Ph.D. received on: 17/4/1998

E-mail: : farina@dis.uniroma1.it

Tutor: Prof. S. Monaco, Università di Roma ‘La Sapienza’
 
 

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Positive systems and the positive realization problem  __________________________________________________________________________________________________________

Advisor:

Prof. S. Monaco, Università di Roma ‘La Sapienza’

Summary:

Positive systems are systems in which the state and output variables remain nonnegative for any non negative input sequence. Positive systems are quite common in applications in which input, output and state variables represent non negative quantities such as, for example, when dealing with animal or human populations, goods consumption, densities of chemicals and so forth. Positive systems are, just to cite a few, compartmental systems used, for example, to model drugs dynamics in humans, partially observed Markov chains  (hidden Markov models) mainly adopted for speech recognition, age-structured population models (Leslie models) and the Leontieff models used in the analysis of prices in economic systems.
The positive realization problem is a long standing problem (the oldest reference seems to be H. Furstenberg, Stationary Processes and Prediction Theory, Annals of Mathematical Studies, Princeton University Press, 1960) with a significant theoretical value both in a deterministic and stochastic setting. The relevance of this problem is well documented in many diverse applicative fields such as, for example, in the identification of compartmental systems, in the filtering of data generated by hidden Markov sequences and in the design of digital filters using a recent CCD-MOS technology (Charge Routing Networks). The basic reference of the thesis is a geometric interpretation of the positive realization problem in term of invariant cones given by H. Maeda and S. Kodama, Positive realization of difference equation, IEEE Trans. on Circuits and Systems, 28, 39-47 (1981).
The thesis is organized as follows. The first section describes the most important known results on positive systems. The notions of positivity, influence graph, irreducibility, stability and invariance are introduced. Such concepts are often tightly linked with the theory of non negative matrices (i.e. with nonnegative entries). Therefore, the results presented in this section, are not due to the author but are taken from the literature. The second section contain the results of the research work on this topic. Section 2.2.1 contains the geometric conditions for the existence of a positive realization in terms of polyhedral invariant cones. In Section 2.2.2 necessary and sufficient conditions for the polyhedrality of such cones in terms of the spectrum of the dynamic matrix, are stated. The consequences of this property on the finite time reachability problem, are also discussed. The complete solution to the positive realization problem is then presented in Section 2.2.3. Finally, Section 2.3 deals with the minimality problem and provides necessary and sufficient conditions for minimality basing on the Hankel matrix and, for the third order case, directly on the transfer function. It is worth noting that all the proofs presented in throuout the thesis are constructive, that is, once the conditions of theorems are fulfilled, a positive realization can be found in a straightforward manner.

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