Design method of Lagrange system adaptive security tracking controller under spoofing attack

Abstract

The invention belongs to the field of security control of information physical systems, and particularly relates to a design method of a Lagrange system adaptive security tracking controller under spoofing attacks, which comprises the following steps: constructing a network attack model and a nonlinear Euler-Lagrange system model; introducing an error variable according to the model and designing a virtual controller by using a backstepping method; designing an adaptive controller and a parameter adaptive law according to the virtual controller; the distributed adaptive control algorithm based on the projection operator solves the influence of preventing signal transmission in the system caused by network attacks, and overcomes the difficulty in controller design caused by mutual coupling between time-varying characteristics of attacks, uncertainty of a non-linear multi-agent system and consistent tracking targets. The method ensures that a plurality of uncertain Euler-Lagrange systems are stable and consistent tracking errors are bounded under the influence of network attacks.

Description

Technical Field

[0001] This invention belongs to the field of security control of Eulerian-Lagrange systems, specifically involving the design method of an adaptive controller for realizing leader-follower consistency in multiple Eulerian-Lagrange systems and for uncertain nonlinear systems under network attacks. Background Technology

[0002] In the context of networked systems, with the rapid development of multi-agent systems, distributed control of Eulerian-Lagrange systems has gradually become an important research direction. The coordinated control problem of multi-agent systems, especially the realization of consensus control, has become one of the research hotspots in this field. Meanwhile, the engineering application value of distributed control in Eulerian-Lagrange systems is increasingly prominent. On the one hand, it can achieve group collaboration relying only on local neighbor information, which helps reduce communication pressure and improve system scalability; on the other hand, distributed control is more likely to maintain system stability and robustness under communication topology changes, node failures, or environmental uncertainties. Distributed control methods based on Eulerian-Lagrange modeling have shown good application potential in scenarios such as collaborative robotic arms, UAV swarms, mobile robot clusters, and space robots, providing effective technical support for the efficient collaboration and safe operation of complex electromechanical systems.

[0003] Significant progress has been made in tracking control methods for Euler-Lagrange systems. For example, patent CN113759711A proposes a design method for independent distributed controllers for multiple Euler-Lagrange systems; patent CN115179295A proposes a method for designing robust binary consensus tracking controllers for multiple Euler-Lagrange systems; and patent CN113625781A proposes a tracking control design method for event-triggered uncertain Euler-Lagrange systems. However, these methods do not consider the security control problem of the system under attack. Therefore, how to simultaneously cope with sensor and actuator attacks and ensure the stability of consensus control in complex nonlinear systems remains an important issue that urgently needs to be addressed.

[0004] However, in complex nonlinear systems, simultaneously addressing sensor and actuator attacks while ensuring the stability of consistent control presents numerous challenges. First, malicious attacks can inject false information by hijacking or tampering with signals between sensors and actuators, misleading the control system and affecting the achievement of consistent control. Attackers can also manipulate communication links to distort transmitted control signals, leading to incoordination between agents and ultimately impacting the stability and performance of the entire multi-agent system. Second, due to the unpredictability of attacks, existing controllers typically exhibit strong conservatism to ensure system stability in worst-case scenarios. However, this conservative design significantly reduces system performance and limits the advantages of cooperation between agents, especially during attacks, where agents may fail to adjust in time, resulting in sluggish and inefficient system responses. Finally, the inherently high nonlinearity of Eulerian-Lagrange systems makes incorporating network attack factors into control design even more complex. Traditional linear control methods are ill-suited for such complex nonlinear systems, and existing distributed control methods often face even greater challenges in this environment. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention specifically relates to a design method for an adaptive secure tracking controller for a Lagrange system under deception attacks. The method includes: constructing a network attack model and a nonlinear Euler-Lagrange system model; designing a virtual controller based on error variables from the model and using a backstepping method; designing an adaptive controller and parameter adaptive law based on the virtual controller; and using a distributed adaptive control algorithm based on projection operators to solve the impact of network attacks on signal transmission in the system. This overcomes the difficulties in controller design caused by the time-varying nature of attacks, the uncertainty of nonlinear multi-agent systems, and the coupling between consistent tracking targets. It ensures the stability of multiple uncertain Euler-Lagrange systems under the influence of network attacks and that the consistent tracking error is bounded.

[0006] The network attack model is constructed as follows: We assume that signals from the controller terminal to the actuator transmission channel and from the sensor to the controller receiving channel are both vulnerable to attack. The attacked sensor data is represented as follows: Where λ i (t)=1+w i (t),w i (t) represents the unknown time-varying weights, It is spurious data that occupies the sensor-to-controller communication channel and needs to satisfy w i (t)≠-1, otherwise x i,q =0 is meaningless. The dynamic description of a nonlinear Euler-Lagrange system is as follows:

[0007] The dynamic description of a nonlinear Euler-Lagrange system is as follows:

[0008]

[0009] in and It represents the generalized coordinates, velocity, and acceleration vectors. It is a symmetric and positive definite inertial matrix. It is a Coriolis centrifugal matrix. It is the gravity vector. It is a vector of unknown parameters. It is the control input. For ease of representation, we define... and Therefore, the subsystem can be rewritten as:

[0010]

[0011] in as well as

[0012] Preferably, the process of introducing coordinate transformations based on the model and designing a virtual controller using the backstepping method includes:

[0013] Step 1: For each subsystem (1≤i≤N), introduce the following error variables:

[0014]

[0015] z i,2 =x i,2 -α i

[0016] 1 κ It is a κ-dimensional vector with all elements being 1; then we get We use μ i =1 to represent y r The i-th subsystem was visited. (Undirected graph) This can represent information interaction between multi-agent systems. Here, υ = {1,...N} is a set of nodes, which is not empty; This represents the set of edges between two subsystems. (Connection matrix) Representation diagram The weighted adjacency matrix, where if (j,i)∈ε, then a ij =1; otherwise a ij =0. α i The virtual controller designed for this purpose.

[0017] Step 2, Definition We can get Where L = [l ij ] N×N=DA, D = diag{d1,d2,...,d N} is a diagonal matrix, and its diagonal elements are defined as follows: when i∈υ, therefore And l ij =-a ij (When i ≠ j). Ι κ It is a κ×κ identity matrix. B = diag{μ1,μ2,...,μ N}, δ=[δ1,δ2,...,δ N ] T δ i =x i,1 -λy r l κ .

[0018] Step 3: Set the first Lyapunov candidate function based on coordinate changes. Then, taking the derivative with respect to V1, a virtual controller is designed based on the derived Lyapunov candidate function. in Yes estimate, Yes estimate, Yes The estimation. Y1, Y2, and Y3 are all positive factors that need to be selected; substitute the expression of the virtual controller into

[0019] From the derivative of the Lyapunov candidate function, we obtain

[0020]

[0021] in γ is a constant greater than 1, z 1,1 This represents the first virtual control error, z. 2,1 This represents the second virtual control error;

[0022] Step 4: Select the second Lyapunov function

[0023]

[0024] Where θ = diag{1 / θ n ,1 / θ g ,1 / θ c ,1 / θ d ,1 / θ h}, They are and The estimated value of V2. We take the derivative of V2:

[0025]

[0026] in The preferred method for designing an adaptive controller for an uncertain nonlinear system is characterized by the fact that the signals from the controller terminal to the actuator transmission channel and from the sensor to the controller receiving channel are susceptible to attack, and the adaptive controller and parameter adaptive law settings are as follows:

[0027] Step 1: Define the Lyapunov function and find its derivative;

[0028] Step 2: Obtain the error variable under no-attack conditions based on the derivative of the Lyapunov function;

[0029] Step 3: Design an adaptive controller and parameter adaptive law under no-attack conditions based on the error variable.

[0030] Furthermore, a design method for an adaptive controller for an uncertain nonlinear system under cyberattacks is proposed. This method uses the designed controller and parameter adaptive laws to verify the stability of the uncertain nonlinear system under cyberattacks and to ensure that the stabilization error tends to be arbitrarily small. The method includes:

[0031] Step 1: Obtain the Lyapunov function inequality based on the designed attack-adaptive controller and parameter law;

[0032] Step 2: Obtain the Lyapunov function inequalities under the attack based on the designed control signals;

[0033] Step 3: Based on the Lyapunov function inequality obtained during the network attack, further update the Lyapunov function inequality to verify the stability of the uncertain nonlinear system under network attack and the tracking error tending to be arbitrarily small.

[0034] The beneficial effects of this invention are:

[0035] The distributed adaptive control algorithm based on projection operators solves the problem of network attacks blocking signal transmission in the system. It overcomes the difficulties in controller design caused by the time-varying nature of attacks, the uncertainty of nonlinear multi-agent systems, and the mutual coupling between the consistent tracking target. It ensures the stability of multiple uncertain Euler-Lagrange systems under the influence of network attacks and the bounded consistency tracking error. Detailed Implementation

[0036] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0037] A design method for an adaptive secure tracking controller for a Lagrange system under deception attacks is disclosed. The method includes: constructing a system model and a nonlinear Euler-Lagrange system model under network attacks; introducing coordinate transformation based on the model and designing a virtual controller using the backstepping method; designing an adaptive controller and parameter adaptive law based on the virtual controller; and verifying the stability of the Euler-Lagrange system under network attacks and the stabilization error tending to arbitrarily small by using the designed controller and parameter adaptive law.

[0038] The specific implementation steps of a design method for an adaptive secure tracking controller for a Lagrange system under deception attack are as follows:

[0039] Step 1: Establish a network attack model and a nonlinear Eulerian-Lagrange cyber-physical system model.

[0040] The purpose of a cyberattack is to maliciously compromise all subsystems. When the signals from the controller terminal to the actuator transmission channel and from the sensor to the controller receiving channel are attacked, the sensors and actuators can only receive false data that occupies the sensor-to-controller communication channel.

[0041] The network attack model is constructed as follows: We assume that signals from the controller terminal to the actuator transmission channel and from the sensor to the controller receiving channel are both vulnerable to attack. The attacked sensor data is represented as follows: Where λ i (t)=1+w i (t),w i (t) represents the unknown time-varying weights, It is spurious data that occupies the sensor-to-controller communication channel and needs to satisfy w i (t)≠-1, otherwise x i,q =0 is meaningless. The dynamic description of a nonlinear Euler-Lagrange system is as follows:

[0042]

[0043] in and It represents the generalized coordinates, velocity, and acceleration vectors. It is a symmetric and positive definite inertial matrix. It is a Coriolis centrifugal matrix. It is the gravity vector. It is a vector of unknown parameters. It is the control input. For ease of representation, we define... and Therefore, the subsystem can be rewritten as:

[0044]

[0045] in as well as

[0046] Now, we assume that the signals from the controller terminal to the actuator transmission channel and from the sensor to the controller receiving channel are both vulnerable to attack. The attacked data from the sensor is represented as follows:

[0047]

[0048] Where λ i (t)=1+w i (t), w i (t) represents the unknown time-varying weights. It is necessary to satisfy w i (t)≠-1, otherwise x i,q =0 is meaningless. This is false data occupying the sensor-to-controller communication channel, therefore the sensor terminal's output signal... In terms of actual position and speed Inconsistent. The data transmitted to the executor after the attack is represented as follows:

[0049]

[0050] in It is a time-varying, uncertain attack weight. It is a non-linear function. i It is the actual controller that needs to be designed.

[0051] Step 2: Introduce error variables based on the model and design a virtual controller using the backstepping method. Specifically, to counteract the disturbances caused by the attack by designing the controller using the backstepping method, it is first necessary to solve the problem of the virtual controller becoming non-differentiable due to the attack hindering the transmission of information from the sensor to the controller channel. Obtain the continuous state information of the uncertain nonlinear cyber-physical system model and use the continuous state information to replace the information destroyed by the network attack model. First, for each subsystem (1≤i≤N), introduce the following error variation:

[0052]

[0053] z i,2 =xi,2 -α i

[0054] Where z i,1 Let x represent the i-th virtual control error. i,1 x i,2 α represents the system state. i This represents the i-th virtual controller that needs to be designed; 1 κ It is a κ-dimensional vector with all elements equal to 1; then we get We use μ i =1 to represent y r The i-th subsystem was visited. (Undirected graph) This can represent information interaction between multi-agent systems. Here, υ = {1,...N} is a set of nodes, which is not empty; This represents the set of edges between two subsystems. (Connection matrix) Representation diagram The weighted adjacency matrix, where if (j,i)∈ε, then a ij =1; otherwise a ij =0. α i For the design of the virtual controller. Definition available Where L = [l ij ] N×N =DA, D = diag{d1,d2,...,d N} is a diagonal matrix, and its diagonal elements are defined as follows: when i∈υ, therefore And l ij =-a ij (When i ≠ j). Ι κ It is a κ×κ identity matrix. B = diag{μ1, μ2, ..., μ...} N}, δ=[δ1,δ2,...,δ N ] T δ i =x i,1 -λy r l κ .

[0055] Differentiate the Lyapunov function with respect to the error variable and the set parameters, and design a virtual controller. Specifically, this includes the following steps: Differentiate the variable δ... i Differentiate:

[0056]

[0057] The virtual controller α in this step i It can be designed as:

[0058]

[0059] in Yes The estimate, Yes The estimate, Yes The estimate is then defined as μ = diag{μ1,...,μ}. N},

[0060] Preferably, the first Lyapunov candidate function is set according to the coordinate changes. Then, differentiate with respect to V1 and substitute the expression for the virtual controller into...

[0061] From the derivative of the Lyapunov candidate function, we obtain

[0062]

[0063] in Using Young's inequality, further calculations yield the following:

[0064]

[0065] After updating the algorithm, we can obtain:

[0066]

[0067] Step 3: Design an adaptive controller and parameter adaptation law based on the virtual controller. This includes the process of setting up the adaptive controller and parameter adaptation law, where signals from the controller terminal to the actuator transmission channel and from the sensor to the controller receiving channel are vulnerable to attack.

[0068] Step 4, we provide and Design an update algorithm, as follows:

[0069]

[0070] in

[0071]

[0072] and exist and Under the following circumstances, we can obtain:

[0073]

[0074] in

[0075] and These are all constants that need to be designed. O1 = O p+1 ∈R p+1 , O2 = O (p+1)×(v+1) ∈R (p+1)×(v+1) And O3 = O v+1 ∈R v+1 Proj[·] is the parametric projection operator.

[0076] Step 5, based on the error change:

[0077]

[0078] in

[0079]

[0080] Choose the second Lyapunov function:

[0081]

[0082] Where θ = diag{1 / θ n ,1 / θ g ,1 / θ c ,1 / θ d ,1 / θ h}, exist In the case of, and define They are and The estimated value. From the previous formula, we can obtain:

[0083]

[0084] in From F i From the definition of the previous formula, we can obtain:

[0085]

[0086] Let's simplify using the most complex term as an example:

[0087]

[0088]

[0089] Similarly, we can also handle and therefore, It can be rewritten as:

[0090]

[0091] in

[0092]

[0093] Therefore, controller u i The design is as follows:

[0094] u i =N i (χ i )ω i

[0095]

[0096] In this embodiment, the design process of the adaptive controller and parameter adaptive law in the presence of network attacks includes:

[0097] Step 1: Define the Lyapunov function and find its derivative;

[0098] In the final step of the backstepping method, the appropriate Lyapunov function is chosen as follows:

[0099]

[0100] Where θ = diag{1 / θ n ,1 / θ g ,1 / θ c ,1 / θ d ,1 / θ h}, They are and The estimated value of V2. We take the derivative of V2:

[0101]

[0102] in

[0103] Step 2: Reassign control signals according to the attack model. During this time interval, obtain the error variable during the attack based on the differentiated Lyapunov function. According to the previous expression, we can obtain:

[0104]

[0105] in

[0106]

[0107] Step 3: Based on the error variables under the attack, design an adaptive controller and parameter adaptation law for a denial-of-service attack. First, we will... and Design an update algorithm, as follows:

[0108]

[0109] in

[0110]

[0111] and exist and Under the following circumstances, we can obtain:

[0112]

[0113] in

[0114] and These are all constants that need to be designed. O1 = O p+1 ∈R p+1 , O2 = O (p+1)×(v+1) ∈R (p+1)×(v+1) And O3 = O v+1 ∈R v+1 Proj[·] is the parametric projection operator.

[0115] Step 4: Verify the stability of the nonlinear Euler-Lagrange system under denial-of-service attacks and the stabilization error tending to arbitrarily small by using the designed controller and parameter adaptive law.

[0116] Step 5: Obtain the Lyapunov function inequality based on the designed attack-free controller and parameter adaptive law. Then, call the attack-free controller and parameter adaptive law to... From this, we can obtain:

[0117]

[0118] exist In this case, there will be 1 p =1 N(p+1) 1 υ =1 N(p+1)(υ+1) 1 h =1 N(h+1) , and in Y1, Y2, and Y3 are all positive factors that need to be selected. Taking the derivative with respect to V1, we can obtain...

[0119]

[0120] in

[0121]

[0122] The above-described embodiments further illustrate the purpose, technical solution, and advantages of the present invention. It should be understood that the above-described embodiments are merely preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made to the present invention within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A design method for an adaptive secure tracking controller for a Lagrange system under deception attacks, comprising: Construct system models under network attacks and nonlinear Eulerian-Lagrange system models; Based on the model, a consistent error variable is introduced, and a virtual controller is designed using the backstepping method. Based on the virtual controller, a distributed adaptive controller and parameter adaptive law are designed. The designed controller and parameter adaptive law are used to verify the stability of the Euler-Lagrange system under deception attacks and the tracking error tending to arbitrarily small.

2. The design method of an adaptive secure tracking controller for a Lagrange system under deception attack as described in claim 1, characterized in that, The process of constructing a nonlinear Euler-Lagrange system model and a deception attack model includes: The dynamic system of nonlinear Euler-Lagrange is described as follows: in and It represents the generalized coordinates, velocity, and acceleration vectors. It is a symmetric and positive definite inertial matrix. It is a Coriolis centrifugal matrix. It is the gravity vector. It is a vector of unknown parameters. It is the control input. For ease of representation, we define... and Therefore, the subsystem can be rewritten as: in as well as Next, we construct a spoofing attack model. We assume that the signals from the controller terminal to the actuator transmission channel and from the sensor to the controller receiving channel are both susceptible to attack. The data received by the sensor after the attack is represented as follows: Where λ i (t)=1+w i (t),w i (t) represents the unknown time-varying weights. It is false data that intrudes into the sensor-to-controller communication channel, and it needs to meet w i (t)≠-1, otherwise x i,q =0 is meaningless.

3. The design method for an adaptive secure tracking controller of a Lagrange system under deception attack as described in claim 1, the process of introducing a consistent error variable based on the model and designing a virtual controller using the backstepping method includes: Step 1: For each subsystem (1≤i≤N), introduce the following error variables: 1 κ It is a κ-dimensional vector with all elements being 1; then we get We use μ i =1 to represent y r The i-th subsystem was visited. (Undirected graph) This can represent information interaction between multi-agent systems. Here, υ = {1,...N} is a set of nodes, which is not empty; This represents the set of edges between two subsystems. (Connection matrix) Representation diagram The weighted adjacency matrix, where if (j,i)∈ε, then a ij =1; otherwise a ij =0. α i For the design of the virtual controller. Definition We can get Where L = [l ij ] N×N =DA, D = diag{d1,d2,...,d N } is a diagonal matrix, and its diagonal elements are defined as follows: when i∈υ, therefore l ij =-a ij Ι κ It is a κ×κ identity matrix, B=diag{μ1,μ2,...,μ N }, δ=[δ1,δ2,...,δ N ] T δ i =x i,1 -λy r l κ . Step 2: Set the Lyapunov function and design the virtual controller based on the error variable and the Lyapunov function. Define μ = diag{μ1,...,μ N }, The first Lyapunov candidate function is set based on the error variable. Then, taking the derivative with respect to V1, a virtual controller is designed based on the derived Lyapunov candidate function. in Yes estimate, Yes estimate, Yes The estimates are as follows. Y1, Y2, and Y3 are all positive factors that need to be selected. Substituting the expression for the virtual controller into the derivative of the Lyapunov candidate function, we obtain V1. in 4. The design method of an adaptive secure tracking controller for a Lagrange system under deception attack according to claim 1, characterized in that, The process of designing a distributed adaptive controller and parameter adaptive law based on a virtual controller includes: Step 1: Define the Lyapunov function and find its derivative; Step 2: Obtain the error variable based on the derivative of the Lyapunov function; Step 3: Design a distributed adaptive controller and parameter adaptive law based on the error variables.

5. The design method of an adaptive secure tracking controller for a Lagrange system under deception attack as described in claim 1, characterized in that, The stability of the Euler-Lagrange system under deception attacks and the tracking error tending to arbitrarily small are verified using the designed controller and parameter adaptive law, including: Step 1: Obtain the Lyapunov function inequality based on the designed attack-adaptive controller and parameter law; Step 2: Obtain the Lyapunov function inequalities under the attack based on the designed control signals; Step 3: Based on the obtained Lyapunov function inequality, verify the stability of the Euler-Lagrange system under deception attack and the fact that the tracking error tends to be arbitrarily small.

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