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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.

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.

Fiber-optic filters can be used for processing wideband radio-frequency, microwave, and millimetric signals directly in the optical domain avoiding costly electrooptic and optoelectronic conversions. Because these filters sum signals on an intensity basis, they are bound to be positive and, hence, heavily restricted in their achievable performance. Nevertheless, such filters are the only choice because positivity is a consequence of the underlying physical mechanism. In this paper, the authors present a new design methodology to realize an arbitrary filter as a difference of two positive filters, thus overcoming any performance limitation.

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.

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.

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.