Robust H8 Fuzzy Control Design for Nonlinear Two-Time Scale System with Markovian Jumps based on LMI Approach
This paper examines the problem of designing a robust H8 state-feedback controller for a class of nonlinear two-time scale systems with Markovian Jumps described by a Takagi-Sugeno (TS) fuzzy model. Based on a linear matrix inequality (LMI) approach, LMI-based sufficient conditions for the uncertain Markovian jump nonlinear two-time scale systems to have an H8 performance are derived. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard nonlinear two-time scale systems. A numerical example is provided to illustrate the design developed in this paper.
[1] H. J. Kushner, Stochastic Stability and Control, New York: Academic
Press, 1967.
[2] E. B. Dynkin, Markov Process, Berlin: Springer-Verlag, 1965.
[3] W. M. Wonham, "On a matrix Riccati equation of stochastic control,"
SIAM J. Contr., vol. 6, pp. 681-697, 1968.
[4] X. Feng, K. A. Loparo, Y. Ji, and H. J. Chizeck, "Stochastic stability
properties of jump linear system," IEEE Tran. Automat. Contr., vol. 37,
pp. 38-53, 1992.
[5] C. E. de Souza and M. D. Fragoso, "H1 control for linear systems with
Markovian jumping parameters," Control-Theory and Advanced Tech.,
vol. 9, pp. 457-466, 1993.
[6] E. K. Boukas and Z. K. Liu, "Suboptimal design of regulators for jump
linear system with time-multiplied quadratic cost," IEEE Tran. Automat.
Contr., vol. 46, pp. 944-949, 2001.
[7] E. K. Boukas and H. Yang , "Exponential stabilizability of stochastic systems
with Markovian jump parameters," Automatica, vol. 35, pp. 1437-
1441, 1999.
[8] M. A. Rami and L. Ei Ghaoui, "H1 state-feedback control of jump
linear systems," Proc. Conf. Decision and Contr., pp. 951-952, 1995.
[9] P. Shi and E. K. Boukas, "H1 control for Markovian jumping linear
system with parametric uncertainty," J. of Opt. Theory and App., vol. 95,
pp. 75-99, 1997.
[10] K. Benjelloun, E. K. Boukas, and O. L. V. Costa, "H1 control for linear
time delay with Markovian jumping parameters," J. of Opt. Theory and
App., vol. 105, pp. 73-95, 1997.
[11] E. K. Boukas, Z. K. Liu, and G. X. Liu, "Delay-dependent robust
stability and H1 control of jump linear systems with time-delay," Int.
J. of Contr., 2001.
[12] V. Dragan, P. Shi, and E. K. Boukas, "Control of singularly perturbed
system with Markovian jump parameters: An H1 approach," Automatica,
vol. 35, pp. 1369-1378, 1999.
[13] E. Fridman, "State-feedback H
1 control of nonlinear singularly perturbed
systems," Int. J. Robust Nonlinear Contr., vol. 6, pp. 25-45, 2001.
[14] P. Shi and V. Dragan, "Asymptotic H1 control of singularly perturbed
system with parametric uncertainties," IEEE Trans. Automat. Contr.,
vol. 44, pp. 1738-1742, 1999.
[15] P. V. Kokotovic, H. K. Khalil, and J. O-Reilly, Singular Perturbation
Methods in Control: Analysis and Design, London: Academic Press,
1986.
[16] S. K. Nguang and P. Shi, "H1 fuzzy output feedback control design
for nonlinear systems: An LMI approach," Proc. IEEE Conf. Decision
and Contr., pp. 4352-4357, 2001.
[17] Z. X. Han and G. Feng, "State-feedback H
1 controller design of fuzzy
dynamic system using LMI techniques," Fuzzy-IEEE-98, pp. 538-544,
1998.
[18] K. Tanaka, T. Ikeda, and H. O. Wang, "Robust stabilization of a class of
uncertain nonlinear systems via fuzzy control: Quadratic stability, H1
control theory, and linear martix inequality," IEEE Trans. Fuzzy. Syst.,
vol. 4, pp. 1-13, 1996.
[19] H. O. Wang, K. Tanaka, and M. F. Griffin, "An approach to fuzzy control
of nonlinear systems: Stability and design issues," IEEE Trans. Fuzzy
Syst., vol. 4, no. 1, pp. 14-23, 1996.
[20] D. P. de Farias, J. C. Geromel, J. B. R. do Val, and O. L. V. Costa,
"Output feedback control of Markov jump linear systems in continuoustime,"
IEEE Trans. Automat. Contr., vol. 45, pp. 944-949, 2000.
[21] S. K. Nguang and P. Shi, "H
1 fuzzy output feedback control design
for nonlinear systems: An LMI approach ," IEEE Trans. Fuzzy Syst.,
vol. 11, pp. 331-340, 2003.
[22] S. K. Nguang, "Robust nonlinear H1 output feedback control," IEEE
Trans Automat. Contr., vol. 41, pp. 1003-1008, 1996.
[23] W. Assawinchaichote and S. K. Nguang "Fuzzy control design for
singularly perturbed nonlinear systems: An LMI approach," ICAIET,
(Kota Kinabalu, Malaysia), pp. 146-151, 2002.
[24] W. Assawinchaichote and S. K. Nguang, "Fuzzy observer-based controller
design for singularly perturbed nonlinear systems: An LMI approach,"
Proc. IEEE Conf. Decision and Contr., (Las Vegas), pp. 2165-
2170, 2002.
[25] P. Gahinet, A. Nemirovski, A.J. Laub, and M. Chilali, LMI Control
Toolbox - For Use with MATLAB, MA: The MathWorks, Inc., 1995.
[26] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix
Inequalities in Systems and Control Theory, vol. 15, Philadelphia: SIAM,
1994.
[27] S. Mehta and J. Chiasson, "Nonlinear control of a series dc motor:
Theory and experiment," IEEE Trans. Ind. Electron., vol. 45, pp. 134-
141, 1998.
[1] H. J. Kushner, Stochastic Stability and Control, New York: Academic
Press, 1967.
[2] E. B. Dynkin, Markov Process, Berlin: Springer-Verlag, 1965.
[3] W. M. Wonham, "On a matrix Riccati equation of stochastic control,"
SIAM J. Contr., vol. 6, pp. 681-697, 1968.
[4] X. Feng, K. A. Loparo, Y. Ji, and H. J. Chizeck, "Stochastic stability
properties of jump linear system," IEEE Tran. Automat. Contr., vol. 37,
pp. 38-53, 1992.
[5] C. E. de Souza and M. D. Fragoso, "H1 control for linear systems with
Markovian jumping parameters," Control-Theory and Advanced Tech.,
vol. 9, pp. 457-466, 1993.
[6] E. K. Boukas and Z. K. Liu, "Suboptimal design of regulators for jump
linear system with time-multiplied quadratic cost," IEEE Tran. Automat.
Contr., vol. 46, pp. 944-949, 2001.
[7] E. K. Boukas and H. Yang , "Exponential stabilizability of stochastic systems
with Markovian jump parameters," Automatica, vol. 35, pp. 1437-
1441, 1999.
[8] M. A. Rami and L. Ei Ghaoui, "H1 state-feedback control of jump
linear systems," Proc. Conf. Decision and Contr., pp. 951-952, 1995.
[9] P. Shi and E. K. Boukas, "H1 control for Markovian jumping linear
system with parametric uncertainty," J. of Opt. Theory and App., vol. 95,
pp. 75-99, 1997.
[10] K. Benjelloun, E. K. Boukas, and O. L. V. Costa, "H1 control for linear
time delay with Markovian jumping parameters," J. of Opt. Theory and
App., vol. 105, pp. 73-95, 1997.
[11] E. K. Boukas, Z. K. Liu, and G. X. Liu, "Delay-dependent robust
stability and H1 control of jump linear systems with time-delay," Int.
J. of Contr., 2001.
[12] V. Dragan, P. Shi, and E. K. Boukas, "Control of singularly perturbed
system with Markovian jump parameters: An H1 approach," Automatica,
vol. 35, pp. 1369-1378, 1999.
[13] E. Fridman, "State-feedback H
1 control of nonlinear singularly perturbed
systems," Int. J. Robust Nonlinear Contr., vol. 6, pp. 25-45, 2001.
[14] P. Shi and V. Dragan, "Asymptotic H1 control of singularly perturbed
system with parametric uncertainties," IEEE Trans. Automat. Contr.,
vol. 44, pp. 1738-1742, 1999.
[15] P. V. Kokotovic, H. K. Khalil, and J. O-Reilly, Singular Perturbation
Methods in Control: Analysis and Design, London: Academic Press,
1986.
[16] S. K. Nguang and P. Shi, "H1 fuzzy output feedback control design
for nonlinear systems: An LMI approach," Proc. IEEE Conf. Decision
and Contr., pp. 4352-4357, 2001.
[17] Z. X. Han and G. Feng, "State-feedback H
1 controller design of fuzzy
dynamic system using LMI techniques," Fuzzy-IEEE-98, pp. 538-544,
1998.
[18] K. Tanaka, T. Ikeda, and H. O. Wang, "Robust stabilization of a class of
uncertain nonlinear systems via fuzzy control: Quadratic stability, H1
control theory, and linear martix inequality," IEEE Trans. Fuzzy. Syst.,
vol. 4, pp. 1-13, 1996.
[19] H. O. Wang, K. Tanaka, and M. F. Griffin, "An approach to fuzzy control
of nonlinear systems: Stability and design issues," IEEE Trans. Fuzzy
Syst., vol. 4, no. 1, pp. 14-23, 1996.
[20] D. P. de Farias, J. C. Geromel, J. B. R. do Val, and O. L. V. Costa,
"Output feedback control of Markov jump linear systems in continuoustime,"
IEEE Trans. Automat. Contr., vol. 45, pp. 944-949, 2000.
[21] S. K. Nguang and P. Shi, "H
1 fuzzy output feedback control design
for nonlinear systems: An LMI approach ," IEEE Trans. Fuzzy Syst.,
vol. 11, pp. 331-340, 2003.
[22] S. K. Nguang, "Robust nonlinear H1 output feedback control," IEEE
Trans Automat. Contr., vol. 41, pp. 1003-1008, 1996.
[23] W. Assawinchaichote and S. K. Nguang "Fuzzy control design for
singularly perturbed nonlinear systems: An LMI approach," ICAIET,
(Kota Kinabalu, Malaysia), pp. 146-151, 2002.
[24] W. Assawinchaichote and S. K. Nguang, "Fuzzy observer-based controller
design for singularly perturbed nonlinear systems: An LMI approach,"
Proc. IEEE Conf. Decision and Contr., (Las Vegas), pp. 2165-
2170, 2002.
[25] P. Gahinet, A. Nemirovski, A.J. Laub, and M. Chilali, LMI Control
Toolbox - For Use with MATLAB, MA: The MathWorks, Inc., 1995.
[26] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix
Inequalities in Systems and Control Theory, vol. 15, Philadelphia: SIAM,
1994.
[27] S. Mehta and J. Chiasson, "Nonlinear control of a series dc motor:
Theory and experiment," IEEE Trans. Ind. Electron., vol. 45, pp. 134-
141, 1998.
@article{"International Journal of Information, Control and Computer Sciences:54762", author = "Wudhichai Assawinchaichote and Sing Kiong Nguang", title = "Robust H8 Fuzzy Control Design for Nonlinear Two-Time Scale System with Markovian Jumps based on LMI Approach", abstract = "This paper examines the problem of designing a robust H8 state-feedback controller for a class of nonlinear two-time scale systems with Markovian Jumps described by a Takagi-Sugeno (TS) fuzzy model. Based on a linear matrix inequality (LMI) approach, LMI-based sufficient conditions for the uncertain Markovian jump nonlinear two-time scale systems to have an H8 performance are derived. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard nonlinear two-time scale systems. A numerical example is provided to illustrate the design developed in this paper.
", keywords = "TS fuzzy, Markovian jumps, LMI, two-time scale systems.", volume = "2", number = "4", pages = "1109-6", }
