on the stability of interval type-2 tsk fuzzy logic control systems

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS 1

On the Stability of Interval Type-2 TSKFuzzy Logic Control Systems

Mohammad Biglarbegian, Student Member, IEEE, William W. Melek, Senior Member, IEEE, andJerry M. Mendel, Life Fellow, IEEE

Abstract—Type-2 fuzzy logic systems have recently been uti-lized in many control processes due to their ability to modeluncertainties. This paper proposes a novel inference mechanismfor an interval type-2 Takagi–Sugeno–Kang fuzzy logic controlsystem (IT2 TSK FLCS) when antecedents are type-2 fuzzy setsand consequents are crisp numbers (A2-C0). The proposed infer-ence mechanism has a closed form which makes it more feasibleto analyze the stability of this FLCS. This paper focuses oncontrol applications for the following cases: 1) Both plant andcontroller use A2-C0 TSK models, and 2) the plant uses type-1Takagi–Sugeno (TS) and the controller uses IT2 TS models. Inboth cases, sufficient stability conditions for the stability of theclosed-loop system are derived. Furthermore, novel linear-matrix-inequality-based algorithms are developed for satisfying the sta-bility conditions. Numerical analyses are included which validatethe effectiveness of the new inference methods. Case studies revealthat an IT2 TS FLCS using the proposed inference engine clearlyoutperforms its type-1 TSK counterpart. Moreover, due to the sim-ple nature of the proposed inference engine, it is easy to implementin real-time control systems. The methods presented in this paperlay the mathematical foundations for analyzing the stability andfacilitating the design of stabilizing controllers of IT2 TSK FLCSsand IT2 TS FLCSs with significantly improved performance overtype-1 approaches.

Index Terms—Adaptive control, modular and reconfigurablerobots, robot manipulators, Takagi–Sugeno–Kang (TSK), type-2fuzzy logic control (T2 FLC).

I. INTRODUCTION

EVEN THOUGH fuzzy logic was originally developed tomodel linguistic terms, interpretations, and human per-

ceptions, most implementations of fuzzy logic systems (FLSs)have been in control applications [1], [2]. Up to now, fuzzylogic control (FLC) has been implemented with great successin many real-world applications and has also been shown insome cases to outperform traditional control systems [1], [3].One of the most well known model structures of fuzzy systemsused for control applications is Takagi–Sugeno–Kang (TSK)[4], [5]. The method presented in [4] requires the design ofconsequent parameters of a general fuzzy TSK model based ona least squares method. Later, Sugeno and Kang [5] presented

Manuscript received December 30, 2008; revised April 28, 2009. This paperwas recommended by Associate Editor C.-T. Lin.

M. Biglarbegian and W. W. Melek are with the Department of Mechanicaland Mechatronics Engineering, University of Waterloo, Waterloo, ON N2L3G1, Canada (e-mail: [email protected]; [email protected]).

J. Mendel is with the Ming Hsieh Department of Electrical Engineering,University of Southern California, Los Angeles, CA 90089 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSMCB.2009.2029986

the identification structure of fuzzy systems (premise and con-sequents) that has made it possible to design and analyze fuzzycontrol systems rigorously. Hence, in this paper, we adopt theTSK model structure.

Recently, there has been a growing interest in usingtype-2 FLSs (T2 FLSs) in many applications as well as incontrol processes due to their ability to model uncertainties[3], [6]–[19]. The major operation in an interval T2 FLS (IT2FLS) is type reduction which reduces the T2 FLS output to atype-1 fuzzy set. The most commonly adopted IT2 FLSs utilizeKarnik–Mendel (KM) algorithms for type reduction [20], [21].KM algorithms compute the left and right end points neededto characterize interval type-2 fuzzy sets (IT2 FSs). In orderto bypass the computational effort of KM algorithms, Wu andMendel [22] developed uncertainty bounds for IT2 FSs toapproximate type reduction while achieving similar results. Inthis paper, we modify the Wu–Mendel uncertainty bounds (WMUBs) to design stable interval type-2 TSK fuzzy logic controlsystems (IT2 TSK FLCS).

With the development of T2 FLSs and their ability to handleuncertainty, utilizing type-2 FLCSs (T2 FLCSs) has attracted alot of interest in recent years. Although, to date, only IT2 FLSshave been applied for control applications, promising resultshave been reported, e.g., Wu and Tan [9] designed an IT2 FLCSfor a coupled-tank liquid-level system and showed that whenthe level of uncertainty increases, the IT2 FLCS outperformsits type-1 counterpart. In addition, Hagras [15] applied IT2 FLCto mobile robot navigation in dynamic unstructured indoor andoutdoor environments. All the IT2 FLCSs implemented in [15]used much smaller rule bases than their type-1 counterparts, andit was concluded that IT2 FLCSs provide a faster computationplatform as well as enhanced performance results over theirtype-1 counterpart. Recently, Lam and Seneviratne [23] inves-tigated stability analysis of IT2 Takagi–Sugeno (TS) FLCSs.Their approach requires several assumptions to be made aboutthe membership functions in order to enable the derivation ofstability conditions, which makes the approach applicable onlyin specific situations. In addition, no systematic method is intro-duced to identify the membership function parameters requiredto satisfy the inequalities defined by those assumptions, andtheir model structure produces linear matrix inequalities (LMIs)that cannot be easily simplified or evaluated to examine theexistence of stability criteria.

Due to the sophisticated mathematical structure of T2 FLSs,to date, no systematic analyses have been published for designof stabilizing controllers; hence, this may be why control de-signers have distanced themselves from adopting those systemson a wider scale. Existing IT2 FLCSs do not provide anygeneralized methodology to help guarantee the stability of

1083-4419/$26.00 © 2009 IEEE

2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS

the control system. Designing stabilizing controllers is accom-plished through simulations or by imposing ad hoc assumptionsto derive conditions for the stability of closed-loop control sys-tems. In an attempt to address the stability of IT2 TSK FLCSs,we introduce a new inference mechanism and propose novelmethods to design its parameters. The new inference enginefor IT2 TSK FLCSs has the following advantages: 1) a closedmathematical form that can be easily used for control design,and 2) the conditions necessary to guarantee the asymptoticstability of IT2 TSK FLCs that use this inference engine canbe easily analyzed.

In Section II of this paper, IT2 TSK FLSs are reviewed,and necessary preliminaries are established. In Section III, anovel inference mechanism for IT2 TSK A2-C0 models isproposed. In Section IV, a model is proposed for single-inputsingle-output (SISO) IT2 TSK FLCS, and sufficient stabilityconditions are presented for it. In Section V, stability condi-tions for multi-input multi-output (MIMO) IT2 TS FLCSs arederived. In Section VI, illustrative examples are provided whichdemonstrate the details of the stability analysis of IT2 TSKFLSs. Finally, in Section VI, conclusions are presented.

II. IT2 TSK A2-C0 MODELS OF DYNAMIC SYSTEMS

A. Discrete IT2 TSK A2-C0 Model

The general structure of an IT2 TSK A2-C0 model is givenas follows [20]:

If x(k) is F i1 and x(k − 1) is F i

2 and · · · and x(k − n + 1) is F in

Then yi = ai1x(k) + · · · + ai

nx(n − k + 1) (1)

where i = 1, . . . ,M ; F ij represents the IT2 FS of input state

j in rule i, namely, x(k − j); ai1, . . . , a

in are the coefficients

of the output function for rule i (and hence are crisp numbers,i.e., type-0 FSs); yi is the output of the ith rule; and M is thenumber of rules. The aforementioned rules allow us to modelthe uncertainties encountered in the antecedents. In an IT2 TSKA2-C0 model, lower and upper firing strengths of the ith rule,

i.e., f i and fi, are given by

f i(x) = μF i

1(x(k)) ∗ · · · ∗ μ

F in

(x(k − n + 1)) (2)

fi(x) = μ

F i1(x(k)) ∗ · · · ∗ μ

F in

(x(k − n + 1)) (3)

where μF i

j

and μF i

j

represent the jth (j = 1, . . . , n) lower and

upper membership functions of rule i, respectively, and “∗” is at-norm operator. State vector x is defined as

x = [x(k), x(k − 1), . . . , x(k − n + 1)]T . (4)

The final output of the IT2 TSK A2-C0 model is givenas [20]

YTSK/A2-C0(x)= [yl(x), yr(x)]

=∫

f1∈[f1,f1]

· · ·∫

fM∈[fM ,f

M] 1

/∑Mi=1 f i(x)yi∑Mi=1 f i(x)

(5)

where yi is given by the consequent portion of (1). YTSK/A2-C0

is an interval type-1 set and only depends on its left and rightend points yl, yr, which can be computed using the iterativeKM algorithms, similar to the type-reduction method explainedin [20], and its final output is given as

Youtput(x) =yl(x) + yr(x)

2. (6)

For development of IT2 FLCs, the following are keyrequirements.

1) An analytical methodology is preferred to guarantee astable control design.

2) The control structure must be suited for real-timeimplementation.

Therefore, a closed-form I/O inference engine relationshipis preferred particularly for Lyapunov-based control design.Unfortunately, (5) does not provide such a closed-form rela-tionship. Moreover, to satisfy the second requirement, iterativeKM inference algorithms may not be suitable. Hence, we turnnext to an alternative approach.

B. WM UBs

As an alternative to computing Youtput(x) using (2)–(6),we use WM UBs [22]. Background on WM UBs and theirgeneral form as stated in [24]1 are given in Appendix I-A. Wesubsequently apply the general form of WM UBs to (1)–(6).Since we are dealing with IT2 A2-C0 TSK models, yi

l = yir =

yi, the boundary T1 FLSs defined by (A.1)–(A.4) reduce to thefollowing two equations:

y(0)(x) =

∑Mi=1 f i(x)yi∑Mi=1 f i(x)

(7)

y(M)(x) =∑M

i=1 fi(x)yi∑M

i=1 fi(x)

. (8)

Without loss of generality, assume y(M)(x) > y(0)(x)[YWM(x) in (A.9) is invariant to y(M)(x) > y(0)(x)];therefore, (A.5)–(A.8) can be written as

yl(x) = y(0)(x) =

∑Mi=1f

i(x)yi∑Mi=1f

i(x)(9)

yr(x) = y(M)(x) =

∑Mi=1f

i(x)yi∑M

i=1fi(x)

(10)

yl(x) =

∑Mi=1f

i(x)yi∑Mi=1f

i(x)

−

⎡⎣ ∑Mi=1

(f

i(x)−f i(x)

)∑M

i=1fi(x) ·

∑Mi=1f

i(x)

×∑M

i=1fi(x)(yi−y1) ·

∑Mi=1f

i(x)(yM−yi)∑M

i=1fi(x)(yi−y1)+

∑Mi=1f

i(x)(yM−yi)

](11)

1In [24], a Mamdani rule is used in which the consequent is an IT2 FS.

BIGLARBEGIAN et al.: ON THE STABILITY OF INTERVAL TYPE-2 TSK FUZZY LOGIC CONTROL SYSTEMS 3

yr(x) =∑M

i=1fi(x)yi∑M

i=1fi(x)

+

⎡⎣ ∑Mi=1

(f

i(x)−f i(x)

)∑M

i=1fi(x) ·

∑Mi=1f

i(x)

×∑M

i=1fi(x)(yi−y1) ·

∑Mi=1f

i(x)(yM−yi)∑Mi=1f

i(x)(yi−y1)+

∑Mi=1f

i(x)(yM−yi)

].

(12)

By using (9)–(12), it is straightforward to show that YWM(x)in (A.9) can be expressed as (13), shown at the bottom of thepage. YWM(x) given by (13) represents the final output of theIT2 TSK A2-C0 system (1). It is easy to see that YWM(x)can be computed without having to perform TR, and therefore,YWM(x) can be considered a viable alternative to using (5) and(6) for real-time control.

Now, we apply YWM(x) to YTSK/A2-C0(x) using the follow-ing discrete-time model that appears in the consequent of rule iin (1):

yi =n∑

p=1

aipx(k − p + 1). (14)

It follows that

yi − y1 =n∑

p=1

(ai

p − a1p

)x(k − p + 1)

≡n∑

p=1

vi,paipx(k − p + 1) (15)

yM − yi =n∑

p=1

(aM

p − aip

)x(k − p + 1)

≡n∑

p=1

wi,paipx(k − p + 1) (16)

where

vi,p ≡ai

p − a1p

aip

(17)

wi,p ≡aM

p − aip

aip

. (18)

Substituting (14)–(16) into (13), it is straightforward to showthat YWM(x) can be expressed as

YWM(x) =12

∑Mi=1 f i(x)

(∑np=1 ai

px(k − p + 1))

∑Mi=1 f i(x)

+12

∑Mi=1 f

i(x)

(∑np=1 ai

px(k − p + 1))

∑Mi=1 f

i(x)

+ α(x) + β(x) (19)

where α(x) and β(x) are defined in (20) and (21), shown at thebottom of the next page.

Equation (19) has been used recently for control design[13]; however, no information is available to date on how tosystematically design IT2 FLCSs using YWM(x). We tried toobtain stability analysis for (19) but were unsuccessful.

III. NEW INFERENCE METHOD FOR IT2 TSKCONTROL DESIGNS

To obtain stability conditions for an FLCS using rigorousmathematical analyses, closed-form equations are required;hence, in this section, we introduce the following new inferenceengine:

YTSK/NEW(x) = m

∑Mi=1 f i(x)yi∑Mi=1 f i(x)

+ n

∑Mi=1 f

i(x)yi∑M

i=1 fi(x)

(22)

where yi is given by (1), and f i(x) and fi(x) are given by

(2) and (3), respectively (if M = 1, then m + n = 1). Observethat m and n are design parameters that weight the sharingof lower and upper firing levels of each fired rule and canbe tuned during the design of this new TSK system. Observe

also that if all uncertainty disappears so that f i(x) = fi(x),

then (22) reduces to a T1 TSK FLCS in which we can setm + n = 1. There is also a connection between YTSK/NEW(x)and YWM(x).

Proposition 1: If m and n are independent parameters thatdo not depend on the inference process, then YTSK/NEW(x) isderivable from YWM(x) and is a simplified version of YWM(x).

Proof: See Appendix I-B. �When (22) is used to model the plant, a procedure to obtain

the TSK consequent parameters and the tuning parameters mand n is given next. First, we derive bounds for the tuningparameters of the plant. Then, for a given m and n, we

YWM(x) =12

(∑Mi=1 f i(x)yi∑Mi=1 f i(x)

+∑M

i=1 fi(x)yi∑M

i=1 fi(x)

)

− 14

⎡⎣ ∑Mi=1

(f

i(x) − f i(x)

)∑M

i=1 f i(x) ·∑M

i=1 fi(x)

×∑M

i=1 f i(x)(yi − y1) ·∑M

i=1 fi(x)(yM − yi)∑M

i=1 f i(x)(yi − y1) +∑M

i=1 fi(x)(yM − yi)

⎤⎦

+14

⎡⎣ ∑Mi=1

(f

i(x) − f i(x)

)∑M

i=1 f i(x) ·∑M

i=1 fi(x)

×∑M

i=1 fi(x)(yi − y1) ·

∑Mi=1 f i(x)(yM − yi)∑M

i=1 fi(x)(yi − y1) +

∑Mi=1 f i(x)(yM − yi)

⎤⎦ (13)


mathematically explain how to identify the IT2 TSK conse-quent parameters, and finally, we present an algorithm to obtainsuitable tuning parameters for the plant.

We prove in Appendix I-B that m and n are given by

m =12

+ g1 (23)

n =12

+ g2 (24)

where g1 and g2 are given by (A.12) and (A.13), respectively.Since we are dealing with stability analysis, it is assumedthat the parameters of the membership functions are known(identifying the membership functions is not within the scopeof this paper). When IT2 TSK is used for practical controldesign, states x(k), x(k − 1), . . . , x(k − n + 1) are physicalquantities, e.g., displacement, velocity, and acceleration. There-fore, for a specific problem, the lower and upper bounds ofthese states can be determined by the designer. Moreover,yi =

∑np=1 ai

px(k − p + 1) corresponds to the output of rulei and represents a physical quantity (similar arguments canbe made for the two terms

∑np=1 vi,pa

ipx(k − p + 1) and∑n

p=1 wi,paipx(k − p + 1) in g1 and g2). Hence, regardless of

whether aip is known or not, the designer can establish the

lower and upper bounds on yi as well as another similar termsin g1 and g2 (the range of variation is known). The lowerand upper bounds on g1 and g2 for all rules can therefore bedetermined, i.e.,

gmin1 ≤ g1 ≤ gmax

1 (25)

gmin2 ≤ g2 ≤ gmax

2 . (26)

Then, using (23) and (24) in (25) and (26), the bounds on mand n are given as

mmin ≡ 12

+ gmin1 ≤ m ≤ 1

2+ gmax

1 ≡ mmax (27)

nmin ≡ 12

+ gmin2 ≤ n ≤ 1

2+ gmax

2 ≡ nmax. (28)

Recently, for type-2 fuzzy systems, fuzzy clustering and sub-tractive clustering have been proposed for finding the systemparameters. Subsequently, similar to the method in [25], inour method, it is assumed that the parameters of the inputmembership functions are known by using a predefined cluster-ing method, and we identify the TSK consequent parameters.Assume that the plant is modeled as

If x1 is F i1 and x2 is F i

2 and · · · and xn is F in

Then yi = ai0 + ai

1x1 + ai2x2 + · · · + ai

nxn (29)

where i = 1, . . . , M . Suppose that p input–output data (trainingdata) for the plant are given as{[

xi1, x

i2, . . . , x

in

], Y i

}p

i=1(30)

where [xi1, x

i2, . . . , x

in] is the ith input vector consisting of n

inputs and Y i is the corresponding output. Define Y ∈ Rp

containing the training outputs

Y ≡ [Y 1, Y 2, . . . , Y p]T. (31)

Using (22) and applying the method described in [26], Y i

can be expressed as

Y i = m

∑Mj=1 f j

i

(aj0 + aj

1xi1 + aj

2xi2 + · · · + aj

nxin

)∑M

j=1 f ji

+n

∑Mi=1 fi

j(aj0 + aj

1xi1 + aj

2xi2 + · · · + aj

nxin

)∑M

j=1 fij

(32)

where i = 1, . . . , p. Let

vji =

f ji∑M

j=1 f ji

(33)

vij =

fij∑M

j=1 fij. (34)

α(x) = − 14

∑Mi=1

(f

i(x) − f i(x)

)∑M

i=1 f i(x) ·∑M

i=1 fi(x)

×∑M

i=1 f i(x)(∑n

p=1 vi,paipx(k − p + 1)

)·∑M

i=1 fi(x)

(∑np=1 wi,pa

ipx(k − p + 1)

)∑M

i=1 f i(x)(∑n

p=1 vi,paipx(k − p + 1)

)+∑M

i=1 fi(x)

(∑np=1 wi,pai

px(k − p + 1)) (20)

β(x) =14

∑Mi=1

(f

i(x) − f i(x)

)∑M

i=1 f i(x) ·∑M

i=1 fi(x)

×∑M

i=1 fi(x)

(∑np=1 vi,pa

ipx(k − p + 1)

)·∑M

i=1 f i(x)(∑n

p=1 wi,paipx(k − p + 1)

)∑M

i=1 fi(x)

(∑np=1 vi,pai

px(k − p + 1))

+∑M

i=1 f i(x)(∑n

p=1 wi,paipx(k − p + 1)

) (21)


By using (32), (33) and (34), can be rewritten as

Y i = mM∑

j=1

vji

(aj0 + aj

1xi1 + aj

2xi2 + · · · + aj

nxin

)+n

M∑j=1

vij(aj0 + aj

1xi1 + aj

2xi2 + · · · + aj

nxin

). (35)

Define φ, φ, and θ as in (36)–(38), respectively, shown at thebottom of the page.

By using (36) and (37), Y can be expressed as

Y = Aθ (39)

where A = mφ + nφ is a known matrix consisting of theparameters of the input membership functions.

Finally, the error vector is defined as e ≡ Y − Aθ, and thetotal error et, which is the sum of squares of the components ofe, is defined by

et ≡p∑

i=1

e2i . (40)

Our proposed algorithm to obtain the tuning parameters mand n of the plant inference engine is as follows:

Algorithm 1 Finding the plant tuning parameters.m ← mmin and n ← nmin, and calculate the initial errorusing (31)–(40)repeat

repeat1. n ← nmin

2. Solve for θ from (39), and find the total errorfrom (40)

3. If the new error is less than the error found in theprevious step, save m, n, and θ

4. Increment n, i.e., n←n+Δn (whereΔn=0.05n)until n ≤ nmax

Let m ←← m + Δm (where Δm = 0.05m)until m ≤ mmax

Fig. 1. Closed-loop IT2 TSK A2-C0 fuzzy control system.

IV. STABILITY OF SISO IT2 TSK FLCSs

In this section, we introduce a model for stability analysis ofSISO2 IT2 TSK FLCS. SISO systems are considered because ofthe variety of applications in computing systems and bioengi-neering [27], [28]. To begin, a controller structure is introduced;then, a model is introduced for a closed-loop control system, af-ter which mathematical analyses are established for the designof stable IT2 TSK FLCSs.

A. Controller

Fig. 1 shows a controller in which the inputs are the statesx(k) and the output is u(k). For this system, the general ithrule has the following form:

ith controller rule:If x(k) is Ci

1 and x(k − 1) is Ci2 and · · ·

and x(k − n + 1) is Cin

Then ui(k + 1) = ci1x(k) + ci

2x(k − 1) + · · ·+ ci

nx(k − n + 1) (41)

where i = 1, 2, . . . , Q, Cij represents the T2 FS of input state j

of the ith rule, and cij is the jth coefficient of the output function

2When we refer to SISO, “input” is considered the controller output signaland “output” is the plant output (with both input and output being scalars).

φ ≡

⎡⎢⎢⎢⎢⎢⎣v1

1 · · · vM1 v1

1x11 · · · vM

1 x11 v1

1x1n · · · vM

1 x1n

v12 · · · vM

2 v12x

21 · · · vM

2 x21 v1

2x2n · · · vM

2 x2n

· · · · · ·...

......

......

......

......

v1p · · · vM

p v1px

p1 · · · vM

p xp1 v1

pxpn · · · vM

p xpn

⎤⎥⎥⎥⎥⎥⎦ (36)

φ ≡

⎡⎢⎢⎢⎢⎢⎣v1

1 · · · vM1 v1

1x11 · · · vM

1 x11 v1

1x1n · · · vM

1 x1n

v12 · · · vM

2 v12x

21 · · · vM

2 x21 v1

2x2n · · · vM

2 x2n

· · · · · ·...

......

......

......

......

v1p · · · vM

p v1px

p1 · · · vM

p xp1 v1

pxpn · · · vM

p xpn

⎤⎥⎥⎥⎥⎥⎦ (37)

θ ≡[a10, . . . , a

M0 , a1

1, . . . , aM1 , . . . , a1

n, . . . , aMn

]T. (38)


for rule i. Applying (22) to (41), the controller output u(k) canbe expressed as

u(k) = m′∑Q

i=1 vi(x)ui(k + 1)∑Qi=1 vi(x)

+ n′∑Q

i=1 v i(x)ui(k + 1)∑Qi=1 v i(x)

(42)where

vi(x) = μCi

1(x(k)) ∗ · · · ∗ μ

Cin

(x(k − n + 1)) (43)

v i(x) = μCi

1(x(k)) ∗ · · · ∗ μ

Cin

(x(k − n + 1)) (44)

and m′ and n′ are tuning parameters of the controller. Substi-tuting the consequent part of (41) into (42), (42) can be writ-ten as

u(k) = m′∑Q

i=1

∑nj=1 vi(x)ci

jx(k − j + 1)k1

+n′∑Q

i=1

∑nj=1 v i(x)ci

jx(k − j + 1)k2

(45)

where {k1 ≡

∑Qi=1 vi(x)

k2 ≡∑Q

i=1 v i(x).(46)

Note that parameters k1 and k2 are short for k1(x) and k2(x),respectively.

B. Closed-Loop System

Consider the feedback control system shown in Fig. 1, wherethe plant and the controller are each IT2 TSK A2-C0 models.For a closed-loop system, the controller signal u(k) is incor-porated as an input to the plant. The general ith rule for theplant is

If x(k) is F i1 and x(k − 1) is F i

2 and · · ·and x(k − n + 1) is F i

n and u(k) is Bi

Then xi(k + 1)=ai1x(k) + · · · + ai

nx(n − k + 1) + biu(k)

(47)

where i = 1, . . . , M , xi(k + 1) is the output of the ith plantrule, F i

j represents the T2 FS of input state j of the ith rule,

Bi represents the T2 FS of the plant input, and aij is the jth

coefficient of the output function for rule i. The control rules arethe same as (41) with Q being the number of rules, and u(k) isgiven by (45). Substituting u(k) from (45) into the consequentof (47), the output of the ith plant rule, i.e., xi(k + 1), isgiven by

xi(k + 1) =n∑

j=1

[ai

jx(k − j + 1)]

+ bim′∑Q

l=1

∑nj=1 vlc l

jx(k − j + 1)k1

+ bin′∑Q

l=1

∑nj=1 v lc l

jx(k − j + 1)k2

. (48)

By using (22), the output of the closed-loop system, i.e., x(k +1), can be expressed as

x(k + 1) =m∑M

i=1 f ixi(k + 1)∑Mi=1 f i

+n∑M

i=1 fixi(k + 1)∑M

i=1 fi

(49)

where m and n are the tuning parameters of the plant and f i

and fi

[short for f i(x) and fi(x)] are given by

f i(x) = μF i

1(x(k)) ∗ · · · ∗ μ

F in

(x(k − n + 1)) ∗ μBi (u(k))

(50)

fi(x) = μ

F i1(x(k)) ∗ · · · ∗ μ

F in

(x(k − n + 1)) ∗ μBi (u(k)) .

(51)

Applying (48) to (49), x(k + 1) can be expressed as

x(k + 1) =m∑M

i=1

∑nj=1 f iai

jx(k − j + 1)∑Mi=1 f i

+ mm′∑M

i=1

∑Ql=1

∑nj=1 f ivlbic l

jx(k − j + 1)

k1

∑Mi=1 f i

+ mn′∑M

i=1

∑Ql=1

∑nj=1 f iv lbic l

jx(k − j + 1)

k1

∑Mi=1 f i

+n∑M

i=1

∑nj=1 f

iai

jx(k − j + 1)∑Mi=1 f

i

+ nm′∑M

i=1

∑Ql=1

∑nj=1 f

ivlbic l

jx(k − j + 1)

k2

∑Mi=1 f

i

+ nn′∑M

i=1

∑Ql=1

∑nj=1 f

iv lbic l

jx(k − j + 1)

k2

∑Mi=1 f

i.

(52)

Next, define n × n matrices Ai and Bi,l as follows:

Ai =

⎛⎜⎜⎜⎜⎝ai1 ai

2 · · · ain−1 ai

n

1 0 · · · 0 00 1 · · · 0 0...

.... . .

......

0 0 · · · 1 0

⎞⎟⎟⎟⎟⎠

Bi,l =

⎛⎜⎜⎜⎜⎝bic l

1 bic l2 · · · bic l

n−1 bic ln

1 0 · · · 0 00 1 · · · 0 0...

.... . .

......

0 0 · · · 1 0

⎞⎟⎟⎟⎟⎠ (53)

where i = 1, 2, . . . ,M and l = 1, 2, . . . , Q. Define the outputvector as

x(k + 1) = [x(k + 1), x(k), . . . , x(k − n + 2)]T . (54)

By using (52) and (53), it is straightforward to show that x(k +1) in (54) can be written as

x(k + 1) = Cx(k) (55)


where

C =m∑M

i=1 f iAi∑Mi=1 f i

+n∑M

i=1 fiAi∑M

i=1 fi

+ mm′∑M

i=1

∑Ql=1 f ivlBi,l

k1

∑Mi=1 f i

+ mn′∑M

i=1

∑Ql=1 f iv lBi,l

k1

∑Mi=1 f i

+ nm′∑M

i=1

∑Ql=1 f

ivlBi,l

k2

∑Mi=1 f

i

+ nn′∑M

i=1

∑Ql=1 f

iv lBi,l

k2

∑Mi=1 f

i. (56)

Although (55) may look like a linear system, it is not because

C depends on x through the dependences of f i and fi

on x.

C. Stability of Closed-Loop System

The stability of T1 TSK FLCSs using fuzzy Lyapunov func-tion (FLF) has been addressed in several works [29], [30]. Mostnotably, Tanaka et al. [29] proposed an FLF method composedof multiple Lyapunov functions to obtain the stability condi-tions for a T1 TSK FLCS. They also presented a new designcontrol methodology using parallel-distributed compensation.The proposed FLF methodology in [29] provided relaxed stabil-ity conditions for a T1 TSK FLCS. However, the design processrequired the time derivatives of premise membership functions,and it is not always possible to derive such derivatives fromthe system states, which limits the use of this method. Becausethe same drawback will limit the use of this method for IT2TSK FLCSs, in this paper, we introduce a quadratic Lyapunovfunction to derive stability conditions for IT2 TSK FLCSs.

Our Lyapunov function is V (x(k)) = xT(k)Px(k),where P is a positive-definite matrix [31]. ΔV (x(k)) isgiven by

ΔV (x(k)) = xT(k + 1)Px(k + 1) − xT(k)Px(k). (57)

By using (55), ΔV (x(k)) can be expressed as

ΔV (x(k)) = xT(k)Zx(k) (58)

where

Z ≡ CTPC − P (59)

and C is given by (56). Z has 36 components and can beexpressed as

Z ≡ Z1 + Z2 + Z3 (60)

where Z1, Z2, and Z3 are given in Appendix I-C.To ensure stability in a Lyapunov sense, it is required that

ΔV (x(k)) < 0. Hence, if all the components of ΔV (x(k)) aremade negative (equivalently, all the components of Z are madenegative definite), the result will be an asymptotically stablesystem.

Let the first bracketed term of Z1 in (A.16) be denoted asZ1,1, i.e.,

Z1,1 =m∑M

i=1 f iATi∑M

i=1 f iP

m∑M

j=1 f jAj∑Mj=1 f j

− 136

P . (61)

It can be expressed as

Z1,1

⎛⎝M∑i=1

f iM∑

j=1

f j

⎞⎠=M∑

i,j=1

f if j

[m2AT

i PAj−136

P

]. (62)

By using the fact that f i and f j are positive, for Z1,1 < 0,the expression inside the bracket in (62) must be negative defi-nite. Thus

m2ATi PAj −

136

P < 0 (63)

where i, j = 1, 2, . . . ,M . It is straightforward to demonstratethat similar conditions are obtained for the remaining threeterms of Z1, i.e.,

aATi PAj −

136

P < 0 (64)

where

a = {m2,mn, n2}. (65)

Applying the same method to Z2 and Z3, similar condi-tions can be obtained (some of the details are provided inAppendix I-D). The resulting conclusions are

bATi PBj,l + bBT

i,lPAj −118

P < 0 (66)

where

b = {m2m′,mnm′,m2n′,mnn′, n2m′, n2n′} (67)

cBTi,lPBj,q −

136

P < 0 (68)

with

c = {m2m′2, n2n′2,m2m′n′,mnm′n′,m2n′2,mnn′2,mnm′2, n2m′2, n2m′n′}. (69)

In (64), (66), and (68), i, j = 1, 2, . . . ,M , and l, q = 1, 2,. . . , Q. For each i, j, l, and q combination, only 18 inequali-ties are given by (64)–(68), because half of the 36 inequalityconditions for ΔV (x(k)) < 0 are repetitive. If there exists acommon positive-definite matrix P that satisfies the inequal-ities in (64)–(68), then the closed-loop system is globallyasymptotically stable.

Note that combining the terms of C in (56) will not lead tosimpler stability conditions, because when common terms arecombined, the expressions inside the resulting multiple summa-tions include several combinations of the Ai and Bj,l matricesand hence require a larger number of inequalities to be satisfied.

Next, we formulate (64)–(68) into the LMI problems that canbe solved using numerical techniques such as the interior pointmethod [32]. Consider, e.g., one of the inequalities given by(64). By multiplying both sides of it by −36, it is straightfor-ward to show that it can be rewritten as

P − 36aATi PAj > 0. (70)


Since (70) can be nonsymmetric, it follows that its symmetricpart must satisfy

12

{[P − 36aAT

i PAj

]T+[P − 36aAT

i PAj

]}> 0. (71)

Equation (71) can be reexpressed as [i, j = 1, 2, . . . ,M ;a in (65)]

P − 36a

2(AT

j PAi + ATi PAj

)> 0. (72)

It can be similarly shown that for (66) and (68), the followingequivalent LMIs can be obtained [i, j = 1, 2, . . . ,M , and l, q =1, 2, . . . , Q; b in (67) and c in (69)]:

P − 182

b(BT

j,lPAi + ATi PBj,l

)− 18

2b(AT

j PBi,l + BTi,lPAj

)> 0 (73)

P − 362

c(BT

i,lPBj,q + BTj,lPBi,q

)> 0. (74)

The stability conditions in (72)–(74) can be evaluated usingstandard software tools, such as the Matlab LMI toolbox3 orthe CVX4 that have been developed to efficiently solve LMIproblems.

If the LMIs given by (72)–(74) have a positive-definitesolution for P , then (52) is globally asymptotically stable.

D. Bounds for the Controller Tuning Parameters

In this section, a method is proposed for deriving the boundsfor the controller tuning parameters m′ and n′ in (42). Thesebounds are used to find the controller tuning parameters in thenext section. Paralleling the derivation in Appendix I-B, it isstraightforward to show that the controller tuning parametersm′ and n′ are given by

m′ =12

+ g′1 (75)

n′ =12

+ g′2 (76)

3Matlab LMI toolbox solves semidefinite programming and LMI problems.4CVX is a Matlab-based modeling system for convex optimization [33], [34].

where g′1 and g′2 are defined in (77) and (78), respectively,shown at the bottom of the page, with ri,p and si,p given as

ri,p ≡cip − c1

p

cip

(79)

si,p ≡cQp − ci

p

cip

. (80)

When IT2 TSK is used for control design, bounds on g′1 andg′2 from (77) and (78) can be determined, i.e.,

g′min1 ≤ g′1 ≤ g′max

1 (81)g′min2 ≤ g′2 ≤ g′max

2 . (82)

It follows from (75) and (76), (81) and (82) that the boundson m′ and n′ are given as

m′min ≡ 12

+ g′min1 ≤ m′ ≤ 1

2+ g′max

1 ≡ m′max (83)

n′min ≡ 12

+ g′min2 ≤ n′ ≤ 1

2+ g′max

2 ≡ n′max. (84)

E. Algorithm to Find the Controller Tuning Parameters

Our algorithm to find the controller tuning parameters m′ andn′ is given as follows:

Algorithm 2 Finding the controller tuning parameters.m′ ← m′min

repeatrepeat1. n′ ← n′min

2. Solve the LMIs given by (72)–(74), and deter-mine the feasibility/infeasibility5 of P

3. If the LMIs are feasible, save m′, n′, and P , andexit the loop (inner loop)

4. Increment n′, i.e., n′ ← n′ + Δn′ (whereΔn′ = 0.05n′)

until n′ <= n′max

Let m′ ← m′ + Δm′ (where Δm′ = 0.05m′)until m′ <= m′max

5If a positive definite P is found, the LMI is called feasible; otherwise, it isinfeasible.

g′1 = − 14

∑Qi=1

(v i(x) − vi(x)

)[∑Qi=1 vi(x)

∑np=1 ci

px(k − p + 1)]∑Q

i=1 v i(x)

×∑Q

i=1

[vi(x)

∑np=1 ri,pc

ipx(k − p + 1)

]∑Qi=1

[v i(x)

∑np=1 si,pc

ipx(k − p + 1)

]∑Q

i=1

[vi(x)

∑np=1 ri,pci

px(k − p + 1)]

+∑Q

i=1

[v i(x)

∑np=1 si,pci

px(k − p + 1)] (77)

g′2 =14

∑Qi=1

(v i(x) − vi(x)

)[∑Qi=1 v i(x)

∑np=1 ci

px(k − p + 1)]∑Q

i=1 vi(x)

×∑Q

i=1

[v i(x)

∑np=1 ri,pc

ipx(k − p + 1)

]∑Qi=1

[vi(x)

∑np=1 si,pc

ipx(k − p + 1)

]∑Q

i=1

[v i(x)

∑np=1 ri,pci

px(k − p + 1)]

+∑Q

i=1

[vi(x)

∑np=1 si,pci

px(k − p + 1)] (78)


At the end of this algorithm, a set of feasible (m′, n′,P ) isfound. The designer chooses the specific set that achieves thebest transient performance. An advantage of this algorithm isthat there is no need to change the controller TSK parameters,as opposed to its T1 TSK counterpart. By using this algorithm,when m′ and n′ are chosen by tuning only two controllerparameters, stabilizing the system is easy to achieve.

V. STABILITY OF MIMO IT2 TS FLCSs

In this section, unlike Section IV where IT2 TSK was usedfor plant to derive the stability conditions, we simplify the plantmodel to a T1 TSK. This simplifies the process of stabilityanalysis for MIMO IT2 control systems. This assumption isreasonable because T1 TSK FLSs have been proven to beuniversal approximators [35] and can model nonlinear plantsrelatively well [36]. Hence, the focus on this section will be thestability analysis of IT2 TS FLCSs that use T1 TS plant model.To begin, we review the structure of the hybrid system for a T1TS FLCS.

A. T1 TS FLCS

The general ith rule for the plant is now given as [36]

Rip: If x(k) is P i

1 and · · · and x(k − n + 1) is P in

Then xi(k + 1) = Aix(k) + biu(k), i = 1, 2, . . . , r

(85)

where r is the number of rules, xi(k + 1) is the output of eachrule, P i

j represents the T1 FS of input state j of rule i, x(k)is the state vector and is given by (4), and Ai ∈ R

n×n, bi ∈R

n×m, u(k) ∈ Rm. The output of the system, i.e., x(k + 1), is

given by

x(k + 1) =∑r

i=1 qi(k) {Aix(k) + biu(k)}∑ni=1 qi(k)

(86)

where

qi(k) = μP i1(x(k)) ∗ · · · ∗ μP i

n(x(k − n + 1)) . (87)

The ith control rule is [36]

Ric: If x(k) is Ci

1 and · · · and x(k − n + 1) is Cin

Then ui(k) = F ix(k), i = 1, 2, . . . , r (88)

where F j is the jth feedback gain matrix of the consequent partand Ci

j represents the T1 FS of input state j of rule i. Note thatthe number of rules for the controller is also r. The controller

output u(k) for a system that uses a T1 TS FLCS is givenby [36]

u(k) =

∑rj=1 wj(k)F jx(k)∑r

j=1 wj(k)(89)

where

wi(k) = μCi1(x(k)) ∗ · · · ∗ μCi

n(x(k − n + 1)) . (90)

The complete closed-loop modified T1 TS FLCS is obtainedby substituting (89) into (86).

B. IT2 TS FLCS

As mentioned at the start of Section V, the IT2 TS FLSutilizes an IT2 TSK FLS for the controller and the T1 TS FLSin (86) for the plant. The rule structure for the IT2 TS FLCSis kept the same as (88) except that Ci

j’s are replaced with IT2

FSs, i.e., Cji . Now, however, u(k) has the same structure as (42)

and is given as

u(k) = m′∑r

j=1 wj(k)F jx(k)∑rj=1 wj(k)

+ n′∑r

j=1 wj(k)F jx(k)∑rj=1 wj(k)

(91)

where

wj(k) =μCj

1(x(k)) ∗ · · · ∗ μ

Cjn

(x(k − n + 1)) (92)

wj(k) =μCj

1(x(k)) ∗ · · · ∗ μ

Cjn

(x(k − n + 1)) . (93)

Substituting (91) into (86), the output of the system, i.e.,x(k + 1), can be expressed as (94), shown at the bottom ofthe page, which can be further expressed in a more compactform as

x(k + 1) =

∑ri,j,l=1 gijl(k)Gijl∑r

i,j,l=1 gijl(k)x(k) (95)

where

gijl(k) = qi(k)wj(k)wl(k) (96)

Gijl =Ai + m′biF j + n′biF l. (97)

It is straightforward to show that∑r

i,j,l=1 Gijl can beexpressed as

r∑i,j,l=1

Gijl =r∑

i=1

Giii +r∑

i�=j

r∑j=1

Gijj +r∑

i=1

r∑j �=l

r∑l=1

Gijl.

(98)

x(k + 1) =

∑ri=1 qi(k)

{Aix(k) + bim

′∑r

j=1w

j(k)F jx(k)∑r

j=1wj(k)

+ n′∑r

j=1wj(k)F jx(k)∑r

j=1wj(k)

}∑n

i=1 qi(k)

=

∑ri,j,l=1 qi(k)wj(k)wl(k) {Aix(k) + m′biF jx(k) + n′biF lx(k)}∑r

i,j,l=1 qi(k)wj(k)wl(k)(94)


Observe that

r∑i�=j

r∑j=1

Gijj =r∑

i<j

r∑j=1

Gijj +r∑

i>j

r∑j=1

Gijj

=r∑

i<j

r∑j=1

Gijj +r∑

t<p

r∑t=1

Gptt

= 2r∑

i<j

r∑j=1

[Gijj + Gjii

2

](99)

r∑i=1

r∑j �=l

r∑l=1

Gijl =r∑

i=1

r∑j<l

r∑l=1

Gijl +r∑

i=1

r∑j>l

r∑l=1

Gijl

=r∑

i=1

r∑j<l

r∑l=1

Gijl +r∑

i=1

r∑p>t

r∑l=1

Gipt

=r∑

i=1

r∑j<l

r∑l=1

Gijl +r∑

i=1

r∑l>j

r∑l=1

Gilj

= 2r∑

i=1

r∑j<l

r∑l=1

[Gijl + Gilj

2

]. (100)

Substituting (99) and (100) into (98), (98) is obtained.Next, define Ht and vt(k) as

Ht ≡

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

Gijl+Gilj

2 , t = i + r(j − 1 + (l−1)(l−2)

2

)and j < l

Gijj+Gjii

2 , t = i + j(j−1)2 + r2(r−1)

2and j = l, i < j

Giii, t = i + i(i−1)2 + r2(r−1)

2and i = j = l

(101)

vt(k) ≡

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

2gijl(k), t = i + r(j − 1 + (l−1)(l−2)

2

)and j < l

2gijj(k), t = i + j(j−1)2 + r2(r−1)

2and j = l, i < j

giii(k), t = i + i(i−1)2 + r2(r−1)

2and i = j = l.

(102)

Note that the number of Ht matrices given in (101) iscalculated as

r(r + 1)2

+r2(r − 1)

2=

r(r2 + 1)2

(103)

where r(r + 1)/2 is the number of Ht for the case when i ≤ jand j = l, and r2(r − 1)/2 is the number of Ht matrices forthe case when j < l. Observe that Ht matrices are functions ofthe Gijl matrices that are dependent on the control parametersm′ and n′ in (97).

By using (101) and (102), (95) can be reexpressed, using asingle summation, as

x(k + 1) =∑ r(r2+1)

2s=1 vs(k)Hs∑ r(r2+1)

2s=1 vs(k)

x(k). (104)

It is well known [36] that (104) is globally asymptoticallystable if there exists a common positive-definite matrix Psuch that

HTs PHs − P < 0 (105)

for s = 1, 2, . . . , r(r2 + 1)/2.Observe, from (95), that r3 LMIs (note the three summa-

tions) must be satisfied to ensure the stability of the IT2 TSFLCS; however, by introducing the Ht matrices, only (r(r2 +1)/2) < r3 LMIs need to be satisfied to arrive at the sameresult.

Before assessing the feasibility of the LMIs in (105), we needto identify the bounds on the controller tuning parameters m′

and n′ in (91) because Hs depends on them [Ht matricesare functions of the Gijl matrices (97)]. When WM UB in(19) and (20) is used as an inference engine, the controlleroutput is uWM(x), and hence, each of its components uj

WM(x)for j = 1, . . . , r can be expressed as [[F ix(k)]j replaces∑n

p=1 aipx(k − p + 1) in (A.14)]

ujWM(x) =

12

∑ri=1 wi(x)

([F ix(k)]j

)∑r

i=1 wi(x)

+12

∑ri=1 w i(x)

([F ix(k)]j

)∑r

i=1 w i(x)

+ g′j1 ×∑r

i=1 wi(x) [F ix(k)]j∑ri=1 wi(x)

+ g′j2 ×∑r

i=1 w i(x) [F ix(k)]j∑ri=1 w i(x)

(106)

where g′j1 and g′j2 are given by (107) and (108), respectively,shown at the bottom of the page.

ujWM(x) can therefore be expressed as

ujWM(x) =

∑ri=1 wi(x)

m′j︷︸︸︷(

12

+ g′j1

)[F ix(k)]j∑r

i=1 wi(x)

+

∑ri=1 w i(x)

n′j︷︸︸︷(

12

+ g′j2

)[F ix(k)]j∑r

i=1 w i(x). (109)

g′j1 = − 14

∑ri=1

(w i(x) − wi(x)

)∑ri=1 w i(x) ·

∑ri=1 wi(x) [F ix(k)]j

·∑r

i=1 wi(x) [(F i − F 1)x(k)]j ·∑r

i=1 w i(x) [(F r − F i)x(k)]j∑ri=1 wi(x) [(F i − F 1)x(k)]j +

∑ri=1 w i(x) [(F r − F i)x(k)]j

(107)

g′j2 =14

∑ri=1

(w i(x) − wi(x)

)∑ri=1 wi(x)

∑ri=1 w i(x) [F ix(k)]j

·∑r

i=1 w i(x) [(F i − F 1)x(k)]j ·∑r

i=1 wi(x) [(F r − F i)x(k)]j∑ri=1 w i(x) [(F i − F 1)x(k)]j +

∑ri=1 wi(x) [(F r − F i)x(k)]j

(108)


Finally, the bounds of m′ and n′ are given by

m′min ≡ minj

(12

+ g′j min1

)≤ m′ ≤ max

j

(12

+ g′j max1

)≡m′max (110)

n′min ≡ minj

(12

+ g′j min2

)≤ n′ ≤ max

j

(12

+ g′j max2

)≡n′max (111)

where j = 1, . . . , r.Next, observe that (105) can be transformed into the follow-

ing LMIs (s = 1, 2, . . . , r(r2 + 1)/2):

P − HTs PHs > 0. (112)

Let X ≡ P−1, and multiply both sides of (112) by X from leftand right. It is straightforward to see that (112) becomes

X − XHTs X−1HsX

−1 > 0 (113)

which is equivalent to the following LMIs (s = 1, 2, . . . ,r(r2 + 1)/2): [

X XHTs

HsX X

]> 0. (114)

By using the bounds of m′ and n′ found in (110) and (111)and a procedure similar to the algorithm in Section IV-E[(72)–(74) are replaced with (114)], the feasibility of (114) canbe investigated using the Matlab LMI toolbox or the CVX. If apositive-definite X exists, then the closed-loop system will beasymptotically stable.

VI. EXAMPLES

This section introduces two examples. The first presentsapplications of the developed theory to analyze the stability ofSISO and MIMO systems, respectively. The second examplepresents case studies demonstrating the effectiveness of theIT2 TS FLCS in tracking applications and control of nonlinearsystems.

Example 1: This example presents two case studies. The firstdeals with a SISO system defined in Section IV-C. The seconddeals with a MIMO system described in Section V.

Case Study A: Consider the following SISO IT2 TSK FLS.

1) If x(k) is F 1 and x(k − 1) is F 2, then x1(k + 1) =2.3x(k) − 2x(k − 1) + 0.7u(k).

2) If x(k) is F 2 and x(k − 1) is F 1, then x2(k + 1) =1.5x(k) − 1x(k − 1) + 0.01u(k).

This system has two control rules.

1) If x(k) is C1 and x(k − 1) is C2, then u1(k + 1) =−0.9x(k) − 1.08x(k − 1).

2) If x(k) is C2 and x(k − 1) is C1, then u2(k + 1) =1.4x(k) − 2.1x(k − 1).

The membership functions for the plant, F 1 and F 2, and thecontroller, C1 and C2, are shown in Fig. 2. The Bij and Ai

matrices, according to (53), are

A1 =[

2.3 −21 0

]A2 =

[1.5 −11 0

](115)

Fig. 2. Membership functions for Example 1, Case Study A.

TABLE ISOME SELECTED CONTROLLER TUNING PARAMETERS

AND THEIR CORRESPONDING P

B11 =[−0.63 −0.756

1 0

]B12 =

[−0.009 −0.011

1 0

](116)

B21 =[

0.980 −1.4701 0

]B22 =

[0.014 −0.021

1 0

].

(117)

Note that, in this example, m and n in (49), which are plantparameters, are assumed given as m = 0.1 and n = 0.1. Thecontroller tuning parameters m′and n′ were designed basedon the method introduced in Sections IV-D and E (note thatthis is a stabilizing controller design problem). Bounds on thestates were assumed to be [−3, 1]. By using the algorithm ofSection IV-E, the bounds for m′and n′ were obtained as [0.176,0.5] and [0.149, 1.299], respectively. Table I summarizes someof the selected values for m′ and n′ and their correspondingmatrix P .

Tuning parameters m′and n′ were selected, which resultedin the best output transient response, e.g., when the initialconditions are x(1) = 0.1 and x(2) = 0.01, the response of thesystem for m′ and n′ in Table I is shown in Fig. 3. The output ofthe system shown with solid line (m′ = n′ = 0.2) has the besttransient response.

Case Study B: In this case study, we apply an IT2 TS FLCto stabilize an inverted pendulum which is an example of abenchmark problem often used in the design of controllers.Fig. 4 shows an inverted pendulum located on a cart. The con-trol problem is to stabilize the inverted pendulum by applying ahorizontal force to the system (control action).


Fig. 3. Closed-loop system response for different controller tuningparameters.

Fig. 4. Inverted pendulum.

Tanaka and Wang [37] demonstrated that, for certain initialconditions (angles), a linear controller is not capable of stabiliz-ing the system. Hence, they introduced a T1 TS FLS that can beused to model as well as control this inverted pendulum. In thiscase study, we show that the proposed IT2 TS FLC in Section Vis capable of stabilizing the inverted pendulum while achievinga better performance compared to its T1 counterpart.

The inverted pendulum system has nonlinear dynamics, andthe equations of motion are given as (118) and (119), shown atthe bottom of the page, where x1(t) and x2(t) are the angularposition and velocity of the pendulum, respectively, u(t) is thecontrol input, m is the pendulum mass, M is the cart mass, 2lis the length of the pendulum, and a ≡ 1/(m + M).

We also compare the performance of our IT2 TS FLC witha nonlinear controller [38] as well as a linear controller. Thestructure of the nonlinear controller is given by

u(t) =g

atan (x1(t)) +

4le1e2

3aln [sec (x1(t)) + tan (x1(t))]

− e1e2ml sin (x1(t))

− (e1 + e2)x2(t)a

[4l

3sec (x1(t)) − aml cos (x1(t))

](120)

Fig. 5. Type-2 membership functions for Example 1, Case Study B.

where e1 and e2 are specified closed-loop eigenvalues. To com-pare the performance of the IT2 TS FLC with the T1 controller,we keep the model of the plant as a T1 TS and only replace thecontroller with an IT2 TS model. To make a fair comparison,the parameters of the plants and controllers are kept unchangedfor both control systems, and only the membership functionsfor the IT2 controller are designed.

Define x(t) ≡ [x1(t), x2(t)]T, where x1(t) and x2(t) arethe state variables, i.e., angular position and velocity of thependulum. The structure of the plant and the controllers is givenin the following.Plant rules (see Fig. 5 for antecedent membership functions)6

If x1(t) is “about 0,” then x = A1x(t) + b1u(t).If x1(t) is “about π/2 or −(π/2),” then x = A2x(t) +

b2u(t).Control rules

If x1(t) is “about 0,” then u(t) = f1x(t).If x1(t) is “about π/2 or −(π/2),” then u(t) = f2x(t).

From the aforementioned rules, the following are defined:

A1 =[

0 117.3118 0

]A2 =

[0 1

9.3696 0

](121)

b1 =[

0−0.1765

]b2 =

[0

−0.0052

](122)

f1 = [ 120.6667 22.6667 ] f2 = [ 2551.6 0.7640 ].(123)

Note that in order to make an unbiased comparison, f1

and f2 are adopted from [36]. Those gains were chosen bya pole-placement method. The linear controller is given byu(t) = f1x(t). The values of the parameters used in this casestudy are given as follows: m = 2 kg, M = 8 kg, l = 0.5 m,a = 0.1 kg−1, g = 9.81 m/s2, e1 = e2 = −2, and the tuning pa-rameters for the IT2 controllers are m′ = 1 and n′ = 0.9. Notethat this example deals with a continuous system. Following thesame approach explained in Section V, it is very straightforward

6x1 is given in radians.

x1(t) = x2(t) (118)

x2(t) =g sin (x1(t)) − amlx2

2(t) sin (2x1(t)) /2 − a cos (x1(t)) u(t)4l/3 − aml cos2 (x1(t))

(119)


Fig. 6. Outputs of different controllers for the initial angle of x1(0) =0.105 rad.

TABLE IIRISE AND SETTLING TIMES OF DIFFERENT CONTROLLERS

FOR DIFFERENT INITIAL ANGLES

to show that the stability conditions for the continuous systemcan be simply written as

HTs P + PHs < 0 (124)

where Hs matrices are defined by (101) and P is a positive-definite matrix. By using the Matlab LMI toolbox, it is easy toobtain the following P matrix satisfying (124):

P =[

0.6219 0.08520.0852 0.0324

]. (125)

Fig. 6 shows the performance of the different controllerssimulated for the initial angle of x1(0) = 0.105 rad (notethat the responses of the linear and nonlinear controllers arealmost the same for this specific initial angle). Clearly, theIT2 controller not only stabilizes the system but also resultsin enhanced transient performances, i.e., reduced rise time andfaster settling time. The rise and settling times for the T1controller are tr = 0.64 s and ts = 1.16 s, respectively. For thenonlinear controller, the corresponding values are tr = 0.49 sand ts = 0.9 s. Finally, for the IT2 controller, these values aretr = 0.49 s and ts = 0.90 s (see Table II).

Fig. 7 compares the response of each controller for differentinitial angles. The nonlinear, T1, and IT2 controllers are capableof stabilizing the system for all initial angles x1(0) ∈ (0, π/2).For initial angles x1(0) > 0.7854 rad, the linear controller,however, fails to stabilize the system. Furthermore, the IT2controller is consistently outperforming other controllers interms of transient response. The IT2 controller offers a simplerstructure that does not have the complexity of the nonlinearcontroller in (120), yet it performs considerably better whileconcurrently ensuring the stability of the nonlinear system.

Example 2: This example offers two more case studies thatdemonstrate the effectiveness of the proposed IT2 TS FLC fortracking applications and control of nonlinear dynamic systemssuch as chaotic oscillators.

Fig. 7. Outputs of different controllers for different initial angles.

Fig. 8. Coordinate system used to describe the car position and orientation.

Case Study C (Tracking Application): This control exampleis adopted from the study in [36], where a T1 TS controller wasdesigned to track a predefined trajectory of a model car. Thespecific problem is to control a computer-simulated model carfrom an arbitrary initial position by manipulating the steeringangle and allowing only forward movements. The car is theplant and is modeled by a T1 TS FLCS. In [36], it was verifiedthat the dynamics of the approximated fuzzy model agree withthe original model. In Fig. 8, observe that x0 is the angle thatthe car makes with the horizontal axis and that x1 is the verticalposition of the rear end of the car. The control objective is totrack the car from a given initial position to the position wherex0 = x1 = 0 with no backward movement.

This example has two parts. First, stability conditions arederived for a system that utilizes an IT2 TS FLCS in controllerdesign. Second, the performance of the designed IT2 TS FLCSis compared with its type-1 counterpart.

a) Stability: As mentioned earlier, a T1 TS model is avalid approximation for the plant. Moreover, in order to makean unbiased comparison of the performance of T1 TS FLCSand IT2 TS FLCS, the plant is considered as a T1 TS modeland only the controller is redesigned.

Plant and control rules are given as follows.Plant rules (see Fig. 9 for antecedent membership functions)7

If x0(k) is “about 0,” then x(k + 1) = A1x(k) + b1u(k).

7x0 is given in radians.


Fig. 9. Type-2 membership functions for Example 2, Case Study C.

If x0(k) is “about π or −π,” then x(k + 1) = A2x(k) +b2u(k).

Control rulesIf x0(k) is “about 0,” then u(k) = f1x(k).If x0(k) is “about π or −π,” then u(k) = f2x(k).


A1 =[

1 01 1

]A2 =

[1 0

0.003183 1

](126)

b1 =[

0.3571431

]b2 =

[0.357143

1

](127)

f1 = [−0.4212 −0.02933] f2 =[−0.0991 − 0.00967].(128)

Note that in order to make an unbiased comparison, f1

and f2 are adopted from the study in [36], where they wereobtained by a pole-placement method. Similar to Example 1,the bounds for m′and n′ are obtained as [−6.207, 1.275]and [−2.235, 4.935], respectively. In this example, r = 2, andhence, the number of LMIs to be satisfied is 5. By using (101),the Hi matrices are calculated as

H1 =[

0.8495 −0.01051 1

]H2 =

[0.9071 −0.0070

1 1

]H3 =

[0.9071 −0.00700.5016 1

]H4 =

[0.9071 −0.00700.0032 1

]H5 =

[0.9646 −0.0035

1 1

](129)

from which it follows that, P , computed from the MatlabLMI toolbox, is

P =[

699.6386 57.376657.3766 11.7997

]. (130)

It can be verified that P satisfies the stability conditions forall five LMIs in (105), i.e.,

HT1 PH1 − P =

[−85.369 −3.659−3.659 −1.125

]< 0

HT2 PH2 − P =

[−8.077 1.6501.650 −0.765

]< 0

HT3 PH3 − P =

[−68.788 −4.033−4.033 −0.765

]< 0

Fig. 10. Trajectories of the car model for the two controllers.

Fig. 11. Angular position of the car for type-1 and type-2 fuzzy controllers.

HT4 PH4 − P =

[−123.637 −9.714−9.714 −0.765

]< 0

HT5 PH5 − P =

[−48.296 −4.325−4.325 −0.3880

]< 0. (131)

Therefore, the closed-loop system is asymptotically stable.b) Performance evaluation: Here, we compare the per-

formance of T1 TS FLC and the IT2 TS FLC given in part (a) ofthis example. The rules and consequent parameters of T1 TS arekept the same as in part (a), and only the membership functionsof the antecedents for the proposed type-2 inference engineare redesigned from T1 to IT2. The membership functions areshown in Fig. 9, and the initial conditions are x0(0) = π andx1(0) = 20. To make the comparison more realistic, a steeringangle threshold has been set to ±(π/3) which is the thresholdof a steering angle of a typical car.

Fig. 10 compares the performance of the T1 TS and IT2 TSsystems. Observe that the T1 TS controller has a noticeableovershoot8 and a slower convergence to the set point in com-parison to the IT2 TS controller. Clearly, the IT2 TS controllerhas less overshoot. It is easy to see that the IT2 TS controllerhas a faster settling time than the T1 TS. Moreover, the T1TS controller has a noticeably undesirable undershoot betweent = 57 s and t = 80 s, where the IT2 TS controller does not.

Fig. 11 compares the angular positions of the two controllers.Angular position with the IT2 TS controller has a slightly larger

8“Overshoot” occurs when the system’s response overshoots the starting tra-jectory (initial position), and “undershoot” occurs when the system’s responseundershoots the zero trajectory.


Fig. 12. Controller outputs for type-1 and type-2 fuzzy systems.

negative slope that helps the car to get to x0 = 0 and x1 = 0faster. This is noticeable by comparing the time required forboth controllers to reach the final angle, i.e., 0◦. To furtherexplain this, the control efforts (steering angles) of both con-trollers are compared in Fig. 12. As can be seen, initially, thecontrol effort of the IT2 TS controller is significantly greaterthan that of the T1 TS controller. This results in less overshootin the transient response of the IT2 TS controller. In addition,the IT2 TS control effort reveals a large slope in decreasingthe steering angle, rendering less undershoot compared to theT1 TS controller. This is attributed to the shape of the type-2membership functions that allows more uncertainty in the con-troller structure.

This example demonstrates that the proposed IT2 TS with theproposed inference, YTSK/NEW, is capable of outperforming awell-tuned T1 TS FLCS.

Case Study D (Control of a Nonlinear/Chaotic System): Inthe second part of this example, we develop an IT2 TS FLC andapply it to another popular nonlinear system. This case studyshows that the IT2 controller can stabilize a chaotic systemwhile simultaneously achieving enhanced results compared toits T1 counterpart. The system considered in this problem isan electrical circuit, known as Chua’s circuit [39], that exhibitschaotic output behaviors. The Chua’s circuit consists of oneinductor (L), two capacitors (C1, C2), one linear resistor (R),and one piecewise linear resistor (g(vc1)). This circuit is de-scribed by the following equations [40]:

vc1 =1C1

(1R

(vc2 − vc1) − g(vc1))

+ u1 (132)

vc2 =1C2

(1R

(vc1 − vc2) + iL

)+ u2 (133)

iL =1L

(−vc2 − R0iL) + u3 (134)

where g(vc1) is given by

g(vc1) =

⎧⎨⎩Gbvc1 + (Ga − Gb)E, vc1 ≥ EGavc1, −E < vc1 < EGbvc1 − (Ga − Gb)E, vc1 ≤ −E

(135)

where vc1, vc2, iL are state variables, Ga, Gb, E are thecharacteristics of the piecewise linear resistor, and u1, u2,

Fig. 13. Type-2 membership functions for Example 2, Case Study D.

u3 are the control inputs. For more information about Chua’scircuit, see [39] and [41]. It is well known that Chua’s circuithas nonlinear dynamics such as bifurcation and chaos. Chaoticsystems can reveal large oscillations/motions, and hence, thereis a growing interest in the controllers that can effectivelyhandle such systems. Wang and Tanaka [40] developed a T1TS model that represents this circuit well. Hence, similar toprevious examples, we keep the plant as a T1 TS and replacethe controller with our proposed IT2 TS FLC. To make anunbiased comparison, we only redesign the IT2 membershipfunctions while keeping all other parameters the same forboth control systems. Let x(t) ≡ [x1(t), x2(t), x3(t)]T, wherexi(t)’s are the state variables, i.e., x1 = vc1, x2 = vc2, x3 = iL.The values for the parameters used in this example are givenas follows: R = 1.4286, R0 = 0 Ω, C1 = 0.1, C2 = 0.2, L =0.1429, Ga = −2, Gb = 0.1, and E = 1.

Plant and control rules are given as follows.Plant rules

If x1(t) is M1, then x = A1x(t) + bu(t).If x1(t) is M2, then x = A2x(t) + bu(t).

Fig. 13 shows the antecedent membership functions M1

and M2.Control rules

If x1(t) is M1, then u(t) = F 1x(t).If x1(t) is M2, then u(t) = F 2x(t).


A1 =

⎡⎣ 5.7143 14.2857 00.7143 −0.7143 0.5

0 −7 0

⎤⎦

A2 =

⎡⎣−12.0190 14.2857 00.7143 −0.7143 0.5

0 −7 0

⎤⎦ (136)

F 1 =

⎡⎣−33.3333 −31.6202 −1.796124.2702 0.0167 −1.98081.7961 8.4808 −0.3333

⎤⎦

F 2 =

⎡⎣ 3.0667 21.4379 −3.7158−28.7879 0.0167 −20.27223.7158 26.7722 −0.3333

⎤⎦ (137)


Fig. 14. Chua’s circuit response to the T1 and IT2 controllers.

Fig. 15. Chua’s circuit response to the T1 and IT2 controllers (controllers areinvoked at t = 30 s).

and b is a 3 by 3 identity matrix. We adopt F 1 and F 2

from [40]. It is very easy to show that for m′ = n′ = 0.8, thefollowing P matrix satisfies the LMIs in (124):

P =

⎡⎣ 2.2240 0.0112 0.17010.0112 2.5747 −0.01980.1701 −0.0198 2.0743

⎤⎦ . (138)

Fig. 14 shows the response of the Chua’s circuit for theduration of 50 s when both T1 and IT2 controllers are applied.The initial conditions considered for simulations are x(0) =[1, 0, 0]T. Note that the controllers are invoked at t = 30 s.Before the controllers are activated, the system’s output isoscillating. As shown in Fig. 14, both controllers stabilize thesystem. However, examining the performance after 30 s usingFig. 15 (enlarged plot of the controllers output in Fig. 14 whent ≥ 30 s), it is easy to observe that the IT2 controller hasa much better transient response. The rise and settling timesfor the IT2 and T1 controllers, respectively, are tr = 0.15 s,ts = 0.19 s and tr = 0.22 s, ts = 0.25 s. Moreover, the IT2

controller produces much less overshoot compared to the T1controller when 30 s ≤ t ≤ 31 s.

VII. CONCLUSION

In this paper, we have modified the WM UBs to developa new inference mechanism for IT2 TSK A2-C0 and IT2 TSFLSs. The inference engine was formulated in closed form andhence does not require using the iterative KM algorithms. Dueto the simple structure of the proposed inference engine, it canbe adopted to design IT2 TSK and IT2 TS FLCSs for real-timecontrol applications.

By using the proposed inference mechanism, LMI stabilityconditions for IT2 TSK FLCSs and IT2 TS FLCSs were derivedand transformed into the standard formats that can be easilysolved using software tools such as the Matlab LMI toolbox.Consequently, the stability of a control system can be provedanalytically when the proposed inference mechanism is used todesign both IT2 TSK FLCSs and IT2 TS FLCSs.

To evaluate the performance of IT2 TS FLCSs with theproposed inference engine in control and tracking applications,two benchmark examples were adopted from the literature.It was shown that a well-tuned IT2 TS FLCS significantlyoutperforms its type-1 counterpart.

In conclusion, using the proposed model for IT2 TSK A2-C0 or IT2 TS FLSs will enable control engineers to designand implement stable IT2 FLCSs with enhanced performance.Future work will focus on the following: 1) developing stabilityanalysis for MIMO IT2 TS FLCSs when both plant and con-troller are IT2 TS, and 2) developing uncertainty bounds, notconsidered by the proposed YTSK/NEW, to assist in the processof designing stable IT2 TSK FLCSs.

APPENDIX IDERIVATIONS AND PROOFS

A. Background on WM UBs

WM UBs use the following four centroids (also calledboundary T1 FLSs):

{LMFs, left} : y(0)l (x) =

∑Mi=1 f i(x)yi

l∑Mi=1 f i(x)

(A.1)

{LMFs, right} : y(M)r (x) =

∑Mi=1 f i(x)yi

r∑Mi=1 f i(x)

(A.2)

{UMFs, left} : y(M)l (x) =

∑Mi=1 f

i(x)yi

l∑Mi=1 f

i(x)

(A.3)

{UMFs, right} : y(0)r (x) =

∑Mi=1 f

i(x)yi

r∑Mi=1 f

i(x)

(A.4)

where yil and yi

r are the left and right end points of the

centroid of the ith consequent IT2 FS and f i(x) and fi(x) are

computed using (2) and (3). The WM UBs are lower and upperbounds for yl and yr and are defined as in (A.5)–(A.8), shownat the bottom of the next page.


By using the WM UBs, the final output of an IT2 FLS, i.e.,YWM(x), is computed as [24]

YWM(x) =12

[y

l(x) + yl(x)

2+

yr(x) + yr(x)

2

]. (A.9)

B. Proof of Proposition

By using (20) and (21), α(x) and β(x) can be expressed asnonlinear functions of the upper and lower firing levels of eachrule as well as the input states, i.e.,

α(x) = g1

(f i(x), f

i(x),x, vi,p, wi,p

)

×∑M

i=1 f i(x)[∑n

p=1 aipx(k − p + 1)

]∑M

i=1 f i(x)(A.10)

β(x) = g2

(f i(x), f

i(x),x, vi,p, wi,p

)

×∑M

i=1 fi(x)

[∑np=1 ai

px(k − p + 1)]

∑Mi=1 f

i(x)

(A.11)

where functions g1and g2 are given by9 (A.12) and (A.13),shown at the bottom of the page.

By using (A.10)–(A.13), YWM(x) in (19) can be written as

YWM(x) =

∑Mi=1 f i(x)

(12

∑np=1 ai

px(k − p + 1))

∑Mi=1 f i(x)

+

∑Mi=1 f

i(x)

(12

∑np=1 ai

px(k − p + 1))

∑Mi=1 f

i(x)

+ g1 ×∑M

i=1 f i(x)[∑n

p=1 aipx(k − p + 1)

]∑M

i=1 f i(x)

+ g2 ×∑M

i=1 fi(x)

[∑np=1 ai

px(k − p + 1)]

∑Mi=1 f

i(x)

.

(A.14)

9In order to simplify the notation, in the rest of the derivation, g1 and g2

are short for g1(f i(x), fi(x), x, vi,p, wi,p) and g2(f i(x), f

i(x), x, vi,p,

wi,p).

yl(x) = min{

y(0)l (x), y(M)

l (x)}

(A.5)

yr(x) = max

{y(0)

r (x), y(M)r (x)

}(A.6)

yl(x) = yl(x) −

⎡⎣ ∑Mi=1

(f

i(x) − f i(x)

)∑M

i=1 f i(x) ·∑M

i=1 fi(x)

×∑M

i=1 f i(x)(yi

l − y1l

)∑Mi=1 f

i(x)

(yM

l− yi

l

)∑Mi=1 f i(x)

(yi

l − y1l

)+∑M

i=1 fi(x)

(yM

l− yi

l

)⎤⎦ (A.7)

yr(x) = yr(x) +

⎡⎣ ∑Mi=1

(f

i(x) − f i(x)

)∑M

i=1 f i(x) ·∑M

i=1 fi(x)

×∑M

i=1 fi(x)

(yi

r − y1r

)·∑M

i=1 f i(x)(yM

r− yi

r

)∑Mi=1 f

i(x) (yi

r − y1r) +

∑Mi=1 f i(x)

(yM

r− yi

r

)⎤⎦ (A.8)

g1 = − 14

∑Mi=1

(f

i(x) − f i(x)

)[∑M

i=1 f i(x)∑n

p=1 aipx(k − p + 1)

]∑Mi=1 f

i(x)

×∑M

i=1

[f i(x)

∑np=1 vi,pa

ipx(k − p + 1)

]∑Mi=1

[f

i(x)

∑np=1 wi,pa

ipx(k − p + 1)

]∑M

i=1

[f i(x)

∑np=1 vi,pai

px(k − p + 1)]

+∑M

i=1

[f

i(x)

∑np=1 wi,pai

px(k − p + 1)] (A.12)

g2 =14

∑Mi=1

(f

i(x) − f i(x)

)[∑M

i=1 fi(x)

∑np=1 ai

px(k − p + 1)]∑M

i=1 f i(x)

×∑M

i=1

[f

i(x)

∑np=1 vi,pa

ipx(k − p + 1)

]∑Mi=1

[f i(x)

∑np=1 wi,pa

ipx(k − p + 1)

]∑M

i=1

[f

i(x)

∑np=1 vi,pai

px(k − p + 1)]

+∑M

i=1

[f i(x)

∑np=1 wi,pai

px(k − p + 1)] (A.13)


Combining the first and the third terms and the second and thefourth terms of YWM(x), (A.14) can be rewritten as

YWM(x) =

∑Mi=1 f i(x)

⎛⎜⎜⎝∑np=1

m︷︸︸︷(12

+g1

)ai

px(k−p+1)

⎞⎟⎟⎠∑M

i=1 f i(x)

+

∑Mi=1 f

i(x)

⎛⎜⎜⎝∑np=1

n︷︸︸︷(12

+g2

)ai

px(k−p+1)

⎞⎟⎟⎠∑M

i=1 fi(x)

. (A.15)

Comparing (A.15) and (22) [remember that yi is given by (1)],it can be seen that m and n correspond to ((1/2) + g1) and((1/2) + g2), respectively. Under the assumption that m and nare adjustable parameters that do not depend on the inferenceprocess, YWM(x) simplifies to YTSK/NEW(x).

C. Components of Z in (60)

Components of Z, namely, Z1, Z2, and Z3, are given asin (A.16), (A.17), and (A.18), respectively. Equation (A.18), isshown at the bottom of the next page.

Z1 =

[m∑M

i=1fiAT

i∑Mi=1f

iP

m∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[m∑M

i=1fiAT

i∑Mi=1f

iP

n∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[n∑M

i=1fiAT

i∑Mi=1f

iP

m∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[n∑M

i=1fiAT

i∑Mi=1f

iP

n∑M

j=1fjAj∑M

j=1fj

− 136

P

](A.16)

Z2 =

[m∑M

i=1fiAT

i∑Mi=1f

iP

mm′∑Mj=1

∑Ql=1f

jvlBj,l

k1

∑Mj=1f

j− 1

36P

]

+

[m∑M

i=1fiAT

i∑Mi=1f

iP

mn′∑Mj=1

∑Ql=1f

jv lBj,l

k1

∑Mj=1f

j− 1

36P

]

+

[m∑M

i=1fiAT

i∑Mi=1f

iP

nm′∑Mj=1

∑Ql=1f

jvlBj,l

k2

∑Mi=1f

j− 1

36P

]

+

[m∑M

i=1fiAT

i∑Mi=1f

iP

nn′∑Mj=1

∑Ql=1f

jv lBj,l

k2

∑Mj=1f

j− 1

36P

]

+

[n∑M

i=1fiAT

i∑Mi=1f

iP

mm′∑Mj=1

∑Ql=1f

jvlBj,l

k1

∑Mj=1f

j− 1

36P

]

+

[n∑M

i=1fiAT

i∑Mi=1f

iP

mn′∑Mj=1

∑Ql=1f

jv lBj,l

k1

∑Mj=1f

j− 1

36P

]

+

[n∑M

i=1fiAT

i∑Mi=1f

iP

nm′∑Mj=1

∑Ql=1f

jvlBj,l

k2

∑Mj=1f

j− 1

36P

]

+

[n∑M

i=1fiAT

i∑Mi=1f

iP

nn′∑Mj=1

∑Ql=1f

jv lBj,l

k2

∑Mj=1f

j− 1

36P

]

+

[mm′

∑Mi=1

∑Ql=1f

ivlBTi,l

k1

∑Mi=1f

iP

m∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[mm′

∑Mi=1

∑Ql=1f

ivlBTi,l

k1

∑Mi=1f

iP

n∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[mn′∑M

i=1

∑Ql=1f

iv lBTi,l

k1

∑Mi=1f

iP

m∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[mn′∑M

i=1

∑Ql=1f

iv lBTi,l

k1

∑Mi=1f

iP

n∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[nm′∑M

i=1

∑Ql=1f

iv lBTi,l

k1

∑Mi=1f

iP

m∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[nm′∑M

i=1

∑Ql=1f

iv lBTi,l

k1

∑Mi=1f

iP

n∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[nn′∑M

i=1

∑Ql=1f

iv lBTi,l

k1

∑Mi=1f

iP

m∑M

j=1fjAj∑M

j=1fj

− 136

P

]

+

[nn′∑M

i=1

∑Ql=1f

iv lBTi,l

k1

∑Mi=1f

iP

n∑M

j=1fjAj∑M

j=1fj

− 136

P

](A.17)

D. Supplementary Details of Stability Derivations

In this section, we present additional details of the stabilityconditions (66) and (68). Consider the first bracketed term ofZ2 in (A.17), and denote it as Z2,1, i.e.,

Z2,1≡m∑M

i=1 f iATi∑M

i=1 f iP

mm′∑Mj=1

∑Qq=1 f jvqBj,q

k1

∑Mj=1 f j

− 136

P .

(A.19)

Similarly, denote the ninth bracketed term of Z2 as Z2,9, i.e.,

Z2,9 ≡ mm′∑M

i=1

∑Ql=1 f ivlBT

i,l

k1

∑Mi=1 f i

Pm∑M

j=1 f jAj∑Mj=1 f j

− 136

P .

(A.20)

Multiply both sides of (A.19) and (A.20) byk1

∑Mi=1 f i ∑M

j=1 f j . It is easy to see that

(Z2,1 + Z2,9)

⎛⎝k1

M∑i=1

f iM∑

j=1

f j

⎞⎠= m

M∑i=1

f iATi Pmm′

M∑j=1

Q∑l=1

f jvlBj,l


+ mm′M∑i=1

Q∑l=1

f ivlBi,lPmM∑

j=1

f jATj

− 118

P

⎛⎝k1

M∑i=1

f iM∑

j=1

f j

⎞⎠=

M∑i,j=1

f if j

×[

Q∑l=1

m2m′[vlATi PBj,l+vlBj,lPAT

i

]−

Q∑l=1

vl 118

P

]

=M∑

i,j=1

f if j

×{

Q∑l=1

vl

[m2m′ (AT

i PBj,l+Bj,lPATi

)− 1

18P

]}.

(A.21)

Z3 =

[mm′∑M

i=1

∑Ql=1 f ivlBT

i,l

k1

∑Mi=1 f i

Pmm′∑M

j=1

∑Qq=1 f jvqBj,q

k1

∑Mj=1 f j

− 136

P

]

+

[mm′∑M

i=1

∑Ql=1 f ivlBT

i,l

k1

∑Mi=1 f i

Pmn′∑M

j=1

∑Qq=1 f jv qBj,q

k1

∑Mj=1 f j

− 136

P

]

+

[mm′∑M

i=1

∑Ql=1 f ivlBT

i,l

k1

∑Mi=1 f i

Pnm′∑M

j=1

∑Qq=1 f

jvqBj,q

k2

∑Mj=1 f

j− 1

36P

]

+

[mm′∑M

i=1

∑Ql=1 f ivlBT

i,l

k1

∑Mi=1 f i

Pnn′∑M

j=1

∑Qq=1 f

jv qBj,q

k2

∑Mj=1 f

j− 1

36P

]

+

[mn′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pmm′∑M

j=1

∑Qq=1 f jvqBj,q

k1

∑Mj=1 f j

− 136

P

]

+

[mn′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pmn′∑M

j=1

∑Qq=1 f jv qBj,q

k1

∑Mj=1 f j

− 136

P

]

+

[mn′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pnm′∑M

j=1

∑Qq=1 f

jvqBj,q

k2

∑Mj=1 f

j− 1

36P

]

+

[mn′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pnn′∑M

j=1

∑Qq=1 f

jv qBi,q

k2

∑Mj=1 f

i− 1

36P

]

+

[nm′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pmm′∑M

j=1

∑Qq=1 f jvqBi,q

k1

∑Mj=1 f j

− 136

P

]

+

[nm′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pmn′∑M

j=1

∑Qq=1 f jv qBj,q

k1

∑Mj=1 f j

− 136

P

]

+

[nm′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pnm′∑M

j=1

∑Qq=1 f

jvqBj,q

k2

∑Mi=1 f

j− 1

36P

]

+

[nm′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pnn′∑M

j=1

∑Qq=1 f

jv qBj,q

k2

∑Mi=1 f

j− 1

36P

]

+

[nn′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pmm′∑M

j=1

∑Qq=1 f jvqBj,q

k1

∑Mj=1 f j

− 136

P

]

+

[nn′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pmn′∑M

j=1

∑Qq=1 f jv qBj,q

k1

∑Mi=1 f j

− 136

P

]

+

[nn′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pnm′∑M

j=1

∑Qq=1 f

jvqBj,q

k2

∑Mi=1 f

j− 1

36P

]

+

[nn′∑M

i=1

∑Ql=1 f iv lBT

i,l

k1

∑Mi=1 f i

Pnn′∑M

j=1

∑Qq=1 f

jv qBj,q

k2

∑Mj=1 f

j− 1

36P

](A.18)


It can be seen that, for the remaining terms of Z2, similarexpressions can be obtained. Hence, to make all terms of Z2

negative definite, the following criteria must be satisfied:

bATi PBj,l + bBT

i,lPAj −118

P < 0 (A.22)

where i, j = 1, 2, . . . ,M ; l = 1, 2, . . . , Q; and

b = {m2m′,mnm′,m2n′,mnn′, n2m′, n2n′}. (A.23)

Next, consider the first term of Z3 in (A.18) and denote it asZ3,1, i.e.,

Z3,1 ≡mm′∑M

i=1

∑Ql=1 f ivlBT

i,l

k1

∑Mi=1 f i

× Pmm′∑M

j=1

∑Qq=1 f jvqBj,q

k1

∑Mj=1 f j

− 136

P . (A.24)

Multiply both sides of (A.24) by (k21

∑Mi=1 f i ∑M

j=1 f j). It isstraightforward to show [using (46)]⎛⎝k2

1

M∑i=1

f iM∑

j=1

f j

⎞⎠Z3,1

=mm′M∑i=1

Q∑l=1

f ivlBTi,lPmm′

M∑j=1

Q∑q=1

f jvqBi,l

− 136

P

⎛⎝k21

M∑i=1

f iM∑

j=1

f j

⎞⎠=

M∑i,j=1

f if j

⎡⎣ Q∑l,q=1

m2m′2vlvqBTi,lPBj,q−

Q∑l,q=1

vlvq 136

P

⎤⎦=

M∑i,j=1

f if j

⎧⎨⎩Q∑

l,q=1

vlvq

[m2m′2BT

i,lPBj,q−136

P

]⎫⎬⎭ .

(A.25)

In a similar fashion, it can be seen that, for the remaining termsof Z3, similar expressions can be obtained. Hence, to make allterms of Z3 negative definite, the following criteria must besatisfied:

cBTi,lPBj,q −

136

P < 0 (A.26)

where i, j = 1, 2, . . . ,M ; l, q = 1, 2, . . . , Q; and

c = {m2m′2, n2n′2,m2m′n′,mnm′n′,m2n′2,

mnn′2,mnm′2, n2m′2, n2m′n′}. (A.27)

REFERENCES

[1] E. Mamdani, “Fuzzy control—A misconception of theory andapplication,” IEEE Expert, vol. 9, no. 4, pp. 27–28, Aug. 1994.

[2] D. Wu and W. Tan, “Type-2 fuzzy logic system modeling capa-bility analysis,” in Proc. IEEE FUZZ Conf., Reno, NV, May 2005,pp. 242–247.

[3] H. Hagras, “Type-2 fuzzy control: A new generation of fuzzy controllers,”IEEE Comput. Intell. Mag., vol. 2, no. 1, pp. 30–43, Feb. 2007.

[4] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its ap-plications to modeling and control,” IEEE Trans. Syst., Man, Cybern.,vol. SMC-15, no. 1, pp. 116–132, Feb. 1985.

[5] M. Sugeno and G. T. Kang, “Structure identification of fuzzy model,”Fuzzy Sets Syst., vol. 28, no. 1, pp. 15–33, Oct. 1988.

[6] J. M. Mendel, “Type-2 fuzzy sets and systems: An overview,” IEEEComput. Intell. Mag., vol. 2, no. 2, pp. 20–29, May 2007.

[7] J. Figueroa, J. Posada, J. Soriano, M. Melgarejo, and S. Rojas, “A type-2fuzzy controller for tracking mobile objects in the context of roboticsoccer games,” in Proc. IEEE FUZZ Conf., Reno, NV, May 2005,pp. 359–364.

[8] Q. Ren, L. Baron, and M. Balazinski, “Type-2 Takagi–Sugeno–Kangfuzzy logic modeling using subtractive clustering,” in Proc. NAFIPS,Montreal, QC, Canada, Jun. 2006, pp. 120–125.

[9] D. Wu and W. Tan, “A type-2 fuzzy logic controller for the liquid-levelprocess,” in Proc. IEEE FUZZ Conf., Budapest, Hungary, Jul. 2004,pp. 953–958.

[10] C. Hwang and F. C.-H. Rhee, “Uncertain fuzzy clustering: Intervaltype-2 fuzzy,” IEEE Trans. Fuzzy Syst., vol. 15, no. 4, pp. 107–120,Feb. 2007.

[11] B. Wu and X. Yu, “Fuzzy modelling and identification with geneticalgorithm based learning,” Fuzzy Sets Syst., vol. 113, no. 3, pp. 351–365,Aug. 2000.

[12] J. M. Mendel, “Computing derivatives in interval type-2 fuzzy logicsystems,” IEEE Trans. Fuzzy Syst., vol. 12, no. 1, pp. 84–98, Feb. 2004.

[13] C. Wagner and H. Hagras, “A genetic algorithm based architecturefor evolving type-2 fuzzy logic controllers for real world autonomousmobile robots,” in Proc. IEEE FUZZ Conf., London, U.K., Jul. 2007,pp. 193–198.

[14] O. Castillo, L. T. Aguilar, N. R. Cazarez-Castro, and S. Cardenas,“Systematic design of a stable type-2 fuzzy logic controller,” J. Appl.Soft Comput., vol. 8, no. 3, pp. 1274–1279, Jun. 2008.

[15] H. Hagras, “A hierarchical type-2 fuzzy logic control architecture forautonomous mobile robots,” IEEE Trans. Fuzzy Syst., vol. 2, no. 4,pp. 524–539, Aug. 2004.

[16] H. Hagras, “Developing a type-2 FLC through embedded type-1 FLCs,”in Proc. IEEE FUZZ Conf., Hong Kong, Jun. 2008, pp. 148–155.

[17] H. Chaoui and W. Gueaieb, “Type-2 fuzzy logic control of a flexible-jointmanipulator,” J. Intell. Robot. Syst., vol. 51, no. 2, pp. 159–186, Feb. 2008.

[18] M. Y. Hsiao, T. H. Li, J. Z. Lee, C. H. Chao, and S. H. Tsai, “Design ofinterval type-2 fuzzy sliding-mode controller,” Inf. Sci., vol. 178, no. 6,pp. 1696–1716, Mar. 2008.

[19] O. Castillo, P. Melin, O. Montiel, A. Rodriguez-Diaz, and R. Sepulveda,“Handling uncertainty in controllers using type-2 fuzzy logic,” J. Intell.Syst., vol. 14, no. 3, pp. 237–262, Aug. 2005.

[20] J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introductionand New Directions. Upper Saddle River, NJ: Prentice-Hall, 2001.

[21] N. N. Karnik and J. M. Mendel, “Centroid of a type-2 fuzzy set,” Inf. Sci.,vol. 132, no. 1–4, pp. 195–220, Feb. 2001.

[22] H. Wu and J. M. Mendel, “Uncertainty bounds and their use in the designof interval type-2 fuzzy logic systems,” IEEE Trans. Fuzzy Syst., vol. 10,no. 5, pp. 622–639, Oct. 2002.

[23] H. K. Lam and L. D. Seneviratne, “Stability analysis of interval type-2fuzzy-model-based control systems,” IEEE Trans. Syst., Man, Cybern. B,Cybern., vol. 38, no. 3, pp. 617–628, Jun. 2008.

[24] J. M. Mendel, “Advances in type-2 fuzzy sets and systems,” Inf. Sci.,vol. 177, no. 1, pp. 84–110, Jan. 2007.

[25] L. Wang and R. Langari, “Complex systems modeling via fuzzy logic,”IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 26, no. 1, pp. 100–106,Feb. 1996.

[26] M. B. Begian, W. W. Melek, and J. M. Mendel, “Parametric design of sta-ble type-2 TSK fuzzy systems,” in Proc. North Amer. Fuzzy Inf. Process.Soc., New York, May 2008, pp. 1–6.

[27] S. Keshav, “A control-theoretic approach to flow control,” SIGCOMMComput. Commun. Rev., vol. 25, no. 1, pp. 188–201, Jan. 1995.

[28] C. V. Hollot, V. Misra, D. Towsley, and W. B. Gong, “A control theo-retic analysis of RED,” in Proc. IEEE Infocom Conf., Anchorage, AK,Apr. 2001, pp. 1510–1519.

[29] K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov functionapproach to stabilization of fuzzy control systems,” IEEE Trans. FuzzySyst., vol. 11, no. 4, pp. 582–589, Aug. 2003.

[30] J.-D. Hwang, Z.-R. Tsai, and C.-J. Chang, “A fuzzy Lyapunov functionapproach to stabilize uncertain nonlinear systems using improved randomsearch method,” in Proc. IEEE Conf. Syst., Man, Cybern., Taipei, Taiwan,Oct. 2006, pp. 3091–3096.


[31] K. H. Khalil, Nonlinear Systems., 2nd ed. Englewood Cliffs, NJ:Prentice-Hall, 1996.

[32] S. P. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory., 1st ed. Philadelphia, PA:SIAM, 1994.

[33] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined ConvexProgramming (Web Page and Software), 2008. [Online]. Available:http://stanford.edu/boyd/cvx

[34] M. Grant and S. Boyd, “Graph implementations for nonsmoothconvex programs,” in Recent Advances in Learning and Control. Berlin,Germany: Springer-Verlag, 2008. (A Tribute to M. Vidyasagar).

[35] H. Ying, Fuzzy Control and Modeling: Analytical Foundations andApplications., 1st ed. Piscataway, NJ: IEEE Press, 2000.

[36] K. Tanaka and M. Sano, “Trajectory stabilization of a model car via fuzzycontrol,” Fuzzy Sets Syst., vol. 70, no. 2/3, pp. 155–170, Mar. 1995.

[37] K. Tanaka and H. O. Wang, Fuzzy Control Systems Designs and Analysis.,1st ed. New York: Wiley, 2001.

[38] W. T. Baumann and W. J. Rugh, “Feedback control of nonlinear systemsby extended linearization,” IEEE Trans. Autom. Control, vol. AC-31,no. 1, pp. 40–46, Jan. 1986.

[39] L. Chua, M. Komuro, and T. Matsumoto, “The double scrollfamily,” IEEE Trans. Circuits Syst., vol. CAS-33, no. 11, pp. 1073–1118,Nov. 1986.

[40] H. O. Wang and K. Tanaka, “An LMI-based stable fuzzy control of non-linear systems and its application to control of chaos,” in Proc. 5th IEEEInt. Conf. Fuzzy Syst., New Orleans, LA, Sep. 1996, pp. 1433–1438.

[41] L. Chua, C. W. Wu, A. Hunang, and G. Q. Zhong, “A universal circuit forstudying and generating chaos. I. Routes to chaos,” IEEE Trans. CircuitsSyst. I, Fundam. Theory Appl., vol. 40, no. 10, pp. 732–744, Oct. 1993.

Mohammad Biglarbegian (S’07) received the B.Sc.degree (with honors) in mechanical engineeringfrom the University of Tehran, Tehran, Iran, in 2002and the M.A.Sc. degree in mechanical engineeringfrom the University of Toronto, Toronto, ON,Canada, in 2005. He is currently working toward thePh.D. degree in the Department of Mechanical andMechatronics Engineering, University of Waterloo,Waterloo, ON, under NSERC and Ontario Graduatescholarships.

His research interests include intelligent control,type-2 fuzzy logic systems and control, and nonlinear control with applicationsto mechatronics systems.

William W. Melek (M’02–SM’06) received theM.A.Sc. and Ph.D. degrees in mechanical engineer-ing from the University of Toronto, Toronto, ON,Canada, in 1998 and 2002, respectively.

Between 2002 and 2004, he was an ArtificialIntelligence Division Manager with Alpha Global IT,Inc., Toronto. He is currently an Assistant Professorwith the Department of Mechanical and Mechatron-ics Engineering, University of Waterloo, Waterloo,ON. His current research interests include mecha-tronics applications, robotics, industrial automation

and the application of fuzzy logic, neural networks, and genetic algorithms formodeling and control of dynamic systems.

Dr. Melek is a member of the American Society of Mechanical Engineers.

Jerry M. Mendel (S’59–M’61–SM’72–F’78–LF’04) received the Ph.D. degree in electricalengineering from the Polytechnic Institute ofBrooklyn, Brooklyn, NY.

Since 1974, he has been with the Universityof Southern California, Los Angeles, where he iscurrently a Professor of electrical engineering andsystems architecting engineering. He has publishedover 470 technical papers and is the author and/orEditor of eight books, including Uncertain Rule-based Fuzzy Logic Systems: Introduction and New

Directions (Prentice-Hall, 2001). His current research interests include type-2fuzzy logic systems and their applications to a wide range of problems,including smart oil field technology and computing with words.

Dr. Mendel is a distinguished member of the IEEE Control Systems Societyand a fellow of the International Fuzzy Systems Association (2009). He wasthe President of the IEEE Control Systems Society in 1986. He is a member ofthe Administrative Committee of the IEEE Computational Intelligence Societyand was the Chairman of its Fuzzy Systems Technical Committee. Among hisawards are the 1983 Best Transactions Paper Award of the IEEE Geoscienceand Remote Sensing Society, the 1992 Signal Processing Society Paper Award,the 2002 Transactions on Fuzzy Systems Outstanding Paper Award, a 1984IEEE Centennial Medal, an IEEE Third Millenium Medal, a Fuzzy SystemsPioneer Award (2008) from the IEEE Computational Intelligence Society, anda Pioneer Award from the IEEE Granular Computing Conference, May 2006,for outstanding contributions in type-2 fuzzy systems.

on the stability of interval type-2 tsk fuzzy logic control systems

Documents