# zbMATH — the first resource for mathematics

Global stabilization of nonlinear systems with a class of unmatched uncertainties. (English) Zbl 0759.93060
The paper studies global stability properties of dynamical systems described by $\dot x(t)=f(x,t)+\tilde\xi(x,t)+B(x,t)[\xi(x,t)+u],\;x(t_ 0)=x_ 0,$ where $$x(t)\in\mathbb{R}^ n$$ denotes the state, $$u(t)\in\mathbb{R}^ n$$ the control, $$\tilde\xi(x,t)$$ the unmatched constraints and $$\xi(x,t)$$ the matched constraints. Under the assumption of global asymptotic stability of $$\dot x(t)=f(x,t)$$ and under certain unboundedness assumptions on the uncertainties of the constraints as well as of $$f$$ and $$B$$, the author constructs a state feedback controller $$u(x,t)$$ which stabilizes the closed loop system in the large. The techniques make use of the fact that Lyapunov functions are in general not unique.

##### MSC:
 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93C10 Nonlinear systems in control theory 70K20 Stability for nonlinear problems in mechanics 93C15 Control/observation systems governed by ordinary differential equations 93D09 Robust stability 93D30 Lyapunov and storage functions
Full Text:
##### References:
 [1] Barmish, B.R.; Corless, M.J.; Leitmann, G., A new class of stabilizing controllers for uncertain dynamical systems, SIAM J. control optim., 21, 246-255, (1983) · Zbl 0503.93049 [2] Barmish, B.R.; Leitmann, G., On ultimate boundedness control of uncertain systems in the absence of matching assumptions, IEEE trans. automat. control, 27, 153-158, (1982) · Zbl 0469.93043 [3] Chen, Y.H., On the robustness of mismatched uncertain dynamical systems, Trans. ASME J. dynamic systems measurement control, 109, 29-35, (1987) · Zbl 0637.93020 [4] Chen, Y.H.; Leitmann, G., Robustness of uncertain systems in the absence of matching assumptions, Internat. J. control, 45, 1527-1542, (1987) · Zbl 0623.93023 [5] Corless, M.J.; Leitmann, G., Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE trans. automat. control, 26, 1139-1144, (1981) · Zbl 0473.93056 [6] Corless, M.J.; Leitmann, G., Adaptive control for systems containing uncertain functions and uncertain functions with uncertain bounds, J. optim. theory appl., 41, 155-168, (1983) · Zbl 0497.93028 [7] Gutman, S., Uncertain dynamical systems — A Lyapunov MIN-MAX approach, IEEE trans. automat. control, 24, 437-443, (1979) · Zbl 0416.93076 [8] Hale, J.K., Ordinary differential equations, (1980), R.E. Krieger Malabar, FL · Zbl 0186.40901 [9] Michel, A.N.; Miller, R.K., Qualitative analysis of large scale dynamic systems, (1977), Academic Press New York · Zbl 0358.93028 [10] Qu, Z.; Dorsey, J., Robust control of generalized dynamic systems without matching conditions, Trans. ASME J. dynamic systems measurement control, 113, 4, (1991) · Zbl 0745.93063 [11] Z. Qu, Asymptotic stability of controlling uncertain dynamical systems, Automatica (submitted). · Zbl 0806.93044 [12] Qu, Z.; Dawson, D.M., Continuous feedback control guaranteeing exponential stability uncertain dynamical systems, (), 2636-2638 [13] Slotine, J.J.; Li, W., Applied nonlinear control, (1991), Prentice-Hall Englewood Cliffs, NJ [14] Vidyasagar, M., Nonlinear systems analysis, (1978), Prentice-Hall Englewood Cliff, NJ · Zbl 0407.93037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.