Dr Shahkar Ahmad Navhi

2010-2012
Current Position: Asst. Prof, Islamic University of Science & Technology, Srinagar, J&K.

Dynamical systems are used to model diverse physical and artificial processes. Realistic and accurate description of many such systems tends to be of large dimensions, and their simulation is not possible without expenditure of considerable amounts of computational resources and time. Approximation is thus crucial for cost-effective simulation. Model order reduction (MOR) techniques help realize this objective in a computationally efficient way. They result in a dimensionally reduced system with input-output mapping similar to the large system being approximated. MOR methods for large Linear time invariant systems are well researched and have been extensively covered in literature. Many such techniques have been extended to nonlinear systems, but research in nonlinear model reduction is in its infancy. Though strongly motivated and having a broad scope of applications, nonlinear model reduction also poses a different set of problems to be solved. This thesis represents an effort in developing better reduced order modelling techniques for nonlinear dynamical systems. The focus in this thesis is on two popular techniques for nonlinear model reduction, Proper Orthogonal Decomposition (POD) and the Trajectory piece-wise linear (TPWL) approximation. It aims at improving the applicability and computational performance of these strategies and the approximation qualities and robustness of the reduced-order models obtained using them. Although TPWL is a popular and widely applied tool for nonlinear model reduction, recent works have shown that TPWL models can under-perform for some applications. Questions have also been raised about the heuristic choices to be made during the course of the TPWL procedure. The endeavour in this work is to study the efficacy of the TPWL approximation, and this involves investigating the implicit assumptions, the dependence on heuristics and the applicability of the TPWL method. Another point of focus is the method of basis extraction in POD, which constitutes a significant part of the total computational cost of the model reduction procedure. In TPWL, in the offline stage, linearisations on the nonlinear system trajectory are done and stored. A superposition strategy interacts with this database to approximate the nonlinear system dynamics in the online phase. In this thesis, new methods are introduced to improve both these stages of the TPWL procedure. For linearisation point selection, new strategies that seek to address the important concerns regarding previously reported methods are proposed and validated. For the online phase, a new strategy is proposed for improving the superposition of linear systems in TPWL. A separate issue investigated in TPWL is its applicability to a wider class of nonlinear systems, the issues arising out of which are analysed, and in the process, a new method for nonlinear model reduction is introduced. Finally, a new alternative to extract basis functions for POD is illustrated and it is shown that the need for computationally heavy, a-priori simulations of the nonlinear system can be circumvented. Chapter-$1$ consists of the introduction to the thesis, it explains the motivation and builds the background for the subsequent chapters. Chapter-$2$ contains a review of important works in the field of MOR, with emphasis on nonlinear dynamical systems. Chapter-$3$ deals with the issue of linearisation point selection in TPWL. Chapter-$4$ investigates the limitations of conventional superposition schemes in TPWL and in Chapter-$5$ the applicability of TPWL to more general nonlinear systems is analysed. Chapter-$6$ deals with POD and proposes the strategy to reduce the computational costs of the POD procedure. Finally, in the last chapter a summary of the thesis is presented and future directions of research are identified.

Publications

Journals

• Shahkar Ahmad Nahvi, Mashuq-un-Nabi, S. Janardhanan, "Piece-wise Quasi-linear Approximation for Nonlinear Model Reduction", IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Volume 32, Issue 12, pp. 2009-2013, Dec. 2013.


• S.A. Nahvi, M. Nabi, S. Janardhanan, "Nonlinearity-aware sub-model combination in trajectory based methods for nonlinear Mor", Mathematics and Computers in Simulation, Elsevier, Volume 94, pp. 127144, August 2013.


• Shahkar Ahmad Nahvi, Mashuq-un-Nabi, S. Janardhanan, "Trajectory Piece-wise quasi-linear approximation of large nonlinear dynamic systems", International Journal of Modeling, Identi_cation and Control, Inderscience, Vol. 19, No. 4, pp. 369-377, Aug. 2013.


• Shahkar Ahmad Nahvi, Mashuq-un-Nabi, S. Janardhanan, "AFAS - Adaptive Fast Approximate Simulation for nonlinear model reduction", International Journal of Modeling, Identi_cation and Control, Inderscience, Vol. 19, No. 2, pp. 113-124, May 2013.

Conferences

• Mohammad Abid Bazaz, Shahkar Ahmad Nahvi, Mashuq-un- Nabi, S Janardhanan, "Adaptive Parameter Space Sampling in Matrix Interpolatory pMOR", IEEE-RDCAPE, Noida, U.P. , India, March 2015.


• Shahkar Ahmad Nahvi, Mohammad Abid Bazaz, Mashuq-un- Nabi, S Janardhanan, "Fast Simulation of Nonlinear Dynamical Systems for Application in Reduced Order Modelling ", Proc. of the European control conference (ECC-2014), Strasbourgh, France, July 2014.


• Shahkar Ahmad Nahvi, Mohammad Abid Bazaz, Mashuq-un- Nabi, S Janardhanan, "Approximate Snapshot-ensemble Generation for Basis Extraction in Proper Orthogonal Decomposition", Proc. Of the Third International Conference on Advances in control and optimization of dynamical systems (ACODS2014), IIT Kanpur, Kanpur, India, March 2014.


• S. A. Nahvi, M. Nabi and S. Janardhanan, "Adaptive sampling of nonlinear system trajectory for Model Order Reduction", Proc. International Conference on Modelling, Identi_cation and Control, Wuhan, China, June 2012, pp. 1249-1255.


• S. A. Nahvi, M. Nabi and S. Janardhanan, "A Quasi-linearisation approach to trajectory based methods for nonlinear MOR", Proc. International Conference on Modelling, Identi_cation and Control, Wuhan, China, June 2012, pp. 217-222.


• Shahkar Ahmad Nahvi, Mashuq-un-Nabi, S. Janardhanan, "Trajectory based methods for nonlinear MOR: Review and Perspectives", Proc. of the 2012 IEEE International Conference on Signal Processing, Computing and Control, Shimla, India, March 2012, pp. 1-6.


• S.A. Nahvi, Mashuq-un-Nabi, "Optimal control of a heat conduction problem using its low order approximation", Proceedings of IEEE International Conference on Power, Signals, Controls and Computation, Thrissur, India, Jan. 2012, pp.1-6.


• P. Guha, S. A. Nahvi & M. Nabi, "Optimal Control of Temperature Profile for a Distributed Parameter Heat Conduction Problem with Generic Nontrivial geometry using Krylov Projection based Model Order Reduction", Proc. of 3rd International Conference on Control and Optimization with Industrial Applications, Ankara, Turkey, August 2011.