Multi-actuator cooperative vibration suppression control method based on linearization model prediction strategy

Abstract

The invention discloses a multi-actuator cooperative vibration suppression control method based on a linearization model prediction strategy, belongs to the technical field of vibration control and flexible structure control, and constructs a mathematical model capable of describing the dynamics of a flexible beam containing a plurality of motion actuators or motion loads. Designing model predictive control based on a linearized state equation; simulation results show that the control method enables the vibration of the beam to be effectively restrained, each actuator is smoothly transferred to an expected position, and the situation that the performance of the space rigid-flexible coupling system is reduced when the space rigid-flexible coupling system executes a task due to vibration is avoided.

Description

Technical Field

[0001] The present invention belongs to the technical field of vibration control and flexible structure control, and particularly relates to a multi-actuator collaborative vibration suppression control method based on a linearized model prediction strategy. Background Technique

[0002] In new space missions such as variable baseline observations, there are some moving components on flexible space structures, and the system thus has strong rigid-flexible coupling effects, which have attracted the close attention of many scientific researchers. For example, Choura et al. (Choura S, Yigit A S. Control of a two-link rigid–flexible manipulator with a moving payload mass[J]. Journal of Sound and Vibration, 2001, 243(5):883-897.) developed a proportional-derivative type controller based on partial differential equations to suppress the vibration of a two-link rigid-flexible manipulator carrying a moving load. Yong et al. (Yong T, Xiang Y, Yan F, et al. Vibration behaviour analysis and vibration suppression studies of the space robot[J]. International Journal of Aerospace Engineering, 2022, 2022(1):3641051.) designed a controller that combines trajectory tracking and vibration suppression for a rigid-flexible coupling robotic arm. Moradi et al. (Moradi S, Azam S E, Mofid M. On Bayesian active vibration control of structures subjected to moving inertial loads[J]. Engineering Structures, 2021, 239:112313. Ma Z.Q., Sun G.H. Adaptive sliding mode control of tethered satellite deployment with input limitation. Acta Astronautica, 2016, 127(1):67-75.) introduced a new Bayesian framework to design a controller, which successfully suppressed the vibration of a flexible beam caused by multiple masses moving at a constant speed.Stancioiu et al. (Stancioiu D, Ouyang H. Optimal vibration control of beams subjected to a mass moving at constant speed[J]. Journal of Vibration and Control, 2016, 22(14): 3202-3217.) designed an optimal control vibration suppression scheme for flexible beams with a mass moving at a constant speed. Cai et al. (Cai G P, Hong J Z, Yang S X. Model study and active control of a rotating flexible cantilever beam[J]. International Journal of Mechanical Sciences, 2004, 46(6): 871-889.) designed a linear quadratic regulator for the rigid-flexible coupling system of a central rigid body-cantilever beam and achieved vibration control of the system. [5] And trajectory tracking. Sharma et al. (Sharma R, Tewari A. Optimal nonlinear tracking of spacecraft attitude maneuvers[J]. IEEE Transactions on Control Systems Technology, 2004, 12(5): 677-682.) designed a spacecraft attitude controller using a nonlinear optimal control method to complete the attitude adjustment task of a flexible spacecraft and achieved good results.

[0003] Existing research shows that for the vibration suppression of flexible structures with moving loads, optimal control is a common idea and has significant advantages in control effects compared with traditional controllers. However, in practice, due to the large computational complexity of multi-constrained high-dimensional nonlinear systems, it is difficult to implement; moreover, very few scholars have conducted in-depth research on the form of movable actuators. Summary of the Invention

[0004] The present invention provides a multi-actuator collaborative vibration suppression control method based on a linearized model prediction strategy, which can completely suppress the vibration of a flexible beam during the movement of the actuator, avoiding the performance degradation of the space rigid-flexible coupling system during task execution caused by vibration.

[0005] To achieve the above objectives, the present invention adopts the following technical solutions:

[0006] A multi-actuator collaborative vibration suppression control method based on a linearized model prediction strategy, comprising the following steps:

[0007] Select the horizontal position of the actuator and the modal coordinates of the cantilever beam as the generalized coordinates, and obtain the dynamic equation of the system based on the second Lagrange equation;

[0008] Discretize the system according to the system state at the current moment and linearize the system state at the current moment to obtain a discrete linear time-invariant state space equation for predictive control;

[0009] Define a quadratic performance index, substitute the discrete linear time-invariant state space equation into the quadratic performance index and solve the quadratic programming problem to obtain the optimal control sequence of the system at the current moment;

[0010] At the next moment, repeat the above process, re-linearize the system according to the system state at the current moment to obtain a new state matrix and input matrix, establish and solve a new quadratic objective function, and realize the rolling optimization of the system.

[0011] In the above steps, the optimal control sequence of the system at the current moment is obtained. The optimal control sequence is a 2n×N p matrix, and each column in the matrix corresponds to the optimal control input at a moment within the prediction horizon. Only the first column of the matrix is applied to the actual system.

[0012] Beneficial effects: The present invention provides a multi-actuator collaborative vibration suppression control method based on a linearized model prediction strategy, constructs a mathematical model capable of describing the dynamics of a flexible beam containing multiple moving actuators or moving loads, designs a model prediction based on a linearized state equation, can effectively suppress the vibration of the beam, and further avoid the performance degradation of the spatial rigid-flexible coupling system during task execution caused by vibration. It can be seen from the numerical simulation results that by applying the control method of the present invention, the vibration of the cantilever beam is effectively suppressed, and each actuator smoothly transfers to the expected position; the control method proposed by the present invention is practical and feasible, and can effectively control the flexible beam system with moving actuators. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 Schematic diagram of the moving actuator, control force and torque in the embodiment of the present invention;

[0014] Figure 2 Schematic diagram of the deflection of the beam at the position of any actuator at a certain moment in the embodiment of the present invention;

[0015] Figure 3 Curves of the deformation at the end of the beam and the change of the horizontal position of the actuator without applying predictive control. Among them, a is the curve of the deflection at the end of the beam changing with time, and b is the curve of the position change of each actuator on the beam;

[0016] Figure 4 This is the effect diagram of applying predictive control in the embodiments of the present invention, where a is the curve of the end deflection of the beam varying with time, and b is the curve of the position change of each actuator on the beam. Detailed implementation manners

[0017] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments:

[0018] As Figure 1 shown, there is a cantilever beam in the horizontal direction. The cantilever beam is equipped with n movable actuators. The mass of each actuator cannot be ignored relative to the cantilever beam. Each actuator can output a horizontal control force u along the x-axis and a control torque τ pointing out of the page. Therefore, the actuator can be controlled to slide on the central axis of the beam by the control force u. The control torque τ can be transmitted to the beam through the actuator to control the dynamic deformation of the beam. It is assumed that the system under study operates in deep space, so the influence of gravity can be ignored. It is assumed that the cantilever beam is an Euler-Bernoulli beam and the deformation of the beam is limited to a small range. Figure 2 shows the deflection at the position of any actuator on the beam. The horizontal position between the i-th moving actuator and the coordinate origin can be represented by x i (t), and w(x i (t),t) represents the deflection of the beam at the i-th moving actuator; ρ, EI, l, and A are respectively the unit volume density of the cantilever beam, the uniform flexural rigidity of the beam, the horizontal length of the beam, and the cross-sectional area. In addition, it is assumed that the mass of each movable actuator is the same, all being m, and x i is used to simplify the representation of x i (t) in the subsequent derivation.

[0019] By selecting the horizontal position of the actuator and the modal coordinates of the cantilever beam as the generalized coordinates, the following dynamic equations of the system can be directly obtained based on the second Lagrange equation:

[0020]

[0021] where p = [x T q T T = [x1 x2 … x n q1 q2 q3] T is the generalized coordinate vector, M, C, K, and D are respectively the mass matrix, Coriolis force matrix, stiffness matrix, and control matrix, U 1×2n = [u T τ T T = [u1 u2 … u n τ1 τ2 … τ n T ​​is the generalized force vector composed of the horizontal control force and the control moment pointing out of the plane on each motion actuator; x represents the actuator horizontal position vector composed of all motion actuators in the Figure 1 numbering order in

[0022] Since the actuators are in motion, the matrices M and C have non-diagonal time-varying elements; in addition, the expression form of the Coriolis force matrix C is not unique, and an appropriate C matrix is selected to satisfy being skew-symmetric. In the simulation of the present invention, the M, C, K, and D matrices of the system are deduced for different numbers of actuators. In this embodiment, for the general case of installing n motion actuators on the beam, specific expressions are given.

[0023] According to the characteristics of each matrix, they can all be divided into four parts, as shown in Equations (2)-(16).

[0024]

[0025] The K matrix has only the sub-matrix K 22 containing non-zero elements:

[0026]

[0027] where φ jxx represents φ j represents the j-th order mode of the system, and x is the horizontal position at a certain place on the flexible beam. In the present invention, for the deflection w at each place on the flexible beam, the modal truncation method is used to approximately represent it as a linear combination of the first three modes w(x,t) = φ1(x(t))q1(t) + φ2(x(t))q2(t) + φ3(x(t))q3(t), where q j (t) is the modal coordinate corresponding to the mode.

[0028] The three non-zero sub-matrices of the M matrix are in the following forms:

[0029]

[0030]

[0031] where wx represents xi(i = 1 2… n) represents the horizontal position of the i-th actuator on the beam.

[0032] The Coriolis force matrix C is expressed in the following form

[0033]

[0034]

[0035] Among them, φ jx represents is:

[0036]

[0037] Obviously, the values of the elements in matrix C are related to the horizontal velocity of the actuator the change rate of the modal coordinate and the mass m of a single actuator. It can be seen that when the mass of the actuator cannot be ignored relative to the overall flexible structure, the position and velocity of the moving mass will significantly change the matrices C and M of the system, thereby affecting the control effect of the controller.

[0038] The non-zero sub-matrix of the control matrix D is shown as follows:

[0039]

[0040] The continuous system is discretized by the Euler method into the form, where p(k) is the value of the generalized coordinate vector at t k moment, and T s represents the discrete time interval. Linearize the system at the current moment, and the elements in the M, C, and K matrices in Equation (1) are approximated by the true values at t k moment and remain unchanged during a single prediction. Thus, the discrete linear time-invariant state-space equation for predictive control can be written as:

[0041] p(k + 1) = Ap(k) + BU(k) (17)

[0042] Among them, U(k) represents the value of the generalized force vector at t k moment, p(k) is the value of the generalized coordinate vector at t k moment, and A and B are the state matrix and input matrix after linearizing the system at the current moment, which are composed of constant elements.

[0043] Subsequently, the quadratic performance index is defined as:

[0044]

[0045] Among them, S, Q, and R respectively represent the weight diagonal matrices of the terminal cost, running cost, and control quantity cost of the system, and N p represents the set prediction step. The larger the value of an element in the weight matrix relative to other elements, the higher the importance of the corresponding variable in the optimization problem. P(k) represents the difference between the control target p d and the current state quantity, P(k) = pd -p(k).

[0046] Substituting Equation (17) into Equation (18) and solving it using an optimization algorithm applicable to linear quadratic programming problems, the optimal control sequence U k of the system at time t * (k) can be obtained. This sequence is represented as a 2n×N p matrix, where each column in the matrix corresponds to the optimal control input at a moment within the prediction horizon; only the first column of the matrix is applied to the actual system. At the next moment, repeat the above process. According to the system state at the current moment, re-linearize the system to obtain a new state matrix A and input matrix B, establish and solve a new quadratic objective function to achieve the rolling optimization of the system.

[0047] The predictive control strategy of this embodiment is verified through the following numerical simulation results:

[0048] The system parameters are defined as follows: Assume that there are three actuators installed on the beam. The mass of a single actuator is m = 0.1 kg, the length of the flexible beam is l = 1.4 m, and the linear density is ρ = 0.3 kg·m -1 , and the flexural rigidity EI = 0.7 N·m 2 . The initial positions of the actuators are 1 m, 0.9 m, and 0.8 m on the beam respectively.

[0049] Without applying predictive control, assume that the system has an initial deformation, which can be expressed in the form of modal coordinates as q(0) = [0.1 0.02 0] T , as shown in Figure 3 (a). The vibration at the end of the beam is affected by multiple modes. As can be seen from Figure 3 (b), the accelerations of the actuators are not constant. The reason for the change in the actuator accelerations is that the mass matrix M and the Coriolis force matrix C in Equations (4) and (5) contain coupling terms related to each mode and the actuator positions. When the system vibration includes multiple modes, these coupling terms bring complex effects to the horizontal movement of the actuators.

[0050] Apply the predictive control method of this embodiment. Assume that the initial state of the system is the same as that in the experiment without applying predictive control, and set the desired positions of the actuators to 0.5 m, 0.4 m, and 0.3 m. The control results are as shown in Figure 4 . It can be clearly seen that the vibration of the cantilever beam is effectively suppressed, and each actuator smoothly moves to the expected position; indicating that the above predictive control method is practical and feasible and can effectively control the flexible beam system with moving actuators.

[0051] The above are only the preferred embodiments of the present invention. It should be noted that those skilled in the art can make corresponding changes and adjustments to the technology of the present invention without departing from the basic principles of the present invention, and these changes and adjustments all fall within the protection scope of the present invention.

Claims

1. A multi-actuator collaborative vibration suppression control method based on a linearized model prediction strategy, characterized in that It includes the following steps: Select the horizontal position of the actuator and the modal coordinates of the cantilever beam as the generalized coordinates, and obtain the dynamic equation of the system based on the second kind of Lagrange equation; Discretize the system according to the system state at the current moment and linearize the system state at the current moment to obtain the discrete linear time-invariant state space equation for predictive control; Define the quadratic performance index, substitute the discrete linear time-invariant state space equation into the quadratic performance index and solve the quadratic programming problem to obtain the optimal control sequence of the system at the current moment; At the next moment, repeat the above process, re-linearize the system according to the system state at the current moment to obtain a new state matrix and input matrix, establish and solve a new quadratic objective function, and realize the rolling optimization of the system.

2. The multi-actuator collaborative vibration suppression control method based on the linearized model prediction strategy according to claim 1, characterized in that The dynamic equation of the system is: where p 1×(n+3) = [x T q T T = [x1 x2…x n q1 q2 q3] T is the generalized coordinate vector, M, C, K, and D are the mass matrix, Coriolis force matrix, stiffness matrix, and control matrix respectively, and U 1×2n = [u T τ T T = [u1 u2…u n τ1 τ2…τ n T is the generalized force vector composed of the horizontal control force and the out-of-plane control moment on each motion actuator.​​​ 3. The multi-actuator collaborative vibration suppression control method based on the linearized model prediction strategy according to claim 2, characterized in that Due to the actuator motion, matrices M and C have non-diagonal time-varying elements; is skew-symmetric.

4. The multi-actuator collaborative vibration suppression control method based on the linearized model prediction strategy according to claim 2 or 3, characterized in that The value of the element in matrix C is related to the horizontal velocity of the actuator the rate of change of the modal coordinate and the mass m of a single actuator 5. The multi-actuator collaborative vibration suppression control method based on the linearized model prediction strategy according to claim 1, characterized in that, The discrete linear time-invariant state space equation for predictive control is: p(k + 1) = Ap(k) + BU(k) where p(k) is the value of the generalized coordinate vector at time t k and U(k) represents the value of the generalized force vector at time t k The matrices A and B are the linearized state matrix and input matrix, respectively, and are composed of constant elements.

6. The multi-actuator collaborative vibration suppression control method based on the linearized model prediction strategy according to claim 1, wherein The quadratic performance index is: Among them, S, Q, and R respectively represent the weight diagonal matrices of the terminal cost, operating cost, and control quantity cost of the system, and P(k) represents the difference between the control target p d and the current state quantity, P(k) = p d - p(k).

7. The multi-actuator collaborative vibration suppression control method based on the linearized model prediction strategy according to claim 1, characterized in that, The optimal control sequence of the system at the current moment is a 2n×N p matrix, and each column in the matrix corresponds to the optimal control input at a moment within the prediction horizon.

8. The multi-actuator collaborative vibration suppression control method based on the linearized model prediction strategy according to claim 7, characterized in that Only apply the optimal control input of the first column of the matrix to the system.

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