CN111723442B - Design method of rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping - Google Patents

Design method of rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping Download PDF

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CN111723442B
CN111723442B CN202010515163.9A CN202010515163A CN111723442B CN 111723442 B CN111723442 B CN 111723442B CN 202010515163 A CN202010515163 A CN 202010515163A CN 111723442 B CN111723442 B CN 111723442B
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rolling mill
vibration
lyapunov
controller
vertical vibration
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CN111723442A (en
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张柳柳
钱承
李亚峰
华长春
白振华
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Yanshan University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
    • B21B—ROLLING OF METAL
    • B21B37/00—Control devices or methods specially adapted for metal-rolling mills or the work produced thereby
    • B21B37/007—Control for preventing or reducing vibration, chatter or chatter marks
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
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Abstract

The invention provides a design method of a rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping, which comprises the steps of firstly, establishing a four-degree-of-freedom machine-liquid coupling nonlinear model of rolling mill vertical vibration according to the dynamic principle of a rolling mill vibration system; then, the working roll is jumped up and down according to the vertical vibration of the rolling mill, so that the control target is set when the vertical vibration displacement of the working roll of the rolling mill approaches zero; and finally, selecting a proper Lyapunov function in combination with a nonlinear model of the vertical vibration of the rolling mill, and solving a virtual controller, an actual controller and a self-adaptive law which enable V (t) to be zero when t tends to be infinite, so as to finally obtain a design method of the controller with the preset performance for inhibiting the vertical vibration of the rolling mill. The invention establishes a rolling mill vertical vibration nonlinear model which is more in line with the actual working condition, considers the characteristic of dead zone of a servo valve and the limitation on the vibration displacement of the roller, designs a rolling mill vertical vibration suppressor, realizes the rapid active suppression of vertical vibration in the high-speed rolling process, and ensures the stability of the high-speed rolling process of a strip.

Description

Design method of rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping
Technical Field
The invention relates to the technical field of metallurgical rolling control, in particular to a design method of a rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping.
Background
When rolling extremely thin steel strip at high speed, vibration perpendicular to the rolling direction, or vertical vibration, often occurs. The vertical vibration of the rolling mill not only affects the dimensional accuracy and surface quality of the strip, but also can cause damage to rolling equipment if the vertical vibration is serious. At present, the emergency measure for solving the problem of vertical vibration of the rolling mill on site is to reduce the rolling speed, but the reduction of the rolling speed influences the production efficiency of the plate strip and cannot be used as an effective treatment measure for treating the vertical vibration of the rolling mill. Therefore, how to effectively solve the problem of vertical vibration in the high-speed process of the plate strip becomes the key point and the difficulty of field technical attack. In the past, the vibration of the rolling mill is mainly treated from the aspects of machinery and technology, such as eliminating equipment clearance, optimizing rolling rules and inhibiting the vibration of the rolling mill by measures of improving lubricating parameters of a rolling roll gap.
With the development of control technology, a reasonably designed vibration suppression controller has become a focus of attention from the viewpoint of active suppression of vibration. In practice the mill vibration system is a mechanical hydraulic coupling nonlinear system. For the nonlinear system control problem, backstepping control method can be used for processing. In addition, functions and parameters in the coupled vibration system of the rolling mill are unknown, and the self-adaptive fuzzy technology has a good effect on the unknown functions and parameters in the processing system. In addition, the roll run-out displacement and the valve core of an electro-hydraulic servo valve need to be limited in an actual rolling mill coupled vibration system, so that the dead zone problem exists. Therefore, the research on how to design the control method to ensure the effective and quick suppression of the vibration of the rolling mill has strong practical significance.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a design method of a rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping, which establishes a rolling mill vertical vibration nonlinear model more conforming to the actual working condition by considering the limitation of preset performance on the vibration displacement of a roller, realizes the rapid active suppression of vertical vibration in the high-speed rolling process and ensures the stability of the high-speed rolling process of a plate strip.
The invention provides a design method of a rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping, which comprises the following steps of firstly, establishing a nonlinear model of rolling mill vertical vibration according to the dynamic principle of a rolling mill vibration system; then, determining a control target for inhibiting the vibration of the rolling mill according to the actual working condition; and finally, designing a self-adaptive backstepping controller by combining a nonlinear model of vertical vibration of the rolling mill and a control target for inhibiting vibration of the rolling mill, wherein the specific design method of the controller comprises the following steps:
s1, collecting parameters of a mechanical-hydraulic coupling vibration system of the rolling mill and characteristic parameters of a dead zone of an electro-hydraulic servo valve;
s2, establishing a four-degree-of-freedom mechanical-hydraulic coupling nonlinear model of vertical vibration of the rolling mill according to Newton' S second theorem:
Figure BDA0002529811990000021
wherein z is1For the oscillating displacement of the working rolls, z2Is the work roll vibration speed, z3For the oscillatory displacement of the intermediate roll, z4Is the intermediate roll oscillation speed, z5For supporting the roll vibrational displacement, z6To the vibration speed of the support roller, z7For oscillating displacement of the cylinder, z8Is the cylinder vibration speed, z9＝P1Is the pressure at the rodless cavity, miTo an equivalent mass, kiTo equivalent stiffness, ciFor equivalent damping, A1Is the area of the rodless cavity, A2Is the area of the rod cavity, ctIs the leakage coefficient, P, in the cylinder2Pressure of the rod chamber, V initial volume of the control chamber, betaeIs the bulk modulus, k, of the oilqIs a process coefficient, u is a comprehensive expression of a characteristic parameter of the dead zone of the servo valve, Fz(z1,z2) Is a function expression related to the vibration displacement and the vibration speed of the working roll;
s3, determining a target for restraining the vibration of the rolling mill;
s31, setting the vertical vibration displacement of the working roll of the rolling mill close to zero as a control target according to the fact that the vertical vibration of the rolling mill is caused by the vertical jumping of the working roll;
s32, controlling the vertical vibration attenuation rate and the maximum allowable displacement of the rolling mill within a set range, wherein the target for restraining the vibration of the rolling mill can be expressed as follows:
Figure BDA0002529811990000031
wherein xi is1＝z1Indicating the displacement of the work rolls during vibration of the mill, mu (t) ═ mu0e-kt+μ∞，μ0，k，μ∞Is a positive real number that is specified,δand
Figure BDA0002529811990000032
are given positive real numbers;
s4, giving a controller and a parameter adaptive law according to Lyapunov stability criterion, and designing an adaptive fuzzy vibration suppressor;
s41, selecting a proper Lyapunov function according to the four-degree-of-freedom mechanical-hydraulic coupling nonlinear model of the vertical vibration of the rolling mill established in the step S2;
s42, derivation is carried out on the Lyapunov function, and when t tends to be infinite, V is solved1(t) a vanishing virtual controller and adaptive law;
and S43, finally obtaining the design method of the preset performance controller for inhibiting the vertical vibration of the rolling mill.
Preferably, nine lyapunov functions are selected according to the nine subsystems included in step S2, and step S4 specifically includes the following steps:
s411, selecting a first Lyapunov function:
Figure BDA0002529811990000033
wherein the content of the first and second substances,
Figure BDA0002529811990000034
s421, deriving the Lyapunov function, and solving to make V tend to be infinite1(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000035
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002529811990000036
ε1and a1Is a positive parameter being designed;
s412, selecting a second Lyapunov function:
Figure BDA0002529811990000041
wherein xi is2＝z2-α1，σ21、σ22Is a positive parameter that is being designed for,
Figure BDA0002529811990000042
is ρ2The error of the estimation of (2) is,
Figure BDA0002529811990000043
is theta2The estimation error of (2);
s422, derivation is carried out on the Lyapunov function, and when t tends to be infinite, V is solved2(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000044
Figure BDA0002529811990000045
Figure BDA0002529811990000046
wherein the content of the first and second substances,
Figure BDA0002529811990000047
is ρ2Is estimated byThe value of the one or more of the one,
Figure BDA0002529811990000048
ε22is a given normal number, σ21，τ2，l21，l22Is a positive parameter of the design and,
Figure BDA0002529811990000049
is the vector of the fuzzy basis function of the second step,
Figure BDA00025298119900000410
is theta2An estimated value of (d);
s413, selecting a third Lyapunov function:
Figure BDA00025298119900000411
wherein xi is3＝z3-α2，σ32Is a positive parameter that is designed to be,
Figure BDA00025298119900000412
is θ3The estimation error of (2);
s423, deriving the Lyapunov function, and solving to make V tend to be infinite3(t) the vanishing virtual controller and adaptation law is:
Figure BDA00025298119900000413
Figure BDA00025298119900000414
wherein, a3，l32，σ32Is a positive parameter that is being designed for,
Figure BDA00025298119900000415
is that
Figure BDA00025298119900000416
The method (2) is implemented by the following steps,
Figure BDA00025298119900000417
is the vector of the fuzzy basis function of the third step,
Figure BDA00025298119900000418
is θ3An estimated value of (d);
s414, selecting a fourth Lyapunov function:
Figure BDA0002529811990000051
wherein xi is4＝z4-α3，σ41、σ42Is a positive parameter that is being designed for,
Figure BDA0002529811990000052
is ρ4The error of the estimation of (2) is,
Figure BDA0002529811990000053
is theta4The estimated error of (2);
s424, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution4(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000054
Figure BDA0002529811990000055
Figure BDA0002529811990000056
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002529811990000057
is ρ4Is determined by the estimated value of (c),
Figure BDA0002529811990000058
ε42is a given normal number; sigma41，τ4，l41，l42Is a positive parameter of the design and,
Figure BDA0002529811990000059
is the vector of the fuzzy basis function of the fourth step,
Figure BDA00025298119900000510
is theta4An estimated value of (d);
s415, selecting a fifth Lyapunov function:
Figure BDA00025298119900000511
wherein ξ5＝z5-α4，σ52Is a positive parameter that is being designed for,
Figure BDA00025298119900000512
is θ5The estimated error of (2);
s425, deriving the Lyapunov function, and solving to enable V to be equal to zero when t is close to infinity5(t) the control law and the adaptive law of the vanishing virtual controller are:
Figure BDA00025298119900000513
Figure BDA00025298119900000514
wherein, a5，l52，σ52Is a positive parameter that is being designed for,
Figure BDA00025298119900000515
is that
Figure BDA00025298119900000516
The method (2) is implemented by the following steps,
Figure BDA00025298119900000517
is the vector of the fuzzy basis function of the fifth step,
Figure BDA00025298119900000518
is θ5An estimated value of (d);
s416, selecting a sixth Lyapunov function:
Figure BDA0002529811990000061
wherein ξ6＝z6-α5，σ61、σ62Is a positive parameter that is designed to be,
Figure BDA0002529811990000062
is ρ6The error of the estimation of (2) is,
Figure BDA0002529811990000063
is theta6The estimated error of (2);
s426, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution6(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000064
Figure BDA0002529811990000065
Figure BDA0002529811990000066
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002529811990000067
is ρ6Is determined by the estimated value of (c),
Figure BDA0002529811990000068
ε62is a given normal number; sigma61，τ6，l61，l62Is a positive parameter of the design and,
Figure BDA0002529811990000069
is the vector of the fuzzy basis function of the sixth step,
Figure BDA00025298119900000610
is theta6An estimated value of (d);
s417, selecting a seventh Lyapunov function:
Figure BDA00025298119900000611
wherein xi is7＝z7-α6，σ72Is a parameter that is being designed for,
Figure BDA00025298119900000612
is θ7The estimation error of (2);
s427, derivation is carried out on the Lyapunov function, and when t tends to be infinite, V is solved7(t) the vanishing virtual controller and adaptation law is:
Figure BDA00025298119900000613
Figure BDA00025298119900000614
wherein, a7，l72，σ72Is a positive parameter that is being designed for,
Figure BDA00025298119900000615
is that
Figure BDA00025298119900000616
The transpose of (a) is performed,
Figure BDA00025298119900000617
is the vector of the fuzzy basis function of the seventh step,
Figure BDA00025298119900000618
is θ7An estimated value of (d);
s418, selecting an eighth Lyapunov function:
Figure BDA0002529811990000071
wherein xi is8＝z8-α7，σ81，σ82Is a positive parameter of the design and,
Figure BDA0002529811990000072
is that
Figure BDA0002529811990000073
The transpose of (a) is performed,
Figure BDA0002529811990000074
s428, derivation is carried out on the Lyapunov function, and when t tends to be infinite, V is obtained through solution8(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000075
Figure BDA0002529811990000076
Figure BDA0002529811990000077
wherein the content of the first and second substances,
Figure BDA0002529811990000078
is ρ8Is determined by the estimated value of (c),
Figure BDA0002529811990000079
ε82is a given normal number;
σ81，σ82，l81，l82is a positive parameter of the design and,
Figure BDA00025298119900000723
is the vector of the fuzzy basis function of the eighth step,
Figure BDA00025298119900000710
is θ8An estimated value of (d);
s419, selecting a ninth Lyapunov function:
Figure BDA00025298119900000711
wherein ξ9＝z9-α8，σ91，σ92Is a positive parameter of the design and,
Figure BDA00025298119900000712
is that
Figure BDA00025298119900000713
The method (2) is implemented by the following steps,
Figure BDA00025298119900000714
Figure BDA00025298119900000715
is ρ9The estimated error of (2);
s429, derivation and solving of Lyapunov functionThe solution is such that when t tends to infinity, V9(t) the actual controller and adaptation law going to zero is:
Figure BDA00025298119900000716
Figure BDA00025298119900000717
Figure BDA00025298119900000718
wherein the content of the first and second substances,
Figure BDA00025298119900000719
is ρ9Is determined by the estimated value of (c),
Figure BDA00025298119900000720
ε92is a given normal number; sigma91，σ92，l91，l92Is a positive parameter of the design and,
Figure BDA00025298119900000721
is the fuzzy basis function vector of the ninth step,
Figure BDA00025298119900000722
is theta9An estimate of (d).
Preferably, in step S2, m1，m2，m3，m4Equivalent mass of a working roll and a bearing, an intermediate roll and a bearing, a supporting roll and a bearing, a hydraulic cylinder and a piston rod respectively; k is a radical of1，k2，k3Equivalent rigidity between a working roll and a middle roll, between the middle roll and a supporting roll, and between the supporting roll and a hydraulic cylinder; c. C1，c2，c3The equivalent damping is respectively between the working roll and the middle roll, between the middle roll and the supporting roll, between the supporting roll and the hydraulic cylinder.
Preferably, in step S2, u is a comprehensive expression of the servo valve dead zone characteristic parameter, and the specific form thereof may be expressed as:
Figure BDA0002529811990000081
wherein etar、ηl、brAnd blAre all servo valve dead zone characteristic parameters.
Preferably, in step S2, kqFor process coefficients, the specific form can be expressed as:
Figure BDA0002529811990000082
wherein, CdIs a flow coefficient of a valve port of the servo valve; w is the area gradient of the valve port of the servo valve, rho is the density of hydraulic oil in the hydraulic cylinder, and P issFor supply pressure, PtAs oil return pressure, xvFor servo valve spool displacement, kvIs the gain factor.
Preferably, in order to ensure that the designed nine lyapunov functions are all positive, the coefficients in the nine lyapunov functions need to be designed as positive numbers according to the item in which the coefficients are located; to ensure the effectiveness of the controller, nine Lyapunov functions V are requirediThe derived results respectively satisfy the Lyapunov stabilization criterion, the coefficients in the Lyapunov functions need to be designed as positive numbers according to the item where the coefficients are located, and simultaneously, each Lyapunov function satisfies the requirement after derivation
Figure BDA0002529811990000083
Wherein c, Δ are both normal numbers; after each lyapunov function is derived, the parameters to be designed are different.
Preferably, according to the formal similarity of the neutron system in the nonlinear model created in step S2, the design parameters in the third, fifth and seventh lyapunov functions are identical, the design parameters in the second, fourth and sixth lyapunov functions are identical, and the design parameters in the eighth and ninth lyapunov functions are identical.
Compared with the prior art, the invention has the following advantages:
1. the invention establishes a vibration nonlinear model which is more in line with the reality, considers the characteristic of dead zones of servo valves, designs a vertical vibration suppressor of the rolling mill, realizes the rapid active suppression of vertical vibration in the high-speed rolling process, and ensures the stability of the high-speed rolling strip process;
2. aiming at the problem of vibration suppression control of the rolling mill, the invention considers the limitation of preset performance on the vibration displacement of the roller, designs the fuzzy self-adaptive vibration suppressor, improves the vibration attenuation performance of the rolling mill, such as vibration attenuation speed, maximum allowable vibration displacement, steady-state error and the like, and is improved compared with the traditional control method.
Drawings
FIG. 1 is a flow chart of a design method of a rolling mill sag damping controller based on adaptive fuzzy backstepping in accordance with the present invention;
FIG. 2 is a schematic diagram of hydraulic vibration of rolling mill machinery in the design method of the self-adaptive backstepping-based rolling mill vertical vibration suppression controller of the present invention;
FIG. 3 is a comparison graph of the response curves of the vibration displacement of the working rolls under the condition of no performance constraint in the design method of the self-adaptive fuzzy backstepping-based vertical vibration suppression controller of the rolling mill of the invention; and
FIG. 4 shows the effect of the vibration suppression controller on vibration suppression after the vibration suppression controller is put into use in the design method of the rolling mill vertical vibration suppression controller based on the adaptive fuzzy backstepping.
Detailed Description
The invention will be described in detail with reference to the accompanying drawings for describing the technical content, the achieved purpose and the efficacy of the invention.
A design method of a rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping is disclosed, as shown in figure 1, firstly, a nonlinear model of rolling mill vertical vibration is established according to the dynamic principle of a rolling mill vibration system; then, determining a control target for inhibiting the vibration of the rolling mill according to the actual working condition; and finally, designing a self-adaptive backstepping controller by combining a nonlinear model of vertical vibration of the rolling mill and a control target for inhibiting vibration of the rolling mill, wherein the specific design method of the controller comprises the following steps:
s1, as shown in FIG. 2, collecting parameters of the mechanical-hydraulic coupling vibration system of the rolling mill and the characteristic parameters of the electro-hydraulic servo valve dead zone, as shown in Table 1.
S2, establishing a four-degree-of-freedom mechanical-hydraulic coupling nonlinear model of the vertical vibration of the rolling mill according to Newton' S second theorem:
Figure BDA0002529811990000101
wherein z is1For the oscillating displacement of the working rolls, z2Is the work roll vibration speed, z3For oscillating displacement of the intermediate roll, z4Is the intermediate roll oscillation speed, z5For the vibrational displacement of the supporting roller, z6For the vibration speed of the supporting roller, z7For cylinder vibrational displacement, z8Is the cylinder oscillation speed, z9＝P1Is the pressure at the rodless cavity, miTo equivalent mass, kiTo equivalent stiffness, ciFor equivalent damping, A1Is the area of the rodless cavity, A2Is the area of the rod cavity, ctIs the leakage coefficient, P, in the cylinder2Pressure of the rod chamber, V initial volume of the control chamber, betaeIs the bulk modulus, k, of the oilqIs a process coefficient, u is a comprehensive expression of the characteristic parameter of the dead zone of the servo valve, Fz(z1,z2) Is a function expression related to the vibration displacement and the vibration speed of the working roll.
S3, determining a target for restraining the vibration of the rolling mill;
s31, setting the vertical vibration displacement of the working roll of the rolling mill approaching zero as a control target according to the fact that the vertical vibration of the rolling mill is caused by the up-down run-out of the working roll;
s32, controlling the vertical vibration attenuation rate and the maximum allowable displacement of the rolling mill within a set range, wherein the target for restraining the vibration of the rolling mill can be expressed as follows:
Figure BDA0002529811990000111
wherein xi is1＝z1Indicating the displacement of the work rolls during vibration of the mill, with μ (t) being μ0e-kt+μ∞，μ0，k，μ∞Is a positive and real number that is specified,δand
Figure BDA0002529811990000112
are given positive real numbers.
S4, giving a controller and a parameter adaptive law according to the Lyapunov stability criterion, and designing an adaptive fuzzy vibration suppressor;
s41, selecting a proper Lyapunov function according to the four-degree-of-freedom mechanical-hydraulic coupling nonlinear model of the vertical vibration of the rolling mill established in the step S2;
s42, solving so that
Figure BDA0002529811990000117
As the time t goes to infinity,
Figure BDA0002529811990000116
the control law and the adaptive law of the virtual controller which tend to be zero;
and S43, finally obtaining the design method of the controller with the preset performance for restraining the vertical vibration of the rolling mill.
The parameters to be designed in the controller are determined according to the designed Lyapunov function, and if the designed parameters can meet the Lyapunov stability criterion, the stability of each subsystem can be ensured, namely the controller is good. Therefore, nine lyapunov functions are selected according to the nine subsystems included in step S2, and step S4 specifically includes the following detailed steps:
s411, selecting a first Lyapunov function:
Figure BDA0002529811990000113
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002529811990000114
s421, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is solved1(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000115
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002529811990000121
ε1and a1Is a positive parameter being designed;
s412, selecting a second Lyapunov function:
Figure BDA0002529811990000122
wherein xi is2＝z2-α1，σ21、σ22Is a positive parameter that is being designed for,
Figure BDA0002529811990000123
ρ2the error of the estimation of (2) is,
Figure BDA0002529811990000124
is theta2The estimated error of (2);
s422, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution2(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000125
Figure BDA0002529811990000126
Figure BDA0002529811990000127
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002529811990000128
is ρ2Is determined by the estimated value of (c),
Figure BDA0002529811990000129
ε22is a given normal number, σ21，τ2，l21，l22Is a positive parameter of the design and,
Figure BDA00025298119900001210
is the vector of the fuzzy basis function of the second step,
Figure BDA00025298119900001211
is theta2An estimated value of (d);
s413, selecting a third Lyapunov function:
Figure BDA00025298119900001212
wherein xi is3＝z3-α2，σ32Is a positive parameter that is being designed for,
Figure BDA00025298119900001213
is θ3The estimated error of (2);
s423, deriving the Lyapunov function, and solving to ensure that V tends to be infinite when t is close to infinity3(t) vanishing virtual controller and adaptive law：
Figure BDA00025298119900001214
Figure BDA00025298119900001215
Wherein, a3，l32，σ32Is a positive parameter that is being designed for,
Figure BDA0002529811990000131
is that
Figure BDA0002529811990000132
The method (2) is implemented by the following steps,
Figure BDA0002529811990000133
is the fuzzy basis function vector of the third step,
Figure BDA0002529811990000134
is theta3An estimated value of (d);
s414, selecting a fourth Lyapunov function:
Figure BDA0002529811990000135
wherein xi is4＝z4-α3，σ41、σ42Is a positive parameter that is being designed for,
Figure BDA0002529811990000136
is ρ4The error of the estimation of (2) is,
Figure BDA0002529811990000137
is theta4The estimated error of (2);
s424, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution4(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000138
Figure BDA0002529811990000139
Figure BDA00025298119900001310
wherein the content of the first and second substances,
Figure BDA00025298119900001311
is ρ4Is determined by the estimated value of (c),
Figure BDA00025298119900001312
ε42is a given normal number; sigma41，τ4，l41，l42Is a positive parameter of the design and,
Figure BDA00025298119900001313
is the vector of the fuzzy basis function of the fourth step,
Figure BDA00025298119900001314
is theta4An estimated value of (d);
s415, selecting a fifth Lyapunov function:
Figure BDA00025298119900001315
wherein ξ5＝z5-α4，σ52Is a positive parameter that is being designed for,
Figure BDA00025298119900001316
is θ5The estimated error of (2);
s425, derivation is carried out on the Lyapunov function, and the operation is carried out so that the operation becomes trend at tAt infinite time, V5(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA00025298119900001317
Figure BDA00025298119900001318
wherein, a5，l52，σ52Is a positive parameter that is being designed for,
Figure BDA0002529811990000141
is that
Figure BDA0002529811990000142
The method (2) is implemented by the following steps,
Figure BDA0002529811990000143
is the vector of the fuzzy basis function of the fifth step,
Figure BDA0002529811990000144
is θ5An estimated value of (d);
s416, selecting a sixth Lyapunov function:
Figure BDA0002529811990000145
wherein ξ6＝z6-α5，σ61、σ62Is a positive parameter that is designed to be,
Figure BDA0002529811990000146
ρ6the error of the estimation of (2) is,
Figure BDA0002529811990000147
is theta6The estimated error of (2);
s426, derivation is conducted on the Lyapunov function, and the calculation is conducted so that t tends to be zeroIn poor times, V6(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000148
Figure BDA0002529811990000149
Figure BDA00025298119900001410
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00025298119900001411
is ρ6Is determined by the estimated value of (c),
Figure BDA00025298119900001412
ε62is a given normal constant. Sigma61，τ6，l61，l62Is a positive parameter of the design and,
Figure BDA00025298119900001413
is the vector of the fuzzy basis function of the sixth step,
Figure BDA00025298119900001414
is theta6An estimated value of (d);
s417, selecting a seventh Lyapunov function:
Figure BDA00025298119900001415
wherein ξ7＝z7-α6，σ72Is a parameter that is being designed for,
Figure BDA00025298119900001416
is θ7The estimation error of (2).
S427, derivation is carried out on the Lyapunov function, and when t tends to be infinite, V is solved7(t) the vanishing virtual controller and adaptation law is:
Figure BDA00025298119900001417
Figure BDA00025298119900001418
wherein, a7，l72，σ72Is a positive parameter that is designed to be,
Figure BDA0002529811990000151
is that
Figure BDA0002529811990000152
The transpose of (a) is performed,
Figure BDA0002529811990000153
is the vector of the fuzzy basis function of the seventh step,
Figure BDA0002529811990000154
is theta7An estimated value of (d);
s418, selecting an eighth Lyapunov function:
Figure BDA0002529811990000155
wherein ξ8＝z8-α7，σ81，σ82Is a positive parameter of the design and,
Figure BDA0002529811990000156
is that
Figure BDA0002529811990000157
The method (2) is implemented by the following steps,
Figure BDA0002529811990000158
s428, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is solved8(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000159
Figure BDA00025298119900001510
Figure BDA00025298119900001511
wherein the content of the first and second substances,
Figure BDA00025298119900001512
is ρ8Is determined by the estimated value of (c),
Figure BDA00025298119900001513
ε82is a given normal number.
σ81，σ82，l81，l82Is a positive parameter of the design and,
Figure BDA00025298119900001522
is the fuzzy basis function vector of the eighth step,
Figure BDA00025298119900001514
is theta8An estimated value of (d);
s419, selecting a ninth Lyapunov function:
Figure BDA00025298119900001515
wherein ξ9＝z9-α8，σ91，σ92Is provided withThe positive parameter of the meter is measured,
Figure BDA00025298119900001516
is that
Figure BDA00025298119900001517
The transpose of (a) is performed,
Figure BDA00025298119900001518
Figure BDA00025298119900001519
is ρ9The estimation error of (2);
s429, derivation is conducted on the Lyapunov function, and when t tends to be infinite,
Figure BDA00025298119900001520
the actual controller and adaptation law going to zero is:
Figure BDA00025298119900001521
Figure BDA0002529811990000161
Figure BDA0002529811990000162
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002529811990000163
is ρ9Is determined by the estimated value of (c),
Figure BDA0002529811990000164
ε92is a given normal constant. Sigma91，σ92，l91，l92Is a positive parameter of the design and,
Figure BDA0002529811990000165
is the fuzzy basis function vector of the ninth step,
Figure BDA0002529811990000166
is theta9An estimate of (d).
In step S2, m1，m2，m3，m4Equivalent masses of a working roll and a bearing, a middle roll and a bearing, a supporting roll and a bearing, a hydraulic cylinder and a piston rod are respectively; k is a radical of1，k2，k3Equivalent rigidity between a working roll and a middle roll, between the middle roll and a supporting roll, and between the supporting roll and a hydraulic cylinder; c. C1，c2，c3The equivalent damping between the working roll and the middle roll, between the middle roll and the supporting roll, between the supporting roll and the hydraulic cylinder.
In step S2, u is a comprehensive expression of the servo valve dead band characteristic parameter, which can be expressed in a specific form as:
Figure BDA0002529811990000167
wherein etar、ηl、brAnd blAre all servo valve dead zone characteristic parameters.
In step S2, kqFor process coefficients, the specific form can be expressed as:
Figure BDA0002529811990000168
wherein, CdIs a flow coefficient of a valve port of the servo valve; w is the area gradient of the valve port of the servo valve, rho is the density of hydraulic oil in the hydraulic cylinder, and P issFor supply pressure, PtAs oil return pressure, xvFor servo valve spool displacement, kvIs the gain factor.
In order to ensure that the nine designed Lyapunov functions are all positive, the coefficients in the nine Lyapunov functions need to be designed as positive numbers according to the item where the coefficients are located; in order to ensure the effectiveness of the controller,it is necessary to make nine Lyapunov functions ViThe derived results respectively satisfy the Lyapunov stabilization criterion, the coefficients in the Lyapunov functions need to be designed as positive numbers according to the item where the coefficients are located, and simultaneously, each Lyapunov function satisfies the requirement after derivation
Figure BDA0002529811990000171
Wherein c, Δ are both normal numbers; after each lyapunov function is derived, the parameters to be designed are different.
The first Lyapunov function does not contain a nonlinear function, so that a fuzzy basis function vector is not needed; in the second to ninth lyapunov functions, a fuzzy basis function vector needs to be applied to approximate a nonlinear function.
According to the similarity in the form of the subsystem in the nonlinear model created in step S2, the design parameters in the third lyapunov function, the fifth lyapunov function, and the seventh lyapunov function are identical, the design parameters in the second lyapunov function, the fourth lyapunov function, and the sixth lyapunov function are identical, and the design parameters in the eighth lyapunov function and the ninth lyapunov function are identical.
The design method of the rolling mill vertical oscillation suppression controller based on the self-adaptive fuzzy backstepping of the invention is further described by combining the following embodiments:
s1, collecting parameters of the mechanical-hydraulic coupling vibration system of the rolling mill and the characteristic parameters of the dead zone of the electro-hydraulic servo valve, as shown in the table 1.
S2, establishing a four-degree-of-freedom mechanical-hydraulic coupling nonlinear model of vertical vibration of the rolling mill according to Newton' S second theorem:
Figure BDA0002529811990000172
u is a comprehensive expression of the characteristic parameter of the dead zone of the servo valve, and the specific form of the comprehensive expression can be expressed as follows:
Figure BDA0002529811990000181
s3, determining a target for restraining the vibration of the rolling mill;
s31, setting the vertical vibration displacement of the working roll of the rolling mill close to zero as a control target according to the fact that the vertical vibration of the rolling mill is caused by the vertical jumping of the working roll;
s32, controlling the damping rate of the vertical vibration of the rolling mill and the maximum allowable displacement within a set range, and the objective function of inhibiting the vibration of the rolling mill can be expressed as:
-2(5e-2t+0.1)<ξ1<2(5e-2t+0.1)。
s4, giving a controller and a parameter adaptive law according to Lyapunov stability criterion, and designing an adaptive fuzzy vibration suppressor;
s411, selecting a first Lyapunov function:
Figure BDA0002529811990000182
wherein the content of the first and second substances,
Figure BDA0002529811990000183
s421, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is solved1(t) the vanishing virtual controller and adaptation law is:
Figure BDA0002529811990000184
wherein the content of the first and second substances,
Figure BDA0002529811990000185
s412, selecting a second Lyapunov function:
Figure BDA0002529811990000186
wherein xi is2＝z2-α1，
Figure BDA0002529811990000187
Figure BDA0002529811990000188
Is ρ2The error of the estimation of (2) is,
Figure BDA0002529811990000189
is theta2The estimation error of (2);
s422, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution2(t) the vanishing virtual controller and adaptation law is:
Figure BDA0002529811990000191
Figure BDA0002529811990000192
Figure BDA0002529811990000193
wherein the content of the first and second substances,
Figure BDA0002529811990000194
is ρ2Is determined by the estimated value of (c),
Figure BDA0002529811990000195
Figure BDA0002529811990000196
is the vector of the fuzzy basis function of the second step,
Figure BDA0002529811990000197
is theta2An estimated value of (d);
s413, selecting a third Lyapunov function:
Figure BDA0002529811990000198
wherein xi is3＝z3-α2，
Figure BDA0002529811990000199
Is theta3The estimation error of (2);
s423, deriving the Lyapunov function, and solving to ensure that V tends to be infinite when t is close to infinity3(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA00025298119900001910
Figure BDA00025298119900001911
wherein the content of the first and second substances,
Figure BDA00025298119900001912
is that
Figure BDA00025298119900001913
The transpose of (a) is performed,
Figure BDA00025298119900001914
is the fuzzy basis function vector of the third step,
Figure BDA00025298119900001915
is theta3An estimated value of (d);
s414, selecting a fourth Lyapunov function:
Figure BDA00025298119900001916
wherein ξ4＝z4-α3，
Figure BDA00025298119900001920
Is ρ4The error of the estimation of (2) is,
Figure BDA00025298119900001918
is theta4The estimation error of (2);
s424, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution4(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA00025298119900001919
Figure BDA0002529811990000201
Figure BDA0002529811990000202
wherein the content of the first and second substances,
Figure BDA0002529811990000203
is ρ4Is determined by the estimated value of (c),
Figure BDA0002529811990000204
is the vector of the fuzzy basis function of the fourth step,
Figure BDA0002529811990000205
is theta4An estimated value of (d);
s415, selecting a fifth Lyapunov function:
Figure BDA0002529811990000206
wherein ξ5＝z5-α4，
Figure BDA0002529811990000207
Is theta5The estimation error of (2);
s425, deriving the Lyapunov function, and solving to enable V to be equal to zero when t is close to infinity5(t) the vanishing virtual controller and adaptation law is:
Figure BDA0002529811990000208
Figure BDA0002529811990000209
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00025298119900002010
is that
Figure BDA00025298119900002011
The transpose of (a) is performed,
Figure BDA00025298119900002012
is the fuzzy basis function vector of the fifth step,
Figure BDA00025298119900002013
is θ5An estimated value of (d);
s416, selecting a sixth Lyapunov function:
Figure BDA00025298119900002014
wherein xi is6＝z6-α5，
Figure BDA00025298119900002015
Is ρ6The error of the estimation of (2) is,
Figure BDA00025298119900002016
is theta6The estimation error of (2);
s426, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution6(t) the vanishing virtual controller and adaptation law is:
Figure BDA00025298119900002017
Figure BDA00025298119900002018
Figure BDA00025298119900002019
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002529811990000211
is ρ6Is determined by the estimated value of (c),
Figure BDA0002529811990000212
is the fuzzy basis function vector of the sixth step,
Figure BDA0002529811990000213
is theta6An estimated value of (d);
s417, selecting a seventh Lyapunov function:
Figure BDA0002529811990000214
wherein xi is7＝z7-α6，
Figure BDA0002529811990000215
Is θ7The estimation error of (2).
S427, derivation is carried out on the Lyapunov function, and when t tends to be infinite, V is solved7(t) the vanishingly close virtual controller and adaptive law are:
Figure BDA0002529811990000216
Figure BDA0002529811990000217
wherein the content of the first and second substances,
Figure BDA0002529811990000218
is that
Figure BDA0002529811990000219
The transpose of (a) is performed,
Figure BDA00025298119900002110
is the vector of the fuzzy basis function of the seventh step,
Figure BDA00025298119900002111
is θ7An estimated value of (d);
s418, selecting an eighth Lyapunov function:
Figure BDA00025298119900002112
wherein xi is8＝z8-α7，
Figure BDA00025298119900002113
Is that
Figure BDA00025298119900002114
The method (2) is implemented by the following steps,
Figure BDA00025298119900002115
s428, derivation is carried out on the Lyapunov function, and when t tends to be infinite, V is obtained through solution8(t) the vanishing virtual controller and adaptation law is:
Figure BDA00025298119900002116
Figure BDA00025298119900002117
Figure BDA00025298119900002118
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00025298119900002119
is ρ8Is determined by the estimated value of (c),
Figure BDA00025298119900002120
is the vector of the fuzzy basis function of the eighth step,
Figure BDA00025298119900002121
is theta8An estimated value of (d);
s419, selecting a ninth Lyapunov function:
Figure BDA0002529811990000221
wherein ξ9＝z9-α8，
Figure BDA0002529811990000222
Is that
Figure BDA0002529811990000223
The method (2) is implemented by the following steps,
Figure BDA0002529811990000224
is ρ9The estimated error of (2);
s429, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution9(t) the actual controller and adaptation law going to zero is:
Figure BDA0002529811990000225
Figure BDA0002529811990000226
Figure BDA0002529811990000227
wherein the content of the first and second substances,
Figure BDA0002529811990000228
is ρ9Is determined by the estimated value of (c),
Figure BDA0002529811990000229
is the fuzzy basis function vector of the ninth step,
Figure BDA00025298119900002210
is theta9An estimate of (d).
And S43, finally obtaining the design method of the preset performance controller for inhibiting the vertical vibration of the rolling mill.
After the designed rolling mill vibration suppression controller is applied, the response curve comparison of the rolling mill working roll vibration displacement under the control of the existence of the preset performance is shown in figure 3, and as can be seen from the figure, the vibration attenuation rate, the steady-state error and the overshoot of the rolling mill system after the vibration suppression controller designed by the method is used are obviously improved. Meanwhile, a vibration acceleration sensor is used for collecting and comparing vibration signals, and the situation of the actual site vibration alarm times after the vibration suppressor designed by the method is put into use is shown in fig. 4. As can be seen from the figure, the vibration suppressor has obvious effect on the vibration suppression of the rolling mill, and the effectiveness of the vibration suppressor on the rolling mill is shown.
TABLE 1 Rolling Mill mechanical-Hydraulic coupling vibration System parameters
Figure BDA00025298119900002211
Figure BDA0002529811990000231
The above-mentioned embodiments are merely illustrative of the preferred embodiments of the present invention, and do not limit the scope of the present invention, and various modifications and improvements made to the technical solution of the present invention by those skilled in the art without departing from the spirit of the present invention shall fall within the protection scope defined by the claims of the present invention.

Claims (5)

1. A design method of a rolling mill vertical vibration suppression controller based on self-adaptive fuzzy backstepping is characterized by firstly establishing a non-linear model of rolling mill vertical vibration according to the dynamic principle of a rolling mill vibration system; then, determining a control target for inhibiting the vibration of the rolling mill according to the actual working condition; and finally, designing an adaptive backstepping controller by combining a nonlinear model of vertical vibration of the rolling mill and a control target for inhibiting vibration of the rolling mill, wherein the specific design method of the controller comprises the following steps:
s1, collecting parameters of a mechanical-hydraulic coupling vibration system of the rolling mill and characteristic parameters of a dead zone of an electro-hydraulic servo valve;
s2, establishing a four-degree-of-freedom mechanical-hydraulic coupling nonlinear model of the vertical vibration of the rolling mill according to Newton' S second theorem:
Figure FDA0003670319510000011
wherein z is1For the oscillating displacement of the working rolls, z2Is the work roll vibration speed, z3For the oscillatory displacement of the intermediate roll, z4Is the intermediate roll oscillation speed, z5For supporting the roll vibrational displacement, z6For the vibration speed of the supporting roller, z7For oscillating displacement of the cylinder, z8Is the cylinder vibration speed, z9Is the pressure at the rodless cavity, m1，m2，m3，m4Equivalent mass of a working roll and a bearing, an intermediate roll and a bearing, a supporting roll and a bearing, a hydraulic cylinder and a piston rod respectively; k is a radical of formula1，k2，k3Equivalent rigidity between a working roll and a middle roll, between the middle roll and a supporting roll, and between the supporting roll and a hydraulic cylinder; c. C1，c2，c3Equivalent damping between a working roll and a middle roll, between the middle roll and a supporting roll, between the supporting roll and a hydraulic cylinder respectively; a. the1Is the area of the rodless cavity, A2Is the area of the rod cavity, ctIs the leakage coefficient, P, in the cylinder2Pressure of the rod chamber, V initial volume of the control chamber, betaeIs the bulk modulus, k, of the oilqIs a process coefficient, u is a comprehensive expression of the characteristic parameter of the dead zone of the servo valve, Fz(z1,z2) Is a function expression related to the vibration displacement and the vibration speed of the working roll;
s3, determining a target for restraining the vibration of the rolling mill;
s31, setting the vertical vibration displacement of the working roll of the rolling mill close to zero as a control target according to the fact that the vertical vibration of the rolling mill is caused by the vertical jumping of the working roll;
s32, controlling the vertical vibration attenuation rate and the maximum allowable displacement of the rolling mill within a set range, wherein the target for restraining the vibration of the rolling mill can be expressed as follows:
Figure FDA0003670319510000021
wherein ξ1Indicating the displacement of the work rolls during vibration of the mill, mu (t) ═ mu0e-kt+μ∞，μ0，k，μ∞Is a positive real number that is specified,δand
Figure FDA0003670319510000022
are given positive real numbers;
s4, giving a controller and a parameter adaptive law according to the Lyapunov stability criterion, and designing an adaptive fuzzy vibration suppressor;
s41, selecting a proper Lyapunov function according to the four-degree-of-freedom mechanical-hydraulic coupling nonlinear model of the vertical vibration of the rolling mill established in the step S2;
s42, derivation is carried out on the Lyapunov function, so that when t tends to be infinite, V tends to be zeroi(t) a vanishingly close virtual controller and adaptive law;
s43, finally obtaining a design method of the controller with the preset performance for restraining the vertical vibration of the rolling mill;
selecting nine Lyapunov functions according to the nine subsystems included in the step S2, wherein the step S4 specifically comprises the following steps:
s411, selecting a first Lyapunov function:
Figure FDA0003670319510000023
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003670319510000024
s421, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is solved1(t) the vanishing virtual controller and adaptation law is:
Figure FDA0003670319510000031
wherein the content of the first and second substances,
Figure FDA0003670319510000032
ε1and a1Is a positive parameter being designed;
s412, selecting a second Lyapunov function:
Figure FDA0003670319510000033
wherein xi is2＝z2-α1，σ21、σ22Is a positive parameter that is designed to be,
Figure FDA0003670319510000034
Figure FDA0003670319510000035
is ρ2The error of the estimation of (2) is,
Figure FDA0003670319510000036
is theta2The estimated error of (2);
s422, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution2(t) the vanishingly close virtual controller and adaptive law are:
Figure FDA0003670319510000037
Figure FDA0003670319510000038
Figure FDA0003670319510000039
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA00036703195100000310
is ρ2Is determined by the estimated value of (c),
Figure FDA00036703195100000311
ε22is a given normal number, σ21，τ2，l21，l22Is a positive parameter of the design and,
Figure FDA00036703195100000312
is the vector of the fuzzy basis function of the second step,
Figure FDA00036703195100000313
is theta2An estimated value of (d);
s413, selecting a third Lyapunov function:
Figure FDA00036703195100000314
wherein ξ3＝z3-α2，σ32Is a positive parameter that is designed to be,
Figure FDA00036703195100000315
is theta3The estimation error of (2);
s423, deriving the Lyapunov function, and solving to make V tend to be infinite3(t) the vanishing virtual controller and adaptation law is:
Figure FDA00036703195100000316
Figure FDA0003670319510000041
wherein, a3，l32，σ32Is a positive parameter that is being designed for,
Figure FDA0003670319510000042
is that
Figure FDA0003670319510000043
The transpose of (a) is performed,
Figure FDA0003670319510000044
is the vector of the fuzzy basis function of the third step,
Figure FDA0003670319510000045
is theta3An estimated value of (d);
s414, selecting a fourth Lyapunov function:
Figure FDA0003670319510000046
wherein xi is4＝z4-α3，σ41、σ42Is a positive parameter that is designed to be,
Figure FDA0003670319510000047
Figure FDA0003670319510000048
is ρ4The error of the estimation of (2) is,
Figure FDA0003670319510000049
is theta4The estimation error of (2);
s424, derivation is carried out on the Lyapunov function, and solution is carried out so that
Figure FDA00036703195100000410
When t tends to infinity, V4(t) the vanishingly close virtual controller and adaptive law are:
Figure FDA00036703195100000411
Figure FDA00036703195100000412
Figure FDA00036703195100000413
wherein the content of the first and second substances,
Figure FDA00036703195100000414
is ρ4Is determined by the estimated value of (c),
Figure FDA00036703195100000415
ε42is a given normal number; sigma41，τ4，l41，l42Is a positive parameter of the design and,
Figure FDA00036703195100000416
is the vector of the fuzzy basis function of the fourth step,
Figure FDA00036703195100000417
is theta4An estimated value of (d);
s415, selecting a fifth Lyapunov function:
Figure FDA00036703195100000418
wherein ξ5＝z5-α4，σ52Is a positive parameter that is designed to be,
Figure FDA00036703195100000419
is theta5The estimated error of (2);
s425, deriving the Lyapunov function, and solving to ensure that V is obtained when t tends to be infinite5(t) the control law and the adaptive law of the vanishing virtual controller are:
Figure FDA00036703195100000420
Figure FDA0003670319510000051
wherein, a5，l52，σ52Is a positive parameter that is designed to be,
Figure FDA0003670319510000052
is that
Figure FDA0003670319510000053
The transpose of (a) is performed,
Figure FDA0003670319510000054
is the vector of the fuzzy basis function of the fifth step,
Figure FDA0003670319510000055
is θ5An estimated value of (d);
s416, selecting a sixth Lyapunov function:
Figure FDA0003670319510000056
wherein xi is6＝z6-α5，σ61、σ62Is a positive parameter that is designed to be,
Figure FDA0003670319510000057
Figure FDA0003670319510000058
is ρ6The error of the estimation of (2) is,
Figure FDA0003670319510000059
is theta6The estimation error of (2);
s426, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution6(t) the vanishingly close virtual controller and adaptive law are:
Figure FDA00036703195100000510
Figure FDA00036703195100000511
Figure FDA00036703195100000512
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA00036703195100000513
is ρ6Is determined by the estimated value of (c),
Figure FDA00036703195100000514
ε62is a given normal number; sigma61，τ6，l61，l62Is a positive parameter of the design and,
Figure FDA00036703195100000515
is the vector of the fuzzy basis function of the sixth step,
Figure FDA00036703195100000516
is theta6An estimated value of (d);
s417, selecting a seventh Lyapunov function:
Figure FDA00036703195100000517
wherein ξ7＝z7-α6，σ72Is a parameter that is designed to be,
Figure FDA00036703195100000518
is θ7The estimation error of (2);
s427, derivation of the Lyapunov function, and solving so that when t tends to be infinite, V is7(t) the vanishingly close virtual controller and adaptive law are:
Figure FDA00036703195100000519
Figure FDA0003670319510000061
wherein, a7，l72，σ72Is a positive parameter that is designed to be,
Figure FDA0003670319510000062
is that
Figure FDA0003670319510000063
The transpose of (a) is performed,
Figure FDA0003670319510000064
is the vector of the fuzzy basis function of the seventh step,
Figure FDA0003670319510000065
is θ7An estimated value of (d);
s418, selecting an eighth Lyapunov function:
Figure FDA0003670319510000066
wherein ξ8＝z8-α7，σ81，σ82Is a positive parameter of the design and,
Figure FDA0003670319510000067
is that
Figure FDA0003670319510000068
The transpose of (a) is performed,
Figure FDA0003670319510000069
is θ8The error of the estimation of (2) is,
Figure FDA00036703195100000610
s428, derivation is carried out on the Lyapunov function, and when t tends to be infinite, V is obtained through solution8(t) the vanishingly close virtual controller and adaptive law are:
Figure FDA00036703195100000611
Figure FDA00036703195100000612
Figure FDA00036703195100000613
wherein the content of the first and second substances,
Figure FDA00036703195100000614
is ρ8Is determined by the estimated value of (c),
Figure FDA00036703195100000615
ε82is a given normal number;
σ81，σ82，l81，l82is a positive parameter of the design and,
Figure FDA00036703195100000616
is the fuzzy basis function vector of the eighth step,
Figure FDA00036703195100000617
is θ8An estimated value of (d);
s419, selecting a ninth Lyapunov function:
Figure FDA00036703195100000618
wherein xi is9＝z9-α8，σ91，σ92Is a positive parameter of the design and,
Figure FDA00036703195100000619
is that
Figure FDA00036703195100000620
The method (2) is implemented by the following steps,
Figure FDA00036703195100000621
is θ9The error of the estimation of (2) is,
Figure FDA00036703195100000622
Figure FDA00036703195100000623
is ρ9The estimated error of (2);
s429, derivation is conducted on the Lyapunov function, and when t tends to be infinite, V is obtained through solution9(t) the actual controller and adaptation law going to zero is:
Figure FDA0003670319510000071
Figure FDA0003670319510000072
Figure FDA0003670319510000073
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003670319510000074
is ρ9Is determined by the estimated value of (c),
Figure FDA0003670319510000075
ε92is a given normal number; sigma91，σ92，l91，l92Is a positive parameter of the design and,
Figure FDA0003670319510000076
is the fuzzy basis function vector of the ninth step,
Figure FDA0003670319510000077
is theta9An estimate of (d).
2. The design method of the adaptive fuzzy backstepping based rolling mill droop suppression controller as claimed in claim 1, wherein in step S2, u is a comprehensive expression of the servo valve dead zone characteristic parameter, and the specific form thereof can be expressed as:
Figure FDA0003670319510000078
wherein eta isr、ηl、brAnd blAre all servo valve dead zone characteristic parameters.
3. The design method of the adaptive fuzzy backstepping based rolling mill droop suppression controller according to claim 1, wherein in step S2, kqThe process coefficient can be expressed in a specific form as:
Figure FDA0003670319510000079
wherein, CdIs a flow coefficient of a valve port of the servo valve; w is the area gradient of the valve port of the servo valve, rho is the density of hydraulic oil in the hydraulic cylinder, PsFor supply pressure, PtAs oil return pressure, xvFor servo valve spool displacement, kvIs a gain factor.
4. The design method of the vertical oscillation suppression controller of the rolling mill based on the adaptive fuzzy backstepping as claimed in claim 1, wherein in order to ensure that the nine designed lyapunov functions are all positive, according to the item where the coefficients are located, the coefficients in the nine lyapunov functions need to be respectively designed as positive numbers; to ensure the effectiveness of the controller, nine Lyapunov functions V are requirediThe derived results respectively satisfy the Lyapunov stability criterion, the coefficients in the Lyapunov functions need to be respectively designed as positive numbers according to the item where the coefficients are located, and simultaneously, each Lyapunov function after derivation satisfies the criterion
Figure FDA0003670319510000081
Wherein c, Δ are both normal numbers; after each lyapunov function is derived, the parameters to be designed are different.
5. The design method of the adaptive fuzzy backstepping based rolling mill vertical vibration suppression controller according to claim 1, wherein according to the formal similarity of the neutron system in the nonlinear model established at step S2, the design parameters in the third, fifth and seventh lyapunov functions are consistent, the design parameters in the second, fourth and sixth lyapunov functions are consistent, and the design parameters in the eighth and ninth lyapunov functions are consistent.
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