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Control of nonlinear underactuated systems. (English) Zbl 1022.70010

From the abstract: “We introduce a new method to design control laws for nonlinear, underactuated systems. Our method produces an infinite-dimensional family of control laws…”
It is not clear that this new method always produces any non-trivial control law. By means of Riemannian geometry there is given an equation of motion containing a potential function. The authors try to find admissible (“underactuated system”) control forces by constructing a new metric tensor \(g\) and a new potential \(V\). For \(g\) and \(V\) they derive systems of partial differential equations which merely are necessary conditions. Proposition 1.4 partially deals with existence, but the proof is not understandable.

MSC:

70H03 Lagrange’s equations
70Q05 Control of mechanical systems
37N35 Dynamical systems in control
93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
93B29 Differential-geometric methods in systems theory (MSC2000)
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References:

[1] ; ; Stabilization of mechanical systems using controlled Lagrangians. Proc. IEEE Conf. Dec. Contr., 2356-2361. San Diego, Calif., 1997.
[2] ; ; Matching and stabilization by the method of controlled Lagrangians. Proc. IEEE Conf. Dec. Contr., 1446-1451. Tampa, Fla., 1998.
[3] ; ; Controlled Lagrangians and a stabilization of mechanical systems I: The first matching theorem. Preprint, 1999.
[4] ; ; Potential shaping and the method of controlled Lagrangians. Preprint, 1999.
[5] ; ; Stabilization of the pendulum on a rotor arm by the method of controlled Lagrangians. Proc. IEEE Int. Conf. on Robotics and Automation, 500-505. Detroit, Mich., 1999.
[6] ; ; ; ; Exterior differential systems. Mathematical Sciences Research Institute Publications, 18. Springer, New York, 1991. · Zbl 0726.58002
[7] Notes on differential geometry. Van Nostrand Mathematical Studies, No. 3. D. Van Nostrand, Princeton, N.J.-Toronto-London, 1965.
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