# A Survey on Dynamic Corrective Control of Asynchronous Sequential Machines
###### Abstract  
As a feedback control methodology exclusively targeting asynchronous sequential machines (ASMs), corrective control has been rapidly developing for the past two decades. This paper presents a comprehensive survey on the theory and application of dynamic corrective control in which the controller also has the form of an ASM. First, basic notions and principles of dynamic corrective control, including models of ASMs and configurations of closed-loop systems, are reviewed. Next, assorted dynamic corrective control schemes are presented aiming at solving specific control problems of ASMs, such as model matching and fault-tolerant control. Variations of control aspects are classified according to modeling formalisms of controlled ASMs--input/state,  
correct the faulty behaviors of general sequential machines caused by corrupted inputs [12], incomplete knowledge of models [13], and external disturbances [14]. These studies can be said to serve as the foundation for initiating the study of corrective control for ASMs. Beginning from a preliminary study [16], the research on corrective control has greatly progressed over the past two decades in both theoretical development and application to real-world systems. As will be shown later, the subject of corrective control encompasses a wide range of control problems including model matching (or model following control), robust control with respect to model uncertainty and design flaws, fault diagnosis and fault-tolerant control against attacks of random faults or intelligent adversaries, etc. The objective of this paper is to provide a comprehensive survey of the present state of the art for dynamic corrective control and to consider the direction of future research. Dynamic corrective controllers represent a class of corrective controllers, the structure of which is also an ASM having inner states and asynchronous transitions in response only to changes in the input. This survey will not consider static controllers [17; 18; 19], another class of corrective controllers which consist of purely logic components with no inner states. Hereafter, \"corrective controllers\" implies only dynamic ones. For research on static corrective control, readers are referred to related studies (e.g., [20]). Before proceeding with our discussion, it is worthwhile to compare corrective control with the supervisory control of discrete-event systems (DESs) [21; 22; 23; 24; 25; 26], a representative control theory for event-driven systems based on formal languages. 1. Supervisors for DESs receive traces of inputs (or events) so as to generate commands specifying whether the current (controllable) input be enabled or disabled. Hence a supervisor cannot enlarge the controlled behavior of a DES; the supervisor can only curtail it to meet a given specification. On the other hand, since a corrective controller generates its own control input sequences while suppressing the current external input, it can provide the controlled ASM with new stable-state behaviors that otherwise could not be displayed. 2. Rather than using the method of formal languages, corrective control theory exploits transition equivalence in evaluating the success of control goals. In the case of model matching, for example, the stable-state behavior of the closed-loop system is deemed to be matched with that of the reference model if, staying at the same state, they move to the same next stable state in response to a common input. In this sense, the control specification of corrective control is stricter than supervisory control. 3. Although having a conceptual similarity to corrective control, supervisory control cannot be applied to controlling ASMs since it does not consider intrinsic characteristics of ASMs such as discrimination between stable and transient states and abiding by the principle of fundamental mode operations. The remainder of this article is organized as follows.