CN114296355A - Adaptive event trigger control method for system with dynamic anti-saturation compensator - Google Patents

Adaptive event trigger control method for system with dynamic anti-saturation compensator Download PDF

Info

Publication number
CN114296355A
CN114296355A CN202210000999.4A CN202210000999A CN114296355A CN 114296355 A CN114296355 A CN 114296355A CN 202210000999 A CN202210000999 A CN 202210000999A CN 114296355 A CN114296355 A CN 114296355A
Authority
CN
China
Prior art keywords
saturation
dynamic
adaptive event
matrix
dynamic anti
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Granted
Application number
CN202210000999.4A
Other languages
Chinese (zh)
Other versions
CN114296355B (en
Inventor
李红超
邓惠敏
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Hebei University of Technology
Original Assignee
Hebei University of Technology
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Hebei University of Technology filed Critical Hebei University of Technology
Priority to CN202210000999.4A priority Critical patent/CN114296355B/en
Publication of CN114296355A publication Critical patent/CN114296355A/en
Application granted granted Critical
Publication of CN114296355B publication Critical patent/CN114296355B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
    • Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
    • Y02P90/02—Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]

Abstract

The invention relates to a self-adaptive event triggering control method of a system with a dynamic anti-saturation compensator, which comprises the steps of firstly establishing a system model with the dynamic anti-saturation compensator, then designing a self-adaptive event triggering condition, and updating a signal received by a dynamic controller only when the self-adaptive event triggering condition is met; then, establishing a system model under the trigger control of the self-adaptive event, and designing the trigger condition of the self-adaptive event under the condition of setting a dynamic anti-saturation compensator to enable the saturation system to be gradually stable; under the condition that a dynamic anti-saturation compensator is not given, designing the dynamic anti-saturation compensator and an adaptive event triggering condition at the same time; finally, the Zeno phenomenon of the system can not occur under the designed self-adaptive event trigger control. The method simultaneously introduces a dynamic anti-saturation compensator and a self-adaptive event triggering mechanism into the system, ensures the asymptotic stability of the system, and can reduce the adverse effect of the saturation of the actuator on the system and save network communication resources.

Description

Adaptive event trigger control method for system with dynamic anti-saturation compensator
Technical Field
The invention belongs to the technical field of event trigger control design, and particularly relates to a self-adaptive event trigger control method for a system with a dynamic anti-saturation compensator.
Background
The saturation problem is one of the non-linear problems common in the actual system, the output of the actuator is often limited due to the physical condition limitation of the actuator, and if the actuator saturation is not considered in designing the controller, the system performance can be reduced, and even the system is unstable. At present, two methods for processing actuator saturation are mainly used, one is a direct method, namely, the influence of actuator saturation is directly considered when a controller is designed, so that a system works in a linear region of an actuator; the other is a two-step method which first ignores the saturation nonlinearity to design the controller and then adds an anti-saturation compensator to the system to reduce the adverse effect of the saturation nonlinearity on the saturated system. In addition, compared with a static anti-saturation compensator, the dynamic anti-saturation compensator has more coefficient matrixes, so that more degrees of freedom can be provided for the whole system, and better control performance is realized.
The event trigger mechanism has the advantage of saving network communication resources, and attracts people's extensive attention. Specifically, in event-triggered control, a control task is executed once a designed event-triggered condition is satisfied. The self-adaptive event triggering mechanism comprises an auxiliary dynamic variable meeting a certain differential equation, can achieve the aim of saving communication resources by adjusting the triggering threshold value on line, and is more effective than the traditional event triggering mechanism. Therefore, the adaptive event triggering mechanism can more flexibly and effectively reduce network communication resources, thereby improving the system performance.
The anti-saturation compensator is introduced into the saturation system based on the event trigger mechanism, so that not only can network communication resources be saved, but also the adverse effect of actuator saturation on the system can be greatly reduced.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention aims to solve the technical problem of providing an adaptive event triggering control method containing a dynamic anti-saturation compensator system.
The technical scheme adopted by the invention for solving the technical problems is as follows:
an adaptive event-triggered control method for a system including a dynamic anti-saturation compensator, the method comprising the steps of:
the method comprises the following steps that firstly, a system model containing a dynamic anti-saturation compensator is established, and the system model contains a continuous system with actuator saturation, a dynamic controller and the dynamic anti-saturation compensator;
secondly, designing a self-adaptive event triggering condition as follows:
e(t)TΦ1e(t)≤δ1(t)yp(t)TΦ2yp(t)+δ1(t)∈e-∈t (5)
where e (t) is dynamic error, e (t) yp(tk)-yp(t)，yp(tk) Indicating the transmission time tkData successfully transmitted to the dynamic controller, yp(t) is the output vector of the controlled object,. phi1、Φ2Is a free weight matrix, epsilon is a positive scalar quantity, and T represents matrix transposition; delta1(t) is an adaptive event trigger threshold, and the condition is satisfied: delta1(t)＝max{δ,η(t)}，δ∈(0,1]Represents the lower bound of the adaptive event trigger threshold, η (t) represents the threshold function, the initial value η (0) > 0 and η (t) satisfies equation (6):
Figure BDA0003454063500000011
wherein the content of the first and second substances,
Figure BDA0003454063500000012
representing the first derivative of a threshold function eta (t), theta representing an adjustment parameter of the convergence rate of the threshold function, theta > 1;
thirdly, establishing a system model under the trigger control of the self-adaptive event;
fourthly, under the condition of setting a dynamic anti-saturation compensator, designing a self-adaptive event triggering condition to ensure that the system is gradually stable; constructing an optimization problem, and maximizing an estimation attraction domain of a system by solving the optimization problem;
when delta < eta (t), delta1Substituting equation (5) to obtain the adaptive event triggering condition: e (t)TΦ1e(t)≤η(t)yp(t)TΦ2yp(t)+η(t)∈e-∈t；
When delta is greater than or equal to eta (t), delta1When (t) is δ, the adaptive event triggering condition obtained by substituting formula (5) is as follows: e.g. of the type(t)TΦ1e(t)≤δyp(t)TΦ2yp(t)+δ∈e-∈t；
Fifthly, under the condition that a dynamic anti-saturation compensator is not given, designing an anti-saturation compensator and a self-adaptive event triggering condition simultaneously to ensure that the system is asymptotically stable; constructing an optimization problem, and maximizing an estimation attraction domain of a system by solving the optimization problem;
and sixthly, calculating the minimum event trigger interval to prove that the Zeno phenomenon does not occur in the system with the dynamic anti-saturation compensator under the designed self-adaptive event trigger control.
In the first step, the continuous system with actuator saturation is:
Figure BDA0003454063500000021
wherein, t represents the time of day,
Figure BDA0003454063500000022
is the state vector of the continuous system, np、
Figure BDA0003454063500000023
Are respectively a state vector xpThe dimension and the first derivative of (t),
Figure BDA0003454063500000024
is the output of the dynamic controller, m is the dimension, Ap、Bp、CpAre coefficient matrices, sat (u (t)) [ sat (u)) ]1(t))，…，sat(um(t))]TRepresenting a vector saturation function;
the dynamic controller is as follows:
Figure BDA0003454063500000025
wherein the content of the first and second substances,
Figure BDA0003454063500000026
representing the state vector of a dynamic controller, nc、
Figure BDA0003454063500000027
Are respectively a state vector xcThe dimension and the first derivative of (t),
Figure BDA0003454063500000028
representing the output of the dynamic anti-saturation compensator, Ac、Bc、Cc、DcAre coefficient matrices;
the dynamic anti-saturation compensator comprises:
Figure BDA0003454063500000029
wherein the content of the first and second substances,
Figure BDA00034540635000000210
is the state vector of the dynamic anti-saturation compensator, naw、
Figure BDA00034540635000000211
Are respectively a state vector xaw(t) dimensions and first derivatives; phi (t) sat (u (t)) u (t) represents a dead-zone function, which is the input to the dynamic anti-saturation compensator;
in the third step, after introducing the adaptive event triggering mechanism, the dynamic controller is:
Figure BDA00034540635000000212
without considering the dynamic anti-saturation compensator, the system model is:
Figure BDA0003454063500000031
wherein the content of the first and second substances,
Figure BDA0003454063500000035
is xclFirst derivative of (t), xcl(t) is the state vector of the system without taking into account the dynamic anti-saturation compensator; a. thecl、Bφcl、Becl、Cucl、Duecl、CyclAre coefficient matrices;
considering the dynamic anti-saturation compensator, the system model is:
Figure BDA0003454063500000032
where x (t) is the state vector of the system in view of the dynamic anti-saturation compensator,
Figure BDA0003454063500000037
is the first derivative of x (t); A. b isφ、Be、Cu、Duφ、Due、CyAre coefficient matrices.
In the fourth step, the optimization problem of the structure is as follows:
Figure RE-GDA0003497935380000035
Figure RE-GDA0003497935380000036
wherein rho is a positive definite scalar quantity introduced for maximizing the estimation system attraction domain, Q is a positive definite matrix, R is a diagonal positive matrix, N isiIs the ith row of matrix N, I is the identity matrix, K is the matrix introduced to maximize the estimated system attraction domain, αiIs the absolute value of the saturation bound of the ith dimension control input.
In the fifth step, the optimization problem of the structure is as follows:
Figure BDA0003454063500000034
wherein, Y1Is a sub-matrix of matrix Y, N1 is a sub-matrix of matrix N,
Figure BDA0003454063500000036
N1iis a matrix NiL is a matrix, K11Is a sub-matrix of matrix K.
Compared with the prior art, the invention has the beneficial effects that:
1. the invention adopts a self-adaptive event triggering mechanism and a dynamic anti-saturation compensator in the system at the same time, designs a new self-adaptive event triggering condition, delta1(t) max (δ, η (t)), the threshold function of the existing adaptive event trigger condition contains δ alone1(t) and the adaptive event trigger threshold δ is used in the present invention1The form of (t) max (δ, η (t)) is such that the adaptive event triggers a threshold δ1(t) there is a lower bound δ so that the system can save more communication resources.
2. For the problem of actuator saturation in a system, although a general static anti-saturation compensator is convenient to design and can improve the performance of a saturated system to a certain extent, the performance of the saturated system cannot meet the higher requirement of the actual system on the performance.
3. Different from the event trigger control based on the system state in the prior art, the adaptive event trigger control provided by the invention is based on the output of the system, and can be widely applied to the situation that the system state cannot be completely measured.
4. By designing the free weight matrix phi in the adaptive event trigger condition1，Φ2And coefficient matrix A of the dynamic anti-saturation compensatoraw、Baw、Caw1、Caw2、Daw1、Daw2Etc., the system can be made asymptotically stable. In addition, compared with a related event trigger mechanism with constant parameters, the self-adaptive event trigger mechanism can save network communication resources to a greater extent, and the introduction of the anti-saturation compensator greatly reduces the adverse effect of actuator saturation on the system.
Drawings
FIG. 1 is a block diagram of the system architecture of the present invention;
FIG. 2 is a system input of the present invention including a dynamic anti-saturation compensator;
FIG. 3 is the output of a system of the present invention including a dynamic anti-saturation compensator;
FIG. 4 is an adaptive event trigger interval including a dynamic anti-saturation compensator of the present invention;
FIG. 5 is a system input of the present invention including a static anti-saturation compensator;
FIG. 6 is the output of a system of the present invention including a static anti-saturation compensator;
FIG. 7 is an adaptive event trigger interval including a static anti-saturation compensator of the present invention;
FIG. 8 is a state diagram of a system incorporating a dynamic anti-saturation compensator of the present invention;
FIG. 9 is a state diagram of a system of the present invention including a static anti-saturation compensator;
FIG. 10 is a graph of adaptive event trigger condition parameters over time under the action of a dynamic anti-saturation compensator;
FIG. 11 is a graph of adaptive event trigger condition parameters over time under the action of a static anti-saturation compensator;
FIG. 12 is an estimated attraction domain for a system with a dynamic anti-saturation compensator, a static anti-saturation compensator, and no anti-saturation compensator under adaptive event-triggered control.
Detailed Description
The technical solution of the present invention is described in detail below with reference to the specific embodiments and the accompanying drawings, but the scope of the present invention is not limited thereto.
The invention relates to a self-adaptive event triggering control method of a system with a dynamic anti-saturation compensator, which comprises the steps of firstly establishing a system model with the dynamic anti-saturation compensator, then designing a self-adaptive event triggering condition, and updating a signal received by a dynamic controller only when the self-adaptive event triggering condition is met; then, establishing a system model under the trigger control of the self-adaptive event, and designing the trigger condition of the self-adaptive event under the condition of setting a dynamic anti-saturation compensator to enable the saturation system to be gradually stable; under the condition that a dynamic anti-saturation compensator is not given, designing the dynamic anti-saturation compensator and an adaptive event triggering condition at the same time; finally, the Zeno phenomenon of the system can not occur under the designed self-adaptive event trigger control; the method comprises the following specific steps:
firstly, establishing a system model containing a dynamic anti-saturation compensator:
1-1, establishing a continuous system with actuator saturation as shown in formula (1):
Figure BDA0003454063500000051
wherein, t represents the time of day,
Figure BDA0003454063500000052
is the state vector of the continuous system, np、
Figure BDA0003454063500000053
Are respectively a state vector xpThe dimension and the first derivative of (t),
Figure BDA0003454063500000054
is the output of the dynamic controller, m is the dimension,
Figure BDA0003454063500000055
is a p-dimensional output vector of the controlled object, Ap、Bp、CpAre coefficient matrices, sat (u (t)) [ sat (u)) ]1(t))，…，sat(um(t))]TRepresenting vector saturationFunction, T represents matrix transposition; in addition, the controlled object is considered to be controllable and measurable;
each component in the vector saturation function satisfies equation (2):
Figure BDA0003454063500000056
wherein alpha isi＞0,i＝1,...,m，ui(t) denotes the system ith dimension control input, αiIs the absolute value of the saturation bound of the ith dimension control input;
1-2, a vertical dynamic controller (3):
Figure BDA0003454063500000057
wherein the content of the first and second substances,
Figure BDA0003454063500000058
representing the state vector of a dynamic controller, nc、
Figure BDA0003454063500000059
Are respectively a state vector xcThe dimension and the first derivative of (t),
Figure BDA00034540635000000510
representing the output of the dynamic anti-saturation compensator, Ac、Bc、Cc、DcAre coefficient matrices;
1-3, building a vertical (4) dynamic anti-saturation compensator:
Figure BDA00034540635000000511
wherein the content of the first and second substances,
Figure BDA00034540635000000512
is the state vector of the dynamic anti-saturation compensator, naw、
Figure BDA00034540635000000513
Are respectively a state vector xaw(t) dimensions and first derivatives; phi (t) sat (u (t)) u (t) represents a dead-zone function, which is the input to the dynamic anti-saturation compensator; a. theaw、Baw、 Caw1、Caw2、Daw1And Daw2All the coefficients are coefficient matrixes, and the system is asymptotically stable by designing the matrixes;
step two, designing a self-adaptive event triggering condition:
in order to save network communication resources, the invention provides a self-adaptive event triggering mechanism, and only when the self-adaptive event triggering condition is met, a signal received by a dynamic controller is updated; defining dynamic error e (t) ═ yp(tk)-yp(t)，yp(tk) Representing the transmission time tkThe data successfully transmitted to the dynamic controller is designed with the adaptive event triggering conditions as follows:
e(t)TΦ1e(t)≤δ1(t)yp(t)TΦ2yp(t)+δ1(t)∈e-∈t (5)
wherein the content of the first and second substances,
Figure BDA00034540635000000514
is a free weight matrix, e is a positive scalar quantity; delta1(t) is a self-adaptive event trigger threshold, which can be self-adaptively adjusted according to the dynamic performance of the system and meets the following conditions: delta1(t)＝max{δ,η(t)}，δ∈(0,1]Represents the lower bound of the adaptive event trigger threshold, η (t) represents the threshold function, the initial value η (0) > 0 and η (t) satisfies equation (6):
Figure BDA0003454063500000061
wherein the content of the first and second substances,
Figure BDA0003454063500000062
representing the first derivative of the threshold function η (t), theta representing the threshold functionNumber η (t) convergence rate adjustment parameter, θ > 1;
and step three, establishing a system model under the trigger control of the self-adaptive event:
after introducing the adaptive event triggering mechanism, the dynamic controller is:
Figure BDA0003454063500000063
3-1, under the condition of not considering the dynamic anti-saturation compensator, establishing a vertical (8) system model:
Figure BDA0003454063500000064
wherein the content of the first and second substances,
Figure BDA0003454063500000065
is xclFirst derivative of (t), xcl(t) is the state vector of the system without taking into account the dynamic anti-saturation compensator; a. thecl、Bφcl、Becl、Cucl、Duecl、CyclAre coefficient matrices; wherein the content of the first and second substances,
Figure BDA0003454063500000066
then the following equation is present:
Figure BDA0003454063500000067
3-2, under the condition of considering the dynamic anti-saturation compensator, establishing a vertical (9) system model:
Figure BDA0003454063500000068
where x (t) is the state vector of the system in view of the dynamic anti-saturation compensator,
Figure BDA0003454063500000069
is the first derivative of x (t); A. b isφ、Be、Cu、Duφ、Due、CyAre coefficient matrices; wherein the content of the first and second substances,
Figure BDA00034540635000000610
then the following equation is present:
Figure BDA00034540635000000611
wherein the content of the first and second substances,
Figure BDA00034540635000000612
Figure BDA00034540635000000613
Figure BDA0003454063500000071
Figure BDA0003454063500000072
Im×mare all identity matrixes;
fourthly, under the condition of setting a dynamic anti-saturation compensator, designing a self-adaptive event triggering condition to ensure that the system of the formula (9) is asymptotically stable; constructing an optimization problem, and estimating a system attraction domain in a maximized manner by solving the optimization problem;
4-1, defining the Lyapunov function:
Figure BDA00034540635000000710
wherein, P is a positive definite matrix;
case 1: when delta < eta (t), delta1(t) max (δ, η (t) } η (t), δ1The formula (5) is substituted by η (t) to obtain the adaptive event triggering condition: e (t)TΦ1e(t)≤η(t)yp(t)TΦ2yp(t)+η(t)∈e-∈t；
The following inequality can be obtained from the sector condition, J is an arbitrary diagonal positive definite matrix;
φ(t)TJ(φ(t)+u(t)+Gx(t))≤0 (11)
the lyapunov function of the formula (10) is subjected to derivation, and is further transformed by combining the formula (11):
Figure BDA00034540635000000711
wherein
Figure BDA0003454063500000074
To ensure the asymptotic stability of the system of equation (9), it is necessary to satisfy:
Figure BDA0003454063500000075
by Schur supplementation, we can obtain:
Figure BDA0003454063500000076
case 2: when delta is greater than or equal to eta (t), delta1(t) max (δ, η (t) } δ, δ1(t) represents δ substitution equation (5), and the adaptive event triggering conditions are obtained as follows: e (t)TΦ1e(t)≤δyp(t)TΦ2yp(t)+δ∈e-∈t；
To ensure asymptotic stabilization of the system of equation (9), the derivative of the Lyapunov function
Figure BDA0003454063500000077
It should satisfy:
Figure BDA0003454063500000078
-2φT(t)J(φ(t)+u(t))+∈(δ-1)e-∈t＜0
that is to say, require
Figure BDA0003454063500000079
Obviously, equation (12) is a sufficient condition for equation (13), so that asymptotic stability of the system can still be ensured for case 2 when equation (12) is satisfied;
for the left and right co-multiplication matrixes diag { Q, R, I } of inequality (12), let Q be P-1,R＝J-1,N＝GQTThe following inequality is available:
Figure BDA0003454063500000081
wherein, if there is a free weight matrix
Figure BDA0003454063500000082
Positive definite matrix
Figure BDA0003454063500000083
And diagonal positive definite matrix
Figure BDA0003454063500000084
Inequality (14) is satisfied, then the system of equation (9) satisfies
Figure BDA0003454063500000085
Namely, the gradual stabilization of the system is realized;
making x (t) belong to a set according to sector conditions
Figure BDA0003454063500000086
GiRepresenting the ith row of the matrix G, i.e. the ellipse
Figure BDA0003454063500000087
For the system attraction domain to be estimated,
Figure BDA0003454063500000088
can also be expressed as
Figure BDA0003454063500000089
Further transformation gives the formula (15):
Figure BDA00034540635000000810
wherein I is an identity matrix, Q ═ P-1；NiIs the ith row of the matrix N;
4-2, constructing an optimization problem of the formula (16), and solving the optimization problem to obtain a system estimation attraction domain epsilon (P, 1);
Figure RE-GDA00034979353800000811
where inf represents the minimization, ρ is a positive scalar introduced to maximize the estimated system attraction domain, and s.t. represents the constraint condition; matrix array
Figure BDA00034540635000000812
Is a matrix introduced to maximize the estimated system attraction domain;
fifthly, under the condition that a dynamic anti-saturation compensator is not given, designing an anti-saturation compensator and a self-adaptive event triggering condition simultaneously to ensure that the system is asymptotically stable; constructing an optimization problem, and estimating a system attraction domain in a maximized manner by solving the optimization problem;
according to the matrix decomposition theorem, if the following set of inequalities is satisfied:
Figure BDA00034540635000000813
rank represents the rank of the matrix;
then the positive definite matrix Q can be decomposed into:
Figure BDA00034540635000000814
order to
Figure BDA00034540635000000815
The inequality group (17) is multiplied by the matrix L to the matrix Y to obtain the product
Figure BDA0003454063500000091
Further, according to the projection theorem, the inequality (15) can be expressed as follows:
Ψ+FTΛH+HTΛF＜0(19)
let N be [ N1N 2 ]]，
Figure BDA0003454063500000092
Each matrix in equation (19) can be obtained by substituting the decomposed matrix Q, that is, equation (18), into inequality (15):
Figure BDA0003454063500000093
Figure BDA0003454063500000094
wherein the content of the first and second substances,
Figure BDA0003454063500000095
according to the projection theorem, equation (19) is equivalent to:
Figure BDA0003454063500000096
wherein, WFAnd WHAre matrices composed of the basis vectors of the null spaces of matrices F and H, respectively, and are therefore obtained according to equation (20):
Figure BDA0003454063500000097
further, substituting the equations (20) and (22) into the inequality group (21) can obtain the following equation:
Figure BDA0003454063500000098
Figure BDA0003454063500000099
further, the matrix N in the formula (15)i＝[N1i N2i]＝[N1i N1iY1 -1Y12 N1iY1 -1U1 T]By Schur complement (15) can be represented as:
Figure BDA0003454063500000101
further transformation can obtain:
Figure BDA0003454063500000102
formula (18) may be substituted for formula (23):
Figure BDA0003454063500000103
further, alpha can be obtainedi 2I-N1iY1 -1N1i TNot less than 0, and the Schur supplement is used again to obtain:
Figure BDA0003454063500000104
further, the constraint conditions in the problem formula (16) are optimized
Figure BDA0003454063500000105
The constraint is written as:
ρK-Q-1≥0(25)
order to
Figure BDA0003454063500000106
Equation (25) can thus be expressed as:
Figure BDA0003454063500000107
let ρ K12＝VT,ρK21＝V,
Figure BDA0003454063500000108
Equation (25) can be further expressed as ρ K11-L-1Not less than 0; therefore, when no dynamic anti-saturation compensator is given, the optimization problem of the design formula (27) can realize the asymptotic stability of the system;
Figure BDA0003454063500000109
but the inequality rank (Y-L) in the formula (27) is not more than nawNon-linear, which is not conducive to further calculations, therefore, consider naw≥npAnd nawTwo more widely used cases are 0;
when n isaw≥npWhen it is, let Y12＝L12,Y22＝L22Then, then
Figure BDA00034540635000001010
The optimization problem of equation (27) at this time translates into:
Figure BDA0003454063500000111
let UTU＝YL-1Y-Y，W＝I+UY-1UT，Na＝[N1 N1Y1 -1Y12]，Nb＝N1Y1 -1U1 TThe free weight matrix Φ in the adaptive event trigger condition of equation (5) can be solved by equation (28)1、Φ2Then, obtaining a coefficient matrix of the dynamic anti-saturation compensator by solving the formula (29);
Figure BDA0003454063500000112
wherein the content of the first and second substances,
Figure BDA0003454063500000113
Figure BDA0003454063500000114
Figure BDA0003454063500000115
if there is a matrix
Figure BDA0003454063500000116
Figure BDA0003454063500000117
And the positive scalar ρ satisfies equation (28), then the system of equation (9) can guarantee asymptotic stability;
solving the inequality group of the formula (29) to obtain each coefficient matrix of the dynamic anti-saturation compensator of the formula (4) as follows:
Figure BDA0003454063500000118
up to this point in naw≥npIn time, the design of a dynamic anti-saturation compensator and a self-adaptive event triggering condition is completed, and the gradual stabilization of the system is realized;
when n isawAt 0, the anti-saturation compensator is static, in which case Y-L should be satisfied, so the following optimization problem is designed:
Figure BDA0003454063500000121
let Q be Y, U be W be 0, phi1，Φ2The coefficient matrix of the anti-saturation compensator can be obtained through the calculation (30) and then through the calculation (31);
Figure BDA0003454063500000122
wherein the content of the first and second substances,
Figure BDA0003454063500000123
Figure BDA0003454063500000124
Figure BDA0003454063500000125
if there is a positive matrix
Figure BDA0003454063500000126
And the positive scalar ρ satisfies equation (31), then the system of equation (9) may asymptotically stabilize;
solving the inequality set of the formula (31) to obtain a coefficient matrix of the anti-saturation compensator of
Figure BDA0003454063500000127
Thus completing nawThe design of the anti-saturation compensator and the self-adaptive event triggering condition when the time is equal to 0 achieves the gradual stabilization of the system;
sixthly, calculating a minimum event trigger interval, and proving that a Zeno phenomenon does not occur in a system containing the dynamic anti-saturation compensator under the designed self-adaptive event trigger control;
at time t e [ t ∈k,tk+1) In this case, the dynamic error e (t) is derived to obtain:
Figure BDA0003454063500000128
let a be | λmax(Ap)|，b(tk)＝|λmax(Ap)|·‖yp(tk)‖+‖Cp‖·‖BpII, solving the formula (32) to obtain:
Figure BDA0003454063500000129
wherein λ ismax(Ap) Representation matrix ApThe largest feature root of;
and because of the adaptive event triggering condition e (t)TΦ1e(t)≤δ1(t)yp(t)TΦ2yp(t)+δ1(t)∈e-∈tCan be prepared from e (t)TΦ1e(t)≤δyp(t)TΦ2yp(t)+δ∈e-∈tIs guaranteed, so that it is possible to obtain:
Figure BDA0003454063500000131
wherein λ is1＝λmin(Φ1),λ2＝λmax(Φ2)，
Figure BDA0003454063500000132
Figure BDA0003454063500000133
Is that
Figure BDA0003454063500000134
The lower bound of (c);
from equations (33) and (34), the minimum trigger event interval can be derived as:
Figure BDA0003454063500000135
wherein, tk,tk+1Respectively representing the time when the kth event trigger and the (k + 1) th event trigger occur;
as can be seen from the formula (35), T (T)k,tk+1) > 0, indicating that control is triggered at an adaptive event: (5) And then, the Zeno phenomenon does not occur in the system.
Examples
In the adaptive event trigger control method of the system including the dynamic anti-saturation compensator of the present embodiment, a system structure diagram is shown in fig. 1, and the system includes a dynamic controller, an actuator, a controlled object, an event generator, the dynamic anti-saturation compensator, and a zero-order keeper; the specific steps of this embodiment are:
firstly, establishing a system model containing a dynamic anti-saturation compensator; wherein, the coefficient matrix of the continuous system model with actuator saturation is designed as follows:
Figure BDA0003454063500000136
Cp＝[11]；
assuming that the saturation boundary alpha of the actuator is 0.3, and on the basis of ensuring the stability of the system, designing a coefficient matrix of the dynamic controller as follows:
Figure BDA0003454063500000137
Cc＝[0.0091 0.0438]，Dc＝1.5933；
secondly, designing a self-adaptive event trigger mechanism;
in order to save network communication resources, the invention provides a self-adaptive event trigger mechanism, wherein only when the self-adaptive event trigger condition is met, the signal received by a dynamic controller is updated, and a dynamic error e (t) is defined as yp(tk)-yp(t)，yp(tk) Representing the transmission time tkThe data successfully transmitted to the controller is designed with the adaptive event triggering conditions as follows:
e(t)TΦ1e(t)≤δ1(t)yp(t)TΦ2yp(t)+δ1(t)∈e-∈t (5)
wherein the content of the first and second substances,
Figure BDA0003454063500000138
is a free weight matrix, e is a positive scalar quantity; delta1(t)The self-adaptive event triggering threshold value can be self-adaptively adjusted according to the dynamic performance of the system, and meets the following conditions: delta1(t)＝max{δ,η(t)}，δ∈(0,1]η (t) represents a threshold function, the initial value η (0) > 0 and η (t) satisfies the formula (6):
Figure BDA0003454063500000139
wherein the content of the first and second substances,
Figure BDA0003454063500000141
a first derivative representing a threshold function eta (t), theta > 1 and theta representing an adjustment parameter of the convergence rate of the threshold function eta (t);
thirdly, establishing a system model under the trigger control of the self-adaptive event;
fourthly, under the condition of setting a dynamic anti-saturation compensator, designing a self-adaptive event triggering condition to ensure that the system of the formula (9) is asymptotically stable, and maximizing the attraction domain estimated by the system by solving an optimization problem;
fifthly, under the condition that a dynamic anti-saturation compensator is not given, designing an anti-saturation compensator and a self-adaptive event triggering condition simultaneously to ensure that the system is asymptotically stable; maximizing the attraction domain estimated by the system by solving an optimization problem;
let θ equal to 1.2, δ20.1, 0.2, 0.99, 1; when n isaw≥npThen, solving the inequality group of equation (29) to obtain:
Figure BDA0003454063500000142
N1＝[-0.2581 0.0917]，Φ1＝71.8853，Φ2＝10.8463，
Figure BDA0003454063500000143
N＝[-0.2581 0.0917 16.7638 -3.2489 0.3135 0.0780]
Figure BDA0003454063500000144
when n isawWhen the value is 0, solving the inequality group of equation (31) yields:
ρ＝0.5003，N1＝[-0.1275 -0.0773]，
Figure BDA0003454063500000145
Φ1＝70.7461，Φ2＝11.2614，N＝[-0.1275 -0.0773 -2.2113 0.1110]，
Figure BDA0003454063500000146
Figure BDA0003454063500000147
and sixthly, calculating a minimum event trigger interval to prove that the Zeno phenomenon does not occur in the system with the dynamic anti-saturation compensator under the designed self-adaptive event trigger control.
In order to verify the feasibility of the method, numerical simulation is carried out in an MATLAB environment, and the simulation result is shown in FIGS. 2-12; FIGS. 2-4 respectively depict system inputs, outputs, and adaptive event trigger intervals including dynamic anti-saturation compensators, and FIGS. 5-7 respectively depict system inputs, outputs, and adaptive event trigger intervals including static anti-saturation compensators; it can be seen from the figure that the time for the input and output containing the dynamic anti-saturation compensator to converge to the linear region of the actuator is significantly shorter than for the system containing the static anti-saturation compensator, and the Zeno phenomenon is avoidable.
TABLE 1 statistics of settling time and adaptive event trigger times for two types of anti-saturation compensators
Anti-saturation compensator type Regulating time Adaptive event trigger times
Dynamic anti-saturation compensator 2.93 seconds 23 times of
Static anti-saturation compensator 5.68 seconds 34 times of
Fig. 8 and 9 are diagrams of states of a system including a dynamic anti-saturation compensator and a static anti-saturation compensator, respectively, and it can be seen from fig. 8 that under the adaptive event triggering control proposed herein, the state of the system can converge to the origin, i.e. the asymptotic stability of the system is ensured. In addition, compared with the system state of fig. 5 including the static anti-saturation compensator, the system state of fig. 4 including the dynamic anti-saturation compensator has smaller overshoot and faster convergence speed, which indicates that the dynamic anti-saturation compensator can improve the system performance to a greater extent than the static anti-saturation compensator. Table 1 shows statistical results of the adjustment time and the adaptive event triggering times of two types of anti-saturation compensators, and the adjustment time and the adaptive event triggering times of the system including the dynamic anti-saturation compensator are significantly less than those of the system including the static anti-saturation compensator, so that the system including the dynamic anti-saturation compensator is more advantageous than the system including the static anti-saturation compensator in saving communication resources.
Fig. 10 and 11 show the time-dependent variation of the adaptive event trigger condition parameters for systems with dynamic anti-saturation compensators and systems with static anti-saturation compensators, respectively.
Fig. 12 illustrates the attraction domain of the system including the dynamic anti-saturation compensator, the static anti-saturation compensator, and the system without the anti-saturation compensator under the adaptive event triggered control, and the system including the dynamic anti-saturation compensator has a larger attraction domain than the system including the static anti-saturation compensator and the system without the anti-saturation compensator, and the larger attraction domain is indicative of better system performance, so that the system including the dynamic anti-saturation compensator can not only ensure system stability, but also effectively save communication resources under the adaptive event triggered control.
TABLE 2 statistics of trigger times under different trigger mechanisms
Figure BDA0003454063500000151
As can be seen from table 2, compared with the related event trigger mechanism in which the variable is a constant, the system has fewer trigger times under the adaptive event trigger control, which can save more communication resources.
Nothing in this specification is said to apply to the prior art.

Claims (4)

1. An adaptive event-triggered control method for a system including a dynamic anti-saturation compensator, the method comprising the steps of:
the method comprises the following steps that firstly, a system model containing a dynamic anti-saturation compensator is established, and the system model contains a continuous system with actuator saturation, a dynamic controller and the dynamic anti-saturation compensator;
secondly, designing a self-adaptive event triggering condition as follows:
e(t)TΦ1e(t)≤δ1(t)yp(t)TΦ2yp(t)+δ1(t)∈e-∈t (5)
where e (t) is dynamic error, e (t) yp(tk)-yp(t)，yp(tk) Indicating the transmission time tkData successfully transmitted to the dynamic controller, yp(t) is a controlled pairOutput vector of image, [ phi ]1、Φ2Is a free weight matrix, epsilon is a positive scalar quantity, and T represents matrix transposition; delta1(t) is an adaptive event trigger threshold, and the condition is satisfied: delta1(t)＝max{δ,η(t)}，δ∈(0,1]Represents the lower bound of the adaptive event trigger threshold, η (t) represents the threshold function, the initial value η (0) > 0 and η (t) satisfies equation (6):
Figure FDA0003454063490000011
wherein the content of the first and second substances,
Figure FDA0003454063490000012
representing the first derivative of a threshold function eta (t), theta representing an adjustment parameter of the convergence rate of the threshold function, theta > 1;
thirdly, establishing a system model under the trigger control of the self-adaptive event;
fourthly, under the condition of setting a dynamic anti-saturation compensator, designing a self-adaptive event triggering condition to ensure that the system is asymptotically stable; constructing an optimization problem, and maximizing an attraction domain estimated by a system by solving the optimization problem;
when delta < eta (t), delta1Substituting equation (5) to obtain the adaptive event triggering condition: e (t)TΦ1e(t)≤η(t)yp(t)TΦ2yp(t)+η(t)∈e-∈t；
When delta is greater than or equal to eta (t), delta1When (t) is δ, the adaptive event triggering condition obtained by substituting formula (5) is as follows: e (t)TΦ1e(t)≤δyp(t)TΦ2yp(t)+δ∈e-∈t；
Fifthly, under the condition that a dynamic anti-saturation compensator is not given, designing an anti-saturation compensator and a self-adaptive event triggering condition simultaneously to ensure that the system is asymptotically stable; constructing an optimization problem, and estimating a system attraction domain in a maximized manner by solving the optimization problem;
and sixthly, calculating the minimum event trigger interval to prove that the Zeno phenomenon does not occur in the system with the dynamic anti-saturation compensator under the designed self-adaptive event trigger control.
2. The adaptive event-triggered control method for a system including a dynamic anti-saturation compensator according to claim 1, wherein in the first step, the continuous system with actuator saturation is:
Figure FDA0003454063490000013
wherein, t represents the time of day,
Figure FDA0003454063490000014
is the state vector of the continuous system, np、
Figure FDA0003454063490000015
Are respectively a state vector xpThe dimension and the first derivative of (t),
Figure FDA0003454063490000016
is the output of the dynamic controller, m is the dimension, Ap、Bp、CpAre coefficient matrices, sat (u (t)) [ sat (u)) ]1(t))，…，sat(um(t))]TRepresenting a vector saturation function;
the dynamic controller is as follows:
Figure FDA0003454063490000017
wherein the content of the first and second substances,
Figure FDA0003454063490000021
representing the state vector of a dynamic controller, nc、
Figure FDA0003454063490000022
Are respectively a state vector xcThe dimension and the first derivative of (t),
Figure FDA0003454063490000023
representing the output of the dynamic anti-saturation compensator, Ac、Bc、Cc、DcAre coefficient matrices;
the dynamic anti-saturation compensator comprises:
Figure FDA0003454063490000024
wherein the content of the first and second substances,
Figure FDA0003454063490000025
is the state vector of the dynamic anti-saturation compensator, naw、
Figure FDA0003454063490000026
Are respectively a state vector xaw(t) dimensions and first derivatives; phi (t) sat (u (t)) u (t) represents a dead-zone function, which is the input to the dynamic anti-saturation compensator;
in the third step, after introducing the adaptive event triggering mechanism, the dynamic controller is:
Figure FDA0003454063490000027
without considering the dynamic anti-saturation compensator, the system model is:
Figure FDA0003454063490000028
wherein the content of the first and second substances,
Figure FDA0003454063490000029
is xclFirst derivative of (t), xcl(t) is the state of the system without taking into account the dynamic anti-saturation compensatorAn amount; a. thecl、Bφcl、Becl、Cucl、Duecl、CyclAre coefficient matrices;
considering the dynamic anti-saturation compensator, the system model is:
Figure FDA00034540634900000210
where x (t) is the state vector of the system in view of the dynamic anti-saturation compensator,
Figure FDA00034540634900000212
is the first derivative of x (t); A. b isφ、Be、Cu、Duφ、Due、CyAre coefficient matrices.
3. The adaptive event-triggered control method for a system including a dynamic anti-saturation compensator according to claim 2, wherein in the fourth step, the optimization problem is constructed as follows:
Figure RE-FDA00034979353700000212
wherein rho is a positive definite scalar quantity introduced for maximizing the estimation system attraction domain, Q is a positive definite matrix, R is a diagonal positive definite matrix, NiIs the ith row of matrix N, I is the identity matrix, K is the matrix introduced to maximize the estimated system attraction domain, αiIs the absolute value of the saturation bound of the ith dimension control input.
4. The adaptive event-triggered control method for a system including a dynamic anti-saturation compensator according to claim 2, wherein in the fifth step, the optimization problem is constructed as follows:
Figure FDA0003454063490000031
wherein, Y1Is a sub-matrix of matrix Y, N1 is a sub-matrix of matrix N,
Figure FDA0003454063490000032
N1iis a matrix NiL is a matrix, K11Is a sub-matrix of matrix K.
CN202210000999.4A 2022-01-04 2022-01-04 Self-adaptive event trigger control method containing dynamic anti-saturation compensator system Active CN114296355B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202210000999.4A CN114296355B (en) 2022-01-04 2022-01-04 Self-adaptive event trigger control method containing dynamic anti-saturation compensator system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202210000999.4A CN114296355B (en) 2022-01-04 2022-01-04 Self-adaptive event trigger control method containing dynamic anti-saturation compensator system

Publications (2)

Publication Number Publication Date
CN114296355A true CN114296355A (en) 2022-04-08
CN114296355B CN114296355B (en) 2023-07-07

Family

ID=80975108

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202210000999.4A Active CN114296355B (en) 2022-01-04 2022-01-04 Self-adaptive event trigger control method containing dynamic anti-saturation compensator system

Country Status (1)

Country Link
CN (1) CN114296355B (en)

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5394322A (en) * 1990-07-16 1995-02-28 The Foxboro Company Self-tuning controller that extracts process model characteristics
EP1950636A1 (en) * 2007-01-23 2008-07-30 General Electric Company Multivariable controller design method for multiple input/output systems with multiple input/output constraints
CN103207568A (en) * 2013-03-18 2013-07-17 哈尔滨工程大学 Steering engine saturation resistant self-adaptive control method for ship courses
CN110456681A (en) * 2019-07-01 2019-11-15 天津大学 The output feedback controller of neutral stability saturation system based on event triggering

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5394322A (en) * 1990-07-16 1995-02-28 The Foxboro Company Self-tuning controller that extracts process model characteristics
EP1950636A1 (en) * 2007-01-23 2008-07-30 General Electric Company Multivariable controller design method for multiple input/output systems with multiple input/output constraints
CN103207568A (en) * 2013-03-18 2013-07-17 哈尔滨工程大学 Steering engine saturation resistant self-adaptive control method for ship courses
CN110456681A (en) * 2019-07-01 2019-11-15 天津大学 The output feedback controller of neutral stability saturation system based on event triggering

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
刘晨;董朝阳;王青;冉茂鹏;: "基于扩张状态观测器的动态抗饱和补偿器设计方法", 控制与决策, no. 11 *

Also Published As

Publication number Publication date
CN114296355B (en) 2023-07-07

Similar Documents

Publication Publication Date Title
Fei et al. Event-triggered dynamic output feedback control for switched systems with frequent asynchronism
Peng et al. Network-based H∞ control for T–S fuzzy systems with an adaptive event-triggered communication scheme
Gu et al. Adaptive event-triggered control of a class of nonlinear networked systems
Hua et al. Model following controller design for large-scale systems with time-delay interconnections and multiple dead-zone inputs
CN108490765B (en) Design method of active disturbance rejection controller with anti-integral compensation
Leonessa et al. Adaptive control for nonlinear uncertain systems with actuator amplitude and rate saturation constraints
Liu et al. Fixed-Time $\mathcal {H} _ {\infty} $ Control for Port-Controlled Hamiltonian Systems
Zhang et al. Prescribed performance adaptive neural output feedback dynamic surface control for a class of strict‐feedback uncertain nonlinear systems with full state constraints and unmodeled dynamics
Liu et al. Fixed-time stabilization control for port-Hamiltonian systems
Li et al. An improved event-triggered control for systems subject to asymmetric actuator saturation
CN111221311B (en) Complex network distributed pulse synchronization method and system based on parameter variational method
Mi et al. Fixed-time consensus tracking for multi-agent systems with a nonholomonic dynamics
CN115454127A (en) Multi-aircraft cooperative control method in weak communication network
Xu et al. Observer-based robust fuzzy control of nonlinear networked systems with actuator saturation
Li et al. Event-based distributed fuzzy load frequency control for multiarea nonlinear power systems with switching topology
Lei et al. Adaptive asymptotic tracking control of constrained multi‐input multi‐output nonlinear systems via event‐triggered strategy
Lei et al. Event-triggered fixed-time stabilization of two time scales linear systems
Wang et al. Exponential synchronization of delayed neural networks involving unmeasurable neuron states via impulsive observer and impulsive control
Lin et al. Stabilization of discrete-time switched singular systems with state, output and switching delays
CN114296355A (en) Adaptive event trigger control method for system with dynamic anti-saturation compensator
CN107390527B (en) Fuzzy self-adaptive compensation control method for time lag and failure of actuator
Zhang et al. Adaptive fuzzy sliding exact tracking control based on high-order log-type time-varying blfs for high-order nonlinear systems
Ansarifar et al. An adaptive-dynamic sliding mode controller for non-minimum phase systems
Madjidian et al. $ H_ {2} $ Optimal Coordination of Homogeneous Agents Subject to Limited Information Exchange
CN113515066A (en) Nonlinear multi-agent system dynamic event trigger control method

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant