CN116627042B - Distributed collaborative tracking method for asymmetric saturated multi-self-body system of actuator - Google Patents
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Abstract
The invention belongs to the field of self-adaptive controllers, and discloses a distributed collaborative tracking method of an asymmetric saturated multi-self-body system of an actuator, which comprises the following steps: the method comprises the steps of converting an original multi-autonomous system model by using a part of supersaturation and external disturbance of a radial basis function neural network estimation system, designing a dimension-reducing observer to estimate an undetectable state track of each intelligent agent in the multi-autonomous system according to the converted system model, constructing an adaptive protocol of the multi-autonomous system based on the dimension-reducing observer according to output information and dimension-reducing observer information, and providing full conditions for driving the asymmetric saturated multi-autonomous system of the actuator to realize semi-global robust collaborative tracking according to the designed adaptive control algorithm. The method expands the self-adaptive neural network controller from a single system to a multi-self-body system, and solves the problems of dimension reduction observer design and robust distributed collaborative tracking of the network multi-self-body system with asymmetric saturation executors.
Description
Technical Field
The invention belongs to the field of self-adaptive controllers, and particularly relates to a distributed collaborative tracking method of an asymmetric saturated multi-self-body system of an actuator.
Background
In recent years, with rapid development of distributed networks and multi-agent systems, distributed cooperative control has become a hot spot in research in the control field. The main purpose of research on multi-agent systems is to replace expensive single systems by large-scale cooperation and collaboration among agents, so that complex tasks are expected to be completed, and consistency problems are taken as the basis of cooperation and collaboration among agents, and are focused by researchers in various fields, so that the research on multi-agent systems becomes an important research subject of system and control direction.
In practical application, due to physical condition limitation of an executing mechanism and consideration of system operation safety, the phenomenon of saturation of the executing mechanism inevitably occurs in a multi-main system, for example, the amplitude and frequency of output current and voltage of a controller are limited, the opening of a valve is limited by a certain range, and the water level of a water tank has a fixed maximum value. The saturation compensation strategy is sought, so that the actuator can rapidly exit saturation after entering saturation or avoid the actuator from entering saturation, and the method becomes a key for realizing high-precision control of a saturation system, and considerable research results are obtained in the aspect. It should be noted that, these studies are all directed to a control method for symmetric saturation of an actuator, however, in practical applications, asymmetric saturation of an actuator often occurs, for example, when the actuator is partially damaged, the asymmetric saturation is very easy to occur. Research into the robust consistency of multi-self-body systems of asymmetric saturation actuators presents a significant challenge.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a distributed collaborative tracking method of an asymmetric saturated multi-self-body system of an actuator, which expands a self-adaptive neural network controller from a single system to the multi-self-body system and solves the problems of the design of a dimension-reducing observer and the consistency of robust distributed collaborative tracking of the network multi-self-body system with the asymmetric saturated actuator.
In order to achieve the above purpose, the invention is realized by the following technical scheme:
the invention relates to a distributed collaborative tracking method of an actuator asymmetric saturation multi-self-body system, which comprises the following steps:
step 1, constructing a multi-self-body system influenced by asymmetric saturation and external disturbance of an actuator;
step 2, for multiple autonomous system statesDecomposing a special coordinate base to obtain a corresponding dynamic equation of the corresponding sub-state;
step 3, designing a dimension-reduction observer to estimate the value of an unmeasurable state;
step 4, estimating the oversaturation and the external disturbance part by using a radial basis function neural network;
and 5, designing an adaptive control algorithm based on a dimension reduction observer, and providing a condition for realizing robust consistency tracking of an asymmetric saturation system.
The invention further improves that: the implementation process of the step 1 is as follows:
for input saturation with asymmetry and external disturbancesThe continuous time system of the following agent and the autonomous system of the 1 leading agent are specifically:
，
，
wherein ：denoted as +.>Derivatives following the state variables of the agent, +.>Denoted as +.>Measurable output variable of individual following agent, < >>Expressed as what number of following agents, +.>Denoted as +.>State variables of the following agent, +.>A state variable denoted as leader agent, < +.>Input expressed as a multi-self subject system and output of an asymmetric saturation actuator, +.>Denoted as +.>External disturbance of following agent, +.>Expressed as a measurable output variable of the lead agent,
the follower is marked asThe subscript of the leader is 0, < ->As a state variable of the agent,for measurable output variable, +.>Is the derivative of the state variable>N represents the dimension of each agent state variable, p represents the dimension of each agent measurable output variable,indicate->External perturbation of individual agents and the presence of a bounded perturbation function +.>So thatThe relationship is established and the method is that,the input value of the system and the output value of the asymmetric saturation actuator are specifically:
wherein ， andRespectively->Unknown upper and lower limit, < ->Is a control input variable irrespective of input saturation, < +.>For a matrix of dimensional matching, use +.>Representing the dimension in Euclidean space as +.>Matrix of->Representation->Identity matrix of dimension>A zero matrix or constant zero representing the corresponding dimension;
to facilitate the design of the controller, define a new function asAndthus->The conversion dynamics model of the following agent is as follows:
，
wherein For the part exceeding saturation->Representing the sum of supersaturation and external disturbance +.>Representing the control input variables irrespective of input saturation.
The invention further improves that: the step 2 specifically comprises the following steps:
step 21, matrix pairs in an actuator asymmetric saturation multi-self-body systemSatisfies the conditions of calm, detectable, left reversible and minimum phase, and all elements in the left reversible and infinite zero point structures in the system are 1, namely, the order is 1, the conversion dynamics model of the following agent is decomposed by utilizing a special coordinate base decomposition technology, and the following agent is subjected to the conversion by a nonsingular state conversion matrix->And output conversion matrix->Wherein the inverse of the state transition matrix is +.>, and for a matrix with corresponding dimensions, decomposing the state and output of each following agent and leading agent into +.>And, wherein， andThe three sub-states are respectively a converted state vector and an output vector +.>， and, whereinRepresenting subsystems without direct inputs and outputs, displaying a finite zero structure for a given system;Representing subsystems without direct input, displaying the left reversibility structure of a given system;Representing a subsystem with direct inputs and outputs, showing the infinite zero structure of a given system, and +.>，，，，The specific decomposition vector is expressed in the following form:
，
wherein All represent real numbers;
step 22, providing corresponding dynamics equation of corresponding sub-state for sub-stateThere is
，
Matrix pairs in a multiple self-body system due to asymmetric saturation of actuatorsIs the assumption of the minimum phase, therefore +.>Is Hurwitz, +.> andFor a matrix having corresponding dimensions;
for sub-statesEach element of (3)There is
，
For sub-statesEach element of (3)There is
，
wherein ,for solving by linear system toolkit in MATLAB with corresponding matching dimensionsSubsystem parameter matrix,/->Control input variable indicating no consideration of input saturation, < ->Representing the sum of supersaturation and external perturbation,
from the above, the sub-state can be known andBy measuring the output as a measurable part, sub-state +.>Is an undetectable part and is not affected by an unknown input signal.
The invention further improves that: the step 3 of designing the dimension-reduction observer to estimate the value of the unmeasurable state specifically comprises the following steps:
as can be seen from step 21, the special coordinate-based method divides the state into three sub-states，Andfor step 22 sub-state->Designing an observer which is not affected by the unknown input signal to observe the state of the non-measurable part, wherein +.>And an observer is designed for the asymmetric saturated multi-autonomous system of the executor to estimate the state track of the intelligent agent, specifically:
，
wherein , andRespectively-> andIs->Dimension-reducing observer for the system to follow the state of the agent,/->Conversion matrix for output->Inverse matrix of>Representing the converted output vector, +.>Indicate->And a measurable output variable that follows the agent.
The invention further improves that: the sum of supersaturation and external disturbance in the step 4The estimation is specifically as follows:
for the unknown function of supersaturation and external disturbance, it is necessary to perform the partAnd (5) estimating. The radial basis function neural network can approach the tight set +.>Any continuous function above, thus using a radial basis function neural network to estimate +.>Continuous function of individual agent->The concrete form is as follows:
，
wherein: wherein the node number of the neural network isThe more nodes, the more accurate the approximation, +.>Is an ideal weight vector +.>Transpose of ideal weight vector, +.>Is at->Upper bounded approximation error,/->Is radial basis function vector, ++>Representing the input vector +.>The expression using a general gaussian function is:The node number of the neural network is +.>The more nodes, the more accurate the approximation, +.>Is European norm, ++> andThe center and variance of the Gaussian function are respectively, and the ideal weight vector is the artificial value required by theoretical analysis:, wherein andIs an estimate of the ideal weight vector and its transpose, therefore +.>It needs to be estimated by a function approximation.
The invention further improves that: said step 5 comprises the steps of:
step 51, using a kit comprisingDirected graph of the communication of a following agent and 1 leader agent +.>Describing the information interaction topological relation between the agents, wherein the leading agent is the root node of the directed spanning tree, all following agents can acquire the directed information of the leading agent, and the adjacent matrix and the Laplace matrix are defined as respectively and, wherein ,Representing the specific element value in the adjacency matrix, which value is 0 or 1,/for>Representing the specific element values in the Laplace matrix, < >>Representing the topological relation between the lead agent and the following agent,/for>Satisfy nonsingular->Definition of matrix, its characteristic value isAll have a positive real part and are present in a positive definite diagonal matrixSo that->Is true, whereinDefinitions->Is +.>；
Step 52, solving algebraic Li-Ka equation to obtain：
；
Step 53, according to the obtained andDefinitions->Andthere is->, whereinCombine->， andDefinitions->Is->，，Related variables, and designing an adaptive control algorithm:
，
wherein ：is->The time-varying coupling weight related to the following agent is as follows, and the initial value thereof satisfies the following condition，Is a time-varying function>And each value is a positive number, +.>Is->Estimated value of ∈10->Representing a very small positive constant, define +.>，，，，，，Each following agent writes a compact form of adaptive control algorithm based on a reduced order observer:
，
wherein the neural network self-adaptive law is
，
andIs an adaptive gain matrix-> andIs a positive constant;
in the semi-global collaborative consistency result, the initial state values of all agents are selected from an arbitrarily large bounded setTherefore, according to the Lasal invariant set principle, the coupling state +.>Kinetic parameters->Andare all bounded, deriving +.> andIs bounded and because of the Gaussian function +.> andFunction->Is bounded and is further->Is bounded, so the control input +.>Is bounded, known as +.2 in combination with step 2>Is bounded and because of +.> andAre all bounded by ∈>， andRepresentation, thus define->The specific form of (a) is；
Then, for a given tight setAny initial value of the above can be used to construct a ratio +.>Big tight set->Whereby the radial basis function neural network approximately estimates +.>Is effective in selecting proper +.>，，， andAnd the parameters meet->，When the system realizes semi-global robust consistent tracking, and +.>Converging to a residual set
，
wherein ：the expression of (2) is +.>，，For transpose of subsystem parameter matrix with corresponding matching dimension obtained by linear system toolkit solution in MATLAB, < >>Representation->Is to be used in the present invention,
，and->Is an equation ofIs a solution to (a).
The beneficial effects of the invention are as follows: the distributed collaborative tracking method is mainly aimed at a multi-self-body system of which the system is affected by an asymmetric saturation actuator and external disturbance. Under the precondition that the system input and the external disturbance are bounded and the state is not measurable, the method utilizes a radial basis function neural network to estimate the oversaturation and the external disturbance part and converts the original multi-autonomous system model; according to the converted system model, a dimension-reducing observer is designed to estimate the unmeasurable state track of each intelligent agent in the multi-self-body system; according to the output information and the dimension-reducing observer information, constructing an adaptive protocol of the multi-self-body system based on the dimension-reducing observer; and then, giving out sufficient conditions for driving the asymmetric saturated multi-autonomous system of the actuator to realize semi-global robust collaborative tracking according to a designed self-adaptive control algorithm.
The method expands the self-adaptive neural network controller from a single system to a multi-self-body system, from undirected topology to directed topology, achieves the purposes of algorithm design, parameter selection and full condition all in distributed characteristics, improves the variability and flexibility of centralized characteristics, can be applied to large-scale complex networks, and has larger fault tolerance to system agents.
The scheme breaks through the complex challenges between nonlinearity caused by asymmetric saturation and interference of the actuator and coupling caused by information exchange between the actuators.
Meanwhile, the method solves the problems of dimension reduction observer design and robust distributed collaborative tracking of a network multi-self-body system with an asymmetric saturation actuator.
According to the robust distributed collaborative tracking method for the asymmetric saturation multi-self-body system of the actuator, asymmetric saturation and external disturbance factors are considered based on output feedback in collaborative tracking control research of the multi-self-body system, the prior knowledge of an external disturbance function and the upper and lower bounds of the asymmetric saturation actuator is not needed to be known, dependence on system state information is eliminated, and sufficient conditions for achieving robust collaborative tracking of the system are provided.
Drawings
FIG. 1 is a schematic flow chart of the method of the present invention.
FIG. 2 is a state trace of a leading agent and a following agent of the multi-agent system of the present invention.
Fig. 3 is a trace of an adaptive control algorithm based on a dimension-reduction observer.
Fig. 4 is an overall variation trace of the adaptive gain in the control algorithm.
Fig. 5 is an estimated trajectory process for external disturbances and supersaturation.
Detailed Description
Embodiments of the invention are disclosed in the drawings, and for purposes of explanation, numerous practical details are set forth in the following description. However, it should be understood that these practical details are not to be taken as limiting the invention. That is, in some embodiments of the invention, these practical details are unnecessary.
As shown in fig. 1, the invention is a distributed collaborative tracking method of an asymmetric saturated multi-self-body system of an actuator, by which an adaptive neural network controller is expanded from a single system to the multi-self-body system, and meanwhile, the problems of dimension-reducing observer design and robust distributed collaborative tracking of the network multi-self-body system with the asymmetric saturated actuator are solved. The method comprises the following steps:
step 1, constructing a multi-self-body system influenced by asymmetric saturation and external disturbance of an actuator, wherein the specific implementation process is as follows:
for input saturation with asymmetry and external disturbancesThe continuous time system of the following agent and the autonomous system of the 1 leading agent are specifically:
，
，
wherein ：denoted as +.>Derivatives following the state variables of the agent, +.>Denoted as +.>Measurable output variable of individual following agent, < >>Expressed as what number of following agents, +.>Denoted as +.>State variables of the following agent, +.>A state variable denoted as leader agent, < +.>Input expressed as a multi-self subject system and output of an asymmetric saturation actuator, +.>Denoted as +.>External disturbance of following agent, +.>Expressed as a measurable output variable of the lead agent,
the follower is marked asThe subscript of the leader is 0, < ->As a state variable of the agent,for measurable output variable, +.>Is the derivative of the state variable>N represents the dimension of each agent state variable, p represents the dimension of each agent measurable output variable,indicate->External perturbation of individual agents and the presence of a bounded perturbation function +.>Make->The relationship is established and the method is that,the input value of the system and the output value of the asymmetric saturation actuator are specifically:
wherein ， andRespectively->Unknown upper and lower limit, < ->Is a control input variable irrespective of input saturation, < +.>For a matrix of dimensional matching, use +.>Representing the dimension in Euclidean space as +.>Matrix of->Representation->Identity matrix of dimension>A zero matrix or constant zero representing the corresponding dimension;
to facilitate the design of the controller, define a new function asAndthus->The conversion dynamics model of the following agent is as follows:
，
wherein For the part exceeding saturation->Representing the sum of supersaturation and external disturbance +.>Representing the control input variables irrespective of input saturation.
Step 2, for multiple autonomous system statesAnd (5) decomposing a special coordinate base to obtain a corresponding dynamic equation of the corresponding sub-state.
The method specifically comprises the following steps:
step 21, matrix pairs in an actuator asymmetric saturation multi-self-body systemSatisfies the conditions of calm, detectable, left reversible and minimum phase, and all elements in the left reversible and infinite zero point structures in the system are 1, namely, the order is 1, the conversion dynamics model of the following agent is decomposed by utilizing a special coordinate base decomposition technology, and the following agent is subjected to the conversion by a nonsingular state conversion matrix->And output conversion matrix->Wherein the inverse of the state transition matrix is +.>, and for a matrix with corresponding dimensions, decomposing the state and output of each following agent and leading agent into +.>And, wherein，Andthe three sub-states are respectively a converted state vector and an output vector +.>， and, wherein Representing subsystems without direct inputs and outputs, displaying a finite zero structure for a given system;Representing subsystems without direct input, displaying the left reversibility structure of a given system;Representing a subsystem with direct inputs and outputs, showing the infinite zero structure of a given system, and +.>，，，，The specific decomposition vector is expressed in the following form:
，
wherein All represent real numbers;
step 22, providing corresponding dynamics equation of corresponding sub-state for sub-stateThere is->
，
Matrix pairs in a multiple self-body system due to asymmetric saturation of actuatorsIs the assumption of the minimum phase, therefore +.>Is Hurwitz, +.> andFor a matrix having corresponding dimensions;
for sub-statesEach element of (3)There is
，
For sub-statesEach element of->There is
，
wherein ,solving for linear system toolkits in MATLABThe resulting subsystem parameter matrix with corresponding matching dimensions,/->Control input variable indicating no consideration of input saturation, < ->Representing the sum of supersaturation and external perturbation,
from the above, the sub-state can be known andBy measuring the output as a measurable part, sub-state +.>Is an undetectable part and is not affected by an unknown input signal.
And 3, designing a dimension-reduction observer to estimate the value of the unmeasurable state.
As can be seen from step 21, the special coordinate-based method divides the state into three sub-states， andFor step 22 sub-state->Designing an observer which is not affected by the unknown input signal to observe the state of the non-measurable part, wherein +.>And an observer is designed for the asymmetric saturated multi-autonomous system of the executor to estimate the state track of the intelligent agent, specifically:
，
wherein , andRespectively-> andIs->Dimension-reducing observer for the system to follow the state of the agent,/->Conversion matrix for output->Inverse matrix of>Representing the converted output vector, +.>Indicate->And a measurable output variable that follows the agent.
And 4, estimating the oversaturation and external disturbance part by utilizing a radial basis function neural network, wherein the method specifically comprises the following steps of:
as can be seen from the step 2,this portion needs to be estimated as an unknown function of supersaturation and external disturbances. The radial basis function neural network can approach the tight set +.>Any continuous function above, thus using a radial basis function neural network to estimate +.>Continuous function of individual agent->The concrete form is as follows:
，
wherein: wherein the node number of the neural network isThe more nodes, the more accurate the approximation,is an ideal weight vector +.>Transpose of ideal weight vector, +.>To at the same timeUpper bounded approximation error,/->Is radial basis function vector, ++>Representing the input vector +.>The expression using a general gaussian function is:the node number of the neural network is +.>The more nodes, the more accurate the approximation, +.>Is European norm, ++> andThe center and variance of the Gaussian function are respectively, and the ideal weight vector is the artificial value required by theoretical analysis:, wherein andIs an estimate of the ideal weight vector and its transpose, therefore +.>It needs to be estimated by a function approximation.
Step 5, designing an adaptive control algorithm based on a dimension reduction observer, and providing conditions for realizing robust consistency tracking of an asymmetric saturation system, wherein the method comprises the following steps:
step 51, using a kit comprisingDirected graph of the communication of a following agent and 1 leader agent +.>Describing the information interaction topological relation between the agents, wherein the leading agent is the root node of the directed spanning tree, all following agents can acquire the directed information of the leading agent, and the adjacent matrix and the Laplace matrix are defined as respectively and, wherein ,Representing the specific element value in the adjacency matrix, which value is 0 or 1,/for>Representing the specific element values in the Laplace matrix, < >>Representing the topological relation between the lead agent and the following agent,/for>Satisfy nonsingular->Definition of matrix with characteristic value +.>All have positive real parts and there is positive diagonal matrix +.>So thatIs true, wherein->Definition ofIs +.>；
Step 52, solving algebraic Li-Ka equation to obtain：/>
；
Step 53, according to the obtained andDefinitions->Andthere is->, wherein Combine->， andDefinitions->Is->，，Related variables, and designing an adaptive control algorithm:
，
wherein ：is->The time-varying coupling weight related to the following agent is as follows, and the initial value thereof satisfies the following condition，Is a time-varying function>And each value is a positive number, +.>Is->Estimated value of ∈10->Representing a very small positive constant, define +.>，，，，，，Each following agent writes a compact form of adaptive control algorithm based on a reduced order observer:
，
wherein the neural network self-adaptive law is
，
andIs an adaptive gain matrix-> andIs a positive constant;
in the semi-global collaborative consistency result, the initial state values of all agents are selected from an arbitrarily large bounded setTherefore, according to the Lasal invariant set principle, the coupling state +.>Kinetic parameters->Andare all bounded, deriving +.> andIs bounded and because of the Gaussian function +.> andFunction->Is bounded and is further->Is bounded, so the control input +.>Is bounded, known as +.2 in combination with step 2>Is bounded and because of +.> andAre all bounded by ∈>， andRepresentation, thus define->In the specific form->；
Then, for a given tight setAny initial value of the above can be used to construct a ratio +.>Big tight set->Whereby the radial basis function neural network approximately estimates +.>Is effective in selecting proper +.>，，， andAnd the parameters meet->，When the system realizes semi-global robust consistent tracking, and +.>Converging to a residual set
，
wherein ：the expression of (2) is +.>，，For transpose of subsystem parameter matrix with corresponding matching dimension obtained by linear system toolkit solution in MATLAB, < >>Representation->Is to be used in the present invention,
，and->Is equation->Is a solution to (a).
The following is a detailed description of specific examples.
Consider a multi-self-principal system consisting of 6 follower agents and 1 leader agent, wherein the directed communication topology of the 6 follower agents is represented by an adjacency matrixAnd only the first following agent can directly obtain the directed information of the leading agent.
In the example, a specific form of the dimension reduction observer is given by utilizing special coordinate basis decomposition, and a corresponding self-adaptive control algorithm based on the dimension reduction observer is designed by combining a radial basis function neural network to drive the state track of each agent to achieve consistency.
Giving a system matrix satisfying the condition asIt is evident that matrix pairs in the actuator asymmetric saturation multi-self-body system +.>Has been satisfied with the form of special coordinate base decomposition in whichFor the undetectable sub-state->The observer is designed asThe dimension-reduction observer of the system state is designed as. Initial state of the leader agent +.>Randomly selected from->Following intelligent agentIs +.>Randomly selected from->. The lower and upper limits of the asymmetric saturation actuator are +.>The external disturbance function of each following agent is chosen to be +.>。
In this example, the solution to the algebraic Rick equation yieldsAnd select. Selecting proper->，，，，，。
FIG. 2 depicts a lead agent and a following agentIs a motion trajectory of (a).
Fig. 3 shows a motion track of a control input, which shows that the designed adaptive control algorithm based on the dimension reduction observer can realize robust consistency tracking of a multi-self-body system with an asymmetric saturation actuator, and the control input is maintained in the upper and lower bounds of the asymmetric saturation actuator, so that the performance of the multi-self-body system is ensured.
FIG. 4 depicts adaptive gainTo illustrate that it is bounded and gradually goes towards 0.
FIG. 5 depicts an estimation functionThe trajectory of (2) indicates that the radial basis function neural network can better estimate the unknown external disturbance and the supersaturation in the control input.
The method expands the self-adaptive neural network controller from a single system to a multi-self-body system, and simultaneously solves the problems of dimension reduction observer design and robust distributed collaborative tracking of the network multi-self-body system with asymmetric saturation executors.
The foregoing description is only illustrative of the invention and is not to be construed as limiting the invention. Various modifications and variations of the present invention will be apparent to those skilled in the art. Any modification, equivalent replacement, improvement, or the like, which is within the spirit and principles of the present invention, should be included in the scope of the claims of the present invention.
Claims (5)
1. A distributed collaborative tracking method of an asymmetric saturated multi-self-body system of an actuator is characterized by comprising the following steps of: the method comprises the following steps:
step 1, constructing a multi-self-body system influenced by asymmetric saturation and external disturbance of an actuator;
step 2, for multi-autonomous system state x i Decomposing a special coordinate base to obtain a corresponding dynamic equation of the corresponding sub-state;
step 3, designing a dimension-reduction observer to estimate the value of an unmeasurable state;
step 4, estimating the oversaturation and the external disturbance part by using a radial basis function neural network;
step 5, designing an adaptive control algorithm based on a dimension reduction observer, providing a condition for realizing robust consistency tracking of an asymmetric saturation system,
the implementation process of the step 1 is as follows:
for an autonomous system with N following agents with asymmetric input saturation and 1 leader agent, specifically:
wherein ：expressed as the i-th derivative of the following agent state variable, y i A measurable output variable denoted as the i-th following agent, i being denoted as the number of following agents, x i A state variable, x, denoted as the ith following agent 0 State variables, u, denoted as leader agents i (v i ) Represented as the input to the multi-principal system and the output of the asymmetric saturation actuator,expressed as the i-th external disturbance following the agent, y 0 Expressed as measurable output variables of the leader agent, the follower is marked 1-N, the subscript of the leader is 0,/o>Is a state variable of the agent, +.>For measurable output variable, +.>S=0, 1, …, N represents the dimension of each agent state variable, p represents the dimension of each agent measurable output variable, +.>Representing the exterior of the ith agentDisturbance, and there is a bounded disturbance function w i Make->The relationship is established and the method is that,the input value of the system and the output value of the asymmetric saturation actuator are specifically:
where k=1, 2, …, m, andU respectively ik Unknown upper and lower bounds, v ik Is a matrix with dimension matching, A, B, C is a control input variable which does not consider input saturation, and +.>Representing a matrix of dimension m x n in Euclidean space, I n An identity matrix representing n dimensions, 0 representing a zero matrix or constant zero of the corresponding dimension;
defining a new function as sigma i (v i )＝u i (v i )-v i and σi (v i )+w i ＝δ i The conversion kinetics model for the N following agents is therefore:
wherein σi (v i ) To exceed saturated part, delta i Indicating supersaturation and outsideSum of partial disturbances, v i The representation represents a control input variable that does not take into account input saturation.
2. The method for distributed collaborative tracking of an actuator asymmetric saturation multi-self-body system according to claim 1, wherein: the step 2 specifically comprises the following steps:
step 21, decomposing the conversion dynamics model of the following intelligent agent by using a special coordinate base decomposition technology under the precondition that matrix pairs (A, B, C) in the actuator asymmetric saturation multi-self-body system meet the conditions of being stable, detectable, left reversible and minimum phases and all elements in the left reversible and infinite zero point structures in the system are 1, namely the orders are 1, and decomposing the conversion dynamics model by using a non-singular state conversion matrix Γ s ＝[Γ sa Γ sb Γ sd ]And output conversion matrix Γ o Wherein the inverse of the state transition matrix is andDecomposing the state and output of each following agent and leading agent into andWherein i=0, 1, …, N,> andRespectively a converted state vector and an output vector, and three sub-states and whereinRepresenting subsystems without direct inputs and outputs, displaying a finite zero structure for a given system;Representing subsystems without direct input, displaying the left reversibility structure of a given system;Representing a subsystem with direct inputs and outputs, showing the infinite zero structure of a given system, an The specific decomposition vector is expressed in the following form:
wherein
All represent real numbers;
step 22, providing corresponding dynamics equation of corresponding sub-state for sub-stateHas the following components
Since the matrix pair (A, B, C) in the actuator asymmetric saturation multi-self-body system is the assumption of minimum phase, thereforeIs Hurwitz, L ab and Lad For a matrix having corresponding dimensions;
for sub-statesEach element of->Has the following components
For sub-statesEach element of->Has the following components
wherein ,Lbbk ,，L dbk ,L ddk ，E ka ,E kb ,E kd For subsystem parameter matrix with corresponding matching dimension obtained by solving linear system toolkit in MATLAB, v ik Representing a control input variable delta irrespective of input saturation ik Representing the sum of supersaturation and external perturbation,
from the above, the sub-state can be known andBy measuring the output as a measurable part, sub-stateIs an undetectable part and is not affected by an unknown input signal.
3. The distributed collaborative tracking method for an actuator asymmetric saturation multi-self-body system of claim 2, wherein: the step 3 of designing the dimension-reduction observer to estimate the value of the unmeasurable state specifically comprises the following steps:
for the step 22 sub-stateDesigning an observer which is not influenced by an unknown input signal to observe the state of an undetectable part, wherein i=0, 1, … and N, and designing the observer for the asymmetric saturated multi-autonomous system of the actuator to estimate the state track of the intelligent agent, specifically:
wherein , andRespectively-> and xia Is->Dimension-reducing observer for the system to follow the state of the agent,/->For outputting the conversion matrix Γ o Inverse matrix of>Representing the transformed output vector, y i Representing a measurable output variable representing the ith following agent.
4. The method for distributed collaborative tracking of an actuator asymmetric saturation multi-self-body system according to claim 3, wherein: the step 4 is to sum delta of supersaturation and external disturbance i The estimation is specifically as follows:
estimating continuous function delta of ith agent using radial basis function neural network i I=1, …, N, in the specific form:
wherein ：is an ideal weight vector +.>Transpose of ideal weight vector, E i To approximate errors bounded on pi,as a vector of the radial basis functions,representing the input vector +.>The expression using a general gaussian function is:The node number of the neural network is l more than or equal to 1, and the more the node number is, the more accurate the approximation value is,/-the neural network is>Is European norm, mu it and kit The center and variance of the Gaussian function are respectively, and the ideal weight vector is the artificial value required by theoretical analysis: wherein andIs an estimate of the ideal weight vector and its transpose, therefore +.>It needs to be estimated by a function approximation.
5. The method for distributed collaborative tracking of an actuator asymmetric saturation multi-self-body system according to claim 4, wherein: said step 5 comprises the steps of:
step 51, describing the information interaction topological relation between the agents by using a directed graph g containing N connected following agents and 1 leading agent, wherein the leading agent is the root node of the directed spanning tree, and all the following agents can acquire the directed information of the leading agent, and define the adjacent matrix and the Laplacian matrix as and wherein ,aij Representing the value of a specific element in the adjacency matrix, which is 0 or 1, l ij l ij Representing specific element values in the Laplace matrix, L 0 Represents L 0 Representing the topological relationship between a leading agent and a following agent, L 1 Satisfies the definition of a non-singular M matrix, and has a characteristic value of 0=lambda 1 ≤λ 2 ≤...≤λ N All have a positive real part and there is a positive definite diagonal matrix g=diag { G 1 ，...，g N }∈R N×N So that->Is true, whereinDefinitions->Is lambda 0 ；
Step 52, solving algebraic licarpa equation to obtain Q > 0:
A T Q+QA-2QBB T Q+Ⅰ＝0；
step 53, according to the obtained andDefinitions->Andthere is-> wherein Bonding sigma i (v i )，w i and ∈i Definition of eta i Sum of sigma i (v i )，w i ，∈ i Related variables, and designing an adaptive control algorithm:
wherein ：di Is a time-varying coupling weight associated with the ith following agent and whose initial value satisfies d i (0)＞1，ρ i As a function of the time-variation,and each value is a positive number, +.>Is eta i Estimated value of ∈10->Representing a small positive constant, defining
Each following agent writes a compact form of adaptive control algorithm based on a reduced order observer:
wherein the neural network self-adaptive law is
Γ＝Γ T＞0 and Ξ＝ΞT > 0 is the adaptive gain matrix, k w and kη Is a positive constant;
in the semi-global collaborative consistency result, the initial state values of all agents are selected from an arbitrarily large bounded setTherefore, according to the Lasal invariant set principle, the coupling state xi i (t) kinetic parameter d i(t) and ρi (t) are all bounded, deriving +.> andIs bounded and because of the Gaussian function +.>And tanh function->Is bounded and then v i Is bounded, so the control input v i Is bounded, sigma i (v i ) Is bounded and because of w i and ∈i Are all bounded by andRepresentation, thus define eta i In the specific form->
Then, for a given tight set pi v Any initial value can be used to construct a ratio pi v Large close-fitting pi, whereby radial basis function neural networks approximate δ i Is effective in selecting proper Γ, xi, k w ，k η Andparameter satisfaction When the system realizes semi-global robust consistent tracking, and zeta converges to a residual set
wherein ：Ea The expression of (2) is For transpose of subsystem parameter matrix with corresponding matching dimension obtained by linear system toolkit solution in MATLAB, < >>Representation E a Is to be used in the present invention, and P is aa > 0 is equation->Is a solution to (a).
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