On the Robust Stability of Homogeneous Perturbed Large-Scale Bilinear Systems with Time Delays and Constrained Inputs
The stability test problem for homogeneous large-scale perturbed bilinear time-delay systems subjected to constrained inputs is considered in this paper. Both nonlinear uncertainties and interval systems are discussed. By utilizing the Lyapunove equation approach associated with linear algebraic techniques, several delay-independent criteria are presented to guarantee the robust stability of the overall systems. The main feature of the presented results is that although the Lyapunov stability theorem is used, they do not involve any Lyapunov equation which may be unsolvable. Furthermore, it is seen the proposed schemes can be applied to solve the stability analysis problem of large-scale time-delay systems.
[1] M. Bacic, M. Cannon, and B. Kouvaritakis, "Constrained control of SISO
bilinear system," IEEE Transactions on Automatic Control, vol. 48, pp.
1443-1447, 2003.
[2] L. Berrahmoune, "Stabilization and decay estimate for distributed bilinear
systems," Systems & Control Letters, vol. 36, pp. 167-171, 1999.
[3] C. Bruni, G. D. Pillo, and G. Koch, "Bilinear system: an appealing class of
nearly linear systems in theory and applications," IEEE Transactions on
Automatic Control, vol. 19, pp. 334-348, 1974.
[4] R. Chabour and A. Ferfera, "Noninteracting control with singularity for a
class of bilinear systems," Nonlinear Analysis, vol. 33, pp. 91-96, 1998.
[5] O. Chabour and J. C. Vivalda, "Remark on local and global stabilization of
homogeneous bilinear systems," Systems & Control Letters, vol. 41, pp.
141-143, 2000.
[6] M.S. Chen and S. T. Tsao, "Exponential stabilization of a class of unstable
bilinear systems," IEEE Transactions on Automatic Control, vol. 45, pp.
989-992, 2000.
[7] Y. P. Chen, J. L. Chang, and K. M. Lai, "Stability analysis and bang-bang
sliding control of a class of single-input bilinear systems," IEEE
Transactions on Automatic Control, vol. 45, pp. 2150-2154, 2000.
[8] L. K. Chen and R. R. Mohler, "Stability analysis of bilinear systems," IEEE
Transactions on Automatic Control, vol. 36, pp. 1310-1315, 1991.
[9] O. Chabour and J. C. Vivalda, "Remark on local and global stabilization of
homogeneous bilinear systems," Systems & Control Letters, vol. 41, pp.
141-143, 2000.
[10] C. L. Chiang and F. C. Kung, "Local stability region in large-scale
bilinear systems with decentralized control," Journal of Control Systems
and Technology, vol. 1, pp. 227-234, 1993.
[11] J. S. Chiou, F. C. Kung, and T. H. S. Li, "Robust stabilization of a class of
singular perturbed discrete bilinear systems," IEEE Transactions on
Automatic Control, vol. 45, pp. 1187-1191, 2000.
[12] J. Guojun, "Stability of bilinear time-delay systems," IMA Journal of
Mathematical Control and Information, vol. 18, pp. 53-60, 2001.
[13] L. Guoping and Daniel W.C. Ho, "Continuous stabilization controllers
for singular bilinear systems: The state feedback case," Automatica, vol.
42, pp. 309-314, 2006
[14] J. Hamadi, "Global feedback stabilization of new class of bilinear
systems," Systems & Control Letters, vol. 42, pp. 313-320, 2001.
[15] D. W. C. Ho, G. Lu, and Y. Zheng, "Global stabilization for bilinear
systems with time delay," IEE Proceedings on Control Theory
Application, vol. 149, pp. 89-94, 2002.
[16] H. Jerbi, "Global feedback stabilization of new class of bilinear
systems," Systems & Control Letters, vol. 42, pp. 313-320, 2001.
[17] S. Kotsios, "A note on BIBO stability of bilinear systems," Journal of the
Franklin Institute, vol. 332B, pp. 755-760, 1995.
[18] Z. G. Li, C. Y. Wen and Y. C. Soh, "Switched controllers and their
applications in bilinear systems," Automatica, vol. 37, pp. 477-481,
2001.
[19] C. H. Lee, "On the stability of uncertain homogeneous bilinear systems
subjected to time-delay and constrained inputs," Journal of the Chinese
Institute of Engineers, vol. 31, no. 3, pp. 529-534, 2008.
[20] C. S. Lee and G. Leitmann, "Continuous feedback guaranteeing uniform
ultimate boundness for uncertain linear delay systems: An application to
river pollution control," Computer Mathematical Applications, vol. 16,
pp. 929-938, 1983
[21] G. Lu and D. W. C. Ho, "Global stabilization controller design for
discrete-time bilinear systems with time-delays," Proceedings of the 4th
World Congress on intelligent Control and Automation, pp. 10-14, 2002.
[22] R. R. Mohler, Bilinear Control Processes. NY: Academic, 1973.
[23] S. I. Niculescu, S. Tarbouriceh, J. M. Dion, and L. Dugard, "Stability
criteria for bilinear systems with delayed state and saturating actuators,"
Proceedings of the 34th Conference on Decision & Control, pp.
2064-2069, 1995.
[24] H. Shigeru and M. Yoshihiko, "Output feedback stabilization of bilinear
systems using dead-beat observers," Automatica, vol. 37, pp. 915-920,
2001.
[25] D. D. Siljak and M. B. Vukcevic, "Decentrally stabilizable linear and
bilinear large-scale systems," International Journal of Control, vol. 26,
pp. 289-305, 1977.
[26] C. W. Tao, W. Y. Wang, and M. L. Chan, "Design of sliding mode
controllers for bilinear systems with time varying uncertainties," IEEE
Transactions on Systems, Man, and Cybernetics-Part B, vol. 34, pp.
639-645, 2004.
[27] S. Weissenberger, "Stability region of large-scale systems," Automatica,
vol. 9, pp. 653-663, 1973.
[28] C.-H. Lee, K.-H. Chien and C.-Y. Chen, "A Simple Approach for
Estimating Solution Bounds of the Continuous Lyapunov Equation",
Proceeding of the 4nd IEEE Conference on Industrial Electronics and
Applications, pp. 3458-3463, 2009.
[1] M. Bacic, M. Cannon, and B. Kouvaritakis, "Constrained control of SISO
bilinear system," IEEE Transactions on Automatic Control, vol. 48, pp.
1443-1447, 2003.
[2] L. Berrahmoune, "Stabilization and decay estimate for distributed bilinear
systems," Systems & Control Letters, vol. 36, pp. 167-171, 1999.
[3] C. Bruni, G. D. Pillo, and G. Koch, "Bilinear system: an appealing class of
nearly linear systems in theory and applications," IEEE Transactions on
Automatic Control, vol. 19, pp. 334-348, 1974.
[4] R. Chabour and A. Ferfera, "Noninteracting control with singularity for a
class of bilinear systems," Nonlinear Analysis, vol. 33, pp. 91-96, 1998.
[5] O. Chabour and J. C. Vivalda, "Remark on local and global stabilization of
homogeneous bilinear systems," Systems & Control Letters, vol. 41, pp.
141-143, 2000.
[6] M.S. Chen and S. T. Tsao, "Exponential stabilization of a class of unstable
bilinear systems," IEEE Transactions on Automatic Control, vol. 45, pp.
989-992, 2000.
[7] Y. P. Chen, J. L. Chang, and K. M. Lai, "Stability analysis and bang-bang
sliding control of a class of single-input bilinear systems," IEEE
Transactions on Automatic Control, vol. 45, pp. 2150-2154, 2000.
[8] L. K. Chen and R. R. Mohler, "Stability analysis of bilinear systems," IEEE
Transactions on Automatic Control, vol. 36, pp. 1310-1315, 1991.
[9] O. Chabour and J. C. Vivalda, "Remark on local and global stabilization of
homogeneous bilinear systems," Systems & Control Letters, vol. 41, pp.
141-143, 2000.
[10] C. L. Chiang and F. C. Kung, "Local stability region in large-scale
bilinear systems with decentralized control," Journal of Control Systems
and Technology, vol. 1, pp. 227-234, 1993.
[11] J. S. Chiou, F. C. Kung, and T. H. S. Li, "Robust stabilization of a class of
singular perturbed discrete bilinear systems," IEEE Transactions on
Automatic Control, vol. 45, pp. 1187-1191, 2000.
[12] J. Guojun, "Stability of bilinear time-delay systems," IMA Journal of
Mathematical Control and Information, vol. 18, pp. 53-60, 2001.
[13] L. Guoping and Daniel W.C. Ho, "Continuous stabilization controllers
for singular bilinear systems: The state feedback case," Automatica, vol.
42, pp. 309-314, 2006
[14] J. Hamadi, "Global feedback stabilization of new class of bilinear
systems," Systems & Control Letters, vol. 42, pp. 313-320, 2001.
[15] D. W. C. Ho, G. Lu, and Y. Zheng, "Global stabilization for bilinear
systems with time delay," IEE Proceedings on Control Theory
Application, vol. 149, pp. 89-94, 2002.
[16] H. Jerbi, "Global feedback stabilization of new class of bilinear
systems," Systems & Control Letters, vol. 42, pp. 313-320, 2001.
[17] S. Kotsios, "A note on BIBO stability of bilinear systems," Journal of the
Franklin Institute, vol. 332B, pp. 755-760, 1995.
[18] Z. G. Li, C. Y. Wen and Y. C. Soh, "Switched controllers and their
applications in bilinear systems," Automatica, vol. 37, pp. 477-481,
2001.
[19] C. H. Lee, "On the stability of uncertain homogeneous bilinear systems
subjected to time-delay and constrained inputs," Journal of the Chinese
Institute of Engineers, vol. 31, no. 3, pp. 529-534, 2008.
[20] C. S. Lee and G. Leitmann, "Continuous feedback guaranteeing uniform
ultimate boundness for uncertain linear delay systems: An application to
river pollution control," Computer Mathematical Applications, vol. 16,
pp. 929-938, 1983
[21] G. Lu and D. W. C. Ho, "Global stabilization controller design for
discrete-time bilinear systems with time-delays," Proceedings of the 4th
World Congress on intelligent Control and Automation, pp. 10-14, 2002.
[22] R. R. Mohler, Bilinear Control Processes. NY: Academic, 1973.
[23] S. I. Niculescu, S. Tarbouriceh, J. M. Dion, and L. Dugard, "Stability
criteria for bilinear systems with delayed state and saturating actuators,"
Proceedings of the 34th Conference on Decision & Control, pp.
2064-2069, 1995.
[24] H. Shigeru and M. Yoshihiko, "Output feedback stabilization of bilinear
systems using dead-beat observers," Automatica, vol. 37, pp. 915-920,
2001.
[25] D. D. Siljak and M. B. Vukcevic, "Decentrally stabilizable linear and
bilinear large-scale systems," International Journal of Control, vol. 26,
pp. 289-305, 1977.
[26] C. W. Tao, W. Y. Wang, and M. L. Chan, "Design of sliding mode
controllers for bilinear systems with time varying uncertainties," IEEE
Transactions on Systems, Man, and Cybernetics-Part B, vol. 34, pp.
639-645, 2004.
[27] S. Weissenberger, "Stability region of large-scale systems," Automatica,
vol. 9, pp. 653-663, 1973.
[28] C.-H. Lee, K.-H. Chien and C.-Y. Chen, "A Simple Approach for
Estimating Solution Bounds of the Continuous Lyapunov Equation",
Proceeding of the 4nd IEEE Conference on Industrial Electronics and
Applications, pp. 3458-3463, 2009.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:60488", author = "Chien-Hua Lee and Cheng-Yi Chen", title = "On the Robust Stability of Homogeneous Perturbed Large-Scale Bilinear Systems with Time Delays and Constrained Inputs", abstract = "The stability test problem for homogeneous large-scale perturbed bilinear time-delay systems subjected to constrained inputs is considered in this paper. Both nonlinear uncertainties and interval systems are discussed. By utilizing the Lyapunove equation approach associated with linear algebraic techniques, several delay-independent criteria are presented to guarantee the robust stability of the overall systems. The main feature of the presented results is that although the Lyapunov stability theorem is used, they do not involve any Lyapunov equation which may be unsolvable. Furthermore, it is seen the proposed schemes can be applied to solve the stability analysis problem of large-scale time-delay systems.
", keywords = "homogeneous bilinear system, constrained input, time-delay, uncertainty, transient response, decay rate.", volume = "4", number = "5", pages = "594-6", }
