Robust Stabilization of Linear Time-Delay Systems under Denial-of-Service Attacks

This research examines new methods for stabilizing linear time-delay systems that are subject to denial-of-service (DoS) attacks. The study takes into account the different effects that a DoS attack can have on the system, specifically delay-independent and -dependent behaviour. The traditional proportional-integral-derivative (PID) acts on the error signal, which is the difference between the reference input and the measured output. The approach in this paper uses what we call the PID state feedback strategy, where the controller acts on the state signal. Our proposed strategy uses the Lyapunov–Krasovskii functional (LKF) to develop new linear matrix inequalities (LMIs). The study considers two scenarios where the time delay is either a continuous bounded function or a differentiable and time-varying function that falls within certain bounds. In both cases, new LMIs are derived to find the PID-like state feedback gains that will ensure robust stabilization. The findings are illustrated with numerical examples.


Introduction
Many researchers have been drawn to studying the stabilization and control of systems that are infinite-dimensional or have a time delay. Time delays are considered among the main causes of instability and poor performance in dynamic systems. A study of robust stability analysis and robust control design for time-delay systems (TDS) has been reported in [1].
The choice of an appropriate Lyapunov-Krasovskii functional (LKF) with additional parameters is important in developing sufficient stability conditions in the form of linear matrix inequalities (LMIs). The choice of LKF for linear state-delay systems is based on the type of the time-delay independent, applicable to delays of arbitrary size, or dependent, where the size of the delay is included.
State-derivative feedback methods have been used to design controllers for physical systems, such as mechanical systems [2], bridge systems with cable vibration [3], and car suspension systems [4][5][6][7], but the effect of the time delay was not taken into consideration in these applications. The time delay appears in many other physical systems such as water quality in streams [8], power systems [9], combustion in motor chambers [10], industrialscale polymerization [11], and many others.
A denial-of-service (DoS) attack is a type of cyber-attack that aims to make a network resource unavailable to its intended users by overwhelming it with a flood of traffic or requests. This can cause delays in the network control system (NCS), as the system is unable to process legitimate requests due to the overwhelming amount of traffic. The delays can lead to slow or unresponsive network performance, which can impact the ability of the system to access the network and the services it provides. Additionally, the NCS may disturbances. The study designed the hybrid-triggered controller by merging time-and event-triggered controllers, aided by a Bernoulli random variable. A non-linear function was used to represent the deception attack signal, with the probability of attack occurrence determined by the Bernoulli random variable. To mitigate the effects of disturbances, an H ∞ performance was employed. The study applied a Lyapunov-Krasovskii functional (LKF) to analyse the stabilization of the selected PDE under the proposed controller, with the stabilization conditions derived in terms of linear matrix inequalities (LMIs). This paper will examine the advancement of robust stabilization techniques for systems that are vulnerable to DoS attacks. Specifically, it will address the issue of robust stabilization through the use of a PID-like state feedback controller for DoS attacks that result in both delay-independent and -dependent effects. A new approach for obtaining PID-like parameters will be presented, involving formulating and solving LMIs based on the appropriate choice of the Lyapunov-Krasovskii functional (LKF). The time delay caused by DoS attacks is assumed to be continuous and bounded for delay-independent cases, and differentiable and time-varying with upper-bound relations for delay-dependent cases. A numerical example will be provided to demonstrate the theoretical developments proposed.
The paper is organized as follows. Section 2 presents some preliminary results with the formulation of the PID-like state feedback controller. The closed-loop system is formulated. The main results of this work are stated in Section 3 and its subsections. Section 4 presents the simulation results, while the conclusion and future work are given in Section 5.

Notations:
W t , W −1 and W denote the transpose, inverse, and induced norm of any square matrix W, respectively. W > 0 (W < 0) stands for an asymmetrical and positive (negative) definite matrix W. The n-dimensional Euclid n space is denoted by R n and I stands for the identity matrix with appropriate dimension. The symbol * is used in some matrix expressions to induce a symmetrical structure.

Problem Definition
The following class of linear time-delay (LTD) systems subjected to DoS attacks will be considered: (1) where x(t) ∈ n , u(t) ∈ p , z(t) ∈ q and w(t) ∈ q are the state vector, control input, observed output and disturbance input, respectively. It is assumed that the disturbance w(t) belongs to L 2 [0, ∞). τ(t) > 0 is a time delay caused by DoS attacks. w(φ) is the initial condition which is assumed to be differentiable in [−τ, 0]. A 0 ∈ nxn , B 0 ∈ nxp , G o ∈ qxn , G d0 ∈ qxn , A do ∈ nxn and Γ o ∈ nxq , Φ o ∈ qxq are real and known constant matrices. The problem that will be solved is to find the PID-like constant gains K D ∈ pxn , K I ∈ pxn , K P ∈ pxn such that the following conditions hold: (2) The following PID-like state feedback controller is proposed where ρ is the upper bound on the time delay produced by DoS attacks, K p is a proportional gain designed to ensure internal stability, and K D and K I are to meet the control objectives.
Then, from (1) and (2), the closed-loop system is written as is well defined, the closed-loop system (4) has the following form. .
The following changes of variables are introduced to deal with the system given in (4) Then .
Appending (6) to system (4) and define ζ(t) =: x t (t) υ t (t) t , the following augmented system, involving two time-delay variables produced by DoS attacks is obtained where In the following, the controller gains K P , K D and K I will be determined for two DoS attack behaviours: where the bounds ρ and µ are known. From ref. [11], the usual bounding relation µ < 1, but in this work it is expanded to µ < 3. This new upper bound on µ is shown later in the proof of Theorem 2, contributing to others' work.

DoS Attacks Causing Unknown Time-Delay Design
Let the time delay produced by DoS attack τ(t) be continuous and unknown, as described in Case 1. The next theorem identifies a delay-independent LMI-based condition for PID-like state feedback stabilization of system (1) with H ∞ performance bound γ to overcome the effectiveness of the DoS attacks.

Theorem 1.
For the DoS behaviour defined in Case 1, system (1) is delay-independent and asymptotically stable with performance bound γ under a PID-like state feedback controller has a feasible solution, where Moreover, the feedback gains are given by K P = W X −1 x , K D = −Y X −1 x and K I is computed directly from element (1,2) of E 11 .
Proof. First, the asymptotic stability of the closed-loop system in (7) is stabilized when Define the selective LKF are positive definite unknown matrices. Differentiating V 1 (t) along the solutions of (11) and with some algebraic manipulation, we obtain where then making use of the S-procedure [12] with some algebraic manipulations, after expanding and simplifying its elements in (12) Ω in (13) becomes Using Schur's complement on Π 12 in (14), the asymptotic stability of system (4) follows (14) is bi-linear in X x , Y ,Z x and Q x . Next, it is converted into an LMI to be able to obtain the controller gains. To do this, the performance measure is included as follows. Let the performance measure J be defined as Assume w(t) ∈ L 2 (0, ∞) = 0 and the initial condition x(0) = 0, we thus have ds First, from (16), we evaluate where (13) is modified by adding the coefficient of w(s) from (7). In terms of η, it becomes as Applying the congruent transformation diag T 1 I I I on Ω, and making use of the S-procedure [12] with some algebraic manipulations, (19) becomes Now, expanding (18) and combining it with (20) to obtain Using Schur's complement on Π 12 , (10) is obtained with the controller gains. Since Ξ s < 0, it follows that J < 0 and z(t) 2 < γ w(t) 2 , and the proof of the H ∞ performance bound is achieved.

Remarks:
• The solution to inequality (10) will result in a sub-optimal one. The optimal gains of the delay-independent asymptotically stabilized controller can be determined by solving the following convex minimization problem • The conventional state feedback stabilization controller is obtained as stated by the next lemma.

Lemma 1.
For the DoS behaviour defined in Case 1, system (1) with state feedback u(t) = K P x(t) is delay-independent and asymptotically stabilized with H ∞ performance bound γ if there exist matrices X x > 0, Z x > 0, Q x > 0, N 0 >0 and W such that the following LMI has a feasible solution, whereΠ Moreover, the feedback gain is given by K P = W X −1 x .
Proof. The proof of this lemma is obtained by setting

Remark 1.
The following linear-controlled delay-less system is .
is obtained by setting A do = 0, G do = 0, and K I = 0 in Theorem (1). This special result is stated in the following corollary.
, and W such that the following LMI has a feasible solution. Moreover, the feedback gains of the PD controller are given by K D = −Y X −1 x , K P = W X −1 .

DoS Attacks Causing Time-Varying Delay Design
In this section, we will address the DoS behaviour described in Case 2, i.e., τ(t) is continuous and differentiable. The results presented will be in the form of new LMI characterization for delay-dependent stabilization by the PID-like state feedback controller. The following Leibniz-Newton formula will be used Considering the transformed closed-loop systems (7) and (8), the following theorem is established: Furthermore, the controller parameters are obtained as K P = W X −1 x , K D = −Y X −1 x and K I is obtained directly from element (1,2) of Ω 11 .
Proof. First, the stability of the considered system is proven. Consider the following LKF: where P, Q and Z are as in (12). Using the Leibniz-Newton formula in (25), setting w(.) = 0, evaluating · V 2 (t) along the solutions to (7) and after some algebraic manipulations, it is easy to show that · V 2 (t) can be written as The relaxation matrices Θ and Ψ are introduced to facilitate the delay-dependent analysis. When Ξ s < 0, then · V 2 (t) < 0 for any X (t, φ) = 0 and all τ ≤ ρ.
Let us write (29) as Using Schur's complement, (30) can be written for all 0 < τ(t) < ρ as Substituting the upper bound of τ(t) into (31), we obtain where Φ = P M −1 P. For · V 2 (t) to be less than zero, Ξ s 2 in (32) should be less than zero. By applying the congruent transformation diag T 1 I I I T 1 , to (32), expanding its elements and simplifying, the following inequality is obtained.

Accordingly, (29) is expanded and modified into
Separating (38) as was performed in (30), then using Schur's complement and applying the congruent transformation diag T 1 I I I I T 1 to Ξ s , with some algebraic manipulations, (38) becomes Incorporating Σ ∞ and Ξ s , The term Ω 11 includesΠ 12 , defined in (33) aŝ Using Schur's complement for this term, we obtain Therefore, (26) is obtained.

Simulation
In this section, we demonstrate the application of the foregoing analytical results on two operating points of a typical system. The results of Theorem 2 are reported here. Implementation of the developed theorems was accomplished using the MATLAB LMIsolver. The LIM-solver was used to find the unknown quantities in LMI (26). Then the PID parameters were calculated as stated in the theorem.

Model 1:
The time-delay pattern caused by the DoS attack is τ(t) = 0.1cos(2π f t), where f = 0.75 Hz. The simulation parameters for operating point 1: ρ = 0.1 s, µ = 0.5, and γ = 0.35. The PID-like controller's gains for operating point 1 are as follows: The simulation parameters for operating point 2: ρ = 0.1 s, µ = 0.5, and γ = 0.3. The PID-like controller's gains for operating point 2 are as follows: From the time-delay pattern caused by the DoS attacks, the maximum bounds, ρ and µ, can easily be verified. For both operating points, random noises with a maximum magnitude of 0.1 are taken as disturbances. The initial values of the states are also taken as random numbers between 0 and 1. Figures 1 and 2 show the state trajectories under the proposed controller.
The obtained results show that the proposed strategy yield less performance bound γ, thus providing improved stabilization for the combustion in the rocket motor chambers model with two operating modes.

Conclusions
This paper presented a new approach for stabilizing a class of linear time-delay systems that are subject to DoS attacks. The method takes into account the different ways that a DoS attack can impact the system, specifically its delay-independent and -dependent behaviour. To overcome these behaviours, the authors employed an LKF and derive new linear matrix inequalities. They demonstrated the effectiveness of their method through numerical examples for both delay-independent and -dependent robust stabilization. In general, the results of this study are expected to be useful for understanding and controlling NCSs in the presence of DoS attacks. An extension of the presented results could be to include state observers to estimate the states when they are not measurable. Furthermore, a mathematical model for the cyber-attack may be obtained and augmented with the time-delay system. Data Availability Statement: The data generated during the current study are available from the corresponding author upon reasonable request.