Ulrich Oberst

A Behavioral Approach to the Pole Structure of One-Dimensional and Multidimensional Linear Systems

We use the tools of behavioral theory and commutative algebra to produce a new definition of a (finite) pole of a linear system. This definition agrees with the classical one and allows a direct dynamical interpretation. It also generalizes immediately to the case of a multidimensional (nD) system. We make a natural division of the poles into controllable and uncontrollable poles. When the behavior in question has latent variables, we make a further division into observable and unobservable poles. In the case of a one-dimensional (1D) state-space model, the uncontrollable and unobservable poles correspond, respectively, to the input and output decoupling zeros, whereas the observable controllable poles are the transmission poles. Most of these definitions can be interpreted dynamically in both the 1D and nD cases, and some can be connected to properties of kernel representations. We also examine the connections between poles, transfer matrices, and their left and right matrix fraction descriptions (MFDs). We find behavioral results which correspond to the concepts that a controllable system is precisely one with no input decoupling zeros and an observable system is precisely one with no output decoupling zeros. We produce a decomposition of a behavior as the sum of subbehaviors associated with various poles. This is related to the integral representation theorem, which describes every system trajectory as a sum of integrals of polynomial exponential trajectories.

The canonical Cauchy problem for linear systems of partial difference equations with constant coefficients over the completer-dimensional integral lattice ℕ2r

We canonically define and algorithmically solve the problem of the title. Such algorithms are of great significance for the method of finite differences for the solution of partial differential equations and for many technical applications such as image processing. In contrast to the wide (system theoretic) literature for ordinary difference equations and in spite of the great theoretical and practical significance of this problem, until now, there was no systematic theory of these systems and in particular of the corresponding Cauchy problem, let alone an algorithm. In this paper, we give both. The method consists in a transformation of this problem into a naturally associated problem which is defined over the 2r-dimensional natural number lattice ℕ2r (the upper ‘quadrant’ in ℤ2r) and for which the canonical initial value or Cauchy problem was defined and constructively solved by the second author.

The Constructive Solution of Linear Systems of Partial Difference and Differential Equations with Constant Coefficients

This paper gives a survey of past work in the treated subject and also contains several new results. We solve the Cauchy problem for linear systems of partial difference equations on general integral lattices by means of suitable transfer operators and show that these can be easily computed with the help of standard implementations of Gröbner basis algorithms. The Borel isomorphism permits to transfer these results to systems of partial differential equations. We also solve the Cauchy problem for the function spaces of convergent power series and for entire functions of exponential type. The unique solvability of the Cauchy problem implies that the considered function spaces are large injective cogenerators for which the duality between finitely generated modules and behaviours holds. Already in the beginning of the last century C. Riquier considered and solved problems of the type discussed here.

Transfer Operators and State Spaces for Discrete Multidimensional Linear Systems

This paper treats multidimensional discrete input-output systems from the constructive point of view. We adapt and improve recursive algorithms, derived earlier by E. Zerz and the second author from standard Gröbner basis algorithms, for the solution of the canonical Cauchy problem for linear systems of partial difference equations with constant coefficients on the lattices N = ℕr1 × ℤr2. These recursive algorithms, in turn, furnish four other solution methods for the initial value problem, namely by transfer operators, by canonical Kalman global state equations, by parametrizations of controllable systems and, for systems with proper transfer matrix and left bounded input signals, by convolution with the transfer matrix. In the 2D-case N = ℤ2 the last method was studied by S. Zampieri. Minimally embedded systems are studied and give rise to especially simple Kalman equations. The latter also imply a useful characterization of the characteristic or polar variety of the system by eigenvalue spectra. For N = ℕr we define reachability of a system and prove that controllability implies reachability, but not conversely. Moreover we solve, in full generality, the modelling problem which was introduced and partially solved by F. Pauer and S. Zampieri. Various algorithms have been implemented by the first author in axiom, and examples are demonstrated by means of computer generated pictures. Related work on state space representations has been done by the Padovian and Groningian system theory schools.

Linear Recurring Arrays, Linear Systems and Multidimensional Cyclic Codes over Quasi-Frobenius Rings

This paper generalizes the duality between polynomial modules and their inverse systems (Macaulay), behaviors (Willems) or zero sets of arrays or multi-sequences from the known case of base fields to that of commutative quasi-Frobenius (QF) base rings or even to QF-modules over arbitrary commutative Artinian rings. The latter generalization was inspired by the work of Nechaev et al. who studied linear recurring arrays over QF-rings and modules. Such a duality can be and has been suggestively interpreted as a Nullstellensatz for polynomial ideals or modules. We also give an algorithmic characterization of principal systems. We use these results to define and characterize n-dimensional cyclic codes and their dual codes over QF rings for n>1. If the base ring is an Artinian principal ideal ring and hence QF, we give a sufficient condition on the codeword lengths so that each such code is generated by just one codeword. Our result is the n-dimensional extension of the results by Calderbank and Sloane, Kanwar and Lopez-Permouth, Z. X. Wan, and Norton and Salagean for n=1.

Stability and Stabilization of Multidimensional Input/Output Systems

In this paper we discuss stability and stabilization of continuous and discrete multidimensional input/output (IO) behaviors (of dimension

Time-autonomy and time-controllability of discrete multidimensional behaviours

r

On the minimal number of trajectories determining a multidimensional system

) which are described by linear systems of complex partial differential (resp., difference) equations with constant coefficients, where the signals are taken from various function spaces, in particular from those of polynomial-exponential functions. Stability is defined with respect to a disjoint decomposition of the

Multidimensional Discrete Stability by Serre Categories and the Construction and Parametrization of Observers via Gabriel Localizations

r

Multidimensional Constant Linear Systems

-dimensional complex space into a stable and an unstable region, with the standard stable region in the one-dimensional continuous case being the set of complex numbers with negative real part. A rational function is called stable if it has no poles in the unstable region. An IO behavior is called stable if the characteristic variety of its autonomous part has no points in the unstable region. This is equivalent to the stability of its transfer matrix and an additional condition. The system is called stabilizable if there is a compensator IO system such that the output feedback system is well-posed and stable. We characterize stability and stabilizability and construct all stabilizing compensators of a stabilizable IO system (parametrization). The theorems and proofs are new but essentially inspired and influenced by and related to the stabilization theorems concerning multidimensional IO maps as developed, for instance, by Bose, Guiver, Shankar, Sule, Xu, Lin, Ying, Zerz, and Quadrat and, of course, the seminal papers of Vidyasagar, Youla, and others in the one-dimensional case. In contrast to the existing literature, the theorems and proofs of this paper do not need or employ the so-called fractional representation approach, i.e., various matrix fraction descriptions of the transfer matrix, thus avoiding the often lengthy matrix computations and seeming to be of interest even for one-dimensional systems (at least to the author). An important mathematical tool, new in systems theory, is Gabriel’s localization theory which, only in the case of ideal-convex (Shankar, Sule) unstable regions, coincides with the usual one. Algorithmic tests for stability, stabilizability, and ideal-convexity, and the algorithmic construction of stabilizing compensators, are addressed but still encounter many difficulties; see in particular the open problems listed by Xu e