CN113625647A - Joint Design Method of Event Driver and DOFFSS Controller for Nonlinear Systems - Google Patents

Joint Design Method of Event Driver and DOFFSS Controller for Nonlinear Systems Download PDF

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CN113625647A
CN113625647A CN202110962406.8A CN202110962406A CN113625647A CN 113625647 A CN113625647 A CN 113625647A CN 202110962406 A CN202110962406 A CN 202110962406A CN 113625647 A CN113625647 A CN 113625647A
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CN113625647B (en
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李富强
郜丽赛
祁诗阳
谷小青
张益维
郑宝周
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Henan Agricultural University
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Abstract

The invention discloses a nonlinear system event driver and DOFSS controller joint design method, which comprises the following steps: a, establishing a nonlinear object model, an actuator saturation model and an event driver based on nonlinear object measurement output; b, establishing a random deception attack model, a DOFSS controller model and a closed-loop system model organically integrating random deception attack, an event driver, actuator saturation and network induced delay parameters; and C, designing joint design conditions of the nonlinear system event driver and the DOFSS controller under the influence of random deception attack, actuator saturation and network induced delay, solving parameters of the event driver and a gain matrix of the equivalent DOFSS controller, and finally obtaining the event driver and the DOFSS controller. The invention can solve the problem that the existing nonlinear system cannot be stable under the influence of random deception attack, actuator saturation and network induced delay.

Description

非线性系统事件驱动器与DOFFSS控制器联合设计法Joint Design Method of Event Driver and DOFFSS Controller for Nonlinear Systems

技术领域technical field

本发明涉及网络化控制系统领域，尤其涉及一种随机欺骗攻击下非线性系统事件驱动器与动态输出反馈模糊饱和安全(dynamic output feedback fuzzy saturatedsecurity,DOFFSS)控制器联合设计方法。The invention relates to the field of networked control systems, in particular to a joint design method for a nonlinear system event driver and a dynamic output feedback fuzzy saturated security (DOFFSS) controller under random spoofing attacks.

背景技术Background technique

网络化控制系统将共享通信网络引入控制闭环，具有柔性高、成本低及安装维护方便等优点，广泛应用于智能电网、智慧交通等领域。网络化控制系统通常采用发展成熟的周期采样控制策略，为了在最坏情形下保证系统性能，采样率通常设置较高。然而，实际中最坏情形很少发生，高采样率产生的大量冗余数据导致网络带宽等系统受限资源浪费，极大地影响了系统性能。The networked control system introduces the shared communication network into the control loop, which has the advantages of high flexibility, low cost, convenient installation and maintenance, etc., and is widely used in smart grid, smart transportation and other fields. The networked control system usually adopts a well-developed periodic sampling control strategy. In order to ensure the system performance in the worst case, the sampling rate is usually set higher. However, the worst situation rarely occurs in practice, and the large amount of redundant data generated by high sampling rate leads to the waste of limited system resources such as network bandwidth, which greatly affects the system performance.

不同于周期控制策略忽略系统动态进行按时控制，事件驱动控制策略仅当系统动态满足事件驱动条件时进行按需控制，从而能够节约网络带宽等系统受限资源。然而，现有事件驱动器通常采用连续时间事件驱动机制，该机制需要增加专用监测硬件，且需要复杂前期计算以避免芝诺现象(即有限时间内的无穷多次驱动采样)。Different from periodic control strategy ignoring system dynamics to perform on-time control, event-driven control strategy performs on-demand control only when system dynamics meet event-driven conditions, thereby saving limited system resources such as network bandwidth. However, existing event drivers usually use a continuous-time event-driven mechanism, which requires the addition of dedicated monitoring hardware and complex up-front calculations to avoid Zeno phenomenon (ie, infinitely many driving samples in a finite time).

虽然共享通信网络为网络化控制系统带来了诸多便利，但也引入了使系统性能变差的网络诱导延时，也使系统面临网络攻击威胁。网络攻击大致分为拒绝服务攻击和欺骗攻击，拒绝服务攻击通过阻断通信网络使数据包无法送达；欺骗攻击通过数据篡改产生虚假数据包，欺骗攻击隐蔽性强且危害大，为本发明研究的攻击类型。然而，现有成果重点关注如何设计事件驱动机制以节约更多系统资源，较少同时考虑欺骗攻击和网络诱导延时影响。另外，现有事件驱动控制系统分析与综合方法通常假设不存在攻击威胁，此类方法通常不能直接适用于欺骗攻击影响下的事件触发控制系统分析。Although the shared communication network brings many conveniences to the networked control system, it also introduces a network-induced delay that deteriorates the system performance, and also makes the system face the threat of network attacks. Network attacks are roughly divided into denial of service attacks and spoofing attacks. Denial of service attacks prevent data packets from being delivered by blocking the communication network; spoofing attacks generate false data packets through data tampering, and spoofing attacks have strong concealment and great harm. type of attack. However, existing results focus on how to design event-driven mechanisms to save more system resources, and less consider both spoofing attacks and network-induced delay effects. In addition, existing event-driven control system analysis and synthesis methods usually assume that there is no attack threat, and such methods are usually not directly applicable to event-triggered control system analysis under the influence of spoofing attacks.

现实中，执行器饱和是控制系统中普遍存在的非线性现象。如果执行器输入量超过饱和阈值，执行器进入饱和状态。在饱和状态下，进一步增加执行器输入量对执行器输出不能产生任何影响。然而，现有成果较少同时考虑执行器饱和及欺骗攻击影响。此外，现有成果通常假设系统状态完全可测并设计状态反馈控制器，然而实际中系统状态通常不能直接获取。In reality, actuator saturation is a ubiquitous nonlinear phenomenon in control systems. If the actuator input exceeds the saturation threshold, the actuator goes into saturation. In the saturated state, further increasing the actuator input cannot have any effect on the actuator output. However, few existing works consider both the effect of actuator saturation and spoofing attacks. In addition, the existing works usually assume that the system state is completely measurable and design a state feedback controller, however, the system state cannot be directly obtained in practice.

为了解决上述问题，同时考虑随机欺骗攻击、执行器饱和、网络诱导延时及对象状态不能直接测量影响，本发明提出了非线性系统事件驱动器与DOFFSS控制器联合设计方法。In order to solve the above problems, while considering the influence of random spoofing attack, actuator saturation, network induced delay and object state cannot be measured directly, the present invention proposes a joint design method of nonlinear system event driver and DOFFSS controller.

发明内容SUMMARY OF THE INVENTION

本发明的目的是提供一种非线性系统事件驱动器与DOFFSS控制器联合设计法，能够解决现有非线性系统在随机欺骗攻击、执行器饱和及网络诱导延时影响下不能稳定的问题，并能够有效节约网络带宽等系统受限资源，而且能够解除对系统状态完全可测的假设限制。The purpose of the present invention is to provide a joint design method of nonlinear system event driver and DOFFSS controller, which can solve the problem that the existing nonlinear system cannot be stabilized under the influence of random spoofing attack, actuator saturation and network induced delay, and can It can effectively save system-limited resources such as network bandwidth, and can remove the assumption that the system state is completely measurable.

本发明采用下述技术方案：The present invention adopts following technical scheme:

一种非线性系统事件驱动器与DOFFSS控制器联合设计法，包括以下步骤：A non-linear system event driver and DOFFSS controller joint design method, including the following steps:

A:建立非线性对象模型和执行器饱和模型，并设计基于非线性对象测量输出的事件驱动器；A: Establish nonlinear object model and actuator saturation model, and design an event driver based on nonlinear object measurement output;

B:建立随机欺骗攻击模型及DOFFSS控制器模型，并建立有机融合随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时参数的闭环系统模型；B: Build a random spoofing attack model and a DOFFSS controller model, and build a closed-loop system model that organically integrates random spoofing attacks, event drivers, actuator saturation, and network-induced delay parameters;

C:设计随机欺骗攻击、执行器饱和及网络诱导延时影响下非线性系统事件驱动器与DOFFSS控制器联合设计条件，求出满足非线性系统通信和控制需求的事件驱动器参数

Figure BDA0003222564250000021
和等价DOFFSS控制器的增益矩阵
Figure BDA0003222564250000022
最终得到满足非线性系统通信和控制需求的事件驱动器和DOFFSS控制器。C: Design the joint design conditions of the nonlinear system event driver and DOFFSS controller under the influence of random spoofing attack, actuator saturation and network-induced delay, and find the event driver parameters that meet the communication and control requirements of the nonlinear system
Figure BDA0003222564250000021
and the gain matrix of the equivalent DOFFSS controller
Figure BDA0003222564250000022
Finally, an event driver and DOFFSS controller that meet the communication and control requirements of nonlinear systems are obtained.

所述的步骤A中，非线性对象模型为：In the described step A, the nonlinear object model is:

Figure BDA0003222564250000031
Figure BDA0003222564250000031

其中，

Figure BDA0003222564250000032
为x(t)的导数，x(t)表示对象状态，x(t)为n维实数，
Figure BDA0003222564250000033
表示受执行器饱和影响的控制输入，
Figure BDA00032225642500000316
为nu维实数，y(t)表示测量输出，y(t)为ny维实数，t 表示时间，Ai,Bi和Ci表示增益矩阵；对象模糊规则数目为r，i表示对象模糊规则序号；替代式
Figure BDA0003222564250000034
且
Figure BDA0003222564250000035
Figure BDA0003222564250000036
和θg(t)分别表示第1个、第
Figure BDA0003222564250000037
个和第g个前件变量，g表示前件变量数目，
Figure BDA0003222564250000038
表示前件变量序号,
Figure BDA0003222564250000039
表示前件变量
Figure BDA00032225642500000310
隶属于模糊集
Figure BDA00032225642500000311
的隶属度函数，Σ和Π分别表示累加和累乘运算。in,
Figure BDA0003222564250000032
is the derivative of x(t), x(t) represents the state of the object, x(t) is an n-dimensional real number,
Figure BDA0003222564250000033
represents the control input affected by actuator saturation,
Figure BDA00032225642500000316
is n u -dimensional real number, y(t) represents measurement output, y(t) is n y -dimensional real number, t represents time, A i , B i and C i represent gain matrix; the number of object fuzzy rules is r, i represents object Fuzzy rule number; alternative
Figure BDA0003222564250000034
and
Figure BDA0003222564250000035
Figure BDA0003222564250000036
and θ g (t) represent the first, the first
Figure BDA0003222564250000037
and the g-th antecedent variable, where g represents the number of antecedent variables,
Figure BDA0003222564250000038
Indicates the serial number of the antecedent variable,
Figure BDA0003222564250000039
represents the antecedent variable
Figure BDA00032225642500000310
belongs to fuzzy set
Figure BDA00032225642500000311
The membership function of , Σ and Π represent the accumulation and accumulation operations, respectively.

所述的步骤A中，基于非线性对象测量输出的事件驱动器为：In the described step A, the event driver based on the nonlinear object measurement output is:

Figure BDA00032225642500000312
Figure BDA00032225642500000312

式中，δ∈(0,1)为驱动器阈值参数，

Figure BDA00032225642500000317
为正定矩阵，h表示采样周期，tkh表示第k个驱动时刻，tkh为采样周期的tk倍，tk+1h表示第k+1个驱动时刻，tk+1h为采样周期的tk+1倍，下角标k表示驱动时刻序号，
Figure BDA00032225642500000313
表示当前采样时刻，
Figure BDA00032225642500000314
为tkh 后第
Figure BDA00032225642500000318
个采样周期，y(tkh)和
Figure BDA00032225642500000319
分别表示tkh和
Figure BDA00032225642500000320
对应的非线性对象测量输出，min{}表示最小值函数，矩阵的右上角标T表示矩阵的转置。where δ∈(0,1) is the driver threshold parameter,
Figure BDA00032225642500000317
is a positive definite matrix, h represents the sampling period, t k h represents the kth driving moment, t k h is t k times the sampling period, t k+1 h represents the k+1 th driving moment, and t k+1 h is t k+1 times the sampling period, the subscript k represents the drive time sequence number,
Figure BDA00032225642500000313
represents the current sampling time,
Figure BDA00032225642500000314
is the first after t k h
Figure BDA00032225642500000318
sampling period, y(t k h) and
Figure BDA00032225642500000319
represent t k h and
Figure BDA00032225642500000320
The corresponding nonlinear object measurement output, min{} represents the minimum function, and the upper right corner of the matrix T represents the transpose of the matrix.

所述的步骤A中，执行器饱和模型为：In the described step A, the actuator saturation model is:

Figure BDA00032225642500000315
Figure BDA00032225642500000315

其中，

Figure BDA00032225642500000321
表示u(t)的第
Figure BDA00032225642500000322
维分量，u(t)表示不考虑执行器饱和影响的对象控制输入，u(t)为nu维实数，
Figure BDA00032225642500000323
表示u(t)的维数序号，
Figure BDA00032225642500000324
和
Figure BDA00032225642500000325
分别表示执行器最大允许输出值和最小允许输出值，
Figure BDA00032225642500000326
表示
Figure BDA00032225642500000327
对应的执行器饱和函数值，sat()表示执行器饱和函数。所述的步骤B中，随机欺骗攻击模型为：in,
Figure BDA00032225642500000321
represents the first order of u(t)
Figure BDA00032225642500000322
dimensional component, u(t) represents the object control input without considering the effect of actuator saturation, u(t) is a n u -dimensional real number,
Figure BDA00032225642500000323
represents the dimension number of u(t),
Figure BDA00032225642500000324
and
Figure BDA00032225642500000325
respectively represent the maximum allowable output value and the minimum allowable output value of the actuator,
Figure BDA00032225642500000326
express
Figure BDA00032225642500000327
The corresponding actuator saturation function value, sat() represents the actuator saturation function. In the described step B, the random spoofing attack model is:

Figure BDA0003222564250000041
Figure BDA0003222564250000041

其中，，

Figure BDA0003222564250000042
表示欺骗攻击函数，a(t)∈{0,1}表示伯努利分布随机变量，当 a(t)＝1时，欺骗攻击激活，控制器输入被篡改；当a(t)＝0时，欺骗攻击休眠，控制器输入未被篡改；
Figure BDA0003222564250000043
G为攻击能量限定矩阵；
Figure BDA00032225642500000416
y(t-η(t))表示时刻t-η(t)对应的非线性对象测量输出，t-η(t)＝tkh+nkh，e(t)＝y(tkh)-y(tkh+nkh)，y(tkh+nkh)表示采样时刻 tkh+nkh对应的非线性对象测量输出。in,,
Figure BDA0003222564250000042
represents the spoofing attack function, a(t)∈{0,1} represents a Bernoulli distributed random variable, when a(t)=1, the spoofing attack is activated, and the controller input is tampered with; when a(t)=0 , the spoofing attack is dormant, and the controller input has not been tampered with;
Figure BDA0003222564250000043
G is the attack energy limit matrix;
Figure BDA00032225642500000416
y(t-η(t)) represents the nonlinear object measurement output corresponding to time t-η(t), t-η(t)=t k h+n k h, e(t)=y(t k h )-y(t k h+n k h), y(t k h+n k h) represents the nonlinear object measurement output corresponding to the sampling time t k h+n k h.

所述的步骤B中，DOFFSS控制器模型为：In the described step B, the DOFFSS controller model is:

Figure BDA0003222564250000044
Figure BDA0003222564250000044

其中，

Figure BDA00032225642500000417
为xc(t)的导数，xc(t)表示控制器状态，xc(t)为n维实数，xc(t-η(t))表示t-η(t)对应的控制器状态，
Figure BDA0003222564250000045
和
Figure BDA0003222564250000046
为增益矩阵；控制器模糊规则数目为r，j表示控制器模糊规则序号，替代式
Figure BDA0003222564250000047
且
Figure BDA0003222564250000048
Figure BDA0003222564250000049
表示前件变量
Figure BDA00032225642500000410
隶属于模糊集
Figure BDA00032225642500000411
的隶属度函数。in,
Figure BDA00032225642500000417
is the derivative of x c (t), x c (t) represents the controller state, x c (t) is an n-dimensional real number, and x c (t-η(t)) represents the controller corresponding to t-η(t) state,
Figure BDA0003222564250000045
and
Figure BDA0003222564250000046
is the gain matrix; the number of fuzzy rules of the controller is r, j represents the sequence number of the fuzzy rules of the controller, and the substitution formula
Figure BDA0003222564250000047
and
Figure BDA0003222564250000048
Figure BDA0003222564250000049
represents the antecedent variable
Figure BDA00032225642500000410
belongs to fuzzy set
Figure BDA00032225642500000411
membership function.

所述的步骤B中，有机融合随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时参数的闭环系统模型为：In the described step B, the closed-loop system model that organically integrates random spoofing attacks, event drivers, actuator saturation and network-induced delay parameters is:

Figure BDA00032225642500000412
Figure BDA00032225642500000412

式中，

Figure BDA00032225642500000413
为χ(t)的导数，
Figure BDA00032225642500000414
表示闭环系统状态，χ(t-η(t))表示t-η(t)对应的闭环系统状态,
Figure BDA00032225642500000415
和
Figure BDA0003222564250000051
表示闭环系统增益矩阵，替代式
Figure BDA0003222564250000052
Figure BDA0003222564250000053
和
Figure BDA0003222564250000054
分别表示u1,
Figure BDA00032225642500000514
和
Figure BDA0003222564250000055
对应的执行器死区函数值，u1,
Figure BDA00032225642500000515
和
Figure BDA00032225642500000513
分别表示u(t)的第1维、第
Figure BDA00032225642500000517
维和第nu维分量，
Figure BDA00032225642500000516
表示执行器死区函数。In the formula,
Figure BDA00032225642500000413
is the derivative of χ(t),
Figure BDA00032225642500000414
represents the state of the closed-loop system, χ(t-η(t)) represents the state of the closed-loop system corresponding to t-η(t),
Figure BDA00032225642500000415
and
Figure BDA0003222564250000051
represents the closed-loop system gain matrix, the alternative
Figure BDA0003222564250000052
Figure BDA0003222564250000053
and
Figure BDA0003222564250000054
respectively represent u 1 ,
Figure BDA00032225642500000514
and
Figure BDA0003222564250000055
Corresponding actuator dead zone function value, u 1 ,
Figure BDA00032225642500000515
and
Figure BDA00032225642500000513
represent the first dimension and the first dimension of u(t), respectively.
Figure BDA00032225642500000517
dimension and the n-th u -dimension component,
Figure BDA00032225642500000516
Represents the actuator deadband function.

所述的步骤C包括以下具体步骤：Described step C includes following concrete steps:

C1：基于李雅普诺夫稳定性理论，确定随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时影响下的非线性系统渐近稳定条件；C1: Based on Lyapunov stability theory, determine the asymptotic stability conditions of nonlinear systems under the influence of random spoofing attacks, event drivers, actuator saturation, and network-induced delays;

C2:基于步骤C1得出的非线性系统渐近稳定条件，利用线性矩阵不等式技术，得到事件驱动器与DOFFSS控制器联合设计条件，即求出满足非线性系统通信和控制需求的事件驱动器参数

Figure BDA00032225642500000518
和等价DOFFSS控制器的增益矩阵
Figure BDA0003222564250000056
最终得到满足非线性系统通信和控制需求的事件驱动器和DOFFSS 控制器。C2: Based on the asymptotic stability conditions of the nonlinear system obtained in step C1, using the linear matrix inequality technique, the joint design conditions of the event driver and the DOFFSS controller are obtained, that is, the event driver parameters that meet the communication and control requirements of the nonlinear system are obtained.
Figure BDA00032225642500000518
and the gain matrix of the equivalent DOFFSS controller
Figure BDA0003222564250000056
Finally, an event driver and DOFFSS controller that meet the communication and control requirements of nonlinear systems are obtained.

所述的随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时影响下的非线性系统渐近稳定条件为：The asymptotic stability conditions of nonlinear systems under the influence of random spoofing attacks, event drivers, actuator saturation, and network-induced delays are:

给定采样周期h,网络诱导延时下界τ和上界

Figure BDA00032225642500000519
执行器饱和参数ε,攻击能量限定矩阵G和事件驱动器阈值参数δ∈(0,1),如果存在正定矩阵
Figure BDA00032225642500000520
R＞0,S＞0,Q1＞0,Q2＞0,Q3＞0，及矩阵U2,U3，满足
Figure BDA0003222564250000057
以及Given the sampling period h, the lower bound τ and the upper bound of the network-induced delay
Figure BDA00032225642500000519
The actuator saturation parameter ε, the attack energy limiting matrix G and the event driver threshold parameter δ∈(0,1), if there is a positive definite matrix
Figure BDA00032225642500000520
R>0, S>0, Q 1 >0, Q 2 >0, Q 3 >0, and matrices U 2 , U 3 , satisfying
Figure BDA0003222564250000057
as well as

Figure BDA0003222564250000058
Figure BDA0003222564250000058

则随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时影响下闭环系统是渐近稳定的；上述稳定性条件中，使用了替代式如下：Then the closed-loop system is asymptotically stable under the influence of random spoofing attacks, event drivers, actuator saturation, and network-induced delays; in the above stability conditions, the following alternatives are used:

Figure BDA0003222564250000059
Figure BDA0003222564250000059

Figure BDA00032225642500000510
Figure BDA00032225642500000510

Figure BDA00032225642500000511
Figure BDA00032225642500000511

Figure BDA00032225642500000512
Figure BDA00032225642500000512

Figure BDA0003222564250000061
Figure BDA0003222564250000061

Figure BDA0003222564250000062
Figure BDA0003222564250000062

Figure BDA0003222564250000063
Figure BDA0003222564250000063

Figure BDA0003222564250000064
Figure BDA0003222564250000064

其中，

Figure BDA00032225642500000613
表示a(t)的数学期望，
Figure BDA00032225642500000614
为数学期望函数。He{}表示矩阵与其转置矩阵的和，*表示对称矩阵中的对称项，
Figure BDA00032225642500000615
表示零矩阵，矩阵右上角标-1表示逆矩阵，I为单位矩阵，col{}表示列矩阵，diag{}表示对角矩阵。in,
Figure BDA00032225642500000613
represents the mathematical expectation of a(t),
Figure BDA00032225642500000614
is the mathematical expectation function. He{} represents the sum of the matrix and its transposed matrix, * represents the symmetric term in the symmetric matrix,
Figure BDA00032225642500000615
Represents a zero matrix, the upper right corner of the matrix -1 represents the inverse matrix, I is the identity matrix, col{} represents the column matrix, and diag{} represents the diagonal matrix.

所述的事件驱动器与DOFFSS控制器联合设计条件为：The joint design conditions of the event driver and the DOFFSS controller are:

给定采样周期h，网络诱导延时下界τ和上界

Figure BDA00032225642500000616
执行器饱和参数ε和攻击能量限定矩阵G，实数∈1＞0,∈2＞0,∈3＞0，如果存在实数
Figure BDA00032225642500000617
正定矩阵
Figure BDA00032225642500000618
Figure BDA0003222564250000065
及矩阵
Figure BDA0003222564250000066
满足
Figure BDA0003222564250000067
Figure BDA0003222564250000068
以及
Figure BDA0003222564250000069
Given the sampling period h, the network-induced delay lower bound τ and upper bound
Figure BDA00032225642500000616
Actuator saturation parameter ε and attack energy limit matrix G, real numbers ∈ 1 > 0, ∈ 2 > 0, ∈ 3 > 0, if there are real numbers
Figure BDA00032225642500000617
positive definite matrix
Figure BDA00032225642500000618
Figure BDA0003222564250000065
and matrix
Figure BDA0003222564250000066
Satisfy
Figure BDA0003222564250000067
Figure BDA0003222564250000068
as well as
Figure BDA0003222564250000069

则闭环系统是渐进稳定的，同时得到事件驱动器参数

Figure BDA00032225642500000610
和等价 DOFFSS控制器的增益矩阵
Figure BDA00032225642500000611
如下Then the closed-loop system is asymptotically stable, and the event-driven parameters are obtained at the same time
Figure BDA00032225642500000610
and the gain matrix of the equivalent DOFFSS controller
Figure BDA00032225642500000611
as follows

Figure BDA00032225642500000612
Figure BDA00032225642500000612

上述条件中，使用的替代式如下In the above conditions, the alternatives used are as follows

Figure BDA0003222564250000071
Figure BDA0003222564250000071

Figure BDA0003222564250000072
Figure BDA0003222564250000072

Figure BDA0003222564250000073
Figure BDA0003222564250000073

Figure BDA0003222564250000074
Figure BDA0003222564250000074

Figure BDA0003222564250000075
Figure BDA0003222564250000075

Figure BDA0003222564250000076
Figure BDA0003222564250000076

Figure BDA0003222564250000077
Figure BDA0003222564250000077

Figure BDA0003222564250000078
Figure BDA0003222564250000078

Figure BDA0003222564250000079
Figure BDA0003222564250000079

Figure BDA00032225642500000710
为正定矩阵,N为n×n维实数矩阵。
Figure BDA00032225642500000710
is a positive definite matrix, and N is an n×n-dimensional real number matrix.

本发明能够解决现有非线性系统在随机欺骗攻击、执行器饱和及网络诱导延时影响下不能稳定的问题，并能够有效节约网络带宽等系统受限资源，而且能够解除对系统状态完全可测的假设限制。The invention can solve the problem that the existing nonlinear system cannot be stabilized under the influence of random spoofing attack, actuator saturation and network induced delay, and can effectively save system limited resources such as network bandwidth, and can eliminate the completely measurable system state. hypothetical limit.

附图说明Description of drawings

图1为本发明中欺骗攻击下非线性系统事件驱动输出反馈控制示意图；1 is a schematic diagram of a nonlinear system event-driven output feedback control under spoofing attack in the present invention;

图2为本发明的流程示意图。FIG. 2 is a schematic flow chart of the present invention.

具体实施方式Detailed ways

以下结合附图和实施例对本发明作以详细的描述：Below in conjunction with accompanying drawing and embodiment, the present invention is described in detail:

随机欺骗攻击下非线性系统事件驱动输出反馈控制系统如图1所示，传感器对非线性对象测量输出进行周期采样，传感器采样数据发送至事件驱动器，事件驱动器判断是否满足事件驱动条件：若满足，则发送采样数据；否则，丢弃采样数据。事件驱动器发送数据经由通信网络发送至零阶保持器，通信网络受随机欺骗攻击影响，DOFFSS控制器接收零阶保持器数据并计算控制信号，执行器根据控制信号调整对象状态，并考虑执行器饱和影响。The nonlinear system event-driven output feedback control system under random spoofing attack is shown in Figure 1. The sensor periodically samples the measurement output of the nonlinear object, and the sensor sampling data is sent to the event driver. The event driver determines whether the event-driven condition is met: if so, The sampled data is sent; otherwise, the sampled data is discarded. The event driver sends data to the zero-order keeper via the communication network. The communication network is affected by random spoofing attacks. The DOFFSS controller receives the zero-order keeper data and calculates the control signal. The actuator adjusts the object state according to the control signal and considers actuator saturation. influences.

如图2所示，本发明所述的非线性系统事件驱动器与DOFFSS控制器联合设计法，包括以下步骤：As shown in Figure 2, the non-linear system event driver and DOFFSS controller joint design method of the present invention includes the following steps:

A:建立非线性对象模型和执行器饱和模型，并设计基于非线性对象测量输出的事件驱动器；A: Establish nonlinear object model and actuator saturation model, and design an event driver based on nonlinear object measurement output;

其中，在将非线性对象模型描述为T-S(Takagi-Sugeno)模糊系统时：Among them, when describing the nonlinear object model as a T-S (Takagi-Sugeno) fuzzy system:

设非线性对象第i个模糊规则表示如下：若θ1(t)为Mi1,...,

Figure BDA0003222564250000085
为
Figure BDA0003222564250000086
..., θg(t)为Mig，则Let the ith fuzzy rule of nonlinear object be expressed as follows: If θ 1 (t) is M i1 ,...,
Figure BDA0003222564250000085
for
Figure BDA0003222564250000086
..., θ g (t) is Mig , then

Figure BDA0003222564250000081
Figure BDA0003222564250000081

式中，

Figure BDA0003222564250000087
为x(t)的导数，x(t)表示对象状态，x(t)为n维实数，
Figure BDA0003222564250000088
表示受执行器饱和影响的控制输入，
Figure BDA0003222564250000089
为nu维实数，y(t)表示测量输出，y(t)为ny维实数，t 表示时间，Ai,Bi和Ci表示增益矩阵；对象模糊规则数目为r，i表示对象模糊规则序号；θ1(t),
Figure BDA00032225642500000810
和θg(t)分别表示第1个、第k个和第g个前件变量，g表示前件变量数目，
Figure BDA00032225642500000811
表示前件变量序号,Mi1,
Figure BDA00032225642500000812
和Mig分别表示对象第i个模糊规则下第1个、第
Figure BDA00032225642500000813
个和第g个模糊集。In the formula,
Figure BDA0003222564250000087
is the derivative of x(t), x(t) represents the state of the object, x(t) is an n-dimensional real number,
Figure BDA0003222564250000088
represents the control input affected by actuator saturation,
Figure BDA0003222564250000089
is n u -dimensional real number, y(t) represents measurement output, y(t) is n y -dimensional real number, t represents time, A i , B i and C i represent gain matrix; the number of object fuzzy rules is r, i represents object Fuzzy rule number; θ 1 (t),
Figure BDA00032225642500000810
and θ g (t) represent the 1st, kth and gth antecedent variables, respectively, and g represents the number of antecedent variables,
Figure BDA00032225642500000811
Indicates the serial number of the antecedent variable, M i1 ,
Figure BDA00032225642500000812
and M ig respectively represent the first and third objects under the i-th fuzzy rule of the object.
Figure BDA00032225642500000813
and the gth fuzzy sets.

使用乘积模糊推理、中心平均解模糊器和单点模糊器，得到T-S模糊系统表示的非线性对象模型如下Using product fuzzy inference, center-average defuzzifier, and single-point fuzzer, the nonlinear object model represented by the T-S fuzzy system is obtained as follows

Figure BDA0003222564250000082
Figure BDA0003222564250000082

式中，替代式

Figure BDA0003222564250000083
且
Figure BDA0003222564250000084
Figure BDA00032225642500000814
表示前件变量
Figure BDA00032225642500000816
隶属于模糊集
Figure BDA00032225642500000815
的隶属度函数，Σ和Π分别表示累加和累乘运算。In the formula, the alternative
Figure BDA0003222564250000083
and
Figure BDA0003222564250000084
Figure BDA00032225642500000814
represents the antecedent variable
Figure BDA00032225642500000816
belongs to fuzzy set
Figure BDA00032225642500000815
The membership function of , Σ and Π represent the accumulation and accumulation operations, respectively.

在设计基于非线性对象测量输出的事件驱动器时：When designing event drivers that measure outputs based on nonlinear objects:

基于非线性对象的测量输出，设计事件驱动器为：Based on the measured output of the nonlinear object, the design event driver is:

Figure BDA0003222564250000091
Figure BDA0003222564250000091

式中，δ∈(0,1)为驱动器阈值参数，

Figure BDA00032225642500000911
为正定矩阵，h表示采样周期，tkh表示第k个驱动时刻，tkh为采样周期的tk倍，tk+1h表示第k+1个驱动时刻，tk+1h为采样周期的tk+1倍，下角标k表示驱动时刻序号。
Figure BDA0003222564250000092
表示当前采样时刻，
Figure BDA0003222564250000093
为tkh 后第
Figure BDA00032225642500000912
个采样周期，y(tkh)和
Figure BDA00032225642500000913
分别表示tkh和
Figure BDA00032225642500000914
对应的非线性对象测量输出，min{}表示最小值函数，矩阵的右上角标T表示矩阵的转置。where δ∈(0,1) is the driver threshold parameter,
Figure BDA00032225642500000911
is a positive definite matrix, h represents the sampling period, t k h represents the kth driving moment, t k h is t k times the sampling period, t k+1 h represents the k+1 th driving moment, and t k+1 h is The sampling period is t k+1 times, and the subscript k represents the driving time sequence number.
Figure BDA0003222564250000092
represents the current sampling time,
Figure BDA0003222564250000093
is the first after t k h
Figure BDA00032225642500000912
sampling period, y(t k h) and
Figure BDA00032225642500000913
represent t k h and
Figure BDA00032225642500000914
The corresponding nonlinear object measurement output, min{} represents the minimum function, and the upper right corner of the matrix T represents the transpose of the matrix.

本发明中，事件驱动器在每个周期采样点判断对式(3)中的事件驱动条件进行判断，若满足条件，则发送采样数据；若不满足条件，则丢弃采样数据。因此，事件驱动器仅发送满足事件驱动条件的采样数据，从而能够节约网络带宽等系统受限资源。另外，事件驱动器仅使用非线性对象测量输出的周期采样值，易于软件实现，从原理上避免了芝诺现象，且解除了多数成果对对象状态完全可测的假设限制。In the present invention, the event driver judges the event driving condition in formula (3) at each cycle sampling point. If the condition is satisfied, the sampled data is sent; if the condition is not satisfied, the sampled data is discarded. Therefore, the event driver only sends the sampled data that meets the event-driven conditions, thereby saving system-limited resources such as network bandwidth. In addition, the event driver only uses the periodic sampling value of the nonlinear object measurement output, which is easy to implement in software, avoids the Zeno phenomenon in principle, and removes the assumption that the state of the object is completely measurable in most achievements.

建立执行器饱和模型如下：The actuator saturation model is established as follows:

Figure BDA0003222564250000094
Figure BDA0003222564250000094

式中，

Figure BDA00032225642500000915
表示u(t)的第
Figure BDA00032225642500000916
维分量，u(t)表示不考虑执行器饱和影响的对象控制输入，u(t)为nu维实数，
Figure BDA00032225642500000917
表示u(t)的维数序号，
Figure BDA00032225642500000918
和
Figure BDA00032225642500000919
分别表示执行器最大允许输出值和最小允许输出值，
Figure BDA00032225642500000920
表示
Figure BDA00032225642500000921
对应的执行器饱和函数值，sat()表示执行器饱和函数。In the formula,
Figure BDA00032225642500000915
represents the first order of u(t)
Figure BDA00032225642500000916
dimensional component, u(t) represents the object control input without considering the effect of actuator saturation, u(t) is a n u -dimensional real number,
Figure BDA00032225642500000917
represents the dimension number of u(t),
Figure BDA00032225642500000918
and
Figure BDA00032225642500000919
respectively represent the maximum allowable output value and the minimum allowable output value of the actuator,
Figure BDA00032225642500000920
express
Figure BDA00032225642500000921
The corresponding actuator saturation function value, sat() represents the actuator saturation function.

由执行器饱和模型(4)可知，当

Figure BDA0003222564250000095
时，执行器饱和函数输出为执行器最大允许输出值，即
Figure BDA0003222564250000096
当
Figure BDA0003222564250000097
时，执行器饱和函数输出为执行器最小允许输出值，即
Figure BDA0003222564250000098
当
Figure BDA0003222564250000099
时，执行器饱和函数输出为
Figure BDA00032225642500000923
According to the actuator saturation model (4), when
Figure BDA0003222564250000095
When , the output of the actuator saturation function is the maximum allowable output value of the actuator, that is
Figure BDA0003222564250000096
when
Figure BDA0003222564250000097
When , the output of the actuator saturation function is the minimum allowable output value of the actuator, namely
Figure BDA0003222564250000098
when
Figure BDA0003222564250000099
When , the output of the actuator saturation function is
Figure BDA00032225642500000923

使用执行器饱和模型(4)，将非线性对象模型(2)中受执行器饱和影响的控制输入

Figure BDA00032225642500000922
表示为：Using the actuator saturation model (4), the control inputs in the nonlinear object model (2) that are affected by actuator saturation
Figure BDA00032225642500000922
Expressed as:

Figure BDA00032225642500000910
Figure BDA00032225642500000910

式中，替代式

Figure BDA0003222564250000101
u1和
Figure BDA0003222564250000102
分别表示u(t)的第 1维和第nu维分量，sat(u1)和
Figure BDA0003222564250000103
分别表示u1和
Figure BDA0003222564250000104
对应的执行器饱和函数值。替代式
Figure BDA0003222564250000105
和
Figure BDA0003222564250000106
分别表示u1,
Figure BDA00032225642500001022
和
Figure BDA0003222564250000107
对应的执行器死区函数值，
Figure BDA00032225642500001018
表示执行器死区函数。In the formula, the alternative
Figure BDA0003222564250000101
u 1 and
Figure BDA0003222564250000102
Represent the 1st and nth u -dimensional components of u(t), sat(u 1 ) and
Figure BDA0003222564250000103
denote u 1 and
Figure BDA0003222564250000104
Corresponding actuator saturation function value. Alternative
Figure BDA0003222564250000105
and
Figure BDA0003222564250000106
respectively represent u 1 ,
Figure BDA00032225642500001022
and
Figure BDA0003222564250000107
Corresponding actuator dead zone function value,
Figure BDA00032225642500001018
Represents the actuator deadband function.

由执行器饱和模型(4)，可得下式成立From the actuator saturation model (4), the following formula can be obtained

Figure BDA0003222564250000108
Figure BDA0003222564250000108

式中，实数

Figure BDA0003222564250000109
表示
Figure BDA00032225642500001019
的最大值，max{}表示最大值函数。In the formula, the real number
Figure BDA0003222564250000109
express
Figure BDA00032225642500001019
The maximum value of , max{} represents the maximum value function.

B:建立随机欺骗攻击模型及DOFFSS控制器模型，并建立有机融合随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时参数的闭环系统模型；B: Build a random spoofing attack model and a DOFFSS controller model, and build a closed-loop system model that organically integrates random spoofing attacks, event drivers, actuator saturation, and network-induced delay parameters;

首先，不考虑随机欺骗攻击影响，在零阶保持器作用下，DOFFSS控制器输入

Figure BDA00032225642500001023
表示为First, without considering the effect of random spoofing attacks, under the action of the zero-order retainer, the DOFFSS controller input
Figure BDA00032225642500001023
Expressed as

Figure BDA00032225642500001010
Figure BDA00032225642500001010

式中，[tkh+τk,tk+1h+τk+1)表示零阶保持器保持时间，

Figure BDA00032225642500001011
和
Figure BDA00032225642500001012
分别表示tkh和tk+1h对应的网络诱导延时，τ和
Figure BDA00032225642500001020
分别表示网络诱导延时的下界和上界。where [t k h+τ k ,t k+1 h+τ k+1 ) represents the holding time of the zero-order keeper,
Figure BDA00032225642500001011
and
Figure BDA00032225642500001012
are the network-induced delays corresponding to t k h and t k+1 h, respectively, τ and
Figure BDA00032225642500001020
denote the lower and upper bounds of the network-induced delay, respectively.

定义

Figure BDA00032225642500001021
划分零阶保持器保持时间如下definition
Figure BDA00032225642500001021
Divide the zero-order keeper holding time as follows

Figure BDA00032225642500001013
Figure BDA00032225642500001013

式中，nk表示划分子区间序号，

Figure BDA00032225642500001014
为nk最大值，∪为并集符号，子区间
Figure BDA00032225642500001015
表示如下In the formula, n k represents the number of the divided sub-intervals,
Figure BDA00032225642500001014
is the maximum value of n k , ∪ is the union symbol, subinterval
Figure BDA00032225642500001015
expressed as follows

Figure BDA00032225642500001016
Figure BDA00032225642500001016

在划分子区间

Figure BDA00032225642500001017
上，定义函数如下in dividing subintervals
Figure BDA00032225642500001017
On, define the function as follows

e(t)＝y(tkh)-y(tkh+nkh),η(t)＝t-(tkh+nkh) (10)；e(t)=y(t k h)-y(t k h+n k h), η(t)=t-(t k h+n k h) (10);

式中，y(tkh+nkh)表示采样时刻tkh+nkh对应的非线性对象测量输出。In the formula, y(t k h+n k h) represents the nonlinear object measurement output corresponding to the sampling time t k h+n k h.

使用式(10)，将DOFFSS控制器输入(7)表示为：Using equation (10), the DOFFSS controller input (7) is expressed as:

Figure BDA0003222564250000111
Figure BDA0003222564250000111

式中，y(t-η(t))表示时刻t-η(t)对应的非线性对象测量输出，由(10)得到 t-η(t)＝tkh+nkh。In the formula, y(t-η(t)) represents the nonlinear object measurement output corresponding to time t-η(t), and t-η(t)=t k h+n k h is obtained from (10).

其次，建立随机欺骗攻击模型如下Secondly, the random deception attack model is established as follows

Figure BDA0003222564250000112
Figure BDA0003222564250000112

式中，

Figure BDA0003222564250000113
表示欺骗攻击函数，a(t)∈{0,1}表示伯努利分布随机变量，当 a(t)＝1时，欺骗攻击激活，控制器输入被篡改；当a(t)＝0时，欺骗攻击休眠，控制器输入未被篡改；由于实际中攻击能量通常受限，即
Figure BDA00032225642500001111
满足下式In the formula,
Figure BDA0003222564250000113
represents the spoofing attack function, a(t)∈{0,1} represents a Bernoulli distributed random variable, when a(t)=1, the spoofing attack is activated, and the controller input is tampered with; when a(t)=0 , the spoofing attack is dormant, and the controller input is not tampered with; since the attack energy is usually limited in practice, that is
Figure BDA00032225642500001111
satisfy the following formula

Figure BDA0003222564250000114
Figure BDA0003222564250000114

式中，G为攻击能量限定矩阵。In the formula, G is the attack energy limit matrix.

使用式(11)和(12)，得到随机欺骗攻击下控制器输入

Figure BDA00032225642500001112
如下Using equations (11) and (12), the controller input under random spoofing attack is obtained
Figure BDA00032225642500001112
as follows

Figure BDA0003222564250000115
Figure BDA0003222564250000115

随后，建立DOFFSS控制器模型如下：Subsequently, the DOFFSS controller model is established as follows:

设控制器第j个模糊规则表示为：若θ1(t)为Mj1,...,

Figure BDA00032225642500001113
为
Figure BDA00032225642500001114
...,θg(t)为 Mjg，则Let the jth fuzzy rule of the controller be expressed as: if θ 1 (t) is M j1 ,...,
Figure BDA00032225642500001113
for
Figure BDA00032225642500001114
...,θ g (t) is M jg , then

Figure BDA0003222564250000116
Figure BDA0003222564250000116

式中，

Figure BDA0003222564250000117
为xc(t)的导数，xc(t)表示控制器状态，xc(t)为n维实数，xc(t-η(t))表示t-η(t)对应的控制器状态，
Figure BDA0003222564250000118
和
Figure BDA0003222564250000119
为增益矩阵；控制器模糊规则数目为r，j表示控制器模糊规则序号，Mj1,
Figure BDA00032225642500001115
和Mjg分别表示控制器第j个模糊规则下第1个、第k个和第g个模糊集。In the formula,
Figure BDA0003222564250000117
is the derivative of x c (t), x c (t) represents the controller state, x c (t) is an n-dimensional real number, and x c (t-η(t)) represents the controller corresponding to t-η(t) state,
Figure BDA0003222564250000118
and
Figure BDA0003222564250000119
is the gain matrix; the number of fuzzy rules of the controller is r, j represents the sequence number of the fuzzy rules of the controller, M j1 ,
Figure BDA00032225642500001115
and Mjg respectively represent the 1st, kth and gth fuzzy sets under the jth fuzzy rule of the controller.

使用乘积模糊推理、中心平均解模糊器和单点模糊器，得到DOFFSS控制器模型如下Using product fuzzy inference, center-average defuzzifier, and single-point fuzzer, the DOFFSS controller model is obtained as follows

Figure BDA00032225642500001110
Figure BDA00032225642500001110

式中，替代式

Figure BDA0003222564250000121
且
Figure BDA0003222564250000122
Figure BDA0003222564250000123
表示前件变量
Figure BDA0003222564250000124
隶属于模糊集
Figure BDA0003222564250000125
的隶属度函数。In the formula, the alternative
Figure BDA0003222564250000121
and
Figure BDA0003222564250000122
Figure BDA0003222564250000123
represents the antecedent variable
Figure BDA0003222564250000124
belongs to fuzzy set
Figure BDA0003222564250000125
membership function.

综上，联立非线性对象模型(2)及DOFFSS控制器模型(16)，建立闭环系统模型如下：In summary, the nonlinear object model (2) and the DOFFSS controller model (16) are simultaneously established to establish a closed-loop system model as follows:

Figure BDA0003222564250000126
Figure BDA0003222564250000126

式中，

Figure BDA0003222564250000127
表示闭环系统状态，
Figure BDA0003222564250000128
为χ(t)的导数，χ(t-η(t))表示t-η(t)对应的闭环系统状态,
Figure BDA0003222564250000129
和
Figure BDA00032225642500001210
表示闭环系统增益矩阵。In the formula,
Figure BDA0003222564250000127
represents the closed-loop system state,
Figure BDA0003222564250000128
is the derivative of χ(t), χ(t-η(t)) represents the closed-loop system state corresponding to t-η(t),
Figure BDA0003222564250000129
and
Figure BDA00032225642500001210
represents the closed-loop system gain matrix.

C:设计随机欺骗攻击、执行器饱和及网络诱导延时影响下非线性系统事件驱动器与DOFFSS控制器联合设计条件，求出满足非线性系统通信和控制需求的事件驱动器参数

Figure BDA00032225642500001211
和等价DOFFSS控制器的增益矩阵
Figure BDA00032225642500001212
最终得到满足非线性系统通信和控制需求的事件驱动器和DOFFSS控制器。C: Design the joint design conditions of the nonlinear system event driver and DOFFSS controller under the influence of random spoofing attack, actuator saturation and network-induced delay, and find the event driver parameters that meet the communication and control requirements of the nonlinear system
Figure BDA00032225642500001211
and the gain matrix of the equivalent DOFFSS controller
Figure BDA00032225642500001212
Finally, an event driver and DOFFSS controller that meet the communication and control requirements of nonlinear systems are obtained.

所述的步骤C，包括以下两个具体步骤：Described step C, comprises the following two specific steps:

C1：基于李雅普诺夫稳定性理论，确定随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时影响下的非线性系统渐近稳定条件。C1: Based on Lyapunov stability theory, determine the asymptotic stability conditions of nonlinear systems under the influence of random spoofing attacks, event drivers, actuator saturation, and network-induced delays.

给定采样周期h,网络诱导延时下界τ和上界

Figure BDA00032225642500001215
执行器饱和参数ε,攻击能量限定矩阵G和事件驱动器阈值参数δ∈(0,1),如果存在正定矩阵
Figure BDA00032225642500001216
R＞0,S＞0,Q1＞0,Q2＞0,Q3＞0，及矩阵U2,U3，满足
Figure BDA00032225642500001213
以及Given the sampling period h, the lower bound τ and the upper bound of the network-induced delay
Figure BDA00032225642500001215
The actuator saturation parameter ε, the attack energy limiting matrix G and the event driver threshold parameter δ∈(0,1), if there is a positive definite matrix
Figure BDA00032225642500001216
R>0, S>0, Q 1 >0, Q 2 >0, Q 3 >0, and matrices U 2 , U 3 , satisfying
Figure BDA00032225642500001213
as well as

Figure BDA00032225642500001214
Figure BDA00032225642500001214

则随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时影响下闭环系统 (17)是渐近稳定的。Then the closed-loop system (17) is asymptotically stable under the influence of random spoofing attacks, event drivers, actuator saturation, and network-induced delays.

上述稳定性条件中，使用了替代式如下：In the above stability conditions, the following alternatives were used:

Figure BDA0003222564250000131
Figure BDA0003222564250000131

Figure BDA0003222564250000132
Figure BDA0003222564250000132

Figure BDA0003222564250000133
Figure BDA0003222564250000133

Figure BDA0003222564250000134
Figure BDA0003222564250000134

Figure BDA0003222564250000135
Figure BDA0003222564250000135

Figure BDA0003222564250000136
Figure BDA0003222564250000136

Figure BDA0003222564250000137
Figure BDA0003222564250000137

Figure BDA0003222564250000138
Figure BDA0003222564250000138

其中，

Figure BDA0003222564250000139
表示a(t)的数学期望，
Figure BDA00032225642500001310
为数学期望函数。He{}表示矩阵与其转置矩阵的和，*表示对称矩阵中的对称项，
Figure BDA00032225642500001311
表示零矩阵，矩阵右上角标-1表示逆矩阵，I为单位矩阵，col{}表示列矩阵，diag{}表示对角矩阵。in,
Figure BDA0003222564250000139
represents the mathematical expectation of a(t),
Figure BDA00032225642500001310
is the mathematical expectation function. He{} represents the sum of the matrix and its transposed matrix, * represents the symmetric term in the symmetric matrix,
Figure BDA00032225642500001311
Represents a zero matrix, the upper right corner of the matrix -1 represents the inverse matrix, I is the identity matrix, col{} represents the column matrix, and diag{} represents the diagonal matrix.

证明：构造李雅普诺夫泛函如下Proof: Construct the Lyapunov functional as follows

Figure BDA00032225642500001312
Figure BDA00032225642500001312

式中，替代式

Figure BDA00032225642500001313
ρ＝0.5(η3-η1)，
Figure BDA00032225642500001314
表示积分变量，χ(s), χ(s-η1)和χ(s-η2)分别表示s,s-η1和s-η2对应的闭环系统状态，
Figure BDA00032225642500001315
表示χ(s)的导数。In the formula, the alternative
Figure BDA00032225642500001313
ρ=0.5(η 3 -η 1 ),
Figure BDA00032225642500001314
represents the integral variable, χ(s), χ(s-η 1 ) and χ(s-η 2 ) represent the closed-loop system states corresponding to s, s-η 1 and s-η 2 , respectively,
Figure BDA00032225642500001315
represents the derivative of χ(s).

对李雅普诺夫泛函求导得到Derivation of the Lyapunov functional gives

Figure BDA0003222564250000141
Figure BDA0003222564250000141

式中，

Figure BDA0003222564250000142
为V(t)的导数，
Figure BDA0003222564250000143
和
Figure BDA0003222564250000144
分别表示t和t-ρ对应的替代式
Figure BDA0003222564250000145
χ(t-η1)表示t-η1对应的闭环系统状态，替代式ζ1,ζ2,ζ3表示如下：In the formula,
Figure BDA0003222564250000142
is the derivative of V(t),
Figure BDA0003222564250000143
and
Figure BDA0003222564250000144
represent the alternatives corresponding to t and t-ρ, respectively
Figure BDA0003222564250000145
χ(t-η 1 ) represents the closed-loop system state corresponding to t-η 1 , and the substitution formulas ζ 1 , ζ 2 , ζ 3 are expressed as follows:

Figure BDA0003222564250000146
Figure BDA0003222564250000146

式中，

Figure BDA0003222564250000147
表示积分变量
Figure BDA00032225642500001418
对应的闭环系统状态导数。In the formula,
Figure BDA0003222564250000147
represents the integral variable
Figure BDA00032225642500001418
The corresponding closed-loop system state derivative.

考虑如下两种情况：Consider the following two cases:

(1)若η(t)∈[η1,η2)，对(21)中ζ1,ζ2,ζ3使用琴生不等式，并使用

Figure BDA0003222564250000148
进一步对ζ2使用互凸方法得到(1) If η(t)∈[η 1 ,η 2 ), use Qinsheng's inequality for ζ 1 , ζ 2 , ζ 3 in (21), and use
Figure BDA0003222564250000148
Further use the mutual convex method for ζ 2 to get

Figure BDA0003222564250000149
Figure BDA0003222564250000149

式中，替代式

Figure BDA00032225642500001410
和
Figure BDA00032225642500001411
和χ(t-η3) 分别表示t-η2和t-η3对应的闭环系统状态。In the formula, the alternative
Figure BDA00032225642500001410
and
Figure BDA00032225642500001411
and χ(t-η 3 ) represent the closed-loop system states corresponding to t-η 2 and t-η 3 , respectively.

(2)如果η(t)∈[η2,η3]，对(21)中ζ1,ζ2,ζ3使用琴生不等式，并使用

Figure BDA00032225642500001412
进一步对ζ3使用互凸方法得到(2) If η(t)∈[η 2 ,η 3 ], use Genson's inequality for ζ 1 , ζ 2 , ζ 3 in (21), and use
Figure BDA00032225642500001412
Further use the mutual convex method for ζ 3 to get

Figure BDA00032225642500001413
Figure BDA00032225642500001413

式中，使用了替代式

Figure BDA00032225642500001414
和
Figure BDA00032225642500001415
In the formula, using the alternative
Figure BDA00032225642500001414
and
Figure BDA00032225642500001415

使用(22)和(23)，对李雅普诺夫泛函导数(20)求数学期望得到Using (22) and (23), the mathematical expectation for the derivative of the Lyapunov functional (20) gives

Figure BDA00032225642500001416
Figure BDA00032225642500001416

式中，

Figure BDA00032225642500001417
为替代式。In the formula,
Figure BDA00032225642500001417
for the alternative.

使用式(10)，由事件驱动器(3)及攻击能量受限条件(13)得到Using equation (10), it is obtained from the event driver (3) and the limited attack energy condition (13)

Figure BDA0003222564250000151
Figure BDA0003222564250000151

使用式(25)，由式(24)得到Using Equation (25), it is obtained from Equation (24)

Figure BDA0003222564250000152
Figure BDA0003222564250000152

式中，使用了如下替代式In the formula, the following alternatives are used

Figure BDA0003222564250000153
Figure BDA0003222564250000153

对式(18)使用舒尔补引理得到Using the Schur complement lemma for equation (18), we get

Figure BDA0003222564250000154
Figure BDA0003222564250000154

将式(27)代入式(26)，得到

Figure BDA0003222564250000155
根据李雅普诺夫稳定性理论，闭环系统(17)是渐近稳定的。证毕。Substituting equation (27) into equation (26), we get
Figure BDA0003222564250000155
According to the Lyapunov stability theory, the closed-loop system (17) is asymptotically stable. Certificate completed.

在上述系统稳定条件中，存在事件驱动器正定矩阵

Figure BDA0003222564250000156
的逆矩阵
Figure BDA0003222564250000157
且DOFFSS 控制器增益矩阵
Figure BDA0003222564250000158
与正定矩阵P耦合，因此，该条件不能直接用于事件驱动器和控制器设计。为了解决此问题，本发明将进一步给出事件驱动器与 DOFFSS控制器联合设计方法。In the above system stable condition, there is an event-driven positive definite matrix
Figure BDA0003222564250000156
the inverse of
Figure BDA0003222564250000157
and the DOFFSS controller gain matrix
Figure BDA0003222564250000158
coupled to a positive definite matrix P, therefore, this condition cannot be used directly in event driver and controller designs. In order to solve this problem, the present invention will further provide a joint design method of event driver and DOFFSS controller.

C2:基于步骤C1得出的非线性系统渐近稳定条件，利用线性矩阵不等式技术，得到事件驱动器与DOFFSS控制器联合设计条件，即求出满足非线性系统通信和控制需求的事件驱动器参数

Figure BDA0003222564250000159
和等价DOFFSS控制器的增益矩阵
Figure BDA00032225642500001510
最终得到满足非线性系统通信和控制需求的事件驱动器和DOFFSS 控制器。C2: Based on the asymptotic stability conditions of the nonlinear system obtained in step C1, using the linear matrix inequality technique, the joint design conditions of the event driver and the DOFFSS controller are obtained, that is, the event driver parameters that meet the communication and control requirements of the nonlinear system are obtained.
Figure BDA0003222564250000159
and the gain matrix of the equivalent DOFFSS controller
Figure BDA00032225642500001510
Finally, an event driver and DOFFSS controller that meet the communication and control requirements of nonlinear systems are obtained.

首先，给出步骤C2中使用的现有技术(定理1)如下：First, the prior art used in step C2 (Theorem 1) is given as follows:

定理1.如果给定正定矩阵Q＞0，矩阵L及实数∈＞0，则不等式 (L-∈Q)Q-1(L-∈Q)≥0成立，即不等式-LQ-1L≤∈2Q-2∈L成立。Theorem 1. If a positive definite matrix Q>0, matrix L and a real number ∈>0 are given, then the inequality (L-∈Q)Q -1 (L-∈Q)≥0 holds, that is, the inequality -LQ -1 L≤∈ 2 Q-2∈L holds.

然后，给出事件驱动器与DOFFSS控制器联合设计条件如下：Then, the joint design conditions of event driver and DOFFSS controller are given as follows:

给定采样周期h，网络诱导延时下界τ和上界

Figure BDA00032225642500001511
执行器饱和参数ε和攻击能量限定矩阵G，实数∈1＞0,∈2＞0,∈3＞0，如果存在实数
Figure BDA0003222564250000161
正定矩阵
Figure BDA0003222564250000162
Given the sampling period h, the network-induced delay lower bound τ and upper bound
Figure BDA00032225642500001511
Actuator saturation parameter ε and attack energy limit matrix G, real numbers ∈ 1 > 0, ∈ 2 > 0, ∈ 3 > 0, if there are real numbers
Figure BDA0003222564250000161
positive definite matrix
Figure BDA0003222564250000162

Figure BDA0003222564250000163
及矩阵
Figure BDA0003222564250000164
满足
Figure BDA0003222564250000165
Figure BDA0003222564250000166
以及
Figure BDA0003222564250000163
and matrix
Figure BDA0003222564250000164
Satisfy
Figure BDA0003222564250000165
Figure BDA0003222564250000166
as well as

Figure BDA0003222564250000167
Figure BDA0003222564250000167

则闭环系统(17)是渐进稳定的，同时得到事件驱动器参数

Figure BDA0003222564250000168
和等价DOFFSS控制器(34)的增益矩阵
Figure BDA0003222564250000169
如下Then the closed-loop system (17) is asymptotically stable, and the event-driven parameters are obtained
Figure BDA0003222564250000168
and the gain matrix of the equivalent DOFFSS controller (34)
Figure BDA0003222564250000169
as follows

Figure BDA00032225642500001610
Figure BDA00032225642500001610

上述条件中，使用的替代式如下In the above conditions, the alternatives used are as follows

Figure BDA00032225642500001611
Figure BDA00032225642500001611

Figure BDA00032225642500001612
Figure BDA00032225642500001612

Figure BDA00032225642500001613
Figure BDA00032225642500001613

Figure BDA00032225642500001614
Figure BDA00032225642500001614

Figure BDA00032225642500001615
Figure BDA00032225642500001615

Figure BDA00032225642500001616
Figure BDA00032225642500001616

Figure BDA00032225642500001617
Figure BDA00032225642500001617

Figure BDA00032225642500001618
Figure BDA00032225642500001618

Figure BDA00032225642500001619
Figure BDA00032225642500001619

Figure BDA00032225642500001620
X＞0,Y＞0为正定矩阵,N为n×n维实数矩阵。
Figure BDA00032225642500001620
X>0, Y>0 is a positive definite matrix, and N is an n×n-dimensional real matrix.

证明：将正定矩阵P分解如下Proof: Decompose a positive definite matrix P as follows

Figure BDA0003222564250000171
Figure BDA0003222564250000171

式中，X＞0,Y＞0为正定矩阵,N为n×n维实数矩阵。由舒尔补引理得到，正定矩阵P＞0等价于

Figure BDA0003222564250000172
In the formula, X>0, Y>0 is a positive definite matrix, and N is an n×n-dimensional real number matrix. According to Schur's complement lemma, a positive definite matrix P>0 is equivalent to
Figure BDA0003222564250000172

对步骤C1中系统稳定条件变换如下The transformation of the system stability conditions in step C1 is as follows

Figure BDA0003222564250000173
Figure BDA0003222564250000173

Figure BDA0003222564250000174
Figure BDA0003222564250000174

式中，使用了以下替代式In the formula, the following alternatives are used

Ψ1＝diag{Φ1,Φ1},Ψ2＝diag{Φ1,Φ1,Φ1,Φ1,Φ1,I,I,I,Ψ3,I,I,I},Ψ 1 =diag{Φ 1 ,Φ 1 },Ψ 2 =diag{Φ 1 ,Φ 1 ,Φ 1 ,Φ 1 ,Φ 1 ,I,I,I,Ψ 3 ,I,I,I},

Figure BDA0003222564250000175
Figure BDA0003222564250000175

Figure BDA0003222564250000176
Figure BDA0003222564250000176

对(32)中

Figure BDA0003222564250000177
和
Figure BDA0003222564250000178
使用定理1，得到(28)中
Figure BDA0003222564250000179
和
Figure BDA00032225642500001710
Right (32)
Figure BDA0003222564250000177
and
Figure BDA0003222564250000178
Using Theorem 1, we get (28)
Figure BDA0003222564250000179
and
Figure BDA00032225642500001710

因此，如果满足给定条件，闭环系统(17)是渐近稳定的。同时得到事件驱动器参数

Figure BDA00032225642500001711
和DOFFSS控制器(16)的增益矩阵如下Therefore, the closed-loop system (17) is asymptotically stable if the given conditions are met. Also get event driver parameters
Figure BDA00032225642500001711
and the gain matrix of the DOFFSS controller (16) is as follows

Figure BDA00032225642500001712
Figure BDA00032225642500001712

为了处理(33)中未知矩阵N，使用线性变换

Figure BDA00032225642500001713
得到等价 DOFFSS控制器如下To process the unknown matrix N in (33), use a linear transformation
Figure BDA00032225642500001713
The equivalent DOFFSS controller is obtained as follows

Figure BDA00032225642500001714
Figure BDA00032225642500001714

式中，

Figure BDA00032225642500001715
表示等价DOFFSS控制器状态，
Figure BDA00032225642500001716
为n维实数，
Figure BDA00032225642500001717
为
Figure BDA00032225642500001718
的导数，
Figure BDA0003222564250000181
表示t-η(t)对应的等价DOFFSS控制器状态，增益矩阵
Figure BDA0003222564250000182
由(29)获得。证毕。In the formula,
Figure BDA00032225642500001715
represents the equivalent DOFFSS controller state,
Figure BDA00032225642500001716
is an n-dimensional real number,
Figure BDA00032225642500001717
for
Figure BDA00032225642500001718
the derivative of ,
Figure BDA0003222564250000181
Represents the equivalent DOFFSS controller state corresponding to t-η(t), the gain matrix
Figure BDA0003222564250000182
Obtained from (29). Certificate completed.

通过本发明所述的事件驱动器与DOFFSS控制器联合设计方法，用户可结合具体设计要求，逐一确定各个参数，按所述步骤求得事件驱动器及DOFFSS控制器，事件驱动器能够降低数据发送率，从而节约网络带宽等系统受限资源；DOFFSS 控制器使非线性系统在随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时影响下能够渐近稳定。Through the joint design method of the event driver and the DOFFSS controller of the present invention, the user can determine each parameter one by one according to the specific design requirements, and obtain the event driver and the DOFFSS controller according to the steps. The event driver can reduce the data transmission rate, thereby Save system-constrained resources such as network bandwidth; DOFFSS controller enables nonlinear systems to be asymptotically stable under the influence of random spoofing attacks, event drivers, actuator saturation, and network-induced delays.

本发明应用场景举例：近年来，针对实际工业控制系统的网络攻击频发，如：2015年，乌克兰电力系统被恶意软件攻击，约140万人受停电影响。针对上述场景，应用本发明相关方法，将上述核电站离心机系统及电力系统建模为非线性对象，设计事件驱动器及执行器饱和模型，建立随机欺骗攻击模型、DOFFSS控制器模型，以及有机融合随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时参数的闭环系统模型，推导闭环系统渐近稳定条件，给出事件驱动器与DOFFSS控制器联合设计条件，同时得到满足要求的事件驱动器及DOFFSS控制器。Examples of application scenarios of the present invention: In recent years, network attacks against actual industrial control systems have occurred frequently. For example, in 2015, the Ukrainian power system was attacked by malware, and about 1.4 million people were affected by power outages. Aiming at the above scenario, the related method of the present invention is applied to model the above-mentioned nuclear power plant centrifuge system and power system as nonlinear objects, an event driver and actuator saturation model are designed, a random spoofing attack model, a DOFFSS controller model, and an organic fusion random Close-loop system model of spoofing attack, event driver, actuator saturation and network-induced delay parameters, deduce the asymptotic stability condition of the closed-loop system, give the joint design conditions of event driver and DOFFSS controller, and obtain the event driver and DOFFSS controller that meet the requirements at the same time device.

以下结合实施例对本发明做详细的描述：Below in conjunction with embodiment, the present invention is described in detail:

步骤A：建立非线性对象模型，并设计基于非线性对象测量输出的事件驱动器：Step A: Model the nonlinear object and design an event driver based on the measurement output of the nonlinear object:

其中，非线性对象以质量弹簧阻尼系统为例，其动力学方程描述为Among them, the nonlinear object takes the mass-spring damping system as an example, and its dynamic equation is described as

Figure BDA0003222564250000183
Figure BDA0003222564250000183

式中，

Figure BDA0003222564250000184
表示距离参考点的位移，
Figure BDA0003222564250000185
和
Figure BDA0003222564250000186
分别表示
Figure BDA0003222564250000187
的一阶和二阶导数，m为质量，
Figure BDA0003222564250000188
表示摩擦力，
Figure BDA0003222564250000189
表示弹簧弹力，
Figure BDA00032225642500001810
表示受执行器饱和影响的控制输入，
Figure BDA00032225642500001811
为实数。参数设置为
Figure BDA00032225642500001812
c＝2牛·米/秒,
Figure BDA00032225642500001813
Figure BDA00032225642500001814
In the formula,
Figure BDA0003222564250000184
represents the displacement from the reference point,
Figure BDA0003222564250000185
and
Figure BDA0003222564250000186
Respectively
Figure BDA0003222564250000187
The first and second derivatives of , m is the mass,
Figure BDA0003222564250000188
represents friction,
Figure BDA0003222564250000189
represents the spring force,
Figure BDA00032225642500001810
represents the control input affected by actuator saturation,
Figure BDA00032225642500001811
is a real number. parameter is set to
Figure BDA00032225642500001812
c = 2 N m/s,
Figure BDA00032225642500001813
Figure BDA00032225642500001814

定义对象状态

Figure BDA00032225642500001815
则质量弹簧阻尼系统(35)能够描述为非线性对象(2)，其中μ1(θ(t))＝-(θ1(t)+8)/2.88,μ2(θ(t))＝1-μ1(θ(t)),
Figure BDA0003222564250000191
r＝2,g＝1,增益矩阵如下：define object state
Figure BDA00032225642500001815
Then the mass-spring damping system (35) can be described as a nonlinear object (2), where μ 1 (θ(t))=−(θ 1 (t)+8)/2.88, μ 2 (θ(t))= 1-μ 1 (θ(t)),
Figure BDA0003222564250000191
r=2, g=1, the gain matrix is as follows:

Figure BDA0003222564250000192
Figure BDA0003222564250000192

设计基于非线性对象测量输出的事件驱动器(3)，其中采样周期h＝100毫秒，驱动器阈值参数δ及正定矩阵

Figure BDA00032225642500001913
由步骤C2中事件驱动器与DOFFSS控制器联合设计条件得到。Design an event driver (3) based on the non-linear object measurement output, where the sampling period h=100 ms, the driver threshold parameter δ and the positive definite matrix
Figure BDA00032225642500001913
Obtained from the joint design condition of event driver and DOFFSS controller in step C2.

建立执行器饱和模型(4)：其中，执行器最大允许输出值

Figure BDA0003222564250000193
ui的最大值
Figure BDA0003222564250000194
实数
Figure BDA0003222564250000195
Establish the actuator saturation model (4): where, the maximum allowable output value of the actuator
Figure BDA0003222564250000193
the maximum value of ui
Figure BDA0003222564250000194
real numbers
Figure BDA0003222564250000195

步骤B：建立随机欺骗攻击及DOFFSS控制器模型，并建立有机融合随机欺骗攻击、事件驱动器，执行器饱和及网络诱导延时参数的闭环系统模型；Step B: Build random spoofing attack and DOFFSS controller models, and build a closed-loop system model organically integrating random spoofing attack, event driver, actuator saturation and network-induced delay parameters;

首先，建立随机欺骗攻击模型为

Figure BDA0003222564250000196
First, establish a random spoofing attack model as
Figure BDA0003222564250000196

其中，

Figure BDA0003222564250000197
为欺骗攻击函数，tanh表示非线性双曲正切函数，a(t)∈{0,1}为伯努利分布随机变量，数学期望为
Figure BDA0003222564250000198
攻击能量限定矩阵为G＝0.1(一维矩阵等同为实数)。in,
Figure BDA0003222564250000197
is a spoofing attack function, tanh represents a nonlinear hyperbolic tangent function, a(t)∈{0,1} is a random variable with Bernoulli distribution, and the mathematical expectation is
Figure BDA0003222564250000198
The attack energy limit matrix is G=0.1 (a one-dimensional matrix is equivalent to a real number).

其次，建立DOFFSS控制器模型(16)。Next, build the DOFFSS controller model (16).

然后，建立有机融合随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时参数的闭环系统模型(17)，其中，网络诱导延时下界τ＝10毫秒和上界

Figure BDA0003222564250000199
毫秒。Then, build a closed-loop system model (17) that organically integrates random spoofing attacks, event drivers, actuator saturation, and network-induced delay parameters, where the network-induced delay lower bound τ = 10 ms and upper bound
Figure BDA0003222564250000199
millisecond.

步骤C：设计随机欺骗攻击、执行器饱和及网络诱导延时影响下事件驱动器与DOFFSS控制器联合设计条件，求得事件驱动器参数

Figure BDA00032225642500001910
和等价DOFFSS控制器的增益矩阵
Figure BDA00032225642500001911
最终得到满足系统需求的事件驱动器及DOFFSS控制器。Step C: Design the joint design conditions of the event driver and the DOFFSS controller under the influence of random spoofing attack, actuator saturation and network-induced delay, and obtain the event driver parameters
Figure BDA00032225642500001910
and the gain matrix of the equivalent DOFFSS controller
Figure BDA00032225642500001911
Finally, the event driver and DOFFSS controller that meet the system requirements are obtained.

其中，由步骤C1得到非线性系统渐近稳定条件，进而，由步骤C2得到事件驱动器与DOFFSS控制器联合设计条件，其中∈1＝∈2＝∈3＝1。求解此联合设计条件，得到事件驱动器参数

Figure BDA00032225642500001912
及等价DOFFSS控制器增益矩阵如下Wherein, the asymptotically stable condition of the nonlinear system is obtained from step C1, and further, the joint design condition of the event driver and the DOFFSS controller is obtained from step C2, where ∈ 1 =∈ 2 =∈ 3 =1. Solving this joint design condition yields the event-driven parameters
Figure BDA00032225642500001912
and the equivalent DOFFSS controller gain matrix is as follows

Figure BDA0003222564250000201
Figure BDA0003222564250000201

本实施例中，在设计的DOFFSS控制器作用下，随机欺骗攻击、事件驱动器、执行器饱和及网络诱导延时影响下质量弹簧阻尼系统能够渐近稳定。另外，在仿真时间[0,10秒]内，传感器周期采样100个数据，其中事件驱动器发送71个数据，数据发送率为71％。与周期采样器数据发送率100％比较，事件驱动器在保证系统性能前提下，节约了29％的系统资源。此外，事件驱动器发送的71个数据中，15个数据被随机欺骗攻击篡改，攻击率为21％。In this embodiment, under the action of the designed DOFFSS controller, the mass-spring damping system can be asymptotically stable under the influence of random spoofing attacks, event drivers, actuator saturation and network-induced delay. In addition, in the simulation time [0,10 seconds], the sensor cycle samples 100 data, of which the event driver sends 71 data, the data sending rate is 71%. Compared with the 100% data transmission rate of the periodic sampler, the event driver saves 29% of system resources under the premise of ensuring system performance. Additionally, 15 of the 71 data sent by the event driver were tampered with by random spoofing attacks, an attack rate of 21%.

实施例表明，一方面，事件驱动器能够将数据发送率降为71％，节约了29％网络带宽等系统受限资源。另一方面，虽然高达21％的事件驱动器发送数据被随机欺骗攻击篡改，但是，在DOFFSS控制器作用下，非线性系统仍然能够渐近稳定。另外，本发明方法基于非线性对象测量输出进行设计，解除了多数成果对对象状态完全可测的假设限制。The embodiment shows that, on the one hand, the event driver can reduce the data transmission rate to 71%, and save 29% of system-limited resources such as network bandwidth. On the other hand, although up to 21% of the data sent by event drivers are tampered with by random spoofing attacks, the nonlinear system can still be asymptotically stable under the action of the DOFFSS controller. In addition, the method of the present invention is designed based on the non-linear object measurement output, which relieves the assumption that the state of the object is completely measurable in most achievements.

Claims (10)

1. A method for joint design of a nonlinear system event driver and a dofss controller, comprising the steps of:
a, establishing a nonlinear object model and an actuator saturation model, and designing an event driver based on nonlinear object measurement output;
b, establishing a random deception attack model and a DOFSS controller model, and establishing a closed-loop system model organically integrating random deception attack, an event driver, actuator saturation and network induced delay parameters;
designing the joint design conditions of the nonlinear system event driver and the DOFS controller under the influence of random deception attack, actuator saturation and network induced delay to solve the event driver parameters meeting the communication and control requirements of the nonlinear system
Figure FDA0003222564240000011
And gain matrix of equivalent DOFSS controller
Figure FDA0003222564240000012
And finally obtaining the event driver and the DOFSS controller which meet the communication and control requirements of the nonlinear system.
2. The nonlinear system event driver and dofss controller joint design method according to claim 1, wherein in the step a, the nonlinear object model is:
Figure FDA0003222564240000013
wherein,
Figure FDA0003222564240000014
is the derivative of x (t), x (t) represents the object state, x (t) is an n-dimensional real number,
Figure FDA0003222564240000015
representing a control input affected by actuator saturation,
Figure FDA0003222564240000016
is nuDimensional real number, y (t) representing measurement output, y (t) being nyDimensional real number, t represents time, Ai,BiAnd CiRepresenting a gain matrix; the number of the object fuzzy rules is r, and i represents the serial number of the object fuzzy rules; substitution type
Figure FDA0003222564240000017
And is
Figure FDA0003222564240000018
Figure FDA0003222564240000019
And thetag(t) represents the 1 st and the 1 st, respectively
Figure FDA00032225642400000110
The individual and the g-th antecedent variables, g representing the number of antecedent variables,
Figure FDA00032225642400000111
the sequence number of the front-piece variable is shown,
Figure FDA00032225642400000112
representing a front-part variable
Figure FDA00032225642400000113
Membership in fuzzy sets
Figure FDA00032225642400000114
Represents the accumulation and multiplication operations, respectively.
3. The nonlinear system event driver and dofss controller joint design method according to claim 2, wherein in step a, the event driver based on the nonlinear object measurement output is:
Figure FDA0003222564240000021
wherein, delta epsilon (0,1) is a driver threshold parameter,
Figure FDA0003222564240000022
is a positive definite matrix, h denotes the sampling period, tkh denotes the kth drive time, tkh is t of the sampling periodkMultiple, tk+1h denotes the (k + 1) th driving time, tk+1h is t of the sampling periodk+1The lower subscript k denotes the drive time number,
Figure FDA0003222564240000023
which is indicative of the current sampling instant,
Figure FDA0003222564240000024
is tkAfter h is first
Figure FDA0003222564240000025
One sampling period, y (t)kh) And
Figure FDA0003222564240000026
respectively represent tkh and
Figure FDA0003222564240000027
and (3) outputting corresponding nonlinear object measurement, wherein min { } represents a minimum function, and a right upper corner mark T of the matrix represents the transposition of the matrix.
4. The nonlinear system event driver and dofss controller joint design method according to claim 3, wherein in step a, the actuator saturation model is:
Figure FDA0003222564240000028
wherein,
Figure FDA0003222564240000029
denotes the number u (t)
Figure FDA00032225642400000210
Dimensional components, u (t) representing object control inputs without regard to actuator saturation effects, u (t) being nuThe number of the dimensional real number is,
Figure FDA00032225642400000211
the number of dimensions of u (t) is shown,
Figure FDA00032225642400000212
and
Figure FDA00032225642400000213
respectively representing the maximum and minimum allowable output values of the actuator,
Figure FDA00032225642400000214
to represent
Figure FDA00032225642400000215
The corresponding actuator saturation function value, sat (), represents the actuator saturation function.
5. The nonlinear system event driver and dofss controller joint design method according to claim 4, wherein in the step B, the stochastic spoofing attack model is:
Figure FDA00032225642400000216
wherein,
Figure FDA00032225642400000217
representing a spoofing attack function, wherein a (t) epsilon {0,1} represents a Bernoulli distribution random variable, and when a (t) is 1, the spoofing attack is activated and the controller input is tampered; when a (t) is 0, the spoofing attack sleeps, and the controller input is not tampered;
Figure FDA00032225642400000218
g is an attack energy limiting matrix;
Figure FDA00032225642400000219
y (t- η (t)) represents a nonlinear object measurement output corresponding to time t- η (t), where t- η (t) is tkh+nkh，e(t)＝y(tkh)-y(tkh+nkh)，y(tkh+nkh) Watch (A)Indicating the sampling time tkh+nkh corresponding to the non-linear object measurement output.
6. The nonlinear system event driver and dofss controller joint design method according to claim 5, wherein in the step B, the dofss controller model is:
Figure FDA0003222564240000031
wherein,
Figure FDA0003222564240000032
is xcDerivative of (t), xc(t) denotes controller status, xc(t) is an n-dimensional real number, xc(t- η (t)) represents the controller state for t- η (t),
Figure FDA0003222564240000033
and
Figure FDA0003222564240000034
is a gain matrix; the number of fuzzy rules of the controller is r, j represents the serial number of the fuzzy rules of the controller, and the alternative formula
Figure FDA0003222564240000035
And is
Figure FDA0003222564240000036
Figure FDA0003222564240000037
Figure FDA0003222564240000038
Representing a front-part variable
Figure FDA0003222564240000039
Membership in fuzzy sets
Figure FDA00032225642400000310
Membership function of (c).
7. The nonlinear system event driver and dofss controller joint design method according to claim 6, wherein in the step B, the closed-loop system model organically integrating stochastic spoofing attack, event driver, actuator saturation and network-induced delay parameters is:
Figure FDA00032225642400000311
in the formula,
Figure FDA00032225642400000312
is the derivative of x (t),
Figure FDA00032225642400000313
representing the closed-loop system state, x (t-eta (t)) representing the closed-loop system state corresponding to t-eta (t),
Figure FDA00032225642400000314
and
Figure FDA00032225642400000315
representing closed-loop system gain matrices, alternatively
Figure FDA00032225642400000316
Figure FDA00032225642400000317
And
Figure FDA00032225642400000318
respectively represent
Figure FDA00032225642400000319
And
Figure FDA00032225642400000320
the corresponding function value of the dead zone of the actuator,
Figure FDA00032225642400000321
and
Figure FDA00032225642400000322
respectively represent the 1 st and the 1 st dimensions of u (t)
Figure FDA00032225642400000323
And nuThe component of the dimension(s) is,
Figure FDA00032225642400000324
representing the actuator dead band function.
8. The nonlinear system event driver and dofss controller joint design method according to claim 1, wherein the step C comprises the following specific steps:
c1: determining a nonlinear system asymptotic stability condition under the influence of random deception attack, an event driver, actuator saturation and network induced delay based on the Lyapunov stability theory;
c2 obtaining the combined design condition of the event driver and the DOFSS controller by utilizing the linear matrix inequality technology based on the nonlinear system asymptotic stable condition obtained in the step C1, namely obtaining the event driver parameters meeting the communication and control requirements of the nonlinear system
Figure FDA0003222564240000041
And gain matrix of equivalent DOFSS controller
Figure FDA0003222564240000042
And finally obtaining the event driver and the DOFSS controller which meet the communication and control requirements of the nonlinear system.
9. The nonlinear system event driver and dofss controller joint design method in accordance with claim 8, wherein: the nonlinear system asymptotic stability condition under the influence of random deception attack, event driver, actuator saturation and network induced delay is as follows:
given a sampling period h, the lower bound of network induced delayτAnd upper bound
Figure FDA0003222564240000043
An actuator saturation parameter ε, an attack energy definition matrix G, and an event driver threshold parameter δ ∈ (0,1), if there is a positive definite matrix
Figure FDA0003222564240000044
P＞0,R＞0,S＞0,Q1＞0,Q2＞0,Q3> 0, and matrix U2,U3Satisfy the following requirements
Figure FDA0003222564240000045
And
Figure FDA0003222564240000046
the closed loop system is asymptotically stable under the influence of random deception attack, event drivers, actuator saturation and network induced delay; in the above stability conditions, the following alternative formula is used:
Figure FDA0003222564240000047
Π21＝col{η10Λ1,η21Λ1,η32Λ1,η10Λ2,η21Λ2,η32Λ2},Π31＝CiE1e3+e6,
Figure FDA0003222564240000048
Π51＝G(CiE1e3+e6),
Figure FDA0003222564240000049
Figure FDA00032225642400000410
Π44＝-ε-1,Π55＝-I,
Figure FDA00032225642400000411
Figure FDA00032225642400000412
Figure FDA00032225642400000413
η10＝η1-η0,η21＝η2-η1,η32＝η3-η2,η0＝0,η1＝τ,η2＝0.5(η3+η1),
Figure FDA0003222564240000051
Figure FDA0003222564240000052
Figure FDA0003222564240000053
wherein,
Figure FDA0003222564240000054
the mathematical expectation for a (t) is shown,
Figure FDA0003222564240000055
is a mathematical expectation function; he { } denotes the sum of the matrix and its transpose, denotes the symmetric terms in the symmetric matrix,
Figure FDA0003222564240000056
the matrix represents a zero matrix, the upper right corner mark-1 of the matrix represents an inverse matrix, I is an identity matrix, col represents a column matrix, and diag represents a diagonal matrix.
10. The nonlinear system event driver and dofss controller joint design method in accordance with claim 9, wherein: the joint design conditions of the event driver and the DOFSS controller are as follows:
given a sampling period h, the lower bound of network induced delayτAnd upper bound
Figure FDA0003222564240000057
The saturation parameter epsilon of the actuator and the attack energy limit matrix G, and the real number epsilon1＞0,∈2＞0,∈3> 0, if real numbers are present
Figure FDA0003222564240000058
Positive definite matrix
Figure FDA0003222564240000059
Figure FDA00032225642400000510
And a matrix
Figure FDA00032225642400000511
Satisfy the requirement of
Figure FDA00032225642400000512
Figure FDA00032225642400000513
And
Figure FDA00032225642400000514
the closed loop system is asymptotically stable while obtaining event driver parameters
Figure FDA00032225642400000515
And gain matrix of equivalent DOFSS controller
Figure FDA00032225642400000516
As follows
Figure FDA00032225642400000517
Under the above conditions, the following alternative formulae were used
Figure FDA00032225642400000518
Figure FDA00032225642400000519
Figure FDA00032225642400000520
Figure FDA00032225642400000521
Figure FDA00032225642400000522
Figure FDA0003222564240000061
Figure FDA0003222564240000062
Figure FDA0003222564240000063
Figure FDA0003222564240000064
Ψ1＝diag{Φ1,Φ1},
Figure FDA0003222564240000065
X is greater than 0, Y is greater than 0 and is positive definite matrix, N is N X N dimension real number matrix.
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