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40
Noncommutative Power Series and Formal Liealgebraic Techniques in Nonlinear Control Theory
, 1997
"... In nonlinear control, it is helpful to choose a formalism well suited to computations involving solutions of controlled differential equations, exponentials of vector fields, and Lie brackets. We show by means of an example  the computation of control variations that give rise to the LegendreCleb ..."
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Cited by 34 (12 self)
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In nonlinear control, it is helpful to choose a formalism well suited to computations involving solutions of controlled differential equations, exponentials of vector fields, and Lie brackets. We show by means of an example  the computation of control variations that give rise to the LegendreClebsch condition  how a good choice of formalism, based on expanding diffeomorphisms as products of exponentials, can simplify the calculations. We then describe the algebraic structure underlying the formal part of these calculations, showing that it is based on the theory of formal power series, Lie series, the Chen series  introduced in control theory by M. Fliess  and the formula for the dual basis of a PoincareBirkhoffWitt basis arising from a generalized Hall basis of a free Lie algebra.
Algebraic Differential Equations And Rational Control Systems
 SIAM JOURNAL ON CONTROL AND OPTIMIZATION
"... An equivalence is shown between realizability of input/output operators by rational control systems and high order algebraic differential equations for input/output pairs. This generalizes, to nonlinear systems, the equivalence between autoregressive representations and finite dimensional linear rea ..."
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Cited by 15 (3 self)
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An equivalence is shown between realizability of input/output operators by rational control systems and high order algebraic differential equations for input/output pairs. This generalizes, to nonlinear systems, the equivalence between autoregressive representations and finite dimensional linear realizability.
Chronological algebras: combinatorics and control
, 1999
"... This article investigates the geometric and algebraic foundations of exponential product expansions in nonlinear control. Asurvey of historic developments in geometric control theory on one side, and algebraic combinatorics on the other side exhibits parallel developments and demonstrates how respec ..."
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Cited by 13 (5 self)
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This article investigates the geometric and algebraic foundations of exponential product expansions in nonlinear control. Asurvey of historic developments in geometric control theory on one side, and algebraic combinatorics on the other side exhibits parallel developments and demonstrates how respective ndings translate into powerful tools on the other side. Chronological algebras are shown to provide the fundamental structure
Generating series and nonlinear systems: Analytic aspects, local realizability, and I/O representations
 IPV6 TABLES, 12TH ANNUAL IEEE SYMPOSIUM ON HIGH PERFORMANCE INTERCONNECTS
, 1991
"... This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analyti ..."
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Cited by 11 (7 self)
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This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analytic equations.
Chronological Algebras and Nonlinear Control
 PROC. ASIAN CONF. CONTROL
, 1994
"... The Fliess series of nonlinear control theory is closely related to shuffle algebras considered by algebraic combinatorists. An infinite product expansion of this series underlies several path planning algorithms, and is the basis for polynomial realizations of nilpotent control systems. This pro ..."
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Cited by 10 (5 self)
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The Fliess series of nonlinear control theory is closely related to shuffle algebras considered by algebraic combinatorists. An infinite product expansion of this series underlies several path planning algorithms, and is the basis for polynomial realizations of nilpotent control systems. This product expansion is based on explicit knowledge of the dual bases to Poincar'e Birkhoff Witt bases built on Hall sets. We demonstrate that a chronological algebra, a structure more fundamental than shuffle algebras underlies the product expansion. Moreover, it is very closely related to the structure of general Hall sets.
The formal LaplaceBorel transform of Fliess operators and the composition product
 Inter. J. Math. Math. Sci., Article ID
, 2006
"... Abstract — In this paper, the formal LaplaceBorel transform of an analytic nonlinear inputoutput system is defined, specifically, an inputoutput system that can be represented as a Fliess operator. Using this concept and the composition product, an explicit relationship is then derived between th ..."
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Cited by 10 (8 self)
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Abstract — In this paper, the formal LaplaceBorel transform of an analytic nonlinear inputoutput system is defined, specifically, an inputoutput system that can be represented as a Fliess operator. Using this concept and the composition product, an explicit relationship is then derived between the formal LaplaceBorel transforms of the input and output signals. This provides an alternative interpretation of the symbolic calculus introduced by Fliess to compute the output response of such systems. Finally, it is shown that the formal LaplaceBorel transform provides an isomorphism between the semigroup of all well defined Fliess operators under composition and the semigroup of all locally convergent formal power series under the composition product. I.
GENERATING SERIES FOR INTERCONNECTED ANALYTIC NONLINEAR SYSTEMS
, 2005
"... Given two analytic nonlinear inputoutput systems represented as Fliess operators, four system interconnections are considered in a unified setting: the parallel connection, product connection, cascade connection, and feedback connection. In each case, the corresponding generating series is produced ..."
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Cited by 10 (8 self)
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Given two analytic nonlinear inputoutput systems represented as Fliess operators, four system interconnections are considered in a unified setting: the parallel connection, product connection, cascade connection, and feedback connection. In each case, the corresponding generating series is produced and conditions for the convergence of the corresponding Fliess operator are given. In the process, an existing notion of a composition product for formal power series has its set of known properties significantly expanded. In addition, the notion of a feedback product for formal power series is shown to be well defined in a broad context, and its basic properties are characterized.
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Cited by 8 (0 self)
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative
Loworder controllability and kinematic reductions for affine connection control systems
 SIAM Journal on Control and Optimization
"... Abstract. Controllability and kinematic modelling notions are investigated for a class of mechanical control systems. First, loworder controllability results are given for the class of mechanical control systems. Second, a precise connection is made between those mechanical systems which are dynami ..."
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Cited by 8 (5 self)
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Abstract. Controllability and kinematic modelling notions are investigated for a class of mechanical control systems. First, loworder controllability results are given for the class of mechanical control systems. Second, a precise connection is made between those mechanical systems which are dynamic (i.e., have forces as inputs) and those which are kinematic (i.e., have velocities as inputs). Interestingly and surprisingly, these two subjects are characterised and linked by a certain intrinsic vectorvalued quadratic form that can be associated to an affine connection control system.