Behavioral Systems Theory: A Survey
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paper_title: Paradigms and puzzles in the theory of dynamical systems

paper_content:

A self-contained exposition is given of an approach to mathematical models, in particular, to the theory of dynamical systems. The basic ingredients form a triptych, with the behavior of a system in the center, and behavioral equations with latent variables as side panels. The author discusses a variety of representation and parametrization problems, in particular, questions related to input/output and state models. The proposed concept of a dynamical system leads to a new view of the notions of controllability and observability, and of the interconnection of systems, in particular, to what constitutes a feedback control law. The final issue addressed is that of system identification. It is argued that exact system identification leads to the question of computing the most powerful unfalsified model. >
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paper_title: Multidimensional constant linear systems

paper_content:

A continuous resp. discrete r-dimensional (r≥1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).
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paper_title: Paradigms and puzzles in the theory of dynamical systems

paper_content:

A self-contained exposition is given of an approach to mathematical models, in particular, to the theory of dynamical systems. The basic ingredients form a triptych, with the behavior of a system in the center, and behavioral equations with latent variables as side panels. The author discusses a variety of representation and parametrization problems, in particular, questions related to input/output and state models. The proposed concept of a dynamical system leads to a new view of the notions of controllability and observability, and of the interconnection of systems, in particular, to what constitutes a feedback control law. The final issue addressed is that of system identification. It is argued that exact system identification leads to the question of computing the most powerful unfalsified model. >
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paper_title: When are linear differentiation-invariant spaces differential?

paper_content:

It is shown that a linear differentiation-invariant subspace of a C∞-trajectory space is differential (i.e., can be represented as the kernel of a linear constant-coefficient differential operator) if and only if its McMillan degree is finite.
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paper_title: Multidimensional constant linear systems

paper_content:

A continuous resp. discrete r-dimensional (r≥1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).
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paper_title: Modules and Behaviours in nD Systems Theory

paper_content:

This paper is intended both as an introduction to the behavioural theory of nD systems, in particular the duality of Oberst and its applications, and also as a bridge between the behavioural theory and the module-theoretic approach of Fliess, Pommaret and others. Our presentation centres on Pommaret's notion of a system observable, and uses this concept to provide new interpretations of known behavioural results. We discuss among other subjects autonomous systems, controllable systems, observability, transfer matrices, computation of trajectories, and system complexity.
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paper_title: Multidimensional constant linear systems

paper_content:

A continuous resp. discrete r-dimensional (r≥1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).
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paper_title: Controllable and Autonomous nD Linear Systems

paper_content:

AbstractThe theory of multidimensional systems suffers in certain areas from a lack of development of fundamental concepts. Using the behavioural approach, the study of linear shift-invariant nD systems can be encompassed within the well-established framework of commutative algebra, as previously shown by Oberst. We consider here the discrete case. In this paper, we take two basic properties of discrete nD systems, controllability and autonomy, and show that they have simple algebraic characterizations. We make several non-trivial generalizations of previous results for the 2D case. In particular we analyse the controllable--autonomous decomposition and the controllable subsystem of autoregressive systems. We also show that a controllable nD subsystem of ::: $$(k^q )^{(Z^n )} $$ ::: is precisely one which is minimal in its transfer class.
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paper_title: A New Perspective on Controllability Properties for Dynamical Systems

paper_content:

In this paper we study the properties of weak and strong controllability as newly defined in (Rocha, 1995) for delay-differential systems in a behavioural setting, now for the multidimensional case. Further we give an overview of the relationships between these properties and the original behavioural controllability concept introduced in (Willems, 1988).
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paper_title: A Behavioral Approach To Control Of Distributed Systems

paper_content:

This paper develops a theory of control for distributed systems (i.e., those defined by systems of constant coefficient partial differential operators) via the behavioral approach of Willems. The study here is algebraic in the sense that it relates behaviors of distributed systems to submodules of free modules over the polynomial ring in several indeterminates. As in the lumped case, behaviors of distributed ARMA systems can be reduced to AR behaviors. This paper first studies the notion of AR controllable distributed systems following the corresponding definition for lumped systems due to Willems. It shows that, as in the lumped case, the class of controllable AR systems is precisely the class of MA systems. It then shows that controllable 2-D distributed systems are necessarily given by free submodules, whereas this is not the case for n-D distributed systems, $n \ge 3$. This therefore points out an important difference between these two cases. This paper then defines two notions of autonomous distributed systems which mimic different properties of lumped autonomous systems. ::: Control is the process of restricting a behavior to a specific desirable autonomous subbehavior. A notion of stability generalizing bounded input--bounded output stability of lumped systems is proposed and the pole placement problem is defined for distributed systems. This paper then solves this problem for a class of distributed behaviors.
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paper_title: Structure and representation of 2-D systems

paper_content:

A closure cap of the screw-on or similar type having exertior knurls or the like at the periphery thereof, in combination with a safety overcap having complementary interior knurls or projections for selective engagement with the exterior knurls of the closure cap for turning the latter by rotation of the overcap, said overcap including a yieldable resilient depressed base engaging the base of the closure cap normally maintaining the overcap unengaged with respect to the closure cap. The interior knurls or projections on the interior of the overcap engage the exterior knurls at the periphery of the closure cap only upon the application of downward pressure on the safety overcap relative to the closure cap, said overcap rising to non-engaged position upon release of the pressure.
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paper_title: Autonomicity and the absence of free variables for behaviors over finite rings

paper_content:

The equivalence between autonomicity and the absence of free variables is a well-known result for linear, shift-invariant and complete behaviors over a field. However, this no longer holds for a behavior that does not satisfy one of those properties. In this paper we consider linear, shift-invariant and complete behaviors over $\mathbb{Z}_{p^r}$ and study under which conditions such behaviors are autonomous and/or have no free variables.
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paper_title: Linear Recurring Arrays, Linear Systems and Multidimensional Cyclic Codes over Quasi-Frobenius Rings

paper_content:

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.
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paper_title: Reed-Solomon list decoding from a system-theoretic perspective

paper_content:

In this paper, the Sudan-Guruswami approach to list decoding of Reed-Solomon (RS) codes is cast in a system-theoretic framework. With the data, a set of trajectories or time series is associated which is then modeled as a so-called behavior. In this way, a connection is made with the behavioral approach to system theory. It is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the Sudan-Guruswami approach. The concept of "weighted row reduced" is introduced and used to achieve minimality. Two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given.
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paper_title: Autonomy properties of multidimensional linear systems over rings

paper_content:

Based on the notions of rank and reduced rank from commutative algebra, we discuss several aspects of the concept of autonomy for multidimensional discrete linear systems over finite rings of the form Z/mZ. We review several algebraic characterizations of autonomy that are equivalent for systems over fields and investigate their relationship in the ring case. The strongest of these notions turns out to be equivalent to the non-existence of trajectories with finite support (besides the zero trajectory), and the weakest one amounts to the fact that the system has no free variables (inputs).
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paper_title: Stability and Stabilization of Multidimensional Input/Output Systems

paper_content:

In this paper we discuss stability and stabilization of continuous and discrete multidimensional input/output (IO) behaviors (of dimension $r$) 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 $r$-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
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paper_title: Representations and structural properties of periodic systems

paper_content:

We consider periodic behavioral systems as introduced in [Kuijper, M., & Willems, J. C. (1997). A behavioral framework for periodically time-varying systems. In Proceedings of the 36th conference on decision & control (Vol. 3, pp. 2013-2016). San Diego, California, USA, 10-12 December 1997] and analyze two main issues: behavioral representation/controllability and autonomy. More concretely, we study the equivalence and the minimality of kernel representations, and introduce latent variable (and, in particular, image) representations. Moreover we relate the controllability of a periodic system with the controllability of an associated time-invariant system known as lifted system, and derive a controllability test. Further, we prove the existence of an autonomous/controllable decomposition similar to the time-invariant case. Finally, we introduce a new concept of free variables and inputs, which can be regarded as a generalization of the one adopted for time-invariant systems, but appears to be more adequate for the periodic case.
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paper_title: On the Parametrization of All Regularly Implementing and Stabilizing Controllers

paper_content:

In this paper we deal with problems of controller parametrization in the context of behavioral systems. Given a full plant behavior, a subbehavior of the manifest plant behavior is called regularly implementable if it can be achieved as the controlled behavior resulting from the interconnection of the full plant behavior with a suitable controller behavior, in such a way that the controller does not impose restrictions that are already present in the plant. We establish a parametrization of all controllers that regularly implement a given behavior. We also obtain a parametrization of all stabilizing controllers.
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paper_title: State maps for linear systems

paper_content:

Modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential-algebraic equations. As such, state-space models are not a natural starting point for modeling, while they have utmost importance in the simulation and control phase. The paper addresses the problem of computing state variables for systems of linear differential-algebraic equations of various forms. The point of view from which the problem is considered is the behavioral one, as put forward in [J. C. Willems, Automatica J. IFAC, 22 (1986), pp. 561--580; Dynamics Reported, 2 (1989), pp. 171--269; IEEE Trans. Automat. Control, 36 (1991), pp. 259--294].
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paper_title: Observer synthesis in the behavioral approach

paper_content:

Analyzes the observer design problem in the behavioral context. Observability and detectability notions are first introduced and fully characterized. Necessary and sufficient conditions for the existence of an observer, possibly an asymptotic or an exact one, are introduced, and a complete parameterization of all admissible observers is given. The problem of obtaining observers endowed with a (strictly) proper transfer matrix and the design of observer-based controllers are later addressed and solved. Finally, the above issues are particularized to the case of state-space systems, thus showing they naturally generalize well-known theorems of traditional system theory.
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paper_title: On the state of behaviors

paper_content:

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].
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Title: Behavioral Systems Theory: A Survey

Section 1: Introduction
Description 1: Provide an overview and background of behavioral systems theory, highlighting the contributions of J.C. Willems and the current state of research in this field.

Section 2: Starting point
Description 2: Introduce the reader to the basic concepts of viewing a system, explain Willems' definition of a dynamical system, and set the foundation for understanding behavioral systems.

Section 3: E. Zerz
Description 3: Discuss the modifications and alternative definitions of systems proposed by researchers following Willems' original definition, including key concepts and terminologies.

Section 4: First steps
Description 4: Explain the initial steps in the study of behavioral systems, including linearity, time-invariance, and the concept of kernel representations in both discrete and continuous time.

Section 5: Milestones
Description 5: Highlight significant contributions and fundamental theorems in the development of behavioral systems theory, focusing on concepts such as input-output relationships, controllability, and autonomy.

Section 6: Further paths
Description 6: Examine extensions of the basic theory into more complex areas such as multidimensional systems, continuous time-varying systems, and discrete systems over finite rings, illustrating their unique challenges and results.

Section 7: Where to go from here
Description 7: Provide a short outlook on recent developments, potential future research directions, and important topics within behavioral systems theory that were not covered in the survey.