CN108052001B - Translation self-adaptive performance-guaranteed multi-agent formation control algorithm - Google Patents
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Abstract
The invention discloses a translation self-adaptive performance-guaranteed formation control algorithm, which comprises the following steps: step A: setting system parameters; and B: setting a formation vector; and C: c, judging the feasibility of formation, if feasible, continuing the step D, and if not feasible, returning to the step A to set system parameters and form vectors again; step D: solving a positive definite matrix; step E: solving a gain matrix; step F: solving the performance maintaining value, and finishing the design of relevant parameters of the formation control protocol; 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. The beneficial effects of the invention are: designing a translation self-adaptive guaranteed performance formation control protocol, solving a completely distributed guaranteed performance formation control criterion, and finally designing a completely distributed guaranteed performance 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 individuals to be formed is large, the data processing is too complex to effectively realize the formation control. In the process of formation control, not only whether formation control can be realized but also the performance of the formation control needs to be 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 performance-guaranteed 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;
and C: judging formation feasibility, and solving one satisfaction
Matrix 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 to
The 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 guaranteeing 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, step B: setting a formation vector h (t) to be realized;
step C: judging formation feasibility, and solving a requirement
Matrix 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 matrix
For a given δ and Q, solving satisfies the inequality
and E, step E: solving the gain matrix, will
Substitution into
And
solving the gain matrix K u And K w ；
Step F: solving for a guaranteed performance value based on
The performance maintaining value is solved by the expression of (3), and the design of the relevant 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 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 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 queuing 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 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 control is only effected when the formation status is not nearly zero, i.e. moreThe intelligent system does not implement control when the formation is not realized, and once the formation state difference is zero, the system is not controlled when the required formation is realized; weight of action w ik (t) is adaptively varied with time, from
It 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 vector
If there is any bounded initial state x i (0) (i =1,2, \8230;, N), there is a vector function r (t) and a normal number
So that lim t→+∞ (x i (t)-h i (t) -r (t)) =0 (i =1,2, \8230;, N) and
and 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, such 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 the situation isUnder the condition of gain matrix K u ＝B T P，K w ＝PBB T P, guaranteed Performance value of
Wherein, K h Satisfy the requirements of
h (t) is a formation vector.
Further, for any given adjustment factor δ > 0, if BB is present T Satisfies lambda as the maximum eigenvalue of max (BB T ) Less than or equal to 1 and with a translation factor gamma > 0 and a positive definite matrix
So 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 matrix
Guarantee performance value satisfaction
Wherein, K h Satisfy the requirement of
h (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, the completely distributed criterion conditions are obtained, meanwhile, the guarantee performance value, namely the performance function upper bound, is calculated, the self-adaptive guarantee performance formation control is effectively realized, the completely distributed guarantee performance formation control criterion is solved by designing a translation self-adaptive guarantee performance formation control protocol, and finally, the completely distributed guarantee performance formation control algorithm is designed, so that the method has good economic and social benefits and is suitable for popularization and use.
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 satisfaction
Matrix 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: solving for a guaranteed performance value based on
The 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.
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 satisfaction
Matrix 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 matrix
For a given δ and Q, solving satisfies the inequality
and E, step E: solving the gain matrix, will
Substitution into
And
solving the gain matrix K u And K w ；
Step F: the performance-preserving value is solved according to
The 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 guaranteeing 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 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 For the formation of 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 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 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 varied with time, from
It 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 vector
If there is any bounded initial state x i (0) (i =1,2, \8230;, N), there is a vector function r (t) and a normal number
So that lim t→+∞ (x i (t)-h i (t) -r (t)) =0 (i =1,2, \8230;, N) and
when true, then the multi-agent system (1) is called in the protocol (2)The performance-guaranteed formation control determined by the formation vector h (t) is realized under the action of the control unit.
For any given translation factor γ > 0, if there is a positive definite matrix P T = P > 0, such 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 satisfaction
Wherein, K h Satisfy the requirements of
h (t) is a formation vector. In the proving process of the conclusion, a translation factor gamma is more than 0 is introduced into the Lyapunov function, the translation factor has the function of 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 a 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 achieved, 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 completely 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 condition
For checking whether the formation is feasible, if feasible, calculating the value of the formation vector gain matrix by the condition, if not, resetting the system parameters or formingA team vector.
For any given adjustment factor δ > 0, if BB T Satisfies lambda as the maximum eigenvalue of max (BB T ) 1 or less and a translation factor gamma > 0 and a positive definite matrix
So 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 matrix
Guarantee performance value satisfaction
Wherein, K h Satisfy the requirement of
h (t) is the formation vector in (2).
First of all, a regulating factor delta > 0 is introduced such that P ≦ delta I, which means that if the condition lambda is satisfied max (BB T ) Less than or equal to 1, 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 positive definite matrix can be adjusted by adjusting delta
To achieve the adjustment gain matrix K u And K w The purpose of (1). For a given formation vector, as long as 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 (3)
1. A translation self-adaptive performance-guaranteeing multi-agent formation control algorithm is characterized in that: 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 one satisfaction
Matrix K of h If there is K satisfying the condition h Continuing with step D, if not, the system (1) is unable to implement h (t) under the action of the protocol (2)B, formation, returning to the step A to perform system parameter setting and formation vector setting again; wherein A and B are system matrixes;
the system (1) 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 system matrices;
protocol (2) 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;
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; wherein γ is a translation factor; q is a performance function gain matrix;
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: the performance-preserving value is solved according to
Expression ofSolving the performance guarantee value according to the formula, and finishing the design of related parameters of the formation control protocol;
upper bound for guaranteed performance value;
for any given translation factor γ > 0, if there is a positive definite matrix P T = P > 0, such 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 of
h (t) is a formation vector;
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.
2. A translation self-adaptive performance-guaranteeing multi-agent formation control algorithm is characterized in that: 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 one satisfaction
Matrix K of h If there is K satisfying the condition h Continuing with step D, if not storingIf so, the system (1) can not realize the formation determined by h (t) under the action of the protocol (2), and returns to the step A to perform the system parameter setting and the formation vector setting again; wherein A and B are system matrixes;
the system (1) 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 system matrices;
protocol (2) 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;
step D: solving translation factor gamma and positive definite matrix
For a given δ and Q, solving satisfies the inequality
step E: solving the gain matrix, will
Substitution into
And
solving the gain matrix K u And K w ；
Step F: solving for a guaranteed performance value based on
The 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;
upper bound for guaranteed performance value;
for any given adjustment factor δ > 0, if BB T Satisfies lambda as the maximum eigenvalue of max (BB T ) 1 or less and a translation factor gamma > 0 and a positive definite matrix
So 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 matrix
Guaranteed performance value fulfillment
Wherein, K h Satisfy the requirements of
h (t) is the formation vector in (2);
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 multi-agent formation control algorithm as claimed in claim 1 or 2, wherein: the definition of the achievable guaranteed-performance formation control is as follows:
for a formation vector
If there is any bounded initial state x i (0) (i =1,2, \8230;, N), there is a vector function r (t) and a normal number
So that lim t→+∞ (x i (t)-h i (t) -r (t)) =0 (i =1,2, \8230;, N) and
and 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).
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