# State-Bust Feedback Control for Fault Recovery of Input/State Asynchronous Sequential Machines
## 4 Controller Synthesis
Next, the suffix of \(t\) is substituted by the corresponding part of \(r\), i.e., we induce an alternative input string \(t^{\prime}\) from \(t\) and \(r\) as follows:  
\[\forall x_{j}\in\Gamma(x_{i}),\exists t_{j,i}\in A_{n}^{+}:\ s(x_{j},t_{j,i}) =x_{i}. \tag{5}\]0  
Since \(s(x_{j},t^{\prime})=x_{i}\) by the definition of \(t\) and \(r\), \(t^{\prime}\) can be used instead of \(t\) for fault recovery from \(x_{j}\) to \(x_{i}\). It is clear that \(C\) is implementable with respect to the derived input sequence. Since \(m\) is distinctive with respect to \(x_{i}\), the latter implies the implementability of \(C\). In the above proof, the suffix of the previous input sequence with a longer length (\(|t|-p\)) is replaced by the shorter one (\(|r|-q\)). This is intended to reduce the computational load of the controller by taking a shorter feedback path. Given the configuration of Figure 1, assume that \(W(x_{i})\
eq\emptyset\) for all \(x_{i}\in\hat{X}\subset X\) of \(\Sigma\). Then, \(C\) achieving fault recovery against transient faults by \(A_{d}\) exists if and only if every unauthorized transition is fault detectable and \(\forall x_{i}\in\hat{X},\forall x_{j}\in\Gamma(x_{i})\), \(x_{i}\) is stably reachable from \(x_{j}\). Proof. : (If) Since every unauthorized transition is fault detectable, we can design \(G\) by applying (3). In addition, since \(x_{i}\) is stably reachable from every state of \(\Gamma(x_{i})\), by Proposition 3 we can find a set of control input sequences with which \(C\) is implementable and can design \(C\) by referring to (6)-(8). According to the foregoing discussions, \(\Sigma_{c}\) is immune against any unauthorized transitions by \(A_{d}\). (Only if) In view of Figure 1, the existence of \(C\) implies that the next stable state of every stable transition is identified solely by observing \(z\). Hence every unauthorized transition is fault detectable by Proposition 2. Since \(C\) has access to \(\Sigma\) only by changing \(u\), fault recovery ensures that for all \(x_{i}\) with \(W(x_{i})\
eq\emptyset\) and for all \(x_{j}\in\Gamma(x_{i})\), \(t_{j,i}\in A_{n}^{+}\) exists which takes \(\Sigma\) from \(x_{j}\) to \(x_{i}\) via a chain of stable transitions. Figure 2 illustrates the flowchart elucidating the execution of the proposed static corrective controller for achieving fault tolerance. Compared with dynamic controllers [31], the improvement of computational load in controller synthesis is obvious. The dynamic controller needs to define maximum \(n+1\) states for the correction procedure of each unauthorized transition. Since there may be maximally \(n(n-1)\) unauthorized transitions, the size (or number of states) of the overall controller has the complexity of \(O(n^{3})\). On the other hand, the proposed static controller is much more efficient, as it needs no states. Of course, \(G\) requires memory elements as the state burst and the previous output \(m_{o}\) must be recorded. Since the maximum length of the state burst is \(n-1\), the construction of \(G\) is computed in \(O(n)\), which is a mild degradation of resource usage. Note that a symbolic computation algorithm for inducing feasible control input sequences addressed in (5) is presented in the prior work [1,2]. Further, numerical algorithms avoiding tedious symbolic computations are found in the recent results [37,38].