This paper proposes a convex embedding-based homogeneous finite-time controller design method for a class of nonlinear systems. The controlled system is first converted into a convex combination of ...linear systems. By utilizing a parallel distributed compensation approach, we derive a homogeneous stabilization condition for the control systems. The condition is expressed in the form of a set of linear matrix inequalities (LMIs). Finite-time stability is guaranteed by using a canonical homogeneous norm as a Lyapunov function. The homogeneous finite-time controller is obtained by solving the LMIs. The hybrid Van der Pol–Rayleigh system with an unstable limit cycle is used as an example to demonstrate how effectively the controller design performs.
This work aims at enlarging the sampling intervals in several state feedback control situations by designing a sampling map in the state space. We consider the case of linear time invariant (LTI) ...systems with state-bounded perturbations, and guarantee their exponential stability for a chosen decay-rate. The approach is based on linear matrix inequalities (LMIs) obtained thanks to Lyapunov–Razumikhin stability conditions and convexification arguments. First, it enables to optimize the lower-bound of the sampling maps by computing the adequate Lyapunov–Razumikhin function. This result can be interpreted as a robust stability analysis with respect to arbitrary time-varying sampling intervals, which may be useful in the case of uncontrolled sampling, or in the presence of phenomenon such as sampling jitter. Then, the obtained results are extended to design the sampling map in three dynamic sampling control situations: event-triggered control, self-triggered control, and state-dependent sampling. The results are illustrated with a numerical example from the literature.
In this article, we study a convex embedding for the Euler problem of two fixed centers for energies below the critical energy level. We prove that the doubly-covered elliptic coordinates provide a ...2-to-1 symplectic embedding such that the image of the bounded component near the lighter primary of the regularized Euler problem is convex for any energy below the critical Jacobi energy. This holds true if the two primaries have equal mass, but does not hold near the heavier body.
A Systematic Method for Backstepping via Linear Matrix Inequalities Ibarra, Jorge; Diaz, Jesus Alonso; Marquez, Raymundo ...
2021 18th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE),
2021-Nov.-10
Conference Proceeding
In this paper, a systematic method for implementing backstepping via linear matrix inequalities is developed. It is based on using the direct Lyapunov method at each step of the backstepping ...procedure in order to find a control law as a nonlinear convex sum of gains. These gains are obtained via commercially available software for solving linear matrix inequalities, which guarantees numerical readiness of the proposal. Moreover, dealing with MIMO subsystems at each step can be done without further adjustments. Examples of academic and practical interest illustrate the advantages of the novel methodology over former approaches.
In this work, we present a novel self-triggered control which aims at decreasing the number of sampling instants for the state feedback control of perturbed linear time invariant systems. The ...approach is based on convex embeddings that allow for designing a state-dependent sampling function guaranteeing the system's exponential stability for a desired decay-rate and norm-bounded perturbations. One of the main contributions of this paper is an LMI based algorithm that optimizes the choice of the Lyapunov function so as to enlarge the lower-bound of the sampling function while taking into account both the perturbations and the decay-rate. The advantages of the approach are illustrated with a numerical example from the literature.
Greedy Convex Embeddings for Sensor Networks Berchenko, Y.; Teicher, M.
2009 International Conference on Parallel and Distributed Computing, Applications and Technologies,
2009-Dec.
Conference Proceeding
Recent advances in systems of networked sensors have set the stage for smart environments which will have wide-ranging applications from intelligent wildlife monitoring to social applications such as ...health and elderly care service provisioning. Perhaps the most natural problem in sensor systems is the ¿efficient¿ propagation of a sensed local event. In order to address this problem, the notion of greedy embedding was defined by Papadimitriou and Ratajczak, where the authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, the greedy embedding conjecture was proved by Leighton and Moitra. However, their algorithm does not result in a drawing that is planar and convex in the Euclidean plane for all 3-connected planar graphs. Here we give a random algorithm for embedding 3-connected planar graphs a greedy convex embedding. Our convex embedding is especially useful for the case of sensor networks, where the position assigned to each sensor is the midpoint of the positions of its neighbors.
Greedy Convex Embeddings for Ad-Hoc Networks Berchenko, Y.; Teicher, M.
2009 International Conference on Parallel and Distributed Computing, Applications and Technologies,
2009-Dec.
Conference Proceeding
Recent advances in networked systems and wireless communications have set the stage for applications with wide-ranging benefits. Perhaps the most natural problem in such systems is the ¿efficient¿ ...propagation of locally stored data. In order to address this problem, the notion of greedy embedding was defined by Papadimitriou and Ratajczak, where the authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, the greedy embedding conjecture was proved by Leighton and Moitra. However, their algorithm does not result in a drawing that is planar and convex in the Euclidean plane for all 3-connected planar graphs. Here we consider the planar convex greedy embedding conjecture and give a probabilistic proof for the existence of such embeddings. In addition, we discuss a second proof which is almost immediate in the case of an embedding into the 3-dimensional sphere.
This article presents basic concepts and recent research directions about the stability of sampled-data systems with aperiodic sampling. We focus mainly on the stability problem for systems with ...arbitrary time-varying sampling intervals which has been addressed in several areas of research in Control Theory. Systems with aperiodic sampling can be seen as time-delay systems, hybrid systems, Input/Output interconnections, discrete-time systems with time-varying parameters, etc. The goal of the article is to provide a structural overview of the progress made on the stability analysis problem. Without being exhaustive, which would be neither possible nor useful, we try to bring together results from diverse communities and present them in a unified manner. For each of the existing approaches, the basic concepts, fundamental results, converse stability theorems (when available), and relations with the other approaches are discussed in detail. Results concerning extensions of Lyapunov and frequency domain methods for systems with aperiodic sampling are recalled, as they allow to derive constructive stability conditions. Furthermore, numerical criteria are presented while indicating the sources of conservatism, the problems that remain open and the possible directions of improvement. At last, some emerging research directions, such as the design of stabilizing sampling sequences, are briefly discussed.
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