Adaptive Neural Control of Hypersonic Vehicles with Actuator Constraints

An adaptive neural control method is proposed in this paper for the flexible air-breathing hypersonic vehicle (AHV) with constraints on actuators. This scheme firstly converts the original control problem with input constraints into a new control problem without input constraints based on the control input saturation function. Secondly, on the basis of the implicit function theorem, the radial basis function neural network (RBFNN) is introduced to approximate the uncertain items of the model. And the minimal-learning-parameter (MLP) technique is adopted to design the adaptive law for the norm of network weight vector, which significantly reduces calculations. Meanwhile, the finite-time convergence differentiator (FD) is introduced, through which the model state variables and their derivatives are accurately estimated to ensure the control effect. Finally, it is theoretically proved that the closed-loop control system is stable. And the effectiveness of the designed controller is verified by simulation.


Introduction
Air-breathing hypersonic vehicle (AHV) is a new type of aircraft flying in the near space at a speed of more than Mach 5.It has outstanding advantages in terms of high speed, high maneuverability, and large flight envelope that traditional aircrafts do not have, which possesses potential applications in both civilian and military fields [1][2][3].But it also has more complex coupling, stronger nonlinearity, more severe elastic vibration, and tighter control input constraints than ordinary aircraft due to the flat and slender body design of the AHV and the integration of scramjet engine, which make the flight control of AHV a frontier issue in today's control field [4,5].
In the design process of the control law for AHV, by introducing a neural network to approximate the model's unknown dynamics or control laws that are difficult to be directly implemented, the nonlinear, strong coupling, and uncertainties of the model can be handled well to ensure the robust performance of the control law [6,7].For the rigid body model of AHV, the continuous and discrete adaptive neural controllers are designed, respectively, in [7,8] by expressing original model as a strict feedback form and introducing RBFNN to approximate the model's unknown function, which guarantee that the closed-loop signals are uniformly ultimately bounded but fail to consider the influence of flexible states.Different from references [7,8], a nonsingular direct neural control is proposed in [9] based on backstepping method by firstly designing the ideal backstepping control law for each subsystem of AHV and then using the RBFNN to approximate the designed ideal control law online instead of the unknown function of the model.However, the design process and form of the control law in [9] are cumbersome and complex, which restrict its applications in engineering.Two novel neural backstepping control strategies are proposed in [10,11], which are designed with improved backstepping methods, respectively.Simulations show that the proposed methods in [10,11] have strong robustness and superior control effects, but both of them assume that the model is affine for control input.
On the other hand, considering that AHV controls the height and attitude of longitudinal motion with elevators, as the flying height increases, the efficiency of the elevator will decrease significantly [12].In addition, the AHV can be affected by unknown airflows such as gust and turbulence during the flight process.Therefore, it is easy to encounter the phenomenon of elevator saturation when flying at high altitude.Once the actuators reach saturation, it may cause failure of the control system [13].Thence, it requires urgent development of antisaturation control studies for AHV.However, the related research in this respect is still relatively little.Only considering the throttle setting constraint of the engine, a neural control method based on time-scale decomposition is designed in [14].In [13], the neural network is used to approximate the saturation characteristics of the control law, which effectively solves the problem of control input constraint.However, the neural network weight parameters strongly depend on the model and are difficult to select.The tracking error is corrected through the designed auxiliary error compensation system in [15].Although the simulation result shows that it has certain feasibility in dealing with the problem of actuator constraints, it cannot theoretically guarantee that the tracking error is bounded.The method of [15] is further extended to AHV flight control where both control input and flight attitudes are constrained.However, in the actual project, AHV has no corresponding actuators to directly limit its flight attitude.
There are two shortcomings in the above studies.First, forcing AHV's nonaffine motion model into an affine model will inevitably result in the loss of certain key dynamic characteristics.The control law designed based on the simplified affine model will have the risk of partial or complete failure.The second is that the difficulty of the controller design is increased to some extent by introducing the auxiliary system or using the neural network to approach the actuator saturation characteristic when dealing with the actuator constraints of the AHV.
Based on the above shortcomings, it is imperative to design a simple and effective nonaffine control law to solve the control problem of AHV when the actuators are constrained.This paper studies the control problem of AHV with actuator constraints and proposes an adaptive neural control method.By designing the hyperbolic tangent input saturation function, the original control problem with input constraints is transformed into a control problem without input constraints.Different from the above references, the AHV model is regarded as a pure feedback system with nonaffine control input that is closer to the actual situation of AHV.Based on the implicit function theorem, the RBFNN is introduced to accurately approximate the unknown function of the model.The MLP technique is adopted to adaptively adjust the norm of the weight vector, which greatly reduces the amount of adaptive calculations.And through employing FD to achieve effective estimation of the system state variables, the control accuracy is ensured.Simulation examples verify the effectiveness and superiority of the design method.dynamics and weak coupling of AHV, establishes the following AHV longitudinal motion model [17]: where where velocity V, altitude h, flight-path γ, pitch angle θ, and pitch rate Q are the five rigid body states; m and I yy represent vehicle mass and moment of inertia, respectively; the four flexible modes η 1 , η 1 , η 2 , and η 2 denote the first two bending modes of the fuselage; ζ i and ω i i = 1, 2 represent the damping ratio and natural frequency for flexible modes, respectively; L f and L a represent the length of forward beam and aft beam; mf and ma are the mass distribution of forward beam and aft beam, respectively; and ϕ f ⋅ and ϕ a ⋅ are structural mode shapes [16].The force map of an AHV model is shown in Figure 1.The approximations of thrust T, drag D, lift L, pitching moment M, and the generalized forces N i i = 1, 2 are expressed as [17] T where α = θ − γ is the angle of attack; fuel equivalence ratio Φ and elevator angular deflection δ e are control inputs; c and S are the aerodynamic chord and reference area, respectively; q represents dynamic pressure; ρ is the air density at height h; z T means thrust moment arm; c e is the elevator coefficient; h 0 and ρ 0 represent nominal altitude and air density at the altitude h 0 , respectively; 1/h s is the air density decay rate; M,α i = 0, 1, 2 represents the ith order coefficient of α in M; N α i j i = 0, 1, 2 ; j = 1, 2 is the ith order contribution of α to N j ; and N δ e 2 is the contribution of δ e to N 2 ; β i h, q i = 1, 2, … , 8 is the ith trust fit parameter.For more detailed definitions of the model geometric parameters and coefficients, the reader could refer to [17].

Input Constraint Problem and Model Conversion.
According to [18] and combining (1), ( 2), (3), (4), (5), and (9), we know that velocity V is mainly controlled by equivalence ratio Φ since the thrust T is directly affected by Φ.On the other hand, altitude h is mainly controlled by elevator angular deflection δ e since δ e directly affects the pitch rate Q and then affects the pitch angle θ and flight-path γ, ultimately controlling the change of h.Therefore, we can firstly decompose the AHV model into a velocity subsystem (1) and altitude subsystem ((2), ( 3), (4), and ( 5)) and then design the control law separately [19].
Taking the actual situation into account, the executable ranges of Φ and δ e have certain limits, which are described as follows: where Φ min and Φ max ≥ 0 are the respective upper and lower bound of Φ; δ e min and δ e max δ emin = −δ e max stand for the upper and lower bound of δ e , respectively.In the above situation of actuator constraints, for velocity subsystem and altitude subsystem, the control objective is to design the respective control law Φ and δ e under the constraints of (10) such that V and h track their reference commands V ref and h ref .
Obviously, Φ and δ e have saturation characteristics under the constraints of (10).In order to convert the original control problem with input constraints into a new control problem without input constraints, here we use hyperbolic tangent function to approximate Φ and δ e where Φ * and δ * e ∈ R are the actual input of actuator saturation loop.By (11) and the character of hyperbolic tangent function, we can know that no matter what values Φ * and δ * e take, both Φ and δ e satisfy the constraint of (10).At this point, the original control objective can be converted to design unconstrained Φ * and δ * e such that V and h track their reference commands V ref and h ref .
Remark 1.In order to maintain the normal operating mode of scramjet engine, a restriction is imposed on the value range 3 International Journal of Aerospace Engineering of Φ, which is generally taken as Φ ∈ 0 05, 1 5 ; taking the physical limit of the deflection angle of elevator into account, usually δ e ∈ −20 ∘ , 20 ∘ [20].
Remark 2. Although the problem of actuator saturation is often considered when planning the trajectory, AHV can be affected by unknown airflows such as gusts and turbulences as well as sudden actuator failures during the flight process, which will cause the actuator to be saturated instantly such that the actual executable range of it will be even smaller than its theoretical value [13].

Adaptive Neural Controller Design
3.1.RBFNN Approximation.RBFNN has the characteristics of simple structure, strong learning, and fault tolerance and has the ability of globally approximating nonlinear continuous functions [21].It can be expressed as a mapping from input to output.
Here, h i X is chosen as the following Gaussian function: represents the center of the ith Gaussian function), which can be expressed as follows: For any unknown nonlinear continuous function F X , using the RBFNN and by selecting enough nodes (selecting a sufficiently large N), there must be an ideal weight vector where μ ∈ R is the approximation error and μ M ∈ R + stands for the upper bound of approximation error.When taking N large enough, μ M can be arbitrarily small.
Remark 3. Since exp ⋅ is a strictly monotonically increasing and positive function and Therefore, there must be a bounded constant h ∈ R + such that h X ≤ h.

Controller Design for Velocity Subsystem.
Based on the research conclusion of [7], here velocity subsystem is further expressed as a more general nonaffine form of control input Combined with (11), the above formula can be rewritten as where F V V, Φ * is a completely unknown continuously differentiable function.
In order to design the control law Φ * , the following theorem is given firstly.Theorem 1.For any V, Φ * ∈ Ω V × R, the following inequality is established: where Ω V is a controllable area.
Proof 1. From ( 11), (16), and ( 17), we can know According to [17], there is Therefore, Define velocity tracking error Taking time derivative along (22) and using (17) yield where k V ∈ R + is the design parameter. 4 International Journal of Aerospace Engineering Combine ( 23) and ( 24) Design the control law Φ * as where k V1 and k V2 ∈ R + stand for design parameters and Φ * 1 is the neural control law to be designed to counteract the influence of uncertain term F * V V, Φ * .In order to facilitate the subsequent design process, the implicit function theorem is introduced here [22].
Ψ ς 0 , σ 0 = 0 and the Jacobian matrix ∂Ψ/∂ς ς 0 , σ 0 is nonsingular.Then, for any σ ∈ G, a neighborhood U ⊂ R l of ς 0 and a neighborhood G ⊂ R r of σ 0 can make the equation Ψ ς, σ = 0 which has a unique solution ς ∈ U, and the solution can be expressed as ς = g 0 σ , where g 0 ⋅ is a continuously differentiable function on σ = σ 0 .Remark 4. Theorem 2 shows that once the implicit function Ψ ς, σ satisfies all conditions in the theorem, ς can be expressed as a continuously differentiable function of σ, that is, ς = g 0 σ .At this time, using RBFNN to approximate Ψ ς, σ , only σ, instead of ς, is used as the input signal of the neural network to obtain a satisfactory approximation effect.This is where the special meaning of the implicit function theorem lies.
To illustrate that Γ V, Φ * 0 , Φ * 1 satisfies the implicit function theorem, the following theorem is given.
Then there are a controllable area Ω V ⊂ R and a unique According to [23], a sufficient condition for the existence of Φ * 1 is that the following inequality holds: Consider ( 18), ( 24), (26), and ( 29), there are 18) and ( 33) According to Theorem 3 and (34), Γ V, Φ * 0 , Φ * 1 satisfies the implicit function theorem.Therefore, Φ * 1 can be regarded as a function of Φ * 0 and V, and further F * V V, Φ * can be regarded as a function of Φ * 0 and V. Define X 1 = V, Φ * 0 T ∈ R 2 as the input vector of RBFNN and introduce RBFNN to approximate where vector, N 1 is the number of nodes, and ε and ε M are approximation errors and its upper bounds, respectively.And is the activation function; here, h i X 1 is chosen as the Gaussian function.

International Journal of Aerospace Engineering
The following is based on the MLP technique to adaptively adjust the norm of the RBFNN weight vector.
Define ω V = W * V 2 and design Φ * 1 as where ω V is the estimation of ω V , and its adaptive law is designed as where μ V ∈ R + is the design parameter.Substitute ( 27) and ( 36) into (26) and finally get the control law Remark 5. Different from [24] in which the weight vector of the neural network is directly adjusted online, this paper regards W * V as a whole based on the MLP technique.Adaptive adjustment of ω V requires only one online learning parameter ω V , and the computational complexity of the approximation algorithm is significantly reduced.

Controller Design for Altitude Subsystem. Define altitude tracking error
Taking time derivative along (39) and using (2) yield The reference trajectory of γ is chosen as where k γ1 , and known by ( 40) and (41), the dynamic response of h is Performing Laplace transformation on both sides of (42) can get its characteristic equation The two characteristic roots −k γ1 − k γ1 2 − 4k γ2 /2 and −k γ1 + k γ1 2 − 4k γ2 /2 of (43) are negative real numbers, so the system (42) is stable and h is exponentially convergent.
Therefore, as long as γ → γ d , h can track h ref stably.In this way, the control objective of the altitude subsystem turns to ensure that γ tracks γ d .Define x 1 = γ, x 2 = θ, and x 3 = Q.Based on the research conclusion of [7], the remaining part of the altitude subsystem ((3), (4), and ( 5)) is further expressed as the following more general nonaffine form: Considering (11), the above formula can be rewritten as x 2 and f 3 x, δ * e are completely unknown continuously differentiable functions.Remark 6. Due to the strong coupling between the rigid body states and the flexible modes in the AHV model ((1), ( 2), ( 3), ( 4), ( 5), (6), and ( 7)), here the dynamic characteristics of strong nonlinearity and strong coupling in the original model are regarded as completely unknown continuous differentiable functions (( 16) and (44)) by referring to the method of [25].The proposed method in this paper will use RBFNN to accurately approximate these continuous differentiable functions and then complete the design of the control law and ensure the stability of the closed-loop control system.
In order to design the control law δ * e , the following theorem is given firstly.Theorem 4. For any x, δ * e ∈ Ω x × R, the following inequalities are established: where Ω x is a controllable area.
Proof 3.According to the results in [17], we can know In order to avoid the cumbersome and complicated design process of the traditional backstepping method in designing the control law, the following equivalent transformation is performed on the system (45).

51
where f h2 x, δ * e is a completely unknown continuously differentiable function.

53
where λ ∈ R + stands for the design parameter.Since s + λ 3 is a Hurwitz polynomial, when E is bounded, e must be bounded.
According to (51) and (53), we obtain Considering that z 2 and z 3 are unknown, we can know that z 2 = γ and z 3 = γ from the previous model transformation process.A new finite-time convergence differentiator (FD) will be used below to accurately estimate differential signal.By taking γ as the input signal of FD (take n = 4), we can get the estimated values of z 2 and z 3 , which are expressed as ẑ2 and ẑ3 , respectively.Similarly, the estimations of γ d , γ d , and ⃛ γ d can be obtained via taking γ d as the input signal of FD (take n = 4), which are expressed as γ d , γ d , and ⃛ γ d .Therefore, the estimations of the first three derivatives of e can be expressed as From (53) and (55), we obtain the estimation of E where k h ∈ R + is the design parameter.Combine (55), (56), and (57) where Consider ( 52), ( 59) and ( 61), there are Further, where T ∈ R N 2 is the weight vector, N 2 is the number of nodes, and ι and ι M stand for approximation errors and its upper bounds, respectively.
is the activation function which is chosen as the Gaussian function here. Define where ω h is the estimation of ω h , and its adaptive law is designed as where μ h ∈ R + is the design parameter.

Finite-Time Convergence Differentiator (FD)
Theorem 6.Consider the following FD [26]: where R, a i i = 1, 2, … , n ∈ R + stand for the design parameters.There are ϕ > 0 and ιϕ > n so that where O 1/R ιϕ−i+1 denotes the approximation of 1/R τϕ−i+1 order between ξ i and 8 International Journal of Aerospace Engineering Proof 5.The detailed proof process of Theorem 6 can be found in [26].
Remark 8.In (70), ξ i i = 1, 2, … , n is the state variable of the system, ξ 1 is the estimation of υ t , and ξ i i = 2, 3, … , n is the estimation of the i − 1th derivative of υ t .At the same time, (71) shows that the estimation error is a high-order infinitesimal of 1/R τϕ−i+1 .
Use the above FD to estimate z 2 , z 3 , γ d , γ d , and ⃛ γ d of (54)

Stability Analysis
Theorem 7.For the velocity subsystem of AHV (17), considering the saturation characteristic of Φ (11), and adopting the control law (26) and the adaptive law (37) under the premise of Theorem 1, the closed-loop control system is semiglobally uniformly ultimately bounded.Proof 6. Define the estimation error of ω Substitute ( 26), ( 27), (35), and (36) into ( 25) Choose the following Lyapunov function candidate: Taking time derivative of (75) and invoking (37), ( 73) and (74) yield Further, Then (76) becomes 9 International Journal of Aerospace Engineering Let k V1 > 1/4, and define the following compact sets: , Therefore, the closed-loop control system is semiglobally uniformly ultimately bounded.Further, these error signals V and ω V are semiglobally uniformly ultimately bounded and can be invariant to the following sets Ω V and Ω ω V .The radiuses of Ω V and Ω ω V can be made arbitrarily small by choosing k V1 that is big enough and μ V that is small enough, and the tracking errors V and ω V can also be arbitrarily small.Theorem 8.For the altitude subsystem of AHV (51), considering the saturation characteristic of δ e (11) and adopting the control law (59), the adaptive law (69) and FD (72) under the premise of Theorem 4, the closed-loop control system is semiglobally uniformly ultimately bounded.

Proof 7. Define the estimation error of ω
Then, define the estimation error of FD According to Theorem 6, there is Substitute (59), (67), and (68) into (58) Choose the following Lyapunov function candidate: Taking time derivative of (86) and invoke (69), (82), and (85) yield According to Cauchy-Schwarz inequality, Further, Then (87) becomes Let k h1 > 1/4, and define the following compact sets: then L h < 0. Thus, these error signals Ê and ω h are semiglobally uniformly ultimately bounded and can be invariant to the following sets Ω Ê and Ω ω h .The radiuses of Ω Ê and Ω ω h can be made arbitrarily 10 International Journal of Aerospace Engineering small by choosing big enough k h1 and small enough μ h , and the tracking errors Ê and ω h can also be arbitrarily small.Combine (53), ( 54), ( 55) and ( 56) and ( 83) Consider (84), then (93) can become Therefore, E and e are also bounded, then the closed-loop control system is semiglobally uniformly ultimately bounded.

Simulation Results
With AHV longitudinal motion model as the controlled object, the tracking simulation of velocity and altitude reference commands is performed.The initial velocity and altitude of AHV are taken as V = 2500 m/s and h = 27000 m.Velocity step and altitude step are chosen as ΔV = 200 m/s and Δh = 200 m.Both the velocity and altitude reference inputs are given by a second-order reference model with a damping ratio of 0.9 and a natural frequency of 0 1 rad/s.The design parameters of the controller are chosen as     Remark 9.In the above stability analysis of the control law, in order to ensure the stability of the closed-loop control system, the range of values of related parameters (such as k V1 > 1/4, k h1 > 1/4) is given.At the same time, in order to ensure tracking accuracy, the selection principle of related parameters is also analyzed (e.g., parameters k V1 and k h1 should be taken large enough, and parameters μ V and μ h should be taken small enough).The selection of parameters in the simulation is based on the above principles and is determined by repeated debugging.
Case 1. Considering Remark 1, we assume that actuators are constrained as Φ ∈ 0 05, 1 5 and δ e ∈ −20 ∘ , 20 ∘ .Simultaneously, to verify the robustness of the controller, it is    13 International Journal of Aerospace Engineering assumed that there is a perturbation of ±40% for the aerodynamic coefficient of the AHV model, which is expressed as C = C 0 1 + 0 4 sin 0 1πt , where C 0 represents the nominal value and C stands for the simulation value.And after 50 s, add external disturbances d 1 = 2 sin 0 1πt and d 2 = 0 02 sin 0 1πt to (1) and ( 5) in the AHV model, respectively.The subscript "1" in the figures shows the result obtained by the proposed method, and the subscript "2" shows the result obtained by the method of [27].
Case 2. Combining Remark 2, we further suppose that actuators are subject to stricter constraints, resulting in a smaller executable range than the theoretical value, which are taken as Φ ∈ 0 05, 1 2 and δ e ∈ −18 ∘ , 18 ∘ .Other simulation conditions are the same as Case 1.
Case 3. The control problem of nonlinear systems with actuator constraints has been considered extensively in the existing results, such as [28,29].In order to compare with the method in this paper, the adaptive dynamic surface control method considering actuator constraints proposed in [29] is applied to the AHV model.All conditions in the simulation are the same as in Case 2. The subscript "1" represents the result of the proposed method.The subscript "2" represents the result of the method proposed in [29].
When the control inputs are constrained as described in Case 1 (see Figures 2(h) and 2(i)), both the proposed method and method of [26] can guarantee that V and h track their reference commands V ref and h ref (see Figures 2(a) and 2(d)).However, the tracking accuracy and anti-interference ability of the proposed method in this paper are significantly improved than those of the method in [26] (see Figures 2(b) and 2(e)).At the same time, the changes in the flight-path angle and flexible states are also smoother than [26] (see Figures 2(c), 2(f), and 2(g)).When actuators of AHV are subject to stricter constraints (see Figures 3(h) and 3(i)), the stability of the closed-loop system cannot be guaranteed with the method of [26].But through using the proposed method in this paper, the control objective can still be achieved (see Figures 3(a), 3(b), 3(d), and 3(e)).And the proposed method can effectively inhibit flexible states (see Figures 3(c) and 3(f)).In Case 3, the method proposed in [29] can guarantee that V and h track their reference commands when the actuators are constrained (see Figures 4(a) and 4(d)).However, since the method does not use the relevant means to estimate the uncertain dynamics of the model, the tracking error is significantly larger than the method proposed in this paper (see Figures 4(b) and 4(e)).At the same time, compared with the method in [29], the control curve obtained by the method is smoother (see Figures 4(h) and 4(i)), which is more suitable for engineering applications.

Conclusion
(1) An adaptive neural control law based on input saturation function is designed in this paper, which guarantees the stability of the closed-loop system and the boundedness of tracking error when the actuator reaches saturation.
(2) When the AHV model encounters perturbation of aerodynamic coefficient and external disturbances, the proposed method can still ensure good tracking accuracy, which proves that the algorithm has strong robustness.
(3) The controller designed in this paper can still effectively suppress the flexible states when the actuator is constrained.Simulation examples verify the effectiveness and superiority of the design method.
(4) In this paper, only the most common amplitude saturation problem of the actuator is considered, and the rate and bandwidth constraint problem is not discussed.These will serve as a further research direction.

Figure 1 :
Figure 1: Force map of an AHV model.
30, and λ = 7.The design parameters of the adaptive law are taken as μ V = μ h = 0 05.The design parameters of FD are chosen as R 1 = R 2 = 0 05, a 11 = a 13 = a 21 = a 23 = 0 5, and a 12 = a 14 = a 22 = a 24 = 0 1.The node number of RBFNN is chosen as N 1 = N 2 = 20.In the velocity subsystem, the center c 1 of the Gaussian function is evenly spaced in [2500 m/s, The responses of flexible states η 1