CN110673473B - Error symbol integral robust self-adaptive control method of biaxial coupling tank gun system - Google Patents

Error symbol integral robust self-adaptive control method of biaxial coupling tank gun system Download PDF

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CN110673473B
CN110673473B CN201910865903.9A CN201910865903A CN110673473B CN 110673473 B CN110673473 B CN 110673473B CN 201910865903 A CN201910865903 A CN 201910865903A CN 110673473 B CN110673473 B CN 110673473B
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姚建勇
马翔
邓文翔
杨国来
袁树森
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Nanjing University of Science and Technology
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Abstract

The invention discloses an error sign integral robust self-adaptive control method of a two-axis coupling tank gun system, which can be used for controlling systems with strong uncertainty and nonlinearity by using an error sign integral robust self-adaptive control algorithm. The present invention is set forth in the following background: because the development of the tank gun at present has higher requirements on the control of the precision of the tank gun, the previous modeling methods of pitch and direction are not enough to have good control effect on the whole system, and the modeling scheme of two-axis coupling can effectively reduce the actual motion situation of the tank gun. However, for such two-axis coupled tank gun systems, the control difficulty of such systems is greatly increased due to the coupling characteristic and nonlinearity of the two-axis coupled tank gun systems. The control method disclosed by the invention effectively solves the problem of a two-axis coupling tank gun system, improves the control precision of the whole system and obtains good tracking performance.

Description

Error symbol integral robust self-adaptive control method of biaxial coupling tank gun system
Technical Field
The invention relates to the technical field of motion control, in particular to an error symbol integral robust self-adaptive control method of a two-axis coupling tank gun system.
Background
In modern land-based weapon systems, tanks remain one of the major battle weapons on the ground. The main attack mode of the tank comes from tank weapons carried by the tank. With the trend of the new generation of armored weaponry toward integrated systematization, automation, light weight and unmanned, an unmanned turret system is competitively developed and researched in a plurality of military strong countries in the development process, but the unmanned turret system is inferior to a manned turret in overall performance. The unmanned turret is also called an overhead weapon turret, and means that a weapon system of a tank or an armored vehicle is arranged at the top of a vehicle body, passengers are not arranged in the turret, and all the passengers are positioned in the vehicle body. The development of the tank unmanned turret has not little promotion in the maintenance effect of the safety of tank carrying personnel and the battlefield regardless of the internal environment of passengers. Therefore, unmanned turret control carried by the roadbed mobile platform is very important in the future army wars, and especially plays a key role in the research, development and manufacture of five generations of tanks. However, for such two-axis coupled tank gun systems, the control difficulty of such systems is greatly increased due to factors such as uncertainty, unmodeled errors and nonlinearity. Therefore, the traditional control method is likely to fail to achieve the target of control, and even cause the system to be unstable if the disturbance amount is large.
Therefore, control methods for the two-axis coupling system of the tank gun are continuously and successively proposed. The PID algorithm can achieve certain control on the whole system, but the control performance of the PID algorithm on the whole control system cannot reach a very good state due to the tracking precision, the error convergence time and especially the influence of external interference on the whole system. It has been proposed to adopt a sliding mode variable structure control algorithm (SMC) to control such a system with large external interference. However, due to the discontinuity in SMC control, the actual discontinuity appears to be bounded by a switching plane above which the synovial track drive direction is opposite to and intersects the switching plane below which the synovial track drive direction is intersecting, which intersection is discontinuous. Therefore, in a practical system, especially in a tank gun two-axis coupling system adopted in the invention, the defect can be amplified continuously, so that the system chattering effect is more obvious, and the jitter of a gun muzzle can be increased in the specific adjustment of the gun angle, so that the overall control precision is greatly influenced. The Active Disturbance Rejection Control (ADRC) can achieve a good control for the control system, but the control performance of the ADRC depends on the setting of the algorithm gain parameter, and if the modeling parameter of the system is inaccurate, or many factors are not taken into consideration in the actual system, the control performance of the ADRC is greatly reduced. The state expansion observer (ESO) adopted in the ADRC control system is mainly based on the output error of the model, so that the external error cannot be observed, and in the actual system of the model, the external disturbance is also a factor which cannot be ignored, so that the control performance of the ADRC control algorithm is also reduced.
Disclosure of Invention
The invention aims to provide an error sign integral robust self-adaptive control method of a two-axis coupling tank gun system, which is based on the traditional RISE control method and integrates the thought of self-adaptive control, thereby not only effectively solving the problem of modeling uncertainty, but also obtaining continuous control input and asymptotic tracking performance. The influence of the uncertain coefficient of the model can be better solved in the self-adaptive part, so that the control performance is better, the problems of nonlinearity, model uncertainty, external interference and the like are solved, and the requirement on the control precision of the system is met.
The technical solution for realizing the purpose of the invention is as follows: an error symbol integral robust self-adaptive control method of a two-axis coupling tank gun system comprises the following steps:
step 1, establishing a mathematical model of dynamics of a two-axis coupling tank gun system;
step 2, designing an error symbol integral robust self-adaptive controller;
and 3, performing stability verification by using the Lyapunov stability theory, and introducing a Barbalt theorem to obtain a global asymptotic stability result of the system.
Compared with the prior art, the invention has the following remarkable advantages: the problem of the traditional control method for the two-axis coupling tank gun is effectively solved, the control precision and the error convergence time are reduced, and the tracking performance and the anti-interference capability for external errors are greatly improved; the simulation result verifies the effectiveness of the test paper.
Drawings
FIG. 1 is a schematic diagram of a principle of an error sign integral robust adaptive (RISEA) control method of a two-axis coupling tank gun system of the invention;
FIG. 2 is a schematic diagram of a two-axis coupled tank gun system;
FIG. 3 is a diagram of the process of tracing the desired command by the system in the high and low direction output under the action of the RISEA controller designed by the present invention;
FIG. 4 is a diagram of the process of tracking the horizontal direction output of the system to the desired command under the action of the RISEA controller designed by the present invention;
FIG. 5 is a graph of the tracking error in the elevation direction of the system over time with the RISEA controller;
FIG. 6 is a graph of tracking error in the horizontal direction of the system over time with the RISEA controller;
FIG. 7 shows system interference of
Figure BDA0002201261730000031
Comparing the tracking error of the system in the horizontal direction under the action of the three controllers of RISEA, RISE and PID respectively;
FIG. 8 shows system interference as
Figure BDA0002201261730000032
Comparing the tracking error of the system in the horizontal direction under the action of the three controllers of RISEA, RISE and PID respectively;
FIG. 9 is parameter θ in RISEA controller11A time-varying plot of the estimated value of (a);
FIG. 10 is parameter θ in RISEA controller12A graph of the estimated value of (c) over time;
FIG. 11 is parameter θ in RISEA controller 13A time-varying plot of the estimated value of (a);
FIG. 12 is a RISEA controller parameter θ21A graph of the estimated value of (c) over time;
FIG. 13 is parameter θ in RISEA controller22A graph of the estimated value of (c) over time;
FIG. 14 is parameter θ in RISEA controller23A graph of the estimated value of (c) over time;
FIG. 15 shows system interference of
Figure BDA0002201261730000033
Respectively acting input diagrams of the system in the height direction by the RISEA controller;
FIG. 16 shows system interference of
Figure BDA0002201261730000034
The RISEA controller respectively acts on the input diagrams of the system in the horizontal direction.
Detailed Description
The invention is described in further detail below with reference to the figures and the embodiments.
With reference to fig. 1-2, the method for controlling the error sign integral robustness self-adaptation of the two-axis coupled tank gun system comprises the following steps:
step 1, establishing a mathematical model of dynamics of a two-axis coupled gun system, which comprises the following steps:
step 1.1, considering a modeling thought of a dynamic model modeling thought comprehensive mechanical arm, and establishing a dynamic model of the tank gun by adopting a Lagrange-Euler method:
therefore, according to the Lagrange-Euler method, the kinetic equation of the two-axis coupled tank gun system is as follows:
Figure BDA0002201261730000035
in the formula (1), the rotation angle is q ═ q1 q2]TWherein q is 1For the rotation angle, q, of the two-axis coupling tank gun system in the horizontal direction2The two shafts are coupled with the rotation angle of the tank gun system in the high and low directions; m is a group ofa∈R2×2Is an inertia symmetric positive definite matrix; mb∈R2 ×2Is a Coriolis force centrifugation matrix; mg∈R2×1Is a moment of gravity vector, Mg＝[Tg1 Tg2]T，Tg1The two-axis coupled tank gun system is the gravity moment, T, in the horizontal directiong1＝0，Tg2Coupling the gravity moment of the tank gun system in the height direction for two shafts; tank gun system input T ═ T1 T2]TWherein T is1For horizontal input, T, of a two-axis coupled tank gun system2The input torque in the height direction of the two-axis coupling tank gun system is obtained; friction torque Tf＝[Tf1 Tf2]，Tf1Moment of resistance, T, generated by friction in the horizontal direction of a two-axis coupled tank gun systemf2Resistance moment generated by friction force in high and low directions of two-shaft coupled tank gun system, TfAdopting a lugre model to carry out fitting approximation:
Figure BDA0002201261730000041
i is 1,2, wherein lijFor the friction parameters, i is 1,2, j is 1,2,3, vjIs a friction shape parameter; total interference d ═ d of two-axis coupling tank gun system1 d2]TWherein d is1For horizontal interference of a two-axis coupled tank gun system, where d2The two shafts are coupled with the interference of the tank gun system in the high and low directions;
Figure BDA0002201261730000042
wherein A is11，A12，A21And A22An inertia term of an inertia positive definite matrix;
Figure BDA0002201261730000043
wherein B is12，B21And B 22A coriolis centrifuge term that is a coriolis centrifuge matrix; let sinq bei＝si，cosqi＝ci，i＝1,2；
Therefore MaAnd MbWherein the parameters are expressed by the formula:
Figure BDA0002201261730000044
Figure BDA0002201261730000045
Figure BDA0002201261730000046
wherein Iyy1，Ixx2，Iyy2And Izz2Is the moment of inertia; i isxz2，Iyz2，Ixy2Is the inertia tensor.
Step 1.2, defining state variables:
Figure BDA0002201261730000047
and let u be Ti+TgiAnd i is 1,2, the equation of motion of formula (1) is converted into an equation of state:
Figure BDA0002201261730000051
in the formula (2), the friction function parameter is defined as
Figure BDA0002201261730000052
i is 1,2, j is 1,2,3 and the defining parameters are estimated as
Figure BDA0002201261730000053
j＝1,2,3，
Figure BDA0002201261730000054
Is for thetajThe parameter estimation value of (2); the functional part of the friction function is defined as:
Figure BDA0002201261730000055
Figure BDA0002201261730000056
x1a column vector, x, representing the horizontal and directional angles of rotation of the tank gun2A column vector which represents the horizontal rotation angular velocity and the direction rotation angular velocity of the tank gun;
for the controller design, assume the following:
suppose that the total interference d of a 1 two-axis coupling tank gun system is ═ d1 d2]TIs sufficiently smooth to make
Figure BDA0002201261730000057
Are present and bounded i.e.:
Figure BDA0002201261730000058
upper bound parameter δ in equation (3)1i,δ2iI is 1 and 2 is unknown normal number, i.e.
Figure BDA0002201261730000059
With an uncertain upper bound on the basis of,
and (5) transferring to the step 2.
The total interference of the tank gun system comprises interference caused by external load interference, unmodeled friction, unmodeled dynamic state and deviation of system actual parameters and modeled parameters.
Step 2, designing an error symbol integral robust self-adaptive controller, comprising the following steps:
Step 2.1, definition of z1＝x1-x1dTracking error of tank gun system, x1dIs a position instruction which is expected to be tracked by the tank gun system and the second order of the instruction can be continuously microminiaturized according to a first equation in an equation (2)
Figure BDA00022012617300000510
Selecting x2For virtual control, let equation
Figure BDA00022012617300000511
Tends to a stable state; let x be2eqDesired value, x, for virtual control2eqAnd the true state x2Has an error of z2＝x2-x2eqTo z is to1The derivation can be:
Figure BDA0002201261730000061
designing a virtual control law:
Figure BDA0002201261730000062
adjustable gain in formula (5)
Figure BDA0002201261730000063
k11、k12Are positive numbers, then:
Figure BDA0002201261730000064
due to z1(s)＝G(s)z2(s) wherein G(s) is 1/(s + k)1) Is a stable transfer function when z2When going to 0, z1Also inevitably tends to 0;
step 2.2, to obtain an additional degree of freedom of controller design, defining an auxiliary error signal r:
Figure BDA0002201261730000065
adjustable gain in formula (7)
Figure BDA0002201261730000066
k21、k22Are all positive numbers;
according to equations (2) and (7), there is an expansion of r as follows:
Figure BDA0002201261730000067
according to equation (8), the model-based controller can be designed to:
Figure BDA0002201261730000068
formula (9)
Figure BDA0002201261730000069
Wherein k isr1，kr2Are all made ofPositive feedback gain, ImIs a unit diagonal matrix, uaFor model-based compensation terms, usIs a robust control law and in which us1For a linear robust feedback term, us2For a nonlinear robust term to overcome the influence of modeling uncertainty on system performance, the residual error of parameter estimation is defined as
Figure BDA00022012617300000610
j is 1,2,3, and formula (9) is substituted into formula (8):
Figure BDA00022012617300000611
The parameter adaptation law is designed in equation (10) as follows:
Figure BDA00022012617300000612
Γiare adaptive gains, all are constants; since the state of r is unknown, the state is processed by a fractional integration method, and an actual adaptive law is obtained:
Figure BDA0002201261730000071
according to the design method of an error sign integral robust controller, an integral robust term us2The design is as follows:
Figure BDA0002201261730000072
controller gain in equation (11)
Figure BDA0002201261730000073
Beta should satisfy the following condition:
Figure BDA0002201261730000074
wherein beta is1For gain in the horizontal direction, β2High and low directional gains.
The two-sided derivation of the equation of equation (10) and the use of equations (7), (12) and (13) can be obtained:
Figure BDA0002201261730000075
in the formula, the term is not estimated
Figure BDA0002201261730000076
Defining an error parameter as Z ═ Z1 z2 r]TFrom
Figure BDA0002201261730000077
The structure can be obtained, and a global reversible non-subtractive function rho (| | Z |) is formed for R+Such that:
Figure BDA0002201261730000078
and (5) turning to the step 3.
Step 3, stability is proved by applying Lyapunov stability theory, and a global asymptotic stability result of the system is obtained by introducing barbalt theorem, which is specifically as follows:
defining an auxiliary function L (t), P (t):
Figure BDA0002201261730000079
Figure BDA00022012617300000710
z2(0)、
Figure BDA00022012617300000711
respectively represents z2And
Figure BDA00022012617300000712
an initial value of (1);
is proved to be when
Figure BDA0002201261730000081
If P (t) is greater than or equal to 0.
Proof of this lemma:
integrating the two sides of equation (19) and applying equation (7) to obtain:
Figure BDA0002201261730000082
the equation (20) is divided into parts and integrated to obtain:
Figure BDA0002201261730000083
therefore, it is
Figure BDA0002201261730000084
As can be seen from equation (22), if β is selected to satisfy equation
Figure BDA0002201261730000085
Under the conditions shown, P (t) is greater than or equal to 0, which is the case.
According to the above reasoning, it can be known
Figure BDA0002201261730000086
P (t) ≧ 0, so the Lyapunov function is defined as follows:
Figure BDA0002201261730000087
the derivation of equation (23) and substitution of equations (6), (7), (16), and (22) can be obtained:
Figure BDA0002201261730000091
and because of
Figure BDA0002201261730000092
Then:
Figure BDA0002201261730000093
wherein the parameters
Figure BDA0002201261730000094
To ensure
Figure BDA0002201261730000095
The semi-negative definite row of (2) needs r to be more than or equal to 0, namely
Figure BDA0002201261730000099
From the formula (25)
Figure BDA0002201261730000096
V (t). ltoreq.V (0), whereby V.epsilon.L∞Norm, and thus z can be derived1，z2And r are bounded.
Integration of equation (25) yields:
Figure BDA0002201261730000097
z is represented by formula (25)1，z2，r∈L2Norm, and is obtained according to equations (6), (7), (13) and hypothesis 1:
Figure BDA0002201261730000098
norm, and therefore W, is consistently continuous, as can be seen by the barbalt theorem: t → ∞ time, W → 0. Therefore, t → ∞ time, z1→0。
It is therefore concluded that: an error sign integral robust controller designed for a two-axis coupling tank gun (2) can enable the system to obtain fullThe result of the local asymptotic stabilization is to adjust the gain k1、k2、krAnd beta can make the tracking error of the system tend to zero under the condition that the time tends to infinity. A schematic diagram of the error sign integral robust self-adjustment (RISEA) control principle of a two-axis coupled tank gun system is shown in FIG. 2.
Examples
In order to assess the performance of the designed controller, the following parameters are taken in simulation to model the two-axis coupling tank gun system:
the inertia tensor matrix parameters are: i isyy1＝2547kg·m2、Ixx2＝5400kg·m2、Iyy2＝5443kg、Izz2＝224kg·m2、Ixy2＝-2.8kg·m2、Iyz2＝13.7kg·m2、Izx2＝0.8kg·m2(ii) a The shape parameters in the lugre friction model were used: v. of 1＝200、v2＝10、v3＝160。
The expected instruction of a given system is
Figure BDA0002201261730000101
The system working condition of the simulation is time-varying disturbance:
Figure BDA0002201261730000102
the following controls were taken to compare:
an error sign integral robust self-adjusting (RISEA) controller: taking a controller parameter k11＝150，k12＝15，kr＝50000；β1＝100，k11＝150，k12＝10，kr＝50000，β 2100; adaptive gain of gamma1＝diag[80 300]，Γ2＝diag[8 25]，Γ1＝diag[1.8 8]。
A PID controller: the PID controller parameter selection steps are as follows: firstly, a set of controller parameters is obtained through a PID parameter self-tuning function in Matlab under the condition of neglecting the nonlinear dynamics of a two-axis coupling tank gun system, and then a system is determinedAnd after the nonlinear dynamics of the system is added, the obtained self-tuning parameters are subjected to fine tuning, so that the system obtains the optimal tracking performance. The selected controller parameter in the high-low direction is kP＝318000,kI＝100,kD120000; the selected controller parameter in the horizontal direction is kP＝220000,kI＝100,kD＝850000；
The system output is tracked to the expected instruction under the action of the RISEA controller as shown in FIGS. 3 and 4, and it can be seen that the expected instruction and the system output are basically overlapped and have good tracking performance; the tracking error of the RISEA controller, the tracking error of the RISE controller compared to the PID controller are shown in FIGS. 5, 6, 7 and 8, respectively. As can be seen from FIG. 5 and FIG. 6, under the action of the RISEA controller, the position output of the direct drive motor system has high tracking precision to the command, and the amplitude of the steady-state tracking error is about 1 × 10 -4(rad), it can be seen from the comparison of the tracking errors of the two controllers in fig. 7 and fig. 8 that the tracking error of the RISEA controller proposed by the present invention is much smaller compared to the PID controller, the magnitude of the steady state tracking error of the PID controller is about 2.1 × 10-2(rad), and its tremor profile is much reduced compared to RISE algorithm.
Fig. 9-14 are graphs of the estimated value of the RISEA controller gain θ according to the present invention, which are time-varying curves, and it can be seen from the graphs that although the initial value of the gain is empirically determined, the gain value automatically converges to an appropriate value with the time-varying function due to the adaptive law, so that the friction parameter of the model can be more improved based on the traditional RISE algorithm, and the performance of the system can be improved for practical application.
FIGS. 15 and 16 illustrate system interference of
Figure BDA0002201261730000111
And (3) a curve graph of the control input of the two-axis coupled tank gun along with the change of time under the action of the time RISEA controller. As can be seen from the figure, the obtained control input is a low-frequency continuous signal, which is more beneficial to be implemented in practical application.

Claims (2)

1. An error sign integral robust self-adaptive control method of a two-axis coupling tank gun system is characterized by comprising the following steps of:
Step 1, establishing a mathematical model of dynamics of a two-axis coupling gun system, which comprises the following steps:
step 1.1, considering a dynamic model modeling thought, integrating a modeling thought of a mechanical arm, and establishing a dynamic model of the tank gun by adopting a Lagrange-Euler method:
therefore, according to the Lagrange-Euler method, the kinetic equation of a two-axis coupled tank gun system is as follows:
Figure FDA0003566941260000011
in the formula (1), the rotation angle is q ═ q1 q2]TWherein q is1For the horizontal rotation angle, q, of a two-axis coupled tank gun system2The two shafts are coupled with the rotation angle of the tank gun system in the height direction; m is a group ofa∈R2×2Is an inertia symmetric positive definite matrix; mb∈R2×2Is a Coriolis force centrifugation matrix; mg∈R2×1Is a moment of gravity vector, Mg＝[Tg1 Tg2]T，Tg1The two-axis coupled tank gun system is the gravity moment, T, in the horizontal directiong1＝0，Tg2Coupling the gravity moment of the tank gun system in the height direction for two shafts; tank gun system input T ═ T1 T2]TWherein T is1For horizontal input, T, of a two-axis coupled tank gun system2The input torque in the height direction of the two-axis coupling tank gun system is obtained; friction torque Tf＝[Tf1 Tf2]，Tf1Moment of resistance, T, generated by friction in the horizontal direction of a two-axis coupled tank gun systemf2Resistance moment generated by friction force in high and low directions of two-shaft coupled tank gun system, TfAdopting a lugre model to carry out fitting approximation:
Figure FDA0003566941260000012
Wherein lijThe friction parameter is i-1, 2, j-1, 2,3, the friction shape parameter vj(ii) a Total interference d ═ d of system error of two-axis coupling tank gun1 d2]TWherein d is1For horizontal interference of a two-axis coupled tank gun system, where d2The two shafts are coupled with the interference of the tank gun system in the high and low directions;
Figure FDA0003566941260000013
wherein A is11、A12、A21And A22The inertia terms are inertia positive definite matrixes;
Figure FDA0003566941260000014
wherein B is12、B21And B22A coriolis centrifuge term that is a coriolis centrifuge matrix; let sin qi＝si，cos qi＝ci，i＝1,2；
Therefore MaAnd MbWherein the parameters are expressed by the formula:
Figure FDA0003566941260000021
A12(q)＝A21(q)＝s2Iyz2-c2Ixy2，A22＝Iyy2；
Figure FDA0003566941260000022
Figure FDA0003566941260000023
wherein Iyy1、Ixx2、Iyy2And Izz2Are rotational inertia; i isxz2、Iyz2、Ixy2Are all inertia tensors;
step 1.2, defining state variables:
Figure FDA0003566941260000024
and let u be Ti+TgiAnd i is 1,2, the equation of motion of formula (1) is converted into an equation of state:
Figure FDA0003566941260000025
in the formula (2), the friction function parameter is defined as
Figure FDA0003566941260000026
And defining a parameter estimate of
Figure FDA0003566941260000027
Figure FDA0003566941260000028
Is for thetajThe parameter estimation value of (2); the functional part of the friction function is defined as:
Figure FDA0003566941260000029
Figure FDA00035669412600000210
Figure FDA00035669412600000211
x1a column vector, x, representing the horizontal and directional angles of rotation of the tank gun2A column vector which represents the horizontal rotation angular velocity and the direction rotation angular velocity of the tank gun;
for the controller design, assume the following:
assume that 1: total interference d ═ d of two-axis coupling tank gun system1 d2]TIs sufficiently smooth to make
Figure FDA00035669412600000212
Are present and bounded i.e.:
Figure FDA00035669412600000213
The upper bound parameter in formula (3) is δ1i、δ2iI is 1 and 2 are unknown normal numbers, i.e.
Figure FDA00035669412600000214
If the upper bound is uncertain, the step 2 is carried out;
step 2, designing an error symbol integral robust self-adaptive controller, comprising the following steps:
step 2.1, defining the tracking error z of the tank gun system1＝x1-x1d，x1dIs a position instruction which is expected to be tracked by the tank gun system and the second order of the instruction can be continuously microminiaturized according to the first equation in the formula (2)
Figure FDA0003566941260000031
Selecting x2For virtual control, let equation
Figure FDA0003566941260000032
Tends to a stable state; let x2eqFor desired values of virtual control, x2eqAnd the true state x2Has an error of z2＝x2-x2eqTo z is to1The derivation can be:
Figure FDA0003566941260000033
designing a virtual control law:
Figure FDA0003566941260000034
adjustable gain in formula (5)
Figure FDA0003566941260000035
k11、k12Are positive numbers, then:
Figure FDA0003566941260000036
due to z1(s)＝G(s)z2(s) wherein G(s) is 1/(s + k)1) Is a stable transfer function when z2When going to 0, z1Also necessarily tends to 0;
step 2.2, to obtain an additional degree of freedom of controller design, defining an auxiliary error signal r:
Figure FDA0003566941260000037
adjustable gain in formula (7)
Figure FDA0003566941260000038
k21、k22Are all positive numbers;
according to equations (2) and (7), there is an expansion of r as follows:
Figure FDA0003566941260000039
according to equation (8), the model-based controller is designed to:
Figure FDA00035669412600000310
in the formula (9)
Figure FDA0003566941260000041
Wherein k isr1，kr2All are positive feedback gains, ImIs a unit diagonal matrix, uaFor model-based compensation terms, usIs a robust control law, and u s1For a linear robust feedback term, us2Is a nonlinear robust term for overcoming the influence of modeling uncertainty on the system performance, and the residual error of parameter estimation is defined as
Figure FDA0003566941260000042
Substituting formula (9) into formula (8) to obtain:
Figure FDA0003566941260000043
the parameter adaptation law is designed in equation (10) as follows:
Figure FDA0003566941260000044
Γiare adaptive gains, all are constants; since the state of r is unknown, the state is processed by a fractional integration method, and an actual adaptive law is obtained:
Figure FDA0003566941260000045
according to the design method of an error sign integral robust controller, an integral robust term us2The design is as follows:
Figure FDA0003566941260000046
controller gain in equation (11)
Figure FDA0003566941260000047
Beta should satisfy the following condition:
Figure FDA0003566941260000048
wherein beta is1For the controller gain in the horizontal direction, beta2Gain is the high and low direction of the controller;
the two-sided derivation of the equation of equation (10) and the use of equations (7), (12) and (13) can be obtained:
Figure FDA0003566941260000049
in the formula, the term is not estimated
Figure FDA00035669412600000410
Defining an error parameter as Z ═ Z1 z2 r]TFrom
Figure FDA00035669412600000411
The structure can be obtained, and a global reversible non-subtractive function rho (| | Z |) is formed for R+Such that:
Figure FDA00035669412600000412
turning to the step 3;
step 3, stability is proved by applying Lyapunov stability theory, and a global asymptotic stability result of the system is obtained by introducing barbalt theorem, which is concretely as follows:
defining the auxiliary functions L (t), P (t):
Figure FDA0003566941260000051
Figure FDA0003566941260000052
z2(0)、
Figure FDA0003566941260000053
respectively represents z2And
Figure FDA0003566941260000054
an initial value of (1);
is proved to be when
Figure FDA0003566941260000055
When P (t) ≧ 0, the Lyapunov function is thus defined as follows:
Figure FDA0003566941260000056
The Lyapunov stability theory is applied to carry out stability verification, and the Barbalt theorem is applied to obtain the global asymptotic stability result of the system, so that the gain k is adjusted1、k2、krThe tracking error of the tank gun system tends to zero under the condition that the time tends to infinity.
2. The adaptive control method for the error sign integral robustness of the two-axis coupling tank gun system according to claim 1, characterized in that: in step 1.2, the total interference of the tank gun system comprises the interference caused by external load interference, unmodeled friction, unmodeled dynamic state and the deviation of the actual parameters and the modeling parameters of the system.
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