State Space Isomorphism Theorem Research Articles

The State Space Isomorphism Theorem for Discrete-Time Infinite-Dimensional Systems

It is well-known that the state space isomorphism theorem fails in infinite-dimensional Hilbert spaces: there exist minimal discrete-time systems (with Hilbert space state spaces) which have the same impulse response, but which are not isomorphic. We consider discrete-time systems on locally convex topological vector spaces which are Hausdorff and barrelled and show that in this setting the state space isomorphism theorem does hold. In contrast to earlier work on topological vector spaces, we consider a definition of minimality based on dilations and show how this necessitates certain definitions of controllability and observability.

On the state of behaviors

The theme of the present paper is the study of the concept of state and the corresponding state maps in the context of Willems’ behavioral theory. We concentrate on Markovian system and their representation in terms of first order difference or differential systems. We follow by a full analysis of the special case of state systems, the embedding of a linear system in a state system via the use of state maps arriving at state representations or, equivalently, to a realization theory for behaviors. Minimality is defined and characterized and a state space isomorphism theorem is established. Realization procedures based on the shift realization are developed as well as a rigorous analysis of the construction of state maps. The paper owes much to Rapisarda and Willems [P. Rapisarda, J.C. Willems, State maps for linear systems, SIAM J. Contr. Optim. 35 (1997) 1053–1091].

Structural identifiability of non-linear systems using linear/non-linear splitting

In this paper, the local state space isomorphism theorem for non-linear systems is used to analyse structural identifiability. In particular it is shown that, under certain restrictions, it is possible to perform a linear/non-linear splitting of the analysis. The relatively straightforward linear analysis then restricts the class of local diffeomorphic transformations as given by the non-linear state space isomorphism theorem. This, in turn, leads to possible simplifications to the subsequent non-linear analysis by providing an efficient means for calculating the local state diffeomorphism.

Impulsive-smooth behavior in multimode systemss part II: Minimality and equivalence

This is the second part of a two-part study of linear multimode systems. In the first part, it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant, and both first-order and polynomial representations were introduced that satisfy this requirement. Here we determine the conditions under which first-order representations are minimal. We also show how two minimal representations of the same behavior are related; this leads in particular to an appropriate state-space isomorphism theorem. The minimality conditions are given a dynamic interpretation.

A state space isomorphism theorem for minimal dynamical systems

Consider a discrete time dynamical systemxk+1=f(x k ) on a compact metric spaceM, wheref:M→M is a continuous map. Leth:M→B k be a continuous output function. Suppose that all of the positive orbits off are dense and that the system is observable. We prove that any output trajectory of the system determinesf andh andM up to a homeomorphism. IfM is a compact Abelian topological group andf is an ergodic translation, then any output trajectory determines the system up to a translation and a group isomorphism of the group.

Operator Measures, Selfadjoint Operators and Dynamical Systems

The paper studies multiplication operators in $L^2$-spaces of matrix measures as models for self -adjoint operators of finite multiplicity. The module theoretic aspects are emphasized and an analysis of intertwining maps, that is, module homomorphisms relative to the algebra of multiplication operators by bounded Borel functions, is given. Finally the machinery is applied to the study of dynamical systems withself-adjoint generators. Controllability aspects are studied and a version of the state space isomorphism theorem is derived.

State space theory for resolution space operators

The notion of state has played a fundamental role in modern systems theory both from a theoretical point of view and for the solution of problems in filtering, estimation, and optimal control. The first study of state for abstract inputoutput operators on Resolution Space was made by Saeks [7]. There most of the basic ideas for the construction of a state representation for Hilbert space operators were introduced. However, despite the fact that a Hilbert space structure was used, most of the notions were algebraic and had no topological meaning. Also, the relationship between the state space and the space of state trajectories was not clear. These problems were noticed by two authors, Schnure [9] and Steinberger [IO]. Both attempted to rigorize the theory of Saeks by introducing topological content into Saeks’ ideas. The work presented here, while owing a debt to both these authors, leans much more heavily on that of Schnurc. One basic difference between the notions of state of Schnure on the one hand and of Saeks and Steinberger on the other hand is that for time-varying systems, for Schnure the state space varies with time while for Saeks and Steinberger it is fixed. We adopt the approach of Schnure, This allows us to construct the space of state trajectories using the notion of a direct integral of Hilbert spaces due to von Neumann. As is expected, the space of state trajectories for an infinite time system will turn out to be an extended space [13, I l] of some Hilbert space. The Hilbert space will be precisely the direct integral integral of the state spaces. A fairly complete state space theory is presented here including a state space isomorphism theorem and a realization theorem and the existence of a family of transition operators. In the last part of the paper we consider the time-invar-iant case and show the connection between the state space theory considered here and the theory of C, semigroups.

Realization theory for symmetric systems

Systems of the type { A, KJ, 2iK∗, I} are studied where J  J∗ is unitary and KJK∗  Im A . A complete realization theory as well as state space isomorphism theorem are given. Coupling problems are considered.

Strictly Cyclic Linear Systems

The state space isomorphism theorem is proved for a class of infinite dimensional linear systems. The spectral minimality property is discussed for this class. An example shows that a result of Wonham on controllability cannot be generalized to the infinite dimensional case.

Normal Symmetric Dynamical Systems

In this paper we establish a form of the state space isomorphism theorem for linear differentiable dynamical systems in a Hilbert space and make some application of these results. The methods used are based on spectral representations and suggest a close connection between the state space isomorphism theorem and certain classical representation theorems in analysis. We also give a class of counterexamples which illuminate the difficulties in extending the finite-dimensional theory thus justifying, in part, the stronger hypothesis used here.

System Theory on Group Manifolds and Coset Spaces

The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a cosec space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. These results parallel, and in part specialize to, results available for the familiar case described by $\dot x(t) = Ax(t) + Bu(t)$, $y(t) = Cx(t)$. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.