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Controlled Lagrangian and Hamiltonian Systems


Chang, Dong Eui (2002) Controlled Lagrangian and Hamiltonian Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3DRR-ZV53.


any control systems are mechanical systems. The unique feature of mechanical systems is the notion of energy, which gives much information on the stability of equilibria. Two kinds of forces are associated with the energy: dissipative force and gyroscopic force. A dissipative force is, by definition, a force which decreases the energy, and a gyroscopic force is, by definition, a force that does not change the energy. Gyroscopic forces add couplings to the dynamics. In this thesis, we develop a control design methodology which makes full use of these three physical notions: energy, dissipation, and coupling.

First, we develop the method of controlled Lagrangian systems. It is a systematic procedure for designing stabilizing controllers for mechanical systems by making use of energy, dissipative forces, and gyroscopic forces. The basic idea is as follows: Suppose that we are given a mechanical system and want to design a controller to asymptotically stabilize an equilibrium of interest. We look for a feedback control law such that the closed-loop dynamics can be also described by a new Lagrangian with a dissipative force and a gyroscopic force where the energy of the new Lagrangian has a minimum at the equilibrium. Then we check for asymptotic stability by applying the Lyapunov stability theory with the new energy as a Lyapunov function.

Next, we show that the method of controlled Lagrangian systems and its Hamiltonian counterpart, the method of controlled Hamiltonian systems, are equivalent for simple mechanical systems where the underlying Lagrangian is of the form kinetic minus potential energy. In addition, we extend both the Lagrangian and Hamiltonian sides of this theory to include systems with symmetry and discuss the relevant reduction theory.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Control and Dynamical Systems
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Control and Dynamical Systems
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marsden, Jerrold E.
Thesis Committee:
  • Marsden, Jerrold E. (chair)
  • Burdick, Joel Wakeman
  • Low, Steven H.
  • Mabuchi, Hideo
  • Murray, Richard M.
Defense Date:30 April 2002
Record Number:CaltechTHESIS:10112010-161816245
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6128
Deposited On:12 Oct 2010 14:54
Last Modified:15 Jan 2022 00:28

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