A new method for the rational approximation of stable/unstable single-input, single-output (SISO)/multiple-input, multiple-output (MIMO) fractional-order systems is proposed. The objective of the proposed algorithm is to obtain an integer-order approximant of an SISO/MIMO fractional-order system. The developed method utilizes the concept of matching of an appropriate number of approximate generalized time moments and approximate generalized Markov parameters of squared magnitude function of fractional-order system to those of approximant. The proposed method preserves the stability/instability property and minimum phase/non-minimum phase characteristics of fractional-order system in the approximant. The method also incorporates a provision for matching the steady-state response of the approximant to that of fractional-order system. Numerical examples consider three cases of approximation, while fractional-order system has the characteristics of (a) stable non-minimum phase SISO, (b) stable non-minimum phase MIMO, and (c) unstable SISO which are presented to validate the efficiency of the proposed method.
ChenCPTsayYT (1976) A squared magnitude continued fraction expansion for stable reduced models. International Journal of Systems Science7(6): 625–634.
2.
ChenTCChangCYHanKW (1980) Model reduction using the stability-eq33 method and the continued-fraction method. International Journal of Control32(1): 81–94.
3.
DamodaranSSunil KumarTKSudheerAP (2019) Model-matching fractional-order controller design using AGTM/AGMP matching technique for SISO/MIMO linear systems. IEEE Access7: 41715–41728.
4.
DhabaleASDiveRAwareMV, et al. (2015) A new method for getting rational approximation for fractional order differintegrals. Asian Journal of Control17(6): 2143–2152.
5.
DjouambiACharefABesançonA (2007) Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function. International Journal of Applied Mathematics and Computer Science17(4): 455–462.
6.
DuBWeiYLiangS, et al. (2017) Rational approximation of fractional order systems by vector fitting method. International Journal of Control, Automation and Systems15(1): 186–195.
7.
GenMChengR (2000) Genetic Algorithms and Engineering Optimization. Hoboken, NJ: John Wiley & Sons.
8.
HeQYPuYFYuB, et al. (2020) Arbitrary-order fractance approximation circuits with high order-stability characteristic and wider approximation frequency bandwidth. IEEE/CAA Journal of Automatica Sinica7(5): 1425–1436.
9.
HuttonMFriedlandB (1975) Routh approximations for reducing order of linear, time-invariant systems. IEEE Transactions on Automatic Control20(3): 329–337.
10.
HwangCChenMY (1988) Stable linear system reduction via a multipoint squared-magnitude continued-fraction expansion. IEE Proceedings D: Control Theory and Applications135(6): 441–444.
11.
KhanraM (2014) Rational approximation of fractional order systems and design of fractional order controller. PhD Thesis, IIT Kharagpur, Kharagpur, India.
12.
KhanraMPalJBiswasK (2013a) Rational approximation and analog realization of fractional order transfer function with multiple fractional powered terms. Asian Journal of Control15(3): 723–735.
13.
KhanraMPalJBiswasK (2013b) Reduced order approximation of MIMO fractional order systems. IEEE Journal on Emerging and Selected Topics in Circuits and Systems3(3): 451–458.
14.
KhanraMPalJBiswasK (2013c) Use of squared magnitude function in approximation and hardware implementation of SISO fractional order system. Journal of the Franklin Institute350(7): 1753–1767.
15.
KoseogluM (2022) Time response optimal rational approximation: Improvement of time responses of MSBL based approximate fractional order derivative operators by using gradient descent optimization. Engineering Science and Technology, an International Journal35: 101167.
16.
LiangYLuJ (2017) Direct low order rational approximations for fractional order systems in narrow frequency band: A fix-pole method. Journal of Circuits, Systems and Computers26(4): 1750065.
17.
LucasTN (2004) Shift-and-scale model reduction: An alternative stability-preserving approach. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering218(3): 237–243.
18.
LucasTNDavidsonAM (1983) Frequency-domain reduction of linear systems using Schwarz approximation. International Journal of Control37(5): 1167–1178.
19.
MahataSSahaSKKarR, et al. (2019) Approximation of fractional-order low-pass filter. IET Signal Processing13(1): 112–124.
20.
MaioneG (2002) Laguerre approximation of fractional systems. Electronics Letters38(20): 1234–1236.
21.
MaioneG (2008) Continued fractions approximation of the impulse response of fractional-order dynamic systems. IET Control Theory and Applications2(7): 564–572.
22.
MalekHChenY (2016) Fractional order extremum seeking control: Performance and stability analysis. IEEE/ASME Transactions on Mechatronics21(3): 1620–1628.
23.
MishraSKUpadhyayDKGuptaM (2021) Approximation of fractional-order Butterworth filter using pole-placement in W-plane. IEEE Transactions on Circuits and Systems II: Express Briefs68(10): 3229–3233.
24.
MonjeCAChenYVinagreBM, et al. (2010) Fractional-Order Systems and Controls: Fundamentals and Applications. London: Springer Science+Business Media.
25.
NasiriHHaeriM (2014) How BIBO stability of LTI fractional-order time delayed systems relates to their approximated integer-order counterparts. IET Control Theory and Applications8(8): 598–605.
26.
OldhamKSpanierJ (1974) The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Amsterdam: Elsevier.
27.
OustaloupALevronFMathieuB, et al. (2000) Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications47(1): 25–39.
28.
PakzadMAPakzadSNekouiMA (2015) Exact method for the stability analysis of time delayed linear-time invariant fractional-order systems. IET Control Theory and Applications9(16): 2357–2368.
29.
PalJ (1986) An algorithmic method for the simplification of linear dynamic scalar systems. International Journal of Control43(1): 257–269.
30.
PalJSarveshBGhoshMK (1995) A new method for model order reduction. IETE Journal of Research41(5–6): 305–311.
31.
PodlubnyI (1998) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Amsterdam: Elsevier.
32.
RachidMMaamarBSaidD (2010) Multivariable fractional system approximation with initial conditions using integral state space representation. Computers & Mathematics with Applications59(5): 1842–1851.
33.
RommesJMartinsN (2006) Efficient computation of multivariable transfer function dominant poles using subspace acceleration. IEEE Transactions on Power Systems21(4): 1471–1483.
34.
StanisławskiRLatawiecKJ (2013) Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems. Bulletin of the Polish Academy of Sciences: Technical Sciences61(2): 363–370.
35.
SunHHAbdelwahabAOnaralB (1984) Linear approximation of transfer function with a pole of fractional power. IEEE Transactions on Automatic Control29(5): 441–444.
36.
TavazoeiMS (2016) Criteria for response monotonicity preserving in approximation of fractional order systems. IEEE/CAA Journal of Automatica Sinica3(4): 422–429.
37.
TolbaMFSaidLAMadianAH (2019) FPGA implementation of the fractional order integrator/differentiator: Two approaches and applications. IEEE Transactions on Circuits and Systems I: Regular Papers66(4): 1484–1495.
38.
VenkatramanSYenGG (2005) A generic framework for constrained optimization using genetic algorithms. IEEE Transactions on Evolutionary Computation9(4): 424–435.
39.
XueDZhaoCChenY (2006) A modified approximation method of fractional order system. In: Proceedings of the 2006 international conference on mechatronics and automation, Luoyang, China, 25–28 June, pp. 1043–1048. New York: IEEE.
40.
YüceADenizFNTanN, et al. (2020) Revisiting four approximation methods for fractional order transfer function implementations: Stability preservation, time and frequency response matching analyses. Annual Reviews in Control49: 239–257.
