An adaptive sliding mode control method for nonlinear time-varying systems

Abstract

The application discloses a kind of adaptive sliding mode control methods of nonlinear time-varying system, belong to automatic control technical field.For the problems such as high-frequency chattering of existing sliding mode control, parameter is difficult to adapt, the distance between system state point and sliding surface is quantified by boundary detection function in the present application, and a saturation function is introduced in the boundary layer to suppress chattering;Design disturbance observer with adaptive adjustment of loop iteration times, accurately estimate uncertainty and disturbance online;Using deep expansion network, the forward-backward splitting algorithm is mapped to a learnable reaching law, and the multi-objective optimization index of internal control energy and convergence speed is trained offline, and the ideal control reaching term is output online.The present application realizes the adaptive setting of key parameters of sliding mode control, without prior knowledge of disturbance upper bound, effectively suppresses chattering, improves robustness and optimizes the use of computing resources.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology, and in particular to an adaptive sliding mode control method for nonlinear time-varying systems. Background Technology

[0002] Sliding mode control, with its strong robustness to system parameter perturbations and external disturbances, has been widely used in robotics, aircraft, and motor drives. However, traditional sliding mode control still has many shortcomings when applied to highly nonlinear time-varying systems: it often uses high-gain switching control to deal with system uncertainties, which can easily lead to high-frequency switching chattering. This not only excites unmodeled dynamics in the system but also reduces control accuracy and shortens the lifespan of actuators. Key parameters such as controller switching gain and reaching law coefficients are mostly tuned offline by human experience, which is difficult to adapt to the characteristics of large-scale real-time time-varying system parameters, and can easily lead to conservative design or degraded control performance. At the same time, most of these methods require prior knowledge of the upper bound of system uncertainties and disturbances. In actual working conditions, this upper bound is difficult to determine accurately, and the estimation deviation will further aggravate chattering or weaken robustness. In addition, the existing algorithm has a fixed computational structure and cannot adaptively adjust the iteration depth based on the system operating state and available computing resources, resulting in low resource utilization efficiency.

[0003] Therefore, it is necessary to design an adaptive sliding mode control method that can cope with the strong nonlinearity and time-varying characteristics of the system, tune key parameters online, suppress chattering, and dynamically allocate computing resources according to control requirements. Summary of the Invention

[0004] This invention addresses the shortcomings of existing technologies by proposing an adaptive sliding mode control method for nonlinear time-varying systems. The aim is to solve problems such as chattering, difficulty in parameter adaptation, and computational rigidity in existing methods when dealing with strongly nonlinear time-varying systems. This method achieves online accurate estimation of system uncertainties and adaptive optimization of the control law through three core mechanisms: boundary awareness, iterative optimization, and deep unfolded learning.

[0005] This invention provides an adaptive sliding mode control method for nonlinear time-varying systems, comprising the following steps:

[0006] Step A1: System State Acquisition and Preprocessing: Real-time acquisition of the state variables of the controlled object, followed by processing to generate a standardized system state vector; the corresponding point of this vector in the state space is defined as the system state point; the reference trajectory is defined as the desired motion trajectory of the controlled object; the control quantity is defined as the command signal ultimately output by this method, and the actuator is a physical device that receives the control quantity and drives the motion of the controlled object, whose physical constraints include at least saturation constraints and rate constraints; the control quantity at the previous moment is recorded, and its initial value is set to zero when the system starts.

[0007] Step A2: Boundary Awareness: Calculate the tracking error vector based on the system state vector and the reference trajectory, and construct the sliding mode function vector; calculate the norm of the sliding mode function vector as a distance metric from the system state point to the preset sliding surface; define the boundary layer as the region near the preset sliding surface where the distance metric is less than or equal to a preset boundary layer thickness threshold, and design a boundary detection function to quantify the degree to which the system state point is located within the boundary layer, thereby generating a boundary feature signal; the boundary feature signal includes the norm of the sliding mode function vector and the calculated value of the boundary detection function;

[0008] Step A3: Online Estimation of Uncertainty and Disturbance: Construct a forward computation graph for the disturbance observer, which includes an input processing head, a parameter-sharing intermediate iteration block, and an output processing tail. Transform the system state vector and the control quantity from the previous time step into an initial implicit state via the input processing head. Perform K iterations on the parameter-sharing intermediate iteration block, where K is the number of iterations adaptively adjusted based on the boundary feature signal. The number of iterations K is adaptively adjusted based on the boundary feature signal, and K takes a larger value as the system state point is closer to the boundary layer. When the system state point is outside the boundary layer, a smaller K is used to reduce the computational load; when the system state point is inside the boundary layer, a larger K is used to improve the estimation accuracy. Input the implicit state obtained from the final iteration into the output processing tail to calculate the estimated value of the system's overall uncertainty and disturbance.

[0009] Step A4: Adaptive Sliding Mode Approach Law Calculation: The system state vector, boundary feature signals, and perturbation estimates are input into the adaptive approach law learning module constructed based on a deep unfolded network. This module has a built-in network layer that maps the single iteration step of the discretized forward-backward split algorithm to simulate a forward-backward split iteration calculation. The forward calculation process of this module is equivalent to solving an iterative process of a convex optimization problem with the sliding mode function vector as the variable, and the ideal control approach term is output through the last layer of the network. The learnable parameters of the deep unfolded network include the step size parameter and the regularization strength.

[0010] Step A5: Multi-objective constraint execution: The control quantity equals the ideal control reaching term minus the disturbance estimate; when the system state point enters the boundary layer, a saturation function is introduced to replace the sign function in the ideal sliding mode control to suppress chattering; when the system state point is outside the boundary layer, the reaching law remains dominant; simultaneously, the control quantity is subjected to amplitude and rate limiting processing based on the actuator's saturation and rate constraints, the final control quantity is calculated and output to the actuator; the pre-set multi-objective optimization index includes a weighted combination of control energy consumption and system state convergence speed, which is incorporated into the loss function used during offline training of the deep unfolded network, and this optimization index is not explicitly calculated during online control.

[0011] Furthermore, in step A3, the number of iterations K is adaptively adjusted according to the boundary feature signal: when the system state point is located outside the boundary layer, a smaller K is used to reduce the computational load; when the system state point is located inside the boundary layer, a larger K is used to improve the estimation accuracy; and the number of iterations K is independent of the total number of layers of the depth unfolded network.

[0012] Furthermore, in step A4, the learnable parameters (step size and regularization strength) of the deep unfolded network are fixed in the network after being determined through offline training; during online control, the network parameters remain unchanged, and the network output is adaptively adjusted according to the real-time changes of the input system state vector and boundary feature signals, without the need to update the network parameters online.

[0013] By adopting the above solution, the beneficial effects achieved by the present invention are as follows:

[0014] This invention uses a boundary-aware mechanism to quantify in real time the degree to which a system state point lies within the boundary layer. When the system enters the boundary layer, a saturation function is used instead of the sign function, avoiding high-frequency chattering caused by high-gain switching in traditional sliding mode control. This not only prevents unmodeled dynamics from being excited and improves control accuracy, but also significantly reduces mechanical wear on actuators and extends equipment lifespan.

[0015] This invention utilizes an adaptive reaching law learning module built on a deep unfolded network, treating key parameters such as step size and regularization strength as learnable parameters. Through offline training, it internalizes a multi-objective optimization strategy, enabling adaptive output of the ideal control reaching term during online control without the need for manual experience-based tuning. Simultaneously, this invention achieves accurate online estimation of the system's overall uncertainty through a cyclic iterative mechanism of the disturbance observer. This eliminates the need for prior knowledge of the upper bounds of system uncertainties and external disturbances, avoiding problems such as increased chattering or weakened robustness caused by upper bound estimation deviations. It also exhibits strong adaptability to a wide range of time-varying system parameters.

[0016] This invention adaptively adjusts the number of iterations of the perturbation observer based on boundary feature signals: when the system is outside the boundary layer, the number of iterations is automatically reduced to lower the computational load and ensure real-time performance; after entering the boundary layer, the number of iterations is automatically increased to improve estimation accuracy. Furthermore, this number of iterations is independent of the total number of layers in the depth-unfolded network, achieving decoupling and on-demand allocation of the inference depth of the control algorithm, thus achieving an optimal balance between estimation accuracy and control real-time performance under limited computational resources.

[0017] This invention explicitly incorporates a weighted combination of control energy consumption and system state convergence speed into the offline training loss function of the deep unfolded network. This allows the ideal control convergence term output by the network during online control to implicitly optimize this multi-objective index, simultaneously achieving fast convergence and low energy consumption without explicit calculation. Furthermore, by combining actuator saturation constraints and rate constraints with limiting measures, the control commands are ensured to remain within physically realizable limits, guaranteeing safety and feasibility in practical engineering applications. Attached Figure Description

[0018] Figure 1 This is a flowchart of an adaptive sliding mode control method for a nonlinear time-varying system proposed in this invention.

[0019] Figure 2 This is a schematic diagram of the perturbation observer based on iterative optimization according to the present invention. Detailed Implementation

[0020] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0021] Example 1:

[0022] This embodiment performs simulation verification on a nonlinear time-varying system. The mathematical model of the controlled object is as follows:

[0023] ;

[0024] in, It is the system state vector Regarding time The first derivative; It is the system state vector; It is a control quantity; and These are time-varying parameters; Represents parameter uncertainty; Represents external disturbances; the control objective is to make the system state vector... Tracking a given reference trajectory ;like Figure 1 As shown in the figure, this embodiment provides an adaptive sliding mode control method for a nonlinear time-varying system, and the specific steps are as follows:

[0025] Step A1: System status acquisition and preprocessing.

[0026] Real-time acquisition of the system state vector of the controlled object , and reference trajectory The comparison is performed; the acquired signals are filtered and normalized to generate a standardized system state vector; control variables are defined. This is the final output command signal of this method; the actuator of the controlled object is defined as a physical device (DC servo motor) that receives the control quantity and drives the movement of the controlled object, and its physical constraints include:

[0027] Saturation constraint ;

[0028] in, This represents the minimum permissible physical limit value that the actuator can output. This represents the maximum permissible physical limit value that the actuator can output;

[0029] Rate constraints ;

[0030] in, To control the amount The first derivative with respect to time, The absolute value of the maximum permissible rate of change that the actuator can withstand per unit time; record the control quantity at the previous moment. And its initial value is set to zero when the system starts.

[0031] Step A2: Boundary awareness.

[0032] Based on the system state vector Compared with reference trajectory Calculate the tracking error vector Construct sliding mode function vector Defined as ,in, For tracking error vector The first derivative with respect to time, It is a positive definite diagonal matrix. To control the number of channels; calculate the norm of the sliding mode function vector. As the system state point to the preset sliding surface (i.e. The distance metric of the determined hyperplane; the boundary layer is defined as the area near the preset sliding surface that satisfies... The area The preset boundary layer thickness threshold is greater than 0 (in this embodiment, it is taken as...). Design boundary detection function The degree to which a quantified system state point lies within the boundary layer is calculated using the following formula:

[0033] ;

[0034] in, This is a sensitivity adjustment parameter that controls the steepness of the boundary layer boundary (in this embodiment, it is taken as...). );when Much larger hour, Approaching 0 indicates that the system is in the ultra-short motion phase outside the boundary layer; when much smaller hour, A value approaching 1 indicates that the system has entered the sliding mode stage within the boundary layer; the boundary feature signal is obtained, the boundary feature signal including... and .

[0035] Step A3: Online estimation of uncertainties and disturbances.

[0036] Construct a robust dynamic observer based on a loop iteration mechanism, such as Figure 2 As shown; the forward computation graph of the observer includes: an input processing head, a parameter-shared intermediate iteration block, and an output processing tail; the system state vector Control quantity at the previous moment As input, it is transformed into the initial implicit state by the input processing head. The intermediate iteration block with shared parameters is calculated K times in a loop. The iteration process is as follows:

[0037] , ;

[0038] in, The index of the current iteration step. Indicates the first The hidden state obtained after the next iteration. For the first The hidden state after the next iteration. The mapping function represents the intermediate iteration block with shared parameters; the hidden state obtained from the final iteration. The input-output processing is completed to calculate the system's overall uncertainty and disturbance estimates. ;

[0039] The number of iterations K depends on the boundary feature signal. Adaptive adjustment: When the system state point is outside the boundary layer (approaching the motion phase), When the value is relatively small, a smaller K is used to reduce the computational load; when the system state point is located inside the boundary layer (sliding mode stage), When the value is large, a larger K is used to improve the estimation accuracy; and the number of iterations K is independent of the total number of layers of the depth-unfolded network in step A4; in this embodiment, the maximum number of iterations is set to... According to the formula The number of iterations for this reasoning is dynamically determined, where... This indicates the floor function.

[0040] Step A4: Calculate the adaptive sliding mode reaching law.

[0041] An adaptive reaching law learning module is constructed based on a deep unfolded network. This module maps the single iteration step of the discretized forward split-backward (FBS) algorithm to a network layer, simulating the forward split-backward iterative computation. Its forward computation process is equivalent to solving a problem based on distance... Convex optimization problem with variables:

[0042] ;

[0043] in, The sign for finding the minimum value indicates that we are looking for the variable that minimizes the objective function. ; The ideal control convergent term (optimization variable) to be solved; This is the scaling factor; For the sliding mode function vector Related functions; It is the Euclidean norm; It is an L1 norm; Let be the regularization strength; for the , layer( To further expand the layer index within the network, , (where the total number of network layers is ), the forward propagation formula for network layers is equivalent to performing soft thresholding iteration:

[0044]

[0045] in, For the first Intermediate control values ​​output by the layer. For the first Intermediate control values ​​output by the layer; initial values Determined by the boundary feature signal and the sliding mode function vector; For the first Layer step size parameter; This is a sign function; it takes the value 1 when the independent variable is greater than 0, -1 when it is less than 0, and 0 when it is equal to 0. This is for absolute value operations; To take the larger value between the expression within the parentheses and 0; the output of the last layer of the network. That is, the ideal control approaching term ;

[0046] The learnable parameters of a deep unfolded network include the stride parameter. and regularization strength The parameters were determined through offline training; the loss function used in offline training incorporated multi-objective optimization metrics (a weighted combination controlling energy consumption and convergence speed), for example:

[0047] ;

[0048] in, The value of the loss function. This represents the mathematical expectation, which is the average value of the training samples in batches. To control the energy term, excessive control amounts are penalized. These are the weighting coefficients. The Euclidean norm of the sliding mode function vector measures the convergence speed; this optimization index is not explicitly calculated during online control.

[0049] In a preferred embodiment of the present invention, the learnable parameters of the deep unfolded network are determined through offline training and then fixed in the network; during online control, the network parameters remain unchanged, and the ideal control approach term is adaptively adjusted according to the real-time changes of the system state vector and boundary feature signals; simultaneously, the number of iterations K is determined based on the boundary detection function. Real-time adaptive adjustment; through the above dual adaptive mechanism, the control law is dynamically adjusted according to the changes in system state without the need for online updates of network parameters.

[0050] Step A5: Execution of multi-objective constraints.

[0051] The formula for calculating the control quantity is: When the system state point enters the boundary layer (i.e., satisfies...) When ), the saturation function sat is introduced. Replace the sign function sgn in ideal sliding mode control To suppress chattering, it is defined as:

[0052] sat (when ,

[0053] (when ;

[0054] When the system state point is outside the boundary layer, the reaching law remains dominant; simultaneously, the saturation constraints of the actuator... and rate constraints The control quantity is subjected to amplitude and rate limiting to ensure that the control command is within the physically feasible range before being output to the actuator.

[0055] Constructing Lyapunov candidate functions Taking its time derivative yields Defined by sliding mode function Combined with system dynamics equations and control law When the estimation error of the perturbation observer is bounded, by selecting an appropriate positive definite matrix... The convergence law parameters obtained from training the deep unfolded network can guarantee that there is convergence outside the boundary layer. ,in The saturation function is the convergence rate constant. When the system state enters the boundary layer, the introduction of the saturation function ensures the asymptotic stability of the sliding mode motion. Therefore, the system state converges to the boundary layer in a finite time and remains stable within it. The closed-loop control system satisfies the asymptotic stability in the Lyapunov sense.

[0056] The method of this invention can be applied to various controlled objects exhibiting strong nonlinearity and time-varying characteristics, including but not limited to: trajectory tracking control of multi-joint robot arms, attitude control of quadcopter aircraft, DC servo motor speed control systems, and hydraulic servo systems. Taking a DC servo motor speed control system as an example (see Embodiment 2), the method of this invention can maintain high-precision speed tracking control even when motor parameters (such as armature resistance, inductance, and moment of inertia) change over a wide range with temperature and load variations, while effectively suppressing torque pulsation caused by high-frequency switching in traditional sliding mode control.

[0057] Example 2:

[0058] This embodiment takes a DC motor speed control system as an example to illustrate the specific operating mechanism of the internal loop iteration of the disturbance observer in step A3.

[0059] The discretized state-space model of the motor is as follows:

[0060] ,

[0061] ;

[0062] in, Index for discrete time points; Discrete time The observer state vector, for The rotational speed measurement at that moment. for The overall uncertainty at any given moment; The coefficient matrix is ​​obtained after discretizing the continuous model; for The amount of control at any given moment; for The system's measurable output at any given time;

[0063] At each sampling time The observer executes an internal loop iteration; the initial value of the iteration is set to the final estimate from the previous time step. ;in, This is the estimated rotational speed from the previous moment. This is the uncertainty estimate from the previous time step; for the ... Second internal iteration For internal selection of the number of iterations, ), perform the prediction step and the correction step in sequence:

[0064] Prediction Step: ;

[0065] Calibration step: ;

[0066] in, For the first The state estimate after the next iteration; The next state predicted by the model (uncorrected); The observer gain matrix is ​​used to adjust the correction strength; after K iterations, the final state estimate is taken. The second component is used as an uncertainty estimate. In this embodiment, when Time to take ;when Time to take The intermediate value K is determined by linear interpolation; this mechanism enables the observer to adapt its computational depth without requiring a known upper bound on the perturbation.

[0067] Example 3:

[0068] This embodiment details the offline training process of the adaptive convergence law learning module based on the deep unfolded network in step A4, to demonstrate how the learnable parameters (step size parameter and regularization strength) are determined, and how the multi-objective optimization index is reflected in the loss function of offline training.

[0069] Step B1: Training Dataset Construction. On the controlled object or its high-precision simulation model, an existing basic sliding mode controller is used for closed-loop operation to collect historical operating data under multiple operating conditions; each data sample includes: time... system state vector Sliding mode function vector Boundary feature signals After preprocessing the collected data, it is divided into training set and validation set.

[0070] Step B2: Loss Function Design. To implicitly optimize multi-objective metrics during offline training (controlling energy consumption and convergence speed), the following loss function is designed:

[0071] ;

[0072] in, The value of the loss function; This represents the total number of samples in the training batch. This represents the set of all learnable parameters in a deep unfolded network, including the stride parameters of each layer. and regularization strength , Under the current network parameters, and The ideal control approaching term output after forward propagation from the input network; To control the energy term; The weighting coefficient (in this embodiment, we take...) ); Let be the Euclidean norm of the sliding mode function vector.

[0073] Step B3: Network forward propagation. For each input sample, the network performs forward propagation based on the current set of learnable parameters. Perform the soft threshold iteration formula as described in step A4. Layer forward computation (in this embodiment, the total number of network layers is taken) ), to obtain the predicted output for this sample. .

[0074] Step B4: Parameter backpropagation update. Calculate the loss function. For the set of learnable parameters gradient The network parameters are updated using the gradient descent algorithm, and the update formula is as follows:

[0075] ;

[0076] in, This is the updated set of learnable parameters for the network. The set of learnable parameters of the network before the update; The preset learning rate controls the step size of each parameter update (in this embodiment, it is taken as...). ); For loss function For learnable parameters The gradient of the parameter indicates the direction of parameter adjustment that most quickly reduces the loss function; through multiple epochs of iterative training, until the loss function is reduced... Convergence, determining the final step size parameters. and regularization strength .

[0077] Step B5: Online Deployment. After offline training is complete, the network's learnable parameters... The control is fixed; in actual online control, the real-time acquired system state vector and boundary feature signals are directly input into the trained network, and the ideal control convergent term is output after one forward propagation. This eliminates the need for online calculation of the aforementioned loss function and gradient, thus ensuring real-time control.

[0078] The present invention and its embodiments have been described above. This description is not restrictive. The accompanying drawings are only one embodiment of the present invention, and the actual structure is not limited thereto. In short, if a person skilled in the art is inspired by this description and designs a similar structure and embodiment without departing from the spirit of the present invention, such design should fall within the protection scope of the present invention.

Claims

1. An adaptive sliding mode control method for a nonlinear time-varying system, characterized in that, Includes the following steps: Step A1: System State Acquisition and Preprocessing: Real-time acquisition of the state variables of the controlled object, followed by processing to generate a standardized system state vector; the corresponding point of the system state vector in the state space is defined as the system state point; the reference trajectory is defined as the desired motion trajectory of the controlled object; the control quantity is defined as the command signal finally output by this method, and the actuator is a physical device that receives the control quantity and drives the motion of the controlled object, whose physical constraints include at least saturation constraints and rate constraints; the control quantity at the previous moment is recorded, and its initial value is set to zero when the system starts. Step A2: Boundary Awareness: Calculate the tracking error vector based on the system state vector and the reference trajectory, and construct the sliding mode function vector; calculate the norm of the sliding mode function vector as the distance metric from the system state point to the preset sliding surface; define the boundary layer as the region near the preset sliding surface that satisfies the distance metric being less than or equal to the preset boundary layer thickness threshold, and design a boundary detection function to quantify the degree to which the system state point is located within the boundary layer, generating a boundary feature signal containing the sliding mode function vector norm and the calculated value of the boundary detection function; Step A3: Online estimation of uncertainty and disturbance: Input the system state vector and the control quantity of the previous time step into the disturbance observer based on the cyclic iteration mechanism. After multiple internal cyclic iterations, the estimated value of the system's comprehensive uncertainty and disturbance is output. The number of cyclic iterations is adaptively adjusted according to the boundary feature signal, and the number of iterations is larger when the system state point is closer to the boundary layer. Step A4: Adaptive sliding mode reaching law calculation: Input the system state vector, boundary feature signal and disturbance estimate into the adaptive reaching law learning module constructed based on deep unfolding network. The module solves the convex optimization problem by simulating the iterative process of the forward and backward splitting algorithm and outputs the ideal control reaching term. The convex optimization problem uses the sum of the square term of the sliding mode function vector norm and the regularization term as the objective function. Step A5: Multi-objective constraint execution: The control quantity equals the ideal control approaching term minus the disturbance estimate; when the system state point is within the boundary layer, a saturation function is introduced to replace the sign function to suppress chattering; when it is outside the boundary layer, the approaching law is maintained; the control quantity is output to the actuator after being limited in amplitude and rate based on the actuator's saturation constraints and rate constraints.

2. The method according to claim 1, characterized in that, In step A2, the tracking error vector is denoted as... The first derivative of the tracking error vector with respect to time is denoted as... The sliding mode function vector is ,in It is a positive definite diagonal matrix; the boundary detection function is... ,in The boundary layer thickness threshold. These are the sensitivity adjustment parameters.

3. The method according to claim 1, characterized in that, In step A3, the disturbance observer includes an input processing head, a parameter-sharing intermediate iteration block, and an output processing tail. The system state vector and the control quantity of the previous time step are transformed into the initial implicit state by the input processing head. After cyclic iteration by the intermediate iteration block, the disturbance estimate is calculated by the output processing tail.

4. The method according to claim 3, characterized in that, The system state vector is denoted as The control quantity at the previous moment is recorded as , No. The hidden state obtained after the nth iteration is denoted as . The mapping function of the intermediate iteration block is denoted as The iterative formula is: .

5. The method according to claim 1, characterized in that, The number of iterations is quantized and adjusted based on the calculated value of the boundary detection function: the smaller the boundary detection function value, the fewer the number of iterations; and the larger the boundary detection function value, the more the number of iterations. Furthermore, the number of iterations is independent of the total number of layers in the depthwise unfolded network.

6. The method according to claim 1, characterized in that, In step A4, the objective function of the convex optimization problem is: ； in, For the ideal control approaching term, For regularization strength, For the sliding mode function vector Related functions, To represent the square of the Euclidean norm, L1 norm; the first norm of a deep unfolded network Layer forward propagation is equivalent to using step size parameters One soft threshold iteration is performed, where This is a layer index.

7. The method according to claim 1, characterized in that, In step A5, the saturation function is linearized inside the boundary layer, while the nonlinear characteristics of the sign function are maintained outside the boundary layer.

8. The method according to claim 6, characterized in that, The step size and regularization strength of the deep unfolded network are determined through offline training. The loss function of offline training includes a weighted combination of control energy consumption and convergence rate terms. During online control, the network parameters are fixed, and the network output is adaptively adjusted in real time according to the input state.

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