CN111290015B - Fractional order self-sustaining type electromechanical seismograph system acceleration stability control method with constraint - Google Patents

Fractional order self-sustaining type electromechanical seismograph system acceleration stability control method with constraint Download PDF

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CN111290015B
CN111290015B CN202010157834.9A CN202010157834A CN111290015B CN 111290015 B CN111290015 B CN 111290015B CN 202010157834 A CN202010157834 A CN 202010157834A CN 111290015 B CN111290015 B CN 111290015B
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fractional order
function
control
representing
acceleration
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CN111290015A (en
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罗绍华
李少波
胡雪纯
宋永端
F.路易斯
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Guizhou University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/24—Recording seismic data
    • G01V1/242—Seismographs
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/16—Receiving elements for seismic signals; Arrangements or adaptations of receiving elements
    • G01V1/18—Receiving elements, e.g. seismometer, geophone or torque detectors, for localised single point measurements
    • G01V1/181—Geophones
    • G01V1/182—Geophones with moving coil
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/16—Receiving elements for seismic signals; Arrangements or adaptations of receiving elements
    • G01V1/162—Details
    • G01V1/164—Circuits therefore
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D19/00—Control of mechanical oscillations, e.g. of amplitude, of frequency, of phase
    • G05D19/02—Control of mechanical oscillations, e.g. of amplitude, of frequency, of phase characterised by the use of electric means

Abstract

The invention relates to an acceleration stability control method for a constrained fractional order self-sustaining electromechanical seismograph system, and belongs to the field of seismic exploration. The method comprises the following steps: s1: the method comprises the following steps of system modeling, namely establishing a mathematical model of a fractional order self-sustaining type electromechanical seismograph system according to a Newton second law and a kirchhoff law, and defining constraint conditions; s2: designing an acceleration stability controller, which comprises constructing an acceleration feedforward controller and an optimal feedback controller; the acceleration feedforward controller is integrated by a modeling behavior function based on a fractional order inversion method, a fuzzy wavelet neural network and a tracking differentiator; the optimal feedback controller is formed by fusing a fuzzy wavelet neural network and a self-adaptive dynamic programming strategy. The invention can inhibit chaotic oscillation and realize the minimized cost function while ensuring the boundedness of all signals of the closed-loop system and ensuring the safe operation of the system under the constraint condition.

Description

Fractional order self-sustaining type electromechanical seismograph system acceleration stability control method with constraint
Technical Field
The invention belongs to the field of seismic exploration, and relates to a method for controlling acceleration stability of a constrained fractional order self-sustaining electromechanical seismograph system.
Background
A self-contained electromechanical seismograph system is an electromechanical sensitive instrument and can be used for detecting ground vibration caused by earthquakes. Nonlinear oscillation and chaotic dynamics of a seismograph system have very important influence on stable operation of the seismograph system, and performance indexes are often required to be met in engineering application. Therefore, modeling fractional order self-sustaining electromechanical seismograph systems in an optimal form, dynamics analysis, driving chaotic motion to periodic target trajectories, satisfying system safety constraints, and accelerating stabilization are meaningful and challenging tasks.
There have been sporadic reports of dynamics issues with such seismograph systems over the past several decades. Siewe et al first studied the nonlinear response and chaotic control of the electromechanical seismometer system to the fifth resonance excitation, and they further studied the divergence and control of the co-host orbit of the seismometer system using an analytical method. Hegazy discusses the nonlinear dynamics and vibration control of electromechanical seismometer systems with time-varying stiffness. However, these works are limited only to the dynamics of the integer order electro-mechanical seismograph system and do not accurately describe its operation.
Fractional calculus has received extensive attention in academia since it has proven to be an effective tool for modeling system dynamics with greater accuracy and more design freedom. Later, many research results on fractional order control strategies, such as PI, were reported in successionλDμControl, sliding mode control, robust control, adaptive control and the like. Inversion is well known as a good way to control an uncertain integer order nonlinear system with a triangular structure. Some researchers have further generalized the inversion method to fractional order nonlinear systems, particularly chaotic systems. Liu et al studied the fuzzy inversion control problem of fractional order nonlinear systems and fractional order neural networks with nonlinear inputs and unknown dynamical models. Wei et al discussed the adaptive inversion control problem for fractional order systems and asymmetric fractional order systems using a frequency distribution model. With the increase of the order of the system, the fractional order derivation of the methods can generate 'explosion terms', and meanwhile, the problem of low approximation precision can occur when the general fuzzy logic faces a high-dimensional hyper-chaotic system. Furthermore, regulatory performance control issues, including transient dynamic and steady state responses, cannot be addressed therein.
There are zero and differential countermeasures problems in engineering that involve less resource consumption, and how to approximate the solution of the hamilton-jacobi-isax equation to obtain a nash equilibrium solution becomes somewhat tricky. To address this problem, a learner, such as the Valvouakis, uses data measured along the player trajectory to perform reinforcement learning to solve the multi-player game problem. To solve the zero and derivative countermeasures problem, Modares and Lewis propose an integral reinforcement learning and H ∞ control strategy for constrained input systems. These works are effective for optimal control of integer order systems, not fractional order electro-mechanical seismometer systems. Therefore, how to design an acceleration stability controller with optimal performance for a fractional order electromechanical device remains an open question.
For nonlinear systems given transient and stable behavior, the prescribed performance control has become another interesting but challenging topic. It is worth noting that constraints from safety regulations and physical failures in case of violation of the constraints can lead to degraded performance and unstable operation of the controlled system. The barrier lyapunov function method is used to solve the state constraint problem. In combination with the specified performance control and the barrier Lyapunov function, Zhao et al solve the zero-error tracking problem of the Euler-Lagrange system with full-state constraint and nonparametric uncertainty. Huang et al discusses adaptive neural control for a rigid feedback system with full state constraints at a given performance index. In the fractional order domain, these works are no longer applicable and there is a huge difference between a class of rigid feedback systems and self-sustaining electromechanical seismograph systems with complex dynamics. Furthermore, the optimal energy cost of a non-linear system is not addressed in this type of document.
Disclosure of Invention
In view of the above, the present invention provides a constrained fractional order self-sustaining electromechanical seismograph system acceleration stability control method, which solves the problems of dynamics analysis and acceleration stability control of a fractional order self-sustaining electromechanical seismograph system under an energy mechanism. By using the method, the boundedness of all signals of the closed-loop system is ensured, the safe operation of the system under the condition of meeting the constraint condition is ensured, and the purposes of inhibiting chaotic oscillation and realizing the minimized cost function are achieved.
In order to achieve the purpose, the invention provides the following technical scheme:
a method for controlling acceleration stability of a constrained fractional order self-supporting electromechanical seismograph system is characterized in that a gyro coupling fractional order equation of the control system is established, a phase diagram and a Lyapunov exponent are used for performing dynamics analysis, and strong dependence of chaotic behaviors and periodic behaviors on system physical parameters and fractional order orders is found. The whole controller is composed of an acceleration feedforward controller and an optimal feedback controller. In the acceleration feedforward controller, a modeling behavior function is used for accelerating the convergence of tracking errors in controllable time and speed, unknown items of a transformed fuzzy wavelet neural network approximation system are used, and a tracking differentiator is arranged for solving the complexity problem of the modeling behavior function and fractional order under the framework of a fractional order inversion method. In an optimal feedback controller, an adaptive dynamic programming strategy is proposed to deal with the zero and differential countermeasure solution problem, in which the solution of the constrained hamilton-jacobi-isax equation is solved approximately on-line using a fuzzy wavelet neural network. The method specifically comprises the following steps:
s1: the method comprises the following steps of system modeling, namely establishing a mathematical model of a fractional order self-sustaining type electromechanical seismograph system according to a Newton second law and a kirchhoff law, and defining constraint conditions;
s2: designing an acceleration stability controller, which comprises constructing an acceleration feedforward controller and an optimal feedback controller; the acceleration feedforward controller is integrated by a modeling behavior function based on a fractional order inversion method, a fuzzy wavelet neural network and a tracking differentiator; the optimal feedback controller is formed by fusing a fuzzy wavelet neural network and a self-adaptive dynamic programming strategy.
Further, in step S1, the mathematical model of the fractional order self-sustaining electro-mechanical seismograph system is:
Figure BDA0002404715200000031
where m is the mass, x is the elongation of the spring, k0Is the linear spring constant, k1Is the cubic spring constant, k2Is the fifth order spring coefficient, B represents the magnetic field, l represents the length of the wire; alpha is the order of fractional order, q is the instantaneous charge, C0Is the linear part of a capacitor, I0Is the initial current, aaAnd abIs the coefficient of the nonlinear part of the capacitor;
Figure BDA0002404715200000032
represents the current; f. ofv＝f1+f0cos Ω t represents random vibration due to ground acceleration, f1Represents the critical force and is assumed to be 0, f0And Ω represents the amplitude and frequency of the noise term;
the gyro coupling fractional order control equation of the system is as follows:
Figure BDA0002404715200000033
wherein u isiI-2, 4 denotes an increasing control input; defining dimensionless parameters:
Figure BDA0002404715200000034
Figure BDA0002404715200000035
x1＝y,x3z, C denotes kaputol definition, μ0Representing a damping system, L being a linear inductor, Q0Represents a reference charge, R is a resistance; new dimensionless variables are defined as y ═ x/l, and z ═ Q/Q0,τ＝ωet,
Figure BDA0002404715200000036
Further, in step S1, the defined constraint condition specifically includes:
definition 1: for the real function f (t), the kapton fractional derivative is given as:
Figure BDA0002404715200000037
wherein the content of the first and second substances,
Figure BDA0002404715200000038
n-1 < alpha < n,
Figure BDA0002404715200000039
a gamma function of (a);
taking laplace transform of the above equation:
Figure BDA00024047152000000310
if F(k)(0)＝0,k＝0,1,…,max(pα,pβ) Is formed, wherein pα,pβIf greater than 0, then
Figure BDA00024047152000000311
Definition 2: defining a constrained cost function as:
Figure BDA0002404715200000041
where Q (S) > 0, S and U 'represent a penalty function, a tracking error and a positive definite function, and U' is defined as:
Figure BDA0002404715200000042
wherein R isoRepresenting a symmetric positive definite matrix, λoWhich is a representation of a normal number,
Figure BDA0002404715200000043
representing an optimal control input, and upsilon representing a variable;
definition 3: analogous for kappa (t) ═ 1+ l0(t-t0)2、κ(t)＝exp(l0(t-t0) Or κ (t) ═ 1+ tan (0.5 π tanh (t-t)0) Etc.) of the rate function
Figure BDA0002404715200000044
Has the following characteristics: 1) kappa (t) > 1, where t > t0And κ (t)0)＝1；2)
Figure BDA0002404715200000045
Wherein t is more than or equal to t0；3)
Figure BDA0002404715200000046
Is bounded; wherein l0Denotes a constant, t0Represents an initial time;
suppose that: target trajectory xidI 1,3 and the corresponding fractional order derivative are known and bounded, then
Figure BDA00024047152000000419
Wherein A isi＞0，
Figure BDA00024047152000000420
Further, in the step S2, the constructing the acceleration feedforward controller specifically includes:
to specify steady state error and tracking accuracy in a given time, a modeling behavior function is introduced
Figure BDA0002404715200000047
Wherein etab＞1，
Figure BDA0002404715200000048
And
Figure BDA0002404715200000049
defining the error variables as:
Figure BDA00024047152000000410
wherein alpha isi+1Representing a virtual control;
step 1: to ensure x1And (3) satisfying the constraint condition, and selecting a first Lyapunov candidate function as follows:
Figure BDA00024047152000000411
wherein the content of the first and second substances,
Figure BDA00024047152000000412
the virtual control is selected accordingly as:
Figure BDA00024047152000000413
obtaining V in conjunction with virtual control1The derivative of (c) is:
Figure BDA00024047152000000414
wherein β represents the modeling behavior function, k1＞0，
Figure BDA00024047152000000415
And
Figure BDA00024047152000000416
step 2: selecting a second Lyapunov candidate function as:
Figure BDA00024047152000000417
wherein the content of the first and second substances,
Figure BDA00024047152000000418
an estimate representing a degree of shooting;
approximating on a compact set using a fuzzy wavelet neural network and reconstructing a fractional tracking differentiator based on an integer tracking differentiator:
Figure BDA0002404715200000051
wherein the content of the first and second substances,
Figure BDA0002404715200000052
and
Figure BDA0002404715200000053
indicating the state of the tracking differentiator and,
Figure BDA0002404715200000054
representing the input signal of a tracking differentiator，ciAnd σiIndicates that c is satisfiedi0 and 0 < sigmaiA design constant < 1;
the control inputs for the design adaptation law are:
Figure BDA0002404715200000055
Figure BDA0002404715200000056
combining the fractional order tracking differentiator and the control input to obtain V2The fractional derivative of (a) is:
Figure BDA0002404715200000057
wherein k is2Which is indicative of a normal number of the cells,
Figure BDA0002404715200000058
and step 3: selecting a third Lyapunov candidate function as:
Figure BDA0002404715200000059
wherein
Figure BDA00024047152000000521
The virtual control selection is as follows:
Figure BDA00024047152000000510
wherein k is3Represents a normal number;
determining V in conjunction with virtual control3The derivative of (c) is:
Figure BDA00024047152000000511
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00024047152000000512
and
Figure BDA00024047152000000513
and 4, step 4: the fourth Lyapunov candidate function is selected as:
Figure BDA00024047152000000514
the control inputs and adaptation laws were designed as follows:
Figure BDA00024047152000000515
Figure BDA00024047152000000516
wherein k is4Denotes the normal number, Z4Representing the state of a fractional order tracking differentiator;
combining control inputs and adaptive laws to determine V4The derivative of (c) is:
Figure BDA00024047152000000517
wherein
Figure BDA00024047152000000518
ks＝min{k1 k2 k3 k4}，
Figure BDA00024047152000000519
And
Figure BDA00024047152000000520
further, in step S2, the constructing an optimal feedback controller specifically includes: first, a controlled system is givenStatistics, cost functions and optimal cost functions, the existence of which satisfies
Figure BDA0002404715200000061
Continuous differentiable and unbounded Lyapunov function JaWherein
Figure BDA0002404715200000062
To represent
Figure BDA0002404715200000063
Partial derivatives of (d); defining a symmetric positive definite matrix so that
Figure BDA0002404715200000064
Then, a fuzzy wavelet neural network is adopted to estimate a cost function, and weight approximation errors are introduced
Figure BDA0002404715200000065
The optimal control inputs are obtained as:
Figure BDA0002404715200000066
wherein
Figure BDA0002404715200000067
Therefore, a weight self-adaptation law associated with the fuzzy wavelet neural network is deduced, namely:
Figure BDA0002404715200000068
wherein the content of the first and second substances,
Figure BDA0002404715200000069
korepresenting a design parameter, z1And z2Representing the regulating parameters, the controlled system is:
Figure BDA00024047152000000610
g denotes a fourth order identity matrix.
The invention has the beneficial effects that:
1) the invention establishes a fractional order dynamic model of the self-sustaining electromechanical seismograph system, accurately describes the dynamic characteristics of the system and increases the design freedom of the controller.
2) In order to better detect and record beneficial vibration of the ground, the invention provides an acceleration stability control scheme for converting chaotic motion into conventional motion of a system, so that convergence and stability are accelerated, and the influence of uncertainty and chaotic oscillation is overcome.
3) The overall control strategy of the present invention includes an acceleration feedforward controller and an optimal feedback controller. The acceleration feedforward controller is integrated by a modeling behavior function based on a fractional order inversion method, a fuzzy wavelet neural network and a tracking differentiator, and the optimal feedback controller is formed by fusing the fuzzy wavelet neural network and a self-adaptive dynamic programming strategy. The control strategy not only ensures the boundedness of all signals of the closed-loop system, but also ensures the safe operation of the system under the condition of meeting the constraint condition, and achieves the purposes of inhibiting chaotic oscillation and realizing the minimized cost function.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.
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For the purposes of promoting a better understanding of the objects, aspects and advantages of the invention, reference will now be made to the following detailed description taken in conjunction with the accompanying drawings in which:
FIG. 1 is a schematic diagram of an accelerated stability control method of the present invention;
FIG. 2 is a schematic diagram of a self-contained electromechanical seismograph system;
FIG. 3 is a phase diagram at different fractional orders and sets of parameters 1;
FIG. 4 is a phase diagram at different fractional orders and set 2 parameters;
FIG. 5 is a simulation diagram of the relationship between Lyapunov exponent and time history and fractional order;
FIG. 6 is a simulation diagram of target trajectory tracking under different parameters and fractional orders under the condition of the 1 st scenario;
FIG. 7 is a graph of performance for different parameters and fractional order for case 1;
FIG. 8 is a simulation graph comparing the performance of the method of the present invention with AFBC under a verified embodiment;
reference numerals: 1-spring, 2-seismic mass block, 3-shock absorber, 4-frame, 5-coupling magnet coil, 6-permanent magnet, 7-bolt, 8-base.
Detailed Description
The embodiments of the present invention are described below with reference to specific embodiments, and other advantages and effects of the present invention will be easily understood by those skilled in the art from the disclosure of the present specification. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and examples may be combined with each other without conflict.
Referring to fig. 1 to 8, a method for controlling acceleration and stability of a constrained fractional order self-sustaining electro-mechanical seismograph system is shown in fig. 1, and includes the following steps:
firstly, modeling a system.
The fractional order self-sustaining electromechanical seismograph system shown in FIG. 2 is comprised of electrical and mechanical parts, where the former is comprised of a linear inductor L, a nonlinear capacitor CNLA non-linear resistor RNLAnd electromotive force evAre connected in series with each other and consist of a shock absorber 3 and a seismic mass 2 suspended on springs. The two parts interact with each other through a coupling magnet coil 5 and a permanent magnet 6 to generate a radial magnetic field
Figure BDA0002404715200000071
In case of earthquake, the vibration generated by crust breaking will radiate outwards from the broken positionAnd (4) shooting. Seismographs detect and measure these vibrations from the vertical motion of the seismic mass. Westerlund and Ekstam disclose fractional order experimental values for various dielectric capacitors, in which the current flows
Figure BDA0002404715200000072
And voltage
Figure BDA0002404715200000073
There is a fractional order relationship between
Figure BDA0002404715200000074
Based on this fact and the persistent memory of the fractional order operator, the voltage of the nonlinear resistor and capacitor is expressed as:
Figure BDA0002404715200000075
where α is the fractional order, q is the instantaneous charge, R is the resistance, C0Is the linear part of a capacitor, I0Is the initial current, aaAnd abIs the coefficient of the non-linear part of the capacitor,
Figure BDA0002404715200000076
where I is the current.
Obviously, there is friction and air resistance in the system being controlled, writing a spring force with a non-linear stiffness as
Fk＝k0+k1x2+k2x4, (2)
Wherein x is the elongation of the spring, k0Is the linear spring constant, k1Is the cubic spring constant, k2Is the fifth order spring rate.
Due to the constituent structures of the permanent magnet and the coupling coil, it is necessary to consider the laplace force of the mechanical part and the lorentz electromotive force of the electrical part. Based on Newton's second law and kirchhoff's law, a mathematical model of a fractional order self-sustaining electromechanical seismograph system is designed as follows:
Figure BDA0002404715200000081
wherein m is mass, B is magnetic field, l is length of wire, fv＝f1+f0cost represents the random vibration due to ground acceleration, f1Represents the critical force and is assumed to be 0, f0And represents the amplitude and frequency of the noise term.
Defining new dimensionless variables as:
Figure BDA0002404715200000082
wherein Q0Representing a reference charge.
In the Gr ü nwald-Letnikov, Riemann-Luville and Kapopot fractional definitions, the fractional and integer order systems of the Kapopot definition are adopted since they are the same in the form of the initial conditions in the application. Using equations (3) and (4) and increasing the control input uiI is 2,4, and a control equation of the gyro coupling system is derived
Figure BDA0002404715200000083
Dimensionless parameters are defined as:
Figure BDA0002404715200000084
Figure BDA0002404715200000085
x1＝y,x3z, C denotes the kaputol definition.
Definition 1: for the real function F (t), the fractional order derivative of kapton is given as
Figure BDA0002404715200000086
Wherein the content of the first and second substances,
Figure BDA0002404715200000087
n-1 < alpha < n,
Figure BDA0002404715200000088
the gamma function of (2).
Taking Laplace transform from equation (6) to obtain
Figure BDA0002404715200000089
If F(k)(0)＝0,k＝0,1,…,max(pα,pβ) Is formed, wherein pα,pβIf greater than 0, then
Figure BDA0002404715200000091
Definition 2: a constrained cost function is proposed
Figure BDA0002404715200000092
Where Q (S) > 0, S and U 'represent a penalty function, a tracking error and a positive definite function, and U' is defined as
Figure BDA0002404715200000093
Wherein R isoRepresenting a symmetric positive definite matrix, λoIndicating a normal number.
Definition 3: analogous for kappa (t) ═ 1+ l0(t-t0)2、κ(t)＝exp(l0(t-t0) Or κ (t) ═ 1+ tan (0.5 π tanh (t-t)0) Etc.) of the rate function
Figure BDA0002404715200000094
Has the following characteristics: 1) kappa (t) > 1, where t > t0And κ (t)0)＝1；2)
Figure BDA0002404715200000095
Wherein t is more than or equal to t0；3)
Figure BDA0002404715200000096
Is bounded; wherein l0Denotes a constant, t0Indicating the initial time.
Assume that 1: target trajectory xidI is 1,3 and the corresponding fractional order derivative is known and bounded. This means that
Figure BDA0002404715200000097
Wherein A isi＞0，
Figure BDA00024047152000000910
And II, kinetic analysis.
To reveal the dynamics of the seismometer system, the present example presents two sets of physical parameters of the seismometer system:
group 1: mu.s1＝0.1,μ2＝0.2,ω1＝1,λ1＝0.01,λ2＝-0.7,β1＝0.01,β2＝0.1,γ1＝0.25,γ2＝0.95,ω＝0.5,
Group 2: mu.s1＝0.03,μ2＝0.02,ω1＝1,λ1＝0.5,λ2＝0.6,β1＝0.05,β2＝0.13,γ1＝0.65,γ2＝0.42,ω＝0.25.
And solving the numerical solution of the fractional order differential equation by adopting a multi-step method.
Figure BDA0002404715200000098
Can be approximated by a product integration quadrature formula to produce an explicit Euler method
Figure BDA0002404715200000099
Wherein t ismNh, with a constant step h > 0. The first term on the right in equation (11) represents a taylor polynomial of degree n-1 of a function f (t) centered at the time zero. Numerical solution Fm≈F(tm) In each subinterval [ t ]j,tj+1]In the medium approximation, the vector field y (t, f (t)) is approximated by a first order interpolating polynomial.
Fig. 3 and 4 show phase diagrams under different fractional orders and parameters in two cases, and obviously, different parameter sets including fractional orders and external forces can induce periodic, quasi-periodic and chaotic behaviors. The chaotic attractor undergoes magnification, contraction, twisting, and movement. Fig. 5 reveals the time history and the lyapunov exponent of the fractional order, and it can be seen that the fractional order seismograph system exhibits periodic motion and chaotic oscillations due to the negative and positive values of LE. This oscillation caused by the unbalanced interaction of the potential energy can undermine the stability of the system under the energy mechanism.
The control problem of this embodiment is: an acceleration stability control strategy is found by using a cost function (9) and combining a given mathematical model (5) of a fractional order self-sustaining type electromechanical seismograph system, so that all signals in a closed-loop system are bounded, and constraint conditions are met
Figure BDA0002404715200000101
And
Figure BDA0002404715200000102
and the cost function is minimal. In addition, a target track for detecting and recording earthquake ground vibration is embedded in the chaotic attractor, so that chaotic control is realized.
Designing an acceleration stability controller, which comprises constructing an acceleration feedforward controller and an optimal feedback controller; the acceleration feedforward controller is integrated by a modeling behavior function based on a fractional order inversion method, a fuzzy wavelet neural network and a tracking differentiator; the optimal feedback controller is formed by fusing a fuzzy wavelet neural network and a self-adaptive dynamic programming strategy.
1. Fuzzy wavelet neural network
The fuzzy wavelet neural network has good performance in the aspects of control, prediction and classification. It consists of a series of fuzzy if-then rules:
if x1Is that
Figure BDA0002404715200000103
xnIs that
Figure BDA0002404715200000104
Then
Figure BDA0002404715200000105
Is omegaj,j＝1,…,Ns， (12)
Wherein
Figure BDA0002404715200000106
A j-th membership function, n, representing the i-th inputsRepresenting the input number, NsRepresenting the number of fuzzy rules considered.
Defining the triggering degree of a rule as
Figure BDA0002404715200000107
Where i 1, …, N, j 1, …, N, N denotes the number of inputs, N denotes the number of regular neurons,
Figure BDA0002404715200000108
and
Figure BDA0002404715200000109
representing the center and width of the membership function.
The degree of shooting is defined as follows:
Figure BDA00024047152000001010
let xi ≡ 21,…,ξN]T，w≡[w1,…,wN]TThen, the output of the fuzzy wavelet neural network is as follows:
Figure BDA00024047152000001011
wherein
Figure BDA00024047152000001012
i is 1, …, N and
Figure BDA00024047152000001013
representing vector weights, present
Figure BDA00024047152000001014
Wherein ε and DXRepresenting the approximation error and a compact set with bounded X, an optimal parameter w is set*Is equal to
Figure BDA00024047152000001015
Solution of (2), omegawRepresenting an tight set of w. Introduction of
Figure BDA00024047152000001016
Wherein w*Represents satisfaction
Figure BDA00024047152000001017
And
Figure BDA00024047152000001018
the virtual amount of (a).
To increase the solution speed, the transformations associated with the fuzzy wavelet neural network are derived from (15)
Figure BDA0002404715200000111
Wherein
Figure BDA0002404715200000112
It is true that the first and second sensors,
Figure BDA0002404715200000113
represents ζiAnd biIons > 0.
Since the kapton derivative of the constant is zero, it can be deduced
Figure BDA0002404715200000114
2. Design acceleration feedforward controller
To specify steady state error and tracking accuracy in a given time, a modeling behavior function is introduced
Figure BDA0002404715200000115
Wherein etab＞1，
Figure BDA0002404715200000116
And
Figure BDA0002404715200000117
defining the error variables as:
Figure BDA0002404715200000118
wherein alpha isi+1Representing virtual control.
Based on the principle of a fractional order inversion method, the acceleration feedforward controller consists of four steps.
Step 1: to ensure x1Satisfying the constraint condition, selecting the first Lyapunov candidate function as
Figure BDA0002404715200000119
Wherein
Figure BDA00024047152000001110
Get V1Derivative of (2)Is composed of
Figure BDA00024047152000001111
Wherein
Figure BDA00024047152000001112
And
Figure BDA00024047152000001113
accordingly, the virtual control is selected as
Figure BDA00024047152000001114
Wherein k is1＞0。
According to (22), can convert (21) into
Figure BDA00024047152000001115
Step 2: selecting a second Lyapunov candidate function as
Figure BDA00024047152000001116
V is obtained2Fractional order derivative of
Figure BDA00024047152000001117
Wherein
Figure BDA0002404715200000121
With continuous function
Figure BDA0002404715200000122
Figure BDA0002404715200000123
And
Figure BDA0002404715200000124
involving in system dynamics
Figure BDA0002404715200000125
The term is considered an unknown function. To facilitate controller design, it is approximated on a compact set using a fuzzy wavelet neural network, i.e.
Figure BDA0002404715200000126
Wherein (·) represents (x)1,x2,x3,x4) Abbreviations of (a).
Obviously, due to the complexity of modeling the behavior function and the fractional order, the direct calculation
Figure BDA0002404715200000127
Is very difficult. In order to realize signal estimation of virtual control derivative, a fractional order tracking differentiator is reconstructed based on an integer order tracking differentiator
Figure BDA0002404715200000128
Wherein
Figure BDA0002404715200000129
And
Figure BDA00024047152000001210
indicating the state of the tracking differentiator,
Figure BDA00024047152000001211
representing the input signal of a tracking differentiator, ciAnd σiIndicates that c is satisfiedi0 and 0 & ltsigmaiA design constant of < 1.
Combining equations (26) and (27), equation (25) is rewritten as:
Figure BDA00024047152000001212
then, the control input with the adaptive law is designed as follows
Figure BDA00024047152000001213
Figure BDA00024047152000001214
Wherein k is2Indicating a normal number.
Substituting formula (29) and formula (30) into formula (28):
Figure BDA00024047152000001215
and step 3: third Lyapunov candidate function
Figure BDA00024047152000001216
Wherein
Figure BDA00024047152000001217
The virtual control selection is as follows:
Figure BDA00024047152000001218
wherein k is3Indicating a normal number.
Obtaining V from (31) and (32)3Derivative of (2)
Figure BDA00024047152000001219
Wherein
Figure BDA00024047152000001220
And
Figure BDA00024047152000001221
and 4, step 4: selecting a fourth Lyapunov candidate function
Figure BDA0002404715200000131
Attention is paid to
Figure BDA0002404715200000132
And
Figure BDA0002404715200000133
Figure BDA0002404715200000134
is identified as an unknown function. Once again with fuzzy wavelet neural network
Figure BDA0002404715200000135
Approaching it.
Similarly, step 2, the control input and the adaptation law are directly designed as:
Figure BDA0002404715200000136
Figure BDA0002404715200000137
wherein k is4Denotes the normal number, Z4Indicating the state of the fractional order tracking differentiator.
The derivative of (34) can be derived using equations (35) and (36) as:
Figure BDA0002404715200000138
wherein
Figure BDA0002404715200000139
ks＝min{k1 k2 k3 k4}，
Figure BDA00024047152000001310
And
Figure BDA00024047152000001311
the control strategy consists of an acceleration feedforward controller and an optimal feedback controller. The latter relies on the former rather than being juxtaposed. If it is not
Figure BDA00024047152000001312
And
Figure BDA00024047152000001313
the stability of the closed loop system cannot be guaranteed. In addition, the proposed acceleration feedforward controller does not involve optimality.
3. Designing an optimal feedback controller
Based on equation (37), in order to solve the zero-sum differential countermeasure problem of the seismograph system, an optimal feedback control method combined with an adaptive dynamic programming strategy needs to be developed, namely
Figure BDA00024047152000001314
Wherein G represents a fourth order identity matrix.
From equation (9), the Hamiltonian of equation (38) is defined:
Figure BDA00024047152000001315
wherein
Figure BDA00024047152000001316
To represent
Figure BDA00024047152000001317
Of the gradient of (c).
By solving the theory of Hamilton-Jacobi-Isaxx
Figure BDA00024047152000001318
An optimal cost function J can be calculated*. The optimal control input is written as
Figure BDA00024047152000001319
Wherein
Figure BDA00024047152000001320
Substituting formula (40) into formula (39) to derive Hamilton-Jacobi-Isaxx equation
Figure BDA00024047152000001321
Introduction 1: given a controlled system (38), a cost function (9) and an optimal control (40), there is a satisfaction
Figure BDA0002404715200000141
Continuous differentiable and unbounded Lyapunov function JaWherein
Figure BDA0002404715200000142
To represent
Figure BDA0002404715200000143
Partial derivatives of (a). Defining a symmetric positive definite matrix gamma
Figure BDA0002404715200000144
To solve the problem of like
Figure BDA0002404715200000145
Such unknown systemSystem dynamics problem, using fuzzy wavelet neural network to estimate cost function
Figure BDA0002404715200000146
And the partial derivative is
Figure BDA0002404715200000147
Through Taylor expansion, the optimal control input and Hamilton-Jacobi-Isaxx equation are obtained
Figure BDA0002404715200000148
Figure BDA0002404715200000149
Wherein
Figure BDA00024047152000001410
And
Figure BDA00024047152000001411
given that the weights of the fuzzy wavelet neural network are unknown, it is necessary to introduce currently known weights instead of them:
Figure BDA00024047152000001412
wherein
Figure BDA00024047152000001413
And
Figure BDA00024047152000001414
denotes J and woAn estimate of (d). In addition, the weight approximation error is defined as
Figure BDA00024047152000001415
Calling formula (45), the optimum control input can be rewritten as
Figure BDA00024047152000001416
Wherein
Figure BDA00024047152000001417
Equation (41) becomes:
Figure BDA00024047152000001418
invoke formula (39), need to select
Figure BDA00024047152000001419
Make the square residual error
Figure BDA00024047152000001420
And (4) minimizing. However, during learning, only e is adjustedqThe stability of the controlled system cannot be guaranteed. Therefore, a weight adaptation law associated with the fuzzy wavelet neural network is derived, namely:
Figure BDA00024047152000001421
wherein
Figure BDA00024047152000001422
koRepresents a design parameter, z1And z2Indicating the tuning parameters.
The weight adaptation law consists of three parts. The first part is used for stability analysis, the second part is used for minimizing a Hamiltonian, and the third part is used for guaranteeing the system state to be bounded.
Fourth, stability analysis
Theorem 1: for the acceleration stability control problem of the fractional order self-sustaining electromechanical seismograph system described by the formula (5) and having the cost function formula (9), the acceleration feedforward control input with the adaptive law formula (30) and the formula (36) is designed to be the formula (29) and the formula (35). If the optimal feedback control input is derived as equation (46) by an update law (48) associated with the fuzzy wavelet neural network, all signals in the closed loop system are bounded and do not violate the constraints. At the same time, accelerated stabilization is achieved and the cost function is minimized.
And (3) proving that: selecting the entire Lyapunov candidate function
Figure BDA0002404715200000151
The derivative of equation (49) is:
Figure BDA0002404715200000152
wherein
Figure BDA0002404715200000153
To represent
Figure BDA0002404715200000154
The minimum value of the attribute of (a),
Figure BDA0002404715200000155
equation (50) is further simplified as:
Figure BDA0002404715200000156
wherein
Figure BDA0002404715200000157
λmin(Λ) represents the minimum eigenvalue of Λ.
If the condition is
Figure BDA0002404715200000158
Or
Figure BDA0002404715200000159
Is established, then
Figure BDA00024047152000001510
By adjusting ki,i＝1,…,4、bi,i＝2,4、Ro、λo、ko、ziAnd i is 1,2 and other design parameters, so that satisfactory transient performance and stable results can be obtained. Direct adjustment of kiA lower tracking error can be obtained. However, too large kiThe value will result in a larger control input. Alternative ziTo guarantee the matrix
Figure BDA00024047152000001511
Is a positive definite matrix.
The examples verify that:
the parameters of the acceleration feedforward controller and the tracking differentiator are set to k1＝k2＝10、k3＝k4＝15、b2＝b4＝0.6、l0＝1、c1＝c34 and σ1＝σ30.2. Selecting a parameter for shaping the behavior function as ηb＝1.1、
Figure BDA00024047152000001512
And
Figure BDA00024047152000001513
selecting a target trajectory as x1d1.5sin3t and x3d0.9sin2 t. A is easily obtained1＝1.5，A3＝0.9，
Figure BDA00024047152000001514
And
Figure BDA00024047152000001515
in the optimal feedback controller, the constraint condition of the optimal control input is
Figure BDA00024047152000001516
Penalty function Q equal to
Figure BDA00024047152000001517
Symmetric positive definite matrix RoIs set to I4×4. In addition, z2≡8×I4×4，z1≡[9 9 9 9]TAnd k o5. The center and width of the fuzzy wavelet neural network are defined as
Figure BDA00024047152000001518
And
Figure BDA00024047152000001519
fig. 6 shows target trajectory tracking for different parameters and fractional orders in case 1. The fractional order self-sustaining electromechanical seismograph system can realize high-precision track tracking in a short time. Over time, changes in the external and internal environments to system parameters and fractional order do not affect tracking performance. Compared with fig. 3-4, the controller is designed to completely suppress the chaotic oscillation related to the potential energy. As can be seen from fig. 7, sub-graphs 1 and 5, the Lyapunov function-based control scheme does not violate the state constraint, and the modeling behavior function accelerates the stabilization process regardless of the change of the system parameters and the fractional order.
The 2 nd and 6 th subgraphs of figure 7 reveal the ability of the fuzzy wavelet neural network to approximate unknown system dynamics. It can also be seen that this approximation capability is insensitive to system parameters and sensitive to fractional order. In the 4 th and 8 th sub-graphs of fig. 7, the optimal control input satisfies given constraints regardless of the system parameters and fractional order. The remaining figures of fig. 7 further show the stable control performance of the proposed scheme at different parameters and fractional orders.
In order to verify the superiority of the control method, the scheme designed by the invention is compared with an adaptive fuzzy inversion control (AFBC) method. It can be seen from fig. 8 that the proposed scheme is significantly better than AFBC, both in terms of stable amplitude and response time.
Finally, the above embodiments are only intended to illustrate the technical solutions of the present invention and not to limit the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions, and all of them should be covered by the claims of the present invention.

Claims (4)

1. A constrained fractional order self-sustaining electromechanical seismograph system acceleration stability control method, comprising the steps of:
s1: the method comprises the following steps of system modeling, namely establishing a mathematical model of a fractional order self-sustaining type electromechanical seismograph system according to a Newton second law and a kirchhoff law, and defining constraint conditions;
s2: designing an acceleration stability controller, which comprises constructing an acceleration feedforward controller and an optimal feedback controller; the acceleration feedforward controller is integrated by a modeling behavior function based on a fractional order inversion method, a fuzzy wavelet neural network and a tracking differentiator; the optimal feedback controller is formed by fusing a fuzzy wavelet neural network and a self-adaptive dynamic programming strategy;
in step S1, the mathematical model of the fractional order self-sustaining electromechanical seismograph system:
Figure FDA0003575011870000011
where m is the mass, x is the elongation of the spring, k0Is the linear spring constant, k1Is the cubic spring constant, k2Is the fifth order spring coefficient, B represents the magnetic field, l represents the length of the wire; alpha is the order of fractional order, q is the instantaneous charge, C0Is the linear part of a capacitor, I0Is the initial current, aaAnd abIs the coefficient of the nonlinear part of the capacitor;
Figure FDA0003575011870000012
represents the current; f. ofv＝f1+f0cos Ω t represents random vibration due to ground acceleration, f1Represents the critical force and is assumed to be 0, f0And Ω represents the amplitude and frequency of the noise term;
the gyro coupling fractional order control equation of the system is as follows:
Figure FDA0003575011870000013
wherein u isiI-2, 4 denotes an increasing control input; defining dimensionless parameters:
Figure FDA0003575011870000014
Figure FDA0003575011870000015
x1＝y,x3z, C denotes kaputol definition, μ0Representing a damping system, L being a linear inductor, Q0Represents a reference charge, R is a resistance; new dimensionless variables are defined as y ═ x/l, and z ═ Q/Q0,τ＝ωet,
Figure FDA0003575011870000016
2. The method for controlling acceleration and stabilization of a fractional order self-sustaining electro-mechanical seismograph system with constraints as claimed in claim 1, wherein the defined constraints in step S1 specifically include:
definition 1: for the real function f (t), the kapton fractional derivative is given as:
Figure FDA0003575011870000021
wherein the content of the first and second substances,
Figure FDA0003575011870000022
n-1 < alpha < n,
Figure FDA0003575011870000023
a gamma function of (a);
taking laplace transform of the above equation:
Figure FDA0003575011870000024
if F(k)(0)＝0,k＝0,1,…,max(pα,pβ) Is formed, wherein pα,pβIf greater than 0, then
Figure FDA0003575011870000025
Definition 2: defining a constrained cost function as:
Figure FDA0003575011870000026
where Q (S) > 0, S and U 'represent a penalty function, a tracking error and a positive definite function, and U' is defined as:
Figure FDA0003575011870000027
wherein R isoRepresenting a symmetric positive definite matrix, λoWhich is indicative of a normal number of the cells,
Figure FDA0003575011870000028
representing an optimal control input, and upsilon representing a variable;
definition 3: function of rate
Figure FDA0003575011870000029
Has the following characteristics: 1) kappa (t) > 1, where t > t0And κ (t)0)＝1；2)
Figure FDA00035750118700000210
Figure FDA00035750118700000211
Wherein t is more than or equal to t0；3)
Figure FDA00035750118700000212
Is bounded; wherein l0Denotes a constant, t0Represents an initial time;
suppose that: target trajectory xidI 1,3 and the corresponding fractional order derivative are known and bounded, then | xid|≤Ai＜kζiWherein A isi＞0，kζi＞0。
3. The method for acceleration stability control of a constrained fractional order self-sustaining electro-mechanical seismograph system of claim 2, wherein in step S2, the constructing of the acceleration feedforward controller specifically comprises:
to specify steady state error and tracking accuracy in a given time, a modeling behavior function is introduced
Figure FDA00035750118700000213
Wherein etab＞1，
Figure FDA00035750118700000214
εb< 1 and
Figure FDA00035750118700000215
defining the error variables as:
Figure FDA00035750118700000216
wherein alpha isi+1Representing a virtual control;
step 1: to ensure x1And (3) satisfying the constraint condition, and selecting a first Lyapunov candidate function as follows:
Figure FDA00035750118700000217
wherein the content of the first and second substances,
Figure FDA00035750118700000218
the virtual control is selected accordingly as:
Figure FDA00035750118700000219
obtaining V in conjunction with virtual control1The derivative of (c) is:
Figure FDA0003575011870000031
wherein β represents the modeling behavior function, k1＞0，
Figure FDA0003575011870000032
And
Figure FDA0003575011870000033
step 2: selecting a second Lyapunov candidate function as:
Figure FDA0003575011870000034
wherein the content of the first and second substances,
Figure FDA0003575011870000035
an estimate representing a degree of shooting;
approximating on a compact set using a fuzzy wavelet neural network and reconstructing a fractional tracking differentiator based on an integer tracking differentiator:
Figure FDA0003575011870000036
wherein the content of the first and second substances,
Figure FDA0003575011870000037
and
Figure FDA0003575011870000038
indicating the state of the tracking differentiator,
Figure FDA0003575011870000039
representing the input signal of a tracking differentiator, ciAnd σiIndicates that c is satisfiedi0 and 0 < sigmaiA design constant < 1;
the control inputs for the design adaptation law are:
Figure FDA00035750118700000310
Figure FDA00035750118700000311
combining the fractional order tracking differentiator and the control input to obtain V2The fractional derivative of (a) is:
Figure FDA00035750118700000312
wherein k is2Which is indicative of a normal number of the cells,
Figure FDA00035750118700000313
and step 3: selecting a third Lyapunov candidate function as:
Figure FDA00035750118700000314
wherein
Figure FDA00035750118700000322
The virtual control selection is as follows:
Figure FDA00035750118700000315
wherein k is3Represents a normal number;
determining V in conjunction with virtual control3The derivative of (c) is:
Figure FDA00035750118700000316
wherein the content of the first and second substances,
Figure FDA00035750118700000317
and
Figure FDA00035750118700000318
and 4, step 4: the fourth Lyapunov candidate function is selected as:
Figure FDA00035750118700000319
the control inputs and adaptation laws were designed as follows:
Figure FDA00035750118700000320
Figure FDA00035750118700000321
wherein k is4Denotes the normal number, Z4Representing a tracking differentiator of fractional orderA state;
combining control inputs and adaptive laws to determine V4The derivative of (c) is:
Figure FDA0003575011870000041
wherein
Figure FDA0003575011870000042
ks＝min{k1 k2 k3 k4}，
Figure FDA0003575011870000043
And
Figure FDA0003575011870000044
4. the method for acceleration stability control of a constrained fractional order self-sustaining electromechanical seismograph system of claim 3, wherein in step S2, constructing an optimal feedback controller specifically comprises: first, given a controlled system, a cost function, and an optimal cost function, there is a satisfaction
Figure FDA0003575011870000045
Continuous differentiable and unbounded Lyapunov function JaWherein
Figure FDA0003575011870000046
To represent
Figure FDA0003575011870000047
Partial derivatives of (d); defining a symmetric positive definite matrix gamma
Figure FDA0003575011870000048
Then, a fuzzy wavelet neural network is adopted to estimate a cost function, and weight approximation errors are introduced
Figure FDA0003575011870000049
The optimal control inputs are obtained as:
Figure FDA00035750118700000410
wherein
Figure FDA00035750118700000411
Therefore, a weight self-adaptation law associated with the fuzzy wavelet neural network is deduced, namely:
Figure FDA00035750118700000412
wherein the content of the first and second substances,
Figure FDA00035750118700000413
korepresenting a design parameter, z1And z2Representing the regulating parameters, the controlled system is:
Figure FDA00035750118700000414
g denotes a fourth order identity matrix.
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