Clyde F. Martin

Symmetric linear systems: an application of algebraic systems theory†

Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

Finite escape time for Riccati differential equations

In this paper the finite escape of the Riccati differential equations is studied and it is shown that periodic solutions, if they exist, are ither always finite or always infinite.

A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line

Abstract Let E be an algebraic (or holomorphic) vectorbundle over the Riemann sphere P 1 ( C ). Then Grothendieck proved that E splits into a sum of line bundles E = ⊕ L i and the isomorphism classes of the L i are (up to order) uniquely determined by E . The L i in turn are classified by an integer (their Chern numbers) so that m -dimensional vectorbundles over P 1 C are classified by an m -tuple of integers κ(E) = (κ 1 (E),…,κ m (E)), κ 1 (E)≥⋯≥κ m (E), κ i (E)∈ Z . In this short note we present a completely elementary proof of these facts which, as it turns out, works over any field k .

Lie and Morse theory for periodic orbits of vector fields and matrix Riccati equations, II

In this paper, elementary techniques from linear algebra and elementary properties of the Grassmann manifolds are used to prove the existence of periodic orbits and to study the equilibrium structure of Riccati differential equations.

Symmetric systems with semi-simple structure algebra: The quaternionic case

Linear systems over the quaternions is discussed. The theory of real, semisimple symmetric systems is outlined.

Linear decentralized systems with special structure

Certain physical systems are modelled by linear systems having a high degree of internal symmetry. In this paper it is shown that this internal symmetry can he exploited to yield new system-theoretic invariants and a better understanding of how the underlying structure affects overall system performance.

On families of systems and deformations

Abstract Parameter variations are present in most physical systems. In some cases, such variations can be safely ignored, and one might, for instance, design control loops for some average parameter values. However, in many interesting cases, the variations have to be taken into account in order to design good, or even adequate, control algorithms. Furthermore, the concerns of system reliability demand predictions of possible consequences of large deviations in parameters. Although some of the work in adaptive control is in this spirit, until recently there has.not been any systematic effort towards a theory of systems with parameter variations. We argue here that the concept of families of systems is basic to such an effort. Whereas the necessary tools for the study of individual linear systems with fixed parameters are contained in the theory of differential equations and linear algebra, the techniques of Lie theory, differential geometry and algebraic geometry play an essential role in the study of fam...

Lie and morse theory for periodic orbits of vector fields and matrix riccati equations, I: General lie-theoretic methods

The problem of finding periodic solutions of the matrix Riccati equations of linear control theory is interpreted geometrically as a problem of finding periodic orbits of certain one-parameter transformation groups on Grassmann manifolds. For certain control problems the vector fields which generate these groups can be written as a sum of two commuting vector fields, one a gradient vector field, the other a Killing vector field, i.e., an infinitesimal isometry of a metric on the Grassman manifold. For such vector fields, the methods of Morse theory can be adapted to study the periodic orbits. The topological data that is needed to count periodic orbits, i.e., the Poincare polynomial of certain submanifolds of the Grassmann manifold, can be derived from results proved by A. Borel.

Grassmannian Manifolds, Riccati Equations and Feedback Invariants of Linear Systems

The purpose of these lectures is to present a brief introduction of the role of Grassmannian manifolds in linear control theory. The Riccati equations of linear quadratic optimal control occur naturally as vector fields on the Lagrangian Grassmannian manifolds and exhibit some interesting topological behavior that is discussed in this paper. The feedback structure of linear systems can be deduced through a vector bundle structure on ℙl (ℂ) induced from the “natural bundle” structure on the Grassmannian manifold.

On decentralisation, symmetry and special structure in linear systems

Many systems in nature and engineering (linear dynamical input/output systems in state space form) have a good deal of special structure. They may consist of a collection of identical units connected together in various ways, or there may be large systems consisting of many submits which fall into a small number of types; for example, think of electrical or neural networks. Another example consists of two helicopters connected with a rigid beam, a system which is of considerable importance for practical applications and, as we shall see, which poses many unsolved problems. Other examples arise from the discretization (in space) of partial differential equations [Brockett-Willems, 1974].