Input Vector Field Research Articles

On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems

In the present paper we elaborate on the underlying Hamiltonian structure of interconnected energy-conserving physical systems. It is shown that a power-conserving interconnection of port-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltonian system, and a power-conserving partial interconnection to an implicit port-controlled Hamiltonian system. The crucial concept is the notion of a (generalized) Dirac structure, defined on the space of energy-variables or on the product of the space of energy-variables and the space of flow-variables in the port-controlled case. Three natural representations of generalized Dirac structures are treated. Necessary and sufficient conditions for closedness (or integrability) of Dirac structures in all three representations are obtained. The theory is applied to implicit port-controlled generalized Hamiltonian systems, and it is shown that the closedness condition for the Dirac structure leads to strong conditions on the input vector fields.

Uncertainty compensation for a flexible-link manipulator using nonlinear H control

In a flexible-link manipulator, in general the effect of some parameters such as payload, friction amplitude and damping coefficients cannot be exactly measured. One possibility is to consider the above as parameters with uncertainty. In this paper, constant as well as L2-bounded deviations of parameters from their nominal values are considered as uncertainties. These uncertainties make it difficult for a linear controller to achieve desired closed-loop performance. To remedy this problem, a nonlinear dynamical model of a flexible-link manipulator which has a constant input vector field (g in ẋ=f(x) + g(x)u) is obtained. Based on recent results in nonlinear robust regulation with an H∞ constraint a nonlinear controller is designed for the flexible-link manipulator. The contribution of this paper is in demonstrating that the nonlinear controller has a larger domain of attraction than the linearized controller. In fact, for the single-link flexible manipulator considered in this paper, the linear H∞ controller results in instability for step changes in the desired output of greater than 3.6 rad, whereas the nonlinear H∞ controller yields desired step changes of 2π rad. Simulation results demonstrating the advantages and superiority of the nonlinear H∞ controller over the linear H∞controller are presented.

On Feedback Linearization of Nonlinear SISO Systems

This work discusses the state space exact linearization problems, both locally and globally, for single-input single-output nonlinear systems. First it is shown that if the input vector field has a particular structure, i.e. if it is a constant vector field, then local or global feedback linearizability can be determined by inspection. Then it is shown that, under mild assumptions, there always exists a suitable (local) coordinates transformation which transforms a general nonlinear plant into a system with constant input vector. Using these two facts, simple conditions for local and global feedback linearizability of SISO nonlinear systems are derived. Finally, an explicit formula to locally rectify an affine vector field is presented.

On the Structure of Optimal Oscillatory Trajectories for Two-Input Driftless Smooth Systems in Dimension Three

For two-input three-dimensional driftless smooth systems, we prove an alternate theorem for optimal trajectories and a bang-bang theorem with a bound on the number of switchings for optimal oscillatory trajectories, under the assumption that the two input vector fields and their Lie bracket are linearly independent at each point.

Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems

This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal theorem which arises in the study of exterior differential systems. The results are applied to the problem of finding a set of nilpotent input vector fields for a nonholonomic control system, which can then used to construct explicit trajectories to drive the system between any two points. A kinematic model of a rolling penny is used to illustrate this approach. The methods presented here extend previous work using chained form and cast that work into a coordinate-free setting.

Analysis of two- and three-dimensional rigid and nonrigid motions in the stereokinetic effect

Continued viewing of a rotating ellipse induces three types of percept: the veridical percept (rigid, flat, rotating ellipse), the amoeboid percept (nonrigid, flat, deforming ellipse), and the stereokinetic percept (rigid circular disk rolling in three-dimensional space). A mathematical analysis of the nonveridical percepts is presented, based on the aperture problem and the assumption that the vector field N (the normal component of the rotational velocity field R of the ellipse) is the effective stimulus. Perceived nonrigidity of the amoeboid percept is assessed by an analysis of the geometrical nonrigidity of N. Two analytical tests of geometrical rigidity of velocity fields are described. The first is based on the decomposition of the velocity gradient matrix, and the second is based on an analysis of the temporal derivative of the curvature of moving plane curves. Both tests confirm the geometrical nonrigidity of N, on which the perceived nonrigidity might be based. A limitation of both approaches with respect to the analysis of shape-preserving nonrigidity is noted. For an understanding of the stereokinetic percept and its relation to N, a detailed kinematic analysis is presented of the type of motion that a real circular disk (the physical stereokinetic disk) would have to perform in order to duplicate the perceived motion of the perceptual stereokinetic disk induced by the rotating ellipse. The analysis is based on the Eulerian decomposition of rigid-body rotation into precession, nutation, and spin. It is concluded that the stereokinetic effect cannot be directly based on N because the projected velocity field of the physical stereo-kinetic disk, P differs from N. However, a subsequent transformation with N as the input vector field produces an output vector field whose structure is similar to P, except for scale. This transformation is based on the vector interaction algorithm, which is motivated by psychophysical data on contextual effects in the perception of motion direction. It is also demonstrated that this algorithm predicts qualitative aspects of percepts in several other relevant examples, such as the barber pole effect, translating nonoccluded line segments, and eccentrically rotating circles. An additional analysis shows that, although there is an infinity of figures projectively equivalent to a static ellipse, the added dynamical constraint of projective equivalence of the velocity vector field of a moving figure to P implies that the only rigid figure consistent with stimulation and processing conditions is a circular disk. The bases of the three types of percept induced by the rotating ellipse are assumed to be the neural counterparts of three vector fields: R N, and P.

On Passive Systems: From Linearity to Nonlinearity

In this note we study when the passivity of a nonlinear system can be infered from the passivity of the associated first order approximation. Our contribution is the proof that this is possible provided the input vectorfields and output functions of the nonlinear system satisfy some rank and integrability conditions. Also, we analyze under which conditions a nonlinear system can be made passive via static state feedback once the linearized system is assumed to have this property and furthermore when a linear controller which leaves the linearized system passive, preserves similar input-output properties when applied to the nonlinear plant.

Geometric conditions for local controllability

Small time local controllability of a C 1 affine control system, with a generic drift vector field and n − 1 linearly independent input vector fields on an n-dimensional connected C 1 manifold M, is studied, showing that a necessary condition, namely the drift vector field belonging to the convex hull of the control vector fields at the given point, is also sufficient on an open dense subset H 1 of the set of points where it is verified. Local controllability at the points in the interior of H 1 is also considered when the input vector fields generate an involutive distribution, and a criterion based on the type of contract of H with the hyperplane spanned by the input vector fields is proved. An example shows a situation where the sufficient condition for small time local controllability of H. Sussmann [ SIAM J. Control Option. 25 (1987), 158–194] cannot be applied, but in which the method now presented distinguishes the possible cases.

Sufficient Conditions for Local Controllability with Unbounded Controls

Local controllability of analytic affine control systems $\Sigma $ with an arbitrary number of controls is studied, without any restriction on the dimension of the Lie algebra ${\bf T}'$ generated by the input vector fields.A sufficient condition for local controllability at a point is obtained, first for scalar input systems and then generalized in two different ways to the multi-input case. In all versions, this involves the computation of Lie brackets at the point under consideration and the verification of a criterion of the type considered by Sussmann (this Journal, 16 (1978), pp. 790–802).As an application, controllability of systems defined on $\mathbb{R}^n $ and with constant input vector fields is studied.

Controllability in codimension one

Local and global controlability of analytic affine control systems Σ with an arbitrary number of controls are studied, assuming strong accessibility of Σ and codimension one of the Lie algebra T′ generated by the input vector fields, i.e., dim T′ ( p) = n − 1 at every p ϵ M. Controls are assumed to have no a priori bound. From the study of the set H where a necessary condition for local controllability at a point is verified and assuming some transversality relations, an easy to verify geometric condition is proved: H is a submanifold and Σ is locally controllable at every point of H outside a codimension one submanifold. A geometric sufficient condition for global controllability on simply connected manifolds is then obtained: if every leaf of T′ intersects the manifold H and some transversality relations (including those involved in the local condition) are verified, Σ is globally controllable.

Local controllability of nonlinear systems

The local controllability of control systems with an arbitrary number of controls is considered, first on an open set and then at a given point; necessary conditions are derived concerning the Lie algebra T′ generated by the input vector fields, and applied to gradient systems. The main results are a geometric sufficient condition for local controllability at a point, and an equivalent condition based on the computation of Lie brackets, assuming T′ to be ( n − 1)-dimensional.