# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2017068

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

 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  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.

Citation: Guoshan Zhang, Peizhao Yu. Lyapunov method for stability of descriptor second-order and high-order systems. Journal of Industrial & Management Optimization, 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. [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. [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. [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. [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. [6] G. R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, 2010. [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. [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. [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. [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. [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. [12] H. K. Khalil, Nonlinear Systems, 3$^{rd}$ edition, Publishing House of Electronics Industry, Beijing, 2011. [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. [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. [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. [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. [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. [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. [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. [20] F. Yu and Y. Mohamed, Comprehensive admissibility for descriptor systems, Automatica, 66 (2016), 271-275. doi: 10.1016/j.automatica.2016.01.028. [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. [22] D. Z. Zheng, Linear System Theory, 2$^{nd}$ edition, Tsinghua University Press, Beijing, 2002.

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##### 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. [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. [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. [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. [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. [6] G. R. Duan, Analysis and Design of Descriptor Linear Systems, Springer, 2010. [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. [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. [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. [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. [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. [12] H. K. Khalil, Nonlinear Systems, 3$^{rd}$ edition, Publishing House of Electronics Industry, Beijing, 2011. [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. [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. [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. [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. [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. [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. [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. [20] F. Yu and Y. Mohamed, Comprehensive admissibility for descriptor systems, Automatica, 66 (2016), 271-275. doi: 10.1016/j.automatica.2016.01.028. [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. [22] D. Z. Zheng, Linear System Theory, 2$^{nd}$ edition, Tsinghua University Press, Beijing, 2002.
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