# Nonlinear Adaptive Equivalent Control Based on Interconnection Subsystems for Air-Breathing Hypersonic Vehicles

Authors: Chaofang Hu; Yanwen Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

For the nonminimum phase behavior of the air-breathing hypersonic vehicle model caused by elevator-to-lift coupling, a nonlinear adaptive equivalent control method based on interconnection subsystems is proposed. In the altitude loop, the backstepping strategy is applied, where the virtual control inputs about flight-path angle and attack angle are designed step by step. In order to avoid the inaccurately direct cancelation of elevator-to-lift coupling when aerodynamic parameters are uncertain, the real control inputs, that is, elevator deflection and canard deflection, are linearly converted into the equivalent control inputs which are designed independently. The reformulation of the altitude-flight-path angle dynamics and the attack angle-pitch rate dynamics is constructed into interconnection subsystems with input-to-state stability via small-gain theorem. For the velocity loop, the dynamic inversion controller is designed. The adaptive approach is used to identify the uncertain aerodynamic parameters. Simulation of the flexible hypersonic vehicle demonstrates effectiveness of the proposed method.


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## Body

## 1. Introduction

Hypersonic vehicles have a promising prospect in both military and commercial applications as its flight speed can be more than 5 times of the speed of sound. However, since the model of hypersonic vehicle is nonlinear, multivariable, uncertain, and coupling [1], it is unstable and extremely sensitive to changes in flight condition and parameters. This brings a great challenge to controller design [2]. At present, most researches focus on dealing with nonlinearity and uncertainty of hypersonic vehicles. For example, linear control methods are attempted according to linearized hypersonic vehicle models, such as pole placement techniques [3], LQR method [4], linear output feedback control [5], and LPV control [6]. In addition, nonlinear control strategies are widely used as well, such as feedback linearization approach [7], sliding control [8, 9], and backstepping technique [10]. For uncertainty of hypersonic vehicles, besides adaptive approaches [11], robust strategies are common tools, for example, μ-synthesis, H∞ control [12], stochastic robustness control [13], and nonlinear disturbance observer-based robust control [14]. Although these methods are proven to be effective, they do not usually consider the coupling problems existing in hypersonic vehicles. These problems lead to more difficulties in the flight controller design. In an air-breathing hypersonic vehicle, it is known that there are structural dynamics, flexible effect, elevator-to-lift coupling, and the coupling between thrust and pitch moment, where elevator-to-lift coupling is not neglectable, and it will generate unstable zero dynamics exponentially, that is, the nonminimum phase behavior in pitch rate model, if the controller is designed directly by the inversion.With regard to the elevator-to-lift coupling problem, some strategies have been tried. The basic method usually ignores this coupling, and then the nonminimum phase can be removed from the model during the controller design [2], where this coupling is only regarded as unmodeled dynamics. However, this manner cannot ensure the stability of the control system. The other common approach is to offset the influence of the coupling. For example, a canard is adopted to cancel the influence of elevator on lift, and an adaptive robust controller based on nonlinear sequential loop-closure approach is developed [15, 16]. Nevertheless, the changes of the uncertain parameters are not considered. This may result in the inaccurate cancellation, which means elevator and lift are not decoupled completely. Simultaneously, this approach has an adverse influence on the pitch rate dynamics since its inputs also consist of elevator and canard. In addition, for thermal protection problem resulted from the canard, only the elevator is taken as aerodynamic control surface in reference [17]. The system model is transformed into the interconnection of systems in feedback and feedforward forms to eliminate the nonminimum phase. But the robustness with regard to uncertainty of the hypersonic vehicle model is not addressed totally.From the analysis, we know that adding canard control surface is an effective and simple way to suppress the nonminimum phase behavior, even though the strict cancellation of the elevator-to-lift coupling cannot be realized actually. In this paper, the flexible air-breathing hypersonic vehicle model is considered. For the tracking requirement of altitude and velocity, a nonlinear adaptive equivalent control method based on interconnection subsystems is proposed by incorporating canard. Firstly, in the altitude loop, the virtual control inputs about flight-path angle and attack angle are designed step by step according to the backstepping strategy. Secondly, the terms about the real control inputs, that is, the elevator and canard deflection in the flight-path angle dynamics and the pitch rate dynamics, are linearly converted into the equivalent control inputs instead of direct cancelation of the elevator-to-lift coupling. By designing the new inputs independently, the altitude control loop is reformulated. And the adaptive technique is used to identify the uncertain aerodynamic parameters. Then the interconnection subsystems including the altitude-flight-path angle dynamics and the attack angle-pitch rate dynamics are constructed. Via the small-gain method, the system is proven to be input-to-state stable. In the velocity loop, the adaptive dynamic inversion controller is designed. Simulation results show the power of our approach.In Section2, the air-breathing hypersonic vehicle model is presented. The nonlinear adaptive equivalent control based on interconnection subsystems is introduced in Section 3. Section 4 presents the simulation. The conclusion is drawn in Section 5.


## 2. Air-Breathing Hypersonic Vehicle Model

In this study, the flexible air-breathing hypersonic vehicle model [18] is considered. This model is composed of five rigid-body states, that is, velocity V, altitude h, flight-path angle γ, attack angel α, pitch rate q, and six flexible states, that is, η1,  η˙1,  η2,  η˙2,  η˙3,  and η˙3. The equations of motion are written as
(1)V˙=Tcosα-Dm-gsinγ,h˙=Vsinγ,γ˙=Tsinα+LmV-gcosγV,α˙=Q-γ˙,Q˙=MIyy,η¨i=-2ςmωm,iη˙i-ωm,i2ηi+Ni;i=1,2,3,
where m, Iyy, g represent mass of the aircraft, moment of inertia, gravitational acceleration; damping ratio and natural frequency of the flexible motion are denoted by ςm and ωm,i, respectively; T, D, L, and Ni and M are thrust, drag, lift, generalized forces and moment
(2)L=q-SCL,T=q-(CT,ϕϕ+CT),D=q-SCD,M=zTT+q-Sc-CM,Ni=q-CNi.
The aerodynamic parameters in the above formulation are described as follows:
(3)CL=CLαα+CLδeδe+CLδcδc+CL0+CLΔτ1Δτ1+CLΔτ2Δτ2,CM=CMαα+CMδeδe+CMδcδc+CM0+CMΔτ1Δτ1+CMΔτ2Δτ2,CNi=CNiαα+CNiδeδe+CNiδcδc+CNi0+CNiΔτ1Δτ1+CNiΔτ2Δτ2,CD=CD(α+Δτ1)2(α+Δτ1)2+CD(α+Δτ1)(α+Δτ1)+CDΔτ2Δτ2+CDδe2δe2+CDδeδe+CDαδeαδe+CDδc2δc2+CDδcδc+CDαδcαδc+CD0,CT,ϕ=CT,ϕαα+CT,ϕαM∞-2αM∞-2+CT,ϕM∞-2M∞-2+CT,ϕ0+CT,ϕαΔτ1αΔτ1+CT,ϕΔτ12Δτ12+CT,ϕΔτ1Δτ1,CT=CTαα+CTM∞-2M∞-2+CTAdAd+CTΔτ1Δτ1+CT0,
where the control inputs are fuel-to-air ratio ϕ, elevator deflection δe, and canard deflection δc;  q-,  S,  zT,  c-,  M∞ denote dynamic pressure, reference area, thrust moment arm, mean aerodynamic chord, and Mach number; Δτ1 and Δτ2 are the forebody turn angle and the aftbody vertex angle which are linear mapping of elastic states ηi.In (3), the elevator-to-lift coupling orients from that CL includes the term of δe, which leads to the nonminimum phase behavior. If δe is designed by the dynamic inversion directly, the pitch rate dynamics will become a hyperbolic saddle equilibrium. This unstable zero dynamic brings great difficulties to the controller design.


## 3. Nonlinear Adaptive Equivalent Controllers Design

In order to track the altitude and velocity command signalshref and Vref, two controllers will be designed independently for the altitude loop and the velocity loop. During the controller design, the flexible motion is viewed as external perturbation, and its influence on aerodynamic model (3) is neglected.


### 3.1. Altitude Controller

In the altitude loop, the controller is designed according to the backstepping approach. Then the virtual control inputs about flight-path angle and attack angle are determined, respectively.For the altitude dynamics, leth~=h-href; then its error dynamics is written in the following:
(4)h~˙=Vsinγ-h˙ref≈Vγ-h˙ref.
So the flight-path angle command γd is designed into the following equation:
(5)γd=-kh~h~+h˙refV,
where kh~>0 is the design parameter for h~.Letγ~=γ-γd; the error dynamic of flight-path angle is presented as follows:
(6)γ~˙=Tsinα+LmV-gcosγV-γ˙d.
Here, the thrust is described as the function about the attack angle. Define T=∇Tα+T0, where ∇T=q-(CT,ϕαM∞-2M∞-2ϕ+CT,ϕαϕ+CTα) and T0=q-(CT,ϕM∞-2ϕM∞-2+CT,ϕ0ϕ+CTM∞-2M∞-2+CTAdAd+CT0). As the variation range of the attack angle is small, (6) will be expanded around the final expectation α*.To handle the nonminimum phase problem, the MIMO equivalent method is applied in this paper, which is different from the previous research results [17]. The terms about the elevator and canard deflection are linearly equivalent to the control input vector U=[U1,U2]. The error model of the flight-path angle (6) can be rewritten as
(7)γ~˙=∇Tαsinα+T0sinα+q-SCLααmV+q-SCLδeδemV+q-SCLδCδCmV+q-SCL0-mgcosγ-mVγ˙dmV=∇Tsinα*+∇Tα*cosα*+T0cosα*+q-SCLαmV︸C1α+q-SCLδeδe+CLδcδcmV︸U1+q-SCL0-mgsinγ-mVγ˙dmV+T0sinα*-(∇Tα*cosα*+T0cosα*)α*mV︸β1=C1α+U1+β1,
where C1 and β1 are the terms containing the uncertain aerodynamic parameters. They can be expressed as the following equations:
(8)C1=θ1Tξ1,β1=θ1Tξ2-gcosγV-γ˙d,U1=θ2Tξ3.θ1, θ2 are vectors of the uncertain parameters
(9)θ1=[CT,ϕα;CTα;CT,ϕαM∞-2;CT,ϕM∞-2;CT,ϕ0;CTM∞-2;CTAd;CT0;CLα;CL0],θ2=[CLδe;CLδc],
and ξ1, i=1…3 are regressors
(10)ξ1=q-mV[M∞-2(sinα*+α*cosα*)ϕ;(sinα*+α*cosα*);(sinα*+α*cosα*)M∞-2ϕ;cosα*M∞-2ϕ;cosα*ϕ;cosα*M∞-2;cosα*Ad;cosα*;S;0],ξ2=q-mV[-α*2cosα*ϕ;-α*2cosα*;-α*2cosα*M∞-2ϕ;(sinα*-α*cosα*)M∞-2ϕ;(sinα*-α*cosα*)ϕ;(sinα*-α*cosα*)M∞-2;(sinα*-α*cosα*)Ad;(sinα*-α*cosα*);0;Sα*2],ξ3=q-SmV[δe;δc].
Therefore the dynamics (7) is reformulated as
(11)γ~˙=θ1Tξ1α+θ2Tξ3+θ1Tξ2-gcosγV-γ˙d.
Then the virtual command of the attack angle is chosen as αd=α*-γ~.Letα~=α-αd; the error dynamic of the attack angle is formulated as
(12)α~˙=Q-γ˙-α˙d=Q-γ˙d.
A new variable Z is defined as Z=Q-γ˙d+kα~α~, where  kα~>0 is a design parameter for α~. Then (12) is rewritten as
(13)α~˙=Z-kα~α~.
Using the equivalent control method, the time derivative of Z can be formulated with the new input U2. It includes the pitch rate dynamics
(14)Z˙=zTT+q-Sc-CMIyy+kα~α~˙=q-Sc-CMδeδe+CMδcδcIyy︸U2+zT∇T+q-Sc-CMαIyy︸C2α+zTT0+q-Sc-CM0Iyy+kα~α~˙-γ¨d︸β2=C2α+U2+β2,
where C2 and β2 are similar terms containing the uncertain parameters. They can also be presented by the vectors of the uncertain parameters and the regressors
(15)C2=θ3Tξ4,β2=θ3Tξ5+kα~α~˙-γ¨d,U2=θ4Tξ6,
where
(16)θ3=[CT,ϕα;CTα;CT,ϕαM∞-2;CT,ϕM∞-2;CT,ϕ0;CTM∞-2;CTAd;CT0;CMα;CM0],θ4=[CMδe;CMδc],ξ4=q-Iyy[zTϕ;zT;zTM∞-2ϕ;0;0;0;0;0;Sc-;0],ξ5=q-Iyy[0;0;0;zTM∞-2ϕ;zTϕ;zTM∞-2;zTAd;zT;0;Sc-],ξ6=q-Sc-Iyy[δe;δc].
So (14) can be reformulated as
(17)Z˙=θ3Tξ4α+θ4Tξ6+θ3Tξ5+kα~α~˙-γ¨d.
Due to the uncertainty of the aerodynamic parameters, θi, i=1,…,4 will change with flight of hypersonic vehicles. Therefore it is necessary to estimate their values by the adaptive technique. Let θ^i, θ~i be the estimate vector and the estimate error vector of θi, where θ~i=θi-θ^i, i=1,…,4.Assumption 1.
The aerodynamic parametersθi,  i=1,…,4 are bounded; they lie in a compact convex set.In order to guarantee tracking performance of hypersonic vehicles, the equivalent control inputsU1 and  U2 are designed, respectively, by replacing the uncertain parameter vector θi with its estimate vector and estimate error vector
(18)U^1=θ^2Tξ3=-θ^1Tξ1α*-(θ^1Tξ2-gcosγV-γ˙d)+(θ^1Tξ1-kγ~)γ~-Vh~,U^2=θ^4Tξ6=γ¨d-(θ^3Tξ5+kα~α~˙)-θ^3Tξ4α*-kZZ-(θ^3Tξ4+1)α~,
where kγ~>0,  kZ>0 are the design parameters for γ~ and Z.Letδ=[δe,δc]. There is U=Bδ according to (7) and (14). B is a coefficient matrix and is equal to [(q-S/mV)θ^2T;(q-Sc-/Iyy)θ^4T]. The real inputs of the altitude loop can be obtained as follows:
(19)[δeδc]=B-1[U^1U^2].Combining (18), the state error dynamics about the altitude loop is transformed into the following equations:
(20)h~˙≈-kh~h~+Vγ~,γ~˙=-kγ~γ~-Vh~+yα~+θ~1Tξ1α+θ~1Tξ2+θ~2Tξ3,α~˙=Z-kα~α~,Z˙=-kZZ-α~+yγ~+θ~3Tξ4α+θ~3Tξ5+θ~4Tξ6,
where yα~=θ^1Tξ1α~, yγ~=-θ^3Tξ4γ~.For ensuring the stability of the altitude loop, the new formulation (20) is divided into the altitude-flight-path angle subsystem and the attack angle-pitch rate subsystem. As illustrated in Figure 1, these two subsystems constitute a structure of interconnection. It is seen that yα~  and  yγ~  act as the input and output of the altitude-flight-path angle subsystem and yγ~,  yα~  are the input and output of the attack angle-pitch rate subsystem, respectively.Figure 1
Interconnection subsystem structure.For the above interconnection subsystems, input-to-state stability will be analyzed via small gain theorem. Firstly, the definition of the asymptoticL∞ norm ∥·∥a is given [19]
(21)∥λ∥a:=limt→∞sup|λ|.
Then, define ψ1=h~2+γ~2, and choose the Lyapunov function candidate of the altitude-flight-path angle subsystem as
(22)W1=12(h~2+γ~2)+12θ~1Tτ1-1θ~1+12θ~2Tτ2-1θ~2.
Its time derivative is
(23)W˙1=h~h~˙+γ~γ~˙-θ~1Tτ1-1θ^˙1-θ~2Tτ2-1θ^˙2=-kh~h~2-kγ~γ~2+γ~yα~+θ~1Tτ1-1{τ1γ~(ξ1α+ξ2)-θ^˙1}+θ~2Tτ2-1(τ2γ~ξ3-θ^˙2).
The adaptive laws of θ^1,  θ^2 are designed as
(24)θ^˙1=τ1γ~(ξ1α+ξ2),θ^˙2=τ2γ~ξ3,
where τ1 and τ2 are the adaptive parameters.By (24), (23) becomes
(25)W˙1≤-min{kh~,kγ~}|ψ1|2+|ψ1||yα~|.
As a consequence, it satisfies W1 which is negative definite when |ψ1|≥|yα~|/min{kh~,kγ~}. Then W1 is a input-to-state stable Lyapunov function. According to the lemma in [19], we know that ∥ψ1∥a≤∥yα~∥a/min{kh~,kγ~}. As yγ~=-θ^3Tξ4γ~, the following formulation is obtained:
(26)∥yγ~∥a=∥-θ^3Tξ4γ~∥a≤θ^3Tξ4∥ψ1∥a≤θ^3Tξ4min{kh~,kγ~}∥yα~∥a.
For the attack angle-pitch rate subsystem, ψ2=α~2+Z2 is defined, and the following Lyapunov function candidate is chosen:
(27)W2=12(α~2+Z2)+12θ~3Tτ3-1θ~3+12θ~4Tτ4-1θ~4.
Its time derivative is
(28)W˙2=α~α~˙+ZZ˙-θ~3Tτ3-1θ^˙3-θ~4Tτ4-1θ^˙4=-kα~α~2+kZZ2+Zyγ~+θ~3Tτ3-1{τ3Z(ξ4α+ξ5)-θ^˙3}+θ~4Tτ4-1(τ4Zξ6-θ^˙4),
where the adaptive laws of θ^3, θ^4 are determined as
(29)θ^˙3=τ3Z(ξ4α+ξ5),θ^˙4=τ4Zξ6.
Substituting (29) in (28), we can acquire
(30)W˙2≤-min{kα~,kZ}|ψ2|2+|ψ2||yγ~|.
When |ψ2|≥|yγ~|/min{kα~,kZ}, W˙2≤0. Similarly, ∥ψ2∥a≤∥yγ~∥a/min{kα~,kZ} can be obtained, and W2 is input-to-state stable as well. Because yα~=θ^1Tξ1α~, there is
(31)∥yα~∥a=∥θ^1Tξ1α~∥a≤θ^1Tξ1∥ψ2∥a≤θ^1Tξ1min{kα~,kZ}∥yγ~∥a.
The interconnection formulation (20) is input-to-state stable according to small-gain theorem if we choose proper design parameters to make the following equation holds
(32)θ^3Tξ4min{kh~,kγ~}·θ^1Tξ1min{kα~,kZ}<1.Therefore the tracking errors and estimate errors of the altitude loop can converge to a small neighborhood of origin.


### 3.2. Velocity Controller

Since velocity is controlled byϕ directly, the adaptive dynamic inversion method is used. Let V~=V-Vref; the error dynamics of velocity is written as
(33)V~˙=Tcosα-Dm-gsinγ-V˙ref=q-(CT,ϕϕ+CT)cosα-q-SCDm-gsinγ-V˙ref.
For existence of uncertain parameters, the following vectors and repressors are defined
(34)θ5=[CD(α+Δτ1);CD(α+Δτ1)2;CDδe2;CDδe;CDδc2;CDδc;CDαδe;CDαδc;CD0;CTAd;CTα;CTM∞-2;CT0],θ6=[CT,ϕα;CT,ϕαM∞-2;CT,ϕM∞-2;CT,ϕ0],ξ7=q-[Sα;Sα2;Sδe2;Sδe;Sδc2;Sδc;Sαδe;Sαδc;S;-Adcosα;-αcosα;-M∞-2cosα;-cosα],ξ8=q-cosα[α;αM∞-2;M∞-2;1].
Consequently,
(35)V~˙=θ6Tξ8ϕ-θ5Tξ7m-gsinγ-V˙ref.
Then the control input ϕ is designed as
(36)ϕ=-mkv~V~+mV˙ref+mgsinγ+θ^5Tξ7θ^6Tξ8,
where kv~>0 is a design parameter.Determine the Lyapunov function candidate as(37)W3=12mV~2+12θ~5Tτ5-1θ~5+12θ~6Tτ6-1θ~6.
Its time derivative is
(38)W˙3=V~(θ6Tξ8ϕ-θ5Tξ7-mgsinγ-mV˙ref)-θ~5Tτ5-1θ^˙5-θ~6Tτ6-1θ^˙6=V~(θ~6Tξ8ϕ-mkv~V~-θ~5Tξ7)-θ~5Tτ5-1θ^˙5-θ~6Tτ6-1θ^˙6=-mkv~V~2-θ~5Tτ5-1(τ5V~ξ7+θ^˙5)+θ~6Tτ6-1(τ6ϕξ8-θ^˙6).
The adaptive laws of θ^5, θ^6 are obtained in the following:
(39)θ^˙5=-τ5V~ξ7,θ^˙6=τ6ϕξ8.
Thus (38) becomes W˙3=-mkv~V~2<0; that is, when t→∞, the tracking errors and estimate errors of the velocity subsystem can converge to zero finally.Therefore the accurate tracking performance and the stability of altitude and velocity can be guarranteed by the proposed method.


## 3.1. Altitude Controller

In the altitude loop, the controller is designed according to the backstepping approach. Then the virtual control inputs about flight-path angle and attack angle are determined, respectively.For the altitude dynamics, leth~=h-href; then its error dynamics is written in the following:
(4)h~˙=Vsinγ-h˙ref≈Vγ-h˙ref.
So the flight-path angle command γd is designed into the following equation:
(5)γd=-kh~h~+h˙refV,
where kh~>0 is the design parameter for h~.Letγ~=γ-γd; the error dynamic of flight-path angle is presented as follows:
(6)γ~˙=Tsinα+LmV-gcosγV-γ˙d.
Here, the thrust is described as the function about the attack angle. Define T=∇Tα+T0, where ∇T=q-(CT,ϕαM∞-2M∞-2ϕ+CT,ϕαϕ+CTα) and T0=q-(CT,ϕM∞-2ϕM∞-2+CT,ϕ0ϕ+CTM∞-2M∞-2+CTAdAd+CT0). As the variation range of the attack angle is small, (6) will be expanded around the final expectation α*.To handle the nonminimum phase problem, the MIMO equivalent method is applied in this paper, which is different from the previous research results [17]. The terms about the elevator and canard deflection are linearly equivalent to the control input vector U=[U1,U2]. The error model of the flight-path angle (6) can be rewritten as
(7)γ~˙=∇Tαsinα+T0sinα+q-SCLααmV+q-SCLδeδemV+q-SCLδCδCmV+q-SCL0-mgcosγ-mVγ˙dmV=∇Tsinα*+∇Tα*cosα*+T0cosα*+q-SCLαmV︸C1α+q-SCLδeδe+CLδcδcmV︸U1+q-SCL0-mgsinγ-mVγ˙dmV+T0sinα*-(∇Tα*cosα*+T0cosα*)α*mV︸β1=C1α+U1+β1,
where C1 and β1 are the terms containing the uncertain aerodynamic parameters. They can be expressed as the following equations:
(8)C1=θ1Tξ1,β1=θ1Tξ2-gcosγV-γ˙d,U1=θ2Tξ3.θ1, θ2 are vectors of the uncertain parameters
(9)θ1=[CT,ϕα;CTα;CT,ϕαM∞-2;CT,ϕM∞-2;CT,ϕ0;CTM∞-2;CTAd;CT0;CLα;CL0],θ2=[CLδe;CLδc],
and ξ1, i=1…3 are regressors
(10)ξ1=q-mV[M∞-2(sinα*+α*cosα*)ϕ;(sinα*+α*cosα*);(sinα*+α*cosα*)M∞-2ϕ;cosα*M∞-2ϕ;cosα*ϕ;cosα*M∞-2;cosα*Ad;cosα*;S;0],ξ2=q-mV[-α*2cosα*ϕ;-α*2cosα*;-α*2cosα*M∞-2ϕ;(sinα*-α*cosα*)M∞-2ϕ;(sinα*-α*cosα*)ϕ;(sinα*-α*cosα*)M∞-2;(sinα*-α*cosα*)Ad;(sinα*-α*cosα*);0;Sα*2],ξ3=q-SmV[δe;δc].
Therefore the dynamics (7) is reformulated as
(11)γ~˙=θ1Tξ1α+θ2Tξ3+θ1Tξ2-gcosγV-γ˙d.
Then the virtual command of the attack angle is chosen as αd=α*-γ~.Letα~=α-αd; the error dynamic of the attack angle is formulated as
(12)α~˙=Q-γ˙-α˙d=Q-γ˙d.
A new variable Z is defined as Z=Q-γ˙d+kα~α~, where  kα~>0 is a design parameter for α~. Then (12) is rewritten as
(13)α~˙=Z-kα~α~.
Using the equivalent control method, the time derivative of Z can be formulated with the new input U2. It includes the pitch rate dynamics
(14)Z˙=zTT+q-Sc-CMIyy+kα~α~˙=q-Sc-CMδeδe+CMδcδcIyy︸U2+zT∇T+q-Sc-CMαIyy︸C2α+zTT0+q-Sc-CM0Iyy+kα~α~˙-γ¨d︸β2=C2α+U2+β2,
where C2 and β2 are similar terms containing the uncertain parameters. They can also be presented by the vectors of the uncertain parameters and the regressors
(15)C2=θ3Tξ4,β2=θ3Tξ5+kα~α~˙-γ¨d,U2=θ4Tξ6,
where
(16)θ3=[CT,ϕα;CTα;CT,ϕαM∞-2;CT,ϕM∞-2;CT,ϕ0;CTM∞-2;CTAd;CT0;CMα;CM0],θ4=[CMδe;CMδc],ξ4=q-Iyy[zTϕ;zT;zTM∞-2ϕ;0;0;0;0;0;Sc-;0],ξ5=q-Iyy[0;0;0;zTM∞-2ϕ;zTϕ;zTM∞-2;zTAd;zT;0;Sc-],ξ6=q-Sc-Iyy[δe;δc].
So (14) can be reformulated as
(17)Z˙=θ3Tξ4α+θ4Tξ6+θ3Tξ5+kα~α~˙-γ¨d.
Due to the uncertainty of the aerodynamic parameters, θi, i=1,…,4 will change with flight of hypersonic vehicles. Therefore it is necessary to estimate their values by the adaptive technique. Let θ^i, θ~i be the estimate vector and the estimate error vector of θi, where θ~i=θi-θ^i, i=1,…,4.Assumption 1.
The aerodynamic parametersθi,  i=1,…,4 are bounded; they lie in a compact convex set.In order to guarantee tracking performance of hypersonic vehicles, the equivalent control inputsU1 and  U2 are designed, respectively, by replacing the uncertain parameter vector θi with its estimate vector and estimate error vector
(18)U^1=θ^2Tξ3=-θ^1Tξ1α*-(θ^1Tξ2-gcosγV-γ˙d)+(θ^1Tξ1-kγ~)γ~-Vh~,U^2=θ^4Tξ6=γ¨d-(θ^3Tξ5+kα~α~˙)-θ^3Tξ4α*-kZZ-(θ^3Tξ4+1)α~,
where kγ~>0,  kZ>0 are the design parameters for γ~ and Z.Letδ=[δe,δc]. There is U=Bδ according to (7) and (14). B is a coefficient matrix and is equal to [(q-S/mV)θ^2T;(q-Sc-/Iyy)θ^4T]. The real inputs of the altitude loop can be obtained as follows:
(19)[δeδc]=B-1[U^1U^2].Combining (18), the state error dynamics about the altitude loop is transformed into the following equations:
(20)h~˙≈-kh~h~+Vγ~,γ~˙=-kγ~γ~-Vh~+yα~+θ~1Tξ1α+θ~1Tξ2+θ~2Tξ3,α~˙=Z-kα~α~,Z˙=-kZZ-α~+yγ~+θ~3Tξ4α+θ~3Tξ5+θ~4Tξ6,
where yα~=θ^1Tξ1α~, yγ~=-θ^3Tξ4γ~.For ensuring the stability of the altitude loop, the new formulation (20) is divided into the altitude-flight-path angle subsystem and the attack angle-pitch rate subsystem. As illustrated in Figure 1, these two subsystems constitute a structure of interconnection. It is seen that yα~  and  yγ~  act as the input and output of the altitude-flight-path angle subsystem and yγ~,  yα~  are the input and output of the attack angle-pitch rate subsystem, respectively.Figure 1
Interconnection subsystem structure.For the above interconnection subsystems, input-to-state stability will be analyzed via small gain theorem. Firstly, the definition of the asymptoticL∞ norm ∥·∥a is given [19]
(21)∥λ∥a:=limt→∞sup|λ|.
Then, define ψ1=h~2+γ~2, and choose the Lyapunov function candidate of the altitude-flight-path angle subsystem as
(22)W1=12(h~2+γ~2)+12θ~1Tτ1-1θ~1+12θ~2Tτ2-1θ~2.
Its time derivative is
(23)W˙1=h~h~˙+γ~γ~˙-θ~1Tτ1-1θ^˙1-θ~2Tτ2-1θ^˙2=-kh~h~2-kγ~γ~2+γ~yα~+θ~1Tτ1-1{τ1γ~(ξ1α+ξ2)-θ^˙1}+θ~2Tτ2-1(τ2γ~ξ3-θ^˙2).
The adaptive laws of θ^1,  θ^2 are designed as
(24)θ^˙1=τ1γ~(ξ1α+ξ2),θ^˙2=τ2γ~ξ3,
where τ1 and τ2 are the adaptive parameters.By (24), (23) becomes
(25)W˙1≤-min{kh~,kγ~}|ψ1|2+|ψ1||yα~|.
As a consequence, it satisfies W1 which is negative definite when |ψ1|≥|yα~|/min{kh~,kγ~}. Then W1 is a input-to-state stable Lyapunov function. According to the lemma in [19], we know that ∥ψ1∥a≤∥yα~∥a/min{kh~,kγ~}. As yγ~=-θ^3Tξ4γ~, the following formulation is obtained:
(26)∥yγ~∥a=∥-θ^3Tξ4γ~∥a≤θ^3Tξ4∥ψ1∥a≤θ^3Tξ4min{kh~,kγ~}∥yα~∥a.
For the attack angle-pitch rate subsystem, ψ2=α~2+Z2 is defined, and the following Lyapunov function candidate is chosen:
(27)W2=12(α~2+Z2)+12θ~3Tτ3-1θ~3+12θ~4Tτ4-1θ~4.
Its time derivative is
(28)W˙2=α~α~˙+ZZ˙-θ~3Tτ3-1θ^˙3-θ~4Tτ4-1θ^˙4=-kα~α~2+kZZ2+Zyγ~+θ~3Tτ3-1{τ3Z(ξ4α+ξ5)-θ^˙3}+θ~4Tτ4-1(τ4Zξ6-θ^˙4),
where the adaptive laws of θ^3, θ^4 are determined as
(29)θ^˙3=τ3Z(ξ4α+ξ5),θ^˙4=τ4Zξ6.
Substituting (29) in (28), we can acquire
(30)W˙2≤-min{kα~,kZ}|ψ2|2+|ψ2||yγ~|.
When |ψ2|≥|yγ~|/min{kα~,kZ}, W˙2≤0. Similarly, ∥ψ2∥a≤∥yγ~∥a/min{kα~,kZ} can be obtained, and W2 is input-to-state stable as well. Because yα~=θ^1Tξ1α~, there is
(31)∥yα~∥a=∥θ^1Tξ1α~∥a≤θ^1Tξ1∥ψ2∥a≤θ^1Tξ1min{kα~,kZ}∥yγ~∥a.
The interconnection formulation (20) is input-to-state stable according to small-gain theorem if we choose proper design parameters to make the following equation holds
(32)θ^3Tξ4min{kh~,kγ~}·θ^1Tξ1min{kα~,kZ}<1.Therefore the tracking errors and estimate errors of the altitude loop can converge to a small neighborhood of origin.


## 3.2. Velocity Controller

Since velocity is controlled byϕ directly, the adaptive dynamic inversion method is used. Let V~=V-Vref; the error dynamics of velocity is written as
(33)V~˙=Tcosα-Dm-gsinγ-V˙ref=q-(CT,ϕϕ+CT)cosα-q-SCDm-gsinγ-V˙ref.
For existence of uncertain parameters, the following vectors and repressors are defined
(34)θ5=[CD(α+Δτ1);CD(α+Δτ1)2;CDδe2;CDδe;CDδc2;CDδc;CDαδe;CDαδc;CD0;CTAd;CTα;CTM∞-2;CT0],θ6=[CT,ϕα;CT,ϕαM∞-2;CT,ϕM∞-2;CT,ϕ0],ξ7=q-[Sα;Sα2;Sδe2;Sδe;Sδc2;Sδc;Sαδe;Sαδc;S;-Adcosα;-αcosα;-M∞-2cosα;-cosα],ξ8=q-cosα[α;αM∞-2;M∞-2;1].
Consequently,
(35)V~˙=θ6Tξ8ϕ-θ5Tξ7m-gsinγ-V˙ref.
Then the control input ϕ is designed as
(36)ϕ=-mkv~V~+mV˙ref+mgsinγ+θ^5Tξ7θ^6Tξ8,
where kv~>0 is a design parameter.Determine the Lyapunov function candidate as(37)W3=12mV~2+12θ~5Tτ5-1θ~5+12θ~6Tτ6-1θ~6.
Its time derivative is
(38)W˙3=V~(θ6Tξ8ϕ-θ5Tξ7-mgsinγ-mV˙ref)-θ~5Tτ5-1θ^˙5-θ~6Tτ6-1θ^˙6=V~(θ~6Tξ8ϕ-mkv~V~-θ~5Tξ7)-θ~5Tτ5-1θ^˙5-θ~6Tτ6-1θ^˙6=-mkv~V~2-θ~5Tτ5-1(τ5V~ξ7+θ^˙5)+θ~6Tτ6-1(τ6ϕξ8-θ^˙6).
The adaptive laws of θ^5, θ^6 are obtained in the following:
(39)θ^˙5=-τ5V~ξ7,θ^˙6=τ6ϕξ8.
Thus (38) becomes W˙3=-mkv~V~2<0; that is, when t→∞, the tracking errors and estimate errors of the velocity subsystem can converge to zero finally.Therefore the accurate tracking performance and the stability of altitude and velocity can be guarranteed by the proposed method.


## 4. Numerical Simulation

The feasibility of the proposed method is verified based on a flexible model (1)–(3). The initial trim conditions are h=85000 ft, V=7846 ft/s, α=0.0174 rad, γ=0 rad, q=0 rad/s, η1=0.4588 ft·sulg, η2=-0.08726 ft·sulg, and η3=-0.03671 ft·sulg. Two cases are studied here.Case one is a climbing maneuver with longitudinal acceleration, and the expected equilibrium isα*=0.0219 rad. The increments of altitude and velocity are 2000 ft, and 100 ft/s respectively. Case two is a descending maneuver with velocity reducing gradually, and its corresponding equilibrium is α*=0.0158 rad. The decreasing of altitude is 1000 ft and that of velocity is 100 ft/s.For these two cases, The corresponding reference commands are generated by filtering step reference commands with a second-order profiler withω=0.1 rad/s and ξ=0.9.The simulation results of case one are shown in Figure2.Climbing maneuver with longitudinal acceleration for case one.
(a)
Altitude
(b)
Velocity
(c)
Attack angle
(d)
Flight-path angle
(e)
Flexible states
(f)
Fuel-to-air ratio
(g)
Elevator deflection
(h)
Canard deflectionThe simulation results of case two are shown in Figure3.Descending maneuver with velocity reducing gradually for case two.
(a)
Altitude
(b)
Velocity
(c)
Attack angle
(d)
Flight-path angle
(e)
Flexible states
(f)
Fuel-to-air ratio
(g)
Elevator deflection
(h)
Canard deflectionFrom Figures2(a), 2(b), 3(a), and 3(b), it is seen that the controller can provide stable and accurate tracking of the reference trajectories for the two cases, and the tracking errors of altitude and velocity remain remarkably small. Figures 2(c), 2(d), 3(c), and 3(d) show that both the signals of the flight-path angle and the attack angle can also follow the change of virtual control commands closely.For the flexible dynamics, its effect on aerodynamic model is neglected, and its motion is taken as external perturbation during the control design. That means that the forebody turn angleΔτ1 and the aftbody vertex angle Δτ2 are equal to zero in model (3) when the controller is designed. Simultaneously, the second-order equation about flexible states ηi and η˙i is not considered. From Figures 2(e) and 3(e), we can know that the flexible states can converge to stable states ultimately although the flexible dynamics is not taken into account directly. This denotes that our controller has the strong robustness, and it is suitable to control the flexible hypersonic vehicle.Moreover, the variation ranges of the control inputs that is, fuel-to-air ratio, elevator deflection, and canard deflection are bounded according to Figures2(f), 2(g), 2(h), 3(f), 3(g), and 3(h).In summary, the nonminimum phase behavior is suppressed successfully, and the excellent closed-loop behavior of air-breathing hypersonic vehicle can be obtained by the proposed controller for the cases of maneuver of altitude and velocity.


## 5. Conclusion

For the flexible air-breathing hypersonic vehicle with canard control surface, the controller is designed based on the nonlinear adaptive equivalent control strategy under interconnected structure. The equivalent control inputs are introduced and designed to replace the terms about elevator and canard in the flight-path angle dynamics and the pitch-rate dynamics for eliminating the nonminimum phase. The uncertain aerodynamic parameters are identified online by the adaptive method. And input-to-state stability of the interconnection subsystems is guaranteed by small-gain theorem. Similarly, the adaptive dynamic inversion approach is adopted in the velocity loop. With our approach, the stable and accurate tracking of the hypersonic vehicle model with nonminimum phase can be realized.



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*Source: 560785-2013-08-25.xml*