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Robust semiglobally practical stabilization for nonlinear singularly perturbed systems. (English) Zbl 1161.93022

The paper is devoted to the construction of a state feedback control which provides robust semi-global, practical stabilization for a MIMO nonlinear singularly perturbed system with uncertain variables of the form \[ \dot x= f_1(x,\theta)+ Q_1(x,\theta)z+ g_1(x,\theta)u, \]
\[ \varepsilon\dot z= f_2(x,\theta)+ Q_2(x,\theta)z+ g_2(x,\theta) u, \] where \(x\) is the slow variable, \(z\) the fast variable, \(\theta\) the time-varying uncertain variable (\(\theta\) takes value in a compact set \(\Theta\)). The authors assume that \(Q_2\) is invertible, that the fast subsystem can be stabilized by a linear feedback \(u= k^t(x)z\) for each \((x,\theta)\), that the slow subsystem has a relative degree and that it has an asymptotically stable zero-dynamics. The construction of the feedback depends on two quadratic Lyapunov functions (with coefficients dependent on \((x,\theta)\).

MSC:

93D21 Adaptive or robust stabilization
34H05 Control problems involving ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
93C10 Nonlinear systems in control theory
93C70 Time-scale analysis and singular perturbations in control/observation systems
93B52 Feedback control
Full Text: DOI

References:

[1] Kokotovic, P. V.; Khalil, H. K.; O’Reilly, J., Singular Perturbation Methods in Control: Analysis and Design (1999), SIAM: SIAM Philadelphia · Zbl 0989.93001
[2] O’Malley, R. E., Singular Perturbation Methods for Ordinary Differential Equations (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0743.34059
[3] Corless, M.; Garofalo, F.; Gilelmo, L., New results on composite control of singularly perturbed uncertain linear systems [J], Automatica, 29, 2, 387-400 (1993) · Zbl 0772.93062
[4] Sen, S.; Datta, K. B., Stability bounds of singularly perturbed systems, IEEE Trans. Automat. Control, 38, 2, 302-304 (1993) · Zbl 0774.93053
[5] Garcia, G.; Daafouz, J.; Bernussou, J., \(H_2\) guaranteed cost control for singularly perturbed uncertain systems, IEEE Trans. Automat. Control, 43, 9, 1323-1329C (1998) · Zbl 0957.93057
[6] Shi, P.; Dragan, V., Asymptotic \(H_∞\) control of singularly perturbed systems with parametric uncertainties, IEEE Trans. Automat. Control, 44, 9, 1738-1742 (1994) · Zbl 0958.93066
[7] Singh, H.; Brown, R. H.; Naidu, D. S.; Heinen, J. A., Robust stability of singularly perturbed state feedback systems using unified approach, IEE Proc. Control Theory Appl., 49, 5, 391-396 (2001)
[8] Dragan, V.; Morozan, T.; Shi, P., Asymptotic properties of input-output operators norm associated with singularly perturbed systems with multiplicative white noise, SIAM J. Control Optim., 41, 1, 141-163 (2002) · Zbl 1027.93036
[9] Dragan, V.; Shi, P.; Boukas, E. K., Control of singularly perturbed systems with Markovian jump parameters: An \(H_∞\) approach, Automatica, 35, 8, 1369-1378 (1999) · Zbl 0931.93021
[10] Shao, Z. H., Robust stability of two-time-scale systems with nonlinear uncertainties, IEEE Trans. Automat. Control, 49, 2, 258-261 (2004) · Zbl 1365.93373
[11] Khalil, H. K., Nonlinear Systems (2002), Prentice-Hall: Prentice-Hall NewJersey · Zbl 0626.34052
[12] Son, J. W.; Lim, J. T., Robust stability of nonlinear singularly perturbed system with uncertainties, IEE Proc. Control Theory Appl., 53, 1, 104-110 (2006)
[13] Panagiotis, D. C.; Andrew, R. T.; Daoutidis, P., Robust semi-global output tracking for nonlinear singularly perturbed systems, (Proc. of IEEE conference on Decision and Control (1995), New Orleans: New Orleans LA), 5240-5245
[14] Xiushan, C.; Zhengzhi, H.; Chunhai, Kou, Semiglobally practical stabilization of a class of multivariable nonlinear systems with uncertainty, Acta Automat. Sinica, 30, 6, 1021-1026 (2004) · Zbl 1498.93608
[15] Xiushan, C.; Zhengzhi, H.; Chunhai, Kou, Semiglobally practical stabilization of a class of multivariable nonlinear systems by output feedback, Control Theory Appl., 22, 3 (2005)
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