Self-adaptive coagulation nerve tracking control method for nonlinear space-time system

Abstract

The invention discloses an adaptive coagulation nerve tracking control method for a nonlinear space-time system. The method comprises the following steps: establishing a mathematical model of the nonlinear space-time system and a tracking signal; constructing a coagulation neural network, approximating the nonlinear space-time uncertainty of an open-loop error system into a form of multiplying a coagulation weight by a network base vector, and representing bias input by time-varying disturbance and space-time approximation errors; and constructing a Lyapunov function, counteracting residual bias terms based on a sliding-mode-like smooth function, and designing a virtual control law, a parameter adaptive law and a controller on the basis of a backstepping method. The control scheme designed based on the method can realize effective tracking of any micro-traceable signal by a nonlinear space-time system, and ensures an adjustable bounded tracking error. Meanwhile, when adjustable parameters in the sliding mode function meet the square integrable condition, the tracking error is gradually converged to zero.

Description

Technical Field

[0001] The invention relates to an adaptive condensed neural tracking control method for a nonlinear space-time system, belonging to the technical field of control based on neural networks. Background Art

[0002] In the real world, spatiotemporal systems are common in areas such as robot swarm coordination and autonomous vehicle navigation. These systems are often characterized by nonlinear dynamics and complex interactions between spatial variables and time-dependent factors, making their behavior inherently unpredictable. As the demand for real-time performance and stability continues to increase across industries, effective control strategies that can account for these spatiotemporal uncertainties become critical.

[0003] Neural networks and fuzzy logic systems have become popular nonlinear control schemes due to their universal approximation capabilities. However, they may lose effectiveness when approximating spatiotemporal uncertainties because time-dependent variables are usually unavailable. Meanwhile, although adaptive estimation methods (e.g., variable condensation methods) have made significant progress in dealing with spatiotemporal systems, spatiotemporal uncertainties are often assumed to be parameterized as known basis functions multiplied by unknown parameters. When the basis functions are unknown, the applicability of these adaptive methods will be affected. In summary, spatiotemporal uncertainties pose significant challenges to both classical adaptive estimation and neural network-based approximation methods. Summary of the invention

[0004] Purpose of the invention:

[0005] The purpose of the present invention is to provide an adaptive condensed neural tracking control method for nonlinear spatiotemporal systems to achieve adjustable tracking performance for arbitrary differentiable reference signals.

[0006] Technical solution:

[0007] The adaptive condensed neural tracking control method for nonlinear space-time system described in the present invention comprises the following steps: establishing a mathematical model of nonlinear space-time system and tracking signal; constructing a condensed neural network, approximating the nonlinear space-time uncertainty of the open-loop error system in the form of condensed weight multiplied by the network basis vector, and expressing the bias input with time-varying disturbance and space-time approximation error; constructing a Lyapunov function, offsetting the remaining bias term 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 nonlinear space-time system mathematical model is specifically:

[0009]

[0010] in is an unknown time-varying parameter, for r=1,…,n, is the system state, f r : is a sufficiently smooth map that represents uncertainty across time and space scales, Indicates the output, Represents input.

[0011] The condensed neural network approximates the nonlinear spatiotemporal uncertainty of the open-loop error system, specifically:

[0012] First, define the error variable z r =x r -α r-1 , α0=y d , where α r-1 is the virtual control law, y d is the reference tracking signal. Calculate the open-loop error system, the expression is:

[0013]

[0014] Next, define the function is the spatiotemporal uncertainty, where χ r (t) is the state vector, is a time-varying parameter vector. Using condensed neural network approximation, the final expression is:

[0015]

[0016] in is the condensation weight of the network, is the basis vector, is the unknown time-varying disturbance, is the space-time approximation error.

[0017] The virtual control law, parameter adaptive law and controller are specifically:

[0018] The designed virtual control law is:

[0019]

[0020] where k r >0 is the control gain, χ1=x1, and When r≥2, and is the adaptive parameter. The parameter adaptive law is designed as:

[0021]

[0022]

[0023] in and is the adaptive gain. Finally, the controller is designed as:

[0024] u=α n ;

[0025] The condensation neural network bias term is compensated by using a sliding mode-like smooth function sgm(x,∈), and the function property expression is:

[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] The present invention proposes an adaptive condensed neural tracking control method for nonlinear spatiotemporal systems, which can effectively track any differentiable tracking signal and ensure an adjustable bounded tracking error. It also points out that when the adjustable parameters in the sliding mode-like function in the controller meet the square integrability condition, the tracking error converges to zero asymptotically. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 is a flow chart of the method of the present invention;

[0031] Figure 2 is the system tracking error and adaptive parameter trajectory diagram;

[0032] Figure 3 The figure shows a comparison of the tracking error trajectories of the condensed neural controller and the traditional neural controller. DETAILED DESCRIPTION

[0033] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments, but the content of the present invention is not limited to the scope of the embodiments.

[0034] like Figure 1 As shown, the present invention provides a condensed neural network approximation method for nonlinear spatiotemporal uncertainty, including the following steps: establishing a mathematical model of a nonlinear spatiotemporal system and a tracking signal; constructing a condensed neural network, approximating the nonlinear spatiotemporal uncertainty of an open-loop error system in the form of condensed weights multiplied by a network basis vector, and representing the bias input with a time-varying disturbance and a spatiotemporal approximation error; constructing a Lyapunov function, offsetting 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 establishment principle of the present invention is described as follows:

[0036] 1. Build model and problem description

[0037] In the present invention, a class of uncertain nonlinear space-time system is considered, and the expression is:

[0038]

[0039] Among them, for r=1,…,n, For unknown time-varying parameters satisfying assumption 1, is the system status, Indicates the output, represents input, f r : is a sufficiently smooth map that represents uncertainty across temporal and spatial scales.

[0040] Assumption 1: The time-invariant parameter θ(t) is continuous and belongs to the compact set Ω θ .

[0041] The objective of the present invention is to design an adaptive controller u so that the output y of the system can track the time-varying reference signal y d (t). d The following assumptions are met:

[0042] Assumption 2: y d (t) is a known n-th order differentiable signal, which itself is known but its higher-order derivatives are unknown.

[0043] 2. Design of adaptive condensed neural controller

[0044] First, define the error variable z r =x r -α r-1 , α0=y d , where α r-1 is the virtual control law, y d is the reference tracking signal. Calculate the open-loop error system, the expression is:

[0045]

[0046] Next, define the function is the spatiotemporal uncertainty, where

[0047] Design virtual control law, parameter adaptive law and controller, specifically:

[0048]

[0049] where k r >0 is the control gain, χ1=x1, and When r≥2, and is an adaptive parameter.

[0050] The design parameter adaptive law is designed as:

[0051]

[0052] in and is the adaptive gain.

[0053] Finally, the controller is designed as:

[0054] u=α n ;

[0055] The controller design in the present invention is now completed.

[0056] 3. System stability analysis

[0057] Consider the following Lyapunov function

[0058]

[0059] in, is the estimation error. The derivative of V is calculated as

[0060]

[0061] in, And for i=2,…,n,

[0062]

[0063] definition Using condensed neural network, function It can be expressed as:

[0064]

[0065] in is the condensation weight of the network, is the basis vector, is the unknown time-varying disturbance, is the space-time approximation error.

[0066] Substituting into

[0067]

[0068] Using the properties, we can get

[0069]

[0070] Adopt, and finally get

[0071]

[0072] Indicates that hour, holds, implying that z1(t) is bounded for any t ≥ 0. Solving the differential inequality on the interval [t1, t2), for t2>t1≥0, we obtain

[0073]

[0074] So it is not difficult to get,

[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 Tracking Error In this case, the designed controller can achieve asymptotic tracking.

[0078] The effectiveness of this method is verified based on the following second-order mass-spring-damper system design simulation experiment.

[0079]

[0080] in and q represent acceleration, velocity and position respectively; M, C and K represent the mass, damping coefficient and stiffness of the spring respectively; F represents the external force. Define x1 = q, as well as Set M = 1kg, in θ2(t)=1+0.2cos(t). The above system can be transformed into, in and θ(t)=[θ1(t),θ2(t)] T The control goal 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 to k1=k2=10. and ∈ = 0.1. The initial conditions are set to x1(0) = x2(0) = 5, In order to approximate the spatiotemporal uncertainty function The neural network basis function is constructed as S(χ)=[s1(χ),…,s 21 (χ)], where and The input vector is

[0082] Figure 2 The trajectories of the tracking error and the adaptive parameters are plotted. Figure 3 The tracking error trajectories of the traditional neural network control method and the proposed adaptive condensed neural control method are shown. It should be noted that in order to make a fair comparison, the initial values ​​of all states and the selection of controller gain parameters are consistent in the simulation experiment. 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 presented results further prove the effectiveness of the method. Compared with the traditional neural network control method, the proposed method has better performance for the tracking control problem of nonlinear spatiotemporal systems.

[0083] The specific implementation modes of the present invention have been described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the above implementation modes, and various changes can be made within the knowledge scope of ordinary technicians in this field without departing from the purpose of the present invention.

Claims

1. An adaptive condensed neural tracking control method for nonlinear spatiotemporal systems, characterized in that: The following steps are involved: S1. Establish mathematical models of nonlinear space-time systems and tracking signals; S2. Construct a condensed neural network to approximate the nonlinear spatiotemporal uncertainty of the open-loop error system in the form of condensed weights multiplied by the network basis vectors, and the bias input is represented by time-varying disturbances and spatiotemporal approximation errors; S3. Construct the Lyapunov function, offset the remaining bias terms based on a sliding mode-like smooth function, and design the virtual control law, parameter adaptive law and controller based on the backstepping method.

2. The adaptive condensed neural tracking control method for nonlinear spatiotemporal systems according to claim 1, characterized in that: The mathematical model of nonlinear space-time system is as follows: in is an unknown time-varying parameter, for r=1,…,n, is the system status, is a sufficiently smooth map that represents uncertainty across time and space scales, Indicates the output, Represents input.

3. The adaptive condensed neural tracking control method for nonlinear spatiotemporal systems according to claim 2, characterized in that: The condensed neural network approximates the nonlinear spatiotemporal uncertainty of the open-loop error system, specifically: First, define the error variable z r =x r -α r-1 , α0=y d , where α r-1 is the virtual control law, y d is the reference tracking signal; calculate the open-loop error system, the expression is: Next, define the function is the spatiotemporal uncertainty, where χ r (t) is the state vector, is a time-varying parameter vector; using condensed neural network approximation, the final expression is: in is the condensation weight of the network, is the basis vector, is the unknown time-varying disturbance, is the space-time approximation error.

4. The adaptive condensed neural tracking control method for nonlinear spatiotemporal systems according to claim 3, characterized in that: Design virtual control law, parameter adaptive law and controller, specifically: The designed virtual control law is: where k r >0 is the control gain, χ1=x1, and When r≥2, and is the adaptive parameter; the parameter adaptive law is designed as: in and is the adaptive gain. Finally, the controller is designed as: u=α n .

5. The adaptive condensed neural tracking control method for nonlinear spatiotemporal systems according to claim 4 is characterized in that a sliding mode-like smooth function sgm(x,∈) is used to compensate the condensed neural network bias term, and the function property expression is: |x|-xsgm(x,∈)≤ρ(x); where ∈ is a parameter greater than zero and ρ(·) is a κ-like function.

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