CN109426146B - High-order nonsingular Terminal sliding mode control method of hypersonic aircraft - Google Patents

Abstract

The invention discloses a high-order nonsingular Terminal sliding mode control method of a hypersonic aircraft, which is used for processing a nonlinear model of the hypersonic aircraft based on a feedback linearization method, and compensating modeling errors and external disturbance existing in a system by adopting a RBF neural network control strategy. And then based on the linearized longitudinal model, a novel neural network sliding mode controller based on a recursive structure sliding mode surface controls the speed and the height of the aircraft by controlling a command signal set by an engine throttle valve of the high-speed and overspeed aircraft and an elevator deflection signal under a nominal cruising flight condition. The controller has good robustness to pneumatic nonlinearity, pneumatic interference and system parameter uncertainty. Simulation results show that the method can realize good tracking of the instruction signals and has higher response speed.

Description

High-order nonsingular Terminal sliding mode control method of hypersonic aircraft

Technical Field

The invention belongs to the technical field of aircraft control. In particular to a high-order nonsingular Terminal sliding mode control method of a hypersonic aircraft.

Background

The hypersonic aircraft has the advantages of both aircrafts and spacecrafts, integrates the leading-edge technologies of a plurality of human aerospace, is an important direction for the development of future aircrafts, has important significance for providing future space exploration capability and military warfare capability, and has been widely concerned by scholars at home and abroad. The key technologies of the hypersonic aircraft comprise a propelling technology, an aircraft integrated design technology, a hypersonic aerodynamic/thermodynamic technology, a structural material process technology, a flight control technology and the like. Flight control is one of the core problems of hypersonic flight technology and is also one of the hot spots of current control research. The hypersonic flight vehicle is very sensitive to changes of the shape, air dynamic parameters and atmospheric conditions under the influence of the design of high flying altitude and Mach number and the flying conditions. The high coupling of the fuselage to the engine during flight, the special cruising environment, the complex dynamics, the elastic deformation of the aircraft structure and the strong aerodynamic characteristics of the aircraft during cruising, present a serious challenge to the control of the aircraft. When designing a hypersonic aircraft controller, the coupling effect between the pneumatic distribution and the propulsion system and the control system is fully considered. For example, the compressed airflow of the front body air inlet channel will generate lift force and head-up moment, and the nozzle airflow of the rear body will also generate lift force and head-down moment; the conventional hypersonic aircraft is often a blunt leading edge, the phenomenon of non-linearity of aerodynamic force is serious when the hypersonic aircraft flies with a large attack angle in the flying process, and the aerodynamic force is non-linear even if the hypersonic aircraft flies with a small attack angle, and the non-linearity of the aerodynamic force brings great difficulty for the design of a flight control system of the hypersonic aircraft.

Compared with the conventional aircraft, the hypersonic aircraft adopts the integrated design of the fuselage engine in the overall layout, so that the subsystems have stronger coupling and nonlinearity. In order to meet the requirement that the hypersonic aircraft still has stable flight performance and good flight quality under complex flight conditions, a brand-new control means must be adopted. According to the characteristics of high order and large parameter change of a hypersonic aircraft model, the hypersonic aircraft of a NASA Lanli laboratory is taken as a research object, and a novel high-order nonsingular Terminal sliding mode control system based on a sliding mode surface with a recursive structure is designed, so that the hypersonic aircraft can control the speed and the height of the hypersonic aircraft by designing an instruction signal set by an engine throttle valve and an elevator deflection signal under the condition of nominal cruising flight. The designed flight control system can generate continuous control quantity and higher control precision, so that the state of the system is converged to a balance point within limited time, and the problems of singularity of Terminal sliding mode control and buffeting of sliding mode control are solved in sliding mode motion.

Chinese patent CN105653827 adopts a Terminal sliding mode design method to design a controller of a hypersonic aircraft, and the sliding mode surface is nonlinear, so that the response speed of the system is high, but the Terminal sliding mode has a singular problem.

Chinese patents CN102880053 and CN102880056 both adopt sliding mode design methods to design controllers of hypersonic flight vehicles, but the problem of buffeting exists in sliding mode control.

Disclosure of Invention

The invention aims to solve the technical problem of disclosing a high-order nonsingular Terminal sliding mode control method of a hypersonic aircraft, so that the flight control system has a good tracking effect on speed and height step response.

The nonlinear model of the hypersonic aircraft is processed based on a feedback linearization method, and a RBF neural network control strategy is adopted to compensate modeling errors and external disturbance of the system. And then based on the linearized longitudinal model, a novel neural network sliding mode controller based on a recursive structure sliding mode surface controls the speed and the height of the aircraft by controlling a command signal set by an engine throttle valve of the high-speed and overspeed aircraft and an elevator deflection signal under a nominal cruising flight condition. The controller has good robustness to pneumatic nonlinearity, pneumatic interference and system parameter uncertainty. Simulation results show that the method can realize good tracking of the instruction signals and has higher response speed.

The invention discloses a high-order nonsingular Terminal sliding mode control method of a hypersonic aircraft, which comprises the following steps of:

step A: modeling longitudinal motion dynamics of the hypersonic aircraft;

the nonlinear equation set of the longitudinal dynamic model of the hypersonic aircraft is described as follows on a speed coordinate system according to the stress condition:

wherein V is the flying speed; gamma is the inclination angle of the flight channel; h is the flying height; alpha is a flight attack angle; q is a pitch angle velocity; mu is a universal gravitation constant; m and I_yyRespectively high overspeed aircraft mass and its moment of inertia along the y-axis; l, D, T and M_yyRespectively lift force, resistance, thrust and pitching moment, and the calculation expression is as follows:

L＝ρV²sC_L/2 (6)

D＝ρV²sC_D/2 (7)

T＝ρV²sC_T/2 (8)

r＝h+R_E (10)

in the formula (I), the compound is shown in the specification,

and R_EAir density, reference area, average aerodynamic chord length and earth radius; r is the radial distance from the center of the earth; c_L,C_DAnd C_TRespectively a lift coefficient, a drag coefficient and a thrust coefficient; c_M(α),C_M(δ_e) And C_M(q) the coefficients of the pitching moment due to the angle of attack, the deviation of the elevator, and the pitch angle, respectively, then C_L,C_D,C_T,C(α),C_M(δ_e) And C_MThe expressions of (q) are:

wherein, c_e0.0292 is the coefficient;

with throttle opening beta, which is related to engine combustion rate and thrust coefficient, elevator yaw angle delta_eFor control input, the second-order system of the engine dynamic model is as follows:

in the formula (I), the compound is shown in the specification,

as second derivative of throttle opening beta, ξ and w_nRespectively representing the damping ratio and the undamped natural frequency of a second-order system model of the engine; beta is a_cA command signal for the throttle opening;

and B: the hypersonic speed aircraft dynamic model is subjected to differential linearization processing of a speed channel and an altitude channel;

differentiating the speed and the height of an output channel of the aircraft longitudinal model according to a complete feedback input/output linearization theory; defining vector x ═ V, γ, α, β, h]^TAnd control vector u ═ beta_c,δ_e]^TAnd continuously differentiating V for three times and h for four times to obtain:

in the formula (I), the compound is shown in the specification,

in formula (13)

And

expressed as:

regarding the second derivatives of α and β as being composed of two parts, namely a control-related part and a control-unrelated part, the expression is:

in the formula (I), the compound is shown in the specification,

definition of

The output dynamics of V and h can be expressed as explicitly containing the control quantity β_cAnd delta_eIn the form of (a);

in the formula (I), the compound is shown in the specification,

and C: establishing a nonsingular Terminal sliding mode surface of a recursive structure;

establishing the following sliding mode surface with a recursive structure on the basis of the step B:

in the formula, epsilon_ij,η_ijIs a normal number; s_v0＝V_d-V，S_h0＝h_d-h，V_dIs a step of speed, h_dIs a height step;

leading: for the following non-linear slip-form surfaces:

wherein X is the system state, epsilon and eta are normal numbers, the state X can be converged to zero in a limited time and is maintained at the zero point after reaching the zero point, and the convergence time satisfies T < 1/(epsilon eta);

and (3) proving that: defining a Lyapunov function

The above formula can be derived over time:

therefore, the Lyapunov function converges asymptotically, and the system state X can converge to zero and is maintained at the zero point after reaching the zero point;

from the formula (18), a

Further obtain

Integrating the two sides of the above formula to obtain

In the above, X (0) represents an initial value of the system state X;

thus, if the control system can make S_h3At a finite time T₂₄Internally converges to zero and > T at time T₂₄Can still be kept at zero; then, by theory, S_h2Will be in a limited time T₂₃(T₂₃＜1/(ε₂₃β₂₃) Converge to zero, and S_h1And e_hCan be respectively in limited time T₂₂(T₂₂＜1/(ε₂₂η₂₂) ) and T₂₁(T₂₁＜1/(ε₂₁η₂₁) Converge to zero); the h channel of the aircraft reaches the equilibrium point (e)_h0) total time T₂Comprises the following steps:

wherein, T₂₁～T₂₄Different times when the system on the aircraft altitude channel reaches the sliding mode surface are respectively set; epsilon₂₁～ε₂₃And η₂₁～η₂₃Are respectively T₂₁～T₂₃The corresponding normal number;

the same principle can prove that the total time T for the speed channel of the aircraft to converge to the equilibrium point for any system state₁Is composed of

Wherein, T₁₃The time of the system on the speed channel of the aircraft to reach the sliding mode surface is calculated; thus, after the system reaches the slip surface, e is given an arbitrary initial state_v(0) And e_h(0) The system will stabilize and converge to an equilibrium point in a finite time;

a high-order nonsingular Terminal sliding mode controller of the hypersonic aircraft;

designing a sliding mode controller for a longitudinal model formula (16) of the hypersonic aircraft on the basis of establishing a nonsingular Terminal sliding mode surface of a recursive structure in the step C; setting the expected speed and the expected altitude of the aircraft as Vd and hd respectively;

let x₁₁＝V，

x₂₁＝h，

Equation (13) can be transformed to take into account modeling errors and external disturbances of the system

In the formula,. DELTA.f_v,Δf_hThe centralized uncertainty item of the system comprises modeling errors and external disturbance of the system; the controller design target at this time is: let x be₁＝[x₁₁,x₁₂,x₁₃]^TAnd x₂＝[x₂₁,x₂₂,x₂₃,x₂₄]^TTracking respectively desired trajectories

And

theorem 1: for the speed channel subsystem in equation (26), the following control law is designed:

wherein, itMiddle epsilon_v4,η_v4Respectively are normal numbers, W is the output weight of the RBF neural network, and phi (X) is a function;

the output weight value self-adaptation law of the uncertain compensation RBF neural network is as follows:

robust term for eliminating neural network approximation error

The adaptive law is:

wherein epsilon_v4,η_v4Are normal numbers respectively; eta_W,η_ζAn adaptive learning rate for the parameter; w is the output weight of the RBF neural network, and phi (X) is a function; controlling a robust term for

Eliminating approximation error zeta of the neural network; the speed channel subsystem state can then be stable and reach the slip form face, i.e. S, within a limited time_v2→0；

And (3) proving that: defining the Lyapunov function:

robust term for control to eliminate approximation error ζ of neural network

The difference between the approximation error and the approximation error,

modeling error of output weight for RBF neural network, S_v2The sliding mode surface is a sliding mode surface which can be reached in a stable and limited time in the speed channel subsystem state;

the time derivative of equation (30) can be given as:

in the formula, epsilon₁₂、_η12、ε_i+1、_ηi+1、ε_v4、_ηv4The normal numbers corresponding to different sliding mode surfaces are all obtained by substituting adaptive law equations (28) and (29) for equations (31):

according to the theory of Lyapunov stability, the state of the system can reach the sliding mode surface within a limited time, namely S_v2＝0；

Theorem 2: for the height channel subsystem in system equation (26), the following control law can be obtained in the same way:

wherein epsilon_h5And η_h5Is a normal number, S_h3The sliding mode surface is a sliding mode surface which can be stably reached in a limited time in the state of the height channel subsystem;

RBF neural network output weight self-adaptation law W for uncertain compensation_hIs composed of

In the formula eta_WhFor adaptive learning rate of parameter, phi_h(X) is a function;

robust term for eliminating neural network approximation error

The adaptive law is:

in the formula eta_ζhAn adaptive learning rate for the parameter;

from a combination of formula (27) and formula (33)

In the above formula, ∈_1,i+1、η_1,i+1、ε_2,i+1、η_2,i+1All are corresponding normal numbers of different sliding mode surfaces; therefore, if the control law equation (36) is applied to the system equation (26), the whole closed-loop control system is asymptotically stable;

simulating a control system;

in the numerical simulation, under the cruise flight conditions of V15060 ft/s (1ft 0.3048m), h 11000ft, α 0.0315rad, q 0rad/s, and γ 0, it is assumed that the flight speed command V is given from time 0_d200ft/s, flight height command h_d3000 ft; the controller parameter is selected to be epsilon₂₁＝ε₁₁＝5，ε₂₂＝ε₁₂＝2，ε₂₃＝0.01，η₂₁＝η₁₁＝2，η₂₂＝η₁₂＝1，η₂₄＝0.01，ε_v4＝0.1，η_v4＝0.1，ε_h5＝0.1，η_h50.1; the adaptive learning rate of the neural network is selected as eta_Wh＝0.001，η_ζh＝0.002；

The control process ends.

The control method of the invention is divided into five steps. Firstly, establishing a longitudinal motion dynamic model of a hypersonic aircraft; secondly, carrying out differential linear processing on a speed and altitude channel on a longitudinal motion dynamics model of the hypersonic aircraft; thirdly, designing a nonsingular Terminal sliding mode surface with a recursive structure; fourthly, designing a high-order nonsingular Terminal sliding mode controller and carrying out stability analysis on the controller; and fifthly, simulating the control system.

The invention has the advantages that aiming at the singularity problem of the Terminal sliding mode, according to the characteristics of high order and large parameter change of a hypersonic aircraft model, the hypersonic aircraft in a NASA Lanli laboratory is taken as a research object, and a novel high-order nonsingular Terminal sliding mode control system based on neural network compensation is designed, so that the hypersonic aircraft can control the speed and the height of the aircraft through an instruction signal set by an engine throttle valve and an elevator deflection signal under the nominal cruising flight condition. The designed flight control system can generate continuous control quantity and higher control precision, so that the state of the system is converged to a balance point within limited time, and the obvious advantages of buffeting and singularity do not exist in sliding mode motion.

Drawings

FIG. 1 is a geometric outline view (a top view and b side view) of a hypersonic aerocraft;

FIG. 2 is a high-order nonsingular Terminal sliding mode control principle diagram of the hypersonic aircraft of the invention;

FIG. 3 is a control flow of a high-order nonsingular Terminal sliding mode of the hypersonic flight vehicle of the invention;

FIG. 4 is a graph of the response of a hypersonic aircraft to a 200ft/s speed step command at cruise flight conditions with the speed rapidly converging to a desired value in an embodiment of the invention;

FIG. 5 is a graph showing the rapid convergence of altitude to a desired value for a hypersonic aircraft response to a 3000ft altitude step command at cruise flight conditions in an embodiment of the present invention;

FIG. 6 is a response curve of the hypersonic aircraft trajectory angle γ in an embodiment of the present invention;

FIG. 7 is a response curve of hypersonic aircraft angle of attack α in accordance with an embodiment of the present invention;

FIG. 8 is a response curve for pitch rate q of a hypersonic vehicle in an embodiment of the invention.

Detailed Description

The present invention will be described in further detail with reference to the accompanying drawings and examples.

The reference numerals, symbols, lines, etc. in FIGS. 1 to 8 illustrate: the abscissa in fig. 4-8 represents the simulation time in seconds; the ordinate in fig. 4 represents the velocity in meters per second; the ordinate in fig. 5 represents height in meters; the ordinate in fig. 6 represents the track angle γ in rad; in fig. 7 the ordinate represents the angle of attack α in rad; the ordinate in fig. 8 represents the pitch angle rate q in rad/s.

FIG. 3 is a design method of the high-order nonsingular Terminal sliding mode control of the hypersonic flight vehicle. The method comprises the following steps:

step 210: modeling longitudinal motion dynamics of the hypersonic aircraft;

the nonlinear equation set of the longitudinal dynamic model of the hypersonic aircraft is described as follows on a speed coordinate system according to the stress condition:

wherein V is the flying speed; gamma is the inclination angle of the flight channel; h is flyA line height; alpha is a flight attack angle; q is a pitch angle velocity; mu is a universal gravitation constant; m and I_yyRespectively high overspeed aircraft mass and its moment of inertia along the y-axis; l, D, T and M_yyRespectively lift force, resistance, thrust and pitching moment, and the calculation expression is as follows:

L＝ρV²sC_L/2 (6)

D＝ρV²sC_D/2 (7)

T＝ρV²sC_T/2 (8)

r＝h+R_E (10)

in the formula (I), the compound is shown in the specification,

and R_EAir density, reference area, average aerodynamic chord length and earth radius; r is the radial distance from the center of the earth; c_L,C_DAnd C_TRespectively a lift coefficient, a drag coefficient and a thrust coefficient; c_M(α),C_M(δ_e) And C_M(q) the coefficients of the pitching moment due to the angle of attack, the deviation of the elevator, and the pitch angle, respectively, then C_L,C_D,C_T,C(α),C_M(δ_e) And C_MThe expressions of (q) are:

wherein, c_e0.0292 is the coefficient;

with throttle opening beta, which is related to engine combustion rate and thrust coefficient, elevator yaw angle delta_eFor control input, the second-order system of the engine dynamic model is as follows:

in the formula (I), the compound is shown in the specification,

as second derivative of throttle opening beta, ξ and w_nRespectively representing the damping ratio and the undamped natural frequency of a second-order system model of the engine; beta is a_cA command signal for the throttle opening;

step 220: the hypersonic speed aircraft dynamic model is subjected to differential linearization processing of a speed channel and an altitude channel;

differentiating the speed and the height of an output channel of the aircraft longitudinal model according to a complete feedback input/output linearization theory; defining vector x ═ V, γ, α, β, h]^TAnd control vector u ═ beta_c,δ_e]^TAnd continuously differentiating V for three times and h for four times to obtain:

in the formula (I), the compound is shown in the specification,

in formula (13)

And

expressed as:

regarding the second derivatives of α and β as being composed of two parts, namely a control-related part and a control-unrelated part, the expression is:

in the formula (I), the compound is shown in the specification,

definition of

The output dynamics of V and h can be expressed as explicitly containing the control quantity β_cAnd delta_eIn the form of (a);

in the formula (I), the compound is shown in the specification,

230: establishing a nonsingular Terminal sliding mode surface of a recursive structure;

establishing the following sliding mode surface with a recursive structure on the basis of the step B:

in the formula, epsilon_ij,η_ijIs a normal number; s_v0＝V_d-V，S_h0＝h_d-h，V_dIs a step of speed, h_dIs a height step;

leading: for the following non-linear slip-form surfaces:

wherein X is the system state, epsilon and eta are normal numbers, the state X can be converged to zero in a limited time and is maintained at the zero point after reaching the zero point, and the convergence time satisfies T < 1/(epsilon eta);

and (3) proving that: defining a Lyapunov function

The above formula can be derived over time:

therefore, the Lyapunov function converges asymptotically, and the system state X can converge to zero and is maintained at the zero point after reaching the zero point;

from the formula (18), a

Further obtain

Integrating the two sides of the above formula to obtain

In the above, X (0) represents an initial value of the system state X;

thus, if the control system can make S_h3At a finite time T₂₄Internally converges to zero and > T at time T₂₄Can still be kept at zero; then, by theory, S_h2Will be in a limited time T₂₃(T₂₃＜1/(ε₂₃β₂₃) Converge to zero, and S_h1And e_hCan be respectively in limited time T₂₂(T₂₂＜1/(ε₂₂η₂₂) ) and T₂₁(T₂₁＜1/(ε₂₁η₂₁) Converge to zero); the h channel of the aircraft reaches the equilibrium point (e)_h0) total time T₂Comprises the following steps:

wherein, T₂₁～T₂₄Different times when the system on the aircraft altitude channel reaches the sliding mode surface are respectively set; epsilon₂₁～ε₂₃And η₂₁～η₂₃Are respectively T₂₁～T₂₃The corresponding normal number;

the same principle can prove that the total time T for the speed channel of the aircraft to converge to the equilibrium point for any system state₁Is composed of

Wherein, T₁₃The time of the system on the speed channel of the aircraft to reach the sliding mode surface is calculated; thus, after the system reaches the slip surface, e is given an arbitrary initial state_v(0) And e_h(0) The system will stabilize and converge to an equilibrium point in a finite time;

step 240: a high-order nonsingular Terminal sliding mode controller of the hypersonic aircraft;

designing a sliding mode controller for a longitudinal model formula (16) of the hypersonic aircraft on the basis of establishing a nonsingular Terminal sliding mode surface of a recursive structure in the step C; let V be the desired speed and altitude of the aircraft_dAnd h_d；

Let x₁₁＝V，

x₂₁＝h，

Equation (13) can be transformed to take into account modeling errors and external disturbances of the system

In the formula,. DELTA.f_v,Δf_hThe centralized uncertainty item of the system comprises modeling errors and external disturbance of the system; the controller design target at this time is: let x be₁＝[x₁₁,x₁₂,x₁₃]^TAnd x₂＝[x₂₁,x₂₂,x₂₃,x₂₄]^TTracking respectively desired trajectories

And

theorem 1: for the speed channel subsystem in equation (26), the following control law is designed:

wherein epsilon is_v4,η_v4Respectively are normal numbers, W is the output weight of the RBF neural network, and phi (X) is a function;

the output weight value self-adaptation law of the uncertain compensation RBF neural network is as follows:

robust term for eliminating neural network approximation error

The adaptive law is:

wherein epsilon_v4,η_v4Are normal numbers respectively; eta_W,η_ζAn adaptive learning rate for the parameter; w is the output weight of the RBF neural network, and phi (X) is a function; controlling a robust term for

Eliminating approximation error zeta of the neural network; the speed channel subsystem state can then be stable and reach the slip form face, i.e. S, within a limited time_v2→0；

And (3) proving that: defining the Lyapunov function:

robust term for control to eliminate approximation error ζ of neural network

The difference between the approximation error and the approximation error,

modeling error of output weight for RBF neural network, S_v2The sliding mode surface is a sliding mode surface which can be reached in a stable and limited time in the speed channel subsystem state;

the time derivative of equation (30) can be given as:

in the formula, epsilon₁₂、_η12、ε_i+1、_ηi+1、ε_v4、_ηv4The normal numbers corresponding to different sliding mode surfaces are all obtained by substituting adaptive law equations (28) and (29) for equations (31):

according to the theory of Lyapunov stability, the state of the system can reach the sliding mode surface within a limited time, namely S_v2＝0；

Theorem 2: for the height channel subsystem in system equation (26), the following control law can be obtained in the same way:

wherein epsilon_h5And η_h5Is a normal number, S_h3The sliding mode surface is a sliding mode surface which can be stably reached in a limited time in the state of the height channel subsystem;

RBF neural network output weight self-adaptation law W for uncertain compensation_hIs composed of

In the formula eta_WhFor adaptive learning rate of parameter, phi_h(X) is a function;

robust term for eliminating neural network approximation error

The adaptive law is:

in the formula eta_ζhAn adaptive learning rate for the parameter;

from a combination of formula (27) and formula (33)

In the above formula, ∈_1,i+1、_η1,i+1、ε_2,i+1、_η2,i+1All are corresponding normal numbers of different sliding mode surfaces;

therefore, if the control law equation (36) is applied to the system equation (26), the whole closed-loop control system is asymptotically stable;

step 250: simulating a control system;

in the numerical simulation, under the cruise flight conditions of V15060 ft/s (1ft 0.3048m), h 11000ft, α 0.0315rad, q 0rad/s, and γ 0, it is assumed that the flight speed command V is given from time 0_d200ft/s, flight height command h_d3000 ft; the controller parameter is selected to be epsilon₂₁＝ε₁₁＝5，ε₂₂＝ε₁₂＝2，ε₂₃＝0.01，η₂₁＝η₁₁＝2，η₂₂＝η₁₂＝1，η₂₄＝0.01，ε_v4＝0.1，η_v4＝0.1，ε_h5＝0.1，η_h50.1; the adaptive learning rate of the neural network is selected as eta_Wh＝0.001，η_ζh＝0.002；

Step 260: the control process ends.

Claims (1)

1. The high-order nonsingular Terminal sliding mode control method of the hypersonic aircraft is characterized by comprising the following steps of:

step A: modeling longitudinal motion dynamics of the hypersonic aircraft;

the nonlinear equation set of the longitudinal dynamic model of the hypersonic aircraft is described as follows on a speed coordinate system according to the stress condition:

wherein V is the flying speed; gamma is the inclination angle of the flight channel; h is the flying height; alpha is a flight attack angle; q is a pitch angle velocity; mu is a universal gravitation constant; m and I_yyRespectively high overspeed aircraft mass and its moment of inertia along the y-axis; l, D, T and M_yyRespectively lift force, resistance, thrust and pitching moment, and the calculation expression is as follows:

L＝ρV²sC_L/2 (6)

D＝ρV²sC_D/2 (7)

T＝ρV²sC_T/2 (8)

r＝h+R_E (10)

in the formula (I), the compound is shown in the specification,

and R_EAir density, reference area, average aerodynamic chord length and earth radius; r is the radial distance from the center of the earth; c_L,C_DAnd C_TRespectively a lift coefficient, a drag coefficient and a thrust coefficient; c_M(α),C_M(δ_e) And C_M(q) the coefficients of the pitching moment due to the angle of attack, the deviation of the elevator, and the pitch angle, respectively, then C_L,C_D,C_T,C(α),C_M(δ_e) And C_MThe expressions of (q) are:

C_L＝0.6203α, (11)

C_D＝0.6405α²+0.0043378α+0.003772,

C_M(α)＝-0.035α²+0.036617α+5.3216×10^-6,

C_M(δ_e)＝c_e(δ_e-α),

wherein, c_e0.0292 is the coefficient;

with throttle opening beta, which is related to engine combustion rate and thrust coefficient, elevator yaw angle delta_eFor control input, the second-order system of the engine dynamic model is as follows:

in the formula (I), the compound is shown in the specification,

as second derivative of throttle opening beta, ξ and w_nRespectively representing the damping ratio and the undamped natural frequency of a second-order system model of the engine; beta is a_cA command signal for the throttle opening;

and B: the hypersonic speed aircraft dynamic model is subjected to differential linearization processing of a speed channel and an altitude channel;

differentiating the speed and the height of an output channel of the aircraft longitudinal model according to a complete feedback input/output linearization theory; defining vector x ═ V, γ, α, β, h]^TAnd control vector u ═ beta_c,δ_e]^TAnd continuously differentiating V for three times and h for four times to obtain:

in the formula (I), the compound is shown in the specification,

in formula (13)

And

expressed as:

regarding the second derivatives of α and β as being composed of two parts, namely a control-related part and a control-unrelated part, the expression is:

in the formula (I), the compound is shown in the specification,

definition of

The output dynamics of V and h can be expressed as explicitly containing the control quantity β_cAnd delta_eIn the form of (a);

in the formula (I), the compound is shown in the specification,

and C: establishing a nonsingular Terminal sliding mode surface of a recursive structure;

establishing the following sliding mode surface with a recursive structure on the basis of the step B:

in the formula, epsilon_ij,η_ijIs a normal number; s_v0＝V_d-V，S_h0＝h_d-h，V_dIs a step of speed, h_dIs a height step;

leading: for the following non-linear slip-form surfaces:

wherein X is the system state, epsilon and eta are normal numbers, the state X can be converged to zero in a limited time and is maintained at the zero point after reaching the zero point, and the convergence time satisfies T < 1/(epsilon eta);

and (3) proving that: defining a Lyapunov function

The above formula can be derived over time:

therefore, the Lyapunov function converges asymptotically, and the system state X can converge to zero and is maintained at the zero point after reaching the zero point;

from the formula (18), a

Further obtain

Integrating the two sides of the above formula to obtain

In the above, X (0) represents an initial value of the system state X;

thus, if the control system can make S_h3At a finite time T₂₄Internally converges to zero and > T at time T₂₄Can still be kept at zero; then, by theory, S_h2Will be in a limited time T₂₃(T₂₃＜1/(ε₂₃β₂₃) Inner convergence toZero, and S_h1And e_hCan be respectively in limited time T₂₂(T₂₂＜1/(ε₂₂η₂₂) ) and T₂₁(T₂₁＜1/(ε₂₁η₂₁) Converge to zero); the h channel of the aircraft reaches the equilibrium point (e)_h0) total time T₂Comprises the following steps:

wherein, T₂₁～T₂₄Different times when the system on the aircraft altitude channel reaches the sliding mode surface are respectively set; epsilon₂₁～ε₂₃And η₂₁～η₂₃Are respectively T₂₁～T₂₃The corresponding normal number;

the same principle can prove that the total time T for the speed channel of the aircraft to converge to the equilibrium point for any system state₁Is composed of

Wherein, T₁₃The time of the system on the speed channel of the aircraft to reach the sliding mode surface is calculated; thus, after the system reaches the slip surface, e is given an arbitrary initial state_v(0) And e_h(0) The system will stabilize and converge to an equilibrium point in a finite time;

step D: a high-order nonsingular Terminal sliding mode controller of the hypersonic aircraft;

designing a sliding mode controller for a longitudinal model formula (16) of the hypersonic aircraft on the basis of establishing a nonsingular Terminal sliding mode surface of a recursive structure in the step C; let x₁₁＝V，

x₂₁＝h，

Equation (13) can be transformed to take into account modeling errors and external disturbances of the system

In the formula,. DELTA.f_v,Δf_hThe centralized uncertainty item of the system comprises modeling errors and external disturbance of the system; the controller design target at this time is: let x be₁＝[x₁₁,x₁₂,x₁₃]^TAnd x₂＝[x₂₁,x₂₂,x₂₃,x₂₄]^TTracking respectively desired trajectories

And

theorem 1: for the speed channel subsystem in equation (26), the following control law is designed:

wherein epsilon is_v4,η_v4Respectively are normal numbers, W is the output weight of the RBF neural network, and phi (X) is a function;

the self-adaptive law of the output weight W of the uncertain compensation RBF neural network is as follows:

robust term for eliminating neural network approximation error

The adaptive law is:

wherein epsilon_v4,η_v4Are normal numbers respectively; eta_W,η_ζAn adaptive learning rate for the parameter; w is the output weight of the RBF neural network, and phi (X) is a function; controlling a robust term for

Eliminating approximation error zeta of the neural network; the speed channel subsystem state can then be stable and reach the slip form face, i.e. S, within a limited time_v2→0；

And (3) proving that: defining the Lyapunov function:

robust term for control to eliminate approximation error ζ of neural network

The difference between the approximation error and the approximation error,

modeling error of output weight for RBF neural network, S_v2The sliding mode surface is a sliding mode surface which can be reached in a stable and limited time in the speed channel subsystem state;

the time derivative of equation (30) can be given as:

in the formula, epsilon₁₂、_η12、ε_i+1、_ηi+1、ε_v4、_ηv4The normal numbers corresponding to different sliding mode surfaces are all obtained by substituting adaptive law equations (28) and (29) for equations (31):

according to the theory of Lyapunov stability, the state of the system can reach the sliding mode surface within a limited time, namely S_v2＝0；

Theorem 2: for the height channel subsystem in system equation (26), the following control law can be obtained in the same way:

wherein epsilon_h5And η_h5Is a normal number, S_h3The sliding mode surface is a sliding mode surface which can be stably reached in a limited time in the state of the height channel subsystem;

RBF neural network output weight W for uncertain compensation_hLaw of adaptation of

In the formula eta_WhFor adaptive learning rate of parameter, phi_h(X) is a function;

robust term for eliminating neural network approximation error

The adaptive law is:

in the formula eta_ζhAn adaptive learning rate for the parameter;

from a combination of formula (27) and formula (33)

In the above formula, ∈_1,i+1、η_1,i+1、ε_2,i+1、η_2,i+1All are corresponding normal numbers of different sliding mode surfaces;

therefore, if the control law equation (36) is applied to the system equation (26), the whole closed-loop control system is asymptotically stable;

step E: simulating a control system;

in the numerical simulation, under the cruise flight conditions of V15060 ft/s (1ft 0.3048m), h 11000ft, α 0.0315rad, q 0rad/s, and γ 0, it is assumed that the flight speed command V is given from time 0_d200ft/s, flight height command h_d3000 ft; the controller parameter is selected to be epsilon₂₁＝ε₁₁＝5，ε₂₂＝ε₁₂＝2，ε₂₃＝0.01，η₂₁＝η₁₁＝2，η₂₂＝η₁₂＝1，η₂₄＝0.01，ε_v4＝0.1，η_v4＝0.1，ε_h5＝0.1，η_h50.1; the adaptive learning rate of the neural network is selected as eta_Wh＝0.001，η_ζh＝0.002；

The control process ends.

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