Models for dynamics.

*(English)*Zbl 0685.93002
Dyanamics reported, Vol. 2, 171-269 (1989).

[For the entire collection see Zbl 0659.00009.]

This paper develops an axiomatic framework for a mathematical theory of dynamical systems. The following definition is proposed: A dynamical system is a triple \(\sum =(T,W,{\mathfrak B})\) with \(T\subseteq {\mathbb{R}}\) the time axis, W a set called the signal space, and \({\mathfrak B}\subseteq W^ T\) the behavior. In this paper attention is restricted to the case \(T={\mathbb{Z}}\) or \({\mathbb{R}}\) and \(\sigma^ t{\mathfrak B}={\mathfrak B}\) for all \(t\in T\) \((\sigma^ t\) denotes the t-shift). This last condition is called time-invariance. Moreover in much of the paper it is assumed that the system is linear, in particular that \(W={\mathbb{R}}^ q\) and that \({\mathfrak B}\) is a linear subspace of \(({\mathbb{R}}^ q)^ T\). The behavior \({\mathfrak B}\) is often described by behavioral equations, usually differential or difference equations. Finally, in models obtained from first principles the laws of a dynamical system will often contain latent variables. It is the interplay of these three notions: the behavior, behavioral equations, and latent variables which constitutes the basic premise of this paper.

Within this framework, a number of issues are addressed. The first one is the construction of the state of a dynamical system. The upshot of this development is the proof that the belief that the state of a dynamical system is uniquely defined up to relabeling and barring hidden variables is, in general, not true. However, it is shown to be correct for important classes of systems, in particular for linear and for autonomous dynamical systems.

The paper contrasts the definition used here with the more classical notions of flows on manifolds used in topological dynamics as the basic notion in dynamics and of an input/output system used in control theory.

Whereas the first parts of this paper have a somewhat philosophical flavor, the last part studies a concrete class of dynamical systems: the discrete-time systems which are linear time-invariant and complete. This class of systems can be described by difference equations \[ R(\sigma,\sigma^{-1})\omega \quad =\quad M(\sigma,\sigma^{-1})a \] with \(\omega\) : \({\mathbb{Z}}\to R^ q\) the signal trajectory, \(a:{\mathbb{Z}}\to R^ d\) the latent variable trajectory, and \(R(s,s^{-1})\in R^{g\times q}[s,s^{-1}]\), \(M(s,s^{-1})\in R^{g\times q}[s,s^{- 1}]\) polynomial matrices which specify the dynamical laws of the system. For this class of systems the following problems are studied: elimination of latent variables, controllability, observability, autonomous systems, input/output representations, state space representations, and, finally, input/state/output representations.

This paper develops an axiomatic framework for a mathematical theory of dynamical systems. The following definition is proposed: A dynamical system is a triple \(\sum =(T,W,{\mathfrak B})\) with \(T\subseteq {\mathbb{R}}\) the time axis, W a set called the signal space, and \({\mathfrak B}\subseteq W^ T\) the behavior. In this paper attention is restricted to the case \(T={\mathbb{Z}}\) or \({\mathbb{R}}\) and \(\sigma^ t{\mathfrak B}={\mathfrak B}\) for all \(t\in T\) \((\sigma^ t\) denotes the t-shift). This last condition is called time-invariance. Moreover in much of the paper it is assumed that the system is linear, in particular that \(W={\mathbb{R}}^ q\) and that \({\mathfrak B}\) is a linear subspace of \(({\mathbb{R}}^ q)^ T\). The behavior \({\mathfrak B}\) is often described by behavioral equations, usually differential or difference equations. Finally, in models obtained from first principles the laws of a dynamical system will often contain latent variables. It is the interplay of these three notions: the behavior, behavioral equations, and latent variables which constitutes the basic premise of this paper.

Within this framework, a number of issues are addressed. The first one is the construction of the state of a dynamical system. The upshot of this development is the proof that the belief that the state of a dynamical system is uniquely defined up to relabeling and barring hidden variables is, in general, not true. However, it is shown to be correct for important classes of systems, in particular for linear and for autonomous dynamical systems.

The paper contrasts the definition used here with the more classical notions of flows on manifolds used in topological dynamics as the basic notion in dynamics and of an input/output system used in control theory.

Whereas the first parts of this paper have a somewhat philosophical flavor, the last part studies a concrete class of dynamical systems: the discrete-time systems which are linear time-invariant and complete. This class of systems can be described by difference equations \[ R(\sigma,\sigma^{-1})\omega \quad =\quad M(\sigma,\sigma^{-1})a \] with \(\omega\) : \({\mathbb{Z}}\to R^ q\) the signal trajectory, \(a:{\mathbb{Z}}\to R^ d\) the latent variable trajectory, and \(R(s,s^{-1})\in R^{g\times q}[s,s^{-1}]\), \(M(s,s^{-1})\in R^{g\times q}[s,s^{- 1}]\) polynomial matrices which specify the dynamical laws of the system. For this class of systems the following problems are studied: elimination of latent variables, controllability, observability, autonomous systems, input/output representations, state space representations, and, finally, input/state/output representations.

Reviewer: J.C.Willems

##### MSC:

93A10 | General systems |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

70Gxx | General models, approaches, and methods |

93C55 | Discrete-time control/observation systems |