CN109683477B - Design method and system of finite time controller of random multi-agent system - Google Patents
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Abstract
The invention discloses a design method and a system of a finite time controller of a random multi-agent system, which researches a non-strict feedback model, establishes a novel random finite time stability criterion and successfully utilizes a back-stepping method to constructThe finite time controller utilizes the radial basis function neural network approximation theory, and solves the difficulty brought by the unknown function in the system model to the design of the controller. The experimental result can show that the output of the whole system model can well follow the given value y in a limited timerA change in (c).
Description
Technical Field
The invention relates to the field of intelligent agents, in particular to a design method and a system of a random multi-agent system limited time controller.
Background
In recent years, the problem of consistent stability control of random multi-agent systems has attracted considerable attention due to its wide range of civilian and military uses, and its application fields relate to traffic control, sensor network control, mobile robots, social networks, and so on. Firstly, various problems of a single random intelligent agent system are discussed, and finally, the single random system is expanded to a random multi-intelligent agent system, and the discussion is made according to various conditions of the system.
The system includes a switching situation, and the scholars design the system by using different methods. Some scholars explore a switching random nonlinear system with unknown parameters and time delay, and design a self-adaptive output feedback controller by introducing a power integrator technology and a backstepping method. Next, some scholars discuss how to design a controller for a stochastic nonlinear switching system with unknown parameters. An effective adaptive neural state feedback controller design algorithm is provided by combining an average residence time scheme and an adaptive backstepping design. A controller design method based on a radial basis function neural network is provided for a lower triangular structure single-input single-output nonlinear random switching system with output constraint. In addition, some scholars combine the backstepping method and the radial basis neural network to study the following problem of the stochastic nonlinear system. Part of researchers apply the Lyapunov function method, the backstepping technology and the neural network approximation method to the switching random nonlinear system of asymmetric input saturation. Besides the Lyapunov function method and the backstepping technology, the partial researchers also utilize the structural characteristics of the fuzzy system and the robust adaptive control technology to research the non-affine random non-linear switching system.
Event triggering is also a hot issue of research. Some scholars consider an interconnected random nonlinear system based on events, estimate unknown functions in the system by using an approximate theory of a fuzzy logic system, and design an adaptive controller by using a backstepping method. Some researchers research the problem of adaptive event triggering control of uncertain random nonlinear systems, consider a random nonlinear system with actuator faults, adopt adaptive fuzzy control technology aiming at uncertain system parameters and random disturbance, and design a reasonable controller by introducing adaptive back-stepping control. Other researchers next investigated more complex problems for stochastic nonlinear systems, including unknown state variables, actuator failure, and input quantization. Due to the existence of unknown state variables, nonlinear function authors in the system use an approximation method for constructing a fuzzy logic system. In order to deal with the influence of input quantization and actuator faults, a damping term with unknown boundary estimation and a variable integral function with timing are introduced. Finally, the self-adaptive controller is designed by utilizing a back-stepping method.
Due to the limitations of single stochastic nonlinear systems, some researchers are gradually turning to the study of stochastic multi-agent systems. The stability analysis conclusion of the random delay system is applied to the multi-agent system by some authors, the authors not only consider the delay but also add random disturbance, a plurality of stability criteria of the random delay system are established by introducing an improved comparison principle, and the obtained result is utilized to process the consistency problem of the multi-agent system with transmission delay and time-varying topology. Partial scholars use a distributed event trigger control strategy to research the mean square consistency problem of leading to follow a random multi-agent system. Partial scholars study the leader following control problem of a random multi-agent system containing heterogeneous nonlinear dynamics and unknown disturbance, apply a fuzzy logic system to the approximation of the unknown nonlinear dynamics, design a self-adaptive parameter to attenuate the influence of external disturbance, and design a self-adaptive controller by utilizing a backstepping method idea.
In addition to the following problem of the first-order stochastic multi-agent system, some scholars also have generated interest in the higher-order stochastic multi-agent system. For a random high-order nonlinear system, part of documents completely eliminate the power order limitation aiming at the system, the nonlinear growth condition is greatly relaxed, and a weaker sufficient condition is given. The problem of adaptive stability is solved by a back-stepping method. Aiming at an uncertain high-order random nonlinear system with input limitation, partial scholars process nonlinear functions and random disturbance in the system by using a radial basis function network, and a distributed adaptive neural tracking control method is provided by using a backstepping method and a Lyapunov function.
Disclosure of Invention
In the random multi-agent system, the consistency problems of the random multi-agent system are analyzed, wherein the consistency problems comprise first-order and high-order conditions, and most scholars consider the conditions of input quantization, output limitation, nonlinear functions and the like, but the output of the system is changed along with the given value under the condition of limited time in the random multi-agent system; the nonlinear function in the system model contains all the state variables. In addition, the present invention contemplates approximating unknown functions in the system using a radial basis function neural network approach. Therefore, how to design a simpler and more effective controller through a neural network function, an adaptive backstepping method and a Lyapunov function, and the controller can make the output of the system follow the change of a given value in a limited time becomes the research direction of the invention.
The invention provides a design method of a random multi-agent system finite time controller for solving the technical problem, which comprises the following steps:
(1) acquiring a dynamic equation of a random multi-agent system consisting of N followers and a leader;
where i represents the ith agent equation set, i 1, 2.. and N +1, j represents the jth equation in each agent equation set, j 1.. and N-1, N and N are positive integers, and N is a positive integer>1；xi,jIs a state variable of the system, uiIs the input of the ith agent, yiIs the output of the ith agent, and w is a standard randomBrownian motion, fi,j(xi)，
Is an unknown non-linear function, xi＝[xi,1,...,xi,n]；
(2) By the acquired input signal y of the random multi-agent systemrDefining error variables
Wherein etaiNot less than 0, only when the ith intelligent agent receives the input signal yrTime, etai>0, otherwise, ηi＝0，ai,mRepresents the connection weight between node i and node m; to zi,1Differentiating dzi,1Combining the first equation of equation (3) with the error variable zi,2＝xi,2-τi,1To obtain an error variable zi,1Regarding the form of equation (a); selecting Lyapunov function Vi,1Then for the selected Vi,1Differentiating operator LVi,1Obtaining LV according to equation (b), Yang inequality and theorem 2i,1In the simplest form, and in LVi,1The simplest form of (1) selects a virtual control quantity tau containing a power of betai,1To obtain LVi,1In the final simplest form, at zi,2When 0, LV is obtainedi,1Is solved to obtain tau after the final simplest form of (1) is related to the form of theorem 1i,1Make the system stable for a limited time;
(3) according to the error variable zi,j＝xi,j-τi,j-1Wherein the control signal xi,jEqual to the virtual control quantity taui,j-1To obtain an error variable zi,jRegarding the form of equation (a), the Lyapunov function V is choseni,jThen for the selected Vi,jDifferentiating operator LVi,jObtaining LV according to equation (b), Yang inequality and theorem 2i,jIn its simplest form, and LVi,jIs selected from the simplest formula (c), the virtual control quantity tau containing the power of beta is selectedi,j-1To obtain LVi,jIn the final simplest form, at zi,j+1When 0, LV is obtainedi,jIs solved to obtain tau after the final simplest form of (1) is related to the form of theorem 1i,1Make the system stable for a limited time; wherein j is 2.., n-1;
(4) according to the error variable zi,n＝xi,n-τi,n-1Wherein the control signal xi,nEqual to the virtual control quantity taui,n-1To obtain an error variable zi,nRegarding the form of equation (a), the Lyapunov function V is choseni,nThen for the selected Vi,nDifferentiating operator LVi,nAnd obtaining LV according to equation (b), Yang inequality and theorem 2i,nIn the simplest form, and in LVi,nIn the simplest form, an adaptive control rate u containing the power of beta is selectediTo obtain LVi,nSolving after form about lemma 1 to obtain self-adaptive control rate uiMaking the system stable for a limited time.
Further, in the method for designing a finite time controller of a stochastic multi-agent system of the present invention, when the node i can obtain information of the node m, ai,m>0, otherwise, ai,m＝0。
Further, in the method for designing a random multi-agent system finite time controller of the present invention, the formulas (a) and (b) are defined as follows:
the random nonlinear system is:
dx＝f(x)dt+g(x)dw (a)
wherein x represents the state of the system, w is a standard random brownian motion, f (·), g (·) is a continuous function, and f (0) ═ 0, g (0) ═ 0 is satisfied;
for any given V (x) in combination with a random nonlinear system, the differential operator is defined as follows
Wherein Tr { A } is a trace of the matrix A, and h represents a vector formed by any unknown linear function.
The invention also provides a design system of the finite time controller of the random multi-agent system, and the design method of the finite time controller of the random multi-agent system is adopted to design the finite time controller of the random multi-agent system.
The invention researches a non-strict feedback model, establishes a novel random finite time stability criterion, successfully constructs a finite time controller by using a back-stepping method, and solves the difficulty brought by an unknown function in a system model to the design of the controller by using a radial basis function neural network approximation theory. The experimental result can show that the output of the whole system model can well follow the given value y in a limited timerA change in (c).
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a diagram of a communication topology;
fig. 2 is a flowchart for solving the virtual control amount.
Detailed Description
For a more clear understanding of the technical features, objects and effects of the present invention, embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
Algebraic graph theory plays a key role in the consistency analysis of multi-agent systems. The method includes the steps that an information interaction topological graph g-j among n agents is set, wherein a fixed point set v-1. The edge (i, j) of g indicates that an interaction of information may occur between agent i and agent j. Adjacency matrix a of fig. g ═ aij]Wherein a isijMore than or equal to 0 represents the weight of the connection between the node i and the node j, and a is used when the node i can obtain the information of the node jij>0, otherwise, aij0. The laplacian matrix of the graph g is defined as L ═ D-a, where the matrix D ═ diag { D ═ D1,d2,...,dnI.e. have
For an undirected graph g, the laplace matrix L is a symmetric matrix.
Definition 1: considering random non-linear systems
dx＝f(x)dt+g(x)dw, (1)
Where x denotes the state of the system, w is a standard random brownian motion, f (·), g (·) is a continuous function, and f (0) ═ 0, g (0) ═ 0 is satisfied. For the system, define
For any given V (x) in combination with a random nonlinear system, the differential operator is defined as follows
Tr { A } is referred to herein as the trace of matrix A.
Introduction 1: if the random nonlinear system has a globally unique solution, if a positive definite bounded second-order continuous micro Lyapunov function V (x) exists, and the real number c is more than 0, 0 is more than beta and less than 1, and rho is more than 0, the conditions are met
LV(x)≤-cVβ(x)+ρ
For all x ∈ RnThe solution of the random nonlinear system (1) is globally time-limited and randomly stable.
2, leading: the radial basis function is used to approximate a continuous function f (Z). This function can be described in the following form:
where W is an ideal known weight, Z is an input vector, (Z) is an approximation error, and | (Z) | <, q >1 is satisfied.
Considering a stochastic multi-agent system consisting of N followers and one leader, the kinetic equations of the stochastic multi-agent system are described as follows:
where i represents the ith agent equation set, i 1, 2.. and N +1, j represents the jth equation in each agent equation set, j 1.. and N-1, N and N are positive integers, and N is a positive integer>1；xi,jIs a state variable of the system, uiIs an input to the system, yiIs the output of the system, w is a standard random Brownian motion, fi,j(xi)， 
Is an unknown non-linear function, xi＝[xi,1,...,xi,n]The present system model is therefore a non-rigorous feedback model. Corresponding to h, h in the formula (2)i＝[ψi,1ψi,2ψi,3…ψi,n]。
FIG. 1 shows a leader-follower topology diagram:
the goal of designing the controller is to let the output y of the system model (3) beiCapable of following an input signal yrA change in (c).
From the system model, N agents are studied and adaptive control rate u only occurs in the nth equationiAnd the expression can not be directly designed, so that a self-adaptive control design method based on a back-stepping method is provided for the system (3). The specific design steps are as follows:
(1) acquiring a dynamics equation of a random multi-agent system consisting of N followers and a leader, specifically referring to a formula (3);
(2) referring to fig. 2, input signal y by the acquired random multi-agent systemrDefining error variables
Wherein etaiNot less than 0, only when the ith intelligent agent receives the input signal yrTime, etai>0, otherwise, ηi＝0，ai,mRepresents the connection weight between node i and node m; to zi,1Differentiating dzi,1Combining the first equation of equation (3) with the error variable zi,2＝xi,2-τi,1To obtain an error variable zi,1Regarding the form of equation (1); selecting Lyapunov function Vi,1Then for the selected Vi,1Calculus operator LVi,1Obtaining LV according to equation (2), Yang inequality and theorem 2i,1In the simplest form, and in LVi,1Is the simplest formula to choose the virtual control quantity tau containing beta poweri,1To obtain LVi,1In the final simplest form, at zi,2When 0, LV is obtainedi,1Is solved to obtain tau after the final simplest form of (1) is related to the form of theorem 1i,1Make the system stable for a limited time; wherein the control signal x is generated based on the thought of a backstepping methodi,2Viewed as a virtual control quantity τi,1；
(3) According to the error variable zi,j＝xi,j-τi,j-1Wherein the control signal xi,jEqual to the virtual control quantity taui,j-1To obtain an error variable zi,jRegarding the form of equation (1), the Lyapunov function V is choseni,jThen for the selected Vi,jDifferentiating operator LVi,jObtaining LV according to equation (2), Yang inequality and theorem 2i,jIn its simplest form, and LVi,jIs selected from the simplest formula (c), the virtual control quantity tau containing the power of beta is selectedi,j-1To obtain LVi,jIn the final simplest form, at zi,j+1When 0, LV is obtainedi,jIs solved to obtain tau after the final simplest form of (1) is related to the form of theorem 1i,1Make the system stable for a limited time; wherein j is 2.., n-1;
(4) according to the error variable zi,n＝xi,n-τi,n-1Wherein the control signal xi,nEqual to the virtual control quantity taui,n-1To obtain an error variable zi,nAboutIn the form of equation (1), a Lyapunov function V is selectedi,nThen for the selected Vi,nDifferentiating operator LVi,nAnd obtaining LV according to equation (2), Yang inequality and theorem 2i,nIn the simplest form, and in LVi,nIn the simplest form, an adaptive control rate u containing the power of beta is selectediTo obtain LVi,nSolving after form about lemma 1 to obtain self-adaptive control rate uiMaking the system stable for a limited time.
While the present invention has been described with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, which are illustrative and not restrictive, and it will be apparent to those skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope of the invention as defined in the appended claims.
Claims (3)
1. A method for designing a finite time controller of a random multi-agent system is characterized by comprising the following steps:
(1) acquiring a dynamic equation of a random multi-agent system consisting of N followers and a leader;
(c)
where i represents the ith agent equation set, i 1, 2.. and N +1, j represents the jth equation in each agent equation set, j 1.. and N-1, N and N are positive integers, and N is a positive integer>1；xi,jIs a state variable of the system, uiIs the input of the ith agent, yiIs the output of the ith agent, w is a standard random Brownian motion, fi,j(xi)，
Is an unknown non-linear function, xi＝[xi,1,...,xi,n]；
(2) By the acquired input signal y of the random multi-agent systemrDefining error variables
Wherein etaiNot less than 0, only when the ith intelligent agent receives the input signal yrTime, etai>0, otherwise, ηi＝0，ai,mRepresents the connection weight between node i and node m; to zi,1Differentiating dzi,1Combining the first equation of equation (c) with the error variable zi,2＝xi,2-τi,1To obtain an error variable zi,1Regarding the form of equation (a); selecting Lyapunov function Vi,1Then for the selected Vi,1Differentiating operator LVi,1Obtaining LV according to equation (b), Yang inequality and theorem 2i,1In the simplest form, and in LVi,1Is the simplest formula to choose the virtual control quantity tau containing beta poweri,1To obtain LVi,1In the final simplest form, at zi,2When 0, LV is obtainedi,1Is solved to obtain tau after the final simplest form of (1) is related to the form of theorem 1i,1Make the system stable for a limited time;
(3) according to the error variable zi,j＝xi,j-τi,j-1Wherein the control signal xi,jEqual to the virtual control quantity taui,j-1To obtain an error variable zi,jRegarding the form of equation (a), the Lyapunov function V is choseni,jThen for the selected Vi,jDifferentiating operator LVi,jObtaining LV according to equation (b), Yang inequality and theorem 2i,jIn its simplest form, and LVi,jIs selected from the simplest formula (c), the virtual control quantity tau containing the power of beta is selectedi,j-1To obtain LVi,jIn the final simplest form, at zi,j+1When 0, LV is obtainedi,jIs solved to obtain tau after the final simplest form of (1) is related to the form of theorem 1i,1Make the system stable for a limited time; wherein j is 2.., n-1;
(4) according to the error variable zi,n＝xi,n-τi,n-1Wherein the control signal xi,nIs equal to virtualControl quantity taui,n-1To obtain an error variable zi,nRegarding the form of equation (a), the Lyapunov function V is choseni,nThen for the selected Vi,nDifferentiating operator LVi,nAnd obtaining LV according to equation (b), Yang inequality and theorem 2i,nIn the simplest form, and in LVi,nIn the simplest form, an adaptive control rate u containing the power of beta is selectediTo obtain LVi,nSolving after form about lemma 1 to obtain self-adaptive control rate uiMake the system stable for a limited time;
wherein the content of the first and second substances,
introduction 1: if the random nonlinear system has a globally unique solution, if a second-order continuous micro-Lyapunov function V (x) with definite boundary exists and real numbers c >0, 0< beta <1 and rho >0 satisfy
LV(x)≤-cVβ(x)+ρ
For all x ∈ RnThe solution of the random nonlinear system (1) is globally time-limited and randomly stable;
2, leading: the radial basis function is used for approximating a continuous function f (Z); this function can be described in the following form:
wherein W is an ideal known weight, Z is an input vector, (Z) is an approximation error, and satisfies | (Z) | <, q > 1;
the formulas (a) and (b) are defined as follows:
the random nonlinear system is:
dx＝f(x)dt+g(x)dw (a)
wherein x represents the state of the system, w is a standard random brownian motion, f (·), g (·) is a continuous function, and f (0) ═ 0, g (0) ═ 0 is satisfied;
for any given V (x) in combination with a random nonlinear system, the differential operator is defined as follows
(b)
Wherein Tr { A } is a trace of the matrix A, and h represents a vector formed by any unknown linear function.
2. The method of claim 1, wherein a is a when node i can obtain information of node mi,m>0, otherwise, ai,m＝0。
3. A system for designing a random multi-agent system finite time controller, characterized in that the method for designing a random multi-agent system finite time controller according to any one of claims 1-2 is used for designing a random multi-agent system finite time controller.
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