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44
Formations of vehicles in cyclic pursuit
 IEEE Transactions on Automatic Control
, 2004
"... Abstract—Inspired by the socalled “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 ..."
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Cited by 77 (1 self)
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Abstract—Inspired by the socalled “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.
Univariate polynomials: nearly optimal algorithms for factorization and rootfinding
 In Proceedings of the International Symposium on Symbolic and Algorithmic Computation
, 2001
"... To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zerofree annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of t ..."
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Cited by 37 (11 self)
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To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zerofree annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the nth degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new rootfinder incorporates the earlier techniques of Schönhage, Neff/Reif, and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the computational complexity of both complete factorization and rootfinding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of wellconditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, whose roots can be illconditioned, forming
Solving strict polynomial inequalities by Bernstein expansion
 In: Symbolic Methods in Control System Analysis and Design
, 1999
"... 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 timeinvariant systems, important control issues such as robust stability and robust performance can be reduce ..."
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Cited by 20 (1 self)
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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 timeinvariant 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 axisparallel 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
LINEARIZATION OF MATRIX POLYNOMIALS EXPRESSED IN POLYNOMIAL BASES
"... Companion matrices of matrix polynomials L(λ) (with possibly singular leading coefficient) are a familiar tool in matrix theory and numerical practice leading to socalled “linearizations ” λB − A of the polynomials. Matrix polynomials as approximations to more general matrix functions lead to the s ..."
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Cited by 16 (1 self)
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Companion matrices of matrix polynomials L(λ) (with possibly singular leading coefficient) are a familiar tool in matrix theory and numerical practice leading to socalled “linearizations ” λB − A of the polynomials. Matrix polynomials as approximations to more general matrix functions lead to the study of matrix polynomials represented in a variety of classical systems of polynomials, including orthogonal systems and Lagrange polynomials, for example. For several such representations, it is shown how to construct (strong) linearizations via analogous companion matrix pencils. In case L(λ) has Hermitian or alternatively complex symmetric coefficients, the determination of linearizations λB −A with A and B Hermitian or complex symmetric is also discussed.
Zeros of Reliability Polynomials and fVectors of Matroids
 Combin. Probab. Comput
"... 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 conjectur ..."
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Cited by 12 (3 self)
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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 seriesparallel network. The proof is by an application of the HermiteBiehler Theorem and development of a theory of higherorder interlacing for polynomials with only real nonpositive zeros. We conclude by establishing some new inequalities which are satisfied by the fvector of any matroid without coloops, and by discussing some stronger inequalities which would follow (in the cographic case) from the BrownColbourn Conjecture, and are hence true for cographic matroids of seriesparallel networks.
A dynamic systems perspective on qualitative simulation
 Artificial Intelligence
, 1990
"... 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 longterm behavior. I recast transitio ..."
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Cited by 11 (1 self)
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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 longterm behavior. I recast transition analysis as a search problem in phase space and replace the assorted transition rules with two algebraic conditions. The first condition determines transitions between arbitrarily shaped regions in phase space, as opposed to QS which only handles ndimensional rectangles. It also provides more accurate results by considering only the boundaries between regions. The second condition 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 interpretation 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
NONLINEAR DYNAMICAL SYSTEM IDENTIFICATION FROM UNCERTAIN AND INDIRECT MEASUREMENTS
, 2002
"... We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regressio ..."
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Cited by 10 (0 self)
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We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the timecontinuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errorsinvariables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.
Application of Bernstein Expansion to the Solution of Control Problems
 University of Girona
, 1999
"... 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 t ..."
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Cited by 7 (0 self)
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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.
Rankone LMIs and Lyapunov's Inequality
 Proceedings of the IEEE Conference on Decision and Control
, 2000
"... We describe a new proof of the wellknown 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 semidefinite programming. Links are established between the Lyapunov matrix, rankon ..."
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Cited by 5 (3 self)
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We describe a new proof of the wellknown 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 semidefinite programming. Links are established between the Lyapunov matrix, rankone 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. semidefinite). The location of the eigenvalues of A can be characterized as follows. 1 Corresponding author. FAX: +33 5 6...