FUZZY FREQUENCY RESPONSE FOR UNCERTAIN DYNAMIC

SYSTEMS

Carlos Cesar Teixeira Ferreira and Ginalber Luiz de Oliveira Serra

Federal Institute of Education, Science and Technology of Maranh˜ao (IFMA)

Av. Get´ulio Vargas, 04, Monte Castelo, 65025-001, S˜ao Lu´ıs, MA, Brazil

Keywords:

Takagi-Sugeno fuzzy control, Uncertain dynamic systems, Frequency response analysis.

Abstract:

This paper focuses on the Fuzzy Frequency Response: Deﬁnition and Analysis for Uncertain Dynamic Sys-

tems. In terms of transfer function, the uncertain dynamic system is partitioned into several linear sub-models

and it is organized into Takagi-Sugeno (TS) fuzzy structure. The main contribution of this approach is demon-

strated, from a Theorem, that fuzzy frequency response is a boundary in the magnitude and phase Bode plots.

Low and high frequency analysis of fuzzy dynamic model is obtained by varying the frequency ω from zero

to inﬁnity.

1 INTRODUCTION

The design of control systems is currently driven by

a large number of requirements posed by increas-

ing competition, environmental requirements, energy

and material costs, the demand for robust and fault-

tolerant systems. These considerations introduce ex-

tra needs for effective process control techniques. In

this context, the analysis and synthesis of compen-

sators are completely related to each other. In the

analysis, the characteristics or dynamic behaviour of

the control system are determined. In the design,

the compensators are obtained to attend the desired

characteristics of the control system from certain per-

formance criteria. Generally, these criteria may in-

volve disturbance rejection, steady-state erros, tran-

sient response characteristics and sensitivity to pa-

rameter changes in the plant.

Test input signals is one way to analyse the dy-

namic behaviour of real world system. Many test sig-

nals are available, but a simple and useful signal is the

sinusoidal wave form because the system output with

a sinusoidal wave input is also a sinusoidal wave, but

with a different amplitude and phase for a given fre-

quency. This frequency response analysis describes

how a dynamic system responds to sinusoidal inputs

in a range of frequencies and has been widely used in

academy, industry and considered essential for robust

control theory (Serra and Ferreira, 2010).

The frequency response methods were devel-

oped during the period 1930 − 1940 by Harry

Nyquist (1889 − 1976) (Nyquist, 1932), Hendrik

Bode (1905 − 1982) (Bode, 1940), Nathaniel B.

Nichols (1914− 1997) (James et al., 1947) and many

others. Since, frequency response methods are among

the most useful techniques and available to analyse

and synthesise the compensators. In (Jr, 1973), the

U.S. Navy obtains frequency responses for aircraft

by applying sinusoidal inputs to the autopilots and

measuring the resulting position of the aircraft while

the aircraft is in ﬂight. In (Lascu et al., 2009), four

current controllers for selective harmonic compensa-

tion in parallel Active Power Filters (APFs) have been

compared analytically in terms of frequency response

characteristics and maximum operational frequency.

Most real systems, such as circuit components (in-

ductor, resistor, operational ampliﬁer, etc.) are often

formulated using differential/integral equations with

uncertain parameters (Kolev, 1993). The uncertain

about the systems arises from aging, temperature vari-

ations, etc. These variations do not follow any of

the known probability distributions and are most of-

ten quantiﬁed in terms of boundaries. The classical

methods of frequency response do not explore these

boundaries for uncertain dynamic systems. To over-

come this limitation, this paper proposes the deﬁni-

tion of Fuzzy Frequency Response (FFR) and its ap-

plication for analysis of uncertain dynamic systems.

209

Cesar Teixeira Ferreira C. and Luiz de Oliveira Serra G. (2010).

FUZZY FREQUENCY RESPONSE FOR UNCERTAIN DYNAMIC SYSTEMS.

In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 209-212

DOI: 10.5220/0002815402090212

 SciTePress

2 FORMULATION PROBLEM

This section presents some essentials concepts for the

formulation and development of this paper Fuzzy Fre-

quency Response for Uncertain Dynamic Systems.

2.1 Uncertain Dynamic Systems

In terms of transfer function, the general form of an

uncertain dynamic systems is given by Eq. 1, as de-

picted in Fig.1.

G(s, )

Figure 1: TS fuzzy model.

G(s, ν) =

Y(s, ν)

X(s)

(ν)s

+ b

α−1

(ν)

α−1

+ . . . + b

(ν)s

+ b

(ν)s+ b

(ν)

+ a

β−1

(ν)s

β−1

+ . . . + a

(ν)s+ a

(ν)

(1)

where: X(s) and Y(s, ν) represents the input and the

output of uncertain dynamic systems; a

∗

(ν) and b

∗

(ν)

are the varying parameters; ν(t) is the time varying

scheduling variable; s is the Laplace operator; α and

β are the orders of the numerator and denominator of

the transfer function, respectively (with β ≥ α). The

scheduling variable ν belongs to a compact set ν ∈ V,

with its variation limited by |

ν| ≤ d

max

, with d

max

≥ 0.

This formulation is very efﬁcient and the fuzzy fre-

quency response of (1) can be used for stability anal-

ysis and robust control design.

2.2 Takagi-Sugeno Fuzzy Dynamic

Model

The inference system TS, originally proposed in (Tak-

agi and Sugeno, 1985), presents in the consequent a

dynamic functional expression of the linguistic vari-

ables of the antecedent. The i



[i=1,2,...,l]

-th rule, where

l is the rules numbers, is given by

Rule

(i)

IF ˜x

is F

{1,2,...,p

˜x

AND. . . AND ˜x

is F

{1,2,...,p

˜x

THEN y

= f

(˜x) (2)

where the total number of rules is l = p

˜x

× . . . ×

˜x

. The vector ˜x = [ ˜x

, . . . , ˜x

]

∈ ℜ

contain-

ing the linguistics variables of antecedent, where T

represents the operator for transpose matrix. Each

linguistic variable has its own discourse universe

˜x

, . . . , U

˜x

, partitioned by fuzzy sets representing

its linguistics terms, respectively. In i-th rule, the

variable ˜x

{1,2,...,n}

belongs to the fuzzy set F

{ ˜x

,..., ˜x

}

with a membership degree µ

{ ˜x

,..., ˜x

}

deﬁned by a

membership function µ

{ ˜x

,..., ˜x

}

: ℜ → [0, 1], with

{ ˜x

,..., ˜x

}

∈ {µ

1|{ ˜x

,..., ˜x

}

, µ

2|{ ˜x

,..., ˜x

}

, . . . , µ

p|{ ˜x

,..., ˜x

}

where p

{ ˜x

,..., ˜x

}

is the partitions number of the dis-

course universe associated with the linguistic vari-

able ˜x

, . . . , , ˜x

. The output of the TS fuzzy dynamic

model is a convex combination of the dynamic func-

tional expressions of consequent f

(˜x), without lost of

generality for the bidimensional case, as illustrated in

Fig. 2, given by Eq. 3.

...

| x 1

x 1

x2x 1,( )f1

x2x 1,( )2f

x2x 1,( )

2x 1,( )f4

x2x 1,( )f5

x2x 1,( )f3

Rule

Polytope

1 3 2 p

...

Consequent space

Rule

...

Antecedent space

...

Figure 2: Fuzzy dynamic model: A TS model can be re-

garded as a mapping from the antecedent space to the space

of the consequent parameters one.

y(˜x, γ) =

∑

i=1

(˜x) f

(˜x) (3)

where γ is the scheduling variable of the TS fuzzy dy-

namic model. It can be observed that the TS fuzzy

dynamic system, which represents any uncertain dy-

namic model, may be considered as a class of sys-

tems where γ

(˜x) denotes a decomposition of linguis-

tic variables [ ˜x

, . . . , ˜x

]

∈ ℜ

for a polytopic geo-

metric region in the consequent space from the func-

tional expressions f

(˜x).

3 FUZZY FREQUENCY

RESPONSE (FFR): DEFINITION

This section will present how a TS fuzzy model of

an uncertain dynamic system responds to sinusoidal

inputs, which in this paper is proposed as the deﬁni-

tion of fuzzy frequency response. The response of a

TS fuzzy model to a sinusoidal input of frequency ω

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

210

in both amplitude and phase, is given by the transfer

function evaluated at s = jω

, as illustrated in Fig. 3.

E(s)

i = 1

i W (s)

Y(s)

Figure 3: TS fuzzy transfer function.

For this TS fuzzy model,

Y(s) =

∑

i=1

(s)

E(s) (4)

Consider

∑

i=1

( jω) a complex number for a

given ω, as

∑

i=1

( jω) =



∑

i=1

( jω)



jφ(ω)



∑

i=1

( jω)



∠φ(ω) =



∑

i=1

( jω)



∠arctan

∑

i=1

( jω)

(5)

Then, for the case that the input signal e(t) is si-

nusoidal, that is,

e(t) = Asinω

t (6)

the output signal y

(t), in the steady state, is given

(t) = A



∑

i=1

( jω)



sin[ω

t + φ(ω

)] (7)

As result of the fuzzy frequency response deﬁni-

tion, it is proposed the following theorem:

Theorem 3.1. Fuzzy frequency response is a region

in the frequency domain, deﬁned by the consequent

sub-models and from the operating region of the an-

tecedent space.

Proof. Considering that the parameter ν(t) is uncer-

tain and can be represented by linguistic terms, once

known its discurse universe, as shown in Fig. 4, the

activation degrees, h

(

ν)|

i=1,2,...,l

, are also uncertain,

since it dependes of the dynamic system:

Membership degree

| }{

ν 1,..., ν n

1,...,

nν ν

}{ nν 1,..., ν

}{

1,..., nνν

}{|

}

− Discurse universe

}{ n}{

1,..., n1,..., n

~ ~

1, ... ,

ν 1,..., ννν

ν ν

1 n

,...,ν∗ ν∗

{

1,..., nν ν

U U

...

Fuzzy sets representing linguistics terms

Figure 4: Functional description of the linguistic variables:

linguistic terms, discurse universes and membership de-

grees.

(

ν) = µ

∗

⋆ µ

∗

⋆ . . . ⋆ µ

∗

(8)

where

∗

{1,2,...,n}

∈ U

{1,2,...,n}

, respectively, and ⋆ is

a fuzzy logic operator.

So, the normalized activation degrees

(ν)|

i=1,2,...,l

, are also uncertain, as shown in:

(

ν) =

(

ν)

∑

r=1

(

ν)

(9)

This normalization implies

∑

k=1

(

ν) = 1 (10)

The output of the TS fuzzy model is a weighted

sum of the consequent functional expression, e.g.,

a linear convex combination of the local functions

(

ν), and is given by

ν) =

∑

i=1

(

ν) f

(

ν) (11)

Let F(

ν) a vectorial space of transfer functions

with degree ≤ l and f

(s), f

(s), . . . , f

(s) transfer

functions which belongs to this vectorial space. A

transfer function f(s) ∈ F(

ν) must be a linear convex

combination of the vectors f

(s), f

(s), . . . , f

(s). So

f(s) = γ

(s) + γ

(s) + . . . + γ

(s) (12)

f(s) =

∑

i=1

(

ν) f

(

ν) (13)

The TS fuzzy model must attend the polytope

property. So, the sum of the normalized activation

degree must be equal to 1, as shown in Eq (10). To

satisfy this property, each rule must be singly acti-

vated. This condition is called boundary condition. In

this way, the following results are obtained:

If just the rule 1 is activated, it has (γ

= 1, γ

0, γ

= 0, . . . , γ

= 0). Hence,

FUZZY FREQUENCY RESPONSE FOR UNCERTAIN DYNAMIC SYSTEMS

211

f(s) = 1f

(s) + 0f

(s) + . . . + 0 f

(s) = f

(s)

(14)

From (5), it has

f( jω) =



( jω)



∠ f

( jω) (15)

If just the rule 2 is activated, it has (γ

= 0, γ

1, γ

= 0, . . . , γ

= 0). Hence,

f(s) = 0f

(s) + 1f

(s) + . . . + 0 f

(s) = f

(s)

(16)

From (5), it has

f( jω) =



( jω)



∠ f

( jω) (17)

If just the rule l is activated, it has (γ

= 0, γ

0, γ

= 0, . . . , γ

= 1). Hence,

f(s) = 0f

(s) + 0f

(s) + . . . + 1 f

(s) = f

(s)

(18)

From (5), it has

f( jω) =



( jω)



∠ f

( jω) (19)

Note that



( jω)



∠ f

( jω) and



( jω)



∠ f

( jω) deﬁne a boundary region. Under

such circumstances, it seems plausible that the fuzzy

frequency response for uncertain dynamic systems

converges to a boundary in the frequency response,

as shown in Fig.5.

4 CONCLUSIONS

The Fuzzy Frequency Response: Deﬁnition and Anal-

ysis for Uncertain Dynamic Systems is proposed in

this paper. It was shown that the fuzzy frequency re-

sponse is a region in the frequency domain, deﬁned

by the consequent linear sub-models G

(s), from op-

erating regions of the uncertain dynamic system, ac-

cording to the proposed Theorem 3.1. This formula-

tion is very efﬁcient and can be used for robust stabil-

ity analysis and control design for uncertain dynamic

systems.

Upper limit

Lower limit

Polytope

1 3 2 p

...

Rule

...

Antecedent space

...

Consequent space

MagnitudePhase

Frequency Response

Frequency

Mapping

Rule

| 1

ν 1 | νν

( ) 1, 22

( ) 1, 2

f4 ( ) 1, 2

ν 2 1, ν( )

ν ν

( ) 1, 2

f5 ( ) 1, 2

ν ν

Upper limit

Lower limit

Figure 5: Fuzzy frequency response: mapping from the

consequent space to the region in the frequency domain.

ACKNOWLEDGEMENTS

The authors wish to express their gratitude for

FAPEMA and CAPES by support of this research.

REFERENCES

Bode, H. W. (1940). Feedback ampliﬁer design. Bell Sys-

tems Technical Journal, 19:42.

James, H. M., Nichols, N. B., and Phillips, R. S. (1947).

Theory of servomechanisms. McGraw-Hill, MIT Ra-

diation Laboratory Series. New York.

Jr, A. P. S. (1973). Determination of aircraft response car-

acteristics in approach/landing conﬁguration for mi-

crowave landing system program. In Report FT-61R-

73, Naval Air Test Center. Patuxent River, MD.

Kolev, L. V. (1993). Interval methodsfor circuit analysis.

Singapore: World Scienhﬁc.

Lascu, C., Asiminoaei, L., Boldea, I., and Blaabjerg, F.

(2009). Frequency response analysis of current con-

trollers for selective harmonic compensation in ac-

tive power ﬁlters. IEEE Transactions on Industrial

Eletronics, 56(2).

Nyquist, H. (1932). Regeneration theory. Bell Systems

Technical Journal.

Serra, G. L. O. and Ferreira, C. C. T. (2010). Fuzzy fre-

quncy response: Deﬁnition and analysis for nonlinear

dynamic systems. IEEE International Symposium on

Industrial Electronics.

Takagi, T. and Sugeno, M. (1985). Fuzzy identiﬁcation of

systems and its applications to modeling and control.

IEEE Trans. Syst. Man. Cyber, 15:116–132.

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

212