123 research outputs found
Canonical forms for nonlinear systems
Necessary and sufficient conditions for transforming a nonlinear system to a controllable linear system have been established, and this theory has been applied to the automatic flight control of aircraft. These transformations show that the nonlinearities in a system are often not intrinsic, but are the result of unfortunate choices of coordinates in both state and control variables. Given a nonlinear system (that may not be transformable to a linear system), we construct a canonical form in which much of the nonlinearity is removed from the system. If a system is not transformable to a linear one, then the obstructions to the transformation are obvious in canonical form. If the system can be transformed (it is called a linear equivalent), then the canonical form is a usual one for a controllable linear system. Thus our theory of canonical forms generalizes the earlier transformation (to linear systems) results. Our canonical form is not unique, except up to solutions of certain partial differential equations we discuss. In fact, the important aspect of this paper is the constructive procedure we introduce to reach the canonical form. As is the case in many areas of mathematics, it is often easier to work with the canonical form than in arbitrary coordinate variables
Control bifurcations
A parametrized nonlinear differential equation can have multiple equilibria as the parameter is varied. A local bifurcation of a parametrized differential equation occurs at an equilibrium where there is a change in the topological character of the nearby solution curves. This typically happens because some eigenvalues of the parametrized linear approximating differential equation cross the imaginary axis and there is a change in stability of the equilibrium. The topological nature of the solutions is unchanged by smooth changes of state coordinates so these may be used to bring the differential equation into Poincare/spl acute/ normal form. From this normal form, the type of the bifurcation can be determined. For differential equations depending on a single parameter, the typical ways that the system can bifurcate are fully understood, e.g., the fold (or saddle node), the transcritical and the Hopf bifurcation. A nonlinear control system has multiple equilibria typically parametrized by the set value of the control. A control bifurcation of a nonlinear system typically occurs when its linear approximation loses stabilizability. The ways in which this can happen are understood through the appropriate normal forms. We present the quadratic and cubic normal forms of a scalar input nonlinear control system around an equilibrium point. These are the normal forms under quadratic and cubic change of state coordinates and invertible state feedback. The system need not be linearly controllable. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations
Geometric structure of the equivalence classes of a controllable pair
Given a pair of matrices representing a
controllable linear system, we study its equivalence
classes by the single or combined action of feedbacks
and change of state and input variables, as well as
their intersections. In particular, we prove that they
are differentiable manifolds and we compute their
dimensions. Some remarks concerning the effect of
different kinds of feedbacks are derived.Postprint (published version
Geometric structure of single/combined equivalence classes of a controllable pair
Given a pair of matrices representing a controllable linear system, its equivalence
classes by the single or combined action of feedbacks, change of state and input variables, as well as
their intersection are studied. In particular, it is proved that they are differentiable manifolds and
their dimensions are computed. Some remarks concerning the effect of different kinds of feedbacks
are derived.Postprint (published version
State and Feedback Linearizations of Single-Input Control Systems
In this paper we address the problem of state (resp. feedback) linearization of nonlinear single-input control systems using state (resp. feedback) coordinate transformations. Although necessary and sufficient geometric conditions have been provided in the early eighties, the problems of finding the state (resp. feedback) linearizing coordinates are subject to solving systems of partial differential equations. We will provide here a solution to the two problems by defining algorithms allowing to compute explicitly the linearizing state (resp. feedback) coordinates for any nonlinear control system that is indeed linearizable (resp. feedback linearizable). Each algorithm is performed using a maximum of steps ( being the dimension of the system) and they are made possible by explicitly solving the Flow-box or straightening theorem. We illustrate with several examples borrowed from the literature
Dynamic Feedback Linearization of Control Systems with Symmetry
Control systems of interest are often invariant under Lie groups of
transformations. Given such a control system, assumed to not be static feedback
linearizable, a verifiable geometric condition is described and proven to
guarantee its dynamic feedback linearizability. Additionally, a systematic
procedure for obtaining all the system trajectories is shown to follow from
this condition. Besides smoothness and the existence of symmetry, no further
assumption is made on the local form of a control system, which is therefore
permitted to be fully nonlinear and time varying. Likewise, no constraints are
imposed on the local form of the dynamic compensator. Particular attention is
given to those systems requiring non-trivial dynamic extensions; that is,
beyond augmentation by chains of integrators. Nevertheless, the results are
illustrated by an example of each type. Firstly, a control system that can be
dynamically linearized by a chain of integrators, and secondly, one which does
not possess any linearizing chains of integrators and for which a dynamic
feedback linearization is nevertheless derived. These systems are discussed in
some detail. The constructions have been automated in the Maple package
DifferentialGeometry.Comment: 41 pages, minor revisions and error correction
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