A hierarchy of LMI inner approximations of the set of stable polynomials

Exploiting spectral properties of symmetric banded Toeplitz matrices, we describe simple sufficient conditions for positivity of a trigonometric polynomial formulated as linear matrix inequalities (LMI) in the coefficients. As an application of these results, we derive a hierarchy of convex LMI inner approximations (affine sections of the cone of positive definite matrices of size $m$) of the nonconvex set of Schur stable polynomials of given degree $n < m$. It is shown that when $m$ tends to infinity the hierarchy converges to a lifted LMI approximation (projection of an LMI set defined in a lifted space of dimension quadratic in $n$) already studied in the technical literature

Stability Criteria for SIS Epidemiological Models under Switching Policies

We study the spread of disease in an SIS model. The model considered is a time-varying, switched model, in which the parameters of the SIS model are subject to abrupt change. We show that the joint spectral radius can be used as a threshold parameter for this model in the spirit of the basic reproduction number for time-invariant models. We also present conditions for persistence and the existence of periodic orbits for the switched model and results for a stochastic switched model

Positivity of discrete singular systems and their stability: An LP-based approach

In this paper we present an efficient approach to the analysis of discrete positive singular systems. One of our main objectives is to investigate the problem of characterizing positivity of such systems. Previously, this issue was not completely addressed. We provide easily checkable necessary and sufficient conditions for such problem to be solved. On the other hand, we study the stability of discrete positive singular systems. Note that this is not a trivial problem since the set of admissible initial conditions is not the whole space but it is represented by a special cone. All the conditions we provide are necessary and sufficient, and are based on a reliable computational approach via linear programming

Discrete-time positive periodic systems with state and control constraints

The aim of this paper is to provide an efficient control design technique for discrete-time positive periodic systems. In particular, stability, positivity and periodic invariance of such systems are studied. Moreover, the concept of periodic invariance with respect to a collection of boxes is introduced and investigated with connection to stability. It is shown how such concept can be used for deriving a stabilizing state-feedback control that maintains the positivity of the closed-loop system and respects states and control signals constraints. In addition, all the proposed results can be efficiently solved in terms of linear programming

A hybrid camera- and ultrasound-based approach for needle localization and tracking using a 3D motorized curvilinear ultrasound probe

Three-dimensional (3D) motorized curvilinear ultrasound probes provide an effective, low-cost tool to guide needle interventions, but localizing and tracking the needle in 3D ultrasound volumes is often challenging. In this study, a new method is introduced to localize and track the needle using 3D motorized curvilinear ultrasound probes. In particular, a low-cost camera mounted on the probe is employed to estimate the needle axis. The camera-estimated axis is used to identify a volume of interest (VOI) in the ultrasound volume that enables high needle visibility. This VOI is analyzed using local phase analysis and the random sample consensus algorithm to refine the camera-estimated needle axis. The needle tip is determined by searching the localized needle axis using a probabilistic approach. Dynamic needle tracking in a sequence of 3D ultrasound volumes is enabled by iteratively applying a Kalman filter to estimate the VOI that includes the needle in the successive ultrasound volume and limiting the localization analysis to this VOI. A series of ex vivo animal experiments are conducted to evaluate the accuracy of needle localization and tracking. The results show that the proposed method can localize the needle in individual ultrasound volumes with maximum error rates of 0.7 mm for the needle axis, 1.7° for the needle angle, and 1.2 mm for the needle tip. Moreover, the proposed method can track the needle in a sequence of ultrasound volumes with maximum error rates of 1.0 mm for the needle axis, 2.0° for the needle angle, and 1.7 mm for the needle tip. These results suggest the feasibility of applying the proposed method to localize and track the needle using 3D motorized curvilinear ultrasound probes

Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius

We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the JSR is Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR

Optimal control of risk process in a regime-switching environment

This paper is concerned with cost optimization of an insurance company. The surplus of the insurance company is modeled by a controlled regime switching diffusion, where the regime switching mechanism provides the fluctuations of the random environment. The goal is to find an optimal control that minimizes the total cost up to a stochastic exit time. A weaker sufficient condition than that of (Fleming and Soner 2006, Section V.2) for the continuity of the value function is obtained. Further, the value function is shown to be a viscosity solution of a Hamilton-Jacobian-Bellman equation.Comment: Keywords: Regime switching diffusion, continuity of the value function, exit time control, viscosity solutio

A unified method for optimal arbitrary pole placement

We consider the classic problem of pole placement by state feedback. We offer an eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing feedback matrix that can deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. This parametric formula is then exploited to introduce an unconstrained nonlinear optimisation algorithm to obtain a feedback matrix that delivers the desired pole placement with optimal robustness and minimum gain. Lastly we compare the performance of our method against several others from the recent literature