# A Survey on Dynamic Corrective Control of Asynchronous Sequential Machines
## 5 Fault-Tolerant Control
### Transient Faults  
Let us first review the study on fault-tolerant corrective control against transient faults. In general, transient faults are defined as temporary violations of the system's normal behaviour while having no correlation with each another [63; 64; 65]. In the dynamics of ASMs, transient faults usually mean the situation where ASMs undergo unauthorized state transitions, namely, they are forced to transfer to faulty next stable states by external adversarial entities charging the control input channel or internal malfunctions. Note that fault tolerance against transient faults is frequently studied in corrective control. The main reason for this popularity is that transient faults are the most common event happening to asynchronous digital systems working in hazardous environments such as space and nuclear power plants [66; 67; 68; 69]. To describe fault-tolerant corrective control overcoming transient faults, suppose that \(\Sigma\) has been staying at a stable state \(x_{i}\) and \(C\) at the transition state \(\xi_{t}\) (see also Figure 2). Suppose further that an adversarial input \(w_{a}\) enters \(\Sigma\) such that \(s(x_{i},w_{a})=x_{j}\). Then \(\Sigma\) undergoes the unauthorized transition from \(x_{i}\) to \(x_{j}\). \(C\) can identify an occurrence of the transient fault by observing that the state feedback changes to \(x_{j}\) while both the external input \(v\) and the control input \(u\) remain unchanged. To represent the outcome of each transient fault, define the adversarial skeleton matrix as follows. **Definition 5**. : _Given \(\Sigma=(A,X,\xi,f)\), \(K^{a}(\Sigma)\in\{0,1\}^{n\times n}\) is the adversarial skeleton matrix of which \((i,j)\) entry is defined as_  
\[K^{a}_{i,j}(\Sigma):= \begin{cases}1&\exists w_{a}\in A_{d}:s(x_{i},w_{a})=x_{j }\\ 0&\text{otherwise}\end{cases} \]  
_where \(A_{d}\) denotes the set of adversarial inputs._  
To overcome the transient fault, \(C\) must initiate the correction procedure immediately after \(\Sigma\) reaches \(x_{j}\) so that \(\Sigma\) can be controlled back to \(x_{i}\). The existence condition for such a corrective controller is derived in a similar way to model matching as follows. \[K^{a}(\Sigma)\leq(K(\Sigma))^{T},\]  
where \((K(\Sigma))^{T}\) indicates the transpose of \(K(\Sigma)\). According to Definition 5, the unauthorized transition from \(x_{i}\) to \(x_{j}\) is quantified by \(K^{a}_{i,j}(\Sigma)=1\). If the above existence condition is valid, \(K^{a}_{i,j}(\Sigma)=1\) leads to \(K_{j,i}(\Sigma)=1\) and thus an input string, say \(t^{\prime}\in A^{+}\), exists such that \(s(x_{j},t^{\prime})=x_{i}\). A fault-tolerant corrective controller can be designed utilizing \(t^{\prime}\) in the identical framework of model matching control depicted in Figure 2.  
Fault-tolerant corrective control against transient faults was first studied in [70]. It considers the problem with respect to input/state ASMs, while it differs from other related results in the accessibility of the adversarial input. Some publications assume that the controller can identify the value of the adversarial input, whereas [70] stipulates that the controller does not receive any information about the adversarial input, which is a more general and practical problem setting. Following [70], a wide variety of approaches has been presented in the literature on fault-tolerant corrective control against transient faults. A study exists which applies the corrective controller of [70] to an asynchronous triple modular redundancy (TMR) memory embedded in space-borne digital systems. On the other hand, another report proposes fault-tolerant corrective control with bounded delays wherein if complete fault recovery is impossible, the controller is designed so that the control goal is achieved after elapse of a number of input changes. There also exists a novel approach to fault-tolerant corrective control which supposes that transient faults may occur in both fundamental and non-fundamental modes.