Irinel-Constantin Morărescu

Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains

The paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin. The controller that we propose, uses multiple delayed measurements instead of derivative terms. Doing so, we increase the performances of the closed loop in presence of system uncertainties and/or noisy measurements. The problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally. On one hand, we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension-three triple zero bifurcation. On the other hand, we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium. The aim of this analysis is to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed.

Reset strategy for consensus in networks of clusters

This paper addresses the problem of consensus in networks structured in several clusters. The clusters are represented by fixed, directed and strongly connected graphs. They are composed by a number of agents which are able to interact only with other agents belonging to the same cluster. To every agent we associate a scalar real value representing its state. The states continuously evolve following a linear consensus protocol and approach local agreements specific to each cluster. In order to enforce a global agreement over the whole network, we consider that each cluster contains an agent that can be exogenously controlled. The state of this agent, called leader, will be quasi-periodically reseted by a local master controller that receives information from some neighboring leaders. In order to control the consensus value we have to firstly characterize it. Precisely we show that it depends only on the initial condition and the interaction topologies. Secondly, we provide sufficient Linear Matrix Inequality (LMI) conditions for the global uniform exponential stability of the consensus in presence of a quasi-periodic reset rule. The study of the network behavior is completed by a decay rate analysis. Finally we design the interaction network of the leaders which allows to reach a prescribed consensus value. Numerical implementation strategy is provided before illustrating the results by some simulations.

Inverted Pendulum Stabilization Via a Pyragas-Type Controller: Revisiting the Triple Zero Singularity

Abstract The use of Pyragas-Type controller proved interest in the stabilization of unstable periodic orbits. The stabilization problem of a balancing inverted pendulum on an horizontally moving cart by the use of such a controller is considered. The main objective of the paper is to propose delayed control law containing only proportional gains able to stabilize the inverted pendulum by avoiding the existence of a triple zero eigenvalue at the origin. We analyze the center dynamics described by a three dimensional system of ordinary differential equations (ODEs) with a codimension-three triple zero bifurcation. Furthermore, the stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium is also proposed. This analysis is done in order to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed. Time-Delay, Stability, Delayed Feedback, Control, Center Manifold Theorem, Normal Forms, Local Bifurcation, Inverted Pendulum.

Control design with guaranteed cost for synchronization in networks of linear singularly perturbed systems

Abstract This work presents the design of a decentralized control strategy that allows singularly perturbed multi-agent systems to achieve synchronization with global performance guarantees. The study is mainly motivated by the presence of two features that characterize many physical systems. The first is the complexity in terms of interconnected subsystems and the second is that each subsystem involves processes evolving on different time-scales. In the context of interconnected systems, the decentralized control is interesting since it considerably reduces the communication load (and the associated energy) which can be very important when dealing with centralized policies. Therefore, the main difficulty that we have to overcome is that we have to avoid the use of centralized information related to the interconnection network structure. This problem is solved by rewriting the synchronization problem in terms of stabilization of a singularly perturbed uncertain linear system. The singularly perturbed dynamics of subsystems generates theoretical challenges related to the stabilizing controller design but also numerical issues related to the computation of the controller gains. We show that these problems can be solved by decoupling the slow and fast dynamics. Our theoretical developments are illustrated by some numerical examples.

Topology Preservation for Multi-agent Networks: Design and Implementation

We consider a network of interconnected systems with discrete-time dynamics. Each system is called agent and we assume that two agents can interact as far as their states are close in a sense defined by an algebraic relation. In this work, we present several implementation strategies answering to different classical problems in multiagent systems . The primary goal of our methodology is to characterize the controllers that preserve a given interconnection subgraph that makes possible the global coordination. The second goal is to choose among these controllers those that ensure an agreement. This is done by solving a convex optimization problem associated to the minimization of a well-chosen cost function. Examples concerning full or partial consensus of agents with double-integrator dynamics illustrate the implementation of the proposed methodology.

Design and Analysis of Reset Strategy for Consensus in Networks with Cluster Pattern

This chapter addresses the problem of consensus in networks partitioned in several disconnected clusters. Each cluster is represented by a fixed, directed, and strongly connected graphs. In order to enforce the consensus, we assume that each cluster poses a leader that can reset its state by taking into account other leaders state. First, we characterize the consensus value of this model. Second, we provide sufficient condition in LMI form for the stability of the consensus. Finally, we perform a decay rate analysis and design the interaction network of the leaders which allows to reach a prescribed consensus value.

Decentralized Formation Control in Fleets of Nonholonomic Robots with a Clustered Pattern

In this work we consider a fleet of non-holonomic robots that has to realize a formation in a decentralized and collaborative manner. The fleet is clustered due to communication or energy-saving constraints. We assume that each robot continuously measures its relative distance to other robots belonging to the same cluster. Due to this, the robots communicate on a directed connected graph within each cluster. On top of this, in each cluster there exists one robot called leader that receives information from other leaders at discrete instants. In order to realize the formation we propose a two-step strategy. First, the robots compute reference trajectories using a linear consensus protocol. Second, a classical tracking control strategy is used to follow the references. Overall, formation realization is obtained. Numerical simulations with robot teams illustrate the effectiveness of this approach.

Smith Predictor-Based Control with Distance Feedback for Haptic Systems Under Distributed Time-Delays

This chapter proposes a Smith predictor-based control with distance feedback until a possible collision for network based haptic systems. The main idea is to use a predictor just on the haptic side in order to compensate the viscosity effect and to provide an accurate feeling in case of contacts. The critical problem is that in haptics there are two situations: free and restricted motion, which implies a model changing on the predictor side. For solving this problem, a new approach is presented by using the available information on the distance from the virtual reality simulator and introducing it in the predictor in order to maintain the similitude between the real and the predicted model. The three most common delay distributions (uniform, gamma with gap and normal) are analyzed from the stability point of view and are tested on a 3-dof real-time experimental platform.