Control and Stabilization of a Class of Nonlinear Systems with Symmetry

View/ Open
Date
1998Author
Manikonda, Vikram
Advisor
Krishnaprasad, P.S.
Metadata
Show full item recordAbstract
The focus of this dissertation is to study issues related to controllability and stabilization of a class of underactuated mechanical systems with symmetry. In particular we look at systems whose configuration can be identified with a Lie group and the reduced equations are of the Lie-Poisson type. Examples of such systems include hovercraft, spacecraft and autonomous underwater vehicles. We present sufficient conditions for the controllability of affine nonlinear control systems where the drift vector field is a Lie-Poisson reduced Hamiltonian vector field. In this setting we show that depending on the existence of a radially unbounded Lyapunov type function, the drift vector field of the reduced system is weakly positively Poisson stable. The weak positive Poisson stability along with the Lie algebra rank condition is used to show controllability. These controllability results are then extended to the unreduced dynamics. Sufficient conditions for controllability are presented in both cases where the symmetry group is compact and noncompact. We also present a constructive approach to design feedback laws to stabilize relative equilibria of these systems. The approach is based on the observation that, under certain hypotheses the fixed points of the Lie-Poisson dynamics belong to an immersed equilibrium submanifold. The existence of such equilibrium manifolds, along with the center manifold theory is used to design stabilizing feedback laws.
Collections
Related items
Showing items related by title, author, creator and subject.
-
Control of Large Actuator Arrays Using Pattern-Forming Systems
Justh, Eric W. (1998)Pattern-forming systems are used to model many diverse phenomena from biology,chemistry and physics. These systems of differential equations havethe property that as a bifurcation (or control) parameter passes through ... -
Efficient Implementation of Controllers for Large Scale Linear Systems via Wavelet Packet Transforms
Kantor, George A.; Krishnaprasad, Perinkulam S. (1998)In this paper we present a method of efficiently implementing controllers for linear systems with large numbers of sensors and actuators. It is well known that singular value decomposition can be used to diagonalize any ... -
Computing Balanced Realizations for Nonlinear Systems
Newman, Andrew J.; Krishnaprasad, Perinkulam S. (2000)This paper addresses the problem of computability pertaining to the Scherpen(1994) theory and procedure for balancing of nonlinear systems. In contrastto Moore's (1981) balancing method for linear systems, the Scherpen ...