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Performance Analysis of Positive Systems and Optimization Algorithms with Time-delays
KTH, School of Electrical Engineering (EES), Automatic Control.ORCID iD: 0000-0003-1149-4715
2016 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

Time-delay dynamical systems are used to model many real-world engineering systems, where the future evolution of a system depends not only on current states but also on the history of states. For this reason, the study of stability and control of time-delay systems is of theoretical and practical importance. In this thesis, we develop several stability analysis frameworks for dynamical systems in the presence of communication and computation time-delays, and apply our results to different challenging engineering problems.

The thesis first considers delay-independent stability of positive monotone systems. We show that the asymptotic stability of positive monotone systems whose vector fields are homogeneous is independent of the magnitude and variation of time-varying delays. We present explicit expressions that allow us to give explicit estimates of the decay rate for various classes of time-varying delays. For positive linear systems, we demonstrate that the best decay rate that our results guarantee can be found via convex optimization. We also derive a set of necessary and sufficient conditions for asymptotic stability of general positive monotone (not necessarily homogeneous) systems with time-delays. As an application of our theoretical results, we discuss delay-independent stability of continuous-time power control algorithms in wireless networks.

The thesis continues by studying the convergence of asynchronous fixed-point iterations involving maximum norm pseudo-contractions. We present a powerful approach for characterizing the rate of convergence of totally asynchronous iterations, where both the update intervals and communication delays may grow unbounded. When specialized to partially asynchronous iterations (where the update intervals and communication delays have a fixed upper bound), or to particular classes of unbounded delays and update intervals, our approach allows to quantify how the degree of asynchronism affects the convergence rate. In addition, we use our results to analyze the impact of asynchrony on the convergence rate of discrete-time power control algorithms in wireless networks.

The thesis finally proposes an asynchronous parallel algorithm that exploits multiple processors to solve regularized stochastic optimization problems with smooth loss functions. The algorithm allows the processors to work at different rates, perform computations independently of each other, and update global decision variables using out-of-date gradients. We characterize the iteration complexity and the convergence rate of the proposed algorithm, and show that these compare favourably with the state of the art. Furthermore, we demonstrate that the impact of asynchrony on the convergence rate of the algorithm is asymptotically negligible, and a near-linear speedup in the number of processors can be expected.

Abstract [sv]

Tidsfördröjningar uppstår ofta i tekniska system: det tar tid för två ämnen attblandas, det tar tid för en vätska att rinna från ett kärl till ett annat, och det tar tid att överföra information mellan delsystem. Dessa tidsfördröjningar lederofta till försämrad systemprestanda och ibland även till instabilitet. Det är därförviktigt att utveckla teori och ingenjörsmetodik som gör det möjligt att bedöma hur tidsfördröjningar påverkar dynamiska system.

I den här avhandlingen presenteras flera bidrag till detta forskningsområde. Fokusligger på att karaktärisera hur tidsfördröjningar påverkar konvergenshastigheten hos olinjära dynamiska system. I kapitel 3 och 4 behandlar vi olinjära system varstillstånd alltid är positiva. Vi visar att stabiliteten av dessa positiva system är oberoende av tidsfördröjningar och karaktäriserar hur konvergenshastigheten hos olinjära positiva system beror på tidsfördröjningarnas storlek. I kapitel 5 betraktar vi iterationer som är kontraktionsavbildningar, och analyserar hur deras konvergens påverkas av begränsade och obegränsade tidsfördröjningar. I avhandlingens sistakapitel föreslår vi en asynkron algoritm för stokastisk optimering vars asymptotiska konvergenshastighet är oberoende av tidsfördröjningar i beräkningar och i kommunikation mellan beräkningselement.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. , xv, 146 p.
TRITA-EE, ISSN 1653-5146 ; 2015:83
Keyword [en]
Optimization, Delay, Positive system, Asynchronous
National Category
Engineering and Technology
Research subject
Electrical Engineering
URN: urn:nbn:se:kth:diva-177651ISBN: 978-91-7595-790-6OAI: diva2:873946
Public defence
2016-01-15, Q2, Osquldas väg 10, KTH, Stockholm, 10:00 (English)

QC 20151204

Available from: 2015-12-04 Created: 2015-11-25 Last updated: 2015-12-04Bibliographically approved

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