A method for suppressing vibration in a flexible structure domain

Abstract

The application discloses a flexible structure in-domain collaborative vibration suppression method, comprising the following steps: using a spatial one-dimensional partial differential equation model to describe the space-time dynamic behavior of vibration evolution of a flexible structure; wherein, actuators and sensors for vibration suppression are discretely arranged on the axial direction of the flexible structure; the actuators and the sensors are regarded as intelligent agents, a multi-agent consensus protocol is combined, and vibration measurement information obtained from the sensors is used to design a flexible structure in-domain collaborative vibration suppression strategy; and the collaborative vibration suppression strategy is applied to the one-dimensional partial differential equation model corresponding to the flexible structure, so that the flexible structure in-domain collaborative vibration suppression is realized. Compared with a traditional boundary vibration suppression mode, the application introduces a multi-agent collaborative control thought, so that the constructed collaborative vibration suppression algorithm is more flexible and robust, not only effectively improves the vibration suppression performance in the time and space dimensions, but also opens up a brand-new flexible structure collaborative vibration suppression research direction.

Description

Technical Field

[0001] This invention relates to the field of vibration suppression technology for large flexible structures, and particularly to a method for synergistic vibration suppression within a flexible structure domain. Background Technology

[0002] Large flexible structures have wide applications in practical engineering, such as satellite solar panels, large antennas, bridges, and skyscrapers. Most of these flexible structures can be simplified into flexible beams (e.g.,...). Figure 2 The simply supported beam shown is susceptible to vibration due to its low rigidity, making it vulnerable to external disturbances. This vibration affects the structural performance, accelerates fatigue damage, and reduces its service life. For example, large suspension bridges are subject to visible deformation due to thermal expansion and contraction, the weight of vehicles on the bridge deck, and natural wind, threatening bridge safety. Therefore, research on vibration suppression of flexible structures has significant practical application value.

[0003] The vibration evolution dynamics of flexible structures are more complex, encompassing dynamic information in both time and space dimensions, and are typically characterized by partial differential equations. Early research on vibration suppression in flexible structures largely employed the hypothetical modal method to obtain modal truncation models, and then designed vibration suppression algorithms based on these models. However, for flexible structures, due to their dense modalities and low damping, vibration suppression based on modal truncation models easily leads to control overflow problems. In recent years, designing vibration suppression strategies directly based on the structural partial differential equation model has received increasing attention. Considering the non-intrusive nature of control and measurement for specific structures, existing research has largely focused on boundary vibration suppression frameworks.

[0004] Of course, engineering practice also presents a type of vibration suppression problem for flexible structures, such as suspension bridges and skyscrapers. Typically, actuators and sensors are discretely placed along the axial direction to obtain more accurate and comprehensive vibration information and suppression effects. For boundary vibration suppression of such flexible structures, actuators and sensors can only be placed on the structure's boundaries. This limits the vibration suppression capability of proposed control algorithms for flexible structures to some extent, especially in cases with distributed external disturbances. To effectively suppress the vibration of flexible structures and improve their safety, actuators and sensors are placed inside the flexible structure (e.g., along the axial direction), such as... Figure 3 As shown, their quantity and placement can be adjusted according to actual needs (such as installation process, usage environment, etc.). This is completely different from existing boundary vibration suppression methods, belonging to the category of control within the distributed parameter system domain, and can fundamentally improve the flexibility and robustness of vibration suppression strategies. Literature review reveals that research on vibration suppression problems within flexible structure domains based on partial differential equation models remains open.

[0005] Although multi-agent cooperative control has been extensively and deeply studied and has yielded fruitful results, the question of how to introduce the concept of multi-agent cooperative control and construct a cooperative vibration suppression algorithm to fundamentally improve the vibration suppression performance of flexible structures in both time and space dimensions remains open, especially the corresponding analysis of consistent space realization characterized by vibration measurement output. Summary of the Invention

[0006] This invention provides a method for cooperative vibration suppression within a flexible structure domain, addressing the technical problem that existing technologies lack effective solutions to introduce the concept of multi-agent cooperative control and construct a cooperative vibration suppression algorithm to fundamentally improve the vibration suppression performance of flexible structures in both time and space dimensions.

[0007] To solve the above-mentioned technical problems, the present invention provides the following technical solution:

[0008] A method for synergistic vibration suppression within a flexible structure domain includes:

[0009] The spatiotemporal dynamic behavior of vibration evolution of flexible structures is characterized by a one-dimensional partial differential equation model; where actuators and sensors for vibration suppression are discretely arranged along the axial direction of the flexible structure.

[0010] By treating actuators and sensors as intelligent agents, combining a multi-agent consensus protocol, and utilizing vibration measurement information obtained from sensors, a collaborative vibration suppression strategy within the flexible structure domain is designed.

[0011] The cooperative vibration suppression strategy within the flexible structure domain is applied to the one-dimensional partial differential equation model corresponding to the flexible structure to achieve cooperative vibration suppression within the flexible structure domain.

[0012] Furthermore, the method for synergistic vibration suppression within the flexible structure domain also includes:

[0013] By constructing an energy function method and combining it with integral by parts, the problem of cooperative vibration suppression in the domain of flexible structures is transformed into a feasibility problem constrained by linear matrix inequalities.

[0014] Furthermore, the equations of the one-dimensional partial differential equation model are as follows:

[0015]

[0016] Constrained by the following boundary conditions

[0017] w(0,t)=w(L,t=0,w x (x,t)| x＝0,L =0,t>0 (2a)

[0018] or

[0019]

[0020] and initial conditions

[0021] w(x,0)=w0(x),w t (x,0)=w1(x),x∈[0,L] (3)

[0022] Where w(x,t) represents the vibration offset of the flexible structure at position x at time t, the subscripts x and t represent its partial derivatives with respect to position x and time t, respectively, ρ represents the density of the structure, A represents the cross-sectional area of ​​the structure, and EI represents the bending stiffness of the structure. The control input used for vibration suppression is provided by m actuators; L represents the axial length of the flexible structure.

[0023] Furthermore, the actuators are discretely distributed along the axial direction of the structure, and their distribution is determined by a function. Characterization, function The specific definition is:

[0024]

[0025] in, and If let Then expression (4) describes the case where the actuator acts only at a certain point, corresponding to the point control case of the distributed parameter system;

[0026] w0(x) and w1(x) are defined to represent the initial vibration state of the flexible structure; boundary condition (2a) describes the spatiotemporal vibration dynamics of the simply supported beam, while boundary condition (2b) describes the spatiotemporal vibration behavior of the cantilever beam.

[0027] Indicates the length of the actuator-sensor pair's operating area. This indicates the left boundary of the actuator-sensor interaction area. This indicates the right boundary of the actuator-sensor's area of ​​action;

[0028] Vibration measurement output of flexible structure for:

[0029]

[0030] In this configuration, the sensors and actuators are placed in the same location. The placement of the sensors within the space (0,L) and the placement of the actuators within the spatial domain are determined by a function. To depict.

[0031] Furthermore, by treating actuators and sensors as intelligent agents, and combining a multi-agent consensus protocol with vibration measurement information obtained from sensors, a cooperative vibration suppression strategy within the flexible structure domain is designed, including:

[0032] Treating actuators and sensors as intelligent agents, the communication topology between these agents is represented by an m-order directed / undirected time-varying weighted graph. Description, in which, It is a set of agent IDs. It is a set of edges, and It is ω ij (t) is the directed / undirected time-varying weighted adjacency matrix of the elements; the set of neighboring agents that communicate with agent i. Defined as If agent j is a neighbor of agent i, then agent i can communicate with agent j; for time-varying directed / undirected weighted graphs. Its Laplace matrix Defined as and It is a degree matrix, and its elements are ω ij (t) represents the time-varying communication topology connection weight between agents i and j;

[0033] For time-varying directed / undirected weighted graphs We introduce the following assumptions:

[0034] Assumption 1: If G(t) is a time-varying undirected graph, then it is assumed to be connected; if G(t) is a time-varying directed graph, then it is assumed to be strongly connected and balanced; where balanced means that the out-degree and in-degree of any node in the graph are equal.

[0035] If the time-varying directed / undirected weighted graph G(t) satisfies Assumption 1, then its Laplace matrix... satisfy:

[0036]

[0037] Combining a multi-agent consensus protocol, the following cooperative vibration suppression strategy is constructed:

[0038]

[0039] in, β(t)≥0 is the undetermined time-varying control gain, and β(t)≥0 is the given time-varying cooperative control gain.

[0040] Furthermore, under the aforementioned cooperative vibration suppression strategy, the closed-loop form of the partial differential equation is:

[0041]

[0042] Furthermore, the method for constructing the energy function, combined with integration by parts, transforms the problem of cooperative vibration suppression within the flexible structure domain into a feasibility problem constrained by linear matrix inequalities, including:

[0043] First, Lemma 1 gives the integration by parts technique used in the derivation:

[0044] Lemma 1: Let f(x) and g(x), x∈[0,L], be two functions of x, each having continuous derivatives df(x) / dx and dg(x) / dx, and integrals... If it exists, then There exists and there is a formula for integration by parts.

[0045] Theorem 1 Consider a class of flexible structures whose spatiotemporal vibration dynamics can be characterized by partial differential equations (1)-(3) and the time-varying communication topology G(t) between agents satisfies Assumption 1; for a given time-varying cooperative control gain β(t)>0, and If control gain exists The following conditions must be met:

[0046]

[0047] Then there exists a cooperative vibration suppression strategy that ensures the closed-loop equation (8) constrained by boundary conditions (2) and initial conditions (3) is stable and asymptotically realized by the equation y1(t)=y2(t)=…=y m (t) describes a one-dimensional uniform space;

[0048] 1) Stability analysis based on energy function

[0049]

[0050] Taking the derivative of equation (7) along the solution locus of the closed-loop equation (8), we get:

[0051]

[0052] Using integration by parts and considering the boundary conditions (2), we get:

[0053]

[0054] Applying the integration by parts technique again and considering the boundary conditions (2), we get:

[0055]

[0056] Substituting equations (12) and (13) into equation (11), we get:

[0057]

[0058] in,

[0059] Using inequality (9), for y(t) ≠ 0, we have:

[0060]

[0061] That is, the energy function defined by equation (10) along the solution trajectory of the closed-loop equation (8) decays;

[0062] 2) From the equation y1(t)=y2(t)=…=y m (t) represents the asymptotic realization of the consistency space.

[0063] make Then the following inequality holds.

[0064]

[0065] Among them, the inequality sign is based on facts. get, It is a matrix The largest eigenvalue;

[0066] According to equation (14), by contradiction, we obtain: v(t)→0, as t→∞; then according to the definition of v(t), from the equation y1(t)=y2(t)=…=y m The one-dimensional uniform space described by (t) is asymptotically realized, Q.E.D.

[0067] The beneficial effects of the technical solution provided by this invention include at least the following:

[0068] 1. The cooperative vibration suppression strategy design in the flexible structure domain cooperative vibration suppression method provided by the present invention fully considers the evolution process of flexible structure vibration in both time and space dimensions, and can effectively overcome the problem of control overflow caused by vibration suppression based on the modal truncation model.

[0069] 2. The collaborative vibration suppression strategy in the flexible structure domain collaborative vibration suppression method provided by the present invention adopts an intra-domain control method, which can easily introduce the concept of multi-agent collaborative control, improve vibration suppression performance in both time and space dimensions, and enhance the safety of flexible structures.

[0070] 3. The partial differential equation model based on flexible structures proposed in this invention, combined with multi-agent cooperative control technology, designs a cooperative vibration suppression strategy, which can not only effectively improve the vibration suppression performance in both time and space dimensions, but also the algorithm has better flexibility and robustness.

[0071] 4. By leveraging the observer output feedback control technology, the cooperative vibration suppression strategy in the flexible structure domain cooperative vibration suppression method provided by this invention can be easily extended to non-co-located vibration measurement scenarios. Attached Figure Description

[0072] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0073] Figure 1 This is a flowchart of the method for suppressing cooperative vibration within a flexible structure domain provided in an embodiment of the present invention;

[0074] Figure 2 This is a schematic diagram of a type of flexible beam, where L is the length of the flexible beam and w(x,t) is the vibration offset.

[0075] Figure 3 This is a schematic diagram of piezoelectric transducer patches discretely arranged along the axial direction of a flexible beam. The upper side is the actuator, and the lower side is the sensor. The patches share vibration information through communication.

[0076] Figure 4 This is a spatiotemporal evolution profile of flexible beam vibration under the synergistic vibration suppression strategy of the present invention.

[0077] Figure 5 This is a schematic diagram of the closed-loop evolution trajectory of the vibration offset of a flexible beam driven by the cooperative vibration suppression strategy of this invention and the traditional non-cooperative vibration suppression strategy;

[0078] Figure 6 It is the closed-loop vibration offset of the flexible beam driven by the cooperative vibration suppression strategy and the traditional non-cooperative vibration suppression strategy: the vibration offset w(x,t) on the spatial axis at different times t=0.3, t=1, t=1.5. Detailed Implementation

[0079] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0080] This embodiment provides a method for intra-domain cooperative vibration suppression of flexible structures, which improves vibration suppression performance by introducing the concept of multi-agent cooperative control. This method not only considers the dynamic information of flexible structure vibration in both time and space dimensions, but also employs intra-domain control to incorporate the concept of multi-agent cooperative control, ultimately achieving cooperative vibration suppression of flexible structures. This provides effective technical support and theoretical guidance for enhancing the safety performance of flexible structures, such as spacecraft, bridges, and super high-rise buildings.

[0081] This method targets a class of flexible structures with multiple actuators and sensors discretely configured along their axial direction. Treating the actuators and sensors as intelligent agents, it proposes an intra-domain cooperative vibration suppression method by introducing the concept of multi-agent cooperative control. First, based on a spatial one-dimensional partial differential equation model characterizing the vibration of the flexible structure, vibration measurement information obtained from sensors is used, combined with a multi-agent consensus protocol, to construct an intra-domain cooperative vibration suppression algorithm, aiming to simultaneously improve the vibration suppression performance in both time and space dimensions. By constructing a suitable energy function and utilizing integration by parts, sufficient conditions for the existence of this intra-domain cooperative vibration suppression algorithm are given, transforming the cooperative vibration suppression problem of flexible structures into a feasibility problem constrained by linear matrix inequalities. For the more general case of non-co-location measurement, this cooperative vibration suppression strategy can be easily extended using observer output feedback control technology.

[0082] Specifically, the execution flow of this method is as follows: Figure 1 As shown, it includes the following steps:

[0083] S1 uses a one-dimensional partial differential equation model to characterize the spatiotemporal dynamic behavior of the vibration evolution of the flexible structure; where actuators and sensors for vibration suppression are discretely arranged along the axial direction of the flexible structure.

[0084] S2 treats actuators and sensors as intelligent agents, combines a multi-agent consensus protocol and utilizes vibration measurement information obtained from sensors to design a cooperative vibration suppression strategy within the flexible structure domain.

[0085] It should be noted that, for more general non-co-located vibration measurement scenarios, the cooperative vibration suppression strategy presented in this embodiment can be easily extended by leveraging observer output feedback control technology.

[0086] S3, The cooperative vibration suppression strategy within the flexible structure domain is applied to the one-dimensional partial differential equation model corresponding to the flexible structure to achieve cooperative vibration suppression within the flexible structure domain.

[0087] Furthermore, the method for synergistic vibration suppression within the flexible structure domain also includes:

[0088] By constructing an energy function method and combining it with integral by parts, the problem of cooperative vibration suppression in the domain of flexible structures is transformed into a feasibility problem constrained by linear matrix inequalities.

[0089] Specifically, the design steps for the cooperative vibration suppression strategy within the flexible structure domain are as follows:

[0090] 1) Partial differential equation model of vibration of flexible structure

[0091] Suppose a class of flexible structures whose spatiotemporal dynamic evolution behavior of vibration can be characterized by the following partial differential equation:

[0092]

[0093] Constrained by the following boundary conditions

[0094] w(0,t)=w(L,t=0,w x (x,t)| x＝0,L =0,t>0 (2a)

[0095] or

[0096]

[0097] and initial conditions

[0098] w(x,0)=w0(x),w t (x,0)=w1(x),x∈[0,L] (3)

[0099] Where w(x,t) represents the vibration offset of the flexible structure at position x at time t, the subscripts x and t represent its partial derivatives with respect to position x and time t, respectively, ρ represents the density of the structure, A represents the cross-sectional area of ​​the structure, and EI represents the bending stiffness of the structure. This represents the control input used for vibration suppression, which is provided by m actuators. L represents the axial length of the flexible structure;

[0100] These actuators are discretely distributed along their axial direction, and their distribution is determined by a function. Characterization. This function. The specific definition is:

[0101]

[0102] in, and If let Then expression (4) describes the case where the actuator acts only at a certain point, corresponding to the point control case of the distributed parameter system; w0(x) and w1(x) are defined to represent the initial vibration state of the flexible structure; boundary condition (2a) describes the spatiotemporal vibration dynamics of a simply supported beam (such as a bridge), while boundary condition (2b) describes the spatiotemporal vibration behavior of a cantilever beam (such as a skyscraper or a flexible robotic arm); Indicates the length of the actuator-sensor pair's operating area. This indicates the left boundary of the actuator-sensor interaction area. This indicates the right boundary of the actuator-sensor's area of ​​action;

[0103] Vibration measurement output of flexible structure for:

[0104]

[0105] The placement of the sensors within the space (0,L) is determined by a function. Characterization. Note that in measurement equation (5), the sensor deployment function is characterized. It is also used to characterize the placement of actuators in the spatial domain, which means that the sensors and actuators are placed in the same location.

[0106] 2) Collaborative vibration suppression strategy

[0107] Treating actuators and sensors as intelligent agents, the communication topology between these agents is derived from a more general m-order directed / undirected time-varying weighted graph. Description, in which, It is a set of agent IDs. It is a set of edges, and It is ω ij (t) is the directed / undirected time-varying weighted adjacency matrix of the elements; the set of neighboring agents that communicate with agent i. Defined as If agent j is a neighbor of agent i, then agent i can communicate with agent j; for time-varying directed / undirected weighted graphs. Its Laplace matrix Defined as and It is a degree matrix, and its elements are ω ij (t) represents the time-varying communication topology connection weight between agents i and j;

[0108] For time-varying directed / undirected weighted graphs We introduce the following assumptions:

[0109] Assumption 1: If G(t) is a time-varying undirected graph, then it is assumed to be connected; if G(t) is a time-varying directed graph, then it is assumed to be strongly connected and balanced; where balanced means that the out-degree and in-degree of any node in the graph are equal.

[0110] If the time-varying directed / undirected weighted graph G(t) satisfies Assumption 1, then its Laplace matrix... satisfy:

[0111]

[0112] Combining a multi-agent consensus protocol, the following cooperative vibration suppression strategy is constructed:

[0113]

[0114] in, β(t)≥0 is the undetermined time-varying control gain, and β(t)≥0 is the given time-varying cooperative control gain. If β(t)=0, the cooperative vibration suppression strategy (7) degenerates into the traditional non-cooperative vibration suppression strategy.

[0115] Under the action of the cooperative vibration suppression strategy (7), the closed-loop form of the partial differential equation (1) is:

[0116]

[0117] 3) Construct an energy function method and combine it with integration by parts to obtain sufficient conditions for the existence of the cooperative vibration suppression strategy. The cooperative vibration suppression problem within the flexible structure domain is then transformed into a feasibility problem constrained by linear matrix inequalities. The steps are as follows:

[0118] First, Lemma 1 gives the integration by parts technique used in the derivation:

[0119] Lemma 1: Let f(x) and g(x), x∈[0,L], be two functions of x, each having continuous derivatives df(x) / dx and dg(x) / dx, and integrals... If it exists, then There exists and there is a formula for integration by parts.

[0120] Theorem 1 Consider a class of flexible structures whose spatiotemporal vibration dynamics can be characterized by partial differential equations (1)-(3) and the time-varying communication topology G(t) between intelligent agents (actuators and sensors) satisfies Assumption 1; for a given time-varying cooperative control gain β(t) > 0, and If control gain exists The following conditions must be met:

[0121]

[0122] Then there exists a cooperative vibration suppression strategy (7) that ensures that the closed-loop equation (8) constrained by the boundary conditions (2) and the initial conditions (3) is stable and asymptotically realized by the equation y1(t)=y2(t)=…=y m (t) describes a one-dimensional uniform space;

[0123] (1) Stability analysis based on energy function

[0124]

[0125] By taking the derivative of equation (7) along the solution locus of the closed-loop equation (8), we can obtain:

[0126]

[0127] By applying the integration by parts technique (i.e., Lemma 1) and considering the boundary condition (2), we can obtain:

[0128]

[0129] By applying the integration by parts technique again and considering the boundary conditions (2), we can obtain:

[0130]

[0131] Substituting equations (12) and (13) into equation (11), we get:

[0132]

[0133] in,

[0134] Using inequality (9), for y(t) ≠ 0, we have:

[0135]

[0136] That is, the energy function defined by equation (10) along the solution trajectory of the closed-loop equation (8) decays;

[0137] (2) From the equation y1(t)=y2(t)=…=y m (t) represents the asymptotic realization of the consistency space.

[0138] make Then the following inequality holds.

[0139]

[0140] Among them, the inequality sign is based on facts. We obtain (according to inequality (6)) here, It is a matrix The largest eigenvalue;

[0141] According to equation (14), by contradiction, it is easy to obtain the following conclusion: v(t)→0, as t→∞; and according to the definition of v(t), from the equation y1(t)=y2(t)=…=y m The one-dimensional uniform space described by (t) is asymptotically realized, Q.E.D.

[0142] The effectiveness of the method in this embodiment will be explained below with reference to practical application examples.

[0143] Consider a class of simply supported beams, let ρ = 2729.5, A = 1.471 × 10⁻⁶. -3 E = 66.26 × 10 9 I = 1.14 × 10 -8 L = 1. For the placement of the co-position sensor-actuator along the beam axis (i.e., within the spatial region [0,1]), select... Based on inequality (9), we select time-varying control gains k1(t) = 0.8, k2(t) = 1.2, k3(t) = 0.9, time-varying cooperative control gain β(t) = 5, and the undirected Laplace matrix of the time-varying communication topology G between the three pairs of co-actuator-sensor pairs.

[0144]

[0145] To fully demonstrate the effectiveness and robustness of the cooperative control strategy, a distributed disturbance d(x,t) is added to the partial differential equation (1). In this case, equation (1) can be modified as follows:

[0146]

[0147] Assume the initial offset of the simply supported beam is w0(x)=sin(πcos(πx)), x∈[0,1], the initial offset velocity is w1(x)=0, x∈[0,1], and the distributed disturbance value is d(x,t)=sin(πx / L)sin(t). Applying the intra-domain cooperative vibration suppression strategy (7) with the above parameter values ​​to the differential equation (17), the spatiotemporal evolution profile of the vibration offset w(x,t) of the simply supported beam and the closed-loop evolution trajectory of |w(·,t)|2 are as follows. Figure 4 and 5 . Figure 6 The closed-loop vibration offsets of the flexible beam driven by the cooperative vibration suppression strategy of this embodiment and the traditional non-cooperative vibration suppression strategy are given respectively: the vibration offsets w(x,t) on the spatial axis at different times t=0.3, t=1, and t=1.5. Meanwhile, Figure 5The closed-loop evolution trajectory of the vibration offset of the flexible beam driven by the traditional non-cooperative vibration suppression strategy is also presented. The above simulation results reveal the effectiveness and robustness of the cooperative control strategy constructed in this embodiment, as well as the significant improvement in both time and space dimensions compared with the traditional non-cooperative vibration suppression performance.

[0148] In summary, the method in this embodiment directly designs a vibration suppression algorithm for the partial differential equation model of the flexible structure, effectively overcoming the problem of control overflow that easily occurs when vibration suppression is based on a modal truncation model. Unlike existing boundary vibration suppression methods, the actuators and sensors used for vibration suppression are discretely placed along the axial direction of the flexible structure. These actuators and sensors are treated as intelligent agents, and a cooperative vibration suppression strategy within the flexible structure domain is proposed using a multi-agent consensus protocol and vibration measurement information obtained from the sensors, simultaneously improving vibration suppression performance in both time and space dimensions. By constructing a suitable energy function and applying integral-from-parts techniques, sufficient conditions for the existence of this cooperative vibration suppression strategy within the domain are given, ultimately reducing the cooperative vibration suppression problem within the flexible structure domain to a feasibility problem constrained by linear matrix inequalities. Furthermore, with the help of observer output feedback control technology, this cooperative vibration suppression strategy can be easily extended to non-co-located vibration measurement scenarios. Therefore, it can provide effective technical support and theoretical guidance for scenarios such as enhancing the safety performance of flexible structures, including spacecraft, bridges, and super high-rise buildings.

[0149] Furthermore, it should be noted that the present invention can be provided as a method, apparatus, or computer program product. Therefore, embodiments of the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Moreover, embodiments of the present invention can take the form of a computer program product implemented on one or more computer-usable storage media containing computer-usable program code.

[0150] Embodiments of the present invention are described with reference to flowchart illustrations and / or block diagrams of methods, terminal devices (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, embedded processor, or other programmable data processing terminal device to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing terminal device, generate instructions for implementing the flowchart illustrations. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0151] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing terminal device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The functions specified in one or more boxes. These computer program instructions may also be loaded onto a computer or other programmable data processing terminal equipment to cause a series of operational steps to be performed on the computer or other programmable terminal equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable terminal equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0152] It should also be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or terminal device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or terminal device. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or terminal device that includes said element.

[0153] Finally, it should be noted that the above description represents a preferred embodiment of the present invention. It should be pointed out that although preferred embodiments have been described, those skilled in the art, once they understand the basic inventive concept of the present invention, can make various improvements and modifications without departing from the principles described herein. These improvements and modifications should also be considered within the scope of protection of the present invention. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the embodiments of the present invention.

Claims

1. A method for suppressing cooperative vibration within a flexible structure domain, characterized in that, include: The spatiotemporal dynamic behavior of vibration evolution of flexible structures is characterized by a one-dimensional partial differential equation model; where actuators and sensors for vibration suppression are discretely arranged along the axial direction of the flexible structure. By treating actuators and sensors as intelligent agents, combining a multi-agent consensus protocol, and utilizing vibration measurement information obtained from sensors, a collaborative vibration suppression strategy within the flexible structure domain is designed. The cooperative vibration suppression strategy within the flexible structure domain is applied to the one-dimensional partial differential equation model corresponding to the flexible structure to achieve cooperative vibration suppression within the flexible structure domain. The equations of the one-dimensional partial differential equation model are as follows: Constrained by the following boundary conditions w(0,t)=w(L,t)=0,w x (x,t)| x＝0,L =0,t>0 (2a) or and initial conditions w(x,0)=w0(x),w t (x,0)=w1(x),x∈[0,L] (3) Where w(x,t) represents the vibration offset of the flexible structure at position x at time t, the subscripts x and t represent its partial derivatives with respect to position x and time t, respectively, ρ represents the density of the structure, A represents the cross-sectional area of ​​the structure, and EI represents the bending stiffness of the structure. The control input for vibration suppression is provided by m actuators; L represents the axial length of the flexible structure. The actuators are discretely distributed along the axial direction of the structure, and their distribution is determined by the function g. i Characterize (x), i∈M, function The specific definition is: in, and If let Then expression (4) describes the case where the actuator acts only at a certain point, corresponding to the point control case of the distributed parameter system; w0(x) and w1(x) are defined to represent the initial vibration state of the flexible structure; boundary condition (2a) describes the spatiotemporal vibration dynamics of the simply supported beam, while boundary condition (2b) describes the spatiotemporal vibration behavior of the cantilever beam. Indicates the length of the actuator-sensor pair's operating area. This indicates the left boundary of the actuator-sensor interaction area. This indicates the right boundary of the actuator-sensor's area of ​​action; Vibration measurement output of flexible structure for: In this configuration, the sensors and actuators are placed in the same location. The placement of the sensors within the space (0,L) and the placement of the actuators within the spatial domain are determined by a function. Depiction; The method of treating actuators and sensors as intelligent agents, combining multi-agent consensus protocols, and utilizing vibration measurement information obtained from sensors to design a cooperative vibration suppression strategy within the flexible structure domain includes: Treating actuators and sensors as intelligent agents, the communication topology between these agents is represented by an m-order directed / undirected time-varying weighted graph. Description, in which, It is a set of agent IDs. It is a set of edges, and It is ω ij (t) is the directed / undirected time-varying weighted adjacency matrix of the elements; the set of neighboring agents that communicate with agent i. Defined as If agent j is a neighbor of agent i, then agent i can communicate with agent j; for time-varying directed / undirected weighted graphs. Its Laplace matrix Defined as and It is a degree matrix, and its elements are ω ij (t) represents the time-varying communication topology connection weight between agents i and j; For time-varying directed / undirected weighted graphs We introduce the following assumptions: Assumption 1: If G(t) is a time-varying undirected graph, then it is assumed to be connected; if G(t) is a time-varying directed graph, then it is assumed to be strongly connected and balanced; where balanced means that the out-degree and in-degree of any node in the graph are equal. If the time-varying directed / undirected weighted graph G(t) satisfies Assumption 1, then its Laplace matrix... satisfy: Combining a multi-agent consensus protocol, the following cooperative vibration suppression strategy is constructed: in, y is the undetermined time-varying control gain, where β(t)≥0 is the given time-varying cooperative control gain; i (t) represents the vibration measurement output of the flexible structure.

2. The method for suppressing coordinated vibration within a flexible structure domain as described in claim 1, characterized in that, The method for coordinated vibration suppression within the flexible structure domain also includes: By constructing an energy function method and combining it with integral by parts, the problem of cooperative vibration suppression in the domain of flexible structures is transformed into a feasibility problem constrained by linear matrix inequalities.

3. The method for suppressing coordinated vibration within a flexible structure domain as described in claim 2, characterized in that, Under the aforementioned collaborative vibration suppression strategy, the closed-loop form of the partial differential equation is:

4. The method for suppressing coordinated vibration within a flexible structure domain as described in claim 3, characterized in that, The method for constructing the energy function, combined with the integration by parts technique, transforms the problem of cooperative vibration suppression within the flexible structure domain into a feasibility problem constrained by linear matrix inequalities, including: First, Lemma 1 gives the integration by parts technique used in the derivation: Lemma 1: Let f(x) and g(x), x∈[0,L], be two functions of x, each having continuous derivatives df(x) / dx and dg(x) / dx, and integrals... If it exists, then There exists and there is a formula for integration by parts. Theorem 1 Consider a class of flexible structures whose spatiotemporal vibration dynamics can be characterized by partial differential equation (1) and whose time-varying communication topology G(t) between agents satisfies Assumption 1; for a given time-varying cooperative control gain β(t) > 0, and If control gain exists The following conditions must be met: Then there exists a cooperative vibration suppression strategy that ensures the closed-loop equation (8) constrained by boundary conditions (2a), (2b) and initial condition (3) is stable and asymptotically realized by the equation y1(t)=y2(t)=…=y m (t) describes a one-dimensional uniform space; 1) Stability analysis based on energy function Taking the derivative of equation (10) along the solution locus of the closed-loop equation (8), we get: Using integration by parts and considering boundary conditions (2a) and (2b), we get: Applying the integration by parts technique again and considering the boundary conditions (2a) and (2b), we get: Substituting equations (12) and (13) into equation (11), we get: in, Using inequality (9), for y(t) ≠ 0, we have: That is, the energy function defined by equation (10) along the solution trajectory of the closed-loop equation (8) decays; 2) From the equation y1(t)=y2(t)=…=y m (t) represents the asymptotic realization of the consistency space. make Then the following inequality holds. Among them, the inequality sign is based on facts. get, It is a matrix The largest eigenvalue; According to equation (14), by contradiction, we obtain: v(t)→0, as t→∞; then according to the definition of v(t), from the equation y1(t)=y2(t)=…=y m The one-dimensional uniform space described by (t) is asymptotically realized, Q.E.D.

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