Decoupling of cubic polynomial matrix systems

Peizhao Yu; Guoshan Zhang; Yi Zhang

doi:10.3934/naco.2020012

Previous Article
On the GSOR iteration method for image restoration
NACO Home
This Issue
Next Article
Fault-tolerant control against actuator failures for uncertain singular fractional order systems

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 and 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.
[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.
[3]	G. R. Duan, Analysis and Design of Descriptor Linear Systems , Springer, 2010. doi: 10.1007/978-1-4419-6397-0.
[4]	I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. doi: 10.1137/1.9780898719024.
[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.
[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.
[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.
[8]	P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, 1985.
[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.
[10]	P. Lancaster, Linearization of regular matrix polynomials, Electronic Journal of Linear Algebra, 17 (2008), 21-27. doi: 10.13001/1081-3810.1246.
[11]	P. Lancaster and I. Zaballa, Diagonalizable quadratic eigenvalue problems, Mechanical Systems and Signal Processing, 23 (2009), 1134-1144.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	A. I. Vardulakis, Linear Multivariable Control: Algebraic Analysis and Synthesis Methods , Wiley, Chichester, U. K. 1991.
[21]	D. Z. Zheng, Linear System Theory, 2$^nd$ edition, Tsinghua University Press, Beijing, 2002.
[22]	J. C. Zúñiga Anaya, On diagonalizable quadratic eigenvalue problems , Note#2. Technical Note, March 2009.
[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.

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.
[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.
[3]	G. R. Duan, Analysis and Design of Descriptor Linear Systems , Springer, 2010. doi: 10.1007/978-1-4419-6397-0.
[4]	I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. doi: 10.1137/1.9780898719024.
[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.
[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.
[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.
[8]	P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, 1985.
[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.
[10]	P. Lancaster, Linearization of regular matrix polynomials, Electronic Journal of Linear Algebra, 17 (2008), 21-27. doi: 10.13001/1081-3810.1246.
[11]	P. Lancaster and I. Zaballa, Diagonalizable quadratic eigenvalue problems, Mechanical Systems and Signal Processing, 23 (2009), 1134-1144.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	A. I. Vardulakis, Linear Multivariable Control: Algebraic Analysis and Synthesis Methods , Wiley, Chichester, U. K. 1991.
[21]	D. Z. Zheng, Linear System Theory, 2$^nd$ edition, Tsinghua University Press, Beijing, 2002.
[22]	J. C. Zúñiga Anaya, On diagonalizable quadratic eigenvalue problems , Note#2. Technical Note, March 2009.
[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.

[1]	Christian Pötzsche, Evamaria Russ. Topological decoupling and linearization of nonautonomous evolution equations. Discrete and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and 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]	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 and Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691
[11]	Xiuxiang Zhou. Approximations of infinite dimensional disturbance decoupling and almost disturbance decoupling problems. Mathematical Control and Related Fields, 2014, 4 (3) : 381-399. doi: 10.3934/mcrf.2014.4.381
[12]	V. Casarino, K.-J. Engel, G. Nickel, S. Piazzera. Decoupling techniques for wave equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575
[15]	Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025
[16]	Misha Guysinsky, Boris Hasselblatt, Victoria Rayskin. Differentiability of the Hartman--Grobman linearization. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 979-984. doi: 10.3934/dcds.2003.9.979
[17]	David L. Russell. Control via decoupling of a class of second order linear hybrid systems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1321-1334. doi: 10.3934/dcdss.2014.7.1321
[18]	Lian Zhang, Mingchao Cai, Mo Mu. Decoupling PDE computation with intrinsic or inertial Robin interface condition. Electronic Research Archive, 2021, 29 (2) : 2007-2028. doi: 10.3934/era.2020102
[19]	H. A. Erbay, S. Erbay, A. Erkip. On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3111-3122. doi: 10.3934/dcds.2017133
[20]	Livio Flaminio, Miguel Paternain. Linearization of cohomology-free vector fields. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1031-1039. doi: 10.3934/dcds.2011.29.1031

American Institute of Mathematical Sciences

Decoupling of cubic polynomial matrix systems

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Decoupling of cubic polynomial matrix systems

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors