx^˙t+But+Bddt+Lyt,where L∈Rn×M is a gain matrix such that A-LC is stable, which is possible, and (AC) is assumed to be detectable. Hence, we have (54)ut=Φ1Ttω1t+Φ2Ttω2t+Φ3dTtω3t+K2trt+Φ3tft,where Φ1Tt, Φ2Tt, Φ3dTt, K2t, and Φ3t are adaptive estimates of the corresponding nominal controller parameters and (55)ω1t=asΛsut,ω2t=asΛsyt,ω3t=bsΛsft,with as=[IMsIM⋯sn-1IM]T, bs=[IqsIq⋯sn-1Iq]T, and Λs being a chosen monic stable polynomial of degree n, which has the same eigenvalues with A-LC. Then, introducing the fictitious filters for the plant yt=CsI-A-1But+CsI-A-1Bddt and using series transformations, the control input described as in (54) is transformed into the form (56)ut=G11s,·y¯t+G12s,·rt+G13s,·ft+G14s,·fpt,where y¯t=hsyt (hs is given below (43)) and G11s,·, G12s,·, G13s,·, and G14s,· are proper stable operators with finite gains. Furthermore, a filtered version of the output signal yt is expressed in a feedback framework: (57)y¯t≤x0+β1∫0te-α1t-τx1τ∫0τe-α2τ-ωy¯ω¯dω¯dτ,for some β1,α1,α2>0, and x1t=Φ˙t+εtmt∈L2∩L∞. Applying the small gain lemma to (57), we conclude that y¯t∈L∞, and so yt,ut∈L∞. Thus, the signals satisfy ω¯t,ζt,ξt,mt,ε∈L∞. Furthermore, θi,εt/mt,Φ˙t,Ψ˙t∈L2 (Lemma 12) are satisfied, and in turn ξt and et=yt-ymt, such that et=yt-ymt converges to zero.


## 3.1. Problem Formulation

Consider the linear time-invariant system as(12)x˙t=Axt+But+Bddtyt=Cxt,where A∈Rn×n, B∈Rn×M, Bd∈RM×p, and C∈RM×n are constant and unknown; xt∈Rn, ut∈RM, and yt∈RM are the system state vector, input vector, and output vector, respectively; dt=d1t,…,dptT∈Rp is the disturbance vector. The disturbance signal dt is unmatched with the control input ut, in the sense that B and Bd are not linearly dependent, Bd≠Bα for any matrix α∈RM×p.The control objective is to design an adaptive state-feedback control signalut for (12), to make closed-loop signal boundedness and the output yt track a chosen reference signal ymt generated from a reference model: (13)ymt=Wmsrt∈RM,where Wms∈RM×M is a stable transfer function matrix and rt∈RM is an external reference input signal for defining a desired ymt. Note that, in this paper, we use the notation yt=Gsut to represent the output yt of a system whose transfer matrix is Gs and input is ut, a convenient notation for adaptive control systems.


## 3.2. Preliminaries and Assumptions

Lemma 1 (see [8]).
For anyM×M strictly proper and full rank rational transfer matrix Gs, there exists a lower triangular polynomial matrix ξms, defined as the left interactor matrix of Gs, of the form (14)ξms=d1s000h21msd2s00⋮⋮⋮⋮hM1mshM2ms⋯dMs,where hijm(s),j=1,2,…,M-1,i=2,…,M, are some polynomials and dis are any chosen monic stable polynomials such that the high-frequency gain matrix of Gs defined as Kp=lims→∞ξmsGs.Lemma 2 (see [24]).
EveryM×M real matrix Kp with nonzero leading principal minors Δ1,Δ2,…,ΔM can be uniquely factored as(15)Kp=LDU,where L∈RM×M is unity lower triangular, U∈RM×M is unity upper triangular, and (16)D=diag⁡d1,d2,…,dm=diag⁡Δ1,Δ2Δ-1,…,ΔmΔm-1-1.Assumption 3.
All zeros ofG0s=CsI-A-1B are stabilizable and detectable.Assumption 4.
G 0 s is strictly proper with full rank and has a known modified interactor matrix ξms such that Kp=lims→∞ξmsG0s is finite and nonsingular (so that Wm(s)=ξm-1s can be chosen as the transfer matrix for the reference model system).Assumption 5.
The leading principal minors of the high-frequency gain matrixKp are nonzero, and their signs are known.
From plant (12), the control and disturbance transfer functions are obtained as G0s=CsI-A-1B and Gds=CsI-A-1Bd and are expressed in their left coprime polynomial matrix decompositions: G0s=Pl-1sZ0s and Gds=Pl-1sZds, where Pls,Z0s∈RM×M and Zds∈RM×p are some polynomial matrices.Assumption 6.
The transfer matrixZ0-1sZds is proper.Remark 7.
Assumption3 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 ut and the disturbance input dt to the output yt 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(17)u∗t=K1∗Txt+K2∗rt+K3∗t,where the nominal parameters K1∗T∈RM×n and K2∗T∈RM×M are for the plant-model output matching, and K3∗t∈RM is used to cancel the effect of the disturbance dt.Lemma 8 (see [25]).
The matrixKd=lims→∞⁡ξmsGds is finite if Z0-1sZds is proper.Based on Lemma8, the existence of a nominal controller (17) is established a follows.Theorem 9.
For plant (12) with the unmatched disturbances, under Assumptions 3 and 6, there exists a state-feedback control law, to make the boundedness of all closed-loop signals, disturbance rejection, and output tracking of the reference ymt.
From plant (12), the input-output form is obtained as (18)yt=G0sut+y¯t,with y¯t=G0sdt=CsI-A-1Bddt.Operate the interactor matrix (a polynomial matrix)ξms on plant (12), x˙t=Axt+But+Bddt, yt=Cxt, to reach an expression of ξmsymt in a possible form: (19)ξmsyt=-K¯0xt+K¯put+K¯p1u˙+⋯+K¯pl0ul0t+K¯ddt+K¯d1d˙t+⋯+K¯dl1dl1t,with some constant matrices K¯0∈RM×n, K¯p∈RM×M, K¯pj∈RM×M, j=1,2,…,l0, K¯d∈RM×p, and K¯di∈RM×p, i=1,2,…,l1, for some integers l0,l1≥0. From (12) and (18), we have (20)xs=sI-A-1Bus+sI-A-1Bdds.Expressing (19) in s domain and using (20), we have (21)ξmsys=-K¯0sI-A-1Bus+K¯pus+K¯p1sus+⋯+K¯pl0sl0us-K¯0sI-A-1Bdds+K¯dds+K¯d1sds+⋯+K¯dl1sl1ds.From Assumption 4, Kp=lims→∞ξmsG0s is finite and nonsingular and from Assumption 6, Kpj=0, j=1,…,l0, K¯P=Kp, and Kdj=0, j=1,…,l0, K¯d=Kd. Hence, we have (22)ξmsyt=-K¯0xt+Kput+Kddt.From (17), the control law can be designed as (23)ut=u∗t=K1∗Txt+K2∗rt+K3∗t,where K1∗T=Kp-1K¯0, K2∗=Kp-1, and K3∗t=K3ddt with K3dt=-Kp-1Kd, which leads to the output matching: ξmsyt=rt. We applied (23) to plant (12); we have (24)yt=CsI-A-BK1∗T-1BK2∗rt+CsI-A-BK1∗T-1BK3∗t+CsI-A-BK1∗T-1Bdds=Wmsrt=ymt.Remark 10.
From (24), we can conclude that the plant-model matching conditions are (25)CsI-A-BK1∗T-1BK2∗=WmsWmsK2∗-1K3∗s+CsI-A-BK1∗T-1Bdds=0.For the plant-model matching condition (25), there exist constant matrices K1∗T and K2∗ and K3∗t, if and only if the relative degree condition in Assumption 6 is satisfied.


## 3.4. Parameterizations of the TermK3∗t

For the disturbance vectordt∈Rp, each element djt in (12) can be expressed as (26)djt=dj0+∑k=1qjdjkfjkt=μj∗Tfjt,j=1,2,…,p,with some unknown constants dj0, djk and some known bounded continuous signals fjk,j=1,2,…,p,k=1,2,…,qj.Remark 11.
For model (26), if qj=0, then djt=dj0,j=1,2,…,p, representing a constant disturbance signal. If we choose qj=1 and fj1t=sin⁡wj1t, then djt=dj0+dj1sin⁡wj1t,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 fjkt in (26).The parameter matrix and the disturbance signal components are(27)μj∗=dj0,dj1,…,djqjT∈Rqj+1,fjt=1,fj1t,…,fjqjtT∈Rqj+1,j=1,2,…,p.Hence, the disturbancedt is expressed as (28)dt=N∗Tft,N∗T=μ1T0q2+1T0q3+1T⋯0qp+1T0q1+1Tμ2∗T0q3+1T⋯0qp+1T⋮⋮⋮⋮⋮0q1+1T0q2+1T0q3+1T⋯μp∗T∈Rp×q,ft=f1Ttf2Tt⋯fpTtT∈Rqq=q1+q2+⋯+qp+p.WithK3d∗=k3d1∗,k3d2∗,…,k3dp∗, k3dj∗∈RM, j=1,2,…,p, the disturbance rejection term K3∗t is parameterized as (29)K3∗t=K3d∗dt=K3d∗N∗Tft=Φ3∗ft,where the parameter matrix is (30)Φ3∗=ϕ31∗,ϕ32∗,…,ϕ3p∗∈RM×q,q=q1+q2+⋯+qp+p,ϕ3j∗=k3dj∗μj∗T∈RM×qj+1,j=1,2,…,p.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 closed-loop system becomes (31)x˙t=A+BK1∗Txt+BK2∗rt+BK3∗t+Bddt+But-K1∗Txt-K2∗rt-K3∗tyt=Cxt.In view of (24), (25), and (31), the output tracking error equation is (32)et=yt-ymt=WmsKp∗u-K1∗Tx-K2∗r-K3∗t+fpt,Kp∗=K2∗-1,where fpt=CeA+BK1∗Ttx0 converges to zero exponentially fast due to the stability of A+BK1∗T and Wm(s)=ξm-1(s). Hence, we have (33)ξmset=Kput-K1∗Txt-K2∗rt-K3∗t.To deal with the uncertainty of the high-frequency gain matrixKp, the LDU decomposition of Kp is used in (33), so that we have (34)L-1ξmset=DUut-K1∗Txt-K2∗rt-K3∗t.We have the following equation: (35)Uut=ut-I-Uut.With (34) and (35), we have (36)L-1ξmset=Dut-I-Uut-UK1∗Txt-K2∗rt-K3∗t.The new equation is (37)L-1ξmset=Dut-Φ0∗ut-Φ1∗Tωt,where Φ1∗T=UK1∗T,UK2∗,UK3f∗ and ωt=xTt,rTt,fTtT. This new equation has a new controller structure: (38)ut=Φ0ut+Φ1Tωt,where Φ0 and Φ1T are the estimates of Φ0∗ and Φ1∗T and Φ0 is upper triangular with zero diagonal elements (only its nonzero elements are estimated). The matrix Φ0 has the same strictly form as that of Φ0∗=I-U: (39)Φ0=0ϕ12ϕ13⋯ϕ1M00ϕ23⋯ϕ2M⋮⋮⋮⋮00⋯0ϕM-1M00⋯00∈RM×M.From (37) and (38), we obtain a new error model: (40)ξmset+Θ0∗ξmset=DΦ~Ttω¯t,where the parameter error is Φ~(t)=Φ(t)-Φ∗ and ΦT(t)=[Φ0(t),Φ1T(t)] is the estimate of unknown parameter matrix Φ∗T=[Φ0∗,Φ1∗T], ω¯(t)=[uT(t),ωT(t)]T, and ω(t)=xT(t),rT(t),fT(t)T, where Θ0∗=L-1-I is introduced to parameterize the unknown matrix L, which has a special form:(41)Θ0∗=00⋯0θ21∗0⋯0θ31∗θ32∗⋯0⋮⋮⋮0θM-11∗⋯00θM1∗⋯θMM-1∗0∈RM×M.For such a matrix Θ0∗, the parameter vectors are defined as (42)θ2∗=θ21∗∈R,θ3∗=θ31∗,θ32∗T∈R2,⋮θM-1∗=θM-11∗,…,θM-1M-2∗∈RM-2,θM∗=θM1∗,…,θMM-2∗∈RM-1,and their estimates are (43)θ2t=θ21t∈R,θ3t=θ31t,θ32tT∈R2,⋮θM-1t=θM-11t,…,θM-1M-2t∈RM-2,θMt=θM1t,…,θMM-2t∈RM-1.We introduce a filter hs=1/fs, where fs is chosen as a stable and monic polynomial whose degree is equal to the maximum degree of the modified interactor ξms. Operating both sides of (40) by hsIM leads to (44)ξmshset+Θ0∗ξmshset=DhsΦ~Tω¯t.We define (45)e¯t=ξmshset=e¯1t,…,e¯MtT,(46)ηit=e¯1t,…,e¯i-1tT∈Ri-1,i=2,3,…,M.From (45) and (46) in (44), we obtained (47)e¯t+0,θ2∗Tη2t,θ3∗Tη3t,…,θM∗TηMtT=DhsΦ~Tω¯t.Based on the parameterized error equation (47), an estimation error signal is introduced: (48)εt=e¯t+0,θ2Tη2t,θ3Tη3t,…,θMTηMtT+Ψtξt∈RM,where Ψ(t)∈RM×M is the estimate of Ψ∗=D and (49)ζt=hsω¯t,i=1,2,…,M,ξt=ΦTtζt-hsΦTω¯t,i=1,2,…,M.It then follows from (44) (47) and (48) that(50)εt=0,θ~2Tη2t,θ~3Tη3t,…,θ~MTηMtT+DΦ~T+Ψ~tξt,where Ψ~(t)=Ψ(t)-Ψ∗ and θ~i=θi-θi∗ are the parameter errors. This error model is choice for update laws.


## 3.6. Adaptive Laws

Based on the error model (50), the adaptive laws are chosen as (51)θ˙it=-Γθiεitηitm2t,i=2,…,M,Φ˙Tt=-DεtζTtm2t,Ψ˙t=-ΓεtξTtm2t,where Γθi=ΓθiT>0,i=2,3,…,M, and Γ=ΓT>0 are adaptive gains; D is defined in (15); ε(t)=ε1(t),ε2(t),…,εM(t)T is calculated from (50); and (52)m2t=1+ζTtζt+ξTtξt+∑i=2MηiTηit.


## 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)
θi(t)∈L∞, i=2,3,…,M, Φ(t)∈L∞, Ψ(t)∈L∞, and ε(t)/m(t)∈L2∩L∞;(ii)
θ˙i(t)∈L2∩L∞, i=2,3,…,M, Φ˙(t)∈L2∩L∞, and Ψ˙(t)∈L2∩L∞.Based on Lemma12, 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, 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 limt→∞⁡e(t)=0 with e(t)=y(t)-ym(t).Proof (outline).
The proof of this stability theorem can be established through using a unified framework. Because the control inputut described in (38) depends on the state xt, it first needs to be expressed by using the system output yt through establishing the state observer of the plant: (53)x^˙t=A-LCx^˙t+But+Bddt+Lyt,where L∈Rn×M is a gain matrix such that A-LC is stable, which is possible, and (AC) is assumed to be detectable. Hence, we have (54)ut=Φ1Ttω1t+Φ2Ttω2t+Φ3dTtω3t+K2trt+Φ3tft,where Φ1Tt, Φ2Tt, Φ3dTt, K2t, and Φ3t are adaptive estimates of the corresponding nominal controller parameters and (55)ω1t=asΛsut,ω2t=asΛsyt,ω3t=bsΛsft,with as=[IMsIM⋯sn-1IM]T, bs=[IqsIq⋯sn-1Iq]T, and Λs being a chosen monic stable polynomial of degree n, which has the same eigenvalues with A-LC. Then, introducing the fictitious filters for the plant yt=CsI-A-1But+CsI-A-1Bddt and using series transformations, the control input described as in (54) is transformed into the form (56)ut=G11s,·y¯t+G12s,·rt+G13s,·ft+G14s,·fpt,where y¯t=hsyt (hs is given below (43)) and G11s,·, G12s,·, G13s,·, and G14s,· are proper stable operators with finite gains. Furthermore, a filtered version of the output signal yt is expressed in a feedback framework: (57)y¯t≤x0+β1∫0te-α1t-τx1τ∫0τe-α2τ-ωy¯ω¯dω¯dτ,for some β1,α1,α2>0, and x1t=Φ˙t+εtmt∈L2∩L∞. Applying the small gain lemma to (57), we conclude that y¯t∈L∞, and so yt,ut∈L∞. Thus, the signals satisfy ω¯t,ζt,ξt,mt,ε∈L∞. Furthermore, θi,εt/mt,Φ˙t,Ψ˙t∈L2 (Lemma 12) are satisfied, and in turn ξt and et=yt-ymt, such that et=yt-ymt converges to zero.


## 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 (58)A=-0.0385818.984-0.001028-0.632537.8601E-5-0.7590500-0.04362-249.760-32.1391.3233E-410.0056123.7553E-6-0.5183-0.0079341-3.0808E-71000249.760,B=10.10-1.5446E-400.024656-0.010770000,C=1000000001.The turbulence disturbances are described in [22, 23]; we can get(59)Bd=Bd1,Bd2,Bd1=-0.03860.0017.8601E-500T,Bd2=-0.052620.00251-0.0030400T.


### 4.2. Adaptive Control Design

For the aircraft system, the transfer function,G0s=CsI-A-1B, has stable zeros, s1=-4.507, s2=-0.91, and s3=-0.5685, and is strictly proper and full rank. The interactor matrix is chosen as (60)ξms=diag⁡s+1s+12.The high-frequency matrix is (61)Kp=lims→∞⁡ξmsG0s=2.26738.8455-0.0902-0.3521,and it is is finite and nonsingular and the matrix (62)Kd=lims→∞⁡ξmsGds=-0.0386-0.0526-0.2481-0.6246is finite. From the specified left coprime polynomial matrix decompositions, G0s=Pl-1sZ0s and Gds=Pl-1sZds, we can obtain (63)lims→∞⁡Z0-1sZds=-0.017-0.00642.74931.7739,which means that the relative degree condition in Assumption 6 can be ensured.The related gain parameters in adaptive laws (51) are chosen as Γθ=100, D=diag⁡{22}, and Γ=diag⁡{1010}.


### 4.3. Simulation Results

For this simulations study, the initial state is chosen asx0t=[240000295], 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.Figure 3
Final landing phase altitude for the aircraft.Figure 4
Final landing phase tracking error for the aircraft.Figure 5
Final landing phase altitude for the aircraft.Figure 6
Final landing phase tracking error for the aircraft.Figure 7
Final landing phase control signal.


## 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 (58)A=-0.0385818.984-0.001028-0.632537.8601E-5-0.7590500-0.04362-249.760-32.1391.3233E-410.0056123.7553E-6-0.5183-0.0079341-3.0808E-71000249.760,B=10.10-1.5446E-400.024656-0.010770000,C=1000000001.The turbulence disturbances are described in [22, 23]; we can get(59)Bd=Bd1,Bd2,Bd1=-0.03860.0017.8601E-500T,Bd2=-0.052620.00251-0.0030400T.


## 4.2. Adaptive Control Design

For the aircraft system, the transfer function,G0s=CsI-A-1B, has stable zeros, s1=-4.507, s2=-0.91, and s3=-0.5685, and is strictly proper and full rank. The interactor matrix is chosen as (60)ξms=diag⁡s+1s+12.The high-frequency matrix is (61)Kp=lims→∞⁡ξmsG0s=2.26738.8455-0.0902-0.3521,and it is is finite and nonsingular and the matrix (62)Kd=lims→∞⁡ξmsGds=-0.0386-0.0526-0.2481-0.6246is finite. From the specified left coprime polynomial matrix decompositions, G0s=Pl-1sZ0s and Gds=Pl-1sZds, we can obtain (63)lims→∞⁡Z0-1sZds=-0.017-0.00642.74931.7739,which means that the relative degree condition in Assumption 6 can be ensured.The related gain parameters in adaptive laws (51) are chosen as Γθ=100, D=diag⁡{22}, and Γ=diag⁡{1010}.


## 4.3. Simulation Results

For this simulations study, the initial state is chosen asx0t=[240000295], 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.Figure 3
Final landing phase altitude for the aircraft.Figure 4
Final landing phase tracking error for the aircraft.Figure 5
Final landing phase altitude for the aircraft.Figure 6
Final landing phase tracking error for the aircraft.Figure 7
Final landing phase control signal.


## 5. Conclusions

In this paper, a multivariable disturbance rejection scheme is presented to solve the wind turbulence problem. The state-feedback 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.



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*Source: 7345056-2016-08-02.xml*