# An Overview on Denial-of-Service Attacks in Control Systems: Attack Models and Security Analyses
## 3 Deterministic and Game-Theoretical Approaches to Denial-of-Service Attack Modeling
### Deterministic Attack Models with Average Duration and Frequency Constraints
: _There exist scalars \(\kappa_{\mathrm{F}}\geq 0\) and \(\rho_{\mathrm{F}}\in[0,1)\) such that_  
\[\mathcal{I}(0,t)\leq\kappa_{\mathrm{F}}+\rho_{\mathrm{F}}t,\quad t\geq 0, \tag{14}\]  
_where \(\mathcal{I}(\tau,t)\in\mathbb{N}_{0}\) denotes the number of attack intervals in the time frame \([\tau,t]\)._  
The scalar \(\rho_{\mathrm{F}}\) in Equation (14) provides an upper-bound on the frequency of attacks in the long run. To achieve stabilization of networked control systems with periodic transmissions, attack frequency needs to be bounded by a scalar \(\rho_{\mathrm{F}}\) small enough. To see this, first consider periodic transmission attempts at every \(\Delta\) units of time. A strategic attacker who knows the transmission period \(\Delta\) can concentrate his/her attacks to pinpoint the periodic transmission times with attacks that have very short durations (which satisfy Assumption 1). Thus, for such settings, all transmission attempts may fail if \(\rho_{\mathrm{F}}\geq\frac{1}{\Delta}\). In [48], it is mentioned that Assumptions 1 and 2 do not suffice when one considers system dynamics with disturbance. This is because, under Assumptions 1 and 2, the attacker is allowed to attack continuously for arbitrarily long intervals as long as Equations (13) and (14) are satisfied. The attacker can achieve this by initially waiting for a long duration. This scenario is particularly dangerous for systems with disturbance, because even though the control packets may successfully be delivered in the initial attack-free period, the state never reaches zero due to disturbance. When the attack-free period ends, the attacker can attack continuously and cause the state to grow to very large values. To avoid such issues, [PERSON] and [PERSON] [48] proposed further restrictions of the average attack duration and the average attack frequency so that the maximum length of a continuous attack is bounded. These new restrictions are described by the inequalities  
\[|\mathcal{A}(\tau,t)| \leq\kappa_{\mathrm{D}}+\rho_{\mathrm{D}}(t-\tau), \tag{15}\] \[\mathcal{I}(\tau,t) \leq\kappa_{\mathrm{F}}+\rho_{\mathrm{F}}(t-\tau), \tag{16}\]which are required to be satisfied for all \(\tau\geq 0\) and \(t\geq\tau\). Notice that Equations (15) and (16) imply Equations (13) and (14), but not vice versa. In [48; 50; 51], average duration and frequency conditions in Equations (15) and (16) are utilized for modeling attacks in a networked control problem. It is shown in [48] that asymptotic stabilization with an event-triggered controller can be achieved under any attack strategy that satisfies Assumptions 1 and 2 with sufficiently small scalars \(\rho_{\mathrm{D}}\) and \(\rho_{\mathrm{F}}\). In particular, the sufficient condition given in [48] for asymptotic stability is  
\[\Delta^{*}\rho_{\mathrm{F}}+\rho_{\mathrm{D}}<\omega,\]  
where \(\omega>0\) is a scalar that depends on system dynamics, and \(\Delta^{*}>0\) is scalar that upper-bounds the intervals between network transmission instants. It is shown in [48] that this condition guarantees input-to-state stability for systems with disturbance on the condition that \(\rho_{\mathrm{D}}\) and \(\rho_{\mathrm{F}}\) satisfy Equations (15) and (16) (in addition to satisfying Equations (13) and (14)). The authors of [50; 51] considered periodic sampled-data controllers and showed that stability condition in this case can be made less conservative by using predictor and buffer mechanisms (see also Section 5.1.2). As noted in [48; 50; 51], \(\kappa_{\mathrm{D}}\) and \(\kappa_{\mathrm{F}}\) used in the inequalities in Equations (13) and (14) (or Equations (15) and (16)) affect bounds on state trajectories and the performance, but they do not affect the stability properties of linear systems. In the case of nonlinear systems, \(\kappa_{\mathrm{D}}\) and \(\kappa_{\mathrm{F}}\) play a role also in stability properties.