Lev Kapitanski

Control of nonlinear underactuated systems

In this paper we introduce a new method to design control laws for nonlinear, underactuated systems. Our method produces an infinite-dimensional family of control laws, whereas most control techniques only produce a finite-dimensional family. These control laws each come with a natural Lyapunov function. The inverted pendulum cart is used as an example. In addition, we construct an abstract system that is open-loop unstable and cannot be stabilized using any linear control law and demonstrate that our method produces a stabilizing control law.

On the

We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely characterized by a linear system of first order partial differential equations for one set of variables (

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-Equations for Matching Control Laws

) followed by a linear system of first order partial differential equations for the second set of variables (

Shape and Morse theory of attractors

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Brief Matching, linear systems, and the ball and beam

,

On the fundamental solution of a perturbed harmonic sscillator

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Holonomy and Skyrme's Model

). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the

Matching and digital control implementation for underactuated systems

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Does a Quantum Particle Know the Time

-equations. We give four examples illustrating different features of matching control laws. The last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations. There are systems with many matching control laws where only basic solutions are potentially useful. We introduce a rank condition indicating when this is likely to be the case.