José-Job Flores-Godoy

CONTROL-ORIENTED PROPERTIES PRESERVATION IN LINEAR SYSTEMS WHEN APPLYING PR0 SUBSTITUTIONS

Abstract We present in this paper some results concerning the preservation of control-oriented properties in Linear Time-Invariant systems (LTI systems) when applying substitutions of the Laplace variable s by a particular class of positive real functions, the socalled Positive Real functions of zero relative degree (PR0 functions). In particular, we show that the families of Bounded Real (BR), Strictly Bounded Real (SBR), and Positive Real (PR), rational functions are closed under compositions with the specified class of positive real functions. We also give some results concerning the preservation of stability in proper real rational transfer functions (in fact this is our main result), as well as the preservation of the H∞-norm bound for this class of systems. Our study is restricted to Single-Input Single Output (SISO) systems.

Formation Tracking of Heterogeneous Mobile Agents Using Distance and Area Constraints

This paper presents two formation tracking control strategies for a combined set of single and double integrator agents with an arbitrary undirected communication topology. The first approach is based on the design of distance-based potential functions with interagent collision avoidance using local information about the distance and orientation between agents and the desired trajectory. The second approach adds signed area constraints to the desired formation specification and a control strategy that uses distance as well as area terms is designed to achieve tracking convergence. Numerical simulations show the performance from both control laws.

An exactly soluble model of a shallow double well

Abstract Shallow one-dimensional double-well potentials appear in atomic and molecular physics and other fields. Unlike the “deep” wells of macroscopic quantum coherent systems, shallow double wells need not present low-lying two-level systems. We argue that this feature, the absence of a low-lying two-level system in certain shallow double wells, may allow the finding of new test grounds for quantum mechanics in mesoscopic systems. We illustrate the above ideas with a family of shallow double wells obtained from reflectionless potentials through the Darboux–Backlund transform.

A toy model of a macroscopic quantum coherent system

This paper deals with macroscopic quantum coherence while using only basic quantum mechanics. A square double well is used to illustrate Leggett–Caldeira oscillations. The effect of thermal radiation on two-level systems is discussed. The concept of decoherence is introduced at an elementary level. Reference values are deduced for the energy, temperature and time scales involved in macroscopic quantum coherence.

Marching control based on the leader-followers scheme and formation graphs

This paper analyzes the case of marching control, where a leader robot follows a desired trajectory, while the whole group reaches a formation pattern established by a formation graph. To guarantee that the group preserves the desired formation and follows the marching path simultaneously, two approaches are given, based on the feedback of the velocity of the desired trajectory or the velocity of some robots. The main result establishes that for both approaches and any well-defined formation graph, there is convergence to the formation and the marching path. The analysis addresses the cases of omnidirectional robots and the extension to unicycle-type robots. The performance of the control strategies is shown some numerical simulations and real-time experiments.

Properties of strictly positive real functions: products and compositions

In this work necessary conditions on one side are sufficient conditions on another side given such that the product of strictly positive real (SPR) functions to be an SPR0 function. Also, sufficient conditions are given for the composition of SPR functions to be an SPR function. As a consequence, a result on robust SPR transfer functions, under non-linearly correlated perturbations in the coefficients, induced by the composition and product with SPR functions is presented. Based on these facts, we give an application on preservation of robust and simultaneous stabilisation of single-input single-output (SISO), linear time-invariant (LTI) systems. The applicability of the results is shown through two examples.

Synchronization preservation under linear polynomial modifications

Robustness and preservation of stability and synchronization in the presence of structural changes is an important issue in the study of chaotic dynamical systems. In this work we present a methodology to establish conditions for preservation of stability in dynamical system in terms of linear matrix polynomial evaluation. The idea is to construct a group of dynamical transformations under which stability is retained along the stable, unstable and synchronization manifolds using simultaneous Schur decompositions.

On Operators on Positive Real Functions and Related Issues

In this paper, we proved that the sets of strict positive real functions with zero relative degree (SPR0) and positive real (PR) single-input-single-output functions are closed under linear operators preserving stability and for rational functions by differentiation of the numerator and denominator. Also, we show that any PR system can be approximated by an SPR0 system. These results are extended to strict bounded real and bounded real functions when the numerator and denominator polynomials are transformed by a class of linear operators. Additionally, for functions in RfrH infin, the H infin-norm is preserved under linear operators applied on numerator and denominator polynomials. Three possible applications are given.

Stabilization and passification of distributed-order fractional linear systems using methods of preservation

Abstract This paper studies the stabilization and passification of a class of distributed-order linear time-invariant systems, by using methods of preservation in the frequency domain. Results about preservation of stability and passivity of classical linear time-invariant systems are extended to one more general family of matrix functions. Based on these results, a new approach to the problems of stabilization and passification of distributed-order linear time-invariant systems is presented. Also a result that extends the known techniques for pole placement of classical linear time-invariant systems to the new class of distributed-order linear time-invariant systems is given. Examples are given to show the validity of theoretical results.

The Attractors in the Complex Lorenz Model

Abstract We address the question of finding the attractors of the complex Lorenz model (CLM), which is obtained by extending the space from ℝ 3 to ℂ 3 , and defining the model by the same equations as the classical Lorenz model (LM). We have numerical evidence of 2 strong attractors un-related to the Lorenz attractor. We calculate its Lyapunov exponents and show that 2 of them are 0, and the other 4 are double and negative. Hence the attractors are non-chaotic. We show that they have a quasi-periodic nature. To decipher the structure of these attractors, we introduce the imaginary Lorenz model (ILM), which is defined in the same space ℂ 3 by multiplying with √−1 the Lorenz equations. Both models locally commute, and with its help we account for the double Lyapunov exponent 0 and show that the basin of attraction of each attractor is a big open set of ℂ 3 . The chaotic limit set L ℂ ⊂ ℂ 3 obtained from the classical Lorenz attractor L 0 of (LM) by moving it with the (ILM) has 2 positive Lyapunov exponents, but only captures a set of 6D-volume 0 in its basin of attraction.