A data-driven predictive control method for anti-interference control of nonlinear systems

Abstract

The application provides a data-driven predictive control method for nonlinear system anti-interference control, relates to Koopman fine modeling, accurate interference estimation and MPC precise control, and can significantly improve the control precision and robustness of the system. In the method, first, a dimension increasing function is designed based on time delay variables, so as to finely construct a Koopman model of the nonlinear system; on this basis, the nonlinear and external interference which are difficult to model are uniformly regarded as lumped interference, an interference model is established, and an interference observer is designed to realize interference estimation; further, based on the idea of active anti-interference, the interference estimation is embedded into a rolling horizon optimization framework to realize unbiased MPC control. The application not only simplifies the nonlinear control design process, but also significantly enhances the anti-interference ability of the system in an uncertain environment. Therefore, the application provides an efficient and reliable solution for the field of nonlinear system anti-interference control, and has a wide application prospect and significant practical value.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear system control technology, specifically relating to a data-driven predictive control method for anti-interference control of nonlinear systems under uncertain environments. Background Technology

[0002] The basic approach to modeling and controlling complex nonlinear systems typically involves: first, constructing a nonlinear analytical model based on the system's physical laws or mechanistic analysis; then, designing a nonlinear controller based on the model to achieve system stability. Mechanism modeling and nonlinear control design are often complex and demanding, requiring a high level of theoretical knowledge from engineers. In recent years, data-driven predictive control methods based on Koopman operator theory have been proposed and widely applied to nonlinear system control, such as soft robots, rigid robotic arms, and mobile robots. This method, by designing a Koopman dimension-upgrading function, can map the nonlinear system to a high-dimensional linear space, achieving global linearization of the nonlinear system and enabling the application of advanced linear control theories such as Model Predictive Control (MPC). Although this method has good versatility and simplifies control design, it has shortcomings in handling external disturbances and model uncertainties. Its robustness in practical applications needs improvement, and it is difficult to achieve high-performance anti-interference control.

[0003] To enhance the anti-interference performance of data-driven predictive control methods based on the Koopman operator in uncertain environments, improvements need to be made to the classic methods in three aspects: fine modeling of Koopman, accurate estimation of disturbances, and precise control of MPC. Specifically: (1) It is challenging to establish a fine Koopman model of a nonlinear system under unperturbed conditions, and the key lies in optimizing the design of the up-dimensional function; (2) The influence of nonlinearities that are difficult to model and external disturbances is not considered, and disturbances can be estimated by using disturbance observer technology; (3) Lumped disturbances are difficult to completely eliminate by the feedback mechanism of MPC, and unbiased MPC control can be achieved based on the idea of ​​active disturbance rejection to enhance the robustness of the system.

[0004] In summary, the application of data-driven predictive control methods based on the Koopman operator to nonlinear systems remains an open topic, especially in achieving anti-interference control under uncertain environments, which presents challenges and requires further in-depth research and innovation. Summary of the Invention

[0005] To address the aforementioned problems, this invention discloses a data-driven predictive control method for anti-interference control of nonlinear systems, involving Koopman fine modeling, accurate disturbance estimation, and MPC precise control, to enhance anti-interference control performance. This method has broad application prospects and significant practical value.

[0006] To achieve the above objectives, the technical solution of the present invention is as follows:

[0007] A data-driven predictive control method for anti-disturbance control of nonlinear systems includes the following steps:

[0008] Step S1: Finely establish the Koopman model of the nonlinear system;

[0009] Step S2: Accurately estimate lumped disturbances;

[0010] Step S3: MPC precise control based on disturbance estimation.

[0011] Preferably, in step S1:

[0012] Based on the optimization of the time-delay variable for the design of the dimension-upgrading function, the extended dynamic mode decomposition algorithm is used to approximate the Koopman operator in a data-driven manner, and the Koopman model of the nonlinear system is finely constructed.

[0013] Step S11: Optimize the design of the dimension-upgrading function based on time-delay variables, wherein the time-delay variables include state time-delay variables and input time-delay variables;

[0014] Assume the discrete dynamic equations of the nonlinear system are expressed as:

[0015] x(k+1)=f(x(k),u(k))

[0016] Where x(k) and u(k) are the state and input of the system at time step k, and f represents the nonlinear dynamics of the system;

[0017] The dimension-upgrading function Ψ based on time-delay variables can be expressed as:

[0018] Ψ=[x(k) x(k-1) x(k-2) … x(ki) u(k) u(k-1) u(k-2) … u(kj) 1]

[0019] Where x(ki) is the i-th time delay of the system state at time k, u(kj) is the j-th time delay of the system input at time k, and 1 is the bias term;

[0020] Introducing the state time delay variable x(ki) into the dimensionality-upgrading function allows the higher-order derivative of the state to be inferred based on the sampling period, thus enabling the dimensionality-upgrading state to describe higher-order systems. In addition, the input time delay variable can also be introduced to infer the rate of change of the input, thereby reflecting the rate-dependent nonlinearity of the system.

[0021] Step S12: Data-driven acquisition of the Koopman model. Based on the above-mentioned dimension-increasing function designed based on time-delay variables, the extended dynamic mode decomposition algorithm is used to approximate the Koopman operator in a data-driven manner, and the Koopman model of the nonlinear system is finely constructed.

[0022] A random input excitation system or simulation model is used to fully stimulate the nonlinear dynamics of the system. Input and state data are collected, and the state data is organized into two matrices with a one-sampling-step evolution relationship:

[0023] X1 = [x(1) x(2) … x(p)]

[0024] X2 = [x(2) x(3) … x(p+1)]

[0025] Based on the dimension-upgrading function Ψ designed in step S11, X1 and X2 are increased as follows:

[0026] X 1lift =[Ψ(x(1),u(1)) Ψ(x(2),u(2)) … Ψ(x(p),u(p))]

[0027] X 2lift =[Ψ(x(2),u(2)) Ψ(x(3),u(3)) … Ψ(x(p+1),u(p+1))]

[0028] Further, an input term, X, is introduced. 1lift ,X 2lift Expand to

[0029] Y 1lift =[X 1lift U] Τ

[0030] Y 2lift =[X 2lift U] Τ

[0031] Where U=[u[1] u[2] ··· u[p]];

[0032] Solve the following minimization problem to obtain a finite-dimensional approximation of the Koopman operator:

[0033]

[0034] Furthermore, from The Koopman model is partitioned into smaller, isolated sections:

[0035]

[0036] z(k+1)=Az(k)+Bu u(k)

[0037] y(k)=Cz(k)

[0038] Where z(k)=Ψ(x(k),u(k)) represents the upgraded state at time step k, and y(k)=x(k) and C=[IO] are defined to allow mapping back from the upgraded state space to the original state space.

[0039] Preferably, in step S2:

[0040] We treat nonlinearities that are difficult to model and external disturbances as lumped disturbances, analyze the characteristics of the disturbances and construct a disturbance model, and design a disturbance observer to achieve disturbance estimation.

[0041] S21: Analyze the interference characteristics and construct an interference model;

[0042] Considering model uncertainties and external disturbances, the perturbed Koopman model can be used to describe the actual operating conditions of the nonlinear system:

[0043] z(k+1)=Az(k)+B u u(k)+B d d(k)

[0044] y(k)=Cz(k)

[0045] Where d(k) is the lumped disturbance experienced by the Koopman model, including model uncertainty and external disturbances, B d The channels and interference gains that exist in the upgraded state have been identified, and they are usually designed as identity matrices.

[0046] Nonlinear systems are often subjected to unpredictable time-varying disturbances in actual operating conditions. The time-varying disturbance model is defined as follows:

[0047]

[0048] Where h j (k) is the j-th derivative of the lumped disturbance d(k) at time step k, T s Representing the sampling time, this interference model assumes that the (j+1)th derivative of the interference remains constant. The above time-varying interference model can be written in matrix form:

[0049] D(k+1)=A w D(k)

[0050] d(k)=C w D(k)

[0051] in C w =[I 0 - 0 0];

[0052] S22: Based on the above interference model, the following interference observer is designed:

[0053] q(k+1)=(A w -L w B d C w )(q(k)+L w z(k))-L w (Az(k)+B u u(k))

[0054]

[0055] in It is an auxiliary variable, L w It is the observer gain; by designing the observer gain, (A) w -L w B d C w By arranging the eigenvalues ​​of a function within the unit circle, the estimation error can be asymptotically converged.

[0056] Preferably, in step S3:

[0057] The interference observer in step S22 is embedded into the rolling time domain optimization to achieve unbiased MPC control effect.

[0058] To achieve unbiased tracking, the penalty function in the rolling temporal optimization is defined as follows:

[0059]

[0060] Where δu(k)=u(k)-u(k-1) represents the control increment, and e(k) is the value relative to the reference trajectory y. r The tracking error is represented by Q, R, and F, which are the penalty weights for the tracking error, control increment, and terminal tracking error, respectively.

[0061] Solve the following optimization problem at time step k:

[0062]

[0063] y(i|k)=Cz(i|k)

[0064] e(i|k)=y(i|k)-y r (i|k)

[0065]

[0066] Where N p In the prediction time domain, the solution to this optimization problem at time step k can be expressed as:

[0067] δu * (k)=[δu * (k|k)δu * (k+1|k)…δu * (k+N p -1|k)]

[0068] The control input at time step k is the optimal sequence δu. * The sum of the first value of (k) and the control input at time step k-1:

[0069] u(k)=δu * (k|k)+u(k-1)

[0070] Beneficial technical effects of the present invention:

[0071] This invention provides a data-driven predictive control method for anti-interference control of nonlinear systems, significantly improving the system's control accuracy and robustness. By introducing state and input time-delay variables to optimize the up-dimensional function, a precise Koopman model of the nonlinear system is constructed in a data-driven manner, achieving an accurate description of the system's hysteresis behavior and higher-order characteristics, ensuring high-fidelity global linearization of the nonlinear system. Furthermore, a time-varying disturbance model is established and a disturbance observer is designed to accurately estimate model uncertainties and external disturbances. Finally, the disturbance estimation is embedded into a rolling time-domain optimization design of an unbiased MPC controller, effectively eliminating the impact of lumped disturbances on the system. This invention not only simplifies the nonlinear control design process but also significantly enhances the system's anti-interference capability under uncertain environments. Therefore, this invention provides an efficient and reliable solution for the field of anti-interference control of nonlinear systems, with broad application prospects and significant practical value. Attached Figure Description

[0072] Figure 1 This is a control block diagram of a data-driven predictive control method for anti-interference control of nonlinear systems according to the present invention.

[0073] Figure 2 This is a schematic diagram of the internal structure of the robust Koopman-MPC controller described in this invention;

[0074] Figure 3 This is a flowchart illustrating the implementation of the robust Koopman-MPC controller described in this invention.

[0075] Figure 4 This is a flowchart illustrating the implementation of a data-driven predictive control method for anti-interference control of nonlinear systems, as described in this invention, applied to the load control of a soft robot. Detailed Implementation

[0076] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0077] Example 1

[0078] This embodiment describes a specific implementation structure of a data-driven predictive control method for anti-interference control of nonlinear systems. (Refer to...) Figure 1 , Figure 1 This is a control block diagram of a data-driven predictive control method for anti-interference control of nonlinear systems, including a robust Koopman-MPC controller, actuators, a nonlinear system, and sensors. The robust Koopman-MPC controller solves for the control input based on a reference trajectory and the actual trajectory. Its solution process comprehensively considers the influence of external disturbances and internal model uncertainties on the nonlinear system, and sends the control output command to the actuators. Common actuators include electro-proportional valves, flow valves, and motors, the specific type depending on the driving method of the nonlinear system. The actuators are responsible for transmitting power and driving the nonlinear system to achieve preset motion or behavior. The sensors measure the kinematic or mechanical signals of the nonlinear system, including but not limited to pose, angle, output force, and output torque, and feed them back to the robust Koopman-MPC controller, forming a closed-loop control system.

[0079] The internal structure of the robust Koopman-MPC controller is referenced. Figure 2 The system comprises a Koopman fine modeling module, a rolling time-domain optimization module, and a disturbance observation module. The Koopman fine modeling module employs state and input time-delay variables to optimize the design of an upgraded function, precisely describing the hysteresis behavior and higher-order system characteristics of the nonlinear system, thereby constructing a high-precision Koopman model through data-driven analysis. The disturbance observation module comprehensively considers external disturbances and internal model uncertainties to construct a time-varying disturbance model and designs a disturbance observer. This observer accurately estimates the lumped disturbance based on the control output and the actual trajectory. The rolling time-domain optimization module, based on the system model constructed by the Koopman fine modeling module and the lumped disturbance estimated by the disturbance observation module, establishes a disturbed Koopman model. This model fully considers the lumped disturbance caused by external disturbances and internal model uncertainties to achieve accurate prediction. Rolling time-domain optimization is then performed based on this disturbed model to solve for the optimal control quantity. Furthermore, the implementation process of the robust Koopman-MPC controller is as follows: Figure 3 .

[0080] This embodiment describes the specific implementation structure of a data-driven predictive control method for anti-disturbance control of nonlinear systems, and proposes a detailed design including the system control block diagram and its internal modules. The robust Koopman-MPC controller, through the collaborative work of the Koopman fine modeling module, disturbance observation module, and rolling time-domain optimization module, can effectively cope with external disturbances and internal model uncertainties experienced by nonlinear systems, improving the system's control accuracy and robustness. This method not only achieves accurate modeling of the complex dynamic characteristics of nonlinear systems, but also significantly enhances the system's anti-disturbance capability under uncertain environments through disturbance estimation and unbiased control strategies, providing an effective solution for achieving high-performance control of nonlinear systems.

[0081] Example 2

[0082] This embodiment describes the specific implementation details of a data-driven predictive control method for anti-interference control of nonlinear systems, specifically addressing the load control problem of a nonlinear soft robot system. The implementation flowchart of this method is as follows: Figure 4 As shown, it includes the following steps:

[0083] Step S1: Establish a detailed Koopman model of the soft robot under no-load conditions. This step specifically includes:

[0084] Step S11: Optimize the design of the dimension-upgrading function based on time-delay variables, wherein the time-delay variables include state time-delay variables and input time-delay variables;

[0085] The discrete dynamic equations of the soft robot system are expressed as follows:

[0086] x(k+1)=f(x(k),u(k))

[0087] Where x(k) is the state of the system at time step k, usually the bending angle, elongation or end position, u(k) is the input of the system at time step k, usually air pressure or voltage, and f represents the nonlinear dynamics of the system.

[0088] The dimension-upgrading function Ψ based on time-delay variables can be expressed as:

[0089] Ψ=[x(k) x(k-1) x(k-2) … x(ki) u(k) u(k-1) u(k-2) … u(kj) 1]

[0090] Where x(ki) is the i-th time delay of the system state at time step k, u(kj) is the j-th time delay of the system input at time step k, and 1 is the bias term.

[0091] Considering that the nonlinearity of soft robot systems is mainly manifested in hysteresis nonlinearity, specifically as history correlation and rate correlation, and that the system dynamics typically exhibit second-order-like characteristics, introducing a state delay variable x(ki) into the up-dimensional function is equivalent to introducing historical information about the state, thus reflecting the history correlation of the hysteresis. Simultaneously, the first derivative of the state can be inferred based on the single-step delay and sampling period; the first derivative in the discrete system state can reflect the second-order characteristics of the system. Furthermore, introducing an input delay variable can also infer the rate of change of the input, thus corresponding to the rate-dependent nonlinearity of the system. Therefore, the introduction of state and input delay variables allows the up-dimensional function to comprehensively describe the typical nonlinear characteristics of soft robots, including the history correlation, rate correlation, and second-order-like characteristics of the hysteresis, which helps in the precise construction of the Koopman model of the soft robot.

[0092] Step S12: Data-driven acquisition of the Koopman model of the soft robot under no-load conditions. Based on the aforementioned dimensionality-increasing function designed with time-delay variables, the extended dynamic mode decomposition algorithm is used to approximate the Koopman operator in a data-driven manner, from which the Koopman model of the system is extracted.

[0093] Under no-load conditions, random input is used to excite the soft robot system to fully stimulate its nonlinear dynamics. Simultaneously, input and state data are collected, and the state data is organized into two matrices with a one-sampling-step evolution relationship:

[0094] X1 = [x(1) x(2) … x(p)]

[0095] X2 = [x(2) x(3) … x(p+1)]

[0096] Based on the dimension-upgrading function Ψ designed in step S11, X1 and X2 are increased as follows:

[0097] X 1lift =[Ψ(x(1),u(1)) Ψ(x(2),u(2)) … Ψ(x(p),u(p))]

[0098] X 2lift =[Ψ(x(2),u(2)) Ψ(x(3),u(3)) … Ψ(x(p+1),u(p+1))]

[0099] Further, an input term, X, is introduced. 1lift ,X 2lift Expand to

[0100] Y 1lift =[X 1lift U] Τ

[0101] Y2lift =[X 2lift U] Τ

[0102] Where U = [u[1] u[2] ··· u[p]], note that the input terms do not need to be nonlinearly upgraded in order to ensure that the constructed Koopman model has linear input characteristics.

[0103] A finite-dimensional approximation of the Koopman operator is obtained by solving the following minimization problem:

[0104]

[0105] Furthermore, from The Koopman model is partitioned into smaller, isolated sections:

[0106]

[0107] z(k+1)=Az(k)+B u u(k)

[0108] y(k)=Cz(k)

[0109] Where z(k)=Ψ(x(k),u(k)) represents the upgraded state at time step k, and y(k)=x(k) and C=[IO] are defined to allow mapping back from the upgraded state space to the original state space.

[0110] Step S2: Establish a time-varying interference model and design an interference observer.

[0111] Nonlinearity and external load disturbances that are difficult to model are treated as lumped disturbances. The characteristics of the disturbances are analyzed and a disturbance model is constructed. A disturbance observer is designed to achieve disturbance estimation.

[0112] Step S21: Analyze the interference characteristics and construct an interference model.

[0113] Considering model uncertainties and external disturbances, a perturbed Koopman model can be used to describe the actual working conditions of the soft robot system:

[0114] z(k+1)=Az(k)+B u u(k)+B d d(k)

[0115] y(k)=Cz(k)

[0116] Where d(k) represents the lumped disturbances experienced by the Koopman model, including model uncertainty and external load disturbances, and B d Once the channels with interference and the interference gain in the upgraded state are identified, they are usually designed as identity matrices to simplify the analysis.

[0117] Soft robots are often subjected to unpredictable time-varying disturbances in actual working conditions. The time-varying disturbance model is defined as follows:

[0118]

[0119] Where h j (k) is the j-th derivative of the lumped disturbance d(k) at time step k, T s Representing the sampling time, this interference model assumes that the (j+1)th derivative of the interference remains constant. The above time-varying interference model can be written in matrix form:

[0120] D(k+1)=A w D(k)

[0121] d(k)=C w D(k)

[0122] in C w =[I 0 … 0 0];

[0123] Step S22: Based on the above interference model, design the following interference observer:

[0124] q(k+1)=(A w -L w B d C w )(q(k)+L w z(k))-L w (Az(k)+B u u(k))

[0125]

[0126] in It is an auxiliary variable, L w It is the observer gain; by designing the observer gain, (A) w -L w B d C w By arranging the eigenvalues ​​of a function within the unit circle, the estimation error can be asymptotically converged.

[0127] Step S3: Design an unbiased MPC controller based on the interference observer. The interference observer from step S22 is embedded into rolling time-domain optimization to achieve unbiased MPC control.

[0128] To achieve unbiased tracking, the penalty function in the rolling temporal optimization is defined as follows:

[0129]

[0130] Where δu(k)=u(k)-u(k-1) represents the control increment, and e(k) is the value relative to the reference trajectory y. r The tracking error is denoted by Q, R, and F, which are the penalty weights for the tracking error, control increment, and terminal tracking error, respectively. Note that penalizing the control increment helps achieve unbiased tracking.

[0131] Solve the following optimization problem at time step k:

[0132]

[0133] y(i|k)=Cz(i|k)

[0134] e(i|k)=y(i|k)-y r (i|k)

[0135]

[0136] Where N p This is a prediction in the time domain. To solve this optimization problem, the perturbation-influenced Koopman model is first rewritten in incremental form:

[0137]

[0138] y[k]=Cz[k]

[0139] Where z(k)=[z i (k)u(k-1)d(k-1)] Τ ,

[0140] C = [C il×N O l×(N+m) ] l*(2N+m)

[0141] Based on this model, linear extrapolation is performed to obtain the output prediction equation.

[0142]

[0143] in,

[0144]

[0145] Furthermore, the penalty function for the above optimization problem can be expressed in matrix form:

[0146]

[0147] in,

[0148] Furthermore, the above optimization problem can be transformed into the standard quadratic programming (QP) form:

[0149]

[0150] stlb≤δU k ≤ub

[0151] Where lb and ub represent the lower and upper bounds of the constraint, respectively. Using the standard QP algorithm, the solution to this optimization problem in time step k can be expressed as:

[0152] δu * (k)=[δu * (k|k)δu * (k+1|k)…δu * (k+N p -1|k)]

[0153] The control input at time step k is the optimal sequence δu. * The sum of the first value of (k) and the control input at time step k-1:

[0154] u(k)=δu * (k|k)+u(k-1)

[0155] This embodiment addresses the problem of precise control of nonlinear soft robot systems under load conditions. It details the implementation of a data-driven predictive control method for anti-interference control of nonlinear systems, proposing a complete solution including refined modeling, disturbance estimation, and unbiased control. By introducing time-delay variables to optimize the design of the upgraded function, constructing a time-varying disturbance model and designing a disturbance observer, and designing an unbiased MPC controller based on the disturbance observer, the anti-interference performance of the Koopman operator-based data-driven predictive control method under uncertain environments is significantly improved.

[0156] It should be noted that the above content merely illustrates the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. For those skilled in the art, various improvements and modifications can be made without departing from the principle of the present invention, and all such improvements and modifications fall within the scope of protection of the claims of the present invention.

Claims

1. A data-driven predictive control method for anti-interference control of nonlinear systems, characterized in that: Includes the following steps: Step S1: Finely establish the Koopman model of the nonlinear system, as follows; Based on the optimization of the time-delay variable for the design of the dimension-upgrading function, the extended dynamic mode decomposition algorithm is used to approximate the Koopman operator in a data-driven manner, and the Koopman model of the nonlinear system is finely constructed. Step S11: Optimize the design of the dimension-upgrading function based on time-delay variables, wherein the time-delay variables include state time-delay variables and input time-delay variables; Assume the discrete dynamic equations of the nonlinear system are expressed as: ； in, For the system in The state and input of the time step. Represents the nonlinear dynamics of the system; Dimensional Upgrading Function Designed Based on Time Delay Variables Represented as: ； in, For the system state in the th... The first step of time Step delay, For system input at the first The first step of time Step delay, 1 is the bias term; Introducing the state delay variable into the dimension-upgrading function The higher-order derivatives of the state are inferred based on the sampling period, thus enabling the upgraded state to describe higher-order systems; in addition, the rate of change of the input is inferred by introducing the input time delay variable, thereby reflecting the rate-dependent nonlinearity of the system. Step S12: Data-driven acquisition of the Koopman model. Based on the above-mentioned dimension-increasing function designed with time-delay variables, the extended dynamic mode decomposition algorithm is used to approximate the Koopman operator in a data-driven manner, and the Koopman model of the nonlinear system is finely constructed. A random input excitation system or simulation model is used to fully stimulate the nonlinear dynamics of the system. Input and state data are collected, and the state data is organized into two matrices with a one-sampling-step evolution relationship: ； The dimension-upgrading function designed based on step S11 promote : ； Further input items are introduced. Expand to ； in ; Solve the following minimization problem to obtain a finite-dimensional approximation of the Koopman operator: ； Furthermore, from The Koopman model is partitioned into smaller, isolated sections: ； ； in represent The upgraded state of the time step is defined. This allows mapping back from the higher-dimensional state space to the original state space; Step S2: Accurately estimate the lumped disturbance, as follows; We treat nonlinearities that are difficult to model and external disturbances as lumped disturbances, analyze the characteristics of the disturbances and construct a disturbance model, and design a disturbance observer to achieve disturbance estimation. S21: Analyze the interference characteristics and construct an interference model; Considering model uncertainties and external disturbances, a perturbed Koopman model is used to describe the actual operating conditions of the nonlinear system: ； in, This refers to the lumped disturbances experienced by the Koopman model, including model uncertainty and external disturbances. The channels and interference gain that exist in the upgraded state were identified, and they were designed as identity matrices. Nonlinear systems are subjected to unpredictable time-varying disturbances in actual operating conditions. The time-varying disturbance model is defined as follows: ； in It is centralized interference In the time step First derivative, Representing the sampling time, this interference model assumes the interference is... The first derivative remains constant; the above time-varying disturbance model can be written in matrix form: ； in ; S22: Based on the above interference model, the following interference observer is designed: ； in It is an auxiliary variable. It is the observer gain; by designing the observer gain, The eigenvalues ​​are arranged inside the unit circle, which achieves asymptotic convergence of the estimation error; Step S3: MPC precise control based on disturbance estimation, as detailed below; The interference observer in step S22 is embedded into the rolling time domain optimization to achieve unbiased MPC control effect. To achieve unbiased tracking, the penalty function in the rolling temporal optimization is defined as follows: ； in, Represents control increment, It is relative to the reference trajectory Tracking error, These are the penalty weights for tracking error, control increment, and terminal tracking error, respectively. exist Time steps, solve the following optimization problem: ； in It is a prediction time domain problem, and the optimization problem is in The solution for each time step is expressed as: ； exist The control input for each time step is the optimal sequence. The first value and Sum of time step control inputs: 。

Images