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.

We have presented a complete framework for modeling and identifying multi-dimensional systems by extending the class of linear state space and linear descriptor systems. Proposing a model was the easy part of this work, afterwards all relevant properties and the related applications must be described. This was especially difficult for the class of commuting descriptor systems, for which the literature is spares and far between. No efforts of analyzing the system properties where attempted in the past.

In this first part of this thesis, one-dimensional system theory was presented, with the main purpose to serve as a stepping stone and introduction to the more advanced concepts. Although most of this content was well known, we took an alternative approach to the introduction and analysis of descriptor systems and introduced the generalized state vector, making abstraction of the initial and final state. This resulted in a more elegant formulation and analysis of the model properties, which in turn lead to a novel realization algorithm and the discovery of a small mistake in the current literature. Both results where unexpected and are a consequence of the more abstract and general way descriptor systems where described.

After the introduction of one-dimensional systems theory we attempted, with success, to generalize the presented concepts to multi-dimensional systems in part two of this thesis. One of the key elements in this analysis was the generalized theorem of Cayley–Hamilton. Starting from this theorem, the difference equation and observability/controllability matrices could be defined. A second important result was the introduction of the generalized eigenbasis for commuting matrices. Although commuting matrices can not always be simultaneously diagonalized, a generalized eigenbasis can always be constructed. Using this eigenbasis, concepts like the Popov–Belevitch–Hautus lemma and the modal solutions were obtained.

By far the most challenging part in this thesis, and tedious to read through, was to derive the well-posedness condition for commuting descriptor systems. We proved that a multi-dimensional descriptor system is only well-posed if and only if all system matrices can be transformed to a commuting matrix family. This result was key to unlock all further properties, like the classification and description of the gap, which results from canonical form.

In the third part of this work we shifted our focus on the identification of multi-dimensional systems. Central to the presented algorithm and analysis lies the generalized Hankel matrix and the recursive Hankel matrix. The link between both matrices is discussed and we proof that both matrices can be factorized as the outer-product of an observability-like matrix and a controllability-like matrix. We also got side tracked and discuss the recursive Hankel tensor, which is an interesting idea but is not further used in the rest of this thesis.

After the introduction of the generalized Hankel matrices, we introduce a novel realization algorithm for one-dimensional descriptor systems. This algorithm skips one previously important step and uses the more abstract evolution operator in stead of the Weierstrass canonical form of regular matrix pencils. This approach leads to a mathematically more elegant algorithm to calculate a suitable system realization.

To end this thesis we described the realization of multi-dimensional commuting state space models and descriptor systems. One algorithm is presented for commuting state space models and two for commuting descriptor systems. The presented algorithms may appear simple on the surface, as they consist of only a couple of steps, and equally few lines to implement them. However to really understand every one of these steps you have to write a complete PhD dissertation, so to speak.

The main result of this thesis is a single, simple and concise algorithm that can identifying one, two, three or more dimensional regular and singular systems. Every single step in this algorithm is explained and proven in this work and all details are worked out.

Future Work

Continuous-time models

In this thesis only a small subset of dynamic systems has been described, and we failed to mention continuous systems. When talking about continuous multi-dimensional dynamic systems we enter the territory of partial differential equations, like the heat equation. The heat equation has been mentioned a couple of times in this work, but the link between state space models and partial differential equations has not been studied. Partial differential equations can be transformed to multivariate polynomials by using the Laplace transformation~\cite{debnath1989theorems}. A single partial differential equation will result in a single multivariate polynomial, for which it is known that in general the solution to this equation is a manifold instead of isolated roots. When specifying a correct number of boundary conditions and initial condition, the continuous manifold can be reduced to singular isolated roots, which can be modeled by a continuous time state space model. It is still unknown how this state space model can that be used to construct the original manifold of the partial differential equation.

Input-output models

The next logical extension of the results formulated in this thesis, is the introduction of input-output models. Some preliminary results have been derived by me during me research but are not included in this text. This problem can be tackled by studying the superposition principle, which states that the sum of solutions is a valid solution. This analysis has led me to introduce the following model class

\begin{equation} \begin{aligned} Ex[k+1,l] = & A x[k,l] + \sum_{i=0}^{n_2} \mathcal{B}_{n_2}^{i-l} D u[k,l] \\ Fx[k,l+1] = & B x[k,l] \\ y[k,l] = & C x[k,l], \end{aligned} \end{equation}

where the state \(x\), the output \(y\) and the input \(u\) is defined on the rectangular domain of size \(n_1+1 \times n_2+1\) and \(\mathcal{B}_{n_2}^{l} = B^l F^{n_2-l}\). The system can model input-output dynamic processes where the first independent variable \(k\) represents time, or an other causal variable. The output as a function of the generalized state and the input is equal to

\begin{equation} y[k,l] = C \mathcal{A}_{n_1}^k \mathcal{B}_{n_2}^l x + (\mathcal{A}_{n_1}^k\sum_{i=0}^{n_2} \mathcal{B}_{n_2}^l D u[k,i]), \end{equation}

which is the sum of an autonomous part and a term which depends on the input signal. This equation matches the solution to linear partial differential equations and the model properties must be further investigated.

Stochastic systems

All presented theory is completely deterministic, meaning that the measured signals are exact and there are no model errors. This is of course an enormous simplification of how real processes behave. At the moment of writing this thesis, I have no idea how to incorporate noise into the presented model formulations. This is still an interesting open problem.

Realization theory for distributed systems

Together with noise there exist the concept of model errors. A model is, as the name suggests, just a model and nothing more. When modeling an unknown processes, modeling errors will be made because a model can only reflect the true system behavior till a certain degree. By nature partial differential equations and distributed systems are examples of distributed systems, which in general do not have a finite state. In contrasts, all presented models do have a finite state. Modeling distributed systems will thus always lead to model errors. One open question is how we can work with these modeling errors, and what is the best way to truncate the state.

Numerical algorithms and large scale implementation

All algorithms in this thesis are presented in a naive way;

  • rank calculations are exact
  • matrix size are not discussed
  • sparsity is not exploited
  • efficient data storage in the Hankel matrices is not discussed

and so on. What happens when one of these rank calculations fails, or when an algorithm introduce floating point errors. These are still open problems and not considered in this thesis. A lot can be gained when working out these computational details.

Tensor methods

Last but not least, is the `tensorization’ of the presented results. I alluded a couple of times to the fact that my work is riddled with tensors. Multi-dimensional time series are tensors, there also exists a tensor representation for the recursive Hankel matrix, which I have studied but not included in this thesis. Factorizing the recursive Hankel matrix gives the recursive observability matrix, which, as you could imagine can be represented as a tensor. The blocks and blocks of blocks, which I indicate with the slightly suggestive lines in all my matrices hint to this tensor structure. Extending my presented work to a tensor framework is in my opinion a low hanging fruit, but outside the scope of my work.