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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.

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.

We study flatness of two-input control-affine systems, defined on an n-dimensional state-space. We give a complete geometric characterization of systems that become static feedback linearizable after a one-fold prolongation of a suitably chosen control. They form a particular class of flat systems: they are of differential weight n + 3. We provide a system of first order PDE's to be solved in order to find all minimal flat outputs. We illustrate our results by two examples: the induction motor and the polymerization reactor.

We discuss linearization, via a diffeomorphism, of mechanical control systems and study the problem of whether both structures, linear and mechanical ones, are compatible. The first problem we consider is: given a mechanical control systems that is linearizable, can we linearize it preserving, simultaneously, its given mechanical structure. The second problem is whether a general control-affine system that is linearizable and admits a mechanical control structure can be transformed into a linear mechanical structure. Finally we discuss equivalence to a subclass of linear mechanical control systems, namely those subject to positional forces only.

We propose a kinematic model of a system moving in lm+1 and consisting of n rigid bars attached successively to each other and subject to the nonholonomic constraints that the velocity of the source point of each bar is parallel to that bar. We prove that the associated control system is controllable and feedback equivalent to the m-chained form around any regular configuration. Hence we deduce that the n-bar system is flat and the cartesian position of the source point of the last bar is a flat output. The n-bar system is a natural generalization of the n-trailer system and we provide a comparison of flatness properties of both systems.

We study the geometry of mechanical control systems. A special class is distinguished: the class of geodesically accessible mechanical systems, for which the uniqueness of the mechanical structure is guaranteed (up to an extended point transformation). We characterise nonlinear control systems that are state equivalence to a system from that class and describe the canonical mechanical structure attached to it.

We consider the Euler equations on the Lie algebra so ( 4 , C ) with a diagonal quadratic Hamiltonian. It is known that this system always admits three functionally independent polynomial first integrals. We prove that if the system has a rational first integral functionally independent of the known three ones (so-called fourth integral), then it has a polynomial fourth first integral. This is a consequence of a more general fact that for these systems the existence of a Darboux polynomial with non vanishing cofactor implies the existence of a polynomial fourth integral.

The main purpose of this paper is to identify a class of SISO systems, which are relatively easy linearizable. These systems are in the so called Linear - Non-Linear (L-NL) form. For systems in the L-NL form it is easy to derive a linearizing diffeomorphism, which is the identity on a part of the state space, and feedback. Verifiable necessary and sufficient conditions are given in order to check if a system is in the L-NL form, followed by a constructive procedure (which does not require solving a system of first-order partial differential equations) to linearize the system. We illustrate this procedure with a model of turbine.

This paper follows a recent classification, obtained by the authors, of generic bifurcations of 1-parameter families of planar control systems. In this paper, we analyze all bifurcations of planar systems with quadratic nonlinearities, in particular those not appearing among generic ones. To illustrate our results, we discuss briefly bifurcations of a control version of the parameterized Volterra-Lotka system.

The main result of the paper states that almost any analytic single-input control system, which is truly nonlinear, that is not feedback linearizable, with controllable linearization at an equilibrium point, does not admit any symmetry preserving that point. By almost any system, we mean that we exclude a small class of odd systems, that admit just one nontrivial symmetry conjugated to minus identity. The obtained results are based on a recent classification of nonlinear single-input systems under formal feedback. We also describe symmetries of feedback linearizable systems.

We study the feedback classification problem for single-input nonlinear control systems with uncontrollable linearization. We attach different weights to components of the homogeneous terms corresponding to the controllable and uncontrollable parts of the system and obtain a normal form of weighted homogeneous systems. We also describe invariants of the action of weighted homogeneous feedback. A successive normalization of weighted homogeneous terms of the system leads to a normal form for any single-input system, which is a main contribution of the paper. We illustrate our result by two examples.

In this paper we consider the problem of observer design for dynamical systems with scalar output by linearization of the error dynamics via coordinate change, output injection, and time scaling. We present necessary and sufficient conditions which guarantee the existence of a coordinate change and/or output dependent time scaling such that in the new coordinates and with respect to the new time the system has linear error dynamics.

Goursat structures are the dual objects of E. Cartan's “ systems of class zero ” (also called “ en drapeau ” in [7]). We prove that the growth vector is not the only local invariant for a Goursat structure on ℝ n , n ≥ 9; and that there exist infinitely many locally non-equivalent Goursat structures on ℝ n , n ≥ 10.

We study the class of nonholonomic systems given by Goursat structures and discuss a family of their nilpotent canonical forms, called Kumpera-Ruiz normal forms. It is always possible to bring locally a given Goursat structure to this form using regular feedback and coordinates changes. We give a description of rigid curves of Goursat structures. Finally, using the concept of nice distribution, we derive necessary conditions for global conversion of Goursat structures on R 4 to the simplest Kumpera-Ruiz normal form, i.e., chained form. This gives necessary conditions for global x -flatness on R 4 . All results are illustrated by a classical benchmark system: the car towing n trailers.

The L2-gain almost disturbance decoupling problem for SISO nonlinear systems is formulated. Sufficient conditions are identified for the existence of a parametrized state feedback controller such that the L2-gain from disturbances to output can be made arbitrarily small by increasing its gain. The controller is explicitly constructed using a Lyapunov-based recursive scheme. Sufficient conditions for the solvability of the H∞ almost disturbance decoupling problem and the explicit construction of the controller are given for a more restrictive class of nonlinear systems.

We introduce the notion of dynamic controllability distributions for nonlinear control systems. we show the usefulness of this concept by applying it. to study the left and right invertibility problem

In this paper the problem is studied when the system ẋ =ƒ(x) + ∑ik=1 utgt(x), x(0) = x0, defined on an n-dimensional analytic manifold, can be decomposed into either independent subsystems or a cascade of subsystems. Necessary and sufficient conditions for the existence of such local decompositions are given in terms of Lie algebras generated by the system.