CN113064347B - 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 PDFInfo
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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
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 to be excellent tools for solving such problems. By fusing a given performance barrier Lyapunov function and a tracking differentiator) into a traditional 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 ]. Recent adv.Electron.Electron.Eng. (Formery Recent 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," Recent adv.Electron.Eng. (Former Recent Patents sources Electron. Vol.13,2020, 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 research and design an effective strategy for guaranteeing the input and 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 the adaptive inversion control scheme. 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, Sun shine, Xiehua. Surface vessels with Actuator Saturation and failure Error Constrained LOS Path tracking [ J ]. IEEE Trans.Syst. Man, Cybern.Syst., 2018, 48 (10): 1794-1805.Z.Zheng, L.Sun, and L.Xie, "Error-Constrained Path Following of a Surface Vessel with Actuator failure and efficiency," IEEE Trans.Syst. Man, Cybern.Syst., LOS 48, No.10, pp.1794-1805,2018). 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 a Surface Vessel with Actuator failure and failures and" IEEE Trans.Syst. Man, Cybern.Syst., vol.48, No.10, pp.1794-1805,2018), the results are obtained ignoring the low accuracy due to the disparity between the calculation inputs and the constraint inputs. Based on this, the literature (congratulatory soldiers, durolin, never-honor, a class of Command filtering robust adaptive neural network control of uncertain strict 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 systems of uncertain strict feedback nonlinear systems," j.franklin instrument, vol.355, No.15, 7548-7569,2018) proposes adaptive neural control of strict feedback nonlinear systems with actuator saturation 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 Inst.,2018,355(15):7548-7569.G.Zhu, J.Du, and Y.Kao, "Command filtered robust adaptive NN control for a class of infinite real-feedback systems, input maintenance," J.Franklin Inst., 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, so far, the research results of the design of the controller based on the asymmetric input saturated PMSM chaotic system are still few. 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 scenarios 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 suitable for handling 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, "learning-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 sarbasic, murmerd munichite, Adaptive finite time neural 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 control of non-linear feedback systems subzero-linear control, un-linear control direction, and input nonlinearities" inf.sci. (Ny), 520.vol. 291,2020) has studied the inverse control function based on unknown control block functions by combining them. However, the Lyapunov function transform type nonlinearity of the segmented barrier may add extra 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, radial basis function Neural networks with arbitrary estimation capabilities are commonly used in adaptive inversion control Design for many practical Systems, such as those in the literature [ J ] physics frontiers, 2020,8:1-8.R.Luo, Y.Deng, Y.Xie, Uncertain Permanent Magnet Synchronous Motor Drive Chaotic system Neural Network inversion Controller Design [ J ] physics frontiers, 2020,8:1-8.R.Luo, Y.Deng, and Y.Xie, "Neural Network backing Controller Design for Uncertain Permanent Magnet Synchronous Motor Drive magnetic Systems video Command Filter," front.Phys., vol.8, June, Lin.1-8,2020 ], PMSM and literature (J.Yu, P.35, S.Member, W.Dong, B.Chenn, C.Lin. Synchronous Motor [ J ] 26, P.2015.26, P.26, M.J.26, P.31, P.32, P.3532, S.M.M. Men., P.26, M. 12, M. J.26, M. PyO. J.26, M. S.26, M. PyO.26, P.26, M.26, M.S.S.S.S.26, M. PyO.26, M.S.S.26, M.S.S.S.26, M.S.S.26, M.S.S.S.S.S.S.S.26, M.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.26, P.S.103. and P.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.103, C.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.26, P.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.22, C.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.. 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. For the convenience of application of the controller, the chebyshev neural network is widely applied in adaptive control design as a single-layer neural network designed by expanding an input pattern by introducing a chebyshev polynomial basis function. The advantages of the chebyshev neural network based controller were further verified by comparison with the controller performance of the radial basis function neural network based scheme. 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: the self-adaptive control method of the PMSM chaotic system 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 dynamic equation of the permanent magnet synchronous motor system is established as follows:
in the formula (I), the compound is shown in the specification,
and
representing the d-axis and q-axis currents,
and
representing the d-axis and q-axis voltages as system inputs, L,
R,
ψ r b, J and n p Respectively 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 L d ＝L q Definition of
And
n p ＝1，x 1 ＝ω,x 2 ＝i q ，x 3 ＝i d ,L＝L d ＝L q and (3) obtaining a simplified dimensionless model of formula (1) by considering uncertain external interference and asymmetric input saturation:
in the formula (I), the compound is shown in the specification,
σ 1 ＝BL/(JR)，σ 2 ＝-n p ψ r 2 /(BR)，
and
Δ i i is 1,2,3 is uncertain external interference;
in the formula, x 1 Representing nominal angular velocity, x 2 Representing the q-axis current, x 3 Representing d-axis current, T time, T L Represents the load, u d Denotes the d-axis voltage, u q Representing the q-axis voltage, σ 1 And σ 2 Representing an unknown parameter.
Asymmetric input saturation is expressed as:
in the formula u max And u min Representing the amplitude, v, of an asymmetrically saturated input g And u g Respectively 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 x 1 Following the desired signal y d ；
(c) Control input constraints are not violated;
(d) output x 1 Is defined as
Setting 1: variable sigma i I is 1,2 and δ i I is unknown but bounded, i.e. 1,2,3
σ im ≤σ i ≤σ iM ,|δ i |≤δ M In the formula (4), sigma im ,σ iM I is 1,2 and δ M ,(δ M >0) Is a real number, δ i Is the estimation error;
setting 2: there is a desired trajectory
And time derivative thereof
And
satisfy inequality
Wherein
And X is a positive real number;
setting a reference value 3: presence of real number c i >0, e.g. | Δ i |≤c i ,i＝1,2,3；
Introduction 1: for the
Obtaining:
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:
P i+1 (x)＝2xP i (x)-P i-1 (x),P 0 (x)＝1 (6)
wherein x ∈ R and P 1 (x) Denoted by x,2x,2x-1 or 2x +1, where the first term x is used, and x ═ x (x) of the chebyshev polynomial 1 ,...,x m ) T ∈R m The enhancement mode is given by:
φ(x)＝[1,P 1 (x 1 ),...,P n (x 1 ),...,P 1 (x m ),...,P n (x m )] T (7)
in the formula, φ (x) represents the vector of the basis function of the Chebyshev polynomial, P i (x j ) 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 *T φ(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
Wherein W is [ omega ] 1 ,ω 2 ,...,ω 3 ] T ∈R l Is a weight vector;
in the nth step of the self-adaptive neural inversion control scheme, a Chebyshev neural network W is adopted i T φ i I-1, 2,3 approximate unknown uncertainty f i * (x) Existence of
f i * (x)＝W i T φ i +δ i ,i＝1,2,3 (10)
In the formula, W i ＝W i * 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
θ i ＝||W i || 2 ＝W i T W i ,i＝1,2,3 (11)
Wherein | | · | | and θ i Respectively represent W i And 2-norm of unknown variable;
due to σ in (2) 1 The sign of (2) leads to the problem of unknown control direction 1 Introducing a Nussbaum type function,
definition 1: if the continuous even function N (χ) satisfies:
the continuous even function is called Nussbaum-type function, and many functions satisfy both (12) and (13), such as χ 2 cos (x) and
here, χ is used 2 cos(χ)。
2, leading: if a non-negative smoothing function V (t) satisfies:
wherein χ (t)30 is defined as [0, t f ) A smoothing function of c 0 Is a real number and c 0 >0, N (-) is an even number Nussbaum type function, g is defined in the set
Variables of, V (t), χ (t) and
at [0, t f ) An upper bound;
for asymmetric output constraints, a new conversion formula is defined as
In the formula, positive real number
And
as an original boundary, λ 1 (t) is the tracking error given later, μ and s (t) are the transition boundary and transition error, respectively;
derived from t ∈ [0, ∞))
And
using the barrier Lyapunov function of equation (15) and the logarithmic model, a uniform barrier Lyapunov function is created as
Wherein log (-) is the natural logarithm of (-);
the resulting formula (16) satisfies the Lyapunov design principle
It ensures that S is restricted to set P S :＝{-μ<S<μ }. Based on equation (15), the tracking error λ is further derived 1 Is limited to a set
Performing the following steps;
3, management: for
And
exist of
In the formula, K S ＝S/(μ 2 -S 2 )；
Definition 2: the unified barrier Lyapunov function formula (16) is proposed by fusing a new transformation formula (15) into a conventional logarithmic barrier Lyapunov function to bypass the complex derivation caused by the segmentation formula existing in the general segmented barrier Lyapunov function, and has a greater potential versatility in terms of the design of a constraint controller for a nonlinear system than the 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
In the formula, beta 2 Representing a virtual controller, with the real number C being x 3 An initial value of (1);
definition 3: error control plane lambda conventional to conventional inversion controller technology 2 In contrast, by letting λ 3 ＝x 3 -input u of C design (2) d The 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 adopted i Is defined as
In the formula, auxiliary power system
The joint type (2) and (17) are combined to deduce lambda 1 And zi, i is 2,3, the time derivative of
In the formula (I), the compound is shown in the specification,
defining error variables
Is composed of
In the formula, variable
Is composed of
An estimated value of (d);
designing a controller based on a traditional inversion controller framework:
first, selecting a barrier Lyapunov function
Wherein r1 is a real number and r 1 >0；
V in formula (22) is derived from formulas (15), (16) and (21) 1 Is a time derivative of
By the formula (20), there are obtained
In the formula, k 1 >0 is a design parameter and uncertainty is unknown
Using Nussbaum-type functions and Chebyshev neural networks, respectively
To estimate the unknown gain sigma 1 And unknown uncertainty f 1 * ；
According to formulae (4), (5), (10) and (11), there are obtained
In the formula, a 1 Is a real number and a 1 >0；
By substituting formula (24) for formula (25)
Designing a virtual input beta 2 And new law of adaptation
Is composed of
In the formula, gamma>0 and l 1 >0 and are all real numbers,
representing the secondary controller, χ representing a variable of a Nussbaum-type function;
in the formulae (27) to (30) to (26), the derivation is carried out
The second step: establishing a Lyapunov function as
In the formula, r 2 >0 and is a real number;
the time derivative V in equation (32) is determined using equation (21) 2 Is composed of
Designing an auxiliary power system
Is composed of
In the formula, k 2 >0 and is a real number;
from formulae (20) and (34), yield
To overcome the defect caused by the calculation formula (35)
The complexity explosion problem caused by the difference of virtual controllers, the concept of tracking differentiators is introduced:
in the formula, an input signal beta 2 Is obtained by the method of the formula (27),
and
are all real, v 1 V and v 2 Are each beta 2 And
an estimated value of (d);
and (4) introduction: if the initial deviation is
In
And is real, then v 2 Satisfy the requirement of
In the formula I ν2 >0 and is an unknown real number;
by substituting formulae (31), (35) and (37) for formula (33)
In the formula, the uncertainty will not be known
Is defined as
Definition 4: using Chebyshev neural networks
Evaluation of
Analogously to formula (25), obtaining
In the formula, a 2 >0 and is a real number;
by substituting formula (39) for formula (38) to give
Design control input u q And law of adaptation
Is composed of
In the formula I 2 >0 and is a real number;
the formula (40) is re-expressed as the formula (41) and the formula (42)
The third step: designing a Lyapunov function as
In the formula, r 3 >0 and is a real number;
similar to equation (34), consider an auxiliary power system
In the formula, k 3 >0 and is a real number;
combined formula (20) to obtain
Then, V in the formula (44) is obtained 3 Is a time derivative of
The formula (46) is re-expressed as the formula (43) and the formula (46)
In the formula, uncertainty is unknown
It is clear that the uncertainty is unknown
Are adversely affected by external disturbances and systematic errors. To overcome the above-mentioned adverse effects, the Chebyshev neural network is utilized
To approach
Analogously to formula (25), obtaining
In the formula, a 3 >0 and is a real number;
by the formula (49), the formula (48) is simplified to
Design control input u d And new control law
Is composed of
In the formula I 3 >0 and is a real number;
the expression (50) is re-expressed as
By using the formula (5) and the formula (21), it is deduced
Then obtain
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 i q And i d A trajectory diagram of (a);
FIG. 7 shows a practical controller u q And u d The response graph of (a);
FIG. 8 is a graph comparing output traces;
FIG. 9 is a tracking error trajectory comparison graph;
FIG. 10 is an input u q Compare the figures.
Detailed Description
The invention is further described with reference to the following figures and specific embodiments.
Example 1: inspired by problem analysis in the background art, the invention focuses on self-adaptive neural inversion control design for PMSM (permanent magnet synchronous motor), so as to inhibit chaotic oscillation, ensure asymmetric input and output constraint and ensure that all closed-loop signals are bounded. First, the chaotic attractor and phase diagram are presented to illustrate chaotic oscillation of a 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. The chebyshev neural network is used to identify the integration uncertainty consisting of parameter variations and external disturbances. By integrating a minimum learning parameterization technique into the detailed design, the computational load of the chebyshev neural network can be further reduced. Meanwhile, Adaptive finite time neural control [ J ] Inf.Sci ] (Ny) & 2020,520:271-291.A. Kamalamiri, M.Shahrokhi, and M.Mohit, "Adaptive fine-time neural control of non-linear feedback system of unknown control direction and input nonlinearity, and the like, and the functions of the type of phosphor used in the literature (ZJ, Maramily, Morama, Zjc.S.S.P.A. Zy, 2020,520:271-291.A. Kamalamii, M.Shahroughhi, and M.Mohit," Adaptive fine-time neural control of non-linear system of phosphor systems of sub-phosphor of phosphor system, and input nonlinear systems, "Inf.Sci. (Ny) & p.520, 271-291,2020) and the functions of the type of phosphor, Zm.P.S.S.S.P.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.A. of the chaos.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.S.A. of the characteristics of zero. of zero. of zero. of zero. of zero. of zero. zero, tracking differentiators adopted in int.j.electric.power Energy 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 work is directed to 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, 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) 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 Command filtering robust adaptive neural network control of uncertain strict 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 infinite linear system units and input transmission," j.nklin instrument, vol.355, No.15, pp.7548-7569,2018). Such a design helps to ensure that the control scheme has sufficient accuracy 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 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:
in the formula (I), the compound is shown in the specification,
and
representing the d-axis and q-axis currents,
and
representing the d-axis and q-axis voltages as system inputs, L,
R,
ψ r b, J and n p Respectively 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)
Simplifying the formula (1), and selecting L as L d ＝L q Definition of
And
n p ＝1，x 1 ＝ω,x 2 ＝i q ，x 3 ＝i d ,L＝L d ＝L q and (3) obtaining a simplified dimensionless model of formula (1) by considering uncertain external interference and asymmetric input saturation:
in the formula (I), the compound is shown in the specification,
σ 1 ＝BL/(JR)，σ 2 ＝-n p ψ r 2 /(BR)，
and
D i i is 1,2 and 3 are uncertain external interference;
in the formula, x 1 Representing nominal angular velocity, x 2 Representing the q-axis current, x 3 Representing d-axis current, T time, T L Represents the load, u d Denotes the d-axis voltage, u q Representing the q-axis voltage, σ 1 And σ 2 Representing an unknown parameter.
Asymmetric input saturation is expressed as:
in the formula u max And u min Representing the amplitude of asymmetric input saturation, v g And u g Input 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 set 1 (0)＝0.1,x 2 (0)＝0.9,x 3 (0) 20 and u q ＝u d ＝T L Chaos analysis was performed as 0. FIG. 1 shows the conditions and variable parameter σ as described above 1 And σ 2 Chaotic 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 may lead to poor performance of PMSM, it is highly desirable 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, the problem from asymmetric input-output constraints needs to be solved.
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 x 1 Following the desired signal y d ；
(c) Does not violate control input constraints;
(d) output x 1 Is defined as
To design the adaptive neural inversion control of step (3), the following assumptions and reasoning are given:
assume that 1: variable sigma i I is 1,2 and δ i I is unknown but bounded, i.e. 1,2,3
σ im ≤σ i ≤σ iM ,|δ i |≤δ M In the formula (4), sigma im ,σ iM I is 1,2 and delta M ,(δ M >0) Is a real number, δ i Is an estimation error, which will be explained later;
assume 2: there is a desired trajectory
And time derivative thereof
And
satisfy the inequality
Wherein
And XIs a positive real number;
assume that 3: presence of real number c i >0, e.g. | D i |≤c i ,i＝1,2,3；
Introduction 1: for the
Obtaining:
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
P i+1 (x)＝2xP i (x)-P i-1 (x),P 0 (x)＝1 (6)
Wherein x ∈ R and P 1 (x) Denoted by x,2x,2x-1 or 2x +1, where the first term is used. Chebyshev polynomial x ═ x (x) 1 ,...,x m ) T ∈R m The enhancement mode is given by:
φ(x)＝[1,P 1 (x 1 ),...,P n (x 1 ),...,P 1 (x m ),...,P n (x m )] T (7)
in the formula, φ (x) represents the vector of the basis function of the Chebyshev polynomial, P i (xj), i 1, n, j 1, m being the order of the chebyshev polynomials and n representing the order;
thus f will be * (x) Is defined as
f * (x)＝W *T φ(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
Wherein W is [ omega ] 1 ,ω 2 ,...,ω 3 ] T ∈R l Is a weight vector;
in the step of designing the self-adaptive neural inversion control scheme, a Chebyshev neural network W is adopted i T φ i I ═ 1,2,3 approximately unknown uncertainties f i * (x) Existence of
f i * (x)＝W i T φ i +δ i ,i＝1,2,3 (10)
In the formula, W i ＝W i * And f i * (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
θ i ＝||W i || 2 ＝W i T W i ,i＝1,2,3 (11)
Wherein | | · | | and θ i Respectively represent W i And 2-norm of unknown variable;
nussbaum type function:
due to sigma in the formula (2) 1 The sign of (3) causes an unknown control direction problem, and thus, is directed to σ in equation (2) 1 The Nussbaum-type function is introduced,
definition 1: if the continuous even function N (χ) satisfies:
the continuous even function is called the type NussbaumFunction, many of which satisfy both the formula (12) and the formula (13), e.g. χ 2 cos (χ) and
here, χ is adopted 2 cos(χ)。
2, introduction: if a non-negative smoothing function V (t) satisfies:
wherein χ (t) ≧ 0 is defined as [0, t ≧ 0 f ) A smoothing function of c 0 Is a real number and c 0 >0, N (-) is an even number Nussbaum type function, and g is defined in the set
Variables of, V (t), χ (t) and
at [0, t f ) An upper bound;
creating a uniform barrier Lyapunov function:
to handle asymmetric output constraints, a new conversion formula is defined as
In the formula, positive real number
And
as an original boundary, λ 1 (t) is the tracking error given later, μ and s (t) are the transition boundary and transition error, respectively;
from t ∈ [0, ∞))
And
using the barrier Lyapunov function of equation (15) and the logarithmic model, a uniform barrier Lyapunov function is created as
Wherein log (-) is the natural logarithm of (-);
the resulting formula (16) satisfies the Lyapunov design principle
It ensures that S is restricted to set P S :＝{-μ<S<μ }. Based on equation (15), the tracking error λ is further derived 1 Is limited to a set
Performing the following steps; for clarity of illustration, a schematic diagram is shown in FIG. 2.
3, management: for the
And
exist of
In the formula, K S ＝S/(μ 2 -S 2 )；
Definition 2: the unified barrier Lyapunov function formula (16) is proposed by fusing a new transformation formula (15) into a conventional logarithmic barrier Lyapunov function to bypass the complex derivation caused by the segmentation formula existing in the general segmented barrier Lyapunov function, and has a greater potential versatility in terms of the design of a constraint controller for a nonlinear system than 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
In the formula, beta 2 Representing a virtual controller, with the real number C being x 3 The initial value of (1);
definition 3: error control plane lambda conventional to conventional inversion controller techniques 2 In contrast, by letting λ 3 ＝x 3 -input u of C design (2) d The 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 adopted i Is defined as
In the formula, auxiliary power system
This will be given later;
the two are connected (2) and (17), and lambda is deduced 1 And z i The time derivative of i ═ 2,3 is
In the formula (I), the compound is shown in the specification,
defining an error variable
Is composed of
In the formula, variable
Is theta i An estimated value of (d);
then, the controller design steps based on the traditional inversion controller framework are given:
first, selecting a barrier Lyapunov function
In the formula, r 1 Is a real number and r 1 >0；
V in formula (22) is derived from formulas (15), (16) and (21) 1 Is a time derivative of
By the formula (20), obtaining
In the formula, k 1 >0 is a design parameter and uncertainty is unknown
It can be seen that the uncertainty f is unknown 1 * From an uncertainty parameter σ 1 And external disturbance D 1 And a load torque T L The 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 are used, respectively
To estimate the unknown gain sigma 1 And unknown uncertainty f 1 * ；
According to formulae (4), (5), (10) and (11), there are obtained
In the formula, a 1 Is a real number and a 1 >0；
By substituting formula (24) for formula (25)
Designing a virtual input beta 2 And a new law of adaptation
Is composed of
In the formula, gamma>0 and l 1 >0 and are all real numbers, and the number of the bits is zero,
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
The second step is that: establishing a Lyapunov function as
In the formula, r 2 >0 and is a real number;
by means of the formula (21), the time derivative V in the formula (32) is determined 2 Is composed of
Designing an auxiliary power system
Is composed of
In the formula, k 2 >0 and is a real number;
from the formulae (20) and (34) gives
To overcome the defect caused by the calculation formula (35)
The complexity explosion problem caused by the difference of virtual controllers, the concept of tracking differentiators is introduced:
in the formula, an input signal beta 2 Is obtained by the method of the formula (27),
and
are all real, v 1 V and v 2 Are each beta 2 And
an estimated value of (d);
and (4) introduction: if the initial deviation is
In
And is real, then v 2 Satisfy the requirement of
In the formula I ν2 >0 and is an unknown real number;
by substituting formulae (31), (35) and (37) for formula (33)
In the formula, the uncertainty will not be known
Is defined as
Definition 4: different from the former tool based on a first-order filter, a tracking differentiator is designed to obtain beta 2 Can improve the tracking accuracy of the proposed solution.
Uncertainty of unknown
Is a complex nonlinear function with effects due to coupling terms between velocity and current, unknown system dynamics and errors. For facilitating subsequent design, Chebyshev neural network is adopted
Evaluation of
Analogously to formula (25), obtain
In the formula, a 2 >0 and is a real number;
by substituting formula (39) for formula (38) to give
Design control input u q And law of adaptation
Is composed of
In the formula I 2 >0 and is a real number;
the formula (40) is re-expressed as the formula (41) and the formula (42)
The third step: designing a Lyapunov function as
In the formula, r 3 >0 and is a real number;
similar to equation (34), consider an auxiliary power system
In the formula, k 3 >0 and is a real number;
combined formula (20) to obtain
Then, V in the formula (44) is obtained 3 Is a time derivative of
The formula (46) is re-expressed as the formula (43) and the formula (46)
In the formula, uncertainty is unknown
As can be clearly seen, the uncertainty is unknown
Are adversely affected by external disturbances and systematic errors. To overcome the above-mentioned adverse effects, the Chebyshev neural network is utilized
Come to approach
Analogously to formula (25), obtaining
In the formula, a 3 >0 and is a real number;
by the formula (49), the formula (48) is simplified to
Design control input u d And new control law
Is composed of
In the formula I 3 >0 and is a real number;
the expression (50) is re-expressed as
Using the equations (5) and (21), the derivation is made
Then obtain
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 [ J ] of the fractional order nonlinear system with time lag and input saturation in the literature (Song comma, Park Ju Hyun, Zhang Baoyong)]Applied, matrix, composite, 2020,364, S, Song, J, H, park, B, Zhang, and X, Song, "Adaptive hybrid feedback output feedback control for reactive-order nonlinear systems with time-varying delay and input consumption," applied, matrix, composite, vol.364, p, 124662,2020) g And v g G ═ d, q inconsistency, and can bypass literature (congratulatory, duculol, scout]Franklin Instrument, 2018,355(15):7548-7569, G.Zhu, J.Du, and Y.Kao, "Command filtered robust adaptive NN control for a class of uncoordinated stress-feedback non-linear systems input maintenance," J.Franklin Instrument, vol.355, No.15, pp.7548-7569,2018).
Definition 6: since the Chebyshev neural network can only be determined by the order of the Chebyshev polynomial, this work utilizes the Chebyshev neural network to determineApproximating the unknown uncertainty f i * I-1, 2,3 instead of the most common radial basis function neural network determined based on the center and width of the gaussian function, connected (25), (39) and (49), the minimum learning parameter technique is inserted in the design of the controller and the adaptive law, which benefits from the reduction of design parameters and 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
Theorem 1: on the premise of satisfying assumptions 1-3, for chaotic PMSM with unknown uncertainty and asymmetric input-output constraints, control law expressions (27), (41), (51) and adaptive laws (30), (42) and (52) are designed, and when conditions are satisfied
And
and when the requirements are met, the scheme of the invention can ensure the realization.
And (3) proving that: constructing the whole Lyapunov function
From formula (55)
From the formula (17) can be obtained
Namely that
Where ρ ═ min {2k } 1 ,2k 2 ,2k 3 ,l 1 ,l 2 ,l 3 }，
By integrating equation (60) over the [0, t ] interval, one can obtain
By factoring 2, the functions V (t), χ and
has an interval of [0t]。
Equation (61) may be re-expressed as
Wherein C is 0 Is that
The upper limit of (2).
Thus can obtain
In addition, can obtain
From formula (15) can be obtained
The expressions (64) and (65) denote S and lambda, respectively 1 Is bounded. Likewise, z can be obtained 2 、z 3 And
is bounded. The compound represented by the formula (21) can be further obtained
Is bounded. By means of the formulae (3), (27), (41) and (51), the variable β can be deduced 2 、ν g And u g And g is bounded by a and d. Accordingly, Du g ＝u g -v g And g is d and q is bounded.
In addition, to ensure error control surface
Convergence of i 2,3, also needs to be considered
Is well-defined. To this end, we designed a suitable Lyapunov function as
Passing formula (34), formula (45) and derivative
Can obtain
Wherein | Δ u | ═ max { | Δ u |) d |,|Δu q |}。
From formula (5) can be obtained
Can then obtain
Wherein a is 0 ＝min{2k i -1}, wherein k i >1/2,i＝2,3，b 0 ＝Δu 2 。
By solving the formula (69), the
Analogously to formula (64), can be obtained
By the formula (71), it can be understood that
And
is bounded. In formula (19), represented by i And
can infer the error control plane lambda 2 And λ 3 Is bounded. To sum up, all variables of the PMSM are bounded.
Further, it can be seen from the expressions (15) and (16)
When the temperature of the water is higher than the set temperature,
to pair
The guarantee is ensured. In addition, due to
And
can obtain
Then, by
And
can deduce
Thus, the demonstration of stability analysis was completed.
Definition 7: tracking error lambda 1 Is a visual index for controlling performance, and can be selected properly
And ρ is arbitrarily set small. We can get specific adjustment criteria to increase the parameter k i ,a i ,r i γ, wherein k 2 >1/2,k 3 >1/2, decreasing the parameter
It should be noted that v q Subject to amplitude v 2 The influence of (c). For this purpose, first of all by selecting suitable ones
k 1 ,r 1 ,a 1 ,l 1 Adjusting a suitable value to v 2 And then adjust other parameters.The values of the control parameters may be tested repeatedly using the specific adjustment criteria described above. In practice, k is used to achieve a predetermined target i ,a i ,r i γ and l i The value of i-1, 2,3 needs to be adjusted explicitly.
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 y d Sin (t). The upper and lower bounds of the asymmetric input constraint are u max 25 and u min -3. For equation (2), the output variable satisfies the constraint condition-1.2<x 1 <1.24. Can obtain
And
initial conditions are x 1 (0)＝0.1∈(-1.2,1.24)，x 2 (0)＝0.9,x 3 (0)＝20,χ(0)＝1.55,ν 1 (0)＝1 , 
Then the parameter is chosen to be k 1 ＝97,r 1 ＝0.001,k i ＝r i ＝1，i＝2,3,a 1 ＝51,a 2 ＝121,a 3 ＝71,T L ＝3,l 1 ＝0.2,l 2 ＝6.2,l 3 ＝4.8,
And gamma is 0.1 external interference
The order of equation (7) is designated as 2 using a single-layer chebyshev neural network. The basis function of the Chebyshev polynomial can be described as
φ i (x)＝[1,P 1 (x 1 ),P 2 (x 1 ),...,P 1 (x 3 ),P 2 (x 3 )] 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 λ 1 May remain unchanged within a certain range. State variable i q .i d And a real controller u q ,u d The 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). Neglecting the asymmetric input-output control and considering ud as 0, the actual proportional-integral-derivative control is
Wherein k is P ,k I ,k D Are 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
β 2f (0)＝β 2 (0) Definition of lambda 2 ＝x 2 -β 2f Wherein beta is 2f Tau for the stability controller to be designed 2f >0 is a design real number. Then the corresponding controller v designed by the formula (41) q Is modified to
Also, the following indices are defined for comparison
Where N is the number of samples, M λ 、μ λ And σ λ Respectively represent λ 1 (i) Maximum, mean and standard deviation values of;
the simulation tests were performed under different unknown external disturbances, case 1:
case 2:
in all comparisons, k is selected P ＝-140,k I ＝-0.08,k D -160 and τ 2f 0.05. The remaining parameters and conditions are provided in subsection A. In the range of 0to 30s, we calculated the simulation and quantification indicators.
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 seen q Beyond the boundary value. In contrast, the other two controllers constrain u q The 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 schemes λ Are 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 adaptive neural inversion control is the best for the PMSM. Therefore, the conclusion can be drawn that the designed scheme has high precision in the aspect of controlling the PMSM chaotic system by using unknown external interference and asymmetric input and output constraintsAnd (4) degree.
TABLE II Performance index comparison results
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 direction, complexity explosion, unknown uncertainty and heavy calculation burden generated in the design are solved through a Nussbaum type function, 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) providing 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 integrated 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 an embodiment 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 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:
in the formula (I), the compound is shown in the specification,
and
representing the d-axis and q-axis currents,
and
representing the d-axis and q-axis voltages as system inputs, L,
R,
ψ r b, J and n p Respectively 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 L d ＝L q Definition of
And
n p ＝1，x 1 ＝ω,x 2 ＝i q ，x 3 ＝i d ,L＝L d ＝L q considering unknown external interference and asymmetric input saturation, a simplified dimensionless model of formula (1) is obtained:
in the formula (I), the compound is shown in the specification,
σ 1 ＝BL/(JR)，σ 2 ＝-n p ψ r 2 /(BR)，
and
Δ i i ═ 1,2,3 is unknown external interference;
in the formula, x 1 Representing nominal angular velocity, x 2 Representing the q-axis current, x 3 Denotes d-axis current, T denotes time, T L Represents the load, u d Denotes the d-axis voltage, u q Representing the q-axis voltage, σ 1 And σ 2 Representing an unknown parameter;
asymmetric input saturation is expressed as:
in the formula u max And u min Representing the amplitude, v, of the asymmetrical input saturation g And u g Input and output representing asymmetric input saturation, respectively;
(2) setting a control object:
(a) all variables in the PMSM chaotic system are bounded;
(b) output x 1 Following the desired signal y d ；
(c) Does not violate control input constraints;
(d) output x 1 Is defined as
Setting 1: variable sigma i I is 1,2 and δ i I is unknown but bounded, i.e. 1,2,3
σ im ≤σ i ≤σ iM ,|δ i |≤δ M (4)
In the formula, σ im ,σ iM I is 1,2 and δ M ,(δ M > 0) is a real number, δ i Is the estimation error;
setting 2: there is a desired trajectory
And time derivative thereof
And
satisfy inequality
Wherein
And xi are positive real numbers;
setting a reference value 3: presence of real number c i >0, e.g. | Δ i |≤c i ,i＝1,2,3；
Introduction 1: for
Obtaining:
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) Deriving the Chebyshev polynomial by the following formula
P i+1 (x)＝2xP i (x)-P i-1 (x),P 0 (x)＝1 (6)
Wherein x ∈ R and P 1 (x) Is x, x ═ x (x) of chebyshev polynomial 1 ,...,x m ) T ∈R m The enhancement mode is given by:
φ(x)＝[1,P 1 (x 1 ),...,P n (x 1 ),...,P 1 (x m ),...,P n (x m )] T (7)
in the formula, phi (x) represents the vector of the Chebyshev polynomial basis function, P i (x j ) 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 *T φ(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
Wherein W is [ omega ] 1 ,ω 2 ,...,ω l ] T ∈R l Is a weight vector;
using Chebyshev neural networks, W i T φ i I 1,2,3 is approximately unknown uncertainty, there is
f i * (x)＝W i T φ i +δ i ,i＝1,2,3 (10)
In the formula, W i ＝W i * And f i * (x)；
Definition of
Wherein | | · | | and
respectively represent W i And 2-norm of unknown variable;
for σ in formula (2) 1 Introducing a Nussbaum type function;
definition 1: if the continuous even function N (χ) satisfies:
the continuous even function is called Nussbaum-type function, and many functions satisfy both (12) and (13), such as χ 2 cos (χ) and
here, χ is used 2 cos(χ)；
2, leading: if a non-negative smoothing function V (t) satisfies:
wherein χ (t) ≧ 0 is defined as [0, t ≧ 0 f ) A smoothing function of c 0 Is a real number and c 0 >0, N (. cndot.) is an even NF function, g is defined in the set
Variables of, V (t), χ (t) and
at [0, t f ) An upper bound;
for asymmetric output constraints, a new conversion formula is defined as
In the formula, positive real number
And
as an original boundary, λ 1 (t) is the tracking error given later, μ and s (t) are the transition boundary and transition error, respectively;
derived from t ∈ [0, ∞))
And
using equation (15) and a logarithmic-type performance barrier Lyapunov function, a uniform barrier Lyapunov function is created as
Where log (-) is the natural logarithm of (-); s is a conversion error;
the obtained formula (16) satisfies the design principle of Lyapunov function
Based on equation (15), the tracking error λ is further derived 1 Is limited to a set
Performing the following steps;
and 3, introduction: for
And
exist of
In the formula, K S ＝S/(μ 2 -S 2 ) (ii) a S is a conversion error;
(2) building an adaptive inversion controller
Defining the error control plane as
In the formula, beta 2 Representing a virtual controller, with the real number C being x 3 An initial value of (1);
definition 3: enhanced dynamic error z i Is defined as
In the formula, auxiliary power system
As will be given below;
the two are connected (2) and (17), and lambda is deduced 1 And z i The time derivative of i ═ 2,3 is
In the formula (I), the compound is shown in the specification,
defining error variables
Is composed of
In the formula, variable
Is composed of
An estimated value of (d);
a controller design step based on an inversion controller framework:
first, a performance barrier Lyapunov function is selected:
in the formula, r 1 Is a real number and r 1 ＞0；
V in formula (22) is derived from formulas (15), (16) and (21) 1 Is a time derivative of
By the formula (20), obtaining
In the formula, k 1 0 is a design parameter and uncertainty is unknown
Using Nussbaum-type functions and Chebyshev nerves, respectivelyNetwork W 1 T φ 1 To estimate the unknown gain sigma 1 And unknown uncertainty f 1 * ；
According to formulae (4), (5), (10) and (11), there are obtained
In the formula, a 1 Is a real number and a 1 ＞0；
By substituting formula (24) for formula (25)
Designing a virtual input beta 2 And new law of adaptation
Is composed of
In the formula, upsilon >0 and l 1 Is greater than 0 and is a real number,
representing the secondary controller, χ representing a variable of a Nussbaum-type function; in the formulae (27) to (30) to (26), the derivation is carried out
The second step is that: establishing a Lyapunov function as
In the formula, r 2 Is greater than 0 and is a real number;
by means of the formula (21), the time derivative V in the formula (32) is determined 2 Is composed of
Designing an auxiliary power system
Is composed of
In the formula, k 2 Is greater than 0 and is a real number;
from the formulae (20) and (34) gives
The concept of a tracking differentiator is introduced:
in the formula, an input signal beta 2 Is obtained by the method of the formula (27),
and
are all real, v 1 V and v 2 Are each beta 2 And
an estimated value of (d);
and (4) introduction: if the initial deviation is
In
And is real, then v 2 Satisfy the requirement of
In the formula (I), the compound is shown in the specification,
and is an unknown real number;
by substituting formulae (31), (35) and (37) for formula (33)
In the formula, the uncertainty will not be known
Is defined as
Definition 4: using Chebyshev neural networks
Evaluation of
Analogously to formula (25), obtaining
In the formula, a 2 Is greater than 0 and is a real number;
by substituting formula (39) for formula (38) to give
Design control input u q And law of adaptation
Is composed of
In the formula I 2 Is greater than 0 and is a real number;
the formula (40) is re-expressed as the formula (41) and the formula (42)
The third step: designing a Lyapunov function as
In the formula, r 3 Is greater than 0 and is a real number;
similar to equation (34), consider an auxiliary power system
In the formula, k 3 Is greater than 0 and is a real number;
combined formula (20) to obtain
Then, V in the formula (44) is obtained 3 Is a time derivative of
The formula (46) is re-expressed as the formula (43) and the formula (46)
In the formula, uncertainty is unknown
Using Chebyshev neural networks
To approach
Analogously to formula (25), obtaining
In the formula, a 3 Is greater than 0 and is a real number;
by the formula (49), the formula (48) is simplified to
Design control input u d And new control law
Is composed of
In the formula I 3 Is greater than 0 and is a real number;
the expression (50) is re-expressed as
Using the equations (5) and (21), the derivation is made
Then obtain
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