Decentralised control for complex systems - an invited survey <s> Fundamental concept <s> To control the large scale systems is important. In this paper, a multi variable non-linear system (two inverted pendulum coupled by a spring) is output feedback linearized and the system is generalized in two subsystems and decentralized dynamic output feedback basis on lyapunov equation is applied. Using this model, the large scale system can be formulated, designed and generalized to be controlled. <s> BIB001 </s> Decentralised control for complex systems - an invited survey <s> Fundamental concept <s> In this paper, time delay interconnected systems are considered where the nominal isolated subsystems are fully nonlinear. The interconnections and the matched and mismatched disturbances are nonlinear and time-delayed. A decentralised static output feedback control strategy is proposed to stabilise the resulting interconnected system uniformly asymptotically. A novel approach is proposed to deal with the interconnections by introducing artificial interconnections, such that accessible information is fully exploited in the control design to reduce conservatism. Simulation results show the effectiveness of the proposed approach. <s> BIB002
From the mathematical point of view, a nonlinear large scale interconnected system composed of N n i th order subsystems can be described bẏ where is a neighbourhood of the origin) and u i ∈ R mi , y i ∈ R pi are the states, inputs and outputs of the i th subsystem respectively for i = 1, 2, . . . , N . All the matrix functions g i (·) ∈ R ni×mi and the nonlinear vectors f i (·) ∈ R ni and h i (·) ∈ R pi with h i (0) = 0, are known. The terms ∆g i (·) and ∆f i (·) represent the matched and the mismatched uncertainties respectively. The term ζ ij (t, x j ) represents the interconnection of the i th subsystem. It is assumed that all the nonlinear functions are smooth enough such that the unforced systems have unique continuous solutions. Firstly, the following basic definitions are introduced for the system (1). Definition 1: Consider system (1). The systeṁ is called the i th isolated subsystem of system (1), and the systeṁ is called the i th nominal isolated subsystem of system (1). It is well known that one of the main problems for interconnected systems is establishing under what conditions the interconnected system (1) exhibits the desired performance if all the isolated subsystems (2) or all the isolated nominal subsystems (3) have the performance. Therefore, how to deal with interconnection is one of the main concerned problems in decentralised control. Definition 2: Consider system (1). If there exist controllers for each subsystems depending upon the state of the subsystems only, i.e., such that the corresponding closed-loop system formed by applying the control u i in (4) to system (1), has the desired performance, then the control (4) forms a decentralised static state feedback control scheme. If the controllers in (4) have the form that is, each local controller depends upon the output of the subsystem only, then they are called decentralised static output feedback controllers. Furthermore, if there exist dynamical systemṡ where controllers are such that the corresponding closed-loop system formed by applying the control u i in (7) withx given by dynamical system (6) to the system (1), has the desired performance, then the control (6) and (7) is called a decentralised dynamic output feedback control scheme. Specifically, if (6) is an observer of the system (1), then a decentralised observer-based control results. It is straightforward to see, according to Definition 2 above, that it is required that the dynamical systems (6) are decoupled in a decentralised dynamical output feedback scheme. It should be mentioned that in some of the existing work, see for example BIB001 and , the designed dynamical systems (6) are not decoupled (in fact they are interconnected systems). In this case, the developed controller is sometimes still called a decentralised control. However, in precise terms, such a class of controllers is not decentralised because there exists information transfer between subsystems in the dynamical system design (see e.g., BIB001 . Several decades ago, most work on decentralised control focused on linear interconnected systems due to the limitation of available control paradigms that were able to deal with nonlinearity. However, the dynamics of large scale natural and engineered interconnected systems are usually highly nonlinear. It is not only the structure of the system and interconnections which produce complexity but also the nonlinearity of the dynamics themselves. It is clear that although a linear dynamics may approximate the orbit of a nonlinear system locally, it does not permit the existence of the multiple states observed in real networks and does not accommodate global properties of the system. Such global properties can be crucial because they may become significant when the system is perturbed or a subsystem enters a failure state. Increasing requirements on system performance coupled with the ability to mode and simulate reality by means of complex, possibly nonlinear, interconnected systems models has motivated increasing contributions in the study of such systems. This interest has been further stimulated by the simultaneous development of nonlinear systems theory and the emergence of powerful mathematical and computational tools which render the formal and constructive study of nonlinear large scale systems increasingly possible (see BIB002 . In order to explain the decentralised control concept, the following schematic diagram in which the interconnected system has three subsystems, is produced to show that in static decentralised output feedback control, the local control u i of the i th subsystem only uses local output information y i ; no output information y j (j ̸ = i) is involved in the design. From Figure 1 , it is clear to see that there is no local output information transfer between the local controllers u i and u j (i ̸ = j) for i, j = 1, 2, 3.