# The Investigation onL1 Adaptive Control of the Tilt-Rotor Aircraft

Authors: Quanlong Chen; Shanyong Zhao; Ke Lu; Senkui Lu; Chunsheng Liu; Renliang Chen
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

Considering the uncertainty of the flight dynamics model of the tilt-rotor aircraft in different flight modes, anL1 adaptive controller for full flight modes control system of tilt-rotor aircraft is designed. Taking advantage of the separation of robustness and adaptive design of the L1 adaptive controller, adaptive gain, and low-pass filter are designed to achieve the desired control performance and meet the requirements of flight quality. The simulations of XV-15 tilt-rotor aircraft in helicopter mode and airplane mode are carried out. Then, the simulation of conversion mode is further carried out. The results show that the tilt-rotor aircraft can track the reference signal well under the L1 control system. In addition, the changes of states as well as controls in conversion mode flight are quite smooth which is very meaningful for engineering application.


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## Body

## 1. Introduction

A tilt-rotor aircraft has the advantages of both fixed-wing and rotorcraft such as high speed and long-range flight, vertical take-off, and landing ability. It can cover the flight envelope of conventional helicopters and propeller planes and has broad applications in both civil and military fields. However, due to the integration of flight modes and control modes of fixed-wing aircraft and helicopters, tilt-rotor aircraft has helicopter mode, conversion mode, and airplane mode, as shown in Figure1. It is a typical morphing aircraft. Therefore, the mathematical model of flight dynamics is highly nonlinear. It is much more complicated in flight control system design compared with other aircraft.Figure 1
Tilt-rotor aircraft three flight modes.
(a)
Helicopter modein=90∘(b)
Conversion mode0∘<in<90∘(c)
Airplane modein=0∘Since the promulgation of ADS-33E-PRF, preliminary studies on flight control of tilt-rotor aircraft mainly focus on stability enhancement system design and quality improvement [1–4]. Juhasz developed a high order rotorcraft mathematical mode against a large civil tilt-rotor concept. At the same time, the corresponding control system is designed to meet the quality requirements [5]. In Ref. [6], neural network controller was used to accomplish the neural network path-tracking and real-time multitask flight simulation for the automatic transition of tilt-rotor aircraft. The flight performance of conversion mode is evaluated based on the linearized model. Rysdyk and Calise [7] propose an inverse control method based on neural network model to control civil tilt-rotor aircraft, which can ensure satisfied dynamic performance in the whole flight envelope. The stability of the system and the boundary of tracking error are strictly proved based on Lyapunov theory. Choi et al. [8] divide the nacelle angle into 90-60 degrees for helicopter mode, 60-30 degrees for conversion mode, and 30-0 degrees for airplane mode in the flight control system of unmanned tilt-rotor aircraft. The way of control scheduling is used, which is easy to be implemented in engineering. Lee et al. [9] divide the tilt-rotor aircraft into different operating points based on discretization and optimize the control law parameters at different operating points using particle swarm optimization algorithm. In Ref. [10], based on an optimal control concept, an online optimization control method was developed for the tilt-rotor UAV. In Ref. [11], a smooth switching control scheme is provided for the tilt-rotor aircraft. However, the flight mechanic model in the above literature has been simplified to a large extent. As a result, the simulation results cannot reflect the real performance of the controllers. For a class of singularly perturbed Markov jump descriptor systems with nonlinear perturbation, a stochastic integral sliding mode control strategy is developed in [12, 13]. In aeronautical engineering application field, the L1 adaptive control has attracted wide attention in recent years. It is modified on the basis of the classical model reference adaptive control [14–17]. A low pass filter is introduced to decouple the adaptive and robustness of the control system, a larger adaptive gain can be adopted to improve the parameter estimation speed and obtain better control performance. The corresponding test work has been carried out, especially in the field of aerospace [18–23].Motivated by the discussions above, we introduce theL1 adaptive control method to the tilt-rotor aircraft in the flight control system. The major limitations for its applications can be summarized as follows:
(1)
The dynamics of tilt-rotor aircraft are usually highly nonlinear and coupled, and it has multiple flight modes. It is still a challenging problem to get desired dynamic performance for the controlled system in the full envelope(2)
The performance of tilt-rotor aircraft control system should meet the related requirements in flight quality specification(3)
The design of controller parameters should fully consider the physical realizability of the aircraftThe paper is organized as follows. Section2 introduces the mathematical model. Section 3 gives the L1 adaptive controller design. Section 4 presents simulation studies. Finally, Section 5 draws the conclusion.


## 2. Mathematical Model Description and Verification

### 2.1. Model Description

The flight dynamics model is the basis for the analysis of flight characteristics of the aircraft and flight control system design. XV-15 tilt-rotor aircraft is chosen forth research object, and its basic parameters are shown in Table1. The flight dynamics model of tilt-rotor aircraft is established.Table 1
Basic parameters of XV-15 tilt-rotor.
Parameters valueWeight (m)5897 kgRotor radius (R)3.81 mBlade number (Nb)3×2Blade twist (θt)-41°Rotor speed (Ω)589 rpm (helicopter mode)517 rpm (airplane mode)Lock number (γ)3.83Solidity (σ)0.089Pitch-flap coupling angle (δ3)15°Flapping hinge restraintKβ17476 Nm/radFlap moment of inertia (Iβ)139 kg·m2Nacelle length (d)1.4 mWing area (Sw)16.82 m2Horizontal tail areas (Sht)4.67 m2Vertical tail areas (Svt)2.35 m2In the modeling of flight mechanics, tilt-rotor aircraft adopts component aerodynamic modeling method, and the rotor aerodynamic modeling is the focus of the entire modeling work.The main aeromechanic features in the XV-15 model are listed below:(1)
Rigid prop-rotor blades with nonlinear, quasisteady aerodynamics in table look-up form as functions of the angle of attack and Mach number(2)
Finite-state rotor inflow model with 13 inflow states in total for each rotor(3)
The wing/flap lift, drag, and pitching moment coefficients are defined as functions of the angle of attack, nacelle angle, and flap setting obtained by matching experimental data(4)
Rotor-wing-empennage interaction modeled was determined from wind tunnel experiment(5)
Nonlinear fuselage aerodynamics is functions of the angle of attack and sideslip(6)
Empennage aerodynamics is modeled in a similar manner to the main wingThe XV-15 tilt-rotor aircraft uses both helicopter and airplane control strategies to control the aircraft. In helicopter flight mode, longitudinal cyclic, differential collective, and differential longitudinal cyclic are used to pitch, roll, yaw, and heave control, respectively. As the tilt-rotor aircraft converts from helicopter flight mode to airplane flight mode, the helicopter rotor control surfaces are washed out as a function of nacelle angle and flight speed.The tilt-rotor aircraft flight dynamics model could be given by(1)Ẋ=fX,t+gX,tu.The state variables are given in the form of a vectorX∈R47×1, which includes 26 rotor inflow states, 12 flapping motion states, and 9 aircraft states. u∈R4×1 denotes a vector constituting the control inputs. The model detail can refer to Ref [24] (Ke Lu and C. S. Liu).


### 2.2. Mode Validation

#### 2.2.1. Trim Result Validation

In the trim validation, the control inputs at different nacelle incidence angles are compared with simulation results from GTRS reports [25]. Figures 2 and 3, respectively, show the comparison of collective pitch and longitudinal control.Figure 2
Collective pitch trim validation.Figure 3
Longitudinal stick trim validation.According to Figures2 and 3, in helicopter mode, the variation trend of collective pitch is similar to the conventional helicopter. The collective pitch has the characteristic bucket profile as a function of flight speed. In airplane mode, the function of the rotor is providing forward pull force to overcome fuselage drag. That is to say, the wing is able to generate enough lift to overcome gravity. The collective pitch is much larger than that of the helicopter mode. In addition, longitudinal control input increases with speed in all flight modes. Overall, the calculation results are quite consistent with GTRS results in all flight modes under the trim condition.


#### 2.2.2. Linearized Result Validation

With the application of a linearization algorithm, the rotor and inflow modes residualized out via quasistatic reduction, and then the nonlinear equation can be reduced to the form of(2)ẋ=Ax+Bu,(3)x=u,v,w,p,q,r,ϕ,θ,ψ,(4)u=δcoll,δlong,δlat,δped.To linearization validation, a comparison of the eigenvalues for matrixA in equation (2) with results from the flight test and GTRS is shown in Tables 2 and 3. As shown in Table 2, the characteristic in helicopter mode of the eigenvalue distribution between calculation results in this paper and flight test are very similar. In particular, for phugoid and dutch roll modes, calculation results in this paper and flight test have given the conclusion of instability. The airplane mode eigenvalues are shown in Table 3. In general, the calculation results in this paper are closer to GTRS, especially the prediction of short period modes.Table 2
Hover mode eigenvalues validation.
Hover modeNatural modeCalculatedFlight dataLongitudinalPhugoid0.2981±0.7555i0.2681±0.5132iPitch subsidence-1.0-1.32Heave subsidence-0.17-0.105LateralDutch roll0.1866±1.0826i0.1868±0.4061iSpiral subsidence-0.1609-0.102Roll subsidence-2.6539-1.23Table 3
Airplane mode (V=200kt) eigenvalues validation.
Hover modeNatural modeCalculatedGTRSLongitudinalPhugoid−0.2111±0.2625i−0.2115±0.1576iShort-period pitch−1.7402±3.9172i−1.6948±3.4555iLateralDutch roll−0.6525±2.2092i−0.4989±3.4555iSpiral subsidence-0.1384-0.1226Roll subsidence-0.7128-1.0649


#### 2.2.3. Control Response Validation

The control response verification is mainly to verify whether the response of tilt-rotor aircraft conforms to the laws of physics under the excitation of the control input. In this section, GTRS data is used to verify the control response of the tilt-rotor aircraft.In the helicopter mode, the longitudinal stick in the hovering state is pushed forward by 5%, and the time response of the pitch angle and rate is shown in Figures4 and 5. It can be seen from the figure that when the longitudinal stick is pushed forward and causes the aircraft head down, which conforms to the basic physical characteristics of the aircraft. In addition, the calculation model in this paper has the same trend and similar amplitude with the GTRS data.Figure 4
Pitch angle step response in helicopter mode (hover).Figure 5
Pitch rate step response in helicopter mode (hover).In conclusion, the calculated values in this paper are in good agreement with those in the GTRS. In brief, the XV-15 tilt-rotor flight dynamics model is proved to be valid. So we have enough confidence in the following controller design.


## 2.1. Model Description

The flight dynamics model is the basis for the analysis of flight characteristics of the aircraft and flight control system design. XV-15 tilt-rotor aircraft is chosen forth research object, and its basic parameters are shown in Table1. The flight dynamics model of tilt-rotor aircraft is established.Table 1
Basic parameters of XV-15 tilt-rotor.
Parameters valueWeight (m)5897 kgRotor radius (R)3.81 mBlade number (Nb)3×2Blade twist (θt)-41°Rotor speed (Ω)589 rpm (helicopter mode)517 rpm (airplane mode)Lock number (γ)3.83Solidity (σ)0.089Pitch-flap coupling angle (δ3)15°Flapping hinge restraintKβ17476 Nm/radFlap moment of inertia (Iβ)139 kg·m2Nacelle length (d)1.4 mWing area (Sw)16.82 m2Horizontal tail areas (Sht)4.67 m2Vertical tail areas (Svt)2.35 m2In the modeling of flight mechanics, tilt-rotor aircraft adopts component aerodynamic modeling method, and the rotor aerodynamic modeling is the focus of the entire modeling work.The main aeromechanic features in the XV-15 model are listed below:(1)
Rigid prop-rotor blades with nonlinear, quasisteady aerodynamics in table look-up form as functions of the angle of attack and Mach number(2)
Finite-state rotor inflow model with 13 inflow states in total for each rotor(3)
The wing/flap lift, drag, and pitching moment coefficients are defined as functions of the angle of attack, nacelle angle, and flap setting obtained by matching experimental data(4)
Rotor-wing-empennage interaction modeled was determined from wind tunnel experiment(5)
Nonlinear fuselage aerodynamics is functions of the angle of attack and sideslip(6)
Empennage aerodynamics is modeled in a similar manner to the main wingThe XV-15 tilt-rotor aircraft uses both helicopter and airplane control strategies to control the aircraft. In helicopter flight mode, longitudinal cyclic, differential collective, and differential longitudinal cyclic are used to pitch, roll, yaw, and heave control, respectively. As the tilt-rotor aircraft converts from helicopter flight mode to airplane flight mode, the helicopter rotor control surfaces are washed out as a function of nacelle angle and flight speed.The tilt-rotor aircraft flight dynamics model could be given by(1)Ẋ=fX,t+gX,tu.The state variables are given in the form of a vectorX∈R47×1, which includes 26 rotor inflow states, 12 flapping motion states, and 9 aircraft states. u∈R4×1 denotes a vector constituting the control inputs. The model detail can refer to Ref [24] (Ke Lu and C. S. Liu).


## 2.2. Mode Validation

### 2.2.1. Trim Result Validation

In the trim validation, the control inputs at different nacelle incidence angles are compared with simulation results from GTRS reports [25]. Figures 2 and 3, respectively, show the comparison of collective pitch and longitudinal control.Figure 2
Collective pitch trim validation.Figure 3
Longitudinal stick trim validation.According to Figures2 and 3, in helicopter mode, the variation trend of collective pitch is similar to the conventional helicopter. The collective pitch has the characteristic bucket profile as a function of flight speed. In airplane mode, the function of the rotor is providing forward pull force to overcome fuselage drag. That is to say, the wing is able to generate enough lift to overcome gravity. The collective pitch is much larger than that of the helicopter mode. In addition, longitudinal control input increases with speed in all flight modes. Overall, the calculation results are quite consistent with GTRS results in all flight modes under the trim condition.


### 2.2.2. Linearized Result Validation

With the application of a linearization algorithm, the rotor and inflow modes residualized out via quasistatic reduction, and then the nonlinear equation can be reduced to the form of(2)ẋ=Ax+Bu,(3)x=u,v,w,p,q,r,ϕ,θ,ψ,(4)u=δcoll,δlong,δlat,δped.To linearization validation, a comparison of the eigenvalues for matrixA in equation (2) with results from the flight test and GTRS is shown in Tables 2 and 3. As shown in Table 2, the characteristic in helicopter mode of the eigenvalue distribution between calculation results in this paper and flight test are very similar. In particular, for phugoid and dutch roll modes, calculation results in this paper and flight test have given the conclusion of instability. The airplane mode eigenvalues are shown in Table 3. In general, the calculation results in this paper are closer to GTRS, especially the prediction of short period modes.Table 2
Hover mode eigenvalues validation.
Hover modeNatural modeCalculatedFlight dataLongitudinalPhugoid0.2981±0.7555i0.2681±0.5132iPitch subsidence-1.0-1.32Heave subsidence-0.17-0.105LateralDutch roll0.1866±1.0826i0.1868±0.4061iSpiral subsidence-0.1609-0.102Roll subsidence-2.6539-1.23Table 3
Airplane mode (V=200kt) eigenvalues validation.
Hover modeNatural modeCalculatedGTRSLongitudinalPhugoid−0.2111±0.2625i−0.2115±0.1576iShort-period pitch−1.7402±3.9172i−1.6948±3.4555iLateralDutch roll−0.6525±2.2092i−0.4989±3.4555iSpiral subsidence-0.1384-0.1226Roll subsidence-0.7128-1.0649


### 2.2.3. Control Response Validation

The control response verification is mainly to verify whether the response of tilt-rotor aircraft conforms to the laws of physics under the excitation of the control input. In this section, GTRS data is used to verify the control response of the tilt-rotor aircraft.In the helicopter mode, the longitudinal stick in the hovering state is pushed forward by 5%, and the time response of the pitch angle and rate is shown in Figures4 and 5. It can be seen from the figure that when the longitudinal stick is pushed forward and causes the aircraft head down, which conforms to the basic physical characteristics of the aircraft. In addition, the calculation model in this paper has the same trend and similar amplitude with the GTRS data.Figure 4
Pitch angle step response in helicopter mode (hover).Figure 5
Pitch rate step response in helicopter mode (hover).In conclusion, the calculated values in this paper are in good agreement with those in the GTRS. In brief, the XV-15 tilt-rotor flight dynamics model is proved to be valid. So we have enough confidence in the following controller design.


## 2.2.1. Trim Result Validation

In the trim validation, the control inputs at different nacelle incidence angles are compared with simulation results from GTRS reports [25]. Figures 2 and 3, respectively, show the comparison of collective pitch and longitudinal control.Figure 2
Collective pitch trim validation.Figure 3
Longitudinal stick trim validation.According to Figures2 and 3, in helicopter mode, the variation trend of collective pitch is similar to the conventional helicopter. The collective pitch has the characteristic bucket profile as a function of flight speed. In airplane mode, the function of the rotor is providing forward pull force to overcome fuselage drag. That is to say, the wing is able to generate enough lift to overcome gravity. The collective pitch is much larger than that of the helicopter mode. In addition, longitudinal control input increases with speed in all flight modes. Overall, the calculation results are quite consistent with GTRS results in all flight modes under the trim condition.


## 2.2.2. Linearized Result Validation

With the application of a linearization algorithm, the rotor and inflow modes residualized out via quasistatic reduction, and then the nonlinear equation can be reduced to the form of(2)ẋ=Ax+Bu,(3)x=u,v,w,p,q,r,ϕ,θ,ψ,(4)u=δcoll,δlong,δlat,δped.To linearization validation, a comparison of the eigenvalues for matrixA in equation (2) with results from the flight test and GTRS is shown in Tables 2 and 3. As shown in Table 2, the characteristic in helicopter mode of the eigenvalue distribution between calculation results in this paper and flight test are very similar. In particular, for phugoid and dutch roll modes, calculation results in this paper and flight test have given the conclusion of instability. The airplane mode eigenvalues are shown in Table 3. In general, the calculation results in this paper are closer to GTRS, especially the prediction of short period modes.Table 2
Hover mode eigenvalues validation.
Hover modeNatural modeCalculatedFlight dataLongitudinalPhugoid0.2981±0.7555i0.2681±0.5132iPitch subsidence-1.0-1.32Heave subsidence-0.17-0.105LateralDutch roll0.1866±1.0826i0.1868±0.4061iSpiral subsidence-0.1609-0.102Roll subsidence-2.6539-1.23Table 3
Airplane mode (V=200kt) eigenvalues validation.
Hover modeNatural modeCalculatedGTRSLongitudinalPhugoid−0.2111±0.2625i−0.2115±0.1576iShort-period pitch−1.7402±3.9172i−1.6948±3.4555iLateralDutch roll−0.6525±2.2092i−0.4989±3.4555iSpiral subsidence-0.1384-0.1226Roll subsidence-0.7128-1.0649


## 2.2.3. Control Response Validation

The control response verification is mainly to verify whether the response of tilt-rotor aircraft conforms to the laws of physics under the excitation of the control input. In this section, GTRS data is used to verify the control response of the tilt-rotor aircraft.In the helicopter mode, the longitudinal stick in the hovering state is pushed forward by 5%, and the time response of the pitch angle and rate is shown in Figures4 and 5. It can be seen from the figure that when the longitudinal stick is pushed forward and causes the aircraft head down, which conforms to the basic physical characteristics of the aircraft. In addition, the calculation model in this paper has the same trend and similar amplitude with the GTRS data.Figure 4
Pitch angle step response in helicopter mode (hover).Figure 5
Pitch rate step response in helicopter mode (hover).In conclusion, the calculated values in this paper are in good agreement with those in the GTRS. In brief, the XV-15 tilt-rotor flight dynamics model is proved to be valid. So we have enough confidence in the following controller design.


## 3.L1 Adaptive Controller Design

### 3.1. Problem Description

The mathematical model of the pitch channel of the tilt-rotor aircraft can be described by the following formula(5)ẋt=Amxt+bωut+ft,xt,ξt,x0=x0,ẋξt=gt,xξt,xt,xξ0=xξ0,ξt=ht,xξt,yt=c⊤xt,where xt=θ,qT is the pitch angle and pitch angle velocity, respectively; Am∈ℝ2×2 is a known Hurwitz matrix whose characteristics meet the specifications of flight quality; b,c∈ℝ2 is known constant vectors; ut∈ℝ is control input; ω is control efficiency; ft,xt,ξt is unknown nonlinear dynamic mapping; xξt and ξt are the state and output of unmodeled dynamics, respectively, and g⋅ and h⋅ are nonlinear mappings of corresponding dimensions.


### 3.2.L1 Adaptive Control System Design

The system above is subject to the following assumptions:Assumption 1.
(Boundedness offit,0) There exists Bi>0, such that fit,0∞≤Bi holds for t≥0 and for i=1,2.Assumption 2.
(Semiglobal Lipschitz condition) For arbitraryδ>0, there exist positive K1δ,K2δ, such that fit,x1−fit,x2∞≤Kiδx1−x2∞, i=1,2, for all xj∞≤δ,j=1,2, uniformly in t.Assumption 3.
(Stability of unmodeled dynamics) Thexξ-dynamics are BIBO stable with respect to both initial conditions xξ0 and inputxt.In this study, the uncertainties are mainly caused by aerodynamic parameters. They are always uniformly bounded and limited in how fast they can change. Thus, these assumptions are reasonable.The state prediction system, adaptive law, and control law of theL1 adaptive control system of the tilt-rotor aircraft are designed as follows.


#### 3.2.1. State Prediction System

(6)x^̇t=Amx^t+bω^tut+λ^txtL∞+σ^t,x^0=x0,y^t=c⊤x^t,where x^t∈ℝn is the state estimate of the system, and ω^t∈ℝ,λ^t∈ℝ,σ^t∈ℝ is the adaptive parameter estimates.


#### 3.2.2. Adaptive Law

(7)ω^̇t=ΓProjω^t,−x~⊤tPbut,ω^0=ω^0,λ^̇t=ΓProjλ^t,−x~⊤tPbxtL∞,λ^0=λ^0,σ^̇t=ΓProjσ^t,−x~⊤tPb,σ^0=σ^0,where x~t≜x^t−xt; Γ∈ℝ+ is the adaptive gain; P=P⊤>0 is the solution of the Lyapunov equation Am⊤P+PAm=−Q, and Q=Q⊤>0; Proj is the projection operator.


#### 3.2.3. Control Law

(8)us=−kDsη^s−kgrs,where rs is the Laplace transform of rt, kg≜−1/c⊤Am−1b, k is the design parameter of the low-pass filter, Ds generally takes the form of integral, where η^s is the Laplace transform of the following formula
(9)η^t≜ω^tut+λ^txt∞+σ^t.The adaptive control signal passes through the low-pass filter shown in the following equation before entering the actual system.(10)Cs≜ωkDs1+ωkDs,where Cs is a filter that satisfies C0=1 and satisfies the following L1 adaptive stability conditions.Theorem 4.
For system (5), given a ρ0, there is ρr>ρin. When the system adopts the abovementioned state prediction system, adaptive law, and control law, the system is closed-loop stable when the following norm inequality is satisfied.
(11)GsL1<ρr−HsCskgL1rL∞−ρinLρrρr+B,where Gs≜Hs1−Cs,Hs≜sI−Am−1b,ρin≜ssI−Am−1L1ρ0,Lρr≜ρ¯rρr/ρrdfxρ¯rρr, and ρ¯rρr≜maxρr+γ¯1,L1ρr+γ¯1+L2.


### 3.3.L1 Adaptive Control System Analysis

In order to analyze the closed-loop characteristics of the system, the following closed-loop reference system is defined(12)ẋreft=Amxreft+bωureft+ft,xreft,ξt,xref0=x0,urefs=Csωkgrs−ηrefs,yreft=c⊤xreft,where ηreft≜ft,xreft,ξt.Lemma 5.
When the closed-loop reference system satisfies the stability condition, for anyτ>0(13)ξτL∞≤L1xrefτL∞+γ1+L2,(14)xrefL∞≤ρr,urefL∞≤ρur.
That is, the closed-loop reference system is BIBS stable.Lemma 6.
For system (5), when the system adopts the state prediction system, adaptive law, and control law and satisfies the stability condition (14), if the adaptive gain Γ is chosen to verify the lower bound
(15)Γ≥ρmρ,ρu,ρu̇λminPγ02.Then, the following inequality holds
(16)x~L∞<γ0.Then, we have
(17)xref−xL∞<γ1,(18)uref−uL∞≤γ2,where γ1 and γ2 are dependent on fit,x bounds and L1 controller design parameters.Remark 1.
As the adaptive gainΓ goes to infinity, the system (5) will follow the reference system (12) arbitrarily closely. It implies that the performance limitations are consistent with the hardware limitations.


### 3.4. Command System Design

ADS-33E-PRF flight quality specification specifies a variety of response types. These response types are the guarantee for the helicopter to complete high-quality flight missions. The most basic is the angular rate response type and the attitude response type. In general, the aircraft basically has the angular rate response type through the preliminary stabilization system. In order to reduce the pilot workload, Attitude Command/Attitude Hold response type control system is designed for the longitudinal pitch channel.According to ADS-33E-PRF flight quality specification, the flight quality constraint parameters of the attitude command type are the bandwidth and phase delay index for small-scale high-frequency attitude changes and the quickness index for medium attitude changes.The following transfer function is used to design the pitch attitude command model.(19)θcomsδlongs=KAC1/ω2s2+2ζ/ωs+1e−τSLs.According to the definition of bandwidth and phase delay in the ADS-33E-PRF flight quality specification, the relationship between the phase bandwidthωBW of the attitude command model and the command model parameters ω, ζ, and τSL are as follows.
(20)tanπ4+τSLωBW−2ζωBWωωBW2−ω2=0.Phase delayτP(21)τp=2ω180τSL−arctan4ζω180ω/4ω1802−ω22ω180,where ω180 is the frequency corresponding to equation (19) when the phase of the transfer function is 180 deg, which can be obtained by solving the following equation.
(22)tanτSLω180−2ζω180ωω1802−ω2=0.According to the corresponding relationship between response types and mission task recommended in the ADS-33E-PRF flight quality specification, the attitude response type is generally the response type used in complex situations such as degraded visual environment. At the same time, in the case of divided attention operation, the specification has made stricter requirements on the damping ratio of the medium period response. Therefore, the damping ratio of 1 is chosen. In this case, the quickness index of equation (19) is as follows
(23)qpkθpk=−KAC−θminθminTln1−θminKAC,θmin<KAC1−e−1,KACe−1θminT,θmin≥KAC1−e−1.Through the above analysis, the parameters in equation (19) could be determined by equations (20)–(23).


## 3.1. Problem Description

The mathematical model of the pitch channel of the tilt-rotor aircraft can be described by the following formula(5)ẋt=Amxt+bωut+ft,xt,ξt,x0=x0,ẋξt=gt,xξt,xt,xξ0=xξ0,ξt=ht,xξt,yt=c⊤xt,where xt=θ,qT is the pitch angle and pitch angle velocity, respectively; Am∈ℝ2×2 is a known Hurwitz matrix whose characteristics meet the specifications of flight quality; b,c∈ℝ2 is known constant vectors; ut∈ℝ is control input; ω is control efficiency; ft,xt,ξt is unknown nonlinear dynamic mapping; xξt and ξt are the state and output of unmodeled dynamics, respectively, and g⋅ and h⋅ are nonlinear mappings of corresponding dimensions.


## 3.2.L1 Adaptive Control System Design

The system above is subject to the following assumptions:Assumption 1.
(Boundedness offit,0) There exists Bi>0, such that fit,0∞≤Bi holds for t≥0 and for i=1,2.Assumption 2.
(Semiglobal Lipschitz condition) For arbitraryδ>0, there exist positive K1δ,K2δ, such that fit,x1−fit,x2∞≤Kiδx1−x2∞, i=1,2, for all xj∞≤δ,j=1,2, uniformly in t.Assumption 3.
(Stability of unmodeled dynamics) Thexξ-dynamics are BIBO stable with respect to both initial conditions xξ0 and inputxt.In this study, the uncertainties are mainly caused by aerodynamic parameters. They are always uniformly bounded and limited in how fast they can change. Thus, these assumptions are reasonable.The state prediction system, adaptive law, and control law of theL1 adaptive control system of the tilt-rotor aircraft are designed as follows.


### 3.2.1. State Prediction System

(6)x^̇t=Amx^t+bω^tut+λ^txtL∞+σ^t,x^0=x0,y^t=c⊤x^t,where x^t∈ℝn is the state estimate of the system, and ω^t∈ℝ,λ^t∈ℝ,σ^t∈ℝ is the adaptive parameter estimates.


### 3.2.2. Adaptive Law

(7)ω^̇t=ΓProjω^t,−x~⊤tPbut,ω^0=ω^0,λ^̇t=ΓProjλ^t,−x~⊤tPbxtL∞,λ^0=λ^0,σ^̇t=ΓProjσ^t,−x~⊤tPb,σ^0=σ^0,where x~t≜x^t−xt; Γ∈ℝ+ is the adaptive gain; P=P⊤>0 is the solution of the Lyapunov equation Am⊤P+PAm=−Q, and Q=Q⊤>0; Proj is the projection operator.


### 3.2.3. Control Law

(8)us=−kDsη^s−kgrs,where rs is the Laplace transform of rt, kg≜−1/c⊤Am−1b, k is the design parameter of the low-pass filter, Ds generally takes the form of integral, where η^s is the Laplace transform of the following formula
(9)η^t≜ω^tut+λ^txt∞+σ^t.The adaptive control signal passes through the low-pass filter shown in the following equation before entering the actual system.(10)Cs≜ωkDs1+ωkDs,where Cs is a filter that satisfies C0=1 and satisfies the following L1 adaptive stability conditions.Theorem 4.
For system (5), given a ρ0, there is ρr>ρin. When the system adopts the abovementioned state prediction system, adaptive law, and control law, the system is closed-loop stable when the following norm inequality is satisfied.
(11)GsL1<ρr−HsCskgL1rL∞−ρinLρrρr+B,where Gs≜Hs1−Cs,Hs≜sI−Am−1b,ρin≜ssI−Am−1L1ρ0,Lρr≜ρ¯rρr/ρrdfxρ¯rρr, and ρ¯rρr≜maxρr+γ¯1,L1ρr+γ¯1+L2.


## 3.2.1. State Prediction System

(6)x^̇t=Amx^t+bω^tut+λ^txtL∞+σ^t,x^0=x0,y^t=c⊤x^t,where x^t∈ℝn is the state estimate of the system, and ω^t∈ℝ,λ^t∈ℝ,σ^t∈ℝ is the adaptive parameter estimates.


## 3.2.2. Adaptive Law

(7)ω^̇t=ΓProjω^t,−x~⊤tPbut,ω^0=ω^0,λ^̇t=ΓProjλ^t,−x~⊤tPbxtL∞,λ^0=λ^0,σ^̇t=ΓProjσ^t,−x~⊤tPb,σ^0=σ^0,where x~t≜x^t−xt; Γ∈ℝ+ is the adaptive gain; P=P⊤>0 is the solution of the Lyapunov equation Am⊤P+PAm=−Q, and Q=Q⊤>0; Proj is the projection operator.


## 3.2.3. Control Law

(8)us=−kDsη^s−kgrs,where rs is the Laplace transform of rt, kg≜−1/c⊤Am−1b, k is the design parameter of the low-pass filter, Ds generally takes the form of integral, where η^s is the Laplace transform of the following formula
(9)η^t≜ω^tut+λ^txt∞+σ^t.The adaptive control signal passes through the low-pass filter shown in the following equation before entering the actual system.(10)Cs≜ωkDs1+ωkDs,where Cs is a filter that satisfies C0=1 and satisfies the following L1 adaptive stability conditions.Theorem 4.
For system (5), given a ρ0, there is ρr>ρin. When the system adopts the abovementioned state prediction system, adaptive law, and control law, the system is closed-loop stable when the following norm inequality is satisfied.
(11)GsL1<ρr−HsCskgL1rL∞−ρinLρrρr+B,where Gs≜Hs1−Cs,Hs≜sI−Am−1b,ρin≜ssI−Am−1L1ρ0,Lρr≜ρ¯rρr/ρrdfxρ¯rρr, and ρ¯rρr≜maxρr+γ¯1,L1ρr+γ¯1+L2.


## 3.3.L1 Adaptive Control System Analysis

In order to analyze the closed-loop characteristics of the system, the following closed-loop reference system is defined(12)ẋreft=Amxreft+bωureft+ft,xreft,ξt,xref0=x0,urefs=Csωkgrs−ηrefs,yreft=c⊤xreft,where ηreft≜ft,xreft,ξt.Lemma 5.
When the closed-loop reference system satisfies the stability condition, for anyτ>0(13)ξτL∞≤L1xrefτL∞+γ1+L2,(14)xrefL∞≤ρr,urefL∞≤ρur.
That is, the closed-loop reference system is BIBS stable.Lemma 6.
For system (5), when the system adopts the state prediction system, adaptive law, and control law and satisfies the stability condition (14), if the adaptive gain Γ is chosen to verify the lower bound
(15)Γ≥ρmρ,ρu,ρu̇λminPγ02.Then, the following inequality holds
(16)x~L∞<γ0.Then, we have
(17)xref−xL∞<γ1,(18)uref−uL∞≤γ2,where γ1 and γ2 are dependent on fit,x bounds and L1 controller design parameters.Remark 1.
As the adaptive gainΓ goes to infinity, the system (5) will follow the reference system (12) arbitrarily closely. It implies that the performance limitations are consistent with the hardware limitations.


## 3.4. Command System Design

ADS-33E-PRF flight quality specification specifies a variety of response types. These response types are the guarantee for the helicopter to complete high-quality flight missions. The most basic is the angular rate response type and the attitude response type. In general, the aircraft basically has the angular rate response type through the preliminary stabilization system. In order to reduce the pilot workload, Attitude Command/Attitude Hold response type control system is designed for the longitudinal pitch channel.According to ADS-33E-PRF flight quality specification, the flight quality constraint parameters of the attitude command type are the bandwidth and phase delay index for small-scale high-frequency attitude changes and the quickness index for medium attitude changes.The following transfer function is used to design the pitch attitude command model.(19)θcomsδlongs=KAC1/ω2s2+2ζ/ωs+1e−τSLs.According to the definition of bandwidth and phase delay in the ADS-33E-PRF flight quality specification, the relationship between the phase bandwidthωBW of the attitude command model and the command model parameters ω, ζ, and τSL are as follows.
(20)tanπ4+τSLωBW−2ζωBWωωBW2−ω2=0.Phase delayτP(21)τp=2ω180τSL−arctan4ζω180ω/4ω1802−ω22ω180,where ω180 is the frequency corresponding to equation (19) when the phase of the transfer function is 180 deg, which can be obtained by solving the following equation.
(22)tanτSLω180−2ζω180ωω1802−ω2=0.According to the corresponding relationship between response types and mission task recommended in the ADS-33E-PRF flight quality specification, the attitude response type is generally the response type used in complex situations such as degraded visual environment. At the same time, in the case of divided attention operation, the specification has made stricter requirements on the damping ratio of the medium period response. Therefore, the damping ratio of 1 is chosen. In this case, the quickness index of equation (19) is as follows
(23)qpkθpk=−KAC−θminθminTln1−θminKAC,θmin<KAC1−e−1,KACe−1θminT,θmin≥KAC1−e−1.Through the above analysis, the parameters in equation (19) could be determined by equations (20)–(23).


## 4. Simulations

The following parameters are selected through analysis.Am=01−4−4, adaptive gain Γ=10000, low-pass filter parameter k=5.In order to examine the performance of the controller for tilt-rotor aircraft, three simulation scenarios are considered.


### 4.1. Scenario 1: Comparison with MRAC

In this case, the MRAC method was compared to verify the effectiveness of the proposed method. The simulation results for the MRAC and the proposed control methods are shown in Figures6 and 7. It should be mentioned that the MRAC method cannot realize the attitude control of tilt-rotor aircraft in both flight modes, as shown in Figures 6(a) and 7(a). Meanwhile, the MRAC controller in experiment and simulation showed highly oscillatory behavior throughout the stimulations as observed in Figures 6(b) and 7(b), which was, of course, physically not possible.
(1)
Helicopter Mode Comparison Results(2)
Airplane Mode Comparison ResultsFigure 6
The pitch angle and longitudinal cycle for helicopter mode.
(a)
Pitch angle change curve(b)
Longitudinal cycle pitch change curveFigure 7
The pitch angle and elevator for airplane mode.
(a)
Pitch angle change curve(b)
Elevator change curve


### 4.2. Scenario 2: Pitch Angle Capture in Certain Flight Mode

In this scenario, pitch angle capture simulations in helicopter flight mode and airplane flight mode are carrier out.


#### 4.2.1. Pitch Angle Capture Simulation in Helicopter Mode (Hover)

In this case, the head-down 10 deg task in helicopter mode is performed. The change curves of pitch rate and pitch angle are shown in Figure8. From the figure, it can be seen that the aircraft can track the reference signal well under the action of the control system. The control signal of the aircraft is shown in Figure 9. It can be seen from the figure the low-frequency components of the control signal flow into the real system under the action of the L1 adaptive control system. The state prediction error is shown in Figure 10. It can be found that the prediction error quickly converges under the action of the L1 adaptive control system.Figure 8
The pitch rate and pitch angle of step response.
(a)
Pitch rate change curve(b)
Pitch angle change curveFigure 9
Longitudinal cycle pitch change curve.Figure 10
The state prediction error change curves of step response.
(a)
Pitch rate prediction error(b)
Pitch angle prediction error


#### 4.2.2. Pitch Angle Capture Simulation in Airplane Mode

In this case, the head-up 10 deg task in airplane mode is performed. Figure11 shows the variations with time of the pitch rate and pitch angle for the experiment. From Figure 11(b), it can be seen that the pitch angle response can track the reference signal well. The control signal of the aircraft is shown in Figure 12. It can be seen from the figure that the frequency of the elevator signal is significantly reduced under the action of the L1 adaptive control system. The state prediction error is shown in Figure 13, and it can be found that the prediction error quickly converges under the action of the L1 adaptive control system.Figure 11
The pitch rate and pitch angle of step response.
(a)
Pitch rate change curve(b)
Pitch angle change curveFigure 12
Elevator change curve.Figure 13
The state prediction error change curve of step response.
(a)
Pitch rate prediction error(b)
Pitch angle prediction error


### 4.3. Scenario 3: Automatic Conversion Simulation

Conversion mode is the most important flight mode of tilt-rotor aircraft, which has a great impact on flight safety. In order to further verify the performance of the flight control system, an automatic conversion simulation was implemented. The nacelles were assumed to rotate with a constant angular speed, as shown in Figure14. The pitch angle command is shown in Figure 15(a) which is determined by the trim characteristics. As we can know from Figure 15, the pitch angle response can track the reference signal well and the pitch rate changes smoothly. The change curves of its velocity and height are shown in Figure 16. It can be seen that it is an accelerated process with a certain altitude climb from helicopter mode to airplane mode. From Figure 17, the conversion path is completely in the conversion corridor, which meets the flight condition constraints of the conversion mode. The control signal of the aircraft is shown in Figure 18. It can be seen from the figure that the signals of the longitudinal cycle pitch and elevator could be physically implemented. In general, under the action of the L1 adaptive control system, the states and control signals in the whole conversion process are reasonable.Figure 14
Nacelle angle change curve.Figure 15
The pitch angle and pitch rate change curves.
(a)
Pitch angle change curve(b)
Pitch rate change curveFigure 16
The airspeed and height change curves in conversion mode.
(a)
Airspeed change curve(b)
Height change curveFigure 17
Conversion path curves.Figure 18
The control signal change curves.
(a)
Longitudinal cycle pitch change curve(b)
Elevator change curveIn this section, three sets of simulation tests have been completed. It can be obtained from the first set simulation that the control method in this paper can track pitch commend in different flight modes, while MRAC cannot complete this simulation task. The second test shows that the control method in this paper can achieve good flight quality, and the control signal can be physically realized. The following conclusion can be drawn from the third set of tests is that the proposed method can realize automatic transition flight.In addition, in the three sets of simulation tests, exactly the same controller architecture and parameter setting are selected, which avoids the repeated design of the control law and reduces the workload of the control system design.


## 4.1. Scenario 1: Comparison with MRAC

In this case, the MRAC method was compared to verify the effectiveness of the proposed method. The simulation results for the MRAC and the proposed control methods are shown in Figures6 and 7. It should be mentioned that the MRAC method cannot realize the attitude control of tilt-rotor aircraft in both flight modes, as shown in Figures 6(a) and 7(a). Meanwhile, the MRAC controller in experiment and simulation showed highly oscillatory behavior throughout the stimulations as observed in Figures 6(b) and 7(b), which was, of course, physically not possible.
(1)
Helicopter Mode Comparison Results(2)
Airplane Mode Comparison ResultsFigure 6
The pitch angle and longitudinal cycle for helicopter mode.
(a)
Pitch angle change curve(b)
Longitudinal cycle pitch change curveFigure 7
The pitch angle and elevator for airplane mode.
(a)
Pitch angle change curve(b)
Elevator change curve


## 4.2. Scenario 2: Pitch Angle Capture in Certain Flight Mode

In this scenario, pitch angle capture simulations in helicopter flight mode and airplane flight mode are carrier out.


### 4.2.1. Pitch Angle Capture Simulation in Helicopter Mode (Hover)

In this case, the head-down 10 deg task in helicopter mode is performed. The change curves of pitch rate and pitch angle are shown in Figure8. From the figure, it can be seen that the aircraft can track the reference signal well under the action of the control system. The control signal of the aircraft is shown in Figure 9. It can be seen from the figure the low-frequency components of the control signal flow into the real system under the action of the L1 adaptive control system. The state prediction error is shown in Figure 10. It can be found that the prediction error quickly converges under the action of the L1 adaptive control system.Figure 8
The pitch rate and pitch angle of step response.
(a)
Pitch rate change curve(b)
Pitch angle change curveFigure 9
Longitudinal cycle pitch change curve.Figure 10
The state prediction error change curves of step response.
(a)
Pitch rate prediction error(b)
Pitch angle prediction error


### 4.2.2. Pitch Angle Capture Simulation in Airplane Mode

In this case, the head-up 10 deg task in airplane mode is performed. Figure11 shows the variations with time of the pitch rate and pitch angle for the experiment. From Figure 11(b), it can be seen that the pitch angle response can track the reference signal well. The control signal of the aircraft is shown in Figure 12. It can be seen from the figure that the frequency of the elevator signal is significantly reduced under the action of the L1 adaptive control system. The state prediction error is shown in Figure 13, and it can be found that the prediction error quickly converges under the action of the L1 adaptive control system.Figure 11
The pitch rate and pitch angle of step response.
(a)
Pitch rate change curve(b)
Pitch angle change curveFigure 12
Elevator change curve.Figure 13
The state prediction error change curve of step response.
(a)
Pitch rate prediction error(b)
Pitch angle prediction error


## 4.2.1. Pitch Angle Capture Simulation in Helicopter Mode (Hover)

In this case, the head-down 10 deg task in helicopter mode is performed. The change curves of pitch rate and pitch angle are shown in Figure8. From the figure, it can be seen that the aircraft can track the reference signal well under the action of the control system. The control signal of the aircraft is shown in Figure 9. It can be seen from the figure the low-frequency components of the control signal flow into the real system under the action of the L1 adaptive control system. The state prediction error is shown in Figure 10. It can be found that the prediction error quickly converges under the action of the L1 adaptive control system.Figure 8
The pitch rate and pitch angle of step response.
(a)
Pitch rate change curve(b)
Pitch angle change curveFigure 9
Longitudinal cycle pitch change curve.Figure 10
The state prediction error change curves of step response.
(a)
Pitch rate prediction error(b)
Pitch angle prediction error


## 4.2.2. Pitch Angle Capture Simulation in Airplane Mode

In this case, the head-up 10 deg task in airplane mode is performed. Figure11 shows the variations with time of the pitch rate and pitch angle for the experiment. From Figure 11(b), it can be seen that the pitch angle response can track the reference signal well. The control signal of the aircraft is shown in Figure 12. It can be seen from the figure that the frequency of the elevator signal is significantly reduced under the action of the L1 adaptive control system. The state prediction error is shown in Figure 13, and it can be found that the prediction error quickly converges under the action of the L1 adaptive control system.Figure 11
The pitch rate and pitch angle of step response.
(a)
Pitch rate change curve(b)
Pitch angle change curveFigure 12
Elevator change curve.Figure 13
The state prediction error change curve of step response.
(a)
Pitch rate prediction error(b)
Pitch angle prediction error


## 4.3. Scenario 3: Automatic Conversion Simulation

Conversion mode is the most important flight mode of tilt-rotor aircraft, which has a great impact on flight safety. In order to further verify the performance of the flight control system, an automatic conversion simulation was implemented. The nacelles were assumed to rotate with a constant angular speed, as shown in Figure14. The pitch angle command is shown in Figure 15(a) which is determined by the trim characteristics. As we can know from Figure 15, the pitch angle response can track the reference signal well and the pitch rate changes smoothly. The change curves of its velocity and height are shown in Figure 16. It can be seen that it is an accelerated process with a certain altitude climb from helicopter mode to airplane mode. From Figure 17, the conversion path is completely in the conversion corridor, which meets the flight condition constraints of the conversion mode. The control signal of the aircraft is shown in Figure 18. It can be seen from the figure that the signals of the longitudinal cycle pitch and elevator could be physically implemented. In general, under the action of the L1 adaptive control system, the states and control signals in the whole conversion process are reasonable.Figure 14
Nacelle angle change curve.Figure 15
The pitch angle and pitch rate change curves.
(a)
Pitch angle change curve(b)
Pitch rate change curveFigure 16
The airspeed and height change curves in conversion mode.
(a)
Airspeed change curve(b)
Height change curveFigure 17
Conversion path curves.Figure 18
The control signal change curves.
(a)
Longitudinal cycle pitch change curve(b)
Elevator change curveIn this section, three sets of simulation tests have been completed. It can be obtained from the first set simulation that the control method in this paper can track pitch commend in different flight modes, while MRAC cannot complete this simulation task. The second test shows that the control method in this paper can achieve good flight quality, and the control signal can be physically realized. The following conclusion can be drawn from the third set of tests is that the proposed method can realize automatic transition flight.In addition, in the three sets of simulation tests, exactly the same controller architecture and parameter setting are selected, which avoids the repeated design of the control law and reduces the workload of the control system design.


## 5. Conclusions

In this paper, aL1 adaptive controller for tilt-rotor aircraft has been proposed. The effectiveness of the controller is verified through simulation. The main conclusions are summarized as follows.
(1)
Tilt-rotor aircraft has three flight modes in the flight envelope, which results in a large uncertainty in the flight dynamics model. Considering the modeling uncertainty, tilt-rotor aircraft is described as a form with unmodeled dynamics, and theL1 adaptive control system scheme is formed(2)
The MRAC method was compared to verify the effectiveness of the proposed method. The controller in this paper can track pitch commend in different flight modes, while MRAC task failed(3)
The simulation test of the conversion flight is completed with the same controller architecture and parameters setting. The conversion from helicopter mode to airplane mode is successfully realized, which significantly reduced the workload of control law design(4)
In the future work, we will plan to implement the designed algorithm in practical application by conducting scientific experiments



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*Source: 9912618-2021-06-21.xml*