Control of Dynamical Systems
The 'art of control' is to use the input channels of a dynamical system to achieve a behavior that the original (nominal) system does not exhibit.
Mathematical control theory develops techniques to
- analyze the global behavior of (nonlinear) control systems
- study the behavior under uncertainties (robustness)
- design (feedback) controllers that achieve a desired system behavior
- computer numerically the systems behavior and appropriate controls
Our group is interested in the basic mathematical problems related to control and design for systems with uncertainties and a priori bounded controls. The techniques originate from geometric control theory and the theory of dynamical systems. The figures above show a piecewise constant feedback design for the practical stabilization of a tunnel diode. (Supported by NSF, ONR, DGF.)
- Collaborators
- Dr. Fritz Colonius, University of Augsburg, Germany
- Dr. Lars Grüne, University of Augsburg, Germany
- Dr. Gerhard Häckl, Germany
- Dr. Luiz San Martin, University of Campinas, Brazil
- Students
- Dr. Lisa Joseph
- Dr. Shan Lin
- Dr. Ruey-Gang Lai
- Dr. Chung-Ming Ou
- Hualin Wang
- Recent Publications
- Chapters
- Kliemann, W., Analysis of Nonlinear Stochastic Systems, in Analysis and Estimation of
Stochastic Mechanical Systems, (Schiehlen, W. and W. Wedig, eds.), CISM Courses and
Lecture, No. 303, Springer, New York (1988), 43-102.
- Papers, refeered
- Colonius, F. and W. Kliemann, Stability radii and Lyapunov exponents, in: Control of
Uncertain Systems, D. Hinrichsen and B. Martensson (eds.), Birkhäuser (1990), 19-56.
- Colonius, F. and W. Kliemann, Remarks on Ergodic Theory of Stochastic Flows and Control
Flows, in: Diffusion Processes and Related Problems in Analysis, Vol. II, (M. Pinsky and V.
Wihstutz, eds.), Birkähuser (1992), 203-240.
- Colonius, F. and W. Kliemann, Lyapunov Exponents of Control Flows, in: Arnold, L., H.
Crauel, J.-P. Eckmann (eds), 'Lyapunov Exponents,' Springer Lecture Notes in Mathematics no.
1486 (1991), 331-365.
- Colonius, F. and W. Kliemann, Stabilization of Linear Uncertain Systems, in: Modeling,
Estimation and Control of Systems with Uncertainty (A. B. DiMasi, A. Gombani, A. B.
Kurzhansky, eds.), Birkhäuser (1991), 76-90.
- Colonius, F. and W. Kliemann, Minimal and maximal Lyapunov exponents of bilinear control
systems, J. Diff. Equations 101 (1993), 232-275.
- Colonius, F. and W. Kliemann, Stabilization of Uncertain Linear Systems via Lyapunov
Exponents, in Proceedings of IEEE Conference on Decision and Control 1991, Brighton, England
(1991), Vol. I. 887-893.
- Colonius, F. and W. Kliemann, Controlling the Dynamics of a Random System, in: Nonlinear
Stochastic Mechanics (N. Bellomo, F. Casciati, eds.), Springer (1992), 333-346.
- Colonius, F. and W. Kliemann, Limit Behavior and Genericity for Nonlinear Control Systems,
J. Diff. Equations 109 (1994), 8-41.
- Colonius, F. and W. Kliemann, On control sets and feedback for nonlinear systems, in
Proceedings of IFAC NOLCOS '92, Bordeaux, France (1992), 49-56.
- Colonius, F. and W. Kliemann, A Dynamical Systems Approach to Control, in Proceedings of
IFAC NOLCOS '92, Bordeaux, France (1992), 361-367.
- Colonius, F., G. Häckl and W. Kliemann, Controllability near a Hopf bifurcation, Proceedings
of IEEE Conference and Decision and Control 1992, Tucson (1992), 2113-2118.
- Colonius, F. and W. Kliemann, Control properties of linear semigroups on projective spaces, J.
Dynamics and Differential Equations 5 (1993), 495-528.
- Colonius, F. and W. Kliemann, Some aspects of control systems as dynamical systems, J.
Dynamics and Differential Equations 5 (1993), 469-494.
- Colonius, F. and W. Kliemann, Feedback stabilization of one dimensional systems near
bifurcation points, in Proceedings of 2., European Control Conference 1993, Groningen, The
Netherlands (1993), 45-49.
- Colonius, F. and W. Kliemann, Controllability and stabilization of one dimensional systems near
bifurcation points, Systems and Control Letters 24 (1995), 87-95.
- Colonius, F. and W. Kliemann, The Morse spectrum of linear flows on vector bundles, Transa.
AMS. 348(1996), 4355-4388.
- Colonius, F. and W. Kliemann, The Lyapunov spectrum of families of time-varying matrices,
Transa. AMS. 348(1996), 4389-4408.
- Colonius, F. and W. Kliemann, Asymptotic null-controllability of bilinear systems in: Geometry
in Nonlinear Control and Differential Inclusions, Banach Center Publications Vol. 32, Warsaw
(1995), 139-148.
- Colonius, F. and W. Kliemann, A stability radius for nonlinear differential equations subject to
time varying perturbations, in: Proceedings of IFAC NOLCOS '95, (1995) 44-46.
- Lin, C.-M., V. Vittal, W. Kliemann, and A. A. Fouad, Investigation of modal interaction and
its effects on control performance in stressed power systems using normal forms of vector fields,
to appear in IEEE Transactions on PWRS, Paper # 95SM522-3.
- Papers, not refereed
- Colonius, F., W. Kliemann, and S. Krull, Stability and stabilization of linear, uncertain systems
-A Lyapunov exponents approach, Report No. 372 of the Schwerpunktprogramm der Deutschen
Forschungsgemeinschaft 'Anwendungsbezogene Optimierung und Steuerung' (1992), 38 pp.
- Lin, C.-M., W. Kliemann, V. Vittal, and A. A. Fouad, Interaction between excitation control
modes and inertial modes in stressed power systems, in: Proceedings ofthe 26th North American
Power Symposium (1994), 669-678.
- Lin, S., V. Ajjarapu, B. Lee, V. Vittal, and W. Kliemann, Control of voltage collapse in an
electrical power system using center manifold theory, in: Proceedings of the Midwest Power
Symposium 95, 4 pp.