Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> The decentralized robust servomechanism problem [1] for a linear time-invariant system described by: Tx = Ax + Bu + Ew y = Cx + Du + Fw e = y - yref where E may be singular, is considered in this paper, where u, y, e, w denotes the input, output, error and disturbance respectively for the system. In particular, necessary and sufficient conditions to solve the robust decentralized control problem are obtained and a constructive controller synthesis procedure is given for this class of systems. Some simple examples are given to illustrate the result obtained. <s> BIB001 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> Abstract In the first part of this paper, some necessary and sufficient conditions for r polynomials to have to common zeros are derived, one of which is stated by a controllability matrix and is very useful to control theory. In the second part, a necessary and sufficient condition independent of eigenvalues for descriptor systems to have no generalized decentralized fixed modes is developed, which may be thought of as an answer to an issue proposed by Anderson and Clements [1]. <s> BIB002 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> Decentralized stabilization of singular and time-delay large-scale control systems with impulsive solutions which has not been studied up to now is introduced and studied. The control problem considered is the construction of a decentralized controller which ensures that a certain singular, impulsive, and time-delay closed-loop large-scale system is exponentially stable. By means of lumped Picard and Gauss-Seidel iteration methods which enable the authors to avoid the difficulties of constructing Lyapunov functions, explicit algebraic criteria of stabilization for such systems are established. An example is given to illustrate the effectiveness of results obtained. > <s> BIB003 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> The internal properness and stability of singular decentralized control systems are discussed, and a sufficient and necessary condition that singular systems can be properly stabilized by normal decentralized compensators is presented, i.e. there do not exist any unstable finite fixed modes or impulsive fixed modes in the singular system. <s> BIB004 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> A fuzzy state model for nonlinear interconnected descriptor systems is presented by applying a T-S fuzzy descriptor model. Based on the model, a robust fuzzy decentralized controller design method is given. Then, a sufficient condition is obtained in terms of a LMI. In addition, a less conservative LMI design approach is employed to find a stable fuzzy decentralized feedback controller. The effectiveness and the design procedures of the controller presented are illustrated by an example. <s> BIB005 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> Some algebraic properties of linear singular systems at infinity under static decentralized output feedback are studied in this note. New concepts of algebraic multiplicity and geometric multiplicity of the impulsive decentralized fixed modes, and the impulsive decentralized cycle index of the singular system are defined. These concepts indicate how much a singular system can be made close to being impulse-free by decentralized output feedback, and the results are shown to be generic. The geometric multiplicity and the impulsive decentralized cycle index are determined in terms of the system matrices explicitly. The number of impulsive modes that can be eliminated is given in terms of these indexes. The impulsive decentralized cycle index is shown to characterize generic properties on controllability and observability of the closed-loop system through an individual channel (or an external channel). Illustrative examples are provided. <s> BIB006 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> In this paper, a method for design of linear decentralized robust controllers for a class of uncertain large-scale systems in general form is presented. For a given large-scale system, an equivalent descriptor system in input–output decentralized form is defined. Using this representation, closed-loop diagonal dominance sufficient conditions are derived. It is shown that by appropriately minimizing the weighted sensitivity function of each isolated subsystem, these conditions are achieved. Solving the appropriately defined H∞ local problem for each isolated uncertain subsystem, the interactions between the subsystems are reduced, and the overall stability and robust performance are achieved in spite of uncertainties. The designs are illustrated by a practical example. <s> BIB007 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> Regularizability conditions for linear time-invariant descriptor systems under decentralized output feedback are presented in this paper. The regularizability is characterized by rank conditions that involve the original system matrices only. New rank conditions for regularity of the system are also derived. <s> BIB008 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> Abstract This paper considers a decentralized H2 control problem for multi-channel linear time-invariant (LTI) descriptor systems. Our interest is to design a low order dynamic output feedback controller. The control problem is reduced to a feasibility problem of a bilinear matrix inequality (BMI) with respect to variables of a coefficient matrix defining the controller, a Lyapunov matrix and a matrix related to the descriptor matrix. Under a matching condition between the descriptor matrix and the measurement output matrix (or the control input matrix), we propose to set the Lyapunov matrix in the BMI as block diagonal appropriately so that the BMI is reduced to LMIs. <s> BIB009 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> This paper addresses the problem of robust stabilization for uncertain discrete-time singular large-scale systems with parameter uncertainties. The system under consideration is not necessarily regular. The parameter uncertainties are assumed to be time invariant, but norm-bounded. The purpose of the robust stabilization problem is to design state feedback controllers such that, for all admissible uncertainties, the closed-loop system is regular, causal and stable. In terms of strict LMIs, sufficient conditions for the solvability of the problem is presented, and the parameterization of desired state feedback controllers is also given. A numerical example is given to demonstrate the applicability of the proposed design approach. <s> BIB010 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> This paper considers a robust decentralized H∞ control problem for interconnected descriptor systems. The uncertainties are assumed to be time-invariant, norm-bounded, and existing in both the system and control input matrices. Our interest is focused on dynamic output feedback. A sufficient condition for an uncertain interconnected descriptor system to be robustly stabilizable H∞ control with a specified disturbance attenuation level is derived in terms of a nonlinear matrix inequality (NMI). A two-stage homotopy method is employed to solve the NMI iteratively. First, a decentralized controller for the nominal descriptor system is computed by imposing block-diagonal constraints on the coefficient matrices of the controller gradually. Then, the decentralized controller is gradually modified from the nominal descriptor system (without uncertainties) to the original system with uncertainties. On each stage, groups of variables are fixed alternately at the iterations to reduce the NMI to linear matrix inequalities (LMIs). An example is given to show the usefulness of this method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society <s> BIB011 </s> Decentralised control for complex systems - an invited survey <s> Decentralised control of singular systems <s> Purpose – The purpose of this paper is to propose a decentralized observer for large‐scale singular systems.Design/methodology/approach – In this paper, the authors investigate the problem of observers' design for large‐scale singular systems. The proposed decentralized observer is based on a new parameterization of the generalized Sylvester equation solution. The considered system is partitioned into small‐sizes interconnected subsystems with unknown interconnections.Findings – A decentralized observer based on new parameterization of generalized Sylvester equation. The performance of the proposed approach is illustrated by a numerical example.Originality/value – The proposed approach unites the full‐order, the reduced‐order and the minimal order observer design for large‐scale system. The conditions of the existence of this observer are given in the linear matrix inequalities (LMI) form. <s> BIB012
A linear singular control system with N channels can be described by where x(t) ∈ R n denotes the system state; u i (t) ∈ R mi and y i (t) ∈ R li are the input and output of the i th channel, respectively, for i = 1, 2, · · · , N . The matrices E, A, B i and C i are constant with appropriate dimensions. Specifically, the matrix E is singular and 0 < rank(E) < n. It is assumed that the system (11) is regular, i.e., det(sE − A) ̸ = 0. Then, the existence and uniqueness of the solution to system (11) is guaranteed and this solution may take the form of an exponent solution, impulsive solution and/or static solution. System (11) has been studied extensively where most of the research work focuses on decentralised output feedback control design, which is to design a control where K i for i = 1, 2, . . . , N are design parameters, such that the corresponding closed-loop system has the desired performance BIB003  . Compensator-based decentralised control was studied for singular interconnected systems in BIB004 . Such linear singular control systems (11) were first represented and studied by BIB001 , where the concepts of generalised decentralised finite and impulsive fixed modes were proposed to derive conditions under which the systems can be stabilised by decentralised compensator-based feedback control. The decentralised robust servomechanism problem was solved simultaneously. Generalised decentralised finite fixed modes were further characterised in BIB002 , and the concepts of decentralised impulsive controllability and observability were presented and studied in  where it was shown that if the considered system is complete, then the system does not have decentralised fixed modes if and only if it is decentralised controllable and observable. Based on the work in BIB002 and  , the decentralised quadratic stabilisation problem for a class of interconnected singular systems with uncertainty is investigated in  in which it was shown that quadratic stability requires that the considered system has neither decentralised unstable finite fixed modes nor decentralised impulsive fixed modes. The internal properness and stability of system (11) were discussed, and a set of necessary and sufficient conditions was presented in BIB004 such that the system can be stabilised by decentralised compensator-based feedback control. Decentralised stabilisation of singular and time-delay large-scale systems with impulsive solutions was discussed in BIB003 where the designed controllers are independent of time delay. Regularisation and impulse eliminating problems are considered using decentralised controllers in BIB008 and BIB006 respectively. Concerning decentralised H 2 and H ∞ control for singular systems, some recent work is available in BIB009 and BIB011 and the references therein. As for the discrete time case with parametric uncertainties, robust stabilising controllers were designed in BIB010 . Recently, a decentralised observer for the system (11) was considered in BIB012 using LMI techniques, and a fuzzy decentralised control scheme was proposed for a class of nonlinear singular interconnected systems in BIB005 . It has been found in BIB007 that a singular system representation can be used as a suitable tool to solve some control problems such as H ∞ control problem. Some earlier work for singular systems (11) was summarised in  .