# State-Bust Feedback Control for Fault Recovery of Input/State Asynchronous Sequential Machines
## 2 Preliminaries
Referring to Figure 1, \(C\) receives the input triplet \((x,v,m)\) and generates the control input \(u\) as the output. Being a static controller, \(C\) is represented by the following function:  
\[C:X\times A_{n}\times N_{0}\to A_{n}.\]  
During the normal behavior of \(\Sigma\), \(C\) just relays the external input to \(\Sigma\) without modification. When a transient fault is diagnosed, \(C\) provides appropriate control input sequences. The control objective is to achieve immediate fault recovery, namely, to take \(\Sigma\) from the faulty state to the original state at which the fault occurs before further change in the external input. Since neither \(C\) nor \(G\) is governed by a synchronizing clock, their operations are also conducted instantaneously under asynchrony. Hence the procedure of fault diagnosis and fault tolerant control can be completed before the external input changes to the next value, rendering \(\Sigma_{c}\) to show the normal input/state behavior as if no fault has happened. \(\Sigma_{c}\) is assumed to preserve fundamental mode operations [32], wherein no two variables change simultaneously. Under the fundamental mode, \(w\) is supposed to occur to \(\Sigma\) only when \(\Sigma\) stays at a stable state. This is not a burdensome constraint since the speed of transient transitions is so fast that the possibility of fault occurrences during the transitions is negligible. Hence the stable state at which the fault occurs serves as the goal state for the corresponding fault tolerant control. In a similar sense, it is also supposed that \(v\) is not altered during the correction procedure. **Remark 1**. : _Whereas the present study focuses on transient faults of which influence on the machine vanishes right after its occurrence, there exist other kinds of faults differing in the durability of their influences. If the adverse effect persists for a finite time after initial occurrence, the fault is termed an intermittent fault [33]; if the effect remains indefinitely (or irreversible), it is termed a permanent fault [34]. For input/state ASMs, fault recovery is impossible for either intermittent or permanent faults since immediate return to the original state cannot be implemented. The latter problem can be tackled for input/output ASMs, where the output differs from the present state [35], or switched ASMs that possess redundant states which may substitute faulty states [8]._  
**Remark 2**. : _In the field of DESs, the stability and stabilizability of the system under static feedback control means that starting from any arbitrary initial state, the system can (or can be controlled to) go to a \"legal state\" and stay there after a finite number of transitions [36]. In our problem setting, the original state at which a fault occurs can be regarded as a legal state. With no infinite cycles, further, fault recovery implies that \(\Sigma\) must be controlled to the original state in finite steps of stable transitions. In this sense, the fault tolerant controllability of \(\Sigma\) is equivalent to the stabilizability of DESs in supervisory control. Note that a stable pair \((x,v)\in X\times A\) just implies that \(x\) is a fixed point of \(f\), irrelevant to the stability of \(\Sigma\)._