A new energy vehicle brake pressure control method based on a cuppman operator

Abstract

A braking pressure control method for new energy vehicles based on the Koopman operator includes: establishing the dynamic equilibrium equations and nonlinear model of the braking system; constructing an observation function set, identifying the Koopman operator matrix using an extended dynamic mode decomposition algorithm, constructing a global linear prediction model, and correcting the Koopman operator matrix; constructing an augmented state equation in the discrete-time domain, designing a linear extended state observer based on an improved Kalman filter algorithm, and outputting a lumped disturbance estimate; injecting the lumped disturbance estimate into the prediction equation of the linear model predictive control, and obtaining the optimal control sequence by solving the objective function of a quadratic programming problem; converting the first control increment of the optimal control sequence into a motor drive signal, which is applied to the master cylinder motor of the electro-hydraulic braking system for physical pressure build-up. This invention can solve the problems of low hydraulic pressure control accuracy, poor anti-interference ability, and high computational load in existing technologies.

Description

Technical Field

[0001] This invention relates to the field of new energy vehicle technology, and more specifically to a new energy vehicle braking pressure control method based on the Koopman operator. Background Technology

[0002] Against the backdrop of the rapid development of new energy vehicles and intelligent driving technology, electro-hydraulic braking systems, with their characteristics of fast response and high-precision control, are gradually replacing traditional vacuum-assisted braking systems and becoming an indispensable core execution unit in intelligent chassis architecture.

[0003] During long-term driving, rapid changes in brake fluid temperature, aging of seals, and mechanical wear can cause significant drift in the system's physical parameters. At the same time, sudden changes in the road surface adhesion coefficient can also cause load disturbances to the braking system, requiring effective control of braking pressure.

[0004] Current research on braking pressure control in electro-hydraulic braking systems mostly focuses on single control algorithms, such as sliding mode control (SMC), PID control, and fuzzy control. However, these algorithms can only achieve low-precision hydraulic pressure control, have poor anti-interference capabilities, and impose a large computational load on the system. Summary of the Invention

[0005] In view of this, the present invention provides a new energy vehicle braking pressure control method based on the Koopman operator to solve the problems of low hydraulic pressure control accuracy, poor anti-interference ability and large computational load in the prior art.

[0006] A braking pressure control method for new energy vehicles based on the Koopman operator includes: Step S1: Based on the nonlinear volume characteristics of the master cylinder hydraulic system, the equivalent load torque at the motor shaft end, the electromagnetic torque of the motor, and the friction characteristics of the transmission mechanism, establish the dynamic balance equation of the braking system, and then construct a nonlinear model of the electronic hydraulic braking system of new energy vehicles. Step S2: Based on the nonlinear model and braking system dynamic equilibrium equation constructed in step S1, the input and output data sequences of the electro-hydraulic braking system are collected in real time, an observation function set containing the system state and its nonlinear transformation is constructed, the extended dynamic mode decomposition algorithm is used to identify the Koopman operator matrix, a global linear prediction model of the electro-hydraulic braking system in a high-dimensional feature space is constructed, and the Koopman operator matrix is ​​corrected by the recursive least squares method with an adaptive forgetting factor. Step S3: Based on the global linear prediction model in the high-dimensional feature space constructed in step S2, construct the augmented state equation in the discrete time domain, design a linear extended state observer based on the improved Kalman filter algorithm, observe the high-dimensional state of the system in real time, and output the lumped disturbance estimate including the Koopman fitting residual and external physical disturbance. Step S4: Based on the lumped disturbance estimate output in step S3, calculate the deviation norm between the actual observed state at the current time and the predicted state at the previous time. When the deviation norm exceeds the preset dynamic trigger threshold, generate an optimized trigger signal. At the trigger time, inject the lumped disturbance estimate into the prediction equation of the linear model predictive control. Obtain the optimal control sequence by solving the objective function of the quadratic programming problem. Step S5: The first control increment of the optimal control sequence is converted into a motor drive signal, which is applied to the master cylinder motor of the electro-hydraulic braking system to physically build up pressure. The current master cylinder pressure and motor speed are collected in real time and fed back to step S3 to form a hydraulic pressure closed-loop control.

[0007] The braking pressure control method for new energy vehicles based on the Koopman operator provided by the present invention has the following beneficial effects: (1) This invention introduces the Koopman operator theory. By constructing an observation function set containing the nonlinear transformation of the system, the strong nonlinear dynamics of the hydraulic braking system of new energy vehicles is mapped to a global linear model in a high-dimensional feature space. This avoids control oscillations caused by switching between multiple models and can achieve high-precision hydraulic pressure control.

[0008] (2) Based on the traditional static gain linear state observer, this invention designs a linear extended state observer based on an improved Kalman filter algorithm, which effectively overcomes the defect of the traditional static gain linear state observer being prone to amplifying high-frequency sensor noise under strong interference. It can also achieve high-speed tracking and real-time compensation of sudden system disturbances, eliminate the steady-state error of hydraulic pressure tracking, and make braking operation safer and more reliable.

[0009] (3) The present invention designs a dynamic design event triggering mechanism based on the high-dimensional state prediction deviation norm. Combined with the dual judgment logic of relative threshold and absolute threshold limit, the system only wakes up the optimization solver to update the control sequence when the actual state deviates from the predicted trajectory and exceeds the tolerance threshold. This triggering mechanism significantly reduces the average occupancy rate of the central processor, frees up computing power space, and improves driving safety while ensuring high control accuracy of hydraulic pressure. Attached Figure Description

[0010] Figure 1 A flowchart illustrating the braking pressure control method for new energy vehicles based on the Koopman operator provided in an embodiment of the present invention; Figure 2 This is a comparison chart of the master cylinder hydraulic pressure obtained by the present invention with the hydraulic pressure and ideal hydraulic force obtained by the traditional sliding mode algorithm. Detailed Implementation

[0011] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain embodiments of the present invention, and should not be construed as limiting the present invention.

[0012] Please see Figure 1 The braking pressure control method for new energy vehicles based on the Koopman operator provided by this invention includes steps S1 to S5: Step S1: Based on the nonlinear volume characteristics of the master cylinder hydraulic system, the equivalent load torque at the motor shaft end, the electromagnetic torque of the motor, and the friction characteristics of the transmission mechanism, establish the dynamic balance equation of the braking system, and then construct a nonlinear model of the electronic hydraulic braking system of new energy vehicles.

[0013] The nonlinear volumetric characteristics of the master cylinder hydraulic system satisfy the following equation:

[0014] in, It is a nonlinear volume function of the master cylinder hydraulic pressure. Displacement of the master cylinder piston. Main cylinder dead zone displacement, , represents the fitting coefficient for the equivalent volumetric stiffness of the hydraulic system.

[0015] The equivalent load torque at the motor shaft end satisfies the following formula:

[0016] in, This is the equivalent load torque at the motor shaft end. This refers to the angular displacement of the motor rotor. For the lead of the ball screw, The effective cross-sectional area of ​​the master cylinder piston. The mechanical transmission efficiency of the ball screw.

[0017] The frictional characteristics of the transmission mechanism satisfy the following equation:

[0018] in, For frictional torque, The angular velocity of the motor. The frictional torque is Coulomb torque. For the maximum static friction torque, For the switching speed threshold, The shape index, It is a symbolic function.

[0019] The established dynamic equilibrium equations for the braking system are as follows:

[0020] in, This is the equivalent moment of inertia at the motor shaft end. For motor acceleration, This is the system equivalent viscous damping coefficient at the motor shaft end. This refers to the electromagnetic torque of the motor.

[0021] The nonlinear model of the constructed new energy vehicle electro-hydraulic braking system satisfies the following equation: Selecting the angular displacement of the motor rotor and motor angular velocity As a state variable, the quadrature-axis current of the motor is selected. As control input Based on the above physical equations, a nonlinear model of the electro-hydraulic braking system is constructed, and its expression is:

[0022] in, This is the motor torque constant.

[0023] Step S2: Based on the nonlinear model and braking system dynamic equilibrium equations constructed in step S1, the input and output data sequences of the electro-hydraulic braking system are collected in real time, an observation function set containing the system state and its nonlinear transformation is constructed, the Koopman operator matrix is ​​identified using the extended dynamic mode decomposition algorithm, a global linear prediction model of the electro-hydraulic braking system in a high-dimensional feature space is constructed, and the Koopman operator matrix is ​​corrected by the recursive least squares method with an adaptive forgetting factor.

[0024] Specifically, step S2 includes steps S201 to S206: Step S201: Based on the input / output data sequence of the new energy vehicle's electronic hydraulic braking system during operation, obtain... The original state vector of the electro-hydraulic braking system at any given moment and control input , Represents the set of real numbers. and They represent and The vector dimension; In this embodiment, the input-output data sequence refers to the time-discrete set of the motor control voltage continuously recorded with a fixed sampling period as the control input data, and the corresponding master cylinder hydraulic pressure and motor speed of the new energy vehicle electronic hydraulic braking system as the output data of the physical state.

[0025] Step S202: Construct a set of observation functions containing the system state and its nonlinear transformations. Its mathematical representation is a lifting function vector, specifically expressed as:

[0026] in, Represents the set of observation functions. , , These respectively represent the 1st, 2nd, and 3rd observation functions set within the observation function set. One nonlinear transformation basis function , This represents transpose, mapping the original state to a higher-dimensional space. ; Step S203: Identify the Koopman operator matrix using the extended dynamic mode decomposition algorithm, and construct a global linear prediction model of the electro-hydraulic braking system in a high-dimensional feature space, expressed as:

[0027] in, for The high-dimensional state vector at time t, for The high-dimensional state vector at time t, and These are the Koopman operator matrix and the input matrix obtained by the extended dynamic mode decomposition algorithm, respectively. Step S204: Introduce the adaptive forgetting factor of the recursive least squares method, expressed as:

[0028] in, for The adaptive forgetting factor at any given time. for The predicted residual at time, This is the preset minimum forgetting factor limit. This is the sensitivity coefficient; Step S205: Update the error covariance matrix and Koopman gain matrix based on the adaptive forgetting factor, expressed as follows:

[0029]

[0030] in, for The Koopman gain matrix at time t. For recursive least squares method in The error covariance matrix at time t. for Augmented data vector at time step, , For recursive least squares method in The error covariance matrix at time t. It is an identity matrix.

[0031] Step S206: Finally, update the Koopman operator matrix based on the Koopman gain matrix, with the expression:

[0032] in, for The Koopman operator matrix at time t. for The Koopman operator matrix at time t.

[0033] Step S3: Based on the global linear prediction model in the high-dimensional feature space constructed in step S2, construct the augmented state equation in the discrete time domain, design a linear extended state observer based on the improved Kalman filter algorithm, observe the high-dimensional state of the system in real time, and output the lumped disturbance estimate including the Koopman fitting residual and external physical disturbance.

[0034] Specifically, based on the global linear prediction model constructed in the high-dimensional feature space in step S2, an augmented state equation in the discrete-time domain is constructed, including: The Koopman fitting residual and external physical disturbances are defined as the lumped disturbances of the system, which are then extended to become new state variables of the system. Based on a global linear prediction model in a high-dimensional feature space, the augmented state equation in the discrete-time domain is defined as follows:

[0035] in, Indicates in The estimate of the high-dimensional state at time step. Indicates in The estimated value of the high-dimensional state at time step. Indicates in The estimated value of the lumped disturbance at time . Indicates in The estimated value of the lumped disturbance at time . Let be the perturbation matrix. The result is calculated by the improved Kalman filter algorithm. The Kalman gain matrix at time 10:00. for The actual measurement output of the time system It is a measurement matrix used to establish the mapping relationship between high-dimensional observation states and actual physical output space.

[0036] In step S3, the update process of the gain matrix in the improved Kalman filter algorithm is as follows: First according to Optimal augmented state estimate at time 1 and the error covariance matrix of Kalman filtering ,calculate Prior state estimate at time 1 and prior error covariance matrix :

[0037]

[0038] in, and These are the augmented state matrix and the augmented input matrix, respectively. , , for Time-based control input, The process noise covariance matrix is... , Indicates in The estimated value of the high-dimensional state at time step. Indicates in The estimated value of the lumped disturbance at any given time; Then calculate Kalman gain matrix at time step :

[0039] in, To augment the output matrix, , To measure the noise covariance matrix; Combined Actual measurement output of the time system get Optimal augmented state estimate at time 1 :

[0040] Finally, the error covariance matrix is ​​updated for the calculation of the next time step, and the expression is:

[0041] in, for The error covariance matrix of the Kalman filter at time step 1.

[0042] Step S4: Based on the lumped disturbance estimate output in step S3, calculate the deviation norm between the actual observed state at the current time and the predicted state at the previous time. When the deviation norm exceeds the preset dynamic trigger threshold, generate an optimized trigger signal. At the trigger time, inject the lumped disturbance estimate into the prediction equation of the linear model predictive control. Obtain the optimal control sequence by solving the objective function of the quadratic programming problem.

[0043] When the deviation norm exceeds the preset dynamic trigger threshold, the following equation is satisfied during the generation of the optimized trigger signal:

[0044] in, for Time prediction The state trajectory at any given moment. It is a positive definite symmetric weighted matrix. This is a relative threshold. This is the absolute threshold limit.

[0045] In step S4, the objective function of the quadratic programming problem is:

[0046] in, Let be the objective function. for Control increment at any time, for Control increment at any time, To predict the time domain, To control the time domain, for Predicting the future The state trajectory at any given moment. For the future Reference pressure trajectory at any moment Here is the error weight matrix. This is the energy weighting matrix.

[0047] In this embodiment, the lumped disturbance estimate output from step S3 is introduced into the prediction equation in real time for compensation, and the expression is:

[0048] And it satisfies the control constraints:

[0049] in, for Predicting the future The state trajectory at any given moment. and They represent in At any given time, the Kopman matrix and the input matrix are corrected using the recursive least squares method. for Time-based control input, for Time-based control input, for Control increment at any given moment.

[0050] Step S5: The first control increment of the optimal control sequence is converted into a motor drive signal, which is applied to the master cylinder motor of the electro-hydraulic braking system to physically build up pressure. The current master cylinder pressure and motor speed are collected in real time and fed back to step S3 to form a hydraulic pressure closed-loop control.

[0051] Specifically, the current master cylinder pressure and motor speed are collected in real time and fed back to step S3. Optimal augmented state estimate at time 1 In the calculation, a closed-loop control of hydraulic pressure is formed.

[0052] Figure 2 This is a comparison chart of the master cylinder hydraulic pressure obtained by this invention with the hydraulic pressure and ideal hydraulic force obtained by the traditional sliding mode algorithm. Figure 2 It can be seen that the method proposed in this invention reduces the response time by 40% compared with the traditional sliding mode algorithm, suppresses chattering to within 0.02MPa, achieves high-precision hydraulic pressure control, improves anti-disturbance robustness, and significantly reduces the online computing load of the system.

[0053] In summary, the braking pressure control method for new energy vehicles based on the Koopman operator according to the above embodiments has the following beneficial effects: (1) This invention introduces the Koopman operator theory. By constructing an observation function set containing the nonlinear transformation of the system, the strong nonlinear dynamics of the hydraulic braking system of new energy vehicles is mapped to a global linear model in a high-dimensional feature space. This avoids control oscillations caused by switching between multiple models and can achieve high-precision hydraulic pressure control.

[0054] (2) Based on the traditional static gain linear state observer, this invention designs a linear extended state observer based on an improved Kalman filter algorithm, which effectively overcomes the defect of the traditional static gain linear state observer being prone to amplifying high-frequency sensor noise under strong interference. It can also achieve high-speed tracking and real-time compensation of sudden system disturbances, eliminate the steady-state error of hydraulic pressure tracking, and make braking operation safer and more reliable.

[0055] (3) The present invention designs a dynamic design event triggering mechanism based on the high-dimensional state prediction deviation norm. Combined with the dual judgment logic of relative threshold and absolute threshold limit, the system only wakes up the optimization solver to update the control sequence when the actual state deviates from the predicted trajectory and exceeds the tolerance threshold. This triggering mechanism significantly reduces the average occupancy rate of the central processor, frees up computing power space, and improves driving safety while ensuring high control accuracy of hydraulic pressure.

[0056] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.

Claims

1. A method for controlling braking pressure in new energy vehicles based on the Koopman operator, characterized in that, include: Step S1: Based on the nonlinear volume characteristics of the master cylinder hydraulic system, the equivalent load torque at the motor shaft end, the electromagnetic torque of the motor, and the friction characteristics of the transmission mechanism, establish the dynamic balance equation of the braking system, and then construct a nonlinear model of the electronic hydraulic braking system of new energy vehicles. Step S2: Based on the nonlinear model and braking system dynamic equilibrium equation constructed in step S1, the input and output data sequences of the electro-hydraulic braking system are collected in real time, an observation function set containing the system state and its nonlinear transformation is constructed, the extended dynamic mode decomposition algorithm is used to identify the Koopman operator matrix, a global linear prediction model of the electro-hydraulic braking system in a high-dimensional feature space is constructed, and the Koopman operator matrix is ​​corrected by the recursive least squares method with an adaptive forgetting factor. Step S3: Based on the global linear prediction model in the high-dimensional feature space constructed in step S2, construct the augmented state equation in the discrete time domain, design a linear extended state observer based on the improved Kalman filter algorithm, observe the high-dimensional state of the system in real time, and output the lumped disturbance estimate including the Koopman fitting residual and external physical disturbance. Step S4: Based on the lumped disturbance estimate output in step S3, calculate the deviation norm between the actual observed state at the current time and the predicted state at the previous time. When the deviation norm exceeds the preset dynamic trigger threshold, generate an optimized trigger signal. At the trigger time, inject the lumped disturbance estimate into the prediction equation of the linear model predictive control. Obtain the optimal control sequence by solving the objective function of the quadratic programming problem. Step S5: The first control increment of the optimal control sequence is converted into a motor drive signal, which is applied to the master cylinder motor of the electro-hydraulic braking system to physically build up pressure. The current master cylinder pressure and motor speed are collected in real time and fed back to step S3 to form a hydraulic pressure closed-loop control.

2. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 1, characterized in that, In step S1, the nonlinear volumetric characteristics of the master cylinder hydraulic system satisfy the following equation: in, It is a nonlinear volume function of the master cylinder hydraulic pressure. Displacement of the master cylinder piston. Main cylinder dead zone displacement, , The fitting coefficients for the equivalent volumetric stiffness of the hydraulic system; The equivalent load torque at the motor shaft end satisfies the following formula: in, This is the equivalent load torque at the motor shaft end. This refers to the angular displacement of the motor rotor. For the lead of the ball screw, The effective cross-sectional area of ​​the master cylinder piston. The mechanical transmission efficiency of the ball screw; The frictional characteristics of the transmission mechanism satisfy the following equation: in, For frictional torque, The angular velocity of the motor. The frictional torque is Coulomb torque. For the maximum static friction torque, For the switching speed threshold, The shape index, It is a symbolic function; The established dynamic equilibrium equations for the braking system are as follows: in, This is the equivalent moment of inertia at the motor shaft end. For motor acceleration, This is the system equivalent viscous damping coefficient at the motor shaft end. This refers to the electromagnetic torque of the motor. The nonlinear model of the constructed new energy vehicle electro-hydraulic braking system satisfies the following equation: Selecting the angular displacement of the motor rotor and motor angular velocity As a state variable, the quadrature-axis current of the motor is selected. As control input A nonlinear model of the electro-hydraulic braking system is constructed, and its expression is: in, This is the motor torque constant.

3. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 2, characterized in that, Step S2 specifically includes: Step S201: Based on the input / output data sequence of the new energy vehicle's electronic hydraulic braking system during operation, obtain... The original state vector of the electro-hydraulic braking system at any given moment and control input , Represents the set of real numbers. and They represent and The vector dimension; Step S202: Construct a set of observation functions containing the system state and its nonlinear transformations, expressed as: in, Represents the set of observation functions. , , These respectively represent the 1st, 2nd, and 3rd observation functions set within the observation function set. One nonlinear transformation basis function , Indicates transpose; Step S203: Identify the Koopman operator matrix using the extended dynamic mode decomposition algorithm, and construct a global linear prediction model of the electro-hydraulic braking system in a high-dimensional feature space, expressed as: in, for The high-dimensional state vector at time t, for The high-dimensional state vector at time t, and These are the Koopman operator matrix and the input matrix obtained by the extended dynamic mode decomposition algorithm, respectively. Step S204: Introduce the adaptive forgetting factor of the recursive least squares method, expressed as: in, for The adaptive forgetting factor at any given time. for The predicted residual at time, This is the preset minimum forgetting factor limit. This is the sensitivity coefficient; Step S205: Update the error covariance matrix and Koopman gain matrix based on the adaptive forgetting factor; Step S206: Finally, update the Koopman operator matrix based on the Koopman gain matrix.

4. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 3, characterized in that, Step S205 satisfies the following formula: in, for The Koopman gain matrix at time t. For recursive least squares method in The error covariance matrix at time t. for Augmented data vector at time step, , For recursive least squares method in The error covariance matrix at time t. It is an identity matrix.

5. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 4, characterized in that, Step S206 satisfies the following formula: in, for The Koopman operator matrix at time t. for The Koopman operator matrix at time t.

6. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 5, characterized in that, In step S3, based on the global linear prediction model in the high-dimensional feature space constructed in step S2, an augmented state equation in the discrete-time domain is constructed, specifically including: The Koopman fitting residual and external physical disturbances are defined as the lumped disturbances of the system. Then, based on the global linear prediction model in the high-dimensional feature space, the augmented state equation in the discrete-time domain is defined as follows: in, Indicates in The estimated value of the high-dimensional state at time step. Indicates in The estimated value of the high-dimensional state at time step. Indicates in The estimated value of the lumped disturbance at time . Indicates in The estimated value of the lumped disturbance at time . Let be the perturbation matrix. The result is calculated by the improved Kalman filter algorithm. The Kalman gain matrix at time 10:

00. for The actual measurement output of the time system It is a measurement matrix used to establish the mapping relationship between high-dimensional observation states and actual physical output space.

7. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 6, characterized in that, In step S3, the update process of the gain matrix in the improved Kalman filter algorithm is as follows: First according to Optimal augmented state estimate at time 1 and the error covariance matrix of Kalman filtering ,calculate Prior state estimate at time 1 and prior error covariance matrix : in, and These are the augmented state matrix and the augmented input matrix, respectively. , , for Time-based control input, The process noise covariance matrix is... , Indicates in The estimated value of the high-dimensional state at time step. Indicates in The estimated value of the lumped disturbance at any given time; Then calculate Kalman gain matrix at time step : in, To augment the output matrix, , To measure the noise covariance matrix; Combined Actual measurement output of the time system get Optimal augmented state estimate at time 1 : Finally, the error covariance matrix is ​​updated for the calculation of the next time step, and the expression is: in, for The error covariance matrix of the Kalman filter at time step 1.

8. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 7, characterized in that, In step S4, when the deviation norm exceeds the preset dynamic trigger threshold, the following equation is satisfied during the generation of the optimized trigger signal: in, for Time prediction The state trajectory at any given moment. It is a positive definite symmetric weighted matrix. This is a relative threshold. This is the absolute threshold limit.

9. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 8, characterized in that, In step S4, the objective function of the quadratic programming problem is: in, Let be the objective function. for Control increment at any time, for Control increment at any time, To predict the time domain, To control the time domain, for Predicting the future The state trajectory at any given moment. For the future Reference pressure trajectory at any moment Here is the error weight matrix. This is the energy weighting matrix.

10. The method for controlling braking pressure of new energy vehicles based on the Koopman operator according to claim 9, characterized in that, In step S4, the lumped disturbance estimate output from step S3 is introduced into the prediction equation in real time for compensation, and the expression is: in, for Predicting the future The state trajectory at any given moment. and They represent in At any given time, the Kopman matrix and the input matrix are corrected using the recursive least squares method. for Time-based control input.

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