Amphithéâtre Darboux, IHP, Paris, France, January 28-29, 2016.
Titles and abstracts
-
Michel Benaïm (Université de Neuchâtel)
Title: On piecewise deterministic Markov processes
-
François Dufour (IMB, Université de Bordeaux and INRIA)
Title: Optimal control of piecewise deterministic Markov processes
Abstract: In this talk, the infinite-horizon expected discounted continuous-time optimal control problem for piecewise deterministic Markov processes (PDMPs) is studied for the unconstrained as well as the constrained cases. For the unconstrained case we provide sufficient conditions to guarantee the existence and uniqueness of the integro-differential optimality equation as well as the existence of an optimal deterministic stationary control strategy.
For the constrained case we show that the values of the constrained control problem and an associated infinite dimensional linear programming problem (LP) are the same. Sufficient conditions are provided to ensure the solvability of the LP problem and the existence of an optimal feasible randomized stationary control strategy.
-
Antoine Girard (L2S, CentraleSupélec - Université Paris Sud- CNRS)
Title: Safety controller synthesis for incrementally stable switched systems using multiscale symbolic models
Abstract: We propose an approach to the synthesis of safety controllers for a class of switched systems, based on the use of multiscale symbolic models that describe transitions of various durations and whose sets of states are given by a sequence of embedded lattices approximating the state-space, the finer lattices
being accessible only by transitions of shorter duration. We prove that these multiscale symbolic models are approximately bisimilar to the original switched system provided it enjoys an incremental stability property attested by the existence of a common Lyapunov function or of multiple Lyapunov functions with a minimal dwell-time. Then, for specifications given by a safety automaton, we present a controller synthesis algorithm that exploits the specificities of multiscale symbolic models. We formalize the notion of maximal lazy safety controller which gives priority to transitions of longer durations; the shorter transitions and thus the finer scales of the symbolic model are effectively explored only when safety cannot be ensured at the coarser level and fast switching is needed. We propose a synthesis algorithm where symbolic models can be computed on the fly, this allows us to keep the number of symbolic states as low as possible. We provide computational evidence that shows drastic improvements of the complexity of controller synthesis using multiscale symbolic models instead of uniform ones.
-
Nicola Guglielmi (Università degli Studi dell'Aquila)
Title: Approximating Lyapunov exponents and most unstable
trajectories of switching systems
Abstract: We discuss a novel approach for constructing polytope Lyapunov
functions for continuous-time linear switching systems.
The method relies on the discretization of the system and
provides - for any given discretization stepsize - a lower
and an upper bound for the Lyapunov exponent and allows to
compute it with an arbitrary precision.
Then we discuss how to approximate most unstable trajectories,
which is related to the construction of Barabanov polytope norms.
Finally we show some illustrative examples.
This talk is inspired by recent joint works with Vladimir Protasov
and with Marino Zennaro.
-
Philippe Jouan (LMRS, CNRS-Université de Rouen)
Title: Stability of uniformly bounded switched systems and observability
Abstract:
Our aim is to give sufficient conditions for a switched linear system defined by a pair of Hurwitz matrices that share a common but not strict quadratic Lyapunov function to be GUAS.
We show that this property is equivalent to the uniform observability on $[0,+\infty)$ of a bilinear system defined on a subspace whose dimension is in most cases much smaller than the dimension of the switched system.
Some sufficient conditions of uniform asymptotic stability are then deduced from the equivalence theorem, and illustrated by examples.
The results are then extended to nonlinear analytic systems.
-
Raphaël Jungers (INMA, UCLouvain)
Title: Path-complete Lyapunov techniques: when Algebra and Combinatorics meet in Control
Abstract:
I will present an overview of 'Path-complete techniques', which have been developed to analyze stability properties of dynamical systems. These tools rely on concepts from Mathematics, Computer Science, and Optimization. Starting with the pioneering works of Bliman, Ferrari-Trecate, Lee, Dullerud, Daafouz, and others, the full nature of path-complete Lyapunov functions has been progressively understood in the last 20 years, and their range of application is still widening. These Algebro-combinatorial techniques allow in favorable cases to obtain provably efficient algorithms (e.g. for switching systems stability analysis), and are promising for tackling much more general problems, and much more general dynamical systems.
If time permits, I will move to applications in the field of Wireless Control Networks, where again, algebraic and combinatorial arguments allow to push classical unfeasibility boundaries for switching systems control.
I will end by presenting some further questions and potential applications.
-
Victor Kozyakin (Kharkevich Institute, Russian Academy of Sciences)
Title: Hourglass alternative and constructivity of spectral characteristics of matrix product
Abstract:
Recently Blondel, Nesterov and Protasov proved that the finiteness
conjecture holds for the generalized and the lower spectral radii of the
sets of non-negative matrices with independent row/column uncertainty.
We show that this result can be obtained as a simple consequence of the
so-called hourglass alternative which can be used also to establish the
minimax relations between the spectral radii of matrix products.
Axiomatization of the statements that constitute the hourglass alternative
makes it possible to define a new class of sets of positive matrices having
the finiteness property, which includes the sets of non-negative matrices
with independent row uncertainty. This class of matrices, supplemented
by the zero and identity matrices, forms a semiring with the Minkowski
operations of addition and multiplication of matrix sets, which gives means
to construct new sets of non-negative matrices possessing the finiteness
property for the generalized and the lower spectral radii.
The Hourglass alternative helps us to describe a new class of positive
linear discrete-time switching systems for which the problems of stability
or stabilizability can be resolved constructively. This class generalizes
the class of systems with independently switching state vector components.
The distinctive feature of this class is that, for any system from this class,
its components or blocks can be arbitrarily connected in parallel or in series
without loss of the `constructive resolvability' property. It is shown also that,
for such systems, it is possible to build constructively the individual
positive trajectories with the greatest or the lowest rate of convergence
to the zero.
-
Michael Margaliot (School of EE-Systems, Tel-Aviv University)
Title: Stability analysis of positive linear switched systems: a variational approach
Abstract: We consider the stability of continuous-time positive linear switched systems (PLSSs) under arbitrary switching.
We embed the PLSS in a positive bilinear control system (PBCS).
For any final time and any control, the transition matrix of a PBCS is a non-negative matrix.
Stability of the PBCS is determined by the spectral radius of this matrix.
A control is "most destabilizing" if it maximizes the spectral radius. We derive a necessary condition for a
control to be most destabilizing, stated in the form of a maximum principle (MP).
The proof of this MP combines the standard needle variation with a basic result from the Perron-Frobenius theory of non-negative matrices. We describe several applications of this MP to the stability analysis of PLSSs under arbitrary switching.
-
Oliver Mason (Hamilton Institute)
Title: Positivity and Monotonicity for Switched Systems: Results and Applications in Biology
Abstract:
In this talk, I will present an overview of recent efforts to extend techniques based on positivity and monotonicity from autonomous systems to non-autonomous and switched systems. In particular, strong forms of stability for switched systems such as D-stability will be discussed, as well as characterisations of monotonicity and implications of the same for state-dependent switched systems. Applications to epidemiology, genetic regulatory networks and rumour propagation will also be described.
-
Marian Mrozek (Institute of Computer Science and Computational Mathematics, Jagiellonian University)
Title: Morse-Conley theory for combinatorial vector fields
Abstract: In late 90' R. Forman introduced the concept of a combinatorial vector field on a CW complex and
presented a version of Morse theory for acyclic combinatorial vector fields.
He also studied combinatorial vector fields without acyclicity assumption, introduced the
concept of a chain recurrent set and proved Morse inequalities in this setting.
In this talk we present the Morse-Conley theory for combinatorial vector fields
and a certain generalization of combinatorial vector fields oriented on applications in sampled dynamics.
In particular, we we study Morse decompositions and Conley-Morse graph in the combinatorial setting.
-
Sriram Sankaranarayanan (Department of Computer Science, University of Colorado)
Title: Controller Synthesis for Switched Nonlinear Systems through Satisfiability-Modulo Theory (SMT) solvers
Abstract:
We present recent work on synthesizing controllers for nonlinear switched systems using
SMT solvers. Our approach considers a switched plant model, and a quantized control input that
selects a mode of the plant as a function of it's current state. First, we formulate restricted control Lyapunov functions and control
Barrier functions for these systems to prove the existence of non-zeno switching controllers with a minimum dwell time requirement.
We then present approaches to synthesizing these functions using Satisfiability Modulo Theory (SMT)
solvers. Finally, we discuss the numerical results on a set of benchmarks. (Joint work with Hadi Ravanbakhsh and Mohamed Amin Ben Sassi.)
