Nomenclature Ak = time-dependent coefŽ cient of the kth proper orthogonal decompositionmode Aest k = estimation of Ak B = matrix containing the bk coefŽ cients bk = coefŽ cients associatedwith the control input ck i j = coefŽ cients of the nonlinear function gk fa = control input resulting form external forcing of the cylinder gk = nonlinear function of the time-dependent low-dimensionalmodes J...

An n-dimensional system of ODEs x′ = f(x) is said to be a cooperative system in canonical form if either it’s Jacobian matrix has nonnegative off-diagonal entries or there is an integer k with 1 ≤ k < n and the sign structure of the Jacobian, throughout its domain, has k×k and (n−k)× (n−k) main diagonal matrices with nonnegative off-diagonal entries and that the rectangular off-diagonal sub-mat...

The proof relies on the proof of Lemma 3.6, below. Let z, δ ∈ R, and let δ → 0. Suppose z + δ converges to a point that falls into one of the first three cases given in Section 2. Then, from the statement and proof of Lemma 3.6, an element JPK∗exp (z + δ) of the generalized Jacobian of the projection onto the dual of the exponential cone at z + δ, is just a matrix with fixed entries, since proj...

This paper presents an image-based visual servo controller for an industrial robot. The simple structure controller is based on a transposed Jacobian which feeds back directly the image feature errors and the joint velocities. Under these hypotheses, the control algorithm does not need to use the inverse kinematics and the inverse jacobian matrix. Experimental results in two different industria...

This work discusses the chaotification of nonchaotic discrete systems based on a procedure that requires the jacobian matrix of the uncontrolled dynamics. The aim is to verify if chaotification is still effective when the jacobian is analytically obtained from identified models (data-driven models) rather than assumed known. This study, that includes two numerical examples, shows that chaotific...

This paper is concerned with the efficient computation of Jacobian matrices of nonlinear vector maps using automatic differentiation (AD). Specificially, we propose the use of a directed cutset method, weighted minimum cut, to exploit the structure of the computional graph of the nonlinear system. This allows for the efficient determination of the Jacobian matrix using AD software. We discuss t...

Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertex coloring problems her...

where the variable xi represents the state of the ith node and fi represents the combined influence of all nodes having connections with the ith node. The fi may be linear or nonlinear functions. The local stability properties of the system about its (possibly multiple) equilibria can be determined using Lyapunov’s indirect method (Khalil, 2002). This involves determining the location of the ei...

In this paper, the coupling between Jacobian learning and task sequencing through the redundancy approach is studied. It is well known that visual servoing is robust to modeling errors in the jacobian matrices. This justifies why jacobian estimation does not usually degrade the system convergence. However, we show that this is not true anymore when the redundancy formalism is used. In this case...