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Motion planning, equivalence, infinite dimensional systems

100%

Rouchon P.

International Journal of Applied Mathematics and Computer Science

2001

tom 11

nr 1

165-188

Motion planning, i.e., steering a system from one state to another, is a basic question in automatic control. For a certain class of systems described by ordinary differential equations and called flat systems (Fliess et al. 1995; 1999a), motion planning admits simple and explicit solutions. This stems from an explicit description of the trajectories by an arbitrary time function, the flat output, and a finite number of its time derivatives. Such explicit descriptions are related to old problems on Monge equations and equivalence investigated by Hilbert and Cartan. The study of several examples (the car with -trailers and the non-holonomic snake, pendulums in series and the heavy chain, the heat equation and the Euler-Bernoulli flexible beam) indicates that the notion of flatness and its underlying explicit description can be extended to infinite-dimensional systems. As in the finite-dimensional case, this property yields simple motion planning algorithms via operators of compact support. For the non-holonomic snake, such operators involve non-linear delays. For the heavy chain, they are defined via distributed delays. For heat and Euler-Bernoulli systems, their supports are reduced to a point and their definition domain coincides with the set of Gevrey functions of order 2.

Differential flatness and defect: an overview

51%

Fliess M., Lévine J., Martin P., Rouchon P.

Banach Center Publications

1995

tom 32

nr 1

209-225

We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems' standpoint, flatness and defect are best defined without distinguishing between input, state, output and other variables. We treat an example of non-flat system, the variable-length pendulum. A high frequency control strategy is proposed such that the averaged system becomes flat.

Ograniczanie wyników

1 Banach Center Publications

1 International Journal of Applied Mathematics and Computer Science

2 Rouchon P.

1 Fliess M.

1 Lévine J.

1 Martin P.

1 2001

1 1995

Wyniki wyszukiwania

Motion planning, equivalence, infinite dimensional systems

Differential flatness and defect: an overview