Output feedback lqr control method of magneto-rheological damper based on newton iteration method

Abstract

The application belongs to the technical field of automobile nonlinear control, and discloses a Newton iteration method-based output feedback LQR control method for a magneto-rheological damper, which comprises the following steps: performing magneto-rheological damper external characteristic testing, obtaining characteristic data, linearizing the system globally into a Koopman high-dimensional linear model by using Koopman operator theory and extended dynamic mode decomposition, designing an output feedback LQR controller based on the model, introducing the Newton iteration method in the LQR process, taking the expected output damping force as the reference input, and applying the obtained control quantity to the magneto-rheological damper through iterative optimization of a preset method; the linearization model of the magneto-rheological damper is constructed in a data-driven manner by combining Koopman operator theory and extended dynamic mode decomposition algorithm, so as to describe the nonlinear magneto-rheological damper output damping force tracking control problem as a linear convex optimization problem, reduce the calculation complexity, and improve the real-time performance of the control algorithm.

Description

Technical Field

[0001] This invention belongs to the field of automotive nonlinear control technology, specifically relating to an output feedback LQR control method for magnetorheological dampers based on Newton's iterative method. Background Technology

[0002] Magnetorheological dampers are intelligent actuators based on magnetorheological fluids, capable of adjusting damping characteristics by changing the magnetic field strength. The underlying technology involves the magnetorheological effect, which states that the rheological properties of a fluid change under the action of an external magnetic field. This technology has been widely used in engineering fields, providing new solutions for intelligent control in automobiles, building structures, and vibration control systems. In particular, it has become a new generation of semi-active controllers in automotive semi-active suspensions.

[0003] However, the strong hysteresis nonlinearity of MRDs significantly impacts their modeling and control accuracy. To fully utilize the performance of MRDs, reliable and efficient control algorithms must be designed. Therefore, designing an effective control method to solve the nonlinear problems of MRDs has become an urgent problem to be solved. Summary of the Invention

[0004] The purpose of this invention is to provide an output feedback LQR control method for magnetorheological dampers based on Newton's iterative method, so as to solve the problems mentioned in the background art.

[0005] To achieve the above objectives, the present invention provides the following technical solution: an output feedback LQR control method for magnetorheological dampers based on Newton's iteration method, comprising the following steps:

[0006] S1. Perform external characteristic tests on the magnetorheological damper, obtain the external characteristic data of the magnetorheological damper, and use the Koopman operator theory and extended dynamic mode decomposition algorithm to globally linearize the magnetorheological damper system into a Koopman high-dimensional linear model.

[0007] S2. An output feedback LQR controller is designed based on the Koopman high-dimensional linear model. Newton's iteration method is introduced into the LQR control process. By solving a Lyapunov equation in each iteration, the convergence of the stable output feedback solution is guaranteed.

[0008] S3. Using the desired output damping force as a reference input, and based on the continuous iterative optimization of the preset method, the control quantity obtained by the output feedback LQR is applied to the magnetorheological damper, and the desired damping force is tracked through the magnetorheological damper.

[0009] Preferably, in S1, the external characteristic test of the magnetorheological damper is based on the semi-active suspension test bench.

[0010] Preferably, in step S1, the external characteristic data of the magnetorheological damper includes experimental data on the output damping force, piston rod speed, piston rod relative displacement, and current flow of the magnetorheological damper under different experimental conditions.

[0011] Preferably, in step S1, the Koopman operator theory and extended dynamic mode decomposition algorithm are used to train a Koopman linear model with the inputs being the piston rod velocity, the piston rod relative displacement, and the current flowing through it, and the output being the damping force of the magnetorheological damper. The Koopman linear model is as follows:

[0012]

[0013] In the formula, k represents the current time. For the state variables of the model in high-dimensional space. For state promotion functions, The predicted output damping force is obtained by solving relevant optimization problems based on the acquired dataset.

[0014] Preferably, in step S2, the process of designing the output feedback LQR controller based on the Koopman high-dimensional linear model and introducing Newton's iteration method is as follows:

[0015] S21: Design the control objective in the output feedback LQR controller;

[0016] S22: Substitute the standard control quantity form obtained by the output feedback LQR into the control objective, transform the control objective, and obtain the transformed formula.

[0017] Preferably, the control objective is:

[0018] The standard control quantity is in the form of u(k) = -Fy(k) = -FCz(k);

[0019] The transformation formula is:

[0020] Where Q = C T C and R are weighted matrices, and Q and R are both symmetric matrices; N is the cross-term weighted matrix, and F is the output feedback gain matrix, which can be expressed as F = R. -1 (B T P+N T C + C was obtained. + The pseudo-inverse of C is represented by matrix P, which is obtained by solving the following Lyapunov matrix:

[0021] (A-BFC) T P+P(A-BFC)

[0022] =-QC T F T RFC+C T F T N T +NFC;

[0023] A, B, and C are the state matrix, input matrix, and output matrix of the Koopman high-dimensional linear model, respectively, while BFC and RFC are related matrix multiplications.

[0024] Preferably, in step S3, the optimal output feedback gain matrix is ​​found by iteratively optimizing the process using a preset method, and the obtained optimal control quantity is applied to the magnetorheological damper.

[0025] Preferably, the preset method is Newton's iteration method.

[0026] Preferably, in step S3, the iterative optimization based on the preset method specifically includes the following steps:

[0027] S31: Initialize the number of iterations and the criterion function value;

[0028] S32: Calculate the optimal output feedback gain F;

[0029] S33: Update P according to the preset equation;

[0030] S34: Update the criterion function value for this iteration, and continuously iterate and optimize until the stopping criterion is met or the maximum number of iterations is reached;

[0031] S35: Returns the optimal output feedback gain F.

[0032] Preferably, the preset equation is the Lyapunov equation.

[0033] Compared with the prior art, the beneficial effects of the present invention are:

[0034] This invention combines Koopman operator theory and extended dynamic mode decomposition algorithm to construct a linearized model of the magnetorheological damper in a data-driven manner. This describes the nonlinear magnetorheological damper output damping force tracking control problem as a linear convex optimization problem, thereby reducing computational complexity and improving the real-time performance of the control algorithm. Furthermore, by introducing Newton's iteration method into LQR control, it solves the high-dimensional problem that traditional LQR control cannot address. By solving a Lyapunov equation in each iteration step, it ensures the convergence of the stable output feedback solution, realizing the tracking control of the magnetorheological damper output damping force to the desired value. This ensures real-time control while continuously iterating and optimizing to reduce tracking errors. Attached Figure Description

[0035] Figure 1This is a schematic diagram of data acquisition for testing the external characteristics of a magnetorheological damper.

[0036] Figure 2 This is a block diagram of the LQR control system for the magnetorheological damper output feedback based on the Newton iteration method in an embodiment of the present invention.

[0037] Figure 3 This is a flowchart illustrating the iterative optimization of the feedback LQR controller based on the Newton-Raphson iteration method in an embodiment of the present invention.

[0038] Figure 4 This is a tracking curve of the system's expected damping force versus actual damping force when the input sinusoidal signal frequency is 0.1Hz in an embodiment of the present invention.

[0039] Figure 5 This is a tracking curve of the system's expected damping force versus actual damping force when the input sinusoidal signal frequency is 0.5Hz in an embodiment of the present invention.

[0040] Figure 6 This is a tracking curve of the system's expected damping force versus actual damping force when the input sinusoidal signal frequency is 1.0Hz in an embodiment of the present invention.

[0041] Figure 7 This is a tracking curve of the system's expected damping force versus actual damping force when the input sinusoidal signal frequency is 5.0Hz in an embodiment of the present invention.

[0042] Figure 8 This is an error curve showing the difference between the expected damping force and the actual damping force of the system when the input sinusoidal signal frequency is 0.1Hz in an embodiment of the present invention.

[0043] Figure 9 This is an error curve of the system's expected damping force versus actual damping force when the input sinusoidal signal frequency is 0.5Hz in an embodiment of the present invention.

[0044] Figure 10 This is an error curve of the system's expected damping force versus actual damping force when the input sinusoidal signal frequency is 1.0Hz in an embodiment of the present invention.

[0045] Figure 11 This is an error curve showing the difference between the expected damping force and the actual damping force of the system when the input sinusoidal signal frequency is 5.0Hz in an embodiment of the present invention. Detailed Implementation

[0046] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0047] Please see Figures 1-11 As shown, the present invention provides the following technical solution:

[0048] The output feedback LQR control method for magnetorheological dampers based on Newton's iteration method includes the following steps:

[0049] S1. Complete the external characteristic test of the magnetorheological damper, obtain the input and output data of the magnetorheological damper, and use the Koopman operator theory and extended dynamic mode decomposition algorithm to globally linearize the magnetorheological damper system into a Koopman linear model.

[0050] S2. The input and output data required by the output feedback LQR control method of the magnetorheological damper described in this invention are obtained online through a magnetorheological damper experimental platform. Based on this platform, the external characteristic experiments of the magnetorheological damper are completed. Specifically, as follows... Figure 1 As shown, a large amount of input and output data of the magnetorheological damper is collected. The inputs are the current I of the magnetorheological damper, the piston rod displacement s and the piston rod velocity v, and the output is the damping force F.

[0051] S3. Based on the acquired dataset, the Koopman operator and the extended dynamic mode decomposition algorithm are used to train the Koopman linear model in a data-driven manner. Specifically, the Koopman linear model can be represented as:

[0052]

[0053] Where k is the current time, For the state variables after dimensionality upgrade, For state promotion functions, The predicted output damping force; A, B, and C can be obtained by solving a least-squares problem.

[0054] S4. Design an output feedback LQR controller based on the Koopman high-dimensional linear model. In the LQR control process, Newton's iterative method is introduced to solve a Lyapunov equation in each iteration to ensure the convergence of the stable output feedback solution.

[0055] The control process of the output feedback LQR control method for the magnetorheological damper described in this invention, based on the improved Newton iteration method, is as follows: Figure 2As shown, specifically:

[0056] (1) Obtain the output damping force r of the magnetorheological damper at the current moment based on the current motion state of the magnetorheological damper test platform;

[0057] (2) In the LQR control process, add a term g that ensures the output y closely follows a desired output r; the specific form of the matrix g term is g = -(A T -I-(PQ)A -1 BR -1 B T ) -1 C T Qr, where I is the identity matrix.

[0058] (3) In the output feedback LQR controller, the control objective is designed as follows:

[0059]

[0060] (4) Substituting the standard control quantity form u(k) = -Fy(k) = -FCz(k) obtained by output feedback LQR into the above formula, the control objective is transformed into:

[0061]

[0062] Where Q = C T C and R are weighted matrices, and Q and R are both symmetric matrices; N is the cross-term weighted matrix, and F is the output feedback gain matrix, which can be expressed as F = R. -1 (B T P+N T C + C was obtained. + The pseudo-inverse of C is represented; matrix P is obtained by solving the following Lyapunov equation:

[0063] (A-BFC) T P+P(A-BFC)

[0064] =-QC T F T RFC+C T F T N T +NFC;

[0065] A, B, and C are the state matrix, input matrix, and output matrix of the Koopman high-dimensional linear model, respectively, while BFC and RFC are related matrix multiplications.

[0066] S5. Using the desired output damping force as a reference input, and based on the continuous iterative optimization of the Newton-Raphson method, the control quantity obtained by the output feedback LQR is applied to the magnetorheological damper to achieve the tracking of the desired damping force by the magnetorheological damper.

[0067] The iterative optimization process of the output feedback LQR control method improved by Newton's iteration method described in this invention is as follows: Figure 3 As shown, the specific algorithm flow is as follows:

[0068] (1) Initialize the number of iterations and the criterion function value;

[0069] (2) Calculate the optimal output feedback gain F;

[0070] (3) Update P according to the Lyapunov equation;

[0071] (4) Update the criterion function value for this iteration, and continuously iterate and optimize until the stopping criterion is met or the maximum number of iterations is reached;

[0072] (5) Return the optimal output feedback gain F.

[0073] The optimal output feedback gain matrix F is found through iterative optimization using Newton's method; the final optimal control quantity u is then determined. * The form is u * =-Fy+(R+B T PB) -1 B T g will control quantity u * It acts on the magnetorheological damper.

[0074] Finally, the effectiveness of the proposed output feedback LQR control method based on the improved Newton iteration method was verified through the control system of the magnetorheological damper test bench.

[0075] Given that the desired sinusoidal signal frequency f of the magnetorheological damper test bench is 0.1Hz, 0.5Hz, 1.0Hz and 5.0Hz, the desired input signal is: r = 2.5sin(2πft).

[0076] like Figure 4 , Figure 5 , Figure 6 and Figure 7 These are the tracking curves of the system's expected damping force versus the actual damping force when the input desired sinusoidal signal is 0.1Hz, 0.5Hz, 1.0Hz, and 5.0Hz, respectively.

[0077] like Figure 8 , Figure 9 , Figure 10 and Figure 11These are the error curves between the expected damping force and the actual damping force when the input expected sinusoidal signal is 0.1Hz, 0.5Hz, 1.0Hz, and 5.0Hz.

[0078] The mean square error (MSE) at the different frequencies and the percentage of the maximum absolute error after stabilization are calculated to better illustrate the control effect of the present invention.

[0079] The table below shows the mean square error and maximum absolute error of the magnetorheological damper for different frequencies of desired sinusoidal signals of 0.1Hz, 0.5Hz, 1.0Hz and 5.0Hz.

[0080] Frequency / Hz <![CDATA[Mean square error / (kN) 2 > Maximum error / kN Error percentage / % 0.1 <![CDATA[2.7028×10 -7 ]]> <![CDATA[7.5462×10 -4 ]]> 0.0302 0.5 <![CDATA[7.7732×10 -6 ]]> 0.0041 0.1634 1.0 <![CDATA[1.7826×10 -5 ]]> 0.0062 0.2472 5.0 <![CDATA[5.7831×10 -5 ]]> 0.0114 0.4545

[0081] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. An output feedback LQR control method for magnetorheological dampers based on Newton's iteration method, characterized in that: Includes the following steps: S1. Perform external characteristic tests on the magnetorheological damper, obtain the external characteristic data of the magnetorheological damper, and use the Koopman operator theory and extended dynamic mode decomposition algorithm to globally linearize the magnetorheological damper system into a Koopman high-dimensional linear model. S2. An output feedback LQR controller is designed based on the Koopman high-dimensional linear model. Newton's iteration method is introduced into the LQR control process. By solving a Lyapunov equation in each iteration, the convergence of the stable output feedback solution is guaranteed. S3. Using the desired output damping force as a reference input, and based on the continuous iterative optimization of the preset method, the control quantity obtained by the output feedback LQR is applied to the magnetorheological damper, and the desired damping force is tracked through the magnetorheological damper.

2. The LQR control method for the magnetorheological damper based on Newton's iteration method according to claim 1, characterized in that: In S1, the external characteristics test of the magnetorheological damper is based on the semi-active suspension test bench.

3. The LQR control method for output feedback of the magnetorheological damper based on Newton's iteration method according to claim 1, characterized in that: In step S1, the external characteristic data of the magnetorheological damper are obtained, including experimental data on the output damping force, piston rod speed, piston rod relative displacement, and current flow of the magnetorheological damper under different experimental conditions.

4. The LQR control method for output feedback of the magnetorheological damper based on Newton's iteration method according to claim 3, characterized in that: In S1, the Koopman operator theory and extended dynamic mode decomposition algorithm are used to train the Koopman linear model, which takes the piston rod velocity, piston rod relative displacement and current as inputs and the damping force of the magnetorheological damper as outputs. The Koopman linear model is as follows: ； In the formula, For the current moment, For the state variables of the model in high-dimensional space. For state promotion functions, For the predicted output damping force; matrix , , The relevant optimization problems are solved based on the obtained dataset.

5. The LQR control method for the magnetorheological damper based on Newton's iteration method according to claim 4, characterized in that: In S2, the process of designing an output feedback LQR controller based on the Koopman high-dimensional linear model and introducing Newton's iteration method is as follows: S21: Design the control objective in the output feedback LQR controller; S22: Substitute the standard control quantity form obtained by the output feedback LQR into the control objective, transform the control objective, and obtain the transformed formula.

6. The LQR control method for the magnetorheological damper based on Newton's iteration method according to claim 5, characterized in that: The control objective is: ; The standard control quantity is in the form of ; The transformation formula is: ; in and It is a weighted matrix, and and All are symmetric matrices; The cross-term weighted matrix, The output feedback gain matrix can be obtained from... get, represent The pseudo-inverse of a matrix; The solution requires the following Lyapunov matrix: ； Among them This refers to the state matrix, input matrix, and output matrix of the Koopman high-dimensional linear model. BFC and RFC are related matrix multiplications.

7. The output feedback LQR control method for the magnetorheological damper based on Newton's iteration method according to claim 1, characterized in that: In step S3, the optimal output feedback gain matrix is ​​found by iteratively optimizing the process through a preset method, and the obtained optimal control quantity is applied to the magnetorheological damper.

8. The LQR control method for the magnetorheological damper based on Newton's iteration method according to claim 7, characterized in that: The preset method is Newton's iteration method.

9. The LQR control method for output feedback of the magnetorheological damper based on Newton's iteration method according to claim 1, characterized in that: In step S3, the iterative optimization based on the preset method specifically includes the following steps: S31: Initialize the number of iterations and the criterion function value; S32: Calculate the optimal output feedback gain ; S33: Update according to the preset equation ; S34: Update the criterion function value for this iteration, and continuously iterate and optimize until the stopping criterion is met or the maximum number of iterations is reached; S35: Return to optimal output feedback gain .

10. The LQR control method for output feedback of the magnetorheological damper based on Newton's iteration method according to claim 9, characterized in that: The preset equation is the Lyapunov equation.

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