Nonlinear Systems

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 ordinary differential equations(ODE) obtained in the controller design may be solved numerically via feedback.

---