New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models

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New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models Jun Chen a, Shengyuan Xu a,∗, Baoyong Zhang a, Yuming Chu b, Yun Zou a

Q1

a b

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR China School of Science, Huzhou Teachers College, Huzhou 313000, PR China

a r t i c l e

i n f o

Article history: Received 22 May 2014 Revised 25 August 2015 Accepted 14 September 2015 Available online xxx Keywords: Stabilization Membership functions Continuous-time Takagi–Sugeno model Linear matrix inequality

a b s t r a c t This paper deals with the problems of stability and stabilization for continuous-time Takagi– Sugeno (T–S) models. By employing the order relation information of the membership functions (MFs), the transformation matrix and the pseudo-MF vector are obtained. The conventional continuous-time T–S fuzzy model is then transformed to a new compact T–S model, based on which relaxed stability and stabilization conditions are derived in the form of linear matrix inequalities. An improved approach to constructing the transformation matrix is presented, which could take full advantage of the order relation information of the MFs and further reduce the conservatism. Numerical examples are provided to illustrate the effectiveness of the proposed approach. © 2015 Published by Elsevier Inc.

1

1. Introduction

2

The T–S fuzzy model provides an effective mathematical platform to describe complex nonlinear systems, which can be easily and systematically obtained by the so-called sector nonlinearity approach [28]. Since the T–S fuzzy model is proposed [27], the stability and stabilization problem, as one of the most foundational research topics of nonlinear systems, has attracted considerable attention and various kinds of results have been reported. The Lyapunov direct method is one of the popular approaches for addressing the aforementioned problem. In the literature, most of early results are obtained based on the quadratic Lyapunov function and the parallel distributed compensation (PDC) control law [28]. In such results, a common positive-definite matrix is required to satisfy a number of conditions, and thus the results are often much conservative. Thereafter, much effort has been devoted to obtaining less conservative conditions by improving the Lyapunov functions [4–6,9–11,17,19,22,29,35], reforming the control laws [3,4,9,10,29] or relaxing fuzzy summations [12,23,30,35]. In [17], less conservative conditions were achieved via fuzzy Lyapunov functions and a new strategy was given to find an inner estimate of the domain of attraction. In [2,10], local non-quadratic stability and stabilization conditions were provided for continuous-time T–S fuzzy models obtained by the sector nonlinearity approach [28], instead of pursuing global stability and stabilization conditions. In [31,32], a new slack variable technique satisfying the homogenous polynomial setting was developed, which could efficiently accelerate the convergence of the asymptotically necessary and sufficient stabilization conditions. In [16], a general framework was established for the use of non-quadratic Lyapunov functions and relaxed stabilization conditions were developed by combining delayed controllers.

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

∗

Corresponding author. Tel.: +86 25 84303027. E-mail address: [email protected], [email protected] (S. Xu).

http://dx.doi.org/10.1016/j.ins.2015.09.036 0020-0255/© 2015 Published by Elsevier Inc.

Please cite this article as: J. Chen et al., New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.09.036

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It is well known that the MFs play a very critical role in stability and stabilization conditions for T–S models [28]. However, the aforementioned conditions only utilize the convex properties of the MFs and are almost independent of the structural information of the MFs. Therefore, these conditions usually fail as long as there exists an unstable subsystem in the T–S fuzzy models. To the best of the authors’ knowledge, there are only a small number of papers studying the T–S model by considering the structural information of the MFs. In [24], the knowledge of the MFs’ shape was taken into account by considering upper bounds of their cross products (interpreted as an overlap measure). Narimani et al. [21] further generalized the results in [24] by considering both upper and lower bounds of their cross products. In [1], the order relation information of the MFs was utilized, and relaxed stability and stabilization conditions were obtained based on the quadratic and piecewise quadratic Lyapunov functions in the whole and local state space, respectively. Inspired by the above-mentioned literature, especially [1,17], the problem of stability and stabilization is further investigated by utilizing the order relation information of the MFs for the continuous-time T–S fuzzy models. In comparison with [1], an improved approach to constructing the transformation matrix F(α ) is proposed to make the best of the order relation information of the MFs. Meanwhile, the non-quadratic Lyapunov functions are employed, instead of the quadratic and piecewise Lyapunov functions. In comparison with [17], the order relation information of the MFs is taken into consideration which transforms the conventional T–S fuzzy model into a compact form and then less conservative conditions are obtained. Generally speaking, the new transformed T–S fuzzy model actually provides a new platform to analyze and synthesize nonlinear systems, in which the pseudo-MFs are closer to zero and the influence of each pseudo-MF on the stability of the whole system is relatively reduced. The remainder of this paper is organized as follows. In Section 2, preliminaries are introduced. In Section 3, an improved approach to constructing the transformation matrix is given. In Section 4, sufficient stability conditions are presented. In Section 5, stabilization conditions are derived with the non-PDC control law. Finally, a conclusion is drawn in Section 6.

42

Notations. Throughout this paper, He(X) denotes X + X T for any square matrix X. Ir denotes the set {1, 2,…, r}. I denotes the    identity matrix of appropriate dimension. Xz(t) and Xzz(t) denote ri=1 hi (z(t ))Xi and ri=1 rj=1 hi (z(t ))h j (z(t ))Xi j , respectively. The notation X ≥ Y (respectively, X > Y) means that the matrix X − Y is positive semi-definite (respectively, positive definite). The notation X  0 (respectively, X  0) means that every entry of matrix or vector X is nonnegative (respectively, positive). Co(·) denotes the convex hull.

43

2. Preliminaries

38 39 40 41

44

Consider the continuous-time T–S fuzzy model:

x˙ (t ) =

r 

hi (z(t ))(Ai x(t ) + Bi u(t )) = Az(t ) x(t ) + Bz(t ) u(t )

(1)

i=1

45 46 47

where r denotes the number of the rules; z(t ) ∈ R p the premise vector; x(t ) ∈ Rn the state vector; u(t ) ∈ Rm the control signal; Ai ∈ Rn×n and Bi ∈ Rn×m the local subsystem matrices; hi (z(t)) the normalized MF of local system Ai x(t ) + Bi u(t ), which satisfies the properties:

hi (z(t )) ≥ 0,

r 

hi (z(t )) = 1,

i=1

48

r 

h˙ i (z(t )) = 0.

(2)

i=1

For convenience, the following notations are defined:

˜ 1, . . . , h ˜  [h ˜1 . . . h ˜ r˜ ]T , Sh  {h1 , . . . , hr }, S ˜  {h ˜ r˜ }, hi  hi (z(t )), h  [h1 . . . hr ]T , h h ˜i  h ˜ i (z(t )), Li  AT P + PAi , X˜z(t )  h i

r˜  i=1

49 50 51 52 53 54

56 57 58 59

r˜ r˜  

˜ i (z(t ))h ˜ j (z(t ))X˜i j h

i=1 j=1

˜ denote the MF and pseudo-MF vectors, respectively. The elements in S ˜ are defined as h − h when hk , hl ∈ Sh where h and h k l h satisfy the relation hk ≥ hl (there is no intermediate hm ∈ Sh such that hk ≥ hm ≥ hl ), or hk0 − hk1 − · · · − hkm when hk0 , . . . , hkm ∈ Sh satisfy the relation hk0 ≥ hk1 + · · · + hkm (there is no other element hkm+1 ∈ Sh such that hk0 ≥ hk1 + · · · + hkm + hkm+1 ), ˜  0 and hi ∈ Co(S ˜ ), ∀hi ∈ Sh . Therefore, there or hk ∈ Sh when hk is no greater than any other MFs in Sh . It can be seen that h h ˜ i.e., hi = r˜ fi j h ˜ j. exists a matrix F = [( fi j )]rטr  0, called the transformation matrix, satisfying h = F h, j=1 ˜ the new transformed T–S model is obtained from (1): Under the transformation h = F h,

x˙ (t ) = A˜ z(t ) x(t ) + B˜z(t ) u(t ) 55

˜ i (z(t ))X˜i , X˜zz(t )  h

(3)

  ˜ j (z(t ))B˜ j , A˜ j = r fi j Ai and B˜ j = r fi j Bi . It is easily seen that in the where A˜ z(t ) = r˜j=1 h˜ j (z(t ))A˜ j , B˜z(t ) = r˜j=1 h i=1 i=1 transformed T–S fuzzy model, the subsystems are linear combinations of the original subsystems, i.e., A˜ j x(t ) + B˜ j u(t ) = r i=1 f i j (Ai x(t ) + Bi u(t )) and that the coefficients are nonnegative. Since the pseudo-MFs are closer to zero than the original MFs, the influence of each pseudo-MF is weakened on the stability of the whole system. In addition, the sum of the pseudo-MFs  ˜ j (z(t )) = 1 usually holds. is no longer equal to one, that is, the inequality r˜j=1 h Please cite this article as: J. Chen et al., New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.09.036

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3

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The main objective of this paper is to investigate the stability and stabilization conditions for the transformed T–S fuzzy model (3) via the non-quadratic Lyapunov functions. At the end of this section, two lemmas are presented, which are useful in the development of the main results.

63

˜ The following equation Lemma 1. Consider the continuous-time T–S fuzzy model (1) under the transformation h = F h.

60 61

r˜ 

˜ j fj = 1 h

j=1

r

64

holds, where f j =

65

˜ we have Proof. With h = F h, r 

i=1 f i j .

r˜ r  

hi =

i=1

˜j = fi j h

i=1 j=1

r˜ 

˜j h

j=1

r 

r˜ 

fi j =

i=1

˜ j f j = 1. h

j=1



66

For simplicity, we define fz(t ) 

67 68 69

r˜

˜ j=1 h j f j .

Lemma 2 ([28]). Suppose symmetric matrices Xij , i, j ∈ Ir of appropriate dimensions. The inequality about the fuzzy summation Xzz(t) < 0 holds if the following inequalities hold:

Xi j + X ji < 0,

i ≤ j,

i, j ∈ Ir .

74

Remark 1. There are various ways [12,23,28,30] to gain relaxed conditions with respect to the negativity of the fuzzy summations. Nevertheless, for fair comparison, Lemma 2 is employed, which preserves simplicity and efficiency without introducing any additional slack variables. Note that Lemma 2 can be further extended to the negativity of the pseudo-MF fuzzy summations. For  ˜ r˜ ˜ ˜ ˜ ˜ ˜ instance, the inequality X˜zz(t ) = ri=1 j=1 hi h j Xi j < 0 holds if the following inequalities Xi j + X ji < 0, i ≤ j, i, j ∈ Ir˜ hold, where



˜ ˜ X s are symmetric matrices and h s are the pseudo-MFs.

75

3. The approach to constructing F(α)

70 71 72 73

76 77 78 79 80 81

ij

i

The transformation matrix plays a very important role in applications of the order relation information of the MFs. In this section, a new approach to constructing the transformation matrix F is given, which improves the method in [1]. When there exist multiple order relations in the MFs, the transformation matrix F is usually not unique. To illustrate the point, an example is provided in the following. Example 1. Suppose in the 3-rule T–S fuzzy model (1), there exist order relations in the MFs such as h1 ≥ h2 , h1 ≥ h3 . According to the method in [1], one has

⎡

⎡ ⎤

h1 ⎣ h = h2 ⎦, h3 82 83 84 85

87

⎡

89 90

0.5 ⎣ 0 F= 0

0.5 0 0

0.5 1 0

⎤

0.5 0 ⎦. 1

(4)

α11

α12

α11

α12

0 0

1 0

0 1

⎤ ⎦

(5)

where α 11 and α 12 are free parameters satisfying α11 + α12 = 1. Specially, when α11 = 0 or α12 = 0, the transformation matrices F1 and F2 are gained, respectively:

⎡

0 F1 = ⎣0 0 88

⎡

Hence the new transformed system matrices are A˜ 1 = 0.5A1 , A˜ 2 = 0.5A1 , A˜ 3 = 0.5A1 + A2 and A˜ 4 = 0.5A1 + A3 . Then according to Theorem 1 in [1], the sufficient stability conditions are 0.5L1 < 0, 0.5L1 < 0, 0.5L1 + L2 < 0 and 0.5L1 + L3 < 0, which are obviously less conservative than the quadratic-based conditions L1 < 0, L2 < 0 and L3 < 0. In fact, F in (4) is a special case of the general transformation matrix F(α ), given by

F (α) = ⎣ 0 0 86

⎤

h1 − h2 ⎢h1 − h3 ⎥ ˜ =⎢ ⎥ h ⎣ h2 ⎦, h3

1 0 0

0 1 0

⎤

1 0⎦ , 1

⎡

1 F2 = ⎣0 0

0 0 0

1 1 0

⎤

0 0⎦. 1

(6)

Then, in the case of α11 = 0, the transformed system matrices are A˜ 1 = 0 (deleted), A˜ 2 = A1 , A˜ 3 = A2 and A˜ 4 = A1 + A3 and the corresponding stability conditions are L1 < 0, L2 < 0 and L1 + L3 < 0; in the case of α12 = 0, the transformed system matrices are A˜ 1 = A1 , A˜ 2 = 0 (deleted), A˜ 3 = A1 + A2 and A˜ 4 = A3 and the corresponding stability conditions are L1 < 0, L1 + L2 < 0 and Please cite this article as: J. Chen et al., New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.09.036

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L3 < 0. Obviously, the conservatism of these three conditions is different from each other. Through this example, it is seen that appropriately constructing the transformation matrix could make best of the order relation information of the MFs and further reduce the conservatism of stability conditions. In the following an improved approach to constructing F(α ) is summarized. Suppose the T–S fuzzy model (1) has r rules and there exist the following equations

hi =

si 

αi j (h˜ i( j,1) + · · · + h˜ i( j,k) + · · · ), i ∈ Ir

j=1

96 97 98

si where hi ∈ Sh , h˜ i( j,k) ∈ Sh˜ , α ij ≥ 0, α = 1 and si is the maximum value of the number of groups {h˜ i( j,1) , . . . , h˜ i( j,k) , . . .}, j=1 i j ˜i ˜i satisfying h + ··· + h + · · · = hi . Therefore, the transformation matrix F (α) = [( fi j )] ∈ Rrטr is obtained, where r˜ ≥ r, fi j ∈ ( j,1) ( j,k) r˜ {0, αi1 , . . . , αisi } and j=1 fi j ∈ {1, 2, · · · }. In a special case of si = 1, αi1 = 1.

100

Remark 2. If αi j = 1/si , this approach is reduced to that in [1]. If f j = 0 in F, the corresponding subsystem A˜ j x(t ) + B˜ j u(t ) could be deleted from the transformed T–S model. Then, the transformed T–S model is simplified and the number r˜ is reduced. For

101

instance in Example 1, the transformation matrix F1 could be simplified to [0

102

sequel we always assume fj > 0, j ∈ Ir˜ without loss of generality.

99

1

0 1 0

0

103 104

˜ is globally asymptotically stable if Theorem 1. The equilibrium of the unforced T–S model (1) under the transformation h = F (α)h there exists a common symmetric matrix P > 0 such that the following inequalities

A˜ Tj P + P A˜ j < 0, 105

1 0] with r˜ = 3 since f1 = 0. Therefore, in the 1

hold, where A˜ j =

j ∈ Ir˜

(7)

r

i=1 f i j Ai .

107

Proof. The stability conditions could be immediately obtained via the quadratic Lyapunov function V (x) = x(t )T Px(t ) for the transformed T–S fuzzy model (3) with u(t ) = 0. 

108

Example 2. Consider the following nonlinear model [1]:

106





ω01 ω02 − 2 x˙ 1 = x˙ 2 8 − 15ω12 ω03 ω01

109 110 111

where ω01 =

1 , 1+exp(−x1 )



A1 =



112 113 114 115 116

−10ω13 − 9ω01

 

x1 , x2

(8)

1 1 1 2 2 3 3 ω03 = 1+exp (x1 ) , ω1 = 1 − ω0 , ω1 = 1 − ω0 , ω1 = 1 − ω0 . 1 2 3 1 2 3 1 2 3 As in [1], the MFs are defined as h1 (x1 ) = ω0 ω0 ω0 , h2 (x1 ) = ω0 ω0 ω1 , h3 (x1 ) = ω0 ω1 ω0 , h4 (x1 ) = ω01 ω12 ω13 , h5 (x1 ) = 1 ω1 ω02 ω03 , h6 (x1 ) = ω11 ω02 ω13 , h7 (x1 ) = ω11 ω12 ω03 , h8 (x1 ) = ω11 ω12 ω13 . Then the 8-rule unforced T–S model (1) is obtained, where

A5 =

ω02 =

−2ω11 − ω02 − 3

1 , 1+exp(−x1 −1)















−1 8

−4 , −9

A2 =

−1 8

−4 , −19

A3 =

−2 −7

−3 , −9

A4 =

−2 8

−3 , −19

−2 8

−6 , 0

−2 A6 = 8

−6 , −10

−2 A7 = 8

−5 , 0

−2 A8 = 8

−5 . −10















˜ 1 = h1 − h4 , h ˜2 = Among the MFs, there exist order relations h1 ≥ h4 , h6 ≥ h4 , h5 ≥ h3 and h5 ≥ h8 . The pseudo-MFs are defined as h ˜ 1 , . . . , h˜ 9 }. ˜ 3 = h5 − h3 , h˜ 4 = h5 − h8 , h ˜ 5 = h4 , h ˜ 6 = h3 , h ˜ 7 = h8 , h˜ 8 = h2 , h˜ 9 = h7 . So one has S = {h1 , . . . , h8 }, S ˜ = {h h6 − h4 , h h h Ordinary stability conditions fail for this model since A3 is unstable. ˜ 3 + h˜ 6 ) + α52 (h ˜ 4 + h˜ 7 ), where h˜ 3 + h ˜ 6 = h5 , h ˜4 + According to the order relation information of the MFs, one has h5 = α51 (h ˜h7 = h5 and α51 + α52 = 1. Then the transformation matrix F(α ) is obtained:

⎡

1 ⎢0 ⎢ ⎢0 ⎢ ⎢0 F (α) = ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

0 0 0 0 0 1 0 0

0 0 0 0

0 0 0 0

α51

α52

0 0 0

0 0 0

1 0 0 1 0 1 0 0

0 0 1 0

0 0 0 0

α51

α52

0 0 0

0 0 1

0 1 0 0 0 0 0 0

⎤

0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ 0⎥ ⎥ 1⎦ 0

(9)

Please cite this article as: J. Chen et al., New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.09.036

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5

Fig. 1. Stability regions for a and b with (◦) for Theorem 1 [1], (◦ ·) for Theorem 1 and (◦ · ) for Theorem 2.

117

For comparison purpose, A3 is parameterized as



−2 + a A3 = −7 + b 118 119 120 121 122 123 124 125 126

−3 −9



(10)

where a and b are two parameters changing in the intervals [−1, 4] and [−1, 3], respectively. The checked results of guaranteeing the stability of system (8) are shown in the a-b plane in Fig. 1 with (◦) for Theorem 1 [1] and (◦ ·) for Theorem 1 in this paper. It can be easily seen that the a-b regions are distinctly expanded by Theorem 1 in this paper. Note that in Fig. 1, α51 = 1 and α52 = 0. Remark 3. Since the transformation matrix F(α ) is so important, a question naturally arises: how to determine the free variables αi j s? There is different conservatism among stability conditions obtained with different values of αi j s. But the conservatism of these conditions cannot be simply compared with each other in ignorance of the system matrices. This point can be seen from Example 1. Therefore, the optimal values of αi j s are dependent on both the order relations in the MFs and the system matrices. Up to now there is not a systematic method of determining the free variables αi j s. Nevertheless, it is not difficult as it seems

127

because the conservatism of these conditions usually changes continuously as αi j s change continuously.

128

4. Stability conditions

129

In Section 3, the quadratic Lyapunov function has been employed to study the stability of (3), which is actually a special case of the non-quadratic Lyapunov functions [9,14,16–18,31]. In this section, stability conditions for (3) are further addressed via the following non-quadratic Lyapunov function:

130 131

V (x) =

r 

hi (z(t ))xT Pi x = xT P˜z(t ) x

(11)

i=1

132 133 134

 ˜ j (z(t ))P˜j and P˜j = r fi j Pi . It is easily seen that P˜j > 0 if Pi > 0. Assume D ∈ Rn is the compact set of the where P˜z(t ) = r˜j=1 h i=1 state variables in which the continuous-time T–S model (1) exactly approximates the original nonlinear system. And define the region R ∈ Rn of the state variables as

R = {x(t ) ∈ Rn

| |h˜˙ ρ | ≤ φ˜ ρ , ρ ∈ Ir˜ }.

(12)

Please cite this article as: J. Chen et al., New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.09.036

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˜ is asympTheorem 2. For given φ˜ ρ > 0, ρ ∈ Ir˜ , the equilibrium of the unforced T–S model (1) under the transformation h = F (α)h totically stable in D R if there exist positive definite matrices Pi , i ∈ Ir , and symmetric matrices Mi j , i, j ∈ Ir such that

γ˜i j ρ + γ˜ji ρ ≥ 0, i ≤ j, i, j, ρ ∈ Ir˜ , 137 138 139 140 141

γ˜i j + γ˜ ji < 0, i ≤ j, i, j ∈ Ir˜ ρ

(14)

˜ i j , γ˜i j = A˜ T P˜j + P˜j A˜ i + = fi f j P˜ρ + fρ M i

hold, where γ˜i j r r k=1 l=1 f ki f l j Mkl .

r˜

r r r ˜ ρ ˜ ˜ ˜ ρ =1 φρ γ˜i j , f j = k=1 fk j , Ai = k=1 fki Ak , Pj = k=1 fk j Pk and Mi j =

Proof. Consider the non-quadratic Lyapunov function candidate (11). Along the trajectory of the unforced T–S model (3), the time derivative of (11) is computed as

V˙ (x) = xT (A˜ Tz(t ) P˜z(t ) + P˜z(t ) A˜ z(t ) + 142

(13)

r˜  ˜˙ ρ P˜ρ )x. h

(15)

ρ =1

˜ one has By considering the slack matrices Mij and the transformation h = F (α)h,

Mzz(t ) =

r  r 

hi h j Mi j =

i=1 j=1

=

r˜  r˜ 

r  r  r˜ 

˜k fik h

i=1 j=1 k=1

˜ kh ˜l h

k=1 l=1

r  r 

˜ l Mi j f jl h

l=1

fik f jl Mi j =

i=1 j=1

r˜ 

r˜  r˜ 

˜ kh ˜lM ˜ kl h

k=1 l=1

˜ zz(t ) . =M 143

From Lemma 1, one has

(16) r˜

˙

˜ ρ =1 hρ fρ = 0. Therefore,

V˙ (x) = xT A˜ Tz(t ) P˜z(t ) + P˜z(t ) A˜ z(t ) +

= xT A˜ Tz(t ) P˜z(t ) + P˜z(t ) A˜ z(t ) + 144 145

r˜

˙

˜ ˜ ρ =1 hρ fρ Mzz(t ) = 0. Then, (15) can be rewritten as

 r˜  ˜˙ ρ (P˜ρ + fρ M ˜ zz(t ) ) x h

ρ =1

 r˜  ˜˙ ρ ( fz(t ) fz(t ) P˜ρ + fρ M ˜ zz(t ) ) x. h

(17)

ρ =1

ρ ˜ zz(t ) ≥ 0. Then in the region D Note that fz(t ) = 1. According to Lemma 2, (13) implies γ˜zz(t ) = fz(t ) fz(t ) P˜ρ + fρ M

rewritten as

V˙ (x) = xT A˜ Tz(t ) P˜z(t ) + P˜z(t ) A˜ z(t ) +

≤ xT A˜ Tz(t ) P˜z(t ) + P˜z(t ) A˜ z(t ) + =

r˜ r˜  



R, (17) can be

 r˜  ˜˙ ρ γ˜ ρ x h zz(t )

ρ =1 r˜ 

ρ =1

 φ˜ ρ γ˜zzρ(t ) x.

˜ ih ˜ j xT (t )γ˜i j x(t ). h

(18)

i=1 j=1

146 147 148 149 150 151 152 153 154 155 156 157 158

From (14), we have γ˜zz(t ) < 0. Thus, V˙ (x) < 0 is guaranteed, ∀x = 0. Therefore, according to the Lyapunov stability theory, the equilibrium of the unforced T–S model (1) is asymptotically stable in D R.    ρ ˜ i j ) and γ˜i j = f j (A˜ T P + P A˜ i ) + rρ˜ =1 φ˜ ρ fρ ( fi f j P + M ˜ i j ). StabilRemark 4. If Pi = P, then P˜ρ = ri=1 fiρ Pi = fρ P, γ˜i j = fρ ( fi f j P + M i ity conditions of Theorem 2 are reduced to those of Theorem 1 with Mkl = −P. On the other hand, conditions of Theorem 1 in [17] could be recovered from those of Theorem 2 with F (α) = I. The relations between the conditions of Theorem 2 in this paper and Theorem 1 in [17] are stated in Proposition 1. From the proof of Proposition 1, it can be easily seen that the conditions in Theorem 2 are linear combinations of those of Theorem 1 in ˜˙ j | are not considered temporarily. [17] if the bounds of |h˙ i | and |h ˜ The following two Proposition 1. Consider the unforced continuous-time T–S fuzzy model (1) under the transformation h = F (α)h. terms hold: ρ

ρ

(1) γ˜i j ρ + γ˜ji ρ ≥ 0, i ≤ j, i, j, ρ ∈ Ir˜ , if γi j + γ ji ≥ 0, i ≤ j, i, j, ρ ∈ Ir ; (2) Y˜i j + Y˜ ji < 0, i ≤ j, i, j ∈ Ir˜ , if Yi j + Y ji < 0, i ≤ j, i, j ∈ Ir ; ρ

where γi j = Pρ + Mi j , γ˜i j ρ is defined in Theorem 2, Yi j = ATi Pj + Pj Ai +

r

r˜ ρ ρ ˜˙ ˙ ˜T ˜ ˜ ˜ ˜ ρ =1 hρ γi j and Yi j = Ai Pj + Pj Ai + ρ =1 hρ γ˜i j .

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159

7

Proof. (1) Since

γ˜i j ρ = fi f j P˜ρ + fρ M˜ i j =

r  r r  

fki fl j ft ρ (Pt + Mkl )

k=1 l=1 t=1

=

r  r r  

fki fl j ft ρ γklt ,

k=1 l=1 t=1

160 161

one has γ˜i j ρ + γ˜ji ρ =

r

r

r

+ γlkt ). Therefore 1) holds. Note that fij ≥ 0.  ˜˙ ρ = h˙ t . Therefore, one has (2) According to the construction of the transformation matrix F(α ), one has rρ˜ =1 ft ρ h k=1

Y˜i j = A˜ Ti P˜j + P˜j A˜ i + =

= =

r r  

l=1

t t=1 f ki f l j ft ρ (γkl

r˜  ˜˙ ρ γ˜ ρ h ij

ρ =1

fki fl j (ATk Pl + Pl Ak +

k=1 l=1

ρ =1

r r  

r  r˜ 

fki fl j (ATk Pl + Pl Ak +

Then Y˜i j =

t=1

k=1 l=1

t=1 ρ =1

r r  

r 

fki fl j (ATk Pl + Pl Ak +

r

r

l=1 f ki f l j Ykl

k=1

˜˙ ρ γ t ) ft ρ h kl

h˙ t γklt ).

t=1

k=1 l=1

162

r r˜   ˜˙ ρ ft ρ γklt ) h

and Y˜i j + Y˜ ji =

r

r k=1

l=1 f ki f l j (Ykl

+ Ylk ). Therefore 2) holds. 

166

Example 3. Let us continue to consider the T–S fuzzy model (8) in Example 2. For given φ˜ ρ = 0.2, ρ ∈ I9 , the checked results of parameters a and b are still depicted in Fig. 1 with (◦ · ) for Theorem 2. From Fig. 1, it is easily found that Theorem 2 significantly expands the a-b regions guaranteed by Theorem 1 in [1] and Theorem 1 in this paper, which implies that Theorem 2 is less conservative.

167

Example 4. Consider the 3-rule unforced T–S model (1) with

163 164 165



A1 = h1 (x1 ) = 168 169



−5 −1

−4 , −2

1 , 2(1 + ex1 )



A2 =

−2 + a 20 + b

h2 (x1 ) =



−4 , −2

1 , 2(1 + ex1 +1 )



A3 =

−5 0



2 , −6

h3 (x1 ) = 1 − h1 (x1 ) − h2 (x1 ).

(19)

The maximum value b∗ of b is computed so as to guarantee the stability of (19) in the interval b ∈ [0, b∗ ] as a changes in [0, ˜ and F(α ) are obtained in the following since there exists an order relation h1 ≥ h2 in the MFs: 10]. h, h

⎡ ⎤

h1 h = ⎣h2 ⎦, h3

⎡

⎤

⎡

h1 − h2 ˜ = ⎣ h2 ⎦, h h3

1 F (α) = ⎣0 0

1 1 0

⎤

0 0⎦. 1

174

For given φρ = 0.15, ρ ∈ Ir and φ˜ ρ = 0.30, ρ ∈ Ir˜ , the checked results are shown in Fig. 2 in the form of bar diagrams. It can be found that the conditions of Theorem 1 in [17] and Theorem 1 are both significantly relaxed by Theorem 2. As expected, Theorem 1 in [17] is out of action as long as an unstable subsystem appears when a increases to a certain value. The simulation results are shown in Fig. 3 in which the two state variables converge to the equilibrium from the initial position [4.0; −1.5] with a = 4 and b = 100.

175

Example 5. Consider the 2-rule unforced T–S model (1) with

170 171 172 173



A1 = h1 (x1 ) = 176 177 178 179



−5 1

−4 , −2 1

(1 + ex1 −2 )

,



A2 =

3 70



−4 , −2

h2 (x1 ) = 1 − h1 (x1 ).

(20)

The T–S fuzzy model (20) has been elaborately “constructed” in such a way that there does not exist an order relation in the MFs in the whole state space and that there does not exist a common positive matrix P satisfying the two subsystems even if there existed an order relation h1 ≥ h2 globally, i.e., Theorem 1 fails. Meanwhile, ordinary conditions also fail since A2 is unstable. Nevertheless, it is noted that the order relation h1 ≥ h2 holds in the subregion shown in Fig. 4 (the left side of the separator line). Please cite this article as: J. Chen et al., New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.09.036

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150 Theorem 1 Theorem 1 in [16] Theorem 2

b

100

50

0

0

1

2

3

4

5 a

6

7

8

9

10

Fig. 2. Stability region for a and b.

4 x1 x2 3

x(t)

2

1

0

−1

−2

0

2

4

6

8

10

time (sec) Fig. 3. State responses for a = 4 and b = 100.

180 181

Hence Theorem 2 can be used in the local region D = {[x1 ; x2 ] ∈ R2 | x1 ∈ [−∞, 2], x2 ∈ R} where |h˙ ρ | ≤ 0.25, ρ ∈ I2 . For given φ˜ ρ = 0.5, ρ ∈ Ir˜ , the corresponding vectors and matrices are solved as follows:



h=

P1 = 182 183

h1 , h2 0.8024 0.1542



˜= h



h1 − h2 , h2

1 F= 0



0.1542 , 0.3346



P2 =





1 , 1

2.2089 −0.1110



−0.1110 . 0.0190

Fig. 5 shows the Lyapunov equipotential lines obtained by (11) with two trajectories from initial positions [−1.5; 2] and [1.5; −2], respectively. Please cite this article as: J. Chen et al., New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.09.036

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9

2 h1 h2 the derivative of h1

1.5

separator line

1

0.5

0

−0.5

−1 −5

0 x1

5

Fig. 4. Membership functions in Example 5.

3

2

x2

1

0

−1

−2

−3 −2

−1.5

−1

−0.5

0 x1

0.5

1

1.5

2

Fig. 5. Lyapunov equipotential lines and two trajectories.

184 185 186

5. Stabilization ˜ via the non-quadratic In this section, sufficient stabilization conditions are studied for (1) under the transformation h = F (α)h Lyapunov function (21) and the non-PDC control law (22) [9,17,35] in the following: T ˜−1 V (x(t )) = x(t )T Pz−1 (t ) x(t ) = x(t ) Pz(t ) x(t ),

187

188 189

˜ ˜−1 u(t ) = Kz(t ) Pz−1 (t ) x(t ) = Kz(t ) Pz(t ) x(t ) r˜

˜ j P˜j , K˜ where P˜z(t ) = j=1 h z(t ) = closed-loop T–S model:

r˜

j=1

h˜ j K˜ j , P˜j =

x˙ (t ) = (A˜ z(t ) + B˜z(t ) K˜z(t ) P˜z−1 (t ) )x(t ).

(21) (22)

r

i=1 f i j Pi

and K˜ j =

r

i=1 f i j Ki . By substituting (22) into (3), we have the following

(23)

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˜ is Theorem 3. For given φ˜ ρ > 0, ρ ∈ Ir˜ , the equilibrium of the continuous-time T–S model (1) under the transformation h = F (α)h asymptotically stable in D R with the non-PDC control law (22) if there exist positive definite matrices Pi , i ∈ Ir , symmetric matrices Mij , i, j ∈ Ir and matrices Ki , i ∈ Ir such that

γ˜i j ρ + γ˜ji ρ ≥ 0, i ≤ j, i, j, ρ ∈ Ir˜ 193 194 195 196 197

(24)

γ˜i j + γ˜ ji < 0, i ≤ j, i, j ∈ Ir˜

(25)

˜ i j , γ˜i j = He(A˜ i P˜j + B˜i K˜ j ) + hold, where γ˜i j ρ = fi f j P˜ρ + fρ M r r k=1 l=1 f ki f l j Mkl .

r˜

˜ ρ ρ =1 φρ γ˜i j , f j =

r

˜ =

k=1 f k j , Ai

r

˜ =

k=1 f ki Ak , Pj

r

k=1 f k j Pk

˜ ij = and M

Proof. Consider the non-quadratic Lyapunov function candidate (21). Along the trajectory of (23) and with the equation P˜˙z−1 = (t ) P˜˙ P˜−1 , the time derivative of (21) is computed as −P˜z−1 (t ) z(t ) z(t )



˜ ˜ ˜ ˜−1 ˜−1 ˜˙ ˜−1 V˙ (x) = xT He(P˜z−1 (t ) (Az(t ) + Bz(t ) Fz(t ) Pz(t ) )) − Pz(t ) Pz(t ) Pz(t ) x



˜ ˜ ˜ ˜−1 ˜ ˜˙ ˜−1 = xT P˜z−1 (t ) He((Az(t ) + Bz(t ) Fz(t ) Pz(t ) )Pz(t ) ) − Pz(t ) Pz(t ) x

= yT He(A˜ z(t ) P˜z(t ) + B˜z(t ) F˜z(t ) ) −

= yT He(A˜ z(t ) P˜z(t ) + B˜z(t ) F˜z(t ) ) −

= yT He(A˜ z(t ) P˜z(t ) + B˜z(t ) F˜z(t ) ) − 198 199

ρ =1

 r˜  ˜˙ ρ (P˜ρ + fρ M ˜ zz(t ) ) y h

ρ =1

r˜  ˜˙ ρ γ˜ ρ h



zz(t )

ρ =1

y

ρ ˜ zz(t ) . Note that y = 0 if and only if x = 0. According to Lemma 2, (24) and (25) imply where y = P˜z−1 x and γ˜zz(t ) = fz(t ) fz(t ) P˜ρ + fρ M (t ) ρ γ˜zz(t ) ≥ 0 and γ˜zz(t ) < 0, respectively. Therefore, in the region D R, one has

V˙ (x) ≤ yT (He(A˜ z(t ) P˜z(t ) + B˜z(t ) F˜z(t ) ) + = yT γ˜zz(t ) y < 0, 200

 r˜  ˜˙ ρ P˜ρ y h

r˜ 

ρ =1

φ˜ ρ γ˜zzρ(t ) )y

∀x(t ) = 0.

Hence, according to the Lyapunov stability theory, the equilibrium of (23) is asymptotically stable in D



R.



203

Remark 5. If F (α) = I, Theorem 3 are reduced to Theorem 5 in [17]. The relations between the conditions of Theorem 3 in this paper and Theorem 5 in [17] are similar to those between the conditions of Theorem 2 in this paper and Theorem 1 in [17], as stated in Proposition 1 and therefore omitted.

204

Example 6. Consider the T–S fuzzy model (1) with

201 202



A1 =

B1 = 205 206 207 208 209 210



−1 8



−4 , −9

−0.45 −3

−2 + a A2 = −7 + b





,

B2 =





−3 , −9

−1 , −3



−4 A3 = 0



B3 =





0 , −1

−1 . −2

(26)

h, h˜ and F(α ) are the same as those in Example 4. For given φρ = 0.15, ρ ∈ Ir and φ˜ ρ = 0.3, ρ ∈ Ir˜ , stabilization conditions are checked by Theorem 5 in [17] and Theorem 3 with a ∈ [0, 20] and b ∈ [0, 20]. The results of the parameters a and b are depicted in Fig. 6 with (◦) for Theorem 5 in [17] and (◦ ·) for Theorem 3 in this paper. It is easily found that the conditions of Theorem 3 are less conservative than those of Theorem 5 in [17]. With a = 10 and b = 15, the simulation results are shown in Fig. 7, in which the state trajectories converge to the equilibrium from the initial position [1; −1.5] as time goes by. The corresponding solutions about the non-PDC control law u(t ) = Kz(t ) Pz−1 x(t ) are given as follows: (t )



K1 = −78.1525





−49.9217 ,

13.3975 P1 = 20.5445





K2 = 20.0595

20.5445 , 37.5701





−46.7546 ,

11.4912 P2 = 17.9452



17.9452 , 30.5567





K3 = −34.1972

−7.2272 ,

15.2499 P3 = 20.2598

20.2598 . 33.6037





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Fig. 6. Stability regions for a and b with (◦) for Theorem 5 in [17] and (◦ ·) for Theorem 3.

2.5 x1 x2

2 1.5

x(t)

1 0.5 0 −0.5 −1 −1.5

0

2

4

6

8

10

time (sec) Fig. 7. State responses with a = 10 and b = 15.

211 212

Example 7. Let us consider a practical problem of balancing an inverted pendulum on a cart [20] to illustrate the effectiveness of the proposed approach. The nonlinear dynamic system could be approximated in the region D by the 2-rule T–S model (1) with

⎡

0 ⎢ (M + m)mgl ⎢ ⎢ a1 A1 = ⎢ 0 ⎢ ⎣ m2 gl 2 − a1

1 f1 (M + m) − a1 0 f1 ml a1

0

0

0 0

⎥ ⎥ ⎥ ⎥, 1 ⎥ f0 ( J + ml 2 ) ⎦ f0 ml a1

0

−

⎤

a1

⎡

⎤

0 ⎢ ⎥ ⎢ − ml ⎥ ⎢ ⎥ a 1 B1 = ⎢ ⎥, 0 ⎢ ⎥ ⎣ ( J + ml 2 ) ⎦ a1

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⎡

0 √ ⎢ 3 3 (M + m)mgl ⎢ a2 ⎢ 2π A2 = ⎢ ⎢ 0 ⎣ √ 3 3 m2 gl 2 cos(π /3) − a2 2π 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228

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1 −

0

f1 (M + m) a2 0

⎤

0

f0 mlcos(π /3) ⎥ ⎥ a2 ⎥ ⎥, ⎥ 1

0

−

0

f1 mlcos(π /3) a2

0

f0 ( J + ml 2 ) a2

⎦

⎡

0 mlcos(π /3) − a2 0 ( J + ml 2 ) a2

⎢ ⎢ ⎢ B2 = ⎢ ⎢ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

where a1 = (M + m)( J + ml 2 ) − m2 l 2 , a2 = (M + m)( J + ml 2 ) − m2 l 2 cos(π /3)2 ; g = 9.8m/s2 is the gravity constant; m = 0.22kg is the mass of the pendulum; M = 1.3282kg is the mass of the cart; l = 0.304m is the length from the center of mass of the pendulum to the shaft axis; J = 0.004963kg m2 is the moment of inertia of the pendulum round its center of mass; f0 = 22.915N/(m s)−1 is the friction factor of the cart; f1 = 0.007056N/(rad s)−1 is the friction factor of the pendulum; u(t) is 1 the force (N) applied to the cart; the two membership functions are defined as h1 (x1 ) = (1 − )( −7.01(x +π /6) ) and −7.0(x −π /6) 1+e

1

1+e

1

h2 (x1 ) = 1 − h1 (x1 ); x1 denotes the angle (rad) of the pendulum from the vertical; x2 is the angular velocity (rad/s); x3 is the displacement (m) of the cart; x4 is the velocity (m/s) of the cart; D = {x(t ) ∈ R4 |x1 (t ) ∈ [−π /3, π /3], x2 (t ) ∈ [−π /2, π /2], x3 (t ) ∈ [−1, 1], x4 (t ) ∈ [−0.6, 0.6]}. For comparison purpose, we suppose f1 = 0.007056 +  f1 , where f1 is the uncertain parameter. The curves of h1 , h2 and the time derivative of h1 are depicted in Fig. 8. It is seen that in the region R = {x(t ) ∈ R4 |x1 (t ) ∈ [−0.5, 0.5]}, there exists an 1 1 order relation h1 > h2 and the inequalities |h˙ ρ | ≤ 1.5, ρ = 1, 2 hold. Therefore, we have the transformation matrix F = [ ] 0

1

˜˙ ρ | ≤ 3, ρ = 1, 2 in D ∩ R. Then, for given φρ = 1.5 and φ˜ ρ = 3, ρ = 1, 2, the maximum value  f ∗ = 1.07 of the uncertainty and |h 1 f1 is computed by Theorem 3 in this paper while  f1∗ = 0.74 is computed by Theorem 5 in [17]. It is easily seen that Theorem 3 in this paper is less conservative than Theorem 5 in [17]. With  f1 = 0.90, the simulation results are shown in Fig. 9, in which the four state variables converge to zero from the initial position [0.2; 1; 1; 0.5]. The corresponding solutions about the non-PDC control law u(t ) = Kz(t ) Pz−1 x(t ) are given as follows: (t )





K1 = −2.8037

−13.4468

−24.8612

83.6345 ,

K2 = −2.4614

−18.8475

−23.3053

80.2667 ,

−0.0100 0.4093 0.1474 −0.2108

−0.0380 0.1474 4.8031 −0.6512

−0.1006 −0.2108 ⎥ ⎥, −0.6512 ⎦ 2.5909



⎡

0.0056 ⎢−0.0100 P1 = ⎢ ⎣−0.0380 −0.1006



⎤

2 h1 h2 the derivative of h1

1.5 1 0.5 0 −0.5 −1 −1.5 −2 −1.5

−1

−0.5

0 x1

0.5

1

1.5

Fig. 8. The membership functions and the time derivative.

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13

x1

1 0 −1

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15 time (sec)

20

25

30

x2

10 0 −10

x3

5 0 −5

x4

20 0 −20

Fig. 9. State responses with  f1 = 0.90.

⎡

0.0042 ⎢ 0.0042 P2 = ⎢ ⎣−0.0353 −0.0865

0.0042 0.2165 0.0713 −0.3297

−0.0353 0.0713 4.6633 −0.6000

⎤

−0.0865 −0.3297 ⎥ ⎥. −0.6000⎦ 2.4240

237

Remark 6. With the above discussions, it is easily found that the order relation information of the MFs plays a very important role in the properties of the T–S fuzzy model. If the order relation information of the MFs is taken into account, the conservatism of the stability and stabilization conditions could be significantly reduced. In fact, based on the transformed T–S fuzzy model (3), many existing conditions can be further relaxed by the proposed approach, e.g., other conditions in [17], the conditions in [19] obtained with the line-integral Lyapunov functions and the conditions in [16,18,35] obtained with the fuzzy Lyapunov functions or the delayed nonquadratic Lyapunov functions, etc. Although it may be a little difficult that there exist order relations in the MFs globally, there usually do exist locally around the equilibrium in practice. Therefore it is very meaningful to study the T–S fuzzy models by considering the order relation information of the MFs. The new transformed T–S fuzzy model provides a new platform to analyze and synthesize nonlinear systems.

238

6. Conclusions

239

243

In this paper, the problem of stability and stabilization has been investigated for nonlinear systems based on a new compact T–S fuzzy model which is obtained from the conventional T–S fuzzy model by considering the order relation information of the MFs. An improved approach to constructing the transformation matrix has been proposed. Based on the non-quadratic Lyapunov functions and the non-PDC control law, relaxed stability and stabilization conditions have been achieved. Undoubtedly, this work has laid a foundation for the future research about robust and H∞ control, filtering and observer design, and other related topics.

244

Uncited references

229 230 231 232 233 234 235 236

240 241 242

Q2 245

[7,8,13,15,25,26,33,34]

246

Acknowledgments

247

This work was supported in part by the National Nature Science Foundation under Grants 61174038, 61174076, 61203048, 61374086, 61374087,61473151, 61403178, the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT13072), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Program for New Century Excellent Talents in University.

248 249 250

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251

References

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Please cite this article as: J. Chen et al., New relaxed stability and stabilization conditions for continuous-time T–S fuzzy models, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.09.036