## 2SL2S - Pre-Conference Talks

5th July 2017, 14h00, CentraleSupelec (Gif-sur-Yvette) Salle du Conseil L2S - B4.40

### Stability Analysis of Control Systems Subject to Delay-Difference Feedback

####
*Chi Jin*, Silviu Niculescu, Islam Boussaada, Keqin Gu

In engineering practice, delay-difference is often used to approximate the derivatives of output signals for feedback control, leading to a closed-loop system with delay both in the states and in the system's coefficients. In this context, our objective is to find all the delay values contained in some interval that guarantee the exponential stability of the closed-loop system subject to the delay-difference approximation. A method for stability analysis of systems with delay-dependent coefficients developed in our previous work is further extended and applied to tackle the particular form of systems considered in this paper. The proposed stability analysis procedure is illustrated through the design of a mobile-robot path-following controller.

###

Geometric vs. Algebraic Approach: A Study of Double Imaginary Characteristic Roots in Time-Delay Systems

#### Dina Alina Irofti, Islam Boussaada, Silviu-Iulian Niculescu

This paper studies double imaginary characteristic roots in the case of time-delay systems with two delays as parameters. We aim to identify the direction in which double roots cross the imaginary axis, when the delay parameters change. To determine what happens when parameters are under a small perturbation, we present two methods: the algebraic and geometric approaches. Taking a theoretical example, we show that, even if the two methods are conceptually different, the provided results are consistent.

### Self-Triggered Control for Sampled-Data Systems Using Reachability Analysis

####
*Mohammad Al Khatib*, Antoine Girard, Thao Dang

In this work, we design the sampling policy in sampled-data systems. It is known that implementing such systems using variable sampling periods, instead of a constant period, is more efficient in terms of performance and resource utilization. Thus, after rewriting the system in the framework of impulsive linear systems, a self-triggered control strategy obtained using reachability analysis is proposed in order to define the sampling period as a function of the state.

### On the Impact of Additive Disturbances on Auto-Steering Systems

####
*Iris Ballesteros-Tolosana*, Sorin Olaru, Pedro Rodriguez-Ayerbe, Morten Hovd, Renaud Deborne, Guillermo Pita-Gil

Several control techniques are available for the automotive systems and their design is often based on the available measurements of different parameters and the integration of the hard input and performance constraints. The objective of the present study is to revisit the closed loop dynamics of an auto-steering system to offer an evaluation of the impact of the curvature of the road. From a theoretical point of view, robust positive invariance theory is presented in order to perform the analysis of the effects that a bounded parameter-varying additive disturbance has on a linear parameter-varying controller used to ensure stability of a model predictive control strategy.

### Observer Based Path Following for Underactuated Marine Vessels in the Presence of Ocean Currents: A Local Approach

####
*Mohamed Maghenem*, Dennis Belleter, Claudio Paliotta Kristin Y. Pettersen,

In this article a solution to the problem of following a curved path in the presence of a constant unknown ocean current disturbance is presented. The path is parametrised by a path variable that is used to propagate a path-tangential reference frame. The update law for the path variable is chosen such that the motion of the path-tangential frame ensures that the vessel remains on the normal of the path-tangential reference frame. As shown in the seminal work Samson [1992] such a parametrisation is only possible locally. A tube is dened in which the aforementioned parametrisation is valid and the path-following problem is solved within this tube. The size of the tube is proportional to the maximum curvature of the path. It is shown that within this tube, the closed-loop system of the proposed controller, guidance law, and the ocean current observer provides exponential stability of the path-following error dynamics. The sway velocity dynamics are analysed taking into account couplings previously overlooked in the literature, and is shown to remain bounded. Simulation results are presented.

### Methodology for Modeling an Oxygen Regulator

####
*Geoffray Battiston*, Dominique Beauvois, Gilles Duc, Emmanuel Godoy

This paper describes a classical methodology used to model and identify the parameters of an oxygen regulator with the goal of increasing its stability performance. After detailing the modeling-identification part to determine the physical parameters of the regulator and laying down the ongoing difficulties for analyzing the stability of this system, a short application of this methodology is shown.

### Design and Parameter Tuning of a Robust Model Predictive Controller for UAVs

####
*Nathan Michel*, Sylvain Bertrand, Sorin Olaru, Giorgio Valmorbida, Didier Dumur

Two formulations of robust model predictive control (MPC) that have robustness properties with respect to bounded additive disturbance over conventional MPC are applied to Unmanned Aerial Vehicle (UAV) translational dynamics. These controllers use results from invariant sets theory. The tuning of the proposed MPC laws is studied, and their performances are compared in simulations.

## 2SL2S - Pre-Conference Talks

7th December 2016, 9h30 at CentraleSupelec (Gif-sur-Yvette) Salle du conseil L2S - B4.40

### Analysis of PWA Control of Discrete-Time Linear Dynamics in the Presence of Variable Time-Delay

####
*Mohammed-Tahar Laraba*, Sorin Olaru, Silviu-Iulian Niculescu

This paper focuses on the robustness problem for a specific class of dynamical systems, namely the piecewise affine (PWA) systems, defined over a bounded region of the state-space X. We will be interested in PWA systems emerging from linear dynamical systems controlled via feedback channels in the presence of varying transmission delays by a PWA controller defined over a polyhedral partition of the state-space. We exploit the fact that the variable delays are inducing some particular model uncertainty. Our objective is to characterize the delay invariance margins: the collection of all possible values of the time-varying delays for which the positive invariance of X is guaranteed with respect to the closed-loop dynamics. These developments can be useful for the analysis of different design methodologies and in particular for predictive control approaches. The proposed delay margins describes the admissible transmission delays for an MPC implementation. From a different perspective, it further provides the fragility margins of an MPC implementation via the on-line optimization and subject to variable computational time.

### Singular-Perturbations-Based Analysis of Synchronization in Heterogeneous Networks: A Case-Study

####
*Mohamed **Adlene *Maghenem, Elena Panteley, Antonio Loria

*Adlene*Maghenem

In recent work, we laid the basis of an analysis framework for the study of heterogeneous networks. In essence, it is postulated that in a heterogeneous network a collective non-trivial behaviour arises, which may be modelled as a dynamical system itself. Then, we say that the networked systems synchronize or, more precisely, achieve dynamic consensus if they adopt this emergent behaviour. In this paper we consider the case-study of coupled Andronov-Hopf oscillators. We establish that the emergent dynamics, which is of the same nature as a single oscillator, is orbitally stable. Then, we show that the trajectories of the individual oscillators tend to a neighbourhood of the stable orbit. For the first time in the study of synchronization, the analysis is based on singular-perturbations theory; we show that the emergent dynamics corresponds to a slow system while the synchronization errors form a fast dynamics.

### Robustness Margin for Piecewise Affine Explicit Control Law

####
*Rajesh Koduri*, Pedro Rodriguez-Ayerbe, Sorin Olaru

Abstract: Classical robustness margin i.e., gain margin and phase margin, considers the gain variation and phase variation of the model for which the stability of the closed loop is preserved. In this paper, an attempt to find the same kind of margin for a piecewise affine (PWA) controller is done. This type of controller obtained for example via explicit model predictive control (MPC) is defined over a convex region of the state space mathcal{X}. Starting from the invariance property of the closed loop obtained involving a discrete dynamic model and PWA controller in a convex region of the state space, we calculate the two robustness margin preserving this invariance property. The first one will be denoted as gain margin corresponding to the variation of the gain of the model guaranteeing the invariance. The second one, denoted, the robustness margin against first order neglected dynamics will correspond to the slowest first order neglected dynamic allowed in the system preserving the invariance property.

## 2SL2S - Pre-Conference Talks

19 July 2016, 11h30 at CentraleSupelec (Gif-sur-Yvette) Salle du conseil du L2S - B4.40

### Lyapunov-based formation-tracking control of nonholonomic systems under persistency of excitation.

####
*Mohamed Adlene Maghenem*, Antonio Loría, Elena Panteley

We present a smooth nonlinear time-varying controller for leader-follower tracking of non-holonomic mobile robots. Our design relies upon the standing assumption that either the rotational or the translational reference velocity is persistently exciting. Then, we extend our results to cover the problem of formation tracking for a swarm of vehicles interconnected under a spanning tree communication topology rooted at the virtual leader. In this case, we propose a simple distributed control law that establishes the convergence of the error coordinate of each agent, relatively to its neighbourhood, under the same condition of persistency of excitation. In addition, our proofs are based on Lyapunov's second method, that is, we provide a strict Lyapunov function.

###

Energy Shaping Control of an Inverted Flexible Pendulum Fixed to a Cart

####
Prasanna Gandhi, *Luis Pablo Borja*, Romeo Ortega

Control of compliant mechanical systems is increasingly being researched for several applications including flexible link robots and ultra-precision positioning systems. The control problem in these systems is challenging, especially with gravity coupling and large deformations, because of inherent underactuation and the combination of lumped and distributed parameters of a nonlinear system. In this paper we consider an ultra-flexible inverted pendulum on a cart and propose a new nonlinear energy shaping controller to keep the pendulum at the upward position with the cart stopped at a desired location. The design is based on a model, obtained via the constrained Lagrange formulation, which previously has been validated experimentally. The controller design consists of a partial feedback linearization step followed by a standard PID controller acting on two passive outputs. Boundedness of all signals and (local) asymptotic stability of the desired equilibrium is theoretically established. Simulations and experimental evidence assess the performance of the proposed controller.

###

A strict Lyapunov function for non-holonomic systems under persistently-exciting controllers.

####
*Mohamed Adlene Maghenem*, Antonio Loría, Elena Panteley

We study the stability of a non linear time-varying skew symmetric systems \dot{x} = A(t, x)x with particular structures that appear in the study problems of non holonomic systems in chained form as well as adaptive control systems. Roughly, under the condition that each non diagonal element of A(t, x) is persistently exciting or uniform persistently exciting with respect x. Although some stability results are known in this area, our main contribution lies in the construction of Lyapunov functions that allows a computation of convergence rate estimates for the class of non linear systems under study.

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2SL2S - Pre-Conference Talks

1st July 2016, 11h30 at CentraleSupelec (Gif-sur-Yvette) Salle du conseil du L2S - B4.40

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Particle Swarm Optimization of Matsuoka's Oscillator Parameters in Human-Like Control of Rhythmic Movements

####
*Guillaume Avrin*, Maria Makarov, Pedro Rodriguez-Ayerbe, Isabelle Anne Siegler

In the field of neuroscience, the Matsuoka's nonlinear neural oscillator is commonly used to model Central Pattern Generator (CPG) in humans/animals. How the parameters of such structure should be selected is not always clear. It was generally done in past studies thanks to a trial-and-error method that needs to be reiterated each time the task changes. Recent studies using a Describing Function Analysis (DFA) of this CPG model provide interesting analytical tuning methods. Nevertheless, as they are based on a linear approximation, they might have a limited efficiency in the particular case of timing-sensitive task, such as the ball-bouncing task considered in this study. A Particle Swarm Optimization (PSO) is thus proposed to select the parameters of a novel neural oscillator-based human-like control architecture able to face disturbances and to adapt to new reference set-points during the ball-bouncing task. The general method presented in the present paper can also be used for other Matsuoka's oscillator tunings and other tasks.

##

2SL2S - Regular Talks

11 February 2016, 13h30 at CentraleSupelec (Gif-sur-Yvette) Room F3.05 (level 3)

### Lyapunov Stability Analysis of Immature Cell Dynamics in Healthy and Unhealthy Hematopoiesis.

####
*Walid Djema*, Frédéric Mazenc, Catherine Bonnet

The knowledge of Lyapunov-Krasovskii functionals offers strong advantages when investigating the local or global stability properties of equilibrium points of a system with delay. Unfortunately, in many cases the construction of these functionals is not an easy task. This is the case of the model describing hematopoiesis, which is a nonlinear system with distributed delays, which, according to some conditions, admits one or two equilibrium points. By contrast with approaches already used to study this model, we analyze the stability properties of its equilibrium points by constructing Lyapunov-Krasovskii functionals of two types.

In a second step, we discuss the case of unhealthy hematopoiesis. The resulting model takes into account the fact that some parameters, identified as control variables (through drug delivery), are now supposed to be time-varying. Via a novel Lyapunov-Krasovskii functional, stability and instability results are derived for the zero equilibrium of the model.

###

Set invariance for Delay Difference Equations

####
*Mohammed T. Laraba*, Sorin Olaru, Silviu Iulian Niculescu.

This paper deals with set invariance for time delay systems. The first goal is to review the known necessary and/or sufficient conditions for the existence of invariant sets with respect to dynamical systems described by discrete-time delay difference equations (DDEs). Secondly, we address the construction of invariant sets in the original state space (also called D-invariant sets) by exploiting the forward mappings. It will be shown that bilevel optimization problems can also be used for D-invariance design problems. The difficulties related to the nonlinearity of the optimization and the complementarity constraints will be discussed as well as the objective functions which can translate additional features of the D-invariant sets. The notion of D-invariance is interesting because it provides a region of attraction, which is difficult to obtain for delay systems without taking into account the delayed states in an extended state space model.

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2SL2S - Pre-Conference Talks

7 December 2015, 9h50 at CentraleSupelec (Gif-sur-Yvette) Amphithéâtre Ampère

### Global Stabilization of Multiple Integrators by a Bounded Feedback with Constraints on Its Successive Derivatives.

####
*Jonathan Laporte*, Antoine Chaillet, Yacine Chitour

In this paper, we address the global stabilization of chains of integrators by means of a bounded static feedback law whose p first time derivatives are bounded. Our construction is based on the technique of nested saturations introduced by Teel. We show that the control amplitude and the maximum value of its p first derivatives can be imposed below any prescribed values. Our results are illustrated by the stabilization of the third order integrator with prescribed bounds on the feedback and its first two derivatives.

###

Migration of double imaginary characteristic roots under small deviation of two delay parameters

####
Keqin Gu, *Dina Alina Irofti*, Islam Boussaada, Silviu-Iulian Niculescu

We study the migration of double imaginary roots of the characteristic equation for systems with two delays when the delay parameters are subjected to small deviations. As the double roots are not differentiable with respect to the delay parameters, Puiseux series is often used in such a situation in the literature. However, we study the ``least degenerate'' case by using a more traditional analysis, without involving Puiseux series. It was found that the local stability crossing curve has a cusp at the point in the parameter space that causes the double root, and it divides the neighborhood of this point into a G-sector and an S-sector. When the parameters move into the G-sector, one of the roots moves to the right half plane, and the other moves to the left half plane. When the parameters move into the S-sector, both roots move either to the left half plane or the right half plane depending on the sign of some value explicitly expressed in terms of derivatives of the characteristic function up to the third order.

### Inverse Parametric Linear/quadratic Programming Problem for Continuous PWA Functions Defined on Polyhedral Partitions of Polyhedra

####
*Ngoc Anh Nguyen*, Sorin Olaru, Pedro Rodriguez-Ayerbe

Constructive solution to inverse parametric linear/quadratic programming problems has recently been investigated

and shown to be solvable via convex liftings. These results were stated and solved starting from polytopic

partitions of a polytope in the parameter space. Therefore, the case of polyhedral partitions of unbounded polyhedra, was not

handled by this method and deserves a complete characterization to address the general inverse optimality problem. This

paper has as main objective to overcome the unboundedness limitation of the given polyhedral partition and to extend the

constructive solution for this omitted case.