Lyapunov-Global-Lanczos Algorithm for Model Order Reduction & Adaptive PI Controller of Large Scale Electrical Systems

Document Type : Research Note


Universit´e de Tunis El Manar, Institut Sup´erieur d’Informatique et de Gestion de Kairouan, LR-11-ES20 Laboratoire Analyse, Conception et Commande des Systemes, BP 37, LE BELVEDERE 1002, Tunis, Tunisie


Mathematical modeling of complex electrical systems, has led us to linear mathematical models of higher order. Consequently, it is difficult to analyse and to design a control strategy of these systems. The order reduction is an important and effective tool to facilitate the handling and designing of a control strategy. In this paper we present, firstly, a reduction method which is based on the Krylov subspace and Lyapunov  techniques, that we call Lyapunov-Global-Lanczos. This method minimizes the H1 norm error, absolute error and preserves the stability of the reduced system. It also provides a better reduced system of order 1, with closer behaviour to the original system. This first order system is used to design PI (Proportional-Integral) controller. Secondly, we implement an adaptive digital PI controller in a microcontroller. It calculates the PI parameters in real time, referring to the error between the desired and measured outputs and the initial values of PI controller, that were determined from the first order system. Two simulation examples and a real time experimentation are presented to show the effectiveness of the proposed algorithms.


Main Subjects

1. Sinani, K., Gugercin, S., and Beattie, C. A structurepreserving
model reduction algorithm for dynamical
systems with nonlinear frequency dependence", IFACPapersOnLine,
49(9), pp. 56{61 (2016).
2. Fan, H.Y., Weng, P.C.Y., and Chu, E.K.W. Numerical
solution to generalized Lyapunov/stein and rational
riccati equations in stochastic control", Numerical
Algorithms, 71(2), pp. 245{272 (2016).
3. Li, T., Weng, C.Y., Chu, E.K.W., and Lin, W.W.
Solving large-scale stein and Lyapunov equations by
doubling", Numerical Algorithms, 63(4), pp. 727{752
4. Wolf, T. and Panzer, K.H. The ADI iteration for
Lyapunov equations implicitly performs h2 pseudooptimal
model order reduction", International Journal
of Control, 89(3), pp. 481{493 (2016).
5. Frangos, M. and Jaimoukha, I. Adaptive rational
interpolation: Arnoldi and Lanczos-like equations",
European Journal of Control, 14, pp. 342{354 (2008).
6. Oh, D.C. and Jeung, E.T. Model reduction for
the descriptor systems by linear matrix inequalities",
International Journal of Control, Automation and
Systems, 8(4), pp. 875{881 (2010).
7. Zhang, Y. and Wong, N. Compact model order
reduction of weakly nonlinear systems by associated
transform", International Journal of Circuit Theory
and Applications, 29(6), pp. 1{18 (2015).
8. Du , I.P., Vuillemin, P., Vassal, C.P., Briat, C., and
Seren, C. Model reduction for norm approximation:
M. Kouki et al./Scientia Iranica, Transactions D: Computer Science & ... 25 (2018) 1616{1628 1627
An application to large-scale time-delay systems", In
Delays and Networked Control Systems, 6, pp. 37{55
9. Antoulas, A.C., Approximation of Large-Scale Dynamical
Systems, R.C. Smith, Ed., 1st Ed. pp. 1{508,
Advances in Design and Control (2005).
10. Kouki, M., Abbes, M., and Mami, A. Svd-aora
method for dynamic linear time invariant model order
reduction", 8th Vienna International Conference on
Mathematical Modelling, Vienna, Austria, pp. 695{696
11. Nasiri, S.H. and Maghfoori, F.M. Chebyshev rational
functions approximation for model order reduction
using harmony search", Scientia Iranica, 20(3), pp.
771{777 (2013).
12. Yuhang, D. and Wu, K.L. Direct mesh-based model
order reduction of PEEC model for quasi-static circuit
problems", IEEE Transactions on Microwave Theory
and Techniques, 64, pp. 2409{2422 (2016).
13. Aridhi, H., Zaki, H.M., and Tahar, S. Enhancing
model order reduction for nonlinear analog circuit
simulation", IEEE Transactions on Very Large Scale
Integration (VLSI) Systems, 24(3), pp. 1036{1049
14. Xiao, Z.H. and Jiang, Y.L. Model order reduction of
mimo bilinear systems by multi-order arnoldi method",
Systems & Control Letters, 94, pp. 1{10 (2016).
15. Jorn, Z. Lei, W., Paul, U., and Rob, R. A Lanczos
model-order reduction technique to eciently simulate
electromagnetic wave propagation in dispersive media",
Journal of Computational Physics, 315, pp. 348{
362 (2016).
16. Gugercin, S. An iterative SVD-Krylov based method
for model reduction of large-scale dynamical systems",
Linear Algebra and Its Applications, 428, pp. 1964{
1986 (2008).
17. Malekshahi, E. and Mohammadi, S.M.A. The model
order reduction using LS, RLS and MV estimation
methods", International Journal of Control, Automation
and Systems, 12(3), pp. 572{581 (2014).
18. Chu, C., Lai, M., and Feng, W. Mode-order reductions
for mimo systems using global Krylov subspace
methods", Mathematics and Computers in Simulation,
79, pp. 1153{1164 (2008).
19. Lee, H., Chu, C., and Feng, W. An adaptive-order
rational arnoldi method for model-order reductions of
linear time-invariant systems", Linear Algebra and Its
Applications, 415, pp. 235{261 (2006).
20. Kouki, M., Abbes, M., and Mami, A. Arnoldi model
reduction for switched linear systems", Int. J. Operational
Research, 27(1/2), pp. 85-112 (2016).
21. Barkouki, H., Bentbib, A.H., and Jbilou, K. An
adaptive rational block Lanczos-type algorithm for
model reduction of large scale dynamical systems",
Journal of Scienti c Computing, 67(1), pp. 221{236
22. Silvia, G., Enyinda, O., Lothar, R., and Giuseppe, R.
On the Lanczos and Golub{Kahan reduction methods
applied to discrete ill-posed problems", Numerical
Linear Algebra with Applications, 23(1), pp. 187{204
23. Gallivan, K., Grimme, E., and Dooren, P. A rational
Lanczos algorithm for model reduction", Numerical
Algorithms, 12, pp. 33{63, 1996.
24. Bonin, T., Fabender, H., Soppa, A., and Zaeh,
M. A fully adaptive rational global arnoldi method
for the model-order reduction of second-order mimo
systems with proportional damping", Mathematics and
Computers in Simulation, 122, pp. 1{19 (2016).
25. Abidi, O. and Jbilou, K. Balanced truncation-rational
Krylov methods for model reduction in large scale dynamical
systems," Computational and Applied Mathematics,
35, pp. 1{16 (2016).
26. Meiling, W.J., Chu, C., Yu, Q., and Kuh, S.E. On
projection-based algorithms for model-order reduction
of interconnects", IEEE Transactions on Circuits and
Systems I: Fundamental Theory and Applications,
49(11), pp. 1563{1585 (2002).
27. Kouki, K., Abbes, M., and Mami, A. Arnoldi model
reduction for switched linear systems", The 5th International
Conference on Modeling, Simulation and
Applied Optimization, Hammamet, Tunisia, pp. 1{6
28. Malwatkara, G., Sonawaneb, S., and Waghmarec, L.
Tuning PID controllers for higher-order oscillatory
systems with improved performance", Automatica, 48,
pp. 347-353 (2009).
29. Alexander, S.A. and Thathan, M. Design and development
of digital control strategy for solar photovoltaic
inverter to improve power quality", Journal of Control
Engineering and Applied Information, 16(4), pp. 20-29
30. Rebai, A., Guesmi, K., and Hemici, B. Design of an
optimized fractional order fuzzy PID controller for a
piezoelectric actuator", Journal of Control Engineering
and Applied Information, 17(3), pp. 41-49 (2015).
31. Isakssona, A.J. and Graebe, S.F. Analytical PID
parameter expressions for higher order systems", Automatica,
35, pp. 1121-1130 (1999).
32. Malwatkar, G.M., Khandekar, A.A., and Nikam, S.D.
Pid controllers for higher order systems based on
maximum sensitivity function", 3th International Conference
on Electronics Computer Technology (ICECT),
Kanyakumari, India, pp. 259{263 (2011).
1628 M. Kouki et al./Scientia Iranica, Transactions D: Computer Science & ... 25 (2018) 1616{1628
33. Franklin, G.F., Powell, J.D., and Workman, M.L.,
Digital Control of Dynamic Systems, Addison and
Wesley (1990).
34. Kovacic, Z. and Bogdan, S., Fuzzy Controller Design:
Theory and Applications, 19, CRC press (2005).
35. Gharib, M. and Moavenian, M. Synthesis of robust
PID controller for controlling a single input single
output system using quantitative feedback theory technique",
Scientia Iranica. Transaction B, Mechanical
Engineering, 21(6), pp. 1861{1869 (2014).
36. Hasankola, M.D., Ehsaniseresht, A., Moghaddam, M.,
and Saba, A. Analysis, modeling, manufacturing
and control of an elastic actuator for rehabilitation
robots", Scientia Iranica. Transaction B, Mechanical
Engineering, 22(5), pp. 1855{1865 (2015).
37. Massimo, B. and David, C. Arduino" (2016). [Online].