# Observer-Based Active Control Strategy for Networked Switched Systems against Two-Channel Asynchronous DoS Attacks

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## Abstract

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## 1. Introduction

- An observer-based active control strategy is proposed for networked switched systems, which exhibits resilience and robustness against two-channel asynchronous DoS attacks and asynchronous switching behaviors. In addition, the buffer size design approach is proposed.
- The switching signals are designed to ensure the ISS of the networked switched systems under active control strategy against two-channel asynchronous DoS attacks and asynchronous switching behaviors, specifically, the quantitative relationship between the frequency and duration of two-channel DoS attacks and the switching frequency is revealed under ISS conditions. The results can be degraded to the non-switched system case.
- Unlike [26,37,38,43] that only consider the case of full-state measurements, the case of partial-state measurements is studied, and a mode-dependent finite-time observer is designed to rapidly and accurately estimate the system state. In addition, external disturbances are also considered in our work.
- In the existing methods [37,38,43], the effectiveness of active control strategies often relies on assuming that the DoS signals adhere to specific patterns. However, in our work, the effectiveness of the active control strategy is only related to the frequency and duration of DoS attacks, without making any assumptions about specific patterns for DoS attacks, which is more general and challenging.

## 2. Problem Statement and Preliminaries

#### 2.1. Networked Switched Linear System

**Assumption A1**.

**Remark 1**.

#### 2.2. Two-Channel Asynchronous DoS Attacks

**Assumption A2**.

**Remark 2**.

**Assumption A3**.

#### 2.3. Control Objective

**Definition 1**.

## 3. Observer-Based Active Control Strategy

#### 3.1. Mode-Dependent Finite-Time Observer Design

#### 3.2. Active Control Strategy

- Step 1: The observer resets the estimate of the system state $\xi \left({z}_{m}\right)$ when it receives $\mu $ consecutive measurement signals with the same mode i from the sensor, and then sends it to the predictor.
- Step 2: Based on the estimate $\xi \left({z}_{m}\right)$, the system’s mode $\sigma \left({z}_{m}\right)$ and the actual control signals, the predictor predicts the system state $\widehat{x}$ and the controller generate the control sequence $\mathcal{U}\left({z}_{m}\right)$ and transmit it to the buffer at ${s}_{m}$ by using one data packet.
- Step 3: The buffer discards the outdated control signals and sequentially sends the control signals to the actuator, one by one, at each sampling instant. The actuator holds the control signal until the next sampling instant. Return to Step 1.

**Remark 3**.

#### 3.3. Buffer Size Design

**Lemma 1**.

**Proof.**

**Remark 4**.

**Lemma 2**.

**Proof.**

**Lemma 3**.

**Proof.**

**Remark 5**.

## 4. Input-to-State Stability Analysis

#### 4.1. Dynamics under Two-Channel DoS Attacks without Asynchronous Switching

**Lemma 4**.

**Proof.**

#### 4.2. Dynamics under Two-Channel DoS Attacks with Asynchronous Switching

**Lemma 5**.

**Proof.**

**Remark 6**.

**Remark 7**.

#### 4.3. Input-to-State Stability Analysis

**Theorem 1**.

**Proof.**

**Remark 8**.

**Corollary 1**.

**Proof**.

## 5. Numerical Example

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof of Lemma**

**5.**

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**Figure 2.**The schematic diagram of the transmission policy under switching behaviors and two-channel asynchronous DoS attacks. The solid arrows represent the successful transmissions.

**Figure 6.**State response under the active control strategy, set the initial state $x\left({t}_{0}\right)={[1,-1]}^{\mathsf{T}}$ and the disturbance magnitude to 0.01.

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Yin, J.; Lu, A.
Observer-Based Active Control Strategy for Networked Switched Systems against Two-Channel Asynchronous DoS Attacks. *Actuators* **2023**, *12*, 335.
https://doi.org/10.3390/act12080335

**AMA Style**

Yin J, Lu A.
Observer-Based Active Control Strategy for Networked Switched Systems against Two-Channel Asynchronous DoS Attacks. *Actuators*. 2023; 12(8):335.
https://doi.org/10.3390/act12080335

**Chicago/Turabian Style**

Yin, Jiayuan, and Anyang Lu.
2023. "Observer-Based Active Control Strategy for Networked Switched Systems against Two-Channel Asynchronous DoS Attacks" *Actuators* 12, no. 8: 335.
https://doi.org/10.3390/act12080335