Minimal positive realizations: a survey of recent results and open problems

Benvenuti, Luca   ; Farina, Lorenzo

Kybernetika, Tome 39 (2003) no. 2, pp. 217-228

Abstract (VO)

In this survey paper some recent results on the minimality problem for positive realizations are discussed. In particular, it is firstly shown, by means of some examples, that the minimal dimension of a positive realization of a given transfer function, may be much “larger” than its McMillan degree. Then, necessary and sufficient conditions for the minimality of a given positive realization in terms of positive factorization of the Hankel matrix are given. Finally, necessary and sufficient conditions for a third order transfer function with distinct real positive poles to have a third order positive realization are provided and some open problems related to minimality are discussed.

Détail

Classification : 06F05, 93B15, 93B27
Keywords: positive systems; positiverealizations

Bibliographie

[1] Anderson B. D. O.: New developments in the theory of positive systems. In: Systems and Control in the Twenty–first Century (Byrnes, Datta, Gilliam, and Martin, eds.), Birkhäuser, Boston 1997 | MR | Zbl

[2] Anderson B. D. O., Deistler M., Farina, L., Benvenuti L.: Nonnegative realization of a linear system with nonnegative impulse response. IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications 43 (1996), 2, 134–142 | MR

[3] Benvenuti L., Farina L.: On the class of linear filters attainable with charge routing networks. IEEE Trans. Circuits and Systems II: Analog and Signal Processing 43 (1996), 2, 618–622 | Zbl

[4] Benvenuti L., Farina L.: A note on minimality of positive realizations. IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications 45 (1998), 6, 676–677 | MR | Zbl

[5] Benvenuti L., Farina L.: An example of how positivity may force realizations of “large” dimension. System Control Lett. 36 (1999), 4, 261–266 | DOI | MR | Zbl

[6] Benvenuti L., Farina L.: The design of fiber optic filters. IEEE/OSA Journal of Lightwave Technology 19 (2001), 6, 1366–1375 | DOI

[7] Benvenuti L., Farina L., Anderson B. D. O., Bruyne F. De: Minimal positive realizations of transfer functions with positive real poles. IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications 47 (2000), 9, 1370–1377 | MR | Zbl

[8] Berman A., Neumann, M., Stern R. J.: Nonnegative Matrices in Dynamic Systems. Wiley, New York 1989 | MR | Zbl

[9] Commault J., Chemla J. P.: An invariant of representations of phase–type distributions and some applications. J. Appl. Probab. 33 (1996), 368–381 | DOI | MR | Zbl

[10] Farina L.: On the existence of a positive realization. System Control Lett. 28 (1997), 219–226 | DOI | MR

[11] Farina L., Rinaldi S.: Positive Linear Systems: Theory And Applications. Wiley Interscience, New York 2000 | MR | Zbl

[12] Foerster K. H., Nagy B.: Nonnegative realizations of matrix transfer functions. Linear Algebra Appl. 311 (2000), 107–129 | MR | Zbl

[13] Gersho A., Gopinath B.: Charge-routing networks. IEEE Trans. Circuits and Systems 26 (1979), 1, 81–92 | DOI | MR | Zbl

[14] Hadjicostis C.: Bounds on the size of minimal nonnegative realization for discrete–time lti systems. System Control Lett. 37 (1999), 1, 39–43 | DOI | MR

[15] Kajiya F., Kodama, S., Abe H.: Compartmental Analysis – Medical Applications and Theoretical Background. Karger, Basel 1984

[16] Karpelevich F. I.: On the characteristic roots of matrices with nonnegative elements (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 361–383 | MR

[17] Leontieff W. W.: The Structure of the American Economy 1919–1939. Oxford Univ. Press, New York 1951

[18] Leslie P. H.: On the use of matrices in certain population mathematics. Biometrika 35 (1945), 183–212 | DOI | MR | Zbl

[19] Luenberger D. G.: Introduction to Dynamic Systems. Wiley, New York 1979 | Zbl

[20] Minc H.: Nonnegative Matrices. Wiley, New York 1987 | MR | Zbl

[21] Niewenhuis J. W.: About nonnegative realizations. System Control Lett. 1 (1982), 4, 283–287 | DOI | MR

[22] O’Cinneide C. A.: Characterization of phase–tipe distributions. Stochastic Models 6 (1990), 1, 1–57 | DOI | MR

[23] O’Cinneide C. A.: Phase–tipe distributions: open problems and a few properties. Stochastic Models 15 (1999), 731–757 | DOI | MR

[24] Ohta Y., Maeda, H., Kodama S.: Reachability, observability and realizability of continuous-time positive systems. SIAM J. Control Optim. 22 (1984), 2, 171–180 | DOI | MR | Zbl

[25] Picci G.: On the Internal Structure of Finite–State Stochastic Processes. (Lecture Notes in Economics and Mathematical Systems 162). Springer–Verlag, Berlin 1978 | MR | Zbl

[26] Rabiner L. R.: A tutorial on hidden markov models and selected applications in speech recognition. Proc. IEEE 77 (1989), 257–286

[27] Saaty T. L.: Elements of Queueing Theory. McGraw–Hill, New York 1961 | MR | Zbl

[28] Valcher M. E.: Controllability and reachability criteria for discrete–time positive systems. Internat. J. Control 65 (1996), 511–536 | DOI | MR | Zbl

[29] Schuppen J. H. van: Stochastic realization problems motivated by econometric modelling. In: Identification and Robust Control (Byrnes and Lindquist, eds.), Elsevier 1986

[30] Picci G., Schuppen J. H. van: Primes in several classes of the positive matrices. Linear Algebra Appl. 277 (1998), 1–3, 149–185 | MR