An adaptive preset time-bounded tracking control method under state constraint

Abstract

This invention discloses an adaptive preset-time bounded tracking control method under state constraints, belonging to the field of complex nonlinear system control technology. First, it proposes an actual preset-time bounded stability criterion, providing a theoretical basis for the artificial preset of the system convergence time. Second, it designs an adaptive actual preset-time bounded filter, achieving preset-time convergence of the filtering error while addressing the computational complexity explosion problem in backstepping control. Third, it maps the constrained system state to unconstrained variables through a state transformation function, naturally integrating state constraints into the controller design. Finally, it combines backstepping control and radial basis function neural networks to construct an adaptive preset-time bounded tracking controller. In this invention, all signals converge to a bounded range within a preset time, the system state always satisfies the preset constraints, and the tracking error reaches bounded stability within a preset time. This method is suitable for controlling high-order strict feedback nonlinear systems with state constraints.

Description

Technical Field

[0001] This invention belongs to the field of complex nonlinear system control technology, and designs an adaptive preset time bounded tracking control strategy for a high-order strict feedback nonlinear system under state constraints. Background Technology

[0002] Since most real-world systems are nonlinear, such as robotic arm systems, drone systems, and multi-agent systems, the control of nonlinear systems has received widespread attention. The states of practical engineering systems often need to operate within constraints determined by hardware conditions. State out-of-bounds operations will directly lead to decreased control accuracy, system instability, and even safety accidents.

[0003] In existing technologies, control methods used to achieve system stability generally have limitations in terms of the determinism and feasibility of convergence time: bounded stability control cannot predetermine the convergence time; the convergence time of finite-time control depends on the initial conditions of the system, and these initial values ​​are often difficult to obtain accurately; although fixed-time control can make its upper bound of convergence time independent of the initial value, it is still constrained by specific parameters and cannot achieve true time pre-setting; although some existing pre-set time control methods can specify the convergence time, they have singularity problems, which may lead to control failure and affect the reliable operation of the system.

[0004] Furthermore, in backstepping control design, to avoid the "computational complexity explosion" caused by repeatedly differentiating the virtual control input, filters are usually introduced for signal processing. Existing technologies, such as common first-order filters, Kalman filters, and even finite-time and fixed-time filters, can alleviate the above problems to some extent, but the convergence time of their filtering errors often still depends on initial conditions or specific parameter settings, failing to achieve completely independent artificial pre-setting and flexible control independent of the system state.

[0005] Given the above, how to design an adaptive preset time-bounded tracking control strategy for high-order strictly feedback nonlinear systems that handle full-state constraints has attracted increasing attention from scholars. Summary of the Invention

[0006] To address the aforementioned problems in existing technologies, and specifically for a class of high-order strictly feedback nonlinear systems, this invention constructs an adaptive preset time bounded tracking control strategy under state constraints.

[0007] This invention designs an adaptive preset-time bounded tracking control method under state constraints, ensuring that the system can track the reference command within a preset time while satisfying all state constraints, and that all signals of the system can converge to a bounded range within the preset time. Compared with common bounded stability control methods, finite-time control methods, fixed-time control methods, and preset-time control methods, the main innovations are in the following two aspects: (1) To ensure that the closed-loop system converges to a bounded range within a preset time while satisfying all state constraints and avoiding singularity problems in the control process, an actual preset-time bounded stability criterion is proposed. (2) To address the computational explosion problem in backstep control, an adaptive actual preset-time bounded filter is designed based on the proposed actual preset-time bounded stability criterion and introduced into the controller design process.

[0008] The technical solution adopted in this invention is an adaptive preset time bounded tracking control strategy under state constraints, comprising the following steps:

[0009] S1. The high-order strict feedback nonlinear system under consideration is:

[0010] (1) In the formula It is the system status. It is system output. It is a known smooth function. It is an unknown function vector. This indicates a control signal.

[0011] S2. A practical preset-time bounded stability criterion is proposed to achieve preset-time bounded stability of constrained closed-loop systems. To address the "computational complexity explosion" problem caused by repeated differentiation of the virtual control input in backstep control, an adaptive practical preset-time bounded filter is designed.

[0012] S3. In order to solve the full-state constraint problem, a constrained high-order strict feedback nonlinear system is constructed by using a state transformation function.

[0013] S4. Based on the constructed high-order strictly feedback nonlinear system with state constraints, an adaptive preset time-bounded tracking control protocol is constructed using the backstepping control method:

[0014] (2)

[0015] (3)

[0016] (4)

[0017] In equations (2)-(4) above, , Positive design parameters For adjustment function, It is a systematic error variable. For system combination variables, To adapt to the actual preset time-bounded filter output value, It is an adaptive law. is the activation function in a radial basis function neural network.

[0018] S5. Using Lyapunov stability theory, verify the adaptive preset time bounded tracking control performance of the nonlinear system under the premise of satisfying the full-state constraint requirements.

[0019] Compared with existing methods, this paper introduces an adaptive actual preset-time bounded filter and estimates the unknown upper bound of the virtual control input, reducing computational complexity while improving convergence speed. Furthermore, by proposing an actual preset-time bounded stability criterion and applying it to controller design, the preset time of the closed-loop system can be given in advance, ultimately achieving bounded stability of the system. Attached Figure Description

[0020] Figure 1 This is a flowchart of the invention;

[0021] Figure 2 This is a control framework diagram based on the present invention;

[0022] Figure 3 These are simulation results of full-state constraint and tracking control according to embodiments of the present invention;

[0023] Figure 4 These are simulation results of the tracking error and control input signal in an embodiment of the present invention;

[0024] Figure 5 These are the simulation results of the system error variables;

[0025] Figure 6 These are simulation results of the adaptive law signal in an embodiment of the present invention. Detailed Implementation

[0026] Existing methods for bounded tracking control, finite-time tracking control, and fixed-time tracking control of nonlinear systems generally suffer from the problem that the actual convergence time of the closed-loop system cannot be accurately set in advance. Furthermore, to address the computational complexity of backstepping methods, the settling time of the filtering errors of the first-order filters, finite-time filters, or fixed-time filters used are also difficult to pre-set, and their convergence speed is limited.

[0027] To address the aforementioned technical problems, this invention provides an adaptive preset-time bounded tracking control scheme under state constraints. Its core lies in constructing a control system comprised of the following functional modules working collaboratively (see framework below). Figure 2 ), and implement it according to the following steps:

[0028] Step 1: Establishment and Setting of the Convergence Time Determination Module: This module operates based on the constructed actual preset time bounded stability criterion. Users can input a desired preset time parameter into this module. The core function of this module is to provide the basis for analyzing and setting the convergence time of the entire control system, ensuring that the system signal converges to the bounded region before the preset time.

[0029] Step 2: Design of the Signal Filtering and Differential Estimation Module: This module is an adaptive, real-time bounded filter. It directly receives the virtual control signal from the controller and outputs the smoothed filtered signal and its differential estimate. The function of this module is to fundamentally avoid the repeated analytical differentiation of the virtual control signal, thereby completely solving the "computational explosion" problem existing in the traditional backstepping method.

[0030] Step 3: Integration of the State Constraint Processing Module: This module processes the original system state signal through a built-in nonlinear state transformation function. Its function is to map the constrained system state into a new set of unconstrained equivalent variables, thereby naturally integrating the state constraints into the controller design process. The boundedness of the equivalent variables ensures that the system's runtime state never exceeds the preset constraint boundaries.

[0031] Step 4: Design and implementation of the adaptive preset time bounded tracking controller: The output signals of the above three modules are combined—namely, the time criterion from the convergence time determination module, the smoothing command and differential signal from the signal filtering and differential estimation module, and the equivalent state variable from the state constraint processing module—and finally the actual control signal that can be directly applied to the controlled object is calculated by the control law synthesis unit, thus completing the entire adaptive preset time bounded tracking control.

[0032] Based on flowchart Figure 1 Following the research steps described above, this invention implements an adaptive preset-time bounded tracking control method under state constraints, the system architecture of which is as follows: Figure 2 As shown, the specific implementation process is as follows:

[0033] I. System Settings and Initialization

[0034] (1) Reference instruction function Set as A continuously differentiable function of order 1.

[0035] (2) Set system status It needs to satisfy the condition that it is strictly bound to the set. In the middle, constrained boundary and It is a positively bounded differentiable function, and the initial state of the system is guaranteed to satisfy... .

[0036] (3) Define the unknown constants used to estimate the upper bound of the virtual control input derivative. .

[0037] (4) For those defined in the interval Unknown functions on Online estimation is performed using a radial basis function neural network, which takes the form of: ,in, Represents the weight matrix. Represents the number of neurons in a neural network. Gaussian activation function , Represents the center of convergence. This represents the width of the Gaussian function and the approximation error. There is an upper realm .

[0038] II. Construction of the Convergence Time Determination Module: To achieve the goal of pre-setting the system convergence time, a convergence time determination module is constructed. This module operates based on the following pre-set time-bounded stability criterion: System , , There exists a Lyapunov function. normal numbers , such that its derivative satisfies the inequality:

[0039] (5)

[0040] Among them, the time-varying adjustment function Defined as

[0041] (6)

[0042] ( For preset time, parameters If the module determines that the system can be completed within a preset time, then the module will be able to complete the task. The internal structure achieves practically bounded stability. This criterion is the core basis for the controller design of this invention.

[0043] III. Design of Signal Filtering and Differential Estimation Module: To address the "computational explosion" problem caused by repeated differentiation of the virtual control input in the backstepping method design, a signal filtering and differential estimation module (i.e., an adaptive actual preset time bounded filter) is designed. The dynamics of this module are described by the following equation:

[0044] (7)

[0045] in, For filtering error, For virtual control input, , This represents the output signal of the module. The module is capable of online estimation. Its changing trend directly provides a smooth filtered signal. This avoids complex analytical differentiation operations.

[0046] IV. Integration of the State Constraint Processing Module: To integrate state constraints into the controller design, a state constraint processing module is constructed. This module uses a nonlinear state transformation function to process the original constrained state. Process the data and output a new equivalent state variable. The constrained high-order strictly feedback nonlinear system constructed is as follows:

[0047] (8)

[0048] Wherein, state transition function Its derivative is , , This module transforms the original state signal into an equivalent signal that is not bounded, eliminating the need for additional handling of state boundary issues in the design of subsequent control laws, while strictly ensuring that the original state always satisfies the constraints.

[0049] V. Design and Implementation of Adaptive Preset-Time Bounded Tracking Controller: Based on the above modules, the final control protocol is constructed using the adaptive backstepping method. First, the system error variable is defined. and ,in Subsequently, the designed virtual control law and actual control input as follows:

[0050] (9)

[0051] (10)

[0052] (11)

[0053] Simultaneously, the unknown function is estimated using a radial basis function neural network. The squared norm of the weights in the neural network is used to estimate the squared norm. The adaptive law is:

[0054] (12)

[0055] VI. System Stability and Performance Analysis: This is achieved by constructing a composite Lyapunov function that incorporates system error, parameter estimation error, and filtering error. Substituting the controller and adaptive law from the above design, the derivatives satisfy the inequality:

[0056] (14)

[0057] In the formula,

[0058] (15)

[0059] (16)

[0060] (17)

[0061] In equation (14), there exists a constant. ,if only

[0062] (18)

[0063] Equation (18) can be further expressed as:

[0064] (19)

[0065] There exists a positive constant. , making

[0066] (20)

[0067] Based on the stability criterion upon which the convergence time determination module is based, it can be concluded that: all signals within the closed-loop system (including state errors) Parameter estimation Filtering error and control input (etc.) are all actually time-bounded and stable. System state Always meet the preset constraints Tracking error ,and , , ,Depend on The boundedness of can be obtained At the preset time It converges inward to a bounded range, i.e.

[0068] (twenty one)

[0069] The convergence range of the tracking error is then:

[0070] (twenty two)

[0071] Thus, this invention achieves adaptive preset time bounded tracking control for high-order nonlinear systems with full-state constraints.

[0072] To verify the effectiveness of the control method proposed in this invention, a first-order robotic arm system is used as an example. The dynamic model of this system is as follows:

[0073] (twenty three)

[0074] In the above formula, , and These represent the angular positions of the connecting rods ( ), angular velocity ( ) and angular acceleration ( ), Representative quality ( ), Represents length ( ), Represents the coefficient of friction. Represents gravitational acceleration ( ), Represents the driving torque acting on the joint ( ).

[0075] definition , Then equation (23) can be rewritten as

[0076] (twenty four)

[0077] In the formula, , , , . It is the system output. The system control objective is to make the output... Tracking expected reference instructions .

[0078] The specific steps for implementing the control scheme of the present invention are as follows:

[0079] 1. System Initialization and Parameter Setting: Set the initial angular position of the first-order robotic arm's links to 0. The initial angular velocity of the connecting rod is 0.1. , The constraint range for the angular position of the connecting rod is set as follows: The constraint range for the angular velocity of the connecting rod is set to ,in, , , , .

[0080] 2. Parameter configuration of functional modules: (1) Convergence time determination module: Set the preset time Seconds, adjust parameters (2) Signal filtering and differential estimation module (filter): , , , , , , (3) Controller and adaptive law parameters: , , , , , , , (4) The activation function of the radial basis function neural network is a Gaussian function, and its center is designed as follows: .

[0081] 3. Control Signal Generation: Based on the control framework constructed according to this invention, the transformed state variables are obtained by the state constraint processing module, and the filtered signal is provided by the signal filtering and differential estimation module, combined with the time-varying function of the convergence time determination module. The virtual control law is ultimately calculated and generated by the controller. With actual control signals Its specific form is:

[0082] (25)

[0083] (26)

[0084] Meanwhile, the adaptive update law for the neural network weights is:

[0085] (27)

[0086] Implementation Results: Simulations were performed under the stated initialization conditions, parameter configurations, and control signals. The results are shown in the corresponding appendix. Figure 3-6 As shown, the angular position and angular velocity of the connecting rod are always constrained to... and In the closed-loop system, all signals, including the driving torque of the linkage, are considered. Error signal and adaptive parameters All systems achieved bounded stability within a preset time of 1 second. The system tracking error converged to a bounded region within the preset time, verifying that the method of the present invention can strictly guarantee the full-state constraint requirements while achieving bounded tracking control within a preset time.

[0087] The present invention has been described above by way of examples. Those skilled in the art should understand that the present invention is not limited to the examples described above, and various changes, modifications and substitutions can be made without departing from the scope of the present invention.

Claims

1. An adaptive preset-time bounded tracking control method under state constraints, characterized in that, Includes the following steps: Establish a mathematical model for a high-order strict feedback nonlinear system; Construct a bounded stability criterion for a preset time to determine whether the system achieves bounded stability within a preset time. Design an adaptive real-preset time-bounded filter to process the differential signal of the virtual control input in backstep control, avoiding the problem of computational complexity explosion; Construct a state transformation function to map the constrained system state into unconstrained equivalent state variables to handle full-state constraint problems; Based on the backstepping control method, combined with the actual preset time bounded stability criterion, the adaptive actual preset time bounded filter, and the system model after state transformation, an adaptive preset time bounded tracking controller is designed. The stability of the closed-loop system is analyzed using Lyapunov stability theory, and it is verified that, under the premise of satisfying all state constraints, all signals converge to the bounded range within a preset time.

2. The method according to claim 1, characterized in that, The actual preset time bounded stability criterion is: there exists a Lyapunov function and positive constants such that the derivative of the Lyapunov function satisfies an inequality relationship, which includes a time-varying adjustment function. This time-varying adjustment function is defined as a function form related to the preset time and is used to determine whether the system can achieve actual bounded stability within the preset time.

3. The method according to claim 1, characterized in that, The dynamic model of the adaptive actual preset time bounded filter includes a filtering error term, a time-varying adjustment function, a virtual control input, and adaptive estimation parameters. This filter can directly provide a smooth filtered signal and its differential estimate, thereby avoiding repeated analytical differentiation of the virtual control signal.

4. The method according to claim 1, characterized in that, The state transformation function is a nonlinear state transformation function. By processing the original constrained state and its derivative, it outputs a new equivalent state variable, constructs a constrained high-order strict feedback nonlinear system, and converts the original state signal into an equivalent signal that is not restricted by the boundary, ensuring that the design of the subsequent control law does not require additional handling of the state out-of-bounds problem.

5. The method according to claim 1, characterized in that, The adaptive preset time bounded tracking controller includes a virtual control law, an actual control input, and an adaptive law. The virtual control law and the actual control input include error variables, design parameters, and the activation function of the radial basis function neural network. The adaptive law is used to estimate the square norm of the neural network weights.

6. The method according to claim 1, characterized in that, By constructing a composite Lyapunov function that includes system error, parameter estimation error, and filtering error, and substituting it into the controller and adaptive law, the derivative is derived and verified to satisfy the actual preset time bounded stability criterion. This proves that all signals in the closed-loop system achieve actual preset time bounded stability, the system state always satisfies the preset constraints, and the tracking error converges to the bounded range within the preset time.

7. An adaptive preset-time bounded tracking control system under state constraints, characterized in that, include: The convergence time determination module is used to receive the preset time parameters set by the user according to the actual preset time bounded stability criterion, and to provide the basis for analysis and setting of the convergence time for the entire control system. The signal filtering and differential estimation module is used to receive the virtual control signal from the controller and output the smoothed filtered signal and its differential estimate. The state constraint processing module is used to process the original system state signal through the built-in nonlinear state transformation function, and map the constrained system state into an unconstrained equivalent variable. The adaptive preset time bounded tracking controller module is used to integrate the time criteria from the convergence time determination module, the smoothing instructions and differential signals from the signal filtering and differential estimation module, and the equivalent state variables from the state constraint processing module to calculate the actual control signal that can be directly applied to the controlled object.

8. The system according to claim 7, characterized in that, The signal filtering and differential estimation module adopts an adaptive actual preset time bounded filter. The dynamic model of the filter includes a filtering error term, a time-varying adjustment function, a virtual control input, and adaptive estimation parameters.

9. The system according to claim 7, characterized in that, The state constraint processing module uses a nonlinear state transformation function to process the original constrained state and its derivatives, and outputs new equivalent state variables to ensure that the system state always meets the preset constraint boundaries.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed by the processor, it implements the adaptive preset time bounded tracking control method under state constraints as described in any one of claims 1 to 6.

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