Abstract
Consider a discrete+time (internally) positive system described by the equations
where \({x_i} \in R_ + ^n,{u_i} \in R_ + ^m,{y_i} \in R_ + ^p\) are the state, input and output vectors, respectively and \( A \in R_ + ^{nxn} ,B \in R_ + ^{nxm} ,C \in R_ + ^{pxn} ,D \in R_ + ^{pxm} . \)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Kaczorek, Theory of Control systems, PWN, Warszawa, 1999 (in Polish)
J.E. Cohen, Nonnegative ranks, decompositions and factorizations of nonnegative matrices, Linear Algebra and Applications, Vol. 190, 1993, 149–168.
T. Kaczorek, Realisation problem for completely singular linear systems, SPETO’98, Ustro 20-22.05. 1998, 315–320.
B.D.O. Anderson, New developments in the theory of positive systems, Systems and Control, 1996, 17–36.
B.D.O. Anderson, M. Deistler, L. Farina and L. Benvenuti, Nonnegative realization of a linear system with nonnegative impulse response, IEEE Trans. on Circiuts and Systems, Vol. 43, No. 2, 1996, 134–142.
L. Benvenuti and L. Farina, A note on minimality of positive realizations, IEEE Trans. Circuits and Syst. I, Vol. 45, No. 6, 1998, 676–677.
L. Benvenuti and L. Farina, discrete-time filtering via charge routing networks, Signal Processing, 49, 1996, 207–215.
R. Eising, 2-D Systems: An Algebraic Approach, Mathematical Centrum, Amsterdam, 1979.
R. Eising, Realization and stabilization of 2-D systems, IEEE Trans. Autom. Contr., vol. AC-23, No. 5, 1978, 795–799.
M.P. Fanti, B. Maione and B. Turchiano, Controllability of linear single-input positive discrete-time systems, Int. J. Control, Vol. 50, No. 6, 1989, 2523–2542.
M.P. Fanti, B. Maione and B. Turchiano, Controllability of multi-input positive discrete-time systems, Int. J. Control, Vol. 51, No. 6, 1990, 1295–1308.
L. Farina, Minimal order realizations for a class of positive linear systems, J. Franklin Inst. Vol. 333B, No. 6, 1996, 893–900.
L. Farina, On the existence of a positive realization, Systems & Control Letters, No. 28, 1996, 219–226.
L. Farina, A note on discrete-time positive realizations, Systems & Control Letters 22, 1994, 467–469.
L. Farina, Necessary conditions for positive realizability of continuous-time linear systems, Systems and Control Lett., 25, 1995, 121–124.
L. Farina and L. Benvenuti, Positive realizations of linear systems, Systems and Control Lett., 26, 1995, 1–9.
L. Farina and L. Benvenuti, Polyedral reachable set with positive controls, Mathematics of Control, Signals and Systems 10, 1997, 364–380.
L. Farina and S. Rinaldi, Positive Linear Systems, Theory and Applications, J. Wiley, New York 2000
E.Fomasini, G.Marchesini, State space realization of two-dimensional filters, IEEE Trans.Autom. Control, AC-21, 1976, 484–491.
E.Fomasini, G.Marchesini, Doubly indexed dynamical systems: State space models and structural properties, Math. Syst. Theory 12, 1978.
T. Hinamoto and F.W. Fairman, Realisation of the Attasi state space model for 2-D filters. Int. J. Systems Sci., Vol. 15, No. 2, 1984, 215–228
J.M. van den Hof, Realization of positive linear systems, Linear Algebra and its Applications, 1997, 287–308.
J.M. van den Hof and J.H. van Schuppen, Realization of positive linear systems using polyhedral cones, Proceedings 33 IEEE Conference on Decision and Control, Lake Buena, Vista, FL, 1994, 3889–3893.
T. Kaczorek, Realization problem for general model of two-dimensional linear systems, Bull. Acad. Pol. Sci. Techn. Sci., Vol. 35, No. 11-l2, 1987, 633–637.
T. Kaczorek, Reachability and controllability of nonnegative 2-D Roesser type models, Bull. Pol. Acad. Techn. Sci., Vol. 44, No. 4, 1996, 405–410.
T. Kaczorek, Reachability and minimum energy control of nonnegative 2-D Roesser type models, 14 Triennial World Congress, IFAC, Peking, China, 5-9.07, 1999, 379–384
T. Kaczorek, Realisation problem for discrete-time positive linear systems, Appl. Math. and Compo Sc., Vol. 7, No. 1, 1997, 117–124.
T. Kaczorek, Linear Control Systems, Vol. 2, Research Studies Press and J. Wiley, New York, 1993.
T. Kaczorek, Realization problem, reachability and minimum energy control of positive 2D Roesser type model, 6th Annual International Conference on Advances in Communication and Control, 23-27 June 1997, Corfu (Greece), 765–776
T. Kaczorek, Two-Dimensional Linear Systems, Springer-Verlag, New York, 1985.
T. Kaczorek, When the local controllability of the general model of 2-D linear systems implies its local reachability, Systems and Control Letters, Vol. 23, 1994, 445–452.
T. Kaczorek, When the local controllability of Roesser model implies its local reachability, Bull. Pol. Acad. Techn. Sci., Vol. 41, 1994, 261–267.
T. Kaczorek, U-Reachability and U-Controllability of 2-D Roesser Model, Bull. Pol. Acad. Techn. Sci., vol. 43, No 1, 1995, pp. 31–37.
T. Kaczorek, Positive realisations of improper transfer matrices of discretetime linear systems, Bull. Pol. Acad. Techn. Sci., Vol.45, No 2, 1997, 277–286.
T. Kaczorek, Positive stable realizations for linear systems, Bull. Pol. Acad. Techn. Sci., Vol. 45, No. 4, 1997, 549–557.
T. Kaczorek, Realization problem for 2-D positive systems, 2nd IFAC Workshop on New Trends in Design of Control Systems, Sept. 7-10. 1997, Smolenice, Slovakia, 502–507.
T. Kaczorek, Positive 2D linear systems, 9th International Symposium on System Modelling Control, Zakopane 27.04.-1.05. 1998, 50–67
T. Kaczorek, Positive realization in canonical form of the 2D Roesser type model, Proc. Control and Decision Conf, San Diego, 1997, 335–336
T. Kaczorek, Realisation problem for singular 2D linear systems, Bull. Pol. Acad. Techn. Sci., Vol. 46 No. 3, 1998, 317–330.
T. Kaczorek, Positive singular discrete linear systems, Bull. Pol. Acad. Techn. Sci., Vol. 45. No. 4, 1997, 619–631
T. Kaczorek, Positive linear systems and their relationship with electrical circuits, XX Seminarium z Podstaw Elektrotechniki i Teorii Obwodów, SPETO’97, Gliwice-Ustro, 21-24.05.1997, 33–41.
T. Kaczorek and J. Klamka, Minimum energy control of 2-D linear systems with variable coefficients, Int. J. Control, Vol. 44, No. 3, 1986, 645–650.
T. Kaczorek and J. Klamka, Minimum energy control for general model of 2-D linear systems, Int. J. Control, Vol. 47, No. 5, 1988, 1555–1562.
T. Kitano and H. Maeda, Positive realization of discrete-time systems by geometric approach, IEEE Trans. Circuits and Syst.-I: Fundamental Theory and Applications.
J. Klamka, M-dimensional nonstationary linear discrete systems in Banach spaces, Proc. 12 World IMACS Congress, Paris, Vol. 4, 1988, 31–33.
J. Klamka, Constrained controllability of 2-D linear systems, Proc. 12 World IMACS Congress, Paris, Vol. 2, 1988, 166–169.
J. Klamka, Complete controllability of singular 2-D system, Proc. 13 IMACS World Congress, Dublin, 1991, 1839–1840.
J. Klamka, Minimum energy control of singular 2-D linear systems with variable coefficients, Proc. IMACS Symp. Line 1991, Vol. 2, 155–159.
J. Klamka, Minimum energy control problem for general linear 2-D systems in Hilbert spaces, Proc. IEEE Mediterranear Symp. on „New Directions in Control Theory and Applications“, 21-23.06. 1993. Chania, Crete, Greece.
J. Klamka, Controllability of Dynamical Systems, Kluwer Academic Publ., Dordrecht, 1991.
J. Klamka, Constrained Controllability of Discrete 2-D Linear Systems, Proc. IMACS Intern. Symp. Signal Processing, Robotics and Neural Networks, April 25-27, 1994, Lille, 166–169.
H. Maeda and S. Kodama, Reachability, observability and realizability of linear systems with positive constraints, Transactions IEEE 63-A, 1980, 688–694.
H. Maeda and S. Kodama, Positive realization of difference equation, IEEE Trans. Circuits and Systems, 28, 1981, 39–47.
H. Maeda, S. Kodama, F. Kajiya, Compartmental system analysis: Realization of a class of linear systems with physical constraints, IEEE Trans. on Circiuts and Systems, Vol. CAS-24, No. 1, 1997, 8–14.
J.W. Nieuwenhuis, About nonnegative realizations, Systems and Control Letters, Vol. 1, No. 5, 1982, 283–287.
Y. Ohta, H. Maeda and S. Kodama, Reachability, observability and realizability of continuous-time positive systems, SIAM J. Control Optim. 22, 1984, 171–180.
G. Picci and J.H. Schuppen, Stochastic realizations of finite-valued processes and primes in the positive matrices, Proceedings of the International Symposium, MTNS-91, Mita Press, 1992, 227–232.
P.R. Roesser, A discrete state-space model for linear image processing, IEEE Trans. Autom. Contr. Vol. AC-20, No. 1, 1975, 1–10.
M.E. Valcher and E. Fornasini, State models and asymptotic behaviour of 2D positive systems, IMA Journal of Mathematical Control and Information, No. 12, 1995, 17–36.
C. Wende and L. Darning, Nonnegative realizations of systems over nonnegative quasi-fields, Acta Mathematicae Applicatae Sinica 5, 1989, 252–261.
B.G. Zaslavsky, Positive realizability of linear control systems, Automath. Telemekh., 6, 1988, 13-22
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag London
About this chapter
Cite this chapter
Kaczorek, T. (2002). Realisation problem of positive 1D systems. In: Positive 1D and 2D Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-0221-2_4
Download citation
DOI: https://doi.org/10.1007/978-1-4471-0221-2_4
Publisher Name: Springer, London
Print ISBN: 978-1-4471-1097-2
Online ISBN: 978-1-4471-0221-2
eBook Packages: Springer Book Archive