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Divide and Conquer: Distributed Optimization and Robustness Analysis
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

As control of large-scale complex systems has become more and more prevalent within control, so has the need for analyzing such controlled systems. This is particularly due to the fact that many of the control design approaches tend to neglect intricacies in such systems, e.g., uncertainties, time delays, nonlinearities, so as to simplify the design procedure.

Robustness analysis techniques allow us to assess the effect of such neglected intricacies on performance and stability. Performing robustness analysis commonly requires solving an optimization problem. However, the number of variables of this optimization problem, and hence the computational time, scales badly with the dimension of the system. This limits our ability to analyze large-scale complex systems in a centralized manner. In addition, certain structural constraints, such as privacy requirements or geographical separation, can prevent us from even forming the analysis problem in a centralized manner.

In this thesis, we address these issues by exploiting structures that are common in large-scale systems and/or their corresponding analysis problems. This enables us to reduce the computational cost of solving these problems both in a centralized and distributed manner. In order to facilitate distributed solutions, we employ or design tailored distributed optimization techniques. Particularly, we propose three distributed optimization algorithms for solving the analysis problem, which provide superior convergence and/or computational properties over existing algorithms. Furthermore, these algorithms can also be used for solving general loosely coupled optimization problems that appear in a variety of fields ranging from control, estimation and communication systems to supply chain management and economics.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. , 330 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1676
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:liu:diva-117503DOI: 10.3384/diss.diva-117503ISBN: 978-91-7519-050-1 (print)OAI: oai:DiVA.org:liu-117503DiVA: diva2:808690
Public defence
2015-06-11, Visionen, Hus B, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2015-05-04 Created: 2015-04-29 Last updated: 2017-01-17Bibliographically approved
List of papers
1. Distributed solutions for loosely coupled feasibility problems using proximal splitting methods
Open this publication in new window or tab >>Distributed solutions for loosely coupled feasibility problems using proximal splitting methods
2015 (English)In: Optimization Methods and Software, ISSN 1055-6788, E-ISSN 1029-4937, Vol. 30, no 1, 128-161 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we consider convex feasibility problems (CFPs) where the underlying sets are loosely coupled, and we propose several algorithms to solve such problems in a distributed manner. These algorithms are obtained by applying proximal splitting methods to convex minimization reformulations of CFPs. We also put forth distributed convergence tests which enable us to establish feasibility or infeasibility of the problem distributedly, and we provide convergence rate results. Under the assumption that the problem is feasible and boundedly linearly regular, these convergence results are given in terms of the distance of the iterates to the feasible set, which are similar to those of classical projection methods. In case the feasibility problem is infeasible, we provide convergence rate results that concern the convergence of certain error bounds.

Place, publisher, year, edition, pages
Taylor & Francis, 2015
Keyword
feasible/infeasible convex feasibility problems, proximal splitting, distributed solution, flow feasibility problem
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-110124 (URN)10.1080/10556788.2014.902056 (DOI)000345371800006 ()
Available from: 2014-09-03 Created: 2014-09-03 Last updated: 2017-12-05
2. A Distributed Primal-dual Interior-point Method for Loosely Coupled Problems Using ADMM
Open this publication in new window or tab >>A Distributed Primal-dual Interior-point Method for Loosely Coupled Problems Using ADMM
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we propose an efficient distributed algorithm for solving loosely coupled convex optimization problems. The algorithm is based on a primal-dual interior-point method in which we use the alternating direction method of multipliers (ADMM) to compute the primal-dual directions at each iteration of the method. This enables us to join the exceptional convergence properties of primal-dual interior-point methods with the remarkable parallelizability of ADMM. The resulting algorithm has superior computational properties with respect to ADMM directly applied to our problem. The amount of computations that needs to be conducted by each computing agent is far less. In particular, the updates for all variables can be expressed in closed form, irrespective of the type of optimization problem. The most expensive computational burden of the algorithm occur in the updates of the primal variables and can be precomputed in each iteration of the interior-point method. We verify and compare our method to ADMM in numerical experiments.

National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-117495 (URN)
Available from: 2015-04-29 Created: 2015-04-29 Last updated: 2015-05-19
3. Distributed primal-€dual interior-point methods for solving tree-structured coupled convex problems using message-passing
Open this publication in new window or tab >>Distributed primal-€dual interior-point methods for solving tree-structured coupled convex problems using message-passing
2016 (English)In: Optimization Methods and Software, ISSN 1055-6788, E-ISSN 1029-4937, 1-35 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we propose a distributed algorithm for solving coupled problems with chordal sparsity or an inherent tree structure which relies on primal–dual interior-point methods. We achieve this by distributing the computations at each iteration, using message-passing. In comparison to existing distributed algorithms for solving such problems, this algorithm requires far fewer iterations to converge to a solution with high accuracy. Furthermore, it is possible to compute an upper-bound for the number of required iterations which, unlike existing methods, only depends on the coupling structure in the problem. We illustrate the performance of our proposed method using a set of numerical examples.

Place, publisher, year, edition, pages
Taylor & Francis, 2016
National Category
Control Engineering Mathematics
Identifiers
urn:nbn:se:liu:diva-133995 (URN)10.1080/10556788.2016.1213839 (DOI)000399480200001 ()
Note

The previous status of this article was Manuscript and the working title was Distributed Primal-dual Interior-point Methods for Solving Loosely Coupled Problems Using Message Passing.

Funding agencies: Swedish Department of Education within the ELLIIT project

Available from: 2017-01-17 Created: 2017-01-17 Last updated: 2017-05-18Bibliographically approved
4. Robust stability analysis of sparsely interconnected uncertain systems
Open this publication in new window or tab >>Robust stability analysis of sparsely interconnected uncertain systems
2014 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 59, no 8, 2151-2156 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we consider robust stability analysis of large-scale sparsely interconnected uncertain systems. By modeling the interconnections among the subsystems with integral quadratic constraints, we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities. We also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness. The sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques, and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem.

Place, publisher, year, edition, pages
IEEE, 2014
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-110125 (URN)10.1109/TAC.2014.2305934 (DOI)000342923700012 ()
Available from: 2014-09-03 Created: 2014-09-03 Last updated: 2017-12-05Bibliographically approved
5. Distributed Robustness Analysis of Interconnected Uncertain Systems Using Chordal Decomposition
Open this publication in new window or tab >>Distributed Robustness Analysis of Interconnected Uncertain Systems Using Chordal Decomposition
2014 (English)In: Proceedings of the 19th IFAC World Congress, 2014 / [ed] Edward Boje and Xiaohua Xia, International Federation of Automatic Control , 2014, 2594-2599 p.Conference paper, Published paper (Refereed)
Abstract [en]

Large-scale interconnected uncertain systems commonly have large state and uncertainty dimensions. Aside from the heavy computational cost of solving centralized robust stability analysis techniques, privacy requirements in the network can also introduce further issues. In this paper, we utilize IQC analysis for analyzing large-scale interconnected uncertain systems and we evade these issues by describing a decomposition scheme that is based on the interconnection structure of the system. This scheme is based on the so-called chordal decomposition and does not add any conservativeness to the analysis approach. The decomposed problem can be solved using distributed computational algorithms without the need for a centralized computational unit. We further discuss the merits of the proposed analysis approach using a numerical experiment.

Place, publisher, year, edition, pages
International Federation of Automatic Control, 2014
Series
World Congress, ISSN 1474-6670 ; Volume 19, Part 1
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-110127 (URN)10.3182/20140824-6-ZA-1003.01649 (DOI)978-3-902823-62-5 (ISBN)
Conference
19th IFAC world congress, The International Federation of Automatic Control, Cape Town, South Africa, August 24-29, 2014
Available from: 2014-09-03 Created: 2014-09-03 Last updated: 2015-05-19Bibliographically approved
6. Distributed Semidefinite Programming with Application to Large-scale System Analysis
Open this publication in new window or tab >>Distributed Semidefinite Programming with Application to Large-scale System Analysis
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Distributed algorithms for solving coupled semidefinite programs (SDPs) commonly require manyiterations to converge. They also put high computational demand on the computational agents. In thispaper we show that in case the coupled problem has an inherent tree structure, it is possible to devisean efficient distributed algorithm for solving such problems. This algorithm can potentially enjoy thesame efficiency as centralized solvers that exploit sparsity. The proposed algorithm relies on predictorcorrectorprimal-dual interior-point methods, where we use a message-passing algorithm to compute thesearch directions distributedly. Message-passing here is closely related to dynamic programming overtrees. This allows us to compute the exact search directions in a finite number of steps. Furthermorethis number can be computed a priori and only depends on the coupling structure of the problem. Weuse the proposed algorithm for analyzing robustness of large-scale uncertain systems distributedly. Wetest the performance of this algorithm using numerical examples.

National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-117496 (URN)
Available from: 2015-04-29 Created: 2015-04-29 Last updated: 2015-05-19
7. Robust finite-frequency H2 analysis of uncertain systems with application to flight comfort analysis
Open this publication in new window or tab >>Robust finite-frequency H2 analysis of uncertain systems with application to flight comfort analysis
Show others...
2013 (English)In: Control Engineering Practice, ISSN 0967-0661, E-ISSN 1873-6939, Vol. 21, no 6, 887-897 p.Article in journal (Refereed) Published
Abstract [en]

In many applications, design or analysis is performed over a finite-frequency range of interest. The importance of the H2 norm highlights the necessity of computing this norm accordingly. This paper provides different methods for computing upper bounds of the robust finite-frequency H2 norm for systems with structured uncertainties. An application of the robust finite-frequency H2 norm for a comfort analysis problem of an aero-elastic model of an aircraft is also presented.

Place, publisher, year, edition, pages
Elsevier, 2013
Keyword
Robust H-2 norm, Uncertain systems, Robust control, Flight comfort analysis
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-94316 (URN)10.1016/j.conengprac.2013.02.003 (DOI)000318327900011 ()
Available from: 2013-06-24 Created: 2013-06-24 Last updated: 2017-12-06

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Khoshfetrat Pakazad, Sina

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