# State-Bust Feedback Control for Fault Recovery of Input/State Asynchronous Sequential Machines
## 2 Preliminaries
In this study, \(s(x,v)=x^{\prime}\) is alternatively described as follows:  
\[(x,v,x^{\prime})\in s\Leftrightarrow s(x,v)=x^{\prime}.\]  
The domain of \(s\) is often extended to \(X\times A_{n}^{+}\) through the following relation:  
\[s(x,v_{1}v_{2}\cdots v_{k}):=s(s(x,v_{1}),v_{2}\cdots v_{k}),\ v_{1}v_{2} \cdots v_{k}\in A_{n}^{+}.\]  
where \(x^{\prime}\) is said to be _stably reachable_ from \(x\) if \(s(x,t)=x^{\prime}\) for some \(t\in A_{n}^{+}\)[1; 9]. With \(|X|=n\), every state of \(\Sigma\) is stably reachable in at most \(n-1\) steps of stable transitions. Thus, the length of \(t\) is bounded by \(1\leqslant|t|\leqslant n-1\). When \(\Sigma\) goes through a stable transition, it generates a _state burst_[31], or a fast state sequence consisting of underlying transient states and next stable state. If each generated state is separately delivered to the controller, the resultant configuration has the state feedback control mechanism. If, on the other hand, the controller has access to the state burst, the closed-loop system is endowed with burst-feedback control. In this study, we utilize the burst feedback control scheme, as it gives more flexibility of controller synthesis, albeit needing an additional resource to record the state burst. The state burst of a valid state-input pair is described as the following mapping:  
\[\alpha:X\times A\to X^{+}.\]  
\(\alpha(x,u)\in X^{+}\) denotes the state burst with respect to a valid pair \((x,u)\in X\times A\), namely, \(\alpha(x,u)\) is generated when \(\Sigma\) takes the stable transition from \((x,u)\). For instance, \((x^{0},a)\) addressed in (1) and (2) leads to the following state burst:  
\[\alpha(x^{0},a)=x^{0}x^{1}\cdots x^{k}.\]  
For a state burst \(z\in X^{+}\), let \(z_{f}\in X\) and \(z_{l}\in X\) be its first and last element, respectively. In terms of the foregoing notations, \(\alpha_{f}(x^{0},a)=x^{0}\) and \(\alpha_{l}(x^{0},a)=x^{k}\). Figure 1 shows the proposed static fault tolerant corrective control system, where \(C\) is the static corrective controller, \(G\) is the diagnoser, \(v\in A_{n}\) is the external input, \(u\in A_{n}\) is the control input provided by \(C\), \(w\in A_{d}\) is the adversarial input, \(z\in X^{+}\) is the state burst, and \(x\in X\) and \(m\in N_{0}\) are respectively the state feedback and the fault indicator signal generated by \(G\). \(\Sigma_{c}\) denotes the closed-loop system consisting of \(\Sigma\), \(C\), and \(G\). When \(w\) enters \(\Sigma_{c}\), it overrides \(u\) and causes a transient fault, forcing \(\Sigma\) to undergo an unauthorized state transition whenever it is valid with respect to the current state. \(w\) represents an external disturbance that infiltrates into \(\Sigma_{c}\) through the control input channel. As addressed before, a typical instance of \(w\) is an SEU fault [14; 15] that upsets the logic value of memory bits expressing the control input. Inherent mechanical or electrical faults to the actuator can be also modeled by \(w\). Figure 1: Static corrective control system with a diagnoser. For \(x\in X\), let  
\[W(x):=T(x)\cap A_{d}\]  
be the set of adversarial inputs that cause unauthorized transitions to \(\Sigma\) when it stays at the stable state \(x\), and let  
\[\Gamma(x):=\{s(x,w)|w\in W(x)\}\]  
be the set of states \(\Sigma\) reaches as a result of a transient fault occurring at \(x\). Note that \(w\) is unobservable from both \(C\) and \(G\), which fits into the characteristics of adversarial entities. \(G\) provides \(C\) with the next stable state \(x\in X\) and the fault indicator signal \(m\in N_{0}\) based on the state burst \(z\) and the control input \(u\). Thus \(G\) has the following mapping:  
\[G:X^{+}\times A_{n}\to X\times N_{0}.\]  
With the state notation \(X=\{x_{1},\ldots,x_{n}\}\), \(m=0\) indicates that \(\Sigma\) has undergone a nominal stable transition. On the other hand, \(m=i\
eq 0\) implies that \(\Sigma\) has undergone an unauthorized transition from \(x_{i}\) caused by an unspecified adversarial input \(w\in W(x_{i})\).