CN113031434A - Fractional order self-adaptive control method and device for time-lag multi-flexible swing arm system - Google Patents

Fractional order self-adaptive control method and device for time-lag multi-flexible swing arm system Download PDF

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CN113031434A
CN113031434A CN202110146374.4A CN202110146374A CN113031434A CN 113031434 A CN113031434 A CN 113031434A CN 202110146374 A CN202110146374 A CN 202110146374A CN 113031434 A CN113031434 A CN 113031434A
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swing arm
flexible swing
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张雄良
郑世祺
张传科
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China University of Geosciences
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Abstract

The invention relates to the field of automatic control, and provides a fractional order self-adaptive control method and a fractional order self-adaptive control device for a time-lag multi-flexible swing arm system, which comprise the following steps: constructing a fractional order nonlinear model aiming at a time-lag multi-flexible swing arm system; constructing a communication network of the time-delay multi-flexible swing arm system through a directed topological graph; introducing tracking errors of each order of each flexible swing arm in a communication network; calculating to obtain a self-adaptive controller through a fractional order nonlinear model and a tracking error; and controlling the motion of each flexible swing arm in the time-delay multi-flexible swing arm system by using the self-adaptive controller. According to the method, the accuracy and the complexity of the model can be comprehensively considered by constructing the fractional order nonlinear model, and a time-lag multi-flexible swing arm system can be better described; a self-adaptive control method is provided, and uncertainty factors in a time-lag multi-flexible swing arm system are processed; an error barrier function and a radial basis function are introduced to compensate time-lag factors in the system, and accurate tracking of the multiple flexible swing arms on the reference signal is achieved.

Description

Fractional order self-adaptive control method and device for time-lag multi-flexible swing arm system
Technical Field
The invention relates to the field of automatic control, in particular to a fractional order self-adaptive control method and device for a time-lag multi-flexible swing arm system.
Background
The integrated circuit chip is the foundation of modern industry, and the microelectronic and semiconductor manufacturing technology formed by the design, manufacture, test and packaging of the integrated circuit chip is the core of the integrated circuit industry and is one of the important cornerstones for the industrial development in China. With the development of science and technology, the electronic manufacturing industry is developing towards high efficiency and high precision, and the requirements on the efficiency and precision of the equipment for testing, sorting and packaging chips are higher and higher. The flexible swing arm is a very key functional component in electronic manufacturing equipment and mainly undertakes tasks such as chip mounting, element sorting and the like. The invention takes the multiple flexible swing arms as research objects and focuses on researching the self-adaptive control problem of the multiple flexible swing arm system with time lag.
The flexible swing arms have the characteristics of high motion frequency, high track repetition degree, flexible system structure, non-constant load and the like, and in addition, when a plurality of flexible swing arms are mutually cooperated in groups, the system has some relevant characteristics of a multi-agent system. Fig. 1 is a simple double-flexible swing arm system, information exchange is performed between swing arms through a communication network, an upper computer determines control input of each swing arm, the control input reaches a controller through a control network, and then the control of the flexible swing arms is realized through an actuator (motor). The control problem of the flexible swing arm is a hot topic with great theoretical and practical significance. As early as this century, researchers have been continuously researching this problem until now. In the last decade, the control problem of flexible swing arms has been receiving increasing attention, and there have been some outstanding research results in the related field. For example, in an article published in 2011 by zai national hei et al of shanghai traffic university, an integer-order linear system model is established by taking a flexible swing arm as a research object, and the output control problem of a class of flexible swing arms is researched by adopting a time delay positive feedback method. Claudia F. et al established an integer order model for a two-degree-of-freedom flexible swing arm of a composite material in 2015, and realized position control of the flexible swing arm by reducing swing arm vibration. Liu jin Kun et al establishes a dynamic model of a flexible swing arm based on partial differential equations, and further designs a flexible swing arm observer, wherein the observer can mainly estimate vibration in the swing arm motion process through an actual measurement value of a swing arm boundary position, and a controller based on the observer can effectively realize gradual stabilization of a closed-loop system.
The above control method is mainly based on an integer order Ordinary Differential Equations (ODE) model or an integer order PDE (PDE) model. However, as the swing arm structure becomes more complex, the swing arm itself and the external environment bring more uncertainties to the system, and a larger error is generated when an integer order ODE is used for modeling, so that the control effect of the designed controller is weakened. Although a relatively accurate model can be established by the integral-order PDE, due to the infinite dimension characteristic, the designed controller is often infinite in dimension, which results in an excessively complex controller structure and is difficult to implement.
The above is only for the purpose of assisting understanding of the technical aspects of the present invention, and does not represent an admission that the above is prior art.
Disclosure of Invention
The invention mainly aims to solve the technical problems that the control effect of a controller is not enough and the structure of the controller is too complex in the prior art.
In order to achieve the purpose, the invention provides a fractional order self-adaptive control method of a time-lag multi-flexible swing arm system, which comprises the following steps:
constructing a fractional order nonlinear model aiming at a time-lag multi-flexible swing arm system;
constructing a communication network of the time-delay multi-flexible swing arm system through a directed topological graph;
introducing tracking errors of each order of each flexible swing arm in the communication network;
calculating to obtain an adaptive controller through the fractional order nonlinear model and the tracking error;
and controlling the motion of each flexible swing arm in the time-delay multi-flexible swing arm system through the self-adaptive controller.
Preferably, the expression of the fractional order nonlinear model is specifically:
Figure BDA0002930580940000021
Figure BDA0002930580940000022
wherein i is 1, …, N represents the number of the flexible swing arm, and N is a positive integer greater than 1; j tableThe system state number, j 1, n-1,
Figure BDA0002930580940000023
representing the state quantity of the corresponding flexible swing arm, wherein n is a positive integer greater than 1;
Figure BDA0002930580940000024
and di,k(t) respectively representing an unknown nonlinear function, an unknown time-varying time-lag function and an unknown external disturbance in the time-lag multi-flexible swing arm system; u. ofiRepresenting control inputs, y, to the flexible swing armi＝xi,1Representing the output, y, of the corresponding flexible swing armrA reference signal representing the time-lag multi-flexible swing arm system, namely an output signal of the leader flexible swing arm; α ∈ (0,1) denotes the order of the fractional order multi-agent system.
Preferably, the expression of each order of tracking error of the flexible swing arm is specifically:
Figure BDA0002930580940000031
wherein, i is 1, …, and N represents the number of the flexible swing arm; k is 2, …, n represents the number of step, n is a positive integer greater than 2; 1, …, N, N is a positive integer greater than 1; a isilRepresenting the communication weight between the sub flexible swing arms; biRepresenting the communication weight of the flexible swing arm of the leader; si,1Representing the first order tracking error, s, of the corresponding flexible swing armi,kRepresenting the tracking error of each other order of the corresponding flexible swing arm; x is the number ofi,kIs a state variable; v isi,k-1The control quantity is a virtual control quantity to be designed and is also an input signal of the fractional order filter;
Figure BDA0002930580940000037
represents the output of a fractional order filter; z is a radical ofi,k-1Representing the fractional order filter error.
Preferably, the complexity of the expression of the virtual controlled variable is simplified by a fractional order filter, and the expression of the fractional order filter is specifically:
Figure BDA0002930580940000032
wherein k is a hierarchical number, k is 2, …, n is a positive integer greater than 2; h isi,k-1＝hi(k-2) and hi,k-11(k 3, …, n) is a new set of parameters, ζi,k-1Is the normal number to be designed for,
Figure BDA0002930580940000033
is an arbitrary normal number that is small enough,
Figure BDA0002930580940000034
is a certain unknown constant χi,k-1An estimate of (d).
Preferably, the stability of each order of tracking error of the flexible swing arm is improved through a neural network algorithm, and the method comprises the following specific steps:
the time lag items in each order of tracking error of the flexible swing arm are limited by designing a barrier function, and the specific formula is as follows: | si,k|<ci,k，i＝1,…,N,k＝1,…,n；
Constructing an unknown function of each order of tracking error of the flexible swing arm, wherein the specific expression of the unknown function is as follows:
Figure BDA0002930580940000035
approximating the unknown function through a radial basis function network to obtain an optimal approximation result, wherein the optimal approximation result specifically comprises the following steps:
Figure BDA0002930580940000036
wherein, thetai,kFor optimal input to the radial basis function neural network, phii,kIs a Gaussian function, ei,kIs an approximation error.
Preferably, the adaptive controller is calculated by using the fractional order nonlinear model and the tracking error, and specifically comprises:
to the firstFirst-order Lyapunov function V of flexible swing arm Ii,1Carrying out derivation of an order alpha, wherein i is 1, …, N represents the number of the flexible swing arm, and N is a positive integer greater than 1;
first-order tracking error s to the flexible swing arm No. ii,1And introducing a second-order state quantity x into the fractional order nonlinear modeli,2In combination with formula si,2＝xi,2-zi,1-νi,1Performing equation transformation to make the first-order Lyapunov function Vi,1Satisfies the condition DαVi,1≤-Li,1Vi,1+hi,1(s,z)+Hi,1Wherein L isi,1And Hi,1Is a normal number;
j-order Lyapunov function V of the ith flexible swing armi,jPerforming an α -order derivation, wherein j is 2, …, n-1, n is a positive integer greater than 2;
step j +1 state variable xi,j+1Considering known control quantity and designing appropriate j-order Lyapunov function Vi,jThe j order Lyapunov function V is controlled according to the fraction order adaptive rate and the virtual control quantityi,jSatisfies the condition DαVi,j≤-Li,jVi,j-hi,j-1(s,z)+hi,j(s,z)+Hi,j；
An n-order Lyapunov function V of the ith flexible swing armi,nCarrying out alpha-order derivation;
designing a suitable fractional order adaptive rate and control input to enable the nth order Lyapunov function Vi,nSatisfies the condition DαVi,j≤-Li,jVi,j-hi,j-1(s,z)+hi,j(s,z)+Hi,j；
Obtaining a closed loop Lyapunov function of the ith flexible swing arm
Figure BDA0002930580940000041
Carrying out alpha-order derivation on the closed-loop Lyapunov function to obtain DαVi≤-LiV+Hi；
Calculating the control input u of the flexible swing arm No. ii；
Repeating the steps for N times to obtain the control input of all the flexible swing arms, namely the self-adaptive controller.
Preferably, the expression of the fractional order adaptation rate is specifically:
Figure BDA0002930580940000042
wherein k is 1, …, n, j is 2, …, n, mi,1,…,mi,nOuter and ni,1,…,ni,nAre all normal number parameters to be designed; p is a radical ofi,1＝hi，pi,2＝…＝pi,n＝1；
Figure BDA0002930580940000043
And
Figure BDA0002930580940000044
are each theta* i,k，Δ* i,kHexix-i,jEstimated values of three sets of unknown constants or vectors;
Figure BDA0002930580940000045
and
Figure BDA0002930580940000046
is the corresponding estimation error.
Preferably, the calculation of the control input u of the ith flexible swing armiThe concrete formula of (1) is as follows:
Figure BDA0002930580940000047
in the formula, betai,1And betai,k(k 2, …, n-1) is the normal number parameter to be designed.
A fractional order self-adaptive control device of a time-lag multi-flexible swing arm system comprises the following modules:
the fractional order nonlinear model building module is used for building a fractional order nonlinear model aiming at the time-lag multi-flexible swing arm system;
the communication network generation module is used for constructing a communication network of the time-delay multi-flexible swing arm system through a directed topological graph;
the tracking error generating module is used for introducing tracking errors of various orders of flexible swing arms in the communication network;
the adaptive controller generating module is used for calculating and obtaining an adaptive controller through the fractional order nonlinear model and the tracking error;
and the control module is used for controlling the motion of each flexible swing arm in the time-delay multi-flexible swing arm system through the self-adaptive controller.
The invention has the following beneficial effects:
1. according to the method, the accuracy and the complexity of the model can be comprehensively considered by constructing the fractional order nonlinear model, and a time-lag multi-flexible swing arm system can be better described;
2. based on a fractional order nonlinear model, a novel fractional order distributed self-adaptive control method is provided, a fractional order self-adaptive law is designed, and uncertainty factors in a time-lag multi-flexible swing arm system are processed;
3. an error barrier function and a radial basis function are introduced to compensate time-lag factors in the system, so that accurate tracking of the multiple flexible swing arms on the reference signal is realized.
Drawings
FIG. 1 is a flow chart of a fractional order adaptive control method for a time-lag multi-flexible swing arm system;
FIG. 2 is a control input for simulating a four sub-flexible swing arm;
FIG. 3 is a graph of output signals simulating a four sub-compliant swing arm;
FIG. 4 is a graph of output signals of two or four sub-flexible swing arms simulated;
FIG. 5 is a graph showing the tracking error of two or four sub-flexible swing arms;
FIG. 6 is a control input for simulating two or four sub-flexible swing arms;
FIG. 7 is a structural diagram of a fractional order adaptive control device of a time-lag multi-flexible swing arm system;
the implementation, functional features and advantages of the objects of the present invention will be further explained with reference to the accompanying drawings.
Detailed Description
It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention provides a fractional order self-adaptive control method of a time-lag multi-flexible swing arm system, which is used for solving the problem of self-adaptive consistency control of the multi-flexible swing arm system consisting of N time-lag flexible swing arms;
referring to fig. 1, the fractional order adaptive control method for the time-lag multi-flexible swing arm system specifically includes the steps of:
s1: constructing a fractional order nonlinear model aiming at a time-lag multi-flexible swing arm system;
s2: constructing a communication network of the time-delay multi-flexible swing arm system through a directed topological graph;
s3: introducing tracking errors of each order of each flexible swing arm in the communication network;
s4: calculating to obtain an adaptive controller through the fractional order nonlinear model and the tracking error;
s5: and controlling the motion of each flexible swing arm in the time-delay multi-flexible swing arm system through the self-adaptive controller.
Further, in step S1, a fractional order nonlinear model of strict state feedback needs to be established for the time-lag multi-flexible swing arm system, and under the condition of strict state feedback, the state quantity in the time-lag multi-flexible swing arm system can be measured and can be used to design an adaptive controller; the expression of the fractional order nonlinear model is specifically as follows:
Figure BDA0002930580940000061
wherein i is 1, …, N represents the number of the flexible swing arm, and N is a positive integer greater than 1; j represents a system state number, j 1., n-1,
Figure BDA0002930580940000062
representing the state quantity of the corresponding flexible swing arm, wherein n is a positive integer greater than 1;
Figure BDA0002930580940000063
and di,k(t) respectively representing an unknown nonlinear function, an unknown time-varying time-lag function and an unknown external disturbance in the time-lag multi-flexible swing arm system; u. ofiRepresenting control inputs, y, to the flexible swing armi＝xi,1Representing the output, y, of the corresponding flexible swing armrA reference signal representing the time-lag multi-flexible swing arm system, namely an output signal of the leader flexible swing arm; α ∈ (0,1) denotes the order of the fractional order multi-agent system.
The alpha fractional order differential for the continuous function h (t) is defined as follows:
Figure BDA0002930580940000064
therefore, a fractional order nonlinear model is established for the time-lag multi-flexible swing arm system, the system modeling is more accurate, the system order is limited, the design difficulty of the self-adaptive controller is reduced to a certain extent, and the control effect of the controller can be effectively improved; then, unknown nonlinear functions and time-lag factors are considered, and nonlinear characteristics and time-lag characteristics of the flexible swing arm are effectively simulated; finally, the impact of environmental factors on the system is simulated in view of bounded external disturbances.
Further, in step S2, a communication structure of the communication network of the time-lag multi-flexible swing arm system is constructed by using the directed topological graph; for N +1 flexible swing arms including the leader flexible swing arm, a directed communication graph is defined as
Figure BDA0002930580940000071
Wherein the content of the first and second substances,
Figure BDA0002930580940000072
is a node set of a communication network, representing all participationA set of communicating flexible swing arms;
Figure BDA0002930580940000073
a boundary known as a flexible swing arm; and one arbitrary set (m, l) satisfying (m, l) epsilon represents that the information can be transmitted to the flexible swing arm m from the flexible swing arm l in a single direction but can not be transmitted in a reverse direction, and at the moment, the flexible swing arm l is positioned in the 'neighbor' range of the flexible swing arm m; it can be seen that the neighborhood of the flexible swing arm m is defined as
Figure BDA0002930580940000074
The flexible swing arm m can only receive the information of the adjacent flexible swing arm; parameter amlThe representation is the communication weight between the sub-flexible swing arms, b is [ b ]1,…,bN]TRepresenting a weight matrix from the leader flexible swing arm to the child flexible swing arm, and, in addition,
Figure BDA0002930580940000075
the invention provides an output tracking problem of a time-lag multi-flexible swing arm system, wherein the communication among all flexible swing arms and between the flexible swing arms and a reference signal has a mutual exclusion phenomenon, namely when one flexible swing arm can directly receive the reference signal, the output information of other flexible swing arms is not received, and only when the reference signal cannot be directly sensed, the state adjustment can be carried out through the output signals of other flexible swing arms; thus, in addition to the communication diagram, the communication method can be used for realizing the communication method
Figure BDA0002930580940000076
When b is greater thanm＝1，aml0; when in use
Figure BDA0002930580940000077
When, if
Figure BDA0002930580940000078
Then aml1, otherwise aml0. In addition, to simplify the design of the distributed adaptive controller, a variable h is introducedm＝bm+dmIn a direction of sight, hm≠0。
Further, in step S3, the expression of each order of tracking error of the flexible swing arm is specifically:
Figure BDA0002930580940000079
wherein, i is 1, …, and N represents the number of the flexible swing arm; k is 2, …, n represents the number of step, n is a positive integer greater than 2; 1, …, N, N is a positive integer greater than 1; a isilRepresenting the communication weight between the sub flexible swing arms; biRepresenting the communication weight of the flexible swing arm of the leader; si,1Representing the first order tracking error, s, of the corresponding flexible swing armi,kRepresenting the tracking error of each other order of the corresponding flexible swing arm; x is the number ofi,kIs a state variable; v isi,k-1The control quantity is a virtual control quantity to be designed and is also an input signal of the fractional order filter;
Figure BDA00029305809400000710
represents the output of a fractional order filter; z is a radical ofi,k-1Representing the fractional order filter error.
Furthermore, the complexity of the function is greatly increased due to the fractional calculus, and the design of the adaptive controller by using a back-stepping method can differentiate some functions for many times, so that the problem of complexity explosion is caused, and the design difficulty of the adaptive controller is greatly increased; in order to solve the problem, the invention simplifies the expression complexity of the virtual control quantity through a fractional order filter, wherein the expression of the fractional order filter is specifically as follows:
Figure BDA0002930580940000081
wherein k is a hierarchical number, k is 2, …, n is a positive integer greater than 2; h isi,k-1＝hi(k-2) and hi,k-11(k 3, …, n) is a new set of parameters, ζi,k-1Is the normal number to be designed for,
Figure BDA0002930580940000082
is an arbitrary normal number that is small enough,
Figure BDA0002930580940000083
is a certain unknown constant χi,k-1An estimated value of (d);
for xi,k-1The introduction of (A) is as follows: from the formulae (3) and (4)
Figure BDA0002930580940000084
In the formula, gammai,k-1(·)＝Dανi,k-1Which is the result of the fractional differentiation of the virtual control quantity, is an unknown, bounded function. Thus, the function exists in the upper bound as an unknown normal χi,k-1Satisfy | γi,k-1(·)|≤χi,k-1. Meanwhile, for processing unknown linear functions and unknown time-varying time-lag functions existing in a system model, a radial basis function is introduced to approximate the unknown functions, so that an upper error bound needs to be set to ensure that the functions are bounded, and the upper error bound is set to be | si,k|≤ci,k(k＝1,…,n)；
And the expression of the radial basis function is:
Figure BDA0002930580940000085
where θ is the weight vector, φ (ω) is a generally selected Gaussian function, and ω is the unknown function and the input vector to the radial basis function neural network.
Further, the stability of each order of tracking error of the flexible swing arm is improved through a neural network algorithm, and the method specifically comprises the following steps:
the time lag items in each order of tracking error of the flexible swing arm are limited by designing a barrier function, and the specific formula is as follows:
|si,k|<ci,k，i＝1,…,N,k＝1,…,n (7)
constructing an unknown function of each order of tracking error of the flexible swing arm, wherein the specific expression of the unknown function is as follows:
Figure BDA0002930580940000086
approximating the unknown function through a radial basis function network to obtain an optimal approximation result, wherein the optimal approximation result specifically comprises the following steps:
Figure BDA0002930580940000087
wherein, thetai,kFor optimal input to the radial basis function neural network, phii,kIs a Gaussian function, ei,kIs an approximation error; therefore, the neural network can be used for carrying out approximation processing on the unknown nonlinear function in the design process of the self-adaptive controller; it is worth noting that the unknown nonlinear function and the time-lag factors which are limited in the system form a new unknown nonlinear function, and the new unknown nonlinear function is approximately estimated through the radial basis function network, so that the calculation amount is greatly simplified.
However, the radial basis function neural network can generate an approximation error when approximating an unknown function, the system can be influenced by external disturbance, in order to solve the influence of the unknown items on the system, firstly, an unknown variable delta is defined as e + d, and because the approximation error and the external disturbance are bounded, the unknown variable delta is bounded, and the condition that the value of the delta is less than or equal to the value of the delta is met*. Furthermore, the following auxiliary inequalities will be used to design the self-implementing controller:
Figure BDA0002930580940000091
in the formula, S is an unknown constant,
Figure BDA0002930580940000092
again an arbitrarily small positive constant.
Further, in step S4, the fractional order nonlinear model is passed throughCalculating the tracking error to obtain a self-adaptive controller; the control aim of the invention is to realize the tracking of the output of the sub-flexible swing arm on the reference signal, namely to realize si,1Converge within some arbitrarily small interval of origin 0; in each step of the design, an error variable s is first introduced as in equation (3)i,k(i-1, …, N, k-1, …, N) while using the state variable x in model (1)i,1,xi,2,…,xi,nTo design the corresponding virtual control volume vi,1,νi,2,…,νi,n-1And in the last step at the design control input uiConstructing a proper adaptive control rate by combining the constructed fractional order filter, the radial basis function for approximation and the auxiliary inequality for error compensation, and realizing the design of an adaptive controller; the method comprises the following specific steps:
first-order Lyapunov function V for No. i flexible swing armi,1Carrying out derivation of an order alpha, wherein i is 1, …, N represents the number of the flexible swing arm, and N is a positive integer greater than 1;
first-order tracking error s to the flexible swing arm No. ii,1And introducing a second-order state variable x into the fractional order nonlinear modeli,2In combination with formula si,2＝xi,2-zi,1-νi,1Performing equation transformation to make the first-order Lyapunov function Vi,1Satisfies the condition DαVi,1≤-Li,1Vi,1+hi,1(s,z)+Hi,1Wherein L isi,1And H,1Is a normal number;
j-order Lyapunov function V of the ith flexible swing armi,jPerforming an α -order derivation, wherein j is 2, …, n-1, n is a positive integer greater than 2;
step j +1 state variable xi,j+1Considering known control quantity and designing appropriate j-order Lyapunov function Vi,jWith fractional order adaptive rate and virtual control quantity to make the j order Vi,jSatisfies DαVi,j≤-Li,jVi,j-hi,j-1(s,z)+hi,j(s,z)+Hi,j；
An n-order Lyapunov function V of the ith flexible swing armi,nCarrying out alpha-order derivation;
designing a suitable fractional order adaptive rate and control input to enable the nth order Lyapunov function Vi,nSatisfies the condition DαVi,j≤-Li,jVi,j-hi,j-1(s,z)+hi,j(s,z)+Hi,j；
The Lyapunov function is selected in each step as follows:
Figure BDA0002930580940000101
Figure BDA0002930580940000102
obtaining a closed loop Lyapunov function of the ith flexible swing arm
Figure BDA0002930580940000103
Carrying out alpha-order derivation on the closed-loop Lyapunov function to obtain DαVi≤-LiV+Hi(ii) a Therefore, all signals in the time-lag multi-flexible swing arm system are bounded, the stability of the system is ensured, and the tracking error si,1Converging in an arbitrarily small interval of 0 to realize the tracking of the output of any sub-flexible swing arm on the reference signal;
calculating the control input u of the flexible swing arm No. ii；
Repeating the steps for N times to obtain the control input of all the flexible swing arms, namely the self-adaptive controller.
Further, the expression of the fractional order adaptation rate is specifically as follows:
Figure BDA0002930580940000104
wherein k is 1, …, n, j is 2, …, n, mi,1,…,mi,nOuter and ni,1,…,ni,nAre all normal number parameters to be designed; p is a radical ofi,1＝hi，pi,2＝…＝pi,n＝1；
Figure BDA0002930580940000105
And
Figure BDA0002930580940000106
are each theta* i,k，Δ* i,kHexix-i,jEstimated values of three sets of unknown constants or vectors;
Figure BDA0002930580940000107
and
Figure BDA0002930580940000108
is the corresponding estimation error.
Further, the control input u of the ith flexible swing arm is calculatediThe concrete formula of (1) is as follows:
Figure BDA0002930580940000109
in the formula, betai,1And betai,k(k 2, …, n-1) is the normal number parameter to be designed. Further, in order to verify the effectiveness and superiority of the fractional order self-adaptive control method of the time-lag multi-flexible swing arm system, the following time-lag multi-flexible swing arm system model is designed for simulation verification; first, the reference signal of the time-lag multi-flexible swing arm system is yrSecondly, a fractional order nonlinear model is established by 4 flexible swing arms, namely N is 4, the dimension of the time-lag multi-flexible swing arm system is N is 2, and the order is alpha is 0.85; the nonlinear function, time-lag function and external disturbance are selected as follows:
Figure BDA0002930580940000111
in addition, the communication weight between the flexible swing arms is b1＝1，a21＝a31＝a41＝a431, the rest are 0. Initial value of state is set as xi＝[0.1,-0.1]TAnd the upper error limit is set to c ═ ci,1,ci,2]T＝[1,1]T；
In this embodiment, two different reference signals are simulated, first, a selected reference signal is simulated as yrObtaining a simulation result when the value is 0; FIG. 2 shows a control input u simulating a four sub-compliant swing armi(ii) a As can be seen from FIG. 3, the output signal y of a four sub-compliant swing arm is simulatediThe signal is well converged in a small enough range of the origin 0, the reference signal is tracked, and the self-adaptive controller provided by the invention has a good control effect; subsequently, in simulation two, the reference signal is selected to be yrSin (t); as can be seen from fig. 4, the output signals of the two and four simulated sub-flexible swing arms track the reference signal well; FIG. 5 shows the tracking error of two or four sub-flexible swing arms, and it can be seen from FIG. 5 that the tracking error is within 5%, which indicates that the designed adaptive controller has good control effect; the control inputs for simulating two or four sub-compliant swing arms are shown in fig. 6.
Further, referring to fig. 7, the present invention provides a fractional order adaptive control device for a time-lag multiple flexible swing arm system, which is characterized by comprising the following modules:
the fractional order nonlinear model building module 10 is used for building a fractional order nonlinear model aiming at the time-lag multi-flexible swing arm system;
the communication network generation module 20 is used for constructing a communication network of the time-lag multi-flexible swing arm system through a directed topological graph;
a tracking error generating module 30, configured to introduce a tracking error of each order of each flexible swing arm in the communication network;
an adaptive controller generating module 40, configured to calculate and obtain an adaptive controller according to the fractional order nonlinear model and the tracking error;
and the control module 50 is used for controlling the motion of each flexible swing arm in the time-delay multi-flexible swing arm system through the self-adaptive controller.
Other embodiments or specific implementation manners of the fractional order adaptive control device of the time-lag multi-flexible swing arm system can refer to the above method embodiments, and are not described herein again.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or system. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other like elements in a process, method, article, or system that comprises the element.
The above-mentioned serial numbers of the embodiments of the present invention are merely for description and do not represent the merits of the embodiments. In the unit claims enumerating several means, several of these means may be embodied by one and the same item of hardware. The use of the words first, second, third and the like do not denote any order, but rather the words first, second and the like may be interpreted as indicating any order.
The above description is only a preferred embodiment of the present invention, and not intended to limit the scope of the present invention, and all modifications of equivalent structures and equivalent processes, which are made by using the contents of the present specification and the accompanying drawings, or directly or indirectly applied to other related technical fields, are included in the scope of the present invention.

Claims (9)

1. A fractional order self-adaptive control method for a time-lag multi-flexible swing arm system is characterized by comprising the following steps of:
constructing a fractional order nonlinear model aiming at a time-lag multi-flexible swing arm system;
constructing a communication network of the time-delay multi-flexible swing arm system through a directed topological graph;
introducing tracking errors of each order of each flexible swing arm in the communication network;
calculating to obtain an adaptive controller through the fractional order nonlinear model and the tracking error;
and controlling the motion of each flexible swing arm in the time-delay multi-flexible swing arm system through the self-adaptive controller.
2. The fractional order adaptive control method of the time-lag multi-flexible swing arm system according to claim 1, wherein the expression of the fractional order nonlinear model is specifically as follows:
Figure FDA0002930580930000011
Figure FDA0002930580930000012
wherein i is 1, …, N represents the number of the flexible swing arm, and N is a positive integer greater than 1; j represents a system state number, j 1., n-1,
Figure FDA0002930580930000013
representing the state quantity of the corresponding flexible swing arm, wherein n is a positive integer greater than 1;
Figure FDA0002930580930000014
and di,k(t) respectively representing an unknown nonlinear function, an unknown time-varying time-lag function and an unknown external disturbance in the time-lag multi-flexible swing arm system; u. ofiRepresenting control inputs, y, to the flexible swing armi＝xi,1Representing the output, y, of the corresponding flexible swing armrA reference signal representing the time-lag multi-flexible swing arm system, namely an output signal of the leader flexible swing arm; α ∈ (0,1) denotes the order of the fractional order multi-agent system.
3. The fractional order adaptive control method of the time-lag multi-flexible swing arm system according to claim 2, wherein the expression of each order tracking error of the flexible swing arm is specifically as follows:
Figure FDA0002930580930000015
wherein, i is 1, …, and N represents the number of the flexible swing arm; k is 2, …, n represents the number of step, n is a positive integer greater than 2; 1, …, N, N is a positive integer greater than 1; a isilRepresenting the communication weight between the sub flexible swing arms; biRepresenting the communication weight of the flexible swing arm of the leader; si,1Representing the first order tracking error, s, of the corresponding flexible swing armi,kRepresenting the tracking error of each other order of the corresponding flexible swing arm; x is the number ofi,kIs a state variable; v isi,k-1The control quantity is a virtual control quantity to be designed and is also an input signal of the fractional order filter;
Figure FDA0002930580930000021
represents the output of a fractional order filter; z is a radical ofi,k-1Representing the fractional order filter error.
4. The fractional order adaptive control method of the time-lag multi-flexible swing arm system according to claim 3, wherein the complexity of the expression of the virtual control quantity is simplified by a fractional order filter, and the expression of the fractional order filter is specifically:
Figure FDA0002930580930000022
wherein k is a hierarchical number, k is 2, …, n is a positive integer greater than 2; h isi,k-1＝hi(k-2) and hi,k-11(k 3, …, n) is a new set of parameters, ζi,k-1Is the normal number to be designed for,
Figure FDA0002930580930000023
is an arbitrary normal number that is small enough,
Figure FDA0002930580930000024
is a certain unknown constant χi,k-1An estimate of (d).
5. The fractional order self-adaptive control method of the time-lag multi-flexible swing arm system according to claim 3, wherein the stability of each order tracking error of the flexible swing arm is improved through a neural network algorithm, and the method comprises the following specific steps:
the time lag items in each order of tracking error of the flexible swing arm are limited by designing a barrier function, and the specific formula is as follows: | si,k|<ci,k，i＝1,…,N,k＝1,…,n；
Constructing an unknown function of each order of tracking error of the flexible swing arm, wherein the specific expression of the unknown function is as follows:
Figure FDA0002930580930000025
approximating the unknown function through a radial basis function network to obtain an optimal approximation result, wherein the optimal approximation result specifically comprises the following steps:
Figure FDA0002930580930000026
wherein, thetai,kFor optimal input to the radial basis function neural network, phii,kIs a Gaussian function, ei,kIs an approximation error.
6. The fractional order adaptive control method of the time-lag multi-flexible swing arm system according to claim 4, wherein an adaptive controller is obtained by calculation through the fractional order nonlinear model and the tracking error, and specifically comprises the following steps:
first-order Lyapunov function V for No. i flexible swing armi,1Carrying out derivation of an order alpha, wherein i is 1, …, N represents the number of the flexible swing arm, and N is a positive integer greater than 1;
first-order tracking error s to the flexible swing arm No. ii,1And the fractional order nonlinearityIntroducing a second-order state variable x into the modeli,2In combination with formula si,2＝xi,2-zi,1-νi,1Performing equation transformation to make the first-order Lyapunov function Vi,1Satisfies the condition DαVi,1≤-Li,1Vi,1+hi,1(s,z)+Hi,1Wherein L isi,1L and Hi,1Is a normal number;
j-order Lyapunov function V of the ith flexible swing armi,jPerforming an α -order derivation, wherein j is 2, …, n-1, n is a positive integer greater than 2;
step j +1 state variable xi,j+1Considering known control quantity and designing appropriate j-order Lyapunov function Vi,jWith fractional order adaptation rate and virtual control quantity such that the j order Vi,j DαVi,j≤-Li,jVi,j-hi,j-1(s,z)+hi,j(s,z)+Hi,j；
An n-order Lyapunov function V of the ith flexible swing armi,nCarrying out alpha-order derivation;
designing a suitable fractional order adaptive rate and control input to enable the nth order Lyapunov function Vi,nSatisfies the condition DαVi,j≤-Li,jVi,j-hi,j-1(s,z)+hi,j(s,z)+Hi,j；
Obtaining a closed loop Lyapunov function of the ith flexible swing arm
Figure FDA0002930580930000031
Carrying out alpha-order derivation on the closed-loop Lyapunov function to obtain DαVi≤-LiV+Hi；
Calculating the control input u of the flexible swing arm No. ii；
Repeating the steps for N times to obtain the control input of all the flexible swing arms, namely the self-adaptive controller.
7. The fractional order adaptive control method of the time-lag multi-flexible swing arm system according to claim 6, wherein the expression of the fractional order adaptive rate is specifically as follows:
Figure FDA0002930580930000032
wherein k is 1, …, n, j is 2, …, n, mi,1,…,mi,nOuter and ni,1,…,ni,nAre all normal number parameters to be designed; p is a radical ofi,1＝hi，pi,2＝…＝pi,n＝1；
Figure FDA0002930580930000033
And
Figure FDA0002930580930000034
are respectively
Figure FDA0002930580930000035
Figure FDA0002930580930000036
Hexix-i,jEstimated values of three sets of unknown constants or vectors;
Figure FDA0002930580930000037
and
Figure FDA0002930580930000038
is the corresponding estimation error.
8. The fractional order adaptive control method for a time-lapse multiple-flexibility swing arm system according to claim 7, wherein the control input u of the ith flexible swing arm is calculatediThe concrete formula of (1) is as follows:
Figure FDA0002930580930000039
in the formula, betai,1And betai,k(k 2, …, n-1) is the normal number parameter to be designed.
9. The fractional order self-adaptive control device for the time-lag multi-flexible swing arm system is characterized by comprising the following modules:
the fractional order nonlinear model building module is used for building a fractional order nonlinear model aiming at the time-lag multi-flexible swing arm system;
the communication network generation module is used for constructing a communication network of the time-delay multi-flexible swing arm system through a directed topological graph;
the tracking error generating module is used for introducing tracking errors of various orders of flexible swing arms in the communication network;
the adaptive controller generating module is used for calculating and obtaining an adaptive controller through the fractional order nonlinear model and the tracking error;
and the control module is used for controlling the motion of each flexible swing arm in the time-delay multi-flexible swing arm system through the self-adaptive controller.
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