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

Abstract

The application 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 DoS attack. The method comprises the following steps: establishing a singular system model and a DoS attack model; constructing a reduced-order observer by solving a Sylvester equation, and designing a controller in combination with the output of the reduced-order observer; establishing a closed-loop augmented system based on the reduced-order observer control; establishing sufficient conditions for the closed-loop augmented system to meet the regular, impulse-free and asymptotically stable; iteratively optimizing by adjusting scalar parameters and solving a linear matrix inequality set, so as to collaboratively design the gain parameters of the reduced-order observer and the gain parameters of the controller, and finally determine a group of scalar parameter values; and calculating the maximum duration of the DoS attack and the minimum running time required by the controller, so as to realize the collaborative optimization of the stability of the singular system and the running cost. The method solves the problems of cost and complexity increase caused by the full-order observer control.

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:

[0007] S1. Establishing a singular system model containing differential-algebraic constraints and a periodic DoS attack model describing the on-off state of DoS attack;

[0008] 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;

[0009] S3. Establishing a closed-loop augmented system for the singular system under DoS attack based on the reduced-order observer control;

[0010] 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 set of linear matrix inequalities, which contain adjustable scalar parameters for quantifying the stability margin and performance of the system;

[0011] 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, 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;

[0012] S6. Based on the optimized scalar parameter values and the attack period, calculating the maximum duration of DoS attack that the singular system can tolerate and the minimum running time required by the controller, and realizing collaborative optimization of the stability of the singular system and the running cost.

[0013] As a further limitation of the present application, S1 specifically comprises:

[0014] S11. Establishing a singular system model:

[0015]

[0016] wherein, x represents the state of the system; y represents the output of the system; u represents the control input of the system; matrix is a singular matrix, and satisfies , is a constant matrix;

[0017] S12. Establishing a DoS attack model:

[0018]

[0019] wherein, is the periodicity, , is the period, is the attack sleep time, , represents no attack signal, and the communication is normal; represents that there is an attack information, and the communication is interrupted.

[0020] The singular system model and the 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 the singular system under DoS attack. The periodic DoS attack model can clearly describe the on-off state of the 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 the DoS attack.

[0021] 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.

[0022] As a further limitation of the technical scheme of the application, S2 specifically comprises:

[0023] 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:

[0024]

[0025] In the formula, is the state of the reduced-order observer, is the estimated value of the system state , is the parameter matrix to be designed;

[0026] S22, determining the parameter matrix of the reduced-order observer by solving the Sylvester equation;

[0027] The Sylvester equation is as follows:

[0028]

[0029] wherein J is an auxiliary matrix determined by matrix equivalent transformation;

[0030] S23, defining an intermediate variable , the error system is constructed based on Sylvester equation constraint;

[0031]

[0032] wherein, is the estimation error;

[0033] S24, a state estimator based on the output of the reduced-order observer and the output of the system is designed to design a controller; the controller is a switching controller based on the state of the reduced-order observer:

[0034]

[0035] wherein, Q , S is the controller parameter matrix to be designed.

[0036] By solving the Sylvester equation to construct the reduced-order observer, the computational complexity of the system can be effectively reduced, the real-time performance and response speed of the system can be improved, and the problem of exponential growth of the computational burden caused by the full-order observer in the prior art is solved, so that the system is more suitable for complex applications in actual engineering. The output information of the reduced-order observer is combined to design the controller, which can more accurately estimate the system state, thereby improving the performance of the controller and the stability of the system, enhancing the control accuracy and stability of the system under DoS attack, and effectively dealing with the damage of DoS attack to the performance of the system. The reduced-order observer can adapt to different system size and complexity, and enhances the adaptability and flexibility of the system.

[0037] As a further limitation of the technical scheme of the application, in S3, the singular system, the error system, the controller and the DoS attack model are combined to obtain a closed-loop augmented system:

[0038]

[0039] wherein, .

[0040] As a further limitation of the technical scheme of the application, S4 specifically comprises:

[0041] 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: 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, ;

[0042] 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 characterized by linear matrix inequalities; the set of linear matrix inequalities specifically includes two parts corresponding to two system modes of no attack period and attack period;

[0043] S43, in the process of constructing the set of linear matrix inequalities, adjustable scalar parameters for quantifying system stability margin and performance are introduced; the scalar parameters include parameters for constraining system state decay rate , parameters for constraining the positive definite matrix proportional relationship , and auxiliary scalar parameters for decoupling and relaxing matrix inequalities to expand the range of feasible solutions;

[0044] S44, the set of constructed linear matrix inequalities and the inequality constraint on attack dormancy time jointly constitute sufficient conditions for ensuring that the closed-loop augmented system has desired stable performance under denial of service attack.

[0045] Based on Lyapunov stability theory, sufficient conditions for the closed-loop augmented system to meet regularity, impulse-free and asymptotic stability are 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 adjustable scalar parameters, 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 conditions are converted into a set of linear matrix inequalities, which can be solved using mathematical tools, efficiently optimizing the design and improving the feasibility and efficiency of the design, providing effective mathematical tools and methods for solving complex control problems of information physical systems under DoS attack.

[0046] As a further limitation of the technical solution of the application, the step S5 includes:

[0047] S51, set the initial value of the introduced adjustable scalar parameter, and configure the convergence tolerance and maximum iteration number of the linear matrix inequality solver;

[0048] 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 set of linear matrix inequalities as convex constraints, and calling a numerical solving tool to solve the convex optimization problem;

[0049] S53, judge whether there is a feasible solution to the convex optimization problem under the current iteration;

[0050] If there is a feasible solution, record the current scalar parameter value and the corresponding matrix variable feasible solution, and perform step S54;

[0051] If there is no feasible solution, adjust the value of the scalar parameter, and return to S52 for the next iteration calculation;

[0052] 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;

[0053] 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.

[0054] By adjusting the scalar parameters 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 parameter optimization design is realized, the system collaborative control ability is improved, and the overall performance of the system under DoS attack is enhanced. The optimized scalar parameter values can make the system have better stability and performance under DoS attack, improve the anti-attack ability of the system, effectively cope with the damage of DoS attack to the performance and stability of the system, and ensure 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.

[0055] As a further limitation of the technical solution of the application, the steps of S6 include:

[0056] S61, obtain a set of optimized scalar parameter values, and obtain the attack period from the established denial of service attack model;

[0057] S62, according to the obtained optimized scalar parameter values and attack period, calculate the maximum attack duration of denial of service attack in each period under the current optimization parameter to ensure system stability;

[0058] 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;

[0059] S64, judge whether the optimization results of the maximum attack duration and the minimum running time meet the predetermined system design index;

[0060] If it is satisfied, output all the current optimized parameters and performance indexes;

[0061] If not satisfied, return to S5, readjust the scalar parameter and iterate optimization again until the system design index meeting the requirements is obtained.

[0062] 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 running 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 running cost on the premise of ensuring stability.

[0063] As a further limitation of the technical scheme of the application, the maximum attack duration The calculation formula is:

[0064]

[0065] The minimum running time .

[0066] 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, so that users can select the most suitable system design on the premise of ensuring system stability, and realize the optimal trade-off between performance and cost.

[0067] As a further limitation of the technical scheme of the application, the method further comprises:

[0068] The calculated maximum attack duration is used as a quantitative index to evaluate the anti-attack robustness of the current system design.

[0069] The calculated minimum running time of the controller is used as a quantitative index to evaluate the minimum running cost required to achieve the level of robustness.

[0070] The maximum attack duration and the minimum running time are used as the final output of the system design, which provides clear design trade-off basis for users: the highest intensity attack scenario that can be tolerated and the minimum control resources required on the premise of ensuring system stability; so that users can make reasonable decisions according to actual demand and resource constraints.

[0071] In a second aspect, the technical scheme of the application further provides a multi-parameter coordinated optimization control system for singular systems under DoS attack, comprising:

[0072] 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 the DoS attack;

[0073] 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;

[0074] a closed-loop augmented system establishment module configured to establish a closed-loop augmented system of the singular system under DoS attack based on the reduced-order observer control;

[0075] a stability condition construction module configured to, based on Lyapunov stability theory, establish sufficient conditions 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, which contain adjustable scalar parameters for quantifying the stability margin and performance of the system;

[0076] a collaborative optimization module configured to, by adjusting the scalar parameters and solving the corresponding set of linear matrix inequalities, perform iterative optimization 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;

[0077] an optimization output module configured to, based on the optimized scalar parameter values and the attack period, calculate the maximum duration of DoS attack that the singular system can tolerate and the minimum running time required for the controller, realize collaborative optimization of the stability of the singular system and the running cost, and output the final optimization parameters and performance indicators.

[0078] As can be seen 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 addressing the network security problems faced by the singular system, and ensuring the performance and stability of the system. The collaborative optimization of the stability of the singular system and the running cost is realized, which can minimize the consumption of control resources under the premise of ensuring the stability of the system, reduce the cost and computational complexity of the sensors, improve the running efficiency and economy of the system, and solve the problem of increased cost and complexity caused by full-order observer control.

[0079] Through the multi-parameter collaborative optimization control method, the robustness of the singular system under DoS attack is enhanced, making the system better cope with complex and variable network attacks, ensuring the stable operation of critical infrastructure, and improving the adaptability and universality of the system, meeting the needs of complex systems in actual engineering. BRIEF DESCRIPTION OF DRAWINGS

[0080] In order to more clearly illustrate the technical solutions of the present application, the drawings required to be used in the description will be briefly introduced as follows. 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 any creative effort on the basis of these drawings.

[0081] Figure 1 The flow chart of the multi-parameter collaborative optimization control method provided by the present application.

[0082] Figure 2 The schematic diagram of the modeling principle of the singular system under DoS attack in the present application.

[0083] Figure 3 The block diagram of the multi-parameter collaborative optimization control system provided by the present application. DETAILED DESCRIPTION

[0084] In order to make the application purposes, features and advantages of the present application more obvious and easy to understand, the technical solutions protected by the present application will be described clearly and completely by using specific embodiments and drawings. Obviously, the following described embodiments are only some of the embodiments of the present application, but not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without any creative effort are within the scope of protection of the present application.

[0085] Unless otherwise defined, all technical and scientific terms used in the present application have the same meanings as those commonly understood by those skilled 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 the specific embodiments and are not intended to limit the present application.

[0086] As shown in Figure 1 The present application embodiment provides a multi-parameter collaborative optimization control method of singular system under DoS attack, which comprises the following steps:

[0087] 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;

[0088] In this step, the data packet loss problem under DoS network attack is considered, and the dynamic model of singular system and the DoS attack model are established. The modeling principle of singular system under DoS attack is shown in Figure 2 .

[0089] 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 Sylvester equation, and a controller is designed in combination with the output information of the reduced-order observer;

[0090] It should be noted that Sylvester equation and non-homogeneous linear equation set solving knowledge 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.

[0091] S3, a closed-loop augmented system of the singular system based on the reduced-order observer control under DoS attack is established;

[0092] S4, based on Lyapunov stability theory, a sufficient condition is established to make the closed-loop augmented system satisfy regular, impulse-free and asymptotically stable; 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;

[0093] In this step, the conditions that the closed-loop augmented system is regular, impulse-free and asymptotically stable are established by Lyapunov stability theory and matrix inequality theory, so that the singular information physical system can still operate safely and stably under DoS network attack.

[0094] S5, by adjusting the scalar parameters and solving the corresponding linear matrix inequality set, 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;

[0095] It should be further noted that the obtained stability condition is used to give the design algorithm of the reduced-order observer and the controller by using the multi-parameter coordinated optimization control method.

[0096] 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.

[0097] In this step, the maximum attack time of the DoS network attack that the singular system can withstand in each period is calculated by optimizing each design parameter , 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 , and finally the dual optimization strategy of coordinated optimization control and reduction of actual running cost is realized.

[0098] In the embodiment of the application, S1 is specifically:

[0099] S11. Establish a singular system model:

[0100]

[0101] in, Indicates the state of the system; Indicates the system output; Represents the system's control input; matrix It is a singular matrix and satisfies , Represents a constant matrix;

[0102] S12. Establish a DoS attack model:

[0103]

[0104] in, It refers to the number of cycles. , It's a cycle. To attack the sleep time, , This indicates that there are no attack signals and communication is normal. This indicates an attack has occurred, and communication has been interrupted.

[0105] In some embodiments, S2 specifically includes:

[0106] S21. Construct a reduced-order observer based on the singular system model to reconstruct the system state information; the reduced-order observer is as follows:

[0107]

[0108] In the formula, For the state of the reduced-order observer, System status The estimated value, The parameter matrix to be designed;

[0109] S22. Determine the parameter matrix of the reduced-order observer by solving the Sylvester equation;

[0110] The Sylvester equation is:

[0111]

[0112] Where J is an auxiliary matrix determined through matrix equivalence transformation;

[0113] S23 includes:

[0114] (a) Define intermediate variables An error system is constructed based on the constraints of the Sylvester equations;

[0115]

[0116] wherein, is the estimation error;

[0117] the definition matrix , wherein is a row full rank matrix;

[0118] from equation can be obtained:

[0119]

[0120]

[0121] (b) from Sylvester equation can be obtained:

[0122]

[0123] solving the above equation can be obtained:

[0124]

[0125] wherein, matrix is the unknown parameter to be designed.

[0126] S24, based on the state estimation value of the reduced order observer output and the output of the system design controller; the controller is a switching controller based on the state of the reduced order observer:

[0127]

[0128] wherein, Q , S is the controller parameter matrix to be designed.

[0129] using can be obtained:

[0130]

[0131] wherein, matrix is the parameter matrix to be designed.

[0132] In some embodiments, S3, combined with singular system, error system, controller and DoS attack model, the closed loop augmented system is obtained:

[0133]

[0134] wherein, .

[0135] In some embodiments, S4 specifically comprises:

[0136] S41, using Lyapunov function, analyzing system dynamics under each attack mode, and deriving constraint conditions to ensure asymptotic stability of the entire switching system; the constraint conditions include: requiring existence of two positive definite matrices to satisfy the proportional boundedness relationship between them, i.e. At the same time, attack dormancy time must be greater than a lower bound calculated by scalar parameter and attack period , i.e. ;

[0137] S42, converting the constraint conditions and the regularity and impulse-free requirements that the closed-loop augmented system needs to meet 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 two system modes of attack-free period and attack period;

[0138] S43, in the process of constructing the set of linear matrix inequalities, introducing adjustable scalar parameters for quantifying system stability margin and performance; the scalar parameters include parameter for constraining system state decay rate, parameter for constraining the proportional relationship of the positive definite matrices, and auxiliary scalar parameters for decoupling and relaxing matrix inequalities to expand the range of feasible solutions;

[0139] S44, the set of constructed linear matrix inequalities and the inequality constraint on attack dormancy time jointly constitute sufficient conditions to ensure that the closed-loop augmented system has desired stability performance under denial-of-service attack.

[0140] 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:

[0141]

[0142]

[0143] When ,

[0144]

[0145] When ,

[0146]

[0147] where, , , denote the symmetric terms in the corresponding positions of the matrices, the subscript denotes the transpose of the matrix, denotes the sum of the matrix and its transpose, and the positive definite matrix , , is a free matrix, and the parameters are positive numbers, , the matrix and satisfy .

[0148] The above inequalities are solved by using the LMI toolbox in MATLAB, when the optimal feasible solution is obtained, the closed-loop augmented system satisfies the regularity, non-impulsiveness and asymptotic stability, and the state feedback gain matrix and the reduced-order observer gain matrix are respectively:

[0149]

[0150] In the embodiments of the present application, the asymptotic stability of the closed-loop augmented system is also proved based on the decoupling lemma of matrix inequality and Lyapunov function, which is specifically as follows:

[0151] S421, the closed-loop augmented system is proved to be regular, non-impulsive and asymptotically stable. First, a decoupling lemma of matrix inequality is given: for a matrix and a constant , the following two inequalities are equivalent:

[0152] (1)

[0153] (2)

[0154] S422, a Lyapunov function is constructed:

[0155]

[0156] wherein, the matrix .

[0157] S423, when , the matrix inequality and the given decoupling lemma are used to calculate:

[0158]

[0159] wherein, The above inequalities continue to use the matrix inequality decoupling lemma, and the following can be obtained:

[0160]

[0161] wherein, , .

[0162] S424, when , the same method as S423 can be obtained:

[0163]

[0164] S425, the derivative of Lyapunov function V(t) is calculated, and the linear matrix inequality obtained by combining steps S423 and S424 is calculated:

[0165]

[0166] Integrating the above inequalities respectively can obtain:

[0167]

[0168] From , it can be concluded that:

[0169]

[0170] Using the constraint condition in S42, the iterative calculation can be obtained:

[0171]

[0172] wherein Using Lyapunov stability theory, it can be obtained that the closed-loop augmented system is asymptotically stable.

[0173] In some embodiments, the steps of S5 include:

[0174] 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;

[0175] 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;

[0176] S53, judge whether there is a feasible solution to the convex optimization problem in the current iteration;

[0177] If there is a feasible solution, record the current scalar parameter value and the corresponding matrix variable feasible solution, and execute step S54;

[0178] If there is no feasible solution, adjust the value of the scalar parameters, and return to S52 for the next iteration calculation;

[0179] 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;

[0180] 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.

[0181] The design algorithm of the reduced order observer and the controller is shown in Algorithm 1:

[0182] Algorithm 1: Design of reduced order observer and controller

[0183] Input: matrix , parameter ;

[0184] Output: reduced order observer gain matrix ; controller gain matrix ;

[0185] Reduced order observer parameters: ;

[0186] Controller parameters: ;

[0187] Step 1: Solve the convex optimization condition given in S42 to obtain a feasible optimal solution ;

[0188] Step 2: If the convex optimization condition has no feasible solution, adjust the parameters , and return to execute Step 1;

[0189] Step 3: Solve the reduced order observer gain matrix , the controller gain matrix ;

[0190] Step 4: The controller parameters can be obtained using S24 , ;

[0191] Step 5: The matrix can be calculated using S23(a) ;

[0192] Step 6: The reduced order observer parameters: are derived using S23(b) respectively

[0193] In some embodiments, the steps of S6 include:

[0194] S61, obtaining a set of optimized scalar parameter values, and obtaining an attack period from the established denial-of-service attack model;

[0195] S62, calculating the maximum attack duration of the denial-of-service attack in each period under the current optimized parameters to ensure system stability according to the obtained optimized scalar parameter values and the attack period;

[0196] S63, calculating the minimum running time of the controller based on the reduced-order observer in each attack period, which must be at least normally running, according to the calculated maximum attack duration and the attack period; the minimum running time is equal to the attack period minus the maximum attack duration;

[0197] S64, judging whether the optimization results of the maximum attack duration and the minimum running time meet the predetermined system design index;

[0198] If yes, output all the current optimized parameters and performance indexes;

[0199] If no, return to S5 to re-adjust the scalar parameters and perform iterative optimization again until the system design index meeting the requirements is obtained.

[0200] Maximum attack duration The calculation formula is:

[0201]

[0202] Minimum running time .

[0203] In some embodiments, the method further comprises:

[0204] The calculated maximum attack duration is used as a quantitative index to evaluate the anti-attack robustness of the current system design;

[0205] 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;

[0206] The maximum attack duration and the minimum running time are used as the final output of the system design to provide 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.

[0207] 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, which comprises:

[0208] a modeling module, configured to establish a singular system dynamics model containing differential-algebraic constraints and a periodic DoS attack model describing on-off states of DoS attacks;

[0209] a design module, configured to construct 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 design a controller in combination with output information of the reduced-order observer;

[0210] a closed-loop augmented system establishment module, configured to establish a closed-loop augmented system of the singular system under DoS attacks based on the reduced-order observer control;

[0211] a stability condition construction module, configured to establish a sufficient condition for the closed-loop augmented system to satisfy regularity, no impulse and asymptotic stability based on Lyapunov stability theory; the sufficient condition is represented by a linear matrix inequality set, which contains adjustable scalar parameters for quantifying a stability margin and performance of the system;

[0212] a collaborative optimization module, configured to perform iterative optimization by adjusting the scalar parameters and solving corresponding linear matrix inequality sets until a set of feasible solutions is obtained, so as to collaboratively design gain parameters of the reduced-order observer and gain parameters of the controller, and finally determine a set of optimized scalar parameter values;

[0213] an optimization output module, configured to calculate a maximum duration of DoS attacks that the singular system can tolerate and a minimum running time required by the controller based on the optimized scalar parameter values and an attack period, so as to realize collaborative optimization of singular system stability and running cost, and output final optimization parameters and performance indicators.

[0214] In some embodiments, the modeling module specifically includes:

[0215] a singular system model establishment unit, configured to establish a singular system model:

[0216]

[0217] wherein, x represents a state of the system; y represents an output of the system; u represents a control input of the system; and matrix is a singular matrix, and satisfies , c represents a constant matrix;

[0218] a DoS attack model establishment unit, configured to establish a DoS attack model:

[0219]

[0220] wherein, is the periodicity, , is the period, is the attack dormancy time, , represents no attack signal, and the communication is normal; represents attack information, and the communication is interrupted.

[0221] In some embodiments, the design module specifically comprises:

[0222] a reduced-order observer construction unit, configured to construct a reduced-order observer according to the singular system model, and used to reconstruct system state information; wherein the reduced-order observer is as follows:

[0223]

[0224] wherein, is the state of the reduced-order observer, is the estimated value of the system state , is a parameter matrix to be designed;

[0225] a Sylvester equation solving unit, configured to determine the parameter matrix of the reduced-order observer by solving a Sylvester equation;

[0226] the Sylvester equation is as follows:

[0227]

[0228] wherein, J is an auxiliary matrix determined through matrix equivalent transformation;

[0229] an error system construction unit, configured to define an intermediate variable , and construct an error system based on Sylvester equation constraints;

[0230]

[0231] wherein, is an estimation error;

[0232] a controller design unit, configured to design 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:

[0233]

[0234] wherein, Q , S is a controller parameter matrix to be designed.

[0235] In some embodiments, the closed-loop augmented system establishment module is configured to establish a closed-loop augmented system by combining a singular system, an error system, a controller and a DoS attack model.

[0236]

[0237] wherein, .

[0238] In some embodiments, the stability condition construction module comprises:

[0239] 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 switched system; the constraint conditions include: requiring the existence of two positive definite matrices to satisfy the proportional boundedness relationship between them, i.e. Meanwhile, the attack dormant time must be greater than a lower bound calculated from the scalar parameter and the attack period , i.e. ;

[0240] a linear matrix inequality construction unit configured to convert the constraint conditions and the regularity and impulse-free requirements that the closed-loop augmented system needs to satisfy 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 two system modes of the attack-free period and the attack period;

[0241] a scalar parameter introduction unit configured to introduce adjustable scalar parameters for quantifying the system stability margin and performance in the process of constructing the set of linear matrix inequalities; the scalar parameters include a parameter for constraining the system state decay rate, a parameter for constraining the proportional relationship of the positive definite matrices, and auxiliary scalar parameters for decoupling and relaxing the matrix inequalities to expand the feasible solution range;

[0242] a stability condition integration unit configured to jointly constitute sufficient conditions for ensuring the closed-loop augmented system to have desired stable performance under denial-of-service attacks from the constructed set of linear matrix inequalities and the inequality constraints on the attack dormant time.

[0243] In some embodiments, the collaborative optimization module is specifically configured to perform the following steps:

[0244] 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;

[0245] S52, taking the current value of the scalar parameter as a fixed condition, constructing a convex optimization problem with the constructed linear matrix inequality set as a convex constraint and a matrix variable as an optimization objective, and calling a numerical solving tool to solve the convex optimization problem;

[0246] S53, judging whether there is a feasible solution to the convex optimization problem in the current iteration;

[0247] If there is a feasible solution, record the current scalar parameter value and the corresponding matrix variable feasible solution, and execute step S54;

[0248] If there is no feasible solution, adjust the value of the scalar parameter, and return to S52 for next iteration calculation;

[0249] S54, calculating the gain parameters of the reduced-order observer and the controller according to the matrix variable feasible solution obtained in S53;

[0250] S55, determining the set of scalar parameter values recorded in S53, which makes the convex optimization problem have a feasible solution, as the optimized scalar parameter values.

[0251] The optimization output module is specifically configured to execute the following steps:

[0252] S61, obtaining a set of optimized scalar parameter values, and obtaining the attack period from the established denial of service attack model;

[0253] S62, calculating the maximum attack duration of the denial of service attack in each period under the current optimized parameter to ensure system stability according to the obtained optimized scalar parameter values and attack period;

[0254] S63, calculating the minimum time length that the controller based on the reduced-order observer must be normally operated in each attack period according to the calculated maximum attack duration and in combination with the attack period; the minimum running time is equal to the attack period minus the maximum attack duration;

[0255] S64, judging whether the optimization results of the maximum attack duration and the minimum running time meet the predetermined system design index;

[0256] If yes, output all the current optimized parameters and performance indexes;

[0257] If not, return to S5 to re-adjust the scalar parameter and perform iterative optimization again until the system performance index meeting the requirements is obtained.

[0258] The calculation formula of the maximum attack duration

[0259] ​

[0260] Minimum run time .

[0261] The foregoing description of the disclosed embodiments enables a person skilled in the art to make or use the application. Numerous modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein can be applied to other embodiments without departing from the spirit or scope of the application. Therefore, the present application is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A multi-parameter cooperative optimization control method for a singular system under a DoS attack, characterized in that, Includes the following steps: S1. Establish a singular system model with differential-algebraic constraints and a periodic DoS attack model describing the on / off state of a DoS attack. 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 the Sylvester equation, and a controller is designed by combining the output information of the reduced-order observer. S3. Establish a closed-loop augmented system based on the reduced-order observer control of the singular system under DoS attack; S4. Based on Lyapunov stability theory, establish sufficient conditions for the closed-loop augmented system to be regular, impulse-free, and asymptotically stable; the sufficient conditions are characterized by a set of linear matrix inequalities, which contain 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 feasible solution 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. S6. Based on the optimized scalar parameter values ​​and attack cycle, calculate the maximum duration of a DoS attack that the singular system can tolerate and the minimum running time required by the controller, thereby achieving synergistic optimization of the stability and operating cost of the singular system.

2. The multi-parameter collaborative optimization control method for a singular system under a DoS attack according to claim 1, characterized in that, S1 specifically includes: S11. Establish a singular system model: in, Indicates the state of the system; Indicates the system output; Represents the system's control input; matrix It is a singular matrix and satisfies , Represents a constant matrix; S12. Establish a DoS attack model: in, It refers to the number of cycles. , It's a cycle. To attack the sleep time, , This indicates that there are no attack signals and communication is normal. This indicates an attack has occurred, and communication has been interrupted.

3. The multi-parameter collaborative optimization control method for a singular system under a DoS attack according to claim 2, characterized in that, S2 specifically includes: S21. Construct a reduced-order observer based on the singular system model to reconstruct the system state information; the reduced-order observer is as follows: In the formula, For the state of the reduced-order observer, System status The estimated value, The parameter matrix to be designed; S22. Determine the parameter matrix of the reduced-order observer by solving the Sylvester equation; The Sylvester equation is: Where J is an auxiliary matrix determined through matrix equivalence transformation; S23. Define intermediate variables An error system is constructed based on the constraints of the Sylvester equations; in, To estimate the error; S24. Design a controller based on the state estimate output of the reduced-order observer and the system output; the controller is a switching controller based on the reduced-order observer state: in, Q , S This is the parameter matrix of the controller to be designed.

4. The multi-parameter collaborative optimization control method for a singular system under a DoS attack according to claim 3, characterized in that, In S3, by combining singular systems, error systems, controllers, and DoS attack models, a closed-loop augmented system is obtained: in, .

5. The multi-parameter collaborative optimization control method for a singular system under a DoS attack according to claim 4, characterized in that, S4 specifically includes: S41. Using Lyapunov functions, analyze the system dynamics under each attack mode and derive the constraints that guarantee the asymptotic stability of the entire switching system; the constraints include: requiring the existence of two positive definite matrices. To satisfy the bounded proportional relationship between the two, that is At the same time, attack the sleep time It must be greater than a scalar parameter and attack cycle The calculated lower bound, i.e. ; S42. The constraints and the regularity and impulsivity requirements that the closed-loop augmented system must satisfy are transformed into a set of sufficient conditions in the form of linear matrix inequalities. This set of linear matrix inequalities specifically includes two parts, corresponding to the two system modes of no-attack period and attack period, respectively. S43. In constructing a system of linear matrix inequalities, adjustable scalar parameters are introduced to quantify the system's stability margin and performance; these scalar parameters include parameters used to constrain the system's state decay rate. Parameters used to constrain the proportional relationships of the positive definite matrices And auxiliary scalar parameters used to decouple and relax matrix inequalities to expand the range of feasible solutions; S44. The constructed set of linear matrix inequalities and the inequality constraints regarding the attack sleep time together constitute sufficient conditions to guarantee that the closed-loop augmented system has the desired stable performance under denial-of-service attacks.

6. The multi-parameter collaborative optimization control method for a singular system under a DoS attack according to claim 5, characterized in that, The steps in S5 include: S51. Set the initial values ​​of the introduced adjustable scalar parameters and configure the convergence tolerance and maximum number of iterations of the linear matrix inequality solver. S52. With the current value of the scalar parameter as a fixed condition, the constructed system of linear matrix inequalities is used as a convex constraint to construct a convex optimization problem with matrix variables as the optimization objective, and the numerical solution tool is called to solve the convex optimization problem. S53. Determine whether the convex optimization problem has a feasible solution in the current iteration; If a feasible solution exists, record the current scalar parameter value and the corresponding feasible solution of the matrix variable, and execute step S54; If no feasible solution exists, adjust the value of the scalar parameter and return to S52 for the next iteration calculation; S54. Based on the feasible solution of the matrix variables 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 for a singular system under a DoS attack according to claim 6, characterized in that, The steps in S6 include: S61. Obtain a set of optimized scalar parameter values, and at the same time, obtain the attack cycle from the established denial-of-service attack model; S62. Based on the obtained optimized scalar parameter values ​​and attack cycle, calculate the maximum attack duration of the denial-of-service attack in each cycle under the current optimized parameters to ensure system stability. S63. Based on the calculated maximum attack duration and in conjunction with the attack cycle, calculate the minimum time length during which the controller based on the reduced-order observer must operate normally in each attack cycle, i.e., the minimum running time; the minimum running time is equal to the attack cycle minus the maximum attack duration. S64. Determine whether the optimization results of the maximum attack duration and minimum running time meet the predetermined system design indicators; If satisfied, output all optimized parameters and performance metrics. If the requirements are not met, return to S5, readjust the scalar parameters, and perform iterative optimization again until the system design metrics that meet the requirements are obtained.

8. The multi-parameter collaborative optimization control method for a singular system under a DoS attack according to claim 7, characterized in that, Maximum attack duration The calculation formula is as follows: Minimum running time .

9. The multi-parameter collaborative optimization control method for a singular system under a DoS attack according to claim 8, characterized in that, The method also includes: The calculated maximum attack duration is used as a quantitative indicator to evaluate the attack robustness of the current system design. The calculated minimum controller runtime is used as a quantitative metric to evaluate the minimum operating cost required to achieve this level of robustness. The maximum attack duration and minimum runtime are used as the final output of the system design to provide users with a basis for design trade-offs: that is, the highest intensity attack scenario that can be withstood and the minimum control resources required while ensuring system stability.

10. A multi-parameter cooperative optimization control system for a singular system under a DoS attack, characterized in that, include: The modeling module is used to build dynamic models of singular systems with differential-algebraic constraints and periodic DoS attack models that describe the on / off states of DoS attacks. The design module is used to construct a reduced-order observer for reconstructing system state information by solving the Sylvester equation based on the established singular system model and DoS attack model, and to design a controller by combining the output information of the reduced-order observer. A closed-loop augmented system establishment module is used to establish a closed-loop augmented system for a singular system under a DoS attack based on the reduced-order observer control. The stability condition construction module is used to establish sufficient conditions for the closed-loop augmented system to be regular, impulse-free, and asymptotically stable based on Lyapunov stability theory. The sufficient conditions are characterized by a set of linear matrix inequalities, which contain adjustable scalar parameters for quantifying the stability margin and performance of the system. The collaborative optimization module is used 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, 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 optimized output module is used to calculate the maximum duration of a DoS attack that the singular system can tolerate and the minimum runtime required by the controller based on the optimized scalar parameter values ​​and attack cycle, thereby achieving coordinated optimization of the singular system's stability and operating costs, and outputting the final optimized parameters and performance indicators.

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