The category of affine connection control systems

Lewis, Andrew D.

doi:10.1109/cdc.2001.914762

Cited by 10 publications

(16 citation statements)

References 18 publications

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“…We give a brief summary of a different perspective of the method of controlled Lagrangians taken by [2] and [23], with a flavor of [29]. This will help us to understand the under-actuation structure and controlled Lagrangians.…”

Section: Appendix I General Discussion On Controlled Lagrangiansmentioning

confidence: 99%

“…As shown above, this form of potential satisfies SM-5 with given by (29). The potential for the controlled Lagrangian is given in the new coordinates by (33) where is an arbitrary function on .…”

Section: Sm-5'mentioning

confidence: 99%

“…Thus, Assumption SM-5 is a necessary and sufficient condition for the integrability of the PDE (16). In particular, when is of the form (28) is satisfied and the solution is given by (29) where is an arbitrary function.…”

Section: A Notationmentioning

confidence: 99%

“…A different perspective on matching is given in [2] and [23]. We give a summary of this perspective in Appendix 1 along with a discussion of the related paper [29].…”

Section: F Matchingmentioning

confidence: 99%

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Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Bloch

Chang

Leonard

et al. 2001

IEEE Trans. Automat. Contr.

302

232

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Abstract-We extend the method of controlled Lagrangians (CL) to include potential shaping, which achieves complete state-space asymptotic stabilization of mechanical systems. The CL method deals with mechanical systems with symmetry and provides symmetry-preserving kinetic shaping and feedback-controlled dissipation for state-space stabilization in all but the symmetry variables. Potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables. The approach also extends the method of controlled Lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline.

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Section: Appendix I General Discussion On Controlled Lagrangiansmentioning

confidence: 99%

Section: Sm-5'mentioning

confidence: 99%

Section: A Notationmentioning

confidence: 99%

“…A different perspective on matching is given in [2] and [23]. We give a summary of this perspective in Appendix 1 along with a discussion of the related paper [29].…”

Section: F Matchingmentioning

confidence: 99%

See 2 more Smart Citations

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Bloch

Chang

Leonard

et al. 2001

IEEE Trans. Automat. Contr.

302

232

View full text Add to dashboard Cite

show abstract

“…The use of category theory for the study of problems in system theory also has a long history which can be traced back to the works of Arbib and Manes (see [2] for an introduction). More recently, several authors have also adopted a categorical approach as in [19], where the category of affine control systems is investigated. We also mention [33], where a categorical approach has been used to provide a general theory of systems.…”

mentioning

confidence: 99%

Quotients of Fully Nonlinear Control Systems

Tabuada¹,

Pappas²

2005

SIAM J. Control Optim.

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Abstract. In this paper, we introduce and study quotients of fully nonlinear control systems. Our definition is inspired by categorical definitions of quotients as well as recent work on abstractions of affine control systems. We show that quotients exist under mild regularity assumptions and characterize the structure of the quotient state/input space. This allows one to understand how states and inputs of the quotient system are related to states and inputs of the original system. We also introduce a notion of projectability which turns out to be equivalent to controlled invariance. This allows one to regard previous work on symmetries, partial symmetries, and controlled invariance as leading to special types of quotients. We also show the existence of quotients that are not induced by symmetries or controlled invariance. Such decompositions have a potential use in a theory of hierarchical control based on quotients.Key words. quotient control systems, control systems category, controlled invariance, symmetries AMS subject classifications. 93A10, 93A30, 93B11, 93C10 DOI. 10.1137/S03630129013990271. Introduction. The analysis and synthesis problems for nonlinear control systems are often very difficult due to the size and complicated nature of the equations describing the processes to be controlled. It is therefore desirable to have a methodology that decomposes control systems into smaller subsystems while preserving the properties relevant for analysis or synthesis. From a theoretical point of view, the problem of decomposing control systems is also extremely interesting since it reveals system structure that must be understood and exploited.In this paper we will focus on the study of quotient control systems since they can be seen as lower dimensional models that may still carry enough information about the original system. We will build on several accumulated results of different authors that in one way or another have made contributions to this problem. One of the first approaches was given in [17] were the analysis of the Lie algebra of a control system lead to a decomposition into smaller systems. At the same time in [35], quotients of control systems induced by observability equivalence relations where introduced in the more general context of realization theory. In [31], Lie algebraic conditions are formulated for the parallel and cascade decomposition of nonlinear control systems, while the feedback version of the same problem was addressed in [24]. A different approach was based on reduction of mechanical systems by symmetries. In [39], symmetries were introduced for mechanical control systems and further developed in [9] for general control systems. The existence of such symmetries was then used to decompose control systems as the interconnection of lower dimensionality subsystems. The notion of symmetry was further generalized in [26], where it was shown that the existence of symmetries implies that a certain distribution associated with the

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An Elementary Result On The Uniform Stability Of A Class Of Continuous Autonomous Systems

Sosa¹,

Martinez‐Rodriguez²,

Pujol³

2014

Asian Journal of Control

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This paper presents an elementary result on the uniform stability of an equilibrium point for a class of autonomous systems that have an exponentially decreasing factor in their dynamics. The mapping that defines the dynamics may be not Lipschitz so uniqueness of solutions is not assured. Such a class of systems may arise in the stability analysis of some closed-loop control systems, as is the case that leads to this problem, namely, the application of the vertically transverse function approach to the practical point-stabilization of underactuated mechanical systems evolving on Lie groups.

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The category of affine connection control systems

Cited by 10 publications

References 18 publications

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Quotients of Fully Nonlinear Control Systems

An Elementary Result On The Uniform Stability Of A Class Of Continuous Autonomous Systems

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