# Structurally Constrained Controllers

## Chapters (5)

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.

... Decentralized control systems appear in different models such as in electric power systems, communication networks, large space structures, robotic systems, economic systems and traffic networks, etc, see e.g. [29,33] and the references therein. These systems are characterized as large-scale which are therefore composed of lower order subsystems. ...

... where ψ(·) is as defined in (29) and τ ∈ (0, 1) is a given constant. Similar to the treatment in the previous sections and instead of considering the constrained problem (30) we rather consider the following unconstrained minimization problem: ...

... where ψ(·) is given by (29). The proposed NM and PSO methods are stopped as soon as the achieved K(0), K(1), . . . ...

In this article, we consider the output feedback eigenvalue assignment problem for continuous and discrete-time control systems. This problem is formulated as un-constrained matrix optimization problems and tackled by the Nelder-Mead simplex method and particle swarm optimization method. The two methods are extended to compute the approximate solutions of the eigenvalue assignment problem for the particular cases of decentralized and periodic control systems. The performance of the methods is demonstrated numerically on several test problems from the benchmark collection [22] as well as other test examples from the system and control literature. AMS subject classification: 93D15, 93B55, 93B60, 90C56

... There is also the concern that the owners of the different DERs are not willing to hand over control of their resources to a third party. Finally, central systems are usually regarded as not being very scalable and system maintenance requires complete shutdown [13]. To overcome these issues, more distributed control architectures are developed, as described in the sections below. ...

In this paper, the motivation to develop microgrids as an effective solution for the control of distribution networks with high level penetration of Distributed Energy Resources (DERs) is discussed. As many different control methods for microgrids can be found in literature, this paper proposes a classification from highly centralized to distributed peer-to-peer control architectures. A peer-to-peer control paradigm is
proposed as a way to control the distribution network with a high penetration of distributed energy resources. Different control algorithms suited for the proposed peer-to-peer control strategy are discussed.

This note considers the design of static output feedback mixed
${H}_2/{H}_\infty $
controllers for linear control systems with certain equality and inequality constraints imposed directly on the feedback matrix. Based on the barrier method, we solve an auxiliary minimization problem to obtain an approximate solution to the original nonconvex constrained optimization problem. Necessary conditions for the optimal solution of the auxiliary minimization problem are derived using the Lagrange multiplier method. Subsequently, an iterative steepest descent algorithm is developed to find an approximate optimal solution. Finally, an example is provided to validate the proposed approach.

In this paper, we design resilient sparse state-feedback controllers for a linear time-invariant (LTI) control system while attaining a pre-specified guarantee on the performance measure of interest. Our method consists of two main steps: first, we leverage a technique from non-fragile control theory to identify a region of quadratically resilient state-feedback controllers, (ii) then, we search for a sparse controller inside the region. We use two different techniques for the sparsification task: the greedy method of sparsification, as well as the re-weighted l1 norm minimization scheme that is solved along with linear matrix inequalities. The method highlights a tradeoff between the sparsity of the controller, performance measure, and fragility of the design.

Control system research is entering a new era, which may be considered a milestone in the history of Systems and Control. The traditional framework of a control system structure — i.e. plant, sensor, actuator and a controller — presents a number of limitations to some current research, i.e. (a) System to be controlled comprise network of subsystems; (b) Control task is achieved by a network of distributed local controllers (agents); (c) New research topics on control of networked behaviour, including consensus, formation and synchronisation, are clearly beyond the scope of the traditional “one controller” framework. It is worth noting that with these new trends some new concepts are emerging, e.g. consensusability, formationability, computability. The intended contributions of this paper are: (1) A new framework of control research represented by a new block diagram and its mathematical treatment. It is pointed out that, many current different studies can be considered as some special cases under this general framework; (2) a brief overview of various research fitting the proposed framework and some open questions. This paper is largely motivated by past, current and future applications of power system control, and is based on a recent invited presentation, of the first author, at a special workshop on “Bridging the Gap — Control Theory and Control Engineering Practice”.

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