An Adaptive Vibration Control Method for a Flexible Plate and Strip Rolling System under Output Constraints

Abstract

The present invention discloses an adaptive vibration control method for a flexible strip rolling system under output constraints, which relates to the technical field of boundary vibration control in the strip rolling process. First, according to Hamilton's principle, a dynamic model of the strip rolling system is established. Then, considering system output constraints, unknown actuator faults, and boundary disturbance factors, based on the Lyapunov stability theorem, a boundary fault-tolerant controller for strip vibration is designed to achieve vibration suppression of the strip, and it has a certain adaptive characteristic for boundary disturbances. At the same time, Matlab simulation is used to verify the proposed control method. The present invention establishes a more practical flexible strip rolling system model, takes into account system output constraints, unknown actuator faults, and unknown boundary disturbance factors, designs a boundary vibration controller, realizes rapid active suppression of the flexible strip during the high-speed rolling process, and ensures the stability of the high-speed rolling strip process.

Description

Technical Field

[0001] The present invention relates to the technical field of boundary vibration control in strip rolling processes, and in particular to an adaptive vibration control method for a flexible strip rolling system under output constraints. Background Art

[0002] The strip rolling system is crucial for the production of metals widely used in different engineering industries such as automobiles, aerospace, semiconductors, and household appliances. Due to specification switching and process parameter fluctuations during strip rolling, the system has uncertainties and external disturbances, which often lead to vibration problems. In the past few decades, research on the vibration of strip rolling systems has mainly focused on the rolls, with little attention paid to the vibration of flexible strips. In fact, due to the flexibility of the strip, the vibration of the strip is inevitable. The vibration of the flexible strip is harmful because it reduces its dimensional accuracy and may even cause fracture. The vibration of the strip rolling system is a difficult problem restricting the improvement of strip quality. Therefore, it is of great significance to study the vibration control of the strip rolling system.

[0003] Currently, the main methods for controlling distributed parameter systems are distributed control and boundary control. Since boundary control can achieve the control effect by simply placing controllers at certain characteristic positions, boundary control has received extensive attention in the vibration control of distributed parameter systems in recent years. More typically, Zhao proposed a robust adaptive vibration suppression control strategy for a riser vessel system with output constraints. He provided a control based on an asymmetric BLF for a string system to avoid violating time-varying constraints. Ren studied the stability of a string system with actuator faults and dead zones using fault-tolerant vibration damping control. Liu proposed an adaptive event-triggered vibration suppression solution for an air refueling hose system with actuator faults. Xing studied the robust adaptive control algorithm for a cascade ordinary differential equation string system with actuator faults.

[0004] Although the above-mentioned boundary control has achieved satisfactory results at present, none of the current vibration control algorithms consider the output constraints and unknown actuator fault problems existing in the flexible strip rolling system. Therefore, the design of an adaptive boundary vibration control considering the output constraints and unknown actuator faults existing in the actual industrial strip rolling system is of practical significance. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide an adaptive vibration control method for a flexible strip rolling system under output constraints. According to Hamilton's principle, a dynamic model of the strip rolling system is established. Then, considering system output constraints, unknown actuator faults, and boundary disturbances, a boundary fault-tolerant controller for strip vibration is designed to achieve vibration suppression of the strip and have a certain adaptive characteristic for boundary disturbances.

[0006] To solve the above technical problems, the technical solution adopted by the present invention is: an adaptive vibration control method for a flexible strip rolling system under output constraints, including the following steps:

[0007] Step S1: Analyze the flexible strip rolling system and establish a dynamic model of the flexible strip rolling system according to Hamilton's principle;

[0008] Step S2: Considering system output constraints, unknown actuator faults, and boundary disturbance factors, design the adaptive characteristics of unknown parameters of the flexible strip rolling system;

[0009] Step S3: Design a boundary controller;

[0010] Step S4: Based on Lyapunov stability theorem, conduct a stability analysis of the flexible strip rolling system;

[0011] Step S5: Judge the boundedness of the state of the flexible strip rolling system;

[0012] Step S6: Simulate the flexible strip rolling system through Matlab simulation software;

[0013] Step S7: View and analyze the simulation effect of the flexible strip rolling system;

[0014] Step S8: Judge whether it is necessary to adjust the gain parameters according to the simulation effect in Step S7.

[0015] A further improvement of the technical solution of the present invention lies in: the specific process of analyzing the flexible strip rolling system in Step S1 is as follows:

[0016] Step S11: Solve the kinetic energy of the flexible strip rolling system:

[0017]

[0018] where ρ is the mass per unit length of the strip, r is the time-varying length of the flexible strip, h = h(y, t) is the vibration displacement of each position of the flexible strip on the y-axis at time t, m1 is the equivalent mass of the roll, and v is the running speed of the flexible strip.

[0019] Step S12: Solve the potential energy of the flexible strip rolling system:

[0020]

[0021] where S is the time-varying tension difference acting on both ends of the rolled strip, E is the elastic coefficient of the strip, I is the cross-sectional moment of inertia of the strip, and A is the cross-sectional area of the strip.

[0022] Step S13: Solve the virtual work of the flexible strip rolling system:

[0023]

[0024] where u(t) is the boundary actuator, h(r, t) is the vibration displacement at the system boundary, d(t) is the boundary disturbance, d c is the boundary damping coefficient, c is the strip motion damping coefficient, and F is the unknown force between the strip and the roll.

[0025] A further improvement of the technical solution of the present invention lies in that: in step S1, according to Hamilton's principle, the dynamic model of the flexible strip rolling system is as follows:

[0026]

[0027]

[0028] h(0, t) = h′(0, t) = h″(r, t) = 0

[0029] where

[0030] A further improvement of the technical solution of the present invention lies in that: in step S2, considering the boundary disturbance d(t) ≤ D existing in the flexible strip rolling system, the approximation error of the system nonlinear term actuator additive fault Let be the composite disturbance, and select the composite disturbance estimator whose adaptive characteristics are designed as:

[0031]

[0032] The adaptive characteristics of the neural network weights are designed as:

[0033]

[0034] The adaptive characteristics of the actuator multiplicative fault coefficient are designed as:

[0035]

[0036] A further improvement of the technical solution of the present invention lies in that: in step S3, the process of designing the boundary controller is as follows:

[0037] The virtual controller is designed as

[0038] x a = -(v + λr)h′(r, t)

[0039] The boundary controller is designed to

[0040]

[0041]

[0042] A further improvement of the technical solution of the present invention lies in that: the process of performing stability analysis on the flexible strip rolling system in step S4 is as follows:

[0043] Define the Lyapunov function V(t) of the flexible strip rolling system as:

[0044] V(t) = V1(t) + V2(t) + V h (t) + V b (t);

[0045] Wherein,

[0046]

[0047]

[0048]

[0049]

[0050] Verify the positive definiteness of the Lyapunov function V(t), and it is obtained that the system is stable in the sense of Lyapunov. Then, verify the negative definiteness of, so as to conclude that the system is bounded stable.

[0051] A further improvement of the technical solution of the present invention lies in that: the specific process of judging the boundedness of the state of the flexible strip rolling system in step S5 is as follows:

[0052] Step S51: The boundary vibration displacement h(r,t) of the flexible strip system can converge to a restricted area after time T s , that is, -w x < h(r,t) < w s , t ≥ T s ;

[0053] Step S52: The longitudinal vibration h(y,t) at each position of the flexible strip at each moment can converge to a compact set Ω, then there is:

[0054]

[0055] A further improvement of the technical solution of the present invention lies in that: in step S8, it is judged whether the vibration amplitude of the flexible strip rolling system meets the requirements according to the simulation results of step S7. If the vibration amplitude does not meet the requirements, return to step S3 to readjust the gain parameter k1 of the boundary controller; if the vibration amplitude meets the requirements, end.

[0056] Due to the adoption of the above technical solution, the technical progress obtained by the present invention is:

[0057] The present invention provides an adaptive vibration control method for a flexible strip rolling system under output constraints. First, according to Hamilton's principle, a dynamic model of the strip rolling system is established. Then, considering system output constraints, unknown actuator faults, and boundary disturbances, based on Lyapunov stability theorem, a boundary fault-tolerant controller for strip vibration is designed to achieve vibration suppression of the strip, and at the same time has a certain adaptive characteristic for boundary disturbances, and uses Matlab simulation to verify the proposed control method. The present invention establishes a more practical flexible strip rolling system model, considers system output constraints, unknown actuator faults, and unknown boundary disturbance factors, designs a boundary vibration controller, realizes rapid active suppression of the flexible strip during high-speed rolling, and ensures the stability of the high-speed rolling strip process. Description of the Drawings

[0058] Figure 1 Flowchart of an adaptive vibration control method for a flexible strip rolling system under output constraints of the present invention;

[0059] Figure 2 Vibration simulation diagram of the strip rolling system without control;

[0060] Figure 3 Vibration simulation diagram of the strip rolling system without output constraint control;

[0061] Figure 4 Vibration simulation diagram of the strip rolling system under the control of the present invention;

[0062] Figure 5 Boundary vibration displacement diagram of the strip rolling system under the first disturbance case;

[0063] Figure 6 Boundary vibration displacement diagram of the strip rolling system under the second disturbance case. Detailed Description of the Invention

[0064] The following further describes the present invention in detail with reference to embodiments:

[0065] An adaptive vibration control method for a flexible strip rolling system under output constraints. In this embodiment, as Figure 1As shown in the figure, an adaptive vibration control method for a flexible strip rolling system under output constraints according to the present invention includes the following steps:

[0066] Step S1: Analyze and model the flexible strip rolling system;

[0067] Kinetic energy of the flexible strip rolling system:

[0068]

[0069] where ρ is the mass per unit length of the strip, r is the time-varying length of the flexible strip, h = h(y, t) is the vibration displacement of the flexible strip at each position on the y-axis at time t, m1 is the equivalent mass of the roll, v is the running speed of the flexible strip,

[0070] Potential energy of the flexible strip rolling system:

[0071]

[0072] where S is the time-varying tension difference acting on both ends of the rolled strip, E is the elastic coefficient of the strip, I is the cross-sectional moment of inertia of the strip, A is the cross-sectional area of the strip,

[0073] Virtual work of the flexible strip rolling system:

[0074]

[0075] where u(t) is the boundary actuator, h(r, t) is the vibration displacement at the system boundary, d(t) is the boundary disturbance, d c is the boundary damping coefficient, c is the damping coefficient of the strip movement, and F is the unknown force between the strip and the roll,

[0076] Then, according to Hamilton's principle, the dynamic model of the flexible strip rolling system is obtained:

[0077]

[0078]

[0079] h(0, t) = h′(0, t) = h″(r, t) = 0

[0080] where,

[0081] Step S2: Considering system output constraints, unknown actuator faults, and boundary disturbance factors, design the unknown parameter adaptability of the flexible strip rolling system

[0082] Considering the boundary disturbance \(d(t)\leq D\) existing in the flexible strip rolling system and the approximation error of the system's non - linear terms Actuator additive fault Let be the composite disturbance, and select the composite disturbance estimator Its adaptive characteristics are designed as:

[0083]

[0084] The adaptive characteristics of the neural network weights are designed as:

[0085]

[0086] The adaptive characteristics of the actuator multiplicative fault coefficient are designed as:

[0087]

[0088] Step S3: Design the boundary controller

[0089] The virtual controller is designed as

[0090] x a =-(v + \(\lambda r\))h′(r,t)

[0091] The boundary controller is designed as

[0092]

[0093]

[0094] Step S4: Based on the Lyapunov stability theorem, conduct a stability analysis of the flexible strip rolling system

[0095] Define the Lyapunov function \(V(t)\) of the flexible strip rolling system as:

[0096] V(t)=V1(t)+V2(t)+V h (t)+V b (t);

[0097] Among them,

[0098]

[0099]

[0100]

[0101]

[0102] Verify the positive definiteness of the Lyapunov function V(t), and it is obtained that the system is stable in the sense of Lyapunov. Then, verify the negative definiteness of, so as to conclude that the system is bounded stable; among them, the method for verifying the positive definiteness of the Lyapunov function V(t) is:

[0103] Let where y ∈ [0, r], t ∈ [0, ∞], is the real number field, and this function also satisfies:

[0104] ψ(0, t) = 0, Then the following inequality holds:

[0105]

[0106] Judge the positive definiteness of the Lyapunov function V(t):

[0107]

[0108] Among them, the parameter θ1 should satisfy

[0109] V1(t) + V2(t) satisfies the following relationship

[0110] 0 ≤ θ2V1(t) ≤ V1(t) + V2(t) ≤ θ3V1(t)

[0111] Among them, the parameters θ2 > 0, θ1 > 1.

[0112] Using the definition of V(t), it can be obtained that:

[0113] 0 ≤ λ1(V1 + V h (t) + V b (t)) ≤ V1(t) + V2(t) + V h (t) + V b (t) ≤ λ2(V1 + V h (t) + V b (t)) Therefore, the positive definiteness of V(t) is verified, where λ1 = min(1, θ2), λ2 = max(1, θ1)

[0114] The above verification The method for negative definiteness is:

[0115] Take the derivative of V1(t) with respect to time

[0116]

[0117] Take the derivative of V2(t) with respect to time

[0118]

[0119] Substitute the virtual control law into to obtain

[0120]

[0121] Take the derivative of V(t) with respect to time and substitute the control law to obtain

[0122]

[0123] where

[0124] It can be obtained that That is The negative definiteness of is verified.

[0125] Step S5: Judge the boundedness of the flexible strip rolling system state

[0126] The boundary vibration displacement h(r, t) of the flexible strip system can converge to the constrained region after time T s That is, -w x <h(r, t)<w s , t≥T s .

[0127] The longitudinal vibration h(y, t) of each position of the flexible strip at each moment can converge to a compact set Ω, then there is:

[0128] Step S6: Simulate the flexible strip rolling system using the Matlab simulation software;

[0129] In this embodiment, the strip vibrations under no control action, the strip vibrations under no output constraint control action, and the strip vibration simulation comparison under the control method proposed by the present invention are respectively compared. Moreover, by selecting different boundary disturbances, the simulation shows that this method has a certain robustness.

[0130] Step S7: View and analyze the simulation effect of the flexible strip rolling system;

[0131] In this embodiment, after the flexible strip system is simulated, its simulation effect is as Figures 2 - 6 shown. Figure 2 is the vibration simulation diagram of the strip rolling system without control, Figure 3 is the vibration simulation diagram of the strip rolling system without output constraint control, Figure 4 is the vibration simulation diagram of the strip rolling system under the control of the present invention. From the three simulation effect diagrams, it can be seen that the present invention can effectively suppress the vibration of the strip; in order to illustrate the robustness of the vibration suppression method of the present invention, two different disturbances are selected for simulation comparison, and the simulation diagrams are asFigures 5 - 6 as shown

[0132] Step S8: Determine whether it is necessary to adjust the gain parameter according to the simulation effect in Step S7;

[0133] Judge whether the vibration amplitude of the flexible strip rolling system meets the requirements according to the simulation results. If the vibration amplitude does not meet the requirements, return to Step S3 to readjust the gain parameter k1 of the boundary controller. If the vibration amplitude meets the requirements, end.

Claims

1. An adaptive vibration control method for a flexible strip rolling system under output constraints, characterized in that: It includes the following steps: Step S1: Analyze the flexible strip rolling system and establish the dynamic model of the flexible strip rolling system according to Hamilton's principle; Step S2: Considering system output constraints, unknown actuator faults, and boundary disturbance factors, design the adaptive characteristics of unknown parameters for the flexible strip rolling system; considering the existing boundary disturbance d(t) ≤ D in the flexible strip rolling system and the approximation error of the system's non-linear terms Actuator additive fault Suppose is the composite disturbance, and select the composite disturbance estimator Its adaptive characteristics are designed as follows: The neural network weight adaptive characteristic is designed as: The actuator multiplicative fault coefficient adaptive characteristic is designed as: Step S3: Design a boundary controller; the process of designing the boundary controller is as follows: The virtual controller is designed as x a = -(v + λr)h′(r, t) The boundary controller is designed as Step S4: Based on Lyapunov stability theorem, conduct stability analysis on the flexible strip rolling system; Step S5: Judge the boundedness of the state of the flexible strip rolling system; Step S6: Simulate the flexible strip rolling system through Matlab simulation software; Step S7: View and analyze the simulation effect of the flexible strip rolling system; Step S8: Judge whether it is necessary to adjust the gain parameters according to the simulation effect in Step S7.

2. The adaptive vibration control method for a flexible strip rolling system under output constraints according to claim 1, wherein: The specific process of analyzing the flexible strip rolling system in Step S1 is as follows: Step S11: Solve the kinetic energy of the flexible strip rolling system: where ρ is the mass per unit length of the strip, r is the time-varying length of the flexible strip, h = h(y, t) is the vibration displacement of each position of the flexible strip in the y-axis at time t, m1 is the equivalent mass of the roll, and v is the running speed of the flexible strip. Step S12: Solve the potential energy of the flexible strip rolling system; Wherein, S is the time-varying tension difference acting on both ends of the rolled strip, E is the elastic coefficient of the strip, I is the cross-sectional moment of inertia of the strip, and A is the cross-sectional area of the strip. Step S13: Solve the virtual work of the flexible strip rolling system; Among them, u(t) is the boundary actuator, h(r, t) is the vibration displacement at the system boundary, d(t) is the boundary disturbance, and d c is the boundary damping coefficient, c is the damping coefficient of the strip movement, and F is the unknown acting force between the strip and the roll.

3. The adaptive vibration control method for a flexible strip rolling system under output constraints according to claim 2, characterized in that: The dynamic model of the flexible strip rolling system obtained in Step S1 according to Hamilton's principle is as follows: Among them, 4. The adaptive vibration control method for a flexible strip rolling system under output constraints according to claim 1, characterized in that: The process of conducting stability analysis on the flexible strip rolling system in Step S4 is as follows: Define the Lyapunov function V(t) of the flexible strip rolling system as: V(t) = V1(t) + V2(t) + V h (t) + V b (t); Among them, Verify the positive definiteness of the Lyapunov function \(V(t)\), and it is obtained that the system is stable in the sense of Lyapunov. Then, verify its negative definiteness, so as to conclude that the system satisfies bounded stability.

5. The adaptive vibration control method for a flexible strip rolling system under output constraints according to claim 2, wherein: The specific process of judging the boundedness of the state of the flexible strip rolling system in Step S5 is as follows: Step S51. The boundary vibration displacement h(r, t) of the flexible plate strip system can converge to a restricted region after time T s , that is, -w x < h(r, t) < w s , t ≥ T s ; Step S52: If the longitudinal vibration h(y, t) of each position of the flexible strip at each moment can converge to a compact set Ω, then there is:

6. An adaptive vibration control method for a flexible strip rolling system under output constraints according to claim 3, characterized in that: In Step S8, judge whether the vibration amplitude of the flexible strip rolling system meets the requirements according to the simulation result in Step S7. If the vibration amplitude does not meet the requirements, return to Step S3 to readjust the gain parameter k1 of the boundary controller; if the vibration amplitude meets the requirements, end.

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