Adaptive predefined time tracking control method for uncertain nonlinear high-order all-wheel drive systems

Abstract

The present invention relates to an adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system. The specific steps are: S1, establishing a mathematical model of the uncertain nonlinear high-order all-wheel drive system; S2, deriving an explicit all-wheel drive controller structure based on auxiliary variables; S3, building a pre-closed-loop control system with adjustable parameters; S4, constructing auxiliary variables with predefined time characteristics; S5, defining parameter setting rules to improve the predefined controller design; S6, using the controller to perform predefined time closed-loop tracking control. The present invention provides a predefined time tracking control method that can effectively track and control an uncertain nonlinear high-order all-wheel drive system. At the same time, the closed-loop system can complete the control task before a user-predefined time node and converge to a predefined accuracy range.

Description

Technical Field

[0001] The present invention belongs to the field of nonlinear system control, and relates to predefined time control technology and adaptive control technology. Specifically, an adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system is designed. Background Art

[0002] Nonlinear control theory has been an important research field for the past few decades, and many excellent research results have been proposed based on first-order nonlinear differential equations. It is worth mentioning that most first-order nonlinear differential equations are derived from the expansion of the actual object model, which makes the transformed system states coupled with each other, resulting in a complex controller design environment. Recently, the high-order fully-actuated system (HOFAS) theory has been proposed as an innovative approach to simplify the design of nonlinear controllers. With the help of HOFAS theory, many nonlinear control problems have been further solved, including but not limited to nonlinear robust control, nonlinear fault-tolerant control, and nonlinear predictive control. However, the control schemes for nonlinear HOFAS have only theoretically evaluated the closed-loop control performance when time tends to infinity, which may result in long convergence time.

[0003] In fact, the convergence time problem has been addressed in first-order differential equations, and several control strategies have been proposed from the perspective of the maximum settling time (MST), typically finite-time control, fixed-time control, and predefined-time control (PTC). Within the finite-time control framework, an MST dependent on the initial conditions is derived to ensure the boundedness of the convergence time. Fixed-time control, as an improved iteration of finite-time control, provides a fixed MST that is independent of the initial conditions, further regulating the convergence rate of the closed-loop system. However, for fixed-time control, solving the MST can be extremely complex or even infeasible. To address this challenge, PTC has emerged as an innovative solution, allowing users to directly predefine the MST without the need for complex MST calculations. Clearly, PTC, as a convenient and accurate method for assigning the MST to closed-loop systems, offers significant advantages in terms of engineering practicality. Unfortunately, for the PTC design of nonlinear HOFAS, it is difficult to directly extract effective solutions from existing nonlinear PTC methods.

[0004] In existing nonlinear PTC design, there are two main control frameworks: sliding mode-based PTC and backstepping-based PTC. The former is suitable for second- and third-order nonlinear systems. However, its scalability is limited by the order of the system, making it unsuitable for nonlinear HOFAs. The latter is suitable for higher-order nonlinear systems, often expressed as multidimensional first-order differential equations. Typically, several outstanding works have been published to address PTC problems in high-order nonlinear systems by combining backstepping with PTC control strategies.

[0005] The paper "An Improved Predefined-Time Adaptive Neural Control Approach for Nonlinear Multiagent Systems" (Pan Yingnan et al., IEEE Transactions on Automation Science and Engineering, 2023) proposes a new backstepping-based predefined-time neural network adaptive tracking control method for a class of uncertain nonlinear multi-agent strict feedback systems. The proposed method uses a backstepping architecture to derive control signals and develops a new predefined-time control mechanism to complete the predetermined control task. In addition, the method uses a radial basis function neural network to approximate the unknown nonlinear dynamics. Finally, two numerical simulation examples are given to verify that the proposed scheme can achieve tracking control of the closed-loop system while achieving system convergence behavior within a predefined time interval.

[0006] The paper "Predefined-Time Adaptive Neural Tracking Control of Switched Nonlinear Systems" (Wang Huanqing et al., IEEE Transactions on Cybernetic, 2022) proposes a predefined-time adaptive neural network tracking control method for a class of uncertain switched nonlinear strict feedback systems. By introducing a finite-time differentiator to estimate the first-order derivative of the virtual controller, and utilizing backstepping and the common Lyapunov function method, a new adaptive predefined-time controller is proposed. Finally, a numerical simulation example is presented, verifying that the proposed scheme can achieve system convergence within a predefined time interval and ensure that the tracking error converges to zero.

[0007] However, the backstepping-based PTC framework is also not suitable for nonlinear HOFAS because backstepping, as a recursive design framework, contradicts the original intention of HOFAS theory to simplify nonlinear controller design. To overcome the shortcomings of HOFAS theory in terms of convergence time, it is necessary to develop a new PTC framework for nonlinear HOFAS. Summary of the Invention

[0008] In response to the above-mentioned problems existing in the prior art, the present invention provides an adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system. In addition to providing the predefined time convergence performance of the closed-loop system, this method can also converge the tracking error to a predefined accuracy range. At the same time, unlike the previous nonlinear HOFAS control design, a parameter-adjustable pre-closed-loop system is generated for PTC design. With the help of the parameter-adjustable pre-closed-loop system, the implementation of PTC only requires an explicit all-wheel drive controller structure, derived auxiliary signals, and a set of parameter setting rules. Compared with the PTC framework based on the backstepping method, the implementation of the predefined control algorithm is significantly simplified.

[0009] In order to achieve the above object, the present invention provides an adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system, the specific steps of which are as follows:

[0010] S1. Establish a mathematical model for uncertain nonlinear high-order all-wheel drive systems;

[0011]

[0012] Where, as well as Represent the state variables of the system, the known control gain matrix, the known nonlinearity, the unknown parameter matrix, the known nonlinearity matrix, the unknown external disturbance, the output signal, and the input signal, respectively. In addition, in this patent, the symbols ||·|| and sgn(·) are used to represent the F norm and the sign function. For any positive integers r, n, and i, this patent adopts the following definitions: and

[0013] S2, derive the explicit all-wheel drive controller structure based on auxiliary variables;

[0014] For a known reference signal Define tracking error z = xy d And derive the following high-level all-wheel drive tracking system:

[0015]

[0016] Where,∈=[z (0～n-1) ,y (0～n-1) ].

[0017] According to the elimination mechanism of high-order all-wheel drive theory, the structure of the explicit all-wheel drive controller based on auxiliary variables is as follows:

[0018]

[0019] Where, is an auxiliary control variable.

[0020] S3. Build a pre-closed-loop control system with adjustable parameters;

[0021] Combining (2) and (3), we can get z (n) =v+Δ T f(∈)+D. Then, define z i =z (i-1) ,

[0022] The following results can be obtained

[0023]

[0024] For i=1,...,n and j=1,...,r, a nonlinear mapping function is introduced in and Then system (4) can be mapped as

[0025]

[0026] Where, At the same time α i and β i is the design parameter to be defined. The above formula is a pre-closed loop control system with adjustable parameters.

[0027] S4. Construct auxiliary variables with predefined time characteristics;

[0028] Introduce the following time-varying function:

[0029]

[0030] Where, t n is a time node, μ is a positive constant and satisfies 0<μ<1. The function satisfies: 1) θ(0)=0; 2) when t∈[0,t n ) is monotonically increasing; 3) when t∈[t n ,∞) when θ(t)=1 / μ. In addition, this function is continuously differentiable, and its derivative is expressed as follows

[0031]

[0032] From the above formula, we can see that there is a parameter satisfy Furthermore, by defining ζ j =s n,j θ constructs the following barrier function

[0033]

[0034] Lyapunov function with obstacles

[0035]

[0036] Where r j is a positive design parameter, is the parameter estimation error, where is an estimate of the unknown parameter vector Δ. Based on the above results, the following auxiliary control variables are constructed

[0037]

[0038] Adaptive update rate with parameters

[0039]

[0040] Where k j with ι j is a positive design parameter.

[0041] S5. defining parameter setting rules to improve the predefined controller design;

[0042] Undefined parameters are designed as follows

[0043]

[0044] Where i=1,...,n-1, ε i >0 and ε n =χμ. Where, t i is the user-defined time node, ε i is the user-defined convergence accuracy threshold. To simplify the notation, for i=1,...,n-1, define So far, the proposed predefined time control algorithm is fully demonstrated in (3), (10) and (11).

[0045] S6. Use the controller to perform closed-loop tracking control for a predefined time.

[0046] Preferably, the setting criterion of the controller parameters is 0<r j , 0<k j ,0<ι j , 0<χ, 0<μ<1 and formula (12).

[0047] Compared with the prior art, the advantages and positive effects of the present invention are:

[0048] The present invention provides an adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system, which can derive an explicit controller structure based on the high-order all-wheel drive theory. Furthermore, a parameter-adjustable pre-closed-loop control system that facilitates the design of predefined time control can be derived through a set of nonlinear mapping functions. Finally, auxiliary variables and parameter design conditions with predefined time characteristics are derived to improve the design of the predefined controller. In addition to providing the predefined time convergence performance of the closed-loop system, the proposed scheme can also converge the tracking error to a predefined accuracy range. At the same time, unlike previous nonlinear HOFAS control designs, with the help of the parameter-adjustable pre-closed-loop system, the implementation of PTC only requires an explicit all-wheel drive controller structure, derived auxiliary signals, and a set of parameter setting rules. Compared with the PTC framework based on the backstepping method, the implementation of the predefined control algorithm is significantly simplified. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 This is a flow chart of an adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system according to an embodiment of the present invention;

[0050] Figure 2 Graph showing the trajectory of states x1, x2 and a reference signal in a numerical simulation according to an embodiment of the present invention;

[0051] Figure 3 Graph showing the trajectories of control inputs u1 and u2 in the numerical simulation of an embodiment of the present invention;

[0052] Figure 4 The estimated vector in the numerical simulation of the embodiment of the present invention is and The trajectory diagram of the norm of ;

[0053] Figure 5 is the system variable s in the numerical simulation of the embodiment of the present invention 3,1 and s 3,2 Trajectory diagram;

[0054] Figure 6 is the system variable s in the numerical simulation of the embodiment of the present invention 2,1 and s 2,2 Trajectory diagram;

[0055] Figure 7 is the tracking error s in the numerical simulation of the embodiment of the present invention 1,1 and s 1,2 Trajectory diagram; DETAILED DESCRIPTION

[0056] The present invention is described in detail below by way of exemplary embodiments. However, it should be understood that the present invention may be beneficially incorporated into other embodiments without further description.

[0057] See also Figure 1 The embodiment of the present invention provides an adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system, the specific steps of which are as follows:

[0058] S1. Establish a mathematical model for uncertain nonlinear high-order all-wheel drive systems;

[0059]

[0060] Where, as well as Represent the state variables of the system, the known control gain matrix, the known nonlinearity, the unknown parameter matrix, the known nonlinearity matrix, the unknown external disturbance, the output signal, and the input signal, respectively. In addition, in this patent, the symbols ||·|| and sgn(·) are used to represent the F norm and the sign function. For any positive integers r, n, and i, this patent adopts the following definitions: and

[0061] S2, derive the explicit all-wheel drive controller structure based on auxiliary variables;

[0062] For a known reference signal Define tracking error z = xy d And derive the following high-level all-wheel drive tracking system:

[0063]

[0064] Where,∈=[z (0～n-1) ,y (0～n-1) ].

[0065] According to the elimination mechanism of high-order all-wheel drive theory, the structure of the explicit all-wheel drive controller based on auxiliary variables is as follows:

[0066]

[0067] Where, is an auxiliary control variable.

[0068] S3. Build a pre-closed-loop control system with adjustable parameters;

[0069] Combining (2) and (3), we can get z (n) =v+Δ T f(∈)+D. Then, define z i =z (i-1) , we can get the following results

[0070]

[0071] For i=1,...,n and j=1,...,r, a nonlinear mapping function is introduced in and Then system (4) can be mapped as

[0072]

[0073] Where, At the same time α i and β i is the design parameter to be defined. The above formula is a pre-closed loop control system with adjustable parameters.

[0074] S4. Construct auxiliary variables with predefined time characteristics;

[0075] Introduce the following time-varying function:

[0076]

[0077] Where, t n is a time node, μ is a positive constant and satisfies 0<μ<1. The function satisfies: 1) θ(0)=0; 2) when t∈[0,t n ) is monotonically increasing; 3) when t∈[t n ,∞) when θ(t)=1 / μ. In addition, this function is continuously differentiable, and its derivative is expressed as follows

[0078]

[0079] From the above formula, we can see that there is a parameter satisfy Furthermore, by defining ζ j =s n,j θ constructs the following barrier function

[0080]

[0081] Lyapunov function with obstacles

[0082]

[0083] Where r j is a positive design parameter, is the parameter estimation error, where is an estimate of the unknown parameter vector Δ. Based on the above results, the following auxiliary control variables are constructed

[0084]

[0085] Adaptive update rate with parameters

[0086]

[0087] Where k j with ι j is a positive design parameter.

[0088] S5. defining parameter setting rules to improve the predefined controller design;

[0089] Undefined parameters are designed as follows

[0090]

[0091] Where i=1,...,n-1, ε i >0 and ε n =χμ. Where, t i is the user-defined time node, ε i is the user-defined convergence accuracy threshold. To simplify the notation, for i=1,...,n-1, define So far, the proposed predefined time control algorithm is fully demonstrated in (3), (10) and (11). The setting criteria of the controller parameters are 0<r j , 0<k j ,0<ι j , 0<χ, 0<μ<1 and formula (12)

[0092] S6. Use the controller to perform closed-loop tracking control for a predefined time.

[0093] The adaptive predefined time tracking control method for the uncertain nonlinear high-order full-drive system described in the embodiment of the present invention can derive an explicit controller structure based on the high-order full-drive theory. Furthermore, a set of nonlinear mapping functions can be used to derive a parameter-adjustable pre-closed-loop control system that facilitates the design of predefined time control. Finally, auxiliary variables and parameter design conditions with predefined time characteristics are derived to improve the design of the predefined controller. In addition to providing the predefined time convergence performance of the closed-loop system, the proposed scheme can also converge the tracking error to a predefined accuracy range. At the same time, unlike the previous nonlinear HOFAS control design, with the help of the parameter-adjustable pre-closed-loop system, the implementation of PTC only requires an explicit full-drive controller structure, derived auxiliary signals, and a set of parameter setting rules. Compared with the PTC framework based on the backstepping method, the implementation of the predefined control algorithm is significantly simplified.

[0094] In order to illustrate the effect of the adaptive predefined time tracking control method of the uncertain nonlinear high-order all-wheel drive system of the present invention, the following numerical simulation experiments are further used to illustrate the present invention.

[0095] Example 1: Consider a two-dimensional third-order non-linear high-order fully actuated system, and establish the following system equations:

[0096]

[0097] where

[0098]

[0099] x = [x1, x2] T , u = [u1, u2] T , D = [d1, d2] T , d1 = 0.1sin(t), d2 = 0.1cos(t),

[0100] Δ1 = [1, 0.5, 1.5] T , Δ2 = [2, 1.5, 0.5] T , and

[0101] The control parameters are designed as μ = 0.02, χ = 5, ε3 = 0.1, ε2 = 0.03, ε1 = 0.01, k1 = k2 = 15, r1 = r2 = 0.1, ι1 = ι2 = 5, t1 = t2 = t3 = 0.3, T1 = 0.3, T2 = 0.6 and T max = 0.9. The reference trajectory is y d = [y d,1 , y d,2 T , where y d,1 = 0.5sin(t) and y d,2 = 0.5cos(t). The initial conditions of the system are x(0) = [0.2, 0.2] T and

[0102] The adaptive predefined-time tracking control method described in the present invention is used to perform tracking control on the above uncertain non-linear high-order fully actuated system, and the results are shown in Figures 2 to 7 , where the tracking trajectory is as shown in Figure 2 . It can be seen that the proposed control method has good tracking performance. Figure 3 and Figure 4 show the trajectories of the control input signal and the norm of the adaptive parameter vector respectively. Obviously, the parameter estimation error is bounded. The trajectories of the variables s 3,1 and s 3,2 are shown in Figure 5 , and it can be seen that they satisfy the preset convergence index, that is, when t ≥ T1 = 0.3 seconds, |s 3,j ​(t)|∈[-ε3,ε3]=[-0.1,0.1]. At the same time, the variable s 2,1 With s 2,2 The predefined convergence characteristics of Figure 6 , that is, when t≥T2=0.6 seconds, |s 2,j (t)|∈[-ε2,ε2]=[-0.03,0.03]. The final tracking error s 1,1 =x1-y d,1 and s 1,2 =x²-y d,2 Shown in Figure 7 , it can be observed that when t≥T max = 0.9 seconds, the tracking error converges to the predefined accuracy range [-0.01, 0.01].

[0103] The above embodiments are used to explain the present invention rather than to limit the present invention. Any modifications and changes made to the present invention within the spirit of the present invention and the protection scope of the claims shall fall within the protection scope of the present invention.

Claims

1. An adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system, characterized in that: The specific steps for implementing the method are: S1. Establish a mathematical model for uncertain nonlinear high-order all-wheel drive systems; Where, as well as They represent the system's state variables, known control gain matrix, known nonlinear dynamics, unknown parameter matrix, known nonlinear function vector, unknown external disturbance, output signal, and input signal respectively. For any positive integers r, n, and i, the following definitions are used: and S2, derive the explicit all-wheel drive controller structure based on auxiliary variables; For a known reference signal Define tracking error z = xy d And derive the following high-level all-wheel drive tracking system: Where,∈=[z (0～n-1) ,y (0～n-1) ]; According to the elimination mechanism of high-order all-wheel drive theory, the structure of the explicit all-wheel drive controller based on auxiliary variables is as follows: Where, is an auxiliary control variable; S3. Build a pre-closed-loop control system with adjustable parameters; Combining (2) and (3), we can get z (n) =v+Δ T f(∈)+D; then, define z i =z (i-1) , we can get the following results For i=1,...,n and j=1,...,r, a nonlinear mapping function is introduced in and Then system (4) can be mapped as Where, At the same time α i and β i is the design parameter to be defined; the above formula is the pre-closed loop control system with adjustable parameters; S4. Construct auxiliary variables with predefined time characteristics; Introduce the following time-varying function: Where, t n is a time node, μ is a positive constant and satisfies 0<μ<1; the function satisfies: 1) θ(0)=0; 2) when t∈[0,t n ) is monotonically increasing; 3) when t∈[t n ,∞) when θ(t)=1 / μ; In addition, the function is continuously differentiable, and its derivative is expressed as follows From the above formula, we can see that there is a parameter satisfy Furthermore, by defining ζ j =s n,j θ constructs the following barrier function Lyapunov function with obstacles Where r j is a positive design parameter, is the parameter estimation error, where is an estimate of the unknown parameter vector Δ; based on the above results, the following auxiliary control variables are constructed Adaptive update rate with parameters Where k j with ι j is a positive design parameter; S5. defining parameter setting rules to improve the predefined controller design; Undefined parameters are designed as follows Where i=1,...,n-1, ε i >0 and ε n =χμ; where t i is the user-defined time node, ε i is the user-defined convergence accuracy threshold; to simplify the symbol design, for i=1,...,n-1, define So far, the proposed predefined time control algorithm is fully demonstrated in (3), (10) and (11); the setting criterion of the controller parameters is 0<r j , 0<k j ,0<ι j , 0<χ, 0<μ<1 and formula (12); S6. Use the controller to perform closed-loop tracking control for a predefined time.

2. The adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system according to claim 1, characterized in that: In step S3, the constructed parameter-adjustable pre-closed-loop control system is designed to meet the parameter design condition of 1<α i , 0<β i and s n When bounded, all variables of the closed-loop system converge in infinite time domain; Consider the following Lyapunov function Taking its derivative we get Based on Young's inequality, we can get Combining (14) and (15) we can derive where a * = min{2(α1 - 1),..., 2(α n-1 - 1)}; For the Lyapunov function Exportable Where, Based on the Lyapunov stability theorem, s i Bounded; further, it can be concluded that all variables of the closed-loop system converge in infinite time domain.

3. The adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system according to claim 2, characterized in that: In step S4, the auxiliary variables with predefined time characteristics are constructed to ensure that the system variables s n It converges within a predefined time and can converge to a predefined accuracy range, that is, s n Satisfaction|s n,j (t)|≤ε n =χ / μ when t≥T1; Consider the following obstacle Lyapunov function Its derivation can be expressed as in, Based on Young's inequality, we can get Combining (19) and (20), we can derive Substituting the designed auxiliary control signal (10) into the above formula, we can get: Using Young's inequality again, we can derive According to the above results, we can get Where a j =min{2k j ,ι j }as well as Based on the Lyapunov stability theorem, γ j and is bounded; according to the Lyapunov barrier theorem, |ζ j (t)|<χ, that is, |s n,j (t)|≤χ / θ(t); so, s n It converges within a predefined time and can converge to a predefined accuracy range, that is, s n Satisfaction|s n,j (t)|≤ε n =χ / μ when t≥T1.

4. The adaptive predefined time tracking control method for an uncertain nonlinear high-order all-wheel drive system according to claim 3, wherein in step S5, a parameter setting rule is defined to ensure that the closed-loop system converges within the predefined time, and the tracking error s 1,j =x j -y d,j It can converge to the specified accuracy range, that is, s 1,j Satisfaction|s 1,j (t)|≤ε1 when t≥T max hour; Consider the following Lyapunov function V n-1,j =|s n-1,j | (25) After taking its derivative, we can get Consider when t≥T1 |s n-1,j (t)|<ε n-1 It can be clearly concluded that once s n-1,j ∈[-ε n-1 ,ε n-1 ],s n-1,j will not jump out of this interval again; then, consider when t≥T1 |s n-1,j (t)|≥ε n-1 The results are as follows: Integrating both ends of the above equation yields Through integration operation, we have the following results Combining (28) to (29), we can derive Further sorting out After certain mathematical operations, it can be derived Substituting parameter definition (12) into (32), the following result holds: From the above formula, we can see that if t≥T1, then Therefore, n-1,j is convergent in a predefined time and satisfies |s n-1,j (t)|≤ε n-1 ,t∈[T2,∞); recursively, for variable s n-2,j , we know that s n-2,j is convergent in a predefined time and satisfies |s n-2,j (t)|≤ε n-2 ,t∈[T3,∞); In general, the variable s i,j is convergent in a predefined time and satisfies |s i,j (t)|≤ε i ,t∈[T n+1-i ,∞); for tracking error s i,j =x j -y d,j , which satisfies |s 1,j (t)|≤ε1,t∈[T max ,∞).

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