CN114428459B - Anti-interference control method considering convergence time synchronization constraint - Google Patents
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Abstract
The invention discloses an anti-interference control method considering convergence time synchronization constraint, which comprises the following steps: aiming at inaccurate modeling information of a multi-input multi-output affine system, an online observation uncertainty item of an interference observer is designed, and a foundation is laid for the design of a controller; on the basis of a designed interference observer, a robust controller for synchronizing convergence time is designed by utilizing the mathematical principle of unit direction vectors, so as to realize the synchronous convergence of all state quantities along with time; and designing the parameters of the controller so that the time for each state of the system to reach stability synchronously is a preset value. The control method has the characteristics of strong robustness, high control precision, low energy consumption and the like, and is suitable for being applied to control tasks with uncertain system items and synchronous and stable states.
Description
Technical Field
The invention relates to an anti-interference control method considering convergence time synchronization constraint, which is mainly applied to rapid maneuvering of spacecraft gestures, cooperative work of double-arm robots and the like, and belongs to the technical field of control.
Background
In recent years, limited time control has been widely studied in various fields. The convergence time of the system based on asymptotically stabilization is infinite, which means that the error state of the system cannot converge to the equilibrium point of the system within a finite time. The limited time control can enable the system to quickly converge to the balance point in a certain time, and the convergence performance can be effectively improved. The essence of finite time control is that it is realized by self-contained fractional power terms, and has a faster convergence speed than the traditional asymptotic steady control. At present, researches on limited time control mainly focus on the aspects of aircraft formation control, incomplete robots and limited time track tracking and path following of single intelligent agents, but related researches on limited time control still have the defects. For example, in a practical mission requirement, the attitude of a spacecraft must arrive at a target in three-axis pointing synchronization within a limited time to prevent the target from being lost, so the design of the controller needs to meet all of the attitude limited time convergence to a steady state. In addition, unavoidable ambient disturbance moments exist in the practical application environment, and most of the disturbances cannot be directly measured, but influence the accuracy of the control variables to a certain extent.
At present, most scholars are designed based on the traditional sign function aiming at the limited time controller. For an under-actuated surface ship, a finite time path tracking output feedback method is designed according to patent CN 202010775077.1. The patent CN106886149B provides a robust limited-time saturation gesture tracking control method of the spacecraft aiming at the problem of limited-time gesture control of the spacecraft, and solves the problems of limited-time convergence and uncertain model in the gesture control process. All the above patents adopt sign functions to be simple combinations of all error directions in physical sense, and are not actual directions of errors, so that moment components can be generated at normal directions of errors by control output, control efficiency is reduced, and unnecessary control energy consumption is caused.
In conclusion, under the complex interference environment, the method realizes state synchronous convergence in a limited time in a high control efficiency way aiming at a nonlinear system, and has very important theoretical significance; because of the wide representativeness, the method is further expanded to practical applications, such as multi-agent tracking, cooperative movement of catamarans and the like, so that the engineering value of the design can be realized.
Disclosure of Invention
The invention solves the technical problems: the anti-interference control method is used for solving the problem of state synchronization stability under the conditions of uncertain models and external interference of the system, and can realize high-precision control without the existence of accurate parameters and external disturbance of the known multi-input multi-output affine system.
The technical proposal of the invention is as follows: according to the robust control method based on time synchronization stability, firstly, an online observation uncertainty item of an interference observer is designed aiming at inaccurate modeling information of an affine system, and a foundation is laid for the design of a controller; then, based on a designed disturbance observer and based on the mathematical principle of unit direction vectors, a novel stable controller of a direction symbol function is designed to realize synchronous convergence of all state quantities along with time; then, the parameters are designed, and the numerical solution and convergence time of the closed-loop system are calculated.
The method specifically comprises the following steps:
s1: aiming at inaccurate modeling information in a multi-input multi-output affine system mathematical model, an online observation uncertainty item of an interference observer is designed, and a foundation is laid for the design of a convergence time synchronous robust controller;
S2: based on the interference observer designed in the step S1, a convergence time synchronous robust controller is designed based on the mathematical principle of unit direction vectors, so as to realize synchronous convergence of all state quantities along with time;
s3: according to the interference observer designed in the S1 and the convergence time synchronous robust controller designed in the S2, the convergence time is used for restricting the time of the controller acting on the multi-input multi-output affine system by designing the global finite state convergence time of the multi-input multi-output affine system, so that the rapid stable control is realized.
In step S1, the mathematical model of the affine system with multiple inputs and multiple outputs is considered as follows:
Wherein the state vector State transfer function for a multiple-input multiple-output affine systemAnd input function/>As a known function,/>Representing control inputs, delta representing unmodeled information and external disturbances of a multiple-input multiple-output affine system;
For unmodeled information and external disturbances of a multiple-input multiple-output affine system, a disturbance observer is designed as follows:
Where k 1,k2,k3,k4 >0 is the observer gain factor, z 0 and z 1 are the observed estimates for x and delta, respectively, AndThe derivatives of z 0 and z 1 with respect to time, respectively.
In the step S2, a convergence time synchronous robust controller is designed as follows:
wherein, the controller gain k 5 is a normal number, and the power coefficient eta is as follows:
in the formula, the exponent parameter g is more than 1, p is E (0, 1), and the direction symbol function is designed as follows:
Wherein the method comprises the steps of For any n-dimensional vector, 0 n is an n-dimensional zero vector.
The S3 is specifically implemented as follows:
first, the finite convergence time of the designed interference observer is T 1, and the upper bound thereof satisfies:
In the middle of Is a positive definite matrix of the matrix and the matrix,
The convergence time coefficient c is designed to be:
The lambda max (·) operation in the above formula is the operator for the corresponding matrix maximum eigenvalue, and Is a positive definite matrix of the matrix and the matrix,Is the disturbance error value.
Secondly, designing the finite convergence time of the convergence time synchronous robust controller to be T 2, constructing a Lyapunov function to be V (x) =kx T x, k >0 to be a positive constant, and then performing a convergence time synchronous robust controllerIf V (x 0)<1(x0 is the initial value of x), then:
the convergence time is calculated by design parameters as follows:
At the position of If V (x 0) >1, then the solution for the multiple-input multiple-output affine system is:
The global finite state convergence time of the multi-input multi-output affine system is calculated by design parameters as follows:
Compared with the prior art, the invention has the advantages that: compared with the traditional limited time control method, the method adopts the state feedback of the construction controller of the direction symbol function, and can realize the synchronous convergence of each state of the system in the limited time. In particular, in the implementation principle: the control resultant force constructed by the direction sign function is enabled to act in the effective direction at maximum efficiency, but the control resultant force constructed by the traditional method cannot act in the most effective direction, so that the control of the invention can effectively optimize the convergence path, obviously reduce energy consumption and effectively improve control performance.
Drawings
FIG. 1 is a schematic flow chart of the method of the present invention;
FIG. 2 is a simulation result of state quantity using the control method of the present invention;
FIG. 3 is a simulation result of a state quantity ratio using the control method of the present invention;
FIG. 4 is a simulation result of a state quantity convergence trajectory employing the control method of the present invention;
FIG. 5 is a simulation result of an observed error using the control method of the present invention;
fig. 6 is a simulation result of the control amount using the control method of the present invention.
Detailed Description
The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is apparent that the described embodiments are merely examples and are not intended to limit the present invention.
As shown in fig. 1, a robust control method based on time synchronization stabilization of the present invention is implemented as follows:
the first step: the mathematical model considering the multiple-input multiple-output affine system is as follows:
Wherein, f 0(x)＝3x,g0(x)＝I3, the initial value of the state vector is set to [2, -7,10] T, The control inputs, unmodeled information of the multiple-input multiple-output affine system and external disturbances Δ= [2,3sin (0.02 pi t), 4sin (0.05 pi t+pi/2) ] T are represented.
For unmodeled information and external disturbances of a multiple-input multiple-output affine system, a disturbance observer is designed as follows:
Wherein z 0 and z 1 are observed estimates of x and delta, respectively, And/>The derivatives of z 0 and z 1 with respect to time, respectively.
Secondly, based on the interference observer in the first step, designing a stable controller for synchronously converging each state quantity along with time into the following form:
u＝-(0.5||x||7/11signn(x)+3x+z1) (4)
The direction symbol function is designed and defined as follows:
Wherein the method comprises the steps of For any 3-dimensional vector, 0 3 is a 3-dimensional zero vector.
And thirdly, designing the global convergence time of the multi-input multi-output affine system, bringing the global convergence time into the design object of the first step, and calculating the convergence time of the system. Lyapunov function was designed as V (x) =kx T x, k=1, atThen:
the convergence time satisfies:
the simulation results of the simulation object state synchronous convergence stabilization control can be obtained based on the implementation method, and are shown in fig. 2-6.
Fig. 2 is a state convergence curve obtained by using the control method designed by the present invention, where x 1,x2 and x 3 represent the 1 st, 2 nd and 3 rd state components of the multiple input multiple output affine system, respectively, and it can be seen that the three components reach a stable state at the same time, and the accuracy reaches the order of 10 -4.
Fig. 3 is a variable ratio curve in the state stabilization process, illustrating that the state components of the input multiple output affine system converge at a fixed ratio, which can reach stabilization at the same time.
In fig. 4, a state quantity convergence trajectory curve obtained using the control method of the present invention, FTSC is the method of the present invention, FTC represents a finite time control based on a conventional sign function,After the disturbance is estimated online, the path of the closed loop system in the three-dimensional space is a straight line under the action of the FTSC, and compared with the FTC, the convergence path is obviously shortened.
FIG. 5 is a diagram showing the observation error curve of the disturbance observer according to the present method, whereinAnd/>The 1 st, 2 nd and 3 rd state components respectively representing the interference observation errors illustrate that the interference observer can accurately estimate unknown information.
FIG. 6 is a control amount output curve obtained using the control method of the present invention, where u 1,u2 and u 3 represent the 1 st, 2 nd, and 3 rd state components of the dry control amount, respectively; it can be seen that after time T 1, the control quantity still has a large fluctuation, indicating that it is counteracting the disturbance.
The simulation result fully shows that the invention can realize the synchronous convergence control of the limited time state of more than 10 -4 orders of magnitude under the condition that the system has unmodeled information.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.
Claims (2)
1. An anti-interference control method considering convergence time synchronization constraint is characterized by comprising the following steps:
s1: aiming at inaccurate modeling information in a multi-input multi-output affine system mathematical model, an online observation uncertainty item of an interference observer is designed, and a foundation is laid for the design of a convergence time synchronous robust controller;
S2: based on the interference observer designed in the S1, a convergence time synchronous robust controller is designed based on the mathematical principle of unit direction vectors, so as to realize the synchronous convergence of all state quantities along with time;
S3: according to the interference observer designed in the S1 and the convergence time synchronous robust controller designed in the S2, designing the global finite state convergence time of the multi-input multi-output affine system, and utilizing the convergence time to restrict the time of the controller acting on the multi-input multi-output affine system so as to realize the rapid stable control;
in the step S1, the mathematical model of the multi-input multi-output affine system is considered as follows:
Wherein the state vector State transfer function/>, of a multiple-input multiple-output affine systemAnd input function/>As a known function,/>Representing control inputs, delta representing unmodeled information and external disturbances of a multiple-input multiple-output affine system;
For unmodeled information and external disturbances of a multiple-input multiple-output affine system, a disturbance observer is designed as follows:
where k 1,k2,k3,k4 > 0 is the observer gain factor, z 0 and z 1 are the observed estimates for x and delta, respectively, And/>The derivatives of z 0 and z 1 with respect to time, respectively;
in the step S2, the designed convergence time synchronous robust controller is in the following form:
wherein, the controller gain k 5 is a normal number, and the power coefficient eta is as follows:
in the formula, the exponent parameter g is more than 1, p is E (0, 1), and the direction symbol function is designed as follows:
Wherein the method comprises the steps of For any n-dimensional vector, 0 n is an n-dimensional zero vector.
2. The method according to claim 1, characterized in that: according to the interference observer designed in the S1 and the convergence time synchronous robust controller designed in the S2, designing the global finite state convergence time of the multi-input multi-output affine system, wherein the specific requirements are as follows:
first, the finite convergence time of the designed interference observer is T 1, and the upper bound thereof satisfies:
In the middle of Is a positive definite matrix of the matrix and the matrix,
The convergence time coefficient c is designed to be:
The lambda max (·) operation in the above formula is the operator for the corresponding matrix maximum eigenvalue, and Is a positive definite matrix,/>Is a disturbance error value;
Secondly, designing the finite convergence time of the convergence time synchronous robust controller to be T 2, constructing a Lyapunov function to be V (x) =kx T x, k >0 to be a positive constant, and then performing a convergence time synchronous robust controller If V (x 0)<1,x0 is the initial value of x:
the convergence time is calculated by design parameters as follows:
At the position of If V (x 0) >1, then the solution for the multiple-input multiple-output affine system is:
The global finite state convergence time of the multi-input multi-output affine system is calculated by design parameters as follows:
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