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We present a new type of feedback linearization that is tailored for mechanical control systems. We call it a mechanical feedback linearization. Its basic feature is preservation of the mechanical structure of the system. For mechanical systems with a scalar control, we formulate necessary and sufficient conditions that are verifiable using differentiations and algebraic operations only. We illustrate our results with several examples.

In this paper, we study the problem of constructing flat inputs for multi-output dynamical systems, in particular, we address the issue of the minimal modification of the initial dynamical system (the measure of modification being the number of equations that have to be changed by adding flat inputs). We show that in the observable case, control vector fields that distort m equations only (where m is the number of measurements) can always be constructed (and this is the minimal possible number of equations that have to be modified by adding flat inputs), while in the unobservable case, the best that we can hope for is that m + 1 equations only are modified such that they involve flat inputs. We discuss when the original output is a flat output of minimal possible differential weight for the minimally modified control system (where by differential weight, we mean the minimal number of derivatives of components of a flat output needed to express all states and controls). We propose a solution for constructing flat inputs leading to a minimally modified control system consistent with the minimal differential weight and, moreover, for which the observable part is affected by the minimal possible number of controls (this last property being important in applications). We show that, in that case, at least 2m − 1 equations have to be affected by the flat inputs.

In this paper, we relate the feedback canonical form \textbf{FNCF} of differential-algebraic control systems (DACSs) with the famous Morse canonical form \textbf{MCF} of ordinary differential equation control systems (ODECSs). First, a procedure called an explicitation (with driving variables) is proposed to connect the two above categories of control systems by attaching to a DACS a class of ODECSs with two kinds of inputs (the original control input $u$ and a vector of driving variables $v$). Then, we show that any ODECS with two kinds of inputs can be transformed into its extended \textbf{MCF} via two intermediate forms: the extended Morse triangular form and the extended Morse normal form. Next, we illustrate that the \textbf{FNCF} of a DACS and the extended \textbf{MCF} of the explicitation system have a perfect one-to-one correspondence. At last, an algorithm is proposed to transform a given DACS into its \textbf{FBCF} via the explicitation procedure and a numerical example is given to show the efficiency of the proposed algorithm., 35 pages, 2 figures

In this article, we propose two normal forms for nonlinear differential-algebraic control systems (DACSs) under external feedback equivalence, using a notion called maximal controlled invariant submanifold. The two normal forms simplify the system structures and facilitate understanding the various roles of variables for nonlinear DACSs. Moreover, we study when a given nonlinear DACS is internally regularizable, that is, when there exists a state feedback transforming the DACS into a differential-algebraic equation (DAE) with internal regularity, the latter notion is closely related to the existence and uniqueness of solutions of DAEs. We also revise a commonly used method in DAE solution theory, called the geometric reduction method. We apply this method to DACSs and formulate it as an algorithm, which is used to construct maximal controlled invariant submanifolds and to find internal regularization feedbacks. Two examples of mechanical systems are used to illustrate the proposed normal forms and to show how to internally regularize DACSs.

International audience; For observed mechanical systems, we study the problem of constructing flat inputs that are consistent with the mechanical structure of the systems. Such inputs will be called mechanical flat inputs. We show that contrary to flat inputs, that exist (almost) everywhere, in general, for the existence of mechanical flat inputs, additional structural conditions are needed. We provide necessary and sufficient conditions for the existence of mechanical inputs for observed mechanical systems with n degrees of freedom, where n is arbitrary, and n − 1 measurements.

For systems that are not feedback linearizable, a natural question is: how to find the largest feedback linearizable subsystem and, if the partial linearization is not unique, what are the control-theoretic properties of various partial linearizations. In this paper, we will consider the problem of how to choose a partially linearizing output that renders the zero dynamics asymptotically stable and when such an output exists.We will state general results solving completely the problem for systems whose linearizability defect is one by identifying and describing two classes of systems. For the first class, all maximal partial linearizations lead to the same zero dynamics. For the second class, any asymptotic behavior of the zero dynamics can be achieved by a suitable choice of a partially linearizing output. In the second part, we apply our results to mechanical systems with two-degrees-of-freedom and provide a detailed study of their partial linearizations.We illustrate the obtained results by examples of Acrobot (which belongs to the second class) and Pendubot (which belongs to the first class).

In this paper, we study two kinds of linearization (internal and external) of nonlinear differential-algebraic equations DAEs of semi-explicit SE form. The difference of external and internal linearization is illustrated by an example of a mechanical system. Moreover, we define different levels of external equivalence for two SE DAEs. The proposed explicitation procedure allows us to treat a given SE DAE as a control system defined up to feedback transformation (a class of control systems). Then sufficient and necessary conditions, expressed via explicitation procedure, are given to describe when a given SE DAE is level-3 externally equivalent to a linear SE DAE of some specific forms. At last, we show by an example that level-2 external linearization of a DAE can be achieved if its explicitation is level-2 input-output linearizable.

In this paper, we study feedback linearization problems for nonlinear differential-algebraic control systems (DACSs). We consider two kinds of feedback equivalences, namely, the external feedback equivalence, which is defined (locally) on the whole generalized state space, and the internal feedback equivalence, which is defined on the locally maximal controlled invariant submanifold (i.e., on the set where solutions exist). Necessary and sufficient conditions are given for the locally internal and the locally external feedback linearizability of DACSs with the help of a notion called the explicitation with driving variables, which attaches a class of ordinary differential equation control systems (ODECSs) to a given DACS. We show that the feedback linearizability of a DACS is closely related to the involutivity of the linearizability distributions of the explicitation systems. Finally, we apply our results of feedback linearization of DACSs to an academical example and a constrained mechanical system., 20 pages

For nonlinear differential-algebraic equations (DAEs), we define two kinds of equivalences, namely, the external and internal equivalence. Roughly speaking, the word "external" means that we consider a DAE (locally) everywhere and "internal" means that we consider the DAE on its (locally) maximal invariant submanifold (i.e., where its solutions exist) only. First, we revise the geometric reduction method in DAEs solution theory and formulate an implementable algorithm to realize that method. Then a procedure named explicitation with driving variables is proposed to connect nonlinear DAEs with nonlinear control systems and we show that the driving variables of an explicitation system can be reduced under some involutivity conditions. Finally, due to the explicitation, we will use some notions from nonlinear control theory to derive two nonlinear generalizations of the Weierstrass form., 38 pages

In this paper, we study the problem of constructing flat inputs for multi-output dynamical systems. The notion of flat inputs has been introduced by Waldherr and Zeitz in [30, 31] and can be seen as dual to that of flat outputs. In the single-output case, a flat input can be constructed if and only if the original dynamical system together with its output is observable. In the multi-output case, the observability is not necessary for the existence of flat inputs. The observable case has been treated in [31], where a system of linear algebraic equations has been proposed in order to determine the control vector fields associated to the flat inputs. The goal of this paper is to treat the unobservable case for multi-output dynamical systems. We start by discussing the case when the dynamical system together with the given output is observable and we present a generalization of the results of [31] by relating them with the notion of minimal differential weight. Then we give our main results. We consider the unobservable case for which locally, on an open and dense subset of $\mathbb R^n$ , we construct control vector fields $g_1 , \ldots, g_m$ such that the associated control system is flat. Finally, we explain how our results can be applied to private communication.

This paper addresses the problem of normal forms and singularities of non-holonomic robotic systems represented by control-affine systems. By means of the concept of the end-point map of the system, and of the system’s Jacobian, the configuration singularities have been defined as the control functions for which the Jacobian is not surjective. The presence of these singularities impairs performance of Jacobian motion planning algorithms. Being the singular optimal controls, the configuration singularities can be examined using the tools from the optimal control theory. The main idea of this paper is to rely the analysis of configuration singularities on normal forms of robotic systems. This idea has been applied to the dynamics of a space manipulator. Normal forms of this manipulator under the feedback equivalence have been obtained, and exploited in the analysis of its configuration singularities.

In this paper, we study invariant control systems that generalise positive systems. A characterisation of linear control systems invariant on polyhedral cones (corner regions) in the state-space, called cone-invariant linear control systems, is established both for the inputs taking values in a polyhedral cone in the control space and for the inputs taking values in an affine polyhedral cone. The problem of equivalence between control systems invariant on corner regions is introduced. For cone-invariant linear control systems, we study invariance-preserving state-equivalence and invariance-preserving feedback-equivalence and present characterisations of both notions of equivalence.

International audience; We present normal forms for nonlinear control systems that are the closest to static feedback linearizable ones, that is, for systems that become feedback linearizable via the simplest dynamic feedback, which is the one‐fold prolongation of a suitably chosen control. They form a particular class of flat systems, namely those of differential weight n + m + 1, where n is the number of states and m is the number of controls. We also show that the dynamic feedback may create singularities in the control space depending on the state and we discuss them. We also address the issue of the normalization of the system only versus that of the system together with a flat output. Finally, we illustrate our results by several examples.

International audience; In this paper we study feedback linearization of multi-input control-affine systems via a particular class of nonregular feedback transformations, namely by reducing the number of controls by one. A complete geometric characterization of systems of that class requires special properties of the linearizability distributions Ɗ0 ⊂ Ɗ1 ⊂ Ɗ 2 ⊂⋯. Contrary to the case of regular feedback linearization, they need not be involutive but the first noninvolutive one has to contain a sufficiently large involutive subdistribution. Recently, we solved the problem of Ɗ0 being noninvolutive. In the present paper, we study the case of Ɗ0 involutive but Ɗ1 noninvolutive. This case is specially interesting because it covers a big class of mechanical control systems. We provide geometric necessary and sufficient conditions describing our class of systems that can be verified by differentiation and algebraic operations only. We illustrate our results by several examples and discuss relations with other linearizability problems (static invertible feedback linearization or dynamic linearization).

The paper addresses the problem of transforming single-output continuous and discrete-time nonlinear dynamical systems into their respective extended observer forms. The output injection term in the extended observer form of degree $I$ depends, besides on the output also on its first $I$ derivatives (in the continuous-time case) or first $I$ past values (in the discrete-time case). Intrinsic necessary and sufficient conditions are given for the existence of the state transformation that transforms the equations into the extended observer form, both for continuous- and discrete-time cases. The conditions can be directly checked from state equations and do not rely on the input-output equation as in the earlier papers.

In this paper, we explain how flatness (in particular, how constructing flat inputs) can be applied to secure communication systems via chaotic models. In order to send confidential messages, we use a transmitter whose dynamics are a flat control system steaming from a dynamical one (with no inputs) originally composed by two independent chaotic subsystems (a Chua and a Rossler circuit, resp.). We explain how from the original dynamical (chaotic) system we can construct a flat control system, the flatness property being achieved by adding control vector fields or, equivalently, inputs. Such inputs are called flat inputs in Waldherr and Zeitz (2008, 2010) since they lead to a flat control system. In our construction, adding inputs is done in a minimal way because they affect only three components of the original dynamical chaotic system. We show numerically that the chaotic behavior of the dynamical system is preserved for the control system (which is crucial for secure communication).

This paper is a survey devoted to the formal approach to the feedback equivalence problem for nonlinear control systems. We show how classical Poincare’s approach, developed for dynamical systems, generalizes to control systems for continuous and discrete-time. We present normal forms and canonical forms for nonlinear control systems (single-input and multi-input, control-affine and general). We use the formal approach to study symmetries of nonlinear control systems as well as discuss special forms: linear and feedforward. We illustrate presented forms by various examples in dimensions 3 and 4.

International audience; In this paper we study the feedback linearization of multi-input control-affine systems via a particular class of nonregu-lar feedback transformations. We give a complete geometric characterization of systems that become static feedback linearizable after a one-fold reduction of a suitably chosen control. That problem can be seen as the dual of the linearization via invertible one-fold prolongation of a suitably chosen control (which is the simplest dynamic feedback). We discuss in detail similarities and differences of both problems. We propose necessary and sufficient conditions describing the class of systems linearizable via a one-fold reduction, and discuss when the proposed conditions can be verified (by differentiation and algebraic operations only). We provide a normal form and illustrate our results by several examples.

We study flatness of two-input mechanical systems with 3 degrees of freedom. Our goal is to describe the geometry of this class of systems from the point of view of differential weight. We completely characterize flatness of differential weight 8 and 9. We show that a mechanical systems with 3 degrees of freedom is flat of differential weight 8 or 9 if and only if it is config-flat or vel-flat of differential weight 8 or 9 and we describe config-flatness of differential weight 10.

We study relations between vakonomically and nonholonomically constrained dynamics for the same set of linear constraints for a Lagrangian system on a Lie group. We obtain necessary and sucient conditions under which a nonholomic trajectory is also a vakonomic one. Concrete example of a two-wheeled carriage with a shifted center of mass is discussed.

International audience; We study the problem of optimal laser-induced population transfer in n-level quantum systems. This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on $SO(n)$ , and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n = 2 and n = 3 the Hamiltonian system associated with the Pontryagin Maximum Principle PMP is integrable. We will show that this changes completely for $n\geqslant 4$ . Specifically, for n = 4, we will prove that the adjoint equation of the PMP does not possess any meromorphic first integral independent of the Hamiltonian on the levels of the Casimir functions. This implies that the system is not B-integrable for any $n\geqslant 4$ . In proving our nonintegrability results we will use differential Galois theory.

Summary A nonlinear control system is said to be orbital feedback linearizable if there exist an invertible static feedback and a change of time scale (depending on the state) which transform the system into a linear system. We give geometric necessary and sufficient conditions describing multi-input control-affine systems that are orbital feedback linearizable out of equilibria and in the case of equal controllability indices. We also describe a construction of the time rescaling needed to orbitally linearize the system. Moreover, we analyze close relations between orbital feedback linearizable control-affine systems and control-linear systems that are feedback equivalent to a multi-chained form comparing geometric structures corresponding to both problems. We illustrate our results by two examples, one being a rigid bar moving in .Copyright © 2014 John Wiley & Sons, Ltd.