Optimal design of dilution experiments under volume constraints

The paper develops methods to construct a one-stage optimal design of dilution experiments under the total available volume constraint typical for bio-medical applications. We consider various design criteria based on the Fisher information both is Bayesian and non-Bayasian settings and show that the optimal design is typically one-atomic meaning that all the dilutions should be of the same size. The main tool is variational analysis of functions of a measure and the corresponding steepest descent type numerical methods. Our approach is generic in the sense that it allows for inclusion of additional constraints and cost components, like the cost of materials and of the experiment itself.Comment: 29 pages, 10 figure

On stabilization of nonlinear systems with drift by time-varying feedback laws

This paper deals with the stabilization problem for nonlinear control-affine systems with the use of oscillating feedback controls. We assume that the local controllability around the origin is guaranteed by the rank condition with Lie brackets of length up to 3. This class of systems includes, in particular, mathematical models of rotating rigid bodies. We propose an explicit control design scheme with time-varying trigonometric polynomials whose coefficients depend on the state of the system. The above coefficients are computed in terms of the inversion of the matrix appearing in the controllability condition. It is shown that the proposed controllers can be used to solve the stabilization problem by exploiting the Chen-Fliess expansion of solutions of the closed-loop system. We also present results of numerical simulations for controlled Euler's equations and a mathematical model of underwater vehicle to illustrate the efficiency of the obtained controllers.Comment: This is the author's version of the manuscript accepted for publication in the Proceedings of the 12th International Workshop on Robot Motion Control (RoMoCo'19

Bit flipping and time to recover

We call `bits' a sequence of devices indexed by positive integers, where every device can be in two states: $0$ (idle) and $1$ (active). Start from the `ground state' of the system when all bits are in $0$-state. In our first Binary Flipping (BF) model, the evolution of the system is the following: at each time step choose one bit from a given distribution $\mathcal{P}$ on the integers independently of anything else, then flip the state of this bit to the opposite. In our second Damaged Bits (DB) model a `damaged' state is added: each selected idling bit changes to active, but selecting an active bit changes its state to damaged in which it then stays forever. In both models we analyse the recurrence of the system's ground state when no bits are active. We present sufficient conditions for both BF and DB models to show recurrent or transient behaviour, depending on the properties of $\mathcal{P}$. We provide a bound for fractional moments of the return time to the ground state for the BF model, and prove a Central Limit Theorem for the number of active bits for both models

Obstacle Avoidance Problem for Second Degree Nonholonomic Systems

In this paper, we propose a new control design scheme for solving the obstacle avoidance problem for nonlinear driftless control-affine systems. The class of systems under consideration satisfies controllability conditions with iterated Lie brackets up to the second order. The time-varying control strategy is defined explicitly in terms of the gradient of a potential function. It is shown that the limit behavior of the closed-loop system is characterized by the set of critical points of the potential function. The proposed control design method can be used under rather general assumptions on potential functions, and particular applications with navigation functions are illustrated by numerical examples.Comment: This is the author's accepted version of the paper to appear in: 2018 IEEE Conference on Decision and Control (CDC), (c) IEE

Partial Stability Concept in Extremum Seeking Problems

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov's direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system with oscillating inputs are derived with the use of Lie bracket approximation techniques. These conditions are exploited to describe a broad class of extremum-seeking controllers ensuring the partial stability of the set of minima of a cost function. The obtained theoretical results are illustrated by the Brockett integrator and rotating rigid body.Comment: This is the author's version of the manuscript accepted for publication in the Proceedings of the Joint 8th IFAC Symposium on Mechatronic Systems and 11th IFAC Symposium on Nonlinear Control Systems (MECHATRONICS & NOLCOS 2019

Stability for random measures, point processes and discrete semigroups

Discrete stability extends the classical notion of stability to random elements in discrete spaces by defining a scaling operation in a randomised way: an integer is transformed into the corresponding binomial distribution. Similarly defining the scaling operation as thinning of counting measures we characterise the corresponding discrete stability property of point processes. It is shown that these processes are exactly Cox (doubly stochastic Poisson) processes with strictly stable random intensity measures. We give spectral and LePage representations for general strictly stable random measures without assuming their independent scattering. As a consequence, spectral representations are obtained for the probability generating functional and void probabilities of discrete stable processes. An alternative cluster representation for such processes is also derived using the so-called Sibuya point processes, which constitute a new family of purely random point processes. The obtained results are then applied to explore stable random elements in discrete semigroups, where the scaling is defined by means of thinning of a point process on the basis of the semigroup. Particular examples include discrete stable vectors that generalise discrete stable random variables and the family of natural numbers with the multiplication operation, where the primes form the basis.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ301 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Nonparametric estimation of infinitely divisible distributions based on variational analysis on measures

The paper develops new methods of non-parametric estimation a compound Poisson distribution. Such a problem arise, in particular, in the inference of a Levy process recorded at equidistant time intervals. Our key estimator is based on series decomposition of functionals of a measure and relies on the steepest descent technique recently developed in variational analysis of measures. Simulation studies demonstrate applicability domain of our methods and how they positively compare and complement the existing techniques. They are particularly suited for discrete compounding distributions, not necessarily concentrated on a grid nor on the positive or negative semi-axis. They also give good results for continuous distributions provided an appropriate smoothing is used for the obtained atomic measure