Final Report Summary - A-DAP (Approximate Solutions of the Determinantal Assignment Problem and distance problems)
The Research Fellow has dealt with a central problem in Control Theory the Determinantal Assignment Problem (DAP) which has emerged as the abstract problem formulation of pole, zero assignment of linear systems. DAP has been previously formulated using tools from exterior algebra and classical algebraic geometry as an intersection problem of varieties (a linear variety and the Grassmann variety) and it is an inherently non-linear problem (solution of linear and nonlinear equations). This work has provided solvability conditions for the problem, based on results guaranteeing real intersections of varieties, and has led to the development of techniques for computing such solutions (a "blow up" type methodology). A major difficulty of the previous results has been their inability to handle model uncertainty and produce approximate solutions when either we have no intersections, or when the intersection points are not real. The current project has overcome such obstacles by extending intersection of varieties to distance problems between varieties and then reducing the overall problem to an optimization. These developments allow the transformation of the DAP framework from a synthesis methodology (exact problems) to a design approach that can handle model uncertainty and capable to develop approximate solutions.
The overall project aim has been the development of the exact algebraic framework to a framework allowing the computation of approximate solutions based on optimization. We have addressed the sensitivity of the Global Linearisation framework (previously defined) by using Homotopy based methodologies and by embedding the overall intersection problem into the framework of constrained optimization. The model uncertainty issues have been addressed by casting exact intersections to distance problems by computing the distance of: (i) a point from the Grassmann variety; (ii) a linear variety from the Grassman variety; (iii) parameterised families of linear varieties from the Grassmann variety; (iv) relating the latter distance problems with properties of the stability domain. The research introduces a new framework for transforming algebraic nature results to a setup of “approximate algebraic computations” by extending intersection theory to distance problems between varieties.
Research Work and Outputs: The research fellow has completed the 100% of the project deliverables in terms of research results either as scientific articles, or software in the two years of the project. He has already produced 5 published Journal papers, 5 Conference Proceedings papers and presentations, 7 Research Reports are close to be finalised for submission and there were 2 presentations in the organised Workshop for the project on Complexity Science. The work done well exceeds the expectations of the project and major breakthroughs have been achieved. Furthermore, the Fellow has developed research in areas linked to the theme of the project, but outside the original plan (network theory) using the developed techniques. In summary, there is significant work done in all tasks of the Work-package as summarised below and described in detail in the attachment, where also the list of publications is given. As it has been mentioned in the two periodic reports the tasks in the Work-packages has been completed and the major contributions have been in the following areas:
-Approximate Solutions of Exterior Equations and Distance Problems: Major results were produced in:
-Approximate Decomposability Problem (ADP): This was solved for the Grassmann variety G(2,n) [J.1]. The same problem was solved for the case G(3,6) [J.5] and generalised for the case G(3,n) [S&CR.3]. An alternative approach to the study of this problem was developed through the notion of partial decomposability [S&CR.2]. The above results provide tools enabling the study of solutions to the Approximate Intersection Problem which is now studied for the general G(m,n) case where we aim to produce a numerical method for both approximate and exact DAP. These results tackle the Approximate Intersection Problem tackled in [J.1] [J.5] [S&CR.3]. New results developed in [J.2] contribute to the solution of the exact and the approximate DAP. These results have applications Control theory to areas such as multidimensional singular value decomposition with applications to areas to image processing, and multidimensional data reduction.
-Properties of Grassmann Varieties and Variety Distance Problem: New results [J.2] characterising decomposability through the Grassmann and the newly defined Hodge-Grassmann matrices have been developed [J.2]. The distance problem between varieties was tackled for the case of G(2,4) in [2], G(2,n) in [J.1] and also in [J.5] and [S&CR.3]. A special optimization approach is utilised through the minimisation of the gap metric between the linear and the Grassmann variety a problem. The calculation of the distance between a linear and the Grassmann variety have been developed in [J.2] in terms of a nonlinear eigenvalue-eigenvector problem. These results provide a new matrix representation for exterior equations which are thus central to the study of DAP. A new Cauchy-Swarch type inequality has been defined that is important to the study of the approximation [S&CR.1]. A special optimization approach has been developed based on the minimisation of the gap metric between the linear and the Grassmann variety a problem.
-Global Linearisation Framework and Homotopy Theory: Major results were produced in:
• Reduced Sensitivity Solutions of DAP using Blow-Up Methodology: We have developed a predictor-corrector numerical method based on Homotopy continuation and subsequently a Quasi-Newton method by tracing the frequency placement curves so that solutions derived are far from degenerate compensators thus have reduced sensitivity [CP.2] [CP.5] [S&CR.6]. This methodology leads to solutions of the DAP of reduced sensitivity and have established convergence of the relevant algorithms.
-Parametrisation of Denerate Compensators: Central to the development of the Global Linearization is the parameterisation of the degenerate compensators. We have developed such a parameterisation [S&CR.4] and this provides essential inputs for the optimization search for insensitive solutions. These results open new directions for searching for approximate solutions.
-Extensions of DAP framework to Systems and Control Problems: Major results were produced in:
The work in this area has been extended from the original plan by examining extensions of the framework to:
-Redesign of Passive Networks: A new line of research within the DAP framework is the redesign of electrical RLC networks. Aspects of the redesign have been formulated as a DAP type problem where the methodologies developed here are highly relevant. The development of the fundamentals of system theory based on network models is a major new research dimension, which is required for the development of solutions to the network redesign problem [CP.1] [CP.3] [CP.4] [S&CR.5].
-Parameterization of Feasible Decentralised Schemes: The parameterisation of degenerate compensators was extended to decentralised control and this has led to a characterisation of feasible structures [S&CR.7]. This opens up the new area of design of decentralization schemes using the DAP framework techniques.
-Development of Additional and New lines of Research: During the Fellowship additional research directions were developed by the Fellow due to interactions with the members of the Host Centre which form future research directions and involve:
-Development of the fundamentals of network type implicit models: This provides the basis for the development of methodology for re-engineering networks using the DAP techniques which is on-going [CP.1] [CP.3] [CP.4] [S&CR.5].
• Approximate Algebraic Computations and GCD: The computation of the distance of a point from the Grassmann variety has motivated a new research line for defining the approximate Greatest Common Divisor (GCD) of a set of polynomials by computing distance from the GCD-variety of a certain degree.
• Hierarchical Modelling and Control of the Credit Risk Problem: As a consequence of the participation in the development of the ICT proposal a novel formulation of the problem as a multilayer, decision making problem [WP.2] has been formulated which has been presented in the Complexity Science Workshop at City.
Impact: The research has been in Mathematical Control Theory and Applied Mathematics. As such its main impact is in the area of Control Engineering by the transformation of theoretical approaches to techniques that can provide engineering solutions, as well as in the area of computing where approximate solutions to algebraic problems are sought. The newly introduced research direction on network redesign provides some fundamental principles to the study of systems re-engineering, a problem with great significance in both engineering, management and finance.
Dissemination: The results (5 Journal papers, 5 Conference proceedings, 7 Internal Reports and 2 presentations) of the project were presented in international conferences such as: International Symposium on Communications, Control, and Signal Processing, ISCCSP 14, Athens May, 2014; 22nd Mediterranean Conference MED’14, June 2014, Palermo, Italy (2 papers); 21st International Symposium on Mathematical Theory of Networks and Systems, MTNS 2014, July 2014, University of Groningen, The Netherlands; 19th IFAC World Congress 2014, August 2014, Cape Town, South Africa. He has given two presentations in the Complexity Science Workshop at City in June 2015. Publications were distributed to the wider research community.
(Please see attachments for detail)
Training: The Fellow was integrated extremely well within the Systems and Control Research Centre and has participated in a number of significant for his career training activities. The fellow was already quite senior and very experienced and the focus of the training was on the leadership skills and interdisciplinary project management. The host institution has learned from him as much as he learned from the host and its networks. He attended the University Seminars and Workshops for HORIZON 2020, on Impact of Research and the Series of Seminars of the Centre on Complexity Science, which he has contributed to organise and deliver two lectures. He has been involved in the setting up and running of a Study Group on Control Theory for research students and has contributed significantly in the supervision of 3 PhD research students, which has exposed him to different lines of research. Training has been provided by the preparation and submission of an ICT-15-2014 HORIZON 2020 proposal on “Prediction of the Evolution of Credit Debt Portfolios”.
It's worth pointing out that since this project has been basic research in the area of Mathematics and Control Theory, there hasn't been any patents or trademarks resulted from this project.
The overall project aim has been the development of the exact algebraic framework to a framework allowing the computation of approximate solutions based on optimization. We have addressed the sensitivity of the Global Linearisation framework (previously defined) by using Homotopy based methodologies and by embedding the overall intersection problem into the framework of constrained optimization. The model uncertainty issues have been addressed by casting exact intersections to distance problems by computing the distance of: (i) a point from the Grassmann variety; (ii) a linear variety from the Grassman variety; (iii) parameterised families of linear varieties from the Grassmann variety; (iv) relating the latter distance problems with properties of the stability domain. The research introduces a new framework for transforming algebraic nature results to a setup of “approximate algebraic computations” by extending intersection theory to distance problems between varieties.
Research Work and Outputs: The research fellow has completed the 100% of the project deliverables in terms of research results either as scientific articles, or software in the two years of the project. He has already produced 5 published Journal papers, 5 Conference Proceedings papers and presentations, 7 Research Reports are close to be finalised for submission and there were 2 presentations in the organised Workshop for the project on Complexity Science. The work done well exceeds the expectations of the project and major breakthroughs have been achieved. Furthermore, the Fellow has developed research in areas linked to the theme of the project, but outside the original plan (network theory) using the developed techniques. In summary, there is significant work done in all tasks of the Work-package as summarised below and described in detail in the attachment, where also the list of publications is given. As it has been mentioned in the two periodic reports the tasks in the Work-packages has been completed and the major contributions have been in the following areas:
-Approximate Solutions of Exterior Equations and Distance Problems: Major results were produced in:
-Approximate Decomposability Problem (ADP): This was solved for the Grassmann variety G(2,n) [J.1]. The same problem was solved for the case G(3,6) [J.5] and generalised for the case G(3,n) [S&CR.3]. An alternative approach to the study of this problem was developed through the notion of partial decomposability [S&CR.2]. The above results provide tools enabling the study of solutions to the Approximate Intersection Problem which is now studied for the general G(m,n) case where we aim to produce a numerical method for both approximate and exact DAP. These results tackle the Approximate Intersection Problem tackled in [J.1] [J.5] [S&CR.3]. New results developed in [J.2] contribute to the solution of the exact and the approximate DAP. These results have applications Control theory to areas such as multidimensional singular value decomposition with applications to areas to image processing, and multidimensional data reduction.
-Properties of Grassmann Varieties and Variety Distance Problem: New results [J.2] characterising decomposability through the Grassmann and the newly defined Hodge-Grassmann matrices have been developed [J.2]. The distance problem between varieties was tackled for the case of G(2,4) in [2], G(2,n) in [J.1] and also in [J.5] and [S&CR.3]. A special optimization approach is utilised through the minimisation of the gap metric between the linear and the Grassmann variety a problem. The calculation of the distance between a linear and the Grassmann variety have been developed in [J.2] in terms of a nonlinear eigenvalue-eigenvector problem. These results provide a new matrix representation for exterior equations which are thus central to the study of DAP. A new Cauchy-Swarch type inequality has been defined that is important to the study of the approximation [S&CR.1]. A special optimization approach has been developed based on the minimisation of the gap metric between the linear and the Grassmann variety a problem.
-Global Linearisation Framework and Homotopy Theory: Major results were produced in:
• Reduced Sensitivity Solutions of DAP using Blow-Up Methodology: We have developed a predictor-corrector numerical method based on Homotopy continuation and subsequently a Quasi-Newton method by tracing the frequency placement curves so that solutions derived are far from degenerate compensators thus have reduced sensitivity [CP.2] [CP.5] [S&CR.6]. This methodology leads to solutions of the DAP of reduced sensitivity and have established convergence of the relevant algorithms.
-Parametrisation of Denerate Compensators: Central to the development of the Global Linearization is the parameterisation of the degenerate compensators. We have developed such a parameterisation [S&CR.4] and this provides essential inputs for the optimization search for insensitive solutions. These results open new directions for searching for approximate solutions.
-Extensions of DAP framework to Systems and Control Problems: Major results were produced in:
The work in this area has been extended from the original plan by examining extensions of the framework to:
-Redesign of Passive Networks: A new line of research within the DAP framework is the redesign of electrical RLC networks. Aspects of the redesign have been formulated as a DAP type problem where the methodologies developed here are highly relevant. The development of the fundamentals of system theory based on network models is a major new research dimension, which is required for the development of solutions to the network redesign problem [CP.1] [CP.3] [CP.4] [S&CR.5].
-Parameterization of Feasible Decentralised Schemes: The parameterisation of degenerate compensators was extended to decentralised control and this has led to a characterisation of feasible structures [S&CR.7]. This opens up the new area of design of decentralization schemes using the DAP framework techniques.
-Development of Additional and New lines of Research: During the Fellowship additional research directions were developed by the Fellow due to interactions with the members of the Host Centre which form future research directions and involve:
-Development of the fundamentals of network type implicit models: This provides the basis for the development of methodology for re-engineering networks using the DAP techniques which is on-going [CP.1] [CP.3] [CP.4] [S&CR.5].
• Approximate Algebraic Computations and GCD: The computation of the distance of a point from the Grassmann variety has motivated a new research line for defining the approximate Greatest Common Divisor (GCD) of a set of polynomials by computing distance from the GCD-variety of a certain degree.
• Hierarchical Modelling and Control of the Credit Risk Problem: As a consequence of the participation in the development of the ICT proposal a novel formulation of the problem as a multilayer, decision making problem [WP.2] has been formulated which has been presented in the Complexity Science Workshop at City.
Impact: The research has been in Mathematical Control Theory and Applied Mathematics. As such its main impact is in the area of Control Engineering by the transformation of theoretical approaches to techniques that can provide engineering solutions, as well as in the area of computing where approximate solutions to algebraic problems are sought. The newly introduced research direction on network redesign provides some fundamental principles to the study of systems re-engineering, a problem with great significance in both engineering, management and finance.
Dissemination: The results (5 Journal papers, 5 Conference proceedings, 7 Internal Reports and 2 presentations) of the project were presented in international conferences such as: International Symposium on Communications, Control, and Signal Processing, ISCCSP 14, Athens May, 2014; 22nd Mediterranean Conference MED’14, June 2014, Palermo, Italy (2 papers); 21st International Symposium on Mathematical Theory of Networks and Systems, MTNS 2014, July 2014, University of Groningen, The Netherlands; 19th IFAC World Congress 2014, August 2014, Cape Town, South Africa. He has given two presentations in the Complexity Science Workshop at City in June 2015. Publications were distributed to the wider research community.
(Please see attachments for detail)
Training: The Fellow was integrated extremely well within the Systems and Control Research Centre and has participated in a number of significant for his career training activities. The fellow was already quite senior and very experienced and the focus of the training was on the leadership skills and interdisciplinary project management. The host institution has learned from him as much as he learned from the host and its networks. He attended the University Seminars and Workshops for HORIZON 2020, on Impact of Research and the Series of Seminars of the Centre on Complexity Science, which he has contributed to organise and deliver two lectures. He has been involved in the setting up and running of a Study Group on Control Theory for research students and has contributed significantly in the supervision of 3 PhD research students, which has exposed him to different lines of research. Training has been provided by the preparation and submission of an ICT-15-2014 HORIZON 2020 proposal on “Prediction of the Evolution of Credit Debt Portfolios”.
It's worth pointing out that since this project has been basic research in the area of Mathematics and Control Theory, there hasn't been any patents or trademarks resulted from this project.