Ettore Fornasini

• Professor Emeritus at University of Padua

In this work we investigate the realization problem of periodic convolutional codes. As convolutional codes are discrete linear systems over a finite field we use systems theory techniques to address our problem. In particular, we aim at deriving and studying state-space realizations of 2-periodic convolutional codes. Although one cannot expect, in...

Insiemi invarianti e insiemi limite. Teoremi di LaSalle e di Krasowskii Traiettorie periodiche dei sistemi continui Traiettorie periodiche dei sistemi discreti Traiettorie periodiche nei sistemi discreti unidimensionali. Teorema di Sarkowski. Proprieta' dell'equilibrio nei sistemi discreti unidimensionali. Mappe di Lyapunov e di Sylvester

Raggiungibilita' e controllabilita' nei sistemi lineari discreti. Raggiungibilita' e controllabilita' nei sistemi lineari continui. Ingressi di controllo e matrici gramiane. Controllo e matrici gramiane su orizzonte infinito. Controllo del movimento su un sottospazio. Rapptesentazione dei sistemi non raggiungibili. Ulteriori criteri di raggiungibil...

Osservabilita' e ricostruibilita' dei sistemi lineari, discreti e continui. Dualita' Decomposizione canonica Stimatori asintotici dello stato, di orine intero e ridotto Sintesi del regolatore

Connessioni elementari di sistemi strettamente propri (paralleo, serie, retroazione). Connessione di sistemi non strettamente propri. Connessione di sistemi discreti e di sistemi continui

Introductory chapter

Discrete time linear models: representation in the time domain, discrete time signals, formal power series, MA, AR and ARMA models, representation in the z domain, modal analysis. Continuous time linear systems: representation in the time domain, exponential matrices, unforced and forced evolution, representation in the s domain, modal analysis. C...

Equilibrium and equilibrium stability in autonomous systems. Linear systems stability. Continuous and discrete time systems: stability and instability criteria Lyapunov equations. Stability analysis via linearization. Stability of unidimensional systems

Invariant sets and limit sets. LaSalle and Krasowskii theorems. Periodic trajectories of an autonomous continuous system. Periodic trajectories of an autonomous discrete time system. Equilibrium properties of unidimensional systems. Lyapunov and Sylvester maps.

Le catene di Markov sono sistemi dinamici a tempo discreto che presentano caratteristi-che alquanto diverse rispetto a quelli considerati finora. Nell'accezionepì u semplice, si tratta di sistemi autonomi a stati finiti 1 nei quali-ecì o rappresenta una essenziale novità rispetto ai modelli analizzati finora-la transizione da uno stato all'altro av...

Index

Capitolo 11 RAPPRESENTAZIONE DEI SISTEMI DISCRETI POSITIVI Chiamiamo "positivi" i sistemi dinamici nei quali tutte le variabili di stato, cos`cos`ı come quelle di ingresso e di uscita, se presenti, possono assumere soltanto valori non negativi. Situazioni nelle quali le grandezze in gioco hanno significato esclusivamente quando ad esse si attribuis...

La stabilità asintotica di un sistema positivo lineare x(t + 1) = F x(t) (12.1) si definisce come per il caso di un sistema lineare generico. Ovviamente lo stato ini-ziale deve essere non negativo ma, sebbene il movimento abbia inizio e si svolga soltanto nell'ortante R n + , la condizione (necessaria e sufficiente per la stabilità asintotica di un...

In questa appendice sono richiamati i concetti fondamentali dell'Algebra Lineare e sono discussi in maggior dettaglio alcuni argomenti, normalmente non inclusi nei corsi di Mate-matica che precedono quello di Teoria dei Sistemi e che risultano critici per la comprensione della dispensa. Per ulteriori approfondimenti si rinvia ai testi citati nella...

Questa appendicè e dedicata alla definizione dei coni in R n e alla discussione di alcune loro proprietà, utili nell'analisi dei sistemi lineari positivi.

In this paper we introduce a special class of 2D convolutional codes, called composition codes, which admit encoders G(d1; d2) that can be decomposed as the product of two 1D encoders, i.e., G(d1; d2) = G2(d2)G1(d1). Taking into account this decomposition, we obtain syndrome formers of the code directly from G1(d1) and G2(d2), in case G1(d1) and G2...

In the paper output feedback control of Boolean control networks (BCNs) is investigated. First, necessary and sufficient conditions for the existence of a time-invariant output feedback (TIOF) law, stabilizing the BCN to some equilibrium point, are given, and constructive algorithms to test the existence of such a feedback law are proposed. Two suf...

Local reachability of two-dimensional (2D) positive systems, by means of positive scalar inputs, is addressed. The combinatorial nature of this property allows for a graph theoretic approach. Indeed, to every 2D positive system of dimension n with scalar inputs one can associate a 2D influence graph with n vertices, one source and two types of arcs...

In this paper we consider a special class of 2D convolutional codes (composition codes) with encoders G(d 1, d 2) that can be decomposed as the product of two 1D encoders, i.e., \(G(d_{1},d_{2}) = G_{2}(d_{2})G_{1}(d_{1})\). In case that \(G_{1}(d_{1})\) and \(G_{2}(d_{2})\) are prime we provide constructions of syndrome formers of the code, direct...

REALIZATION THEORY What a realization is, properties of minimal realizations, constructing a realization for a scalar rational function W(s), constructing a realization for a rational matrix W(s), the impulse content of a generalized linear system (GLS), McMillan degree, transfer functions product, Smith canonical form, Smith McMillan canonical for...

A PRIMER OF LEAST SQUARES THEORY Bilinear forms, inner product vector spaces, real vector spaces with a positive (semi)definite inner product, Riccati differential equation with scalar coefficients, optimum problems and matrix differential Riccati equation, finite horizon optimal control problems, infinite horizon optimal control and algebraic Ricc...

DISSIPATIVE SYSTEMS : General properties of dissipative systems, reachability versus dissipativity, lossless dissipative systems, dissipativity conditions for a linear system, lossless linear dissipative systems, dissipativity in generalized linear systems

Appendix (an outline of Linear Algebra), Index, References

AN OUTLINE OF CONTINUOUS LINEAR SYSTEMS THEORY Definition of dynamical system, state space properties, time invariant continuous linear systems, generalized (= non proper) linear systems, controllability, observability, elementary analysis of electrical networks

SOLVING THE RICCATI EQUATION Solutionof the differential Riccati equation, solution of ARE via Jordan canonical form, iterative techniques for the solution of ARE

LAPLACE DOMAIN ANALYSIS OF THE INPUT/OUTPUT MAP OF LINEAR DISSIPATIVE SYSTEMS Some facts from the theory of analytical functions, hermitian forms and matrices, bounded real (BR) matrices, positive real (PR) matrices, connections between PR and BR matrices, lossless BR and PR matrices, diffusion matrix, dissipativity conditions for a (G)LS in the La...

ALGEBRAIC STRUCTURE OF DISSIPATIVE LINEAR SYSTEMS The positive real lemma and the dissipation function, limiting solutions of the PR lemma, dissipative realizations and non real-reduced matrices, the PR lemma for lossless linar systems, computational aspects, spectral factorization, structure of spectral factors

In this paper we address various forms of identification problems for Boolean networks (BNs) and for Boolean control networks (BCNs). For the former, we assume to have a set of infinite or finite support output trajectories and we want to identify a BN compatible with those trajectories. Conditions for the identified BN to coincide (up to a relabel...

Given a single-input continuous-time positive system, described by a pair (A, b)$, with A a diagonal matrix, we investigate under what conditions there exists a state-feedback law u(t) = c^T x(t) that makes the resulting controlled system positive and asymptotically stable, by this meaning that A+bc^T$ is Metzler and Hurwitz. In the second part of...

In this paper we address the optimal control problem for Boolean control networks (BCNs). We first consider the problem of finding the input sequences that minimize a given cost function over a finite time horizon. The problem solution is obtained by means of a recursive algorithm that represents the analogue for BCNs of the difference Riccati equ...

In this paper we address various forms of identification problems for Boolean networks (BNs) and for Boolean control networks (BCNs). For the former, we assume to have a set of infinite or finite support output trajectories and we want to identify a BN compatible with those trajectories. Conditions for the identified BN to coincide (up to a relabel...

Given a single-input continuous-time positive system, described by a pair (A, b), with A a diagonal matrix, we investigate under what conditions there exist state-feedback laws u(t) = cTx(t) that make the resulting controlled system positive and asymptotically stable, namely A + bcT Metzler and Hurwitz. In the second part of the paper we assume tha...

Continuous-time positive systems, switching among p subsystems whose matrices differ by a rank one matrix, are introduced, and a complete characterization of the existence of a common linear copositive Lyapunov function for all the subsystems is provided. Also, for this class of systems it is proved that a well-known necessary condition for asympto...

In this paper we consider discrete-time positive switched systems, switching among autonomous subsystems, characterized either by monomial matrices or by circulant matrices. Necessary and sufficient conditions are provided guaranteeing either (global uniform) asymptotic stability or stabilizability (i.e. the possibility of driving to zero the state...

The aim of this paper is to introduce and characterize observability and reconstructibility properties for Boolean networks and Boolean control networks, described according to the algebraic approach proposed by D. Cheng and co-authors in the series of papers \cite{Cheng2009, Cheng2010b,Cheng2010a} and in the recent monography \cite{BCNCheng}. A co...

In this note we first characterize the periodic trajectories (or, equivalently, the limit cycles) of a Boolean network, and their global attractiveness. We then investigate under which conditions all the trajectories of a Boolean control network may be forced to converge to the same periodic trajectory. If every trajectory can be driven to such a p...

In this paper we investigate the connections between Boolean Control Networks (BCNs), in particular Boolean Networks (BNs), and symbolic dynamics (SD). We show that the set of state-space trajectories of a BN is a shift space of finite type (SFT). Similarly, the sets of state-space and of input-state trajectories of a BCN are both SFTs. Some of th...

In this paper we address observability and reconstructibility of Boolean networks and of Boolean control networks. By resorting to the algebraic approach recently introduced by D. Cheng and co-authors [2], [4], [5], [6], we provide necessary and sufficient conditions for these properties to hold, based on the Boolean matrices involved in the networ...

In this paper we consider the class of discrete-time switched systems switching between $p$ autonomous positive subsystems. First, sufficient conditions for testing stability, based on the existence of special classes of common Lyapunov functions, are investigated, and these conditions are mutually related, thus proving that if a linear copositiv...

In this paper we consider discrete-time positive switched systems, switching among autonomous subsystems, characterized either by cyclic monomial matrices or by circulant matrices. For these two classes of systems, some interesting necessary and sufficient conditions for stability and stabilizability are provided. Such conditions lead to simple alg...

Continuous-time positive systems, switching among $p$ subsystems, are introduced, and a complete characterization for the existence of a common linear copositive Lyapunov function for all the subsystems is provided. When the subsystems are obtained by applying different feedback control laws to the same continuous-time single-input positive system,...

In this paper we consider the class of discrete-time switched systems switching between two autonomous positive subsystems. It is shown that if these systems are stabilizable, they can be stabilized by means of a periodic switching sequence, which asymptotically drives to zero every positive initial state. Necessary and sufficient conditions for t...

In this paper, monomial reachability and reachability properties for a special class of discrete-time positive switched systems are investigated. Necessary and sufficient conditions for these properties to hold, together with some examples, are provided.

A convolutional code can be decomposed into smaller codes if it admits decoupled encoders. In this paper we show that if a code can be decomposed into smaller codes (subcodes), its column distances re the minimum of the column distances of iys subcodes. Moreover, the j-th column distance of a convolutional code C is equal to the j-th column distanc...

Reachability and observability of two-dimensional (2D) discrete state-space models are introduced in two different forms: a local form, which refers to single local states, and a global form, which pertains to the infinite set of local states lying on a separation set. While local reachability and observability can be naturally characterized by res...

Abstract—When dealing with two-dimensional (2-D) discrete state-space models, controllability properties are introduced in two different forms: a local form, which refers to single local states, and a global form, which instead pertains the infinite set of local states lying on a separation set. In this paper, these concepts are investigated in the...

In this paper, (local/global) reachability and observability are introduced in the context of two-dimensional (2D) positive systems. While local reachability and observability are naturally characterized by resorting to state space techniques, their global versions are better investigated via a polynomial approach. Necessary and sufficient conditio...

In this paper, polynomial matrix fraction descriptions (MFD’s) are used as a tool for investigating the structure of a (linear) convolutional code C and the family of its rational (in particular, polynomial) encoders and syndrome formers. Given a rational causal encoder G(d) of C, if we represent G(d) via an irreducible MFD G(d)= D(d)^{-1}N(d) su...

Positive systems in the behavioural approach are introduced as sets of non-negative trajectories that satisfy a closure condition with respect to linear combinations with non-negative coefficients. Completeness and finite memory properties are discussed and compared with the analogous properties of linear shift invariant behaviours.

The dominant state plays an essential role in the asymptotic analysis of dynamical systems. As global states of a 2D system are bilateral sequences, the existence of a dominant state implies that the free evolution of 2D global states converges to a suitable sequence, up to the multiplication by a normalizing factor. In this contribution the existe...

Positive systems in the behavioural approach are introduced as sets of non-negative trajectories that satisfy a closure condition with respect to linear combinations with non-negative coefficients. Completeness and finite memory properties are discussed and compared with the analogous properties of linear shift invariant behaviours.

The paper discusses the possibility of characterizing some important properties of convolutional codes and its encoders and syndrome formers by means of matrix fraction descriptions and state space models. A complete parametrization is then provided for all minimal encoders and minimal syndrome formers of a given code. Finally state feedback and st...

The paper investigates the possibility of synthesizing a positive system in state space form as a (series, parallel and feedback) interconnection of a finite number of elementary positive systems.

Bilinear systems in input-output form are introduced and represented by means of suitable rational functions in two indeterminates. Necessary and sufficient conditions for BIBO stability are derived, and compared with analogous results on 2D systems stability. Several features of the output evolutions, corresponding to finite support as well as to...

The main features of finite multidimensional behaviors are introduced as properties of the trajectories supports, and connected with the polynomial matrices adopted for their description. Observability and local detectability are shown to be equivalent to the kernel representation of a behavior via some parity check matrix H^T. The main properties...

In this paper the primitivity of a positive matrix pair (A,B) is introduced as a strict positivity constraint on the asymptotic behavior of the associated two-dimensional (2D) state model. The state evolution is first considered under the assumption of periodic initial conditions. In this case the system evolves according to a one-dimensional state...

The aim of this contribution is to discuss some basin issues connected with bilinear input/output maps as described in a paper by E.Fornasini and G.Marchesini in J.Franklin Inst., vol. 301, pp. 143-160 (1976). We consider first stability problems and derive necessary and sufficient conditions guaranteeing that bounded inputs produce bounded outputs...

In this paper, different primeness definitions and factorization properties, arising in the context of nD Laurent polynomial matrices, are investigated and applied to a detailed analysis of nD finite support signal families produced by linear multidimensional systems. As these families are closed w.r.t. linear combinations and shifts along the coor...

Pairs of linear transformations on a finite dimensional vector space are of great relevance in the analysis of two-dimensional (2D) systems evolutions. In this paper, special properties of matrix pairs, such as finite memory, separability, property~L and property~P, as well as their dynamical interpretations, are investigated. Practical criteria fo...

Two-dimensional (2D) positive systems are 2D state space models whose variables take only nonnegative values and, hence, are described by a family (A;B;M;N;C;D) of nonnegative matrices. In the paper the notions of asymptotic and simple stability, corresponding to arbitrary set of nonnegative initial conditions, are introduced and related to the spe...

In the paper the definition and main properties of a 2D-digraph, namely a directed graph with two kinds of arcs, are introduced. Under the assumption of strong connectedness, the analysis of its paths and cycles is performed, basing on an integer matrix whose rows represent the compositions of all circuits, and on the corresponding row-module. Natu...

Two-dimensional (2D) compartmental models are 2D positive systems obeying some conservation law, and hence described by matrix pairs with substochastic sum. A canonical form, to which all 2D compartmental models reduce, is derived, allowing for a completeanalysis of stability properties. The relevance of these models is illustrated by two examples...

Pairs of linear transformations on a finite dimensional vector space are of great relevance in the analysis of two-dimensional (2D) systems evolutions. In this paper, special properties of matrix pairs, such as finite memory, separability, property L and property P, as well as their dynamical interpretations, are investigated. Practical criteria fo...

In the paper the definition and main properties of a 2-digraph, i.e. a directed graph with two kinds of arcs, are introduced. Natural constrains on the composition of the paths connecting each pair of vertices lead to the definition of 2-strongly connected digraph and of 2-imprimitivity classes. Irreducible matrix pairs, that is pairs endowed with...

The dynamics of a 2D positive system depends on the pair of nonnegative square matrices that provide the updating of its local states. In this paper, several spectral properties, like finite memory, separability and property L, which depend on the characteristic polynomial of the pair, are investigated under the nonnegativity constraint and in con...

2D system dynamics depends on matrix pairs which represent the shift operators along coordinate axes. The structure of a matrix pair is analysed referring to its characteristic polynomial and to the traces of suitable matrices in the algebra generated by the elements of the pair. Necessary and sufficient conditions for properties L and P are provid...

Homogeneous 2D positive systems are 2D state space models whose variables are always nonnegative and, consequently, are described by a pair of nonnegative square matrices (A, B). In the paper, the properties of these pairs are discussed both in the general case and under particular assumptions like finite memory, separability and property L. Vario...

In this contribution the local structure of finite support $n$D signals is analysed according to the behavioral approach. Observability and local detectability are introduced, and characterized in terms of polynomial matrix descriptions. The problem of extending into a legal trajectory a set of data that satisfies the parity checks on a suitable s...

The paper analysises the internal properties (controllability, observability, detectability and extendability) of finite support multidimensional behaviors and the relations with their polynomial matrix descriptions.

A 2D positive linear system is a positive state model whose variables depend on two independent integer indices, according to a quarter plane causality law. Here we restrict our investigation to unforced 2D state motions, described by the following updating equation {\bf x}(h+1,k+1) = A {\bf x}(h,k+1) + B {\bf x}(h+1,k), where the doubly indexed...

Several characterizations of finite memory and separability properties for a 2D system are presented, in terms of both the characteristic polynomial and the matrix pair that describes the state evolution. Necessary and sufficient conditions for a finite memory or separable 2D system to have an inverse with the same properties are given; these invol...

Ch.1 INTRODUCTION [Representation problems, structural properties, solutions analysis and dynamics control] Ch.2 2D MODELS STRUCTURE [Preliminary remarks, 2D input/output maps, ARMA models, local state models, models equivalence, global state models, the realization problem] Ch.3 PROPERTIES OF 2D STATE MODELS [Reachability, observability, minimal...

Finite support nD output behaviors are discussed and various connections with primeness notions of nD FIR transfer functions are investigated.

Two-dimensional (2D) codes are introduced as linear shift-invariant spaces of admissible signals on the discrete plane. Convolutional and, in particular, basic codes are characterized both in terms of their internal properties and by means of their input-output representations. The algebraic structure of the class of all encoders that correspond to...