Abstract—Inspired by the so-called “bugs ” problem from mathematics, we study the geometric formations of multivehicle systems under cyclic pursuit. First, we introduce the notion of cyclic pursuit by examining a system of identical linear agents in the plane. This idea is then extended to a system of wheeled vehicles, each subject to a single nonholonomic constraint (i.e., unicycles), which is the principal focus of this paper. The pursuit framework is particularly simple in that the identical vehicles are ordered such that vehicle pursues vehicle CImodulo. In this paper, we assume each vehicle has the same constant forward speed. We show that the system’s equilibrium formations are generalized regular polygons and it is exposed how the multivehicle system’s global behavior can be shaped through appropriate controller gain assignments. We then study the local stability of these equilibrium polygons, revealing which formations are stable and which are not. Index Terms—Circulant matrices, cooperative control, multiagent systems, pursuit problems. I.

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Introduction Many interesting control system design and analysis problems can be recast as systems of inequalities for multivariate polynomials in real variables. In particular, for linear time-invariant systems, important control issues such as robust stability and robust performance can be reduced to such systems. Typically, the variables in the (multivariate) polynomials come from plant (controlled system) and compensator (controller) parameters. In this chapter, we describe a method for solving such systems of inequalities. By solving we mean that we end up with a collection of axis-parallel boxes in the parameter space whose union provides an inner approximation of the solution set, i.e., the polynomial inequalities are fulfilled for each parameter vector taken from such a box. This method is based on the expansion of a multivariate polynomial into Bernstein polynomials. It provides an alternative to symbolic methods like quantifier elimination whose application to control

This paper examines qualitative simulation (QS) from the phase space perspective of dynamic systems theory. QS consists of two steps: transition analysis determines the sequence of qualitative states that a system traverses and global interpretation derives its long-term behavior. I recast transition analysis as a search problem in phase space and replace the assorted transition rules with two algebraic conditions. The first con-dition determines transitions between arbitrarily shaped regions in phase space, as opposed to QS which only handles n-dimensional rectangles. It also provides more accurate results by considering only the boundaries between regions. The second con-dition determines whether nearby trajectories approach a fixed point asymptotically. It obtains better results than QS by exploiting local stability properties. I recast global interpretation as a search for attractors in phase space and present a global interpreta-tion algorithm for systems whose local behavior determines global behavior uniquely. 'This research was performed while I was in the Clinical Decision Making Group oftheM.I.T. Laboratory

For a finite multigraph G, the reliability function of G is the probability RG (q) that if each edge of G is deleted independently with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if q 2 C is such that RG (q) = 0 then jqj 1. We verify that this conjectured property of RG (q) holds if G is a series-parallel network. The proof is by an application of the Hermite-Biehler Theorem and development of a theory of higher-order interlacing for polynomials with only real nonpositive zeros. We conclude by establishing some new inequalities which are satisfied by the f-vector of any matroid without coloops, and by discussing some stronger inequalities which would follow (in the cographic case) from the Brown-Colbourn Conjecture, and are hence true for cographic matroids of series-parallel networks.

We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semi-definite programming. Links are established between the Lyapunov matrix, rank-one LMIs and the Lagrange multiplier arising in duality theory. Keywords Linear Systems, Stability, LMI. 1 Introduction Let A 2 C n\Thetan be a given complex matrix and let D = fs 2 C : 1 s ? a b b ? c 1 s ! 0g denote a given open region of the complex plane, where Hermitian matrix a b b ? c 2 C 2\Theta2 has one strictly negative eigenvalue and one strictly positive eigenvalue, and the star denotes transpose conjugate. In the sequel, the notation P 0 or \GammaP OE 0 (resp. P 0 or \GammaP 0) means that matrix P is positive definite (resp. semi-definite). The location of the eigenvalues of A can be characterized as follows. 1 Corresponding author. FAX: +33 5 6...

In this paper we present a survey on the \Favard theorem" and its extensions. Key words: Favard Theorem, recurrence relations 1 Introduction. Given a sequence fP n g 1 n=0 of monic polynomials satisfying a certain recurrence relation, we are interested in nding a general inner product, if one exists, such that the sequence fP n g 1 n=0 is orthogonal with respect to it. The original \classical" result in this direction is due to J. Favard [10] even though his result seems to be known by dierent mathematicians. The rst who obtained a similar result was Stieltjes in 1894 [23]. In fact, from the point of view of J continued fractions obtained from the contraction of an S continued fraction with positive coecients, Stieltjes proved the existence of a positive linear functional such that the denominators of the approximants are orthogonal with respect to it [23, x11]. Later on, Stone gave another approach using the spectral resolution of a self-adjoint operator associated to a Jacobi...

We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used to solve systems of strict polynomial inequalities.

The established theory of the resultant of two polynomials assumes that they are expressed in the power (monomial) basis, and a basis transformation is therefore necessary if the resultant of two Bernstein polynomials is required. In this paper, a resultant matrix for two scaled Bernstein polynomials (polynomials of degree n whose basis functions are (1 x) ; i = 0 : : : n) is constructed. In particular, a companion matrix M for a scaled Bernstein polynomial r(x) is developed, and this is used to form a resultant matrix s(M ), where s(x) is a scaled Bernstein polynomial. Key words: Bernstein basis, resultants AMS classi cation : 11C08, 11C20 1

A closed form expression for a companion matrix M of a Bernstein polynomial is obtained, and this is used to derive an expression for a resultant matrix of two Bernstein polynomials. It is shown that M diers from its equivalent form for a power basis polynomial because an upper triangular Hankel matrix does not de ne a similarity transformation between M and M . A measure of the numerical condition of a resultant matrix, for polynomials in an arbitrary basis, is reviewed and this is used to compare the stability of two resultant matrices. In particular, computational tests are performed and it is shown that the resultant matrix of two Bernstein polynomials is numerically better conditioned than the resultant matrix that is obtained when a simple parameter substitution is used to transform the polynomials to the power basis.