Selection of Decentralised Control Structures: Structural Methodologies and Diagnostics

The paper aims at formulating an integrated approach for the selection of decentralized control structures using a number of structural criteria aiming at facilitating the design of decentralised control schemes. This requires the selection of decentralisation structure that will allow the generic solvability of a variety of decentralised control problems, such as pole assignment by decentralised output feedback. The approach is based on the use of necessary and sufficient conditions for generic solvability and exact solvability of decentralised control problems. The generic solvability conditions lead to characterisations of inputs and outputs channel partitions. The exact solvability conditions use criteria on avoiding the presence of fixed modes, as well as necessary conditions for pole assignment, expressed in terms of properties of Plϋcker invariants and Markov type matrices. The structural approach provides a classification of desirable input and output partitions based on structural criteria and it is embedded in an overall framework that may involve aspects related to large scale design

Zero assignment of matrix pencils by additive structured transformations

Matrix pencil models are natural descriptions of linear networks and systems. Changing the values of elements of networks, that is redesigning them, implies changes in the zero structure of the associated pencil and this is achieved by structured additive transformations. The paper examines the problem of zero assignment of regular matrix pencils by a special type of structured additive transformations. For a certain family of network redesign problems the additive perturbations may be described as diagonal perturbations and such modifications are considered here. This problem has certain common features with the pole assignment of linear systemsby structured static compensators and thus the new powerful methodology of global linearization [J. Leventides, N. Karcanias, Sufficient conditions for arbitrary Pole assignment by constant decentralised output feedback, Mathematics of Control for Signals and Systems 8 (1995) 222–240; J. Leventides, N. Karcanias, Global asymptotic linearisation of the pole placement map: A closed form solution for the constant output feedback problem, Automatica 31 (1995) 1303–1309] can be used. For regular pencils with infinite zeros, families of structured degenerate additive transformations are defined and parameterized and this lead to the derivation of conditions for zero structure assignment, as well as methodology for computing such solutions. The case of regular pencils with no infinite zeros is also considered and conditions of zero assignment are developed. The results here provide the means for studying problems of linear network redesign by modification of the non-dynamic elements

Zero Assignment Problem in RLC Networks

The paper deals with the problem of zero assignment in RLC networks by selection of appropriate values for the non- dynamical elements, the resistors. For a certain family of network redesign problems by the additive perturbations may be described as diagonal perturbations and such modifications are considered here. This problem belongs to the family of DAP problems (Determinantal Assignment Problem) and has common features with the pole assignment problem by decentralized output feedback and the zero assignment problem via structured additive perturbations. We demonstrate that the sufficient condition for generic zero assignment by selecting the resistors holds true. This condition is related to the rank of the differential of the related map and holds true generically when the degrees of freedom of the matrix of resistors exceeds the number of frequencies to be assigned (n > p + q). Using this result, we show through a generic example that the sufficient condition for the general zero assignment problems in RLC networks is satisfied and thus, zero assignment can be achieved via resistor determination

System Degeneracy and the Output Feedback Problem: Parametrisation of the Family of Degenerate Compensators

The paper provides a new characterisation of constant and dynamic degenerate compensators for proper multivariable systems. The motivation stems from the very important property that degenerate feedback gains may be used for the linearisation of the pole assignment map and enable frequency assignment. The objective is the characterisation and parametrisation of all feedback gains that may allow the asymptotic linearisation of the pole placement map. Such a parametrisation introduces new degrees of freedom for the linearisation of the related frequency assignment map and plays an important role to the solvability of the output feedback pole assignment problem. The paper reviews the Global Asymptotic Linearisation method associated with the core versions of determinantal pole assignment problems and defines the conditions which characterises degenerate solutions of different types. Using the theory of ordered minimal bases, we provide a parametrisation of special families of degenerate compensators according to their degree. Finally, the special properties of degenerate solutions that allow frequency assignment are considered

Selection of Decentralised Schemes and Parametrisation of the Decentralised Degenerate Compensators

The design of decentralised control schemes has two major aspects. The selection of the decentralised structure and then the design of the decentralised controller that has a given structure and addresses certain design requirements. This paper deals with the parametrisation and selection of the decentralized structure such that problems such as the decentralised pole assignment may have solutions. We use the approach of global linearisation for the asymptotic linearisation of the pole assignment map around a degenerate compensator. Thus, we examine in depth the case of degenerate compensators and investigate the conditions under which certain degenerate structures exist. This leads to a parametrisation of decentralised structures based on the structural properties of the system

Approximate solutions of the determinantal assignment problem and distance problems

The paper introduces the formulation of an exact algebrogeometric problem, the study of the Determinantal Assignment Problem (DAP) in the set up of design, where approximate solutions of the algebraic problem are sought. Integral part of the solution of the Approximate DAP is the computation of distance of a multivector from the Grassmann variety of a projective space. We examine the special case of the calculation of the minimum distance of a multivector in ∧2(ℝ5) from the Grassmann variety G 2(ℝ5). This problem is closely related to the problem of decomposing the multivector and finding its best decomposable approximation. We establish the existence of the best decomposition in a closed form and link the problem of distance to the decomposition of multivectors. The uniqueness of this decomposition is then examined and several new alternative decompositions are presented that solve our minimization problem based on the structure of the problem

Reduced sensitivity solutions to global linearisation of the pole assignment map

The problem of pole assignment, by static output feedback controllers has been tackled as far as solvability conditions and the computation of solutions when they exist by a powerful method referred to as global linearisation. This is based on asymptotic linearisation (around a degenerate point) of the pole placement map. The essence of the present approach is to reduce the multilinear nature of the problem to the solution of a linear set of equations. The solution is given in closed form in terms of a one-parameter family of static feedback compensators, for which the closed-loop poles approach the required ones as ε → 0. The use of degenerate compensators makes the method numerically sensitive. This paper develops further the global linearisation framework by developing numerical techniques which make the method less sensitive to the use of degenerate solutions as the basis of the methodology. The proposed new computational framework for finding output feedback controllers improves considerably the sensitivity properties by using a predictor-corrector numerical method based on homotopy continuation. The modified method guarantees the maximum distance from the degenerate point. The current numerical method developed for the constant output feedback extends also to the case of dynamic output feedback

Solution of the determinantal assignment problem using the Grassmann matrices

The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation (Formula presented.) where (Formula presented.) is an n −dimensional vector space over (Formula presented.) which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of (Formula presented.), and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector (Formula presented.) are given in terms of the rank properties of the Grassmann matrix, (Formula presented.) of the vector (Formula presented.), which is constructed by the coordinates of (Formula presented.). It is shown that the exterior equation is solvable ((Formula presented.) is decomposable), if and only if (Formula presented.) where (Formula presented.); the solution space for a decomposable (Formula presented.), is the space (Formula presented.). This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge–Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge–Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist

Matrix pencil representation of structural transformations of passive electrical networks

The paper examines the problem of systems redesign within the context of passive electrical networks by considering the problem of multi-parameter and topology changes, and their representation. This representation may be used to investigate the impact of such changes on properties such as characteristic frequencies. The general problem area is the modelling of systems, whose structure is not fixed but evolves during the system life-cycle. The specific problem we are addressing is the study of effect of changing the topology of an electrical network that is changing individual elements of the network into elements of different type and value, augmenting / or eliminating parts of the network and developing a framework that allows the study of the effect of such transformations on the natural frequencies. This problem is a special case of the more general network redesign problem. We use the Impedance-Admittance models and we establish a representation of the different types of transformations on such models. The representation of the structural transformations is given in terms of the companion pencil that preserves the natural topologies of the RLC network

Implementation of the Basel Accords I and II in Greek Banking System: The Application of Standardized Approach

This paper focuses on the analysis of the main implications of Basel I and Basel II, based on risk sensitiveness due to credit risk, in Greek Banking System and assesses their effect per portfolio and per Bank in order to evaluate capital charges and to measure credit risk exposure