Controllability of nonlinear systems
TL;DR: In this article, the controllability of nonlinear systems described by the equation dx/dt - F(x,u) was discussed and it was shown that strong accessibility implies strong accessibility for a large class of manifolds including Euclidean spaces.
About: This article is published in Journal of Differential Equations.The article was published on 1972-07-01 and is currently open access. It has received 718 citations till now. The article focuses on the topics: Controllability & Manifold.
Citations
More filters
TL;DR: In this paper, the authors introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous feedback, which subsumes the physical properties of a linearizing output and provides another nonlinear extension of Kalman's controllability.
Abstract: 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 where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.
3,025 citations
TL;DR: The properties of controllability, observability, and the theory of minimal realization for linear systems are well-understood and have been very useful in analyzing such systems as discussed by the authors.
Abstract: The properties of controllability, observability, and the theory of minimal realization for linear systems are well-understood and have been very useful in analyzing such systems. This paper deals with analogous questions for nonlinear systems.
2,306 citations
TL;DR: In this article, a class of inherently nonlinear control problems arising directly from physical assumptions about constraints on the motion of a mechanical system is identified and a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested.
Abstract: A class of inherently nonlinear control problems has been identified, the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system. Models are presented for mechanical systems with nonholonomic constraints represented both by differential-algebraic equations and by reduced state equations. Control issues for this class of systems are studied and a number of fundamental results are derived. Although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback, a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested. >
857 citations
TL;DR: In this paper, it was shown that the orbits of D are C' submanifolds of M, and moreover that they are the maximal integral submansions of a certain C9? distribution PD.
Abstract: Let D be an arbitrary set of Cc vector fields on the Cc manifold M. It is shown that the orbits of D are C' submanifolds of M, and that, moreover, they are the maximal integral submanifolds of a certain C9? distribution PD. (In general, the dimension of PD(m) will not be the same for all m EM.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow's theorem to the maximal integral submanifolds of the smallest distribution A such that every vector field X in the Lie algebra generated by D belongs to A (i.e. X(m) e A(m) for every m EM). Their work therefore requires the additional assumption that A be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of A is not assumed in proving the first main result. It turns out that A is integrable if and only if A = PD' and this fact makes it possible to derive a characterization of integrability and Chow's theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.
851 citations
TL;DR: Tangent spaces of a sub-Riemannian manifold are themselves sub-riemannians as mentioned in this paper, and they come with an algebraic structure: nilpotent Lie groups with dilations.
Abstract: Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian, case, they are indeed vector spaces, that is, abelian groups with dilations. Actually, the above is true only for regular points. At singular points, instead of nilpotent Lie groups one gets quotient spaces G/H of such groups G.
749 citations
References
More filters
Book•
01 Jan 1962TL;DR: In this article, the classification of symmetric spaces has been studied in the context of Lie groups and Lie algebras, and a list of notational conventions has been proposed.
Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces On the classification of symmetric spaces Functions on symmetric spaces Bibliography List of notational conventions Symbols frequently used Author index Subject index Reviews for the first edition.
3,013 citations
Book•
01 Jan 1983
TL;DR: In this article, the authors present an algebraic model of transitive differential geometry and the integrability problem for geometrical structures on manifolds, which they call integral calculus on manifold.
Abstract: Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations Differentiable Manifolds: 1. Definitions 2. Differential maps 3. Sard's theorem 4. Partitions of unity, approximation theorems 5. The tangent space 6. The principal bundle 7. The tensor bundles 8. Vector fields and Lie derivatives Integral Calculus on Manifolds: 1. The operator $d$ 2. Chains and integration 3. Integration of densities 4. $0$ and $n$-dimensional cohomology, degree 5. Frobenius' theorem 6. Darboux's theorem 7. Hamiltonian structures The Calculus of Variations: 1. Legendre transformations 2. Necessary conditions 3. Conservation laws 4. Sufficient conditions 5. Conjugate and focal points, Jacobi's condition 6. The Riemannian case 7. Completeness 8. Isometries Lie Groups: 1. Definitions 2. The invariant forms and the Lie algebra 3. Normal coordinates, exponential map 4. Closed subgroups 5. Invariant metrics 6. Forms with values in a vector space Differential Geometry of Euclidean Space: 1. The equations of structure of Euclidean space 2. The equations of structure of a submanifold 3. The equations of structure of a Riemann manifold 4. Curves in Euclidean space 5. The second fundamental form 6. Surfaces The Geometry of $G$-Structures: 1. Principal and associated bundles, connections 2. $G$-structures 3. Prolongations 4. Structures of finite type 5. Connections on $G$-structures 6. The spray of a linear connection Appendix I: Two existence theorems Appendix II: Outline of theory of integration on $E^n$ Appendix III: An algebraic model of transitive differential geometry Appendix IV: The integrability problem for geometrical structures References Index.
1,209 citations
Book•
[...]
01 Jan 1946
TL;DR: Chevalley as mentioned in this paper introduced the notion of a Lie group as a global object in the calculus of exterior differential forms, and showed how to construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group).
Abstract: This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms.The book opens with a short description of the classical groups: unitary groups, orthogonal groups, symplectic groups, etc. These special groups are then used to illustrate the general properties of Lie groups, which are considered later. The general notion of a Lie group is defined and correlated with the algebraic notion of a Lie algebra; the subgroups, factor groups, and homomorphisms of Lie groups are studied by making use of the Lie algebra. The last chapter is concerned with the theory of compact groups, culminating in Peter-Weyl's theorem on the existence of representations. Given a compact group, it is shown how one can construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group). This construction is intimately related to the proof of the generalization given by Tannaka of Pontrjagin's duality theorem for Abelian groups.The continued importance of Lie groups in mathematics and theoretical physics make this an indispensable volume for researchers in both fields.
1,003 citations
Book•
01 Oct 1964TL;DR: Manifolds Lie groups Fibre bundles Differential forms Connexions Affine connexions Riemannian manifolds Geodesics and complete RiemANNian manifoldolds Riemmannian curvature Immersions and the second fundamental form Second variation of arc length Theorems on differential equations Bibliography Subject index as mentioned in this paper
Abstract: Manifolds Lie groups Fibre bundles Differential forms Connexions Affine connexions Riemannian manifolds Geodesics and complete Riemannian manifolds Riemannian curvature Immersions and the second fundamental form Second variation of arc length Theorems on differential equations Bibliography Subject index.
937 citations
Journal Article•
714 citations