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

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.

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 Section 2, 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,  𝜂̇₁,  𝜂₂,  𝜂̇₂,  𝜂̇₃,  and 𝜂̇₃. The equations of motion are written as 𝑉̇=𝑇cos𝛼−𝐷𝑚̇̇−𝑔sin𝛾,ℎ=𝑉sin𝛾,𝛾=𝑇sin𝛼+𝐿−𝑚𝑉𝑔cos𝛾𝑉,̇̇̇𝑀𝛼=𝑄−𝛾,𝑄=𝐼_𝑦𝑦,𝜂̈_𝑖=−2𝜍_𝑚𝜔_𝑚,𝑖𝜂̇_𝑖−𝜔²_𝑚,𝑖𝜂_𝑖+𝑁_𝑖;𝑖=1,2,3, (1) where 𝑚, 𝐼_𝑦𝑦, 𝑔 represent mass of the aircraft, moment of inertia, gravitational acceleration; damping ratio and natural frequency of the flexible motion are denoted by 𝜍_𝑚 and 𝜔_𝑚,𝑖, respectively; 𝑇, 𝐷, 𝐿, and 𝑁_𝑖 and 𝑀 are thrust, drag, lift, generalized forces and moment 𝐿=𝑞𝑆𝐶_𝐿,𝑇=𝑞𝐶_𝑇,𝜙𝜙+𝐶_𝑇,𝐷=𝑞𝑆𝐶_𝐷,𝑀=𝑧_𝑇𝑇+𝑞𝑆𝑐𝐶_𝑀,𝑁_𝑖=𝑞𝐶_𝑁𝑖. (2) The aerodynamic parameters in the above formulation are described as follows: 𝐶_𝐿=𝐶^𝛼_𝐿𝛼+𝐶^𝛿𝑒_𝐿𝛿_𝑒+𝐶^𝛿𝑐_𝐿𝛿_𝑐+𝐶⁰_𝐿+𝐶^Δ𝜏1_𝐿Δ𝜏₁+𝐶^Δ𝜏2_𝐿Δ𝜏₂,𝐶_𝑀=𝐶^𝛼_𝑀𝛼+𝐶^𝛿𝑒_𝑀𝛿_𝑒+𝐶_𝑀^𝛿𝑐𝛿_𝑐+𝐶⁰_𝑀+𝐶^Δ𝜏1_𝑀Δ𝜏₁+𝐶^Δ𝜏2_𝑀Δ𝜏₂,𝐶_𝑁𝑖=𝐶^𝛼_𝑁𝑖𝛼+𝐶^𝛿𝑒_𝑁𝑖𝛿_𝑒+𝐶^𝛿𝑐_𝑁𝑖𝛿_𝑐+𝐶⁰_𝑁𝑖+𝐶^Δ𝜏1_𝑁𝑖Δ𝜏₁+𝐶^Δ𝜏2_𝑁𝑖Δ𝜏₂,𝐶_𝐷=𝐶^{(𝛼+Δ𝜏1)2}_𝐷𝛼+Δ𝜏₁²+𝐶^{(𝛼+Δ𝜏1)}_𝐷𝛼+Δ𝜏₁+𝐶^Δ𝜏2_𝐷Δ𝜏₂+𝐶^𝛿2𝑒_𝐷𝛿²_𝑒+𝐶^𝛿𝑒_𝐷𝛿_𝑒+𝐶^𝛼𝛿𝑒_𝐷𝛼𝛿_𝑒+𝐶^𝛿𝑐2_𝐷𝛿²_𝑐+𝐶^𝛿𝑐_𝐷𝛿_𝑐+𝐶^𝛼𝛿𝑐_𝐷𝛼𝛿_𝑐+𝐶⁰_𝐷,𝐶_𝑇,𝜙=𝐶^𝛼_𝑇,𝜙𝛼+𝐶^{𝛼𝑀∞−2}_𝑇,𝜙𝛼𝑀_∞⁻²+𝐶^𝑀∞−2_𝑇,𝜙𝑀_∞⁻²+𝐶⁰_𝑇,𝜙+𝐶^𝛼Δ𝜏1_𝑇,𝜙𝛼Δ𝜏₁+𝐶^Δ𝜏21_𝑇,𝜙Δ𝜏²₁+𝐶^Δ𝜏1_𝑇,𝜙Δ𝜏₁,𝐶_𝑇=𝐶^𝛼_𝑇𝛼+𝐶^𝑀∞−2_𝑇𝑀_∞⁻²+𝐶^𝐴𝑑_𝑇𝐴_𝑑+𝐶^Δ𝜏1_𝑇Δ𝜏₁+𝐶⁰_𝑇, (3) where the control inputs are fuel-to-air ratio 𝜙, elevator deflection 𝛿_𝑒, and canard deflection 𝛿_𝑐;  𝑞,  𝑆,  𝑧_𝑇,  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 𝜂_𝑖.

In (3), the elevator-to-lift coupling orients from that 𝐶_𝐿 includes the term of 𝛿_𝑒, which leads to the nonminimum phase behavior. If 𝛿_𝑒 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 signals ℎ_ref and 𝑉_ref, 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, let ℎ=ℎ−ℎ_ref; then its error dynamics is written in the following: ̇̇ℎ=𝑉sin𝛾−ℎ_ref≈𝑉𝛾−ℎ̇_ref. (4) So the flight-path angle command 𝛾_𝑑 is designed into the following equation: 𝛾_𝑑=ℎ−𝑘ℎ+ℎ̇_ref𝑉, (5) where 𝑘ℎ>0 is the design parameter for ℎ.

Let 𝛾=𝛾−𝛾_𝑑; the error dynamic of flight-path angle is presented as follows: ̇𝛾=𝑇sin𝛼+𝐿−𝑚𝑉𝑔cos𝛾𝑉−𝛾̇_𝑑. (6) Here, the thrust is described as the function about the attack angle. Define 𝑇=∇𝑇𝛼+𝑇₀, where ∇𝑇=𝑞(𝐶^{𝛼𝑀∞−2}_𝑇,𝜙𝑀_∞⁻²𝜙+𝐶^𝛼_𝑇,𝜙𝜙+𝐶^𝛼_𝑇) and 𝑇₀=𝑞(𝐶^𝑀∞−2_𝑇,𝜙𝜙𝑀_∞⁻²+𝐶⁰_𝑇,𝜙𝜙+𝐶^𝑀∞−2_𝑇𝑀_∞⁻²+𝐶^𝐴𝑑_𝑇𝐴_𝑑+𝐶⁰_𝑇). 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 ̇𝛾=∇𝑇𝛼sin𝛼+𝑇₀sin𝛼+𝑞𝑆𝐶^𝛼_𝐿𝛼+𝑚𝑉𝐶𝑞𝑆^𝛿𝑒𝐿𝛿_𝑒+𝑚𝑉𝐶𝑞𝑆^𝛿𝐶𝐿𝛿_𝐶+𝑚𝑉𝑞𝑆𝐶⁰_𝐿−𝑚𝑔cos𝛾−𝑚𝑉𝛾̇_𝑑=𝑚𝑉∇𝑇sin𝛼^∗+∇𝑇𝛼^∗cos𝛼^∗+𝑇₀cos𝛼^∗+𝑞𝑆𝐶^𝛼_𝐿𝑚𝑉_𝐶1𝛼+𝐶𝑞𝑆^𝛿𝑒𝐿𝛿_𝑒+𝐶^𝛿𝑐_𝐿𝛿_𝑐𝑚𝑉_𝑈1+𝑞𝑆𝐶⁰_𝐿−𝑚𝑔sin𝛾−𝑚𝑉𝛾̇_𝑑+𝑇𝑚𝑉⁰sin𝛼^∗−∇𝑇𝛼^∗cos𝛼^∗+𝑇₀cos𝛼^∗𝛼^∗𝑚𝑉_𝛽1=𝐶₁𝛼+𝑈₁+𝛽₁, (7) where 𝐶₁ and 𝛽₁ are the terms containing the uncertain aerodynamic parameters. They can be expressed as the following equations: 𝐶₁=𝜃^𝑇₁𝜉₁,𝛽₁=𝜃^𝑇₁𝜉₂−𝑔cos𝛾𝑉−𝛾̇_𝑑,𝑈₁=𝜃^𝑇₂𝜉₃. (8)𝜃₁, 𝜃₂ are vectors of the uncertain parameters 𝜃₁=𝐶^𝛼_𝑇,𝜙;𝐶^𝛼_𝑇;𝐶^{𝛼𝑀∞−2}_𝑇,𝜙;𝐶^𝑀∞−2_𝑇,𝜙;𝐶⁰_𝑇,𝜙;𝐶^𝑀∞−2_𝑇;𝐶^𝐴𝑑_𝑇;𝐶⁰_𝑇;𝐶^𝛼_𝐿;𝐶⁰_𝐿,𝜃₂=𝐶^𝛿𝑒_𝐿;𝐶^𝛿𝑐_𝐿, (9) and 𝜉₁, 𝑖=1…3 are regressors 𝜉₁=𝑞𝑚𝑉sin𝛼^∗+𝛼^∗cos𝛼^∗𝜙;sin𝛼^∗+𝛼^∗cos𝛼^∗;sin𝛼^∗+𝛼^∗cos𝛼^∗𝑀_∞⁻²𝜙;cos𝛼^∗𝑀_∞⁻²𝜙;cos𝛼^∗𝜙;cos𝛼^∗𝑀_∞⁻²;cos𝛼^∗𝐴_𝑑;cos𝛼^∗,𝜉;𝑆;0₂=𝑞𝑚𝑉−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^∗𝐴_𝑑;sin𝛼^∗−𝛼^∗cos𝛼^∗,𝜉;0;𝑆₃=𝑞𝑆𝛿𝑚𝑉^𝑒;𝛿_𝑐. (10) Therefore the dynamics (7) is reformulated as ̇𝛾=𝜃^𝑇₁𝜉₁𝛼+𝜃^𝑇₂𝜉₃+𝜃^𝑇₁𝜉₂−𝑔cos𝛾𝑉−𝛾̇_𝑑. (11) Then the virtual command of the attack angle is chosen as 𝛼_𝑑=𝛼^∗−𝛾.

Let 𝛼=𝛼−𝛼_𝑑; the error dynamic of the attack angle is formulated as ̇𝛼=𝑄−𝛾̇−𝛼̇_𝑑=𝑄−𝛾̇_𝑑. (12) A new variable 𝑍 is defined as 𝑍=𝑄−𝛾̇_𝑑+𝑘𝛼𝛼, where  𝑘𝛼>0 is a design parameter for 𝛼. Then (12) is rewritten as ̇𝛼=𝑍−𝑘𝛼𝛼. (13) Using the equivalent control method, the time derivative of 𝑍 can be formulated with the new input 𝑈₂. It includes the pitch rate dynamics 𝑍̇=𝑧_𝑇𝑇+𝑞𝑆𝑐𝐶_𝑀𝐼_𝑦𝑦𝛼̇𝛼=+𝑘𝑞𝑆𝑐𝐶^𝛿𝑒_𝑀𝛿_𝑒+𝐶_𝑀^𝛿𝑐𝛿_𝑐𝑦𝐼_𝑦𝑈2+𝑧_𝑇∇𝑇+𝑞𝑆𝑐𝐶^𝛼_𝑀𝐼_{𝑦𝑦𝐶2}𝛼+𝑧_𝑇𝑇₀+𝑞𝑆𝑐𝐶⁰_𝑀𝐼_𝑦𝑦𝛼̇+𝑘𝛼−_𝑑𝛽2𝛾̈=𝐶₂𝛼+𝑈₂+𝛽₂, (14) 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 𝐶₂=𝜃^𝑇₃𝜉₄,𝛽₂=𝜃^𝑇₃𝜉₅𝛼̇+𝑘𝛼−𝛾̈_𝑑,𝑈₂=𝜃^𝑇₄𝜉₆, (15) where 𝜃₃=𝐶^𝛼_𝑇,𝜙;𝐶^𝛼_𝑇;𝐶^{𝛼𝑀∞−2}_𝑇,𝜙;𝐶^𝑀∞−2_𝑇,𝜙;𝐶⁰_𝑇,𝜙;𝐶^𝑀∞−2_𝑇;𝐶^𝐴𝑑_𝑇;𝐶⁰_𝑇;𝐶^𝛼_𝑀;𝐶⁰_𝑀,𝜃₄=𝐶^𝛿𝑒_𝑀;𝐶^𝛿𝑐_𝑀,𝜉₄=𝑞𝐼_𝑦𝑦𝑧_𝑇𝜙;𝑧_𝑇;𝑧_𝑇𝑀_∞⁻²𝜙;0;0;0;0;0;𝑆,𝜉𝑐;0₅=𝑞𝐼_𝑦𝑦0;0;0;𝑧_𝑇𝑀_∞⁻²𝜙;𝑧_𝑇𝜙;𝑧_𝑇𝑀_∞⁻²;𝑧_𝑇𝐴_𝑑;𝑧_𝑇;0;𝑆𝑐,𝜉₆=𝑞𝑆𝑐𝐼_𝑦𝑦𝛿_𝑒;𝛿_𝑐. (16) So (14) can be reformulated as 𝑍̇=𝜃^𝑇₃𝜉₄𝛼+𝜃^𝑇₄𝜉₆+𝜃^𝑇₃𝜉₅𝛼̇+𝑘𝛼−𝛾̈_𝑑. (17) Due to the uncertainty of the aerodynamic parameters, 𝜃_𝑖, 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 𝜃_𝑖, where 𝜃_𝑖=𝜃_𝑖−𝜃_𝑖, 𝑖=1,…,4.

Assumption 1. The aerodynamic parameters 𝜃_𝑖,  𝑖=1,…,4 are bounded; they lie in a compact convex set.

In order to guarantee tracking performance of hypersonic vehicles, the equivalent control inputs  𝑈₁ and  𝑈₂ are designed, respectively, by replacing the uncertain parameter vector 𝜃_𝑖 with its estimate vector and estimate error vector 𝑈₁=𝜃^𝑇₂𝜉₃=−𝜃^𝑇₁𝜉₁𝛼^∗−𝜃^𝑇₁𝜉₂−𝑔cos𝛾𝑉−𝛾̇_𝑑+𝜃^𝑇₁𝜉₁𝛾𝑈−𝑘𝛾−𝑉ℎ,₂=𝜃^𝑇₄𝜉₆=𝛾̈_𝑑−𝜃^𝑇₃𝜉₅𝛼̇+𝑘𝛼−𝜃^𝑇₃𝜉₄𝛼^∗−𝑘_𝑍𝑍−𝜃^𝑇₃𝜉₄+1𝛼, (18) where 𝑘𝛾>0,  𝑘_𝑍>0 are the design parameters for 𝛾 and 𝑍.

Let 𝛿=[𝛿_𝑒,𝛿_𝑐]. There is 𝐔=𝐵𝛿 according to (7) and (14). 𝐵 is a coefficient matrix and is equal to [(𝑞𝑆/𝑚𝑉)𝜃^𝑇₂;(𝑞𝑆𝑐/𝐼_𝑦𝑦)𝜃^𝑇₄]. The real inputs of the altitude loop can be obtained as follows: 𝛿_𝑒𝛿_𝑐=𝐵⁻¹𝑈₁𝑈₂. (19)

Combining (18), the state error dynamics about the altitude loop is transformed into the following equations: ̇ℎ̇𝛾𝛼+𝜃ℎ≈−𝑘ℎ+𝑉𝛾,𝛾=−𝑘𝛾−𝑉ℎ+𝑦^𝑇₁𝜉₁𝛼+𝜃^𝑇₁𝜉₂+𝜃^𝑇₂𝜉₃,̇𝛼̇𝛼=𝑍−𝑘𝛼,𝑍=−𝑘_𝑍𝑍−𝛼+𝑦𝛾+𝜃^𝑇₃𝜉₄𝛼+𝜃^𝑇₃𝜉₅+𝜃^𝑇₄𝜉₆, (20) where 𝑦𝛼=𝜃^𝑇₁𝜉₁𝛼, 𝑦𝛾=−𝜃^𝑇₃𝜉₄𝛾.

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 asymptotic 𝐿_∞ norm ‖⋅‖_𝑎 is given [19] ‖𝜆‖_𝑎∶=lim_𝑡→∞sup|𝜆|. (21) Then, define 𝜓₁=ℎ²+𝛾², and choose the Lyapunov function candidate of the altitude-flight-path angle subsystem as 𝑊₁=12ℎ²+𝛾²+12𝜃^𝑇₁𝜏₁⁻¹𝜃₁+12𝜃^𝑇₂𝜏₂⁻¹𝜃₂. (22) Its time derivative is 𝑊̇₁=ℎℎ̇+𝛾𝛾̇−𝜃^𝑇₁𝜏₁⁻¹𝜃̇₁−𝜃^𝑇₂𝜏₂⁻¹𝜃̇₂ℎ=−𝑘ℎ²−𝑘𝛾𝛾²+𝛾𝑦𝛼+𝜃^𝑇₁𝜏₁⁻¹𝜏₁𝛾𝜉₁𝛼+𝜉₂−̇𝜃₁+𝜃^𝑇₂𝜏₂⁻¹𝜏₂𝛾𝜉₃−̇𝜃₂. (23) The adaptive laws of 𝜃₁,  𝜃₂ are designed as ̇𝜃₁=𝜏₁𝛾𝜉₁𝛼+𝜉₂,̇𝜃₂=𝜏₂𝛾𝜉₃, (24) where 𝜏₁ and 𝜏₂ are the adaptive parameters.

By (24), (23) becomes 𝑊̇₁𝑘ℎ𝛾≤−min,𝑘𝜓₁²+𝜓₁𝑦𝛼. (25) As a consequence, it satisfies 𝑊₁ which is negative definite when |𝜓₁𝛼ℎ𝛾|≥|𝑦|/min{𝑘,𝑘}. Then 𝑊₁ is a input-to-state stable Lyapunov function. According to the lemma in [19], we know that ‖𝜓₁‖_𝑎≤‖𝑦𝛼‖_𝑎ℎ𝛾/min{𝑘,𝑘}. As 𝑦𝛾=−𝜃^𝑇₃𝜉₄𝛾, the following formulation is obtained: 𝑦𝛾_𝑎=−𝜃^𝑇₃𝜉₄𝛾_𝑎≤𝜃^𝑇₃𝜉₄𝜓_1𝑎≤𝜃^𝑇₃𝜉₄𝑘ℎ𝛾𝑦𝛼min,𝑘^𝑎. (26) For the attack angle-pitch rate subsystem, 𝜓₂=𝛼²+𝑍² is defined, and the following Lyapunov function candidate is chosen: 𝑊₂=12𝛼²+𝑍²+12𝜃^𝑇₃𝜏₃⁻¹𝜃₃+12𝜃^𝑇₄𝜏₄⁻¹𝜃₄. (27) Its time derivative is 𝑊̇₂=𝛼𝛼̇+𝑍𝑍̇−𝜃^𝑇₃𝜏₃⁻¹𝜃̇₃−𝜃^𝑇₄𝜏₄⁻¹𝜃̇₄=−𝑘𝛼𝛼²+𝑘_𝑍𝑍²+𝑍𝑦𝛾+𝜃^𝑇₃𝜏₃⁻¹𝜏₃𝑍𝜉₄𝛼+𝜉₅−̇𝜃₃+𝜃^𝑇₄𝜏₄⁻¹𝜏₄𝑍𝜉₆−̇𝜃₄, (28) where the adaptive laws of 𝜃₃, 𝜃₄ are determined as ̇𝜃₃=𝜏₃𝑍𝜉₄𝛼+𝜉₅,̇𝜃₄=𝜏₄𝑍𝜉₆. (29) Substituting (29) in (28), we can acquire 𝑊̇₂𝑘≤−min𝛼,𝑘_𝑍𝜓₂²+𝜓₂𝑦𝛾. (30) When |𝜓₂|≥|𝑦𝛾|/min{𝑘𝛼,𝑘_𝑍}, 𝑊̇₂≤0. Similarly, ‖𝜓₂‖_𝑎≤‖𝑦𝛾‖_𝑎/min{𝑘𝛼,𝑘_𝑍} can be obtained, and 𝑊₂ is input-to-state stable as well. Because 𝑦𝛼=𝜃^𝑇₁𝜉₁𝛼, there is 𝑦𝛼_𝑎=𝜃^𝑇₁𝜉₁𝛼_𝑎≤𝜃^𝑇₁𝜉₁𝜓_2𝑎≤𝜃^𝑇₁𝜉₁min𝑘𝛼,𝑘_𝑍𝑦𝛾_𝑎. (31) 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 𝜃^𝑇₃𝜉₄𝑘ℎ𝛾⋅𝜃min,𝑘^𝑇1𝜉₁min𝑘𝛼,𝑘_𝑍<1. (32)

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 by 𝜙 directly, the adaptive dynamic inversion method is used. Let 𝑉=𝑉−𝑉_ref; the error dynamics of velocity is written as ̇𝑉=𝑇cos𝛼−𝐷𝑚−𝑔sin𝛾−𝑉̇_ref=𝑞𝐶_𝑇,𝜙𝜙+𝐶_𝑇cos𝛼−𝑞𝑆𝐶_𝐷𝑚−𝑔sin𝛾−𝑉̇_ref. (33) For existence of uncertain parameters, the following vectors and repressors are defined 𝜃₅=𝐶^{(𝛼+Δ𝜏1)}_𝐷;𝐶^{(𝛼+Δ𝜏1)2}_𝐷;𝐶^𝛿2𝑒_𝐷;𝐶^𝛿𝑒_𝐷;𝐶^𝛿𝑐2_𝐷;𝐶^𝛿𝑐_𝐷;𝐶^𝛼𝛿𝑒_𝐷;𝐶^𝛼𝛿𝑐_𝐷;𝐶⁰_𝐷;𝐶^𝐴𝑑_𝑇;𝐶^𝛼_𝑇;𝐶^𝑀∞−2_𝑇;𝐶⁰_𝑇,𝜃₆=𝐶^𝛼_𝑇,𝜙;𝐶^{𝛼𝑀∞−2}_𝑇,𝜙;𝐶^𝑀∞−2_𝑇,𝜙;𝐶⁰_𝑇,𝜙,𝜉₇=𝑞𝑆𝛼;𝑆𝛼²;𝑆𝛿²_𝑒;𝑆𝛿_𝑒;𝑆𝛿²_𝑐;𝑆𝛿_𝑐;𝑆𝛼𝛿_𝑒;𝑆𝛼𝛿_𝑐;𝑆;−𝐴_𝑑cos𝛼;−𝛼cos𝛼;−𝑀_∞⁻²,𝜉cos𝛼;−cos𝛼₈=𝑞cos𝛼𝛼;𝛼𝑀_∞⁻²;𝑀_∞⁻²;1. (34) Consequently, ̇𝜃𝑉=^𝑇6𝜉₈𝜙−𝜃^𝑇₅𝜉₇𝑚−𝑔sin𝛾−𝑉̇_ref. (35) Then the control input 𝜙 is designed as 𝜙=−𝑚𝑘𝑣𝑉̃+𝑚𝑉̇_ref+𝑚𝑔sin𝛾+𝜃^𝑇₅𝜉₇𝜃^𝑇₆𝜉₈, (36) where 𝑘𝑣̃>0 is a design parameter.

Determine the Lyapunov function candidate as 𝑊₃=12𝑚𝑉²+12𝜃^𝑇₅𝜏₅⁻¹𝜃₅+12𝜃^𝑇₆𝜏₆⁻¹𝜃₆. (37) Its time derivative is 𝑊̇₃=𝑉𝜃^𝑇₆𝜉₈𝜙−𝜃^𝑇₅𝜉₇−𝑚𝑔sin𝛾−𝑚𝑉̇_ref−𝜃^𝑇₅𝜏₅⁻¹̇𝜃₅−𝜃₆^𝑇𝜏₆⁻¹̇𝜃₆=𝑉𝜃^𝑇₆𝜉₈𝜙−𝑚𝑘𝑣𝑉̃−𝜃^𝑇₅𝜉₇−𝜃^𝑇₅𝜏₅⁻¹̇𝜃₅−𝜃^𝑇₆𝜏₆⁻¹̇𝜃₆=−𝑚𝑘𝑣𝑉̃²−𝜃^𝑇₅𝜏₅⁻¹𝜏₅𝑉𝜉₇+̇𝜃₅+𝜃^𝑇₆𝜏₆⁻¹𝜏₆𝜙𝜉₈−̇𝜃₆. (38) The adaptive laws of 𝜃₅, 𝜃₆ are obtained in the following: ̇𝜃₅=−𝜏₅𝑉𝜉₇,̇𝜃₆=𝜏₆𝜙𝜉₈. (39) Thus (38) becomes 𝑊̇₃=−𝑚𝑘𝑣𝑉̃²<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 is  𝛼^∗=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 with 𝜔=0.1 rad/s and 𝜉=0.9.

The simulation results of case one are shown in Figure 2.

(a) Altitude

(b) Velocity

(c) Attack angle

(d) Flight-path angle

(e) Flexible states

(f) Fuel-to-air ratio

(g) Elevator deflection

(h) Canard deflection

Figure 2 

Climbing maneuver with longitudinal acceleration for case one.

The simulation results of case two are shown in Figure 3.

(a) Altitude

(b) Velocity

(c) Attack angle

(d) Flight-path angle

(e) Flexible states

(f) Fuel-to-air ratio

(g) Elevator deflection

(h) Canard deflection

Figure 3 

Descending maneuver with velocity reducing gradually for case two.

From Figures 2(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 angle Δ𝜏₁ 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 𝜂_𝑖 and 𝜂̇_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 Figures 2(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.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant 61074064 and Natural Science Foundation of Tianjin 12JCZDJC30300. The authors are grateful to the anonymous reviewers for their helpful comments and constructive suggestions with regard to this paper.