Fuzzy dynamical system approach to the observer design of uncertain systems

Abstract

A fuzzy dynamical system approach is applied to the observer design of uncertain systems. In the fuzzy dynamical system, the uncertainty lies within a fuzzy set. The desirable system performance is twofold: one deterministic and one fuzzy. While the deterministic performance ensures the bottom line, the fuzzy performance enhances the cost consideration. Based on the output, an observer is proposed in order to estimate the state. The observer design is deterministic and is not IF-THEN rules-based. To improve the performance, a constrained optimization problem associated with the design is formulated. The problem can be solved analytically and its solution exists and is unique. The closed forms of the solution and the resulting minimum cost are presented.

Keywords

fuzzy dynamical systems fuzzy set theory observer optimal design

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References

Kumar

P. R.

Varaiya

Stochastic systems: estimation, identification, and adaptive control (Prentice-Hall, Englewood Cliffs, New Jersey 1986)

Kalman

R. E.

‘A new approach to linear filtering and prediction problems.’

Trans. ASME D, J. Basic Engng 82 (1960): 35–35

Kalman

R. E.

‘Contributions to the theory of optimal control.’

Bol. Soci. Maste. Mex. 5 (1960): 102–119

Kalman

R. E.

‘Randomness reexamined.’

Model., Identif. Control 15 (1994): 141–151

Kalman

R. E.

‘Probability in the real world as a system attribute.’

CWI Quart. 9 (1996): 181–204

Pincus

Kalman

R. E.

‘Not all (possibly) “random” sequences are created equal.’

Proc. Natl. Acad. Sci. USA (1967): 3513–3518

Zadeh

L. A.

‘The birth and evolution of fuzzy logic.’

Int. J. Gen. Syst. 17 (1990): 95–105

Zadeh

L. A.

‘Fuzzy sets.’

Inf. Control 8 (1965): 338–353

Yager

R. R.

Zadeh

L. A.

An introduction to fuzzy logic applications in intelligent systems (Kluwer, Norwell, Massachusetts 1992)

10.

Hanss

Applied fuzzy arithmetic: an introduction with engineering applications (Springer-Verlag, Berlin, Germany 2005)

11.

Bede

Rudas

I. J.

Fodor

‘Friction model by using fuzzy differential equations.’ In Foundations of fuzzy logic and soft computing, 2007, pp. 23–32 (Springer-Verlag, Berlin, Germany).

12.

Tanaka

Ikeda

Wang

H. O.

‘Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs.’

IEEE Trans. Fuzzy Syst. 6 (1998): 250–265

13.

Klir

G. J.

Yuan

Fuzzy sets and fuzzy logic: theory and applications (Prentice-Hall, Englewood Cliffs, New Jersey 1995)

14.

Corless

‘State and input estimation for a class of uncertain systems.’

Automatica 34 (1998): 757–764

15.

Hui

Zak

S. H.

‘Observer design for systems with unknown inputs.’

Int. J. Appl. Math. Comput. Sci. 15 (2005): 431–446

16.

Soong

T. T.

Active structural control: theory and practice (Longman, UK 1990)

17.

18.

Corless

‘Control of uncertain nonlinear systems.’

J. Dyn. Syst., Meas. Control 115 (1993): 362–372

19.

Leitmann

‘On one approach to the control of uncertain systems.’

J. Dyn. Syst., Meast, Control 115 (1993): 373–380

20.

Horn

R. A.

Johnson

C. R.

Matrix analysis (Cambridge University Press, Cambridge, UK. 1985)

21.

Chen

Y. H.

Leitmann

‘Robustness of uncertain systems in the absence of matching assumptions.’

Int. J. Control 45 (1987): 1527–1542

22.

Hale

J. K.

Ordinary differential equations (Wiley-Interscience, New York 1969)

23.

Chen

Y. H.

‘Performance analysis of controlled uncertain systems.’

Dyn. Control 6 (1996): 131–142

24.

Cannon

R. H.

Dynamics of physical systems (McGraw-Hill, New York 1967)

25.

de Silva

C. W.

Sensors and actuators: control system instrumentation (CRC Press, Boca Raton, Florida 2007)

26.

Winder

Analog and digital filter design, second edition 2002 (Elsevier Science, Woburn, Massachusetts).

27.

Ayyub

B. M.

McCuen

R. H.

Numerical methods for engineers (Prentice-Hall, New Jersey 1996)