SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models


  • Mahtab Uddin Institute of Natural Sciences, United International University, Dhaka-1212, Bangladesh
  • M. Monir Uddin Department of Mathematics and Physics, North South University, Dhaka-1229, Bangladesh
  • M. A. Hakim Khan Department of Mathematics, Bangladesh University of Engineering & Technology, Dhaka-1000, Bangladesh
  • M. Tanzim Hossain Department of Electrical and Computer Engineering, North South University, Dhaka-1229, Bangladesh



Singular Value Decomposition, Krylov Subspace, Alternative Direction Implicit, Riccati Equation, ℌ2 -norm, Optimal Feedback Stabilization


We propose an efficient sparsity-preserving reduced-order modelling approach for index-1 descriptor systems extracted from large-scale power system models through two-sided projection techniques. The projectors are configured by utilizing Gramian based singular value decomposition (SVD) and Krylov subspace-based reduced-order modelling. The left projector is attained from the observability Gramian of the system by the low-rank alternating direction implicit (LR-ADI) technique and the right projector is attained by the iterative rational Krylov algorithm (IRKA). The classical LR-ADI technique is not suitable for solving Riccati equations and it demands high computation time for convergence. Besides, in most of the cases, reduced-order models achieved by the basic IRKA are not stable and the Riccati equations connected to them have no finite solution. Moreover, the conventional LR-ADI and IRKA approach not preserves the sparse form of the index-1 descriptor systems, which is an essential requirement for feasible simulations. To overcome those drawbacks, the fitting of LR-ADI and IRKA based projectors from left and right sides, respectively, desired reduced-order systems attained. So that, finite solution of low-rank Riccati equations, and corresponding feedback matrix can be executed. Using the mechanism of inverse projection, the Riccati-based optimal feedback matrix can be computed to stabilize the unstable power system models. The proposed approach will maintain minimized ℌ2 -norm of the error system for reduced-order models of the target models.


Hossain, M.S. and Uddin, M.M., 2019. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic equations. Numerical Algebra, Control & Optimization, 9(2), p.173.

Benner, P., Saak, J. and Uddin, M.M., 2016, December. Reduced-order modeling of index-1 vibrational systems using interpolatory projections. In 2016 19th International Conference on Computer and Information Technology (ICCIT) (pp.134-138). IEEE.

Uddin, M.M., 2015. Computational methods for model reduction of large-scale sparse structured descriptor systems (Doctoral dissertation, Otto-von Guericke Universita ̈t Magdeburg).

Benner, P., Saak, J. and Uddin, M.M., 2016. Structure preserving model order reduction of large sparse second-order index-1 systems and application to a mechatronics model. Mathematical and Computer Modelling of Dynamical Systems, 22(6), pp.509-523.

Rahman, M., Uddin, M.M., Andallah, L.S. and Uddin, M., 2020. Interpolatory Projection Techniques for H_2 Optimal Structure-Preserving Model Order Reduction of Second-Order Systems. Advances in Science, Technology and Engineering Systems Journal, 5(4), pp.715-723.

Chu, E.K.W., 2011, August. Solving large-scale algebraic Riccati equations by doubling. In Talk presented at the Seventeenth Conference of the International Linear Algebra Society, Braunschweig, Germany (Vol. 22).

Chen, W. and Qiu, L., 2015. Linear quadratic optimal control of continuous-time LTI systems with random input gains. IEEE Transactions on Automatic Control, 61(7), pp.2008-2013.

Abou-Kandil, H., Freiling, G., Ionescu, V. and Jank, G., 2012. Matrix Riccati equations in control and systems theory. Birkhäuser.

Bänsch, E., Benner, P., Saak, J. and Weichelt, H.K., 2015. Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flows. SIAM Journal on Scientific Computing, 37(2), pp.A832-A858.

Uddin, M., Khan, M.A.H. and Uddin, M.M., 2019, September. Riccati based optimal control for linear quadratic regulator problems. In 2019 5th International Conference on Advances in Electrical Engineering (ICAEE) (pp.290-295). IEEE.

Uddin, M., Khan, M.A.H. and Uddin, M.M., 2019, December. Efficient computation of Riccati-based optimal control for power system models. In 2019 22nd International Conference on Computer and Information Technology (ICCIT) (pp.260-265). IEEE.

Uddin, M., Uddin, M.M., Khan, M.A.H. and Rahman, M.M., 2021. Interpolatory projection technique for Riccati-based feedback stabilization of index-1 descriptor systems. IOP Conference Series: Materials Science and Engineering, 1070(1), pp.12-22.

Li, J.R., 2000. Model reduction of large linear systems via low rank system Gramians (Doctoral dissertation, Massachusetts Institute of Technology).

Hossain, M.S., Omar, S.G., Tahsin, A. and Khan, E.H., 2017, September. Efficient system reduction modeling of periodic control systems with application to circuit problems. In 2017 4th International Conference on Advances in Electrical Engineering (ICAEE) (pp. 259-264). IEEE.

Gugercin, S., Antoulas, A.C. and Beattie, C., 2008. H2 model reduction for large-scale linear dynamical systems. SIAM journal on matrix analysis and applications, 30(2), pp.609-638.

Wyatt, S.A., 2012. Issues in interpolatory model reduction: Inexact solves, second-order systems and DAEs (Doctoral dissertation, Virginia Tech).

Gugercin, S., 2008. An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems. Linear Algebra and its Applications, 428(8-9), pp.1964-1986.

Khatibi, M., Zargarzadeh, H. and Barzegaran, M., 2016, September. Power system dynamic model reduction by means of an iterative SVD-Krylov model reduction method. In 2016 IEEE Power & Energy Society Innovative Smart Grid Technologies Conference (ISGT) (pp.1-6). IEEE.

Li, S., Trevelyan, J., Wu, Z., Lian, H., Wang, D. and Zhang, W., 2019. An adaptive SVD–Krylov reduced order model for surrogate based structural shape optimization through isogeometric boundary element method. Computer Methods in Applied Mechanics and Engineering, 349, pp.312-338.

Lu, A. and Wachspress, E.L., 1991. Solution of Lyapunov equations by alternating direction implicit iteration. Computers & Mathematics with Applications, 21(9), pp.43-58.

Benner, P., Li, J.R. and Penzl, T., 2008. Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems. Numerical Linear Algebra with Applications, 15(9), pp.755-777.

Benner, P., Køhler, M. and Saak, J., 2011. Sparse-dense Sylvester equations in H_2 model order reduction.

Freitas, F.D. and Costa, A.S., 1999. Computationally efficient optimal control methods applied to power systems. IEEE transactions on power systems, 14(3), pp.1036-1045.

Leandro, R.B., e Silva, A.S., Decker, I.C. and Agostini, M.N., 2015. Identification of the oscillation modes of a large power system using ambient data. Journal of Control, Automation and Electrical Systems, 26(4), pp.441-453.



20-08-2021 — Updated on 25-09-2021


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How to Cite

Uddin, M., Uddin, M. M. ., Khan, M. A. H. ., & Hossain, M. T. . (2021). SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models. Journal of Engineering Advancements, 2(03), 125–131. (Original work published August 20, 2021)



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