# An Overview on Denial-of-Service Attacks in Control Systems: Attack Models and Security Analyses
## 5 An Overview of Attack-Resilient Control and Communication Techniques
### Resilient Control Approaches
#### 5.1.2 Control Frameworks with Predictors and Buffers  
To mitigate the effects of DoS attacks in an output-feedback networked control problem, [PERSON] and [PERSON] [51] introduced a controller with a predictor and an impulsive observer. In that work, the output \(y(t)\) of the system in Equations (4) and (5) is measured periodically and transmitted over a network that faces DoS attacks. Moreover, the control input packets are assumed to be transmitted over a secure network.  
In [51], the system dynamics involve disturbance, and the transmitted output measurements can be noisy. The observer and the predictor at the controller side are designed in a way to ensure  
Figure 8: Illustration of the event-triggering approach in [25] for the cases of successful and failed transmissions at time \(t_{i}\). A transmission is triggered at time \(t_{i+1}\) when the state is predicted to make the move indicated with red dotted lines. In the case of successful transmissions (**Left**), the state is guaranteed to stay inside the level set \(\{x\in\mathbb{R}^{n}\colon V(x)=\beta V(x(t_{i}))\}\) in between two event-triggering instants.  
that accurate state estimates are obtained at the controller side after a certain number of successful transmissions. In particular, in the case without disturbance and noise, the state estimate at the controller matches the actual state if \(\mu\) number of consecutive output measurements can be received at the controller side, where \(\mu\in\{1,\ldots,n\}\) is the observability index of the pair \((e^{\Lambda\Delta},C)\) with \(\Delta\) denoting the output measurement/transmission period. When \(\mu\) consecutive measurements are not available, the knowledge of the system dynamics is utilized in the predictor to predict future state values.  
For this framework, it is shown in [51] that the closed-loop system stability can be preserved under any attack strategy satisfying the constraints in Equations (15) and (16), if the scalars \(\rho_{\mathrm{D}},\rho_{\mathrm{F}}\) in those constraints also satisfy \(\rho_{\mathrm{D}}+\Delta\rho_{\mathrm{F}}<1-(\mu-1)\Delta\rho_{\mathrm{F}}\). It is important to note that this condition is independent of the choice of control gain. Note also that if instead of the output measurements, the state measurements are sent to the controller (i.e., \(\mu=1\)), then the condition takes the form  
\[\rho_{\mathrm{D}}+\Delta\rho_{\mathrm{F}}<1. \tag{30}\]  
This inequality shows that the predictor based approach guarantees stability regardless of the dynamics of the open-loop and the closed-loop systems.  
When the control input packets are transmitted over channels that also face DoS, the right-hand side of the condition in Equation (30) is replaced by a term that depends on system dynamics. The work in [50] considers such scenarios, where DoS attacks affect the delivery of both control and measurement packets. There, a buffer is utilized at the plant side. The controller at each time sends the current control input together with future control inputs. When there is no DoS attack, the transmitted control input packets are placed in the buffer so that they can be utilized later when the attacker becomes active and blocks transmissions. It is shown in [50] that for sufficiently large buffer sizes, the closed-loop stability can be guaranteed for attacks that satisfy the condition in Equation (30).