Document Type : Research Note

**Authors**

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

**Abstract**

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.

**Keywords**

**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

(2013).

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

(2016).

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

(2015).

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

(2016).

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 Scientic Computing, 67(1), pp. 221{236

(2016).

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

(2016).

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

(2013).

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

(2014).

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].

Available: https://www.arduino.cc

Transactions on Computer Science & Engineering and Electrical Engineering (D)

May and June 2018Pages 1616-1628