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Exponential Stabilization of Driftless Nonlinear Control Systems

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Robert T. M'Closkey

PhD Dissertation, Caltech, Dec 1994

This dissertation lays the foundation for practical exponential
stabilization of driftless control systems. Driftless systems have
the form $$\dot x = X_1(x)u_1+\cdots +X_m(x)u_m, \quad x\in\real^n$$.
Such systems arise when modeling mechanical systems with nonholonomic
constraints. In engineering applications it is often required to
maintain the mechanical system around a desired configuration. This
task is treated as a stabilization problem where the desired
configuration is made an asymptotically stable equilibrium point. The
control design is carried out on an approximate system. The
approximation process yields a nilpotent set of input vector fields
which, in a special coordinate system, are homogeneous with respect to
a non-standard dilation. Even though the approximation can be given a
coordinate-free interpretation, the homogeneous structure is useful to
exploit: the feedbacks are required to be homogeneous functions and
thus preserve the homogeneous structure in the closed-loop system.
The stability achieved is called {\em $\rho$-exponential stability}.
The closed-loop system is stable and the equilibrium point is
exponentially attractive. This extended notion of exponential
stability is required since the feedback, and hence the closed-loop
system, is not Lipschitz. However, it is shown that the convergence
rate of a Lipschitz closed-loop driftless system cannot be bounded by
an exponential envelope.

The synthesis methods generate feedbacks which are smooth on
\rminus. The solutions of the closed-loop system are proven to be
unique in this case. In addition, the control inputs for many
driftless systems are velocities. For this class of systems it is
more appropriate for the control law to specify actuator forces
instead of velocities. We have extended the kinematic velocity
controllers to controllers which command forces and still
$\rho$-exponentially stabilize the system.

Perhaps the ultimate justification of the methods proposed in this
thesis are the experimental results. The experiments demonstrate the
superior convergence performance of the $\rho$-exponential stabilizers
versus traditional smooth feedbacks. The experiments also highlight
the importance of transformation conditioning in the feedbacks. Other
design issues, such as scaling the measured states to eliminate
hunting, are discussed. The methods in this thesis bring the
practical control of strongly nonlinear systems one step closer.

CDS Technical Report
(PDF, 1428K, 139 pages)

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Richard Murray (murray@cds.caltech.edu)
Last modified: Tue Aug 30 07:42:21 2005