Self-adaptive specified performance PMSM system control method based on disturbance observer

Abstract

The invention discloses an adaptive specified performance PMSM (permanent magnet synchronous motor) system control method based on a disturbance observer, which solves the asymmetric specified performance constraint suffered by the system by designing an effective funnel-shaped performance function without depending on an initial output error, ensures the transient and steady-state performance of the asymmetric funnel constraint, and improves the stability of the system. The defect that the initial tracking error in common funnel control and PPC must be accurately kept within the boundary is overcome; the problem of'complex explosion 'of classical anti-gait control is successfully solved by applying an instruction filter, and filtering errors caused by the instruction filter are effectively eliminated by constructing a proper compensation system; the robustness and the control precision of the control system are ensured by designing a fast convergence DOB, and the minimum learning parameters of the neural network are greatly reduced; by applying a Lyapunov function theory and a backstepping technology, virtual and actual controllers with adaptive laws are constructed, so that the stability of the system is ensured.

Description

Technical Field

[0001] The invention belongs to the technical field of PMSM system neural adaptive control, and relates to an adaptive prescribed performance PMSM system control method based on a disturbance observer. Background Art

[0002] In the past few decades, permanent magnet synchronous motor (PMSM) has become increasingly popular in various industrial applications due to its small size, high power density and high efficiency. However, PMSM is a multivariable, strongly coupled nonlinear system. When the system parameters drop to a certain range, chaos will occur, which will destroy the stability of the controlled system and even cause the system to lose control. To this end, researchers have studied many advanced control techniques to regulate PMSM systems, including feedback control and sliding mode control. Among the above methods, backstepping is one of the most commonly used control methods for nonlinear systems because it can solve the uncertainty in the system by integrating fuzzy logic systems (FLS) and neural networks (NN).

[0003] However, the "complexity explosion" caused by the recursive differentiation of the virtual controller limits the extended application of the backstepping scheme. Therefore, in the controller design process, the derivative of the virtual control is estimated by constructing a first-order filter or a command filter. This filter-based solution fails to consider the impact of the filtering error on the control performance. Therefore, a compensation error signal system needs to be constructed to offset the filtering error. The literature "Zirkohi, MM: Adaptive control method of chaotic permanent magnet synchronous motor based on command filtering and combined with practical factors. International Society of Automation. 114, 120–135 (2021)" proposes a compensation scheme based on command filtering to achieve tracking control of PMSM. However, due to the inability to guarantee the asymmetric output constraints of the PMSM system, the promotion of the above scheme in practical applications is limited. Regarding the output constraint problem, a variety of techniques have been proposed for nonlinear control systems, including barrier Lyapunov function (BLF), prescribed performance control (PPC) and funnel control methods. The authors of the paper “Ngo, KB, Mahony, R., Jiang, Z.-P.: Integrator backstepping control method for multi-state constrained systems using barrier functions. 44th IEEE Conference on Decision and Control. pp.8306–8312. IEEE (2005)” first applied the BLF scheme to handle multi-state constraints of Brunofsky regular expression systems. Subsequently, BLF was used to study nonlinear control systems with output constraints and was also applied to the output constraint design of PMSM systems. The BLF-based scheme does not fully consider the given performance constraints and cannot ensure good transient and steady-state performance of the PMSM system. In order to ensure the output performance indicators, various PPC schemes are constructed by designing various given boundary curves. However, traditional PPC techniques may face singularity problems, resulting in potential instability. To address this challenge, researchers have proposed many enhanced PP-BLF methods to incorporate the specified performance function into the BLF. In “Zhang, J., Wang, S., Zhou, P., Zhao, L., Li, S.: Neural adaptive control of chaotic PMSM system based on backstepping method combined with novel prescribed performance-tangent barrier Lyapunov function. International Journal of Power Energy Systems. 31(3), 732-741. (2020)”, the authors developed a speed tracking control for output-constrained PMSM system using PP-BLF. The authors of the literature “Chen, L.: Adaptive feedback dynamic surface control of nonlinear system with asymmetric preset output constraint combined with performance barrier Lyapunov function. International Adaptive Control and Signal Processing. 32, 1417–1439 (2018)” proposed an asymmetric PP-BLF (APP-BLF) to ensure asymmetric output constraints of nonlinear system. But the BLF must be reorganized according to the complexity of the current system, which requires a lot of and complicated additional work.Therefore, researchers have designed many funnel strategies to limit the transient changes of tracking errors in industrial control systems by adopting appropriate circular funnels. However, the aforementioned funnel methods are only applicable to constrained nonlinear systems with symmetric boundaries, and are not applicable to asymmetric cases. In addition, the traditional funnel-shaped boundary function requires the control parameters to be determined based on precise initial conditions. Therefore, this is not applicable to actual PMSM systems with external disturbances and measurement errors. Therefore, developing a funnel-shaped boundary that does not depend on initial errors is still an urgent task. It is crucial to design a reliable funnel tracking controller for chaotic PMSM systems with asymmetric performance constraints.

[0004] In addition, considering the uncertainty of the operation of the PMSM system, its tracking accuracy will inevitably be affected by external disturbances. In order to evaluate the unknown disturbances, researchers have proposed many disturbance observers (DOBs) by designing appropriate structural parameters. At present, many DOBs have been integrated with other advanced control schemes to solve the instability problem of PMSM systems caused by external disturbances. In "Dai, Y., Ni, S., Xu, D., Zhang, L., Yan, X.-G.: Fuzzy sliding mode control of PMSM for electric vehicles based on disturbance observer. Artificial Intelligence in Engineering Applications. 70(3), 2476-2488(2021)", robust tracking control of PMSM systems was proposed by adopting PPC to limit transient tracking errors and constructing DOB to compensate for load disturbances. In “Li, T., Liu, X.: Non-cascaded fast non-singular terminal sliding mode control of permanent magnet synchronous motor based on disturbance observer. Journal of Electronic Engineering Technology. 17, 1061–1075 (2022)”, nonlinear DOB and dual DOB are adopted to compensate load disturbances and enhance the anti-disturbance capability of sliding mode control of PMSM system. By utilizing active damping pole zero elimination and DOB technology, the authors of “Kim, S.-K., Ahn, CK: Active damping speed tracking technology of permanent magnet synchronous motor with transient performance improvement mechanism. IEEE Transactions on Industrial Information. 18, 2171–2179 (2021)” proposed a robust tracking control solution to enhance the closed-loop stability of PMSM system. Although previous controllers using DOB have shown the potential for enhanced robustness, there has been no report on robust funnel tracking control of chaotic PMSM system based on DOB so far. Summary of the invention

[0005] The technical problem to be solved by the present invention is: to provide a neural adaptive prescribed performance control (NABAPPC) method for a PMSM system based on a command filter, aiming to convert a PMSM system with asymmetric prescribed performance constraints into an unconstrained PMSM system, using a command filter to avoid the "complexity explosion", and constructing an error compensation system to reduce the filter error, and to avoid external interference and uncertain nonlinearity.

[0006] The technical solution adopted by the present invention is: an adaptive specified performance PMSM system control method based on a disturbance observer, which is used for a chaotic PMSM system with asymmetric performance constraints. The method comprises the following steps:

[0007] Step 1: Construct a mathematical model of the PMSM system:

[0008] In the (dq) coordinate system, the mathematical model of the PMSM system is expressed as:

[0009]

[0010] Where θ is the rotor position, ω is the angular velocity, i d and i q are the currents of the d-axis and q-axis, u d and u q are the voltages of the d-axis and q-axis respectively, χ1 and χ2 are unknown positive parameters, T L is the load torque;

[0011] Define x1=θ,x2=ω,x3=i q and x4=i d , then (1) can be simplified to:

[0012]

[0013] Where △b is the total interference consisting of measurement noise and unmodeled dynamics.

[0014] According to the conclusion of the literature "Song, Y., Xia, Y., Wang, J., Li, J., Wang, C., Han, Y., Xiao, K.: Adaptive prescribed performance control method based on barrier Lyapunov function for permanent magnet synchronous motor (PMSM) used in robots with full-state and input constraints, Journal of Vibration Control. 29, 1400-1416 (2022)", when the parameters χ1 and χ2 slip into a certain domain, the system (Formula (2)) will experience chaos. For example, when u q 、u d 、T LWhen χ1=5.5, χ2=21 and the initial state is x1(0)=0.15, x2(0)=0.01, x3(0)=0.01, x4(0)=0.01, the system (Formula (2)) will enter a chaotic state, as shown in Figure 1 shown.

[0015] Remarks 1. In the chaotic PMSM system (Eq. (1)), the parameters χ1, χ2 and the load torque T L The rotor position θ is uncertain during operation, and is limited to a certain range. The chaotic PMSM system is required to have fast and accurate tracking performance and strong robustness. Therefore, it is necessary to design a high-performance control scheme to suppress chaotic oscillations, parameter disturbances and external interference. Based on the above considerations, the present invention aims to design a neural adaptive command filter backstepping funnel control based on a disturbance observer for a chaotic PMSM system with asymmetric prescribed performance constraints to achieve: (a) all signals of the closed-loop system are restricted; (b) the output tracking error can converge to an asymmetric performance region.

[0016] In order to facilitate controller design, the following preliminary knowledge is given:

[0017] Set 1: Set system status i=1,...,4 is measurable;

[0018] Assume 2: Assume the reference signal g d and its derivatives is a continuous bounded function;

[0019] Assumption 3: Assume there exists a positive constant φ M So that |φ i |≤φ M ,i=1,...,3,where φ i Represents the estimation error.

[0020] Theorem 1: According to Young's inequality, for Then we have:

[0021]

[0022] In the formula, and is a constant, and

[0023] Theorem 2: If a continuous function Z(W):R n →R is the target function defined on a compact set, then there exists a radial basis function neural network (RBFNN) such that:

[0024]

[0025] In the formula, the neural network input vector Ω W is a compact set, n is the input dimension of the neural network, is the neural network weight vector, l≥2 is the number of neuron nodes, Q(W)=[q1(W),q2(W),K,q l (W)]∈R n is the basis function vector, q i (W) is a radial basis neuron. φ(W) is the approximation error. Assume that the approximation error φ(W) satisfies |φ(W)|<φ p , where the constant φ p >0, the present invention selects Gaussian kernel function as basis function, which can be expressed as:

[0026]

[0027] Where d i =[d1,…,d l ] T is the basis function q i (W) center, θ i is the basis function q i (W) The width of the receptive field.

[0028] In order to increase the learning ability of RBFNN, the minimum learning parameter is defined as:

[0029]

[0030] In the formula, i is the unknown variable, ||·|| is the 2-norm of ·;

[0031] Step 2: Design of neural specified performance backstepping controller

[0032] Step 2.1: Novel asymmetric funnel-shaped prescribed performance

[0033] In the present invention, the purpose of control objective (b) is to limit the tracking error to an arbitrarily small asymmetric region by designing a controller with a given performance. In this regard, prescribed performance control is one of the effective solutions. The concept of prescribed performance control is that for a specific reference signal g d (t):R + →R, propose a control scheme to make the tracking error e1 = x1-g d Constrained to a given performance function , and the funnel boundary is That is, for all t>0, we have Funnel Boundary The expression is The positive constant κ0 is is the initial value of , and β is the exponential convergence rate. And Where κ ∞ is a positive constant;

[0034] However, using the funnel boundary function The prerequisite is that the defined initial output signal condition must be guaranteed to be within the funnel boundary In reality, due to external interference and measurement interference caused by sensors, it is too complicated for system (2) to obtain accurate initial conditions. and the aforementioned funnel boundary formula of e1 The asymmetric prescribed performance control target cannot be achieved. In view of the above two shortcomings, the present invention studies a new asymmetric funnel-type prescribed performance solution.

[0035] In the present invention, it is assumed that the tracking error e1 (e1=e1(t)) converges to the following asymmetric region:

[0036]

[0037] In the formula, is the performance function, e1=x1-g d , and is a constant, and

[0038] Design performance function for:

[0039]

[0040] Where κ1, κ2 and σ are positive constants, csch(·) represents the hyperbolic cosecant function of (·);

[0041] Remark 2. Since e1(0) is always bounded in real systems, and Therefore, we can choose an appropriate σ so that This is a relaxation of the requirement for accurate initial error under the original system conditions. This means that the main difference between formula (9) and the traditional solution is that it does not need to obtain the accurate initial output error of the system. Therefore, the funnel boundary function formed by formula (8) is More suitable for actual situation.

[0042] right Taking the derivative with respect to time t, we get

[0043]

[0044] Where coth(·) is the hyperbolic cotangent function of (·).

[0045] According to formula (8) and formula (9), we can know that: ① and is bounded; ③ represents the lower limit of transient overshoot of e1(t), Indicates the upper limit; ④ represents the minimum value of the steady-state tracking error e1(t), Represents the maximum value of the steady-state tracking error e1(t).

[0046] Define the constraint boundary conversion function as:

[0047]

[0048] In the formula, S1(t) and μ(t) both represent boundary constraint transfer functions. To ensure that the tracking error e1(t) satisfies the constraint condition (7), the transfer function γ1(t) is constructed as follows:

[0049]

[0050] From formula (10) and formula (11), it can be seen that the boundedness of functions S1(t) and γ1(t) depends entirely on whether the error e1(t) and the boundary μ(t) are bounded. For every initial value of , when t∈[0,+∞), if γ1(t) is bounded, then the boundedness of S1(t) and the constraints and boundedness of e1(t) can be guaranteed.

[0051] Taking the derivative of γ1(t) with respect to time t, we can obtain:

[0052]

[0053] Taking the derivative of S1 and μ with respect to time t, we can obtain:

[0054]

[0055] According to (10)-(13), when t∈[0,+∞), if γ1 is bounded, then and s1 are also bounded.

[0056] Remark 3. According to formula (11), we can get γ1∈R. From the expressions of μ(t), we can see that we only need to adjust the parameters κ1,κ2 σ, the tracking error e1(t) satisfies inequality (7) and avoids measuring the initial error exactly within the funnel boundary. and It can be set by itself, which means that the funnel-shaped function γ1(t) given by formula (10) can achieve symmetrical and asymmetrical prescribed performance constraints. Therefore, compared with the traditional funnel control scheme, the scheme proposed in the present invention is more general.

[0057] Step 2.2: Asymmetric Specified Performance Controller Design

[0058] The construction error is converted as follows:

[0059]

[0060] In the formula, g d As the reference signal, the present invention introduces the following second-order command filter:

[0061]

[0062] Where β ic (t) = u i1 (t) is the output of the second-order command filter, τ i-1 is the virtual control input, ρ>0 and δ∈(0,1) are design parameters, and u i1 (0) = τ i-1 (0),

[0063] In order to eliminate the filtering error, the compensation mechanism is constructed as follows:

[0064]

[0065] where ζ i To compensate the signal, the design is as follows:

[0066]

[0067] The design parameter p i >0,i=1,K,4,ζ i (0) = 0 is ζ i Initial condition, s1 = s1(t);

[0068] Remark 4. The command filter expressed by formula (15) not only overcomes the disadvantage of "complexity explosion" of virtual control in traditional backstepping control, but also reduces the chattering effect of the differentiator. However, since the error generated by the command filter affects the performance of the controlled system, the present invention designs a compensation signal expressed by formula (17). This compensation signal can not only eliminate the filter error u i1 -τ i-1 It not only reduces the influence of the sign function in the compensation signal, but also avoids the chattering effect caused by the sign function in the compensation signal that has been reported so far.

[0069] Combining equations (3), (8)-(14) and (17), we can calculate r in equation (16): i Taking the derivative with respect to time t, we get:

[0070]

[0071] Where, μ = μ(t);

[0072] Defining the estimation error for:

[0073]

[0074] In the formula, For i estimates;

[0075] Step 2.2.1, construct the Lyapunov function V1 as:

[0076]

[0077] Taking the derivative of V1 with respect to time t, we can obtain:

[0078]

[0079] Combining equation (17), equation (18) and equation (21), we get:

[0080]

[0081] According to Young's inequality, we can get:

[0082]

[0083] Substituting formula (23) into formula (22), we get:

[0084]

[0085] Design the virtual control input τ1 as:

[0086]

[0087] In the formula, the design parameter m1>0;

[0088] Substituting formula (25) into formula (24), we get:

[0089]

[0090] Step 2.2.2, construct the Lyapunov function V2 as follows:

[0091]

[0092] In the formula, the design parameter a1>0;

[0093] Taking the derivative of V2 with respect to time t, we can obtain:

[0094]

[0095] Substituting equation (17) and equation (18) into equation (28), we get:

[0096]

[0097] The following disturbance observer is constructed to estimate the disturbance △b:

[0098]

[0099] Among them, μ1>0 and μ2>0 are observer coefficients, is the Lipschitz constant, b and is the intermediate variable, and are △b, x2 and An estimated value of

[0100] Note 5. Because ∆b is bounded and the parameter and It can be approximated as △b,x2 and real value, so by choosing appropriate parameters μ1, μ2 and x2,△b and The estimation error of is reduced to the neighborhood near the zero point. In addition, since the observer introduced in formula (30) can effectively solve the unknown disturbance and control chatter problems, the addition of this observer can significantly improve the robustness and convergence of the chaotic PMSM system.

[0101] consider Where the constant h1>0, then equation (29) can be written as:

[0102]

[0103] According to Young's inequality, we can get:

[0104]

[0105] Substituting formula (32) into formula (31), we can obtain:

[0106]

[0107] Define the unknown function Z1(W1) as:

[0108] Z1(W1)=-χ1x2-T L (34)

[0109] Where W1 = [x2] T ;

[0110] According to Lemma 2, Z1(W1) can be obtained by using RBFNN To approximate, its expression is:

[0111]

[0112] Substituting equation (33) and equation (34) into equation (35), we can obtain:

[0113]

[0114] Combining with Young's inequality, we can get:

[0115]

[0116] Where ν2 is a positive constant, Q1(W1)=Q1;

[0117] Substituting formula (37) into formula (36), we can obtain:

[0118]

[0119] Design of virtual control input τ2 and adaptive law for:

[0120]

[0121] Where m2 and α1 are positive constants;

[0122] Note 6. In the traditional neural network adaptive control method, as the number of nodes and the complexity of the approximate function increase, the computational response capability of the control system weakens. To solve this problem, the unknown constant matrix Combined with the minimum learning parameter mechanism expressed by formula (6), an adaptive adjustment mechanism is proposed It is converted into the calculation of the unknown scalar parameter Ξ1. In the subsequent adaptive scheme design, a similar method is used, which greatly reduces the required calculation amount compared with the traditional scheme.

[0123] Substituting formula (39) into formula (38), we can obtain:

[0124]

[0125] Step 2.2.3, select Lyapunov function V3 as:

[0126]

[0127] In the formula, a2>0 is the design parameter;

[0128] Taking the derivative of V3 in equation (41) with respect to time t, we can obtain:

[0129]

[0130] Substituting equation (17) and equation (18) into equation (42), we can obtain:

[0131]

[0132] According to Young's inequality, we can get:

[0133]

[0134] Substituting formula (44) into formula (43), we can obtain:

[0135]

[0136] Define the unknown nonlinear function Z2(W2) as:

[0137] Z2(W2)=-x3-x2x4+χ2x2 (46)

[0138] Where W2=[x2,x3,x4] T ;

[0139] According to Lemma 2, Z2(W2) can be obtained by using RBFNN To estimate, its expression is:

[0140]

[0141] Then formula (45) can be written as:

[0142]

[0143] Combining with Young's inequality, we can get:

[0144]

[0145] Where ν3>0 is a design parameter, Q2(W2)=Q2;

[0146] Substituting formula (49) into formula (48), we can obtain:

[0147]

[0148] Design actual control input u q and Adaptive Law for:

[0149]

[0150] Where m3 and α2 are positive constants.

[0151] Substituting formula (51) into formula (50), we can obtain:

[0152]

[0153] Step 2.2.4, define the Lyapunov function V4 as:

[0154]

[0155] Where, constant a3>0;

[0156] Taking the derivative of V4 with respect to time t, we can obtain:

[0157]

[0158] Substituting equation (17) and equation (18) into equation (54), we can obtain:

[0159]

[0160] Define the unknown function Z3(W3) as:

[0161] Z3(W3)=-x4+x2x3 (56)

[0162] Where W3=[x2,x3,x4] T ;

[0163] According to Lemma 2, Z3(W3) can be obtained by using RBFNN To estimate, its expression is:

[0164]

[0165] Then formula (55) can be written as:

[0166]

[0167] Combining with Young's inequality, we can get:

[0168]

[0169] Where ν4 is a positive constant, Q3(W3)=Q3;

[0170] Substituting formula (59) into formula (58), we can obtain:

[0171]

[0172] Design the actual controller u d and the law of adaptation for:

[0173]

[0174] Where m4 and α3 are positive constants;

[0175] Substituting formula (61) into formula (60), we can obtain:

[0176]

[0177] Compared with the prior art, the effects of the present invention are as follows:

[0178] 1) A funnel-like prescribed performance backstepping control scheme is designed for a chaotic PMSM system. The scheme relies on constructing a transfer function that imposes asymmetric performance constraints on transient and steady-state behavior. In the transfer function, a funnel-like boundary function is designed to eliminate the dependence on the initial output error. This overcomes the shortcomings of conventional funnel control and PPC where the initial tracking error must be bounded in an explicit manner.

[0179] 2) A command filter is introduced to estimate the derivative of the virtual controller, thereby solving the "complexity explosion" problem. At the same time, the present invention also develops an error compensation system to eliminate the filtering error and improve the control accuracy of the control system.

[0180] 3) A fast-converging nonlinear DOB is introduced to effectively estimate external disturbances and an RBFNN with minimum learning parameters is used to deal with system uncertainties, which enhances the robustness and rapid response capability of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0181] Figure 1 Chaotic behavior diagrams of PMSM system: (a) phase diagram of x2-x3-x4, (b) phase diagram of x1-x2-x3, (c) phase diagram of x1-x2-x4, (d) phase diagram of x1-x3-x4;

[0182] Figure 2 It is the control structure diagram of PMSM system;

[0183] Figure 3 is the expected trajectory g d and output signal x1 graph;

[0184] Figure 4 It is the output error curve e1 graph;

[0185] Figure 5 is the dq axis voltage curve u q and u d picture;

[0186] Figure 6 is the dq axis current curve i qand i d picture;

[0187] Figure 7 g in scene 1 d =The trajectory tracking response diagram of sin(πt / 4);

[0188] Figure 8 g in scene 1 d = tracking error graph of sin(πt / 4);

[0189] Fig. 9 g in scene 2 d =The trajectory tracking response diagram of sin(πt / 4);

[0190] Fig.10 g in scene 2 d = tracking error graph of sin(πt / 4);

[0191] Fig.11 g in scene 1 d =3sin(πt / 16) trajectory tracking response diagram;

[0192] Fig.12 g in scene 1 d =3sin(πt / 16) tracking error diagram;

[0193] Fig.13 g in scene 2 d =3sin(πt / 16) trajectory tracking response diagram;

[0194] Fig.14 g in scene 2 d =3sin(πt / 16) tracking error diagram. DETAILED DESCRIPTION

[0195] The present invention is further described below in conjunction with specific embodiments.

[0196] Example 1: Figure 1-14 As shown, an adaptive specified performance PMSM system control method based on disturbance observer is used for a chaotic PMSM system with asymmetric performance constraints. The method includes the following steps

[0197] Step 1: Construct a mathematical model of the PMSM system:

[0198] In the (dq) coordinate system, the mathematical model of the PMSM system is expressed as:

[0199]

[0200] Where θ is the rotor position, ω is the angular velocity, i d and iq is the dq axis current, u d and u q is the dq axis voltage, χ1 and χ2 are unknown positive parameters, T L is the load torque.

[0201] Define x1=θ,x2=ω,x3=i q and x4=i d , then (1) can be simplified to:

[0202]

[0203] Where △b is the total interference consisting of measurement noise and unmodeled dynamics.

[0204] In order to facilitate controller design, the following preliminary knowledge is given:

[0205] Set 1: Set system status i=1,...,4 is measurable;

[0206] Assume 2: Assume the reference signal g d and its derivatives is a continuous bounded function;

[0207] Assumption 3: Assume there exists a positive constant φ M So that |φ i |≤φ M ,i=1,...,3,where φ i Represents the estimation error.

[0208] Theorem 1: According to Young's inequality, for Then we have:

[0209]

[0210] In the formula and is a constant, and

[0211] Theorem 2: If a continuous function Z(W):R n →R is the target function defined on a compact set, then there exists a radial basis function neural network (RBFNN) such that:

[0212]

[0213] In the formula, the neural network input vector Ω W is a compact set, n is the input dimension of the neural network, is the neural network weight vector, l≥2 is the number of neuron nodes. Q(W)=[q1(W),q2(W),K,q l (W)]∈R n is the basis function vector, q i (W) is a radial basis neuron. φ(W) is the approximation error. Assume that the approximation error φ(W) satisfies |φ(W)|<φ p , where the constant φ p >0. The present invention selects the Gaussian kernel function as the basis function, which can be expressed as:

[0214]

[0215] Where d i =[d1,…,d l ] T is the basis function q i (W) center, θ i is the basis function q i (W) The width of the receptive field.

[0216] In order to increase the x-learning ability of RBFNN, the minimum learning parameter is defined as:

[0217]

[0218] In the formula, i is the unknown variable, ||·|| is the 2-norm of ·.

[0219] Step 2: Design of neural specified performance backstepping controller

[0220] Step 2.1: Novel asymmetric funnel-shaped prescribed performance

[0221] Assume that the tracking error e1 converges to the following asymmetric region:

[0222]

[0223] In the formula is the performance function, e1=x1-g d , and is a constant, and

[0224] Design performance function for:

[0225]

[0226] Where κ1, κ2 and σ are positive constants, and csch(·) represents the hyperbolic cosecant function of (·).

[0227] Remark 2. Since e1(0) is always bounded in real systems, and Therefore, we can choose σ small enough so that This is a relaxation of the requirement for accurate initial error under the original system conditions. This means that the main difference between formula (9) and the traditional solution is that it does not need to obtain the accurate initial output error of the system. Therefore, the funnel boundary function formed by formula (8) is More suitable for actual situation.

[0228] right Taking the derivative with respect to time t, we get

[0229]

[0230] Where coth(·) is the hyperbolic cotangent function of (·).

[0231] According to formula (8) and formula (9), we can know that: ① and is bounded; ③ represents the lower limit of transient overshoot of e1(t), Indicates the upper limit; ④ represents the minimum value of the steady-state tracking error e1(t), Represents the maximum value of the steady-state tracking error e1(t).

[0232] Define the constraint boundary conversion function as:

[0233]

[0234] To ensure that the tracking error e1(t) satisfies the constraint condition (7), the conversion function Υ1(t) is constructed as:

[0235]

[0236] From formula (10) and formula (11), it can be seen that the boundedness of functions S1(t) and Υ1(t) depends entirely on whether the error e1(t) and the boundary μ(t) are bounded. For every initial value of , when t∈[0,+∞), if Υ1(t) is bounded, then the boundedness of S1(t) and the constraints and boundedness of e1(t) can be guaranteed.

[0237] Taking the derivative of Υ1(t) with respect to time t, we can obtain:

[0238]

[0239] Taking the derivative of S1 and μ with respect to time t, we can obtain:

[0240]

[0241] According to (10)-(13), when t∈[0,+∞), if Υ1 is bounded, then and s1 are also bounded.

[0242] Remark 3. According to formula (11), we can conclude that Υ1∈R. From the expressions of Υ1(t), S1(t) and μ(t), we can see that we only need to adjust the parameters κ1, κ2, σ, the tracking error e1(t) can satisfy inequality (7) and avoid measuring the initial error exactly within the funnel boundary. and There is no direct relationship between them, which indicates that the funnel-shaped function Y1(t) given by formula (10) can achieve symmetrical and asymmetrical prescribed performance. Therefore, compared with the traditional funnel control scheme, the scheme proposed in the present invention is more universal.

[0243] Step 2.2: Asymmetric Specified Performance Controller Design

[0244] The construction error is converted as follows:

[0245]

[0246] The present invention introduces the following second-order command filter:

[0247]

[0248] Where β ic (t) = u i1 (t) is the output of the second-order command filter, τ i-1 is the virtual control input, ρ>0 and δ∈(0,1) are design parameters, and u i1 (0) = τ i-1 (0),

[0249] In order to eliminate the filtering error, the compensation mechanism is constructed as follows:

[0250]

[0251] where ζ i To compensate the signal, the design is as follows:

[0252]

[0253] The design parameter p i >0,i=1,K,4,ζ i (0) = 0 is ζ i initial conditions.

[0254] Remark 4. The command filter of formula (15) not only overcomes the disadvantage of "complexity explosion" in previous research results, but also reduces the chattering effect of the differentiator. However, due to the error generated by the command filter, it is difficult for the controlled system to obtain excellent performance. The present invention also designs a compensation signal to eliminate the filter error u i1 -τ i-1 And the compensation signal formula (17) also avoids the chattering effect caused by the ordinary sign function.

[0255] Combining (3), (8)-(14) and (17), we can calculate r in (16) i Taking the derivative about time t, we get:

[0256]

[0257] Defining the estimation error for:

[0258]

[0259] In the formula For i Estimates.

[0260] Step 2.2.1, construct the Lyapunov function V1 as:

[0261]

[0262] Taking the derivative of V1 with respect to time t, we can obtain:

[0263]

[0264] Combining (17), (18) and (21), we can obtain:

[0265]

[0266] According to Young's inequality, we can get:

[0267]

[0268] Substituting (23) into (22), we can obtain:

[0269]

[0270] Design the virtual control input τ1 as:

[0271]

[0272] In the formula, the design parameter m1>0.

[0273] Substituting (25) into (24), we can obtain:

[0274]

[0275] Step 2.2.2, construct the Lyapunov function V2 as follows:

[0276]

[0277] In the formula, the design parameter a1>0;

[0278] Taking the derivative of V2 with respect to time t, we can obtain:

[0279]

[0280] Substituting (17) and (18) into (28), we obtain:

[0281]

[0282] The following disturbance observer is constructed to estimate the disturbance △b:

[0283]

[0284] Where μ1>0 and μ2>0 are observer coefficients, is the Lipschitz constant, b and B are intermediate variables, and are △b,x2 and The estimated value of .

[0285] Remark 5. Because ∆b is bounded and the parameter and can be approximated as △b,x2 and So we can conclude that by choosing appropriate parameters μ1, μ2 and The estimation errors of x2, △b and B can be reduced to the neighborhood near the zero point. In addition, the observer introduced in formula (30) can effectively solve the unknown disturbance and control chatter problems. Therefore, adding an observer with appropriate parameters can significantly improve the robustness and convergence of the chaotic PMSM system.

[0286] consider Where the constant h1>0, then (29) can be written as:

[0287]

[0288] According to Young's inequality, we can get:

[0289]

[0290] Substituting (32) into (31), we can obtain:

[0291]

[0292] Define the unknown function Z1(W1) as:

[0293] Z1(W1)=-χ1x2-T L (34)

[0294] Where W1 = [x2] T .

[0295] According to Lemma 2, Z1(W1) can be obtained by using RBFNN To approximate, its expression is:

[0296]

[0297] Substituting (33) and (34) into (35), we obtain:

[0298]

[0299] Combining with Young's inequality, we can get:

[0300]

[0301] Where ν2 is a positive constant;

[0302] Substituting (37) into (36), we can obtain:

[0303]

[0304] Design of virtual control input τ2 and adaptive law for:

[0305]

[0306] Where m2 and α1 are positive constants.

[0307] Note 6. If the traditional neural network adaptive scheme is used to design the control method, an adaptive algorithm must be developed to calculate the target weight matrix. However, as the number of neural network nodes and the complexity of the approximate function increase, resource-intensive calculations must be performed. To solve this problem, the unknown constant matrix Combined with the minimum learning parameter mechanism formula (6), it is transformed into the unknown scalar parameter Ξ1. Then the adaptive mechanism in formula (39) It is used to estimate the value of Ξ1. In the subsequent adaptive scheme design, only the parameter estimation method is used, and the ideal constant weight matrix is ​​not estimated. Therefore, the present invention greatly reduces the required calculation amount compared with the traditional scheme.

[0308] Substituting (39) into (38), we can obtain:

[0309]

[0310] Step 2.2.3, select Lyapunov function V3 as:

[0311]

[0312] In the formula, a2>0 is the design parameter.

[0313] Taking the derivative of V3 in (41) with respect to time t, we can obtain:

[0314]

[0315] Substituting (17) and (18) into (42), we obtain:

[0316]

[0317] According to Young's inequality, we can get:

[0318]

[0319] Substituting (44) into (43), we can obtain:

[0320]

[0321] Define the unknown nonlinear function Z2(W2) as:

[0322] Z2(W2)=-x3-x2x4+χ2x2 (46)

[0323] Where W2=[x2,x3,x4] T .

[0324] According to Lemma 2, Z2(W2) can be obtained by using RBFNN To estimate, its expression is:

[0325]

[0326] Then (45) can be written as:

[0327]

[0328] Combining with Young's inequality, we can get:

[0329]

[0330] Where ν3>0 is a design parameter.

[0331] Substituting (49) into (48), we can obtain:

[0332]

[0333] Design actual control input u q and Adaptive Law for:

[0334]

[0335] Where m3 and α2 are positive constants.

[0336] Substituting (51) into (50), we can obtain:

[0337]

[0338] Step 2.2.4, define the Lyapunov function V4 as:

[0339]

[0340] In the formula, constant a3>0.

[0341] Taking the derivative of V4 with respect to time t, we can obtain:

[0342]

[0343] Substituting (17) and (18) into (54), we obtain:

[0344]

[0345] Define the unknown function Z3(W3) as:

[0346] Z3(W3)=-x4+x2x3 (56)

[0347] Where W3=[x2,x3,x4] T .

[0348] According to Lemma 2, Z3(W3) can be obtained by using RBFNN To estimate, its expression is:

[0349]

[0350] Then (55) can be written as:

[0351]

[0352] Combining with Young's inequality, we can get:

[0353]

[0354] Where ν4 is a normal number.

[0355] Substituting (59) into (58), we can obtain:

[0356]

[0357] Design the actual controller u d and the law of adaptation for:

[0358]

[0359] Where m4 and α3 are positive constants;

[0360] Substituting (61) into (60), we can obtain:

[0361]

[0362] So far, the design of the neural adaptive command filter backstepping funnel control scheme based on disturbance observer has been completed, which is suitable for chaotic PMSM systems with asymmetric performance constraints. The specific block diagram is as follows Figure 2 shown.

[0363] The present invention also includes a stability analysis of the control method:

[0364] For a given A compact set is defined as:

[0365]

[0366] Where P represents a positive real number parameter.

[0367] Theorem 2. Under assumptions 1-3, for a chaotic PMSM system with asymmetric performance constraints, the proposed scheme consists of controllers τ1, τ2, u q ,u d and Adaptive Law If the initial conditions satisfy the compact set Ω i (i=1,K,4) and Then the set control objectives can be achieved.

[0368] Proof: The Lyapunov function of the selected population is:

[0369]

[0370] According to (62), the derivative of V with respect to time t is:

[0371]

[0372] Combining (19) and (3), we can obtain:

[0373]

[0374] Substituting (66) into (65), we can obtain:

[0375]

[0376] For β 2c and β 3c , there exist ρ and δ∈(0,1) such that |β 2c -τ1|≤Λ and |β 3c -τ2|≤Λ and constant Λ>0, we can get:

[0377]

[0378] Then (67) can be written as:

[0379]

[0380] In the formula, c1=min{2m1,K,2m4,p1-2,p2-3,p3-2,p4-1,α1,α2,α3}∈R + ,

[0381] Integrating (69) in [0, t] yields:

[0382]

[0383] According to (70), we can see that V is asymptotically convergent;

[0384] Combining formula (64), we can get:

[0385]

[0386] From (71), we can know that r i and i ,i=1,K,4 is bounded. According to formula (19), we can deduce and Ξ1, Ξ2, Ξ3 are all bounded. Combining r1=Υ1-ζ1 in (16), we can get:

[0387]

[0388] According to the definition of γ1 in (10), we can deduce:

[0389]

[0390] In the formula

[0391] Combining formulas We can get:

[0392]

[0393] Formula (72) shows that Υ1 is bounded. Based on a similar approach, we can derive e1(t), S1(t), μ(t), μ&(t), and s1 are bounded and ensure the performance constraint of e1(t). Since e1=x1-g d , and g d is bounded, then x1 is bounded. Combining Setting 2 and Formula (25), we can get τ1 is also bounded. Similarly, we can get x2,x3,x4,τ2,u q ,u d is bounded. Finally, it can be obtained that all signals in the closed-loop system are bounded, and e1(t) satisfies the constraints in formula (7). At this point, the stability analysis is proven.

[0394] Note 7. The output tracking error of the system is an intuitive performance indicator of the control system, which can be adjusted by selecting the appropriate Θ0 and c1 to adjust its size.

[0395] In order to prove the effect of the present invention, the following simulation is carried out, and the simulation results and analysis are as follows:

[0396] 1) Simulation settings

[0397] The simulation experiments were conducted on an AMD Ryzen5 3600 6-core processor (with Radeon Graphics 3.60 GHz), 8.00 GB memory, Microsoft Windows 11 OS, and MATLAB-Simulink R2022a platform. All simulation experiments were conducted based on the S-function, and the sampling period ts = [-1, 0] recommended by the S-function module was selected. The simulation time for the performance verification test and the comparative experiment was set to 0 to 60 seconds.

[0398] 2) Performance Analysis

[0399] The following three scenarios are considered in this section to test the effectiveness of the developed control algorithm:

[0400] Scenario 1: The control system is not affected by disturbances, that is, χ1=5.5, χ2=21, △b=0.

[0401] Scenario 2: Based on Scenario 1, the system is subjected to an external disturbance △b=0.02x2x3 sin(2t).

[0402] Scenario 3: Based on Scenario 2, the parameters χ1 and χ2 are changed to χ1=7.5 and χ2=17.5 respectively.

[0403] For the controller (NABAPPC) in the present invention, the initial conditions are set to x1(0) = 0.15, x2(0) = x3(0) = x4(0) = 0.01. The design parameters are selected as a1 = 5, a2 = 1, a3 = 1, b j =1,α k1 =3.8, α k2 =0.5m j1 =10,m j2 =80, (j=1,...,4,k=1,2,3);p 11 =1,p 12 =0.1,p 21 =p 22 =1,p 31 =p 32 =0.5, p 41 =0.1,p 42 =1,v i =0.01(i=2,3,4). The filter coefficients are δ=0.5, ρ=0.9, the number of RBFNN nodes used is 9, the bandwidth is 13, and the center is [-8 8].

[0404] Figure 3-6 The main simulation results are given. Figure 3 It shows that the output signal x1 can accurately track the reference trajectory signal g d and the constraints are not violated. Figure 4 It shows that the system tracking error can converge to the constrained range in a very short time and remain within the preset performance constraint range. This also proves that the proposed control method successfully suppresses chaotic oscillations and achieves accurate tracking of the reference trajectory signal. Figure 5 and Figure 6 The actual controller u is shown respectively q ,u d and state i q ,i d By comparing the curve trajectories under different bounded disturbances and chaotic coefficients in scenarios 1-3, it can be seen that the controller designed by the present invention has good robustness to the uncertain parameter changes and bounded disturbance changes in the PMSM system.

[0405] Solution comparison:

[0406] In order to demonstrate the effectiveness and superiority of the proposed method, the implementation of the dynamic surface controller (NDSC) based on nonlinear mapping feedback technology described in the literature "Gao, S., Dong, H., Ning, B., Tang, T., Li, Y.: Chaotic PMSM dynamic surface control feedback technology based on nonlinear mapping (using neural approximation and parameter identification). IET Control Theory and Applications. 12, 819–827 (2018)" is compared under scenarios 1 and 2. The controller is briefly introduced as follows:

[0407]

[0408] In the formula, and d Nj ,j=1,...,4 is a positive constant, and are the constant coefficient and output state variable of the filter respectively. for The estimated neural weight vector of is the adaptive variable, Γ N and Λ are diagonal constant matrices, η N and λ N is a constant, nonlinear function and As described in the literature “Gao, S., Dong, H., Ning, B., Tang, T., Li, Y.: Chaotic PMSM dynamic surface control feedback technology based on nonlinear mapping (using neural approximation and parameter identification). IET Control Theory and Applications. 12, 819–827 (2018)”, the other parameters in formula (75) are consistent with NABAPPC.

[0409] The design parameters are selected as d N1 =3,d N2 =d N3 =16,d N4 =6, ψ1=ψ2=0.1, η N =4 and λ N =0.1,Γ N =200diagI, The initial condition is The remaining parameters and initial conditions are the same as those of the proposed NABAPPC method.

[0410] To fairly evaluate the robustness and generality of the two control methods, each parameter is calculated by using the reference signal g d =sin(πt / 4) is selected after system trial and error. In addition, the present invention also uses the following three quantitative indicators to quantitatively compare the trajectory tracking capabilities.

[0411] 1) Absolute value error integral

[0412]

[0413] 2) Squared error integral

[0414]

[0415] 3) Time and absolute error integral

[0416]

[0417] Figure 7-10 The reference trajectory of the NABAPPC scheme and the NDSC scheme under scenario 1-2 is given as g d = sin(πt / 4) trajectory tracking and error response curve. Figure 7-10 It can be seen that the tracking error of NDSC violates the preset performance constraints, while the tracking error of NABAPPC remains within the preset performance constraints, which shows that the tracking ability of NABAPPC is better than that of NDSC. At the same time, the comparative analysis of the quantitative indicators of the control performance of the two control algorithms in Table 2 shows that the Π ITAE , Π IAE and Π ISE The indicators are all smaller than those of NDSC. Therefore, the NABAPPC scheme proposed in the present invention has better steady-state capability and is superior to the existing NDSC scheme in terms of trajectory tracking performance.

[0418] In order to further test the performance of the controller under different reference trajectories, the reference trajectory g is selected. d =3sin(πt / 16). Figure 11-14 The experimental results are shown by Figure 11-14 It can be seen that the NDSC scheme has larger state fluctuation and tracking error than the NABAPPC scheme. The results in Table 3 show that the Π IAE , Π ISE and Π ITAE The indicators are all at a smaller level, indicating that the NABAPPC scheme is more robust and general.

[0419] Table 2 gd =Sin(πt / 4) performance index comparison results

[0420]

[0421] Table 3 gd =3sin(πt / 16) performance index comparison results

[0422]

[0423] The present invention aims at a chaotic PMSM system with asymmetric performance constraints and explores the design of a neural adaptive command filter backstepping funnel controller based on a disturbance observer. First, without relying on the initial output error, an effective funnel-shaped performance function is designed to solve the asymmetric performance constraints imposed on the system. This design ensures the transient and steady-state performance of the asymmetric funnel-shaped constraint control, and overcomes the defect that the initial tracking error in ordinary funnel control and PPC must be accurately kept within the boundary. Secondly, the "complexity explosion" problem in classical backstepping control is successfully solved by applying the command filter, and the filtering error caused by the command filter is effectively eliminated by constructing an appropriate compensation system. Third, the robustness and control accuracy of the control system are ensured by designing a fast-converging DOB, and the minimum learning parameters of the neural network are greatly reduced. Finally, by applying the Lyapunov function theory and backstepping technology, a virtual and actual controller with adaptive law is constructed to ensure system stability.

[0424] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. An adaptive specified performance PMSM system control method based on disturbance observer, used for a chaotic PMSM system with asymmetric performance constraints, characterized by: The method comprises the following steps: Step 1: Construct a mathematical model of the PMSM system: In the (dq) coordinate system, the mathematical model of PMSM is expressed as: Where θ is the rotor position, ω is the angular velocity, and i d and i q are the currents of the d-axis and q-axis, u d and u q are the voltages of the d-axis and q-axis respectively, χ1 and χ2 are unknown positive parameters, T L is the load torque; Define x1=θ,x2=ω,x3=i q and x4=i d , then formula (1) is simplified to: Where Δb is the total disturbance consisting of measurement noise and unmodeled dynamics; Set 1: Set system status It is measurable; Assume 2: Assume the reference signal g d and its derivatives is a continuous bounded function; Assumption 3: Assume there exists a positive constant φ M So that |φ i |≤φ M ,i=1,...,3, where φ i represents the estimation error; Theorem 1: According to Young's inequality, for Then we have: In the formula, η, and is a constant, and η>0, Theorem 2: If a continuous function Z(W):R n →R is the objective function defined on a compact set, then there exists a radial basis function neural network such that: In the formula, the neural network input vector Ω W is a compact set, n is the input dimension of the neural network, is the neural network weight vector, l≥2 is the number of neuron nodes, Q(W)=[q1(W),q2(W),...,q l (W)]∈R n is the basis function vector, q i (W) is the radial basis neuron, φ(W) is the approximation error, and the approximation error φ(W) satisfies φ(W)|<φ p , where the constant φ p > 0, select the Gaussian kernel function as the basis function, which is expressed as: Where, d i =[d1,...,d l ] T is the basis function q i (W) center, is the basis function q i (W) The width of the receptive field; Define the minimum learning parameter as: In the formula, i is the unknown variable, ||·|| is the 2-norm of ·; Step 2: Design of adaptive specified performance backstepping controller Step 2.1: Design with asymmetric specified performance constraints Assume that the tracking error e1(t) converges to the following asymmetric region: In the formula, is the performance function, e1(t)=e1=x1-g d , and is a constant, and Design performance function for: Where κ1, κ2 and σ are positive constants, csch(·) represents the hyperbolic cosecant function of (·); right Taking the derivative with respect to time t, we get Where coth(·) is the hyperbolic cotangent function of (·); Define the constraint boundary conversion function as: Where S1(t) and μ(t) both represent boundary constraint conversion functions, and the constructed conversion function Υ1(t) is: Taking the derivative of Υ1(t) with respect to time t, we get: Taking the derivative of S1(t) and μ(t) with respect to time t, we get: According to equations (10) to (13), when t∈[0,+∞), if Υ1(t) is bounded, then and s1(t) are also bounded; Step 2.2: Asymmetric Specified Performance Controller Design The construction error is converted as follows: In the formula, g d As the reference signal, the following second-order command filter is introduced: In the formula, β ic (t) = u i1 (t) is the output of the second-order command filter, τ i-1 is the virtual control input, ρ>0 and δ∈(0,1) are design parameters, and u i1 (0) = τ i-1 (0), The compensation mechanism is constructed as follows: Among them, i To compensate the signal, Υ1=Υ1(t) is designed as follows: In the formula, the design parameter p i >0,i=1,...,4,ζ i (0) = 0 is ζ i Initial condition, s1 = s1(t); Combining equations (3), (8)-(14) and (17), we can calculate r in equation (16): i Taking the derivative with respect to time t, we get: Where, μ = μ(t); Defining the estimation error for: In the formula, For i estimates; Step 2.2.1, construct the Lyapunov function V1 as: Taking the derivative of V1 with respect to time t, we get: Combining equation (17), equation (18) and equation (21), we get: According to Young's inequality, we get: Substituting formula (23) into formula (22), we get: Design the virtual control input τ1 as: In the formula, the design parameter m1>0; Substituting formula (25) into formula (24), we get: Step 2.2.2, construct the Lyapunov function V2 as follows: In the formula, the design parameter a1>0; Taking the derivative of V2 with respect to time t, we get: Substituting equation (17) and equation (18) into equation (28), we get: The following disturbance observer is constructed to estimate the disturbance Δb: Among them, μ1>0 and μ2>0 are observer coefficients, is the Lipschitz constant, b and is the intermediate variable, and are Δb, x2 and An estimated value of consider Where, constant h1>0, then equation (29) is written as: According to Young's inequality, we get: Substituting formula (32) into formula (31), we get: Define the unknown function Z1(W1) as: Z1(W1)=-χ1x2-T L (34) Where W1 = [x2] T ; According to Lemma 2, Z1(W1) is a radial basis function neural network. To approximate, its expression is: Substituting equation (33) and equation (34) into equation (35), we get: Combining Young's inequality, we get: In the formula, ν2 is a positive constant; Substituting formula (37) into formula (36), we get: Design of virtual control input τ2 and adaptive law for: Where m2 and α1 are positive constants; Substituting formula (39) into formula (38), we get: Step 2.2.3, select Lyapunov function V3 as: In the formula, a2>0 is the design parameter; Taking the derivative of V3 in equation (41) with respect to time t, we get: Substituting equation (17) and equation (18) into equation (42), we get: According to Young's inequality, we get: Substituting formula (44) into formula (43), we get: Define the unknown nonlinear function Z2(W2) as: Z2(W2)=-x3-x2x4+χ2x2 (46) Where W2=[x2,x3,x4] T ; According to Lemma 2, Z2(W2) is a radial basis function neural network. To estimate, its expression is: Then equation (45) is written as: Combining Young's inequality, we get: In the formula, ν3>0 is a design parameter; Substituting formula (49) into formula (48), we get: Design actual control input u q and adaptive law for: In the formula, m3 and α2 are positive constants; Substituting formula (51) into formula (50), we get: Step 2.2.4, define the Lyapunov function V4 as: Wherein, constant a3>0; Taking the derivative of V4 with respect to time t, we get: Substituting equation (17) and equation (18) into equation (54), we get: Define the unknown function Z3(W3) as: Z3(W3)=-x4+x2x3 (56) Where W3=[x2,x3,x4] T ; According to Lemma 2, Z3(W3) uses radial basis function neural network To estimate, its expression is: Then equation (55) is written as: Combining Young's inequality, we get: In the formula, ν4 is a positive constant; Substituting formula (59) into formula (58), we get: Design the actual controller u d and the law of adaptation for: In the formula, m4 and α3 are positive constants; Substituting formula (61) into formula (60), we get:

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