# A Survey on Dynamic Corrective Control of Asynchronous Sequential Machines
## 4 Model Matching Control  
### Model Matching for Input/State ASMs  
As a comprehensive control goal in corrective control, model matching ensures that a proper corrective controller \(C\) is designed so that it matches the stable-state behavior of the closed-loop system \(\Sigma_{c}\) to that of a prescribed reference model. A key aspect of model matching is that it is sufficient for the closed-loop system to be stably equivalent with the model, namely, model matching is evaluated only in terms of stable behaviors. Thus even if the machine does not have desirable transitions, one can solve the model matching problem by applying the corrective controller which compensates for the stable-state transition characteristics of the controlled machine. An analogous notion of model matching for general finite state machines (FSMs) is established in [46]. The necessary and sufficient condition for the existence of a proper corrective controller achieving model matching for an input/state ASM \(\Sigma\) is that the machine has stable reachability larger than or equal to that of the model. Assume that the reference model  
\[\Sigma^{\prime}=(A,X,\bar{x},s^{\prime})\]  
is given, where \(s^{\prime}:X\times A\to X\) is the stable recursion function of \(\Sigma^{\prime}\). Then the corrective controller \(C\) in Figure 1a achieving model matching between \(\Sigma_{c}\) and \(\Sigma^{\prime}\) exists if and only if  
\[K(\Sigma^{\prime})\leq K(\Sigma),\]  
where matrix inequality is valid entry by entry. To sketch the operation of the corrective controller for model matching, suppose that the foregoing condition holds true with respect to two input/state ASMs \(\Sigma\) and \(\Sigma^{\prime}\). In particular, assume that \(\Sigma^{\prime}\) has a stable transition \(s^{\prime}(x_{i},a)=x_{j}\) whereas \(s(x_{i},a)\
eq x_{j}\) in \(\Sigma\). In the beginning, \(C\) stays at its initial state \(\xi_{0}\). When \(\Sigma\) reaches the stable state \(x_{i}\), \(C\) moves to the transition state \(\xi_{t}\) to deal with a possible entrance of an input causing model mismatch. If the external input changes to another character causing no model mismatch, \(C\) relays it to the control input channel \(u\) (see Figure 1a) without modification and returns to \(\xi_{0}\). On the other hand, if the external input \(v\) changes to \(a\), \(C\) commences the correction procedure since otherwise the subsequent transition of \(\Sigma\) in response to \(a\) would violate the matched behavior. By Definition 1, \(s^{\prime}(x_{i},a)=x_{j}\) leads to \(K_{i,j}(\Sigma^{\prime})=1\). But since \(K(\Sigma^{\prime})\leq K(\Sigma)\), \(K_{i,j}(\Sigma^{\prime})=1\) implies \(K_{i,j}(\Sigma)=1\) and again by Definition 1, there exists an input string \(v_{1}\cdots v_{k}\in K_{i,j}(\Sigma)\) that takes \(\Sigma\) from \(x_{i}\) to \(x_{j}\) via a chain of stable transitions, i.e., \(s(x_{i},v_{1}\cdots v_{k})=x_{j}\). We design \(C\) utilizing \(v_{1}\cdots v_{k}\). To this end, define \(k\) auxiliary states of \(C\)\(\xi_{1},\ldots,\xi_{k}\in\Xi\). Upon receiving \(a\), \(C\) first transfers to \(\xi_{1}\) and generates \(v_{1}\). In response to \(v_{1}\), \(\Sigma\) transfers to the first intermediate stable state \(s(x_{i},v_{1}):=z^{1}\). Receiving the changed state feedback \(z^{1}\), \(C\) further moves to \(\xi_{2}\) while generating \(v_{2}\), which takes \(\Sigma\) from \(z^{1}\) to the second intermediate stable state \(s(z^{1},v_{2}):=z^{2}\), and so forth. At \(\xi_{k}\), finally, the correction procedure is completed when \(\Sigma\) reaches the desired state \(x_{j}\). Figure 2 illustrates the aforementioned interaction between \(C\) and \(\Sigma\) in the process of model matching control. Although \(\Sigma\) traverses a number of stable states \(z^{1},\ldots,z^{k-1}\) in the process, the asynchronous mechanism makes these stable states transient, namely, \(\Sigma_{c}\) would seem to transfer from \(x_{i}\) directly to \(x_{j}\) in response to the external input \(a\), thus realizing the matched behavior.