Nonlinear Rescaling of Control Laws with Application to Stabilization in the Presence of Magnitude Saturation

Motivated by some recent results on the stabilization of homogeneous systems, we present a gain-scheduling approach for the stabilization of non-linear systems. Given a one-parameter family of stabilizing feedbacks and associated Lyapunov functions, we show how the parameter can be rescaled as a function of the state to give a new stabilizing controller. In the case of homogeneous systems, we obtain generalizations of some existing results. We show that this approach can also be applied to nonhomogeneous systems. In particular, the main application considered in this paper is to the problem of stabilization with magnitude limitations. For this problem, we develop a design method for single-input controllable systems with eigenvalues in the left closed plane

Nonlinear Feedback Control of Axisymmetric Aerial Vehicles

We investigate the use of simple aerodynamic models for the feedback control of aerial vehicles with large flight envelopes. Thrust-propelled vehicles with a body shape symmetric with respect to the thrust axis are considered. Upon a condition on the aerodynamic characteristics of the vehicle, we show that the equilibrium orientation can be explicitly determined as a function of the desired flight velocity. This allows for the adaptation of previously proposed control design approaches based on the thrust direction control paradigm. Simulation results conducted by using measured aerodynamic characteristics of quasi-axisymmetric bodies illustrate the soundness of the proposed approach

Modeling for Control of Symmetric Aerial Vehicles Subjected to Aerodynamic Forces

This paper participates in the development of a unified approach to the control of aerial vehicles with extended flight envelopes. More precisely, modeling for control purposes of a class of thrust-propelled aerial vehicles subjected to lift and drag aerodynamic forces is addressed assuming a rotational symmetry of the vehicle's shape about the thrust force axis. A condition upon aerodynamic characteristics that allows one to recast the control problem into the simpler case of a spherical vehicle is pointed out. Beside showing how to adapt nonlinear controllers developed for this latter case, the paper extends a previous work by the authors in two directions. First, the 3D case is addressed whereas only motions in a single vertical plane was considered. Secondly, the family of models of aerodynamic forces for which the aforementioned transformation holds is enlarged.Comment: 7 pages, 4 figure

Feedback control of the general two-trailers system with the Transverse Function approach

The so-called "general two-trailers system" is a nonholonomic system composed of a controlled unicycle-like vehicle and two passive trailers with off-axle hitching. It is not differentially flat and cannot be transformed into the chained form system. Methods developed for this latter class of systems thus do not apply. The Transverse Function (TF) approach is here used to solve the trajectory tracking problem for this system. The proposed control solution yields practical stabilization of any reference motion, whether it is or is not feasible. Practical stabilization of non-feasible trajectories in the case of non-differently flat systems is of particular interest due partly to the difficulty of planning and calculating desired feasible state reference motions. The method is illustrated by simulation results which show that, in addition to the unconditional practical stabilization property evoked above, asymptotic stabilization of feasible and persistently exciting motions can also be achieved with the same performance as local stabilizers derived from a linear approximation of the tracking-error equations

Control with transverse functions and a single generator of underactuated mechanical systems

The control of a class of underactuated mechanical systems on Lie groups is addressed, with the objective of stabilizing, in a practical sense, any (possibly non-admissible) reference trajectory in the configuration space. The present control design method extends a previous result by the authors to systems underactuated by more than one control. For example, it allows to address the case of a 3D-rigid body immersed in a perfect fluid with only three control inputs. The choice of the control parameters is also discussed in relation to the system's zero-dynamics

Practical Stabilization of Driftless Homogeneous Systems Based on the Use of Transverse Periodic Functions

We address the problem of practical stabilization of driftless nonlinear control systems with homogeneous vector fields. A general feedback design approach, based on the concept of transverse functions recently introduced by the authors, is presented. This approach allows to achieve global stabiliza- tion of any neighborhood of the origin, possibly in the presence of additive - known or measured - perturbations acting on the system

Trajectory tracking for nonholonomic systems. Theoretical background and applications

The problem of stabilizing reference trajectories for nonholonomic systems, often referred to as the trajectory tracking problem in the literature on mobile robots, is addressed. The first sections of this report set the theoretical background of the problem, with a focus on controllable driftless systems which are invariant on a Lie group. The interest of the differential geometry framework here adopted comes from the possibility of taking advantage of ubiquitous symmetry properties involved in the motion of mechanical bodies. Theoretical difficulties and impossibilities which set inevitable limits to what is achievable with feedback control are surveyed, and basic control design tools and techniques are recast within the approach here considered. A general method based on the so-called Transverse Function approach --developed by the authors--, yielding feedback controls which unconditionnally achieve the {\em practical} stabilization of arbitrary reference trajectories, including fixed points and non-admissible trajectories, is recalled. This property singles the proposed solution out of the abundant literature devoted to the subject. It is here complemented with novel results showing how the more common property of asymptotic stabilization of persistently exciting admissible trajectories can also be granted with this type of control. The last section of the report concerns the application of the approach to unicycle-type and car-like vehicles. The versatility and potentialities of the Transverse Function (TF) control approach are illustrated via simulations involving various reference trajectory properties, and a few complementary control issues are addressed. One of them concerns the possiblity of using control degrees of freedom to limit the vehicle's velocity inputs and the number of transient maneuvers associated with the reduction of initially large tracking errors. Another issue, illustrated by the car example, is related to possible extensions of the approach to systems which are not invariant on a Lie group

A characterization of the Lie Algebra Rank Condition by transverse periodic Functions

The Lie Algebra Rank Condition (LARC) plays a central role in nonlinear systems control theory. The present paper establishes that the satisfaction of this condition by a set of smooth control vector fields is equivalent to the existence of smooth transverse periodic functions. The proof here enclosed is constructive and provides an explicit method for the synthesis of such functions

Practical Stabilization of Driftless Systems on Lie Groups

A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of transverse functions, the existence of which is equivalent to the system's controllability. Its outcome is the practical stabilization of any trajectory -i.e. not necessarily a solution of the control system--- in the state space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given