Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> Given a dynamical system whose description includes time-varying uncertain parameters, it is often desirable to design an output feedback controller leading to uniform stability of a given equilibrium point. When designing such a controller, one may consider static (i.e., memoryless) or dynamic compensation. In this paper, we show that solvability of various output feedback design problems is equivalent to existence of a solution to a certain Constrained Lyapunov Problem (CLP). The CLP can be stated in purely algebraic terms. Once the CLP is described, we provide necessary and sufficient conditions for its solution to exist. Subsequently, we consider application of the CLP to a number of robust stabilization problems involving static output feedback and observer-based feedback. <s> BIB001 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> Abstract This paper presents a solution to the decentralized tracking or servomechanism problem for a class of nonlinear, multivariable systems. Decentralized variable structure control is used to decompose the solution into two parts: (1) the design of a set of decentralized, nonlinear controls that force each subsystem's state trajectory onto an interconnected manifold in the composite system state space; and (2) the construction of the manifold such that asymptotic tracking occurs when the subsystem state trajectories are restricted to the manifold. While numerical methods are not explicitly considered, manifold design is relatively straightforward, requiring only the use of known decentralized eigenvalue assignment techniques and the solution of linear algebraic equations. The structure of a simple, decentralized, nonlinear control law is proposed which requires little design effort beyond the construction of the manifold. <s> BIB002 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> In the paper, the problem of decentralised model-reference adaptive control, for linear time-varying large-scale systems, is investigated by using a variable-structure control approach. The global reaching condition of the sliding mode of a composite system is derived and a new scheme of decentralised control is proposed. This scheme can guarantee the stability of a sliding manifold of the composite system. Furthermore, a modified scheme is given which makes the weighted sum of the sliding manifold motions converge to the prospective speed. Finally, a numeral example is calculated. <s> BIB003 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> A tutorial account of variable structure control with sliding mode is presented. The purpose is to introduce in a concise manner the fundamental theory, main results, and practical applications of this powerful control system design approach. This approach is particularly attractive for the control of nonlinear systems. Prominent characteristics such as invariance, robustness, order reduction, and control chattering are discussed in detail. Methods for coping with chattering are presented. Both linear and nonlinear systems are considered. Future research areas are suggested and an extensive list of references is included. > <s> BIB004 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> In the paper a new class of output feedback variable structure controllers and state estimators (observers) for uncertain/nonlinear dynamic systems with bounded uncertainties and/or nonlinearities are proposed. No statistical information about the uncertain elements is assumed. A variable structure systems (VSS) approach together with the geometric approach to the analysis and synthesis of system zeros are employed in the synthesis of the proposed output feedback controllers and state estimators. The role of system zeros in the output feedback stabilisation and state estimation, using the VSS approach, is discussed. Numerical examples included illustrate the feasibility of the proposed stabilisation and state estimation schemes. <s> BIB005 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> This paper reviews the static output feedback problem in the control of linear, time-invariant (LTI) systems. It includes analytical and computational methods and presents in a unified fashion the knowledge gained in the decades of research into this important open problem. The paper shows that although many approaches and techniques exist to approach different versions of the problem, no efficient algorithmic solutions are available. <s> BIB006 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> This paper presents a robust decentralized control for large-scale interconnected systems with series nonlinearities through variable-structure control. The proposed variable-structure control ensures the global reaching condition of the sliding mode of the composite system. The uncertain nonlinear system also possesses insensitivity to uncertainties and disturbances like a linear system does. Furthermore, the sliding mode can converge within a specified exponential speed. <s> BIB007 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> In the formation of any control problem there will be discrepancies between the actual plant and the mathematical model for controller design. Sliding mode control theory seeks to produce controllers to over some such mismatches. This text provides the reader with a grounding in sliding mode control and is appropriate for the graduate with a basic knowledge of classical control theory and some knowledge of state-space methods. From this basis, more advanced theoretical results are developed. Two industrial case studies, which present the results of sliding mode controller implementations, are used to illustrate the successful practical application theory. <s> BIB008 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> In this note, the problem of decentralized output feedback robust stabilization is studied for time-varying nonlinear large-scale systems, in which general interconnection and fully nonlinear nominal subsystems are both considered. The uncertainties dealt with in this note may be not only matched but also unmatched, and they may appear not only in interconnection but also in isolated subsystem. Then, a kind of continuous decentralized output feedback control to stabilize the system is proposed. Finally, simulation shows the effectiveness of our result. <s> BIB009 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> This paper discusses overlapping decentralized sliding mode controller design for large-scale continuous-time systems. Design issues, like connective reachability of the sliding manifold and the stability of the sliding mode equations in the expanded and original state spaces are examined. Application of the results to automatic generation control is also discussed briefly. <s> BIB010 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> The effective application of sliding mode control to mechanical systems is not straightforward because of the sensitivity of these systems to chattering. Higher-order sliding modes can counteract this phenomenon by confining the switching control to the higher derivatives of the mechanical control variable, so that the latter results are continuous. Generally, this approach requires the availability of a number of time derivatives of the sliding variable, and, in the presence of noise, this requirement could be a practical limitation. A class of second-order sliding mode controllers, guaranteeing finite-time convergence for systems with relative degree two between the sliding variable and the switching control, could be helpful both in reducing the number of differentiator stages in the controller and in dealing with unmodelled actuator dynamics. In this paper different second-order sliding mode controllers, previously presented in the literature, are shown to belong to the above cited class, and some cha... <s> BIB011 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> Mathematical Developments and VSC.- Higher Order Sliding Modes.- Sliding Modes for Matched and Un-Matched Uncertainty.- Observation and Identification.- Discrete-Time Sliding Modes.- Applications. <s> BIB012 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> In this paper, we develop a fully adaptive decentralized controller of robot manipulators for trajectory tracking. With high-order and adaptive variable-structure compensations, the proposed scheme makes both position and velocity tracking errors of robot manipulators globally converge to zero asymptotically while allowing all signals in closed-loop systems to be bounded, even without any prior knowledge of robot manipulators. Thus this control scheme is claimed to be fully adaptive. Even when the proposed scheme is modified to avoid the possible chattering in actual implementations, the overall performance will remain appealing. Finally, numerical results are provided to verify the effectiveness of the proposed schemes at the end. <s> BIB013 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> In this paper, a class of interconnected systems with mismatched uncertainties is considered. A sliding surface is proposed in an augmented space formed by the system output and the compensator state variables, and the stability of the corresponding sliding mode is analysed. A robust decentralised reduced-order output feedback sliding mode controller is then designed to drive the system to the composite sliding surface and maintain a sliding motion on it thereafter. The proposed approach allows both the isolated nominal subsystem and the nominal interconnected system to be nonminimum phase. The known interconnections and the bounds on the uncertain interconnections are fully used in the control design to reduce conservatism. <s> BIB014 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> This paper considers system theoretic conditions for the solvability of the so-called constrained Lyapunov problem for nonsquare systems. These problems commonly appear in the control systems literature. Both a static output feedback problem and an observer problem are considered. The basis for the work described in this paper is a new canonical form that simplifies the analysis and deals with the equality constraint in a simple way. <s> BIB015 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> Abstract In this paper, a robust stabilization problem for a class of linear time-varying delay systems with disturbances is studied using sliding mode techniques. Both matched and mismatched disturbances, involving time-varying delay, are considered. The disturbances are nonlinear and have nonlinear bounds. A sliding surface is designed and the stability of the corresponding sliding motion is analysed based on the Razumikhin Theorem. Then a static output feedback sliding mode control with time-delay is synthesized to drive the system to the sliding surface in finite time. Simulation results show the effectiveness of the proposed approach. <s> BIB016 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> In this paper, global decentralised stabilisation of a class of interconnected delay systems is considered, where both known and uncertain interconnections involve time delay. Matched and mismatched interconnections are considered separately to reduce the conservatism. A composite sliding surface is designed and the stability of the associated sliding motion which is governed by a time delay interconnected system, is analysed based on the Razumikhin Lyapunov approach. A decentralised static output feedback variable structure control which is dependent on time-delay is synthesized to drive the interconnected system to the sliding surface globally. Simulation results show the effectiveness of the proposed approach. <s> BIB017 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> Higher order sliding mode (HOSM) control design is considered for systems with a known permanent relative degree. In this paper, we introduce the robust Fuller's problem that is a robust generalization of the Fuller's problem, a standard optimal control problem for a chain of integrators with bounded control. By solving the robust Fuller's problem it is possible to obtain feedback laws that are HOSM algorithms of generic order and, in addition, provide optimal finite-time reaching of the sliding manifold. A common difficulty in the use of existing HOSM algorithms is the tuning of design parameters: our methodology proves useful for the tuning of HOSM controller parameters in order to assure desired performances and prevent instabilities. The convergence and stability properties of the proposed family of controllers are theoretically analyzed. Simulation evidence demonstrates their effectiveness. <s> BIB018 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> Decentralized delay-dependent stability and stabilization methods are developed for a class of linear interconnected continuous-time systems. The subsystems are time-delay plants subjected to convex-bounded parametric uncertainties and the interconnections are time-delay couplings. The delay-dependent dynamics are established at the subsystem level through the construction of appropriate Lyapunov-Krasovskii functional. We characterize decentralized linear matrix inequalities (LMIs)-based delay-dependent stability conditions such that every local subsystem of the linear interconnected delay system is robustly asymptotically stable with an γ–level L 2 –gain. A decentralized statefeedback stabilization scheme is designed such that the family of closed-loop feedback subsystems enjoys the delay-dependent asymptotic stability with a prescribed γ–level L 2 gain for each subsystem. The decentralized feedback gains are determined by convex optimization over LMIs. All the developed results are tested on a representative example. <s> BIB019 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> It is shown that the higher the order r of the homogeneous sliding mode is, the less sensitive it is to the presence of unaccounted-for fast stable actuators. With ? being an infinitesimal actuator time constant, the sliding variable magnitude is proved to be proportional to ?r . The system chattering is shown to be not amplified in the presence of actuators. <s> BIB020 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> Global decentralised stabilisation of a class of interconnected time-varying delay systems is considered, where both known and uncertain interconnections involve time delay. Matched and mismatched interconnections are considered separately to reduce the conservatism. A composite sliding surface is designed and the stability of the associated sliding motion, which is governed by a time delayed interconnected system, is analysed based on the Razumikhin-Lyapunov approach. A decentralised static output feedback variable structure control which is dependent on time delay is synthesised to drive the interconnected system to the sliding surface globally. Simulation results show the effectiveness of the proposed approach. <s> BIB021 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <s> Global decentralized discrete sliding mode control of a class of interconnected system using only output information is considered in this paper. Matched and unmatched uncertainties as well as known and unknown interconnections are treated. Bounded time-varying delays are considered within the subsystems and through interconnections. The stability of the reduced-order interconnected systems is analyzed using the Lyapunov-Kraszovskii approach and a novel reachability condition of the composite sliding surfaces is established. The resultant dynamics are proved to be globally asymptotically stable. Simulations show the efficacy of the proposed scheme. <s> BIB022 </s> Decentralised control for complex systems - an invited survey <s> Decentralised variable structure control <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> BIB023
Variable structure control has been recognised as an efficient tool in control design. The main advantages include (see BIB012 BIB008 : • low sensitivity to system parameters and perturbations, eliminating the requirement for exact modelling • the possibility of decoupling the original system into two subsystems with lower dimension • many controllers ensure finite time convergence to the switching surface and can be straightforwardly implemented. Sliding mode control is a particular type of variable structure control. This approach consists of two steps BIB008  BIB004 : 1 the design of a sliding surface such that the system possesses the desired performance when it is restricted to the surface 2 the design of a variable structure control which drives the system trajectory to the sliding surface in finite time and maintains a sliding motion thereafter. Research on decentralised control using variable structure techniques for large-scale interconnected systems can be traced back to the early 1980s  . It should be noted that there are some decentralised variable structure controllers which may not result in sliding motion (see BIB013 BIB023 BIB009 . Static output feedback controllers are proposed in BIB023 and BIB009 to stabilise interconnected systems where the focus is on applying the nonlinear bounds on the uncertainty and interconnections to enhance the robustness. Specifically both the isolated subsystems and interconnections are fully nonlinear in BIB009 and BIB023 . A tracking problem is considered in BIB013 where decentralised state feedback controller are designed for robot manipulators. Broadly, the literature can be separated into state feedback   BIB007 BIB010 BIB002 , static output feedback BIB021 and dynamical/observer-based output feedback  BIB014 . Since all the system state variables are available in the state feedback control design, the requirement of state feedback control for the considered system is usually relaxed and the developed results have higher robustness when compared with that of output feedback control. It is well known that the fundamental question of the existence of a static output feedback controller for a given triple (A, B, C) is still open -even in the single-input single-output case BIB006 although the standard constrained Lyapunov problem has been well discussed in BIB001 and BIB015 . Therefore, most of the decentralised output feedback schemes focus on dealing with various uncertainties and interconnections instead of the control design for the nominal isolated subsystem. Approaches consider under what conditions the interconnected system has the desired performance if the isolated nominal subsystems have the required performance (see BIB014 . A decentralised sliding-mode-based output-feedback control scheme for discrete time systems is available in BIB022 which is built on the work in BIB017 . As for the dynamical/observer-based feedback control, the results developed usually have higher robustness and lower conservatism when compared with static output feedback because for dynamical output feedback, not only the output information can be employed in the design but the dynamical information can also be employed. It is clear that a number of decentralised variable structure strategies have been obtained. The global reachability condition for interconnected systems was first proposed in BIB003 . Later, it was modified as follows in  and the simple proof was given in BIB007 , where s i (x i ) can be considered as a decentralised sliding function which is usually designed based on the centralised sliding surface design approaches given in BIB008 and BIB005 . The reachability condition (8) has been extensively employed in decentralised sliding mode control design to derive the reachability condition but it may not guarantee that the sliding surface is reached in finite time. In order to get a decentralised control strategies, the decentralised (or local) sliding surface s i (x i ) = 0 is designed based on the nominal system of the i th subsystems in nearly all the existing work. Then the composite sliding function is given by ) T which can be considered as the decentralised sliding function. Whether it is possible to design a sliding surface in a general form as to implement a decentralised control scheme, is still open even for a large scale interconnected system with special structure. Moreover, when global results are sought, strong conditions are required on the nominal subsystems to satisfy the reachability condition globally when a static output feedback controller is designed, as presented in BIB017 BIB021 ) (see Assumption 4 in BIB021 . This condition was later employed in BIB022 for a class of discrete time interconnected systems. How to relax this condition is still an open problem. Finally, it should be pointed out that higher order sliding mode control has been well studied (see, e.g., BIB020 BIB011 BIB018 ). In addition, many results concerning decentralised schemes for interconnected systems have been obtained as mentioned above BIB009 BIB019 , even for fully nonlinear interconnected systems BIB016 . However, the results of applying decentralised higher order sliding mode control to large scale interconnected systems are very limited. Specifically, the results of decentralised control for nonlinear time delay interconnected systems using higher order sliding mode techniques are still not available. Study on this area is at its initial stage and thus could be interesting and full of challenge from both viewpoints of theory and application.