CN113064347A - PMSM chaotic system self-adaptive control method considering asymmetric input and output constraints - Google Patents

PMSM chaotic system self-adaptive control method considering asymmetric input and output constraints Download PDF

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CN113064347A
CN113064347A CN202110276966.8A CN202110276966A CN113064347A CN 113064347 A CN113064347 A CN 113064347A CN 202110276966 A CN202110276966 A CN 202110276966A CN 113064347 A CN113064347 A CN 113064347A
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张钧星
李少波
王时龙
李梦晗
罗绍华
王中禹
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Abstract

The invention discloses a permanent magnet synchronous motor chaotic system neural network self-adaptive control method considering asymmetric input and output constraints, which comprises the following steps: 1) establishing a dynamic model of the PMSM system; 2) setting a control object; 3) a neural network self-adaptive controller is established, and the problems of complexity explosion and unknown control direction are solved by respectively utilizing a Nussbaum type function and a tracking differentiator. In addition, the boundedness of the stability of the designed system can be ensured on the premise of not exceeding the input and output constraint boundary. Finally, the effectiveness of the scheme is proved through a simulation test; embedding a conversion error and a new boundary in a logarithmic barrier Lyapunov function, and providing a unified barrier Lyapunov function to avoid switching type nonlinearity related to a segmented barrier Lyapunov function and ensure that an asymmetric output constraint condition is met.

Description

PMSM chaotic system self-adaptive control method considering asymmetric input and output constraints
Technical Field
The invention relates to a PMSM chaotic system self-adaptive control method considering asymmetric input and output constraints, and belongs to the technical field of permanent magnet synchronous motor control methods.
Background
A Permanent Magnet Synchronous Motor (PMSM) having high reliability and high efficiency is increasingly used in various industrial products such as vehicles, robots and airplanes as an effective power source with the development of manufacturing industry. However, since the PMSM may have a chaotic behavior in which system parameters fall within a certain range, and chaotic oscillation may destroy or even crash the system performance, it is important to design a reasonable controller to ensure stable system operation. In the past decades, the design problem of the tracking controller of the PMSM chaotic system has been widely studied in the control field. Adaptive inversion control methods that employ fuzzy logic systems or neural networks to assess uncertainty are well known as excellent tools for solving such problems. By fusing a given performance barrier Lyapunov function and a tracking differentiator) into a conventional inversion controller, a high-precision controller for a PMSM chaotic system is developed. In the literature (Zhang Jun, Wang Timlong, Li Shao wave, Zhongpeng, Adaptive neurodynamic Surface Control of Chaotic PMSM systems with External Disturbances and Output constraints [ J ]. Recentr Adv.Electron.Electron.Eng. (Formerly Recentr Patents Electron.Eng.,2020,121(13). Z.Junxing, W.Shilong, L.Shaobo, and Z.Peng, "Adaptive Neural Dynamic Surface Control for the Chartic PMSM system with External Disturbances and connected Output," Recentr Adv. Electron.Eng. (Former Recentr Patents Electron. Engine, vol. 2020, by integrating the barrier Lyapunov function and Radial Basis Function Neural Network (RBFNN) into a conventional inversion controller, an adaptive output constraint stabilization scheme for a PMSM chaotic system is proposed, however, therefore, it is urgent to develop an effective strategy for ensuring the input/output constraints and apply the strategy to the control of the PMSM chaotic system.
In various nonlinear systems including PMSM chaotic systems, input saturation is generally considered as a general input constraint. Unfortunately, there is an insurmountable problem in the saturation nonlinearity described above, thereby limiting the performance of adaptive inversion control schemes. To overcome this problem, many smooth functions such as hyperbolic tangent functions and gaussian error functions are used to estimate the saturation nonlinearity. In order to solve the problem of asymmetric saturation nonlinearity of a multi-input multi-output nonlinear system, a segmented hyperbolic tangent function is introduced. Segmented Gaussian Error functions have been successfully used to solve the problem of asymmetric Saturation nonlinearity of spacecraft in the literature (Zhengwei, Sunlong, Xiehua. Surface vessels with Actuator Saturation and failure Error constraint LOS Path tracking [ J ]. IEEE Trans.Syst. Man, Cybern.Syst., 2018, 48 (10): 1794-. While the asymmetric Saturation nonlinearity problem has been addressed in the literature (Zhengwei, Sunlong, Xiehua. Surface vessels with actuator Saturation and failure Error bound LOS Path tracking [ J ]. IEEE Trans.Syst. Man, Cybern.Syst., 2018, 48 (10): 1794-1805.Z.Zheng, L.Sun, and L.Xie, "Error-Constrained LOS Path Following of Surface Vessel with actual results practical and facilities," IEEE Trans.Syst. Man, Cybern.Syst., vol.48, No.10, pp.1794-1805, 2018), the result is obtained ignoring the low accuracy due to the disparity between the calculation and constraint inputs. Based on this, the literature (congratulatory, durolin, never-honor. one class of Command filtering robust adaptive neural network control [ J ]. j.franklin instrument, 2018,355(15) that does not determine a strict feedback nonlinear system 7548-7569.g.zhu, j.du, and y.kao, "Command filtered robust adaptive NN control for a class of not of a simple structure-feedback nonlinear system input specification," J. Franklin instrument, vol.355, No.15, pp.7548-7569,2018) proposes adaptive neural control of a strict feedback nonlinear system with saturation of the executive by designing an auxiliary power system. However, the literature (congratulatory, duralumin, scouting. a class of Command filtering robust adaptive neural network control for uncertain stringent feedback nonlinear systems [ J ]. j.franklin instrument, 2018,355(15):7548-7569.g.zhu, j.du, and y.kao, "Command filtered robust adaptive NN control for a class of not acceptable real-time linear-feedback systems-input analysis," j.franklin instrument, vol.355, No.15, pp.7548-7569,2018) does not take into account design and analysis complexities caused by auxiliary power systems. In addition, it is worth noting that research results of controller design based on asymmetric input saturated PMSM chaotic system have been few so far. Therefore, the problem of asymmetric input saturation of the PMSM chaotic system is still an important subject to be researched.
Another important constraint of practical PMSM from system specification and safety considerations is to limit system output or tracking errors to some extent. For both types of constraints, many possible schemes in different non-linear systems have been extensively studied. It is well known that the various barrier Lyapunov functions are schemes that effectively limit the output constraints of the system. However, the barrier Lyapunov function described above is only applicable to handle constraints with equal upper and lower bounds, and cannot solve asymmetric constraints. In order to limit the system output within the asymmetric range, researchers propose a plurality of segmented barrier Lyapunov functions. In the literature (M.Deng, Li Ching, health. A Learning-Based Human-computer Cooperative Control method for Exoskeleton robots [ J ]. IEEE Trans. cybern, 2020,50(1):112-125.M.Deng, Z.Li, Y.Kang, C.L.P.Chen, and X.Chu, "A left-Based cognitive Control Scheme for an Exoskeleton Robot in Human-Robot Cooperative management," IEEE Trans. cybern, vol.50, No.1, pp.112-125,2020), admittance controllers Based on the Lyapunov function have been developed to Control the operation of robots. For non-rigid nonlinear systems with output constraints, the literature (camamamley, murmerd saro-base, murmerd muncht. Adaptive finite time mental control of non-rigid feedback systems with output constraints and unknown control directions and input nonlinearities [ J ]. inf.sci. (Ny).,2020,520:271-291.a. kamalamii, m.shahrokhi, and m.motif, "Adaptive fine-time neural of non-linear feedback systems sub to output control, un-linear control direction, and input non-linear" inf.sci. (Ny), vol.271-291, 2020 type by combining with ssbaum function to process the unknown control direction function, the problem of controlling the non-linear control based on the unknown control function has been studied 520. However, the aforementioned piecewise-obstacle Lyapunov function-transformed non-linearity may add an additional computational burden to the controller. In addition, the research on the tracking control design problem of the PMSM chaotic system under the asymmetric output constraint is less. Therefore, it is necessary to propose a unified barrier Lyapunov function to simplify the low complexity controller design and ensure the asymmetric output constraint of the PMSM chaotic system.
In addition to the above problems, another notable aspect in controller design is being investigated to further enhance the operational dynamics of PMSM by introducing an excellent intelligent approximator to identify unknown uncertainties. In most of the adaptive inversion control designs mentioned above, a fuzzy logic system or neural network as a general approximator is used to estimate the unknown uncertainty. In particular, the radial basis function neural network with arbitrary estimation capability is widely applied to the adaptive inversion control design of many practical systems, for example, PMSM in literature (J. Physics frontier, 2020,8:1-8.R.Luo, Y.Deng, and any of Y.Xie, "Neural Network synchronization Controller Design for Uncertain magnetic Synchronous Motor Drive Controller control Design for Uncertain magnetic Synchronous Motor Drive Controller Design for arithmetic logic magnetic Drive Systems video Command Filter," front.Phys, vol.8, No. June, pp.1-8,2020) and PMSM in literature (J.Yu, P.Shi, S.Member, W.Dong, B.Chen, and C.Lin. Permanent Magnet Synchronous Motor introduction [ J ]. 26(3):640-645 J.Yu, P.Ying, S.M. 2015, W.Dong, B.Chen, and C.Lin. Permanent Magnet Synchronous Motor introduction [ J ]. 26(3) in chemistry Systems, P.Yu, P.M.Y.M. 640, S.M.M.M.M.P.M.M., M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.M.. Although the aforementioned controllers based on radial basis function neural networks have yielded some excellent approximation results, their results are obtained through extensive neural network parameter design and extensive real-time calculations. In order to facilitate the application of the controller, the chebyshev neural network is commonly used in adaptive control design as a single-layer neural network designed by extending an input pattern by introducing a chebyshev polynomial basis function. The advantages of the controller based on the chebyshev neural network were further verified by comparing the controller performance with the radial basis function neural network based approach. In addition, it is worth noting that few researches are currently conducted to design a PMSM chaotic system based on a Chebyshev neural network controller. Therefore, Chebyshev neural networks were first chosen to approximate the unknown uncertainties generated in the controller design.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the self-adaptive control method of the PMSM chaotic system considering the asymmetric input and output constraints is provided to solve the problems in the prior art.
The technical scheme adopted by the invention is as follows: a PMSM chaotic system self-adaptive control method considering asymmetric input and output constraints comprises the following steps:
(1) establishing a PMSM system dynamic model:
in a rotating (d-q) coordinate system, the kinetic equation of the permanent magnet synchronous motor system is established as follows:
Figure BDA0002977037210000031
in the formula (I), the compound is shown in the specification,
Figure BDA0002977037210000032
and
Figure BDA0002977037210000033
representing the d-axis and q-axis currents,
Figure BDA0002977037210000034
and
Figure BDA0002977037210000035
representing the d-axis and q-axis voltages as system inputs, L,
Figure BDA0002977037210000036
R,
Figure BDA0002977037210000037
ψrb, J and npRespectively representing inductance, rotor angular velocity, stator resistance, load torque, flux linkage, viscous friction coefficient, rotor moment of inertia and magnetic pole pairs;
simplifying the formula (1), and selecting L as Ld＝LqDefinition of
Figure BDA0002977037210000038
And
Figure BDA0002977037210000039
np＝1，x1＝ω,x2＝iq，x3＝id,L＝Ld＝Lqand (3) obtaining a simplified dimensionless model of formula (1) by considering uncertain external interference and asymmetric input saturation:
Figure BDA00029770372100000310
in the formula (I), the compound is shown in the specification,
Figure BDA00029770372100000311
σ1＝BL/(JR)，σ2＝-npψr 2/(BR)，
Figure BDA00029770372100000312
and
Figure BDA00029770372100000313
Δii is 1,2,3 is uncertain external interference;
in the formula, x1Representing nominal angular velocity, x2Representing the q-axis current, x3Representing d-axis current, T time, TLDenotes a load, udDenotes the d-axis voltage, uqRepresenting the q-axis voltage, σ1And σ2Showing unknown ginsengAnd (4) counting.
Asymmetric input saturation is expressed as:
Figure BDA00029770372100000314
in the formula umaxAnd uminRepresenting the amplitude, v, of an asymmetrically saturated inputgAnd ugRespectively representing the input and output of an asymmetric saturated input;
(2) setting a control object:
(a) all variables in the PMSM chaotic system are bounded;
(b) output x1Following the desired signal yd；
(c) Does not violate control input constraints;
(d) output x1Is defined as
Figure BDA00029770372100000315
Setting 1: variable sigmaiI is 1,2 and δiI is unknown but bounded, i.e. 1,2,3
σim≤σi≤σiM,|δi|≤δM, (4)
In the formula, σim,σiMI is 1,2 and δM,(δM> 0) is a real number, δiIs the estimation error;
setting 2: there is a desired trajectory
Figure BDA0002977037210000041
And time derivative thereof
Figure BDA0002977037210000042
And
Figure BDA0002977037210000043
satisfy inequality
Figure BDA0002977037210000044
Wherein
Figure BDA0002977037210000045
And xi are positive real numbers;
setting a reference value 3: presence of real number ci> 0, e.g. | Δi|≤ci,i＝1,2,3；
Introduction 1: for the
Figure BDA0002977037210000046
Obtaining:
Figure BDA0002977037210000047
wherein p > 1, ξ > 0, q > 1 and (p-1) (q-1) ═ 1;
selecting a Chebyshev neural network to approximate an unknown uncertainty f generated in the controller design*(x) The chebyshev polynomial is derived by the following formula:
Pi+1(x)＝2xPi(x)-Pi-1(x),P0(x)＝1 (6)
wherein x ∈ R and P1(x) Denoted by x,2x,2x-1 or 2x +1, where the first term x is used, and x ═ x (x) of the chebyshev polynomial1,…,xm)T∈RmThe enhancement mode is given by:
φ(x)＝[1,P1(x1),…,Pn(x1),…,P1(xm),…,Pn(xm)]T (7)
in the formula, phi (x) represents the vector of the Chebyshev polynomial basis function, Pi(xj) I 1, …, n, j 1, …, m is the order of the chebyshev polynomials and n denotes the order;
thus f will be*(x) Is defined as
f*(x)＝W*Tφ(x)+δ (8)
In the formula, W*Is the optimal weight vector, delta is the estimation error;
optimal weight vector W*Is expressed by the following formula
Figure BDA0002977037210000048
Wherein W is [ omega ]1,ω2,…,ω3]T∈RlIs a weight vector;
in the nth step of the self-adaptive neural inversion control scheme, a Chebyshev neural network W is adoptedi T φ i1,2,3 approximate unknown uncertainty fi *(x) Existence of
fi *(x)＝Wi Tφi+δi,i＝1,2,3 (10)
In the formula, Wi＝Wi *And fi *(x)；
Estimating the weights of the Chebyshev neural network using the 2-norm may reduce the computational burden of the Chebyshev neural network. Thus, define
θi＝||Wi||2＝Wi TWi,i＝1,2,3 (11)
Wherein | | · | | and θiRespectively represent WiAnd 2-norm of unknown variable;
due to σ in (2)1The sign of (2) leads to the problem of unknown control direction1Introducing a Nussbaum type function,
definition 1: if the continuous even function N (χ) satisfies:
Figure BDA0002977037210000049
Figure BDA00029770372100000410
the continuous even function is called Nussbaum-type function, and many functions satisfy both equation (12) and equation (13), such as χ2cos (x) and
Figure BDA0002977037210000051
here, χ is used2cos(χ)。
2, leading: if a non-negative smoothing function V (t) satisfies:
Figure BDA0002977037210000052
wherein χ (t) ≧ 0 is defined as [0, t ≧ 0f) A smoothing function of c0Is a real number and c0> 0, N (-) is an even number Nussbaum type function, g is defined in the set
Figure BDA0002977037210000053
Variables of, V (t), χ (t) and
Figure BDA0002977037210000054
at [0, tf) An upper bound;
for asymmetric output constraints, a new conversion formula is defined as
Figure BDA0002977037210000055
In the formula, positive real number
Figure BDA0002977037210000056
And
Figure BDA0002977037210000057
as an original boundary, λ1(t) is the tracking error given later, μ and s (t) are the transition boundary and transition error, respectively;
derived from t ∈ [0, ∞))
Figure BDA0002977037210000058
And
Figure BDA0002977037210000059
using equation (15) and a logarithmic barrier Lyapunov function, a uniform barrier Lyapunov function is created as
Figure BDA00029770372100000510
Wherein log (-) is the natural logarithm of (-);
the resulting formula (16) satisfies the Lyapunov design principle
Figure BDA00029770372100000511
It ensures that S is limited to the set piS{ - μ < S < μ }. Based on equation (15), the tracking error λ is further derived1Is limited to a set
Figure BDA00029770372100000512
Performing the following steps;
and 3, introduction: for the
Figure BDA00029770372100000513
And
Figure BDA00029770372100000514
exist of
Figure BDA00029770372100000515
In the formula, KS＝S/(μ2-S2)；
Definition 2: the uniform barrier Lyapunov function formula (16) is proposed by fusing a new conversion formula (15) into a conventional logarithmic barrier Lyapunov function to bypass a complex derivation caused by a segmented formula existing in a general segmented barrier Lyapunov function, and has a greater potential versatility in terms of a constrained controller design of a nonlinear system than an existing segmented barrier Lyapunov function.
In the following, for the sake of simplicity, the function parameters are omitted without confusion in context.
(2) Building an adaptive inversion controller
Defining the error control plane as
Figure BDA00029770372100000516
In the formula, beta2Representing a virtual controller, with the real number C being x3An initial value of (1);
definition 3: error control plane lambda conventional to conventional inversion controller technology2In contrast, by letting λ3＝x3-input u of C design (2)dThe error control surface is constructed, and the method has the advantages that the overshoot problem caused by the initial error of the d-axis current is eliminated;
in order to overcome the adverse effect brought by the asymmetric input saturation system, the enhanced dynamic error z is adoptediIs defined as
Figure BDA00029770372100000517
In the formula, auxiliary power system
Figure BDA0002977037210000061
The two are connected (2) and (17), and lambda is deduced1And ziThe time derivative of i ═ 2,3 is
Figure BDA0002977037210000062
In the formula (I), the compound is shown in the specification,
Figure BDA0002977037210000063
defining error variables
Figure BDA0002977037210000064
Is composed of
Figure BDA0002977037210000065
In the formula, variable
Figure BDA0002977037210000066
Is thetaiAn estimated value of (d);
designing a controller based on a traditional inversion controller framework:
first, selecting a barrier Lyapunov function
Figure BDA0002977037210000067
In the formula, r1Is a real number and r1＞0；
V in formula (22) is derived from formulas (15), (16) and (21)1Is a time derivative of
Figure BDA0002977037210000068
By the formula (20), obtaining
Figure BDA0002977037210000069
In the formula, k 10 is a design parameter and uncertainty is unknown
Figure BDA00029770372100000610
Using Nussbaum-type functions and Chebyshev neural networks W, respectively1 Tφ1To estimate the unknown gain sigma1And unknown uncertainty f1 *；
According to formulae (4), (5), (10) and (11), there are obtained
Figure BDA00029770372100000611
In the formula, a1Is a real number and a1＞0；
By substituting formula (24) for formula (25)
Figure BDA00029770372100000612
Designing a virtual input beta2And new law of adaptation
Figure BDA00029770372100000613
Is composed of
Figure BDA00029770372100000614
Figure BDA00029770372100000615
Figure BDA00029770372100000616
Figure BDA00029770372100000617
In the formula, upsilon > 0 and l1Is greater than 0 and is a real number,
Figure BDA00029770372100000618
representing the secondary controller, χ representing a variable of a Nussbaum-type function;
in the equations (27) to (30) to (26), the derivation is made
Figure BDA0002977037210000071
The second step is that: establishing a Lyapunov function as
Figure BDA0002977037210000072
In the formula, r2Is greater than 0 and is a real number;
by means of the formula (21), the time derivative V in the formula (32) is determined2Is composed of
Figure BDA0002977037210000073
Designing an auxiliary power system
Figure BDA0002977037210000074
Is composed of
Figure BDA0002977037210000075
In the formula, k2Is greater than 0 and is a real number;
from formulae (20) and (34), yield
Figure BDA0002977037210000076
To overcome the defect caused by the calculation formula (35)
Figure BDA0002977037210000077
The complexity explosion problem caused by the difference of virtual controllers, the concept of tracking differentiators is introduced:
Figure BDA0002977037210000078
in the formula, an input signal beta2Is obtained by the method of the formula (27),
Figure BDA0002977037210000079
and
Figure BDA00029770372100000710
are all real, v1V and v2Are each beta2And
Figure BDA00029770372100000711
the estimated value of (a);
and (4) introduction: if the initial deviation is
Figure BDA00029770372100000712
In
Figure BDA00029770372100000713
And is real, then v2Satisfy the requirement of
Figure BDA00029770372100000714
In the formula (I), the compound is shown in the specification,
Figure BDA00029770372100000715
and is an unknown real number;
by substituting formulae (31), (35) and (37) for formula (33)
Figure BDA00029770372100000716
In the formula, the unknown uncertainty f2 *Is defined as f2 *＝f2+k2λ2+σ1KS+Δ2；
Definition 4: using Chebyshev neural networks
Figure BDA00029770372100000717
Evaluation f2 *；
Analogously to formula (25), obtaining
Figure BDA00029770372100000718
In the formula, a2Is greater than 0 and is a real number;
by substituting formula (39) for formula (38) to give
Figure BDA00029770372100000719
Design control input uqAnd law of adaptation
Figure BDA00029770372100000720
Is composed of
Figure BDA0002977037210000081
Figure BDA0002977037210000082
In the formula I2Is greater than 0 and is a real number;
the formula (40) is re-expressed as the formula (41) and the formula (42)
Figure BDA0002977037210000083
The third step: designing a Lyapunov function as
Figure BDA0002977037210000084
In the formula, r3Is greater than 0 and is a real number;
similar to equation (34), consider an auxiliary power system
Figure BDA0002977037210000085
Figure BDA0002977037210000086
In the formula, k3Is greater than 0 and is a real number;
combined formula (20) to obtain
Figure BDA0002977037210000087
Then, V in the formula (44) is obtained3Is a time derivative of
Figure BDA0002977037210000088
The formula (46) is re-expressed as the formula (43) and the formula (46)
Figure BDA0002977037210000089
In the formula, the uncertainty f3 *＝f3+k3λ3+Δ3；
It can be clearly seen that the uncertainty f is unknown3 *Are adversely affected by external disturbances and systematic errors. To overcome the adverse effects described above, Chebyshev neural networks were used
Figure BDA00029770372100000810
To approach f3 *；
Analogously to formula (25), obtaining
Figure BDA00029770372100000811
In the formula, a3Is greater than 0 and is a real number;
by the formula (49), the formula (48) is simplified to
Figure BDA00029770372100000812
Design control input udAnd new control law
Figure BDA00029770372100000813
Is composed of
Figure BDA00029770372100000814
Figure BDA00029770372100000815
In the formula I3Is greater than 0 and is a real number;
the expression (50) is re-expressed as
Figure BDA0002977037210000091
Using the equations (5) and (21), the derivation is made
Figure BDA0002977037210000092
Then obtain
Figure BDA0002977037210000093
The invention has the beneficial effects that: compared with the prior art, the invention has the following effects:
(1) aiming at a permanent magnet synchronous motor chaotic system with asymmetric input and output constraints and unknown uncertainty, the invention provides a self-adaptive neural inversion (backstepping) control method, wherein a conversion error and a new boundary are embedded in a logarithmic barrier Lyapunov function, and a uniform barrier Lyapunov function is provided, so that switching nonlinearity related to a segmented barrier Lyapunov function is avoided, and asymmetric output constraint conditions are ensured to be met;
(2) aiming at the problems of irreducibility and low precision of a common smooth function tool in asymmetric input saturation, two suitable auxiliary power systems are designed;
(3) as a single-layer neural network based on chebyshev polynomial orthogonal basis functions, it is generally believed that chebyshev neural networks can identify integration uncertainties including parameter variations and external disturbances, facilitating the development of adaptive controllers with low complexity and fewer parameters. Meanwhile, the calculation burden of the neural network is further reduced by fusing the minimum learning parameterization technology into each step of inversion (backstepping);
(4) in the controller design, a Nussbaum type function and a tracking differentiator are respectively used for solving the problems of complexity explosion and unknown control direction. In addition, the boundedness of the stability of the designed system can be ensured on the premise of not exceeding the input and output constraint boundary.
Drawings
FIG. 1 is a chaotic attractor and phase diagram for a PMSM;
FIG. 2 is a schematic diagram of a conversion equation;
FIG. 3 is a PMSM control schematic;
FIG. 4 is a graph of the x-axis angular displacement trace;
FIG. 5 is a graph of output tracking error;
FIG. 6 is a state variable iqAnd idA trajectory diagram of (a);
FIG. 7 shows a practical controller uqAnd udThe response graph of (a);
FIG. 8 is a graph of output trace comparison;
FIG. 9 is a tracking error trajectory comparison graph;
FIG. 10 is an input uqCompare the figures.
Detailed Description
The invention is further described with reference to the accompanying drawings and specific embodiments.
Example 1: inspired by problem analysis in the background art, the invention focuses on an adaptive neural back-off control design for PMSM to suppress chaotic oscillation and ensure asymmetric input-output constraints, while ensuring that all closed-loop signals are bounded. First, the chaotic attractor and phase diagram are presented to illustrate chaotic oscillation of the PMSM with perturbation parameters. The entire control scheme is then designed based on the inversion framework. In the design of a controller, a unified barrier Lyapunov function is designed by fusing a transformed tracking error and a new boundary into a logarithmic barrier Lyapunov function so as to solve the problem of output constraint, and two auxiliary power systems constructed by two independent first-order differential equations are combined and used in the last two steps of the design of the controller respectively so as to solve the problem of asymmetric input saturation nonlinearity. Chebyshev neural networks are used to identify integration uncertainties consisting of parameter variations and external disturbances. By integrating a minimal learning parameterization technique into the detailed design, the computational load of the Chebyshev neural network can be further reduced. Meanwhile, the self-Adaptive finite time neural control [ J ]. Inf.Sci. (Ny) & 2020,520:271-291.A. Kamalamii, M.Shahrokhi, and M.Mohit, "Adaptive fine-time neural control of non-linear feedback system with output constraints and unknown control directions and input nonlinearities," Adaptive nonlinear control system sub-to-output control, un-control direction, and input nonlinearities, "Adaptive fine-time neural control of non-linear feedback system, Adaptive control system, 271.520, pp.271, 291, J.S.S.A. Zstring, and the self-Adaptive control system, the chaotic system, tracking differentiators adopted in int.j.electric.powerenergy Syst., vol.121, No. September 2019, p.105991,2020) solve the problems of 'complex explosion' and unknown control direction respectively. By embedding the above scheme into a conventional inversion controller program, an adaptive neural inversion control method is developed to ensure desired asymmetric input-output constraints and satisfactory tracking metrics and the bounding of all other closed-loop signals. The key contributions of the present invention are summarized below:
(1) the first item is the work aiming at solving the design problem of the tracking controller with the PMSM chaotic system with asymmetric input and output constraints;
(2) different from the segmented barrier Lyapunov function, the invention designs the uniform Lyapunov barrier function by embedding the conversion tracking error and the new boundary into the logarithmic barrier Lyapunov function, and designs the uniform Lyapunov barrier function so as to bypass the conversion type nonlinearity in the previous segmented Lyapunov barrier function and simultaneously ensure that the asymmetric output constraint is met. Therefore, the uniform Lyapunov barrier function is more suitable for asymmetric output-limited controller design.
(3) In order to solve the asymmetric input saturation problem, two auxiliary power systems represented by first order differential equations are introduced in the last two steps of the inversion respectively, instead of the auxiliary power system given in the literature (congratulatory, dualcolol, congo, class of uncertain strict feedback non-linear systems Command filtering robust adaptive neural network control [ J ]. j.franklin instrument, 2018,355(15):7548-7569.g. Zhu, j.du, and y.kao, "Command filtered robust adaptive NN control for a class of infinite feedback non-linear systems system input analysis," j.franklin instrument, vol.355, No.15, pp.7548 7569, 2018). Such a design helps to ensure that the control scheme has sufficient precision and low complexity.
(4) By incorporating the concept of chebyshev neural networks and the skilled use of tracking differentiators, Nussbaum-type functions and minimum learning parameter techniques into adaptive inversion control, a new control scheme with three adaptive laws is designed, dealing with problems from various uncertainties, high complexity problems and heavy computational load. Therefore, the designed controller is widely applied in practice.
As shown in fig. 1-10, a PMSM chaotic system adaptive control method considering asymmetric input and output constraints includes the following steps:
(1) establishing a dynamic model of a permanent magnet synchronous motor system:
in a rotating (d-q) coordinate system, the kinetic equation of the permanent magnet synchronous motor system is established as follows:
Figure BDA0002977037210000101
in the formula (I), the compound is shown in the specification,
Figure BDA0002977037210000102
and
Figure BDA0002977037210000103
representing the d-axis and q-axis currents,
Figure BDA0002977037210000104
and
Figure BDA0002977037210000105
representing the d-axis and q-axis voltages as system inputs, L,
Figure BDA0002977037210000106
R,
Figure BDA0002977037210000107
ψrb, J and npRespectively representing inductance, rotor angular velocity, stator resistance, load torque, flux linkage, viscous friction coefficient, rotor moment of inertia and magnetic pole pairs, wherein the meaning of each variable is given in a table I;
TABLE I meanings of PMSM parameters (denotation)
Figure BDA0002977037210000108
Figure BDA0002977037210000111
Simplifying the formula (1), and selecting L as Ld＝LqDefinition of
Figure BDA0002977037210000112
And
Figure BDA0002977037210000113
np＝1，x1＝ω,x2＝iq，x3＝id,L＝Ld＝Lqand (3) obtaining a simplified dimensionless model of formula (1) by considering uncertain external interference and asymmetric input saturation:
Figure BDA0002977037210000114
in the formula (I), the compound is shown in the specification,
Figure BDA0002977037210000115
σ1＝BL/(JR)，σ2＝-npψr 2/(BR)，
Figure BDA0002977037210000116
and
Figure BDA0002977037210000117
Δii is 1,2,3 is uncertain external interference;
in the formula, x1Representing nominal angular velocity, x2Representing the q-axis current, x3Representing d-axis current, T time, TLDenotes a load, udDenotes the d-axis voltage, uqRepresenting the q-axis voltage, σ1And σ2Representing an unknown parameter.
Asymmetric input saturation is expressed as:
Figure BDA0002977037210000118
in the formula umaxAnd uminRepresenting the amplitude, v, of the asymmetrical input saturationgAnd ugInput and output representing asymmetric input saturation, respectively;
from the prior art, it is known that equation (1) encounters chaotic vibration and slips into certain areas. To give the calculation result of equation (1), x is set1(0)＝0.1,x2(0)＝0.9,x3(0) 20 and uq＝ud＝TLChaos analysis was performed as 0. FIG. 1 shows the conditions and variable parameter σ in the above1And σ2Chaotic attractors and phase diagrams for the lower PMSM. The conclusion is that the chaotic behavior of PMSM is susceptible to parameter variations. Since complex oscillations and uncertainties and violations of asymmetric input-output constraints can lead to poor performance of PMSM, there is a strong need to propose an adaptive neural inversion control solution to reverse this unfavorable situation
(2) Setting a control object:
definition 1: it is emphasized that this study is the first study aimed at solving the problem of tracking controller design for PMSM chaotic systems with asymmetric input-output constraints and unknown uncertainty. In contrast to the prior art, there is a need to address the problems from asymmetric input-output constraints.
In view of the effects discussed above, the control object may be set as follows:
(a) all variables in the PMSM chaotic system are bounded;
(b) output x1Following the desired signal yd；
(c) Does not violate control input constraints;
(d) output x1Is defined as
Figure BDA0002977037210000121
To design the adaptive neural inversion control of step (3), the following assumptions and reasoning are given:
assume that 1: variable sigmaiI is 1,2 and δiI is unknown but bounded, i.e. 1,2,3
σim≤σi≤σiM,|δi|≤δM, (4)
In the formula, σim,σiMI is 1,2 and δM,(δM> 0) is a real number, δiIs an estimation error, which will be explained later;
assume 2: there is a desired trajectory
Figure BDA0002977037210000122
And time derivative thereof
Figure BDA0002977037210000123
And
Figure BDA0002977037210000124
satisfy inequality
Figure BDA0002977037210000125
Wherein
Figure BDA0002977037210000126
And xi are positive real numbers;
assume that 3: presence of real number ci> 0, e.g. | Δi|≤ci,i＝1,2,3；
Introduction 1: for the
Figure BDA0002977037210000127
Obtaining:
Figure BDA0002977037210000128
wherein p > 1, ξ > 0, q > 1 and (p-1) (q-1) ═ 1;
chebyshev neural network:
using the Chebyshev neural network to approximate the unknown uncertainty on a compact set with arbitrary precision, the present invention selects the Chebyshev neural network to approximate the unknown uncertainty f generated in the controller design*(x) Deriving the Chebyshev polynomial by the following formula
Pi+1(x)＝2xPi(x)-Pi-1(x),P0(x)＝1 (6)
Wherein x ∈ R and P1(x) Denoted by x,2x,2x-1 or 2x +1, where the first term is used. Chebyshev polynomial x ═ x (x)1,...,xm)T∈RmThe enhancement mode is given by:
φ(x)＝[1,P1(x1),...,Pn(x1),...,P1(xm),...,Pn(xm)]T (7)
in the formula, phi (x) represents the vector of the Chebyshev polynomial basis function, Pi(xj) I 1, n, j 1, n, m being the order of the chebyshev polynomials and n representing the order;
thus f will be*(x) Is defined as
f*(x)＝W*Tφ(x)+δ (8)
In the formula, W*Is the optimal weight vector, delta is the estimation error;
optimal weight vector W*Is expressed by the following formula
Figure BDA0002977037210000129
Wherein W is [ omega ]1,ω2,...,ω3]T∈RlIs a weight vector;
in the step of designing the self-adaptive neural inversion control scheme, a Chebyshev neural network W is adoptedi TφiI-1, 2,3 approximate unknown uncertainty fi *(x) Existence of
fi *(x)＝Wi Tφi+δi,i＝1,2,3 (10)
In the formula, Wi＝Wi *And fi *(x) As will be given below;
studies have shown that estimating the weights of the Chebyshev neural network using a 2-norm can reduce the computational burden of the Chebyshev neural network. Thus, define
θi＝||Wi||2＝Wi TWi,i＝1,2,3 (11)
Wherein | | · | | and θiRespectively represent WiAnd 2-norm of unknown variable;
nussbaum type function:
due to sigma in the formula (2)1The sign of (2) leads to an unknown control direction problem, and therefore, for σ in equation (2)1Introducing a Nussbaum type function,
definition 1: if the continuous even function N (χ) satisfies:
Figure BDA0002977037210000131
Figure BDA0002977037210000132
the continuous even function is called Nussbaum-type function, and many functions satisfy both equation (12) and equation (13), such as χ2cos (x) and
Figure BDA0002977037210000133
here, χ is used2cos(χ)。
2, leading: if a non-negative smoothing function V (t) satisfies:
Figure BDA0002977037210000134
wherein χ (t) ≧ 0 is defined as [0, t ≧ 0f) A smoothing function of c0Is a real number and c0> 0, N (-) is an even number Nussbaum type function, g is defined in the set
Figure BDA00029770372100001317
Variables of, V (t), χ (t) and
Figure BDA0002977037210000135
at [0, tf) An upper bound;
creating a uniform barrier Lyapunov function:
to handle asymmetric output constraints, a new conversion formula is defined as
Figure BDA0002977037210000136
In the formula, positive real number
Figure BDA0002977037210000137
And
Figure BDA0002977037210000138
as an original boundary, λ1(t) is the tracking error given later, μ and s (t) are the transition boundary and transition error, respectively;
derived from t ∈ [0, ∞))
Figure BDA0002977037210000139
And
Figure BDA00029770372100001310
using equation (15) and a logarithmic barrier Lyapunov function, a uniform barrier Lyapunov function is created as
Figure BDA00029770372100001311
Wherein log (-) is the natural logarithm of (-);
the resulting formula (16) satisfies the Lyapunov design principle
Figure BDA00029770372100001312
It ensures that S is limited to the set piS{ - μ < S < μ }. Based on equation (15), the tracking error λ is further derived1Is limited to a set
Figure BDA00029770372100001313
Performing the following steps; for clarity of illustration, a schematic diagram is shown in FIG. 2.
And 3, introduction: for the
Figure BDA00029770372100001314
And
Figure BDA00029770372100001315
exist of
Figure BDA00029770372100001316
In the formula, KS＝S/(μ2-S2)；
Definition 2: the uniform barrier Lyapunov function formula (16) is proposed by fusing a new conversion formula (15) into a conventional logarithmic barrier Lyapunov function to bypass the complex derivation caused by the segmented formula existing in the general segmented barrier Lyapunov function, and has a larger potential universality in terms of the design of a constraint controller of a nonlinear system compared with the existing segmented barrier Lyapunov function.
In the following, for the sake of simplicity, the function parameters are omitted without confusion in context.
(3) Building an adaptive inversion controller
Defining the error control plane as
Figure BDA0002977037210000141
In the formula, beta2Representing a virtual controller, with the real number C being x3An initial value of (1);
definition 3: error control plane lambda conventional to conventional inversion controller technology2In contrast, by letting λ3＝x3-input u of C design (2)dThe error control surface is constructed, and the method has the advantages that the overshoot problem caused by the initial error of the d-axis current is eliminated;
in order to overcome the adverse effect brought by the asymmetric input saturation system, the enhanced dynamic error z is adoptediIs defined as
Figure BDA0002977037210000142
In the formula, assistForce system
Figure BDA0002977037210000143
This will be given later;
the two are connected (2) and (17), and lambda is deduced1And ziThe time derivative of i ═ 2,3 is
Figure BDA0002977037210000144
In the formula (I), the compound is shown in the specification,
Figure BDA0002977037210000145
defining error variables
Figure BDA0002977037210000146
Is composed of
Figure BDA0002977037210000147
In the formula, variable
Figure BDA0002977037210000148
Is thetaiAn estimated value of (d);
then, the controller design steps based on the traditional inversion controller framework are given:
first, selecting a barrier Lyapunov function
Figure BDA0002977037210000149
In the formula, r1Is a real number and r1＞0；
V in formula (22) is derived from formulas (15), (16) and (21)1Is a time derivative of
Figure BDA00029770372100001410
By the formula (20), obtaining
Figure BDA00029770372100001411
In the formula, k 10 is a design parameter and uncertainty is unknown
Figure BDA00029770372100001412
It can be seen that the uncertainty f is unknown1 *From an uncertainty parameter σ1And external interference delta1And a load torque TLThe composition is a very complex nonlinear term, and a reliable controller is difficult to design. To solve these problems, a Nussbaum-type function and a Chebyshev neural network W are used, respectively1 Tφ1To estimate the unknown gain sigma1And unknown uncertainty f1 *；
According to formulae (4), (5), (10) and (11), there are obtained
Figure BDA00029770372100001413
In the formula, a1Is a real number and a1＞0；
By substituting formula (24) for formula (25)
Figure BDA0002977037210000151
Designing a virtual input beta2And new law of adaptation
Figure BDA0002977037210000152
Is composed of
Figure BDA0002977037210000153
Figure BDA0002977037210000154
Figure BDA0002977037210000155
Figure BDA0002977037210000156
In the formula, upsilon > 0 and l1Is greater than 0 and is a real number,
Figure BDA0002977037210000157
representing the secondary controller, χ representing a variable of a Nussbaum-type function;
in the equations (27) to (30) to (26), the derivation is made
Figure BDA0002977037210000158
The second step is that: establishing a Lyapunov function as
Figure BDA0002977037210000159
In the formula, r2Is greater than 0 and is a real number;
by means of the formula (21), the time derivative V in the formula (32) is determined2Is composed of
Figure BDA00029770372100001510
Designing an auxiliary power system
Figure BDA00029770372100001511
Is composed of
Figure BDA00029770372100001512
In the formula, k2Is greater than 0 and is a real number;
from formulae (20) and (34), yield
Figure BDA00029770372100001513
To overcome the defect caused by the calculation formula (35)
Figure BDA00029770372100001514
The complexity explosion problem caused by the difference of virtual controllers, the concept of tracking differentiators is introduced:
Figure BDA00029770372100001515
in the formula, an input signal beta2Is obtained by the method of the formula (27),
Figure BDA00029770372100001516
and
Figure BDA00029770372100001517
are all real, v1V and v2Are each beta2And
Figure BDA00029770372100001518
the estimated value of (a);
and (4) introduction: if the initial deviation is
Figure BDA00029770372100001519
In
Figure BDA00029770372100001520
And is real, then v2Satisfy the requirement of
Figure BDA00029770372100001521
In the formula (I), the compound is shown in the specification,
Figure BDA00029770372100001523
and is an unknown real number;
by substituting formulae (31), (35) and (37) for formula (33)
Figure RE-GDA00030348499800001521
In the formula, the unknown uncertainty f2 *Is defined as f2 *＝f2+k2λ2+σ1KS+Δ2；
Definition 4: different from the former tool based on a first-order filter, a tracking differentiator is designed to obtain beta2Can improve the tracking accuracy of the proposed solution.
Unknown uncertainty f2 *Is a complex nonlinear function with effects due to coupling terms between velocity and current, unknown system dynamics and errors. For the convenience of subsequent design, the Chebyshev neural network W is adopted2 Tφ2Evaluation f2 *；
Analogously to formula (25), obtaining
Figure BDA0002977037210000161
In the formula, a2Is greater than 0 and is a real number;
by substituting formula (39) for formula (38) to give
Figure BDA0002977037210000162
Design control input uqAnd law of adaptation
Figure BDA0002977037210000163
Is composed of
Figure BDA0002977037210000164
Figure BDA0002977037210000165
In the formula I2Is greater than 0 and is a real number;
the formula (40) is re-expressed as the formula (41) and the formula (42)
Figure BDA0002977037210000166
The third step: designing a Lyapunov function as
Figure BDA0002977037210000167
In the formula, r3Is greater than 0 and is a real number;
similar to equation (34), consider an auxiliary power system
Figure BDA0002977037210000168
Figure BDA0002977037210000169
In the formula, k3Is greater than 0 and is a real number;
combined formula (20) to obtain
Figure BDA00029770372100001610
Then, V in the formula (44) is obtained3Is a time derivative of
Figure BDA00029770372100001611
The formula (46) is re-expressed as the formula (43) and the formula (46)
Figure BDA00029770372100001612
In the formula, the uncertainty f3 *＝f3+k3λ3+Δ3；
It can be clearly seen that the uncertainty f is unknown3 *Are adversely affected by external disturbances and systematic errors. To overcome the above-mentioned adverse effects, the Chebyshev neural network is utilized
Figure BDA00029770372100001613
To approach f3 *；
Analogously to formula (25), obtaining
Figure BDA0002977037210000171
In the formula, a3Is greater than 0 and is a real number;
by the formula (49), the formula (48) is simplified to
Figure BDA0002977037210000172
Design control input udAnd new control law
Figure BDA0002977037210000173
Is composed of
Figure BDA0002977037210000174
Figure BDA0002977037210000175
In the formula I3Is greater than 0 and is a real number;
the expression (50) is re-expressed as
Figure BDA0002977037210000176
Using the equations (5) and (21), the derivation is made
Figure BDA0002977037210000177
Then obtain
Figure BDA0002977037210000178
Detailed adaptive neural inversion control design has been completed. For clarity of illustration, FIG. 3 is a schematic illustration.
Definition 5: compared with the previous scheme for restraining the asymmetric input constraint from occurring, the designed equations (34) and (45) can not only overcome the adaptive mixed fuzzy output feedback control of the fractional order nonlinear system with time lag and input saturation (Song comma, Park Ju Hyun, Zhang Baoyong)]Applied, Math, Compout, 2020,364, S, Song, J, H, park, B, Zhang, and X, Song, "Adaptive hybrid feedback control for reactive-order-orderliner systems with time-varying delay and input consumption," applied, Math, Compout, vol, 364, p, 124662,2020)gAnd vgG ═ d, q inconsistency, and can bypass literature (congratulatory, duculol, scout]Additional auxiliary power systems in J.Franklin Inst.,2018,355(15), 7548-7569, G.Zhu, J.Du, and Y.Kao, "Command filtered robust adaptive NN control for a class of non-reactive streams system input maintenance," J.Franklin Inst., vol.355, No.15, pp.7548-7569,2018) lead to development complexities.
Definition 6: because the Chebyshev neural network can only be composed of multiple Chebyshev itemsThe order of the equation is determined, which works to approximate the unknown uncertainty f using the Chebyshev neural networki *Instead of the most common radial basis function neural network determined based on the center and width of the gaussian function, the joint models (25), (39) and (49) insert the minimum learning parameter technique in the design of the controller and the adaptive law, which benefits from reducing the design parameters and the system computations.
The technical scheme of the invention is subjected to stability analysis:
for any real number p > 0, the tight set can be considered as
Figure BDA0002977037210000181
Theorem 1: on the premise of satisfying assumptions 1-3, for chaotic PMSM with unknown uncertainty and asymmetric input output constraints, control laws (27), (41), (51) and adaptive laws (30), (42) and (52) are designed, and when the condition omega is satisfiedi,i＝1,2,3,
Figure BDA0002977037210000182
And
Figure BDA0002977037210000183
and when the requirements are met, the scheme of the invention can ensure the realization.
And (3) proving that: constructing the whole Lyapunov function
Figure BDA0002977037210000184
From formula (55)
Figure BDA0002977037210000185
From the formula (17) can be obtained
Figure BDA0002977037210000186
Namely, it is
Figure BDA0002977037210000187
Where ρ is min {2k ═1,2k2,2k3,l1,l2,l3}，
Figure BDA0002977037210000188
By integrating equation (60) over the [0, t ] interval, one can obtain
Figure BDA0002977037210000189
By factoring 2, the functions V (t), χ and
Figure BDA00029770372100001810
has an interval of [0t]。
Equation (61) may be re-expressed as
Figure BDA00029770372100001811
Wherein C is0Is that
Figure BDA00029770372100001812
The upper limit of (3).
Thus can obtain
Figure BDA00029770372100001813
In addition, can obtain
Figure BDA00029770372100001814
From the formula (15) can be obtained
Figure BDA00029770372100001815
The expressions (64) and (65) denote S and lambda, respectively1Is bounded. Likewise, z can be obtained2、z3And
Figure BDA0002977037210000191
is bounded. Further obtained by the formula (21)
Figure BDA0002977037210000192
Is bounded. By means of the formulae (3), (27), (41) and (51), the variable β can be deduced2、νgAnd ugAnd g is bounded by a and d. Therefore,. DELTA.ug＝ug-vgAnd g is d and q is bounded.
In addition, to ensure error control surface
Figure BDA0002977037210000193
The convergence of (2) also needs to be considered
Figure BDA0002977037210000194
Is well-defined. To this end, we designed a suitable Lyapunov function as
Figure BDA0002977037210000195
Passing formula (34), formula (45) and derivative
Figure BDA0002977037210000196
Can obtain
Figure BDA0002977037210000197
Wherein | Δ u | ═ max { | Δ ud|,|Δuq|}。
From the formula (5) can be obtained
Figure BDA0002977037210000198
Can then obtain
Figure BDA0002977037210000199
Wherein a is0＝min{2ki-1}, wherein ki＞1/2,i＝2,3，b0＝Δu2。
By solving the formula (69), the
Figure BDA00029770372100001910
Analogously to formula (64), can be obtained
Figure BDA00029770372100001911
By the formula (71), it can be understood that
Figure BDA00029770372100001912
And
Figure BDA00029770372100001913
is bounded. In formula (19), represented byiAnd
Figure BDA00029770372100001914
can infer the error control plane lambda2And λ3Is bounded. To sum up, all variables of the PMSM are bounded.
Further, it can be seen from the expressions (15) and (16) that
Figure BDA00029770372100001915
When the temperature of the water is higher than the set temperature,
Figure BDA00029770372100001916
to pair
Figure BDA00029770372100001917
The guarantee is ensured. In addition, due to
Figure BDA00029770372100001918
And
Figure BDA00029770372100001919
can obtain
Figure BDA00029770372100001920
Then, by
Figure BDA00029770372100001921
And
Figure BDA00029770372100001922
can deduce
Figure BDA00029770372100001923
Thus, the stability analysis was completed.
Definition 7: tracking error lambda1Is a visual index for controlling performance, and can be selected appropriately
Figure BDA00029770372100001924
And ρ is arbitrarily adjusted small. We can get specific adjustment criteria to increase the parameter ki,ai,riγ, wherein k2＞1/2,k3> 1/2, decreasing the parameter
Figure BDA00029770372100001925
It should be noted that vqSubject to amplitude v2The influence of (c). For this purpose, first of all by selecting suitable ones
Figure BDA00029770372100001926
k1,r1,a1,l1Adjusting a suitable value to v2And then adjust other parameters. The values of the control parameters can be repeatedly tested using the specific adjustment criteria described above. In practice, k is used to achieve a predetermined targeti,ai,riγ and liThe value of i-1, 2,3 needs to be adjusted unambiguously.
To illustrate the beneficial effects of the present invention, the following simulations were performed:
simulation tests are carried out on the scheme of the invention to illustrate the effectiveness and robustness of the created adaptive neural inversion control. Reference signal is ydSin (t). The upper and lower bounds of the asymmetric input constraint are u max25 and umin-3. For the formula (2), the output variable satisfies the constraint condition of-1.2 < x1< 1.24. Can obtain
Figure BDA00029770372100001927
And
Figure BDA00029770372100001928
initial conditions are x1(0)＝0.1∈(-1.2,1.24)，x2(0)＝0.9,x3(0)＝20,χ(0)＝1.55,ν1(0)＝1,
Figure BDA00029770372100001929
Then, the parameter is selected as k1＝97,r1＝0.001,ki＝ri＝1，i＝2,3,a1＝51,a2＝121,a3＝71,TL＝3,l1＝0.2,l2＝6.2,l3＝4.8,
Figure BDA00029770372100001930
Y 0.1 external interference is
Figure BDA0002977037210000201
The order of equation (7) is designated as 2 using a single-layer chebyshev neural network. The Chebyshev polynomial basis function can be described as
φi(x)＝[1,P1(x1),P2(x1),...,P1(x3),P2(x3)]T,i＝1,2,3 (73)
Fig. 4-7 illustrate the response of the system. Fig. 4 shows that the output signal y can follow the reference trajectory without violating its constraints. FIG. 5 shows the tracking error λ1May remain unchanged within a certain range. State variable iq.idAnd a real controller uq,udThe responses of (a) are already given in fig. 6 and 7, respectively. The results show that the designed solution performed well and satisfactorily.
And (3) scheme comparison:
to demonstrate the superiority of the created adaptive neural inversion control, proportional integral derivative and adaptive neural dynamics surface control were used as a comparison of equation (2). Ignoring asymmetric input-output control and considering udWhen the ratio is 0, the actual proportional integral differential is controlled to
Figure BDA0002977037210000202
Wherein k isP,kI,kDAre all real numbers.
The only different design resulting from adaptive neural dynamic surface control is the use of a first order filter in place of the tracking differentiator in the adaptive neural inversion control design. Here, a first order filter is used
Figure BDA0002977037210000203
Definition of lambda2＝x2-β2fWherein beta is2fTau for the stability controller to be designed2f> 0 is a design real number. Then the corresponding controller v designed by the formula (41)qIs modified into
Figure BDA0002977037210000204
Also, the following indices are defined for comparison
Figure BDA0002977037210000205
Figure BDA0002977037210000206
Figure BDA0002977037210000207
Where N is the number of samples, Mλ、μλAnd σλRespectively represent | λ1(i) The maximum, average and standard deviation values of |;
the simulation tests were performed under different unknown external disturbances, case 1:
Figure BDA0002977037210000208
case 2:
Figure BDA0002977037210000209
in all comparisons, k is selectedP＝-140,kI＝-0.08,kD-160 and τ2f0.05. The remaining parameters and conditions are provided in subsection A. In the range of 0 to 30s, we calculated a simulation and a quantitative index.
Fig. 8-10 and table 2 show the results of a comparison under unknown external interference. It can be seen from fig. 8-10 that the tracking performance of the adaptive neural inversion control is superior to the other two controllers. From fig. 10, the pid input u can be seenqBeyond the boundary value. In contrast, the other two controllers constrain uqThe amplitude of (d). Meanwhile, table 2 compares the quantization index values of the three schemes under different situations. It follows that the maximum M of the three schemesλAre almost identical. In other indicators, the adaptive neural inversion control is smaller than the other two controllers. The result shows that the control effect of the three controllers of the self-adaptive neural inversion control is the best for PMSM. It can be concluded that the method was designedThe scheme has high precision in the aspect of controlling the PMSM chaotic system by using unknown external interference and asymmetric input and output constraints.
TABLE II Performance index comparison results
Figure BDA00029770372100002010
Figure BDA0002977037210000211
And (4) conclusion: the invention provides a self-adaptive neural inversion control suitable for PMSM (permanent magnet synchronous Motor), which has chaotic ignition, parameter change, external interference and asymmetric input and output constraints and can be used for vehicles, elevators, compressors, robots, machine tools and airplanes. Chaotic oscillations provide system dynamics with parameter fluctuations. In order to ensure that system output constraints have unequal constraints, a uniform barrier Lyapunov function which is not applicable to a segmented expression is provided. Two auxiliary power systems are embedded in the last two design error control planes to eliminate the damage caused by the double asymmetric input saturation of the PMSM. Unknown control directions, complex explosion, unknown uncertainty and heavy calculation burden generated in design are solved through Nussbaum type functions, a tracking differentiator, a Chebyshev neural network and a minimum learning parameter technology. It is then demonstrated that all variables of the PMSM are bounded and do not exceed the input-output constraints. Future designs will enhance our proposed algorithm by specifying an exponential formula in the performance controller and extend it to induction motors and dc servo motors.
The invention has the following advantages:
(1) the invention provides a self-adaptive neural inversion (backstepping) control method for a permanent magnet synchronous motor chaotic system with asymmetric input and output constraints and unknown uncertainty. And (4) giving out a chaotic attractor and a phase diagram to judge whether the system is in a chaotic excitation state. By integrating various effective measures into backstepping technology, a systematic detailed design process is formed. The core design is as follows, a conversion error and a new boundary are embedded in a logarithmic barrier Lyapunov function, a unified barrier Lyapunov function is provided, so that switching type nonlinearity related to a segmented barrier Lyapunov function is avoided, and meanwhile, asymmetric output constraint conditions are guaranteed to be met;
(2) aiming at the problems of irreducibility and low precision of a common smooth function tool in asymmetric input saturation, two suitable auxiliary power systems are designed;
(3) as a single-layer neural network based on chebyshev polynomial orthogonal basis functions, it is generally believed that chebyshev neural networks can identify integration uncertainties including parameter variations and external disturbances, facilitating the development of adaptive controllers with low complexity and fewer parameters. Meanwhile, the calculation burden of the neural network is further reduced by fusing the minimum learning parameterization technology into each step of inversion (backstepping);
(4) in the controller design, a Nussbaum type function and a tracking differentiator are respectively used for solving the problems of complexity explosion and unknown control direction. In addition, the boundedness of the stability of the designed system can be ensured on the premise of not exceeding the input and output constraint boundary.
The above description is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily conceive of the changes or substitutions within the technical scope of the present invention, and therefore, the scope of the present invention should be determined by the scope of the claims.

Claims (1)

1. The PMSM chaotic system self-adaptive control method considering asymmetric input and output constraints is characterized in that: the method comprises the following steps:
(1) establishing a dynamic model of a permanent magnet synchronous motor system:
in a rotating (d-q) coordinate system, the kinetic equation of the permanent magnet synchronous motor system is established as follows:
Figure FDA0002977037200000011
in the formula (I), the compound is shown in the specification,
Figure FDA0002977037200000012
and
Figure FDA0002977037200000013
representing the d-axis and q-axis currents,
Figure FDA0002977037200000014
and
Figure FDA0002977037200000015
representing the d-axis and q-axis voltages as system inputs, L,
Figure FDA0002977037200000016
R,
Figure FDA0002977037200000017
ψrb, J and npRespectively representing inductance, rotor angular velocity, stator resistance, load torque, flux linkage, viscous friction coefficient, rotor moment of inertia and magnetic pole pairs;
simplifying the formula (1), and selecting L as Ld＝LqDefinition of
Figure FDA0002977037200000018
And
Figure FDA0002977037200000019
np＝1，x1＝ω,x2＝iq，x3＝id,L＝Ld＝Lqconsidering unknown external interference and asymmetric input saturation, a simplified dimensionless model of formula (1) is obtained:
Figure FDA00029770372000000110
in the formula (I), the compound is shown in the specification,
Figure FDA00029770372000000111
σ1＝BL/(JR)，σ2＝-npψr 2/(BR)，
Figure FDA00029770372000000112
and
Figure FDA00029770372000000113
Δii ═ 1,2,3 is unknown external interference;
in the formula, x1Representing nominal angular velocity, x2Representing the q-axis current, x3Representing d-axis current, T time, TLRepresents the load, udDenotes the d-axis voltage, uqRepresenting the q-axis voltage, σ1And σ2Representing an unknown parameter.
Asymmetric input saturation is expressed as:
Figure FDA00029770372000000114
in the formula umaxAnd uminRepresenting the amplitude, v, of the asymmetrical input saturationgAnd ugInput and output representing asymmetric input saturation, respectively;
(2) setting a control object:
(a) all variables in the PMSM chaotic system are bounded;
(b) output x1Following the desired signal yd；
(c) Does not violate control input constraints;
(d) output x1Is defined as
Figure FDA00029770372000000115
Setting 1: variable sigmaiI is 1,2 and δiI is unknown but bounded, i.e. 1,2,3
σim≤σi≤σiM,|δi|≤δM (4)
In the formula, σim,σiMI is 1,2 and δM,(δM> 0) is a real number, δiIs the estimation error;
setting 2: there is a desired trajectory
Figure FDA0002977037200000021
And time derivative thereof
Figure FDA0002977037200000022
And
Figure FDA0002977037200000023
satisfy inequality
Figure FDA0002977037200000024
Wherein
Figure FDA0002977037200000025
And xi are positive real numbers;
setting a reference value 3: presence of real number ci> 0, e.g. | Δi|≤ci,i＝1,2,3；
Introduction 1: for the
Figure FDA0002977037200000026
Obtaining:
Figure FDA0002977037200000027
wherein p > 1, ξ > 0, q > 1 and (p-1) (q-1) ═ 1;
selecting a Chebyshev neural network to approximate an unknown uncertainty f generated in the controller design*(x) The Chebyshev is derived by the following formulaPolynomial equation
Pi+1(x)＝2xPi(x)-Pi-1(x),P0(x)＝1 (6)
Wherein x ∈ R and P1(x) Is x, x ═ x (x) of chebyshev polynomial1,...,xm)T∈RmThe enhancement mode is given by:
φ(x)＝[1,P1(x1),...,Pn(x1),...,P1(xm),...,Pn(xm)]T (7)
in the formula, phi (x) represents the vector of the Chebyshev polynomial basis function, Pi(xj) I 1, n, j 1, n, m is the order of the chebyshev polynomials, n denotes the order;
thus f will be*(x) Is defined as
f*(x)＝W*Tφ(x)+δ (8)
In the formula, W*Is the optimal weight vector, delta is the estimation error;
optimal weight vector W*Is expressed by the following formula
Figure FDA0002977037200000028
Wherein W is [ omega ]1,ω2,...,ω3]T∈RlIs a weight vector;
using Chebyshev neural networks, Wi TφiI 1,2,3 is approximately unknown uncertainty, there is
fi *(x)＝Wi Tφi+δi,i＝1,2,3 (10)
In the formula, Wi＝Wi *And fi *(x)；
Definition of
θi＝||Wi||2＝Wi TWi,i＝1,2,3 (11)
Wherein | | · | | and θiRespectively represent WiAnd 2-norm of unknown variable;
for sigma in formula (2)1Introducing a Nussbaum type function;
definition 1: if the continuous even function N (χ) satisfies:
Figure FDA0002977037200000029
Figure FDA00029770372000000210
the continuous even function is called Nussbaum-type function, and many functions satisfy both (12) and (13), such as χ2cos (x) and
Figure FDA00029770372000000211
here, χ is used2cos(χ)。
2, leading: if a non-negative smoothing function V (t) satisfies:
Figure FDA00029770372000000212
wherein χ (t) ≧ 0 is defined as [0, t ≧ 0f) A smoothing function of c0Is a real number and c0> 0, N (·) is an even NF function, g is defined in the set
Figure FDA0002977037200000031
Variables of, V (t), χ (t) and
Figure FDA0002977037200000032
at [0, tf) An upper bound;
for asymmetric output constraints, a new conversion formula is defined as
Figure FDA0002977037200000033
In the formula, positive real number
Figure FDA0002977037200000034
And
Figure FDA0002977037200000035
as an original boundary, λ1(t) is the tracking error given later, μ and s (t) are the transition boundary and transition error, respectively;
derived from t ∈ [0, ∞))
Figure FDA0002977037200000036
And
Figure FDA0002977037200000037
using equation (15) and a logarithmic-type performance barrier Lyapunov function, a uniform barrier Lyapunov function is created as
Figure FDA0002977037200000038
Wherein log (-) is the natural logarithm of (-); and S is a conversion error.
The obtained formula (16) satisfies the design principle of Lyapunov function
Figure FDA0002977037200000039
Based on equation (15), the tracking error λ is further derived1Is limited to a set
Figure FDA00029770372000000310
Performing the following steps;
and 3, introduction: for the
Figure FDA00029770372000000311
And
Figure FDA00029770372000000312
exist of
Figure FDA00029770372000000313
In the formula, KS＝S/(μ2-S2) (ii) a And S is a conversion error.
(2) Building an adaptive inversion controller
Defining the error control plane as
Figure FDA00029770372000000314
In the formula, beta2Representing a virtual controller, with the real number C being x3An initial value of (1);
definition 3: enhanced dynamic error ziIs defined as
Figure FDA00029770372000000315
In the formula, auxiliary power system
Figure FDA00029770372000000321
As will be given below;
the two are connected (2) and (17), and lambda is deduced1And ziThe time derivative of i ═ 2,3 is
Figure FDA00029770372000000316
In the formula (I), the compound is shown in the specification,
Figure FDA00029770372000000317
defining error variables
Figure FDA00029770372000000318
Is composed of
Figure FDA00029770372000000319
In the formula, variable
Figure FDA00029770372000000320
Is thetaiAn estimated value of (d);
a controller design step based on an inversion controller framework:
first, a performance barrier Lyapunov function is selected:
Figure FDA0002977037200000041
in the formula, r1Is a real number and r1＞0；
V in formula (22) is derived from formulas (15), (16) and (21)1Is a time derivative of
Figure FDA0002977037200000042
By the formula (20), obtaining
Figure FDA0002977037200000043
In the formula, k10 is a design parameter and uncertainty is unknown
Figure FDA0002977037200000044
Using Nussbaum-type functions and Chebyshev neural networks W, respectively1 Tφ1To estimate the unknown gain sigma1And unknown uncertainty f1 *；
According to formulae (4), (5), (10) and (11), there are obtained
Figure FDA0002977037200000045
In the formula, a1Is a real number and a1＞0；
By substituting formula (24) for formula (25)
Figure FDA0002977037200000046
Designing a virtual input beta2And new law of adaptation
Figure FDA0002977037200000047
Is composed of
Figure FDA0002977037200000048
Figure FDA0002977037200000049
Figure FDA00029770372000000410
Figure FDA00029770372000000411
Wherein γ > 0 and l1Is greater than 0 and is a real number,
Figure FDA00029770372000000412
representing the secondary controller, χ representing a variable of a Nussbaum-type function;
in the equations (27) to (30) to (26), the derivation is made
Figure FDA00029770372000000413
The second step is that: establishing a Lyapunov function as
Figure FDA00029770372000000414
In the formula, r2Is greater than 0 and is a real number;
by means of the formula (21), the time derivative V in the formula (32) is determined2Is composed of
Figure FDA00029770372000000415
Designing an auxiliary power system
Figure FDA00029770372000000416
Is composed of
Figure FDA00029770372000000417
In the formula, k2Is greater than 0 and is a real number;
from formulae (20) and (34), yield
Figure FDA0002977037200000051
The concept of a tracking differentiator is introduced:
Figure FDA0002977037200000052
in the formula, an input signal beta2Is obtained by the method of the formula (27),
Figure FDA0002977037200000053
and
Figure FDA0002977037200000054
are all real, v1V and v2Are each beta2And
Figure FDA0002977037200000055
an estimated value of (d);
and (4) introduction: if the initial deviation is
Figure FDA0002977037200000056
In
Figure FDA0002977037200000057
And is real, then v2Satisfy the requirement of
Figure FDA0002977037200000058
In the formula (I), the compound is shown in the specification,
Figure FDA00029770372000000521
and is an unknown real number;
by substituting formulae (31), (35) and (37) for formula (33)
Figure FDA0002977037200000059
In the formula, the unknown uncertainty f2 *Is defined as f2 *＝f2+k2λ2+σ1KS+Δ2；
Definition 4: using Chebyshev neural networks
Figure FDA00029770372000000510
Evaluation of
Figure FDA00029770372000000511
Analogously to formula (25), obtaining
Figure FDA00029770372000000512
In the formula, a2Is greater than 0 and is a real number;
by substituting formula (39) for formula (38) to give
Figure FDA00029770372000000513
Design control input uqAnd law of adaptation
Figure FDA00029770372000000514
Is composed of
Figure FDA00029770372000000515
Figure FDA00029770372000000516
In the formula I2Is greater than 0 and is a real number;
the formula (40) is re-expressed as the formula (41) and the formula (42)
Figure FDA00029770372000000517
The third step: designing a Lyapunov function as
Figure FDA00029770372000000518
In the formula, r3Is greater than 0 and is a real number;
similar to equation (34), consider an auxiliary power system
Figure FDA00029770372000000519
Figure FDA00029770372000000520
In the formula, k3Is greater than 0 and is a real number;
combined formula (20) to obtain
Figure FDA0002977037200000061
Then, V in the formula (44) is obtained3Is a time derivative of
Figure FDA0002977037200000062
The formula (46) is re-expressed as the formula (43) and the formula (46)
Figure FDA0002977037200000063
In the formula, the uncertainty f3 *＝f3+k3λ3+Δ3；
Using Chebyshev neural networks
Figure FDA0002977037200000064
To approach f3 *；
Analogously to formula (25), obtaining
Figure FDA0002977037200000065
In the formula, a3Is greater than 0 and is a real number;
by the formula (49), the formula (48) is simplified to
Figure FDA0002977037200000066
Design control input udAnd new control law
Figure FDA0002977037200000067
Is composed of
Figure FDA0002977037200000068
Figure FDA0002977037200000069
In the formula I3Is greater than 0 and is a real number;
the expression (50) is re-expressed as
Figure FDA00029770372000000610
Using the equations (5) and (21), the derivation is made
Figure FDA00029770372000000611
Then obtain
Figure FDA00029770372000000612
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