American Institute of Mathematical Sciences

Advanced
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  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, 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
[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

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

[1]

Christian Pötzsche, Evamaria Russ. Topological decoupling and linearization of nonautonomous evolution equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1235-1268. doi: 10.3934/dcdss.2016050
[2]

Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
[3]

Fernando Hernando, Diego Ruano. New linear codes from matrix-product codes with polynomial units. Advances in Mathematics of Communications, 2010, 4 (3) : 363-367. doi: 10.3934/amc.2010.4.363
[4]

Peizhao Yu, Guoshan Zhang. Eigenstructure assignment for polynomial matrix systems ensuring normalization and impulse elimination. Mathematical Foundations of Computing, 2019, 2 (3) : 251-266. doi: 10.3934/mfc.2019016
[5]

Ricardo M. Martins, Otávio M. L. Gomide. Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142
[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 31-54. doi: 10.3934/dcds.2003.9.31
[7]

Vladimir Pozdyayev. 2D system analysis via dual problems and polynomial matrix inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 491-504. doi: 10.3934/naco.2016022
[8]

Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Networks & Heterogeneous Media, 2009, 4 (3) : 453-468. doi: 10.3934/nhm.2009.4.453
[9]

Barbara A. Shipman. Compactified isospectral sets of complex tridiagonal Hessenberg matrices. Conference Publications, 2003, 2003 (Special) : 788-797. doi: 10.3934/proc.2003.2003.788
[10]

Xiuxiang Zhou. Approximations of infinite dimensional disturbance decoupling and almost disturbance decoupling problems. Mathematical Control & Related Fields, 2014, 4 (3) : 381-399. doi: 10.3934/mcrf.2014.4.381
[11]

Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691
[12]

V. Casarino, K.-J. Engel, G. Nickel, S. Piazzera. Decoupling techniques for wave equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 761-772. doi: 10.3934/dcds.2005.12.761
[13]

Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397
[14]

Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575
[15]

Misha Guysinsky, Boris Hasselblatt, Victoria Rayskin. Differentiability of the Hartman--Grobman linearization. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 979-984. doi: 10.3934/dcds.2003.9.979
[16]

David L. Russell. Control via decoupling of a class of second order linear hybrid systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1321-1334. doi: 10.3934/dcdss.2014.7.1321
[17]

H. A. Erbay, S. Erbay, A. Erkip. On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3111-3122. doi: 10.3934/dcds.2017133
[18]

David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317
[19]

Livio Flaminio, Miguel Paternain. Linearization of cohomology-free vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1031-1039. doi: 10.3934/dcds.2011.29.1031
[20]

Luis Barreira, Claudia Valls. Hölder Grobman-Hartman linearization. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 187-197. doi: 10.3934/dcds.2007.18.187

 Impact Factor: 

Tools

Metrics

  • PDF downloads (19)
  • HTML views (38)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences