An adaptive coalesced neural tracking control method for nonlinear space-time systems
By adopting an adaptive coagulation neural tracking control method, the spatiotemporal uncertainty problem of nonlinear spatiotemporal systems is solved, and effective tracking and bounded error control of arbitrary differentiable reference signals are achieved, significantly improving the stability and accuracy of the system.
Patent Information
- Application Number
- CN202411861341.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-17
- Publication Date
- 2025-11-25
- Estimated Expiration
- 2044-12-17
AI Technical Summary
Existing technologies struggle to effectively address the spatiotemporal uncertainties of nonlinear spatiotemporal systems, especially when the basis functions are unknown. This impacts the applicability of adaptive estimation methods, and neural networks fail in handling time-dependent variables.
An adaptive condensation neural tracking control method is adopted. By constructing a condensation neural network to approximate nonlinear spatiotemporal uncertainty, a virtual control law, a parameter adaptive law, and a controller are designed. The bias term is canceled by a sliding mode function, and the tracking control of any differentiable reference signal is realized based on the backstepping method.
It achieves efficient tracking of any differentiable reference signal, ensures adjustable bounded tracking error, and asymptotically converges to zero under certain conditions.
Smart Images
Figure CN119644749B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to an adaptive coagulation neural tracking control method for nonlinear spatiotemporal systems, belonging to the field of control technology based on neural networks. Background Technology
[0002] In the real world, spatiotemporal systems are commonly found in fields such as robot swarm coordination and autonomous vehicle navigation. These systems are typically characterized by nonlinear dynamics and complex interactions between spatial variables and time-dependent factors, making their behavior inherently unpredictable. As the demands for real-time performance and stability continue to increase across industries, effective control strategies capable of addressing these spatiotemporal uncertainties have become crucial.
[0003] Neural networks and fuzzy logic systems have become popular nonlinear control schemes due to their general approximation capabilities. However, they may lose effectiveness when approximating spatiotemporal uncertainties because time-dependent variables are often unavailable. Meanwhile, adaptive estimation methods (such as variable condensation methods) have made significant progress in handling spatiotemporal systems, but spatiotemporal uncertainties are often assumed to be parameterized by multiplying known basis functions by unknown parameters. When the basis functions are unknown, the applicability of these adaptive methods is affected. In summary, spatiotemporal uncertainty poses a significant challenge to both classical adaptive estimation and neural network-based approximation methods. Summary of the Invention
[0004] Purpose of the invention:
[0005] The purpose of this invention is to provide an adaptive coagulation neural tracking control method for nonlinear spatiotemporal systems, so as to achieve adjustable tracking performance for any differentiable reference signal.
[0006] Technical solution:
[0007] The adaptive condensation neural network tracking control method for nonlinear spatiotemporal systems described in this invention includes the following steps: establishing a mathematical model of the nonlinear spatiotemporal system and the tracking signal; constructing a condensation neural network, approximating the nonlinear spatiotemporal uncertainty of the open-loop error system as the form of condensation weights multiplied by the network basis vectors, and representing the bias input using time-varying disturbances and spatiotemporal approximation errors; constructing a Lyapunov function, canceling the remaining bias terms based on a sliding mode-like smooth function, and designing a virtual control law, a parameter adaptive law, and a controller based on the backstepping method.
[0008] The mathematical model of the nonlinear spatiotemporal system is as follows:
[0009]
[0010] in For unknown time-varying parameters, for r = 1, ..., n, For the system state, f r : It is a perfectly smooth mapping that represents uncertainty across time and space scales. Indicates the output. This indicates input.
[0011] The nonlinear spatiotemporal uncertainty of the condensed neural network approximating the open-loop error system is specifically as follows:
[0012] First, define the error variable z. r =x r -α r-1 α0=y d , where α r-1 For virtual control laws, y d Using the tracking signal as a reference, the open-loop error system is calculated as follows:
[0013]
[0014] Secondly, define the function For spatiotemporal uncertainty, where χ r (t) is the state vector. Let be a time-varying parameter vector. Using a condensation neural network for approximation, the final expression is:
[0015]
[0016] in For the cohesive weight of the network, As basis vectors, For unknown time-varying perturbations This refers to the spacetime approximation error.
[0017] The virtual control law, parameter adaptive law, and controller are specifically as follows:
[0018] The virtual control law is designed as follows:
[0019]
[0020] Where k r >0 indicates the control gain. χ1=x1, and When r≥2, and For adaptive parameters. The parameter adaptive law is designed as follows:
[0021]
[0022]
[0023] in and For adaptive gain. Finally, the controller is designed as follows:
[0024] u = α n ;
[0025] The method employs a sliding mode-like smoothing function sgm(x,∈) to compensate for the bias terms of the condensation neural network. The function's properties are expressed as follows:
[0026] |x|-xsgm(x,∈)≤ρ(x);
[0027] Where ∈ is a parameter greater than zero, and ρ(·) is a κ-like function.
[0028] Beneficial effects:
[0029] This invention proposes an adaptive coagulating neural tracking control method for nonlinear spatiotemporal systems, which can effectively track any differentiable trackable signal while ensuring an adjustable bounded tracking error. It also points out that when the adjustable parameters in the sliding mode-like function of the controller satisfy the square integrability condition, the tracking error asymptotically converges to zero. Attached Figure Description
[0030] Figure 1 This is a flowchart of the method of the present invention;
[0031] Figure 2 The system tracking error and adaptive parameter trajectory diagram;
[0032] Figure 3 To compare the tracking error trajectories of the coagulation neural controller and the conventional neural controller, respectively. Detailed Implementation
[0033] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments, but the scope of the present invention is not limited thereto.
[0034] like Figure 1 As shown, this invention provides a condensed neural network approximation method for nonlinear spatiotemporal uncertainty, comprising the following steps: establishing a mathematical model of the nonlinear spatiotemporal system and the tracking signal; constructing a condensed neural network to approximate the nonlinear spatiotemporal uncertainty of the open-loop error system as the form of condensed weights multiplied by the network basis vectors, with the bias input represented by time-varying perturbations and spatiotemporal approximation errors; constructing a Lyapunov function, canceling the remaining bias terms based on a sliding mode-like smooth function, and designing a virtual control law, a parameter adaptive law, and a controller based on the backstepping method.
[0035] The principle behind this invention is described as follows:
[0036] 1. Establish the model and describe the problem.
[0037] In this invention, we consider a class of uncertain nonlinear spatiotemporal systems, expressed as:
[0038]
[0039] For r = 1, ..., n, For unknown time-varying parameters, Assumption 1 is satisfied. For system status, Indicates the output. Indicates the input, f r : It is a perfectly smooth mapping that represents uncertainty across time and space scales.
[0040] Assumption 1: The time-invariant parameter θ(t) is continuous and belongs to the compact set Ω. θ .
[0041] The objective of this invention is to design an adaptive controller u such that the system output y can track the time-varying reference signal y. d (t). Where y d The following assumptions are satisfied:
[0042] Assumption 2: y d (t) is a known n-order differentiable signal, which is known itself, but its higher-order derivatives are unknown.
[0043] 2. Adaptive Coagulation Neural Controller Design
[0044] First, define the error variable z. r =x r -α r-1 α0=y d , where α r-1 For virtual control laws, y d Using the tracking signal as a reference, the open-loop error system is calculated as follows:
[0045]
[0046] Secondly, define the function It is a spatiotemporal uncertainty, in which
[0047] The design of virtual control laws, parameter adaptive laws, and controllers is as follows:
[0048]
[0049] Where k r >0 represents the control gain. χ1=x1, and When r≥2, and These are adaptive parameters.
[0050] The adaptive law design parameters are as follows:
[0051]
[0052] in and For adaptive gain.
[0053] Finally, the controller is designed as follows:
[0054] u = α n ;
[0055] This completes the controller design in this invention.
[0056] 3. System stability analysis
[0057] Consider the following Lyapunov function
[0058]
[0059] in, To estimate the error, the derivative of V is calculated as follows:
[0060]
[0061] in, And for i = 2, ..., n,
[0062]
[0063] definition Employing condensation neural networks, functions It can be represented as:
[0064]
[0065] in For the cohesive weight of the network, As basis vectors, For unknown time-varying perturbations This refers to the spacetime approximation error.
[0066] Substituting can yield
[0067]
[0068] By utilizing the properties, we can obtain
[0069]
[0070] By adopting this method, we can ultimately obtain...
[0071]
[0072] Indicates when hour, The validity of this statement implies that z1(t) is bounded for any t ≥ 0. Solving the differential inequality on the interval [t1, t2), for t2 > t1 ≥ 0, we can obtain...
[0073]
[0074] Therefore, it is not difficult to obtain.
[0075]
[0076] in Therefore, the tracking error is adjustable in an average sense around c1.
[0077] Furthermore, since c1 is adjustable, we can choose a bounded time-varying parameter. This makes the tracking error In this case, the designed controller can achieve progressive tracking.
[0078] The effectiveness of this method is verified by simulation experiments based on the design of the following second-order mass-spring-damper system.
[0079]
[0080] in Let and q represent acceleration, velocity, and position, respectively; M, C, and K represent the mass, damping coefficient, and stiffness of the spring, respectively; and F represent the external force. Define x1 = q. as well as Let M = 1 kg. in θ²(t) = 1 + 0.2cos(t). The above system can be transformed into, in and θ(t) = [θ1(t), θ2(t)] T All are unknown. The control objective is to drive the output system's output y = q to track the reference signal y. d (t) = sin(5t).
[0081] In the simulation, the parameters are set as k1 = k2 = 10. And ∈ = 0.1. The initial conditions are set as x1(0) = x2(0) = 5. In order to approximate the spacetime uncertainty function The basis functions of the neural network are constructed as S(χ)=[s1(χ),…,s 21 (χ)], where and The input vector is
[0082] Figure 2 The trajectories of tracking error and adaptive parameters were plotted. Figure 3 The tracking error trajectories of the traditional neural network control method and the proposed adaptive coagulation neural control method are shown. It should be noted that, for a fair comparison, the initial values and controller gain parameters for all states were chosen consistently in the simulation experiments. Figure 2 It can be seen that the proposed control method can track the reference signal with high accuracy, and the adaptive parameters are bounded. Figure 3 The results further demonstrate the effectiveness of the method, showing that the proposed method has better performance than traditional neural network control methods for tracking control problems of nonlinear spatiotemporal systems.
[0083] The specific embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.
Claims
1. An adaptive coalesced neural tracking control method for nonlinear spatiotemporal systems, characterized in that, The method comprises the following steps: S1, establishing a mathematical model of a nonlinear space-time system and a tracking signal; S2, constructing a condensed neural network, approximating nonlinear space-time uncertainty of an open-loop error system to a form of condensed weights multiplied by network base vectors, and expressing bias input as time-varying disturbance and space-time approximation error; S3, constructing a Lyapunov function, canceling a residual bias item based on a kind of smooth function similar to a sliding mode, and designing a virtual control law, a parameter adaptive law and a controller based on a backstepping method; The condensed neural network approximates nonlinear space-time uncertainty of the open-loop error system, and specifically comprises: First, define the error variable , where is the virtual control law, is the reference tracking signal; compute the open-loop error system, expressed as: ； Second, define the function is the space-time uncertainty, where, is the state vector, is the time-varying parameter vector; using a condensed neural network approximation, the final expression is: ； wherein is the condensation weight of the network, is the basis vector, is the unknown time-varying disturbance, is the space-time approximation error; The virtual control law, the parameter adaptive law and the controller are designed, and specifically comprise: The virtual control law is designed as: ； wherein is a control gain, , and when , , and is an adaptive parameter; the parameter adaptive law is designed as: ； ； ； wherein , and is an adaptive gain, finally, the controller is designed as: .
2. The adaptive pinched neural follow-the-leader control method for nonlinear spatiotemporal systems of claim 1, wherein, The mathematical model of the nonlinear space-time system comprises: ； wherein is an unknown time-varying parameter, for , is the system state, is a sufficiently smooth mapping representing uncertainty across time and spatial scales, represents the output, represents the input.
3. The adaptive pinched neural tracking control method for nonlinear spatiotemporal systems according to claim 2, characterized in that a kind of class sliding mode's smooth function is adopted The pinched neural network bias term is compensated, and the functional property expression is: ； wherein is a parameter greater than zero, is a class function.