CN115220472A - Fault-tolerant control method for air-ground heterogeneous formation system under switching topology - Google Patents
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Abstract
The invention discloses a fault-tolerant control method for an air-ground heterogeneous formation system under a switching topology, which comprises the following steps: (1) Designing a distributed topology network structure under a switching topology, designing a leader-follower formation strategy under the structure, and determining a Laplace matrix of a communication topology relation; (2) Considering the influence of a complex uncertain item, establishing a model for describing the dynamic characteristics of the air-ground heterogeneous multi-agent formation system; (3) Respectively designing a fixed observer based on a fixed time theory to carry out fixed time observation on the complex uncertain item of the second model and the complex uncertain item of the third model; (4) And designing a distributed formation fault-tolerant controller of an XY axis and a track fault-tolerant tracking controller of a Z axis which are converged at fixed time by adopting a high-order sliding mode control technology and a backstepping method according to the observed value and the Laplace matrix. The invention solves the multiple requirements of the air-ground heterogeneous formation system on adaptability, maneuvering responsiveness, fault-tolerant reliability, continuity of control quantity and the like of the switching topology network.
Description
Technical Field
The invention relates to the technical field of fault-tolerant formation control of a heterogeneous multi-agent system, in particular to a fault-tolerant control method of an air-ground heterogeneous formation system under a switching topology.
Background
In recent years, with the continuous development of artificial intelligence and control science, the intelligent agent has been developed in the aspects of environmental adaptability, autonomy, control accuracy, high efficiency and the like. However, as industrialization progresses faster, it becomes more and more difficult to achieve a desired complex control task by means of a single agent, and thus a multi-agent system constituted by a plurality of agents cooperating with each other is produced. When the dynamic model of one or several agents present in a multi-agent system is different from other agents, such a multi-agent system is called a heterogeneous multi-agent system. A multi-agent formation system composed of an aerial unmanned aerial vehicle and a ground unmanned vehicle is a typical heterogeneous multi-agent system, has various payload capacities, task configuration capacities, control and data acquisition capacities, and is widely applied to the fields of space exploration, collaborative rescue, resource exploration, path planning and the like.
However, in the air-ground cooperative formation process, due to the high complexity of the unmanned aerial vehicle structure, the constraints of individuals and their interaction characteristics and mechanisms, etc., the heterogeneous multi-agent system is susceptible to actuator faults, and the faults are propagated and spread through the interactive topological network, which further affects the stability and safety of the whole formation system. Therefore, the design of the distributed interactive network architecture also brings new challenges to the design of the fault-tolerant controller while improving the overall cooperativity. Especially, the sensing area of the sensor is limited due to the influence of environment factors such as variable and unreliable information transmission links and obstacles which execute tasks, which can cause the change of a topological network structure, and further cause the traditional fault-tolerant formation control strategy based on fixed topological design to be unusable. Moreover, conventional asymptotic responses and discontinuous control inputs have difficulty meeting anticipated control requirements in the face of critical technology systems such as air-ground unmanned systems. The formation control strategy of the existing air-ground unmanned system can only solve one problem in one way, and cannot solve the problems of quick maneuverability, continuity of control input, fault-tolerant control under a switching topology network and the like of the formation system at the same time, so that a new fault-tolerant control scheme capable of solving the series of problems needs to be researched to further improve the realization of the air-ground heterogeneous multi-agent formation system on complex tasks and the capability of maintaining the performance.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the defects, the invention discloses a fault-tolerant control method of an air-ground heterogeneous formation system under a switching topology, which designs continuous fixed time fault-tolerant tracking controllers of a Z-axis motion space in an XY plane under a switching topology network respectively, simultaneously solves the multiple requirements of the air-ground heterogeneous formation system on the adaptability, the maneuvering responsiveness, the fault-tolerant reliability, the continuity of a control quantity and the like of the switching topology network, and realizes the fault-tolerant formation control target of the air-ground heterogeneous formation system in an XYZ three-dimensional space.
The technical scheme is as follows: in order to solve the above problems, the present invention provides a fault-tolerant control method for a switching topology down-air-ground heterogeneous formation system, which specifically includes the following steps:
(1) Designing a distributed topology network structure under a switching topology aiming at an air-ground heterogeneous multi-agent formation system; designing a leader-follower formation strategy under the structure, selecting a unmanned vehicle as a leader and the other unmanned vehicles and/or unmanned aerial vehicles as followers, and determining a Laplacian matrix for describing a communication topological relation between the leader and the followers;
(2) Considering the influence of a complex uncertain item, establishing a model for describing the dynamic characteristics of the air-ground heterogeneous multi-agent formation system according to the kinematics equation of the unmanned aerial vehicle and the unmanned aerial vehicle, wherein the model comprises the following steps: a first model, a second model, a third model; the first model is a dynamic model of the unmanned vehicle of the leader, the second model is a dynamic model of the unmanned vehicle follower and the unmanned vehicle follower in an XY plane, and the third model is a dynamic model of the unmanned vehicle follower in a Z axis; the complex uncertainty comprises actuator faults, model uncertainty and external interference;
(3) Respectively designing a fixed observer based on a fixed time theory to carry out fixed time observation on the complex uncertainty of the second model and the complex uncertainty of the third model;
(4) Designing a distributed formation fault-tolerant controller of an XY axis converging in continuous fixed time by adopting a high-order sliding mode control technology and a backstepping method according to the first observation value and the Laplace matrix obtained in the step (1); the first observation value is an observation value of a fixed observer designed aiming at the complex uncertainty of the second model; designing a Z-axis track fault-tolerant tracking controller converging in continuous fixed time by adopting a high-order sliding mode control technology and a backstepping method according to a second observed value; the second observation value is an observation value of a fixed observer designed for the complex uncertainty of the third model.
Further, (1.1) defining that the air-ground heterogeneous multi-agent formation system is provided with N +1 agents, wherein the agents comprise unmanned planes and unmanned vehicles; wherein the unmanned vehicle has N 1 +1, unmanned aerial vehicle has N 2 N = N 1 +N 2 N is the number of followers; leader unmanned vehicle is labeled 0, follower unmanned vehicle is labeled i =1, \8230;, N 1 With follower drone marked i = N 1 +1,…,N 1 +N 2 ；
(1.2) communication topology network undirected graph between followers is G = (S, E, A); s = { S = 1 ,s 2 ,…,s N Is a set of N followers,
for the set of communication connections between the followers,
denotes the ith 1 A follower and the ith 2 Communication connection of the individual follower and success of the connection, i 1 ∈i、i 2 ∈i；
A connection matrix representing the communication weight between followers, if
Then
Otherwise
The in-degree matrix of follower i is D = diag { deg. } in (s 1 ),...,deg in (s N )}，
The laplace matrix between followers is L = D-a;
(1.3) the communication connection relation matrix between the leader and the follower is B = diag { B } 1 ,...,b i ,...,b N }；b i Representing a communication connection between the ith follower and the leader; the interaction network of the whole formation system comprising the leader and the follower is
Its laplace matrix is H = L + B;
(1.4) since the communication topological relation between the followers is that the switching topological change occurs along with the time, a set containing all the topological structures is established:
theta represents the number of switching topologies present, and theta is greater than or equal to 1; the signal defining the switching topology is σ (t) [ [0, + ∞) → = {1, 2., Θ }, and the switching time is denoted as 0=t = 0 ＜t 1 ＜...＜t P = t, in the time interval [ t p ,t p+1 ) The topology network between followers in the network is represented as a directed fixed topology network
Its Laplace matrix is defined as L σ(t) The communication matrix of the whole formation system is H σ(t) ＝L σ(t) +B σ(t) 。
Further, the step (2) specifically comprises:
(2.1) the expression of the first model is:
in the formula (I), the compound is shown in the specification,
for the first derivative of the position coordinate of the leader in the XY plane, v 0 (t) is the position speed of the leader in the XY plane at time t,
is v of 0 (t) first derivative; u. u 0 (t) inputting the control quantity of the position of the leader in the XY plane at the time t;
(2.2) the expression of the second model is:
in the formula, x i (t)＝[x i ,y i ] T Is the position coordinate of the i-th follower in the XY plane,
is x i (ii) the second derivative of (t); u. of i xy (t) is the control quantity input of the i-th follower in the XY plane;
a complex uncertainty term for the second model;
an additive fault parameter for the ith follower; f. of i xy (t) values of the nonlinear portion of the system containing the model uncertainty for the ith follower;
external interference for the ith follower;
(2.3) the expression of the third model is:
Δ iz (t)＝τ iz (t)+f iz (t)+d iz (t)-ρ iz u iz (t)/m i -g
in the formula, z i (t) is the position coordinate of the ith unmanned aerial vehicle on the Z axis,
is z i (ii) the second derivative of (t); u. of iz (t) inputting the control quantity of the ith unmanned aerial vehicle on the Z axis; delta iz (t) is the complex uncertainty term of the third model; tau is iz (t) additive fault parameters for the ith drone; f. of iz (t) values for the system non-linearity part of the model uncertainty for the ith drone; d iz (t) is the external interference suffered by the ith unmanned aerial vehicle; rho iz Executing an efficiency loss fault parameter for the ith drone; g is the acceleration of gravity; m is i Is the quality of the ith drone.
Further, the step (3) specifically comprises:
(3.1) designing a fixed time observer aiming at the complex uncertainty of the second model, wherein the formula is as follows:
in the formula, σ i1 、σ i2 Is a state variable of the fixed-time observer,
is composed of i1 The first derivative of (a) is,
is σ i2 The first derivative of (a); m is i1 Is a positive constant; wherein the content of the first and second substances,
in the formula o i1 、o i2 、o i3 Parameters of a fixed time observer;
a first derivative of a complex uncertainty term for the second model; w 1 Is a complex uncertainty term
The upper bound of (c);
is the first derivative of the location coordinate of the ith follower in the XY plane;
convergence time T of observer 1i Satisfies the following conditions:
in the formula, v i1 Is a positive constant of a setting, and
then, the minimum convergence time T is obtained 1i ；
(3.2) designing a fixed time observer aiming at the complex uncertainty of the third model, wherein the formula is as follows:
in the formula, σ i3 、σ i4 Is a state variable of the fixed-time observer,
is σ i3 The first derivative of (a) is,
is σ i4 The first derivative of (a); m is i2 Is a positive constant; wherein
In the formula o i4 、o i5 、o i6 Parameters of a fixed time observer;
a first derivative of a complex uncertainty term for the third model; w 2 Is a complex uncertainty term
The upper bound of (c);
a first derivative of a position coordinate of the ith unmanned aerial vehicle on the Z axis is obtained;
convergence time T of observer 1iz Satisfies the following conditions:
in the formula, v i2 Is provided withIs constant, positive constant, and
then, the minimum convergence time T is obtained 1iz ；
Further, the step (4) specifically includes:
(4.1) defining an expected formation vector containing all followers in the XY plane as
The predetermined track of all the unmanned aerial vehicle followers on the Z axis is
There is a settable time constant T that is not limited to the initial state of the system max So that the following holds:
(4.2) based on the observed value σ i2 Laplace matrix H of communication topological relation among leader and follower σ(t) Designing a distributed fault-tolerant formation controller of an XY axis, wherein the expression is as follows:
the virtual control law is as follows:
in the formula, chi 1 、δ 1 、χ 2 、δ 2 、χ 3 And delta 3 Are all positive constant, c 1 、c 2 、c 3 Are all control parameters, d 1 、d 2 、d 3 Are constants with a value greater than 1; zeta i1 、ζ i2 、ζ i3 A third order system which is the controller;
is gamma i2 The first derivative of (a);
is gamma i1 The first derivative of (a);
is the first derivative of the formation tracking error;
(4.3) from the observed value σ i4 Designing a Z-axis track fault-tolerant tracking controller, wherein the expression is as follows:
and the virtual control amount is designed as:
wherein, mu 1 、ω 1 、μ 2 、ω 2 、μ 3 And ω 3 Is a positive constant, n, set manually 1 、n 2 、n 3 Are all control parameters, m 1 、m 2 、m 3 Are constants with a value greater than 1;
a third order system in which the controllers are all the same;
is composed of
The first derivative of (a);
is composed of
The first derivative of (a);
the first derivative of the trajectory tracking error in the Z-axis.
Further, the step (4.2) specifically comprises:
(4.2.1) designing a fixed time integral sliding mode surface based on the tracking error:
in the formula, e i ＝[e ix ,e iy ] T ，sig l (e i )＝[|e ix | l sgn(e ix ),|e iy | l sgn(e iy )] T Alpha and beta are both positive definite constants and satisfy 0 < alpha < 1 and beta > 1 1 、b 2 Are all constants; e.g. of the type ix For the formation tracking error of the x-axis, e iy A formation tracking error for the y-axis;
taking a first derivative and a second derivative of the sliding mode surface to obtain:
in the formula, λ i1 ＝diag{αb 1 |e ix | α-1 ,αb 1 |e iy | α-1 }λ i2 ＝diag{βb 2 |e ix | β-1 ,βb 2 |e iy | β-1 }；
Obtaining a third-order system of an integral sliding mode surface:
(4.2.2) based on the design scheme of the fault-tolerant controller of the backstepping method, the following transformations are carried out on the three-order system:
ζ i1 ＝S i1
ζ i2 ＝S i2 -γ i1
ζ i3 ＝S i3 -γ i2
(4.2.3) designing a distributed fault-tolerant formation controller in an XY plane under a switching topology network based on the established new third-order system, wherein the distributed fault-tolerant formation controller realizes fixed time convergence.
Further, the step (4.3) specifically comprises:
(4.3.1) designing a fixed time integral sliding mode surface based on the tracking error:
in the formula (I), the compound is shown in the specification,
α z and beta z Are all positive definite constants; b 1z ＞0，b 2z The constants are artificially set when the value is more than 0;
taking a first derivative and a second derivative of the sliding mode surface, and finally obtaining a third-order system of the integral sliding mode surface:
in the formula (I), the compound is shown in the specification,
(4.3.2) the fault-tolerant controller design scheme based on the backstepping method is used for carrying out the following transformation on the three-order system:
(4.3.3) designing a track fault-tolerant tracking controller on the Z axis based on the established new third-order system under a switching topology network, wherein the track fault-tolerant tracking controller realizes fixed time convergence.
Furthermore, the present invention also provides a computer device, comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the processor implements the steps of any of the above methods when executing the computer program. A computer-readable storage medium, having stored thereon a computer program which, when being executed by a processor, carries out the steps of any of the methods described above.
Has the beneficial effects that: compared with the prior art, the fault-tolerant control method for the air-ground heterogeneous formation system under the switching topology has the advantages that: 1. aiming at the difference between an unmanned aerial vehicle model and an unmanned vehicle model in an air-ground heterogeneous formation system, a two-dimensional dynamic model of all followers in an XY axis and a dynamic model of an unmanned aerial vehicle in a Z axis are respectively established, a fixed time observer is respectively established for the two established models to realize effective estimation on a complex uncertain item, observed values and neighbor interaction information are utilized in an XY plane, a distributed continuous fixed time fault-tolerant controller under a switching topology network is designed, the controller is designed to integrate a high-order sliding mode technology and a backstep method, not only is the artificial setting on convergence time realized, but also the output of the controller is ensured to be continuous, the fault tolerance and the robustness of the formation system are effectively improved, and meanwhile, the generalization of the designed controller in practical application is greatly improved; 2. a track fault-tolerant tracking controller based on an observer is designed on a Z axis by utilizing a local tracking error, a fixed time theory, a high-order sliding mode control technology and a backstepping method, and the safe and rapid tracking of an expected track by a follower of the unmanned aerial vehicle on the vertical height is ensured.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is an overall block diagram of the air-ground heterogeneous formation system according to the present invention;
FIG. 3 is a diagram illustrating a switching topology of an air-ground heterogeneous formation system; FIG. 3 (a), FIG. 3 (b), FIG. 3 (c), FIG. 3 (d) are all a topology structure diagram;
FIG. 4 is a diagram showing the formation effect of the XY space air-ground heterogeneous formation system according to the present invention;
FIG. 5 is a diagram showing formation error convergence of the XY-axis air-ground heterogeneous formation system according to the present invention; FIG. 5 (a) is a diagram of the formation error of the follower on the X-axis, and FIG. 5 (b) is a diagram of the formation error of the follower on the Y-axis;
FIG. 6 is a tracking error convergence diagram of the Z-axis air-ground heterogeneous formation system according to the present invention;
FIG. 7 is a graph showing the continuous control output of the air-ground heterogeneous formation system of XY axes according to the present invention;
fig. 7 (a) is a control amount output diagram of the follower on the X axis, and fig. 7 (b) is a control amount output diagram of the follower on the Y axis;
FIG. 8 is a graph showing the continuous control output of the Z-axis air-ground heterogeneous formation system according to the present invention;
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
As shown in fig. 1, the present invention provides a fault-tolerant control method for a switching topology down-to-ground heterogeneous formation system, which specifically includes the following steps:
firstly, designing a distributed topology network structure under a switching topology aiming at an air-ground heterogeneous multi-agent formation system; designing a leader-follower formation strategy under the structure, selecting a unmanned vehicle as a leader and the other unmanned vehicles or unmanned aerial vehicles as followers, and determining a Laplacian matrix describing the communication topological relation between the leader and the followers;
(1) Defining an air-ground heterogeneous multi-agent formation system with N +1 agents (including unmanned aerial vehicles and unmanned vehicles); wherein N is 1 +1 unmanned vehicles, N 2 An unmanned aerial vehicle; leader unmanned vehicle is labeled 0, follower unmanned vehicle is labeled i =1, \8230;, N 1 With follower drone marked i = N 1 +1,…,N 1 +N 2 ；N＝N 1 +N 2 N is the number of followers;
(1.2) communication topology network undirected graph between followers G =(S,E,A)；S＝{s 1 ,s 2 ,…,s N Is the set of N followers,
for the set of communication connections between the followers,
denotes the ith 1 A follower and the ith 2 Communication connection of the individual follower and success of the connection, i 1 ∈i、i 2 ∈i；
A connection matrix representing the communication weight between followers, if
Then the
Otherwise
The in-degree matrix for follower i is D = diag { deg. } in (s 1 ),...,deg in (s N )}，
The laplace matrix between followers is L = D-a;
(3) The communication connection relation between the leader and the follower is expressed by a diagonal matrix as follows: b = diag { B } 1 ,...,b i ,...,b N }; wherein, b i Represents the communication between the ith follower and the leader, if b i > 0 means that the ith follower can obtain information from the leader, otherwise communication between the i followers and the leader is not possible.
The interaction network of the whole formation system comprising the leader and the follower is
Its laplace matrix is H = L + B; that is, H describes the communication connection matrix of the entire formation system topology network, and H describes the communication network in which the leader has at least one path that can reach any follower.
(4) For the case that the communication topology relationship between the followers is the switching topology change over time, it is necessary to establish a set including all the topologies and an expression form of the switching topology.
Define the set of all directed graphs as
Θ represents the number of handover topologies present; the signal defining the switching topology is σ (t): [0, + ∞) → = {1, 2., Θ } and the switching time is denoted 0=t = 0 ＜t 1 ＜...＜t P ＝t，t p For a particular switching instant, p is the switching order, i.e. at t p The P-th switching occurs at the moment, and P is the total switching frequency. In the time interval t p ,t p+1 ) The topological network among followers (P ≦ P) is represented as a directed fixed topological network
Its Laplace matrix is defined as L σ(t) The communication matrix of the whole formation system is H σ(t) ＝L σ(t) +B σ(t) 。
As shown in fig. 2, the embodiment has 5 agents, 3 unmanned vehicles and 2 unmanned vehicles; one unmanned vehicle (i = 0) is taken as a leader, two unmanned vehicles (i =1,2) and two unmanned vehicles (i =3,4) are additionally selected as followers, and the leader and the followers jointly form the air-ground heterogeneous multi-agent formation system.
A specific switching topology distributed topology network structure is designed for the air-ground heterogeneous multi-agent formation system and is shown in figure 3, each switching topology structure is maintained for 5 seconds, namely the communication topology of the formation system in 0 s-5 s is shown in figure 3 (a), and in the period of time
The communication topology between 5s and 10s is shown in FIG. 3 (b), during which time
The communication topology between 10s and 15s is shown in FIG. 3 (c), during which time
The communication topology between 15s and 20s is shown in FIG. 3 (d), during which time
Secondly, establishing a model for describing the dynamic characteristics of the air-ground heterogeneous multi-agent formation system according to the kinematics equation of the unmanned aerial vehicle and the unmanned vehicle;
(1) Defining a dynamic model of the leader unmanned vehicle as a first model, wherein the expression of the first model is as follows:
in the formula, x 0 ∈R 2 For the position coordinates, v, of the leader in the XY plane 0 ∈R 2 Is the position speed of the leader in the XY plane, u 0 ∈R 2 Inputting a control quantity for the position of the leader in the XY plane; r represents a real number domain.
The initial position of the leader in this embodiment is x 0 (0)＝[0,0] T The initial state of the follower is x 1 (0)＝[-0.1,0.2] T ，x 2 (0)＝[0.5,-0.3] T ，x 3 (0)＝[-0.4,0] T And x 4 (0)＝[0,0.6] T The leader's input is u 0 ＝[0.02,0.02] T 。
(2) And (3) fully considering the influences of the fault of the intelligent actuator, model uncertainty and external interference, and then establishing a unified two-dimensional dynamic model of the unmanned aerial vehicle and the unmanned aerial vehicle in the XY plane, wherein the model is defined as a second model. Specifically, the ith follower (i ∈ (1,.., N) is added 1 +N 2 ) The model in the XY plane is simplified to the model expression shown below:
wherein, χ i (t)＝[x i ,y i ] T ∈R 2 And
respectively represents the position coordinate of the ith follower in the XY plane and the input of the control quantity,
representing the actuator efficiency loss factor for the ith follower,
an additive failure parameter, f, representing the ith follower i xy (t)∈R 2 Indicating that the ith follower contains the nonlinear part of the system of the model uncertainty,
represents the external interference suffered by the ith follower, i belongs to (1.. The., N) when the ith follower is an unmanned vehicle 1 ) Then A i ＝I 2 ，I 2 Is a two-dimensional identity matrix. Conversely, when the ith follower is a drone, i.e., i e (N) 1 +1,...,N 1 +N 2 ) Then A i ＝diag{1/m i ,1/m i In which m is i Is the mass of the ith drone.
In order to facilitate the design of the controller, the second model expression is further simplified into the following form by designing a complex uncertainty term containing actuator faults, model uncertainty and external interference:
in the formula (I), the compound is shown in the specification,
is a complex uncertainty term for the second model.
(3) In a similar way, the influence of the fault of the intelligent actuator, the uncertainty of the model and the external interference is fully considered, the dynamic model of the unmanned aerial vehicle on the Z axis is a third model, and the expression of the third model is as follows:
wherein z is i E R and u iz (t) belongs to R and is the position coordinate and the control input quantity of the ith unmanned aerial vehicle on the Z axis respectively, and rho iz E.g. R and tau iz (t) is the ith unmanned aerial vehicle execution efficiency loss fault parameter and the additive fault parameter respectively, f iz (t) epsilon R and d iz (t) is the model nonlinear part of the ith unmanned aerial vehicle and the external interference, and g represents the gravity acceleration.
In order to facilitate the design of the controller, the second model expression is further simplified into the following form by designing complex uncertainties including actuator faults, model uncertainty and external interference:
Δ iz (t)＝τ iz (t)+f iz (t)+d iz (t)-ρ iz u iz (t)/m i -g
in the formula,. DELTA. iz (t) is the complex uncertainty term of the third model.
In this embodiment, the execution efficiency loss module is set as:
and an additive fault module as shown below:
τ 1 ＝[-0.2cos(t),-sin(0.5t)] T ,τ 2 ＝[-0.15sin(1.2t),-0.45sin(2t)] T ,
the external interference suffered by each follower is designed as follows:
d 1 ＝[0.1sin(t),-0.3cos(0.7t)] T ,d 2 ＝[0.45cos(t),0.7sin(2t)] T ,
d 3 ＝[-0.3cos(0.5t),-0.2sin(1.5t),0.27sin(2t)] T ,
d 4 ＝[0.5sin(0.25t),0.55cos(1.5t),0.3sin(2t)] T 。
in addition, considering that each drone follower is affected by model uncertainty, there is an uncertainty change in the design model parameters with 20% amplitude.
Step three, respectively designing a fixed observer pair based on a fixed time theory
Δ iz (t) carrying out fixed time observation, wherein the convergence time of the observation error is not limited by the initial state of the system;
(1) Design of fixed time observer pair
And (3) carrying out fixed time observation, wherein the expression is as follows:
in the formula, σ i1 、σ i2 Are state variables of the fixed time observer.
Wherein m is i1 Is a positive constant set by people, and satisfies the following conditions: m is i1 ＞1，
Observer parameter o i1 、o i2 、o i3 Satisfies the following conditions:
in the formula, W 1 Is a complex uncertainty term
Upper bound of the derivative, i.e. satisfies
Complex uncertainty term
Can be respectively sigma i2 Estimated within a fixed time. As will be described below
A fixed time convergence proof is performed for the example.
By pairs of s 1 Differentiation can yield:
order:
then differentiating it can yield:
to this end, complex uncertainties
The system of observation errors of (a) can be expressed as:
the error system described above can complete its convergence process in a fixed time, i.e. σ i2 Can realize the uncertainty of the complex in a fixed time
And the convergence time T of the observer 1i Satisfies the following conditions:
wherein, v i1 Is greater than 0, and when
A minimum convergence time T can be obtained 1i 。
(2) Design of fixed time observer pair Δ iz (t) performing fixed time observation, wherein the expression is as follows:
in the formula, σ i3 、σ i4 All are fixed time observersIs measured.
Wherein m is i2 ＞1，
Observer parameter o i4 、o i5 And o i6 Satisfies the following conditions:
in the formula, W 2 Is a complex uncertainty term Δ iz (t) upper bound of derivative, i.e. satisfies
Then, the complex uncertainty term Δ iz (t) may be respectively σ i4 Estimated over a fixed time. The complex uncertainty Δ in Z-space can be obtained using similar proofs as in step (3.1) iz (T) may also be at a fixed time T 1iz Accurate estimation is achieved, and T 1iz Satisfies the following conditions:
wherein, v i2 > 0, and when
A minimum convergence time T can be obtained 1iz 。
In this embodiment, each parameter is set as: o. o i1 ＝o i4 ＝5，o i2 ＝o i5 ＝2，o i3 ＝o i6 ＝4。
And step four, after fixed-time observation of complex uncertainty is realized, designing a fault-tolerant formation controller based on an observed value and by utilizing a high-order sliding mode control technology and a backstepping method, wherein the design of the controller is divided into an XY-axis distributed fault-tolerant formation control part and a Z-axis distributed track fault-tolerant tracking control part.
(1) Defining an expected formation vector containing all follower agents in the XY plane
Predetermined trajectory of all Z-axis drone followers
In an XYZ three-dimensional space, an expected formation task is that a settable time constant T which can not be limited to the initial state of a system exists in the condition that an air-ground heterogeneous formation system suffers from actuator faults, model uncertainty and external interference max So that the following holds:
in this embodiment, the formation form expected by each follower is a time-varying vector, specifically:
and
(2) Designing a fixed time integral sliding mode surface by utilizing the formation tracking error in an XY axis and the Z axis track tracking error to ensure that the fixed time convergence can be realized by the tracking error after the sliding mode surface is reached; then, a new three-order error system is constructed by utilizing the fixed time sliding mode surface so as to facilitate the designed controller to realize continuous output; and then designing the XY-axis distributed formation fault-tolerant controller by using a backstepping method and interactive information between agents.
(2.1) designing a distributed fault-tolerant controller of an XY axis:
(2.1.1) first, a fixed time integral sliding mode surface is designed based on the tracking error as follows:
wherein e is i ＝[e ix ,e iy ] T ，sig l (e i )＝[|e ix | l sgn(e ix ),|e iy | l sgn(e iy )] T Alpha and beta are positive constants, and satisfy 0 < alpha < 1 and beta > 1. In addition, b 1 ＞0，b 2 And the constant is artificially set when the value is more than 0. e.g. of a cylinder ix For the formation tracking error of the x-axis, e iy For the formation tracking error of the y-axis, e i Defined as the error in formation as:
the fixed time convergence of the slip form surface is given below, and after reaching the slip form surface, the following lyapunov function is designed:
taking the derivative thereof, we can get:
wherein M is e i The designed integral sliding mode surface can realize the fixed time convergence of the global formation error after reaching the sliding mode surface, and the upper bound of the convergence time meets the following requirements:
(2.2.2) taking the first derivative and the second derivative of the sliding mode surface to obtain:
wherein λ is i1 ＝diag{αb 1 |e ix | α-1 ,αb 1 |e iy | α-1 And λ i2 ＝diag{βb 2 |e ix | β-1 ,βb 2 |e iy | β-1 }。
To this end, the following third order system for the integral sliding mode surface can be obtained:
(2.2.3) based on a back-stepping method, carrying out the following transformation on the third-order system:
ζ i1 ＝S i1
ζ i2 ＝S i2 -γ i1
ζ i3 ＝S i3 -γ i2
wherein, γ i1 And gamma i2 Is a virtual control quantity to be designed.
(2.2.4) based on the established new third-order system, a distributed fault-tolerant formation controller in the XY plane is designed under a switching topology network, and convergence of formation errors in fixed time is realized. The expression of the controller is:
the virtual control law is as follows:
wherein, χ 1 、δ 1 、χ 2 、δ 2 、χ 3 And delta 3 Is a positive constant set by human, control parameter c 1 、c 2 、c 3 E (0, 1), and d 1 ,d 2 ,d 3 ＞1。
In this embodiment: b is a mixture of 1 ＝1，b 2 ＝1.1，α＝0.5，β＝1.1，c 1 ＝c 2 ＝c 3 ＝0.5，d 1 ＝d 2 ＝d 3 ＝1.4，χ 1 ＝11，δ 1 ＝10，χ 2 ＝1，δ 2 ＝0.8，χ 3 ＝1，δ 3 ＝2。
The proof of the fixed time convergence for this designed controller is given below:
first, the following Lyapunov function is designed:
deriving and introducing a control law into the method to obtain:
after the fixed time observer finishes observing the complex uncertain items, the following can be obtained:
therefore, when the time T ≧ T 1 In which T is 1 ＝max{T 11 ,T 12 ,...,T 1N }，ζ i3 Can realize fixed time convergence and can obtain the convergence time upper bound of T 1 +T 3 Wherein:
then, the following lyapunov function is further designed:
based on the previously constructed third order system, the following can be derived:
and because when T is more than or equal to T 1 +T 3 When there is
And ζ i3 =0, at this time, for V 3 The derivation may be:
thus, it can be obtained that when T > T 1 +T 3 +T 4 Time, ζ i2 Converge to zero, and wherein:
then, the following lyapunov function is designed:
at T > T 1 +T 3 +T 4 Then, taking the derivative of it, we can get:
wherein the content of the first and second substances,
λ min (H σ(t) ) Is H σ(t) Is determined by the minimum characteristic value of (c),
λ max (H σ(t) ) Is H σ(t) The maximum eigenvalue of (c). Based on the above analysis, it can be found that 1 Is a fixed time convergence, i.e. for the ith (i ∈ (1.,. N.) 1 +N 2 ) For an agent, the designed fixed-time fault-tolerant controller of the XY plane is at T ≧ T 1 +T 3 +T 4 +T 5 Can reach the sliding form surface S i1 . And the sliding mode surface can be T > T 2 Then realizing the convergence of the formation error, thus obtaining that when T is more than T max ＝T 1 +T 2 +T 3 +T 4 +T 5 The desired formation task may be implemented.
(2.2) designing a Z-axis track fault-tolerant tracking controller:
similar to the design mode of the distributed fault-tolerant formation controller in the XY plane, firstly, a fixed-time sliding mode surface is constructed based on the tracking error, then a three-order system is constructed by utilizing the sliding mode surface, a virtual control quantity is designed to form a new error system, and then the error system is proved to realize fixed-time convergence under the control of the designed fault-tolerant controller based on the observer based on a back stepping method.
(2.2.1) first, a fixed time integral sliding mode surface is designed based on the tracking error of the Z axis as follows:
wherein alpha is z And beta z Is a positive constant and satisfies 0 < alpha z < 1 and beta z Is greater than 1. In addition, b 1z ＞0，b 2z The constant is artificially set when the value is more than 0.
The three-order system about the Z-axis integral sliding mode surface can be arranged as follows:
wherein the content of the first and second substances,
and
(2.2.2) further performing the following transformation on the third-order system:
wherein, the first and the second end of the pipe are connected with each other,
and
is a virtual control quantity to be designed.
(2.2.3) designing a Z-axis track fault-tolerant tracking controller expression as follows:
and the virtual control amount is designed as follows:
wherein, mu 1 、ω 1 、μ 2 、ω 2 、μ 3 And ω 3 Is a positive constant set by human, and a control parameter n 1 ,n 2 ,n 3 E (0, 1), and m 1 ,m 2 ,m 3 Is greater than 1. The controller of this design has a proof of fixed time convergence similar to the proof of the controller described above.
Based on the parameter setting in the above embodiment, simulation verification is performed by applying a Simulink module in Matlab. As shown in fig. 4, the two-dimensional queuing process diagram illustrates that the controller is designed to enable the air-ground queuing system to complete a predetermined fault-tolerant queuing control task under a switching topology network. As shown in fig. 5 and fig. 6, the convergence process of the error-tolerant controller designed by the present invention in the XY axis and Z axis formation error is shown respectively, even under the influence of actuator failure, model uncertainty and external interference, the designed controller can still complete the convergence process quickly and reliably within a specified time, and ensure the formation and maintenance performance of the formation system in switching topology network to the geometric formation configuration within a fixed time, so that the robustness and the error tolerance of the whole formation system can be significantly improved by quickly estimating and effectively compensating the complex uncertainty. As shown in fig. 7 and fig. 8, which respectively show the control quantities of the controller designed by the present invention in the XY axis and the Z axis, it can be found that the controller realizes stable and continuous control output, and can autonomously adjust the amplitude of the control quantity for the sudden occurrence of the actuator fault, and the designed controller has relatively strong adaptive capability and fault compensation capability.
Claims (9)
1. A fault-tolerant control method for an air-ground heterogeneous formation system under switching topology is characterized by comprising the following steps:
(1) Designing a distributed topology network structure under a switching topology aiming at an air-ground heterogeneous multi-agent formation system; designing a leader-follower formation strategy under the structure, selecting a unmanned vehicle as a leader and the other unmanned vehicles and/or unmanned aerial vehicles as followers, and determining a Laplacian matrix for describing a communication topological relation between the leader and the followers;
(2) Considering the influence of a complex uncertainty item, building a model for describing the dynamic characteristics of the air-ground heterogeneous multi-agent formation system according to the kinematic equations of the unmanned aerial vehicle and the unmanned vehicle in the air-ground heterogeneous multi-agent formation system, wherein the model comprises the following steps: a first model, a second model, a third model; the first model is a dynamic model of the unmanned vehicle of the leader, the second model is a dynamic model of the unmanned vehicle follower and the unmanned vehicle follower in an XY plane, and the third model is a dynamic model of the unmanned vehicle follower in a Z axis; the complex uncertainty comprises actuator faults, model uncertainty and external interference;
(3) Respectively designing a fixed observer based on a fixed time theory to carry out fixed time observation on the complex uncertain item of the second model and the complex uncertain item of the third model;
(4) Designing a distributed formation fault-tolerant controller of an XY axis converging in continuous fixed time by adopting a high-order sliding mode control technology and a backstepping method according to the first observation value and the Laplace matrix obtained in the step (1); the first observation value is an observation value of a fixed observer designed aiming at a complex uncertainty item of the second model; designing a Z-axis track fault-tolerant tracking controller which converges in continuous fixed time by adopting a high-order sliding mode control technology and a backstepping method according to a second observation value; the second observation value is the observation value of a fixed observer designed aiming at the complex uncertainty of the third model.
2. The fault-tolerant control method for the air-ground heterogeneous formation system under the switching topology according to claim 1, wherein the step (1) specifically comprises:
(1.1) defining an air-ground heterogeneous multi-agent formation system to be provided with N +1 agents, wherein the agents comprise an unmanned aerial vehicle and an unmanned vehicle; wherein the unmanned vehicle has N 1 +1, unmanned aerial vehicle has N 2 N = N 1 +N 2 N is the number of followers; leader unmanned vehicle label 0, follower unmanned vehicle label i =1, \8230, N 1 Follower drone is marked i = N 1 +1,…,N 1 +N 2 ；
(1.2) communication topology network undirected graph between followers is marked as G = (S, E, A); s = { S = 1 ,s 2 ,…,s N Is a set of N followers,
for the set of communication connections between the followers,
denotes the ith 1 A follower and the ith 2 Communication connection and success of each follower, i 1 ∈i、i 2 ∈i；
A connection matrix representing the communication weight between followers, if
Then
Otherwise
The in-degree matrix for follower i is D = diag { deg. } in (s 1 ),...,deg in (s N )}，
The laplace matrix between followers is L = D-a;
(1.3) the communication connection relation matrix between the leader and the follower is B = diag { B } 1 ,...,b i ,...,b N }，b i Representing a communication connection between the ith follower and the leader; the interaction network of the whole formation system comprising the leader and the follower is
Its laplace matrix is H = L + B;
(1.4) since the communication topological relation between the followers is that the switching topological change occurs along with the time, a set containing all the topological structures is established:
theta represents the number of existing switching topologies, and theta is more than or equal to 1; the signal defining the switching topology is σ (t): [0, + ∞) → = {1, 2., Θ } and the switching time is denoted 0=t = 0 ＜t 1 ＜...＜t P = t, in time interval [ t p ,t p+1 ) The topology network between followers in the network is represented as a directed fixed topology network
Its Laplace matrix is defined as L σ(t) (ii) a The communication matrix of the whole formation system is H σ(t) ＝L σ(t) +B σ(t) 。
3. The fault-tolerant control method for the air-ground heterogeneous formation system under the handover topology according to claim 2, wherein the step (2) specifically comprises:
(2.1) the expression of the first model is:
in the formula (I), the compound is shown in the specification,
for the first derivative, v, of the position coordinate of the leader in the XY plane at time t 0 (t) is the position speed of the leader in the XY plane at time t,
is v of 0 (t) first derivative; u. of 0 (t) inputting the control quantity of the position of the leader in the XY plane at the time t;
(2.2) the expression of the second model is:
wherein, χ i (t) = [ x ] i ,y i ] T Is the position coordinate of the ith follower in the XY plane,
is x i (t) second derivative; u. of i xy (t) is the control quantity input of the i-th follower in the XY plane;
a complex uncertainty term for the second model;
an additive fault parameter for the ith follower; f. of i xy (t) is the value of the nonlinear part of the system that the i-th follower contains the model uncertainty;
external interference suffered by the ith follower;
(2.3) the expression of the third model is:
Δ iz (t)＝τ iz (t)+f iz (t)+d iz (t)-ρ iz u iz (t)/m i -g
in the formula, z i (t) is the position coordinate of the ith unmanned aerial vehicle on the Z axis,
is z i (t) second derivative; u. of iz (t) inputting the control quantity of the ith unmanned aerial vehicle on the Z axis; delta of iz (t) is the complex uncertainty term of the third model; tau. iz (t) additive fault parameters for the ith drone; f. of iz (t) values for the system non-linear part of the model uncertainty for the ith drone; d is a radical of iz (t) external interference suffered by the ith unmanned aerial vehicle; rho iz Executing an efficiency loss fault parameter for the ith drone; g is gravity acceleration; m is a unit of i Is the quality of the ith drone.
4. The fault-tolerant control method for the air-ground heterogeneous formation system under the switching topology according to claim 3, wherein the step (3) specifically comprises:
(3.1) designing a fixed time observer aiming at the complex uncertainty of the second model, wherein the formula is as follows:
in the formula, σ i1 、σ i2 Is a state variable of the fixed-time observer,
is σ i1 The first derivative of (a) is,
is σ i2 The first derivative of (a); m is a unit of i1 Is a positive constant; wherein the content of the first and second substances,
in the formula o i1 、o i2 、o i3 Parameters of a fixed time observer;
a first derivative of a complex uncertainty term for the second model; w is a group of 1 Is a complex uncertainty term
The upper bound of (c);
is the first derivative of the position coordinate of the i-th follower in the XY plane;
convergence time T of observer 1i Satisfies the following conditions:
in the formula, v i1 Is a positive constant of a setting, and
then, the minimum convergence time T is obtained 1i ；
(3.2) designing a fixed time observer aiming at the complex uncertainty of the third model, wherein the formula is as follows:
in the formula, σ i3 、σ i4 Is a state variable of the fixed-time observer,
is σ i3 The first derivative of (a) is,
is σ i4 The first derivative of (a); m is i2 Is a positive constant; wherein
In the formula o i4 、o i5 、o i6 Parameters of a fixed time observer;
a first derivative of a complex uncertainty term for the third model; w 2 Is a complex uncertainty term
The upper bound of (c);
the first derivative of the position coordinate of the ith unmanned aerial vehicle on the Z axis is obtained;
convergence time T of observer 1iz Satisfies the following conditions:
in the formula, v i2 Is a positive constant of a setting, and when
Then, the minimum convergence time T is obtained 1iz 。
5. The fault-tolerant control method for the air-ground heterogeneous formation system under the handover topology according to claim 1, wherein the step (4) specifically comprises:
(4.1) defining an expected formation vector containing all followers in the XY plane as
The predetermined track of all the unmanned aerial vehicle followers on the Z axis is
There is a settable time constant T that is not limited to the initial state of the system max So that the following holds:
(4.2) based on the observed value σ i2 Laplace matrix H of communication topological relation between leader and follower σ(t) Designing a distributed fault-tolerant formation controller of an XY axis, wherein the expression is as follows:
the virtual control law is as follows:
in the formula, x 1 、δ 1 、χ 2 、δ 2 、χ 3 And delta 3 Are all positive constant, c 1 、c 2 、c 3 Are all control parameters, d 1 、d 2 、d 3 Are constants with a value greater than 1; ζ represents a unit i1 、ζ i2 、ζ i3 A third order system in which the controllers are all the same;
a second derivative of the position coordinates of the leader in the XY plane;
is gamma i2 The first derivative of (a);
is gamma i1 The first derivative of (a);
a first derivative of a convoy tracking error;
(4.3) from the observed value σ i4 Designing a Z-axis track fault-tolerant tracking controller, wherein the expression is as follows:
and the virtual control amount is designed as follows:
wherein, mu 1 、ω 1 、μ 2 、ω 2 、μ 3 And ω 3 Is a positive constant, n, set by man 1 、n 2 、n 3 Are all control parameters, m 1 、m 2 、m 3 Are constants with a value greater than 1;
a third order system in which the controllers are all the same;
is composed of
The first derivative of (a);
is composed of
The first derivative of (a);
the first derivative of the trajectory tracking error in the Z-axis.
6. The fault-tolerant control method for the air-ground heterogeneous formation system under the handover topology according to claim 5, wherein the step (4.2) specifically comprises:
(4.2.1) designing a fixed time integral sliding mode surface based on the tracking error:
in the formula, e i ＝[e ix ,e iy ] T ，sig l (e i )＝[|e ix | l sgn(e ix ),|e iy | l sgn(e iy )] T Alpha and beta are both positive definite constants and satisfy 0 < alpha < 1 and beta > 1 1 、b 2 Are all constants; e.g. of the type ix For the formation tracking error of the x-axis, e iy A formation tracking error for the y-axis;
taking a first derivative and a second derivative of the sliding mode surface to obtain:
in the formula, λ i1 ＝diag{αb 1 |e ix | α-1 ,αb 1 |e iy | α-1 }λ i2 ＝diag{βb 2 |e ix | β-1 ,βb 2 |e iy | β-1 }；
Obtaining a third-order system of an integral sliding mode surface:
(4.2.2) based on the design scheme of the fault-tolerant controller of the backstepping method, the following transformations are carried out on the three-order system:
ζ i1 ＝S i1
ζ i2 ＝S i2 -γ i1
ζ i3 ＝S i3 -γ i2
(4.2.3) designing a distributed fault-tolerant formation controller in an XY plane based on the established new third-order system under a switching topology network, wherein the distributed fault-tolerant formation controller realizes fixed time convergence.
7. The fault-tolerant control method for the air-ground heterogeneous formation system under the handover topology according to claim 5, wherein the step (4.3) specifically comprises:
(4.3.1) designing a fixed time integral sliding mode surface based on the tracking error:
in the formula (I), the compound is shown in the specification,
α z and beta z Are all positive definite constants; b is a mixture of 1z ＞0，b 2z The parameters are artificially set constants when the value is more than 0;
taking a first derivative and a second derivative of the sliding mode surface, and finally obtaining a third-order system of the integral sliding mode surface:
in the formula (I), the compound is shown in the specification,
(4.3.2) based on the fault-tolerant controller design scheme of the backstepping method, the following transformations are made to the three-order system:
(4.3.3) designing a track fault-tolerant tracking controller in a Z axis under a switching topology network based on the established new third-order system, wherein the track fault-tolerant tracking controller realizes fixed time convergence.
8. A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the steps of the method of any one of claims 1 to 7 when executing the computer program.
9. A computer-readable storage medium, on which a computer program is stored which, when being executed by a processor, carries out the steps of the method of any one of claims 1 to 7.
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| CN116700355A (en) * | 2023-08-04 | 2023-09-05 | 南京航空航天大学 | Fixed wing unmanned aerial vehicle fault-tolerant control method for facing tracking of unmanned aerial vehicle |
| CN116700355B (en) * | 2023-08-04 | 2023-11-07 | 南京航空航天大学 | Fixed wing unmanned aerial vehicle fault-tolerant control method for facing tracking of unmanned aerial vehicle |
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