On the Λ-equations for Matching Control Laws *

We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely charactarized by a linear system of first order partial differential equations for one set of variables (λ) followed by a linear system of first order PDEs for the second set of variables (g, V). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the λ-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. 1. Introduction. Effective procedures for designing control laws are very important in nonlinear control theory. Explicit analytic formulae for control laws play a role similar to explicit solutions to differential equations. Such formulae exist in only a few special cases, but those that exist serve as simple models to develop and test more general techniques. In this paper we discuss a class of full state feedback control laws for underactu-ated systems. In [5] we showed that this class of matching control laws is completely charactarized by a linear system of first order partial differential equations for one set of variables (λ) followed by a linear system of first order PDEs for the second set of variables (g, V). These equations always have a simple family of solutions which we call basic solutions. The system of equations for the first set of variables (λ-equations) is overdetermined. Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the λ-equations (we call these the ν-equations). If only one degree of freedom is unactuated, the solutions to all these systems of PDEs can be completely analyzed. It is often possible to get explicit formulae for the solutions to these equations. We also provide an example of a system with two unactuated degrees of freedom that has only basic solutions. There are systems with many matching control laws where only basic solutions are potentially useful. We write down a rank condition indicating when this is likely to be the case. During the last few years several researchers have investigated control laws in …