Communication security of autonomous ground vehicles based on networked control systems: The optimized LMI approach

The paper presents a study of networked control systems (NCSs) that are subjected to periodic denial-of-service (DoS) attacks of varying intensity. The use of appropriate Lyapunov–Krasovskii functionals (LKFs) help to reduce the constraints of the basic conditions and lower the conservatism of the criteria. An optimization problem with constraints is formulated to select the trigger threshold, which is solved using the gradient descent algo-rithm (GDA) to improve resource utilization. An intelligent secure event-triggered controller (ISETC) is designed to ensure the safe operation of the system under DoS attacks. The approach is validated through experiments with an autonomous ground vehicle (AGV) sys-tem based on the Simulink platform. The proposed method oﬀers the potential for developing eﬀective defense mechanisms against DoS attacks in NCSs


Introduction
The 21st century has seen rapid development in network communication technology, which has revolutionized numerous fields, including industrial control. The integration of control theory, control technology, computer technology, and network communication technology has facilitated the growth of networked control systems (NCSs) [1,2]. NCSs have been extensively utilized in diverse applications, as illustrated in Figure 1, and have emerged as the preferred technology due to the incorporation of communication and computer technology into the Internet-based TCP/IP protocol [3,4].
The proposal of NCSs has allowed for the organic combination of regional control nodes and devices, breaking the information island phenomenon of traditional control systems. This approach expands the way information is transmitted and enables the diversification of management, monitoring, and control research [18,19]. AGVs consist of multiple systems and technologies, including expert system planning functions, computer vision, autonomous navigation, and advanced parallel processing. AGVs can make independent judgments and plans, accept tasks in natural language, devise task execution methods, and continuously revise their plans. This design concept enables AGVs to complete tasks autonomously, even in complex terrain [20]. AGV control systems, as a new interdisciplinary field, can benefit from the use of NCSs, a novel type of control technology that relies on the Internet after the industrialized control system [21]. Therefore, combining NCSs with AGV control systems is an area of significant importance for research.
Based on the previous discussion, this paper focuses on the basic theory of NCSs and AGVs and conducts research on information security and intelligent secure event-triggered controller (ISETC) design issues for AGVs. The main contributions to this paper are summarized below: (1) The paper proposes a model for NCSs under periodic DoS attacks with varying attack intensity.
Suitable LKFs are constructed, and an optimized Linear Matrix Inequality (LMI) is used to analyze the stability of NCSs. (2) The paper transforms the selection of the trigger threshold into an optimization problem with constraints and employs gradient descent algorithm (GDA) to optimize the threshold and ensure maximum utilization of sampling resources. (3) An ISETC is designed for AGV's network communication. The ISETC is used to analyze the security and stability of the system and ensure that data transmission is not affected by malicious attacks.
Notation: Sym{Q} denotes Q + Q T . R m×n denotes the set of m × n real matrices. I n is the n × n identity matrix. M > 0(≥ 0) indicates M is a positive definite matrix. diag{A 1 , A 2 , . . . , A n } indicates a diagonal matrix and the diagonal elements are A i , i = 1, 2, . . . , n. P −1 indicates the inverse P . P T is the transpose of matrix P . R n is the n-dimensional Euclidean space.

Preliminaries
A. Event-trigger control and design of DoS attacks In this paper, we focus on the study of NCSs that are subject to external disturbances as follows: where x(t) ∈ R n means the current state vector; u(t) ∈ R m is the signal to control the input; the external disturbance is ω(t) ∈ L 2 [0, ∞); A , B, C are constant matrices. In addition to external disturbances, this paper also examines the security of NCSs during network communication transmission. Specifically, we focus on the design of an ISETC to address DoS attacks that occur periodically and with varying levels of intensity. To model these attacks, we assume that the system is targeted by hackers at regular intervals, with t k h representing the instantaneous sampling time point. The DoS attack design is based on prior research [22]: where G is attack intensity; δ(t − t kh ) means Dirac function.
. This paper assumes that x(t) is right continuous, then we get x(t k h) = x(t + k ) and has a left limit and the DoS attack interval is shown in Definition 1.
The ZOH function generates a sequence of control signals where the sampling instant t k h satisfies 0 = t 0 < t 1 h < t 2 h < · · · < t k h < · · · , t k+1 h (k ∈ [0, ∞)). Assuming that the sampling period satisfies 0 ≤ h m < t k+1 h − t k h h k ≤ h M , and ∀k ≥ 0. Then, we assume that x(t k h) is the value of the current state of the system thread; x(t * k h) is the system thread state of the last successful transmission of the system. We have where e(t k h) indicates the error between the current thread state of the system and the system thread state of the system's last successful transmission.

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Security and Safety, Vol. 2, 2023016 Attacks launched by hackers may cause errors in the trigger control of the system, as shown in Figure 2. To capture the impact of such attacks, we assume that x(t + k h + σh) represents the system thread state at the last successful transmission following a DoS attack. The error is defined as follows: where σ ∈ N, e(t k h) represents the error between the current state of the system and the last successful transmission state when the hacker attacks. Based on the above analysis, a new ETC is designed as follows [14]: where and Φ > 0 is a weighting matrix; ρ indicates a threshold parameter; G means attack strength.
Defined the delay at every two successful sampling moments τ t − t k h. Then, the control signal is designed as follows: Based on the analysis of (1)-(5), we can get the following NSCs: where K is a controller gain matrix.

Remark 1.
We considered the vulnerabilities of the ISETC in the presence of external attacks and proposed a novel approach to mitigate the effects of a periodic DoS attack φ t kh with varying strengths G . Unlike the existing methods proposed in [15,23], our approach takes into account the attack's periodicity and strength, which has important implications for developing effective defense mechanisms. By studying the behavior of the system under such attacks, we were able to design a robust and secure ISETC that provides reliable communication in the presence of adversarial interference.

B. Parameter optimization based on gradient descent algorithm
Selecting an appropriate threshold parameter ρ is a crucial aspect of trigger threshold design. The optimization of trigger threshold selection is a complex problem that can be formulated as an optimization problem with constraints. Based on optimization methods in several studies [24,25], we also propose an optimal scheme for designing and optimizing trigger threshold selection. The main objective of the scheme is to maximize the utilization of the available sampling resources, subject to the satisfaction of system performance and stability constraints. The following constraint problem is posed: where ρ is the threshold parameter that needs to be determined. F (ρ) : R n −→ R is the objective function. g(ρ) : R n −→ R m denotes a vector function for solving inequality constraint problems at ρ. ρ l and ρ u represent the upper and lower bounds of ρ, respectively. Then, the gradient descent method is used to optimize the target problem by updating the threshold parameter iteratively. At each iteration, the step length is set as ρ k+1 = ρ k + m k , where m k is the step size and l k ρ k is the descent direction. The optimal threshold parameter is obtained when the objective function reaches its minimum value.
The parameter ρ k is necessary for Pareto optimization, as there is no first-order descending direction for all individual goals. For all individual goals, there is no first-order descending direction as follows: where R n + is said to the pyramid, T H (ρ k ) is H in ρ k of the jacobian matrix. When n = 1, l k = −∇h 1 (ρ k ) for the fastest decline in the direction, which is equivalent to minimizing threshold It is proved that the dual of (10) is a sub-problem where ∆ n = {λ : , · · · , m}} is a simplex set. According to the theory in [25], we get the following Remark 2. In accordance with the approach described in references [24,25], selecting an appropriate threshold parameter ρ is crucial for Pareto maximization. To address this problem, we transform the process into an optimization problem, which enables us to iteratively determine the optimal threshold parameter that satisfies the system requirements. By employing the gradient descent algorithm, we accelerate the search for the threshold parameter, resulting in optimized parameters that reduce the trigger rate and save sampling resources. This method has been proven effective in expediting the search process and enhancing the system's performance.
The Pareto first-order stationary point, denoted as ρ k ∈ P, is obtained by solving the optimization problem in equation (8) using the proximal gradient algorithm. This iterative algorithm updates the estimate of the Pareto front using the gradient of the objective function and the proximal operator of the regularization term. The proximal operator enforces the constraint that the estimate of the Pareto front belongs to the feasible set P. The algorithm continues to update the estimate of the Pareto front until convergence is achieved, which is determined by a stopping criterion based on the norm of the difference between successive estimates of the Pareto front. The algorithm also includes a step size parameter m k , which controls the step size of the gradient descent update. This parameter is chosen using a backtracking line search that ensures the update decreases the objective function. The specific steps of the algorithm are as follows: Security and Safety, Vol. 2,2023016 Algorithm 1: Select the trigger threshold ρ based on the GDA Input: ρ k ∈ [ρ l , ρ u ] ⊆ S and a step size sequence m k Output: ρ k+1 1 begin Solve the objective function Iterative the next updates ρ k+1 The average DoS attacks interval of the attack time sequence ϑ = {t 1 , · · · , t k , · · · } is equal to T a if there exist S 0 ≥ 0 and T a , we can get the DoS attacks interval as follow: where ∀T ≥ t ≥ 0 and N ϑ (T, t) is the total number of times the attack sequence ϑ has been hacked over the time period (t, T ).
And there are the arbitrary matrices N 1 , N 2 and N 3 and the matrices M > 0. We can get the following inequality holds:

Main results
In this section, we consider the scenario where the control gain matrix K is known and establish the asymptotic stability condition of the system under the designed safe trigger mechanism, which is presented in Theorem 1. We then proceed to design and solve the controller gain matrix in Theorem 2. To simplify the notation, we define the following symbols:  (7) are asymptotically stable if there exist symmetric matrices P, H , any matrix of suitable dimension M , Q, and Y n (n = 1, 2, 3) that satisfy the following LMIs: where Proof. Given the LKFs candidate as where We take the derivative of V i (t), and we geṫ Using the integral inequality in Lemma 1, the integral term in (17) can be scaled as follows: Based on the above results,V 3 (t) can be rewritten as follows: The constraints of the unsafe ISETC (5) are considered, and the following inequality is obtained: Based on the system (7), the following equation is got According to (14)-(21), the following equation is had as folloẇ Based on the linear convex combinations method [28], for all ξ T (t)Ξξ(t) < 0 are established. We can get Finally, we can conclude that the NCSs (7) are asymptotically stable if the conditions (11) of Theorem 1 are satisfied and if the inequalityV i (t) ≤ ξ T (t)Ξξ(t) ≤ 0 holds. This inequality ensures that the LKF V (t) is decreasing along the system trajectory, and therefore, the system state will converge to the equilibrium point. Thus, the designed safe trigger mechanism ensures the asymptotic stability of the NCSs in the presence of DoS attacks.
Remark 3. Unlike the method in reference [29], the sampling time information is fully considered in the looped function constructed by V 2 (t). It contains both the date information on x(t k h) and x(t k+1 h), This method introduces more sampling time information based on reducing the initial constraints. Furthermore, increasing the information storage of LKFs reduces the conservatism of the criteria.

Remark 4.
Analyzing the computational complexity of control algorithms is essential. In this paper, a loop function is constructed to reduce the initial constraints and therefore decrease the computational complexity of the control algorithm. The resulting algorithm achieves effective control of AGVs with relatively low computational complexity, specifically 6n 2 + n. Moreover, we were able to verify the results within an acceptable time using an Intel(R) Core(TM) i7-8565U CPU @ 1.80 GHz 1.99 GHz computer.
The control algorithm considers stability analysis and employs an optimization approach to determine the maximum allowable delay and the controller gain matrix. This guarantees the system's stability under DoS attacks while minimizing their impact on the system's performance. Control Algorithm 2 is based on the presented stability analysis, and it aims to calculate the maximum allowable delay τ max and the controller gain matrix K to ensure the system's stability under DoS attacks. The algorithm is outlined as follows: Theorem 2. Let ρ, µ 1 , µ 2 , h m , and h M be positive scalars. Consider the NCSs (7) under the designed safe trigger mechanism. The system is asymptotically stable if there exist symmetric matricesP,H , and any matricesM ,Q, and W that satisfy the following LMIs: whereΞ =Ξ a + Sym Γ∆ +Θ Υ t + k+1 h , Proof. The gain matrix K = W X −1 and Φ = X −TΦ X −1 are defined. Pre-multiplying and postmultiplying (13) by Then, the LMIs (23) can be obtained. The detailed proof process is similar to Theorem 1.

Illustrative example
We conducted simulation experiments on the Simulink joint platform to verify the effectiveness of the proposed control algorithm in this paper, using the data provided in reference [19]. The experimental setup is illustrated in Figure 3, and some data related to the vehicle are shown in Table 1: The dynamic physics equations for AGV (see Figure 3) can be written as follows: = a 11 + a 22 r + b 1 σ n + s 2 , r = a 21 + a 22 r + b 2 σ n + s 3 .
Set the state vector is x(t) = [e, φ, , r] T , the control input signal is u(t) = σ f and the external disturbance ω(t) = [s 1 , −ρ(δ c )v x , s 2 , s 3 ] T . Finally, the physical state space model of AGV is expressed as follows:ẋ Security and Safety, Vol. 2, 2023016 The experimental setup was conducted on the Simulink joint platform to verify the effectiveness of the proposed control algorithm in this paper using the data provided in reference [19]. The physical meanings of the parameters were defined in [19]. Specifically, m denotes the weight of the vehicle, I s is the yaw inertia of the vehicle, l n represents the distance from the rear wheel to the center of gravity, l m indicates the distance from the front wheel to the center of gravity, and C n and C m denote the cornering stiffness of the front and rear tires, respectively. We set the intensity to G = 10, with an attack period of 0.1, and assume that h m = 0 and ρ = 0.5. To evaluate the impact of varying h M on the system, we used the Yalmip toolbox to solve for the maximum acceptable time delay τ max .
As shown in Table 2, the proposed control algorithm in this paper has a maximum acceptable delay limit of 1.3821 when h M = 0.4. In contrast, reference [19] limits the maximum acceptable latency to τ max = 0.04. This comparison clearly demonstrates the superior performance of the proposed algorithm in dealing with system delays and DoS attacks. The impact of DoS attacks on system performance is further studied, and we conduct simulations with different attack strengths and maximum delay constraints. Specifically, we set h M = 0.2 and examined the maximum acceptable delay of the DoS attack system under different attack strengths. The results are presented in Table 3, where we observe that the maximum acceptable time delay of the system changes with varying attack strengths. Notably, when the attack strength is set to G = 10, the maximum transmission time delay of the system is τ max = 0.5963. These results indicate the importance of implementing robust control strategies in NCSs that can handle and mitigate the effects of attacks, especially high-intensity DoS attacks. The proposed control algorithm in this paper has a computational complexity of 6n 2 +n, which means that the system's asymptotic stability can be ensured even with a low number of decision variables. Moreover, the low computational complexity of the control algorithm reduces processing time and energy consumption, making it more feasible for realtime control applications. In summary, the proposed control algorithm not only guarantees the system's stability and security but also provides practical benefits by minimizing the computational burden and optimizing resource allocation.
Then, the control gain matrix K = 10 4 [2.0171 −1.9720 −0.9823 −0.6937] was obtained using the method in Theorem 2 when the parameters µ 1 and µ 2 were set to 1. This control gain matrix was then used in a Simulink joint platform simulation experiment to verify the feasibility of the proposed control design method. The results presented in Figures 4 and 5 demonstrate that the proposed control design method is effective in mitigating the impact of DoS attacks on the system, as the system can still converge smoothly under the designed controller and control algorithm, even when subjected to DoS attacks with high intensity and a short attack period. Furthermore, the study found that the proposed control design method is more effective than the method presented in [19], as it enables the system to tolerate a higher maximum delay limit under DoS attacks, as shown in Table 2. These results provide valuable insights into  the development of robust control algorithms for NCSs that are vulnerable to DoS attacks, highlighting the importance of implementing such algorithms to ensure system stability and security. Additionally, the proposed control algorithm has relatively low computational complexity, making it a practical solution for real-time control applications.
Furthermore, selecting appropriate trigger thresholds is crucial for mitigating DoS attacks in practice. In this paper, we propose a novel approach based on GDA for optimizing trigger thresholds. By formulating the threshold selection as a constrained optimization problem, we can find optimal thresholds that minimize the trigger rate of legitimate traffic while maintaining high mitigation of DoS attacks. First, we iterate through the ρ k values using the Python toolbox and then bring the results into the Yalmip toolbox for solving. This learning algorithm significantly improves resource efficiency by iteratively searching for a suitable value of ρ k . The intelligent trigger threshold search mechanism employs machine learning to find the optimal threshold, denoted by ρ, by iteratively traversing the range [0, 1] as shown in the sequence ρ 1 → · · · → ρ 2 → · · · → ρ k−1 → · · · → ρ k → · · · . In this way, the algorithm iteratively learns and searches for the ρ k with the lowest trigger rate. Additionally, we present the number of system triggers under GDA and traditional algorithms are in Figures 6 and 7, respectively. Our results show that GDA-optimized thresholds can significantly reduce the number of false triggers compared to the conventional method, resulting in a lower trigger rate of 86.62% for GDA versus 88.5% for the traditional algorithm. These findings demonstrate the effectiveness of our proposed approach in reducing the impact of DoS attacks on network performance.
Finally, the optimized trigger thresholds can also provide additional benefits in terms of resource allocation and system resilience. By reducing the number of false triggers, our approach can free up more resources for other tasks or mitigate the impact of DoS attacks on system performance. In a word,

Conclusion
This paper addressed the issue of NCSs under DoS attacks with periodicity and attack intensity. The research on the power of the DoS attacks was significant for establishing suitable defense mechanisms. The paper presented a method to construct appropriate LKFs, reducing the constraints of basic conditions and mitigating criterion conservatism. Additionally, the paper transformed the selection problem of the trigger threshold into an optimization problem with constraints and used the GDA to optimize the threshold, saving sampling resources. An ISETC was designed to ensure the normal operation of AGVs under DoS attacks. Finally, the proposed method's effectiveness was verified by simulating the AGVs system based on the Simulink platform. In the future, further research could focus on developing more sophisticated defense mechanisms to protect NCSs from different types of cyber-attacks and enhancing the performance and robustness of AGVs systems under various adverse conditions.

Conflict of Interest
No conflict of interest exists in the submission of this manuscript, and the manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously and is not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.