Article
Full entry |
PDF
(1.4 MB)
Feedback
PDF
(1.4 MB)
Feedback
Keywords:
positive systems; positiverealizations
positive systems; positiverealizations
Summary:
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.
References:
[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 1427775 | Zbl 0864.93032
[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 1375779
[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 0879.93029
[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 1631075 | Zbl 0951.93013
[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 10.1016/S0167-6911(98)00098-X | MR 1752017 | Zbl 0915.93014
[6] Benvenuti L., Farina L.: The design of fiber optic filters. IEEE/OSA Journal of Lightwave Technology 19 (2001), 6, 1366–1375 DOI 10.1109/50.948284
[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 1803660 | Zbl 1030.93009
[8] Berman A., Neumann, M., Stern R. J.: Nonnegative Matrices in Dynamic Systems. Wiley, New York 1989 MR 1019319 | Zbl 0723.93013
[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 10.2307/3215060 | MR 1385346 | Zbl 0855.60071
[10] Farina L.: On the existence of a positive realization. System Control Lett. 28 (1997), 219–226 DOI 10.1016/0167-6911(96)00033-3 | MR 1411875
[11] Farina L., Rinaldi S.: Positive Linear Systems: Theory And Applications. Wiley Interscience, New York 2000 MR 1784150 | Zbl 0988.93002
[12] Foerster K. H., Nagy B.: Nonnegative realizations of matrix transfer functions. Linear Algebra Appl. 311 (2000), 107–129 MR 1758208 | Zbl 0989.93016
[13] Gersho A., Gopinath B.: Charge-routing networks. IEEE Trans. Circuits and Systems 26 (1979), 1, 81–92 DOI 10.1109/TCS.1979.1084617 | MR 0521656 | Zbl 0394.93054
[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 10.1016/S0167-6911(99)00005-5 | MR 1747805
[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 0043063
[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 10.1093/biomet/33.3.183 | MR 0015760 | Zbl 0060.31803
[19] Luenberger D. G.: Introduction to Dynamic Systems. Wiley, New York 1979 Zbl 0458.93001
[20] Minc H.: Nonnegative Matrices. Wiley, New York 1987 MR 0932967 | Zbl 0638.15008
[21] Niewenhuis J. W.: About nonnegative realizations. System Control Lett. 1 (1982), 4, 283–287 DOI 10.1016/S0167-6911(82)80024-8 | MR 0670213
[22] O’Cinneide C. A.: Characterization of phase–tipe distributions. Stochastic Models 6 (1990), 1, 1–57 DOI 10.1080/15326349908807134 | MR 1047102
[23] O’Cinneide C. A.: Phase–tipe distributions: open problems and a few properties. Stochastic Models 15 (1999), 731–757 DOI 10.1080/15326349908807560 | MR 1708454
[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 10.1137/0322013 | MR 0732422 | Zbl 0539.93005
[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 0582802 | Zbl 0395.93032
[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 0133176 | Zbl 0228.60042
[28] Valcher M. E.: Controllability and reachability criteria for discrete–time positive systems. Internat. J. Control 65 (1996), 511–536 DOI 10.1080/00207179608921708 | MR 1657570 | Zbl 0873.93009
[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 1624540
Search
Partner of
