LMI-based robust control of fractional-order uncertain linear systems. (English) Zbl 1228.93087

Lan, Yong-Hong; Zhou, Yong

Comput. Math. Appl. 62, No. 3, 1460-1471 (2011).

Summary: We are concerned with the method of observer-based control and static output feedback control for fractional-order uncertain systems with the fractional commensurate order \(\alpha\)(\(0<\alpha <1)\) and \(\alpha\)(\(1\leq \alpha <2)\) via linear matrix inequality (LMI) approach, respectively. First, the sufficient conditions for robust asymptotical stability of the closed-loop control systems are presented. Next, by using matrix’s singular value decomposition (SVD) and LMI technics, the existence condition and method of designing a robust stabilizing controller for such fractional-order control systems are derived. Unlike previous methods, the results are obtained in terms of LMI, which can be easily obtained by Matlab’s LMI toolbox. Finally, two numerical examples demonstrate the validity of this approach.

MSC:

93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34A08	Fractional ordinary differential equations
93C42	Fuzzy control/observation systems

Keywords:

fractional-order uncertain systems; stability; observer-based control; output feedback control; linear matrix inequality (LMI)

Software:

Matlab; LMI toolbox

References:

[1]	Podlubny, I., Fractional Differential Equations (1999), Academie Press: Academie Press New York · Zbl 0918.34010
[2]	Hilfer, R., Application of Fractional Calculus in Physics (2000), World Science Publishing: World Science Publishing Singapore · Zbl 0998.26002
[3]	Ortigueira, M. D.; Machado, J. A.T., Special issue on fractional signal processing and applications, Signal Processing, 83, 11, 2285-2480 (2003)
[4]	Zhou, Y.; Jiao, F., Existence of mild solutions for fractional neutral evolution equations, Computer and Mathematics with Applications, 59, 1063-1077 (2010) · Zbl 1189.34154
[6]	Ladaci, S.; Loiseau, J. J.; Charef, A., Fractional order adaptive high-gain controllers for a class of linear systems, Communications in Nonlinear Science and Numerical Simulation, 13, 707-714 (2008) · Zbl 1221.93128
[7]	Hwanga, C.; Cheng, Y. C., A numerical algorithm for stability testing of fractional delay systems, Automatica, 42, 825-831 (2006) · Zbl 1137.93375
[8]	Ahn, H. S.; Chen, Y. Q., Necessary and sufficient stability condition of fractional-order interval linear systems, Automatica, 44, 11, 2985-2988 (2008) · Zbl 1152.93455
[9]	Li, Y.; Chen, Y. Q.; Podlubny, I., Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mitta-Leffler stability, Computers and Mathematics with Applications, 24, 1429-1468 (2009)
[10]	Yakar, C., Fractional differential equations in terms of comparison results and Lyapunov stability with initial time difference, Abstract and Applied Analysis, 10, 1-16 (2010) · Zbl 1196.34010
[11]	Lazarević, M. P.; Spasić, A. M., Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach, Mathematical and Computer Modelling, 49, 475-481 (2009) · Zbl 1165.34408
[12]	Wen, X. J.; Wu, Z. M.; Lu, J. G., Stability analysis of a class of nonlinear fractional-order systems, IEEE Transactions on Circuits and Systems II: Express Briefs, 55, 11, 1178-1183 (2008)
[15]	Tavazoei, M. S.; Haeri, M., A note on the stability of fractional order systems, Mathematics and Computers in Simulation, 79, 1566-1576 (2009) · Zbl 1168.34036
[16]	Sabatier, J.; Moze, M.; Farges, C., LMI stability conditions for fractional order systems, Computers and Mathematics with Applications, 59, 1594-1609 (2010) · Zbl 1189.34020
[17]	Lu, J. G.; Chen, Y. Q., Robust stability and stabilization of fractional-order interval systems with the fractional order: the case \(0 < \alpha < 1\), IEEE Transactions on Automatic Control, 55, 1, 152-159 (2010) · Zbl 1368.93506
[18]	Lu, J. G.; Chen, G. R., Robust stability and stabilization of fractional-order interval systems: an LMI approach, IEEE Transactions on Automatic Control, 54, 6, 1294-1299 (2009) · Zbl 1367.93472
[19]	Zhou, K.; Doyle, J.; Glover, K., Robust and Optimal Control (1996), Prentice Hall: Prentice Hall Englewood Cliffs, New Jersey
[20]	Xie, L., Output feedback \(H_\infty\) control of systems with parameter uncertainty, International Journal of Control, 63, 741-752 (1996)
[21]	Chilali, M.; Gahinet, P.; Apkarian, P., Robust pole placement in LMI regions, IEEE Transactions on Automatic Control, 44, 12, 2257-2270 (1999) · Zbl 1136.93352
[22]	Boyd, S.; Ghaoui, L.; Feron, E.; Balakrishnan, V., (Linear Matrix Inequalities in System and Control Theory. Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, vol. 15 (1994), Pennsylvania: Pennsylvania Philadelphia) · Zbl 0816.93004
[23]	Petersen, I. R., A stabilization algorithm for a class of uncertain linear systems, Systems & Control Letters, 8, 351-357 (1987) · Zbl 0618.93056
[24]	MacDuffee, C. C., The Theory of Matrices (2004), Dover Publications: Dover Publications New York · Zbl 0007.19507
[25]	Oustaloup, A.; Levron, F.; Mathieu, B.; Nanot, F. M., Frequency-band complex non-integer differentiator: characterization and synthesis, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47, 1, 25-39 (2000)