Multi-parameter collaborative optimization control method and system for singular system under DoS attack

Abstract

The invention relates to the field of information system security control, in particular to a multi-parameter collaborative optimization control method and system for a singular system under a DoS attack, and the method comprises the steps: building a singular system model and a DoS attack model; a reduced-order observer is constructed by solving a Sylvester equation, and a controller is designed in combination with the output of the reduced-order observer; establishing a closed-loop augmentation system based on control of a reduced-order observer; establishing sufficient conditions for enabling the closed-loop augmentation system to meet regularity, no pulse and asymptotically stable; performing iterative optimization by adjusting scalar parameters and solving a linear matrix inequality group, so as to cooperatively design gain parameters of a reduced-order observer and gain parameters of a controller, and finally determining a group of scalar parameter values; and calculating the maximum duration of the DoS attack and the minimum operation time required by the controller to realize collaborative optimization of the singular system stability and the operation cost. The problem that cost and complexity are increased due to a full-order observer control problem is solved.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of information physical system security control, in particular to a multi-parameter collaborative optimization control method and system for a singular system under DoS attack. BACKGROUND

[0002] As a new generation of intelligent system deeply integrating computing, communication and physical process, Cyber-Physical System (CPS) connects intelligent manufacturing devices through Internet of Things and information technology, and realizes accurate perception, dynamic control and intelligent decision-making of the physical world. Its convenient installation and strong applicability make it widely used in key fields such as intelligent manufacturing, intelligent transportation and network security. However, deep network integration brings openness and flexibility, but also introduces serious network security problems. Among them, Denial of Service (DoS) attack is one of the most common network attack forms. DoS attack directly destroys the real-time performance of the CPS "perception-decision-execution" closed loop by blocking communication channels, exhausting network resources or interfering with node responses, leading to data packet loss or transmission interruption, and thus may cause serious consequences such as out-of-control of physical devices or failure of critical infrastructure, posing a great threat to the performance and stability of the system.

[0003] In the face of complex and variable DoS attacks, traditional single security protection measures (such as firewalls and intrusion detection) have been insufficient. They usually lack deep collaboration with control systems and are difficult to dynamically adjust control strategies to maintain system stability when attacked. Therefore, the industry urgently needs to develop a new defense method that integrates control theory, network communication and security mechanisms, i.e. through multi-parameter collaborative optimization control, to build an adaptive CPS active defense system.

[0004] In particular, there are a large number of complex systems in actual industrial processes whose dynamic behavior is jointly dominated by differential equations and algebraic constraints. Such systems are modeled as singular systems (or differential-algebraic systems). Research on the security control of singular information physical systems has stronger universality. However, there are two major bottlenecks in existing research: first, existing defense technologies mostly focus on attack detection or passive fault tolerance, lack the ability to collaboratively model and analyze system dynamics and attack patterns, and cannot consider the impact of attacks in the design stage. Second, for controller design of singular systems, existing solutions generally rely on full-order state observers to estimate the internal state of the system. Full-order observers have high dimension and complex structure, which not only significantly increases the cost of sensors and the computational burden of the system, but also reduces the response speed of the system, making it difficult to meet the needs of high real-time industrial scenarios. SUMMARY

[0005] To solve the above problems, the application provides a multi-parameter collaborative optimization control method and system for a singular system under DoS attack.

[0006] In a first aspect, the present application provides a multi-parameter collaborative optimization control method for a singular system under DoS attack, comprising the following steps: S1, a singular system model containing differential-algebraic constraints and a periodic DoS attack model describing the on-off state of DoS attack are established; S2, based on the established singular system model and DoS attack model, a reduced-order observer for reconstructing system state information is constructed by solving a Sylvester equation, and a controller is designed in combination with the output information of the reduced-order observer; S3, a closed-loop augmented system of the singular system under DoS attack based on the reduced-order observer control is established; S4, based on Lyapunov stability theory, sufficient conditions are established for the closed-loop augmented system to satisfy regularity, no impulse and asymptotic stability; the sufficient conditions are represented by a set of linear matrix inequalities, and the set of linear matrix inequalities contains adjustable scalar parameters for quantifying the stability margin and performance of the system; S5, by adjusting the scalar parameters and solving the corresponding set of linear matrix inequalities, iterative optimization is performed until a set of feasible solutions is obtained, so as to collaboratively design the gain parameters of the reduced-order observer and the gain parameters of the controller, and finally determine a set of optimized scalar parameter values; S6, based on the optimized scalar parameter values and the attack period, the maximum duration of DoS attack that the singular system can tolerate and the minimum running time required by the controller are calculated, and the collaborative optimization of the stability of the singular system and the running cost is realized.

[0007] As a further limitation of the technical scheme of the present application, S1 specifically comprises: S11, a singular system model is established:

[0008] wherein, represents the state of the system; represents the output of the system; represents the control input of the system; matrix is a singular matrix, and satisfies , represents a constant matrix; S12, a DoS attack model is established:

[0009] wherein, is the periodicity, , is the period, is the attack dormancy time, , Indicates no attack signal, communication is normal; Indicates that there is an attack information, the communication is interrupted.

[0010] The singular system model and DoS attack model are established, which can more accurately describe the dynamic behavior of the system, improve the observability and controllability of the system, and provide an effective technical means for the problems of unobtainable state information and poor control effect of singular system under DoS attack. The periodic DoS attack model can clearly describe the on-off state of DoS attack, provide a clear attack scene for the design of the controller, make the controller better cope with the attack, and enhance the defense ability of the system to DoS attack.

[0011] Through accurate modeling and design, the dynamic response ability of the singular system under DoS attack is improved, so that it can quickly adjust and recover when the attack occurs, reduce the influence of data packet loss or transmission interruption on the stability of the physical system, and ensure the normal operation of the system.

[0012] As a further limitation of the technical scheme of the application, S2 specifically comprises: S21, constructing a reduced-order observer according to the singular system model, for reconstructing the system state information; wherein the reduced-order observer is as follows:

[0013] In the formula, is the state of the reduced-order observer, is the estimated value of the system state is the estimated value of the system state is the parameter matrix to be designed; S22, determining the parameter matrix of the reduced-order observer by solving the Sylvester equation; The Sylvester equation is as follows:

[0014] Wherein, J is an auxiliary matrix determined by matrix equivalent transformation; S23, defining an intermediate variable , constructing an error system based on the Sylvester equation constraint;

[0015] Wherein, is the estimation error; S24, designing a controller based on the state estimation value output by the reduced-order observer and the output of the system; the controller is a switching controller based on the state of the reduced-order observer:

[0016] Wherein, Q ,S The controller parameter matrix to be designed.

[0017] By solving the Sylvester equation to construct the reduced-order observer, the calculation complexity of the system can be effectively reduced, the real-time performance and response speed of the system are improved, the problem that the calculation burden exponentially increases with the system scale caused by the full-order observer in the prior art is solved, the system is more suitable for complex applications in actual engineering. The controller is designed in combination with the output information of the reduced-order observer, the system state can be more accurately estimated, the performance of the controller and the stability of the system are improved, the control accuracy and stability of the system under the DoS attack are enhanced, and the damage of the DoS attack to the system performance is effectively coped with. The reduced-order observer can adapt to different system scales and complexities, and the adaptability and flexibility of the system are enhanced.

[0018] As a further limitation of the technical scheme of the application, in S3, in combination with the singular system, the error system, the controller and the DoS attack model, the closed-loop augmented system is obtained:

[0019] Wherein, .

[0020] As a further limitation of the technical scheme of the application, S4 specifically comprises: S41, using Lyapunov function, the system dynamics under each attack mode is analyzed, and the constraint condition for ensuring the asymptotic stability of the entire switching system is derived; the constraint condition comprises: requiring the existence of two positive definite matrices To meet the proportional boundedness relationship between them, that is, At the same time, the attack dormancy time Must be greater than the lower bound calculated by the scalar parameter And the attack period , that is, ; S42, the constraint condition and the regularity and impulse-free requirement that the closed-loop augmented system needs to meet are converted into a group of sufficient conditions represented in the form of linear matrix inequalities; the group of linear matrix inequalities specifically comprises two parts corresponding to the two system modes of no attack period and attack period; S43, in the process of constructing the group of linear matrix inequalities, adjustable scalar parameters for quantifying the stability margin and performance of the system are introduced; the scalar parameters comprise a parameter For constraining the state decay rate of the system, A parameter for constraining the proportional relationship of the positive definite matrix, S44, the constructed linear matrix inequality set and the inequality constraint about the attack dormancy time jointly constitute a sufficient condition for guaranteeing that the closed-loop augmented system has desired stability performance under the denial-of-service attack.

[0021] Based on Lyapunov stability theory, a sufficient condition for the closed-loop augmented system to satisfy regularity, no impulse and asymptotic stability is established, which provides a strict theoretical guarantee for the stability of singular systems under DoS attack, effectively solves the system instability problem caused by DoS attack, and guarantees the normal operation of the system. By introducing an adjustable scalar parameter, the stability margin and performance of the system can be flexibly adjusted, so that the system can maintain good performance under different operating conditions, realize the optimization and adjustment of system performance, and improve the overall performance and reliability of the system. The stability condition is converted into a linear matrix inequality set, which is solved by using mathematical tools, so that the optimization design can be efficiently performed, the feasibility and efficiency of the design are improved, and an effective mathematical tool and method are provided for solving the complex control problem of information physical systems under DoS attack.

[0022] As a further limitation of the technical scheme of the application, the step S5 comprises: S51, set the initial value of the introduced adjustable scalar parameter, and configure the convergence tolerance and the maximum number of iterations of the linear matrix inequality solver; S52, taking the current value of the scalar parameter as a fixed condition, taking the constructed linear matrix inequality set as a convex constraint, constructing a convex optimization problem with matrix variables as the optimization objective, and calling a numerical solving tool to solve the convex optimization problem; S53, judge whether there is a feasible solution to the convex optimization problem in the current iteration; If there is a feasible solution, record the current scalar parameter value and the corresponding matrix variable feasible solution, and execute step S54; If there is no feasible solution, adjust the value of the scalar parameter, and return to S52 for next iteration calculation; S54, according to the matrix variable feasible solution obtained in S53, calculate the gain parameters of the reduced-order observer and the controller; S55, determine the scalar parameter value recorded in S53 that makes the convex optimization problem have a feasible solution as the optimized scalar parameter value.

[0023] By adjusting the scalar parameter and solving the linear matrix inequality set, the gain parameters of the reduced-order observer and the gain parameters of the controller can be designed collaboratively, the optimization design of the parameters is realized, the collaborative control ability of the system is improved, and the overall performance of the system under the DoS attack is enhanced. The optimized scalar parameter value can make the system have better stability and performance under the DoS attack, improve the anti-attack ability of the system, effectively cope with the destruction of the system performance and stability caused by the DoS attack, and protect the safe operation of the information physical system. The iterative optimization process can automatically adjust the parameters until the optimal solution is obtained, reducing manual intervention and improving the automation degree and efficiency of the optimization process.

[0024] As a further limitation of the technical solution of the application, the step S6 comprises: S61, obtaining a set of optimized scalar parameter values, and simultaneously obtaining an attack period from the established denial of service attack model; S62, according to the obtained optimized scalar parameter value and the attack period, calculating the maximum attack duration of the denial of service attack in each period under the current optimization parameter to ensure the stability of the system; S63, according to the calculated maximum attack duration and in combination with the attack period, calculating the minimum time length that the controller based on the reduced-order observer must be normally operated in each attack period, that is, the minimum operation time; the minimum operation time is equal to the attack period minus the maximum attack duration; S64, judging whether the optimization results of the maximum attack duration and the minimum operation time meet the predetermined system design index; If yes, output all the current optimized parameters and performance indexes; If not, return to S5, readjust the scalar parameter and perform iterative optimization again until the system design index meeting the requirements is obtained.

[0025] Based on the optimized scalar parameter value and the attack period, the maximum duration of the DoS attack that the singular system can tolerate and the minimum operation time required by the controller are calculated, which provides clear indexes and optimization targets for the operation of the system, so that the system can realize the minimization of the operation cost on the premise of ensuring the stability. By calculating the minimum operation time, the operation time of the controller can be minimized on the premise of ensuring the stability of the system, so as to reduce the operation cost and improve the resource utilization efficiency of the system, solving the problems of control resource waste and high operation cost in the prior art.

[0026] As a further limitation of the technical solution of the application, the maximum attack duration The calculation formula of the maximum attack duration is:

[0027] The minimum operation time .

[0028] By calculating the maximum attack duration and the minimum running time, the anti-attack robustness and the running cost of the current system design can be accurately evaluated, which provides a basis for the design and optimization of the system, and enables the user to select the most appropriate system design under the premise of ensuring the stability of the system, so as to realize the optimal trade-off between performance and cost.

[0029] As a further limitation of the technical scheme of the application, the method further comprises: The calculated maximum attack duration is used as a quantitative indicator for evaluating the anti-attack robustness of the current system design; The calculated minimum running time of the controller is used as a quantitative indicator for evaluating the minimum running cost required to achieve the level of robustness; The maximum attack duration and the minimum running time are used as the final output of the system design, providing clear design trade-off basis for the user: the highest intensity attack scenario that can be tolerated and the minimum control resources required under the premise of ensuring the stability of the system; enabling the user to make reasonable decisions according to actual needs and resource limitations.

[0030] In a second aspect, the technical scheme of the application further provides a multi-parameter collaborative optimization control system for a singular system under DoS attack, comprising: A modeling module for establishing a singular system dynamics model containing differential-algebraic constraints and a periodic DoS attack model describing the on-off state of DoS attack; A design module for constructing a reduced-order observer for reconstructing system state information by solving a Sylvester equation based on the established singular system model and DoS attack model, and designing a controller in combination with the output information of the reduced-order observer; A closed-loop augmented system establishment module for establishing a closed-loop augmented system for the singular system under DoS attack based on the reduced-order observer control; A stability condition construction module for establishing sufficient conditions for the closed-loop augmented system to satisfy regularity, no impulse and asymptotic stability based on Lyapunov stability theory; the sufficient conditions are represented by a set of linear matrix inequalities, which contain adjustable scalar parameters for quantifying the stability margin and performance of the system; A collaborative optimization module for iterative optimization by adjusting the scalar parameters and solving the corresponding set of linear matrix inequalities until a set of feasible solutions is obtained, thereby collaboratively designing the gain parameters of the reduced-order observer and the gain parameters of the controller, and finally determining a set of optimized scalar parameter values; The optimization output module is used for calculating the maximum duration of the DoS attack that can be tolerated by the singular system and the minimum running time required by the controller based on the optimized scalar parameter value and the attack period, realizing the collaborative optimization of the singular system stability and the running cost, and outputting the final optimization parameter and the performance index.

[0031] From the above technical solutions, the present application has the following advantages: by establishing a singular system model containing differential-algebraic constraints and a periodic DoS attack model, the behavior of the singular system under DoS attack can be comprehensively modeled and analyzed, thereby providing more comprehensive protection for the system, effectively dealing with the network security problems faced by the singular system, and guaranteeing the performance and stability of the system. The collaborative optimization of the singular system stability and the running cost is realized, the consumption of control resources is minimized under the premise of guaranteeing the stability of the system, the cost and computational complexity of the sensor are reduced, the running efficiency and economy of the system are improved, and the problem of increased cost and complexity caused by the full-order observer control problem is solved.

[0032] Through the multi-parameter collaborative optimization control method, the robustness of the singular system under DoS attack is enhanced, the system can better cope with complex and variable network attacks, guarantee the stable operation of the key infrastructure, and improve the adaptability and universality of the system, meeting the needs of complex systems in actual engineering. BRIEF DESCRIPTION OF DRAWINGS

[0033] In order to more clearly illustrate the technical solutions of the present application, the drawings required in the description will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.

[0034] Figure 1 The flow chart of the multi-parameter collaborative optimization control method provided by the present application.

[0035] Figure 2 The schematic diagram of the singular system modeling under DoS attack in the present application.

[0036] Figure 3 The block diagram of the multi-parameter collaborative optimization control system provided by the present application. DETAILED DESCRIPTION

[0037] In order to make the application purposes, features, advantages of the present application more obvious and easy to understand, the following will use specific examples and drawings to clearly and completely describe the technical solutions protected by the present application. Obviously, the following described examples are only a part of the examples of the present application, but not all the examples. Based on the examples in the present application, all other examples obtained by those skilled in the art without creative labor are within the scope of protection of the present application.

[0038] Unless otherwise defined, all technical and scientific terms used in the present application have the same meanings as commonly understood by one of ordinary skill in the art to which the present application belongs. The terms used in the specification of the present application are only for the purpose of describing specific examples and are not intended to limit the present application.

[0039] As shown in Figure 1 The embodiment of the present application provides a multi-parameter collaborative optimization control method for a singular system under a DoS attack, which comprises the following steps: S1, a singular system model containing differential-algebraic constraints and a periodic DoS attack model describing the on-off state of a DoS attack are established; In this step, the data packet loss problem under a DoS network attack is considered, and a dynamic model of a singular system and a DoS attack model are established; the modeling principle of a singular system under a DoS attack is as shown in Figure 2 .

[0040] S2, based on the established singular system model and DoS attack model, a reduced-order observer for reconstructing system state information is constructed by solving a Sylvester equation, and a controller is designed in combination with the output information of the reduced-order observer; It should be noted that the Sylvester equation and the knowledge of solving a non-homogeneous linear equation set are used to construct a reduced-order observer for reconstructing system state information, and a controller is designed in combination with the information of the reduced-order observer.

[0041] S3, a closed-loop augmented system of a singular system under a DoS attack based on the reduced-order observer control is established; S4, based on Lyapunov stability theory, a sufficient condition is established for the closed-loop augmented system to satisfy regularity, no impulse and asymptotic stability; the sufficient condition is represented by a linear matrix inequality set, and the linear matrix inequality set contains adjustable scalar parameters for quantifying the stability margin and performance of the system; In this step, the Lyapunov stability theory and the matrix inequality theory are used to establish the conditions for the closed-loop augmented system to be regular, non-impulse and asymptotically stable, so that the singular information physical system can still operate safely and stably under a DoS network attack.

[0042] S5, iteratively optimizing by adjusting the scalar parameters and solving the corresponding linear matrix inequality set until a set of feasible solutions is obtained, so as to cooperatively design the gain parameters of the reduced-order observer and the gain parameters of the controller, and finally determine a set of optimized scalar parameter values; It needs to be further explained that by using the obtained stability condition, a design algorithm of the reduced-order observer and the controller is given by the multi-parameter coordinated optimization control method.

[0043] S6, based on the optimized scalar parameter values and the attack period, calculating the maximum duration of the DoS attack that the singular system can tolerate and the minimum running time required by the controller, so as to realize the coordinated optimization of the singular system stability and the running cost.

[0044] In this step, the design parameters are optimized, and the maximum attack time of the DoS network attack on the singular system in each period is calculated , and the minimum time that the controller based on the reduced-order observer can run under the premise of ensuring the asymptotic stability of the system is determined , so as to realize the dual optimization strategy of coordinated optimization control and reduction of actual running cost.

[0045] In the embodiment of the application, S1 is specifically: S11, establishing a singular system model:

[0046] wherein, represents the state of the system; represents the output of the system; represents the control input of the system; and matrix is a singular matrix, and satisfies , represents a constant matrix; S12, establishing a DoS attack model:

[0047] wherein, is the period number, , is the period, is the attack sleep time, , represents no attack signal, and the communication is normal; represents that there is attack information, and the communication is interrupted.

[0048] In some embodiments, S2 specifically includes: S21, constructing a reduced-order observer according to the singular system model, for reconstructing the system state information; wherein the reduced-order observer is as follows:

[0049] wherein, is the state of the reduced-order observer, is the estimated value of the system state , is a parameter matrix to be designed; S22, determining the parameter matrix of the reduced-order observer by solving a Sylvester equation; The Sylvester equation is:

[0050] wherein, J is an auxiliary matrix determined by matrix equivalence transformation; S23 includes: (a) defining an intermediate variable , constructing an error system based on Sylvester equation constraints;

[0051] wherein, is the estimation error; defining a matrix , wherein is a row full-rank matrix; from the equation can be obtained:

[0052]

[0053] (b) from the Sylvester equation:

[0054] Solving the above equation can obtain:

[0055] wherein, the matrix is an unknown parameter to be designed.

[0056] S24, designing a controller based on the state estimated value of the reduced-order observer output and the output of the system; the controller is a switching controller based on the state of the reduced-order observer:

[0057] wherein, Q , Sa parameter matrix to be designed.

[0058] By using It can be obtained that:

[0059] wherein, the matrix a parameter matrix to be designed.

[0060] In some embodiments, in S3, in combination with the singular system, the error system, the controller and the DoS attack model, the closed-loop augmented system is obtained:

[0061] wherein, .

[0062] In some embodiments, S4 specifically comprises: S41, using Lyapunov function, the system dynamics under each attack mode is analyzed, and the constraint condition for ensuring the asymptotic stability of the entire switching system is derived; the constraint condition includes: requiring the existence of two positive definite matrices to meet the proportional boundedness relationship between them, that is, At the same time, the attack dormant time must be greater than the lower bound calculated by the scalar parameter and the attack period , that is, ; S42, the constraint condition and the regularity and impulse-free requirement that the closed-loop augmented system needs to meet are converted into a set of sufficient conditions represented in the form of linear matrix inequalities; the set of linear matrix inequalities specifically includes two parts, corresponding to the system mode of no attack period and attack period respectively; S43, in the process of constructing the set of linear matrix inequalities, adjustable scalar parameters are introduced for quantifying the stability margin and performance of the system; the scalar parameters include the parameter for constraining the state decay rate of the system, the parameter for constraining the proportional relationship of the positive definite matrix, and auxiliary scalar parameters for decoupling and relaxing the matrix inequality to expand the feasible solution range; S44, the set of constructed linear matrix inequalities and the inequality constraint on the attack dormant time jointly constitute sufficient conditions for ensuring that the closed-loop augmented system has desired stable performance under denial of service attack.

[0063] It should be noted that in S42, the sufficient conditions for the closed-loop augmented system to be regular, impulse-free and asymptotically stable are:

[0064]

[0065] When ,

[0066] When ,

[0067] Wherein, , , Indicate the symmetric items in the corresponding position of the matrix, the subscript "T" represents the transpose of the matrix, The sum of the matrix and its transpose is indicated by The positive definite matrix , , Is a free matrix, and the parameter Is a positive number, The matrix And Satisfy .

[0068] The above inequalities are solved by using the LMI toolbox in MATLAB, when the optimal feasible solution is obtained, then the closed-loop augmented system satisfies the regularity, non-impulse and asymptotic stability, and the state feedback gain matrix And the reduced observer gain matrix Respectively are:

[0069] In the embodiment of the application, the asymptotic stability of the closed-loop augmented system is also proved based on the matrix inequality decoupling lemma and Lyapunov function, which is specifically as follows: S421, prove that the closed-loop augmented system is regular, non-impulse and asymptotically stable. First, a decoupling lemma of matrix inequality is given: for the matrix And the constant The following two inequalities are equivalent: (1)

[0070] (2)

[0071] S422, construct Lyapunov function:

[0072] Wherein, the matrix .

[0073] S423, when At this time, by using the matrix inequality and the given decoupling lemma, the following can be calculated:

[0074] wherein, By using the matrix inequality decoupling lemma on the above inequality, the following can be obtained:

[0075] wherein, , .

[0076] S424, when , the same method as S423 can be obtained:

[0077] S425, the derivative of Lyapunov function V(t) is calculated, and the linear matrix inequality obtained by combining steps S423 and S424 can be obtained:

[0078] Integrating the above inequalities respectively can obtain:

[0079] From , it can be concluded that:

[0080] By using the constraint condition in S42, the following can be obtained by iterative calculation:

[0081] wherein By using Lyapunov stability theory, it can be obtained that the closed-loop augmented system is asymptotically stable.

[0082] In some embodiments, the steps of S5 include: S51, set the initial value of the introduced adjustable scalar parameter, and configure the convergence tolerance and the maximum number of iterations of the linear matrix inequality solver; S52, taking the current value of the scalar parameter as a fixed condition, construct a convex optimization problem with the constructed linear matrix inequality set as a convex constraint and the matrix variable as an optimization objective, and call a numerical solving tool to solve the convex optimization problem; S53, judge whether there is a feasible solution to the convex optimization problem under the current iteration; If there is a feasible solution, record the current scalar parameter value and the corresponding matrix variable feasible solution, and execute step S54; If there is no feasible solution, adjust the value of the scalar parameters, and return to S52 for the next iteration calculation; S54, according to the matrix variable feasible solution obtained in S53, calculate the gain parameters of the reduced order observer and the gain parameters of the controller; S55, the set of scalar parameter values recorded in S53 that make the convex optimization problem have a feasible solution is determined as the optimized scalar parameter value.

[0083] The design algorithm of the reduced order observer and the controller is shown in Algorithm 1: Algorithm 1: Design of reduced order observer and controller Input: matrix , parameter ; Output: reduced order observer gain matrix ; controller gain matrix ; Reduced order observer parameters: ; Controller parameters: ; Step 1: Solve the convex optimization condition given in S42 to obtain a feasible optimal solution ; Step 2: If the convex optimization condition has no feasible solution, adjust the parameters , return to execute Step 1; Step 3: Solve the reduced order observer gain matrix , controller gain matrix ; Step 4: Use S24 to obtain the controller parameters , ; Step 5: Use S23(a) to calculate the matrix ; Step 6: Use S23(b) to derive the reduced order observer parameters: .

[0084] In some embodiments, the steps of S6 include: S61, obtain a set of optimized scalar parameter values, and obtain the attack period from the established denial of service attack model; S62, according to the obtained optimized scalar parameter value and attack period, calculate the maximum attack duration of the denial of service attack in each period under the current optimization parameter to ensure system stability; S63, based on the calculated maximum attack duration, and in combination with the attack period, calculating the minimum time length that the controller based on the reduced-order observer must at least normally run in each attack period, i.e. the minimum running time; the minimum running time is equal to the attack period minus the maximum attack duration; S64, judging whether the optimization result of the maximum attack duration and the minimum running time meets the predetermined system design index; If yes, outputting all the parameters and performance indexes after the current optimization; If no, returning to S5 to readjust the scalar parameter and iteratively optimize again until the system design index meeting the requirement is obtained.

[0085] Maximum attack duration The calculation formula is:

[0086] Minimum running time .

[0087] In some embodiments, the method further comprises: Taking the calculated maximum attack duration as a quantitative index for evaluating the anti-attack robustness of the current system design; Taking the calculated minimum running time of the controller as a quantitative index for evaluating the minimum running cost required to achieve the level of robustness; Taking the maximum attack duration and the minimum running time as the final output of the system design, providing clear design trade-off basis for the user: under the premise of ensuring system stability, the highest intensity attack scenario that can be tolerated and the minimum control resources required.

[0088] As shown in Figure 3 The embodiment of the present application also provides a multi-parameter collaborative optimization control system of a singular system under DoS attack, comprising: A modeling module, configured to establish a singular system dynamics model containing differential-algebraic constraints and a periodic DoS attack model describing the on-off state of DoS attack; A design module, configured to, based on the established singular system model and DoS attack model, construct a reduced-order observer for reconstructing system state information by solving a Sylvester equation, and design a controller in combination with the output information of the reduced-order observer; A closed-loop augmented system establishment module, configured to establish a closed-loop augmented system of a singular system under DoS attack based on the reduced-order observer control; a stability condition construction module, configured to establish sufficient conditions for the closed-loop augmented system to satisfy regularity, no impulse and asymptotic stability based on Lyapunov stability theory, the sufficient conditions being represented by a set of linear matrix inequalities, the set of linear matrix inequalities containing adjustable scalar parameters for quantifying the stability margin and performance of the system; a collaborative optimization module, configured to perform iterative optimization by adjusting the scalar parameters and solving the corresponding set of linear matrix inequalities until a set of feasible solutions is obtained, so as to collaboratively design the gain parameters of the reduced-order observer and the gain parameters of the controller, and finally determine a set of optimized scalar parameter values; an optimization output module, configured to calculate the maximum duration of DoS attack that the singular system can tolerate and the minimum running time required by the controller based on the optimized scalar parameter values and the attack period, so as to realize collaborative optimization of the stability of the singular system and the running cost, and output the final optimized parameters and performance indicators.

[0089] In some embodiments, the modeling module specifically includes: a singular system model establishment unit, configured to establish a singular system model:

[0090] wherein, represents the state of the system; represents the output of the system; represents the control input of the system; and matrix is a singular matrix, and satisfies , represents a constant matrix; a DoS attack model establishment unit, configured to establish a DoS attack model:

[0091] wherein, is the periodicity, , is the period, is the attack dormancy time, , represents a non-attack signal, and normal communication; represents an attack signal, and communication interruption.

[0092] In some embodiments, the design module specifically includes: a reduced-order observer construction unit, configured to construct a reduced-order observer according to the singular system model, for reconstructing system state information; wherein the reduced-order observer is as follows:

[0093] wherein, is the state of the reduced-order observer, an estimated value of a system state, a parameter matrix to be designed; a Sylvester equation solving unit, determining the parameter matrix of the reduced-order observer by solving a Sylvester equation; the Sylvester equation is:

[0094] wherein J is an auxiliary matrix determined by a matrix equivalence transformation; an error system constructing unit, defining an intermediate variable constructing an error system based on Sylvester equation constraints;

[0095] wherein, is an estimation error; a controller designing unit, designing a controller based on the state estimation value output by the reduced-order observer and the output of the system; the controller is a switching controller based on the state of the reduced-order observer:

[0096] wherein, Q , S a controller parameter matrix to be designed.

[0097] In some embodiments, in the closed-loop augmented system establishing module, a singular system, an error system, a controller and a DoS attack model are combined to obtain a closed-loop augmented system:

[0098] wherein, .

[0099] In some embodiments, the stability condition constructing module includes: a Lyapunov function analysis unit, configured to analyze the system dynamics under each attack mode by using a Lyapunov function, and derive constraint conditions for ensuring the asymptotic stability of the entire switching system; the constraint conditions include: requiring the existence of two positive definite matrices to satisfy the proportional boundedness relationship between them, i.e. At the same time, the attack dormancy time must be greater than a lower bound calculated from the scalar parameter and the attack period , i.e. ; ​a linear matrix inequality construction unit, configured to convert the constraint condition and the regularity and impulse-free requirement that the closed-loop augmented system needs to satisfy into a set of sufficient conditions characterized by linear matrix inequalities; a scalar parameter introduction unit, configured to introduce adjustable scalar parameters for quantifying the stability margin and performance of the system in the process of constructing the set of linear matrix inequalities , a parameter for constraining the state decay rate of the system , a parameter for constraining the positive definite matrix proportionality relationship , and an auxiliary scalar parameter for decoupling and relaxing the matrix inequality to expand the range of feasible solutions;

[0100] In some embodiments, the collaborative optimization module is specifically configured to perform the following steps: S51, set the initial value of the introduced adjustable scalar parameter, and configure the convergence tolerance and the maximum number of iterations of the linear matrix inequality solver; S52, taking the current value of the scalar parameter as a fixed condition, constructing a convex optimization problem with the constructed set of linear matrix inequalities as a convex constraint and the matrix variable as an optimization objective, and calling a numerical solving tool to solve the convex optimization problem; S53, determining whether there is a feasible solution to the convex optimization problem in the current iteration; If there is a feasible solution, record the current scalar parameter value and the corresponding matrix variable feasible solution, and perform step S54; If there is no feasible solution, adjust the value of the scalar parameter and return to S52 for next iteration calculation; S54, calculating the gain parameters of the reduced-order observer and the controller according to the matrix variable feasible solution obtained in S53; S55, determining the set of scalar parameter values recorded in S53 that make the convex optimization problem have a feasible solution as the optimized scalar parameter values.

[0101] The optimization output module is specifically configured to perform the following steps: S61, obtaining a set of optimized scalar parameter values, and obtaining the attack period from the established denial-of-service attack model; S62, calculating the maximum attack duration of the denial-of-service attack in each period under the current optimized parameter to ensure the stability of the system. S63, based on the calculated maximum attack duration and in combination with the attack period, calculate the minimum time length that the reduced-order observer-based controller must at least normally run in each attack period; the minimum running time is equal to the attack period minus the maximum attack duration; S64, judge whether the optimization result of the maximum attack duration and the minimum running time meets the predetermined system design index; If yes, output all the current optimized parameters and performance indexes; If not, return to S5, readjust the scalar parameters and perform iterative optimization again until the system performance index meeting the requirement is obtained.

[0102] Maximum attack duration The calculation formula is:

[0103] Minimum running time .

[0104] The above description of disclosed embodiments enables those skilled in the art to carry out or use the present application. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined in this application can be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application will not be limited to these embodiments shown in this application, but will conform to the widest scope consistent with the principles and novel features disclosed in this application.

Claims

1. A method for multi-parameter collaborative optimization control of a singular system under DoS attack, characterized in that, The method comprises the following steps: S1, establishing a singular system model containing differential-algebraic constraints and a periodic DoS attack model describing the on-off state of DoS attack; S2, based on the established singular system model and DoS attack model, constructing a reduced-order observer for reconstructing system state information by solving a Sylvester equation, and designing a controller in combination with the output information of the reduced-order observer; S3, establishing a closed-loop augmented system of the singular system under DoS attack based on the reduced-order observer control; S4, based on Lyapunov stability theory, establishing sufficient conditions for the closed-loop augmented system to satisfy regularity, no impulse and asymptotic stability; the sufficient conditions are represented by a linear matrix inequality group, and the linear matrix inequality group contains adjustable scalar parameters for quantifying the stability margin and performance of the system; S5, by adjusting the scalar parameters and solving the corresponding linear matrix inequality group, iterative optimization is performed until a group of feasible solutions is obtained, so that the gain parameters of the reduced-order observer and the gain parameters of the controller are designed, and a group of optimized scalar parameter values are finally determined; S6, based on the optimized scalar parameter values and the attack period, the maximum duration of DoS attack that the singular system can tolerate and the minimum running time required by the controller are calculated, and the stability of the singular system and the running cost are optimized.

2. The multi-parameter collaborative optimization control method of the singular system under the DoS attack according to claim 1, characterized in that, S1 specifically comprises: S11, establishing a singular system model: wherein represents a state of the system; represents an output of the system; represents a control input to the system; matrix is a singular matrix and satisfies , represents a constant matrix; S12, establishing a DoS attack model: wherein, is the number of cycles, , is the cycle, is the attack sleep time, , represents no attack signal, normal communication; represents attack information, communication interruption.

3. The multi-parameter collaborative optimization control method of the singular system under the DoS attack according to claim 2, characterized in that, S2 specifically comprises: S21, constructing a reduced-order observer according to the singular system model for reconstructing system state information; wherein the reduced-order observer is as follows: wherein is the state of the reduced order observer, is the estimate of the system state is the estimate of the system state is the parameter matrix to be designed; S22, determining the parameter matrix of the reduced-order observer by solving a Sylvester equation; The Sylvester equation is as follows: Wherein J is an auxiliary matrix determined by matrix equivalence transformation; S23, define intermediate variables , construct error system based on Sylvester equation constraints; wherein is the estimation error; S24, designing a controller based on the state estimation value output by the reduced-order observer and the output of the system; the controller is a switching controller based on the state of the reduced-order observer: wherein Q , S is the controller parameter matrix to be designed.

4. The multi-parameter collaborative optimization control method of the singular system under the DoS attack according to claim 3, characterized in that, In S3, the closed-loop augmented system is obtained in combination with the singular system, the error system, the controller and the DoS attack model: wherein .

5. The multi-parameter collaborative optimization control method of the singular system under the DoS attack according to claim 4, characterized in that, S4 specifically comprises: S41, using Lyapunov function, analyzing the system dynamics under each attack mode, and deriving constraint conditions to ensure the asymptotic stability of the entire switching system; the constraint conditions include: requiring the existence of two positive definite matrices To meet the proportional boundedness relationship between them, that is, At the same time, the attack dormancy time Must be greater than the lower bound calculated by the scalar parameter And the attack period , that is, ; S42, the constraint conditions and the regularity and non-impulse requirements required by the closed-loop augmented system are converted into a group of sufficient conditions represented in the form of linear matrix inequalities; the group of linear matrix inequalities specifically comprises two parts corresponding to the two system modes of no attack period and attack period; S43、In the process of constructing the linear matrix inequality set, adjustable scalar parameters for quantifying the stability margin and performance of the system are introduced; the scalar parameters include parameters for constraining the state decay rate of the system , parameters for constraining the proportional relationship of the positive definite matrix , and auxiliary scalar parameters for decoupling and relaxing the matrix inequality to expand the range of feasible solutions; S44, the constructed linear matrix inequality group and the inequality constraint about attack dormancy time jointly constitute sufficient conditions for ensuring that the closed-loop augmented system has desired stable performance under denial of service attack.

6. The multi-parameter collaborative optimization control method of the singular system under the DoS attack according to claim 5, characterized in that, The steps of S5 comprise: S51, setting the initial value of the introduced adjustable scalar parameter, and configuring the convergence tolerance and the maximum number of iterations of the linear matrix inequality solver; S52, taking the current value of the scalar parameter as a fixed condition, constructing a convex optimization problem with matrix variables as the optimization objective by taking the constructed linear matrix inequality group as the convex constraint, and calling a numerical solving tool to solve the convex optimization problem; S53, judging whether there is a feasible solution to the convex optimization problem under the current iteration; If there is a feasible solution, record the current scalar parameter value and the corresponding matrix variable feasible solution, and execute step S54; If there is no feasible solution, adjust the value of the scalar parameter, and return to S52 for the next iteration calculation; S54, according to the matrix variable feasible solution obtained in S53, calculate the gain parameters of the reduced-order observer and the gain parameters of the controller; S55, determine the set of scalar parameter values recorded in S53 that make the convex optimization problem have a feasible solution as the optimized scalar parameter values.

7. The multi-parameter collaborative optimization control method of the singular system under the DoS attack according to claim 6, characterized in that, The steps of S6 include: S61, obtain a set of optimized scalar parameter values, and obtain the attack period from the established denial of service attack model; S62, according to the obtained optimized scalar parameter values and the attack period, calculate the maximum attack duration of the denial of service attack in each period under the current optimization parameter to ensure system stability; S63, according to the calculated maximum attack duration, and combined with the attack period, calculate the minimum time length that the controller based on the reduced-order observer must at least normally run in each attack period, that is, the minimum running time; the minimum running time is equal to the attack period minus the maximum attack duration; S64, judge whether the optimization result of the maximum attack duration and the minimum running time meets the predetermined system design index; If it is satisfied, output all the current optimized parameters and performance indicators; If it is not satisfied, return to S5 to re-adjust the scalar parameter and perform iterative optimization again until the system design index that meets the requirements is obtained.

8. The multi-parameter collaborative optimization control method of the singular system under the DoS attack according to claim 7, characterized in that, Maximum attack duration The formula for calculating: Minimum run time .

9. The multi-parameter collaborative optimization control method of the singular system under the DoS attack according to claim 8, characterized in that, The method further includes: The calculated maximum attack duration is used as a quantitative index to evaluate the anti-attack robustness of the current system design; The calculated minimum running time of the controller is used as a quantitative index to evaluate the minimum running cost required to achieve this level of robustness; The maximum attack duration and the minimum running time are used as the final output of the system design to provide the user with a design trade-off basis: the highest intensity attack scenario that can be tolerated and the minimum control resources required under the premise of ensuring system stability.

10. A multi-parameter collaborative optimization control system for a singular system under DoS attack, characterized in that, It includes: A modeling module for establishing a singular system dynamics model containing differential-algebraic constraints and a periodic DoS attack model describing the on-off state of DoS attack; A design module for constructing a reduced-order observer for reconstructing system state information by solving a Sylvester equation based on the established singular system model and DoS attack model, and designing a controller combined with the output information of the reduced-order observer; A closed-loop augmented system establishment module for establishing a closed-loop augmented system of the singular system controlled based on the reduced-order observer under DoS attack; A stability condition construction module for establishing sufficient conditions for the closed-loop augmented system to satisfy regularity, no impulse and asymptotic stability based on Lyapunov stability theory; the sufficient conditions are represented by a set of linear matrix inequalities, which contain adjustable scalar parameters for quantifying the stability margin and performance of the system; a collaborative optimization module, configured to iteratively optimize by adjusting the scalar parameters and solving corresponding linear matrix inequality sets until a set of feasible solutions is obtained, thereby collaboratively designing gain parameters of the reduced-order observer and gain parameters of the controller and finally determining a set of optimized scalar parameter values; an optimization output module, configured to calculate a maximum duration of a DoS attack that the singular system can tolerate and a minimum running time required by the controller based on the optimized scalar parameter values and the attack period, achieve collaborative optimization of stability of the singular system and running cost, and output final optimization parameters and performance indicators.

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