CN112859949B - Boundary anti-saturation vibration suppression method and system based on Euler-Bernoulli beam - Google Patents
Boundary anti-saturation vibration suppression method and system based on Euler-Bernoulli beam Download PDFInfo
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Abstract
The invention relates to the technical field of vibration suppression of rigid systems, and particularly discloses a boundary anti-saturation vibration suppression method based on Euler-Bernoulli beams, wherein the method comprises the following steps: acquiring boundary state information of the Euler-Bernoulli beam; constructing a boundary anti-saturation controller according to boundary state information of the Euler-Bernoulli beam, wherein the boundary anti-saturation controller can output an actuator control value; judging whether the control value of the actuator reaches the upper limit of the saturation of the actuator or not according to the output of the boundary anti-saturation controller; if so, designing an auxiliary system according to boundary state information of the Euler-Bernoulli beam and a part exceeding the saturation upper limit of the actuator; and acquiring a gain value range of the boundary anti-saturation controller when the stability of the Euler-Bernoulli beam is determined. The invention also discloses a boundary anti-saturation vibration suppression system based on the Euler-Bernoulli beam. The boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam can eliminate the influence of actuator saturation limitation on the stability of the Euler-Bernoulli beam, and realize the stability of a system to ensure the smooth operation.
Description
Technical Field
The invention relates to the technical field of vibration suppression of flexible systems, in particular to a boundary anti-saturation vibration suppression method based on a Euler-Bernoulli beam and a boundary anti-saturation vibration suppression system based on the Euler-Bernoulli beam.
Background
Conventional rigid systems are not practical in many practical applications due to their large mass, high energy consumption, and small operational range. Under the background, the flexible material with the characteristics of light weight, high precision, low energy consumption, flexible operation and the like is widely applied to the fields of aerospace, ocean oil and gas development, industrial production and the like. Modeling of flexible materials is generally based on an Euler-Bernoulli beam model, and in many application contexts, requirements on precision, speed and belt load capacity of the Euler-Bernoulli beam are high, so that the Euler-Bernoulli beam is easy to generate elastic deformation in practical application, and further long-time vibration is caused, and the working quality and the production efficiency of the Euler-Bernoulli beam are influenced.
Therefore, the vibration problem of the Euler-Bernoulli beam is effectively inhibited, and the method has important significance. Due to the limitation of the physical conditions of the actuator, the torque output by the actuator is limited and cannot be output according to the ideal torque, in this case, the vibration suppression of the Euler-Bernoulli beam can be influenced, the stability of the Euler-Bernoulli beam can be damaged due to the failure of realizing the ideal vibration suppression effect, and even the Euler-Bernoulli beam can be in elastic deformation for a long time to damage the flexible rod structure, so that the actual industrial operation process is influenced.
Disclosure of Invention
The invention provides a boundary anti-saturation vibration suppression method based on an Euler-Bernoulli beam and a boundary anti-saturation vibration suppression system based on the Euler-Bernoulli beam, and solves the problem that the stability suppression of the Euler-Bernoulli beam cannot be realized in the related technology.
As a first aspect of the present invention, there is provided a boundary anti-saturation vibration suppression method based on a Euler-Bernoulli beam, including:
acquiring boundary state information of the Euler-Bernoulli beam;
constructing a boundary anti-saturation controller according to the boundary state information of the Euler-Bernoulli beam, wherein the boundary anti-saturation controller can output an actuator control value;
judging whether the control value of the actuator reaches the upper limit of the saturation of the actuator according to the output of the boundary anti-saturation controller;
if so, designing an auxiliary system according to the boundary state information of the Euler-Bernoulli beam and the part exceeding the saturation upper limit of the actuator, wherein the auxiliary system is used for eliminating the saturation nonlinear influence of the actuator;
and acquiring a gain value range of the boundary anti-saturation controller when the stability of the Euler-Bernoulli beam is determined.
Further, the boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam comprises the following steps of:
and obtaining a system equation and boundary conditions of the Euler-Bernoulli beam according to the Lagrange equation and the Hamilton principle.
Further, the expression of the system equation of the Euler-Bernoulli beam is as follows:
ρωtt(x,t)+EIωxxxx(x,t)-Tωxx(x,t)＝0，
the expression of the boundary condition is:
wherein, ω (x, t) < 0, L]X [0, + ∞) → R denotes the transverse displacement of the beam in space x and time T coordinates, p, T, EIM and L represent the mass per unit length of the beam, the tension, the bending stiffness, the mass of the tip load and the length of the beam system, respectively, omegax(x, t) and ωt(x, t) represents the derivative of the transverse displacement ω (x, t) of the rod with respect to space x and with respect to time t, respectively, u (t) represents the boundary controller; sat (u (t)) represents the saturation controller, expressed as:
wherein u ismaxAnd uminRepresenting the upper and lower bounds of the actuator, respectively.
Further, the acquiring boundary state information of the Euler-Bernoulli beam comprises:
obtaining the boundary speed omega in the system equation of the Euler-Bernoulli beamt(L, t), boundary displacement ωx(L, t) and the speed ω of the boundary bendingxt(L,t)。
Further, the expression of the auxiliary system is as follows:
wherein z (t) represents the state of the auxiliary system, kzα, β, ε each represent a constant greater than 0, sgn (z (t)) represents a sign function; Δ u (t) denotes a saturation dead band function, denoted as Δ u (t) sat (u (t)) u (t).
Further, the expression of the boundary anti-saturation controller is as follows:
u(t)＝-kaωt(L,t)+kbωx(L,t)-kcωxt(L,t)+kdz(t)，
wherein k isa,kb,kc,kdBoth represent gains of the boundary anti-saturation controller greater than 0.
Further, the obtaining a gain value range of the boundary anti-saturation controller when determining that the Euler-Bernoulli beam is stable includes:
selecting a Lyapunov function, wherein the expression is as follows:
V(t)＝V1(t)+V2(t)+V3(t)+V4(t)，
wherein both alpha and beta are greater than 0, V1(t) represents an energy term consisting of kinetic and potential energy, V2(t) denotes the cross term, V3(t) represents an energy term, V, associated with the state of the auxiliary system4(t) represents an auxiliary item;
verifying the positive nature of the Lyapunov function to obtain:
0<α3(V1(t)+V3(t)+V4(t))≤V(t)≤α4(V1(t)+V3(t)+V4(t))，
wherein alpha is3＝min{(1-υ),1}，α4＝max{(1+υ),1}，
And verifying the first derivative negative nature of the Lyapunov function to the time t by combining the boundary state information and the boundary anti-saturation controller, and obtaining the gain value range of the boundary anti-saturation controller.
As another aspect of the present invention, there is provided a boundary anti-saturation vibration suppression system based on a Euler-Bernoulli beam, comprising:
the sensor is used for acquiring boundary state information of the Euler-Bernoulli beam;
the boundary anti-saturation controller is used for outputting a control value of the actuator according to the boundary state information of the Euler-Bernoulli beam;
the auxiliary system is used for eliminating the saturation nonlinear influence of the actuator according to the boundary state information of the Euler-Bernoulli beam;
and the actuator is used for receiving the control value of the boundary anti-saturation controller and acting on the Euler-Bernoulli beam.
Further, the sensor includes: the device comprises a laser displacement sensor, an inclinometer and a pressure strain gauge.
According to the boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam, the influence of actuator saturation limitation on the stability of the Euler-Bernoulli beam is eliminated by adopting boundary anti-saturation control, loss caused by elastic vibration of the beam for a long time is prevented, the Euler-Bernoulli beam obtains higher precision in practical engineering application, and accordingly the stability of a system is realized, and the smooth operation is guaranteed.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention.
FIG. 1 is a flow chart of a boundary anti-saturation vibration suppression method based on a Euler-Bernoulli beam provided by the invention.
FIG. 2 is a flow chart of an implementation process of the boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam.
FIG. 3 is a graph of the present invention providing Euler-Bernoulli beam vibration displacement with a boundary anti-saturation controller in the event of actuator saturation.
Fig. 4 is a structural block diagram of a boundary anti-saturation vibration suppression system based on a Euler-Bernoulli beam provided by the invention.
Detailed Description
It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict. The present invention will be described in detail below with reference to the embodiments with reference to the attached drawings.
In order to make those skilled in the art better understand the technical solution of the present invention, the technical solution in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the terms "first," "second," and the like in the description and claims of the present invention and in the drawings described above are used for distinguishing between similar elements and not necessarily for describing a particular sequential or chronological order. It is to be understood that the data so used may be interchanged under appropriate circumstances in order to facilitate the description of the embodiments of the invention herein. Furthermore, the terms "comprises," "comprising," and "having," and any variations thereof, are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those steps or elements expressly listed, but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.
In the embodiment, a boundary anti-saturation vibration suppression method based on a Euler-Bernoulli beam is provided, and fig. 1 is a flowchart of the boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam according to the embodiment of the present invention, as shown in fig. 1, including:
s110, obtaining boundary state information of the Euler-Bernoulli beam;
s120, constructing a boundary anti-saturation controller according to the boundary state information of the Euler-Bernoulli beam, wherein the boundary anti-saturation controller can output an actuator control value;
s130, judging whether the actuator control value reaches an actuator saturation upper limit or not according to the output of the boundary anti-saturation controller;
s140, if so, designing an auxiliary system according to the boundary state information of the Euler-Bernoulli beam and the part exceeding the saturation upper limit of the actuator, wherein the auxiliary system is used for eliminating the saturation nonlinear influence of the actuator;
s150, when the stability of the Euler-Bernoulli beam is determined, obtaining a gain value range of the boundary anti-saturation controller.
According to the boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam, the influence of actuator saturation limitation on the stability of the Euler-Bernoulli beam is eliminated by adopting boundary anti-saturation control, loss caused by elastic vibration of the beam for a long time is prevented, the Euler-Bernoulli beam obtains high precision in practical engineering application, and therefore the stability of a system is achieved, and the practical operation is guaranteed to be smoothly carried out.
Specifically, the boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam comprises the following steps of:
and obtaining a system equation and boundary conditions of the Euler-Bernoulli beam according to the Lagrange equation and the Hamilton principle.
It should be understood that the system equations and boundary conditions for a Euler-Bernoulli beam are derived from lagrangian equations and the Hamilton (Hamilton) principle, as shown in fig. 2; acquiring boundary state information of the beam system; considering that the limit of hardware conditions in actual engineering causes the saturation of an executing mechanism and influences the vibration and accurate operation of a beam, and designing an auxiliary system; constructing a corresponding boundary anti-saturation controller by using the obtained boundary state information; obtaining the value range of the gain of the controller under the condition of ensuring the stability of the system; and judging whether the control value reaches the upper saturation limit of the actuator, and if so, compensating the saturation nonlinearity by combining with a designed auxiliary system.
It should be understood that after the gain value range of the boundary anti-saturation controller is obtained, vibration suppression can be achieved as long as the gain of the boundary anti-saturation controller is ensured to be within the value range.
More specifically, the expression of the system equation of the Euler-Bernoulli beam is as follows:
ρωtt(x,t)+EIωxxxx(x,t)-Tωxx(x,t)＝0，
the expression of the boundary condition is:
wherein, ω (x, t) < 0, L]X [0, + ∞) → R denotes the transverse displacement of the beam in space x and time T coordinates, p, T, EIM and L represent the mass per unit length of the beam, the tension, the bending stiffness, the mass of the tip load and the length of the beam system, respectively, omegax(x, t) and ωt(x, t) respectively represent the lateral displacement ω of the rod(x, t) derivatives over space x and over time t, u (t) representing a boundary controller; sat (u (t)) represents the saturation controller, expressed as:
wherein u ismaxAnd uminRepresenting the upper and lower bounds of the actuator, respectively.
Specifically, the acquiring boundary state information of the Euler-Bernoulli beam includes:
obtaining the boundary speed omega in the system equation of the Euler-Bernoulli beamt(L, t), boundary displacement ωx(L, t) and the speed ω of the boundary bendingxt(L,t)。
The expression of the auxiliary system is as follows:
wherein z (t) represents the state of the auxiliary system, kzα, β, ε each represent a constant greater than 0, sgn (z (t)) represents a sign function; Δ u (t) denotes a saturation dead band function, denoted as Δ u (t) sat (u (t)) u (t).
Specifically, the expression of the boundary anti-saturation controller is as follows:
u(t)＝-kaωt(L,t)+kbωx(L,t)-kcωxt(L,t)+kdz(t)，
wherein k isa,kb,kc,kdBoth represent gains of the boundary anti-saturation controller greater than 0.
Specifically, the obtaining a gain value range of the boundary anti-saturation controller when determining that the Euler-Bernoulli beam is stable includes:
selecting a Lyapunov (Lyapunov) function, wherein the expression is as follows:
V(t)＝V1(t)+V2(t)+V3(t)+V4(t)，
wherein both alpha and beta are greater than 0, V1(t) represents an energy term consisting of kinetic and potential energy, V2(t) denotes the cross term, V3(t) represents an energy term, V, associated with the state of the auxiliary system4(t) represents an auxiliary item;
verifying the positive nature of the Lyapunov function to obtain:
0<α3(V1(t)+V3(t)+V4(t))≤V(t)≤α4(V1(t)+V3(t)+V4(t))，
wherein alpha is3＝min{(1-υ),1}，α4＝max{(1+υ),1}，
And verifying the first derivative negative nature of the Lyapunov function to the time t by combining the boundary state information and the boundary anti-saturation controller, and obtaining the gain value range of the boundary anti-saturation controller.
It should be understood that by verifying the first derivative negativity of the Lyapunov function with respect to time t and obtaining the gain value range of the boundary anti-saturation controller, it can be ensured that the system state of the Euler-Bernoulli beam, i.e., the vibration of the Euler-Bernoulli beam, is suppressed.
The effectiveness of the proposed method is illustrated below with reference to specific parameters.
Firstly, selecting the system parameters of the Euler-Bernoulli beam as follows: L1M, M0.1 kg, EI＝7N·m2ρ is 0.1kg/m, and T is 10N. The initial value of the system is selected as
ωt(x,0)＝0。
Second step, consider actuator saturation limit as umax＝5,uminAnd (5) adopting a boundary anti-saturation controller by the controller, wherein the selected controller gain is as follows: k is a radical ofa＝6,kb＝1,kc＝0.04,kd＝0.1。
FIG. 3 is a graph showing the vibration displacement of a flexible rod using a boundary anti-saturation controller in the event of actuator saturation.
As another embodiment of the present invention, there is provided a boundary anti-saturation vibration suppression system based on a Euler-Bernoulli beam, wherein as shown in fig. 4, the boundary anti-saturation vibration suppression system includes:
the sensor is used for acquiring boundary state information of the Euler-Bernoulli beam;
the boundary anti-saturation controller is used for outputting a control value of the actuator according to the boundary state information of the Euler-Bernoulli beam;
the auxiliary system is used for eliminating the saturation nonlinear influence of the actuator according to the boundary state information of the Euler-Bernoulli beam;
and the actuator is used for receiving the control value of the boundary anti-saturation controller and acting on the Euler-Bernoulli beam.
According to the boundary anti-saturation vibration suppression system based on the Euler-Bernoulli beam, the influence of actuator saturation limitation on the stability of the Euler-Bernoulli beam is eliminated by adopting boundary anti-saturation control, loss caused by elastic vibration of the beam for a long time is prevented, the Euler-Bernoulli beam obtains higher precision in practical engineering application, and therefore the stability of the system is achieved, and the practical operation is guaranteed to be carried out smoothly.
Specifically, the sensor is used for measuring boundary state information of the beam, and comprises a laser displacement sensor, an inclinometer, a pressure strain gauge and the like;
the actuator is used for receiving the control signal transmitted by the controller and acting on the Euler-Bernoulli beam;
the auxiliary system is used for eliminating the influence of actuator input saturation nonlinearity on the system and determining the output of the auxiliary system by judging whether the control input is in a saturation range;
the boundary anti-saturation controller is combined with a designed auxiliary system to eliminate the influence of actuator saturation on the system aiming at the actuator saturation condition
It will be understood that the above embodiments are merely exemplary embodiments taken to illustrate the principles of the present invention, which is not limited thereto. It will be apparent to those skilled in the art that various modifications and improvements can be made without departing from the spirit and substance of the invention, and these modifications and improvements are also considered to be within the scope of the invention.
Claims (4)
1. A boundary anti-saturation vibration suppression method based on an Euler-Bernoulli beam is characterized by comprising the following steps:
acquiring boundary state information of the Euler-Bernoulli beam;
constructing a boundary anti-saturation controller according to the boundary state information of the Euler-Bernoulli beam, wherein the boundary anti-saturation controller can output an actuator control value;
judging whether the control value of the actuator reaches the upper limit of the saturation of the actuator according to the output of the boundary anti-saturation controller;
if so, designing an auxiliary system according to the boundary state information of the Euler-Bernoulli beam and the part exceeding the saturation upper limit of the actuator, wherein the auxiliary system is used for eliminating the saturation nonlinear influence of the actuator;
when the Euler-Bernoulli beam is determined to be stable, acquiring a gain value range of the boundary anti-saturation controller;
wherein the boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam comprises the following steps of:
obtaining a system equation and boundary conditions of the Euler-Bernoulli beam according to a Lagrange equation and a Hamilton principle;
wherein, the expression of the system equation of the Euler-Bernoulli beam is as follows:
the expression of the boundary condition is:
wherein, ω (x, t) < 0, L]X [0, + ∞) → R denotes the transverse displacement of the beam in space x and time T coordinates, p, T, EIM and L represent the mass per unit length of the beam, the tension, the bending stiffness, the mass of the tip load and the length of the beam system, respectively, omegax(x, t) and ωt(x, t) represents the derivative of the transverse rod displacement ω (x, t) with respect to space x and with respect to time t, respectively, u (t) represents the boundary controller; sat (u (t)) represents the saturation controller, expressed as:
wherein u ismaxAnd uminRespectively representing an upper bound and a lower bound of the actuator;
wherein, the obtaining of the boundary state information of the Euler-Bernoulli beam comprises the following steps:
obtaining the boundary speed omega in the system equation of the Euler-Bernoulli beamt(L, t), boundary displacement ωx(L, t) and the speed ω of the boundary bendingxt(L，t)；
Wherein the expression of the auxiliary system is:
wherein z (t) represents the state of the auxiliary system, kzα, β, ε each represent a constant greater than 0, sgn (z (t)) represents a sign function; Δ u (t) denotes a saturation dead band function, denoted Δ u (t) sat (u (t)) u (t);
wherein the expression of the boundary anti-saturation controller is:
u(t)＝-kaωt(L，t)+kbωx(L，t)-kcωxt(L，t)+kdz(t)，
wherein k isa，kb，kc，kdBoth represent gains of the boundary anti-saturation controller greater than 0.
2. The boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam as claimed in claim 1, wherein said obtaining the gain value range of the boundary anti-saturation controller when determining that the Euler-Bernoulli beam is stable comprises:
selecting a Lyapunov function, wherein the expression is as follows:
V(t)＝V1(t)+V2(t)+V3(t)+V4(t)，
wherein both alpha and beta are greater than 0, V1(t) represents an energy term consisting of kinetic and potential energy, V2(t) denotes the cross term, V3(t) represents an energy term, V, related to the state of the auxiliary system4(t) represents an auxiliary item;
verifying the positive nature of the Lyapunov function to obtain:
0＜α3(V1(t)+V3(t)+V4(t))≤V(t)≤α4(V1(t)+V3(t)+V4(t))，
wherein alpha is3＝min{(1-υ)，1}，α4＝max{(1+υ)，1}，
And verifying the first derivative negative nature of the Lyapunov function to the time t by combining the boundary state information and the boundary anti-saturation controller, and obtaining the gain value range of the boundary anti-saturation controller.
3. A boundary anti-saturation vibration suppression system based on a Euler-Bernoulli beam, which is used for realizing the boundary anti-saturation vibration suppression method based on the Euler-Bernoulli beam of claim 1 or 2, and is characterized by comprising the following steps:
the sensor is used for acquiring boundary state information of the Euler-Bernoulli beam;
the boundary anti-saturation controller is used for outputting a control value of the actuator according to the boundary state information of the Euler-Bernoulli beam;
the auxiliary system is used for eliminating the saturation nonlinear influence of the actuator according to the boundary state information of the Euler-Bernoulli beam;
and the actuator is used for receiving the control value of the boundary anti-saturation controller and acting on the Euler-Bernoulli beam.
4. The Euler-Bernoulli beam based boundary anti-saturation vibration suppression system according to claim 3, wherein said sensor comprises: the device comprises a laser displacement sensor, an inclinometer and a pressure strain gauge.
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