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

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

Approximate zero polynomials of polynomial matrices and linear systems

This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials". The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of "approximate decoupling polynomials"

Approximate greatest common divisor of many polynomials and pseudo-spectrum

The paper is concerned with establishing the links between the approximate GCD of a set of polynomials and the notion of the pseudo-spectrum defined on a set of polynomials. By examining the pseudo-spectrum of the structured matrix we will derive estimates of the area of the approximate roots of the initial polynomial set. We will relate the strength of the GCD to the weighted strength of the pseudo-spectra and we investigate under which conditions the roots of the approximate GCDs are a subset of the pseudo-spectra

Nearest common root of a set of polynomials: A structured singular value approach

The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (μ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an “approximate GCD” of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the “approximate GCD” of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations

Parameterisation of degenerate solutions of the determinantal assignment problem

The paper is concerned with defining and parametrising the families of all degenerate compensators (feedback, squaring down etc) emerging in a variety of linear control problems. Such compensators indicate the boundaries of the control design, but they also provide the means for linearising the non-linear nature of the Determinantal Assignment Problems, which provide the unifying description for all frequency assignment problems (pole, zero) under static and dynamic compensation schemes. The conditions provide the means for the selection of appropriate degenerate solutions that allow frequency assignability in the corresponding frequencies

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

The Euclidean Division as an Iterative ERES-based Process

Considering the Euclidean division of two real polynomials, we present an iterative process based on the ERES method to compute the remainder of the division and we represent it using a simple matrix form

A Symbolic-Numeric Software Package for the Computation of the GCD of Several Polynomials

This survey is intended to present a package of algorithms for the computation of exact or approximate GCDs of sets of several polynomials and the evaluation of the quality of the produced solutions. These algorithms are designed to operate in symbolic-numeric computational environments. The key of their effectiveness is the appropriate selection of the right type of operations (symbolic or numeric) for the individual parts of the algorithms. Symbolic processing is used to improve on the conditioning of the input data and handle an ill-conditioned sub-problem and numeric tools are used in accelerating certain parts of an algorithm. A sort description of the basic algorithms of the package is presented by using the symbolic-numeric programming code of Maple

Properties and Classification of Generalized Resultants and Polynomial Combinants

Polynomial combinants define the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the frequency assignment problems in Linear Systems. The theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Dynamic combinants are linked to the theory of “Generalised Resultants”, which provide the matrix representation of polynomial combinants. We consider coprime set polynomials for which assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. The complete parameterization of combinants and coresponding Generalised Resultants is prerequisite to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved