Method for controlling structural vibration based on controller saturation under band active mass damper

Abstract

The application discloses a control method of structural vibration based on a controller saturation lower band active mass damper, and successfully reduces the vibration of a building structure by introducing an active mass damper and applying a control strategy. In order to meet the control input constraint and performance requirements, a state feedback optimal control law under the controller saturation is proposed by using a linear matrix inequality method; example analysis shows that the proposed control strategy can realize the stability of the closed-loop system and play the best performance, while reducing the control energy consumption and the floor level disturbance; compared with the case without considering the controller saturation, the proposed controller has smaller gain, reduces the energy consumption, and shows better anti-interference performance; numerical examples verify the effectiveness and feasibility of the proposed controller within the saturation limit; the application provides a new idea and method for solving the actual building structure control problem, and has important theoretical significance and practical application value.

Description

Technical Field

[0001] This invention relates to the field of seismic technology in buildings, and more specifically, to a method for controlling structural vibrations based on controller saturation with active mass dampers. Background Technology

[0002] In recent years, with the increasing number of high-rise buildings, the demand for vibration reduction and control has become increasingly urgent. High-rise buildings are prone to vibration and even collapse when faced with external excitations such as earthquakes and strong winds. Therefore, research on vibration reduction and control of high-rise buildings is of great significance. In recent years, vibration reduction and control of high-rise buildings has been extensively studied, and its commonly used technologies are mainly divided into two types: passive control and active control.

[0003] Passive control methods were initially used to suppress structural vibrations in buildings and were widely accepted due to their advantages, such as requiring no external energy support, simple construction, ease of maintenance, and low cost. However, passive control cannot adjust and optimize its performance in real time in the face of changes in the building system and environmental disturbances. Practice has shown that passive control is less effective in mitigating vibrations in high-rise buildings, failing to effectively absorb vibration energy and thus no longer meeting the needs of practical production. In contrast to passive control, active control introduces external energy into the damping device to generate control force to suppress structural vibrations in real time; active control has the advantages of strong adaptability and good control effect. Common active control devices include active mass dampers, active tuned mass dampers, active cable systems, and active bracing systems. Among these, active mass dampers (AMD) are widely used in high-rise buildings due to their significant vibration reduction performance and low control cost.

[0004] In practical high-rise building structural systems, the output of controllers is often constrained. When the controller output exceeds the acceptable range of the system, controller saturation may lead to system instability and damage. Therefore, considering structural vibration control under controller saturation can ensure that the controller output is within a reasonable range and matches the actual system, thereby maintaining control stability and improving vibration suppression. However, research on controller saturation constraints is relatively limited in building structure control with active mass dampers.

[0005] This invention studies the structural vibration control design of high-rise building structures under external excitation using active mass dampers (AMD), and considers the saturation characteristics of the controller; it provides an effective control strategy for practical engineering applications to achieve real-time control of building structure vibration and improve stability performance, thereby further improving the safety and comfort of buildings. Summary of the Invention

[0006] This invention provides a method for controlling structural vibration with an active mass damper under controller saturation, and applies... The control strategy, combined with active mass dampers, effectively reduces the vibration of the building structure, ensuring the stability of the closed-loop system and maintaining optimal performance while reducing control energy consumption.

[0007] To achieve the above objectives, the present invention provides the following technical solution: A control method for structural vibration with active mass damper under controller saturation is proposed. The controlled system equations of the building structure under controller saturation are as follows: (2) 1) Among them, It is the state vector of the building structure. State vector Regarding time First derivative; voltage u of the trolley motor ; is u signal Norm, limiting the maximum amplitude of a single channel; It is the maximum input value of the controller, which is the preset upper limit; it includes information such as displacement and velocity. It is the control response output of the building structure; These are external disturbances, mainly caused by factors such as earthquakes and random winds. It is a system parameter matrix that describes the dynamic characteristics of the building structure; x(t)∈Rn, where Rn refers to n-dimensional real Euclidean space. Meaning: The state vector x(t) contains n state variables (displacement and velocity of each floor of the building, with a total of n); z(t)∈Rr, where Rr refers to r-dimensional real Euclidean space. Meaning: The controlled output vector z(t) has a total of r output channels (generally the responses that need to be suppressed, such as relative displacement and acceleration of each layer); w(t)∈Rq: Rq refers to q-dimensional real Euclidean space. Meaning: The external disturbance vector w(t) has a total of q disturbance inputs (earthquake excitation, wind load, etc., with q disturbance channels).

[0008] Design status feedback controller ;in, Let be the state vector of the building structure. The control input vector output by the controller. Let be the state feedback gain matrix to be solved.

[0009] This ensures that the control system (11) has asymptotic stability and satisfies Performance gain metrics The requirements, namely (3) in Disturbance input To the controlled output The closed-loop transfer function matrix; : transfer function Norm, characterizing the system's disturbance suppression performance; The 2-norm of a time-domain signal, representing the signal energy; supremum: the maximum value of the output-to-input energy ratio for all non-zero inputs; : The preset disturbance suppression performance threshold.

[0010] Preferably, under external excitation, the control objective is to reduce the relative displacement and velocity of each floor. The control output equation can be expressed as: (4) Preferably, for external stimuli In its simplified model, the building structure uses a motor-driven trolley mounted on the top of the building to generate a reaction force on the building structure, thereby controlling the building's vibrations; the trolley on the top floor is driven by an internal DC motor. Based on angular velocity and linear velocity Relationship: (5) in These are system parameters; they convert the angular velocity of the trolley's motor into linear velocity.

[0011] Car driving force Voltage of the car motor and the linear velocity of the trolley There is a linear functional relationship between them, and the specific expression is as follows: (6) in , These are system parameters; The method of using the Lagrange equation can be used to derive The dynamic equations of the multi-story building system are as follows: (7) in , , and , They represent the first The mass, equivalent viscous damping coefficient, stiffness, and horizontal deflection of the layer; , , and These represent the mass, equivalent viscous damping coefficient, inertial force, and horizontal displacement relative to the top floor of the active mass damper (AMD) on the roof, respectively. It is the horizontal displacement of the ground, representing the excitation on the building; The augmented state vector is selected as (8) Under external excitation, the state equation of a building structure can be expressed in matrix form, i.e. (9) in , , , , , , ; because When it is reversible, equation (5) can be transformed into: (10) make , , , Equation (6) can be simplified to: (11) Further Then the dynamic model of the final building structure system can be simplified to: = (12) in , , ,in and Represented as the zero matrix and the identity matrix, respectively; The control inputs must satisfy the following constraints: (13) in, It is the maximum input value of the controller, and also the preset upper limit.

[0012] Preferably, for a given symmetric matrix ,in and It is a symmetric matrix whose dimension is compatible with the system state matrix. Let X be a matrix whose dimension is compatible with the system state matrix. If there exist symmetric positive definite matrices X and W whose dimensions are compatible with the system state matrix, then the following three conditions are equivalent: (1) ; (2) ; (3) . Therefore, if given parameters , and initial value If a symmetric positive definite matrix of suitable dimension exists... sum matrix This makes the following matrix inequality hold, where I is an identity matrix that is adapted to the corresponding operation dimension. The main diagonal elements of the identity matrix are 1, and the remaining elements are 0. The corresponding adapted dimension of I at different positions is selected according to the position of the block matrix. (14)

[0013] (15) (16) Then design a state feedback. controller This ensures that, under controller saturation conditions, the closed-loop control system (11) simultaneously satisfies the following conditions: (1) The closed-loop system (11) is asymptotically stable; (2) The system satisfies Performance metrics; (3) Input saturation constraints It always holds true.

[0014] The principle and beneficial effects of this technical solution: (1) Application of the present invention under controller saturation conditions The control strategy, combined with active mass dampers, effectively reduced the vibration of the building structure while meeting control input constraints and Performance requirements: The proposed control strategy ensures the stability and optimal performance of the closed-loop system while reducing control energy consumption. The feasibility and effectiveness of the proposed control strategy are verified through case analysis.

[0015] (2) This invention has important theoretical and practical significance for the field of building structure control. It improves the theoretical results of system stability and robust control under controller saturation. Compared with the controller without considering controller saturation, the proposed controller has a smaller gain, reduces energy consumption, and exhibits better anti-interference performance, providing new ideas and methods for the optimization and control of building structures. Attached Figure Description

[0016] Figure 1 A simplified model of a layered building structure containing AMD; Figure 2 For step stimulation image; Figure 3 For simple harmonic excitation image; Figure 4 For random perturbation input image; Figure 5 For random perturbation input image; Detailed Implementation The present invention will now be described in further detail with reference to the accompanying drawings and embodiments: Example: This invention provides a control method for structural vibration with an active mass damper under controller saturation. The controlled system equations for the building structure considering controller saturation are as follows: (17) in, It is the state vector of the building structure, including information such as displacement and velocity; It is the control response output of the building structure; These are external disturbances, mainly caused by factors such as earthquakes and random winds. It is a system parameter matrix that describes the dynamic characteristics of the building structure; When a building structure is subjected to external excitation, with the goal of reducing the relative displacement and velocity of each floor, the control output equation can be expressed as: (18) in, Indicates time The controlled output vector at time step 1. Indicates time The state vector at time t, It is a weighted coefficient matrix; To better analyze and design controllers that meet system requirements, and to ensure that the system is effectively controlled and achieves performance targets, it is necessary to introduce some essential definitions and fundamental lemmas.

[0017] Definition 1: Design Status Feedback controller This makes the control system (11) asymptotically stable and satisfies Performance gain metrics The requirements, namely (19) in Found through search To obtain the optimal value of the closed-loop control system. Optimal control design achieves optimal suppression of disturbances; like Figure 1 As shown, for external stimulus In its simplified model, the building structure uses a motor-driven trolley mounted on the top of the building to generate a reaction force on the building structure, thereby controlling the building's vibrations; the trolley on the top floor is driven by an internal DC motor. Based on angular velocity and linear velocity Relationship between them: (20) in These are system parameters; they convert the angular velocity of the trolley's motor into linear velocity to more accurately describe and control the trolley's motion.

[0018] Car driving force Voltage of the car motor and the linear velocity of the trolley There is a linear functional relationship between them, and the specific expression is as follows: ,(twenty one) in , These are system parameters; The method of using the Lagrange equation can be used to derive The dynamic equations of the multi-story building system are as follows: (twenty two) in , , and , They represent the first The mass, equivalent viscous damping coefficient, stiffness, and horizontal deflection of the layer; , , and These represent the mass, equivalent viscous damping coefficient, inertial force, and horizontal displacement relative to the top floor of the active mass damper (AMD) on the roof, respectively. It is the horizontal displacement of the ground, representing the excitation on the building; The augmented state vector is selected as (twenty three) Under external excitation, the state equation of a building structure can be expressed in matrix form, i.e. (twenty four) in , , , , , , ; because When it is reversible, equation (5) can be transformed into: (25) make , , , Equation (6) can be simplified to: (26) Further Then the dynamic model of the final building structure system can be simplified to: (27) in , , ,in and Represented as the zero matrix and the identity matrix, respectively; In practical applications, the control force of a controller is typically limited by saturation constraints. These constraints aim to ensure the controller does not overload and effectively conserve input energy. To ensure the stability and reliability of the control system, the control input must meet the following constraints: (28) in, It is the maximum input value of the controller, and also the preset upper limit.

[0019] Lemma 1 Schur's complement: For a given symmetric matrix ,in and It is a symmetric matrix of appropriate dimensions. If the matrix is ​​of appropriate dimensions, then the following three conditions are equivalent: (1) ; (2) ; (3) . To ensure the stability of the control system, and taking into account external disturbances The boundedness of the condition is addressed by employing a state feedback-based approach. A control method is proposed to achieve optimal vibration reduction. For the given system equation (11), a control saturation method is suggested. The controller's methods.

[0020] Therefore, if given parameters , and initial value If a symmetric positive definite matrix of suitable dimension exists... sum matrix This makes the following matrix inequality hold. (29)

[0021] (30) (31) Then design a state feedback. controller This ensures that, under controller saturation conditions, the closed-loop control system (11) simultaneously satisfies the following conditions: (1) The closed-loop system (11) is asymptotically stable; (2) The system satisfies Performance metrics; (3) Input saturation constraints It always holds true.

[0022] Proof: Take the following Lyapunov functional as ,in It is a symmetric positive definite matrix; .right Differentiate:

[0023] And Substituting into the above equation, we get: (32) Integrating from both sides, we get

[0024] Given the initial conditions are ,but , Therefore, we can obtain , Therefore, the closed-loop control system (11) satisfies Performance metrics.

[0025] if only (33) achievable That is, system (11) is asymptotically stable. Equation (17) is equivalent to:

[0026] According to the Schur complement property of Lemma 1, the above equation is equivalent to (34) Use matrix diag Left and right multiplication expressions (18), and let , We can obtain the following formula:

[0027] Equation (13) is proved.

[0028] Depend on and We can obtain: (35) By the Schur complement property of Lemma 1: (36) Equation (14) is proved.

[0029] The following proof (15) is: Because We can obtain:

[0030] From the above formula, we can obtain: (37) According to Lyapunov stability theory, it is known that Combining the assumptions From the above, we can conclude that: , (38) If it can be obtained (39) Therefore, equation (21) is true. Thus, from equation (23), we can deduce: (40) According to the Schur complement property of Lemma 1, equation (23) is equivalent to: (41) Multiply inequality (25) on the left and on the right by diag respectively. That is, matrix inequality (25) is equivalent to: (42) because It is a symmetric positive definite matrix, therefore ;definition From the above, we can deduce that: (42) Theorem (15) is proved.

[0031] In summary, a state feedback mechanism considering controller saturation was designed. The controller can achieve optimized control of structural vibration and suppress external disturbances.

[0032] This embodiment's example analysis and verification: A two-story building structure was chosen as the research object, and numerical analysis was conducted. This experiment allows for a deeper understanding of the controller's performance under saturation conditions, which is of great significance for further research on the controller's performance.

[0033] To ensure the broad applicability of the results, simulations were performed using the system parameters of the experimental platform provided by Quanser. Specific parameter values ​​are shown in Table 1.

[0034] Table 1 Parameters of Building Structure Model

[0035] Based on extensive and repeated practice, and according to the disturbance suppression performance indicators of actual high-rise building systems, a weighting coefficient matrix was qualitatively selected. Then, based on the system model parameter values ​​provided in Table 1, the coefficient matrix of system (11) can be obtained, as shown below: , , , .

[0036] In practical engineering, building systems often face various external stimuli, which can originate from the environment, loads, or other factors. To better simulate and evaluate the performance of building systems under different external stimuli, this paper employs three different signal excitation methods: step signals, harmonic signals, and Gaussian white noise signals. Step signals are used to simulate sudden changes in the system, causing significant changes in the system's state variables within a short period. Harmonic signals are used to simulate periodic external disturbances, such as natural phenomena like earthquakes and wind-induced vibrations. These disturbances have a continuous impact on the system's dynamic performance. Gaussian white noise signals are used to simulate various random disturbances faced by the system, such as airflow fluctuations and seismic waves. These disturbances are irregular in nature and have an uncertain impact on the system's stability.

[0037] Furthermore, considering the reaction force of the controller on the top floor and the restraining effect of the upper floors on the lower floors, the study focuses on the stress vibration behavior of the top floor under these three different signal excitation methods. This research aims to comprehensively evaluate the stability and dynamic performance of multi-story or high-rise building structural systems under various external disturbances, providing a basis for optimizing and improving the design of building systems and enhancing their resistance to disturbances in complex external environments.

[0038] right Controller control input Constraints are applied, and the mincx solver in the LMI toolbox is used to find the optimal solution by minimizing the performance index. , and the corresponding feedback gain: , Corresponding best performance , .

[0039] like Figure 1 As shown, Figure 1 This study compares the vibrations of a rooftop under controlled and uncontrolled step signal excitation. Without control, the rooftop exhibits severe oscillations within 0-5.3 seconds after a step input, with a large overshoot and a long settling time, requiring 3.3 seconds to stabilize. In contrast, considering a saturated controller… The control method can better suppress the rooftop The vibration stabilized in 0.45 seconds, with the overshoot reduced by 60.2%. Overall, the controller under saturation... Control enables A smoother response and faster attainment of equilibrium indicate that the controller is under saturation. Control can effectively reduce vibration and keep it within a controllable range. Figure 2 This demonstrates the steady-state vibration of a building structure under harmonic excitation. Without control, the amplitude of the rooftop vibration approaches the excitation frequency but is not effectively controlled. A saturation controller is then introduced. After control, rooftop vibration was reduced by 73.7%, effectively suppressing the impact of excitation signals on the building. Figure 3 Demonstrates controller saturation under random signal excitation Controlled and Uncontrolled Response curves. The comparison shows that the building structure system is continuously affected by disturbances and cannot reach a stable state. After adopting the control strategy, the displacement and vibration of the building structure... On average, the vibration was reduced by 56.5%, with relatively small fluctuations. This indicates that the control measures adopted significantly reduced the impact of external excitation on the building structure's vibration, improved its dynamic performance, and demonstrated strong anti-interference capabilities.

[0040] like Figure 2 and Figure 3 As shown, under controlled conditions, the vibration amplitude of each floor is relatively small. This indicates that the active mass damper can better absorb and disperse external energy, thereby reducing the vibration of the building structure. Furthermore, the calculated control input shows that... The controller's input was consistently less than the preset value. This indicates that the control input is effective under constraints, achieving the vibration reduction target while simultaneously satisfying the control input's saturation limit. Therefore, the controller considering its saturation characteristics not only achieves the vibration reduction target but also satisfies the control input's saturation constraint, effectively reducing floor vibration and improving the building structure's vibration resistance. This preliminarily verifies the effectiveness of the proposed input constraint strategy and provides a valuable reference for subsequent research.

[0041] like Figure 3 and Figure 4 As shown, in order to comprehensively study the influence of controller saturation characteristics on the control effect of building systems, it is necessary to study the stress vibration of building structures without considering controller saturation characteristics. Therefore, and applying the method described in formula (13) of Theorem 1, a traditional... Controller. The optimal solution is found using the mincx solver in the LMI toolbox. Values ​​are used to determine the feedback gain of the controller: , Corresponding best performance , .

[0042] Calculations yield the following results. This result aligns with the conclusion that smaller control gain is generally better. Therefore, when designing and implementing the controller proposed in Theorem 1, considering controller saturation can more effectively save input energy, thereby achieving a more optimized control effect. This further indicates that when designing controllers for vibration control, controller saturation should be fully considered to achieve a more optimized control effect. By... Figure 4 Building structures under traditional control strategies demonstrated Vibration conditions and Figure 3 By comparing the vibration under controller saturation, it can be observed that the vibration of the building structure was significantly reduced. Therefore, when designing controllers for vibration control, the controller saturation problem should be fully considered to achieve a more optimized control effect.

[0043] This invention is applied under controller saturation conditions. The control strategy, combined with active mass dampers, effectively reduced the vibration of the building structure while meeting control input constraints and Performance Requirements. The proposed control strategy ensures the stability and optimal performance of the closed-loop system while reducing control energy consumption. The feasibility and effectiveness of the proposed control strategy are verified through case studies. This research has significant theoretical and practical implications for the field of building structure control, enriching the theoretical achievements of system stability and robust control under controller saturation, and providing new ideas and methods for the optimization and control of building structures.

[0044] The above descriptions are merely embodiments of the present invention, and common knowledge such as specific technical solutions and / or characteristics are not described in detail here. It should be noted that those skilled in the art can make various modifications and improvements without departing from the technical solutions of the present invention, and these should also be considered within the scope of protection of the present invention. These modifications and improvements will not affect the effectiveness of the implementation of the present invention or the practicality of the patent. The scope of protection claimed in this application should be determined by the content of its claims, and the specific embodiments described in the specification can be used to interpret the content of the claims.

Claims

1. A control method for structural vibration with an active mass damper under controller saturation, characterized in that: The control system equations for a building structure under controller saturation conditions are as follows: in, x(t) represents the state vector of the building structure at time t, which includes information on structural displacement and velocity. z(t) represents the controlled output response vector of the building structure at time t; These are external disturbances, mainly caused by earthquakes and random winds; The system outputs a weighted coefficient matrix, which describes the dynamic characteristics of the building structure. Design status feedback controller This ensures that the control system has asymptotic stability and satisfies... Performance gain metrics The requirements, namely in Found through search To obtain the optimal value of the control system. Optimal control design achieves optimal suppression of disturbances; Under external excitation, with the goal of reducing the relative displacement and velocity of each floor, the control output equation is expressed as: For external incentives In its simplified model, the building structure uses a motor-driven trolley mounted on the top of the building to generate a reaction force on the building structure, thereby controlling the building's vibrations; the trolley on the top floor is driven by an internal DC motor. Based on angular velocity and linear velocity Relationship between them: in, These are system parameters; they convert the angular velocity of the trolley's motor into linear velocity. Car driving force With the voltage of the car motor and linear velocity There is a linear functional relationship between them, and the specific expression is as follows: in, , These are system parameters; The method of using the Lagrange equation can be used to derive The dynamic equations of the multi-story building system are as follows: in, , , and , They represent the first The mass, equivalent viscous damping coefficient, stiffness, and horizontal deflection of the layer; , , and These represent the mass, equivalent viscous damping coefficient, inertial force, and horizontal displacement relative to the top floor of the rooftop active mass damper, respectively. It is the horizontal displacement of the ground, representing the excitation on the building; The augmented state vector is selected as Under external excitation, the state equation of a building structure can be expressed in matrix form, i.e. in, , , ， ， ， ， ； because When reversible, the state equation of the building structure transforms into: make , , , The above transformed equation can be simplified to: Further The final dynamic model of the building structure system simplifies to: = in, , , ,in and Represented as the zero matrix and the identity matrix, respectively; The control inputs must satisfy the following constraints: in, It is the maximum input value of the controller, and also the preset upper limit; For a given symmetric matrix ,in and It is a symmetric matrix of appropriate dimensions. If the matrix is ​​of appropriate dimensions, then the following three conditions are equivalent: （1） ; （2） ; （3） . Therefore, if given parameters , and initial value If a symmetric positive definite matrix of suitable dimension exists... sum matrix This makes the following matrix inequality hold. Then design a state feedback. controller This ensures that, under controller saturation conditions, the control system simultaneously satisfies the following conditions: (1) The control system (11) is asymptotically stable; (2) The system satisfies Performance metrics; (3) Input saturation constraints It always holds true.

Images