CN111474852B - Discrete sliding mode control method for piezoelectric drive deformable wing - Google Patents

Discrete sliding mode control method for piezoelectric drive deformable wing Download PDF

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CN111474852B
CN111474852B CN202010078231.XA CN202010078231A CN111474852B CN 111474852 B CN111474852 B CN 111474852B CN 202010078231 A CN202010078231 A CN 202010078231A CN 111474852 B CN111474852 B CN 111474852B
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袁家信
李达
杨慧
胡志贤
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Shanghai University of Engineering Science
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Abstract

The invention relates to a discrete sliding mode control method of a piezoelectric driving deformable wing, which comprises the following steps: (a) establishing a coupling dynamic model for suppressing wing flutter through piezoelectricity according to a Lagrange equation; (b) and obtaining a discrete sliding mode control law of the piezoelectric driving deformation wing according to the discrete control, the sliding mode control and the coupling dynamics model, wherein the wing can be converted into a stable state from a flutter state within a limited time when the input voltage is carried out according to the control law. The invention also performs numerical simulation on discrete sliding mode control. The invention has the most important characteristics that a novel piezoelectric actuator is adopted, so that better driving strain is realized, and the wing surface of the wing can realize active control, thereby reducing the weight of the wing and improving the reliability of the wing; in addition, a discrete sliding mode controller is designed by adopting a discrete sliding mode control method, and wing bending-torsion coupling flutter is effectively inhibited.

Description

Discrete sliding mode control method for piezoelectric drive deformable wing
Technical Field
The invention belongs to the technical field of flight control, and relates to a discrete sliding mode control method of a piezoelectric driving deformable wing, in particular to a discrete sliding mode control method of a piezoelectric driving deformable wing which is closer to an actual control system and has anti-interference performance.
Background
The wing flutter is that the self-excitation phenomenon of the wing occurs under the interaction of aerodynamic force, elastic force and inertia force under the action of the wing and air flow in the flying process of the aircraft; when a certain critical speed is exceeded, the vibration is dispersed due to slight disturbance, the amplitude of the wing is increased sharply, the resulting effect is very serious, the wing is damaged, even the airplane is disassembled, and the safety of the airplane is threatened. Wing flutter is one of the main forms of aircraft failure, and the flutter problem is also the aerodynamic problem mainly faced by high-speed aircraft. The flutter suppression method of the wing is mainly divided into passive control and active control. The traditional control method mainly depends on a method for restraining flutter by controlling surface deflection of the trailing edge of the wing, although the device is simple in structure, the control method needs multiple energy conversion (mechanical, hydraulic, electric and the like), a large number of parts increase the structural weight and potential failure rate, the hydraulic pipeline network is high in fragility, the frequency bandwidth is limited, the control surface deflection angle is limited, and therefore the output control force is limited. In recent years, the following advantages are provided by the novel piezoelectric material: 1) the response speed is high, the distribution is flexible, and the time lag is small; 2) the structural design is simple, complex hydraulic pipelines and mechanical structural designs are not needed, and the weight and the complexity of the wing can be reduced; 3) maintaining airfoil structural integrity; 4) the driving and bearing are integrated, the distribution is flexible, and the mutual interference is avoided; 5) the integral and continuous deformation of the wing surface can be realized, the lift force is improved while the resistance is reduced, and the aerodynamic performance of the wing is comprehensively improved; but is widely applied to the active control of wings.
In the prior art, piezoelectric materials are used as an actuator and a sensor, a control mode adopts speed feedback control, the method belongs to accurate feedback control, has no anti-interference capability, does not consider the uncertain factors of system modeling, and belongs to continuous control, but in many practical systems, the continuous control is very difficult, even difficult to realize; compared with a common continuous control system, the discrete control system has the following advantages: 1) sufficient accuracy can be ensured; 2) the anti-interference performance is good, and the anti-interference capability of the system can be improved; 3) the control law which is difficult to realize by some analog controllers can be realized, and particularly, the method is suitable for complex control processes such as adaptive control, optimal control, intelligent control and the like; 4) time-sharing control can be adopted, the utilization rate of equipment is improved, and different control laws can be adopted for control.
Therefore, piezoelectric materials with high response speed and stronger acting force are used as actuators to inhibit flutter of the flexible wing; meanwhile, a discrete sliding mode control method based on the design is urgently needed, and a high-quality control effect is obtained.
Disclosure of Invention
The invention aims to provide a discrete sliding mode control method of a piezoelectric driving deformable wing, and particularly relates to a discrete sliding mode control method of a piezoelectric driving deformable wing, which is closer to an actual control system and has anti-interference performance.
The invention is based on the control law of discrete sliding mode control, uses voltage as the control input of a system, uses piezoelectric materials as an actuator, and inhibits the flutter of the wing.
The invention discloses a discrete sliding mode control method of a piezoelectric driving deformation wing, which comprises the following steps: (a) establishing a coupling dynamic model for suppressing flutter of the wing through piezoelectricity; (b) obtaining a discrete sliding mode control law of the piezoelectric driving deformation wing by using the coupling dynamics model according to discrete control and sliding mode control;
the step (a) is specifically as follows: calculating potential energy, kinetic energy and virtual work of the wing, wherein the virtual work consists of two parts of aerodynamic force and work done by a piezoelectric actuator; then discretizing the potential energy of the wing, the kinetic energy of the wing and the virtual work of the wing, and substituting the processing result into an equation to obtain a coupling dynamic model for piezoelectrically inhibiting the flutter of the wing;
the step (b) is specifically as follows: and dispersing the coupling dynamic model of the piezoelectric suppression wing flutter by a dispersion control theory to obtain a dispersion sliding mode surface and an index approach law, and simplifying to obtain a dispersion sliding mode control law.
As a preferable technical scheme:
according to the discrete sliding mode control method for the piezoelectrically driven deformable wing, the coupling dynamic model for suppressing the flutter of the wing through piezoelectricity is
Figure BDA0002379241700000021
Wherein the content of the first and second substances,
Figure BDA0002379241700000022
u in the space state equation is a control input, and x is a state variable; m, K, D and R are respectively a mass matrix, a rigidity matrix, a damping matrix and a control input coefficient matrix of the coupling dynamics model;
discretizing a coupling dynamic model of the piezoelectric suppression wing flutter into x (k +1) ═ ax (k) + Bu (k);
wherein A ═ eA1T,
Figure BDA0002379241700000023
T is the sampling time period and η is the integral variable.
According to the discrete sliding mode control method of the piezoelectric driving morphing wing, in order to prevent the occurrence of buffeting of a discrete sliding mode controller, a saturation function sat(s) is adopted to replace a sign function sgn(s) in the discrete sliding mode control law, so that a final control law is obtained;
wherein the content of the first and second substances,
Figure BDA0002379241700000024
wherein Δ is the boundary layer.
According to the discrete sliding mode control method for the piezoelectric drive deformable wing, the virtual work of the wing is
Figure BDA0002379241700000025
In the formula, L is aerodynamic force; t isaIs a pneumatic moment;
Figure BDA0002379241700000026
b is the half chord length, l is the span length, tpIs the thickness of the piezoelectric plate, tbIs the wing thickness, znIs the neutral plane of the beam to which the piezoelectric actuator is attached, EpIs the modulus of elasticity of the wing, d31Is the piezoelectric strain constant; w is the deflection and α is the elastic twist angle around the elastic axis; y is in the spanwise direction; g "(y) represents the second derivative of G (y), G (y) H (y-l)1)-H(y-l2) H (-) is a step function, l1Starting position for bonding the piezoelectric sheet in the span direction,/2The end point position of the piezoelectric sheet pasted along the wingspan direction is provided; v (t) is voltage;
the potential energy of the wings is as follows:
Figure BDA0002379241700000031
wherein l is the wing length, EI is the bending stiffness, GJ is the torsional stiffness, w is the deflection, and α is the elastic torsion angle about the elastic axis; y is in the spanwise direction;
the kinetic energy of the wing is:
Figure BDA0002379241700000032
defining:
Figure BDA0002379241700000033
rho is the bulk density of the wing, m is the unit span length mass,
Figure BDA0002379241700000034
is the elastic axis position, b is the half chord length, SαMass static moment, x, of unit extended wing to elastic axisαIs the distance from the gravity center line of the wing to the elastic axis, IαIs the mass moment of inertia of the unit extended wing to the elastic shaft,
Figure BDA0002379241700000035
the radius of gyration of the wing to the elastic shaft; z is the plane which coincides with the lift direction of the wing and is perpendicular to the span (in the y-direction) and the chord (in the x-direction)
The discrete sliding mode control method for the piezoelectrically driven deformable wing adopts the discretization treatment of an assumed mode method, the discretization treatment of a finite element, the concentrated mass method, the weighted residue method or the transfer matrix method.
According to the discrete sliding mode control method for the piezoelectric driving deformation wing, potential energy, internal energy and virtual work obtained by discretization processing by adopting an assumed modal method are as follows:
when the response of the continuous system is processed by adopting the modal superposition method, the solution of the continuous system is written as a linear combination of all modal functions:
Figure BDA0002379241700000036
Figure BDA0002379241700000037
phi in the formulai(y) and phii(y) is a function of wing bending and torsional modes; q. q ofwi(t) and qαi(t) is called generalized or modal coordinate, qw，qαIs a corresponding vector compact form;
the expressions for potential energy, internal energy and virtual work are converted into:
Figure BDA0002379241700000041
Figure BDA0002379241700000042
Figure BDA0002379241700000043
according to the discrete sliding mode control method for the piezoelectric driving deformation wing, the process result is substituted into the equation, namely the formula of potential energy, internal energy and virtual work after the discretization process is substituted into the Lagrange equation.
The substituting lagrangian equation into the discrete sliding-mode control method for the piezoelectric-driven deformable wing specifically comprises the following steps:
the lagrange equation is:
Figure BDA0002379241700000044
substituting expressions of potential energy, internal energy and virtual work into a Lagrange's equation to obtain:
Figure BDA0002379241700000045
in the formula (I), the compound is shown in the specification,
Figure BDA0002379241700000051
Figure BDA0002379241700000052
H＝[hij]，
Figure BDA0002379241700000053
i＝1,2,...Nw，j＝1,2,...Nα；
Nwrepresenting the truncation order of the degree of freedom of bending, NαRepresenting a torsional degree of freedom truncation order; q. q.sw1，qw2，
Figure BDA0002379241700000054
qα1，qα2And
Figure BDA0002379241700000055
are the corresponding coordinate components.
The above formula is rewritten into a concise matrix form:
Figure BDA0002379241700000056
in the formula
Figure BDA0002379241700000057
Definition of
Figure BDA0002379241700000058
q1＝q,
Figure BDA0002379241700000059
And u is changed into a continuous state space equation, namely a coupling dynamic model for suppressing wing flutter by piezoelectricity, wherein the equation is written as follows:
Figure BDA00023792417000000510
the discrete sliding mode control method for the piezoelectrically-driven deformable wing comprises the following steps:
s(k)＝Cx(k)；
wherein, C > 0;
in order to ensure that the system has good motion quality in the whole state space, a corresponding index approximation law is designed:
Figure BDA00023792417000000511
both sides are multiplied by T to obtain:
s(k+1)-s(k)＝-qTs(k)-εTsgn(s(k))；
wherein ε >0, q >0,1-qT > 0;
substituting s (k +1) ═ Cx (k +1) ═ cax (k) + cbu (k) into the above approximation law:
-(Tq-1)s(k)-εTsgn(s(k))＝CAx(k)+CBu(k)；
when the controllable condition CB ≠ 0 is established, the equation is simplified into a discrete sliding mode control law as follows:
u(k)＝-(CB)-1[CAx(k)-(1-qT)s(k)+εTsgn(s(k))]。
the discrete sliding mode control method of the piezoelectric driven morphing wing further performs numerical simulation on the discrete sliding mode control, namely verifies the stability of the system by using a Lyapunov function, and comprises the following steps:
selecting a Lyapunov function:
Figure BDA0002379241700000061
since the exponential-based discrete approach law satisfies:
[s(k+1)-s(k)]sgn(s(k))
＝[-qTs(k)-εTsgn(s(k))]sgn(s(k))
＝-qT|s(k)|-εT|s(k)|<0；
meanwhile, when the sampling time T is small, 2-qT 0 has
[s(k+1)+s(k)]sgn(s(k))
＝[(2-qT)s(k)-εTsgn(s(k))]sgn(s(k))
＝(2-qT)|s(k)|-εT|s(k)|>0；
Satisfies the existence and arrival conditions of discrete sliding modes and s2(k+1)<s2(k)；
Satisfy the Lyapunov theorem of stability, i.e.
ΔV(k)＝s2(k+1)-s2(k)<0,s(k)≠0。
V (k) >0, Δ V (k) <0, as known from Lyapunov stability theory, the system meets the stability requirement.
In the discrete sliding mode variable structure control, the movement starting from any state is characterized in that:
(1) starting from any initial condition, the motion approaches the switching surface monotonously and reaches or passes through the switching surface in limited steps.
(2) Once the motion has passed through the switching surface, each of its subsequent steps passes through the switching surface from the other side and continues.
(3) After the crossing starts, the length of each step is non-incremental, and the motion track is limited to a specific band.
(4) The steady state or the origin x is 0, namely the ideal quasi-sliding mode condition is adopted; or jitter around the origin, is a non-ideal quasi-sliding mode.
Advantageous effects
1. The piezoelectric material is adopted as the actuator, compared with a method of controlling the front edge and the tail edge, the structural design is simple, complex hydraulic pipelines and mechanical structural designs are not needed, and the weight and complexity of the wing can be reduced; maintaining airfoil structural integrity; the integral and continuous deformation of the wing surface can be realized, the lift force is improved while the resistance is reduced, and the aerodynamic performance of the wing is comprehensively improved; maintaining airfoil structural integrity; the response speed is high, and the time lag is small;
2. the discrete control theory is applied to a sliding mode control method, so that the controlled system has good motion quality in the whole state space, and the wing bending-torsion coupling flutter is effectively inhibited.
Drawings
Fig. 1 is a flowchart of a discrete sliding mode control method for a novel piezoelectric driven morphing wing according to an embodiment of the present invention;
FIG. 2 is a block diagram of a piezoelectric patch wing embodying the present invention;
FIG. 3 is a schematic diagram of discrete sliding mode control embodying the present invention;
FIG. 4 is a graph of the response of a control input u of a discrete sliding mode control embodying the present invention;
FIG. 5(a) is a graph of the response of a first order generalized coordinate for an implementation of the present invention;
FIG. 5(b) is a graph of the response of a second order generalized coordinate for an implementation of the present invention;
FIG. 5(c) is a graph of the response of the third order generalized coordinates of an embodiment of the present invention;
FIG. 5(d) is a graph of response of fourth order generalized coordinates for an embodiment of the present invention;
Detailed Description
The invention will be further illustrated with reference to specific embodiments. It should be understood that these examples are for illustrative purposes only and are not intended to limit the scope of the present invention. Further, it should be understood that various changes or modifications of the present invention may be made by those skilled in the art after reading the teaching of the present invention, and such equivalents may fall within the scope of the present invention as defined in the appended claims.
The invention discloses a discrete sliding mode control method of a piezoelectric driving deformation wing, which comprises the following steps: (a) establishing a coupling dynamic model for suppressing flutter of the wing through piezoelectricity; (b) obtaining a discrete sliding mode control law of the piezoelectric driving deformation wing by using the coupling dynamics model according to discrete control and sliding mode control; fig. 1 is a flowchart of a discrete sliding mode control method for a novel piezoelectric driven morphing wing according to an embodiment of the present invention; fig. 2 is a block diagram of a piezo patch wing embodying the present invention.
The step (a) is specifically as follows: calculating potential energy, kinetic energy and virtual work of the wing, wherein the virtual work consists of two parts of aerodynamic force and piezoelectric actuator acting; then discretizing the potential energy of the wing, the kinetic energy of the wing and the virtual work of the wing, and substituting the processing result into an equation to obtain a coupling dynamic model for piezoelectrically inhibiting the flutter of the wing;
the step (b) is specifically as follows: and (3) discretizing the coupling dynamic model for suppressing the flutter of the wing by using a discrete control theory to obtain a discrete sliding mode surface and an index approach law, and simplifying to obtain the discrete sliding mode control law.
According to the discrete sliding mode control method for the piezoelectrically driven deformable wing, the coupling dynamic model for suppressing the flutter of the wing through piezoelectricity is
Figure BDA0002379241700000071
Wherein the content of the first and second substances,
Figure BDA0002379241700000072
u in the space state equation is a control input, and x is a state variable; m, K, D and R are respectively a mass matrix, a rigidity matrix, a damping matrix and a control input coefficient matrix of the coupling dynamics model;
discretizing a coupling dynamic model of the piezoelectric suppression wing flutter into x (k +1) ═ ax (k) + Bu (k);
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002379241700000073
t is the sampling time period and η is the integral variable.
According to the discrete sliding mode control method of the piezoelectric driving morphing wing, in order to prevent the occurrence of buffeting of a discrete sliding mode controller, a saturation function sat(s) is adopted to replace a sign function sgn(s) in the discrete sliding mode control law, so that a final control law is obtained;
wherein the content of the first and second substances,
Figure BDA0002379241700000081
wherein Δ is the boundary layer.
According to the discrete sliding mode control method for the piezoelectric drive deformable wing, the virtual work of the wing is
Figure BDA0002379241700000082
In the formula, L is aerodynamic force; t is a unit ofaIs a pneumatic moment;
Figure BDA0002379241700000083
b is the half chord length, l is the span length, tpIs the thickness of the piezoelectric plate, tbIs the wing thickness, znIs the neutral plane of the beam to which the piezoelectric actuator is attached, EpIs the modulus of elasticity of the wing, d31Is the piezoelectric strain constant; w is the deflection and α is the elastic twist angle about the elastic axis; y is in the spanwise direction; g "(y) represents the second derivative of G (y), G (y) H (y-l)1)-H(y-l2) H (-) is a step function, l1Starting position for bonding the piezoelectric sheet in the span direction,/2The end point position of the piezoelectric sheet pasted along the wingspan direction is provided; v (t) is voltage;
the potential energy of the wing is:
Figure BDA0002379241700000084
wherein l is the wing length, EI is the bending stiffness, GJ is the torsional stiffness, w is the deflection, and α is the elastic torsion angle about the elastic axis; y is in the spanwise direction;
the kinetic energy of the wing is:
Figure BDA0002379241700000085
defining:
Figure BDA0002379241700000086
rho is the bulk density of the wing, m is the unit span length mass,
Figure BDA0002379241700000087
is the elastic axis position, b is the half chord length, SαMass static moment, x, of unit extended wing to elastic axisαIs the distance from the gravity center line of the wing to the elastic axis, IαIs the mass moment of inertia of the unit extended wing to the elastic shaft,
Figure BDA0002379241700000091
the turning radius of the wing to the elastic shaft; z is the plane which coincides with the lift direction of the wing and is perpendicular to the span (in the y-direction) and the chord (in the x-direction)
The discrete sliding mode control method for the piezoelectrically driven deformable wing adopts the discretization treatment of an assumed mode method, the discretization treatment of a finite element, the concentrated mass method, the weighted residue method or the transfer matrix method.
According to the discrete sliding mode control method for the piezoelectric driving deformation wing, potential energy, internal energy and virtual work obtained by discretization processing by adopting an assumed modal method are as follows:
when the response of the continuous system is processed by adopting the modal superposition method, the solution of the continuous system is written as a linear combination of all modal functions:
Figure BDA0002379241700000092
Figure BDA0002379241700000093
phi in the formulai(y) and phii(y) is a function of wing bending and torsional modes; q. q ofwi(t) and qαi(t) is called the generalized or modal coordinate, qw，qαIs a corresponding vector compact form;
the expressions for potential energy, internal energy and virtual work are converted into:
Figure BDA0002379241700000094
Figure BDA0002379241700000095
Figure BDA0002379241700000096
according to the discrete sliding mode control method for the piezoelectric driving deformation wing, the process result is substituted into the equation, namely the formula of potential energy, internal energy and virtual work after the discretization process is substituted into the Lagrange equation.
The substituting lagrangian equation into the discrete sliding-mode control method for the piezoelectric-driven deformable wing specifically comprises the following steps:
the lagrange equation is:
Figure BDA0002379241700000101
substituting expressions of potential energy, internal energy and virtual work into a Lagrange's equation to obtain:
Figure BDA0002379241700000102
in the formula (I), the compound is shown in the specification,
Figure BDA0002379241700000103
Figure BDA0002379241700000104
H＝[hij]，
Figure BDA0002379241700000105
i＝1,2,...Nw，j＝1,2,...Nα；
Nwrepresenting the truncation order of the degree of freedom of bending, NαRepresenting a torsional degree of freedom truncation order; q. q.sw1，qw2，
Figure BDA0002379241700000106
qα1，qα2And
Figure BDA0002379241700000107
are the corresponding coordinate components.
The above formula is rewritten into a concise matrix form:
Figure BDA0002379241700000108
in the formula
Figure BDA0002379241700000109
Definition of
Figure BDA00023792417000001010
q1＝q,
Figure BDA00023792417000001011
And u is changed into a continuous state space equation, namely a coupling dynamic model for suppressing wing flutter by piezoelectricity, wherein the equation is written as follows:
Figure BDA00023792417000001012
the discrete sliding mode control method for the piezoelectrically-driven deformable wing comprises the following steps:
s(k)＝Cx(k)；
wherein, C > 0;
in order to ensure that the system has good motion quality in the whole state space, a corresponding index approximation law is designed:
Figure BDA0002379241700000111
both sides are multiplied by T to obtain:
s(k+1)-s(k)＝-qTs(k)-εTsgn(s(k))；
wherein ε >0, q >0,1-qT > 0;
substituting s (k +1) ═ Cx (k +1) ═ cax (k) + cbu (k) into the above approximation law:
-(Tq-1)s(k)-εTsgn(s(k))＝CAx(k)+CBu(k)；
when the controllable condition CB ≠ 0 is established, the equation is simplified into a discrete sliding mode control law as follows:
u(k)＝-(CB)-1[CAx(k)-(1-qT)s(k)+εTsgn(s(k))]。
fig. 3 is a schematic diagram of discrete sliding mode control, explaining the relationship among a sliding mode surface, a discrete sliding mode control law, and a discrete system (controlled object).
The discrete sliding mode control method of the piezoelectric driving deformation wing further performs numerical simulation on the discrete sliding mode control, namely the stability of the system is verified by using a Lyapunov function, and the process is as follows:
selecting a Lyapunov function:
Figure BDA0002379241700000112
since the exponential-based discrete approach law satisfies:
[s(k+1)-s(k)]sgn(s(k))
＝[-qTs(k)-εTsgn(s(k))]sgn(s(k))
＝-qT|s(k)|-εT|s(k)|<0；
meanwhile, when the sampling time T is small, 2-qT 0 has
[s(k+1)+s(k)]sgn(s(k))
＝[(2-qT)s(k)-εTsgn(s(k))]sgn(s(k))
＝(2-qT)|s(k)|-εT|s(k)>0；
Satisfies the existence and arrival conditions of discrete sliding modes and s2(k+1)<s2(k)；
Satisfy the Lyapunov theorem of stability, i.e.
ΔV(k)＝s2(k+1)-s2(k)<0,s(k)≠0。
V (k) >0, Δ V (k) <0, as known from Lyapunov stability theory, the system meets the stability requirement.
In the discrete sliding mode variable structure control, the movement starting from any state is characterized in that:
(1) starting from any initial condition, the motion approaches the switching surface monotonously and reaches or passes through the switching surface in limited steps.
(2) Once the motion has passed through the switching surface, each of its subsequent steps passes through the switching surface from the other side and continues.
(3) After the crossing starts, the length of each step is non-incremental, and the motion track is limited to a specific band.
(4) The steady state or the origin x is 0, namely the ideal quasi-sliding mode condition is adopted; or jitter around the origin, are non-ideal quasi-sliding modes.
And in order to verify the effectiveness and the correctness of the designed control system, relevant parameters are substituted into the established model to carry out numerical simulation. The parameters are set as follows: v24.41 m/s, l 0.432m, Ep＝66.47×109N/m2，E＝69×109N/m2，tp＝0.002m， GJ＝0.3028Nm2，
Figure BDA0002379241700000121
aW＝2*pi，Mα＝-1.2，b＝0.025m，m＝0.2kg/m，e＝-0.6m，tb＝0.005m， EI＝0.2096Nm2，xα＝1.693×10-3m，d31＝-1.8×10-10C/N，ρa＝1.225kg/m3，rα＝5.625×10-8. The initial state of the system is x (0) — [0.1,0.1,0, 0%]T. The control law parameters are: e 5, q 10, C55555551]. The following simulations were obtained: FIG. 4 is a graph illustrating the response of a control input u according to an embodiment of the present invention; FIG. 5(a) is a first order generalized coordinate response plot of an embodiment of the present invention; FIG. 5(b) shows an embodiment of the present inventionA response curve plot of the second order generalized coordinate of formula (la); FIG. 5(c) is a graph of the response of the third order generalized coordinates of an embodiment of the present invention; fig. 5(d) is a response graph of fourth-order generalized coordinates according to an embodiment of the present invention. From the analysis of the simulation diagram, a system controlled by the designed discrete sliding mode control method of the novel piezoelectric drive deformable wing can be rapidly converged, the control system has a high dynamic response speed, and the bending-torsion coupling flutter of the wing is effectively inhibited.

Claims (5)

1. A discrete sliding mode control method of a piezoelectric drive deformable wing is characterized by comprising the following steps: (a) establishing a coupling dynamic model for suppressing flutter of the wing through piezoelectricity; (b) obtaining a discrete sliding mode control law of the piezoelectric driving deformation wing by using the coupling dynamics model according to discrete control and sliding mode control;
the step (a) is specifically as follows: calculating potential energy, kinetic energy and virtual work of the wing, wherein the virtual work consists of two parts of aerodynamic force and work done by a piezoelectric actuator; then discretizing the potential energy of the wing, the kinetic energy of the wing and the virtual work of the wing, and substituting the processing result into an equation to obtain a coupling dynamic model for piezoelectrically inhibiting the flutter of the wing; the process result is substituted into the equation, namely the expressions of potential energy, kinetic energy and virtual work after the discretization process are substituted into a Lagrange equation; the substitution lagrange equation is specifically:
the lagrange equation is:
Figure FDA0003595904610000011
the virtual work of the wing is:
Figure FDA0003595904610000012
in the formula, L is aerodynamic force; t isaIs pneumatic moment;
Figure FDA0003595904610000013
b is half chord length, l is span length, tpIs the thickness of the piezoelectric patch, tbIs the wing thickness, znIs the neutral plane of the beam to which the piezoelectric actuator is attached, EpIs the modulus of elasticity of the wing, d31Is the piezoelectric strain constant; w is the deflection and α is the elastic twist angle around the elastic axis; y is in the spanwise direction; g "(y) represents the second derivative of G (y), G (y) H (y-l)1)-H(y-l2) H (-) is a step function, l1Starting position for bonding the piezoelectric sheet in the span-wise direction,/2The end point position of the piezoelectric sheet pasted along the wingspan direction is provided; v (t) is voltage;
the potential energy of the wing is:
Figure FDA0003595904610000014
wherein l is the wing length, EI is the bending stiffness, GJ is the torsional stiffness, w is the deflection, and α is the elastic torsion angle about the elastic axis; y is in the spanwise direction;
the kinetic energy of the wing is:
Figure FDA0003595904610000015
Figure FDA0003595904610000021
defining:
Figure FDA0003595904610000022
rho is the bulk density of the wing, m is the unit span length mass,
Figure FDA00035959046100000210
is the elastic axis position, b is the half chord length, SαTo extend the wing pair elasticity for unitStatic moment of mass of shaft, xαIs the distance from the gravity center line of the wing to the elastic axis, IαMass moment of inertia, r, of the wing to the elastic axisαThe turning radius of the wing to the elastic shaft; z is a plane coincident with the wing lift direction and perpendicular to the span (in the y-direction) and chord (in the x-direction);
substituting expressions of potential energy, kinetic energy and virtual work into a Lagrange equation to obtain:
Figure FDA0003595904610000023
in the formula (I), the compound is shown in the specification,
Figure FDA0003595904610000024
Figure FDA0003595904610000025
Figure FDA0003595904610000026
Nwrepresenting the truncation order of the degree of freedom of bending, NαRepresenting a torsional degree of freedom truncation order; q. q.sw1，qw2，
Figure FDA0003595904610000027
qα1，qα2And
Figure FDA0003595904610000028
are the corresponding coordinate components;
the above formula is rewritten into a concise matrix form:
Figure FDA0003595904610000029
in the formula
Figure FDA0003595904610000031
Definition of
Figure FDA0003595904610000032
And u is changed into a continuous state space equation, namely a coupling dynamic model for suppressing wing flutter by piezoelectricity, wherein the equation is written as follows:
Figure FDA0003595904610000033
wherein the content of the first and second substances,
Figure FDA0003595904610000034
u in the space state equation is a control input, and x is a state variable; m, K, D and R are respectively a mass matrix, a rigidity matrix, a damping matrix and a control input coefficient matrix of the coupling dynamics model;
the step (b) is specifically as follows: and (3) discretizing the coupling dynamic model for suppressing the flutter of the wing by using a discrete control theory to obtain a discrete sliding mode surface and an index approach law, and simplifying to obtain the discrete sliding mode control law.
2. The discrete sliding-mode control method for the piezoelectrically-driven deformable wing is characterized in that in order to prevent the discrete sliding-mode controller from buffeting, a saturation function sat(s) is adopted to replace a sign function sgn(s) in the discrete sliding-mode control law, and a final control law is obtained;
wherein the content of the first and second substances,
Figure FDA0003595904610000035
wherein Δ is the boundary layer.
3. The discrete sliding-mode control method for the piezoelectrically actuated morphing wing according to claim 2, wherein the discretization process adopts a discretization process of an assumed modal method, a discretization process of finite elements, a lumped mass method, a weighted residue number method or a transfer matrix method.
4. The discrete sliding-mode control method for the piezoelectrically-driven deformable wing is characterized in that potential energy, kinetic energy and virtual work obtained by discretization processing by adopting an assumed modal method are as follows:
when the response of the continuous system is processed by adopting the modal superposition method, the solution of the continuous system is written as a linear combination of all modal functions:
Figure FDA0003595904610000036
Figure FDA0003595904610000037
phi in the formulai(y) and phii(y) is a function of wing bending and torsional modes; q. q.swi(t) and qαi(t) is called generalized or modal coordinate, qw，qαIs a corresponding vector compact form;
the expressions for potential energy, kinetic energy and virtual work translate into:
Figure FDA0003595904610000041
Figure FDA0003595904610000042
Figure FDA0003595904610000043
5. the discrete sliding mode control method of the piezoelectrically-driven deformable wing is characterized in that in the discrete sliding mode variable structure control, the movement from any state is characterized in that:
(1) starting from any initial condition, the motion approaches towards the switching surface monotonously and reaches or passes through the switching surface in limited steps;
(2) once the movement passes through the switching surface, each subsequent step of the movement passes through the switching surface from the other surface and continues;
(3) after the crossing starts, the length of each step is non-incremental, and the motion track is limited to a specific band;
(4) the steady state or the origin x is 0, namely the ideal quasi-sliding mode condition is adopted; or jitter around the origin, is a non-ideal quasi-sliding mode.
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