## MAC Seminar at Banyuls-sur-Mer

The team MAC at LAAS (CNRS) regularly hosts an internal seminar where the goal is to promote scientific interaction between the team members. These seminars include talks from team members, and some externally invited speakers. This year, we will host this seminar at Banyuls-sur-Mer from June 22 to June 24, 2016.#### Contact Information:

Organization: | Didier Henrion | henrion@laas.fr |

Scientific Programme: | Aneel Tanwani | tanwani@laas.fr |

Administration: | Christele Mouclier | csoler@laas.fr |

### Schedule of Scientific Talks:

The seminar will comprise four sessions on topics related to optimization and control. The detailed program appears below. By clicking on the speaker's name or title of the talk, you can see the affiliation of the speaker and the abstract.## Affiliation of Speakers and Abstracts

**Amir Ali Ahmadi, Assistant Professor, Princeton University, USA**

*Title: Computation of the joint spectral radius by optimization techniques*

Abstract: The joint spectral radius (JSR) of a set of matrices characterizes the maximum growth rate that can be achieved by multiplying them in arbitrary order. This concept, which essentially generalizes the notion of the "largest eigenvalue" from one matrix to many, was introduced by Rota and Strang in the early 60s and has since emerged in many areas of application such as stability of switched linear systems, computation of the capacity of codes, convergence of consensus algorithms, tracability of graphs, and many others. The JSR is a very difficult quantity to compute even for a pair of matrices. In this talk, we present optimization-based algorithms (e.g., via semidefinite programming or dynamic programming) that can either compute the JSR exactly in special cases or approximate it with arbitrary prescribed accuracy in the general case.

Based on joint work (in different subsets) with Raphael Jungers, Pablo Parrilo, and Mardavij Roozbehani.

**Georgina Hall, PhD Candidate, Princeton University, USA**

*Title: Linear and second order cone programming relaxations for certifying Lyapunov inequalities*

Abstract: Recently, some LP and SOCP-based alternatives to SOS optimization have been proposed by Ahmadi and Majumdar. In this talk, we consider applications to Lyapunov analysis, in particular, the problem of certifying stability of switched linear systems by upper bounding the Joint Spectral Radius (JSR) of the underlying matrices. We show how the structure of this problem can be exploited to generate new LPs and SOCPs that better approximate the SOS bound.

**Roxana Hess, PhD Candidate, LAAS - CNRS, Toulouse, France**

*Title: Semidefinite approximations of the polynomial abscissa*

Abstract: Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is Holder continuous, and not locally Lipschitz in general, which is a source of numerical difficulties for designing and optimizing control laws. In this paper we propose simple approximations of the abscissa given by polynomials of fixed degree, and hence controlled complexity. Our approximations are computed by a hierarchy of finite-dimensional convex semidefinite programming problems. When their degree tends to infinity, the polynomial approximations converge in norm to the abcissa, either from above or from below.

Ref: R. Hess, D. Henrion, J.B. Lasserre, T.S. Pham. Semidefinite approximations of the polynomial abscissa, arXiv:1507.08463, July 2015.

**Guillaume Davy, PhD Candidate, ONERA, Toulouse, France**

*Title: Credible autocoding of an interior point method algorithm*

Abstract: The efficiency of modern optimization methods, coupled with increasing computational resources, has led to the possibility of real-time optimization algorithms acting in safety critical roles. There is a considerable body of mathematical proofs on on-line optimization programs which can be leveraged to assist in the development and verification of their implementation. To demonstrate this we built an autocoder which generates a C implementation of an interior point algorithm solving a specific linear problem given as input to the autocoder. This autocoder will not only generate C code but also the proof of its correctness as ACSL annotation that can be checked by Frama-C, SMT solver and PVS.

**Mario Sznaier, Dennis Picard Trustee Professor of Elec. and Comp. Engg., Northeastern University, Boston, USA**

*Title: Control Design Subject to Information Flow Sparsity Constraints*

Abstract: This talks considers the problem of synthesizing output feedback controllers subject to sparsity constraints. This problem is known to be generically NP-hard, unless the plant satisfies the Quadratic Invariance property. Our main results show that, even if this property does not hold, tractable convex relaxations with optimality certificates can be obtained by recasting the problem into a polynomial optimization through the use of polyhedral Lyapunov functions. Combining these ideas with rank minimization tools leads to a computationally attractive algorithm. In the second portion of the talk we will show how to extend these ideas to the case where, in addition to sparsity, the controller is subject to set membership constraints (such as bound on the control action). In this case, a computationally efficient algorithm can be obtained through a combination of polyhedral Lyapunov functions, polynomial optimization and the extended Farkas Lemma. These results are illustrated with several examples, where we show the effectiveness of the proposed approaches vis-a-vis existing techniques.

**Hiroyuki Ichihara, Associate Professor, Meiji University of Science and Engineering, Japan.**

*Title: SOS relaxation to test copositivity and its application to control problem*

Abstract: This talk deals with a sum of squares approach to testing copositiviy of polynomials, namely, testing if polynomials are nonnegative on nonnegative orthant. The approach is a natural extension of testing copositivity of symmetric matrices by sum of squares relaxation through copositive quadratic forms. This talk also deals with a possiblity of applying the proposed approach to nonnegative nonlinear dynamical systems. Stability analysis of an equilibrium point of the Lotka-Volterra competition system is illustrated by using polynomial Lyapunov functions as well as linear Lyapunov functions for an extended system.

**Luca Zaccarian, Senior Researcher, LAAS - CNRS, Toulouse, France. Also affiliated with University of Trento**

*Title: Dynamic input allocation*

Abstract: Typical control systems designs concentrate on plant output specifications and do not always take sufficiently into account specifications related to its input (e.g., coming from saturation problems). This fact is especially relevant when there are degrees of freedom in the plant input selection due to some kind of actuator redundancy. In this talk we will present a dynamic input allocation scheme that has been proposed roughly five years ago. The core ideas behind the dynamical allocation scheme will be first discussed and motivated. Then we will overview a number of application experiences where the method has shown desirable performance.

**Laura Dal Col, PhD Candidate, LAAS - CNRS, Toulouse, France**

*Title: Global H∞ consensus of linear multi-agent systems with input saturation*

Abstract: This work deals with the problem of global consensus of a network of agents corresponding to identical linear continuous-time systems within a fully connected network and in the presence of saturation. In particular, each agent exchanges a linear output through the network and is subject to input magnitude saturation and a disturbance signal. The mismatch among the external disturbances is supposed to be bounded in energy and then a saturated H ∞ control design problem is addressed and solved. A decentralized dynamic output feedback control law is designed through linear matrix inequality (LMI) conditions that induces the global consensus, a certified local convergence rate and a certified global nonlinear H∞ performance.

**Alexandre Seuret, Researcher, LAAS - CNRS, Toulouse, France**

*Title: Stability of infinite dimensional systems and polynomial approximation*

Abstract: In this presentation, the recent advances on the stability analysis of time-delays systems developed in the group within the ANR project SCIDiS will be exposed. Time-delays systems represent an important class of infinite dimensional systems arising in many applications from Engineering to Biology. During the last years, we have developed a new paradigm for analyzing the stability of such a class of systems mainly based on the exhibition of accurate integral inequalities. These inequalities are related to the Bessel inequality on Hilbert space and to the polynomial approximation of the infinite dimensional state of the system under consideration. After presenting the basics of the method and its relevance for this analysis, several examples of time delay systems will be exposed. Finally, since time-delay systems refer to a particular class of distributed parameters systems, we will review the potential directions where the same methodology can be applicable.

**Paulo Ricardo Arantes Gilz, PhD Candidate, LAAS - CNRS, Toulouse, France**

*Title: A new approach to describe the space constraints for the spacecraft rendezvous hovering phases*

Abstract: The orbital rendezvous hovering phase consists on bringing a chaser satellite to a box-constrained trajectory relatively to another leader spacecraft. In previous works these space constraints imposed on the relative trajectories are modeled by methods that do not produce well-suited algorithms for airspace applications. In our work we propose a new approach to verify the respect of the space restrictions by computing the roots of real univariate polynomials. The resultant method is then integrated to an algorithm to solve the hovering phase fuel-minimization problem and tested on an LEON3 processor-based board.

**Paolo Frasca, Assistant Professor, University of Twente, The Netherlands**

*Title: Nonsmooth systems in opinion dynamics*

Abstract: The basic assumption in opinion dynamics is that one's opinion is attracted by other's opinions. Provided enough interactions occurs between the individuals, this assumption implies that consensus is asymptotically achieved. However, experience suggests that consensus is not always achieved, but disagreement persists. One explanation for such persistence postulates that individuals do not influence each other if their opinions are too far apart. The simplest way to model this idea--referred to as "bounded confidence"--is based on a fixed threshold: individuals interact if their opinions are closer than the threshold. Another explanation for persistent disagreement is that individuals do not share opinions directly, but communicate by (possibly imprecise) verbalizations or by taking certain actions. These effects can be modelled as quantization effects. Including either bounded confidence or quantization makes the system discontinuous and brings the need for studying the limit behavior of its generalized solutions. In the case of bounded confidence, we provide results of existence, completeness and convergence to clusters of agents sharing a common opinion. In the case of quantization, we show that in general consensus can not be expected and we give an asymptotic estimate of the distance from consensus. However, we show that consensus is actually achieved on complete and on complete bipartite graphs.

This is joint work with Francesca Ceragioli (Politecnico di Torino, IT).

**Sofia Urbina Iglesias, PhD Candidate, LAAS - CNRS, Toulouse, France**

*Title: A hybrid control framework for impulsive control of satellite rendezvous*

Abstract: We focus on the problem of satellite rendezvous between two spacecrafts in elliptic orbits. Using a linearized model of the relative dynamics, we first propose a periodic similarity transformation based on Floquet-Lyapunov theory, leading to a set of coordinates under which the free motion is linear time-invariant. Then we address the problem of impulsive control of satellite rendezvous as a hybrid dynamical system, and we show that the arising elegant representation enables designing impulsive control laws with different tradeoffs between computational complexity and fuel consumption. The adopted hybrid formalism allows us to prove suitable stability properties of the proposed controllers. The results are comparatively illustrated on simulation examples.

**Aneel Tanwani, Researcher, LAAS - CNRS, Toulouse, France**

*Title: Lyapunov-based dynamic sampling algorithms for output feedback control of nonlinear systems*

Abstract: For feedback stabilization of a control system using dynamic output feedback, we consider the problem of finding two different sequences of time instants at which the sampled outputs (respectively, control inputs) must be sent to the controller (resp.~the plant). Instead of static inequalities, the states of so-called norm estimators (dynamic filters) are used to determine sampling instants. Using the tools from Lyapunov theory and stability of cascaded nonlinear systems, it is shown that the closed loop system is globally asymptotically stable. Additional assumptions are required on the controller and system dynamics to guarantee that the proposed sampling routines do not lead to an accumulation of sampling times over a finite interval.

This work is in collaboration with Andrew Teel (UCSB) and Christophe Prieur (Gipsa).

**Isabelle Queinnec, Senior Researcher, LAAS - CNRS, Toulouse, France**

*Title: Some control-theoretic questions on control of anesthesia*

Abstract: This talk provides an overview of our work on control of anesthesia. Using the compartment models, and different indicators for intensity of anesthesia, we are concerned with questions such as variation in models due to patients, multiple time scales in dynamics, the constraints on actuators, the quantization and sampling of signals, and the strategies for inducing anesthesia. These different questions are addressed via design of controllers based on utilizing LMIs.