Introduction

Problem statement Modeling multi-dimensional systems In recent years increasing attention in system theory is paid to multi-dimensional dynamic systems. These so-called \gls{nD} systems are characterized by signals that depend on several independent variables. These variables can be for example time and space, in which case observations are not only correlated in time, but also in space. As an example, imagine a bridge or a mechanical structure. At discrete regular locations, sensors are placed that measure the local displacement (or acceleration) of the bridge over time....

23 min · Bob Vergauwen

Concepts from Linear algebra

In this chapter, some elementary concepts from linear algebra are introduced. The dynamic properties of multi-dimensional state space models are largely determined by a set of commuting matrices. Central to the analysis is the theorem of Schur which states that a set of commuting matrices can simultaneously be triangularized. We also introduce the singular value decomposition as it is used a lot throughout this thesis. Afterwards the eigenvalues and eigenvectors of commuting matrices are discussed by using the theorem of Schur....

14 min · Bob Vergauwen

Regular one-dimensional state space models

State space model This thesis is focused around the state space representation of linear time invariant systems. A regular one-dimensional autonomous state space model is described by \begin{equation}\label{eqn:1R-SS} \begin{aligned} x[k+1] = & A x[k] \\ y[k] = & C x[k]. \end{aligned} \end{equation} The matrix \(A \in \real^{m\times m}\) is called the system matrix and the matrix \(C \in \real^{q\times m}\) the output matrix. The state vector is denoted by the vector \(x \in \real^m\)....

24 min · Bob Vergauwen

Singular one-dimensional state space models

In this chapter we introduce one-dimensional singular system, also known as descriptor systems or differential algebraic equations (\gls{DAE})~\cite{brenan1996numerical}. A descriptor system is an extension of the classical state space model, as presented in the previous chapter and is characterized by a causal and non-causal solution, running forward and backward in time respectively (see e.g.~\cite{moonen1992subspace}). The causal part is linked to the finite roots of the characteristic equation associated with the descriptor system and the non-causal part is linked to roots at infinity....

25 min · Bob Vergauwen

Two-dimensional regular commuting state space models

Lets step up the game and add an extra dimension to this work, literally. From now on, we abandon plain one-dimensional systems and explore there multi-dimensional counter parts. A multi-dimensional system is a system with evolves in multiple dimension and the system quantities, like the output and internal state, are index by multiple indices. These indices can for example be space and time. In this chapter all classical ideas, presented in Chapter~\ref{ch:1DR} are generalized for two-dimensional state space models....

40 min · Bob Vergauwen

Two-dimensional commuting descriptor Systems

The class of commuting state space models presented in Chapter~\ref{ch:2DR} can be extended to cover Differential Algebraic Equations or descriptor systems. As shown in Chapter~\ref{ch:1DS} these systems are characterized by a causal and non-causal part. In this chapter two-dimensional, and to some extend multi-dimensional, descriptor systems are described. It has been shown that the commutativity of the system matrices of a multi-dimensional commuting state space model is important for the well-posedness of this class of state space models....

51 min · Bob Vergauwen

Generalized Hankel matrices

At the core of the presented identification algorithms is the concept of a Hankel matrix. Hankelization of time series is a way of restructuring observed time series data in order to be able to estimate the linear relation between subsequent data points. The resulting matrix is called a Hankel matrix. In this chapter we will first introduce the Hankel matrix for one-dimensional time series data, followed by two extensions covering multi-dimensional time series....

24 min · Bob Vergauwen

One-dimensional realization

In this chapter, realization methods for one-dimensional regular state space models and descriptor systems are described. The presented algorithm is know as the algorithm of Kung~\cite{kung1978new} and uses the singular value decomposition to construct a numerical base of the observability and state sequence matrices. The estimated observability and state sequence matrices are than used to estimate the a priori unknown system parameters such as the order, the system matrices and the state sequence....

17 min · Bob Vergauwen

Multi-dimensional realization

In this chapter we introduce three realization algorithms based on the algorithm of Kung, one for regular two-dimensional commuting state space models and two for two-dimensional commuting descriptor systems. All algorithms follow the same approach as in the one-dimensions case, except multiple shifts are needed to estimate the full system dynamics. We start by formulating the problem statement, followed by the description of three realization algorithms. Next the partial realization condition for two-dimensional commuting state space models and descriptor systems is derived....

25 min · Bob Vergauwen

Conclusions and Outlook

Conclusions System theory is one of the most diverse fields within engineering and applied science. With applications ranging from modeling and controlling chemical reactors~\cite{el1995regional} to the analysis of ground vibration test of an F16 airplane~\cite{noel2017f}. In this thesis, we barely scratched the surface of the world of multi-dimensional systems, only focusing on autonomous, local finite state, time-invariant, linear systems. Analyzing the aforementioned examples exceeds the scope of this work and illustrates the challenges ahead....

7 min · Bob Vergauwen