Decoupling Zeros of Positive Discrete-Time Linear Sys-tems

Author(s)
Tadeusz Kaczorek

ABSTRACT

The notions of decoupling zeros of positive discrete-time linear systems are introduced. The relationships between the decoupling zeros of standard and positive discrete-time linear systems are analyzed. It is shown that: 1) if the positive system has decoupling zeros then the corresponding standard system has also decoupling zeros, 2) the positive system may not have decoupling zeros when the corresponding standard system has decoupling zeros, 3) the positive and standard systems have the same decoupling zeros if the rank of reachability (observability) matrix is equal to the number of linearly independent monomial columns (rows) and some additional assumptions are satisfied.

The notions of decoupling zeros of positive discrete-time linear systems are introduced. The relationships between the decoupling zeros of standard and positive discrete-time linear systems are analyzed. It is shown that: 1) if the positive system has decoupling zeros then the corresponding standard system has also decoupling zeros, 2) the positive system may not have decoupling zeros when the corresponding standard system has decoupling zeros, 3) the positive and standard systems have the same decoupling zeros if the rank of reachability (observability) matrix is equal to the number of linearly independent monomial columns (rows) and some additional assumptions are satisfied.

KEYWORDS

Input-Decoupling Zeros, Output-Decoupling Zeros, Input-Output Decoupling Zeros, Positive, Discrete-Time, Linear, System

Input-Decoupling Zeros, Output-Decoupling Zeros, Input-Output Decoupling Zeros, Positive, Discrete-Time, Linear, System

Cite this paper

nullT. Kaczorek, "Decoupling Zeros of Positive Discrete-Time Linear Sys-tems,"*Circuits and Systems*, Vol. 1 No. 2, 2010, pp. 41-48. doi: 10.4236/cs.2010.12007.

nullT. Kaczorek, "Decoupling Zeros of Positive Discrete-Time Linear Sys-tems,"

References

[1] L. Farina and S. Rinaldi, “Positive Linear Systems; Theory and Applications,” Wiley, New York, 2000.

[2] T. Kaczorek, “Positive 1D and 2D Systems,” Springer Verlag, London, 2001.

[3] R. E. Kalman, “Mathematical Descriptions of Linear Systems,” SIAM Journal on Control, Vol. 1, No. 2, 1963, pp. 152-192.

[4] R. E. Kalman, “On the General Theory of Control Systems,” Proceedings of the First International Congress on Automatic Control, Butterworth, London, 1960, pp. 481-493.

[5] P. J. Antsaklis and A. N. Michel, “Linear Systems,” Birkhauser, Boston, 2006.

[6] T. Kaczorek, “Linear Control Systems,” Vol. 1, Wiley, New York, 1993.

[7] T. Kailath, “Linear Systems,” Prentice-Hall, Englewood Cliffs, New York, 1980.

[8] H. H. Rosenbrock, “State-Space and Multivariable Theory,” Wiley, New York, 1970.

[9] W. A. Wolovich, “Linear Multivariable Systems,” Springer-Verlag, New York, 1974.

[10] T. Kaczorek, “Reachability and Controllability to Zero Tests for Standard and Positive Fractional Discrete-Time Systems,” Journal Européen des Systèmes Automatisés, Vol. 42, No. 6-8, 2008, pp. 770-781.

[11] T. Kaczorek, “Decomposition of the Pairs (A,B) and (A,C) of the Positive Discrete-Time Linear Systems,” Proceedings of TRANSCOMP, Zakopane, 6-9 December 2010.

[12] H. H. Rosenbrock, “Comments on Poles and Zeros of Linear Multivariable Systems: A Survey of the Algebraic Geometric and Complex Variable Theory,” International Journal on Control, Vol. 26, No. 1, 1977, pp. 157-161.

[1] L. Farina and S. Rinaldi, “Positive Linear Systems; Theory and Applications,” Wiley, New York, 2000.

[2] T. Kaczorek, “Positive 1D and 2D Systems,” Springer Verlag, London, 2001.

[3] R. E. Kalman, “Mathematical Descriptions of Linear Systems,” SIAM Journal on Control, Vol. 1, No. 2, 1963, pp. 152-192.

[4] R. E. Kalman, “On the General Theory of Control Systems,” Proceedings of the First International Congress on Automatic Control, Butterworth, London, 1960, pp. 481-493.

[5] P. J. Antsaklis and A. N. Michel, “Linear Systems,” Birkhauser, Boston, 2006.

[6] T. Kaczorek, “Linear Control Systems,” Vol. 1, Wiley, New York, 1993.

[7] T. Kailath, “Linear Systems,” Prentice-Hall, Englewood Cliffs, New York, 1980.

[8] H. H. Rosenbrock, “State-Space and Multivariable Theory,” Wiley, New York, 1970.

[9] W. A. Wolovich, “Linear Multivariable Systems,” Springer-Verlag, New York, 1974.

[10] T. Kaczorek, “Reachability and Controllability to Zero Tests for Standard and Positive Fractional Discrete-Time Systems,” Journal Européen des Systèmes Automatisés, Vol. 42, No. 6-8, 2008, pp. 770-781.

[11] T. Kaczorek, “Decomposition of the Pairs (A,B) and (A,C) of the Positive Discrete-Time Linear Systems,” Proceedings of TRANSCOMP, Zakopane, 6-9 December 2010.

[12] H. H. Rosenbrock, “Comments on Poles and Zeros of Linear Multivariable Systems: A Survey of the Algebraic Geometric and Complex Variable Theory,” International Journal on Control, Vol. 26, No. 1, 1977, pp. 157-161.