Structurally Constrained Controllers

In the past three decades, the problem of decentralized control has been thoroughly investigated in the literature, and a variety of its aspects are studied [1, 2, 3]. More recently, the problem of decentralized overlapping control has attracted several researchers [4, 5].The decentralized overlapping control is fundamentally used in two cases:i) when the subsystems of a system (referred to as overlapping subsystems) share some states [6, 7, 8]. In this case, it is usually desired that the structure of the controller matches the overlapping structure of the system [8]; ii) when there are some limitations on the availability of the states. In this case, only certain outputs of the system are available for constructing each control signal.

Numerous real-world systems can be modeled as the interconnected systems consisting of a number of subsystems. The control of an interconnected system is often carried out by means of a set of local controllers, corresponding to the interacting subsystems [1, 2]. It is sometimes assumed that the local controllers can fully communicate with each other in order to elevate their effectiveness over the entire system cooperatively. However, this design technique is often problematic as the required data transmission between two particular local controllers (or equivalently, two subsystems) can be unjustifiably expensive or occasionally infeasible. Consequently, it is normally desired that the local controllers either exchange partial information or act independently of each other. The latter case, where the overall controller consists of a set of isolated local controllers, is referred to as decentralized control in the literature [3, 4, 5]. The control structure in a decentralized control system is, in fact, block-diagonal. It is to be noted that the decentralized control theory has found applications in large space structures, power systems, communication networks, etc. [6, 7, 8, 9]. A wide variety of properties of the decentralized control systems are extensively studied in the literature and different design techniques are proposed [10, 11, 12, 13].

The focus of this chapter is directed towards the problem of characterizing the information flow structures of all classes of LTI structurally constrained controllers with respect to which a given interconnected system has no fixed modes. Any class of structurally constrained controllers can be described by a set of communication links, which delineates how the local controllers of any controller in that class interact with each other. To achieve the objective, a cost is first attributed for establishing any communication link in the control structure. These costs are part of deign specifications and represent the expenditure of data transmission between different subsystems. A simple graph-theoretic method is then proposed to characterize all the relevant classes of controllers systematically. As a by-product of this approach, all classes of LTI stabilizing structurally constrained controllers with the minimum implementation cost are attained using a novel algorithm. The primary advantages of this approach are its simplicity and computational efficiency. The efficacy and importance of this work are thoroughly illustrated in a numerical example. This chapter integrates the ideas proposed in a recently published work and some original techniques to develop its main results.

Many real-world systems such as communication networks, large-space structures, power systems, and chemical processes can be modeled as interconnected systems with homogeneous or heterogeneous interacting subsystems [1, 2, 3, 4, 5]. The classical control techniques often fail to control such systems, in light of some well-known computation or communication constraints. This has given rise to the emergence of the decentralized control area that aims to design non-classical structurally constrained controllers [6]. A decentralized controller comprises a set of non-interacting local controllers corresponding to disparate subsystems. The analysis and synthesis of a decentralized control system has long been studied by many researchers. In particular, the decentralized control theory has been recently developed for systems with geographically distributed subsystems in the context of distributed control for diverse applications, such as flight formation [7], consensus [8, 9] and Internet congestion control [10].

Numerous real-world systems can be envisaged as interconnected systems consisting of a number of subsystems [1]. Every controller for such a system is often composed of a set of local controllers corresponding to the individual subsystems. In an unconstrained control structure, each local controller has access to the outputs of all the subsystems. This class of controllers is referred to as centralized. However, in many control applications, each local controller can only use the information of a subset of subsystems.