CN113359445A - Distributed output feedback asymptotic consistent control method for multi-agent hysteresis system - Google Patents

Distributed output feedback asymptotic consistent control method for multi-agent hysteresis system Download PDF

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CN113359445A
CN113359445A CN202110609595.0A CN202110609595A CN113359445A CN 113359445 A CN113359445 A CN 113359445A CN 202110609595 A CN202110609595 A CN 202110609595A CN 113359445 A CN113359445 A CN 113359445A
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刘烨
杨朋举
张娅倩
郑贤
王清华
陈剑雪
吴健珍
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Shanghai University of Engineering Science
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Abstract

The invention discloses a distributed output feedback asymptotic consistent control method for a multi-agent hysteresis system, which comprises the following steps: tracking consistency errors of a reference signal track and an actual track in real time; inputting the consistency error of the reference signal track and the actual track into a dynamic surface controller containing a nonlinear filter; the actual control signal output by the dynamic surface controller is used as the input of a hysteresis model and the multi-agent system is controlled according to the output of the hysteresis model; applying a state observer with dynamic high gain to carry out online real-time estimation on the unknown state of the multi-agent system; and inputting the state estimation value of the multi-agent system into a dynamic surface controller, and updating the parameter pre-estimation value on line by combining the self-adaptive rate of the parameter estimation in the controller to realize the distributed output feedback asymptotic consistent control of the multi-agent system. The method of the invention can effectively eliminate the hysteresis effect of the actuating mechanism and realize good tracking control performance, and has wide application prospect.

Description

Distributed output feedback asymptotic consistent control method for multi-agent hysteresis system
Technical Field
The invention belongs to the technical field of multi-agent control, and relates to a distributed output feedback asymptotic consistent control method for a multi-agent hysteresis system.
Background
In recent decades, due to the successful application of the multi-agent coordination control technology in the related fields of cluster control, such as unmanned aerial vehicle group control, underwater cooperative work, robot group formation, satellite cluster control and the like, the multi-agent coordination control problem gradually becomes the key point of wide attention of students. The multi-agent coordination control problem mainly comprises consistency, aggregation, bee crowding, clustering and the like, wherein the consistency problem shows huge engineering application potential in the field of coordination control and is a key for researching the multi-agent coordination control problem. Furthermore, in a multi-agent coordinated control system, the designed controller needs not only the information of the current agent but also the information of the neighboring multi-agents, which makes the design of the distributed controller more challenging. Recently, the first document (Yang R, Zhang H, Feng G, et al, bus cooperative output adjustment of multi-agent system via adaptive event-triggered control [ J ] automation, 2019,102: 129-. On the basis of the first document, a second document (Zhang H, Chen J, Wang Z, et al. distributed Event-Triggered Control for Cooperative Output Regulation of multiple Systems With an Online Estimation Algorithm [ J ]. IEEE transactions on cybernetics,2020.) assumes that only a few system matrices of external Systems of an agent are known and topology information of each agent is unknown, and further designs a novel distributed observer to estimate information related to the topology, thereby realizing the asymptotic Cooperative Control of a multi-agent system. However, the actual physical system often has nonlinear characteristics, and the research on the multi-agent system with nonlinear dynamics is beneficial to better describing the system characteristics so as to achieve better control effect. The problem of distributed consistency control of a second-order nonlinear multi-agent system with dead zone input is researched in the third document (Shen Q, Shi P. output controls of multiple Systems with unknown nonlinear device zone [ J ]. IEEE Transactions on Systems, Man, and Cybernetics: Systems,2015,46(10): 1329-. The problem of distributed consistency control of a random nonlinear multi-agent system with dead zone input is studied in the fourth (Hua C, Zhang L, Guan X. distributed adaptive neural network output tracking of leader-following high-order stored nonlinear multiple agent system with unknown dead zone input [ J ]. IEEE transactions on cybernetics,2015,47(1):177-185.), and the result of actual tracking is obtained under the condition of known state. The fifth document (Wu Z, Wu Y, Yue D. distributed Adaptive tracking Control with MIMO stored nonlinear multiple systems with activator failure and unknown devices [ J ]. International Journal of Adaptive Control and Signal Processing,2018,32(12):1694 1714.) further expands the application range of the system, considers the case of actuator failure, and the designed controller can ensure that the system tracking error is within the preset performance limit range.
In fact, there is a large amount of unknown internal state information of the physical system, and it is usually necessary to construct a state observer to estimate its internal state information and to achieve the control goal by designing an adaptive output feedback controller. In the case of unknown system state, the sixth document (Wang C, Wen C, Wang W, et al. output-feedback adaptive control for a class of high-order non-multi-agent systems 2017,27(18): 4931-4-4948.) studies the problem of consistency control of a class of high-order non-linear multi-agent systems with quantized inputs, and the interaction between agents caused by unknown non-linearity is effectively counteracted with the construction of a high-gain K filter and the introduction of a smoothing function. Based on the sixth document, the seventh document (Li Y, Park J H, Wu L, et al. distributed Output-Feedback Adaptive Fuzzy Leader-Following Consenssens of stored Nonlinear Interconnected Multiagent Systems [ J ]. IEEE Transactions on Systems, Man, and Cybernetics: Systems,2020.) further studies the consistency control problem of the high-order Stochastic Nonlinear multi-agent system with unknown control gain, and proposes a distributed Output Feedback Adaptive consistency Fuzzy control scheme. However, it is not easy to find that the multi-agent system consistency control schemes proposed in the documents six and seven adopt the traditional inverse method to design the controller, which needs to repeatedly differentiate the virtual control rate in the design process to cause the phenomenon of 'differential explosion'. To overcome this deficiency, document eight (Hua C, Liu S, Li Y, et al, distributed adaptive feedback feeder-following control for nonlinear multiple agent Systems [ J ]. IEEE Transactions on Systems, Man, and Cybernetics: Systems,2018.) further studies a class of distributed output feedback consistency control problem of high-order nonlinear multi-agent Systems with unknown parameters and nonlinear terms, and the designed adaptive dynamic surface controller can avoid the problem of "differential explosion" and reduce the computational burden of the system, but can only make the tracking error converge into the neighborhood of arbitrarily small zero but not to zero.
On the other hand, in practical engineering applications, the actuator of the multi-agent system often generates hysteresis due to the use of intelligent materials, which seriously affects the tracking control performance of the system. Therefore, it is a research hotspot to eliminate the hysteresis effect of the multi-agent system actuator to ensure the stability of the system and further achieve good tracking control performance. In the prior art, two methods are commonly used to deal with hysteresis, one of which is to design a smooth adaptive hysteresis inverse to mitigate the effect of hysteresis effects (Liu Z, Lai G, Zhang Y, et al. adaptive neural output feedback control of output-constrained non-linear systems with unknown non-linear [ J ]. IEEE transactions on neural networks and hysteresis systems 2015,26(8): 1789-. Recently, document nine (Zhou Q, Wang W, Ma H, et al. event-triggered Fuzzy adaptive control for nonlinear multi-agent Systems with unknown bouu-Bouc-Wen hysteresis input [ J ]. IEEE Transactions on Fuzzy Systems,2019.) proposed an event-triggered Fuzzy inversion control scheme for stochastic nonlinear multi-agent hysteresis Systems that provides a controller that is conditionally updated only at sampling instants. When the system state is not measurable, the document ten (Wang J, Chen K, Liu Q, et al, observer-based adaptive control for Nonlinear multi-agent systems with actuator hysteresis [ J ]. Nonlinear Dynamics,2019,95(3):2181-2195.) further provides a distributed output feedback neural network control scheme based on a state observer, which can effectively eliminate the adverse effects of the hysteresis input and the external disturbance on the control system. In addition to steady-state tracking performance, the eleventh document (Wu Y, Yue d. predefined performance global stable adaptive passive control of stored multi-agent Systems with dynamics inputs and nonlinearities dynamics [ J ]. International Journal of Systems Science,2018,49(16):3431-3447.) further considers the transient tracking performance of multi-agent Systems, which utilizes a radial basis neural network in combination with specific Nussbaum-type functions to solve the problem of unknown control direction, and the proposed inversion control scheme not only eliminates hysteresis of the system actuator, but also guarantees the preset performance of the consistency errors of followers in the system. However, although many efforts have been made to study the problem of consistency control of nonlinear multi-agent hysteresis systems, the problem of asymptotic consistency control of the dynamic surface of higher order nonlinear multi-agent systems with hysteresis inputs is still less difficult to study, the difficulty being the construction of new filters and the design of controllers capable of compensating for the effects of hysteresis.
Therefore, it is very significant to develop a distributed adaptive output feedback leader-follower asymptotic control method for a nonlinear multi-agent system with hysteresis input and unknown state.
Disclosure of Invention
The invention aims to overcome the defect that the prior art cannot solve the problem of the gradual consistent control of the dynamic surface of a high-order nonlinear multi-agent system with hysteresis input, and provides a distributed self-adaptive output feedback leader-follower gradual consistent control method for a nonlinear multi-agent system with hysteresis input and unknown state.
In order to achieve the purpose, the invention provides the following technical scheme:
a distributed output feedback asymptotic consistent control method of a multi-agent hysteresis system is applied to electronic equipment and comprises the following steps:
(1) acquiring a reference signal track and an actual response track of each follower subsystem in the multi-agent system in real time, and tracking consistency errors of the reference signal track and the actual track in real time;
(2) inputting consistency errors of a reference signal track and an actual track into an adaptive dynamic surface controller to process derivatives of a virtual control rate, wherein the dynamic surface controller comprises a nonlinear filter;
(3) the actual control signal output by the dynamic surface controller is used as the input of a hysteresis model, and the nonlinear multi-agent system is controlled according to the output of the hysteresis model;
(5) applying a state observer with dynamic high gain to perform online real-time estimation on the unknown state of the high-order multi-agent system containing nonlinear items linearly related to the unknown state;
(6) and inputting the estimated state estimation value of the multi-agent system into a dynamic surface controller, and updating the parameter estimated value on line by combining the self-adaptive rate of the parameter estimation in the controller, thereby realizing the distributed output feedback asymptotic consistent control of the multi-agent system.
The invention solves the problem of high-precision tracking control of a multi-agent industrial system with an intelligent material actuator under the condition of measurable input and output by researching distributed self-adaptive output feedback leader-follower asymptotic tracking control of a nonlinear multi-agent hysteresis system with unknown state. Compared with the existing output feedback control system, the control system researched by the invention not only has unknown parameters, but also has nonlinear terms linearly related to unknown states, and considers the influence of hysteresis input on the tracking control performance of the system. Therefore, the system model considered by the invention is wider and more general. The invention constructs an improved dynamic high-gain K-filter (state observer) to estimate the unknown state of a nonlinear multi-agent system, which is able to handle nonlinear terms in the system that are linearly related to the unknown state and is asymptotically stable compared to conventional K-filters. In addition, the invention designs a novel nonlinear filter (dynamic surface controller) with a timing variable integral function, and the filter not only can solve the problem of differential explosion and reduce the calculation load, but also can compensate boundary layer errors existing in the traditional dynamic surface control scheme, so that the tracking error is gradually converged to zero. In theory, the invention can promote the research of distributed output feedback consistent control of a high-order nonlinear multi-agent system in the control field, and in fact, the research result of the invention can be applied to practical engineering application to improve the control performance of the nonlinear multi-agent system.
The present application utilizes graph-theoretic knowledge to solve the control problem of multi-agent systems. First, a directed graph is defined to indicate information transmission conditions between sub-system groups
Figure BDA0003095103760000061
Wherein:
Figure BDA0003095103760000062
is a set of nodes;
Figure BDA0003095103760000063
is a set of edges;
Figure BDA0003095103760000064
is the adjacency matrix of the directed graph G. Assuming that there is no self-circulation in the set graph G, i.e.
Figure BDA0003095103760000065
Furthermore, in the adjacency matrix, if node j can directly transmit information to node i, then
Figure BDA0003095103760000066
If not, then,
Figure BDA0003095103760000067
then, defining the in-degree matrix of the directed graph G as
Figure BDA0003095103760000068
Wherein:
Figure BDA0003095103760000069
indicating the in-degree of node i. The Laplace matrix of the directed graph G is defined as
Figure BDA00030951037600000610
In addition, for a multi-agent system with a leader and a follower,
Figure BDA00030951037600000611
output information y representing that the ith follower can access the leaderrAnd if not, the step (B),
Figure BDA00030951037600000612
in the directed graph G, if there is a directed path connection from a certain node to any other node, it is said that there is a directed spanning tree emanating from the node in the directed graph G.
Consider a series of higher-order nonlinear multi-agent systems (i.e., the multi-agent system of the present invention) with hysteresis inputs, unknown parameters, and nonlinear terms, the system comprising N follower subsystems of order N, as follows:
Figure BDA0003095103760000071
Figure BDA0003095103760000072
wherein:
Figure BDA0003095103760000073
and yiE is the state vector and the output of the ith follower subsystem of the system respectively;
Figure BDA0003095103760000074
are all known nonlinear smooth vector functions; a isi,kIs an unknown real constant;
Figure BDA0003095103760000075
is a known lower triangular nonlinear smooth function matrix;
Figure BDA0003095103760000076
and is
Figure BDA0003095103760000077
Is an unknown non-zero constant; rhoi＝ni-mi>1；uiE.R is the input of a hysteresis model; hi(. epsilon. R) is the hysteresis model output. The model used herein to describe hysteresis is the Bouc-Wen model, which is mathematically defined as follows:
Figure BDA0003095103760000078
wherein: 0 < epsiloni< 1 is the stiffness ratio;
Figure BDA0003095103760000079
is a positive parameter related to the non-linear pseudo-natural frequency; xiiIs an auxiliary variable whose derivative is
Figure BDA00030951037600000710
Wherein:
Figure BDA00030951037600000712
constants describing the hysteresis shape and amplitude, respectively; lambda [ alpha ]i≧ 1 controls the transition smoothness of the hysteresis curve from the initial slope to the asymptote slope.
The input-output relationship of the Bouc-Wen hysteresis model is shown in FIG. 1, wherein: xii(0)＝0，ui(t)＝5sin2t，εi＝0.33，
Figure BDA00030951037600000711
Figure BDA00030951037600000713
λi＝2，φi＝0.5。
To achieve the control objective for a nonlinear multi-agent system (1), the following assumptions and reasoning are given:
assume that 1: leader's output trajectory yrAnd
Figure BDA0003095103760000081
is a known and bounded smooth function,
Figure BDA0003095103760000082
wherein: omega0Which is a known tight set, will be defined at the time of stability analysis.
Assume 2: without loss of generality, unknown constants are assumed
Figure BDA0003095103760000083
Is greater than 0, and
Figure BDA0003095103760000084
is a hervitz polynomial.
Assume that 3: for a directed graph G consisting of a non-linear multi-agent system (1), there is at least one directed spanning tree emanating from the leader.
Assume 4: there is a bounded positive function sigma that is sufficiently smooth and integrablei(t) satisfies
Figure BDA0003095103760000085
Figure BDA0003095103760000086
Wherein: tau isiIs a time constant; sigmai,1And σi,2Are all normal numbers.
Positive time-varying integral function sigmai(t) plays a crucial role in analyzing the stability of the dynamic surface control system and can be selected as
Figure BDA0003095103760000087
Wherein:
Figure BDA0003095103760000088
introduction 1: for any R >0 and x ∈ R, satisfy
Figure BDA0003095103760000089
2, leading: there is a symmetrical positive definite matrix PiAnd normal number ai,
Figure BDA00030951037600000810
And
Figure BDA00030951037600000811
satisfy the requirement of
Figure BDA00030951037600000812
Wherein: di＝diag{0,1,…,ni-1}。
Due to PiIs a symmetric positive definite matrix, then only a large enough a needs to be selectediThe expression (8) can be established.
And 3, introduction:for arbitrary piecewise continuous signals uiAnd
Figure BDA00030951037600000813
the solution set of differential equation (4) satisfies
Figure BDA0003095103760000091
Wherein: xii(0) Is the initial value condition; max {. cndot.) represents the maximum value.
By designing the distributed self-adaptive dynamic surface controller, the output track of the follower is consistent with the output track of the leader, the tracking error is gradually converged to zero, and meanwhile, all signals in the closed-loop control system are guaranteed to be semi-globally consistent and finally bounded.
In addition, the system model considered by the application is wider and more general. According to equations (1) and (3), the nonlinear system under study not only involves unknown parameters, but also considers nonlinear terms that are linearly related to unknown states,
Figure BDA0003095103760000092
from the formula (10): non-linear term Fi(yi)xiRepresenting an unknown state xiAnd a non-linear function Fi(yi) The product of, however, existing output feedback control schemes only consider the output yiAnd a non-linear function
Figure BDA0003095103760000093
Thus, existing output feedback control schemes cannot be used to deal with our system, and designing a state observer that can handle nonlinear terms that are linearly related to unknown states is more difficult and challenging.
Compared with the existing multi-agent control scheme, the control scheme provided by the application adopts a dynamic surface reverse-thrust technology to design the controller, and the controller only needs to know that the follower outputs yiLeader output yrAnd its first derivative
Figure BDA0003095103760000094
Furthermore, the non-linear multi-agent system (1) may be heterogeneous.
State estimation i.e. (applying a state observer with dynamic high gain):
in this section, to estimate the unknown state of the nonlinear multi-agent system (1), a series of dynamic high-gain K-filters are designed as follows:
Figure BDA0003095103760000095
wherein: i ∈ [1, …, N]；
Figure BDA0003095103760000101
Figure BDA0003095103760000102
liIs the dynamic gain of the filter, and
Figure BDA0003095103760000103
in addition, the dynamic gain liGiven by the differential equation:
Figure BDA0003095103760000104
wherein: li≥li(0)；κiIs a positive design parameter; psii(yi) A non-negative smooth function;
Figure BDA0003095103760000105
δi,1,1and deltaq,1,2Will be defined in subsequent design steps. Then, an estimate of the unknown state of the multi-agent system (1) is represented as
Figure BDA0003095103760000106
Thus, the error between the actual state of the system and the state estimate is
Figure BDA0003095103760000107
The derivative of which is
Figure BDA0003095103760000108
For convenience of analysis, note
Figure BDA0003095103760000109
Then, coordinate transformation is performed:
Figure BDA00030951037600001010
wherein: a isiIs a normal number, which has been given in (8). According to formula (15), to
Figure BDA00030951037600001011
Derivative to obtain
Figure BDA00030951037600001012
In addition, a suitable parameter q is selectediTo make
Figure BDA00030951037600001013
For a Helvelz matrix, then there must be a positive definite symmetric matrix PiSatisfy the following requirements
Figure BDA0003095103760000111
And (4) introduction: for the error of the state observer, a Lyapunov function is selected as
Figure BDA0003095103760000112
Then, based on the formulas (8), (12), (16) and (17), the appropriate positive design parameter κ is selectediAnd a non-negative smoothing function psii(yi) To obtain
Figure BDA0003095103760000113
Wherein:
Figure BDA0003095103760000114
representing the euclidean norm.
According to the equation [19], the designed dynamic high-gain K-filter is asymptotically stable, and the error between the estimated value of the measured state and the real state of the filter can be converged to zero.
The design of the novel dynamic surface controller:
the design steps are as follows:
first, the following coordinate transformation is performed:
Figure BDA0003095103760000115
wherein: j 2, …, ρi；zi,1Is the consistency error of the ith follower;
Figure BDA0003095103760000116
and
Figure BDA0003095103760000117
are design parameters related to the communication network, which have been given in the graph theory section; alpha is alphai,j-1And
Figure BDA0003095103760000118
respectively is the virtual control rate of the ith follower before and after filtering in the step (j-1); si,j-1Is the boundary layer error filtered by the ith follower in the step (j-1);
Figure BDA0003095103760000119
is a vector
Figure BDA00030951037600001110
Has been given by equation (11).
Step 1(i ═ 1, …, N): according to formulae (1), (14) and (20), zi,1Can be expressed as
Figure BDA0003095103760000121
Wherein:
Figure BDA0003095103760000122
then, to facilitate subsequent design, definitions are made
Figure BDA0003095103760000123
Wherein:
Figure BDA0003095103760000124
is an unknown non-zero constant which is given in equation (2). Then, the virtual control rate α of the step is selectedi,1Is composed of
Figure BDA0003095103760000125
Wherein: c. Ci,1,δi,1,1,δi,1,2,δi,1,3,δi,1,4And deltai,1,5Are all positive design parameters;
Figure BDA0003095103760000126
are each thetai,θi,q,pi,χiThe parameter estimation of (2). Then, select
Figure BDA0003095103760000127
Respectively, of
Figure BDA0003095103760000131
Wherein:
Figure BDA0003095103760000132
σi(t) is a positive time-varying integration function, which has been given in hypothesis 4;
Figure BDA0003095103760000133
and
Figure BDA0003095103760000134
are all positive design parameters. To avoid the problem of differential explosion which cannot be avoided by the traditional back-pushing method, alpha is usedi,1Passing through a novel nonlinear filter as follows:
Figure BDA0003095103760000135
Figure BDA0003095103760000136
wherein: tau isi,1Is a time constant;
Figure BDA0003095103760000137
is Mi,1Estimation of, Mi,1Will be defined below; sigmai(t) is a positive time-varying integration function, which has been given in hypothesis 4;
Figure BDA0003095103760000138
is a positive design parameter;
Figure BDA0003095103760000139
is the boundary layer error of this step, which leads toNumber is
Figure BDA00030951037600001310
Wherein:
Figure BDA00030951037600001311
is about zi,1,ζi,0,2,ζi,k,2,ζq,0,2,ζq,k,2,li,
Figure BDA00030951037600001312
yr,
Figure BDA00030951037600001313
And σi(t) a smooth continuous function, wherein: k is 1, …, ri. In addition, in tight set omega0×Ω1In the presence of a normal number Mi,1≥|Bi,1(·) |, wherein: omega0And Ω1Will be defined in the stability analysis section. Then, the Lyapunov function of the step is selected as:
Figure BDA0003095103760000141
wherein:
Figure BDA0003095103760000142
an error is estimated for the parameter. Then, according to theorem 1 and hypothesis 4, get
Figure BDA0003095103760000143
Combining the formulas (22) and (23) to obtain
Figure BDA0003095103760000144
Wherein: q is 1, …, N. Then, by using the Young's inequality in combination with the equation (23), it is possible to derive
Figure BDA0003095103760000145
Wherein: q is 1, …, N; deltai,1,1,δi,1,2,δi,1,3,δi,1,4And deltai,1,5Are positive design parameters, which are given in equation (24). Then, the formula (29) is differentiated by the formula (32) in combination with the formulae (21) to (31) to obtain
Figure BDA0003095103760000146
Step j (j-2, …, ρ)i-1, i ═ 1, …, N): according to formulae (1) and (20), zi,jIs a derivative of
Figure BDA0003095103760000151
When j is 2, selecting the virtual control rate alpha of the stepi,2Is composed of
Figure BDA0003095103760000152
When j is more than or equal to 3 and less than or equal to rhoiWhen the virtual control rate alpha is 1, the virtual control rate alpha of the step j is selectedi,jIs composed of
Figure BDA0003095103760000153
Wherein: c. Ci,jIs a positive design parameter, j 2, …, pi-1. Then, to avoid the problem of "differential explosion" that cannot be avoided by the conventional back-pushing method, α is madei,jPassing through a novel nonlinear filter as follows:
Figure BDA0003095103760000154
Figure BDA0003095103760000155
wherein: tau isi,jIs a time constant;
Figure BDA0003095103760000156
is Mi,jEstimation of, Mi,jWill be defined below; sigmai(t) is a positive time-varying integration function, which has been given in hypothesis 4;
Figure BDA0003095103760000157
is a positive design parameter;
Figure BDA0003095103760000158
is the boundary layer error of step j, the derivative of which is
Figure BDA0003095103760000159
Wherein:
Figure BDA00030951037600001510
is about
Figure BDA00030951037600001511
And
Figure BDA00030951037600001512
is a smooth continuous function. In addition, in tight set omega0×ΩiIn (1), there is a normal number Mi,j≥|Bi,j(·) |, wherein: omegaiWill be defined in the stability analysis section. Then, the Lyapunov function of step j is selected as
Figure BDA00030951037600001513
Wherein:
Figure BDA0003095103760000161
an error is estimated for the parameter. Then, according to the theorem 1, obtain
Figure BDA0003095103760000162
Wherein: sigmai(t) is a positive time-varying integration function, which has been defined in hypothesis 4. Then, the formula (40) is derived from the formulas (34) to (41) to obtain
Figure BDA0003095103760000163
Step pi(i ═ 1, …, N): according to formulae (1), (3) and (20),
Figure BDA0003095103760000164
is a derivative of
Figure BDA0003095103760000165
Wherein:
Figure BDA0003095103760000166
according to the introduction 3, etaiIs bounded, let us say | ηiThe upper bound of | is
Figure BDA0003095103760000167
Then, the actual control signal and the adaptive rate are respectively selected as
Figure BDA0003095103760000168
Figure BDA0003095103760000169
Wherein:
Figure BDA00030951037600001610
are all positive design parameters;
Figure BDA00030951037600001611
Figure BDA00030951037600001612
is that
Figure BDA00030951037600001613
The parameter estimation of (2). Then, the Lyapunov function of the step is selected as
Figure BDA00030951037600001614
Wherein:
Figure BDA00030951037600001615
an error is estimated for the parameter. According to the lemma 1, the following inequality can be obtained:
Figure BDA0003095103760000171
wherein: sigmai(t) is a positive time-varying integration function, which has been defined in hypothesis 4. Then, the formula (46) is derived according to the formulas (43) to (45) to obtain
Figure BDA0003095103760000172
And (3) stability analysis:
consider a higher order nonlinear multi-agent system (1) with a hysteresis input (3), a state observer (11), novel nonlinear filters (26) and (37), virtual control rates (24), (35), (36), a controller (44), and adaptation rates (25), (27), (38), and (45) forming a closed loop system. Under the condition of assuming 1-4, if V (0) is less than or equal to R1Then by selecting appropriate design parameters
Figure BDA0003095103760000173
The signals in the closed loop system can be made to be semi-globally consistent and finally bounded and the tracking error can be asymptotically converged to zero.
First, to facilitate the stability performance analysis, let
Figure BDA0003095103760000174
Wherein: i is 1, …, N. Then, a bounded tight set is defined as follows:
Figure BDA0003095103760000175
wherein: 1, …, N; q is 1, …, N; r0,R1Is a normal number. Then, the Lyapunov function of the whole system is selected as
Figure BDA0003095103760000181
And according to equations (19) and (48), the derivative thereof can be obtained as
Figure BDA0003095103760000182
Then, the two sides of the equation (52) are integrated at the same time at [0, t ] to obtain
Figure BDA0003095103760000183
According to formula (53), let
Figure BDA0003095103760000184
And combine hypothesis 4 to obtain
Figure BDA0003095103760000185
From which it can be easily deduced
Figure BDA0003095103760000186
Are bounded signals. Further, from the expressions (11), (12) and (20), it can be seen that
Figure BDA0003095103760000187
Are also bounded signals. Then, based on the formula (44), u is knowniIs also bounded. To this end, all signals in a closed loop system are consistent and ultimately bounded. Further, according to the formula (54), the
Figure BDA0003095103760000188
And combining the barbalt theorem to obtain
Figure BDA0003095103760000189
I.e. asymptotic convergence of the consistency error is achieved. Then, according to the knowledge of graph theory and based on the communication network of the system, selecting proper design parameters
Figure BDA00030951037600001810
And
Figure BDA00030951037600001811
can make the output track y of each followeriAsymptotically tracking the output trajectory y of the upper leaderrAnd the tracking error can converge to zero.
Compared with a self-adaptive asymptotic tracking controller designed by a reverse-extrapolation method adopted by the existing asymptotic tracking control scheme, the self-adaptive dynamic surface control scheme provided by the application overcomes the problem of 'differential explosion' caused by repeated differentiation of a virtual control rate, which cannot be avoided by the traditional reverse-extrapolation method, and greatly reduces the design difficulty and the calculation burden of the controller, so that the asymptotic tracking control scheme provided by the application is simpler and more practical.
In the self-adaptive dynamic surface control scheme provided by the application, a novel nonlinear filter is designed, and each unknown parameter is estimated on line through a self-adaptive law, so that an unknown nonlinear function B caused by boundary layer errors in the traditional dynamic surface control is subjected to online estimationi,jThe effects of (c) are fully compensated for, thereby enabling asymptotic tracking of the control system.
According to the above stability conditions, the
Figure BDA0003095103760000191
Figure BDA0003095103760000192
Greater than zero and satisfy
Figure BDA0003095103760000193
All signals in the whole closed loop system can be enabled to realize semi-global consistency and finally be bounded. Then, using initialization technique to make parameter adjustment analysis to let zi,j＝0,i＝1,…,N,j＝1,…,ρi. From the formulae (26) and (37), it can be easily understood
Figure BDA0003095103760000194
Wherein: i 1, …, N, j 1, …, ρi-1. From the formula (55), a
Figure BDA0003095103760000195
Wherein:
Figure BDA0003095103760000196
Figure BDA0003095103760000197
as is clear from the expressions (57), (58) and (59), the following design parameter c is increasedi,1,
Figure BDA0003095103760000201
Transient tracking performance of the whole closed-loop control system can be effectively improved, wherein: 1, …, N, j 1, pi-1,q＝1,…,N。
The invention also provides an electronic device comprising one or more processors, one or more memories, one or more programs, and an information acquisition device for acquiring the reference signal trajectory and the actual trajectory of the multi-agent system;
the one or more programs are stored in the memory and, when executed by the processor, cause the electronic device to perform a multi-agent hysteresis system distributed output feedback asymptotic consistency control method as described above.
Has the advantages that:
(1) the distributed output feedback asymptotic consistent control method of the multi-agent hysteresis system, disclosed by the invention, designs an asymptotic stable dynamic high-gain K-filter to estimate the unknown state of the nonlinear multi-agent system, and can process nonlinear items linearly related to the unknown state in the system;
(2) the distributed output feedback asymptotic consistent control method of the multi-agent hysteresis system designs a novel nonlinear filter with a timing variable integral function, and the filter not only can avoid the problem of differential explosion which cannot be avoided by the traditional backstepping method, but also can compensate the boundary layer error of the traditional dynamic surface. Theoretical analysis shows that the proposed control scheme can effectively eliminate the influence of hysteresis input on the tracking performance of the system, ensure the stability of a closed-loop system and realize the asymptotic convergence of the tracking error of the system;
(3) the distributed output feedback asymptotic consistent control method of the multi-agent hysteresis system can effectively eliminate the hysteresis effect of the actuating mechanism and realize good tracking control performance, and has great application prospect.
Drawings
FIG. 1 is a schematic diagram of the input-output relationship of the Bouc-Wen hysteresis model;
FIG. 2 is a schematic flow chart of a distributed output feedback asymptotic control method for a multi-agent hysteresis system according to the present invention;
fig. 3 is a schematic structural diagram of a communication network according to embodiment 1;
FIGS. 4-7 are schematic diagrams of a multi-agent system output signal, a consistency error signal, an actual control signal and a status signal under the multi-agent hysteresis system distributed output feedback asymptotic consistency control method of embodiment 1, respectively;
FIGS. 8-9 are schematic diagrams of the multi-agent system output signal and the consistency error signal under the control method of comparative example 1.
Detailed Description
The present invention will be described in more detail with reference to the accompanying drawings, in which embodiments of the invention are shown and described, and it is to be understood that the embodiments described are merely illustrative of some, but not all embodiments of the invention.
Example 1
A distributed output feedback asymptotic consistent control method of a multi-agent hysteresis system is applied to electronic equipment and comprises the following steps (the flow chart is shown in figure 2):
(1) acquiring a reference signal track and an actual response track of each follower subsystem in the multi-agent system in real time, and tracking consistency errors of the reference signal track and the actual track in real time;
(2) inputting the consistency error of the reference signal track and the actual track into a dynamic surface controller to process the derivative of the virtual control rate, wherein the dynamic surface controller comprises a nonlinear filter, and the mathematical model of the nonlinear filter is as follows:
Figure BDA0003095103760000211
Figure BDA0003095103760000212
Figure BDA0003095103760000213
Figure BDA0003095103760000214
wherein: i is more than or equal to 1 and less than or equal to N, and j is more than or equal to 1 and less than or equal to rhoi，αi,jAnd
Figure BDA0003095103760000215
respectively the virtual control rates before and after the ith follower subsystem step j is filtered, N is the total number of the follower subsystems of the multi-agent system,
Figure BDA0003095103760000216
is composed of
Figure BDA0003095103760000217
Derivative of, τi,jIs the time constant of the time at which,
Figure BDA0003095103760000221
is Mi,jIs estimated by the estimation of (a) a,
Figure BDA0003095103760000222
is a positive design parameter, si,jIs the boundary layer error of step j,
Figure BDA0003095103760000223
is s isi,jThe derivative of (a) of (b),
Figure BDA0003095103760000224
is about
Figure BDA0003095103760000225
And
Figure BDA0003095103760000226
of a smooth continuous function, σi(t) is a time-varying integral function, zi,jIs the consistency error of the ith follower subsystem step j;
(3) using the actual control signal output by the dynamic surface controller as the input of a hysteresis model (Bouc-Wen model), and controlling the multi-agent system according to the output of the hysteresis model;
(4) the method comprises the following steps of performing online real-time estimation on an unknown state of a multi-agent system by using a state observer with dynamic high gain, wherein the state observer is a dynamic high gain K-filter, and the formula is as follows:
Figure BDA0003095103760000227
Figure BDA0003095103760000228
Figure BDA0003095103760000229
Figure BDA00030951037600002210
Figure BDA00030951037600002211
Figure BDA00030951037600002212
Figure BDA00030951037600002213
wherein: i ∈ [1, …, N]，
Figure BDA00030951037600002214
Figure BDA00030951037600002215
ζi,0、ζi,kAnd vi,jIs a state observerA state variable of liIs the dynamic gain of the filter,/i≥li(0)，LiTo set a vector and
Figure BDA00030951037600002216
κiis a positive design parameter, #i(yi) A non-negative smooth function, R is a real number set,
Figure BDA00030951037600002217
a set of n-dimensional real numbers representing the ith follower sub-system,
Figure BDA00030951037600002218
are all known non-linear smooth vector functions,
Figure BDA0003095103760000231
is a known lower triangular nonlinear smooth function matrix, xi、yiRespectively the state vector sum output, u, of the ith follower subsystemi、Hi() input and output of the hysteresis model, respectively;
(5) and inputting the estimated state estimation value of the multi-agent system into a dynamic surface controller, and updating the parameter estimated value on line by combining the self-adaptive rate of the parameter estimation in the controller, thereby realizing the distributed output feedback asymptotic consistent control of the multi-agent system.
The above multi-agent system is specifically a non-linear multi-agent system with hysteresis input, unknown parameters and non-linear terms:
Figure BDA0003095103760000232
Figure BDA0003095103760000233
Figure BDA0003095103760000234
wherein: bi,0＝1,ai,1＝1,
Figure BDA0003095103760000235
εi＝0.5,
Figure BDA0003095103760000236
Figure BDA0003095103760000239
λi＝3,φi0.5, i is 1,2, 3. The communication network of the system is shown in fig. 3, where node 0 corresponds to the leader system and the other nodes correspond to the follower systems.
Then, based on the communication network and according to the knowledge of graph theory, respectively selecting design parameters
Figure BDA0003095103760000237
Figure BDA0003095103760000238
As follows:
Figure BDA0003095103760000241
Figure BDA0003095103760000242
Figure BDA0003095103760000243
Figure BDA0003095103760000244
the control objective is to design an adaptive control rate u based on the control scheme proposed hereiniTo make the follower output the track yiCan asymptotically track leader inputGo out track yr＝0.5sint。
The dynamic high-gain K-filter is as follows:
Figure BDA0003095103760000245
wherein: i is 1,2,3, qi＝[2,1]TDynamic gain liGiven by the differential equation:
Figure BDA0003095103760000246
wherein:
Figure BDA0003095103760000247
then, according to the control scheme proposed herein, the virtual control rate and the actual control rate are selected as follows:
Figure BDA0003095103760000248
Figure BDA0003095103760000249
wherein: c. Ci,1＝ci,2＝20,δi,1,1＝δi,1,2＝δi,1,3＝δi,1,4＝δi,1,5＝2，ai＝1,
Figure BDA00030951037600002410
τi,1＝0.05,
Figure BDA00030951037600002411
Parameter estimation
Figure BDA00030951037600002412
Respectively, of
Figure BDA0003095103760000251
Wherein:
Figure BDA0003095103760000252
Figure BDA0003095103760000253
then, when carrying out simulation, the initial state value of each follower is selected as x1,1(0)＝0.1,x1,2(0)＝0.6,x2,1(0)＝0.3,x2,2(0)＝0.4,x3,1(0)＝0.2,x3,2(0) 0.8. Then, the initial parameter estimation values of all follower controllers in the system are all selected to be 0. The simulation results are shown in fig. 4 to fig. 7, which respectively show the response curve diagrams of the system output signal, the consistency error signal, the actual control signal and the state signal of each follower of the nonlinear multi-agent system under the novel dynamic surface consistency control scheme proposed herein.
Comparative example 1
A consistent control method of a multi-agent hysteresis system is basically the same as that of the embodiment 1 (the leader output track, the design parameters of a controller, the initial state value and the like are unchanged), and is different from the adopted scheme, namely the scheme described in the literature (Hua C, Liu S, Li Y, et al.
Simulation results are shown in fig. 8-9, and system output tracks and consistency error response curves of the system under the traditional dynamic surface consistency control scheme are respectively given.
As can be seen from the simulation results of comparative example 1 and comparative example 1, compared with the conventional dynamic surface consistency control scheme [ comparative example 1], the dynamic surface consistency control scheme with the novel nonlinear filter proposed herein has better tracking performance, which can overcome the problem that the conventional dynamic surface consistency control scheme can only make the tracking error converge into the neighborhood of arbitrarily small zero but can not converge into zero. Therefore, the simulation results further verify the effectiveness and superiority of the control scheme proposed herein.
Example 2
An electronic device comprising one or more processors, one or more memories, one or more programs, and an information acquisition device to acquire a reference signal trajectory and an actual trajectory of a multi-agent system;
one or more programs are stored in the memory that, when executed by the processor, cause the electronic device to perform a multi-agent hysteresis system distributed output feedback asymptotic consistency control method as described in example 1.
Although specific embodiments of the present invention have been described above, it will be appreciated by those skilled in the art that these embodiments are merely illustrative and various changes or modifications may be made without departing from the principles and spirit of the invention.

Claims (5)

1. A distributed output feedback asymptotic consistent control method of a multi-agent hysteresis system is applied to electronic equipment and is characterized by comprising the following steps:
(1) acquiring a reference signal track and an actual response track of each follower subsystem in the multi-agent system in real time, and tracking consistency errors of the reference signal track and the actual track in real time;
(2) inputting the consistency errors of the reference signal track and the actual track into a dynamic surface controller to process the derivative of the virtual control rate, wherein the dynamic surface controller comprises a nonlinear filter;
(3) the actual control signal output by the dynamic surface controller is used as the input of a hysteresis model, and the multi-agent system is controlled according to the output of the hysteresis model;
(4) applying a state observer with dynamic high gain to carry out online real-time estimation on the unknown state of the multi-agent system;
(5) and inputting the estimated state estimation value of the multi-agent system into a dynamic surface controller, and updating the parameter estimated value on line by combining the self-adaptive rate of the parameter estimation in the controller, thereby realizing the distributed output feedback asymptotic consistent control of the multi-agent system.
2. A multi-agent hysteresis system distributed output feedback asymptotic consistency control method as recited in claim 1, wherein the mathematical model of the nonlinear filter is as follows:
Figure FDA0003095103750000011
Figure FDA0003095103750000012
Figure FDA0003095103750000013
Figure FDA0003095103750000014
wherein: i is more than or equal to 1 and less than or equal to N, and j is more than or equal to 1 and less than or equal to rhoi，αi,jAnd
Figure FDA0003095103750000015
respectively the virtual control rates before and after the ith follower subsystem step j is filtered, N is the total number of the follower subsystems of the multi-agent system,
Figure FDA0003095103750000016
is composed of
Figure FDA0003095103750000017
Derivative of, τi,jIs the time constant of the time at which,
Figure FDA0003095103750000018
is Mi,jIs estimated by the estimation of (a) a,
Figure FDA00030951037500000112
is a positive design parameter, si,jIs the boundary layer error of step j,
Figure FDA0003095103750000019
is s isi,jThe derivative of (a) of (b),
Figure FDA00030951037500000110
is about
Figure FDA00030951037500000113
And
Figure FDA00030951037500000111
of a smooth continuous function, σi(t) is a time-varying integral function, zi,jIs the consistency error of the ith follower subsystem step j.
3. The multi-agent hysteresis system distributed output feedback asymptotic consistency control method of claim 1, wherein the state observer is a dynamic high-gain K-filter, and the formula is as follows:
Figure FDA0003095103750000021
Figure FDA0003095103750000022
Figure FDA0003095103750000023
Figure FDA0003095103750000024
Figure FDA0003095103750000025
Figure FDA0003095103750000026
Figure FDA0003095103750000027
wherein: i ∈ [1, …, N]，
Figure FDA0003095103750000028
Figure FDA0003095103750000029
ζi，0、ζi,kAnd vi,jIs a state variable of a state observer,/iIs the dynamic gain of the filter,/i≥li(0)，LiTo set a vector and
Figure FDA00030951037500000210
κiis a positive design parameter, #i(yi) A non-negative smooth function, R is a real number set,
Figure FDA00030951037500000211
a set of n-dimensional real numbers representing the ith follower sub-system,
Figure FDA00030951037500000212
are all known non-linear smooth vector functions,
Figure FDA00030951037500000213
is a known lower triangular nonlinear smooth function matrix, xi、yiRespectively the state vector sum output, u, of the ith follower subsystemi、Hi(. cndot.) are the input and output, respectively, of the hysteresis model.
4. A multi-agent hysteresis system distributed output feedback asymptotic consistency control method as defined in claim 3, wherein the hysteresis model is a Bouc-Wen model.
5. An electronic device comprising one or more processors, one or more memories, one or more programs, and an information acquisition device to acquire a reference signal trajectory and an actual trajectory for a multi-agent system;
the one or more programs stored in the memory, when executed by the processor, cause the electronic device to perform a multi-agent hysteresis system distributed output feedback asymptotic control method as recited in any one of claims 1 to 4.
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