zbMATH — the first resource for mathematics

Verified stability analysis of continuous-time control systems with bounded parameter uncertainties and stochastic disturbances. (English) Zbl 1246.93122
Summary: For nonlinear systems, feedback control strategies have to be parameterized in such a way that they guarantee asymptotic stability in a certain neighborhood of desired operating points or desired trajectories. Due to not exactly known initial conditions, parameter uncertainties, and measurement errors characterizing dynamic system models in real applications, interval techniques are taken into consideration in this paper to verify stability properties of nonlinear uncertain systems with continuous-time dynamics. These techniques aim at a computation of guaranteed regions of attraction for asymptotically stable equilibria. The practical applicability is shown for the analysis of tracking controllers for ship motions in an uncertain environment. In this application, we focus on analyzing the effects of parameter uncertainties on the domains in the state-space that can be proven to belong to the region of attraction of the desired equilibrium.

93E15 Stochastic stability in control theory
93C41 Control/observation systems with incomplete information
93C73 Perturbations in control/observation systems
93B52 Feedback control
93C10 Nonlinear systems in control theory
Full Text: DOI
[1] Ackermann J, Blue P, Bünte T, Güvenc L, Kaesbauer D, Kordt M, Muhler M, Odenthal D (2002) Robust control: The parameter space approach, 2nd edn. Springer, London
[2] Bendsten C, Stauning O (2007) FADBAD++, Version 2.1. http://www.fadbad.com
[3] Delanoue N (2006) Algorithmes numériques pour l’analyse topologique –Analyse par intervalles et théorie des graphes. Ph.D. thesis, École Doctorale d’Angers (in French)
[4] Hyodo N, Hong M, Yanami H, Anai H, Hara S (2006) Development of a MATLAB toolbox for parametric robust control–New algorithms and functions. In: Proceedings of SICE annual conference. Busan, Korea, pp 2856–2861
[5] Knüppel O (1994) Profil/BIAS–A fast interval library. Computing 53: 277–287 · Zbl 0808.65055
[6] Krawczyk R. (1969) Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4: 189–201 (in German) · Zbl 0187.10001
[7] Kushner H (1967) Stochastic stability and control. Academic Press, New York · Zbl 0244.93065
[8] Pham QC, Tabareau N, Slotine JJ (2009) A contraction theory approach to stochastic incremental stability. IEEE Trans Autom Control 54(4): 816–820 · Zbl 1367.60073
[9] Rauh A, Minisini J, Hofer EP (2008) Towards the development of an interval arithmetic environment for validated computer-aided design and verification of systems in control engineering. In: Proceedings of Dagstuhl Seminar 08021: Numerical validation in current hardware architectures. Lecture Notes in Computer Science, vol 5492. Springer, Germany, pp 175–188
[10] Rauh A, Minisini J, Hofer EP (2009) Verification techniques for sensitivity analysis and design of controllers for nonlinear dynamic systems with uncertainties. Special issue of the International Journal of Applied Mathematics and Computer Science AMCS. Verified Methods: Applications in Medicine and Engineering 19(3): 425–439 · Zbl 1300.93060
[11] Rohn J (1994) Positive definiteness and stability of interval matrices. SIAM J Matrix Anal Appl 15(1): 175–184 · Zbl 0796.65065
[12] Walter É, Jaulin L (1994) Guaranteed characterization of stability domains via set inversion. IEEE Trans Autom Control 39(4): 886–889 · Zbl 0800.93958
[13] Weinmann A (1991) Uncertain models and robust control. Springer, Wien
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.