An exact inversion method for multi-input-single-output decomposable TS fuzzy systems

Abstract

Almost all exact inversion methods provide inverse solutions for only one input variable of fuzzy systems. These methods have certain limitations on the fuzzy system structure such as monotonic rule bases, singleton rule consequents, and invertibility check. These requirements limit the modeling capabilities of the fuzzy systems and also may result in poor application performances. In this study, an exact analytical inversion method is presented for multi-input-single-output decomposable TS fuzzy systems with either singleton or linear consequents. In the proposed method, fuzzy system structures do not need to have monotonic rule bases, singleton rule consequents, or any invertibility conditions. Thus, more flexible fuzzy systems can be used in inverse model based applications. The proposed method provides a simple and systematic way to obtain unique inverse solutions of all input variables simultaneously with respect to any desired system output value. For this purpose, an inversion trajectory approach that guarantees the existence and uniqueness of the inverse solutions is introduced. The inversion trajectory consists of a set of paths defined on the specific edges of universe of discourses of the decomposed fuzzy subsystems. Using this approach, the inverse definition of the overall fuzzy system can easily be derived only by inverting the related decomposed fuzzy subsystems on this inversion trajectory and then combining their inverse definitions. In this way, the inverse definition of the overall fuzzy system is obtained as consisting of analytical solutions of linear and quadratic equations for singleton and linear consequent cases, respectively. Simulation studies are given for the inversion of two and three-input-single-output fuzzy systems, and the exactness and effectiveness of the proposed method are demonstrated.

Keywords

Fuzzy systems decomposability inversion multivariable systems Takagi-Sugeno fuzzy systems

Get full access to this article

View all access options for this article.

References

Rotshtein

A.P.

and Rakytyanska

H.B.

, Fuzzy evidence in identification, forecasting and diagnosis, Springer Publishing Company, 2012.

Varkonyi-Koczy

A.R.

, Peceli

, Dobrowiecki

T.P.

and Kovacshazy

, Iterative fuzzy model inversion, in: Proc. IEEE Conf. Fuzzy Syst., IEEE World Congress on Computational Intelligence, Alaska, USA, 1998, vol. 1, pp. 561–566.

Varkonyi-Koczy

A.R.

, Almos

and Kovacshazy

, Genetic algorithms in fuzzy model inversion, in: Proc. Int. Conf. Fuzzy Systems, FUZZ-IEEE ’99, Seoul, Korea, 1999, vol. 3, pp. 1421–1426.

Ulu

, Exact analytical inverse mapping of decomposable TS fuzzy systems with singleton and linear consequents, Applied Soft Computing 23 (2014), 202–214.

Ulu

, Güzelkaya

and Eksin

, Exact analytical inversion of TSK fuzzy systems with singleton and linear consequents, Int J Approx Reason 55(6) (2014), 1357–1368.

Ulu

, Exact analytical inversion of interval type-2 TSK fuzzy logic systems with closed form inference methods, Applied Soft Computing 37 (2015), 60–70.

Kien

C.V

, Anh

H.P.H.

and Son

N.N.

, Inverse–adaptive multilayer T–S fuzzy controller for uncertain nonlinear system optimized by differential evolution algorithm. Soft Comput 24 (2020), 14073–14089.

and Shin

Y.C.

, A Fuzzy inverse model construction method for general monotonic multi-input-single-output (MISO) systems, IEEE Transactions on Fuzzy Systems 16(5) (2008), 1216–1231.

Filev

D.P.

, Inversion of fuzzy models—practical issues, in: Proc. 7th IEEE Internat. Conf. on Fuzzy Systems (FUZZ’IEEE 98), Anchorage, USA, 1998, pp. 1658–1663.

10.

Jarrett

, Hüyük

and Schaar

M.V.D.

, Inverse Decision Modeling: Learning Interpretable Representations of Behavior, in: Proc. The 38th International Conference on Machine Learning, PMLR 139 (2021), 4755–4771.

11.

Singh

D.J.

, Verma

N.K.

, Ghosh

A.K.

and Malagaudanavar

, An application of interval type-2 fuzzy model based control system for generic aircraft, Applied Soft Computing 121 (2022), 108721.

12.

Tikk

, Kóczy

L.T.

and Gedeon

T.D.

, A survey on universal approximation and its limits in Soft Computing techniques, International Journal of Approximate Reasoning 33(2) (2003), 185–202.

13.

Tikk

, Baranyi

and Patton

R.J.

, Approximation properties of TP model forms and its consequences to TPDC design framework, Asian Journal of Control 9(3) (2007), 221–231.

14.

, Jiang

and Shi

, Nonlinear actuator fault estimation observer: an inverse system approach via a T-S fuzzy model, International Journal of Applied Mathematics and Computer Science 22(1) (2012), 183–196.

15.

Yang

, Wang

and Wang

, Fuzzy modeling and prediction of combustion layers^′ temperature for power plant, Journal of Intelligent & Fuzzy Systems 41(6) (2021), 7081–7091.

16.

Ying

, Sufficient conditions on uniform approximation of multivariate functions by general Takagi-Sugeno fuzzy systems with linear rule consequent, IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans 28(4) (1998), 515–520.

17.

Ying

, Interval Type-2 Takagi–Sugeno fuzzy systems with linear rule consequent are universal approximators, in: Proc. Annual Meeting of the North American Fuzzy Information Processing Society, NAFIPS’09, 2009, pp. 1–5.

18.

Abonyi

, Andersen

, Nagy

and Szeifert

, Inverse fuzzy-process-model based adaptive control, Mathematics and Computers in Simulation 51(1–2) (1999), 119–132.

19.

Kluska

, Analytical methods in fuzzy modeling and control, Springer Publishing Company, Rzeszow, 2012.

20.

Castro

J.L.

, Fuzzy logic controllers are universal approximators, IEEE Trans SMC 25 (1995), 629–635.

21.

Youssef

K.H.

, Yousef

H.A.

, Sebakhy

O.A.

and Wahba

M.A.

, Adaptive fuzzy APSO based inverse tracking-controller with an application to DC motors, Expert Systems with Applications 36(2) (2009), 3454–3458.

22.

Siminski

, Ridders algorithm in approximate inversion of fuzzy model with parametrized consequences, Expert Systems with Applications 51 (2016), 276–285.

23.

Chen

K.S.

and Yu

C.M.

, Fuzzy Decision-making Model for Process Quality Improvement of Machine Tool Industry Chain, Journal of Intelligent & Fuzzy Systems 42(3) (2022), 1547–1558.

24.

Wang

L.-X.

and Mendel

J.M.

, Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Transactions on Neural Networks 3(5) (1992), 807–814.

25.

Fischer

and Isermann

, Robust hybrid control based on inverse fuzzy process models, in: Proc. 5th IEEE Int. Conf. Fuzzy Systems (FUZZ’IEEE 96), New Orleans, USA, 1996, pp. 1210–1216.

26.

Hussein

M.T.

, Gafer

A.S.

and Fadhel

E.Z.

, Robot manipulator inverse kinematics using adaptive neuro-fuzzy inference system, Journal of Engineering Science and Technology 15(3) (2020), 1984–1998.

27.

Asad

M.U.

, et al., Disturbance observer supported fuzzy model based controller with application to bilateral tele-operation systems, Journal of Intelligent & Fuzzy Systems Pre-press (2022), 1–9.

28.

Baranyi

, Bavelaar

, Koczy

L.T.

and Titli

, Inverse rule base of various fuzzy interpolation techniques, in: Proc. 7th IFSA World Congress, Prague, Czech Republic, 1997, pp. 121–126.

29.

Baranyi

, Bavelaar

, Babuska

, Koczy

L.T.

, Titli

and Verbruggen

H.B.

, A method to invert a linguistic fuzzy model, Internat J Systems Sci 29(7) (1998), 711–721.

30.

Babuska

, Fuzzy modeling for control, Kluwer Academic Publishers, Boston, 1998.

31.

Babuska

, Sousa

and Verbruggen

H.B.

, Model based design of fuzzy control systems, in: Proc. third European congress on intelligent techniques and Soft Computing, EUFIT ’95, Aachen, Germany, 1995, pp. 837–841.

32.

Boukezzoula

, Galichet

and Foulloy

, Nonlinear internal model control: application of inverse model based fuzzy control, IEEE Transactions on Fuzzy Systems 11(6) (2003), 814–829.

33.

Galichet

, Boukezzoula

and Foulloy

, Explicit analytical formulation and exact inversion of decomposable fuzzy systems with singleton consequents, Fuzzy Sets and Systems 146(3) (2004), 421–436.

34.

Kumbasar

, Eksin

, Guzelkaya

and Yesil

, Adaptive fuzzy model based inverse controller design using BB-BC optimization algorithm, Expert Systems with Applications 38(10) (2011), 12356–12364.

35.

Kumbasar

, Eksin

, Guzelkaya

and Yesil

, Exact inversion of decomposable interval type-2 fuzzy logic systems, International Journal of Approximate Reasoning 54(2) (2013), 253–272.

36.

Kumbasar

, Eksin

, Guzelkaya

and Yesil

, Interval type-2 fuzzy inverse controller design in nonlinear IMC structure, Engineering Applications of Artificial Intelligence 24(6) (2011), 996–1005.

37.

Rauma

and Kurki

, Fuzzy logic applications in diagnosing mechatronic systems, in: Fuzzy Systems Design: Social and Engineering Applications, Springer Publishing Company, 1998, pp. 309–322.

38.

Shiwen

, Dianwei

, Xiaoyu

, Guang

and Jianjun

, Design of Networked Control Systems Using the Fuzzy Inverse Model, in Proc. 37th Chinese Control Conf.,Wuhan, 2018, pp. 6303–6308.

39.

Zeng

X.J.

and Singh

M.G.

, Decomposition property of fuzzy systems and its applications, IEEE Trans Fuzzy Systems 4(2) (1996), 149–165.

40.

Rayouf

, Ghorbel

and Braiek

N.B.

, A new Hammerstein model control strategy: feedback stabilization and stability analysis, Int J Dynam Control 7 (2019), 1453–1461.