Noncommutative Power Series and Formal Lie-algebraic Techniques in Nonlinear Control Theory

Kawski, Matthias; Sussmann, Héctor J.

doi:10.1007/978-3-663-09823-2_10

Cited by 86 publications

(73 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Now, we recall some concepts and results concerning the application of the free algebras technique proposed and developed in [4][5][6][7][8][9][10]. Different approaches based on series representations close to (12) can be found in [13][14][15][16][17][18][19]. We consider nonlinear power moments ξ m 1 ···m k (θ, u) as words generated by the letters ξ i (θ, u), i.e., assume that the word ξ m 1 .…”

Section: Approximation In the Sense Of Time Optimalitymentioning

confidence: 99%

Time-Optimal Control Problem for a Special Class of Control Systems: Optimal Controls and Approximation in the Sense of Time Optimality

Sklyar

Ignatovich²,

Shugaryov³

2014

J Optim Theory Appl

View full text Add to dashboard Cite

We consider the time-optimal control problem to the origin for a class of nonlinear systems, called dual-to-linear systems. We obtain the general description of possible optimal controls. In particular, we show that optimal controls take the values −1, 0, and +1 only and have a finite number of points of discontinuity. We describe a class of nonlinear affine control systems which can be approximated by dual-to-linear systems in the sense of time optimality.

show abstract

Section: Approximation In the Sense Of Time Optimalitymentioning

confidence: 99%

Time-Optimal Control Problem for a Special Class of Control Systems: Optimal Controls and Approximation in the Sense of Time Optimality

Sklyar

Ignatovich²,

Shugaryov³

2014

J Optim Theory Appl

View full text Add to dashboard Cite

show abstract

“…Moreover, if is a v.f. on , then the dynamic feedback law defined by (19) yields the satisfaction of the following equation: (20) along any solution of the controlled system. Proof: The proof of (18) is easily obtained by differentiating the equality and using the identities .…”

Section: A Point Stabilizationmentioning

confidence: 99%

“…Proof: The proof of (18) is easily obtained by differentiating the equality and using the identities . By applying the feedback law (19) to (1), one deduces (20) from (18).…”

Section: A Point Stabilizationmentioning

confidence: 99%

See 1 more Smart Citation

Practical stabilization of driftless systems on lie groups: The transverse function approach

Morin

Samson

2003

IEEE Trans. Automat. Contr.

214

176

View full text Add to dashboard Cite

Abstract-A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system's controllability. Its outcome is the practical stabilization of any trajectory, i.e., not necessarily a solution of the control system, in the state-space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given.

show abstract

“…The Lie series methods rely on the notion that a system can be well approximated by a "nilpotent approximation". Contributions to this sort of approach have been made, for example, by Hermes [29][30][31], Kawski [40,42], and by Kawski and Sussmann [43]. A recent paper by Bianchini and Kawski [7] indicates that there may well be some limitations to the approach of using nilpotent approximations to determine conditions for controllability.…”

mentioning

confidence: 99%

Controllability on Infinite-Dimensional Manifolds: A Chow–Rashevsky Theorem

Salehani

Markina

2014

Acta Appl Math

View full text Add to dashboard Cite

One of the fundamental problems in control theory is that of controllability, the question of whether one can drive the system from one point to another with a given class of controls. A classical result in geometric control theory of finite-dimensional (nonlinear) systems is Chow-Rashevsky theorem that gives a sufficient condition for controllability on any connected manifold of finite dimension. In other words, the classical Chow-Rashevsky theorem, which is in fact a primary theorem in subriemannian geometry, gives a global connectivity property of a subriemannian manifold. In this paper, following the unified approach of Kriegl and Michor (The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53, Am. Math. Soc., Providence, 1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Chow-Rashevsky theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable. To indicate an application of our approach to the infinite-dimensional geometric control problems, we conclude the paper with a novel controllability result on the group of orientation-preserving diffeomorphisms of the unit circle.

show abstract

Noncommutative Power Series and Formal Lie-algebraic Techniques in Nonlinear Control Theory

Cited by 86 publications

References 9 publications

Time-Optimal Control Problem for a Special Class of Control Systems: Optimal Controls and Approximation in the Sense of Time Optimality

Time-Optimal Control Problem for a Special Class of Control Systems: Optimal Controls and Approximation in the Sense of Time Optimality

Practical stabilization of driftless systems on lie groups: The transverse function approach

Controllability on Infinite-Dimensional Manifolds: A Chow–Rashevsky Theorem

Contact Info

Product

Resources

About