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

A quasi-Newton optimal method for the global linearisation of the output feedback pole assignment

The paper deals with the problem of output feedback pole assignment by static and dynamic compensators using a powerful method referred to as global linearisation which has addressed both solvability conditions and computation of solutions. The method is based on the asymptotic linearisation of the pole assignment map around a degenerate point and is aiming to reduce the multilinear nature of the problem to the solution of a linear set of equations by using algebro-geometric notions and tools. This novel framework is used as the basis to develop numerical techniques which make the method less sensitive to the use of degenerate solutions. The proposed new computational scheme utilizes a quasi-Newton method modified accordingly so it can be used for optimization goals while achieving (exact or approximate) pole placement. In the present paper the optimisation goal is to maximise the angle between a solution and the degenerate compensator so that less sensitive solutions are achieved

A Grassmann Matrix Approach for the Computation of Degenerate Solutions for Output Feedback Laws

The paper is concerned with the improvement of the overall sensitivity properties of a method to design feedback laws for multivariable linear systems which can be applied to the whole family of determinantal type frequency assignment problems, expressed by a unified description, the so-called Determinantal Assignment Problem (DAP). By using the exterior algebra/algebraic geometry framework, DAP is reduced to a linear problem (zero assignment of polynomial combinants) and a standard problem of multilinear algebra (decomposability of multivectors) which is characterized by the set of Quadratic Plücker Relations (QPR) that define the Grassmann variety of P. This design method is based on the notion of degenerate compensator, which are the solutions that indicate the boundaries of the control design and they provide the means for linearising asymptotically the nonlinear nature of the problems and hence are used as the starting points to generate linearized feedback laws. A new algorithmic approach is introduced for the computation and the selection of degenerate solutions (decomposable vectors) which allows the computation of static and dynamic feedback laws with reduced sensitivity (and hence more robust solutions). This approach is based on alternative, linear algebra type criterion for decomposability of multivectors to that defined by the QPRs, in terms of the properties of structured matrices, referred to as Grassmann Matrices. The overall problem is transformed to a nonlinear maximization problem where the objective function is expressed via the Grassmann Matrices and the first order conditions for optimality are reduced to a nonlinear eigenvalue-eigenvector problem. Hence, an iterative method similar to the power method for finding the largest modulus eigenvalue and the corresponding eigenvector is proposed as a solution for the above problem

Structure assignment problems in linear systems: Algebraic and geometric methods

The Determinantal Assignment Problem (DAP) is a family of synthesis methods that has emerged as the abstract formulation of pole, zero assignment of linear systems. This unifies the study of frequency assignment problems of multivariable systems under constant, dynamic centralized, or decentralized control structure. The DAP approach is relying on exterior algebra and introduces new system invariants of rational vector spaces, the Grassmann vectors and Plücker matrices. The approach can handle both generic and non-generic cases, provides solvability conditions, enables the structuring of decentralisation schemes using structural indicators and leads to a novel computational framework based on the technique of Global Linearisation. DAP introduces a new approach for the computation of exact solutions, as well as approximate solutions, when exact solutions do not exist using new results for the solution of exterior equations. The paper provides a review of the tools, concepts and results of the DAP framework and a research agenda based on open problems

Multi-parameter structural transformations of passive electrical networks and natural frequency assignment

The paper examines the problem of systems redesign within the context of passive electrical networks by considering the problem of multi-parameter changes, their representation and impact 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 lifecycle. 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. For the case of network cardinality preserving transformations, we formulate the natural frequencies assignment problem as a problem of zero assignment of matrix pencils by additive structured transformations and this allows the deployment of the Determinantal Assignment Problem framework for the study of assignment and determination of fixed natural frequencies

System Properties of Implicit Passive Electrical Networks Descriptions

Redesigning systems by changing elements, topology, organization, augmenting the system by the addition of subsystems, or removing parts, is a major challenge for systems and control theory. A special case is the redesign of passive electric networks which aims to change the natural dynamics of the network (natural frequencies) by the above operations leading to a modification of the network. This requires changing the system to achieve the desirable natural frequencies and involves the selection of alternative values for dynamic elements and non-dynamic elements within a fixed interconnection topology and/or alteration of the interconnection topology and possible evolution of the network (increase of elements, branches). The use of state-space or transfer function models does not provide a suitable framework for the study of this problem, since every time such changes are introduced, a new state space or transfer function model has to be recalculated. The use of impedance and admittance modeling, provides a suitable framework for the study of network properties under the process of re-engineering transformations. This paper deals with the fundamental system properties of the impedance-admittance network description which provide the appropriate framework for network re-engineering. We identify the natural topologies expressing the structured transformations linked to the impedance-graph, admittance graph-topology of the network and examine issues such as network regularity, number of finite frequencies and provide characterization of them in terms of the basic network matrices. The implicit network representation introduced provides a natural framework for expressing the different types of re-engineering transformations which can be used for the study of the natural frequencies assignment

The approximate Determinantal Assignment Problem

The Determinantal Assignment Problem (DAP) has been introduced as the unifying description of all frequency assignment problems in linear systems and it is studied in a projective space setting. This is a multi-linear nature problem and its solution is equivalent to finding real intersections between a linear space, associated with the polynomials to be assigned, and the Grassmann variety of the projective space. This paper introduces a new relaxed version of the problem where the computation of the approximate solution, referred to as the approximate DAP, is reduced to a distance problem between a point in the projective space from the Grassmann variety Gm(Rn). The cases G2(Rn) and its Hodge-dual Gn−2(Rn) are examined and a closed form solution to the distance problem is given based on the skew-symmetric matrix description of multivectors via the gap metric. A new algorithm for the calculation of the approximate solution is given and stability radius results are used to investigate the acceptability of the resulting perturbed solutions

Algebrogeometric and topological methods in control theory

The aim of this thesis is to provide a unifying framework and tools for the study of a number of Control Theory problems of the determinantal type. These problems are known as Frequency Assignment Problems (FRT) and they include the constant, dynamic, pole, zero assignment by centralised as well as decentralised output feedback and the zero assignment problems via squaring down. It has been shown [Kar.1],[Gia.2] that all such problems may be formulated under the unifying framework of the Determinantal Assignment Problem (DAP), and it can be studied using tools from exterior algebra and algebraic geometry. The main objective of this thesis is to develop further the DAP framework, unify it with other algebrogeometric approaches and develop issues related to computation and parametrisation of solutions when such solutions exist. The natural setup for the study of solutions of the DAP framework has been the intersection theory of projective varieties. This has been extended by developing the topological properties of the pole, zero placement maps and introducing an equivalent formulation for real intersection based on cohomology theory. The properties of this map, with respect to standard system invariants are also established. This approach allows the derivation of new conditions for constant pole, zero assignment with centralised and decentralised controllers, using conditions based on the height of an appropriate cohomology class. Affine algebraic geometry methods are also used for the derivation of partial results for the dynamic case corresponding to PI and OBD controllers. An entirely new approach for the study of solvability of DAP, as well as computation of solutions is introduced in terms of the notion of global linearisation of the corresponding pole, zero assignment map around a degenerate point. This is based on the special “blow up” property of the pole placement map at degenerate feedbacks and permits the reduction of the overall DAP to a globally linear problem, the solvability of which is defined by the properties of a new local invariant, the “blow up” matrix

Partially Fixed Structure Determinantal Assignment Problems

In this article, we deal with the study of the determinantal assignment problem (DAP) when the parameters of the compensator are not entirely free, but some of them are fixed. The problem is reduced to a restricted form of an exterior algebra problem (decomposability of multivectors), which is referred to as partial decomposability problem. We study this problem and in case that this problem has no solution, we examine the problem of approximate partial decomposability. We treat the problem of exact or partial decomposability into a vector and a multivector of lower dimension. If this procedure is repeated then this results in an approximation of the initial multivector into a decomposable vector. The approximation of a vector by an optimal decomposable multivector is a nonlinear procedure and has been solved completely using the power method. The method developed in this article, although it produces a suboptimal solution, can be used alternatively for the solution of DAP or the approximate DAP, as a shorter and easier approach, because it is based on known tools as the singular value decomposition. We apply these results to treat the restricted approximate decomposability problem, which leads to approximate solutions to the pole placement and zero assignment problems