Search

The area of signal processing is very wide. In this review paper we will focus on how the theory of nonnegative matrices may arise in the field of digital filter design: an area in which the role of nonnegative matrices is increasingly relevant due to the development of new technologies in the field of optical components and charge coupling devices.

This paper is a tutorial on the positive realization problem, that is the problem of finding a positive state-space representation of a given transfer function and characterizing existence and minimality of such representation. This problem goes back to the 1950s and was first related to the identifiability problem for hidden Markov models, then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory. Within this framework, developing some ideas sprang in the 1960s, during the 1980s, the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem. In this paper, the reader is carried through the key ideas which have proved to be useful in order to tackle this problem. In order to illustrate the main results, contributions and open problems, several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided.

Compartmental systems are composed of a finite number of subsystems, called compartments, interacting by exchanging material. We propose a set of constraints ensuring compartmentality in an identification process.

Positive systems are systems in which the input/state/output variables are always positive since they represent quantities. We propose an identification procedure for a class of positive linear systems.

When dealing with compartmental systems, an important question is: given an experiment, i.e., an input-output sequence, and supposing there is no error in the data, is the sequence compatible with the compartmental assumption? If the process under analysis is linear, then the previous question is obviously equivalent to asking whether a given transfer function is that of a compartmental system. In this note we provide an answer to the latter question giving necessary and sufficient conditions for a transfer function to be that of a compartmental system of some finite order (i.e., number of compartments). Another problem tackled in this note originates from the observation that in many cases one wants to determine the number of compartments involved in the process. In this note we report a step toward the solution of this fundamental problem by proving necessary and sufficient conditions for a given third order transfer function with real poles to be that of a compartmental system with three compartments.

The linear filters characterized by a state-variable realization given by matrices with nonnegative entries (called positive filters) are heavily restricted in their achievable performance. Nevertheless, such filters are the only choice when dealing with the charged coupled device MOS technology of charge routing networks (CRN's), since nonnegativity is a consequence of the underlying physical mechanism. In order to exploit the advantages offered by this technology, the authors try to overcome the above-mentioned limitation by realizing an arbitrary transfer function as a difference of two positive filters.

The positive realization problem for linear systems is to find conditions, for a given transfer function with nonnegative impulse response, to have a realization such that the resulting system is a positive system. Recently, it has been shown that, under a mild assumption on the long-term behaviour of the impulse response, this problem is related to the maximum modulus poles only. In this paper necessary and sufficient conditions for positive realizability of discrete-time systems are given. They show that also nondominant poles play a role in the most general case. Positive realizability conditions for the continuous-time case are also given.

Let H(z) be a rational transfer function, with associated nonnegative impulse response sequence. The paper considers the question: When does there exist a triple A/spl isin/R/sup N/spl times/N/, b/spl isin/R/sup N/, c/spl isin/R/sup N/ with all nonnegative entries H(z)=c'(zI-A)/sup -1/b? An essentially complete characterization is given of the H(z) allowing such a realization, in terms of the location of the pole or poles of H(z) with maximum modulus.

The problem of realizing arbitrary transfer functions as combinations of positive filters, or, more in general, of representing arbitrary systems by means of Internally Positive Representations (IPRs), has been widely investigated in the discrete-time framework. An IPR is a positive state-space representation, endowed with input, state and output transformations, that realizes arbitrary input-state-output dynamics. This technical note investigates the problem of the IPR construction for continuous-time systems, and proposes a technique that provides stable IPRs for a particular class of systems.

Positive systems are systems in which the state variables take only nonnegative values for all times. In this paper, we derive eigenvalue regions for discrete and continuous-time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix.

A discrete-time positive linear system has the property that any nonnegative initial state and any nonnegative input produces a nonnegative trajectory in the state space and output for all time. Such systems are R n+ - reachable if every state of the nonnegative orthant R n+ is reachable from origin with a nonnegative input function, in a finite-time interval. A simple necessary and sufficient condition for R n+ - realizability , i.e. for the existence of a nonnegative realization that is also R n+ - reachable , is stated. The realization procedure is straightforward.

The linear filters characterized by a state-variable realization given by matrices with nonnegative entries (called positive filters) are heavily restricted in their achievable performance. Nevertheless, such filters are the only choice when dealing with the charged coupled device MOS technology of charge routing networks (CRNs), because nonnegativity is a consequence of the underlying physical mechanism. In order to exploit the advantages offered by this technology, the authors try to overcome the above-mentioned limitation by realizing an arbitrary transfer function as a difference of two positive filters.

A standard result of linear system theory states that an SISO proper rational transfer function of degree n always has a realization of dimension n. In some applications one is interested in having a realization with nonnegative entries and it is known that, when the dominant poles display a specific pattern, forcing nonnegativity leads to a system which is not jointly reachable and observable. In this paper, we show that the minimal dimension of a positive realization may be ‘large’ even in the case of a single dominant pole. More precisely, we provide a family of transfer functions, each of which is of degree n=3, such that for any integer N⩾3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than N.

SUMMARY In this paper the problem of stabilizing uncertain linear discrete-time systems under state and control linear constraints is studied. Many formulations of this problem have been given in the literature. Here we consider the case of finding a linear state feedback control law making a given polytope in the state space positively invariant while the control remains bounded within prefixed values under the e⁄ect of all the uncertainty sequences belonging to a given polytope in the perturbations space. A necessary and suƒcient condition for the existence of a solution of this problem is first given. This condition leads to a set of linear constraints which can be solved using linear programming tecniques by defining an appropriate objective function. A worked example shows the e⁄ectiveness of the proposed algorithm. ( 1998 John Wiley & Sons, Ltd.

Charge Routing Network (CRN) is a term which indicates a family of functional solid-state electronic devices using MOS charge-coupling technology. Under the application of a sequence of clock pulses, these devices move quantities of electrical charge across a semiconductor substrate thus performing a wide range of electronic functions including image sensing, data storage, signal processing or logic operations. A brief description of a CRN is first given and its input-output relationship is derived. Then, under the assumption of a unique dominant pole, the class of transfer functions realizable using CRNs is characterized, and a procedure to realize a filter of this class is given. Finally, using the technique here developed, a low-pass filter is designed.

This paper deals with the positive realization problem. The problem is to find, from a given transfer function, a state equation in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative. Necessary conditions are investigated and a new one is given, together with some related results.

This paper deals with the realization problem of linear systems in which a sign pattern is imposed on the realization{ A b c T }. This problem arises in the context of positive linear systems, i.e. systems in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative. Using a unitary framework for continuous-time and discrete-time systems we derive necessary and sufficient conditions for positive realizability by means of convex analysis. Moreover, the complete solution in the case of a transfer function of McMillan degree n ⩽ 3 is obtained.

A well-known result from linear system theory states that the minimal inner size of a factorization of the Hankel matrix H of a system gives the minimal order of a realization. In this work it is shown that when dealing with positive linear systems, the existence of a factorization of the Hankel matrix into two nonnegative matrices is only a necessary condition for the existence of a positive realization of order equal to the inner size of the factorization. Necessary and sufficient conditions for the minimality of a positive realization in terms of positive factorization of the Hankel matrix are then derived.