# A Survey on Dynamic Corrective Control of Asynchronous Sequential Machines
## 4 Model Matching Control
### Model Matching for Composite ASMs  
All of the controlled ASMs in the cited publications so far are unit ASMs whether they are of input/state or input/output type. Depending on the assigned operation, however,more than one ASM can be combined into forming a composite ASM. Recently, research concerning model matching corrective control for composite ASMs has been receiving much attention. Composite ASMs are further classified into switched ASMs, serially connected ASMs, and parallel connected ASMs.  
Figure 3 illustrates the corrective control system for the switched ASM \(\Sigma\), which is represented by  
\[\Sigma:=\{\Sigma_{i}|i\in\{1,\dots,m\}\},\ \Sigma_{i}=(A,X,\bar{x},f_{i}),\]  
where \(m\in\mathbb{N}\) is the number of subachines and \(\Sigma_{i}\) denotes the \(i\)th submachine. Among \(m\) subachines, the active one \(\Sigma_{\sigma}\) serves as the current dynamics of \(\Sigma\) in accordance with the value of the switching signal \(\sigma\in\{1,\dots,m\}\). The demultiplexer in Figure 3 relays the control input \(u\) to the active submachine \(\Sigma_{\sigma}\). Similarly, the multiplexer extracts the state feedback of \(\Sigma_{\sigma}\) among \(m\) values and delivers it to \(C\) along with the index \(i\in\{1,\dots,m\}\) of the active submachine. In short, the switched ASM is an assembly of multiple ASMs that share the same state set but have different (while analogous) transition characteristics. Since \(C\) can change the mode of \(\Sigma\) by manipulating the value of \(\sigma\), a switched ASM has larger stable reachability than each submachine, hence facilitating the existence of a model matching corrective controller.  
The modeling of switched ASMs and their model matching problem are first introduced in [56]. Based on the result, [56; 57] present an alternative design procedure for the controller using the STP approach.  
Figure 4 shows the corrective control system for the other kinds of composite ASMs--serially connected ASMs (Figure 4a) and parallel connected ones (Figure 4b). In Figure 4a, \(m\) input/state ASMs \(\Sigma_{1},\dots,\Sigma_{m}\) are combined into one in a serial connection in which the state of a rear machine is transmitted to its front machine as the input. \(w_{1},\dots,w_{m}\) denote adversarial inputs causing faults in each submachine; they will be discussed in the next section. The composite ASM in a serial connection has inherent uncertainty about the state since the current state of each submachine is unavailable to the controller except for the last submachine. Thus unlike unit input/state ASMs, the state estimation must be included in the design and operation of an appropriate corrective controller. The study on model matching for serially connected ASMs is first discussed in [58]. The author of [59] addresses a matrix approach to the same control problem. Not only does [59] present the design procedure of a model matching controller, it also elucidates a matrix scheme to derive the shortest control input for a given state pair.  
Figure 3: Corrective control system for the switched ASM.  
The controlled machine in Figure 4b consists of two input/state ASMs \(\Sigma_{1}\) and \(\Sigma_{2}\) that have parallel connections with each other. When the control input \(u\) is transmitted to both \(\Sigma_{1}\) and \(\Sigma_{2}\) simultaneously, the two ASMs undertake their own state transitions, generating the next stable states \(x\) and \(y\), respectively. The feedback value \(z\) is derived from \(x\) and \(y\) via an output function \(h(x,y):=z\). As in the foregoing case of composite ASMs in serial connection, the controller \(C\) in Figure 4b is faced with uncertainty about the inner states of submachines. The notion of parallel interconnected ASMs is first introduced in a comment in [56]. Later, it becomes enriched so that it presents an efficient algorithm for controller synthesis while accommodating the inherent uncertainty about the state.