# State-Bust Feedback Control for Fault Recovery of Input/State Asynchronous Sequential Machines
## 5 Illustrative Examples  
### Home Security System  
Consider \(\Sigma_{1}=(A,X,\bar{x},f)\) whose state flow diagram is shown in Figure 3, where \(A_{n}=\{a,b,c,d\}\), \(A_{d}=\{w_{1},w_{2}\}\), and \(\bar{x}=x_{1}\). \(\Sigma_{1}\) represents a home security system [12], where \(x_{1}\) is the initial state and \(x_{2}\)-\(x_{4}\) are three alarm states that are reached by break-in events \(d\), \(a\), and \(b\), respectively. \(c\) is the reset signal that is activated only at \(x_{3}\).  
Figure 2: Flowchart of the proposed static corrective control scheme.  
As an example instance, let us take the case of \(x_{1}\) with \(U(x_{1})=\{c\}\) and \(W(x_{1})=\{w_{1},w_{2}\}\). To check the fault detectability, derive the state bursts caused by the elements of \(W(x_{1})\) as \(\alpha(x_{1},w_{1})=x_{1}x_{3}x_{2}\) and \(\alpha(x_{1},w_{2})=x_{1}x_{4}x_{3}\). Since \(\alpha(x_{1},w_{1})\
otin sp(\alpha(x_{1},w_{2}))\) and vice versa, \(\Sigma_{1}\) is fault detectable at \(x_{1}\) by Proposition 2.  
Following (3), we design \(G\) as follows:  
\[G(x_{1}x_{3}x_{2},c) =(x_{2},1)\] \[G(x_{1}x_{4}x_{3},c) =(x_{3},1)\] \[G(z,u) =(z_{l},-),\ (z_{f},u,z_{l})\in s,\ m_{o}=1,\ z_{l}\
eq\ x_{1}\] \[G(z,u) =(z_{l},0),\ \text{otherwise}.\]  
If \(G\) receives \(x_{1}x_{3}x_{2}\) or \(x_{1}x_{4}x_{3}\) while \(u\) is equal to \(c\), the unauthorized transition by \(w_{1}\) or \(w_{2}\) is diagnosed with certainty since \((x_{1},c,x_{2})\
otin s\) and \((x_{1},c,x_{3})\
otin s\).  
Referring to Figure 3, \(x_{1}\) is stably reachable from every state of \(\Gamma(x_{1})=\{x_{2},x_{3}\}\), e.g., \(s(x_{2},ac)=x_{1}\) and \(s(x_{3},c)=x_{1}\). By Theorem 1, \(C\) exists which achieves fault recovery against \(W(x_{1})\). To check the implementability of \(C\) associated with \(ac\) and \(c\), \(X_{2,1}\) and \(X_{3,1}\) defined in (9) are derived as \(X_{2,1}=\{x_{2},x_{3}\}\) and \(X_{3,1}=\{x_{3}\}\). Although \(X_{2,1}\cap X_{3,1}=\{x_{3}\}\) is non-empty, both \(ac\) and \(c\) have the common input \(c\) for \(x_{3}\). Hence they can be used as the control input sequences while guaranteeing the implementability of \(C\).  
The design procedure of \(C\) is straightforward. Since \(s(x_{2},a)=x_{3}\) and \(s(x_{3},c)=x_{1}\), the control inputs with respect to \(m=1\) are assigned in line with (7) and (8) as follows:  
\[C(x_{2},c,1) =a \tag{10}\] \[C(x_{3},c,1) =c.\]  
When an unauthorized transition occurs so that \(m\) changes to \(1\) and the state feedback to \(x_{2}\) or \(x_{3}\), \(C\) activates the recovery procedure by performing the above operations. When \(\Sigma_{1}\) reaches \(x_{1}\), \(m\) is reset to \(0\), and upon receiving \((x_{1},0)\) from \(G\), \(C\) terminates the recovery procedure. To this end, set \(C\) as follows:  
\[C(x_{1},c,0)=c.\]  
Since the fault detectability condition in Proposition 2 and the reachability condition (5) are valid for the other states as well, the design of \(G\) and \(C\) for the rest of states and inputs can be similarly conducted.  
As a comparative study, let us try to achieve the above fault tolerant control by the previous static controller \(C_{p}\) receiving no fault indicator signal (see (4)). Assume that \(w_{2}\) occurs when \(\Sigma_{1}\) has been staying at \((x_{1},c)\) so that \(\Sigma_{1}\) is forced to reach \(x_{2}\). In a similar way to (10), \(C_{p}\) must generate \(a\) to initiate the correction procedure, namely, \(C_{p}(x_{2},c)=a\). However, since \(s(x_{2},c)=c\) in Figure 3, \(C_{p}\) already has the assignment \(C_{p}(x_{2},c)=c\) for  
Figure 3: State flow diagram of a home security system \(\Sigma_{1}\).  
ensuring the nominal transition. As this contradicts the foregoing operation, one cannot design \(C_{p}\) that accomplishes fault tolerant control for \(\Sigma_{1}\).