Document Actions
Home
This site is intended to be a collaborative project showcasing nonlinear control techniques for a series of benchmark problems. Because of this we want to recognize everyone who has contributed to this project. If you are interested in contributing, or would like to see who has helped provide the content for this site, please click on the contributors link at the bottom of this page.
Underactuated mechanical systems provide a challenging area for control system design. Some everyday examples consist of rocket guidance, satellite and underwater vehicle orientation control, vibration damping for cargo transport using overhead cranes, and stabilization of hovering aircraft. The characterizing feature of underactuated systems is that they have fewer actuators than degrees of freedom (DOF). A system's DOF can be thought of as the number of independent drives that can be applied to the system.
The control challenge for these systems stem from the nonlinear dynamic governing equations that do not lend themselves to the standard methods for controlling fully actuated mechanical systems, such as state feedback or Lyapunov. Limited success has been achieved through linearization and state space methods, though these approximations have a limited range of usefulness due to the linear approximations.
The development of stabilizing nonlinear controllers for underactuated mechanical systems have been approached from a variety of directions, the ones that we will consider are:
The controlled Lagrangian
method, which requires structured modifications to be
made to the uncontrolled system Lagrangian, thereby, constructing the
controlled Lagrangian. The Euler-Lagrange equations derived from
the controlled Lagrangian describe the closed-loop dynamics of the
system.
The lambda method, where nonlinear partial
differential equations arising from matching equations are recast as
linear partial differential equations through a transformation.
This allows for a family of stabilizing controllers to be found
based on the mass matrix, potential energy and damping terms for the
system.
The method of interconnection and damping assignment -
passivity based control (IDA-PBC), where a goal of the
stabilizing control is to monotonically reduce the total mechanical
energy. IDA-PBC is a significant development since its authors
have shown that both the method of controlled Lagrangian's and the
lambda method are special cases of IDA-PBC.
The direct
Lyapunov approach is the fourth method for the development
of a nonlinear control law, it utilizes the idea of Lyapunov stability
for the design of a stabilizing controller by solving certain matching
conditions. This method may offer a broader range of application than
the methods mentioned previously as the ODE's obtained in the
controller design may be solved numerically via feedback.
Warren N. White is an associate professor of Mechanical Engineering at Kansas State University
Mikil Foss is an assistant professor of Mathematics at the University of Nebraska-Lincoln