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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 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.

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).

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.

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.

In this paper we find a complete set of equivariants of mechanical control systems that satisfy the geodesic accessibility property. These equivariants are structure functions on the configuration manifold and are used to completely characterize the mechanical state equivalence of two mechanical control systems. Further, with the aid of our equivariants, we also characterize the state equivalence of the corresponding control systems, defined on the tangent bundle of the configuration manifolds, under extended point transformations as well as under arbitrary diffeomorphisms of the tangent bundle.

This paper deals with the problem of convergence of normal forms of control systems. We identify an n-dimensional subclass of control systems, called special strict feedforward form, shortly (SSFF), possessing a normal form which is a smooth (resp. analytic) counterpart of the formal normal form of Kang. We provide a constructive algorithm and illustrate by several examples including the Kapitsa pendulum and the cart–pole system. The second part of the paper is concerned about symmetries of single-input control systems. We show that any symmetry of a smooth system in SSFF is conjugated to a scaling translation and any 1-parameter family of symmetries is conjugated to a family of scaling translations along the first variable. We compute explicitly those symmetries by finding the conjugating diffeomorphism. We illustrate our results by computing the symmetries of the cart–pole system. Copyright © 2009 John Wiley & Sons, Ltd.

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.

We define bifurcations of control-affine systems in the plane and classify all generic 1-parameter bifurcations at control-regular points. More precisely, we classify topological bifurcations of invariants of usual feedback equivalence. Such bifurcations form six different classes: two bifurcations of equilibrium sets, two bifurcations of critical sets, and two bifurcations of pairs of invariants. We also classify all generic 1-parameter families of control-affine systems with respect to orbital feedback equivalence.

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 output-dependent time scaling, such that in the new coordinates and with respect to the new time the system has linear error dynamics.

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 group action on single-input nonlinear control systems. We follow an approach of Kang and Krener based on analyzing, step by step, the action of homogeneous transformations on the homogeneous part of the same degree of the system. We construct a dual normal form and dual invariants with respect to those obtained by Kang. We also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms (resp., their dual canonical forms) coincide. We give an explicit construction of transformations bringing the system to its normal, dual normal, canonical, and dual canonical forms. We illustrate our results by simple examples on $\mathbb{R}^3$ and $\mathbb{R}^4$.

We consider nonlinear control-affine systems with two inputs evolving on three-dimensional manifolds. We study their local classification under static state feedback. Under the assumption that the control vector fields are independent we give complete classification of generic systems. We prove that out of a “singular” smooth curve a generic control system is either structurally stable and thus equivalent to one of six canonical forms (models) or finitely determined and thus equivalent to one of two canonical forms with real parameters. Moreover, we show that at points of the “singular” curve the system is not finitely determined and we give normal forms containing functional moduli. We also study geometry of singularities, i.e., we describe surfaces, curves, and isolated points where the system admits its canonical forms.

This paper discusses the problem of using feedback and coordinates transformation in order to transform a given nonlinear system with outputs into a controllable and observable linear one. We discuss separately the effect of change of coordinates and, successively, the effect of both change of coordinates and feedback transformation. One of the main results of the paper is to show what extra conditions are needed, in addition to those required for input-output-wise linearization, in order to achieve full linearity of both state-space equations and output map.

A direct solution to the almost input-output and disturbance decoupling problem is given. The approach is based on a new canonical form for linear systems—which is obtained in a fully constructive way—together with some elementary approximation results. The solution is given in the form of an explicit feedback law, parameterized by a scalar e. By letting e go to zero, the accuracy of the resulting approximate solution can be improved to any desired degree.

We study control-affine systems with n − 1 inputs evolving on an n-dimensional manifold for which the distribution spanned by the control vector fields is involutive and of constant rank (equivalently, they may be considered as 1-dimensional systems with n − 1 inputs entering nonlinearly). We provide a complete classification of such generic systems and their one-parameter families. We show that a generic family for n > 2 is equivalent (with respect to feedback or orbital feedback transformations) to one of nine canonical forms which differ from those for n = 2 by quadratic terms only. We also describe all generic bifurcations of 1-parameter families of systems of the above form.