CN108052001A - A kind of adaptive guaranteed cost formation control algorithm of translation - Google Patents
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Abstract
The invention discloses a kind of adaptive guaranteed cost formation control algorithms of translation, comprise the following steps：Step A：System parameter settings；Step B：Vector of forming into columns is set；Step C：Formation feasibility judges, if feasible, continues step D, if infeasible, return to step A re-starts system parameter settings and vector setting of forming into columns；Step D：Solve positive definite matrix；Step E：Solve gain matrix；Step F：Guaranteed cost value solves, and the design of formation control protocol related parameters finishes；Step G：Guaranteed cost formation compliance test result, the K that will be acquiredh, KuAnd KwIn substitution system, formation effect and guaranteed cost effect are verified.The beneficial effects of the invention are as follows：Design translates adaptive guaranteed cost formation control agreement, solves fully distributed guaranteed cost formation control criterion, finally designs fully distributed guaranteed cost formation control algorithm.
Description
Technical Field
The invention relates to a control algorithm, in particular to a translation self-adaptive guaranteed performance formation control algorithm, and belongs to the technical field of application of distributed optimized formation control algorithms of multi-agent systems.
Background
The existing research on the formation control algorithm of the multi-agent system mostly needs global information such as an action topology Laplacian matrix or characteristic values thereof, and completely distributed control cannot be realized. When the number of the formation individuals is large, the formation control cannot be effectively realized because the data processing is too complex. In the process of formation control, whether formation control can be realized or not is considered, and the performance of formation control is also considered, and the formation control performance is not considered in the conventional research adopting a distributed formation control algorithm. From the existing research results, no research on the aspect of fully distributed optimization formation control algorithm is seen, and therefore, a translation self-adaptive performance-guaranteed formation control algorithm is provided for solving the problems.
Disclosure of Invention
The invention aims to solve the problems, and provides a translation self-adaptive guaranteed performance formation control algorithm.
The invention realizes the aim through the following technical scheme, and a translation self-adaptive performance-guaranteed formation control algorithm comprises the following steps:
step A: setting system parameters, namely setting values of a system matrix A and a system matrix B and a value of a performance function gain matrix Q;
and B, step B: setting a formation vector h (t) to be realized;
step C: judging formation feasibility, and solving one satisfactionMatrix K of h If there is K satisfying the condition h Continuing to the step D, if the step D does not exist, the system (1) can not realize the formation determined by h (t) under the action of the protocol (2), returning to the step A to re-form the formationSetting system parameters and formation vectors;
step D: solving a positive definite matrix P; for given γ and Q, solving for one satisfies the inequality P (A + BK) h )+(A+BK h ) T P-γPBB T P +2Q is less than or equal to 0;
step E: solving a gain matrix; substituting P into K u ＝B T P and K w ＝PBB T P, solving the gain matrix K u And K w ；
Step F: the performance-preserving value is solved according toThe expression of (2) is used for solving the guaranteed performance value, and the design of the related parameters of the formation control protocol is finished;
g: verifying the formation effect of the protective performance, and determining K h ，K u And K w And substituting the data into the system to verify the formation effect and the performance guaranteeing effect.
A translation adaptive guaranteed performance formation control algorithm comprises the following steps:
step A: setting system parameters, namely setting values of a system matrix A and a system matrix B and a value of a performance function gain matrix Q;
and B, step B: setting a formation vector h (t) to be realized;
step C: judging formation feasibility, and solving a requirementMatrix K of h If there is K satisfying the condition h Continuing to the step D, if the step D does not exist, the system (1) can not realize the formation determined by h (t) under the action of the protocol (2), and returning to the step A to perform system parameter setting and formation vector setting again;
step D: solving translation factor gamma and positive definite matrixFor a given δ and Q, solving satisfies the inequality
γ of (A) and
and E, step E: solving the gain matrix, willSubstitution intoAndsolving the gain matrix K u And K w ；
Step F: the performance-preserving value is solved according toThe performance maintaining value is solved by the expression of (3), and the design of the relevant parameters of the formation control protocol is finished;
step G: verifying the formation effect of the protection performance, and obtaining K h ，K u And K w And substituting the data into the system to verify the formation effect and the performance guarantee effect.
Wherein the system (1) in the step C is as follows:
a multi-agent system comprises N isomorphic agents, each agent dynamic model is described as follows:
wherein x is i (t) and u i (t) are the state quantities and control inputs of the ith agent, respectively, and A and B are the system matrices.
Wherein, the protocol (2) in the step C is as follows:
the translation adaptive guaranteed-performance formation control protocol is described as follows:
wherein h is i (t) is the formation vector corresponding to the ith agent, K h For the formation of vector gain matrix, w ik (t) weight of contribution of agent k to agent i at time t, N i Set of neighbors for agent i, J r For the performance optimization function, Q is the performance function gain matrix, K u And K w Is a gain matrix;
wherein, different formation vectors h are set i (t) different forms of formation formations may be generated, such as triangles, squares or circles; if the difference between the state of each agent and the formation vector is called the formation state difference, the slave control input u i (t) it can be seen that the control is implemented only when the difference in the formation state is not zero, i.e. the control is implemented when the multi-agent system has not implemented formation yet, and the control is not generated to the system once the difference in the formation state is zero, i.e. the required formation is implemented; weight of action w ik (t) is adaptively variable over time, fromIt can be seen from the expression of (a), when the difference of formation states is large, w ik (t) has a relatively large rate of change, and w is a value when the difference in formation state gradually decreases until formation is achieved ik (t) the rate of change gradually decreases until it approaches zero; performance optimization function J r The time integral of a quadratic function of the formation state difference describes that an accumulated value of the quadratic function of the formation state difference, namely a quantized value of the control performance in the control process, realizes the performance optimization in the formation control during the process of the system from the beginning to the final formation.
The definition of the achievable guaranteed-performance formation control is as follows:
for a formation vectorIf there is any bounded initial state x i (0) (i =1,2, \8230;, N), there is a vector function r (t) and a normal numberSo that lim t→+∞ (x i (t)-h i (t) -r (t)) =0 (i =1,2, \8230;, N) andand if so, the multi-agent system (1) is called to realize the guaranteed-performance formation control determined by the formation vector h (t) under the action of the protocol (2).
For any given translation factor γ>, 0, if there is a positive definite matrix P T ＝P>, 0, so that P (A + BK) h )+(A+BK h ) T P-γPBB T P +2Q is less than or equal to 0, so that the multi-agent system (1) can realize the performance-guaranteed formation control under the action of the protocol (2), and under the condition, the gain matrix K u ＝B T P，K w ＝PBB T P, guaranteed Performance value satisfaction
Wherein, K h Satisfy the requirements ofh (t) is a formation vector.
Further, for any given adjustment factor δ>, 0, if BB T Satisfies lambda max (BB T ) Less than or equal to 1 and with a translation factor gamma>, 0 and positive definite matrixSo that
The multi-agent system (1) can realize the guaranteed performance formation control under the action of the protocol (2), in this case, the gain matrixGuaranteed performance value fulfillment
Wherein, K h Satisfy the requirement ofh (t) is the formation vector in (2).
The invention has the beneficial effects that: according to the invention, as can be seen from the obtained consistency criterion and the formation control algorithm, the related criterion conditions do not contain the global information of the characteristic value of the Laplace matrix, are completely distributed criterion conditions, and meanwhile, the performance guarantee value, namely the performance function upper bound, is calculated, so that the self-adaptive performance guarantee formation control is effectively realized.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
A translation self-adaptive performance-guaranteed formation control algorithm comprises the following steps:
step A: setting system parameters, namely setting values of a system matrix A and a system matrix B and a value of a performance function gain matrix Q;
and B: setting a formation vector h (t) to be realized;
and C: judging formation feasibility, and solving one satisfactionMatrix K of h If there is K satisfying the condition h Continuing to the step D, if the step D does not exist, the system (1) can not realize the formation determined by h (t) under the action of the protocol (2), and returning to the step A to perform system parameter setting and formation vector setting again;
step D: solving a positive definite matrix P; for given γ and Q, solving for one satisfies the inequality P (A + BK) h )+(A+BK h ) T P-γPBB T P +2Q is less than or equal to 0;
step E: solving a gain matrix; substituting P into K u ＝B T P and K w ＝PBB T P, solving the gain matrix K u And K w ；
Step F: the performance-preserving value is solved according toThe expression of (2) is used for solving the guaranteed performance value, and the design of the related parameters of the formation control protocol is finished;
step G: verifying the formation effect of the protective performance, and determining K h ，K u And K w And substituting the data into the system to verify the formation effect and the performance guarantee effect.
A translation self-adaptive performance-guaranteed formation control algorithm comprises the following steps:
step A: setting system parameters, namely setting values of a system matrix A and a system matrix B and a value of a performance function gain matrix Q;
and B: setting a formation vector h (t) to be realized;
step C: judging formation feasibility, and solving a requirementMatrix K of h If there is K satisfying the condition h Continuing to the step D, if the step D does not exist, the system (1) can not realize the formation determined by h (t) under the action of the protocol (2), and returning to the step A to perform system parameter setting and formation vector setting again;
step D: solving translation factor gamma and positive definite matrixFor a given δ and Q, solving satisfies the inequality
γ of (A) and
step E: solving the gain matrix, willSubstitution intoAndsolving the gain matrix K u And K w ；
Step F: solving for a guaranteed performance value based onThe expression of (2) is used for solving the guaranteed performance value, and the design of the related parameters of the formation control protocol is finished;
g: verifying the formation effect of the protection performance, and obtaining K h ，K u And K w And substituting the data into the system to verify the formation effect and the performance guarantee effect.
Wherein the system (1) in the step C is as follows:
a multi-agent system comprises N isomorphic agents, and the dynamic model of each agent is described as follows:
wherein x i (t) and u i (t) are the state quantities and control inputs of the ith agent, respectively, and A and B are system matrices.
Wherein, the protocol (2) in the step C is as follows:
the translation adaptive guaranteed-performance formation control protocol is described as follows:
wherein h is i (t) is the formation vector corresponding to the ith agent, K h To form a vector gain matrix, w ik (t) weight of contribution of agent k to agent i at time t, N i Set of neighbors for agent i, J r For the performance optimization function, Q is the performance function gain matrix, K u And K w Is a gain matrix;
wherein, different formation vectors h are set i (t) different forms of formation formations may be generated, such as triangles, squares or circles; if the difference between the state of each agent and the formation vector is called the formation state difference, the slave control input u i (t) it can be seen that the control is implemented only when the difference in the formation state is not zero, i.e. the control is implemented when the multi-agent system has not implemented formation yet, and the control is not generated to the system once the difference in the formation state is zero, i.e. the required formation is implemented; weight of action w ik (t) is adaptively variable over time, fromIt can be seen from the expression of (a), when the difference of formation states is large, w ik The change rate of (t) is relatively large, and when the formation state difference becomes smaller gradually until formation is realized, w ik (t) is changedThe chemical rate gradually becomes smaller until the chemical rate tends to zero; performance optimization function J r The time integral of a quadratic function of the formation state difference is used for describing an accumulated value of the quadratic function of the formation state difference, namely a quantized value of the control performance in the control process, in the process of controlling the formation of the system from the beginning to the end, so that the performance optimization in the formation control is realized.
The definition of the achievable guaranteed-performance formation control is as follows:
for a formation vectorIf the initial state x is bounded arbitrarily i (0) (i =1,2, \8230;, N), there is one vector function r (t) and a positive constantSo that lim t→+∞ (x i (t)-h i (t) -r (t)) =0 (i =1,2, \8230;, N) andand if so, the multi-agent system (1) is called to realize the guaranteed-performance formation control determined by the formation vector h (t) under the action of the protocol (2).
For any given translation factor γ>, 0, if a positive definite matrix P exists T ＝P>, 0, so that P (A + BK) h )+(A+BK h ) T P-γPBB T P +2Q is less than or equal to 0, so that the multi-agent system (1) can realize the performance-guaranteed formation control under the action of the protocol (2), and under the condition, the gain matrix K u ＝B T P，K w ＝PBB T P, guaranteed Performance value of
Wherein, K h Satisfy the requirements ofh (t) is a formation vector. In the course of the proof of this conclusion, we introduced a translation factor γ in the Lyapunov function&0, the translation factor is used for eliminating the influence of the minimum non-zero eigenvalue to obtain a completely distributed performance-preserving formation criterion without any global information, and compared with the scaling self-adaption method used in the existing research result, the translation self-adaption method has the advantages that performance-preserving formation control can be realized, namely a performance optimization function J is determined r Upper bound of (2)And if the scaling self-adaptive method needs to determine the upper bound, the reciprocal of the minimum non-zero characteristic value is needed, namely, the fully distributed guaranteed performance formation control cannot be realized. It is noted that for a given system parameter, not all of the queuing vectors are effective in implementing queuing control, a conditionThe method is used for checking whether the formation is feasible or not, if the formation is feasible, the value of the formation vector gain matrix is obtained through the condition, and if the formation is not feasible, the system parameters or the formation vector are required to be reset.
For any given adjustment factor delta>, 0, if BB T Satisfies lambda as the maximum eigenvalue of max (BB T ) 1 or less and in the presence of a translation factor gamma>, 0 and positive definite matrixSo that
The multi-agent system (1) can realize the guaranteed performance formation control under the action of the protocol (2), in this case, the gain matrixGuarantee performance value satisfaction
Wherein, K h Satisfy the requirement ofh (t) is the formation vector in (2).
Firstly, a regulating factor delta is introduced>, 0 makes P ≦ δ I, which means if the condition λ is satisfied max (BB T ) Less than or equal to 1, then there is PBB T P≤δ 2 BB T . At this time, δ can be considered as the maximum non-zero eigenvalue of P, since γ and P are both unknown variables, the equation P (A + BK) h )+(A+BK h ) T P-γPBB T P +2Q is less than or equal to 0 and is difficult to solve, so a linear matrix inequality technology is adopted, a self-adaptive performance-preserving formation control standard is provided, and the problem of solving multiple unknown variables is solved. The advantage of this conclusion is that the solution value of the positive definite matrix P can be adjusted by adjusting δ, thus achieving the adjustment of the gain matrix K u And K w The purpose of (1). For a given formation vector, provided that the formation feasibility condition is satisfied
It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim concerned.
Furthermore, it should be understood that although the present description refers to embodiments, not every embodiment may contain only a single embodiment, and such description is for clarity only, and those skilled in the art should integrate the description, and the embodiments may be combined as appropriate to form other embodiments understood by those skilled in the art.
Claims (7)
1. A translation self-adaptive performance-guaranteed formation control algorithm is characterized by comprising the following steps: the method comprises the following steps:
step A: setting system parameters, namely setting values of a system matrix A and a system matrix B and a value of a performance function gain matrix Q;
and B, step B: setting a formation vector h (t) to be realized;
and C: judging formation feasibility, and solving a requirementMatrix K of h If there is K satisfying the condition h Continuing to the step D, if the step D does not exist, the system (1) can not realize the formation determined by h (t) under the action of the protocol (2), and returning to the step A to perform system parameter setting and formation vector setting again;
step D: solving a positive definite matrix P; for given γ and Q, solving for one satisfies the inequality P (A + BK) h )+(A+BK h ) T P-γPBB T P +2Q is less than or equal to 0;
step E: solving a gain matrix; bringing P into K u ＝B T P and K w ＝PBB T P, solving the gain matrix K u And K w ；
Step F: solving for a guaranteed performance value based onThe performance maintaining value is solved by the expression of (3), and the design of the relevant parameters of the formation control protocol is finished;
step G: verifying the formation effect of the protection performance, and obtaining K h ，K u And K w Substitution systemAnd verifying the formation effect and the performance protection effect.
2. A translation self-adaptive performance-guaranteed formation control algorithm is characterized by comprising the following steps: the method comprises the following steps:
step A: setting system parameters, namely setting values of a system matrix A and a system matrix B and a value of a performance function gain matrix Q;
and B: setting a formation vector h (t) to be realized;
and C: judging formation feasibility, and solving a requirementMatrix K of h If there is K satisfying the condition h Continuing to the step D, if the step D does not exist, the system (1) can not realize the formation determined by h (t) under the action of the protocol (2), and returning to the step A to perform system parameter setting and formation vector setting again;
step D: solving translation factor gamma and positive definite matrixFor a given δ and Q, solving satisfies the inequality
γ and
step E: solving the gain matrix, willSubstitution intoAndsolving the gain matrix K u And K w ；
Step F: solving for a guaranteed performance value based onThe expression of (2) is used for solving the guaranteed performance value, and the design of the related parameters of the formation control protocol is finished;
step G: verifying the formation effect of the protection performance, and obtaining K h ，K u And K w And substituting the data into the system to verify the formation effect and the performance guarantee effect.
3. A translation adaptive guaranteed-performance queuing control algorithm according to claim 1 or 2, characterized in that: the system (1) in the step C is as follows:
a multi-agent system comprises N isomorphic agents, each agent dynamic model is described as follows:
wherein x i (t) and u i (t) are the state quantities and control inputs of the ith agent, respectively, and A and B are the system matrices.
4. A translation adaptive guaranteed-performance queuing control algorithm according to claim 1 or 2, characterized in that: the protocol (2) in the step C is as follows:
the translation adaptive guaranteed-performance formation control protocol is described as follows:
wherein h is i (t) is the formation vector corresponding to the ith agent, K h To form a vector gain matrix, w ik (t) weight of agent k to agent i at time t, N i Set of neighbors for agent i, J r For optimizing the performance ofNumber, Q is a performance function gain matrix, K u And K w Is a gain matrix;
wherein, different formation vectors h are set i (t) different forms of formation formations may be generated, such as triangles, squares or circles; if the difference between each agent's state and the queuing vector is referred to as the queuing state difference, the slave control input u i (t) it can be seen that the control is implemented only when the difference in the formation state is not zero, i.e. the control is implemented when the multi-agent system has not implemented formation yet, and the control is not generated to the system once the difference in the formation state is zero, i.e. the required formation is implemented; weight of action w ik (t) is adaptively variable over time, fromIt can be seen from the expression of (a), when the difference of formation states is large, w ik The change rate of (t) is relatively large, and when the formation state difference becomes smaller gradually until formation is realized, w ik (t) the rate of change gradually decreases until it approaches zero; performance optimization function J r The time integral of a quadratic function of the formation state difference describes that an accumulated value of the quadratic function of the formation state difference, namely a quantized value of the control performance in the control process, realizes the performance optimization in the formation control during the process of the system from the beginning to the final formation.
5. A translation adaptive guaranteed-performance queuing control algorithm according to claim 1 or 2, characterized in that: the definition of the achievable guaranteed-performance formation control is as follows:
for a formation vectorIf there is any bounded initial state x i (0) (i =1,2, \8230;, N), there is a vector function r (t) and a normal numberSo that lim t→+∞ (x i (t)-h i (t) -r (t)) =0 (i =1,2, \8230;, N) andand if so, the multi-agent system (1) is called to realize the guaranteed-performance formation control determined by the formation vector h (t) under the action of the protocol (2).
6. A translation adaptive guaranteed-performance queuing control algorithm according to claim 1 or 2, characterized in that: for any given translation factor gamma>, 0, if there is a positive definite matrix P T ＝P>, 0, so that P (A + BK) h )+(A+BK h ) T P-γPBB T P +2Q is less than or equal to 0, so that the multi-agent system (1) can realize performance-guaranteed formation control under the action of the protocol (2), and in this case, the gain matrix K u ＝B T P，K w ＝PBB T P, guaranteed Performance value of
Wherein, K h Satisfy the requirement ofh (t) is a formation vector.
7. A translation adaptive guaranteed-performance queuing control algorithm according to claim 1 or 2, characterized in that: for any given adjustment factor delta>, 0, if BB T Satisfies lambda as the maximum eigenvalue of max (BB T ) 1 or less and in the presence of a translation factor gamma>, 0 and positive definite matrixSo that
The multi-agent system (1) can realize the performance-guaranteed formation control under the action of the protocol (2), and in the situation, the gain matrixGuarantee performance value satisfaction
Wherein, K h Satisfy the requirement ofh (t) is the formation vector in (2).
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