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April  2018, 14(2): 673-686. doi: 10.3934/jimo.2017068

Lyapunov method for stability of descriptor second-order and high-order systems

Guoshan Zhang , and  Peizhao Yu

School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China

* Corresponding author: zhanggs@tju.edu.cn.

The reviewing process was handled by Bin Li.

Received  April 2016 Revised  August 2016 Published  April 2018 Early access  June 2017

In this study, the stability problem of descriptor second-order systems is considered. Lyapunov equations for stability of second-order systemsare established by using Lyapunov method. The existence of solutions for Lyapunov equations are discussed and linear matrixinequality condition for stability of second-order systems aregiven. Then, based on the feasible solutions of the linear matrixinequality, all parametric solutions of Lyapunov equations are derived.Furthermore, the results of Lyapunov equations and linear matrixinequality condition for stability of second-ordersystems are extended to high-order systems. Finally, illustratingexamples are provided to show the effectiveness of the proposed method.

Keywords: Stability, Lyapunov equations, second-order systems, linear matrix inequality.
Mathematics Subject Classification: Primary: 93D05, 93D20; Secondary: 93C05.
Citation: Guoshan Zhang, Peizhao Yu. Lyapunov method for stability of descriptor second-order and high-order systems. Journal of Industrial & Management Optimization, 2018, 14 (2) : 673-686. doi: 10.3934/jimo.2017068
References:
[1]

B. D. O. Anderson and R. E. Bitmead, Stability of matrix polynomials, International Journal of Control, 26 (1977), 235-247.  doi: 10.1080/00207177708922306.  Google Scholar

[2]

D. S. Bernstein and S. P. Bhat, Lyapunov stability, semistability, and asymptotic stability of matrix second-order systems, Journal of Vibration and Acoustics, 117 (1994), 145-153.  doi: 10.1109/ACC.1994.752501.  Google Scholar

[3]

D. Y. Chen, R. F. Zhang and X. Z. Liu, Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 4105-4121.  doi: 10.1016/j.cnsns.2014.05.005.  Google Scholar

[4]

L. Colombo, F. Jiménez and D. Martín de Diego, Variational integrators for mechanical control systems with symmetries, Journal of Industrial and Management Optimization, 2 (2015), 193-225.  doi: 10.3934/jcd.2015003.  Google Scholar

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A. M. Diwekar and R. K. Yedavalli, Stability of matrix second-order systems: New conditions and perspectives, IEEE Transaction on Automatic Control, 44 (1999), 1773-1777.  doi: 10.1109/9.788551.  Google Scholar

[6]

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

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Y. Feng, M. Yagoubi and P. Chevrel, Parametrization of extended stabilizing controllers for continuous-time descriptor systems, Journal of The Franklin Institute, 348 (2011), 2633-2646.  doi: 10.1016/j.jfranklin.2011.08.006.  Google Scholar

[8]

L. X. Gao, W. H. Chen and Y. X. Sun, On robust admissibility condition for descriptor systems with convex polytopic uncertainty, Proceeding of the American Control Conference, 6 (2003), 5083-5088.   Google Scholar

[9]

S. H. Huang, R. F. Zhang and D. Y. Chen, Stability of nonlinear fractional-order time varying systems Journal of Computational and Nonlinear Dynamics, w (2016), 031007. doi: 10.1115/1.4031587.  Google Scholar

[10]

S. Johansson, B. Kågström and P. V. 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

[11]

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.  doi: 10.1016/j.jsv.2013.08.005.  Google Scholar

[12]

H. K. Khalil, Nonlinear Systems, 3$^{rd}$ edition, Publishing House of Electronics Industry, Beijing, 2011. Google Scholar

[13]

P. Lancaster and P. Zizler, On the stability of gyroscopic systems, Journal of Applied Mechanics, 65 (1998), 519-522.  doi: 10.1115/1.2789085.  Google Scholar

[14]

P. Lancaster, Stability of linear gyroscopic systems: A review, Linear Algebra and its Applications, 439 (2013), 686-706.  doi: 10.1016/j.laa.2012.12.026.  Google Scholar

[15]

P. Losse and V. Mehrmann, Controllability and observability of second order descriptor systems, SIAM Journal on Control and Optimization, 47 (2008), 1351-1379.  doi: 10.1137/060673977.  Google Scholar

[16]

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.  doi: 10.1016/j.jsv.2010.09.005.  Google Scholar

[17]

N. K. Nichols and J. Kautsky, Robust eigenstructure assignment in quadratic matrix polynomials: Nonsingular case, SIAM Journal on Matrix Analysis and Applications, 23 (2001), 77-102.  doi: 10.1137/S0895479899362867.  Google Scholar

[18]

P. Resende and E. Kaszkurewicz, A sufficient condition for the stability of matrix, IEEE Transaction on Automatic Control, 34 (1989), 539-541.  doi: 10.1109/9.24207.  Google Scholar

[19]

L. S. Shieh, M. M. Mehio and H. M. Dib, Stability of the second-order matrix polynomial, IEEE Transaction on Automatic Control, 32 (1987), 231-233.  doi: 10.1109/TAC.1987.1104572.  Google Scholar

[20]

F. Yu and Y. Mohamed, Comprehensive admissibility for descriptor systems, Automatica, 66 (2016), 271-275.  doi: 10.1016/j.automatica.2016.01.028.  Google Scholar

[21]

J. Zhang, H. Ouyang and J. Yang, Partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedback, Journal of Sound and Vibration, 333 (2014), 1-12.  doi: 10.1016/j.jsv.2013.08.040.  Google Scholar

[22]

D. Z. Zheng, Linear System Theory, 2$^{nd}$ edition, Tsinghua University Press, Beijing, 2002. Google Scholar

show all references

The reviewing process was handled by Bin Li.

References:
[1]

B. D. O. Anderson and R. E. Bitmead, Stability of matrix polynomials, International Journal of Control, 26 (1977), 235-247.  doi: 10.1080/00207177708922306.  Google Scholar

[2]

D. S. Bernstein and S. P. Bhat, Lyapunov stability, semistability, and asymptotic stability of matrix second-order systems, Journal of Vibration and Acoustics, 117 (1994), 145-153.  doi: 10.1109/ACC.1994.752501.  Google Scholar

[3]

D. Y. Chen, R. F. Zhang and X. Z. Liu, Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 4105-4121.  doi: 10.1016/j.cnsns.2014.05.005.  Google Scholar

[4]

L. Colombo, F. Jiménez and D. Martín de Diego, Variational integrators for mechanical control systems with symmetries, Journal of Industrial and Management Optimization, 2 (2015), 193-225.  doi: 10.3934/jcd.2015003.  Google Scholar

[5]

A. M. Diwekar and R. K. Yedavalli, Stability of matrix second-order systems: New conditions and perspectives, IEEE Transaction on Automatic Control, 44 (1999), 1773-1777.  doi: 10.1109/9.788551.  Google Scholar

[6]

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

[7]

Y. Feng, M. Yagoubi and P. Chevrel, Parametrization of extended stabilizing controllers for continuous-time descriptor systems, Journal of The Franklin Institute, 348 (2011), 2633-2646.  doi: 10.1016/j.jfranklin.2011.08.006.  Google Scholar

[8]

L. X. Gao, W. H. Chen and Y. X. Sun, On robust admissibility condition for descriptor systems with convex polytopic uncertainty, Proceeding of the American Control Conference, 6 (2003), 5083-5088.   Google Scholar

[9]

S. H. Huang, R. F. Zhang and D. Y. Chen, Stability of nonlinear fractional-order time varying systems Journal of Computational and Nonlinear Dynamics, w (2016), 031007. doi: 10.1115/1.4031587.  Google Scholar

[10]

S. Johansson, B. Kågström and P. V. 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

[11]

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.  doi: 10.1016/j.jsv.2013.08.005.  Google Scholar

[12]

H. K. Khalil, Nonlinear Systems, 3$^{rd}$ edition, Publishing House of Electronics Industry, Beijing, 2011. Google Scholar

[13]

P. Lancaster and P. Zizler, On the stability of gyroscopic systems, Journal of Applied Mechanics, 65 (1998), 519-522.  doi: 10.1115/1.2789085.  Google Scholar

[14]

P. Lancaster, Stability of linear gyroscopic systems: A review, Linear Algebra and its Applications, 439 (2013), 686-706.  doi: 10.1016/j.laa.2012.12.026.  Google Scholar

[15]

P. Losse and V. Mehrmann, Controllability and observability of second order descriptor systems, SIAM Journal on Control and Optimization, 47 (2008), 1351-1379.  doi: 10.1137/060673977.  Google Scholar

[16]

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.  doi: 10.1016/j.jsv.2010.09.005.  Google Scholar

[17]

N. K. Nichols and J. Kautsky, Robust eigenstructure assignment in quadratic matrix polynomials: Nonsingular case, SIAM Journal on Matrix Analysis and Applications, 23 (2001), 77-102.  doi: 10.1137/S0895479899362867.  Google Scholar

[18]

P. Resende and E. Kaszkurewicz, A sufficient condition for the stability of matrix, IEEE Transaction on Automatic Control, 34 (1989), 539-541.  doi: 10.1109/9.24207.  Google Scholar

[19]

L. S. Shieh, M. M. Mehio and H. M. Dib, Stability of the second-order matrix polynomial, IEEE Transaction on Automatic Control, 32 (1987), 231-233.  doi: 10.1109/TAC.1987.1104572.  Google Scholar

[20]

F. Yu and Y. Mohamed, Comprehensive admissibility for descriptor systems, Automatica, 66 (2016), 271-275.  doi: 10.1016/j.automatica.2016.01.028.  Google Scholar

[21]

J. Zhang, H. Ouyang and J. Yang, Partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedback, Journal of Sound and Vibration, 333 (2014), 1-12.  doi: 10.1016/j.jsv.2013.08.040.  Google Scholar

[22]

D. Z. Zheng, Linear System Theory, 2$^{nd}$ edition, Tsinghua University Press, Beijing, 2002. Google Scholar

Figure 1.  State responses of systems

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Figure 2.  State responses of system (16)

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