American Institute of Mathematical Sciences

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March  2021, 11(1): 13-26. doi: 10.3934/naco.2020012

Decoupling of cubic polynomial matrix systems

Peizhao Yu 1,2, , Guoshan Zhang , and  Yi Zhang 3,

1. 

School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
2. 

School of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China
3. 

School of Science, Shenyang University of Technology, Shenyang 110870, China

* Corresponding author: Guoshan Zhang, zhanggs@tju.edu.cn

Received  May 2019 Revised  October 2019 Published  March 2021 Early access  February 2020

Fund Project: The work is supported by the National Natural Science Foundation of China (grant NO. 61473202 and 61903342) and the Doctor fund project of Zhengzhou University of Light Industry (grant NO. 2017BSJJ009)

The decoupling of polynomial matrix system is to diagonalize its system matrix. In this paper, decoupling problems for cubic polynomial matrix system are considered. The decoupling conditions for a class of cubic polynomial matrix systems are derived under strict equivalence transformation. By using linearization, isospectral decoupling method for cubic polynomial matrix system is proposed. To be specific, necessary and sufficient conditions of isospectral diagonalization for nonsingular cubic polynomial matrix are given. These results are extended to singular cubic polynomial matrix. Solving processes are given to obtain isospectral diagonal cubic polynomial matrix for nonsingular and singular cases. Finally, illustrating examples are provided to verify the main results.

Keywords: Cubic polynomial matrix, decoupling, linearization, isospectral diagonalization.
Mathematics Subject Classification: Primary: 15A18, 15A21; Secondary: 65F15.
Citation: Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
References:
[1]

M. Chu and N. Del Buono, Total decoupling of general quadratic pencils, Part Ⅰ: Theory, Journal of Sound and Vibration, 309 (2008), 96-111.   Google Scholar

[2]

M. Chu and N. Del Buono, Total decoupling of general quadratic pencils, Part Ⅱ: Structure preserving isospectral flows, Journal of Sound and Vibration, 309 (2008), 112-128.   Google Scholar

[3]

G. R. Duan, Analysis and Design of Descriptor Linear Systems , Springer, 2010. doi: 10.1007/978-1-4419-6397-0.  Google Scholar

[4] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.  doi: 10.1137/1.9780898719024.  Google Scholar
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D. Henrion and J. C. Zúñiga, Detecting infinite zeros in polynomial matrices, IEEE Trans. Circuits and systems Ⅱ-Express Briefs, 52 (2005), 744-745.   Google Scholar

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S. Johansson, B. Kågström and P. Van Dooren, Stratification of full rank polynomial matrices, Linear Algebra and its Applications, 439 (2013), 1062-1090.  doi: 10.1016/j.laa.2012.12.013.  Google Scholar

[7]

D. T. Kawano, M. Morzfeld and F. Ma, The decoupling of second-order linear systems with a singular mass matrix, Journal of Sound and Vibration, 332 (2013), 6829-6846.   Google Scholar

[8] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, 1985.   Google Scholar
[9]

P. Lancaster and L. Rodman, Canonical forms for Hermitian matrix pairs under strict equivalence and congruence, SIAM Review, 47 (2005), 407-443.  doi: 10.1137/S003614450444556X.  Google Scholar

[10]

P. Lancaster, Linearization of regular matrix polynomials, Electronic Journal of Linear Algebra, 17 (2008), 21-27.  doi: 10.13001/1081-3810.1246.  Google Scholar

[11]

P. Lancaster and I. Zaballa, Diagonalizable quadratic eigenvalue problems, Mechanical Systems and Signal Processing, 23 (2009), 1134-1144.   Google Scholar

[12]

D. S. Mackey, N. Mackey, C. Mehl and V. Mehrmann, Vector spaces of linearizations for matrix polynomials, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 971-1004.  doi: 10.1137/050628350.  Google Scholar

[13]

M. Morzfeld and F. Ma, The decoupling of damped linear systems in configuration and state spaces, Journal of Sound and Vibration, 330 (2011), 155-161.   Google Scholar

[14]

L. Taslaman, F. Tisseur and I. Zaballa, Triangularizing matrix polynomials, Linear Algebra and Its Applications, 439 (2013), 1679-1699.  doi: 10.1016/j.laa.2013.05.006.  Google Scholar

[15]

F. D. Terán, F. M. Dopico and D. S. Mackey, Linearizations of singular matrix polynomials and the recovery of minimal indices, Electronic Journal of Linear Algebra, 18 (2009), 371-402.  doi: 10.13001/1081-3810.1320.  Google Scholar

[16]

F. D. Terán, F. M. Dopico and D. S. Mackey, Spectral equivalence of matrix polynomials and the index sum theorem, Linear Algebra and Its Applications, 459 (2014), 264-333.  doi: 10.1016/j.laa.2014.07.007.  Google Scholar

[17]

F. Tisseur, S. D. Garvey and C. Munro, Deflating quadratic matrix polynomials with structure preserving transformations, Linear Algebra and Its Applications, 435 (2011), 464-479.  doi: 10.1016/j.laa.2010.06.028.  Google Scholar

[18]

F. Tisseur and I. Zaballa, Triangularizing quadratic matrix polynomials, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 312-337.  doi: 10.1137/120867640.  Google Scholar

[19]

C. Tunc, On the existence of periodic solutions of functional differential equations of the third order, Applied and Computational Mathematics, 15 (2016), 189-199.   Google Scholar

[20]

A. I. Vardulakis, Linear Multivariable Control: Algebraic Analysis and Synthesis Methods , Wiley, Chichester, U. K. 1991.  Google Scholar

[21] D. Z. Zheng, Linear System Theory, 2$^nd$ edition, Tsinghua University Press, Beijing, 2002.   Google Scholar
[22]

J. C. Zúñiga Anaya, On diagonalizable quadratic eigenvalue problems , Note#2. Technical Note, March 2009. Google Scholar

[23]

J. C. Zúñiga Anaya, Diagonalization of quadratic matrix polynomials, System & Control Letters, 59 (2010), 105-113.  doi: 10.1016/j.sysconle.2009.12.005.  Google Scholar

show all references

References:
[1]

M. Chu and N. Del Buono, Total decoupling of general quadratic pencils, Part Ⅰ: Theory, Journal of Sound and Vibration, 309 (2008), 96-111.   Google Scholar

[2]

M. Chu and N. Del Buono, Total decoupling of general quadratic pencils, Part Ⅱ: Structure preserving isospectral flows, Journal of Sound and Vibration, 309 (2008), 112-128.   Google Scholar

[3]

G. R. Duan, Analysis and Design of Descriptor Linear Systems , Springer, 2010. doi: 10.1007/978-1-4419-6397-0.  Google Scholar

[4] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.  doi: 10.1137/1.9780898719024.  Google Scholar
[5]

D. Henrion and J. C. Zúñiga, Detecting infinite zeros in polynomial matrices, IEEE Trans. Circuits and systems Ⅱ-Express Briefs, 52 (2005), 744-745.   Google Scholar

[6]

S. Johansson, B. Kågström and P. Van Dooren, Stratification of full rank polynomial matrices, Linear Algebra and its Applications, 439 (2013), 1062-1090.  doi: 10.1016/j.laa.2012.12.013.  Google Scholar

[7]

D. T. Kawano, M. Morzfeld and F. Ma, The decoupling of second-order linear systems with a singular mass matrix, Journal of Sound and Vibration, 332 (2013), 6829-6846.   Google Scholar

[8] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, 1985.   Google Scholar
[9]

P. Lancaster and L. Rodman, Canonical forms for Hermitian matrix pairs under strict equivalence and congruence, SIAM Review, 47 (2005), 407-443.  doi: 10.1137/S003614450444556X.  Google Scholar

[10]

P. Lancaster, Linearization of regular matrix polynomials, Electronic Journal of Linear Algebra, 17 (2008), 21-27.  doi: 10.13001/1081-3810.1246.  Google Scholar

[11]

P. Lancaster and I. Zaballa, Diagonalizable quadratic eigenvalue problems, Mechanical Systems and Signal Processing, 23 (2009), 1134-1144.   Google Scholar

[12]

D. S. Mackey, N. Mackey, C. Mehl and V. Mehrmann, Vector spaces of linearizations for matrix polynomials, SIAM Journal on Matrix Analysis and Applications, 28 (2006), 971-1004.  doi: 10.1137/050628350.  Google Scholar

[13]

M. Morzfeld and F. Ma, The decoupling of damped linear systems in configuration and state spaces, Journal of Sound and Vibration, 330 (2011), 155-161.   Google Scholar

[14]

L. Taslaman, F. Tisseur and I. Zaballa, Triangularizing matrix polynomials, Linear Algebra and Its Applications, 439 (2013), 1679-1699.  doi: 10.1016/j.laa.2013.05.006.  Google Scholar

[15]

F. D. Terán, F. M. Dopico and D. S. Mackey, Linearizations of singular matrix polynomials and the recovery of minimal indices, Electronic Journal of Linear Algebra, 18 (2009), 371-402.  doi: 10.13001/1081-3810.1320.  Google Scholar

[16]

F. D. Terán, F. M. Dopico and D. S. Mackey, Spectral equivalence of matrix polynomials and the index sum theorem, Linear Algebra and Its Applications, 459 (2014), 264-333.  doi: 10.1016/j.laa.2014.07.007.  Google Scholar

[17]

F. Tisseur, S. D. Garvey and C. Munro, Deflating quadratic matrix polynomials with structure preserving transformations, Linear Algebra and Its Applications, 435 (2011), 464-479.  doi: 10.1016/j.laa.2010.06.028.  Google Scholar

[18]

F. Tisseur and I. Zaballa, Triangularizing quadratic matrix polynomials, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 312-337.  doi: 10.1137/120867640.  Google Scholar

[19]

C. Tunc, On the existence of periodic solutions of functional differential equations of the third order, Applied and Computational Mathematics, 15 (2016), 189-199.   Google Scholar

[20]

A. I. Vardulakis, Linear Multivariable Control: Algebraic Analysis and Synthesis Methods , Wiley, Chichester, U. K. 1991.  Google Scholar

[21] D. Z. Zheng, Linear System Theory, 2$^nd$ edition, Tsinghua University Press, Beijing, 2002.   Google Scholar
[22]

J. C. Zúñiga Anaya, On diagonalizable quadratic eigenvalue problems , Note#2. Technical Note, March 2009. Google Scholar

[23]

J. C. Zúñiga Anaya, Diagonalization of quadratic matrix polynomials, System & Control Letters, 59 (2010), 105-113.  doi: 10.1016/j.sysconle.2009.12.005.  Google Scholar

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