Structure assignment problems in linear systems: Algebraic and geometric methods

Abstract

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.

Introduction

Systems and Control provide a paradigm that introduces many open problems of mathematical nature (Kailath, 1980, Rosenbrock, 1970, Wonham, 1979). We distinguish two main approaches in Control Theory, the design methodologies (based on performance criteria and structural characteristics) are mostly of iterative nature and the synthesis methodologies (based on the use of structural characteristics, invariants) linked to well defined mathematical problems. Of course, there exist variants of the two aiming to combine the best features of the two approaches. The Determinantal Assignment Problem (DAP) is a synthesis method and has emerged as the unified abstract problem formulation of pole, zero assignment of linear systems (Karcanias, Giannakopoulos, 1984, Karcanias, Giannakopoulos, 1989, Karcanias, Laios, Giannakopoulos, 1988). DAP unifies the study of (pole, zero) frequency assignment problems of multivariable systems under constant, dynamic centralized, or decentralized control structures. There are two approaches developed for the study of frequency assignment problems which are: (i) the affine space approach; (ii) the projective geometry approach. The first approach was introduced in (Brockett, Byrnes, 1981, Byrnes, 1989, Hermann, Martin, 1977, Martin, Hermann, 1978), and deals with the formulation of the problem in an affine space as an intersection problem of the Grassmannian with a linear space. The DAP approach, as it has been developed in Karcanias and Giannakopoulos (1984); Karcanias et al. (1988); Leventides and Karcanias (1995) is based on the Plücker embedding (Hodge & Pedoe, 1952) of the Grassmannian of the affine space into an appropriate projective space and then deals with finding solutions as the real intersections of a linear space with the Grassmann variety (Hodge & Pedoe, 1952) of the corresponding projective space. The DAP approach relies on exterior algebra (Marcus, 1973) and on the explicit description of the Grassmann variety, in terms of the Quadratic Plücker Relations (QPR) (Hodge & Pedoe, 1952). There are many approaches dealing with specific frequency assignment problems (pole-zero), but they rely on specific system representations and they cannot be easily extended to deal with the whole family of constant, dynamic, decentralised problems. The affine geometry approach deals with generic cases only and it does not provide computations for exact, as well as, approximate problems. The DAP approach has the advantage of introducing new system invariants of rational vector spaces in terms Grassmann vectors and Plücker matrices (Karcanias & Giannakopoulos, 1984) providing a matrix characterisation of decomposability in terms of the Grassmann matrices (Karcanias, Giannakopoulos, 1988, Karcanias, Leventides, 2015) and developing a novel computational framework based on the technique of Global Linearisation (GL) (Leventides & Karcanias, 1995). GL is based on the notion of degenerate feedback (Brockett & Byrnes, 1981) and apart from establishing solvability conditions (Leventides & Karcanias, 1995), also provides a linearisation of the inherently nonlinear equations and leads to the computation of solutions (when such solutions exist). Within the DAP framework a number of solvability conditions have been established (Karcanias, Leventides, 1996, Leventides, Karcanias, 1992, Leventides, Karcanias, 1993, Leventides, Karcanias, 1995) for the exact and generic frequency assignment problem. The GL methodology has in general high sensitivity leading to high gains in the compensation. Techniques such as, homotopy continuation and Newton-type schemes (Leventides, Meintanis, Karcanias, 2014, Leventides, Meintanis, Karcanias, Milonidis, 2014) have been used in order to be able to achieve solutions with much better sensitivity properties.

The DAP framework has been used for the study of constant and dynamic pole assignment, where low complexity solutions have been established (Leventides & Karcanias, 1998), as well as for problems of zero assignment by squaring down (Karcanias, Giannakopoulos, 1989, Leventides, Karcanias, 2009). Degenerate feedback gains (Karcanias, Meintanis, & Leventides, 2016) are defined for both constant and dynamic assignment problems. Parametrising the family of degenerate feedbacks gives extra degrees of freedom in computing appropriate controllers that linearise asymptotically DAP and enabling the selection of solutions with reduced sensitivity This parametrisation methodology plays a key role in selecting feasible structures for decentralized control problems. The selection of a decentralisation scheme has been handled mostly using conditions derived from the nature and spatial arrangement of sub-process units (Siljak, 1991). DAP can provide an algebraic framework for selection of the desirable decentralisation (Karcanias, Meintanis, & Leventides, 2016) aiming at developing schemes that allow the satisfaction of generic solvability conditions and shaping the parametric invariants linked to solvability of decentralised control problems. DAP framework provides simple tests for avoiding fixed modes by exploiting the relationship of algebraic invariants (Plücker matrices) to decentralised Markov parameters (Leventides & Karcanias, 1998). The overall philosophy is to devise methods for design that facilitate the solvability of decentralised control problems. Amongst the problems considered are: (i) Define the desirable cardinality of input, output structures to permit satisfaction of generic solvability conditions, (ii) Design the structure of input, output maps (matrices B, C) to eliminate the fixed modes and guarantee full rank properties to the decentralised Plücker matrices (Leventides & Karcanias, 2006).

A significant advantage of the DAP framework is that it introduces a new approach for the computation of exact and approximate solutions of DAP. This is based on an alternative, linear algebra type, criterion for decomposability of multivectors to that defined by the QPRs, in terms of the rank properties of structured matrices, referred to as Grassmann matrices (Karcanias, Giannakopoulos, 1988, Karcanias, Leventides, 2015). The development of the new computational framework requires the study of the properties of Grassmann matrices, which are further developed by using the Hodge duality (Hodge & Pedoe, 1952) leading to the definition of the Hodge-Grassmann matrix (Karcanias & Leventides, 2015). Computing solutions (exact, or approximate) to DAP requires the investigation of distance problems, such as: (i) distance of a point from the Grassmann variety; (ii) distance of a linear variety from the Grassmann variety; (iii) parametrisation of families of linear varieties with a given distance from the Grassmann variety; (iv) relating the latter distance problems with properties of the stability domain. The distance problems extend the exact intersection problem between the Grassmann and the linear space varieties and are related to classical problems, such as spectral analysis of tensors (Lathauwer, Moor, & Vandewalle, 2000), homotopy and constrained optimization methods (Absil, Mahony, & Sepulchre, 2008), theory of algebraic invariants etc.

This paper provides a review of the concepts, methodology and results of the DAP framework, as well as relevant results that complement those of the current approach. The review is then completed by providing a number of challenges for the DAP approach which form a research agenda for future activities.

The paper is structured as follows: Section 2, deals with the frequency assignment problems in Control Theory, whereas Section 3 presents the abstract DAP framework which is reduced to the study of polynomial combinants and the problem of decomposability of multivectors. The solvability is equivalent to finding real intersections between a linear variety and the Grassmann variety of a projective space. DAP introduces new invariants of rational vector spaces defined by the Grassmann vectors and the Plücker matrices. Section 4, deals with the decomposability of multivectors where first the Quadratic Plücker Relations and then a new test for decomposability provided by the rank properties of the Grassmann matrix is presented. Section 5, examines the solvability of DAP under the genericity assumption. We focus on determining real solutions and a number of results are reviewed derived from the DAP framework and other related approaches. Section 6, deals with the GL methodology by examining the properties of the pole placement map, the notion of degenerate compensators with their parametrisation and the sensitivity of such solutions. Section 7, extends the GL methodology to decentralized control problems. The parametrisation of the family of decentralized degenerate schemes is linked to the selection of appropriate asymptotic linearising decentralized schemes capable of assigning the closed loop poles. Section 8, deals with the computation of exact and approximate solutions of DAP, in terms of the properties of structured matrices, the Grassmann and Hodge–Grassmann matrices. Finally, Section 9 provides a list of open questions related to the DAP framework which form a future research agenda.

Notation: Throughout the paper the following notation is adopted: If $\mathbb{F}$ is a field, or ring then ${\mathbb{F}}^{m\times n}$ denotes the set of m × n matrices over $\mathbb{F}$. $\mathbb{R}\left[s\right]$ is the ring of polynomials and $\mathbb{R}\left(s\right)$ is the field of rational functions over $\mathbb{R}$ respectively. If H is a map, then $\mathcal{R}\left(H\right),$ ${\mathcal{N}}_r\left(H\right),$ ${\mathcal{N}}_l\left(H\right)$ denote the range, right, left null-spaces respectively. Q_{k, n} denotes the set of lexicographically ordered, strictly increasing sequences of k integers from the set $\overline{n}=\{1,2,\dots ,n\}$. If $\mathcal{V}$ is a vector space and $\{{\underline{v}}_{i_1},\dots ,{\underline{v}}_{i_k}\}$ are vectors of $\mathcal{V}$ then ${\underline{v}}_{i_1}\wedge \dots \wedge {\underline{v}}_{i_k}={\underline{v}}_{\omega }\wedge ,$ $\omega =(i_1,\dots ,i_k)$ denotes their exterior product and ${\wedge }^r\mathcal{V}$ the $r-$th exterior power of $\mathcal{V}$ (Marcus, 1973). If $H\in {\mathbb{F}}^{m\times n}$ and r ≤ min {m, n} then C_r(H) denotes the $r-$th compound matrix of H (Marcus & Minc, 1964). In most of the following, we assume that $\mathbb{F}=\mathbb{R}$ , or $\mathbb{C}$.