A sliding mode control method for hypersonic aircraft

Abstract

This invention relates to a sliding mode control method for hypersonic vehicles, which may include the following steps: S1. Transforming the nonlinear dynamic model of the hypersonic vehicle into a state-dependent linear model; S2. Designing an extreme learning machine adaptive neural network disturbance observer with a single hidden layer feedforward network to approximate the disturbances and parameter uncertainties experienced by the system; and S3. Designing a sliding mode control law based on a power function reaching law to suppress chattering in sliding mode control. This method is easy to implement, does not exhibit chattering, and can achieve zero steady-state error tracking of the output reference signal of the hypersonic vehicle.

Description

Technical Field

[0001] This invention relates to the field of automatic control, and more specifically to a sliding mode control method for hypersonic aircraft. Background Technology

[0002] Research on air-breathing hypersonic vehicles (AHVs) has garnered significant attention due to their numerous potential applications in both civilian and military fields. During flight, the aerodynamic parameters of HVs change dramatically, and the flight environment is constantly evolving, resulting in mathematical models characterized by complexity, parameter uncertainty, and nonlinearity. Furthermore, the unique aerodynamic structure of HVs leads to strong interactions between aerodynamics, propulsion systems, and structural dynamics, making them highly sensitive to uncertainties. Therefore, robustness of the control system is crucial in the controller design for HVs.

[0003] As a nonlinear robust control method, sliding mode control (SMC) can effectively compensate for matching uncertainties, including unmodeled dynamics, parameter uncertainties, and external disturbances. In recent years, various SMC strategies have been applied to the control of hypersonic vehicles (HVs). These methods can achieve robust control of HVs under matched disturbances. However, low-order SMCs in HVs suffer from chattering caused by high-frequency control switching. Although robust tracking performance has been improved by introducing adaptive laws and handling chattering through high-order SMCs, these methods are complex and difficult to implement. Therefore, designing a chatter-free and easily implemented SMC for HVs is a work with practical application value.

[0004] Furthermore, the disturbances experienced by HV controllers include mismatched disturbances that sliding mode control cannot handle. While some researchers have proposed combining backstepping with sliding mode control to address mismatched uncertainties, this requires tedious analysis and calculation of the time derivatives of the virtual controller. Recently, control based on nonlinear disturbance observers (NDOs) has offered a promising approach to handling mismatched disturbances. Based on dynamic inversion control, researchers have proposed a sliding mode disturbance observer to handle mismatched disturbances and parameter uncertainties. In addition, neural approximation has proven to be a powerful tool for improving the uncertainty attenuation capability of HV controllers. Although this method achieves good control performance, it still relies on the backstepping approach, resulting in a complex design process. Considering external disturbances and parameter uncertainties, researchers have proposed an ANNDO to suppress disturbances. This method achieves good tracking performance, but the derivation of the adaptive laws for neural network input weighting and output weighting is overly complex. Summary of the Invention

[0005] This invention aims to provide a sliding mode control method for hypersonic vehicles to solve the aforementioned problems. The specific technical solution adopted by this invention is as follows:

[0006] A sliding mode control method for a hypersonic vehicle may include the following steps:

[0007] S1. Transform the nonlinear dynamic model of the hypersonic vehicle into a state-dependent linear model;

[0008] S2. Design an extreme learning machine adaptive neural network perturbation observer with a single hidden layer feedforward network to approximate the perturbations and parameter uncertainties experienced by the system; and

[0009] S3. Design a sliding mode control law based on the power function approach law to suppress chattering in sliding mode control.

[0010] The present invention adopts the above technical solution and has the following beneficial effects: the method is easy to implement, does not produce jitter, and can achieve zero steady-state error tracking of the output reference signal of hypersonic aircraft. Attached Figure Description

[0011] To further illustrate the various embodiments, the present invention provides accompanying drawings. These drawings are part of the disclosure of the present invention, primarily used to illustrate the embodiments and to explain the operating principles of the embodiments in conjunction with the relevant descriptions in the specification. With reference to these drawings, those skilled in the art should be able to understand other possible implementations and the advantages of the present invention. Components in the drawings are not drawn to scale, and similar component symbols are generally used to represent similar components.

[0012] Figure 1 This is a flowchart of a sliding mode control method for a hypersonic aircraft according to the present invention;

[0013] Figure 2 These are the output curves and angle-of-attack response curves under positive parameter uncertainty, where (a), (b), and (c) show the altitude tracking curve, speed tracking curve, and angle-of-attack response curve, respectively.

[0014] Figure 3 These are curves of the control signals under positive parameter uncertainty; where (a) and (b) show the control signals for the throttle and elevator, respectively.

[0015] Figure 4 These are the output curves and angle-of-attack response curves under negative parameter uncertainty, where (a), (b), and (c) show the height tracking curve, speed tracking curve, and angle-of-attack response curve, respectively.

[0016] Figure 5 These are curves of the control signals under negative parameter uncertainty; where (a) and (b) show the control signals for the throttle and elevator, respectively.

[0017] Figure 6These are the output curves and angle-of-attack response curves under external disturbances, where (a), (b), and (c) show the altitude tracking curve, speed tracking curve, and angle-of-attack response curve, respectively.

[0018] Figure 7 These are curves of the control signals under external interference; where (a) and (b) show the control signals for the throttle and elevator, respectively.

[0019] Figure 8 The graphs show the actual interference and the interference estimate, where (a)-(e) show the velocity loop, track angle loop, altitude loop, angle of attack loop and pitch rate loop, respectively. Detailed Implementation

[0020] The present invention will now be further described in conjunction with the accompanying drawings and specific embodiments.

[0021] like Figure 1 As shown, a sliding mode control method for a hypersonic vehicle may include the following steps:

[0022] S1. Transform the nonlinear dynamic model of the hypersonic vehicle into a state-dependent linear model.

[0023] Consider the longitudinal dynamic model of the hypersonic vehicle developed by NASA Langley Research Center

[0024]

[0025] in,

[0026]

[0027] Aerodynamic coefficients are related to flight conditions. This application considers nominal cruise conditions: V = 15060 ft / s, h = 110000 ft, γ = 0 rad, q = 0 rad / s. Parameter uncertainties are modeled as additive disturbances to the nominal values, and their calculation formulas are as follows:

[0028]

[0029] Where m0, I0, S0, μ0, ρ0, R0 represent the nominal values ​​of the parameters, Δm, ΔI, ΔS, Δρ,Δc e ,ΔC Mα The relevant parameters are uncertain.

[0030] Considering external disturbances and parameter uncertainties, the nonlinear model (1) can be written as

[0031]

[0032] In the formula, u=[β,δ e ] T x = [V, γ, h, α, q] T ,y=[V,h] T ,

[0033] d = [d1, d2, d3, d4, d5] T Let Δf represent the external disturbance, and Δf represent the uncertainty caused by the disturbance of physical and aerodynamic parameters. Treating the external disturbance and parameter uncertainty as a single, unified disturbance, the nonlinear model (1) can be expressed as follows:

[0034]

[0035] Where, d s =Δf+d represents the overall disturbance. System (5) can be converted into a state-space model with the following state correlation coefficients.

[0036]

[0037] in,

[0038]

[0039] Input matrix here, It refers to dynamic pressure. When simplifying A(x) and B(x), we assume sinγ≈γ and β<1. This assumption is reasonable because under balanced cruise conditions, γ is close to 0 and the throttle opening β is less than 1.

[0040] S2. Design an extreme learning machine adaptive neural network perturbation observer with a single hidden layer feedforward network to approximate the perturbations and parameter uncertainties of the system.

[0041] The Extreme Learning Machine (ELM) is a single-hidden-layer feedforward network. Because its input weighting and hidden layer parameters do not require adjustment, its learning speed is much faster than traditional feedforward neural networks. The output of a single-hidden-layer feedforward neural network is...

[0042]

[0043] Where, β i It is the weighted sum from the i-th hidden node to the output, g(x,ω) i ,b i ) is the activation function of the i-th hidden node, ω i and b i It is the parameter of the activation function.

[0044] There are two types of activation functions. For additive hidden nodes, the sigmoid function is generally used as the activation function.

[0045]

[0046] Where, ω i and b i These are the input weights and biases for the i-th hidden node, respectively.

[0047] For RBF hidden nodes, a Gaussian function is generally used as the activation function.

[0048]

[0049] Where, ω i and b i These are the center and influence factor of the i-th RBF node, respectively.

[0050] Next, N sampling points are used to train a single-hidden-layer feedforward neural network. If a single-hidden-layer feedforward network with M hidden nodes can approximate these N sampling points with zero error, then there exists a β... i ,ω i ,b i , making

[0051]

[0052] Equation (10) can be written in the following form

[0053] Hβ=Y(11)

[0054] in,

[0055]

[0056] The ELM learning algorithm randomly generates fixed parameters for the activation function, requiring only the calculation of the output layer's weight coefficients. Training an SLFN is simply equivalent to finding the least-squares solution to system (11), which can be obtained through the following equation.

[0057]

[0058] in, It is the Moore-Penrose generalized inverse of matrix H(x,ω,b).

[0059] To estimate the perturbation of system (6), the following ELM-based adaptive neural network perturbation observer was designed.

[0060]

[0061] Where z represents the state of the disturbance observer, and λ>0 is the designed gain coefficient. Here, an SLFN with parameters determined based on ELM is used to approximate the disturbance. The disturbance estimate is then:

[0062]

[0063] Among them, e d =xz is the perturbation observation error, which serves as the input to the neural network, ω = [ω1, ω2, ..., ω L ] T ∈R L ×n (L is the number of hidden layer nodes, n is the dimension of the input state), b = [b1, b2, ..., b L ]∈R L×1 These are the hidden node parameters, which are randomly generated and then fixed. β∈R n×L This represents the weights of the output layer, which will be calculated using the adaptive law. h∈R L×1 This represents the output of the hidden layer. Assume β... i It is the i-th row vector of β, and the activation function is h(ω). T e d +b)=[h1,h2,…,h L ] T h i =1 / (1+exp(-(ω) i e d +b i Training a SLFN with ELM is equivalent to finding the output weights β. * The least squares solution, thus

[0064]

[0065] Wherein, ε(e d This is the approximation error. An error will occur if the number of hidden layer nodes is much smaller than the number of training samples, but it is bounded. That is:

[0066] |ε(e d )|≤ε N (16)

[0067] From formulas (14) and (15), we can obtain...

[0068] From formulas (6) and (13) we obtain

[0069]

[0070] in, This represents the disturbance estimation error.

[0071] The stable adaptive law of β is derived using Lyapunov's second method.

[0072] Consider the following Lyapunov function:

[0073]

[0074] Where, η i It is a positive constant and is called the learning rate of SLFN.

[0075] Differentiation of Lyapunov functions

[0076]

[0077] The second term of formula (19) equals 0 if we make the following equation true

[0078]

[0079] Therefore, there is

[0080]

[0081] Then, we get

[0082]

[0083] The gain λ can be designed to be large enough to satisfy the condition λ > ε N Then the Lyapunov function satisfies the following condition: This proves that the dynamic system (13) under interference observation is stable.

[0084] S3. Design a sliding mode control law based on the power function approach law to suppress chattering in sliding mode control.

[0085] S31. Establishment of the translational state equation

[0086] First, by moving the system to the setpoint x s ,u s To make the output y follow the reference command y r When the system reaches steady state, equation (23) is satisfied.

[0087]

[0088] For a hypersonic vehicle (4), the number of inputs equals the number of outputs, and the steady-state value can be calculated using the following formula.

[0089]

[0090] For formula (24), matrix A(x) s ),B(x s ) and steady state x s Regarding this, we use A(x) and B(x) to approximate A(x). s),B(x s And the lumped disturbance d s Since it is unknown, perturbation estimation is used. Instead of that, the translation setpoint can be calculated using the following formula.

[0091]

[0092] Define w = xx s v=uu s Then, from formulas (6) and (25), we get...

[0093]

[0094] in, This is the disturbance estimation error, which is ignored when designing the sliding mode control law.

[0095] If the disturbance is estimated If it is accurate enough, then the perturbation estimation error The error is very small and can be ignored. This application uses an ELM-based neural network observer to estimate the disturbance, which can obtain a sufficiently accurate estimate. Therefore, it is entirely reasonable to ignore the estimation error when designing the controller.

[0096] S32. Design the sliding surface

[0097] Ignore the disturbance estimation error in system (26) The following system can be obtained.

[0098]

[0099] Perform a nonsingular transformation on system (27), i.e.:

[0100]

[0101] Then system (27) is transformed into the following controllable standard form.

[0102]

[0103] in,

[0104] Equation (29) can be written in the following form

[0105]

[0106] Design a linear sliding surface

[0107]

[0108] Then on the sliding surface, the following conditions are met:

[0109] s=0(32)

[0110] If σ² is invertible, then

[0111]

[0112] Therefore, the first equation of the system state-space model (30) is:

[0113]

[0114] In equation (34), by designing σ so that the pole of Ω is on the left half of the s plane, the sliding surface (31) is a stable sliding surface.

[0115] S33. Sliding Mode Controller Design

[0116] To suppress chattering in sliding mode control, a power-law-based reaching law is selected:

[0117]

[0118] Where q>0, ε>0, and fal(s,α,δ) is a power function, its expression is:

[0119]

[0120] Where 0 < α < 1, 0 < δ < 1.

[0121] The derivation is obtained from equations (27)(28)(28) and (31)(31).

[0122]

[0123] Furthermore, the sliding mode control law is derived from (35)(35) and (37)(37) as follows:

[0124] v=(σTB(x)) -1 (-qs-εfal(s,α,δ)-σTA(x)w)(38)

[0125] Therefore, the control law applied to hypersonic vehicles is:

[0126] u = v + u s (39)

[0127] As can be seen from equations (25), (38), and (39), we do not need to design a compensation strategy; instead, we directly compensate the control law through the setpoint. Therefore, we call this Direct Feedback Compensation (DFC).

[0128] In summary, the control law design process is summarized as follows:

[0129] 1) Design an ELM-based neural network perturbation observer as shown in equation (13), where the perturbation estimation is calculated by equation (14), and the output weighting coefficient of the neural network is updated by the adaptive law to equation (21).

[0130] 2) Calculate the translation setpoint according to formula (25).

[0131] 3) Design discrete sliding mode control law (38) based on model (27).

[0132] 4) Then, the control law applied to the actual hypersonic nonlinear model (4) is shown in equation (39).

[0133] Simulation Results and Analysis

[0134] To demonstrate the effectiveness of the proposed method, simulations were conducted under the conditions of unknown external disturbances as shown in model (1) and parameter uncertainties as described in (3). In the simulations, the initial velocity and altitude were given as 15,060 ft / s and 110,000 ft, respectively, and the reference step signals for the velocity and altitude were given from time 0 as V. r =100ft / s,h r =100ft. To demonstrate the superiority of the proposed ELM-based NNDO, a nonlinear disturbance observer combined with sliding mode control method was used for comparison in the simulation.

[0135] The parameters of the sliding mode controller are selected as follows: The parameters of the power function are α = 0.5 and δ = 0.1. The coefficients of the ELM-based NNDO are selected according to the system characteristics. The number of nodes in the input layer is n = 5, the number of nodes in the hidden layer is L = 11, the number of nodes in the output layer is equal to the number of nodes in the input layer, and the observer gain is set to λ = 50I. The learning rate of SLFN is η1 = 500, η2 = η4 = η5 = 1000, and η3 = 2000. The simulation step size is set to 0.01, and the fourth-order Runge-Kutta method is used to simulate the nonlinear dynamics (1) of the hypersonic vehicle.

[0136] Simulation under conditions of parameter uncertainty

[0137] To test the robustness of the proposed method, the parameter uncertainty mentioned above was considered. First, the maximum positive value was considered, i.e. The comparison curves of sliding mode control based on ELM-NNDO and sliding mode control based on nonlinear disturbance observer are shown below. Figure 2 and 3 As shown. From Figure 2 As can be seen from a and 2b, the control effects of the two methods are basically equivalent; both speed and altitude can accurately track the reference signal. Figure 3It can be seen that there is no chattering phenomenon in the control laws, and the required rudder deflection amplitude is small when using the method of this application.

[0138] Then, the uncertainty parameter is set to its negative maximum value, i.e., Δ = -0.25. The comparison curves are as follows: Figures 4-5 As shown in the diagram. In this case, the control effects of the two methods are basically the same, both achieving accurate tracking of the reference signal. Furthermore, this method requires a smaller rudder deflection amplitude.

[0139] Simulation under continuous external disturbance

[0140] In the simulation, we consider the unknown persistent external disturbances d1 = -5, d3 = 10 when t ≥ 20 and d2 = 0.001, d4 = 0.05, d5 = 0.08 when t ≥ 30. The comparison curves of the two methods are shown below. Figures 5-7 As shown. From Figure 5 As can be seen from (a) and (b), both methods can accurately track the reference signal for both velocity and altitude. Furthermore, when subjected to external disturbances, this method converges to the given value faster in the velocity channel and exhibits less overshoot in the altitude channel. Figure 7 It can be seen that both observers can accurately estimate the actual interference signal. The magnified image also shows that the method converges faster.

[0141] In summary, this application proposes a novel SMC control scheme combined with an ELM-based NNDO to achieve disturbance suppression control of the high-frequency (HV) signal. The scheme employs an SMC based on a power-law approach and proposes an ELM-based NNDO to estimate the disturbance signal. The hidden node parameters of the SLFN are randomly assigned, and the weights of the output layer are updated using an adaptive law derived from Lyapunov's stability law. Through direct feedback compensation, the proposed combined control strategy achieves zero steady-state error tracking of the output reference signal.

[0142] Although the invention has been specifically shown and described in conjunction with preferred embodiments, those skilled in the art should understand that various changes in form and detail may be made to the invention without departing from the spirit and scope of the invention as defined in the appended claims, all of which shall be within the scope of protection of the invention.

Claims

1. A sliding mode control method for a hypersonic aircraft, characterized in that, Includes the following steps: S1. Transform the nonlinear dynamic model of the hypersonic vehicle into a state-dependent linear model; S2. Design an extreme learning machine adaptive neural network perturbation observer with a single hidden layer feedforward network to approximate the perturbations and parameter uncertainties experienced by the system; and S3. Design a sliding mode control law based on the power function approaching law to suppress chattering in sliding mode control; The specific process of S1 is as follows: S11. Considering external disturbances and parameter uncertainties, the nonlinear model of the hypersonic vehicle is written as follows: y = Cx, The nonlinear model is the longitudinal dynamic model of a hypersonic vehicle developed by NASA's Langley Research Center, u=[β,δ e ] T x = [V,γ,h,α,q] T y = [V, h] T , d = [d1, d2, d3, d4, d5] T Δf represents the uncertainty caused by external disturbances and disturbances in physical and aerodynamic parameters; S12. Treating external disturbances and parameter uncertainties as a single, unified disturbance, the nonlinear model can be expressed as: y = Cx, Where, d s =Δf+d represents the overall disturbance; S13. The system is converted into a state-space model with the following state correlation coefficients: y = Cx, in, Input matrix here, It is dynamic pressure. When simplifying A(x) and B(x), assume sinγ≈γ and β<1; The specific process of S2 is as follows: S21. To estimate the system perturbation, the following adaptive neural network perturbation observer based on extreme learning machine was designed: Where z is the state of the interference observer, and λ>0 is the designed gain coefficient; S22. The perturbation is approximated using a feedforward neural network with parameters determined by an extreme learning machine. The perturbation estimate is: Among them, e d =xz is the perturbation observation error, which serves as the input to the neural network, ω = [ω1, ω2, ..., ω L ] T ∈R L×n L is the number of hidden layer nodes, and n is the dimension of the input state; b = [b1, b2, ..., bn] L ]∈R L×1 These are the parameters of the hidden layer nodes; β∈R n ×L Represents the weights of the output layer; h∈R L×1 This represents the output of the hidden layer; S23.ω=[ω1,ω2,…,ω L ] T ∈R L×n and b = [b1, b2, ..., b L ]∈R L×1 The output weights are randomly set. β∈R n×L The adaptive law update is derived from Lyapunov's stability law; The specific process of S3 is as follows: S31. Establish the translational state equations, specifically, by moving the system to the setpoint x. s ,u s To make the output y follow the reference command y r When the system reaches steady state, the following equations are satisfied: A(x s )x s +B(x s )u s +d s =0 Cx s =y r , For hypersonic vehicles, the number of inputs equals the number of outputs, and the steady-state value is calculated using the following formula: Where, matrix A(x) s ),B(x s ) and steady state x s Regarding the approximation of A(x) by A(x) and B(x),... s ),B(x s ); and the lumped disturbance d s Since it is unknown, perturbation estimation is used. Instead of this, the translation setpoint is calculated using the following formula: Define w = xx s v=uu s Then there is y=C(w+x s ); S32. Design a sliding surface, specifically, neglecting the disturbance estimation error in the system. The following system was obtained Perform a nonsingular transformation on the system, and it is converted into the following controllable canonical form. in, Design a linear sliding surface: On the sliding surface, s = 0 is satisfied; S33. Design a sliding mode controller. Specifically, to suppress chattering in sliding mode control, a power-law-based reaching law is selected: Where q>0, ε>0, and fal(s,α,δ) is a power function, its expression is: Where 0 < α < 1, 0 < δ < 1; Derived from the sliding surface: Therefore, the sliding mode control law based on the power function reaching law is: v=(σTB(x)) -1 (-qs-εfal(s,α,δ)-σTA(x)w), Accordingly, the control law applied to hypersonic vehicles is: u=v+u s 。

Images