CN114995140B - Control method of time-varying system of hypersonic aircraft based on straight/gas combination - Google Patents

Control method of time-varying system of hypersonic aircraft based on straight/gas combination Download PDF

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CN114995140B
CN114995140B CN202210637964.1A CN202210637964A CN114995140B CN 114995140 B CN114995140 B CN 114995140B CN 202210637964 A CN202210637964 A CN 202210637964A CN 114995140 B CN114995140 B CN 114995140B
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周荻
王欢
李君龙
张锐
蔡明春
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Harbin Institute of Technology
Beijing Institute of Electronic System Engineering
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Abstract

A control method of a time-varying system of a hypersonic aircraft based on straight/gas combination belongs to the technical field of aircraft control. The invention solves the problems that the existing aircraft control scheme has low execution efficiency and needs to analyze the pneumatic parameters as fixed values. The technical scheme adopted by the method is as follows: the method comprises the following steps: establishing a state space equation of a longitudinal channel; step two: designing a state feedback control law of the pneumatic rudder of the time-varying system in a longitudinal channel; step three: designing a controller with a direct lateral force system of a longitudinal channel, and designing a sliding mode controller with a boundary layer of the longitudinal channel based on the controller with the direct lateral force system; step four: and designing a state feedback control law of a yaw channel and a sliding mode controller with a boundary layer, and designing a controller of a roll channel to realize control of a time varying system of the hypersonic aerocraft. The method can be applied to the technical field of aircraft control.

Description

Control method of time-varying system of hypersonic aircraft based on straight/gas combination
Technical Field
The invention belongs to the technical field of aircraft control, and particularly relates to a control method of a hypersonic aircraft time-varying system based on straight/gas combination.
Background
The hypersonic aircraft is an aircraft with the flight speed of more than 5Ma, strong penetration capability and important military value and economic value. The traditional air defense missile relies on the pneumatic rudder as an actuating mechanism of a control system, so that the response time delay of the system is large, and the actuating efficiency of the pneumatic rudder is low in an atmospheric environment with low dynamic pressure. Meanwhile, the hypersonic aircraft has the problem that the pneumatic parameters change along with time in the actual flight process, but in the traditional method, the pneumatic parameters of the aircraft model are taken as fixed values at characteristic points for calculation, namely, a time-varying system is converted into a constant system for analysis, so that the control precision of the traditional method is low.
In summary, the existing aircraft control schemes have the problems of low execution efficiency and the need to analyze the pneumatic parameters as fixed values.
Disclosure of Invention
The invention aims to solve the problems that the existing aircraft control scheme is low in execution efficiency and needs to analyze pneumatic parameters as fixed values, and provides a control method of a hypersonic aircraft time-varying system based on straight/gas combination.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a control method of a time-varying system of a hypersonic aerocraft based on straight/gas combination specifically comprises the following steps:
the method comprises the following steps: establishing a longitudinal channel mathematical model of the aircraft compositely controlled by the pneumatic rudder and the switch type engine, and obtaining a state space equation of a longitudinal channel based on the established longitudinal channel mathematical model;
step two: designing a state feedback control law of the pneumatic rudder of the time-varying system in the longitudinal channel based on the state space equation of the longitudinal channel obtained in the step one;
step three: designing a controller with a direct lateral force system of a longitudinal channel by using a sliding mode control method, and designing a sliding mode controller with a boundary layer of the longitudinal channel based on the controller with the direct lateral force system;
step four: and designing a state feedback control law of a yaw channel and a sliding mode controller with a boundary layer by adopting the method from the first step to the third step, and designing a controller of a roll channel so as to realize the control of the time varying system of the hypersonic aircraft.
The invention has the beneficial effects that:
the straight/gas composite control method for the hypersonic aircraft makes up the defect of single pneumatic rudder control, improves the execution efficiency of a control system, and accelerates the quick response of the aircraft to the tracking of the command signal;
in order to take practical significance into consideration, the designed hypersonic aircraft control system is a time-varying control system based on the LQR method, and pneumatic parameters are not taken as fixed values to be controlled, so that the control precision is improved; a linear time-varying system state feedback controller designed based on an LQR method applies a Jacobi polynomial to a system for solving the time-varying controller, and deduces and solves a control law based on the properties and an operation matrix of the Jacobi polynomial. Compared with the traditional method, the method provided by the invention combines the actual situation to design and analyze the aircraft control system, provides the process and method for solving the time-varying controller, and obtains a better result through simulation.
Drawings
FIG. 1 is a diagram of an aircraft tracking an overload command;
FIG. 2 is a diagram of the change in angle of attack of an aircraft on command;
FIG. 3 is a graph of the change in pitch rate of an aircraft;
FIG. 4 is a plot of the change in elevator deflection angle for an aircraft;
FIG. 5 is a graph illustrating engine thrust variation for an aircraft;
FIG. 6 is a graph of the variation of the sliding mode function of the aircraft.
Detailed Description
In a first specific embodiment, a control method for a time-varying system of a hypersonic aircraft based on straight/gas combination in this embodiment specifically includes the following steps:
the method comprises the following steps: establishing a longitudinal channel mathematical model (namely a pitching channel mathematical model) of the aircraft compositely controlled by the pneumatic rudder and the switch type engine, and obtaining a state space equation of a longitudinal channel based on the established longitudinal channel mathematical model;
the aircraft with pneumatic rudder and switch engine combined control is controlled by the direct lateral force controller and the rudder surface to constitute the direct/air combined control system.
Straightening: the invention is embodied in that the engine provides thrust (also understood as direct lateral force, which is provided by a side-injection engine), the engine of the invention is of an on-off type, and does not provide energy continuously, and the on-off type is characterized in that the engine is turned on when the engine is required to work and is turned off when the engine is not required, so that fuel is saved compared with the continuous energy supply.
Gas: the pneumatic rudder is embodied, and air power and moment are generated by the rudder.
The constructed mathematical model not only comprises aerodynamic force and moment, but also comprises thrust of a side-jet engine and moment generated by the thrust;
step two: designing a state feedback control law of the pneumatic rudder of the time-varying system in the longitudinal channel based on the state space equation of the longitudinal channel obtained in the first step;
the method comprises the steps of designing a linear time-varying system state feedback control law, applying an LQR (Linear quadratic regression) method, and solving the control law by applying a Jacobi (Jacobi) polynomial. Namely: applying Jacobi polynomial in a system for solving the LQR time-varying controller, wherein the specific process is to deduce and solve the control law based on the property and the operation matrix of the Jacobi polynomial;
step three: designing a controller with a direct lateral force system of a longitudinal channel by using a sliding mode control method, and designing a sliding mode controller with a boundary layer of the longitudinal channel based on the controller with the direct lateral force system;
step four: and (3) designing a state feedback control law of a yaw (lateral) channel and a sliding mode controller with a boundary layer by adopting the method from the first step to the third step, and designing a controller of a roll channel so as to realize control of a time varying system of the hypersonic aircraft.
The invention adopts a direct force and aerodynamic force composite (direct/pneumatic composite) control method, and after actuating mechanisms such as an attitude and orbit control side jet engine and the like are added, the defect of single pneumatic rudder control is overcome, and the rapid response of the aircraft to the instruction signal tracking is accelerated, so that the accurate guidance is realized. Meanwhile, in order to consider practical significance, the aircraft control system is designed into a time-varying system, and the time-varying aircraft control system is analyzed and solved by adopting a method for solving an LQR (linear quadratic optimal regulator) feedback control law based on a Jacobi polynomial.
The second embodiment is as follows: the difference between this embodiment and the first embodiment is that the specific process of the first step is as follows:
establishing a projectile coordinate system O 0 ′x′ 0 y′ 0 z′ 0 ；
The projectile coordinate system O 0 ′x′ 0 y′ 0 z′ 0 Is a moving coordinate system fixed on the missile body, and the origin O 0 ' on the missile centroid, O 0 ′x′ 0 The axis is on the longitudinal axial plane of the projectile body and points to the head of the projectile body in the positive direction, O 0 ′y′ 0 Shaft and O 0 ′x′ 0 Axis is vertical, and O 0 ′y′ 0 Axis in the longitudinal symmetry plane of the projectile, O 0 ′y′ 0 The positive direction of the axis being directed upwards, O 0 ′z′ 0 Axis perpendicular to O according to the right-hand coordinate system 0 ′x′ 0 y′ 0 Kneading;
the method for establishing the longitudinal channel mathematical model of the aircraft comprises the following steps:
Figure BDA0003681264820000031
wherein, a 2 And a 4 Are all aerodynamic coefficients, and a 2 And a 4 As a function of time, t being time, a 1 、a 3 、a 5 、k y And l z All are the power coefficients of the steel wire,
Figure BDA0003681264820000032
k y ＝1/(mV)，l z ＝-l/J z ，
Figure BDA0003681264820000033
representing pitching moment M z For omega z Partial derivatives of, M z δz Representing the pitching moment M z For delta z Partial derivative of, omega z Representing angular velocity in a missile coordinate system to a ground coordinate system O 0 z 0 Component of the axis, δ z Indicating elevator yaw angle, J z Representing moment of inertia in a projectile coordinate system O' 0 z′ 0 The component of the shaft is that of the shaft,
Figure BDA0003681264820000048
representing lift Y vs. rudder deflection angle delta z M, g and V represent the mass, gravitational acceleration and velocity of the aircraft, respectively, l represents the distance from the center of reaction force to the center of mass of the aircraft, n y Indicating output overload of longitudinal channel, tau 1 Representing the dynamic response time constant, tau, of the steering engine in the longitudinal channel 2 Representing the longitudinal passage attitude control engine dynamic response time constant, F Ty Representing the sum of the thrust of a longitudinal channel side-jet engine, F Tyc Representing longitudinal passage attitude-control engine commands, delta zc Indicating an elevator rudder deflection angle command,
Figure BDA0003681264820000041
is n y The first derivative of (a) is,
Figure BDA0003681264820000042
is omega z The first derivative of (a) is,
Figure BDA0003681264820000043
is delta z The first derivative of (a) is,
Figure BDA0003681264820000044
is F Ty The first derivative of (a);
defining the tracking error e of a vertical channel overload instruction y Comprises the following steps:
e y ＝n yc -n y (2)
wherein n is yc Trace instructions indicating longitudinal channel overload;
the mathematical model of the overload command tracking error control system is:
Figure BDA0003681264820000045
wherein the content of the first and second substances,
Figure BDA0003681264820000046
is e y The first derivative of (a);
in order to improve the tracking accuracy of the overload instruction, an integral term of the tracking error of the overload instruction is introduced, namely a state variable x is defined 1 、x 2 、x 3 、x 4 、x 5 Comprises the following steps:
Figure BDA0003681264820000047
x 2 ＝e y 、x 3 ＝ω z 、x 4 ＝δ z 、x 5 ＝F Ty defining a control variable u 1 ＝δ zc Control variable u 2 ＝F Tyc Then, equation (3) is expressed as a longitudinal channel state space equation of equation (4):
Figure BDA0003681264820000051
wherein the content of the first and second substances,
Figure BDA0003681264820000052
is x 1 The first derivative of (a) is,
Figure BDA0003681264820000053
is x 2 The first derivative of (a) is,
Figure BDA0003681264820000054
is x 3 The first derivative of (a) is,
Figure BDA0003681264820000055
is x 4 The first derivative of (a) is,
Figure BDA0003681264820000056
is x 5 The first derivative of (a).
Other steps and parameters are the same as those in the first embodiment.
The ground coordinate system is an inertial coordinate system, and the ground coordinate system O 0 x 0 y 0 z 0 Is fixedly connected with the earth surface and has an origin O 0 At the center of mass of the missile at the launching point, the trajectory plane intersects with the horizontal plane at the position O 0 x 0 A shaft, and O 0 x 0 The positive direction of the axis pointing towards the target, O 0 y 0 The positive direction of the axis points upwards along the vertical line, O 0 z 0 Axis perpendicular to O according to the right-hand coordinate system 0 x 0 y 0 And (5) kneading.
The third concrete implementation mode: the difference between this embodiment and the first or second embodiment is that the specific process of the second step is:
and (3) performing two-step unfolding analysis on the composite control law design of the control system (4) in the step one, firstly designing a time-varying state feedback controller related to the rudder deflection angle of the continuous control quantity, and then designing a controller related to the direct lateral force of the switching control quantity.
When designing a state feedback controller for rudder deflection angle, first, a control variable u is set 2 ＝F Tyc =0, a 4-order system model of equation (5) is designed:
Figure BDA0003681264820000057
in the system model of equation (5), the state vector X 1 Comprises the following steps: x 1 ＝[x 1 x 2 x 3 x 4 ] T Control amount u 1 Comprises the following steps: u. of 1 ＝δ zc Then the state equation of the system is:
Figure BDA0003681264820000058
wherein:
Figure BDA0003681264820000059
is X 1 The first derivative of (a);
Figure BDA0003681264820000061
because A is 1 (t) and B 1 Rank (A) of the matrix 1 (t),B 1 (t)) =4, so the system shown in equation (6) is fully controllable.
In combination with a linear time-varying control theory, the invention adopts an LQR optimal control method, and a system model design controller aiming at the formula (5) is as follows:
u 1 ＝K(t)X 1 ＝K 1 (t)x 1 +K 2 (t)x 2 +K 3 (t)x 3 +K 4 (t)x 4 (7)
wherein K (t) is a state feedback controller, K (t) = [ K = 1 (t) K 2 (t) K 3 (t) K 4 (t)]K 1 (t)、K 2 (t)、K 3 (t)、K 4 (t) are all elements of K (t);
and deducing and solving based on the properties of the Jacobi polynomial and the operation matrix to obtain the state feedback controller K (t).
Other steps and parameters are the same as those in the first or second embodiment.
The fourth concrete implementation mode: the difference between this embodiment and the first to third embodiments is that the properties and the operation matrix based on Jacobi polynomial are derived and solved to obtain the state feedback controller K (t); the specific process comprises the following steps:
step two, firstly: definition and Properties of Jacobi polynomial
The Jacobi polynomial F (-n, β, γ, z') is defined as:
Figure BDA0003681264820000062
wherein (beta) 0 ＝1，(β) k ＝β(β+1)(β+2)(β+3)...(β+k-1)，(-n) k 、(γ) k Form of definition of (A) and (B) k Similarly, γ is any positive integer, n is any integer, k =0,1, …, n, β > -1,z' e [0,1]；
A common expression form of the Jacobi polynomial is
Figure BDA0003681264820000063
Figure BDA0003681264820000071
Wherein, alpha > -1, (beta + 1) 0 ＝1，
Figure BDA0003681264820000072
Figure BDA0003681264820000073
(α+1) n Form (b) and (β + 1) n In the same way, the first and second,
Figure BDA0003681264820000074
λ＝α+β+1；
in the invention, in order to consider practical significance and facilitate the application of Jacobi polynomial in a time-varying system, an independent variable x is changed into a time variable t, and t is an element [ t ] 0 ,t f ]，t 0 Is the start time, t f Is the end time;
order to
x＝(2t-t 0 -t f )/(t f -t 0 ) (10)
Obtaining a transformed Jacobi polynomial J n (t) is:
Figure BDA0003681264820000075
in the formula: n =0,1,2.;
equation (11) is abbreviated:
Figure BDA0003681264820000076
wherein the content of the first and second substances,
Figure BDA0003681264820000077
the orthogonality property of the Jacobi polynomial is:
Figure BDA0003681264820000081
wherein Γ (·) is a Gamma function;
for any time function f (t), the Jacobi polynomial is expanded into:
Figure BDA0003681264820000082
in practical application, the m' term of the polynomial sequence is selected to approximate, i.e.
Figure BDA0003681264820000083
In the formula:
J(t)＝[J 0 (t) J 1 (t) ··· J m′-1 (t)] T (15)
wherein J (t) is a Jacobi polynomial vector, J 0 (t)J 1 (t)···J m′-1 (t) is a term in J (t); f. of n Jacobi polynomial expansion coefficients of f (t); f = [ f = 0 f 1 ··· f m′-1 ] T A Jacobi polynomial expansion coefficient vector of f (T), T representing transpose;
the integral expression E of expression (16) is minimized by using the orthogonal property expression (13) to obtain the expansion coefficient f of Jacobi polynomial n ：
Figure BDA0003681264820000084
Order to
Figure BDA0003681264820000085
Calculating to obtain Jacobi coefficient f n Comprises the following steps:
Figure BDA0003681264820000086
definition of f in formula (17) n Is r (n), then
Figure BDA0003681264820000087
Step two: operation matrix of Jacobi polynomial
The Jacobian polynomial operation matrix involved in the invention comprises a product operation matrix and an integral operation matrix.
In the formula (15), J (t) is a Jacobian polynomial vector, and in order to obtain J n (t) results of J (t), first calculate J n (t)J j (t) of (d). Order to
Figure BDA0003681264820000088
Is J n (t)J j (t) expansion coefficient of Jacobi polynomial, i.e.
Figure BDA0003681264820000091
From formulae (17) and (18):
Figure BDA0003681264820000092
wherein r (i) is
Figure BDA0003681264820000093
The coefficient of (a);
from the Euler integral (Euler integral), obtaining
Figure BDA0003681264820000094
And then the product is obtained according to the formula (12) and the formula (21):
Figure BDA0003681264820000095
the analysis is combined to obtain:
Figure BDA0003681264820000096
wherein, definition G = [ G = 0 G 1 ... G m′-1 ]A product operation matrix which is Jacobi polynomial;
integral operation matrix P of Jacobi polynomial tf Equation (24) holds:
Figure BDA0003681264820000097
calculated as follows:
Figure BDA0003681264820000098
in the formula:
Figure BDA0003681264820000101
Figure BDA0003681264820000102
Figure BDA0003681264820000103
Figure BDA0003681264820000104
step two and step three: combining analysis and derivation of Jacobi polynomial to obtain state feedback control law
The linear time-varying system dynamic equation is:
Figure BDA0003681264820000105
in the formula: x (t) is a state vector,
Figure BDA0003681264820000106
the first derivative of x (t), u (t) is a control vector, y (t) is an output vector, and A (t), B (t) and C (t) all represent a time-varying matrix of the system;
solving the optimal control law u (t) to minimize the quadratic performance index J of equation (27):
Figure BDA0003681264820000107
wherein, s, Q (t) and R (t) are weighting matrixes, and s, Q (t) and R (t) are diagonal matrixes, and the optimal control law of the system is obtained:
u(t)＝-R -1 (t)B T (t)P(t)x(t)＝-K(t)x(t) (28)
in the formula: k (t) = R -1 (t)B T (t) P (t) is the optimal feedback law of the system;
the matrix P (t) satisfies the Riccati matrix equation:
Figure BDA0003681264820000108
Figure BDA0003681264820000109
is the first derivative of P (t);
wherein:
P(t)＝[Ω 22 (t 0 ,t f )-sΩ 12 (t 0 ,t f )] -1 ·[sΩ 11 (t 0 ,t f )-Ω 21 (t 0 ,t f )] (30)
time-varying matrix
Figure BDA0003681264820000111
Ω 11 (t 0 ,t f ) Is omega (t) 0 ,t f ) Of (1) a first block matrix, Ω 12 (t 0 ,t f ) Is omega (t) 0 ,t f ) Of (1) a second block matrix, Ω 21 (t 0 ,t f ) Is omega (t) 0 ,t f ) Of (1) a third block matrix, Ω 22 (t 0 ,t f ) Is omega (t) 0 ,t f ) The fourth block matrix of (1);
combining the analysis to obtain a state transition matrix of the augmented state equation:
Figure BDA0003681264820000112
in the formula: λ (t) is a co-modal vector; f (t) is a 2n multiplied by 2n order time-varying matrix;
from the properties of the state transition matrix:
Figure BDA0003681264820000113
from the above analysis, the key to solving the optimal control law u (t) is the solution equation (32).
From t to both sides of formula (32) 0 To t f Integrating to obtain:
Figure BDA0003681264820000114
wherein, I 2n Is a 2n multiplied by 2n order identity matrix;
from F (t) = FJ 2n (t)，Ω(t 0 ,t f )＝ΩJ 2n (t) and I 2n ＝[I 2n 0 ... 0]J 2n (t), where F and Ω are Chang Jishu matrices, then equation (33) becomes:
Figure BDA0003681264820000115
easy certificate
Figure BDA0003681264820000116
In the formula, G i For the ith sub-block of the multiplication matrix,
Figure BDA0003681264820000117
represents a Kronecker product, F = [ F = [) 0 F 1 … F m′-1 ]，F i Is the ith matrix in F;
and meanwhile, the method is easy to prove:
Figure BDA0003681264820000118
substituting the formula (35) and the formula (36) into the formula (34) to obtain:
Figure BDA0003681264820000121
finishing to obtain:
Figure BDA0003681264820000122
obtaining an optimal feedback law K (t) by using the formula (38), the formula (28) and the formula (30);
step two, four: adding the obtained optimal feedback law K (t) into the system, and obtaining a closed-loop model as follows:
Figure BDA0003681264820000123
wherein the content of the first and second substances,
Figure BDA0003681264820000124
other steps and parameters are the same as those in one of the first to third embodiments.
The fifth concrete implementation mode: the difference between this embodiment and one of the first to fourth embodiments is that the specific process of the third step is:
when designing the direct lateral force control law, the new system model is as follows:
Figure BDA0003681264820000125
wherein the content of the first and second substances,
Figure BDA0003681264820000126
is X 2 The first derivative of (a) is,
Figure BDA0003681264820000127
Figure BDA0003681264820000128
Figure BDA0003681264820000131
and (4) considering the switch type working characteristics of the side-spraying engine, selecting a sliding mode control mode to design a control law. To X 2 Linear non-singular transformation is performed to obtain:
Figure BDA0003681264820000132
the longitudinal channel (i.e., pitch channel) mathematical model is of the form:
Figure BDA0003681264820000133
wherein the content of the first and second substances,
Figure BDA0003681264820000134
is to X 2 As a result of the linear non-singular transformation,
Figure BDA0003681264820000135
is to x 1 As a result of the linear non-singular transformation,
Figure BDA0003681264820000136
is to x 2 As a result of the linear non-singular transformation,
Figure BDA0003681264820000137
is to x 3 As a result of the linear non-singular transformation,
Figure BDA0003681264820000138
is to x 4 As a result of the linear non-singular transformation,
Figure BDA0003681264820000139
is to x 5 As a result of the linear non-singular transformation,
Figure BDA00036812648200001310
is composed of
Figure BDA00036812648200001311
The first derivative of (a) is,
Figure BDA00036812648200001312
wherein r is 0 As constants, the linear non-singular transformation matrix P is:
Figure BDA00036812648200001313
wherein r is i′ I' =1,2,3,4 is the coefficient of the characteristic polynomial of equation (43);
det(dI-A 2 )＝d 5 +r 4 d 4 +r 3 d 3 +r 2 d 2 +r 1 d+r 0 (43)
where det (-) represents the value of the determinant, d is a variable of the characteristic polynomial;
selecting a switching surface
Figure BDA00036812648200001314
Comprises the following steps:
Figure BDA00036812648200001315
wherein p is i″ I "= 1,2,3,4,5 satisfies the Hurwize condition, and in order to make the switching plane
Figure BDA00036812648200001316
Each point of (1) is a dead point, p i″ I "= 1,2,3,4,5 satisfies the following equation:
(r T -p T G′ 0 )＝ρp T (45)
wherein p = [ p ] 1 ,p 2 ,…,p 5 ] T ，r＝[r 0 ,r 1 ,…,r 4 ] T ，
Figure BDA0003681264820000141
Satisfies equation (46):
ρ 5 -r 4 ρ 4 +r 3 ρ 3 …+(-1) 5 r 0 ＝0 (46)
Figure BDA0003681264820000142
then the designed sliding mode control law u 3 Comprises the following steps:
Figure BDA0003681264820000143
the designed slip form surface belongs to a slip form subspace, wherein F s Is the steady state thrust of a single side injection engine;
in order to eliminate the buffeting phenomenon and save fuel, the design of adding the boundary layer to the controller is as follows:
Figure BDA0003681264820000144
wherein u is 3 For the controller after increasing the boundary layer, ε represents a small positive value;
when in use
Figure BDA0003681264820000145
Or
Figure BDA0003681264820000146
When satisfy the inequality
Figure BDA0003681264820000147
Wherein, b n Is a matrix B 3 Value of the last term in f 1 Is u 3 Upper bound of f 2 Is u 3 The lower bound of (c);
i.e. the state of the system can be reached on the slip-form face. And because the slip form surface meets the Hurwize condition, the movement on the slip form surface can be converged to the origin.
According to the designed sliding mode control law, the current flight state and the instruction requirements, the amplitude of the control force suitable for the system is selected to be switched between a large gear and a small gear, if the state of the current system is far away from the sliding mode surface, the amplitude of the thrust is large, and when the state of the system is close to the sliding mode surface, the amplitude of the thrust is small, the control torque generated by the two gears is necessarily larger than the external disturbance torque, and further, the expression of variable structure control is expanded into:
Figure BDA0003681264820000151
wherein, F s1 ＝F gs ，F gs Represents the steady state thrust of the rail-controlled engine; f s2 ＝F zs l zz /l z ，F zs Representing steady state thrust of the attitude-controlled engine,/ z Arm of force representing rail-controlled engine zz Representing arm of force of attitude-controlled engine, epsilon 1 And ε 2 Are all constants greater than 0, and 1 ＞ε 2 。
other steps and parameters are the same as in one of the first to fourth embodiments.
Simulation part
Simulating the straight/air composite system of the aircraft in the high altitude of 30km, wherein the simulation time is 5s, the overload instruction tracked by the longitudinal channel is in a form of a step signal plus a sinusoidal signal, and the overload instruction is taken as n yc =9+1.5 sin (2 pi t/1.5), relevant simulation parameters of the aircraft are shown in table 1:
TABLE 1
Figure BDA0003681264820000152
In the simulation, a 2 (t) and a 4 (t) there is a certain proportionality relationship, and the proportionality coefficient is-0.042. In fig. 1, simulation results show that the overload instruction rise time of the straight/gas composite control system of the aircraft is 0.4s, and the overshoot is small. The angle of attack response and overload response of the control system of fig. 2 are synchronized and the steady state process remains steady. The pitch rate in fig. 3 has a maximum value of 125/s and stabilizes around the zero axis with a sinusoidal variation after the angle of attack has been established. In the dynamic process of fig. 4, the rudder deflection angle of the elevator of the straight/gas compound control system is saturated to a certain degree, the steady state is basically stable, and the system state is stabilized on the sliding mode surface and converged to the origin. In fig. 5, the attitude and orbit control engine operates in a dynamic process, and is kept in a shutdown state and a substantially steady state. In fig. 6, the sliding mode function of the straight/gas compound control system converges rapidly to the sliding mode surface at 0.4s and remains substantially stable. According to simulation analysis, the feasibility of the designed aircraft control method is verified.
The above-described calculation examples of the present invention are merely to explain the calculation model and the calculation flow of the present invention in detail, and are not intended to limit the embodiments of the present invention. It will be apparent to those skilled in the art that other variations and modifications of the present invention can be made based on the above description, and it is not intended to be exhaustive or to limit the invention to the precise form disclosed, and all such modifications and variations are possible and contemplated as falling within the scope of the invention.

Claims (2)

1. A control method of a time-varying system of a hypersonic aerocraft based on straight/gas combination is characterized by specifically comprising the following steps:
the method comprises the following steps: establishing a longitudinal channel mathematical model of the aircraft compositely controlled by the pneumatic rudder and the switch type engine, and obtaining a state space equation of a longitudinal channel based on the established longitudinal channel mathematical model;
the specific process of the step one is as follows:
establishing a projectile coordinate system O 0 ′x′ 0 y′ 0 z′ 0 ；
The projectile coordinate system O 0 ′x′ 0 y′ 0 z′ 0 Is a moving coordinate system fixed on the missile body, and the origin O 0 ' on the missile centroid, O 0 ′x′ 0 The axis is on the longitudinal axial plane of the projectile body and points to the head of the projectile body in the positive direction, O 0 ′y′ 0 Shaft and O 0 ′x′ 0 Axis is vertical, and O 0 ′y′ 0 Axis in the longitudinal symmetry plane of the projectile, O 0 ′y′ 0 The positive direction of the axis being directed upwards, O 0 ′z′ 0 Axis perpendicular to O according to the right-hand coordinate system 0 ′x′ 0 y′ 0 z′ 0 Kneading;
the method for establishing the longitudinal channel mathematical model of the aircraft comprises the following steps:
Figure FDA0003899982340000011
wherein, a 2 And a 4 Are all aerodynamic coefficients, and a 2 And a 4 As a function of time, t being time, a 1 、a 3 、a 5 、k y And l z All are the power coefficients of the steel wire,
Figure FDA0003899982340000012
k y ＝1/(mV)，l z ＝-l/J z ，
Figure FDA0003899982340000013
representing the pitching moment M z For omega z The partial derivative of (a) of (b),
Figure FDA0003899982340000014
representing the pitching moment M z To delta z Partial derivative of, omega z Representing angular velocity in a missile coordinate system to a ground coordinate system O 0 z 0 Component of the axis, δ z Indicating elevator yaw angle, J z Representing moment of inertia in a projectile coordinate system O' 0 z′ 0 The component of the axis is such that,
Figure FDA0003899982340000019
representing lift Y vs. rudder deflection angle delta z M, g and V represent the mass, gravitational acceleration and velocity of the aircraft, respectively, l represents the distance from the center of reaction force to the center of mass of the aircraft, n y Indicating output overload of longitudinal channel, tau 1 Representing the dynamic response time constant, τ, of the steering engine in the longitudinal channel 2 Representing the longitudinal passage attitude control engine dynamic response time constant, F Ty Representing the sum of the thrust of a longitudinal channel side-jet engine, F Tyc Representing longitudinal passage attitude-control engine commands, delta zc Indicating an elevator rudder deflection angle command,
Figure FDA0003899982340000015
is n y The first derivative of (a) is,
Figure FDA0003899982340000016
is omega z The first derivative of (a) is,
Figure FDA0003899982340000017
is delta z The first derivative of (a) is,
Figure FDA0003899982340000018
is F Ty The first derivative of (a);
defining the tracking error e of a vertical channel overload instruction y Comprises the following steps:
e y ＝n yc -n y (2)
wherein n is yc Trace instructions indicating longitudinal channel overload;
the mathematical model of the overload command tracking error control system is:
Figure FDA0003899982340000021
wherein the content of the first and second substances,
Figure FDA0003899982340000022
is e y The first derivative of (a);
defining a state variable x 1 、x 2 、x 3 、x 4 、x 5 Comprises the following steps:
Figure FDA0003899982340000023
x 2 ＝e y 、x 3 ＝ω z 、x 4 ＝δ z 、x 5 ＝F Ty defining a control variable u 1 ＝δ zc Control variable u 2 ＝F Tyc Then, equation (3) is expressed as a longitudinal channel state space equation of equation (4):
Figure FDA0003899982340000024
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003899982340000025
is x 1 The first derivative of (a) is,
Figure FDA0003899982340000026
is x 2 The first derivative of (a) is,
Figure FDA0003899982340000027
is x 3 The first derivative of (a) is,
Figure FDA0003899982340000029
is x 4 The first derivative of (a) is,
Figure FDA0003899982340000028
is x 5 The first derivative of (a);
step two: designing a state feedback control law of the pneumatic rudder of the time-varying system in the longitudinal channel based on the state space equation of the longitudinal channel obtained in the step one;
the specific process of the second step is as follows:
when designing a state feedback controller for rudder deflection angle, let the control variable u 2 ＝F Tyc =0, a 4-order system model of equation (5) is designed:
Figure FDA0003899982340000031
in the system model of equation (5), the state vector X 1 Comprises the following steps: x 1 ＝[x 1 x 2 x 3 x 4 ] T Control amount u 1 Comprises the following steps: u. of 1 ＝δ zc Then the state equation of the system is:
Figure FDA0003899982340000032
wherein:
Figure FDA0003899982340000033
is X 1 The first derivative of (a);
Figure FDA0003899982340000034
by adopting an LQR optimal control method, aiming at the system model design controller of the formula (5), the method comprises the following steps:
u 1 ＝K(t)X 1 ＝K 1 (t)x 1 +K 2 (t)x 2 +K 3 (t)x 3 +K 4 (t)x 4 (7)
wherein K (t) is a state feedback controller, K (t) = [ K = 1 (t) K 2 (t) K 3 (t) K 4 (t)]K 1 (t)、K 2 (t)、K 3 (t)、K 4 (t) are all elements of K (t);
deducing and solving based on the properties of the Jacobi polynomial and the operation matrix to obtain a state feedback controller K (t);
deducing and solving the properties and the operation matrix based on the Jacobi polynomial to obtain a state feedback controller K (t); the specific process comprises the following steps:
step two, firstly: definition and Properties of Jacobi polynomial
The Jacobi polynomial F (-n, β, γ, z') is defined as:
Figure FDA0003899982340000035
wherein (beta) 0 ＝1，(β) k ＝β(β+1)(β+2)(β+3)...(β+k-1)，(-n) k 、(γ) k Form of definition of (A) and (B) k Similarly, γ is any positive integer, the value of the parameter n in the formula (8) is any integer, k =0,1, …, n, β > -1,z' e [0,1]；
A common expression form of the Jacobi polynomial is
Figure FDA0003899982340000041
Figure FDA0003899982340000042
Wherein, alpha > -1, (beta + 1) 0 ＝1，
Figure FDA0003899982340000043
Figure FDA0003899982340000044
(α+1) n Form (b) and (β + 1) n In the same way, the first and second,
Figure FDA0003899982340000045
λ＝α+β+1；
changing the independent variable x into a time variable t, t ∈ [ t ] 0 ,t f ]，t 0 Is the start time, t f Is the end time;
order to
x＝(2t-t 0 -t f )/(t f -t 0 ) (10)
Obtaining a transformed Jacobi polynomial J n (t) is:
Figure FDA0003899982340000046
in the formula: the value of the parameter n in the formula (11) is n =0,1,2.;
equation (11) is abbreviated:
Figure FDA0003899982340000047
wherein the content of the first and second substances,
Figure FDA0003899982340000048
the orthogonality property of the Jacobi polynomial is:
Figure FDA0003899982340000051
wherein Γ (·) is a Gamma function;
for any time function f (t), the Jacobi polynomial is expanded into:
Figure FDA0003899982340000052
selecting m' terms of the polynomial series for approximate approximation, i.e.
Figure FDA0003899982340000053
In the formula:
J(t)＝[J 0 (t) J 1 (t) ··· J m′-1 (t)] T (15)
wherein J (t) is a Jacobi polynomial vector, J 0 (t) J 1 (t) ··· J m′-1 (t) is a term in J (t); f. of n A Jacobi polynomial expansion coefficient of f (t); f = [ f = [ f ] 0 f 1 ··· f m′-1 ] T Jacobi polynomial expansion coefficient vector for f (T), T representing transpose;
the integral expression E of the expression (16) is minimized by the orthogonal property expression (13) to obtain a Jacobi polynomial expansion coefficient f n ：
Figure FDA0003899982340000054
Order to
Figure FDA0003899982340000055
The value of the parameter n in the formula (16) is n =0,1,.., m' -1, and the Jacobi coefficient f is calculated n Comprises the following steps:
Figure FDA0003899982340000056
definition of f in formula (17) n Is r (n), then
Figure FDA0003899982340000057
Step two: operation matrix of Jacobi polynomial
Order to
Figure FDA0003899982340000058
Is J n (t)J j (t) expansion coefficient of Jacobi polynomial, i.e.
Figure FDA0003899982340000059
From formulae (17) and (18):
Figure FDA0003899982340000061
wherein r (i) is
Figure FDA0003899982340000062
The coefficient of (a);
according to Euler integral, obtain
Figure FDA0003899982340000063
And then according to the formula (12) and the formula (21):
Figure FDA0003899982340000064
the analysis is combined to obtain:
Figure FDA0003899982340000065
wherein, definition G = [ G = 0 G 1 … G m′-1 ]A product operation matrix which is Jacobi polynomial;
integral operation matrix P of Jacobi polynomial tf Equation (24) holds:
Figure FDA0003899982340000066
calculating to obtain:
Figure FDA0003899982340000067
in the formula:
Figure FDA0003899982340000068
Figure FDA0003899982340000069
Figure FDA0003899982340000071
Figure FDA0003899982340000072
step two and step three: combining analysis and derivation of Jacobi polynomial to obtain state feedback control law
The linear time-varying system dynamic equation is:
Figure FDA0003899982340000073
in the formula: x (t) is a state vector,
Figure FDA0003899982340000074
the first derivative of x (t), u (t) is a control vector, y (t) is an output vector, and A (t), B (t) and C (t) all represent a time-varying matrix of the system;
solving the optimal control law u (t) to minimize the quadratic performance index J of equation (27):
Figure FDA0003899982340000075
wherein s, Q (t) and R (t) are weighting matrixes, and s, Q (t) and R (t) are diagonal matrixes, and the optimal control law is obtained:
u(t)＝-R -1 (t)B T (t)P(t)x(t)＝-K(t)x(t) (28)
in the formula: k (t) = R -1 (t)B T (t) P (t) is the optimal feedback law of the system;
the matrix P (t) satisfies the Riccati matrix equation:
Figure FDA0003899982340000076
Figure FDA0003899982340000077
is the first derivative of P (t);
wherein:
P(t)＝[Ω 22 (t 0 ,t f )-sΩ 12 (t 0 ,t f )] -1 ·[sΩ 11 (t 0 ,t f )-Ω 21 (t 0 ,t f )] (30)
time-varying matrix
Figure FDA0003899982340000078
Ω 11 (t 0 ,t f ) Is omega (t) 0 ,t f ) Of (1) a first block matrix, Ω 12 (t 0 ,t f ) Is omega (t) 0 ,t f ) Of (1) a second block matrix, Ω 21 (t 0 ,t f ) Is omega (t) 0 ,t f ) Of (1) a third block matrix, Ω 22 (t 0 ,t f ) Is omega (t) 0 ,t f ) The fourth block matrix of (1);
combining the analysis to obtain a state transition matrix of the augmented state equation:
Figure FDA0003899982340000081
in the formula: λ (t) is a co-modal vector; f (t) is a 2n multiplied by 2n order time-varying matrix;
from the properties of the state transition matrix:
Figure FDA0003899982340000082
from t to both sides of formula (32) 0 To t f Integrating to obtain:
Figure FDA0003899982340000083
wherein, I 2n Is a 2n multiplied by 2n order identity matrix;
from F (t) = FJ 2n (t)，Ω(t 0 ,t f )＝ΩJ 2n (t) and I 2n ＝[I 2n 0 … 0]J 2n (t), where F and Ω are Chang Jishu matrices, then equation (33) becomes:
Figure FDA0003899982340000084
easy certificate
Figure FDA0003899982340000085
In the formula, G i Is the ith sub-block of the product operation matrix,
Figure FDA0003899982340000086
represents the product of Kronecker, F = [) 0 F 1 … F m′-1 ]，F i Is the ith matrix in F;
and simultaneously, the method is easy to prove:
Figure FDA0003899982340000087
substituting the formula (35) and the formula (36) into the formula (34) to obtain:
Figure FDA0003899982340000088
finishing to obtain:
Figure FDA0003899982340000089
obtaining an optimal feedback law K (t) by using the formula (38), the formula (28) and the formula (30);
step two, four: adding the obtained optimal feedback law K (t) into the system, and obtaining a closed-loop model as follows:
Figure FDA0003899982340000091
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003899982340000092
step three: designing a controller with a direct lateral force system of a longitudinal channel by using a sliding mode control method, and designing a sliding mode controller with a boundary layer of the longitudinal channel based on the controller with the direct lateral force system;
step four: and designing a state feedback control law of a yaw channel and a sliding mode controller with a boundary layer by adopting the method from the first step to the third step, and designing a controller of a roll channel so as to realize the control of the time varying system of the hypersonic aircraft.
2. The control method of the time-varying system of the hypersonic flight vehicle based on the straight/gas combination as claimed in claim 1, wherein the specific process of the third step is as follows:
when designing the direct lateral force control law, the new system model is as follows:
Figure FDA0003899982340000093
wherein the content of the first and second substances,
Figure FDA0003899982340000094
is X 2 The first derivative of (a) is,
Figure FDA0003899982340000095
Figure FDA0003899982340000096
Figure FDA0003899982340000097
to X 2 Linear non-singular transformation is performed to obtain:
Figure FDA0003899982340000101
the longitudinal channel mathematical model is then of the form:
Figure FDA0003899982340000102
wherein the content of the first and second substances,
Figure FDA0003899982340000103
is to X 2 As a result of the linear non-singular transformation,
Figure FDA0003899982340000104
Figure FDA0003899982340000105
is to x 1 As a result of the linear non-singular transformation,
Figure FDA0003899982340000106
is to x 2 As a result of the linear non-singular transformation,
Figure FDA0003899982340000107
is to x 3 As a result of the linear non-singular transformation,
Figure FDA0003899982340000108
is to x 4 As a result of the linear non-singular transformation,
Figure FDA0003899982340000109
is to x 5 As a result of the linear non-singular transformation,
Figure FDA00038999823400001010
is composed of
Figure FDA00038999823400001011
The first derivative of (a) is,
Figure FDA00038999823400001012
wherein r is 0 As constants, the linear non-singular transformation matrix P is:
Figure FDA00038999823400001013
wherein r is i′ I' =1,2,3,4 is the coefficient of the characteristic polynomial of equation (43);
det(dI-A 2 )＝d 5 +r 4 d 4 +r 3 d 3 +r 2 d 2 +r 1 d+r 0 (43)
where det (·) represents the value of the determinant, d is a variable of the characteristic polynomial;
selecting a switching surface
Figure FDA00038999823400001014
Comprises the following steps:
Figure FDA00038999823400001015
wherein p is i″ I "= 1,2,3,4,5 satisfies the Hurwize condition, and in order to make the switching plane
Figure FDA00038999823400001016
Each point of (A) is a dead point, p i″ I "= 1,2,3,4,5 satisfy the following equation:
(r T -p T G′ 0 )＝ρp T (45)
wherein p = [ p ] 1 ,p 2 ,…,p 5 ] T ，r＝[r 0 ,r 1 ,…,r 4 ] T ，
Figure FDA00038999823400001017
Satisfies equation (46):
ρ 5 -r 4 ρ 4 +r 3 ρ 3 …+(-1) 5 r 0 ＝0 (46)
Figure FDA0003899982340000111
then the designed sliding mode control law u 3 Comprises the following steps:
Figure FDA0003899982340000112
wherein, F s Is the steady state thrust of a single side injection engine;
the addition of the boundary layer design to the controller is as follows:
Figure FDA0003899982340000113
where ε represents a small positive value;
when in use
Figure FDA0003899982340000114
Or
Figure FDA0003899982340000115
When, inequality is satisfied
Figure FDA0003899982340000116
Wherein, b n Is a matrix B 3 Value of the last term in f 1 Is u 3 Upper bound of f 2 Is u 3 The lower bound of (c);
according to a designed sliding mode control law, a current flight state and an instruction requirement, selecting the amplitude of a control force to switch between a large gear and a small gear, wherein control moments generated by the two gears must be both larger than an external interference moment, and an expression of variable structure control is expanded as follows:
Figure FDA0003899982340000117
wherein, F s1 ＝F gs ，F gs Represents the steady state thrust of the rail-controlled engine; f s2 ＝F zs l zz /l z ，F zs Representing steady state thrust of the attitude-controlled engine,/ z Arm of force representing rail-controlled engine zz Representing arm of force of attitude-controlled engine, epsilon 1 And epsilon 2 Are all constants greater than 0, and 1 ＞ε 2 。
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