CN111474852A - Discrete sliding mode control method for piezoelectric drive deformable wing - Google Patents
Discrete sliding mode control method for piezoelectric drive deformable wing Download PDFInfo
- Publication number
- CN111474852A CN111474852A CN202010078231.XA CN202010078231A CN111474852A CN 111474852 A CN111474852 A CN 111474852A CN 202010078231 A CN202010078231 A CN 202010078231A CN 111474852 A CN111474852 A CN 111474852A
- Authority
- CN
- China
- Prior art keywords
- wing
- mode control
- discrete
- sliding mode
- discrete sliding
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
- 238000000034 method Methods 0.000 title claims abstract description 80
- 230000008878 coupling Effects 0.000 claims abstract description 29
- 238000010168 coupling process Methods 0.000 claims abstract description 29
- 238000005859 coupling reaction Methods 0.000 claims abstract description 29
- 238000005381 potential energy Methods 0.000 claims description 21
- 239000011159 matrix material Substances 0.000 claims description 18
- 230000004044 response Effects 0.000 claims description 17
- 230000008569 process Effects 0.000 claims description 12
- 238000013459 approach Methods 0.000 claims description 9
- 238000005452 bending Methods 0.000 claims description 9
- 238000012545 processing Methods 0.000 claims description 8
- 230000014509 gene expression Effects 0.000 claims description 7
- 230000001629 suppression Effects 0.000 claims description 4
- 238000013016 damping Methods 0.000 claims description 3
- 230000005484 gravity Effects 0.000 claims description 3
- 230000002401 inhibitory effect Effects 0.000 claims description 3
- 230000010354 integration Effects 0.000 claims description 3
- 230000007935 neutral effect Effects 0.000 claims description 3
- 238000005070 sampling Methods 0.000 claims description 3
- 230000003068 static effect Effects 0.000 claims description 3
- 238000012546 transfer Methods 0.000 claims description 3
- 238000004088 simulation Methods 0.000 abstract description 6
- 238000013461 design Methods 0.000 description 5
- 239000000463 material Substances 0.000 description 5
- 238000010586 diagram Methods 0.000 description 4
- 230000002829 reductive effect Effects 0.000 description 4
- 230000000694 effects Effects 0.000 description 3
- RZVHIXYEVGDQDX-UHFFFAOYSA-N 9,10-anthraquinone Chemical compound C1=CC=C2C(=O)C3=CC=CC=C3C(=O)C2=C1 RZVHIXYEVGDQDX-UHFFFAOYSA-N 0.000 description 1
- 230000009471 action Effects 0.000 description 1
- 230000003044 adaptive effect Effects 0.000 description 1
- 238000006243 chemical reaction Methods 0.000 description 1
- 238000005094 computer simulation Methods 0.000 description 1
- 230000003993 interaction Effects 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 238000003908 quality control method Methods 0.000 description 1
- 230000000452 restraining effect Effects 0.000 description 1
Images
Classifications
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
- G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
- G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
- Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation
Landscapes
- Engineering & Computer Science (AREA)
- Health & Medical Sciences (AREA)
- Artificial Intelligence (AREA)
- Computer Vision & Pattern Recognition (AREA)
- Evolutionary Computation (AREA)
- Medical Informatics (AREA)
- Software Systems (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Automation & Control Theory (AREA)
- Feedback Control In General (AREA)
- Micromachines (AREA)
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
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 fine disturbance can make the vibration disperse, the amplitude of the wing is increased sharply, the resulting effect is very serious, the wing is damaged, even the airplane is disintegrated, 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 factors of uncertainty 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 (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.
As a preferred 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
Wherein,
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,
T is the sample time period and η is the integration 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,
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
Wherein L is aerodynamic force, TaIs a pneumatic moment;
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, α is the elastic torsion angle around the elastic axis, y is in the span direction, G' (y) is the second derivative of the G (y) pair, G (y) is 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:
where l is wing length, EI is bending stiffness, GJ is torsional stiffness, w is deflection, α is the elastic twist angle about the elastic axis, y is in the span direction;
the kinetic energy of the wing is:
defining:
rho is the bulk density of the wing, m is the unit span length mass,
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 center of gravity of the wing to the elastic axis, IαIs the mass moment of inertia of the unit extended wing to the elastic shaft,
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:
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, internal energy and virtual work are converted into:
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:
substituting expressions of potential energy, internal energy and virtual work into a Lagrange's equation to obtain:
in the formula,
H＝[hij]，
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，
qα1，qα2And
are the corresponding coordinate components.
The above formula is rewritten into a concise matrix form:
in the formula
Definition of
q1＝q,
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:
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:
both sides are multiplied by T to obtain:
s(k+1)-s(k)＝-qTs(k)-Tsgn(s(k))；
wherein >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)-Tsgn(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)+Tsgn(s(k))]。
the discrete sliding mode control method for the piezoelectric driven morphing wing further performs numerical simulation on the discrete sliding mode control, namely the stability of the system is verified by using L yapunov function, and the process is as follows:
choose L yapunov function:
since the exponential-based discrete approach law satisfies:
[s(k+1)-s(k)]sgn(s(k))
＝[-qTs(k)-Tsgn(s(k))]sgn(s(k))
＝-qT|s(k)|-T|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)-Tsgn(s(k))]sgn(s(k))
＝(2-qT)|s(k)|-T|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 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 (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
Wherein,
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,
t is the sample time period and η is the integration 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,
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
Wherein L is aerodynamic force, TaIs a pneumatic moment;
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, α is the elastic torsion angle around the elastic axis, y is in the span direction, G' (y) is the second derivative of the G (y) pair, G (y) is 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:
where l is wing length, EI is bending stiffness, GJ is torsional stiffness, w is deflection, α is the elastic twist angle about the elastic axis, y is in the span direction;
the kinetic energy of the wing is:
defining:
rho is wingM is the unit length-expanded mass,
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 center of gravity of the wing to the elastic axis, IαIs the mass moment of inertia of the unit extended wing to the elastic shaft,
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:
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, internal energy and virtual work are converted into:
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:
substituting expressions of potential energy, internal energy and virtual work into a Lagrange's equation to obtain:
in the formula,
H＝[hij]，
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，
qα1，qα2And
are the corresponding coordinate components.
The above formula is rewritten into a concise matrix form:
in the formula
Definition of
q1＝q,
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:
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:
both sides are multiplied by T to obtain:
s(k+1)-s(k)＝-qTs(k)-Tsgn(s(k))；
wherein >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)-Tsgn(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)+Tsgn(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 L yapunov function is used for verifying the stability of the system, and the process is as follows:
choose L yapunov function:
since the exponential-based discrete approach law satisfies:
[s(k+1)-s(k)]sgn(s(k))
＝[-qTs(k)-Tsgn(s(k))]sgn(s(k))
＝-qT|s(k)|-T|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)-Tsgn(s(k))]sgn(s(k))
＝(2-qT)|s(k)|-T|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.
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，
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: 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) is a graph of the response of the second order generalized coordinate of an embodiment 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 response graph of fourth-order generalized coordinates according to an embodiment of the present invention. From the above-mentioned simulation chartAnd analysis shows that 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 high dynamic response speed, and wing bending-torsion coupling flutter is effectively inhibited.
Claims (10)
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 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 according to claim 1, wherein the coupling dynamic model for suppressing the flutter of the wing through piezoelectricity is
Wherein,
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,
T is the sample time period and η is the integration variable.
3. 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,
wherein Δ is the boundary layer.
4. The discrete sliding mode control method for the piezoelectrically actuated morphing wing according to claim 1, wherein the virtual work of the wing is
Wherein L is aerodynamic force, TaIs a pneumatic moment;
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, α is the elastic torsion angle around the elastic axis, y is in the span direction, G' (y) is the second derivative of the G (y) pair, G (y) is 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:
where l is wing length, EI is bending stiffness, GJ is torsional stiffness, w is deflection, α is the elastic twist angle about the elastic axis, y is in the span direction;
the kinetic energy of the wing is:
defining:
rho is the bulk density of the wing, m is the unit span length mass,
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 center of gravity of the wing to the elastic axis, IαIs the mass moment of inertia of the unit extended wing to the elastic shaft,
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)
5. The discrete sliding-mode control method for the piezoelectrically actuated morphing wing according to claim 1, 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.
6. The discrete sliding-mode control method of the piezoelectrically-driven deformable wing is characterized in that the potential energy, the internal energy and the virtual work obtained by discretization processing by the assumed mode 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:
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, internal energy and virtual work are converted into:
7. the discrete sliding-mode control method of the piezoelectric driven deformable wing according to claim 1, wherein the step of substituting the processing result into the equation is to substitute expressions of potential energy, internal energy and virtual work after the discretization processing into a Lagrangian equation.
8. The discrete sliding-mode control method for the piezoelectrically-driven deformable wing according to claim 7, wherein the substituting lagrangian equation specifically comprises:
the lagrange equation is:
substituting expressions of potential energy, internal energy and virtual work into a Lagrange's equation to obtain:
in the formula,
Nwrepresenting the truncation order of the degree of freedom of bending, NαRepresenting a torsional degree of freedom truncation order; q. q.sw1，qw2，
qα1，qα2And
are the corresponding coordinate components.
The above formula is rewritten into a concise matrix form:
in the formula
Definition of
q1＝q,
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:
9. the discrete sliding-mode control method for the piezoelectrically actuated morphing wing according to claim 1, wherein the discrete sliding-mode surfaces are:
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:
both sides are multiplied by T to obtain:
s(k+1)-s(k)＝-qTs(k)-Tsgn(s(k))；
wherein >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)-Tsgn(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)+Tsgn(s(k))]。
10. the discrete sliding-mode control method for the piezoelectrically actuated morphing wing according to claim 1, wherein the discrete sliding-mode control is numerically simulated, namely the stability of the system is verified by using L yapunov function, and the process is as follows:
choose L yapunov function:
since the exponential-based discrete approach law satisfies:
[s(k+1)-s(k)]sgn(s(k))
＝[-qTs(k)-Tsgn(s(k))]sgn(s(k))
＝-qT|s(k)|-T|s(k)|<0；
meanwhile, when the sampling time T is small, 2-qT 0, there are
[s(k+1)+s(k)]sgn(s(k))
＝[(2-qT)s(k)-Tsgn(s(k))]sgn(s(k))
＝(2-qT)|s(k)|-T|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.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202010078231.XA CN111474852B (en) | 2020-02-03 | 2020-02-03 | Discrete sliding mode control method for piezoelectric drive deformable wing |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202010078231.XA CN111474852B (en) | 2020-02-03 | 2020-02-03 | Discrete sliding mode control method for piezoelectric drive deformable wing |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN111474852A true CN111474852A (en) | 2020-07-31 |
| CN111474852B CN111474852B (en) | 2022-05-31 |
Family
ID=71747994
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202010078231.XA Active CN111474852B (en) | 2020-02-03 | 2020-02-03 | Discrete sliding mode control method for piezoelectric drive deformable wing |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN111474852B (en) |
Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112180732A (en) * | 2020-10-14 | 2021-01-05 | 燕山大学 | Hydraulic drive unit sliding mode controlled foot type robot driving method and system |
| CN113015279A (en) * | 2021-02-03 | 2021-06-22 | 昆明理工大学 | Temperature control method for microwave heating of silicon carbide ceramic based on sharp point mutation model |
| CN113315413A (en) * | 2021-06-17 | 2021-08-27 | 吉林大学 | Filter type second-order terminal discrete sliding mode controller of piezoelectric linear motor |
| CN114637312A (en) * | 2022-03-18 | 2022-06-17 | 北京航空航天大学杭州创新研究院 | Unmanned aerial vehicle energy-saving flight control method and system based on intelligent deformation decision |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP1901153A1 (en) * | 2006-09-12 | 2008-03-19 | OFFIS e.V. | Control system for unmanned 4-rotor-helicopter |
| CN105607472A (en) * | 2015-11-10 | 2016-05-25 | 江苏科技大学 | Self-adaptive inversion sliding-mode control method and device of nonlinear binary wings |
| CN109521673A (en) * | 2018-10-25 | 2019-03-26 | 北京航空航天大学 | A kind of section sliding formwork suppressing method of two-dimensional wing Flutter Problem |
| CN110362124A (en) * | 2019-07-22 | 2019-10-22 | 扬州大学 | The optimum PID control method of double control face two-dimensional wing flutter system |
-
2020
- 2020-02-03 CN CN202010078231.XA patent/CN111474852B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP1901153A1 (en) * | 2006-09-12 | 2008-03-19 | OFFIS e.V. | Control system for unmanned 4-rotor-helicopter |
| CN105607472A (en) * | 2015-11-10 | 2016-05-25 | 江苏科技大学 | Self-adaptive inversion sliding-mode control method and device of nonlinear binary wings |
| CN109521673A (en) * | 2018-10-25 | 2019-03-26 | 北京航空航天大学 | A kind of section sliding formwork suppressing method of two-dimensional wing Flutter Problem |
| CN110362124A (en) * | 2019-07-22 | 2019-10-22 | 扬州大学 | The optimum PID control method of double control face two-dimensional wing flutter system |
Non-Patent Citations (3)
| Title |
|---|
| 惠俊鹏等: "采用分布式压电驱动器的翼面热颤振主动抑制", 《宇航学报》, no. 12, 30 December 2010 (2010-12-30) * |
| 许进林等: "二元机翼颤振抑制的次最优控制", 《航空兵器》, no. 06, 15 December 2009 (2009-12-15) * |
| 赵志航等: "基于反演滑模的二元机翼颤振控制", 《电子设计工程》, no. 21, 5 November 2015 (2015-11-05) * |
Cited By (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112180732A (en) * | 2020-10-14 | 2021-01-05 | 燕山大学 | Hydraulic drive unit sliding mode controlled foot type robot driving method and system |
| CN112180732B (en) * | 2020-10-14 | 2021-11-30 | 燕山大学 | Hydraulic drive unit sliding mode controlled foot type robot driving method and system |
| CN113015279A (en) * | 2021-02-03 | 2021-06-22 | 昆明理工大学 | Temperature control method for microwave heating of silicon carbide ceramic based on sharp point mutation model |
| CN113015279B (en) * | 2021-02-03 | 2022-12-13 | 昆明理工大学 | Temperature control method for microwave heating of silicon carbide ceramic based on sharp point mutation model |
| CN113315413A (en) * | 2021-06-17 | 2021-08-27 | 吉林大学 | Filter type second-order terminal discrete sliding mode controller of piezoelectric linear motor |
| CN113315413B (en) * | 2021-06-17 | 2022-09-09 | 吉林大学 | Design method of filter type second-order terminal discrete sliding mode controller of piezoelectric linear motor |
| CN114637312A (en) * | 2022-03-18 | 2022-06-17 | 北京航空航天大学杭州创新研究院 | Unmanned aerial vehicle energy-saving flight control method and system based on intelligent deformation decision |
| CN114637312B (en) * | 2022-03-18 | 2024-06-18 | 北京航空航天大学杭州创新研究院 | Unmanned aerial vehicle energy-saving flight control method and system based on intelligent deformation decision |
Also Published As
| Publication number | Publication date |
|---|---|
| CN111474852B (en) | 2022-05-31 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN111474852B (en) | Discrete sliding mode control method for piezoelectric drive deformable wing | |
| Tsushima et al. | Flutter suppression for highly flexible wings using passive and active piezoelectric effects | |
| CN111319036B (en) | Self-adaptive algorithm-based mobile mechanical arm position/force active disturbance rejection control method | |
| Versiani et al. | Gust load alleviation in a flexible smart idealized wing | |
| CN105182742A (en) | Elastic body aircraft adaptive constrained tracking control indirect method | |
| Li et al. | Application of piezoelectric fiber composite actuator to aircraft wing for aerodynamic performance improvement | |
| CN116277036B (en) | Rapid fault-tolerant vibration suppression control method for flexible-base and flexible-arm space robot | |
| Zhang et al. | Nonlinear dynamical behaviors of deploying wings in subsonic air flow | |
| Bilgen et al. | Theoretical and experimental analysis of hysteresis in piezocomposite airfoils using Preisach model | |
| Wang et al. | Theoretical and experimental investigations on modified LQ terminal control scheme of piezo-actuated compliant structures in finite time | |
| Versiani et al. | Aeroelastic behavior of a composite plate-like wing under piezoelectrically induced stresses | |
| Avanzini et al. | L1 adaptive control of flexible aircraft: preliminary results | |
| Liu et al. | Continuous dynamic simulation for morphing wing aeroelasticity | |
| CN107962567A (en) | The distributed vibration control system and its vibration control method of float flexible mechanical arm | |
| Ko et al. | Structured model reference adaptive control for a wing section with structural nonlinearity | |
| Vindigni et al. | Simple adaptive wing-aileron flutter suppression system | |
| Muhammad et al. | Computational simulations and experiments on vibration control of a flexible two-link manipulator using a piezoelectric actuator | |
| Viswamurthy et al. | Effect of piezoelectric hysteresis on helicopter vibration control using trailing-edge flaps | |
| Mohamed et al. | Reduced order model based flight control system for a flexible aircraft | |
| Emad et al. | Modeling and flight control of small UAV with active morphing wings | |
| Sebastijanovic et al. | Panel flutter detection and control using the eigenvector orientation method and piezoelectric layers | |
| Lai et al. | Panel flutter limit-cycle suppression with piezoelectric actuation | |
| Gao et al. | Flutter Fault-Tolerant Control with Observer Considering Actuator Faults and Time Delay | |
| Tsushima et al. | Passive and active piezoelectric effects on flutter suppression of highly flexible wings | |
| CN112464372B (en) | Design sensitivity engineering numerical method for control surface efficiency of aileron of elastic wing |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |