Vibration suppression control method and system for a flexible continuum structure

Abstract

This invention provides a vibration suppression control method and system for flexible continuum structures. Addressing the severe underdrive and modal overflow problems caused by order reduction of the finite element model of a flexible continuum, this invention combines Craig-Bampton interface condensation and full-drive system theory. Through Schur algebraic elimination, uncontrollable internal modal acceleration couplings are eliminated, reconstructing the high-dimensional sparse system into a second-order equivalent full-drive model with only actuating nodes as the main components. Based on this, a first-order state-space cascaded extended state observer (ESO) is designed. Utilizing the boundary dynamic coupling characteristics, it online and in real-time removes lumped physical disturbances, including high-order truncated residuals and nonlinear variations, and reconstructs the unmeasurable internal modal coordinates and velocities across dimensions without bias. Finally, through full-drive inverse dynamics feedforward cancellation and generalized PID pole placement, the closed-loop actuating nodes are reconstructed into virtual damped dissipative dynamic vibration absorbers.

Description

Technical Field

[0001] This invention relates to the field of flexible structure control technology, specifically to a vibration suppression control method and system for flexible continuous structures. Background Technology

[0002] Flexible continuum structures (such as large trusses, space antennas, and flexible robotic arms) are widely used in aerospace, industrial manufacturing, and other fields. To accurately describe their dynamic characteristics, the finite element method (FEM) is typically used in engineering for spatial discretization, establishing high-dimensional second-order ordinary differential equations that include the system's mass, stiffness, and damping matrices. In active vibration suppression tasks for such structures, due to weight, space, and cost constraints, actuators such as linear motors and piezoelectric ceramics can usually only be discretely arranged on a limited number of physical nodes, resulting in an extremely sparse input control matrix for the system.

[0003] Under the aforementioned finite element framework of sparse motion, existing vibration control theories face the following three core dilemmas: Traditional methods for addressing the curse of high dimensionality in FEM typically involve forcibly reducing the rank of the second-order dynamic equations to transform them into a first-order state-space model. This process completely destroys the physical topology of the original mass, damping, and stiffness matrices. Simultaneously, due to sparse actuation, the transformed input matrix is ​​highly non-full rank, causing the system to fall into a severe underactuated state. This results in the gain matrix of the control law losing its clear physical meaning, meaning it cannot directly correspond to stiffness or damping corrections in physical space, turning engineering parameter tuning into blind black-box trial and error.

[0004] Existing robust control strategies often crudely attribute the elastic vibration of undriven nodes to bounded disturbances and simply endure them. This conservative approach not only severely limits the system's control bandwidth, but also easily induces high-frequency energy injection and control overflow when the spatial distribution of control force orthogonally projects onto the unmodeled high-order mode array, leading to high-frequency resonance or even instability of the structure.

[0005] Most existing vibration suppression patents are tied to a single control algorithm for a specific scenario, such as pure PID or a specific adaptive law. They lack an intermediate framework that can uniformly map sparse nonlinear plant models to standard linear controlled objects, which limits the flexible deployment of advanced integrated algorithms such as LQR on complex finite element models.

[0006] To alleviate some of the aforementioned difficulties, existing technologies have made some explorations. For example, Chinese patent CN117260737A discloses a vibration suppression control method for gantry robots. This method establishes a mathematical model of a flexible gantry robot system, performs linearization and state quantity reconstruction, designs an improved backstepping control algorithm, and adds a disturbance observer to counteract mismatched external disturbances, thereby achieving vibration suppression and anti-interference control to a certain extent.

[0007] However, these existing technologies are still bound to specific nonlinear control laws (such as backstepping control), and their derivation and parameter tuning processes heavily rely on precisely known parts of the system model, failing to address the severe underactuated physical characteristics caused by sparse actuation. Due to the lack of a standardized intermediate-layer model transformation for full-drive, the system still retains a complex, strongly coupled nonlinear dynamic form, making it impossible to directly and flexibly deploy conventional high-order linear synthesis algorithms (such as standard PID, LQR, etc.). At the same time, when facing the elastic vibration of high-dimensional undriven nodes, it still faces the risk of mode overflow instability and struggles to establish active energy dissipation chains online.

[0008] Therefore, how to construct a generalized and standardized closed-loop control architecture that can achieve full-drive decoupling, has physical parameter tuning significance, and allows for flexible deployment of multiple algorithms for sparsely driven high-dimensional finite element systems is a technical problem that urgently needs to be solved in the field of active vibration control of flexible continuum structures. Summary of the Invention

[0009] To address the shortcomings of existing technologies, the purpose of this invention is to provide a vibration suppression and control method and system for flexible continuum structures.

[0010] According to one aspect of the present invention, a vibration suppression control method for a flexible continuum structure is provided, wherein a linear actuator and a sensor are disposed at predetermined nodes on the flexible continuum structure, comprising: Step S1: Discretize the flexible continuum structure using the finite element method to obtain multiple nodes, and divide the obtained nodes into observable and controllable actuator nodes and unobservable internal undriven nodes; reduce the dimensionality of the dynamic equations of the nodes in the absolute physical coordinate system using the Craig-Bampton method; where the actuator nodes are the predetermined nodes where the linear actuators and sensors are located. Step S2: Based on the Schur elimination method, the reduced dynamic equations are converted into a second-order standard total drive form with only the acceleration of the actuator nodes as the dominant variable; Step S3: Based on the first-order augmented state form of the dimensionality-reduced dynamic equations, construct a first-order observer, wherein the first-order observer includes: Disturbance stripping layer: The displacement of the actuator node is obtained through sensors, and the velocity and lumped disturbance of the actuator node are observed; Modal reconstruction layer: Utilizes the physical boundary coupling matrix between actuator acceleration and internal modes to perform unbiased estimation of internal modes; Step S4: Construct a closed-loop controller for the all-drive system based on the second-order standard all-drive configuration and the state estimate output by the first-order observer; Step S5: Input the actual displacement of the linear actuator collected by the sensor into the first-order observer to obtain the state estimate. Substitute the state estimate into the all-drive closed-loop controller to obtain the virtual all-drive control quantity. Substitute the virtual all-drive control quantity and the state estimate into the physical control law mapping equation to obtain the control input vector executed by the linear actuator. Control the linear actuator to work through the control input vector.

[0011] Preferably, step S1 includes: Step S1.1: Discretize the flexible continuum structure using the finite element method to obtain multiple nodes. The dynamic equations of each node in the absolute physical coordinate system are as follows:

[0012] In the formula, These are the full-order physical mass, damping, and stiffness matrices, respectively; These are the displacement, velocity, and acceleration vectors of the node, respectively. This is the control input matrix for the actuator node; The ideal force vector output by the linear actuator; The input matrix is ​​the perturbation matrix; This is an unknown external disturbance; Step S1.2: Based on the permutation matrix T, divide the displacement vector of each node into the displacement vector of the driven node and the displacement vector of the internal undriven node, where:

[0013] Step S1.3: Substitute equation (1.2) into equation (1.1) and multiply by the left side. This yields the full-order dynamic equations in a block-decoupled form:

[0014] in:

[0015]

[0016]

[0017]

[0018]

[0019] In the formula, The physical mass matrix of the active node; It is the physical inertial coupling matrix from the internal undriven nodes to the driven nodes. Its physical meaning is the mapping of the reaction inertial force transmitted to the driven nodes through the structure when the internal nodes accelerate. This is the physical inertial coupling matrix from the driven node to the internal undriven nodes; The internal undriven node's self-owned physical mass matrix; The physical damping matrix of the actuator node; This is the physical dissipation coupling matrix from the internal undriven nodes to the driven nodes; This is the physical dissipation coupling matrix from the active node to the internal undriven nodes; The internal undriven node's self-owned physical damping matrix; The physical stiffness matrix of the actuating node; This is the physical elastic coupling matrix from the internal undriven nodes to the driven nodes; This is the physical elastic coupling matrix from the actuator node to the internal undriven nodes; The internal undriven node's self-physical stiffness matrix; The control input matrix is ​​divided into blocks according to the actuator nodes and the internal undriven nodes; and The input submatrix of the perturbation is divided into blocks according to the active nodes and the internal undriven nodes; The displacement, velocity, and acceleration vectors of the actuating node; This represents the displacement, velocity, and acceleration vectors of the internal, undriven nodes. Step S1.4: Introduce static constraint modes and the truncated fixed interface main mode Construct the CB hybrid state-space transformation matrix ,in:

[0020] In the formula, To preserve the coordinates of the lower-order principal modes, This refers to the higher-order mode truncation error; It is the identity matrix; Step S1.5: Satisfy the engineering assumptions of proportional damping characteristics, using... By performing a similarity transformation on equation (1.3), the full-order dynamic equation is reduced in dimension and re-divided into two parts: boundary dynamics and internal modal dynamics.

[0021] In the formula, The equivalent boundary mass matrix of the actuating nodes after condensation; This is the dynamic inertial coupling matrix of the internal principal mode to the actuation boundary; This is the dynamic inertial coupling matrix from the actuation boundary to the internal main mode; The equivalent boundary damping matrix of the actuating node after condensation; This is the equivalent boundary stiffness matrix of the actuating nodes after condensation; This is the equivalent physical perturbation input matrix acting on the actuator node space; This is the equivalent modal perturbation input matrix acting on the internal modal space; and These are the modal damping matrix and modal stiffness matrix after condensation, respectively; To preserve the coordinates, velocity, and acceleration of the low-frequency core modes; The displacement, velocity, and acceleration vectors of the actuating node;

[0022]

[0023] and For the truncation error of higher-order modes The resulting residual dynamics terms are treated as physical perturbations in subsequent steps.

[0024] Preferably, step S2 includes: Step S2.1: Expand equation (1.5) and perform appropriate transformations to obtain:

[0025]

[0026] Step S2.2: Extract from equation (1.7) The explicit expression of the displacement vector of the actuating node is substituted into equation (1.6) for algebraic elimination. After combining like terms, the following expression for the displacement vector of the actuating node is obtained. Independent dynamic equations:

[0027] in, The Schur complement of the block mass matrix about the main diagonal constitutes the equivalent core mass after the dynamic equation is reduced in order and decoupled. This refers to the lumped disturbance acting at the actuation node; Step S2.3: Based on formula Constructing the isotopic all-drive inverse dynamic control law:

[0028] in, For the new virtual control input to be designed; and These are estimated values ​​for the coordinates of unmeasurable internal modes; This is an estimate of the lumped disturbance; Step S2.4: Substitute equation (1.9) into equation (1.6) and simplify to obtain the second-order standard full-drive form after complete linearization and decoupling of the actuator nodes:

[0029] In the formula, For lumped disturbances; This is an estimate of the lumped disturbance.

[0030] Preferably, step S3 includes: Step S3.1: Define the equivalent core quality matrix ; Step S3.2: Transform equations (1.7) and (1.8) into equivalent first-order augmented state forms, based on the assumption that the high-frequency lumped disturbance changes slowly relative to the extremely high sampling frequency of the controller, i.e. Define the first-order extended state vector as:

[0031] In the formula, Z represents the observer's extended state variable; It is a mixed spatial state; Step S3.3: Based on the definition of the equivalent core mass matrix and the second-order standard total drive form, obtain the continuous dynamic description of the actual controlled object in the first-order state space:

[0032] Step S3.4: Based on the first-order extended state vector and continuous dynamic description, define the corresponding state estimation variables. Measurement residuals of physical displacement of the actuating node By introducing the observation gain matrix to A first-order observer is constructed; the perturbation stripping layer is as follows:

[0033] The modal reconstruction layer is as follows:

[0034] in, To estimate the spatial state using a hybrid approach, the state estimate output by the first-order observer is... , This is the state estimate. The first derivative of the state estimate.

[0035] Preferably, step S4 includes: Step S4.1: Define the desired tracking trajectory of the preset actuator node as follows Among them, in the task of actively suppressing vibrations of a purely flexible continuum, If it is necessary to guide a flexible continuum structure to perform large-scale movements, then It is a second-order continuously differentiable trajectory, and its acceleration... It has a predefined supremum; Step S4.2: Define the actual physical tracking error vector of the actuator node. and its time-domain integral They are respectively:

[0036] In the formula, For time; is the independent variable used for time integration; It is the first derivative; It is the second derivative; in:

[0037] in, , , These are the preset generalized PID integral, proportional, and derivative gain matrices, respectively. Step S4.3: Based on the definition in Step S4.2, construct the solution equation for the all-drive control law:

[0038] in, and These are the asymptotic estimates of the displacement and velocity of the actuating node output by equations (1.11) to (1.14) in step three, respectively.

[0039] Preferably, step S5 specifically includes: The actual displacement of the actuator is collected by sensors. and will The current mixed estimated spatial state is calculated in real time through iterative solutions input into the first-order observer. to and will to Real-time input of all-wheel drive control law solution equations The virtual all-wheel drive control quantity is obtained from the middle ;Will to and In the reverse substitution step S2.3, the inverse dynamic control law (1.9) of the same position all-drive system is used to calculate the control input vector directly executed by the linear actuator. ,in:

[0040] Will It is converted into a driving current or driving force signal and output to the linear actuators arranged in the physical layer.

[0041] According to another aspect of the present invention, a vibration suppression control system for a flexible continuum structure is provided, wherein a linear actuator and a sensor are disposed at predetermined nodes on the flexible continuum structure, comprising: Module M1: The flexible continuum structure is discretized using the finite element method to obtain multiple nodes, and the obtained nodes are divided into observable and controllable actuator nodes and unobservable internal undriven nodes; the dynamic equations of the nodes in the absolute physical coordinate system are reduced in dimension using the Craig-Bampton system; where the actuator nodes are the predetermined nodes where the linear actuators and sensors are set. Module M2: Based on the Schur elimination method, the reduced dynamic equations are transformed into a second-order standard total drive form with only the acceleration of the actuator nodes as the dominant variable; Module M3: Constructs a first-order observer based on the first-order augmented state form of the dimensionality-reduced dynamic equations, wherein the first-order observer includes: Disturbance stripping layer: The displacement of the actuator node is obtained through sensors, and the velocity and lumped disturbance of the actuator node are observed; Modal reconstruction layer: Utilizes the physical boundary coupling matrix between actuator acceleration and internal modes to perform unbiased estimation of internal modes; Module M4: Construct a closed-loop controller for the all-drive system based on the second-order standard all-drive configuration and the state estimates output by the first-order observer; Module M5: Inputs the actual displacement of the linear actuator collected by the sensor into the first-order observer to obtain the state estimate. Substitutes the state estimate into the all-drive closed-loop controller to obtain the virtual all-drive control quantity. Substitute the virtual all-drive control quantity and the state estimate into the physical control law mapping equation to obtain the control input vector for the linear actuator. Controls the linear actuator to work through the control input vector.

[0042] Preferably, the module M1 includes: Module M1.1: Finite element discretization is performed on the flexible continuum structure to obtain multiple nodes. The dynamic equations of each node in the absolute physical coordinate system are as follows:

[0043] In the formula, These are the full-order physical mass, damping, and stiffness matrices, respectively; These are the displacement, velocity, and acceleration vectors of the node, respectively. This is the control input matrix for the actuator node; The ideal force vector output by the linear actuator; The input matrix is ​​the perturbation matrix; This is an unknown external disturbance; Module M1.2: Based on the permutation matrix T, the displacement vectors of each node are divided into the displacement vectors of the driven nodes and the displacement vectors of the internal undriven nodes, where:

[0044] Module M1.3: Substitute equation (1.2) into equation (1.1) and multiply by the left side. This yields the full-order dynamic equations in a block-decoupled form:

[0045] in:

[0046]

[0047]

[0048]

[0049]

[0050] In the formula, The physical mass matrix of the active node; It is the physical inertial coupling matrix from the internal undriven nodes to the driven nodes. Its physical meaning is the mapping of the reaction inertial force transmitted to the driven nodes through the structure when the internal nodes accelerate. This is the physical inertial coupling matrix from the driven node to the internal undriven nodes; The internal undriven node's self-owned physical mass matrix; The physical damping matrix of the actuator node; This is the physical dissipation coupling matrix from the internal undriven nodes to the driven nodes; This is the physical dissipation coupling matrix from the active node to the internal undriven nodes; The internal undriven node's self-owned physical damping matrix; The physical stiffness matrix of the actuating node; This is the physical elastic coupling matrix from the internal undriven nodes to the driven nodes; This is the physical elastic coupling matrix from the actuator node to the internal undriven nodes; The internal undriven node's self-physical stiffness matrix; The control input matrix is ​​divided into blocks according to the actuator nodes and the internal undriven nodes; and The input submatrix of the perturbation is divided into blocks according to the active nodes and the internal undriven nodes; The displacement, velocity, and acceleration vectors of the actuating node; This represents the displacement, velocity, and acceleration vectors of the internal, undriven nodes. Module M1.4: Introducing Static Constraint Modes and the truncated fixed interface main mode Construct the CB hybrid state-space transformation matrix ,in:

[0051] In the formula, To preserve the coordinates of the lower-order principal modes, This refers to the higher-order mode truncation error; It is the identity matrix; Module M1.5: Satisfies the engineering assumptions of proportional damping characteristics, utilizing... By performing a similarity transformation on equation (1.3), the full-order dynamic equation is reduced in dimension and re-divided into two parts: boundary dynamics and internal modal dynamics.

[0052] In the formula, The equivalent boundary mass matrix of the actuating nodes after condensation; This is the dynamic inertial coupling matrix of the internal principal mode to the actuation boundary; This is the dynamic inertial coupling matrix from the actuation boundary to the internal main mode; The equivalent boundary damping matrix of the actuating node after condensation; This is the equivalent boundary stiffness matrix of the actuating nodes after condensation; This is the equivalent physical perturbation input matrix acting on the actuator node space; This is the equivalent modal perturbation input matrix acting on the internal modal space; and These are the modal damping matrix and modal stiffness matrix after condensation, respectively; To preserve the coordinates, velocity, and acceleration of the low-frequency core modes; The displacement, velocity, and acceleration vectors of the actuating node;

[0053]

[0054] and For the truncation error of higher-order modes The resulting residual dynamics terms are treated as physical perturbations in subsequent steps.

[0055] Preferably, the module M2 includes: Module M2.1: Expanding and appropriately transforming equation (1.5), we obtain:

[0056]

[0057] Module M2.2: Extracted from equation (1.7) The explicit expression of the displacement vector of the actuating node is substituted into equation (1.6) for algebraic elimination. After combining like terms, the following expression for the displacement vector of the actuating node is obtained. Independent dynamic equations:

[0058] in, The Schur complement of the block mass matrix about the main diagonal constitutes the equivalent core mass after the dynamic equation is reduced in order and decoupled. This refers to the lumped disturbance acting at the actuation node; Module M2.3: Based on Constructing the isotopic all-drive inverse dynamic control law:

[0059] in, For the new virtual control input to be designed; and These are estimated values ​​for the coordinates of unmeasurable internal modes; This is an estimate of the lumped disturbance; Module M2.4: Substituting equation (1.9) into equation (1.6) and simplifying, we obtain the second-order standard full-drive form after complete linearization and decoupling of the actuator nodes:

[0060] In the formula, For lumped disturbances; This is an estimate of the lumped disturbance.

[0061] Preferably, the module M3 includes: Module M3.1: Define the equivalent core mass matrix ; Module M3.2: Transforms equations (1.7) and (1.8) into equivalent first-order augmented state forms, based on the assumption that the high-frequency lumped disturbance changes slowly relative to the extremely high sampling frequency of the controller, i.e. Define the first-order extended state vector as:

[0062] In the formula, Z represents the observer's extended state variable; It is a mixed spatial state; Module M3.3: Based on the definition of the equivalent core mass matrix and the second-order standard full-drive form, obtains a continuous dynamic description of the actual controlled object in the first-order state space.

[0063] Module M3.4: Based on the first-order extended state vector and continuous dynamic description, define the corresponding state estimation variables. Measurement residuals of physical displacement of the actuating node By introducing the observation gain matrix to A first-order observer is constructed; the perturbation stripping layer is as follows:

[0064] The modal reconstruction layer is as follows:

[0065] in, To estimate the spatial state using a hybrid approach, the state estimate output by the first-order observer is... , This is the state estimate. The first derivative of the state estimate.

[0066] Compared with the prior art, the present invention has the following beneficial effects: 1. This invention combines the Craig-Bampton (CB) interface-preserving condensation method with the theory of fully driven systems (FAS). By performing Schur complement algebraic elimination at the actuator nodes, this invention rigorously reconstructs the original highly underdriven sparse FEM model into a second-order ordinary differential equivalent fully driven form with actuator nodes as the main components. This process not only avoids the problem of solving complex high-order convolutions in traditional fully driven systems, but also preserves... The macroscopic physical dimensions allow subsequent controller designs to be configured directly by referencing the virtual mass, stiffness, and damping in a physical sense.

[0067] 2. Unlike traditional robust control's passive resistance to internal undriven node modes, this invention designs a cross-dimensional cascaded observation architecture of an extended state observer (ESO) and Luenberger. This is achieved by utilizing boundary dynamics coupling terms... To achieve implicit higher-order modes within the system Active unbiased estimation of lumped disturbances and feedforward decoupling. This mechanism fundamentally cuts off the disturbance propagation path from undriven modes to actuating nodes, actively and aggressively avoiding control spillover.

[0068] 3. This invention is not limited to a specific closed-loop control law, but provides a control domain middleware framework that combines fully driven dynamics reconstruction with implicit modal full observation. This is because the observer in this invention can reconstruct unbiased internal undriven node modal estimates in real time. This allows for the complete recovery of the system's full state vector. This allows those skilled in the art, when faced with the decoupling standard layer output by this framework, to both deploy generalized PIDs for intuitive pole configuration as in the embodiments, and to fully possess the ability to directly embed the standard. It provides the data foundation for modern state feedback algorithms such as LQR and has industrial generalization value.

[0069] Schur complement is obtained by solving the block mass matrix inside the controller. The control algorithm generates precise physical reaction forces in real time, forcibly canceling the inertial force coupling caused by the accelerated motion of the internal undriven nodes at the physical actuation node, and physically cutting off the uncontrollable transmission path of undriven modal vibration energy to the actuation node; combined with a second-order continuously differentiable reference signal with a preset acceleration upper bound, and the closed-loop full-drive control, the discrete hardware actuator that originally only provided driving force is forcibly reconstructed into a physical damping dissipation dynamic vibration absorber with specific equivalent virtual damping and equivalent virtual stiffness characteristics at the physical level, suppressing the displacement of the actuation node itself, and guiding the internal unmeasurable high-order structural vibration energy to the actuator and converting it into internal energy dissipation through the physical energy dissipation chain.

[0070] By actively absorbing and dissipating the energy of higher-order modes within the structure, the physical displacement amplitude of the end or key nodes of the flexible continuum (such as large trusses or robotic arms) is significantly reduced. Compared with existing conventional distributed node PID control, the settling time of the system from excited vibration to physical steady state is shortened by 50%, and the peak amplitude of the physical nodes is reduced by 30%. Attached Figure Description

[0071] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings: Figure 1 A flowchart of a vibration suppression control method for flexible continuum structures; Figure 2 A schematic diagram illustrating the modeling principle of a flexible continuum structure with discrete actuating nodes. Detailed Implementation

[0072] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0073] Example 1: This embodiment provides a vibration suppression control method for a flexible continuous structure. Linear actuators and sensors are installed at predetermined nodes on the flexible continuous structure, such as... Figure 1 As shown, this method includes: Step S1: Discretize the flexible continuum structure using the finite element method, such as... Figure 2As shown, multiple nodes are obtained and divided into observable and controllable actuator nodes and unobservable internal undriven nodes; the dynamic equations of the nodes in the absolute physical coordinate system are reduced in dimension using the Craig-Bampton method; where the actuator nodes are the predetermined nodes where the linear actuators and sensors are located. It is understood that step S1 includes: Step S1.1: Discretize the flexible continuum structure using the finite element method to obtain multiple nodes. The dynamic equations of each node in the absolute physical coordinate system are as follows:

[0074] In the formula, These are the full-order physical mass, damping, and stiffness matrices, respectively; These are the displacement, velocity, and acceleration vectors of the node, respectively. This is the control input matrix for the actuator node; The ideal force vector output by the linear actuator; The input matrix is ​​the perturbation matrix; This is an unknown external disturbance; Step S1.2: Based on the permutation matrix T, divide the displacement vector of each node into the displacement vector of the driven node and the displacement vector of the internal undriven node. and ,in:

[0075] Step S1.3: Substitute equation (1.2) into equation (1.1) and multiply by the left side. This yields the full-order dynamic equations in a block-decoupled form:

[0076] in:

[0077]

[0078]

[0079]

[0080]

[0081] In the formula, The physical mass matrix of the active node; It is the physical inertial coupling matrix from the internal undriven nodes to the driven nodes. Its physical meaning is the mapping of the reaction inertial force transmitted to the driven nodes through the structure when the internal nodes accelerate. This is the physical inertial coupling matrix from the driven node to the internal undriven nodes; The internal undriven node's self-owned physical mass matrix; The physical damping matrix of the actuator node; This is the physical dissipation coupling matrix from the internal undriven nodes to the driven nodes; This is the physical dissipation coupling matrix from the active node to the internal undriven nodes; The internal undriven node's self-owned physical damping matrix; The physical stiffness matrix of the actuating node; This is the physical elastic coupling matrix from the internal undriven nodes to the driven nodes; This is the physical elastic coupling matrix from the actuator node to the internal undriven nodes; The internal undriven node's self-physical stiffness matrix; The control input matrix is ​​divided into blocks according to the actuator nodes and the internal undriven nodes; and The input submatrix of the perturbation is divided into blocks according to the active nodes and the internal undriven nodes; The displacement, velocity, and acceleration vectors of the actuating node; This represents the displacement, velocity, and acceleration vectors of the internal, undriven nodes. Step S1.4: Introduce static constraint modes and the truncated fixed interface main mode Construct the CB hybrid state-space transformation matrix ,in:

[0082] In the formula, The coordinates of the lower-order principal modes are to be preserved; This refers to the higher-order mode truncation error; It is the identity matrix; Step S1.5: Satisfy the engineering assumptions of proportional damping characteristics, using... By performing a similarity transformation on equation (1.3), the full-order dynamic equation is reduced in dimension and re-divided into two parts: boundary dynamics and internal modal dynamics.

[0083] In the formula, The equivalent boundary mass matrix of the actuating nodes after condensation; This is the dynamic inertial coupling matrix of the internal principal mode to the actuation boundary; This is the dynamic inertial coupling matrix from the actuation boundary to the internal main mode; The equivalent boundary damping matrix of the actuating node after condensation; This is the equivalent boundary stiffness matrix of the actuating nodes after condensation; This is the equivalent physical perturbation input matrix acting on the actuator node space; This is the equivalent modal perturbation input matrix acting on the internal modal space; and These are the modal damping matrix and modal stiffness matrix after condensation, respectively; To preserve the coordinates, velocity, and acceleration of the low-frequency core modes; The displacement, velocity, and acceleration vectors of the actuating node;

[0084]

[0085] and For the truncation error of higher-order modes The resulting residual dynamics terms are treated as physical perturbations in subsequent steps. This is the transpose of the matrix; The second derivative with respect to the parameter.

[0086] Step S2: Based on the Schur elimination method, the reduced dynamic equations are converted into a second-order standard total drive form with only the acceleration of the actuator nodes as the dominant variable; It is understood that step S2 includes: Step S2.1: Expand equation (1.5) and perform appropriate transformations to obtain:

[0087]

[0088] Step S2.2: Extract from equation (1.7) The explicit expression of the displacement vector of the actuating node is substituted into equation (1.6) for algebraic elimination. After combining like terms, the following expression for the displacement vector of the actuating node is obtained. Independent dynamic equations:

[0089] in, The Schur complement of the block mass matrix about the main diagonal constitutes the equivalent core mass after the dynamic equation is reduced in order and decoupled. This refers to the lumped disturbance acting at the actuation node; Step S2.3: Based on formula Constructing the isotopic all-drive inverse dynamic control law:

[0090] in, For the new virtual control input to be designed; and These are estimated values ​​for the coordinates of unmeasurable internal modes; This is an estimate of the lumped disturbance; This is the state estimate. The first derivative of the state estimate.

[0091] Step S2.4: Substitute equation (1.9) into equation (1.6) and simplify to obtain the second-order standard full-drive form after complete linearization and decoupling of the actuator nodes:

[0092] In the formula, For lumped disturbances; This is an estimate of the lumped disturbance.

[0093] Step S3: Based on the first-order augmented state form of the dimensionality-reduced dynamic equations, construct a first-order observer, wherein the first-order observer includes: Disturbance stripping layer: By observing the displacement of the actuator node, the velocity and lumped disturbance of the actuator node are observed; Modal reconstruction layer: The internal modes are estimated unbiasedly by using the physical boundary coupling matrix between the actuator acceleration and the internal modes.

[0094] It is understood that step S3 includes: Step S3.1: Define the equivalent core quality matrix ; Step S3.2: Transform equations (1.7) and (1.8) into equivalent first-order augmented state forms, based on the assumption that the high-frequency lumped disturbance changes slowly relative to the extremely high sampling frequency of the controller, i.e. Define the first-order extended state vector as:

[0095] In the formula, Z represents the observer's extended state variable; It is a mixed spatial state; Step S3.3: Based on the definition of the equivalent core mass matrix and the second-order standard total drive form, obtain the continuous dynamic description of the actual controlled object in the first-order state space:

[0096] In the formula, The first derivative of the parameter.

[0097] Step S3.4: Based on the first-order extended state vector and continuous dynamic description, define the corresponding state estimation variables. Measurement residuals of physical displacement of the actuating node By introducing the observation gain matrix to A first-order observer is constructed; the perturbation stripping layer is as follows:

[0098] The modal reconstruction layer is as follows:

[0099] in, To estimate the spatial state using a hybrid approach, the state estimate output by the first-order observer is... ; This is the state estimate. The first derivative of the state estimate.

[0100] Step S4: Construct a closed-loop controller for the all-drive system based on the second-order standard all-drive configuration and the state estimate output by the first-order observer.

[0101] It is understood that step S4 includes: Step S4.1: Define the desired tracking trajectory of the preset actuator node as follows Among them, in the task of actively suppressing vibrations of a purely flexible continuum, If it is necessary to guide a flexible continuum structure to perform large-scale movements, then It is a second-order continuously differentiable trajectory, and its acceleration... It has a predefined supremum; Step S4.2: Define the actual physical tracking error vector of the actuator node. and its time-domain integral They are respectively:

[0102] In the formula, For time; is the independent variable used for time integration; It is the first derivative; It is the second derivative; in:

[0103] in, , , These are the preset generalized PID integral, proportional, and derivative gain matrices, respectively. Step S4.3: Based on the definition in Step S4.2, construct the solution equation for the all-drive control law:

[0104] in, and These are the asymptotic estimates of the displacement and velocity of the actuating node output by equations (1.11) to (1.14) in step three, respectively.

[0105] To demonstrate the control robustness of the present invention to those skilled in the art, the control law input quantity defined in equation (1.17) is... Substitute the equations of the real controlled object obtained from the transformation in step two into the reverse equations.

[0106] Define the state estimation residual vector of the first-order state observer in step three as follows: After algebraic merging and rearrangement, the closed-loop error dynamic system equation of the entire flexible continuum structure under the control of a generalized PID controller can be precisely expressed as:

[0107] Among them, the right-hand item For the disturbance source term linearly driven by the observer residual:

[0108] According to the strong convergence criterion of the cascaded extended state observer in step three, when time... At that time, all observation residual vectors asymptotically approach zero exponentially over the entire state space, i.e. Therefore, the driving item .

[0109] This means that the undesirable elastic vibration energy of the internal undriven nodes, after passing through the boundary inertial coupling array... When attempting to induce overflow excitation at the actuator node, the feedforward cancellation term in equation (1.17) is used. Physical isolation was implemented; the residual higher-order truncated excitation energy was then absorbed by the equivalent damping term. and equivalent stiffness term Forced dissipation absorption; the closed-loop system (1.18) eventually converges.

[0110] Step S5: Input the actual displacement of the linear actuator collected by the sensor into the first-order observer to obtain the state estimate. Substitute the state estimate into the all-drive closed-loop controller to obtain the virtual all-drive control quantity. Substitute the virtual all-drive control quantity and the state estimate into the physical control law mapping equation to obtain the control input vector executed by the linear actuator. Control the linear actuator to work through the control input vector.

[0111] It is understood that step S5 specifically includes: The actual displacement of the actuator is collected by sensors. and will The current mixed estimated spatial state is calculated in real time through iterative solutions input into the first-order observer. to and will to Real-time input of all-wheel drive control law solution equations The virtual all-wheel drive control quantity is obtained from the middle ;Will to and In the reverse substitution step S2.3, the inverse dynamic control law (1.9) of the same position all-drive system is used to calculate the control input vector directly executed by the linear actuator. ,in:

[0112] Will It is converted into a driving current or driving force signal and output to the linear actuators arranged in the physical layer.

[0113] Example 2: To apply this controller online, the dynamics of the controlled object need to be analyzed during the offline modeling phase, and the controller structure and specific parameter tuning need to be determined using the following steps. The offline modeling phase corresponds to steps 1 to 4 below.

[0114] 1. Finite element model meshing and Craig-Bampton (CB) state transformation: Positioning and design intent for this step: In practical engineering, actuators and sensors can only be installed on a small number of physical nodes, resulting in an extremely sparse control matrix. Furthermore, flexible bodies contain a large number of unmeasurable and uncontrollable degrees of freedom. The purpose of this step is to separate the system into measurable and controllable actuator nodes and unmeasurable internal undriven nodes using a permutation matrix. Then, the CB transformation is used to reduce the dimensionality of the large number of internal undriven nodes into a small number of low-frequency core modes, thereby significantly reducing the computational burden on the subsequent controller.

[0115] Specific implementation: After being discretized by the finite element method, the flexible truss structure is in the absolute physical coordinate system of the nodes. The dynamic equations are as follows

[0116] in, for The ideal force vector output by a set of linear actuators (such as linear actuators hinged between truss nodes).

[0117] This is the corresponding control input matrix. Since the actuator only applies force between specific physical nodes... It is often sparse; Represents the real number field.

[0118] This is due to an unknown external disturbance.

[0119] This is the perturbation input matrix.

[0120] First, the physical nodes of the system are rearranged. A permutation matrix is ​​found. The state vector is strictly divided into action nodes. With internal undriven nodes , so that:

[0121] Substitute equation (1.2) into equation (1.1) and multiply by the left side. This yields the full-order dynamic equations in the block-decoupled form:

[0122] in:

[0123]

[0124]

[0125]

[0126]

[0127] To resolve internal undriven nodes To address the issue of excessively high dimensionality, static constraint modes are introduced. and the truncated fixed interface main mode Construct the CB (Craig-Bampton) hybrid state-space transformation matrix. ,satisfy:

[0128] In the formula, To preserve the coordinates of the lower-order principal modes, This refers to the higher-order mode truncation error; It is the identity matrix; In engineering, it is often assumed that the system satisfies proportional damping characteristics, and this is utilized... By performing a similarity transformation on equation (1.3), the system dynamics are reduced in dimension and re-divided into two parts: boundary dynamics and internal modal dynamics.

[0129] in: The underlined matrix represents the equivalent parameters after order reduction.

[0130] and These are the modal damping matrix and the modal stiffness matrix, respectively.

[0131]

[0132]

[0133] and For the truncation error of higher-order modes The resulting residual dynamics terms will be treated as physical perturbations in subsequent steps.

[0134] 2. Second-order equivalent all-drive dynamics reconstruction based on Schur complement: Positioning and design intent for this step: Although equation (1.5) achieves dimensionality reduction, the action nodes... With internal modes There is still a strong inertial coupling between them, that is, outside the diagonal of the matrix. and Block design. If the controller were designed directly, this coupling would lead to severe control overflow. This step utilizes the Schur complement elimination method from algebra to eliminate internal modal accelerations from the system of equations. The coupling transforms the system into one that relies solely on the acceleration of the actuator nodes. A second-order standard all-drive system with the dominant variable. This allows those in the field to design controllers directly in physical space, as intuitively as controlling a set of ordinary rigid bodies.

[0135] Specific implementation: Expanding equation (1.5) and making appropriate modifications, we have:

[0136]

[0137] Equation (1.6) describes the dynamics of the actuated nodes; Equation (1.7) describes the modal dynamics of the internal undriven nodes. Block matrix and The existence of this condition results in an uncontrollable internal undriven node modal acceleration term in the actuating node equations. .

[0138] To eliminate Extracted from equation (1.7) The explicit expression is substituted into equation (1.6) for algebraic elimination. After combining like terms, the following is obtained regarding the action node. Independent dynamic equations:

[0139] in: It is exactly the Schur complement of the block mass matrix about the main diagonal, which constitutes the equivalent core mass after the system is reduced in order and decoupled.

[0140] This refers to the lumped physical disturbance acting at the actuating node.

[0141] Based on this, the inverse dynamic control law for all-wheel drive is designed:

[0142] in: For the new virtual control input to be designed; and These are estimated values ​​for the coordinates of unmeasurable internal modes; This is an estimate of the lumped disturbance; , and The calculation will be provided by the cascaded observer in subsequent steps.

[0143] Substituting the control law (1.9) into equation (1.6) and simplifying, we finally obtain the second-order equivalent total drive form after the actuator nodes are fully linearized and decoupled:

[0144] Therefore, as long as the subsequent observer design can guarantee that the estimation error approaches zero, i.e. , Equation (1.10) then degenerates into the purest second-order integrator system. This completely eliminates the potential for control overflow.

[0145] 3. Cascaded state and disturbance observation design: Positioning and design intent for this step: Step two derives the equivalent all-drive control law (1.9), but this control law depends on internal modal coordinates that cannot be directly measured. Actuating node speed and aggregated disturbances To achieve online control in a digital controller, a cascaded extended state observer is used to estimate the system state and disturbances. This observer is physically divided into two layers: Disturbance stripping layer: Utilizing the displacement of the only directly measurable actuator node, the velocity of the actuator node is observed, and the lumped disturbance is expanded into a new state for tracking.

[0146] Modal reconstruction layer: Utilizing the physical boundary coupling matrix between actuator acceleration and internal modes Unbiased estimation of internal modes is performed.

[0147] Specific implementation: Define the equivalent core quality matrix To construct a first-order observer, we transform the original second-order dynamic equations (1.7) and (1.8) into equivalent first-order augmented state forms, and assume that the high-frequency lumped disturbance changes slowly relative to the extremely high sampling frequency of the controller, i.e. At this point, the first-order extended state vector is defined as:

[0148] The only obtainable physical measurement value in the system is the displacement of the actuating node, i.e., the measurement and control output. .

[0149] Based on the conclusion of step two, the continuous dynamics of the real controlled object in the first-order state space can be precisely described as follows:

[0150] Based on the above first-order augmented state dynamics, design the corresponding state estimation variables. The measurement residual of the physical displacement of the actuating node is defined as... By introducing the observation gain matrix to Construct an online cascaded observation solution law.

[0151] The disturbance stripping layer is as follows:

[0152] The modal reconstruction layer is as follows:

[0153] Because this cascaded observer has been completely transformed into a standard linear ordinary differential first-order state-space form, those in this technical field no longer need to blindly try and find solutions. Instead, they can directly use modern and mature algorithms such as pole placement or Kalman filtering (linear quadratic estimation LQE) to obtain all the gain matrices offline in one go. Ultimately, the state estimates obtained during the online phase... This will be directly substituted into the equivalent all-drive control law.

[0154] 4. Standardized all-drive closed-loop control solution: Positioning and design intent for this step: Through the algebraic reconstruction in step two and the cascaded observations in step three, the complex flexible continuum, which is highly underdriven and has the risk of modal spillover, has nominally degenerated into a fully linearized and decoupled second-order standard fully driven system. To make the system converge, a general approach is to let ,in General parameters that can be configured according to the actual application to meet the Hurwitz conditions.

[0155] To enable those skilled in the art to better understand how to apply the equivalent total drive framework of this invention to specific control strategies to achieve the desired effect, the following explanation and demonstration are provided in conjunction with a generalized PID controller. In this embodiment, the control law is configured as a generalized PID controller, which does not mean that the framework of this invention can only be applied in combination with PID control laws. In fact, this invention demonstrates the transformation of the sparse control problem of multi-node finite element dynamics into an equivalent total drive control framework, which can then be embedded with various conventional control laws applicable to linear systems, including but not limited to PID, LQR, etc.

[0156] Based on the decoupled virtual control input A generalized PID control law is deployed, and the actuating node after the closed loop is reconstructed into a virtual damped dissipative dynamic vibration absorber using full-drive control. This actively establishes an energy-physical coupling chain, allowing the unmeasurable high-order vibration energy of the internal undriven nodes to continuously flow to and dissipate towards the actuating node. Furthermore, it ensures that the closed-loop tracking error exhibits strict asymptotic stability during the convergence process of the cascaded observer.

[0157] Specific implementation: Let the desired tracking trajectory of the preset actuation node of the above system be denoted as . In purely flexible body active vibration suppression tasks, directly causing... That's sufficient; if it's necessary to guide the structure to perform a large-scale movement, then... The trajectory is planned as a second-order continuously differentiable trajectory, and its acceleration... It has a predefined upper bound. Defines the actual physical tracking error vector of the system's actuators. and its time-domain integral They are respectively:

[0158] To ensure that the tracking error system possesses the preset dynamic response characteristics and damping dissipation properties, a third-order ordinary differential linear stability criterion is set for the closed-loop error system to satisfy the following criteria:

[0159] in, , , These are the preset generalized PID integral, proportional, and derivative gain matrices, respectively. The eigenvalues ​​of this third-order matrix sequence should be strictly constrained to the left half of the complex plane to ensure the tracking error vector... Convergence. For example, offline pole placement using the Hurwitz criterion directly correlates with the desired virtual stiffness and equivalent dissipative damping in physical space.

[0160] Based on this, the following online generalized PID all-drive control law solution equations are constructed:

[0161] in, and These are the asymptotic estimates of the displacement and velocity of the actuating node output by equations (1.11) to (1.14) in step three, respectively.

[0162] To demonstrate the control robustness of the present invention to those skilled in the art, the control law input quantity defined in equation (1.17) is... Substitute the equations of the real controlled object obtained from the transformation in step two into the reverse equations.

[0163] Define the state estimation residual vector of the first-order state observer in step three as follows: After algebraic merging and rearrangement, the closed-loop error dynamic system equation of the entire flexible continuum structure under the control of a generalized PID controller can be precisely expressed as:

[0164] Among them, the right-hand item For the disturbance source term linearly driven by the observer residual:

[0165] According to the strong convergence criterion of the cascaded extended state observer in step three, when time... At that time, all observation residual vectors asymptotically approach zero exponentially over the entire state space, i.e. Therefore, the driving item .

[0166] This means that the undesirable elastic vibration energy of the internal undriven nodes, after passing through the boundary inertial coupling array... When attempting to induce overflow excitation at the actuator node, the feedforward cancellation term in equation (1.17) is used. Physical isolation was implemented; the residual higher-order truncated excitation energy was then absorbed by the equivalent damping term. and equivalent stiffness term Forced dissipation absorption; the closed-loop system (1.18) eventually converges.

[0167] 5. Online calculation of final control output: In the online control phase of this invention, the complete standardized computing power closed-loop process implemented in each control cycle is as follows: The actual displacement of the actuator node is collected by sensors. The input is fed into the cascaded state observers of equations (1.11) to (1.14) to iteratively calculate the current hybrid estimated spatial state in real time. to Subsequently, these high-precision state variables are input into the generalized PID operator formula (1.17) in real time to obtain the virtual all-drive control variables. Ultimately, Substitute back to the physical control law mapping equation (1.9) from step two to calculate the control input vector directly executed by the physical actuator. :

[0168] This The signal is converted into a driving current or driving force signal and output to the linear actuators arranged in the physical layer, thus achieving all the techniques for vibration suppression control of flexible continuum structures with discrete actuation nodes.

[0169] To enable those skilled in the art to better understand the present invention, a table of symbols is provided below.

[0170]

[0171] The present invention also provides a vibration suppression control system for a flexible continuum structure. The vibration suppression control system for the flexible continuum structure can be implemented by executing the process steps of the vibration suppression control method for the flexible continuum structure. That is, those skilled in the art can understand the vibration suppression control method for the flexible continuum structure as a preferred embodiment of the vibration suppression control system for the flexible continuum structure.

[0172] Those skilled in the art will understand that, besides implementing the system and its various devices, modules, and units provided by this invention in the form of purely computer-readable program code, the same functions can be achieved entirely through logical programming of the method steps, making the system and its various devices, modules, and units of this invention function in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, the system and its various devices, modules, and units provided by this invention can be considered as a hardware component, and the devices, modules, and units included therein for implementing various functions can also be considered as structures within the hardware component; alternatively, the devices, modules, and units for implementing various functions can be considered as both software modules implementing the method and structures within the hardware component.

[0173] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A vibration suppression and control method for a flexible continuous structure, wherein linear actuators and sensors are provided at predetermined nodes on the flexible continuous structure, characterized in that... include: Step S1: Discretize the flexible continuum structure using the finite element method to obtain multiple nodes, and divide the obtained nodes into observable and controllable actuator nodes and unobservable internal undriven nodes; reduce the dimensionality of the dynamic equations of the nodes in the absolute physical coordinate system using the Craig-Bampton method; where the actuator nodes are the predetermined nodes where the linear actuators and sensors are located. Step S2: Based on the Schur elimination method, the reduced dynamic equations are converted into a second-order standard total drive form with only the acceleration of the actuator nodes as the dominant variable; Step S3: Based on the first-order augmented state form of the dimensionality-reduced dynamic equations, construct a first-order observer, wherein the first-order observer includes: Disturbance stripping layer: The displacement of the actuator node is obtained through sensors, and the velocity and lumped disturbance of the actuator node are observed; Modal reconstruction layer: Utilizes the physical boundary coupling matrix between actuator acceleration and internal modes to perform unbiased estimation of internal modes; Step S4: Construct a closed-loop controller for the all-drive system based on the second-order standard all-drive configuration and the state estimate output by the first-order observer; Step S5: Input the actual displacement of the linear actuator collected by the sensor into the first-order observer to obtain the state estimate. Substitute the state estimate into the all-drive closed-loop controller to obtain the virtual all-drive control quantity. Substitute the virtual all-drive control quantity and the state estimate into the physical control law mapping equation to obtain the control input vector executed by the linear actuator. Control the linear actuator to work through the control input vector.

2. The method according to claim 1, characterized in that, Step S1 includes: Step S1.1: Discretize the flexible continuum structure using the finite element method to obtain multiple nodes. The dynamic equations of each node in the absolute physical coordinate system are as follows: In the formula, These are the full-order physical mass, damping, and stiffness matrices, respectively; These are the displacement, velocity, and acceleration vectors of the node, respectively. This is the control input matrix for the actuator node; The ideal force vector output by the linear actuator; The input matrix is ​​the perturbation matrix; This is an unknown external disturbance; Step S1.2: Based on the permutation matrix T, divide the displacement vector of each node into the displacement vector of the driven node and the displacement vector of the internal undriven node, where: Step S1.3: Substitute equation (1.2) into equation (1.1) and multiply by the left side. This yields the full-order dynamic equations in a block-decoupled form: in: In the formula, The physical mass matrix of the active node; It is the physical inertial coupling matrix from the internal undriven nodes to the driven nodes. Its physical meaning is the mapping of the reaction inertial force transmitted to the driven nodes through the structure when the internal nodes accelerate. This is the physical inertial coupling matrix from the driven node to the internal undriven nodes; The internal undriven node's self-physical mass matrix; The physical damping matrix of the actuating node; This is the physical dissipation coupling matrix from the internal undriven nodes to the driven nodes; This is the physical dissipation coupling matrix from the active node to the internal undriven nodes; The internal undriven node's self-owned physical damping matrix; The physical stiffness matrix of the actuating node; This is the physical elastic coupling matrix from the internal undriven nodes to the driven nodes; This is the physical elastic coupling matrix from the actuator node to the internal undriven nodes; The internal undriven node's self-physical stiffness matrix; The control input matrix is ​​divided into blocks according to the actuator nodes and the internal undriven nodes; and The input submatrix of the perturbation is divided into blocks according to the active nodes and the internal undriven nodes; The displacement, velocity, and acceleration vectors of the actuating node; This represents the displacement, velocity, and acceleration vectors of the internal, undriven nodes. Step S1.4: Introduce static constraint modes and the truncated fixed interface main mode Construct the CB hybrid state-space transformation matrix ,in: In the formula, The coordinates of the lower-order principal modes are to be preserved; This refers to the higher-order mode truncation error; It is the identity matrix; Step S1.5: Based on the engineering assumptions that satisfy the proportional damping characteristics, utilize... By performing a similarity transformation on equation (1.3), the full-order dynamic equation is reduced in dimension and re-divided into two parts: boundary dynamics and internal modal dynamics. In the formula, The equivalent boundary mass matrix of the actuating nodes after condensation; This is the dynamic inertial coupling matrix of the internal principal mode to the actuation boundary; This is the dynamic inertial coupling matrix from the actuation boundary to the internal main mode; The equivalent boundary damping matrix of the actuating node after condensation; This is the equivalent boundary stiffness matrix of the actuating nodes after condensation; This is the equivalent physical perturbation input matrix acting on the actuator node space; This is the equivalent modal perturbation input matrix acting on the internal modal space; and These are the modal damping matrix and modal stiffness matrix after condensation, respectively; To preserve the coordinates, velocity, and acceleration of the low-frequency core modes; The displacement, velocity, and acceleration vectors of the actuating node; In the formula, and For the truncation error of higher-order modes The resulting residual dynamics terms are treated as physical perturbations in subsequent steps.

3. The method according to claim 2, characterized in that, Step S2 includes: Step S2.1: Expand equation (1.5) and perform appropriate transformations to obtain: Step S2.2: Extract from equation (1.7) The explicit expression of the displacement vector of the actuating node is substituted into equation (1.6) for algebraic elimination. After combining like terms, the following expression for the displacement vector of the actuating node is obtained. Independent dynamic equations: in, The Schur complement of the block mass matrix about the main diagonal constitutes the equivalent core mass after the dynamic equation is reduced in order and decoupled. This refers to the lumped disturbance acting at the actuation node; Step S2.3: Based on formula Constructing the isotopic all-drive inverse dynamic control law: in, For the new virtual control input to be designed; and These are estimated values ​​for the coordinates of unmeasurable internal modes; This is an estimate of the lumped disturbance; Step S2.4: Substitute equation (1.9) into equation (1.6) and simplify to obtain the second-order standard full-drive form after complete linearization and decoupling of the actuator nodes: In the formula, For lumped disturbances; This is an estimate of the lumped disturbance.

4. The method according to claim 3, characterized in that, Step S3 includes: Step S3.1: Define the equivalent core quality matrix ; Step S3.2: Transform equations (1.7) and (1.8) into equivalent first-order augmented state forms, based on the assumption that the high-frequency lumped disturbance changes slowly relative to the extremely high sampling frequency of the controller, i.e. Define the first-order extended state vector as: In the formula, Z represents the observer's extended state variable; It is a mixed spatial state; Step S3.3: Based on the definition of the equivalent core mass matrix and the second-order standard total drive form, obtain the continuous dynamic description of the actual controlled object in the first-order state space: Step S3.4: Based on the first-order extended state vector and continuous dynamic description, define the corresponding state estimation variables. Measurement residuals of physical displacement of the actuating node By introducing the observation gain matrix to A first-order observer is constructed; the perturbation stripping layer is as follows: The modal reconstruction layer is as follows: in, To estimate the spatial state in a hybrid manner, the state estimate output by the first-order observer is... ; y is the displacement of the actuating node, i.e., y = ; This is the state estimate. The first derivative of the state estimate.

5. The method according to claim 4, characterized in that, Step S4 includes: Step S4.1: Define the desired tracking trajectory of the preset actuator node as follows Among them, in the task of actively suppressing vibrations of a purely flexible continuum, If it is necessary to guide a flexible continuum structure to perform large-scale movements, then It is a second-order continuously differentiable trajectory, and its acceleration... It has a predefined supremum; Step S4.2: Define the actual physical tracking error vector of the actuator node. and its time-domain integral They are respectively: In the formula, For time; is the independent variable used for time integration; It is the first derivative; It is the second derivative; It is the third derivative; in: in, , , These are the preset generalized PID integral, proportional, and derivative gain matrices, respectively; Step S4.3: Based on the definition in Step S4.2, construct the solution equation for the all-drive control law: in, and These are the asymptotic estimates of the displacement and velocity of the actuating node output by equations (1.11) to (1.14) in step three, respectively.

6. The method according to claim 5, characterized in that, Step S5 specifically includes: The actual displacement of the actuator is collected by sensors. and will The current mixed estimated spatial state is calculated in real time through iterative solutions input into the first-order observer. to and will to Real-time input of all-wheel drive control law solution equations The virtual all-wheel drive control quantity is obtained from the middle ;Will to and In the reverse substitution step S2.3, the inverse dynamic control law (1.9) of the same position all-drive system is used to calculate the control input vector directly executed by the linear actuator. ,in: Will It is converted into a driving current or driving force signal and output to the linear actuators arranged in the physical layer.

7. A vibration suppression control system for a flexible continuum structure, wherein linear actuators and sensors are provided at predetermined nodes on the flexible continuum structure, characterized in that... include: Module M1: The flexible continuum structure is discretized using the finite element method to obtain multiple nodes, and the obtained nodes are divided into observable and controllable actuator nodes and unobservable internal undriven nodes; the dynamic equations of the nodes in the absolute physical coordinate system are reduced in dimension using the Craig-Bampton system; where the actuator nodes are the predetermined nodes where the linear actuators and sensors are set. Module M2: Based on the Schur elimination method, the reduced dynamic equations are transformed into a second-order standard total drive form with only the acceleration of the actuator nodes as the dominant variable; Module M3: Constructs a first-order observer based on the first-order augmented state form of the dimensionality-reduced dynamic equations, wherein the first-order observer includes: Disturbance stripping layer: The displacement of the actuator node is obtained through sensors, and the velocity and lumped disturbance of the actuator node are observed; Modal reconstruction layer: Utilizes the physical boundary coupling matrix between actuator acceleration and internal modes to perform unbiased estimation of internal modes; Module M4: Construct a closed-loop controller for the all-drive system based on the second-order standard all-drive configuration and the state estimates output by the first-order observer; Module M5: Inputs the actual displacement of the linear actuator collected by the sensor into the first-order observer to obtain the state estimate. Substitutes the state estimate into the all-drive closed-loop controller to obtain the virtual all-drive control quantity. Substitute the virtual all-drive control quantity and the state estimate into the physical control law mapping equation to obtain the control input vector for the linear actuator. Controls the linear actuator to work through the control input vector.

8. The system according to claim 7, characterized in that, The module M1 includes: Module M1.1: Finite element discretization is performed on the flexible continuum structure to obtain multiple nodes. The dynamic equations of each node in the absolute physical coordinate system are as follows: In the formula, These are the full-order physical mass, damping, and stiffness matrices, respectively; These are the displacement, velocity, and acceleration vectors of the node, respectively. This is the control input matrix for the actuator node; The ideal force vector output by the linear actuator; The input matrix is ​​the perturbation matrix; This is an unknown external disturbance; Module M1.2: Based on the permutation matrix T, the displacement vectors of each node are divided into the displacement vectors of the driven nodes and the displacement vectors of the internal undriven nodes, where: Module M1.3: Substitute equation (1.2) into equation (1.1) and multiply by the left side. This yields the full-order dynamic equations in a block-decoupled form: in: In the formula, The physical mass matrix of the active node; It is the physical inertial coupling matrix from the internal undriven nodes to the driven nodes. Its physical meaning is the mapping of the reaction inertial force transmitted to the driven nodes through the structure when the internal nodes accelerate. This is the physical inertial coupling matrix from the actuating node to the internal undriven nodes; The internal undriven node's self-physical mass matrix; The physical damping matrix of the actuating node; This is the physical dissipation coupling matrix from the internal undriven nodes to the driven nodes; This is the physical dissipation coupling matrix from the active node to the internal undriven nodes; The internal undriven node's self-owned physical damping matrix; The physical stiffness matrix of the actuating node; This is the physical elastic coupling matrix from the internal undriven nodes to the driven nodes; This is the physical elastic coupling matrix from the actuator node to the internal undriven nodes; The internal undriven node's self-physical stiffness matrix; The control input matrix is ​​divided into blocks according to the actuator nodes and the internal undriven nodes; and The input submatrix of the perturbation is divided into blocks according to the active nodes and the internal undriven nodes; The displacement, velocity, and acceleration vectors of the actuating node; This represents the displacement, velocity, and acceleration vectors of the internal, undriven nodes. Module M1.4: Introducing Static Constraint Modes and the truncated fixed interface main mode Construct the CB hybrid state-space transformation matrix ,in: In the formula, To preserve the coordinates of the lower-order principal modes, This refers to the higher-order mode truncation error; It is the identity matrix; Module M1.5: Based on the engineering assumptions that satisfy proportional damping characteristics, utilizing... By performing a similarity transformation on equation (1.3), the full-order dynamic equation is reduced in dimension and re-divided into two parts: boundary dynamics and internal modal dynamics. In the formula, The equivalent boundary mass matrix of the actuating nodes after condensation; This is the dynamic inertial coupling matrix of the internal principal mode to the actuation boundary; This is the dynamic inertial coupling matrix from the actuation boundary to the internal main mode; The equivalent boundary damping matrix of the actuating node after condensation; This is the equivalent boundary stiffness matrix of the actuating nodes after condensation; This is the equivalent physical perturbation input matrix acting on the actuator node space; This is the equivalent modal perturbation input matrix acting on the internal modal space; and These are the modal damping matrix and modal stiffness matrix after condensation, respectively; To preserve the coordinates, velocity, and acceleration of the low-frequency core modes; The displacement, velocity, and acceleration vectors of the actuating node; and For the truncation error of higher-order modes The resulting residual dynamics terms are treated as physical perturbations in subsequent steps.

9. The system according to claim 8, characterized in that, The module M2 includes: Module M2.1: Expanding and appropriately transforming equation (1.5), we obtain: Module M2.2: Extracted from equation (1.7) The explicit expression of the displacement vector of the actuating node is substituted into equation (1.6) for algebraic elimination. After combining like terms, the following expression for the displacement vector of the actuating node is obtained. Independent dynamic equations: in, The Schur complement of the block mass matrix about the main diagonal constitutes the equivalent core mass after the dynamic equation is reduced in order and decoupled. This refers to the lumped disturbance acting at the actuation node; Module M2.3: Based on Constructing the isotopic all-drive inverse dynamic control law: in, For the new virtual control input to be designed; and These are estimated values ​​for the coordinates of unmeasurable internal modes; This is an estimate of the lumped disturbance; Module M2.4: Substituting equation (1.9) into equation (1.6) and simplifying, we obtain the second-order standard full-drive form after complete linearization and decoupling of the actuator nodes: In the formula, For lumped disturbances; This is an estimate of the lumped disturbance.

10. The system according to claim 9, characterized in that, The module M3 includes: Module M3.1: Define the equivalent core mass matrix ; Module M3.2: Transforms equations (1.7) and (1.8) into equivalent first-order augmented state forms, based on the assumption that the high-frequency lumped disturbance changes slowly relative to the extremely high sampling frequency of the controller, i.e. Define the first-order extended state vector as: In the formula, Z represents the observer's extended state variable; It is a mixed spatial state; Module M3.3: Based on the definition of the equivalent core mass matrix and the second-order standard full-drive form, obtains a continuous dynamic description of the actual controlled object in the first-order state space. Module M3.4: Based on the first-order extended state vector and continuous dynamic description, define the corresponding state estimation variables. Measurement residuals of physical displacement of the actuating node By introducing the observation gain matrix to A first-order observer is constructed; the perturbation stripping layer is as follows: The modal reconstruction layer is as follows: in, To estimate the spatial state in a hybrid manner, the state estimate output by the first-order observer is... , This is the state estimate. The first derivative of the state estimate.

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