Self-adaptive predefined time tracking control method for uncertain nonlinear high-order all-wheel-drive system

Abstract

The invention relates to a self-adaptive predefined time tracking control method for an uncertain nonlinear high-order all-drive system. The method comprises the following specific steps: S1, establishing a mathematical model of the uncertain nonlinear high-order all-drive system; s2, exporting an explicit all-wheel-drive controller structure based on an auxiliary variable; s3, establishing a pre-closed loop control system with adjustable parameters; s4, constructing an auxiliary variable with a predefined time characteristic; s5, defining a parameter setting rule to perfect a predefined controller design; and S6, performing predefined time closed-loop tracking control by adopting the controller. According to the predefined time tracking control method, effective tracking control can be carried out on an uncertain nonlinear high-order all-wheel-drive system, and meanwhile, a closed-loop system can complete a control task before a time node predefined by a user and converge to a predefined precision range.

Description

Technical Field

[0001] The present invention belongs to the field of nonlinear system control, and relates to a predefined time control technology and an 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 in 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 the complexity of the controller design environment. Recently, the High-Order Fully-Actuated System (HOFAS) theory has been proposed as an innovative way 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 scheme for nonlinear HOFAS only theoretically evaluates the closed-loop control performance when time tends to infinity, which may lead to a long convergence time.

[0003] In fact, the problem of convergence time has been solved in the first-order differential equation, and several control strategies have been proposed from the perspective of maximum settling time (MST), such as finite-time control, fixed-time control and predefined-time control (PTC). In the framework of finite-time control, an MST related to 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 speed of the closed-loop system. However, for fixed-time control, solving the MST may be extremely complex or even infeasible. To meet this challenge, PTC has emerged as an innovative solution that allows users to directly predefine the MST without the need for complex calculations of the MST. Obviously, PTC, as a convenient and accurate MST allocation method for closed-loop systems, has significant advantages in 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 the existing nonlinear PTC design, there are two main control frameworks, namely the sliding mode-based PTC framework and the backstepping-based PTC framework. The former is applicable to second-order / third-order nonlinear systems. The extension of this method is limited by the order of the system, so it is not applicable to nonlinear HOFAS. The latter is applicable to high-order nonlinear systems that are often expressed as multi-dimensional first-order differential equations. Typically, by combining backstepping with PTC control strategies, some excellent works have been published to solve the PTC problem of high-order nonlinear systems.

[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 predefined-time neural network adaptive tracking control method based on backstepping for a class of uncertain nonlinear multi-agent strict feedback systems. The proposed method uses a backstepping architecture to derive the control signal and develops a new predefined time control mechanism to complete the predetermined control task. In addition, the method uses a radial basis neural network to approximate the unknown nonlinear dynamics. Finally, two numerical simulation examples are given to verify that the proposed scheme can complete the system convergence behavior within a predefined time interval while realizing the tracking control of the closed-loop system.

[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 using the backstepping method and the common Lyapunov function method, a new adaptive predefined-time controller is proposed. Finally, a numerical simulation example is given to verify that the proposed scheme can complete the system convergence within a predefined time interval and can ensure that the tracking error converges to zero.

[0007] However, the PTC framework based on backstepping is also not suitable for nonlinear HOFAS because backstepping, as a recursive design framework, contradicts the original intention of HOFAS theory to simplify the design of nonlinear controllers. In order 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 view of 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, the tracking error can also converge 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, and the specific steps are as follows:

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

[0011]

[0012] In the formula, as well as Respectively represent the state variables of the system, known control gain matrix, known nonlinearity, unknown parameter matrix, known nonlinearity matrix, unknown external disturbance, output signal and input signal. 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, deriving 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] In the formula, ∈=[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] In the formula, 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] In the formula, At the same time α i With 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] In the formula, 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 obstacle function

[0033]

[0034] Lyapunov function with obstacles

[0035]

[0036] In the formula, 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] In the formula, 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. Among them, 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 has been 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 is convenient for predefined time control design 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 predefined controller design. 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 all-wheel drive controller structure, a derived auxiliary signal 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 It is a flow chart of an adaptive predefined time tracking control method of an uncertain nonlinear high-order all-wheel drive system according to an embodiment of the present invention;

[0050] Figure 2 It is a trajectory diagram of states x1, x2 and a reference signal in the numerical simulation of an embodiment of the present invention;

[0051] Figure 3 is a trajectory diagram 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 3,2 Trajectory diagram of

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

[0055] Figure 7 is the tracking error s in the numerical simulation of the embodiment of the present invention 1,1 and 1,2 Trajectory diagram of 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 can 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, and the specific steps are as follows:

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

[0059]

[0060] In the formula, as well as Respectively represent the state variables of the system, known control gain matrix, known nonlinearity, unknown parameter matrix, known nonlinearity matrix, unknown external disturbance, output signal and input signal. 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, deriving 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] In the formula, ∈=[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] In the formula, 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] In the formula, At the same time α i With 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] In the formula, 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 obstacle function

[0080]

[0081] Lyapunov function with obstacles

[0082]

[0083] In the formula, 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] In the formula, 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. Among them, 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 criterion of the controller parameters is 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 of the above-mentioned uncertain nonlinear high-order full-drive system 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 is convenient for predefined time control design. Finally, auxiliary variables and parameter design conditions with predefined time characteristics are derived to improve the predefined controller design. 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, a derived auxiliary signal 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 experiment further illustrates the present invention.

[0095] Example 1: Consider a two-dimensional third-order nonlinear high-order all-wheel drive system and establish the following system equation:

[0096]

[0097] In the formula,

[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 condition of the system is x(0)=[0.2,0.2] T and

[0102] The adaptive predefined time tracking control method of the present invention is used to track and control the above uncertain nonlinear high-order all-wheel drive system. The results are shown in Figures 2 to 7 , where the tracking trajectory is Figure 2 As shown in Figure 2, it can be seen that the proposed control method has good tracking performance. Figure 3 and Figure 4 The trajectory of the control input signal and the norm of the adaptive parameter vector are shown respectively. It is clear that the parameter estimation error is bounded. 3,1 With s 3,2 The trajectory is shown in Figure 5 , it can be seen that it meets the pre-set 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 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 modification and change 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 of implementing the method are: S1. Establish a mathematical model of uncertain nonlinear high-order all-wheel drive system; In the formula, as well as Respectively represent the state variables of the system, known control gain matrix, known nonlinearity, unknown parameter matrix, known nonlinearity matrix, unknown external disturbance, output signal and input signal. 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 S2, deriving 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: In the formula, ∈=[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: In the formula, 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 In the formula, At the same time α i With β i is the design parameter to be defined. The above formula is a 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, this 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 In the formula, 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 In the formula, 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. Among them, 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 presented 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 set to be i , 0<β i and 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 In the formula, Based on the Lyapunov stability theorem, we know that s i It can be further 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 1, characterized in that: In step S4, the constructed auxiliary variable with predefined time characteristics can ensure that the system variable 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 In the formula, a j =min{2k j ,ι j }as well as Based on the Lyapunov stability theorem, γ j and is bounded. According to the obstacle Lyapunov 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 of the uncertain nonlinear high-order all-wheel drive system according to claim 1, the bounded conclusion of the closed-loop system according to claim 2, and the s n,j Predefined time convergence characteristics, in step S5, defining parameter setting rules can ensure that the closed-loop system converges within the predefined time, while tracking the 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)|≤ε1when 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. 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 gives Through integration operation, we have the following results Combining (28) to (29), we can derive Further sorting out After some 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 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 the tracking error s i,j =x j -y d,j , which satisfies |s 1,j (t)|≤ε1,t∈[T max ,∞).

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