# A Survey on Dynamic Corrective Control of Asynchronous Sequential Machines
## 4 Model Matching Control
### Model Matching for Input/Output ASMs  
The procedure of model matching corrective control for input/output ASMs is conducted in a similar way to the case of input/state ASMs with the following two additional points to be considered. First, stable reachability of the controlled ASM must be described in terms of the input/output behavior with respect to the reference model. Next, the state feedback to the controller is delivered not by the controlled machine, but by the state observer.  
Suppose that the reference model is given by  
\[\Sigma^{\prime}=(A,Y,\hat{X},\hat{x}_{0},s^{\prime},h^{\prime}),\]  
where \(\hat{X}\) is the state set with \(|\hat{X}|=q\), \(\hat{x}_{0}\in\hat{X}\) is the initial state, and \(s^{\prime}:\hat{X}\times A\rightarrow\hat{X}\) and \(h^{\prime}:\hat{X}\to Y\) are the stable recursion and output function, respectively. Note that \(\Sigma^{\prime}\) has the same input and output set as those of \(\Sigma\), while its state set differs from \(X\). In particular, let \(\hat{X}:=\{\hat{x}_{1},\ldots,\hat{x}_{q}\}\).  
**Definition 3**.: _Given \(\Sigma=(A,Y,X,\bar{x},f,h)\) and \(\Sigma^{\prime}=(A,Y,\hat{X},\hat{x}_{0},s^{\prime},h^{\prime})\), the output equivalence list of \(\Sigma\) with respect to \(\Sigma^{\prime}\) is_  
\[E(\Sigma,\Sigma^{\prime}) :=\{E_{1},\ldots,E_{q}\}\] \[E_{i} :=\{x\in X|h(x)=h^{\prime}(\hat{x}_{i})\},\ i=1,\ldots,q.\]  
\(E_{i}\subset X\) contains all the states of \(\Sigma\) whose outputs are equal to that of \(\hat{x}_{i}\). As we only have to investigate stable equivalence between \(\Sigma\) and \(\Sigma^{\prime}\) in terms of the input/output behavior, it is sufficient to quantify the stable reachability of \(\Sigma\) between different output equivalent states.  
**Definition 4**.: _A non-deficient subordinate list \(\Lambda\) of \(E(\Sigma,\Sigma^{\prime})\), termed \(\Lambda\prec E(\Sigma,\Sigma^{\prime})\), is a collection of states \(\Lambda:=\{x_{\lambda_{1}},\ldots,x_{\lambda_{q}}\}\) such that \(\lambda_{i}\in\{1,\ldots,n\}\) and \(x_{\lambda_{i}}\in E_{i}\), \(i=1,\ldots,q\). The fused skeleton matrix \(\Delta(\Sigma,\Lambda)\) of \(\Lambda\) is a \(q\times q\) Boolean matrix of which \((i,j)\) entry is defined as_  
\[\Delta_{i,j}(\Sigma,\Lambda):=K_{\lambda_{i}\cup\lambda_{j}}(\Sigma),\ i,j\in \{1,\ldots,q\}.\]  
Drawing on the prior result of input/state ASMs and referring to Definition 4, one can derive the existence condition on a model matching corrective controller for the input/output ASM \(\Sigma\) as follows.  
\[\exists\Lambda\prec E(\Sigma,\Sigma^{\prime}):K(\Sigma^{\prime})\leq\Delta( \Sigma,\Lambda).\]  
The design algorithm for the state observer \(B\) and corrective controller \(C\) in Figure 1b is well established in [29; 30; 31]. Compared with the case of input/state ASMs, the study of model matching for input/output ASMs is relatively rare. In [52; 53], the model matching problem is concerned with input/output ASMs with nondeterministic transition features caused by critical races. The approaches of [52; 53] differ from [29; 30; 31] in that, instead of output bursts, unit output characters are delivered as output feedback to the controller. The authors of [54; 55] also tackle model matching for input/output ASMs with nondeterministic behaviors, but fully utilizing output bursts in the operation of the state observer.