阅读说明：本技术 一种多胞胎不确定奇异网络系统的滤波器设计方法 (Filter design method of multi-fetus uncertain singular network system ) 是由 张应奇 刘彩侠 湛妙俊 闫晶晶 李笑 李影 于 2021-09-08 设计创作，主要内容包括：本发明涉及滤波器设计技术领域,公开了一种多胞胎不确定奇异网络系统的滤波器设计方法,包括以下步骤：导出多胞胎不确定奇异网络系统的滤波器的网络模型；通过松弛矩阵变量证明网络模型是性能容许的；通过李雅普诺夫函数方法证明网络模型是渐近稳定的、正则和因果的；给出了误差模型是容许的与具有H-(∞)性能指标的充分性判据；应用滤波器等价方法,设计事件驱动下对应滤波器的增益矩阵,获得事件驱动鲁棒的滤波器,这种滤波器设计方法,大大减少了的通信资源,并且具有一定的抗干扰能力,极大减少了宽带资源和运行成本,进而极大节约了网络通讯成本。(The invention relates to the technical field of filter design, and discloses a filter design method of a multi-fetus uncertain singular network system, which comprises the following steps: deriving a network model of a filter of the multi-fetus uncertain singular network system; the network model is proved to be performance-tolerant through a relaxation matrix variable; the network model is proved to be asymptotically stable, regular and causal by a Lyapunov function method; given that the error model is tolerable and has H ∞ The sufficiency criterion of the performance index; the filter design method greatly reduces communication resources, has certain anti-interference capability, greatly reduces broadband resources and operation cost, and further greatly saves network communication cost.)

1. A filter design method of a multi-fetus uncertain singular network system is characterized by comprising the following steps:

deriving a network model of a filter of the multi-fetus uncertain singular network system;

the network model is proved to be performance-tolerant through a relaxation matrix variable;

the network model is proved to be asymptotically stable, regular and causal through the Lyapunov function;

given that the error model is tolerable and has H_∞The sufficiency criterion of the performance index;

and designing a gain matrix corresponding to the filter under the event drive by using a filter equivalent method to obtain the event-driven robust filter.

2. The method of filter design for a multiple-run indeterminate singular network system as claimed in claim 1, wherein said method of deriving the network model of the filter for a multiple-run indeterminate singular network system comprises:

the following multi-cell type discrete time singular network system (DSNS) with perturbation was employed:

Ex(q+1)＝Ax(q)+Bw(q) (1a)

y(q)＝Hx(q) (1b)

z(q)＝Cx(q)+Dw(q) (1c)

wherein x (q) e Rⁿ，y(q)∈R^p1，z(q)∈R^p2Respectively, state vector, output state and estimated signal; perturbation input w (q) ε R^p3Belong to₂A [0, + ∞) space, E being a space satisfying a rank (E) ═ r < n, and further, the multi-fetal coefficient matrices of DSNS (1a) - (1c) belong to the following convex set:

wherein:to representThe jth vertex of the middle polyhedron domain;

in order to save computing resources and communication cost, an event-driven design strategy is adopted, and information transmission is triggered and transmitted only when the current conditions are met:

(y(q)-y(τ_k))^TΩ(y(q)-y(τ_k))＞μy^T(q)Ωy(q)

wherein: mu is a given constant and mu epsilon [0,1), omega is the positive definite matrix to be designed, y (q) is the current sampling signal, tau_k(k ═ 0,1, …, + ∞) is the release time constant, τ_k+1Indicates the next trigger time,Denotes network induced delay, y (τ)_k) For the most recent transmitted signal, for any q e [ tau ]_k,τ_k+1-1]From the event generator above:

(y(q)-y(τ_k))^TΩ(y(q)-y(τ_k))≤μy^T(q)Ωy(q) (3)

the following full-order filter is designed:

to derive the event-driven error DSNS model, consider the following two cases:

a1: if it isThenDefinition of h (q) ═ q- τ_kAndtherefore, h (q) satisfies

A2 ifThere is a positive integerSatisfy the requirement of

And alsoThe following inequality holds:

(y(τ_k+ν)-y(τ_k))^TΩ(y(τ_k+ν)-y(τ_k))≤μy^T(τ_k+ν)Ωy(τ_k+ν)；

thus, the handle sectionThe method comprises the following steps:

wherein: in this case, the function h (q) is defined as:

and the difference between the current sampling signal and the latest transmission signalAs defined below:

therefore, it is easy to obtainIs provided withIf true;

from the group consisting of B1 and B2,is easy to obtain

By the event generator (3) and the above analysis,comprises the following steps:

thus, the filters (4a) and (4b) in the event-driven regime are represented as follows:

now, defineAndfrom (1a) to (1c) and (7a) and (7b), the following error DSNS is obtained:

wherein Γ ═ I_n 0]， To simplify writing, orderThenAnd e (q) ═ Φ₂η (q), wherein: and

3. the method for designing a filter of a multiple-fetus uncertain singular network system as claimed in claim 2, wherein said network model is proven to be performance-tolerant by relaxation matrix variables by a specific method comprising:

if det (sE-A) is not always equal to zero, thenAnd DSNS (when w (q) is not equal to 0 ≡ 08a) The model is said to be regular;

if deg (det (sE-a) ═ rank (e)), thenAnd w (q) the DSNS (8a) model at [ identical to ] 0 is said to be causal;

if: the DSNS (8a) model is regular, causal and asymptotically stable; DSNS (8a) and (8b) has H_∞The noise indicator γ, i.e. satisfying the constraint | e | at an initial value of zero₂≤γ‖w‖₂The DSNS (8a) and (8b) models are said to have performance index γ tolerance;

introduction 1: given matrix R > 0, M ═ M₁ M₂ M₃ M₄]And a function h (q), whereinAnd M₁＝[M₁₁ M₁₂]Then the following inequality holds:

wherein:

given scalar quantity h> 0 and μ > 0, and if Ω > 0, γ > 0, P > 0,R＞0，A_f，B_f，D_f，M＝[M₁ M₂ M₃ M₄]the following inequality is made true:

the DSNS (8a) and (8b) network models are then allowed with performance index γ, where:

and matrixSatisfy the requirement ofAnd rank

4. The filter design method of the multiple-fetus uncertain singular network system according to claim 3, characterized in that the specific method for proving that the network model is asymptotically stable by the Lyapunov function is:

to demonstrate that DSNS (8a) and (8b) are asymptotically stable and have H_∞And selecting the noise index gamma as the following Lyapunov function:

wherein: let Δ V (q) ═ V (q +1) -V (q), thenIs provided with

Wherein:and

using lemma 1, we get:

byTo obtain

Wherein:

known from (6) and (11) to (16):

wherein: phi₆＝diag{0,μH^TΩH,-Ω,-γ²I}，Applying the schur complement γ < 0 is equivalent to (10), so when w (q) is 0, the errors DSNS (8a) and (8b) Δ v (q) < 0, so the system is asymptotically stable;

when the initial value x (0) is 0, the value is obtained from (17)

Therefore, the temperature of the molten metal is controlled,de | e |₂≤γ‖w‖₂(ii) a On the other hand, from (10)

ByAnd further from (19)Are regular and causal, so the error DSNS (8a) and (8b) models are regular and causal.

5. The method of claim 4, wherein the error model is tolerable and has a H-value_∞The sufficiency criterion of the performance index is as follows:

given scalar quantity h> 0 and μ > 0, then when Ω > 0, γ > 0, P_j＞0，R_j＞0，A_f，B_f，C_f，D_f，G₁，G₂，And M_j＝[M_1j M_2j M_3j M_4j]，So that the following inequality holds:

the DSNS (8a) and (8b) network models are then allowed with performance index γ, where:

and matrixSatisfy the requirement ofAnd rank

The corresponding filter and trigger gain matrix under the drive of the design event is as follows:

given scalar quantity h> 0 and μ > 0, if Ω > 0, γ > 0, R_j＞0，M_11j，M_2j，M_3j，M_4j，L₁，L₂，L_3j，Y₁， and G₁，So that the following inequality holds:

the DSNS (8a) and (8b) models are then of H_∞Performance index γ allowed, where:

wherein: Θ_32j＝-E^TL_3j,Θ_99j＝P_11j-Sys{Y₁},the matrix S ∈ R^n×nSatisfies E^TS-0 and rank (S) -n-r.

6. The method according to claim 5, wherein the filter design method for the multi-fetus uncertain singular network system is to design a gain matrix of the corresponding filter under event driving, and obtain the event-driven robust filter as follows:

event driven robust H_∞The filter gain matrix is designed to:

and (3) proving that: letSys { G } is known from equation (20)₂Is } > 0, thus Y₄Is an invertible matrix, defining:J＝diag{J₁,I,I,I,I,I,I,J₁,I}，

reissue to order

ThenTherefore, equation (21) ensures that the DSNS (8a) and (8b) models are of H_∞Performance index γ allowed, note that the filter is fromToThe transfer function of (a) is:

from equation (22) and equation (23), equation (24) is equivalent to

Thus, equation (22) is the designed event-driven robust H_∞And a filter.

Technical Field

The invention relates to the technical field of filter design, in particular to a filter design method of a multi-fetus uncertain singular network system.

Background

The singular system has been widely used in electric power systems, robot systems, electronic circuits, economic systems, etc., and is classified into a continuous singular system and a discrete singular system. The british scholars rosenblock studied the decoupling zero point of the singular system and the first equivalence of the system in 1974, and put forward the concept of the singular system for the first time. Then, the Luenberger explores the uniqueness of the existence of the solution of the singular linear system; cobb provides the energy control, energy observation and dual theory of the singular system; dai summarized the singular system base theory in 1989, which marks the formation of the singular system theory. The singular system is structurally divided into a linear singular system and a nonlinear singular system. When the differential coefficient matrix is reversible, the singular system is converted into a general normal system; if it is not reversible, the system can be converted into differential and algebraic systems, and the existence, uniqueness, etc. of the solution of the system are different and different from those of the general system.

Recent studies of event-driven control systems by scholars have focused on continuous and discrete linear systems, and mostly employ periodic sampling and self-triggering mechanisms, and the design of triggering control is mostly a given controller, and the design of triggering mechanisms and controllers together is not usually discussed. The filtering design based on event-driven networking is also concentrated on a linear system, and is concentrated on self-triggering and periodic sampling, a sensor collects a lot of unnecessary data and transmits the unnecessary data to a filter through a network, and the collected useless data occupy a large amount of memory, occupy a lot of broadband resources, waste resources, consume energy and reduce the working efficiency. The current singular systems of multiple births are mainly limited to the stability analysis, H, of the system_∞The control and state estimation problem is not reported at present about the state estimation and filtering design of a multi-cell singular system under event driving, the filtering design of the current linear discrete singular network system is mainly focused on the filtering design of the multi-cell singular system under a non-event driving mechanism, according to a periodic sampling strategy, a sensor needs to sample a time sequence according to a period, and samples, stores and filters all dataAnalysis, thus greatly wasting network consumption, increasing network resource consumption and increasing network operation cost. We pre-study the robustness H of the multiple-birth singular system under the event-driven mechanism_∞The filter design problem realizes the common design of the filter and the event driver, has certain anti-interference capability, and greatly reduces broadband resources and operation cost.

The invention researches the robust H of a multi-cell singular system under event driving_∞And (4) a filter design problem. According to an event-driven method, firstly, a discrete multi-cell type singular system network model with network induced time delay is derived; then, by introducing a relaxation matrix variable and a Lyapunov function method, the method gives that an error model is allowable and has H_∞The sufficiency criterion of the performance index; a filter equivalence method is applied, and the event trigger matrix and the filter are jointly designed; the effectiveness of the robust filter design method is verified by applying a direct current motor model example.

Disclosure of Invention

The invention provides a filter design method of a multi-birth uncertain singular network system, which can solve the problems in the prior art.

The invention provides a filter design method of a multi-fetus uncertain singular network system, which comprises the following steps:

deriving a network model of a filter of the multi-fetus uncertain singular network system;

the network model is proved to be performance-tolerant through a relaxation matrix variable;

the network model is proved to be asymptotically stable, regular and causal by a Lyapunov function method;

giving a sufficiency criterion that the error model is allowable and has an H infinite performance index;

and designing a gain matrix corresponding to the filter under the event drive by using a filter equivalent method to obtain the event-driven robust filter.

The method for deriving the network model of the filter of the multi-fetus uncertain singular network system comprises the following steps:

the following multi-cell type discrete time singular network system (DSNS) with perturbation was employed:

Ex(q+1)＝Ax(q)+Bw(q)， (1a)

y(q)＝Hx(q)， (1b)

z(q)＝Cx(q)+Dw(q)， (1c)

wherein x (q) e Rⁿ,y(q)∈R^p1,z(q)∈R^p2Respectively, state vector, output state and estimated signal; perturbation input w (q) ε R^p3Belong to₂A [0, + ∞) space, E being a space satisfying a rank (E) ═ r < n, and further, the multi-fetal coefficient matrices of DSNS (1a) - (1c) belong to the following convex set:

wherein:to representThe jth vertex of the middle polyhedron domain;

in order to save computing resources and communication cost, an event-driven design strategy is adopted, and information transmission is triggered and transmitted only when the current conditions are met:

(y(q)-y(τ_k))^TΩ(y(q)-y(τ_k))＞μy^T(q)Ωy(q)

wherein: mu is a given constant and mu epsilon [0,1), omega is the positive definite matrix to be designed, y (q) is the current sampling signal, tau_k(k ═ 0,1, …, + ∞) is the release time constant, τ_k+1Indicates the next trigger time,Denotes network induced delay, y (τ)_k) For the most recent transmitted signal, for any q e [ tau ]_k,τ_k+1-1]From the event generator above:

(y(q)-y(τ_k))^TΩ(y(q)-y(τ_k))≤μy^T(q)Ωy(q) (3)

the following full-order filter is designed:

to derive the event-driven error DSNS model, consider the following two cases:

a1: if it isThenDefinition of h (q) ═ q- τ_kAndtherefore, h (q) satisfies

A2 ifThere is a positive integerSatisfy the requirement of

And alsoThe following inequality holds:

(y(τ_k+ν)-y(τ_k))^TΩ(y(τ_k+ν)-y(τ_k))≤μy^T(τ_k+ν)Ωy(τ_k+ν)；

thus, the handle sectionThe method comprises the following steps:

wherein:in this case, the function h (q) is defined as:

and the difference between the current sampling signal and the latest transmission signalAs defined below:

therefore, it is easy to obtainIs provided withIf true;

from the group consisting of B1 and B2,is easy to obtain

By the event generator (3) and the above analysis,comprises the following steps:

thus, the filters (4a) and (4b) in the event-driven regime are represented as follows:

now, defineAndfrom (1a) to (1c) and (7a) and (7b), the following error DSNS is obtained:

wherein Γ ═ I_n 0]， To simplify writing, orderThene(q)＝Φ₂η (q), wherein:and

the above proves that the network model is performance-tolerant through the relaxation matrix variable, and the specific method comprises the following steps:

if det (sE-A) is not always equal to zero, thenAnd w (q) the DSNS (8a) model at [ identical to ] 0 is said to be canonical;

if deg (det (sE-a) ═ rank (e)), thenAnd w (q) the DSNS (8a) model at [ identical to ] 0 is said to be causal;

if the DSNS (8a) model is regular, causal and asymptotically stable; and DSNS (8a) and (8b) have H_∞The noise indicator γ, i.e. satisfying the constraint | e | at an initial value of zero₂≤γ‖w‖₂The DSNS (8a) and (8b) models are said to have performance index γ tolerance;

introduction 1: given matrix R > 0, M ═ M₁ M₂ M₃ M₄]And a function h (q), whereinAnd M₁＝[M₁₁M₁₂]Then the following inequality holds:

wherein:

given scalar quantity h> 0 and μ > 0, and if Ω > 0, γ > 0, P > 0,R＞0，A_f，B_f，D_f，M＝[M₁ M₂ M₃ M₄]the following inequality is made true:

the DSNS (8a) and (8b) network models are then allowed with performance index γ, where:

and matrixSatisfy the requirement ofAnd rank

The network model is proved to be a concrete method of asymptotic stability by the Lyapunov function method:

to demonstrate that DSNS (8a) and (8b) are asymptotically stable and have H_∞Noise index γ, candidate lyapunov function as follows:

wherein:

let Δ V (q) ═ V (q +1) -V (q), thenIs provided with

Wherein:and

by using the introduction 1 to obtain

ByCan obtain the productOr

η^T(q)γ₂η(q)＝0， (16)

Wherein:

known from (6) and (11) to (16):

wherein: phi₆＝diag{0,μH^TΩH,-Ω,-γ²I}, The use of schulk's complement γ < 0 is equivalent to (10); therefore, when w (q) is 0, the errors DSNS (8a) and (8b) Δ v (q) < 0, so the system is asymptotically stable;

when the initial value x (0) is 0, the value is obtained from (17)

Therefore, the temperature of the molten metal is controlled,de | e |₂≤γ‖w‖₂(ii) a On the other hand, from (10)

ByAnd further from (19)Are regular and causal, so the error DSNS (8a) and (8b) models are regular and causal.

The above error model is tolerable and has the adequacy criterion of H performance index as follows:

given scalar quantity h> 0 and μ > 0, then when Ω > 0, γ > 0, P_j＞0，R_j＞0，A_f，B_f，C_f，D_f，G₁，G₂，Andso that the following inequality holds:

the DSNS (8a) and (8b) network models are then allowed with performance index γ, where:

and matrixSatisfy the requirement ofAnd rank

The corresponding filter and trigger gain matrix under the drive of the design event is as follows:

given scalar quantity h> 0 and μ > 0, if Ω > 0, γ > 0,R_j＞0，M_11j，M_2j，M_3j，M_4j，L₁，L₂，L_3j，Y₁，and G₁，So that the following inequality holds:

the DSNS (8a) and (8b) models are then of H_∞Performance index γ allowed, where:

wherein:

the matrix S ∈ R^n×nSatisfies E^TS-0 and rank (S) -n-r.

The gain matrix of the corresponding filter under the event drive is designed, and the event drive robust filter is obtained as follows:

event driven robust H_∞The filter gain matrix is designed to:

and (3) proving that: letSys { G } can be found from (20)₂Is } > 0, thus Y₄Is an invertible matrix, defining:

J＝diag{J₁,I,I,I,I,I,I,J₁,I}，

reissue to order

ThenTherefore, equation (21) ensures that the DSNS (8a) and (8b) models are of H_∞Performance index γ allowed, note that the filter is fromToThe transfer function of (a) is:

from equation (22) and equation (23), equation (24) is equivalent to

Thus, equation (22) is the designed event-driven robust H_∞And a filter.

Compared with the prior art, the invention has the beneficial effects that:

the invention realizes the common design of the filter and the event driver by the event-driven design method, greatly reduces communication resources, has certain anti-interference capability, greatly reduces broadband resources and operation cost, and further greatly saves network communication cost.

Drawings

Fig. 1 is a frame of an event-driven filter design in a filter design method of a multiple-birth uncertain singular network system according to the present invention.

Fig. 2 is a schematic diagram of event-driven trigger time and interval in the filter design method of the multiple-birth uncertain singular network system provided by the present invention.

Fig. 3 is a schematic diagram of an output error in the filter design method of the multi-fetus uncertain singular network system provided by the present invention.

Detailed Description

An embodiment of the present invention will be described in detail below with reference to fig. 1-3, but it should be understood that the scope of the present invention is not limited to the embodiment.

The singular system has been widely used in electric power systems, robot systems, electronic circuits, economic systems, etc., and is classified into a continuous singular system and a discrete singular system. The invention researches the robust H of a multi-cell singular system under event driving_∞And (4) a filter design problem. According to an event-driven method, firstly, a discrete multi-cell type singular system network model with network induced time delay is derived; then, by introducing a relaxation matrix variable and a Lyapunov function method, the method gives that an error model is allowable and has H_∞The sufficiency criterion of the performance index; the event trigger matrix and the filter are jointly designed by applying filter equivalent skills; the effectiveness of the robust filter design strategy was verified using a dc model example.

The invention adopts the following multi-cell type discrete time singular network system (DSNS) with disturbance:

Ex(q+1)＝Ax(q)+Bw(q)， (1a)

y(q)＝Hx(q)， (1b)

z(q)＝Cx(q)+Dw(q)， (1c)

wherein x (q) e Rⁿ,y(q)∈R^p1Respectively is z (q) epsilon R^p2State vector, measurable output state, and estimated signal. Disturbance input w: (q)∈R^p3Belong to₂A [0, + ∞) space, E being a space satisfying a rank (E) ═ r < n, and further, the multi-fetal coefficient matrices of DSNS (1a) - (1c) belong to the following convex set:

wherein:to representThe jth vertex of the polyhedron domain.

In order to save computing resources and communication cost, the event-driven design strategy of fig. 1 is adopted by the invention, because of limited network broadband, the influence of network transmission time lag is inevitable, and the information transmission proposed by the invention is triggered and transmitted only when the following conditions are met:

(y(q)-y(τ_k))^TΩ(y(q)-y(τ_k))＞μy^T(q)Ωy(q)，

wherein: mu is a given constant and mu epsilon [0,1), omega is the positive definite matrix to be designed, y (q) is the current sampling signal, tau_k(k ═ 0,1, …, + ∞) is the release time constant, τ_k+1Indicates the next trigger time,Denotes network induced delay, y (τ)_k) The signal is transmitted newly. For any q ∈ [ tau ]_k,τ_k+1-1]Event generators from above are readily available:

(y(q)-y(τ_k))^TΩ(y(q)-y(τ_k))≤μy^T(q)Ωy(q) (3)

the following full-order filter is designed:

to derive the event-driven error DSNS model, consider the following two cases:

a1: if it isThenDefinition of h (q) ═ q- τ_kAndtherefore, h (q) satisfies

A2 ifThere is a positive integerSatisfy the requirement of And alsoThe following inequality holds:

(y(τ_k+ν)-y(τ_k))^TΩ(y(τ_k+ν)-y(τ_k))≤μy^T(τ_k+ν)Ωy(τ_k+ν)。

thus, the handle sectionSegmentationComprises the following steps:

wherein:in this case, the function h (q) can be defined as

And the difference between the current sampling signal and the latest transmission signalCan be defined as follows:

therefore, it is easy to obtainIs provided withThis is true.

From the group consisting of B1 and B2,is easy to obtain

By the event generator (3) and the above analysis,is provided with

Thus, the filters (4a) and (4b) in the event-driven regime are represented as follows:

now, defineAndfrom (1a) to (1c) and (7a) and (7b), the following error DSNS is liable to be obtained:

wherein Γ ═ I_n 0]，

To simplify writing, orderThen

e(q)＝Φ₂η(q)，

Wherein:and

to construct event-based driven finite time H_∞Filters, some necessary definitions and lemmas are given below.

Definitions 1 (canonical and causal)

(i) If det (sE-A) is not always equal to zero, thenAnd w (q) DSNS (8a) model at [ identical to ] 0 is said to be canonical.

(ii) If deg (det (sE-a) ═ rank (e)), thenAnd w (q) 0, the DSNS (8a) model is said to be causal.

Definition 2 if

(i) The DSNS (8a) model is regular, causal and asymptotically stable;

(ii) DSNS (8a) and (8b) has H_∞The noise indicator γ, i.e. satisfying the constraint | e | at an initial value of zero₂≤γ‖w‖₂The filter gain models of DSNS (8a) and (8b) are said to be performance index γ tolerant.

Lemma 1 gives a matrix R > 0, M ═ M₁ M₂ M₃ M₄]And a function h (q), whereinAnd M₁＝[M₁₁ M₁₂]Then the following inequality holds:

wherein:

event-driven filter analysis and design of 2-multocyte singular system

This section gives H for an event-driven multiple-birthmic network-based system_∞Filter analysis and design issues. First, applying lemma 1, relaxation matrix variables and variable replacement techniques, the network model given DSNS (8a) and (8b) is tolerant with performance index γ.

Theorem 1 gives a scalar quantity h> 0 and μ > 0. If omega > 0, gamma > 0, P > 0,R＞0，A_f，B_f，D_f，M＝[M₁ M₂ M₃ M₄]the following inequality is made true:

the DSNS (8a) and (8b) network models are then allowed with performance index γ, where:

and matrixSatisfy the requirement ofAnd rank

And (3) proving that: first, we demonstrate that DSNS (8a) and (8b) are asymptotically stable and have H_∞The noise index γ.

The following Lyapunov functions are candidate:

wherein:

let Δ V (q) ═ V (q +1) -V (q), thenIs provided with

Wherein:

by theory 1, can obtain

ByCan obtain the productOr

η^T(q)γ₂η(q)＝0， (16)

Wherein:

from (6) and (11) to (16)

Wherein: phi₆＝diag{0,μH^TΩH,-Ω,-γ²I}, By using Schulk's complement, it is known that γ < 0 is equivalent to (10). Therefore, when w (q) is 0, the errors DSNS (8a) and (8b) Δ v (q) < 0, and thus the system is asymptotically stable. When the initial value x (0) is 0, it can be obtained from (17)

Therefore, the temperature of the molten metal is controlled,can obtain | e |₂≤γ‖w‖₂。

On the other hand, from (10) can be obtained

ByFurther obtainable from (19)Are both canonical and causal. The error DSNS (8a) and (8b) models are both regularization and causality. After the syndrome is confirmed.

From theorem 1 and the matrix transformation technique, the following results can be obtained,

theorem 2 given scalar h> 0 and μ > 0. Then P is present when Ω > 0, γ > 0_j＞0，R_j＞0，A_f，B_f，C_f，D_f，G₁，G₂，And so that the following inequality holds:

the DSNS (8a) and (8b) network filter gain models are then tolerant with the performance index γ, where:

and matrixSatisfy the requirement ofAnd rank

From theorem 2 and filter equivalence techniques, the following results can be obtained and the corresponding filter and trigger gain matrix under event drive are designed.

Theorem 3 given scalar quantity h> 0 and μ > 0. If omega > 0, gamma > 0, R_j＞0，M_11j，M_2j，M_3j，M_4j，L₁，L₂，L_3j，Y₁，and G₁，So that the following inequality holds:

the DSNS (8a) and (8b) network filter gain models are then tolerant with the performance index γ, where:

the matrix S ∈ R^n×nSatisfies E^TS-0 and rank (S) -n-r; moreover, event driven robustness H_∞The filter gain matrix can be designed as:

and (3) proving that: letSys { G } can be found from (20)₂Is } > 0, thus Y₄Is a reversible momentAnd (5) arraying. Definition ofJ＝diag{J₁,I,I,I,I,I,I,J₁,I}，

Reissue to order

ThenTherefore, equation (21) ensures that the DSNS (8a) and (8b) models are of H_∞Performance index γ allowed, note that the filter is fromToThe transfer function of (a) is:

from equation (22) and equation (23), equation (24) is equivalent to

Thus, equation (22) is the designed event-driven robust H_∞And a filter.

3 numerical calculation example

This section gives a numerical example to verify the validity of the results.

Example 1 consider the following model of a dc motor, which can be described as

y(t)＝Hx(t)， (25b)

z(t)＝Cx(t)+Dw(t)， (25c)

Wherein: h ═ 21],C＝[1 2],D＝0,δJ∈[0.2,0.4],

Thus, (25a) - (25c) can be discretized into the following discrete-time description system:

wherein

If the sampling period is selected to be T-0.1. Let R equal to 4, b equal to 0.5, K_w＝2,J＝3.8,K_t＝1,μ＝0.16,h＝2,From theorem 3, it is easy to obtain

Ω＝1.7903×10³,γ＝1.7040。

Thus, event driven robust H_∞The gain of the filter can be designed as

C_f＝[-0.0001 -0.0410],D_f＝-1.8109

Now given w (q) ═ e^-0.1qAnd initial conditionsNext, the release moments and intervals and the output errors of the DSNS (8a) and (8b) are shown in FIGS. 2 and 3, and the singular system model of the multiple-fetus uncertain error is tolerable with a certain H_∞Performance index. Meanwhile, only 14 times of triggering are performed in 100 times of simulation, so that 86% of communication resources are reduced, network communication cost is greatly saved, and the effectiveness of the filter design method under event driving is verified.

The above disclosure is only for a few specific embodiments of the present invention, however, the present invention is not limited to the above embodiments, and any variations that can be made by those skilled in the art are intended to fall within the scope of the present invention.