Motivated by questions in robust control and switched linear dynamical systems, we consider the p... more Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether all convex combinations of k matrices in R^{n x n} are stable. In particular, we are interested whether there exist algorithms which can solve this problem in time polynomial in n and k. We show that if $k= \lceil n^d \rceil$ for any fixed real d>0, then the problem is NP-hard, meaning that no polynomial-time algorithm in n exists provided that P is not NP, a widely believed conjecture in computer science. On the other hand, when k is a constant independent of n, then it is known that the problem may be solved in polynomial time in n. Using these results and the method of measurable switching rules, we prove our main statement: verifying the absolute asymptotic stability of a continuous-time switched linear system with more than n^d matrices A_i in R^{n x n} satisfying 0 >= A_i + A_i^{T} is NP-hard.
An elementary proof of the Kharitonov theorem is presented. The proof is based on the concept of ... more An elementary proof of the Kharitonov theorem is presented. The proof is based on the concept of a Bezoutian matrix. Generally, exploiting the special structure of such matrices (e.g., Bezoutians, Toeplitz, Hankel or Vandermonde matrices, etc.) can be interesting, e.g., leading to unified approaches in different cases, as well as to further generalizations. Here the concept of the Bezoutian matrix is used to provide a unified derivation of the Kharitonov-like theorems for the continuous-time and discrete-time settings. Finally, the (block) Anderson-Jury Bezoutians are used to propose a possible technique to attack an difficult open problem related to the robust stability in the MIMO case.
We discuss an old distributed algorithm for reaching consensus that has received a fair amount of... more We discuss an old distributed algorithm for reaching consensus that has received a fair amount of recent attention. In this algorithm, a number of agents exchange their values asynchronously and form weighted averages with (possibly outdated) values possessed by their neighbors. We overview existing convergence results, and establish some new ones, for the case of unbounded intercommunication intervals.
We propose a model for deterministic distributed function computation by a network of identical a... more We propose a model for deterministic distributed function computation by a network of identical and anonymous nodes. In this model, each node has bounded computation and storage capabilities that do not grow with the network size. Furthermore, each node only knows its neighbors, not the entire graph. Our goal is to characterize the class of functions that can be computed within this model. In our main result, we provide a necessary condition for computability which we show to be nearly sufficient, in the sense that every function that satisfies this condition can at least be approximated. The problem of computing (suitably rounded) averages in a distributed manner plays a central role in our development; we provide an algorithm that solves it in time that grows quadratically with the size of the network. * The authors are with the
We provide an example proving that there exists no quadratic Lyapunov function for a certain clas... more We provide an example proving that there exists no quadratic Lyapunov function for a certain class of linear agreement/consensus algorithms, a fact that had been numerically verified in . We also briefly discuss sufficient conditions for the existence of such a Lyapunov function.
We derive lower bounds on the convergence speed of a widely used class of distributed averaging a... more We derive lower bounds on the convergence speed of a widely used class of distributed averaging algorithms. In particular, we prove that any distributed averaging algorithm whose state consists of a single real number and whose (possibly nonlinear) update function satisfies a natural smoothness condition has a worst case running time of at least on the order of n 2 on a line network of n nodes. Our results suggest that increased memory or expansion of the state space is crucial for improving the running times of distributed averaging algorithms.
We consider distributed iterative algorithms for the averaging problem over timevarying topologie... more We consider distributed iterative algorithms for the averaging problem over timevarying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We also describe an algorithm within this class whose convergence time is the best among currently available averaging algorithms for time-varying topologies. We then propose and analyze distributed averaging algorithms under the additional constraint that agents can only store and communicate quantized information, so that they can only converge to the average of the initial values of the agents within some error. We establish bounds on the error and tight bounds on the convergence time, as a function of the number of quantization levels. *
We propose three new algorithms for the distributed averaging and consensus problems: two for the... more We propose three new algorithms for the distributed averaging and consensus problems: two for the fixed-graph case, and one for the dynamic-topology case. The convergence rates of our fixed-graph algorithms compare favorably with other known methods, while our algorithm for the dynamic-topology case is the first to be accompanied by a polynomial-time bound on the convergence time
We study the convergence speed of distributed iterative algorithms for the consensus and averagin... more We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.
Motivated by questions in robust control and switched linear dynamical systems, we consider the p... more Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether all convex combinations of k matrices in R^{n x n} are stable. In particular, we are interested whether there exist algorithms which can solve this problem in time polynomial in n and k. We show that if $k= \lceil n^d \rceil$ for any fixed real d>0, then the problem is NP-hard, meaning that no polynomial-time algorithm in n exists provided that P is not NP, a widely believed conjecture in computer science. On the other hand, when k is a constant independent of n, then it is known that the problem may be solved in polynomial time in n. Using these results and the method of measurable switching rules, we prove our main statement: verifying the absolute asymptotic stability of a continuous-time switched linear system with more than n^d matrices A_i in R^{n x n} satisfying 0 >= A_i + A_i^{T} is NP-hard.
An elementary proof of the Kharitonov theorem is presented. The proof is based on the concept of ... more An elementary proof of the Kharitonov theorem is presented. The proof is based on the concept of a Bezoutian matrix. Generally, exploiting the special structure of such matrices (e.g., Bezoutians, Toeplitz, Hankel or Vandermonde matrices, etc.) can be interesting, e.g., leading to unified approaches in different cases, as well as to further generalizations. Here the concept of the Bezoutian matrix is used to provide a unified derivation of the Kharitonov-like theorems for the continuous-time and discrete-time settings. Finally, the (block) Anderson-Jury Bezoutians are used to propose a possible technique to attack an difficult open problem related to the robust stability in the MIMO case.
We discuss an old distributed algorithm for reaching consensus that has received a fair amount of... more We discuss an old distributed algorithm for reaching consensus that has received a fair amount of recent attention. In this algorithm, a number of agents exchange their values asynchronously and form weighted averages with (possibly outdated) values possessed by their neighbors. We overview existing convergence results, and establish some new ones, for the case of unbounded intercommunication intervals.
We propose a model for deterministic distributed function computation by a network of identical a... more We propose a model for deterministic distributed function computation by a network of identical and anonymous nodes. In this model, each node has bounded computation and storage capabilities that do not grow with the network size. Furthermore, each node only knows its neighbors, not the entire graph. Our goal is to characterize the class of functions that can be computed within this model. In our main result, we provide a necessary condition for computability which we show to be nearly sufficient, in the sense that every function that satisfies this condition can at least be approximated. The problem of computing (suitably rounded) averages in a distributed manner plays a central role in our development; we provide an algorithm that solves it in time that grows quadratically with the size of the network. * The authors are with the
We provide an example proving that there exists no quadratic Lyapunov function for a certain clas... more We provide an example proving that there exists no quadratic Lyapunov function for a certain class of linear agreement/consensus algorithms, a fact that had been numerically verified in . We also briefly discuss sufficient conditions for the existence of such a Lyapunov function.
We derive lower bounds on the convergence speed of a widely used class of distributed averaging a... more We derive lower bounds on the convergence speed of a widely used class of distributed averaging algorithms. In particular, we prove that any distributed averaging algorithm whose state consists of a single real number and whose (possibly nonlinear) update function satisfies a natural smoothness condition has a worst case running time of at least on the order of n 2 on a line network of n nodes. Our results suggest that increased memory or expansion of the state space is crucial for improving the running times of distributed averaging algorithms.
We consider distributed iterative algorithms for the averaging problem over timevarying topologie... more We consider distributed iterative algorithms for the averaging problem over timevarying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We also describe an algorithm within this class whose convergence time is the best among currently available averaging algorithms for time-varying topologies. We then propose and analyze distributed averaging algorithms under the additional constraint that agents can only store and communicate quantized information, so that they can only converge to the average of the initial values of the agents within some error. We establish bounds on the error and tight bounds on the convergence time, as a function of the number of quantization levels. *
We propose three new algorithms for the distributed averaging and consensus problems: two for the... more We propose three new algorithms for the distributed averaging and consensus problems: two for the fixed-graph case, and one for the dynamic-topology case. The convergence rates of our fixed-graph algorithms compare favorably with other known methods, while our algorithm for the dynamic-topology case is the first to be accompanied by a polynomial-time bound on the convergence time
We study the convergence speed of distributed iterative algorithms for the consensus and averagin... more We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.
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Papers by Alex Olshevsky