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 TSK Fuzzy Logic Control Systems Mohammad Biglarbegian, Student Member, IEEE, William W. Melek, Senior Member, IEEE, and Jerry M. Mendel, Life Fellow, IEEE

Abstract—Type-2 fuzzy logic systems have recently been utilized in many control processes due to their ability to model uncertainties. This paper proposes a novel inference mechanism for an interval type-2 Takagi–Sugeno–Kang fuzzy logic control system (IT2 TSK FLCS) when antecedents are type-2 fuzzy sets and consequents are crisp numbers (A2-C0). The proposed inference mechanism has a closed form which makes it more feasible to analyze the stability of this FLCS. This paper focuses on control applications for the following cases: 1) Both plant and controller use A2-C0 TSK models, and 2) the plant uses type-1 Takagi–Sugeno (TS) and the controller uses IT2 TS models. In both cases, sufficient stability conditions for the stability of the closed-loop system are derived. Furthermore, novel linear-matrixinequality-based algorithms are developed for satisfying the stability conditions. Numerical analyses are included which validate the effectiveness of the new inference methods. Case studies reveal that an IT2 TS FLCS using the proposed inference engine clearly outperforms its type-1 TSK counterpart. Moreover, due to the simple nature of the proposed inference engine, it is easy to implement in real-time control systems. The methods presented in this paper lay the mathematical foundations for analyzing the stability and facilitating the design of stabilizing controllers of IT2 TSK FLCSs and IT2 TS FLCSs with significantly improved performance over type-1 approaches. Index Terms—Adaptive control, modular and reconfigurable robots, robot manipulators, Takagi–Sugeno–Kang (TSK), type-2 fuzzy logic control (T2 FLC).

I. I NTRODUCTION

E

VEN THOUGH fuzzy logic was originally developed to model linguistic terms, interpretations, and human perceptions, most implementations of fuzzy logic systems (FLSs) have been in control applications [1], [2]. Up to now, fuzzy logic control (FLC) has been implemented with great success in many real-world applications and has also been shown in some cases to outperform traditional control systems [1], [3]. One of the most well known model structures of fuzzy systems used for control applications is Takagi–Sugeno–Kang (TSK) [4], [5]. The method presented in [4] requires the design of consequent parameters of a general fuzzy TSK model based on a least squares method. Later, Sugeno and Kang [5] presented Manuscript received December 30, 2008; revised April 28, 2009. This paper was recommended by Associate Editor C.-T. Lin. M. Biglarbegian and W. W. Melek are with the Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON N2L 3G1, 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 online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMCB.2009.2029986

the identification structure of fuzzy systems (premise and consequents) that has made it possible to design and analyze fuzzy control systems rigorously. Hence, in this paper, we adopt the TSK model structure. Recently, there has been a growing interest in using type-2 FLSs (T2 FLSs) in many applications as well as in control processes due to their ability to model uncertainties [3], [6]–[19]. The major operation in an interval T2 FLS (IT2 FLS) is type reduction which reduces the T2 FLS output to a type-1 fuzzy set. The most commonly adopted IT2 FLSs utilize Karnik–Mendel (KM) algorithms for type reduction [20], [21]. KM algorithms compute the left and right end points needed to characterize interval type-2 fuzzy sets (IT2 FSs). In order to bypass the computational effort of KM algorithms, Wu and Mendel [22] developed uncertainty bounds for IT2 FSs to approximate type reduction while achieving similar results. In this paper, we modify the Wu–Mendel uncertainty bounds (WM UBs) to design stable interval type-2 TSK fuzzy logic control systems (IT2 TSK FLCS). With the development of T2 FLSs and their ability to handle uncertainty, utilizing type-2 FLCSs (T2 FLCSs) has attracted a lot of interest in recent years. Although, to date, only IT2 FLSs have been applied for control applications, promising results have been reported, e.g., Wu and Tan [9] designed an IT2 FLCS for a coupled-tank liquid-level system and showed that when the level of uncertainty increases, the IT2 FLCS outperforms its type-1 counterpart. In addition, Hagras [15] applied IT2 FLC to mobile robot navigation in dynamic unstructured indoor and outdoor environments. All the IT2 FLCSs implemented in [15] used much smaller rule bases than their type-1 counterparts, and it was concluded that IT2 FLCSs provide a faster computation platform as well as enhanced performance results over their type-1 counterpart. Recently, Lam and Seneviratne [23] investigated stability analysis of IT2 Takagi–Sugeno (TS) FLCSs. Their approach requires several assumptions to be made about the membership functions in order to enable the derivation of stability conditions, which makes the approach applicable only in specific situations. In addition, no systematic method is introduced to identify the membership function parameters required to satisfy the inequalities defined by those assumptions, and their model structure produces linear matrix inequalities (LMIs) that cannot be easily simplified or evaluated to examine the existence of stability criteria. Due to the sophisticated mathematical structure of T2 FLSs, to date, no systematic analyses have been published for design of stabilizing controllers; hence, this may be why control designers have distanced themselves from adopting those systems on a wider scale. Existing IT2 FLCSs do not provide any generalized 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 accomplished through simulations or by imposing ad hoc assumptions to derive conditions for the stability of closed-loop control systems. In an attempt to address the stability of IT2 TSK FLCSs, we introduce a new inference mechanism and propose novel methods to design its parameters. The new inference engine for IT2 TSK FLCSs has the following advantages: 1) a closed mathematical form that can be easily used for control design, and 2) the conditions necessary to guarantee the asymptotic stability of IT2 TSK FLCs that use this inference engine can be easily analyzed. In Section II of this paper, IT2 TSK FLSs are reviewed, and necessary preliminaries are established. In Section III, a novel inference mechanism for IT2 TSK A2-C0 models is proposed. In Section IV, a model is proposed for single-input single-output (SISO) IT2 TSK FLCS, and sufficient stability conditions are presented for it. In Section V, stability conditions for multi-input multi-output (MIMO) IT2 TS FLCSs are derived. In Section VI, illustrative examples are provided which demonstrate the details of the stability analysis of IT2 TSK FLSs. Finally, in Section VI, conclusions are presented. II. IT2 TSK A2-C0 M ODELS OF DYNAMIC S YSTEMS

The general structure of an IT2 TSK A2-C0 model is given as follows [20]: If x(k) is F1i and x(k − 1) is F2i and · · · and x(k − n + 1) is Fni Then yi = ai1 x(k) + · · · + ain x(n − k + 1) (1) where i = 1, . . . , M ; Fji represents the IT2 FS of input state j in rule i, namely, x(k − j); ai1 , . . . , ain 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 the number of rules. The aforementioned rules allow us to model the uncertainties encountered in the antecedents. In an IT2 TSK A2-C0 model, lower and upper firing strengths of the ith rule, i i.e., f i and f , are given by (2)

n

1

i

f (x) = μFi (x(k)) ∗ · · · ∗ μFi (x(k − n + 1))

(3)

n

1

where μFi and μFi represent the jth (j = 1, . . . , n) lower and j j upper membership functions of rule i, respectively, and “∗” is a t-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 given as [20] YTSK/A2-C0 (x) = [yl (x), yr (x)]  = ··· 1

f 1 ∈[f 1 ,f ]



 

f M ∈ f M ,f

 M

1

M f i (x)yi i=1 M i i=1 f (x)

Youtput (x) =

yl (x) + yr (x) . 2

(6)

For development of IT2 FLCs, the following are key requirements. 1) An analytical methodology is preferred to guarantee a stable control design. 2) The control structure must be suited for real-time implementation. Therefore, a closed-form I/O inference engine relationship is preferred particularly for Lyapunov-based control design. Unfortunately, (5) does not provide such a closed-form relationship. Moreover, to satisfy the second requirement, iterative KM inference algorithms may not be suitable. Hence, we turn next to an alternative approach. B. WM UBs

A. Discrete IT2 TSK A2-C0 Model

f i (x) = μFi (x(k)) ∗ · · · ∗ μFi (x(k − n + 1))

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 right end points yl , yr , which can be computed using the iterative KM algorithms, similar to the type-reduction method explained in [20], and its final output is given as

As an alternative to computing Youtput (x) using (2)–(6), we use WM UBs [22]. Background on WM UBs and their general form as stated in [24]1 are given in Appendix I-A. We subsequently apply the general form of WM UBs to (1)–(6). Since we are dealing with IT2 A2-C0 TSK models, yli = yri = yi , the boundary T1 FLSs defined by (A.1)–(A.4) reduce to the following two equations: M i f (x)yi (0) y (x) = i=1 (7) M i i=1 f (x) M i f (x)yi . (8) y (M ) (x) = i=1 i M i=1 f (x) 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 M i f (x)yi (0) (9) y l (x) = y (x) = i=1 M i i=1 f (x) M i f (x)yi (10) y r (x) = y (M ) (x) = i=1 i M f (x) i=1 M i i=1 f (x)yi y l (x) = M i i=1 f (x)  ⎡ M i i i=1 f (x)−f (x) − ⎣ M M i i i=1 f (x) · i=1 f (x) M i M i i=1 f (x)(yi −y1 ) · i=1 f (x)(yM −yi ) × M M i i i=1 f (x)(yi −y1 )+ i=1 f (x)(yM −yi ) (11)

(5) 1 In

[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

M i f (x)yi y r (x) = i=1 i M i=1 f (x)  ⎡ M i i (x)−f (x) f i=1 + ⎣ M M i i i=1 f (x) · i=1 f (x) M i M i i=1 f (x)(yi −y1 ) · i=1 f (x)(yM −yi ) × M i . M i i=1 f (x)(yi −y1 )+ i=1 f (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 the page. YWM (x) given by (13) represents the final output of the IT2 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 following discrete-time model that appears in the consequent of rule i in (1): yi =

n

aip x(k − p + 1).

(14)

p=1

It follows that yi − y1 = ≡ yM − yi =

n



 aip − a1p x(k − p + 1)

p=1 n

p=1 n

vi,p aip x(k − p + 1)

≡

(15)



 i aM p − ap x(k − p + 1)

wi,p aip x(k − p + 1)

(16)

p=1

where aip − a1p aip M ap − aip ≡ . aip

vi,p ≡

(17)

wi,p

(18)

1 YWM (x) = 2

 M

i i=1 f (x)yi M i i=1 f (x)

M +

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

  M i n i f (x) a x(k − p + 1) p=1 p 1 i=1 YWM (x) = M i 2 f (x) i=1

  M i n i f (x) a x(k − p + 1) p=1 p 1 i=1 +  i M 2 f (x) i=1

+ α(x) + β(x)

(19)

where α(x) and β(x) are defined in (20) and (21), shown at the bottom of the next page. Equation (19) has been used recently for control design [13]; however, no information is available to date on how to systematically design IT2 FLCSs using YWM (x). We tried to obtain stability analysis for (19) but were unsuccessful. III. N EW I NFERENCE M ETHOD FOR IT2 TSK C ONTROL D ESIGNS To obtain stability conditions for an FLCS using rigorous mathematical analyses, closed-form equations are required; hence, in this section, we introduce the following new inference engine: M i M i f (x)yi i=1 f (x)yi + n i=1 YTSK/NEW (x) = m M i (22) i M i=1 f (x) i=1 f (x) i

p=1 n

3

i i=1 f (x)yi M i i=1 f (x)

where yi is given by (1), and f i (x) and f (x) are given by (2) and (3), respectively (if M = 1, then m + n = 1). Observe that m and n are design parameters that weight the sharing of lower and upper firing levels of each fired rule and can be tuned during the design of this new TSK system. Observe i also that if all uncertainty disappears so that f i (x) = f (x), then (22) reduces to a T1 TSK FLCS in which we can set m + n = 1. There is also a connection between YTSK/NEW (x) and YWM (x). Proposition 1: If m and n are independent parameters that do not depend on the inference process, then YTSK/NEW (x) is derivable 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 m and n is given next. First, we derive bounds for the tuning parameters of the plant. Then, for a given m and n, we



 ⎤ i M i M i i f (x) − f (x) i=1 1⎣ i=1 f (x)(yi − y1 ) · i=1 f (x)(yM − yi ) ⎦ − × 4 M f i (x) · M f i (x) M f i (x)(yi − y1 ) + M f i (x)(yM − yi ) i=1 i=1 i=1 i=1  ⎤ ⎡ M i M i M i i f (x)(y − y ) · f (x)(y − y ) i=1 f (x) − f (x) 1⎣ i 1 M i i=1 i=1 ⎦ + × 4 M f i (x) · M f i (x) M f i (x)(yi − y1 ) + M f i (x)(yM − yi ) i=1 i=1 i=1 i=1 ⎡ M

(13)

4

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

mathematically explain how to identify the IT2 TSK consequent parameters, and finally, we present an algorithm to obtain suitable tuning parameters for the plant. We prove in Appendix I-B that m and n are given by 1 + g1 2 1 n = + g2 2

m=

(23) (24)

where g1 and g2 are given by (A.12) and (A.13), respectively. Since we are dealing with stability analysis, it is assumed that the parameters of the membership functions are known (identifying the membership functions is not within the scope of this paper). When IT2 TSK is used for practical control design, states x(k), x(k − 1), . . . , x(k − n + 1) are physical quantities, e.g., displacement, velocity, and acceleration. Therefore, for a specific problem, the lower and upper bounds of these  states can be determined by the designer. Moreover, yi = np=1 aip x(k − p + 1) corresponds to the output of rule i and represents a physical quantity arguments can n (similar i be made for the two terms v a x(k − p + 1) and p=1 i,p p n i w a x(k − p + 1) in g and g ). Hence, regardless of 1 2 p=1 i,p p i whether ap is known or not, the designer can establish the lower and upper bounds on yi as well as another similar terms in g1 and g2 (the range of variation is known). The lower and upper bounds on g1 and g2 for all rules can therefore be determined, i.e., g1min ≤ g1 ≤ g1max

(25)

≤ g2 ≤

(26)

g2min

g2max .

Then, using (23) and (24) in (25) and (26), the bounds on m and n are given as 1 1 + g1min ≤ m ≤ + g1max ≡ mmax 2 2 1 1 ≡ + g2min ≤ n ≤ + g2max ≡ nmax . 2 2

mmin ≡

(27)

nmin

(28)

Recently, for type-2 fuzzy systems, fuzzy clustering and subtractive clustering have been proposed for finding the system parameters. Subsequently, similar to the method in [25], in our method, it is assumed that the parameters of the input membership functions are known by using a predefined clustering method, and we identify the TSK consequent parameters. Assume that the plant is modeled as If x1 is F1i and x2 is F2i and · · · and xn is Fni Then yi = ai0 + ai1 x1 + ai2 x2 + · · · + ain xn

(29)

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

 p xi1 , xi2 , . . . , xin , Y i i=1

(30)

where [xi1 , xi2 , . . . , xin ] 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  M j j f i a0 + aj1 xi1 + aj2 xi2 + · · · + ajn xin j=1 Yi =m M j j=1 f i

 M j j j i j i j i a f + a x + a x + · · · + a x i n n 0 1 1 2 2 i=1 +n (32) M j j=1 fi where i = 1, . . . , p. Let fj v ji = M i vi j =

j j=1 f i j fi M j j=1 fi

(33) .

 i i f (x) − f (x) i=1 1 α(x) = − M M i i 4 i=1 f (x) · i=1 f (x)

 

  M i n i M n i i f (x) v a x(k − p + 1) · f (x) w a x(k − p + 1) i,p i,p p p i=1 p=1 i=1 p=1

  

  × i M n M n i i i x(k − p + 1) f (x) v a x(k − p + 1) + f (x) w a i,p i,p p p i=1 p=1 i=1 p=1  M i i i=1 f (x) − f (x) 1 β(x) = M M i i 4 i=1 f (x) · i=1 f (x)

 

  n M i M n i i i f (x) v a x(k − p + 1) · f (x) w a x(k − p + 1) i,p i,p p p i=1 p=1 i=1 p=1

  

  × i M n M n i i i x(k − p + 1) f (x) v a x(k − p + 1) + f (x) w a i,p i,p p p i=1 p=1 i=1 p=1

(34)

M

(20)

(21)

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

5

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

M



 v ji aj0 + aj1 xi1 + aj2 xi2 + · · · + ajn xin

j=1

+n

M



 vi j aj0 + aj1 xi1 + aj2 xi2 + · · · + ajn xin . (35)

j=1

Define φ, φ, and θ as in (36)–(38), respectively, shown at the bottom 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 the parameters of the input membership functions. Finally, the error vector is defined as e ≡ Y − Aθ, and the total error et , which is the sum of squares of the components of e, is defined by et ≡

p

e2i .

(40)

i=1

Our proposed algorithm to obtain the tuning parameters m and 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 error using (31)–(40) repeat repeat 1. n ← nmin 2. Solve for θ from (39), and find the total error from (40) 3. If the new error is less than the error found in the previous 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. S TABILITY OF SISO IT2 TSK FLCSs In this section, we introduce a model for stability analysis of SISO2 IT2 TSK FLCS. SISO systems are considered because of the variety of applications in computing systems and bioengineering [27], [28]. To begin, a controller structure is introduced; then, a model is introduced for a closed-loop control system, after which mathematical analyses are established for the design of stable IT2 TSK FLCSs. A. Controller Fig. 1 shows a controller in which the inputs are the states x(k) and the output is u(k). For this system, the general ith rule has the following form: ith controller rule:  i and x(k − 1) is C  i and · · · If x(k) is C 1 2 ni and x(k − n + 1) is C i i i Then u (k + 1) = c1 x(k) + c2 x(k − 1) + · · · + cin x(k − n + 1)

(41)

 i represents the T2 FS of input state j where i = 1, 2, . . . , Q, C j i of the ith rule, and cj is the jth coefficient of the output function 2 When we refer to SISO, “input” is considered the controller output signal and “output” is the plant output (with both input and output being scalars).

1 v 11 · · · v M v 11 x11 · · · v M 1 1 x1 ⎢ v 1 · · · v M v 1 x2 · · · v M x2 2 2 1 2 1 ⎢ 2 ⎢ ······ φ≡⎢ .. .. .. .. .. ⎢ .. ⎣ . . . . . . 1 p M p v 1p · · · v M v x · · · v p p 1 p x1 ⎡ 1 1 v 1 · · · vM v 11 x11 · · · v M 1 1 x1 ⎢ v 1 · · · v M v 1 x2 · · · v M x2 2 2 1 2 1 ⎢ 2 ⎢ ······ φ≡⎢ .. .. .. .. .. ⎢ .. ⎣ . . . . . . 1 p M p v 1p · · · v M v x · · · v p p 1 p x1  1  1 M 1 M T θ ≡ a0 , . . . , aM . 0 , a1 , . . . , a1 , . . . , an , . . . , an

v 11 x1n v 12 x2n

··· ···

.. . v 1p xpn

.. . ···

v 11 x1n v 12 x2n

··· ···

.. . v 1p xpn

.. . ···

1 ⎤ vM 1 xn 2 ⎥ vM 2 xn ⎥ ⎥ ⎥ .. ⎥ . ⎦ M p v p xn 1 ⎤ vM 1 xn 2 ⎥ vM 2 xn ⎥ ⎥ ⎥ .. ⎥ . ⎦

(36)

(37)

p vM p xn

(38)

6

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

for rule i. Applying (22) to (41), the controller output u(k) can be expressed as Q i Q v (x)ui (k + 1) v i (x)ui (k + 1) u(k) = m i=1Q + n i=1Q i i i=1 v (x) i=1 v (x) (42) where

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

v i (x) = μCi (x(k)) ∗ · · · ∗ μCi (x(k − n + 1))

(43)

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

v i (x) = μCi (x(k)) ∗ · · · ∗ μCi (x(k − n + 1))

(44)

f i (x) = μFi (x(k)) ∗ · · · ∗ μFi (x(k − n + 1)) ∗ μBi (u(k))

n

1

n

1



By using (22), the output of the closed-loop system, i.e., x(k + 1), can be expressed as   i i i i m M n M i=1 f x (k + 1) i=1 f x (k + 1) + x(k + 1) = M i M i i=1 f i=1 f (49) i

i

n

1

(50)



and m and n are tuning parameters of the controller. Substituting the consequent part of (41) into (42), (42) can be written as Q n i i i=1 j=1 v (x)cj x(k − j + 1)  u(k) = m k1 Q n i i i=1 j=1 v (x)cj x(k − j + 1) + n (45) k2 where 

 v i (x) k1 ≡ Q i=1 Q k2 ≡ 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, where the plant and the controller are each IT2 TSK A2-C0 models. For a closed-loop system, the controller signal u(k) is incorporated as an input to the plant. The general ith rule for the plant is If x(k) is F1i and x(k − 1) is F2i and · · · i and x(k − n + 1) is Fi and u(k) is B n

Then xi (k + 1) = ai1 x(k) + · · · + ain x(n − k + 1) + bi u(k) (47) where i = 1, . . . , M , xi (k + 1) is the output of the ith plant rule, Fji represents the T2 FS of input state j of the ith rule,  i represents the T2 FS of the plant input, and ai is the jth B j coefficient of the output function for rule i. The control rules are the same as (41) with Q being the number of rules, and u(k) is given by (45). Substituting u(k) from (45) into the consequent of (47), the output of the ith plant rule, i.e., xi (k + 1), is given by n

 i  aj x(k − j + 1) x (k + 1) =

i

+b m

Q n 

l=1

j=1

Q n + bi n

l=1

j=1

n

1

(51) Applying (48) to (49), x(k + 1) can be expressed as  n i i m M i=1 j=1 f aj x(k − j + 1) x(k + 1) = M i i=1 f M Q n i l i l i=1 j=1 f v b cj x(k − j + 1) l=1 + mm M i k1 i=1 f M Q n i l i l i=1 j=1 f v b cj x(k − j + 1) l=1  + mn  i k1 M i=1 f n  i i n M i=1 j=1 f aj x(k − j + 1) + M i i=1 f M Q n i l i l i=1 j=1 f v b cj x(k − j + 1) l=1  + nm  i k2 M i=1 f M Q n i l i l i=1 j=1 f v b cj x(k − j + 1) l=1  + nn .  i k2 M i=1 f (52) Next, define n × n matrices Ai and B i,l as follows: ⎛ i ⎞ a1 ai2 · · · ain−1 ain ⎜ 1 0 ··· 0 0 ⎟ ⎜ ⎟ 0 1 ··· 0 0 ⎟ Ai = ⎜ ⎜ . .. . . .. .. ⎟ ⎝ .. . . . . ⎠ ⎛

0

···

0

bi c1l

⎜ 1 ⎜ 0 B i,l = ⎜ ⎜ . ⎝ .. 0

1

0

0 1 .. .

··· ··· ··· .. .

l bi cn−1

0

···

1

bi c2l

0 0 .. .

⎞ bi cnl 0 ⎟ ⎟ 0 ⎟ .. ⎟ . ⎠

(53)

0

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

v l cjl x(k − j + 1) k1

v l cjl x(k − j + 1) . k2

f (x) = μFi (x(k)) ∗ · · · ∗ μFi (x(k − n + 1)) ∗ μBi (u(k)) .

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

i

j=1

i

(54)

By using (52) and (53), it is straightforward to show that x(k + 1) in (54) can be written as (48)

x(k + 1) = Cx(k)

(55)

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

Let the first bracketed term of Z 1 in (A.16) be denoted as Z 1,1 , i.e.,   j i T m M m M 1 j=1 f Aj i=1 f Ai (61) Z 1,1 = P − P. M i M j 36 f f i=1 j=1

where C=

m

M i=1

M

f i Ai

+

n

M

fi M Q

i=1

+ mm

i=1

l=1

M

i=1

M

i

f Ai

i=1

f

i

f i v l B i,l

k1 i=1 f i M Q i l l=1 f v B i,l  + mn i=1  i k1 M i=1 f M Q i l l=1 f v B i,l + nm i=1  i k2 M i=1 f M Q i l  i=1 l=1 f v B i,l + nn .  i k2 M i=1 f

It can be expressed as ⎛ ⎞  M M M

1 Z 1,1 ⎝ f i f j⎠= P . (62) f if j m2 AT i P Aj − 36 i=1 j=1 i,j=1

(56)

Although (55) may look like a linear system, it is not because i C depends on x through the dependences of f i and f on x. C. Stability of Closed-Loop System The stability of T1 TSK FLCSs using fuzzy Lyapunov function (FLF) has been addressed in several works [29], [30]. Most notably, Tanaka et al. [29] proposed an FLF method composed of multiple Lyapunov functions to obtain the stability conditions for a T1 TSK FLCS. They also presented a new design control methodology using parallel-distributed compensation. The proposed FLF methodology in [29] provided relaxed stability conditions for a T1 TSK FLCS. However, the design process required the time derivatives of premise membership functions, and it is not always possible to derive such derivatives from the system states, which limits the use of this method. Because the same drawback will limit the use of this method for IT2 TSK FLCSs, in this paper, we introduce a quadratic Lyapunov function to derive stability conditions for IT2 TSK FLCSs. Our Lyapunov function is V (x(k)) = xT (k)P x(k), where P is a positive-definite matrix [31]. ΔV (x(k)) is given by ΔV (x(k)) = xT (k + 1)P x(k + 1) − xT (k)P x(k). (57) By using (55), ΔV (x(k)) can be expressed as ΔV (x(k)) = xT (k)Zx(k)

(58)

Z ≡ C TP C − P

(59)

where

and C is given by (56). Z has 36 components and can be expressed as Z ≡ Z1 + Z2 + Z3

7

(60)

where Z 1 , Z 2 , and Z 3 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)) are made negative (equivalently, all the components of Z are made negative definite), the result will be an asymptotically stable system.

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