Robust stabilization for uncertain Markovian jump fuzzy systems based on free weighting matrix method

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Robust stabilization for uncertain Markovian jump fuzzy systems based on free weighting matrix method Min Kook Song a , Jin Bae Park a,∗ , Young Hoon Joo b a Department of Electrical and Electronic Engineering, Yonsei University, Seoul, 120-749, Republic of Korea b School of Electronic and Information Engineering, Kunsan National Univ. Kunsan, Chonbuk 573-701, Republic of Korea

Received 24 February 2014; received in revised form 10 January 2015; accepted 6 February 2015

Abstract This paper presents a new methodology of robust state feedback fuzzy controller based on free weighting matrix method for a class of uncertain Markovian jump nonlinear systems. The class of systems under consideration is represented by the T–S fuzzy model with partly known transition probability matrix. The free weighting matrix method is proposed to obtain a less conservative stochastic stability criterion of the uncertain Markovian jump fuzzy systems (MJFSs) in terms of linear matrix inequalities (LMIs). Furthermore, a sufficient condition for the mode-dependent state feedback fuzzy controller is derived for the MJFSs for all admissible parameter uncertainties. Finally, a simulation example is provided to illustrate the effectiveness of the proposed methodology. © 2015 Published by Elsevier B.V. Keywords: Uncertain systems; Markovian jump fuzzy systems (MJFSs); Partially known transition matrices; Linear matrix inequalities (LMIs); Free weighting matrix method

1. Introduction In practical applications, many dynamical systems may experience random changes in variable structures and parameters, which usually modeled as Markovian jump systems (MJSs). These changes may result from component failures and repairs, changing subsystem interconnection and sudden environmental disturbances. These kinds of MJSs are very common in many numerous physical systems including manufacturing systems, fault-tolerant control systems and electrical power systems, etc. In the MJSs, the jumps in operate modes are governed by Markov process. Over the past decades, much attention for MJSs and some important results have been reported in the [1,2], and references therein. However, little attention has been paid to the stability and stabilization problem for Markovian jump nonlinear systems. The fuzzy-model-based control design techniques have been proved to be a powerful method for the control problem of complex nonlinear systems. * Corresponding author. Tel.: +82 2 2123 2773; fax: +82 2 362 4539.

E-mail addresses: [email protected] (M.K. Song), [email protected] (J.B. Park), [email protected] (Y.H. Joo). http://dx.doi.org/10.1016/j.fss.2015.02.004 0165-0114/© 2015 Published by Elsevier B.V.

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In particular, the so-called Takagi–Sugeno (T–S) fuzzy model has been widely employed for the control design of nonlinear systems [3–31]. Specifically, the T–S fuzzy system, which are composed of smoothly connected local linear time-invariant systems through membership functions, has attracted much attention due to the fact that it provides an efficient approach to apply conventional linear system theory to analysis and controller synthesis of the nonlinear systems. In recent years, this T–S fuzzy-model-based technique has been used to deal with Markovian jump nonlinear systems [3,4]. In particular, when parameter uncertainties appear in Markovian jump fuzzy systems (MJFSs), a robust stabilization condition was derived in terms of linear matrix inequalities (LMIs) by decoupling system matrices from stochastic Lyapunov matrices in [3]. By introducing more slack variables, a less conservative method for stabilizing controller for the discrete case was proposed in [5]. In [6], a new fuzzy model with two levels of structure was introduced such that the Markovian jump nonlinear system modeled in the fuzzy modelling. LMI-based sufficient conditions for output-feedback controllers were derived for singularly perturbed nonlinear MJSs approximated by the T–S fuzzy model [7]. However, it should be noted that partially known transition probabilities in the jumping process were not considered in previous results. Whether in theory or in practice, it is necessary to further consider more general Markovian jump systems with partial known transition probabilities. Sheng investigated the problem of stabilization for MJFSs with partly known transition probabilities [8]. In [4], by making full use of the continuous property of transition matrix, new sufficient conditions are proposed. Although the previous results [4,8,9,16–31] considered the MJFSs with partly unknown transition probabilities, to the best of the author’s knowledge, the problem of robust stability analysis and stabilization is still open so far and there is further room for investigation. When the terms which contained unknown transition probabilities were separated from the others, the fixed connection matrices were selected in the previous results [4,8], which may lead to the conservativeness. The main contribution of this paper is some sufficient conditions in the LMI format and a systematic design procedure for the controller design for a nonlinear system with parametric uncertainties and partially unknown transition probabilities. Motivated by aforementioned observations, this paper presents a novel design methodology for the robust fuzzy control of uncertain MJFSs with partly known transition probabilities based on free weighting matrix method. Differing from the previous results, we derive a less conservative robust stability and mode-dependent robust stabilization conditions are derived for the uncertain MJFSs in terms of LMIs by making full use of the continuity of the transition probability matrix and using free-weighting matrix method. It is proved that these conditions are less conservative or at least the same as those for previous results. The purpose of this paper is focused on the design of a mode-dependent state feedback fuzzy controller so that the MJFSs can be stochastically stabilized for all admissible uncertainties. Using the proposed approach, we obtain a method for design of robust fuzzy controller. Finally, a simulation example is presented to show the effectiveness of the proposed methodology. Notations Rn := n-dimensional real space. Rm×n := Set of all real m by n matrices. AT := Transpose of matrix A. P  0 (resp. P ≺ 0) := Positive (resp., negative)-definite symmetric matrix. A star (*) := The transposed element in the symmetric position. 2. System description Given a probability space (, F, P), where  is the sample space, F is the algebra of events and P is the probability measure define on F . We consider a class of uncertain Markovian jump nonlinear systems over the space (, F, P), which can be represented by the following T–S fuzzy model: Ri : IF z1 (t) is i1 and · · · and zp (t) is ip THEN

x(t) ˙ = (Ai (η(t)) + Ai (η(t), t))x(t) + (Bi (η(t)) + Bi (η(t), t))u(t)

(1)

where x(t) ∈ Rn the state; u(t) ∈ Rm the control input. Ri , i ∈ IR = {1, 2, · · · , r}, denotes the ith fuzzy rule, the scalar r is the number of IF–THEN rules. In the framework of fuzzy systems, zh (t), h ∈ IP = {1, 2, · · · , p} is the premise variable which are assumed to be given or to be only a function of x(t), ih , (i, h) ∈ IR × IP , is the fuzzy set. The system mode η(t), t ≥ 0 is a continuous time Markov process on the probability space taking values in a finite state space N = {1, 2, · · · , N }. The set N comprises the operation modes of the system.

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The system mode η(t) has the following mode transition probabilities:  πls t + o(t), s = l Pr(η(t + t) = s|η(t) = l) = 1 + πls t + o(t), s = l = 0, and πls ≥ 0, for l = s, is the transition probability from mode l at time t to mode s N at time t + t , and πll = − s=1,s=l πls ≤ 0. The system matrices of the ith rule are denoted by Ai (η(t)), Bi (η(t)), which are known real valued matrices with appropriate dimensions. Ai (η(t), t), Bi (η(t), t) are the uncertain matrices of appropriate dimensions representing time-varying parameter uncertainties satisfying     Ai (η(t), t) Bi (η(t), t) = Di (η(t))Fi (η(t), t) E1i (η(t)) E2i (η(t)) (2)

where t > 0, limδ→0

t o(t)

where Fi (η(t), t) is the unknown time-varying matrix satisfying FiT (η(t), t)Fi (η(t), t) ≤ I, ∀t ≥ 0, i ∈ IR , η(t) ∈ N,

(3)

E1i (η(t)) and E2i (η(t)) are real constant matrices that characterize the structure of uncertainties. The parameter uncertainties Ai (η(t), t), Bi (η(t), t) are said to be admissible if both (2) and (3) are hold. In this paper, the transition probabilities of the jumping process η(t) are assumed to be partly unknown. For instance, the Markov process transition rate matrix  for system (1) with N operation modes may be defined by ⎡ ⎤ π11 ? · · · π1N ? ⎥ ⎢ π21 π22 . . . ⎥ (4) =⎢ .. ⎣ ⎦ . ?

?

···

πN N

where “?” represents the unknown transition probabilities. For this example, the elements in the lth row of the matrix  are represented by the following sets: TlK = {s : if πls are known} TlUK

= {s : if πls is unknown}

Moreover, if TlK = 0, it is described as

TlK = El1 , · · · , Elm , 1 ≤ m ≤ N

(5) (6)

(7)

where Elm represents the mth element in the lth row of the matrix . Remark 1. The transition probabilities of the jumping process η(t) is commonly assumed to be completely known TlUK = 0, or completely unknown Tl = TlUK in existing literatures. It is clear that when TlUK = 0 and Tl = TlK , the system (1) becomes the usual assumption case. It is worthwhile to note that choosing this kind of assumption in transition probability matrix is motivated by [10], but in this paper a more generalized case of MJFSs are given. Remark 2. Sheng [8] treated some unknown transition probabilities with lower and upper bounds as completely known. Shen derived sufficient conditions for MJFSs by making full use of the continuity of the transition probability matrix [4]. However, in this paper, the free weighting matrices is employed to express the relation ship among the transition probabilities. Then, a less conservative stability criterion for MJFS is derived in terms of LMIs. By the parallel distributed compensation (PDC) technique, the following mode-dependent state feedback controller for the fuzzy system in (1) is formulated as follows: Ri : IF z1 (t) is i1 and · · · and zp (t) is ip THEN

u(t) = Kj (η(t))x(t)

(8)

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where Kj (η(t)) are the control gain matrices to be determined. Then, the overall fuzzy controller is then represented by u(t) =

r 

  θj (z(t)) Kj (η(t)) x(t)

(9)

j =1

Let the mode at t be l and using the center-average defuzzifer, product inference, and singleton fuzzifier, (1) is inferred as x(t) ˙ =

r 

θi (z(t)) ((Ai (η(t)) + Ai (η(t), t))x(t) + (Bi (η(t)) + Bi (η(t), t))u(t))

i=1

(10) = (Ai,l (z) + Ai,l (z))x(t) + (Bi,l (z) + Bi,l (z))u(t) r r p where θi (z(t)) = wi (z(t))/ i=1 wi (z(t)) and i=1 θi (z(t)) = 1; wi (z(t)) = h=1 μih (zh (t)); μih (zh (t)): Uzh ⊂ R → R[0,1] the membership function of zh (t) on a compact set Uzh . Substituting (9) into (10), the closed-loop MJFS (10) can be described by (11) x(t) ˙ = (Ai,l (z) + Ai,l (z) + (Bi,l (z) + Bi,l (z))Kj,l (z))x(t) r where Kj,l (z) = j =1 θj (z(t))Kj (η(t)). For simplicity, when u(t) = 0, the MJFS (11) is referred to as a free MJFS. In the case Fi (η(t), t) = 0, it is referred to as a nominal MJFS. Throughout this paper, the following definition is essential to give our main results. Definition 1. (See [11].) The MJFS (11) is said to be stochastically stable if, for any initial condition x0 and initial distribution for η0 , the following holds: ⎧∞ ⎫ ⎨ ⎬ E

x 2 (t) dt|x0 , η0 < ∞. (12) ⎩ ⎭ 0

The problem to be addressed in the paper is to design a mode-dependent state feedback fuzzy controller (9) for the uncertain MJFS (10) with partially known transition probabilities such that the resulting closed-loop system (11) is robustly stochastically stable. To give our main results in the next section, the following lemmas well be needed throughout the proof. Lemma 1. (See [3].) Let A and X be real matrices with appropriate dimensions. Then there exists a matrix P = P T > 0 such that P AT + AP + X < 0, if and only if, there exist a scalar μ > 0 and Z such that   −Z − Z T ZT A + P ZT < 0. (13) ∗ −μ−1 P + X 0 ∗ ∗ −μP Lemma 2. (See [4].) Unforced MJFS (11) with partly known transition probabilities (4) is stochastically stable if k, there exist matrices Qk > 0 satisfying the following inequalities: For k ∈ IK ¯ k + π¯ kk Qk + δ¯k Qp < 0, He(Qk Ak (h)) + Q K K k , and for k ∈ IUK

¯ k + π¯ kk Qk < 0, He(Qk Ak (h)) + Q K k ). Qp ≤ Qk (p ∈ IUK

Lemma 3. (See [12].) Given matrices Q, H , and E of appropriate dimensions where Q = QT , then Q + H F (t)E + E T F T (t)H T < 0 for all F (t) satisfying F T (t)F (t) ≤ I , if and only if there exists a scalar λ > 0 such that Q + λH H T + λ−1 E T E < 0.

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Lemma 4. (See [13].) If the following conditions hold: Xii < 0, 1 ≤ i ≤ r, 1 1 Xii + (Xij + Xj i ) < 0, 1 ≤ i = j ≤ r r −1 2 then the following parameterized linear matrix inequality holds: r r  

θi (z(t))θj (z(t))Xij < 0.

i=1 j =1

3. Main results 3.1. Stochastic stability analysis and state feedback stabilization for nominal Markovian jump fuzzy systems First, stochastic stability criterion for free MJFSs with the partly known transition probabilities based on the freeconnection matrix method is presented in the following theorem: Theorem 1. The free MJFS (11) with a partly unknown transition rate matrix (4) is stochastically stable if there exist matrices Pl = PlT > 0, Ps = PsT > 0, Wl = WlT , l, s ∈ T satisfying the following inequalities for i ∈ IR : ATi,l (z)Pl + Pl Ai,l (z) +



πls (Ps − Wl ) + πll (Pl − Wl ) +

s∈TK

ATi,l (z)Pl + Pl Ai,l (z) +





πls (Ps − Wl ) ≺ 0, l ∈ TlK , s = l

(14)

s∈TUK

πls (Ps − Pl − Wl ) ≤ 0, ∀l ∈ TlUK ,

(15)

s∈TK

Ps − Pl − Wl ≤ 0, ∀s ∈ TlUK , s = l.

(16)

Proof. Choose a fuzzy-basis-dependent stochastic Lyapunov functional candidate as follows: V (x(t), t, η(t)) = x T (t)P (η(t))x(t)

(17)

where P (η(t)), η(t) ∈ N are all matrices with appropriate dimensions to be determined. It is known that {(x(t), t, η(t))} is a Markov process with initial state x0 , mode η0 and its weak infinitesimal generator, acting on Lyapunov function V , is defined in [14]. LV (x(t), t, η(t) = l) = lim

→0+

1 [E(V (x(t + ), t + , η(t + )|x(t), η(t) = l) − V (x(t), t, l)]. 

The weak infinitesimal operator L of the stochastic process {(x(t), t, η(t))} along the evolution of V (x(t), t, l) is given as follows: LV (x(t), t, l) = 2x T (t)Pl x(t) ˙ + x T (t)

N 

(18)

πls Ps x(t).

s=l

Taking into account the situation that the information of transition probabilities is not accessible completely, the  following zero equations hold for arbitrary matrices Wl = WlT due to N s=1 πls = 0, −x T (t)

N 

πls Wl x(t) = 0, ∀l ∈ N.

(19)

s=1

Adding the left side of (19) into (18) and using the fact

N

s=1 πls

=



TlK ,s=l πls

+ πll +



s∈TlUK ,s=l πls

yields

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LV (x(t), t, l) = 2x T (t)Pl x(t) ˙ + x T (t)

N 

πls (Ps − Wl )x(t)

s=1

N    πls (Ps − Wl )x(t) = x T (t) Pl Ai,l (z) + ATi,l (z)Pl x(t) + x T (t) s=1

  = x (t) Pl Ai,l (z) + ATi,l (z)Pl T

+

 s∈TlK ,s=l

 πls (Ps − Wl ) x(t)



πls (Ps − Wl ) + πll (Pl − Wl ) +

s∈TlUK ,s=l

= x T (t)ϒi,ls x(t).  Note that πll = − N s=1,s=l πls is a non-positive scalar, ϒi,ls is dealt with in the following two cases.  First we consider the case that πll is known, l ∈ TlK , s = l. Considering the fact s∈Tl ,s=l πls = (−πll − UK  s∈TlK ,s=l πls ), then ϒi,ls is rewritten as follows:      ϒi,ls = Pl Ai,l (z) + ATi,l (z)Pl + πls (Ps − Wl ) + πll (Pl − Wl ) + πls (Ps − Wl ) 

s∈TlUK ,s=l πls

= 



s∈TlK ,s=l πls



+ 

s∈TlUK ,s=l πls s∈TlK ,s=l πls

+



s∈TlUK ,s=l

Pl Ai,l (z) + ATi,l (z)Pl



 πls (Ps − Wl ) + πll (Pl − Wl ) +

s∈TlK ,s=l

= 

s∈TlK ,s=l





πls (Ps − Wl )

s∈TlUK ,s=l

Pl Ai,l (z) + ATi,l (z)Pl

πls (Ps − Wl ) + πll (Pl − Wl ) +

s∈TlK





 πls (Ps − Wl ) .

(20)

s∈TlK

We conclude that ϒi,ls < 0 by (14) and πls ≥ 0 (∀l, s ∈ T, s = l).  Next, we consider the case that πll is unknown, l ∈ TlUK , s = l. We replace πll with (− s∈Tl ,s=l πls − K  s∈TlUK ,s=l πls ). Then ϒi,ls is rewritten as follows:      ϒi,ls = Pl Ai,l (z) + ATi,l (z)Pl + πls (Ps − Wl ) + πll (Pl − Wl ) + πls (Ps − Wl )   = Pl Ai,l (z) + ATi,l (z)Pl +

s∈TlK ,s=l



s∈TlK ,s=l

πls (Ps − Pl − Wl ) +



s∈TlUK ,s=l

 πls (Ps − Pl − Wl ) .

(21)

s∈TlUK ,s=l

 According to (15)–(16) and the fact πls ≥ 0 and πll = − N s=1,s=l πls < 0, it is straightforward that ϒi,ls < 0. Therefore, if (14)–(16) hold, we have that the free MJFS (11) is stochastically stable with the partly unknown transition probabilities. This completes the proof. 2 Remark 3. Compared with the method proposed in [4,8,9], there are two new features in Theorem 1. One is that the free connection matrix method is proposed to study the stability of MJFSs. The free weighting matrices are introduced to express the relation ship among the transition probability πll by using the zero equation, which may l lead to less  conservative or at least the same results for the existing method. The other is that if πls ∈ IUK , the property l∈I πls = 0 is used again. As a result, Theorem 1 is more general stability condition is obtained.

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Remark 4. If the free weighting matrix Wl = 0, Theorem 1 can give the same results as the existing approach described by Theorem 1 in [8]. Therefore, Theorem 1 generalizes the previous results. Remark 5. Compare with the previous stability sufficient conditions proposed by Shen [4], the free weighting matrices are introduced by making use of the relationship of the transition probabilities among various subsystems. When Wl = 0, proposed Theorem 1 in this paper is equivalent to Theorem 1 in [4]. However, the free weighting matrices Wl will provide more freedom to chose, and less conservative or at least same results are derived. As discussed by Sheng [8] and Shen [4], the conditions given in Theorem 1 cannot be used to obtain the fuzzy controller gain Kj,l by the convex framework. To overcome the nonconvexity, the following theorem is proposed to design the mode-dependent stabilizing controller with the form (9) for system (10). Theorem 2. Consider the system (11) with unknown transition probabilities. For a positive scalar μ, if there exist matrices Ql = QTl > 0, Sij l = SijT l , Rl = RlT , Yj,l , Z, i, j ∈ IR , l ∈ N satisfying the following LMIs: For l ∈ TlK 1ii,lsp ≺ 0, s ∈ TlK , p ∈ TlUK , s, p = l  1 1 1 1ii,l + ij,l + 1j i,l ≺ 0, i = j ∈ IR , s ∈ TlK , p ∈ TlUK , s, p = l, r −1 2

(22) (23)

For l ∈ TlUK 2ii,lsp ≺ 0, s ∈ TlK , s = l  1 2 1 2ii,l + ij,l + 2j i,l ≺ 0, i = j ∈ IR , s ∈ TlK , s = l r −1 2 Ql − Qs − Rl ≥ 0, s ∈ TlUK where

(24) (25) (26)

∗ ∗ ∗ ∗ ⎤ −Z − Z T 1 ∗ ∗ ∗ ⎥ ⎢ Ai,l Z + Bi,l Yj,l + Ql 22ij,ls ⎢ ∗ ∗ ⎥ Z 0 −μSij l 1ij,ls = ⎢ ⎥ ⎣ k )T 0 −Q1K ∗ ⎦ 0 (ζ1K k )T 0 −Q2K 0 0 (ζ2K ⎡ ⎤ −Z − Z T ∗ ∗ ∗ 2 ∗ ∗ ⎥ ⎢ A Z + Bi,l Yj,l + Ql 22ij,ls 2ij,ls = ⎣ i,l ⎦ ∗ Z 0 −μSij l k 1 T 0 −QK 0 (ζ1K ) √  √ k ζ1K = −πl1 Ql , · · · , −πl(l−1) Ql , −πl(l+1) Ql , · · · −πl O Ql , , (O = max{s})   Q1K = diag Ql1 , · · · , Ql(l−1) , Ql(l+1) , · · · , Ql O  √ √ k = −πl1 Ql , · · · , −πl(l−1) Ql , −πl(l+1) Ql , · · · −πl P Ql , (P = max{p}) ζ2K   Q2K = diag Ql1 , · · · , Ql(l−1) , Ql(l+1) , · · · , Ql P   with 122ij,ls = μ−1 (Sij l − 2Ql ) + πll (Ql − Rl ) − s πls Rl − p πlp Rl , 222ij,ls = μ−1 (Sij l − 2Ql ) −  s πls (Ql + Rl ), then the system (11) is stochastically stable via fuzzy controller (8). Moreover, the state-feedback fuzzy controller gains are given by ⎡

Kj,l = Yj,l Z −1 , j ∈ IR , l ∈ N

(27)

Proof. Let us consider the stabilization problem of system (11) with control input u(t). Based on Theorem 1, we know that the MJFS (11) is stochastically stable, if the following conditions are satisfied

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ATci,l (z)Pl + Pl Aci,l (z) +



πls (Ps − Wl ) + πll (Pl − Wl ) +

s∈TK

ATci,l (z)Pl

+ Pl Aci,l (z) +





  πlp Pp − Wl ≺ 0

(28)

p∈TUK

πls (Ps − Pl − Wl ) ≤ 0, ∀l

∈ TlUK ,

(29)

s∈TK

Ps − Pl − Wl ≤ 0, ∀s ∈ TlUK , s = l

(30)

where Aci,l (z) = Ai,l (z) + Bi,l (z)Kj,l (z). And adding the null terms μ−1 Sij l (z) − μ−1 Sij l (z) and V − V + V T − V T , where V = Z −1 , to left side of (28), we can obtain ! " −V − V T + μ−1 Sij l (z) ∗ ≺0 (31) Pl Aci,l (z) + V 22 where 22 = −μ−1 Sij l (z) +



πls (Ps − Wl ) + πll (Pl − Wl ) +

s∈TK ,s=l



  πlp Pp − Wl .

p∈TUK ,p=l

We introduce the following new variables Sij l (z) =

r  r 

θi (z(t))θj (z(t))Sij (η(t)),

i=1 j =1

Ql = Pl−1 , Rl = Pl−1 Wl Pl−1 . By applying Schur complement to the above inequality (31) and the congruence transformation with diag{Z T , Ql }, we can see that ! " −Z − Z T + μ−1 Z T Sij l (z)Z ∗ (31) ⇔ −μ−1 Ql Sij l (z)Ql + πll (Ql − Rl ) Aci,l (z)Z + Ql " ! 0 ∗     ≺0 (32) + 0  −1 −1 s∈TK πls Ql Qs Ql − Rl + p∈TUK πlp Ql Qp Ql − Rl Similarly, applying the Schur complement to the above inequality (32), the left side of inequality (32) is rewritten as  −Z − Z T ∗ #∗ (32) = Aci,l (z)Z + Ql ∗ 22 Z 0 −μSij l (z)   0    ∗   ∗ −1 −1 + 0 ∗ s∈TK πls Ql Qs Ql + s∈TUK πls Ql Qs Ql 0 0 0     0 −Z − Z T ∗ ∗ 0 (O −1)×1 (P −1)×1 # k k + ⇔ Aci,l (z)Z + Ql ∗ ζ1K ζ2K 22 0(O−1)×1 0(P −1)×1 Z 0 −μSij l (z) 

−1 −1 −1 −1 −1 −1 −1 × diag{Q−1 1 , · · · , Ql−1 , Ql+1 , · · · , QO , Q1 , · · · , Ql−1 , Ql+1 , · · · , QP }   0(O−1)×1 0(P −1)×1 T k k × ζ1K ζ2K 0(O−1)×1 0(P −1)×1 #   where 22 = −μ−1 Ql Sij l (z)Ql + πll (Ql − Rl ) − s∈TK πls Rl − p∈TUK πlp Rl .

(33)

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9

Applying the Schur complement to the above inequality (33) again, we obtain ⎡ −Z − Z T ∗ ∗ ∗ ∗ ⎤ ⎢ Aci,l (z)Z + Ql ⎢ Z (33) ⇔ ⎢ ⎣ 0

∗ ∗ ∗ ⎥ ∗ ∗ ⎥ −μSij l (z) ⎥<0 1 0 −QK ∗ ⎦ k )T 0 −Q2K 0 (ζ2K   where 22 = −μ−1 Ql Sij l (z)Ql + πll (Ql − Rl ) − s∈TK πls Rl − p∈TUK πlp Rl . Recalling that (S − Q)T S −1 (S − Q) ≥ 0, we have 22 0 k )T (ζ1K 0

(34)

−Ql Sij l (z)−1 Ql ≤ Sij l (z) − 2Ql . Considering this and μ > 0, we have −μ−1 Ql Sij l (z)−1 Ql ≤ μ−1 (Sij l (z) − 2Ql ).

(35)

Combining (35) with (34), matrix inequality (34) is rewritten as follows: r r  

θi (z(t))θj (z(t)) 1ij,ls < 0.

(36)

i=1 j =1

By Lemma 4, we prove that LMIs (22) and (23) imply (36) since r r  

1  1  θi (z(t)) 1ii,ls + θi (z(t))θj (z(t))( 1ij,ls + 1j i,ls ). r −1 2 r

θi (z(t))θj (z(t)) 1ij,ls <

i=1 j =1

i=1

r

r

i=1 j =1

Using similar methods as above for (29) and (30), we can derive (24)–(26). Therefore, if LMIs (22)–(26) hold, we can conclude that the closed-loop system (11) is stochastically stable according to Theorem 1. Then from Theorem 1, nominal MJFS (11) can be stabilized with the state feedback controller (9), and the desired controller gains are given by (27). The proof is completed. 2 3.2. Stochastic stability analysis and state feedback stabilization for uncertain Markovian jump fuzzy systems In this subsection, we are in a position to give the main result on robust fuzzy controller for the uncertain Markovian jump fuzzy system in (10). Because the parameter uncertainties are not considered in previous subsection, Theorem 1 cannot be utilized directly to determine the stability of uncertain MJFSs. The following theorem provides a sufficient criteria for the uncertain MJFSs (10) to be robustly stable. Theorem 3. The free MJFS (10) with a partly unknown transition rate matrix (4) is robustly stochastically stable if there exist matrices Pl = PlT > 0, Wl = WlT and positive scalars σl such that the following inequalities are feasible for i ∈ IR , l ∈ N:   1ils ∗ ∗ (37) ≺ 0, l ∈ TlK , s = l, ∗ Pl Di,l −σl I 0 −σl I σl E1li   2ils ∗ ∗ ≤ 0, ∀l ∈ TlUK , ∀s ∈ TlK (38) Pl Di,l −σl I ∗ 0 −σl I σl E1li (39) Ps − Pl − Wl ≤ 0, ∀l ∈ TlUK , ∀s ∈ TlUK , s = l where 1ils = ATi,l Pl + Pl Ai,l + +

 s∈TlK



πls (Ps − Wl ) + πll (Pl − Wl ) +

s∈TK

πls (Ps − Wl ) ≺ 0,

 s∈TUK

πls (Ps − Wl )

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2ils = ATi,l Pl + Pl Ai,l +



πls (Ps − Pl − Wl ) ≺ 0.

s∈TK

Proof. It follows from Schur complement that the inequalities of (37)–(38) are equivalent to  (37) ⇔ ATi,l (z)Pl + Pl Ai,l + πls (Ps − Wl ) + πll (Pl − Wl ) 

+

s∈TK

πls (Ps − Wl ) +

s∈TUK

(38) ⇔ ATi,l Pl + Pl Ai,l +





T T πls (Ps − Wl )σl E1i,l E1i,l + σl−1 Pl Di,l Di,l Pl ≺ 0,

(40)

s∈TlK T T πls (Ps − Pl − Wl ) + σl E1i,l E1i,l + σl−1 Pl Di,l Di,l Pl ≤ 0.

(41)

s∈TK

From (2), (3) and Lemma 3, it is easily seen that T T T T Pl Di Fi,l (t)E1i,l + E1i,l Fi,l (t)DiT Pl ≤ l Pl Di,l Di,l Pl + l−1 E1i,l E1i,l ,

(42)

where σl = l−1 . By using the above inequality (42), the inequality (40)–(41) is rewritten as follows:  T    Ai,l + Ai,l Pl + Pl Ai,l + Ai,l + πls (Ps − Wl ) + πll (Pl − Wl ) +



πls (Ps − Wl ) +

s∈TUK



s∈TK

πls (Ps − Wl ) ≺ 0,

(43)

s∈TlK

 T    Ai,l + Ai,l Pl + Pl Ai,l + Ai,l + πls (Ps − Pl − Wl ) ≤ 0,

(44)

s∈TK

for all admissible uncertainties Fi,l (t), l ∈ Tl . Therefore, it follows from Theorem 1 that the uncertain MJFS (10) is stochastically stable for all admissible uncertainties, i.e., the uncertain MJFS (10) is robustly stochastically stable. 2 Similar to the Theorem 2, we have the following theorem, which presents an LMI-based method for the design of a mode-dependent robust fuzzy controller for the uncertain MJFS (10). Theorem 4. Consider the uncertain MJFS (11) with unknown transition probabilities. For some positive scalar μ, if T , R = R T , S , i, j ∈ I , l ∈ N, Z and positive scalar  satisfying there exists matrices Ql = QTl > 0, Yij,l = Yij,l l j,l R l l the following LMIs: For l ∈ TlK 1 ϒii,lsp ≺ 0, s ∈ TlK , p ∈ TlUK , s, p = l  1 1 1 1 + ϒii,lsp ϒij,llsp + ϒj1i,lsp ≺ 0, i = j ∈ IR , s ∈ TlK , p ∈ TlUK , s, p = l, r −1 2

(45) (46)

For l ∈ TlUK 2 ϒii,ls ≺ 0, s ∈ TlK , s = l  1 1 2 2 + ϒii,ls ϒij,ls + ϒj2i,ls ≺ 0, i = j ∈ IR , s ∈ TlK , s = l r −1 2

Ql − Qs − Rl ≤ 0, s ∈ TlUK where

(47) (48) (49)

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⎡

−Z − Z T 1 ϒ12 ⎢ ⎢ 0 ⎢ ⎢ 1 ϒij,lsp = ⎢ E1i,l Z + E2i,l Yi,l ⎢ Z ⎢ ⎣ 0 0 ⎡ −Z − Z T 2 ϒij,ls

1 ϒ12 ⎢ ⎢ ⎢ 0 =⎢ ⎢ E1i,l Z + E2i,l Yi,l ⎣ Z 0

∗ 1 ϒ22 l DlT 0 0 k )T (ζ1K 0

∗ ∗ −l I 0 0 0 k )T (ζ2K

∗ 1 ϒ22 l DlT 0 0 k )T (ζ1K

∗ ∗ −l I 0 0 0

∗ ∗ ∗ −l I 0 0 0 ∗ ∗ ∗ −l I 0 0

∗ ∗ ∗ ∗ −μSij l 0 0 ∗ ∗ ∗ ∗ −μSij l 0

∗ ∗ ∗ ∗ ∗ −Q1K 0 ∗ ∗ ∗ ∗ ∗ −Q1K

11

∗ ⎤ ∗ ⎥ ⎥ ∗ ⎥ ⎥ ∗ ⎥, ⎥ ∗ ⎥ ∗ ⎦ −Q2K ⎤ ⎥ ⎥ ⎥ ⎥, ∗⎥ ⎦

with 1 ϒ12 = Ai,l Z + Bi,l Yj,l + Ql 1 ϒ22 = μ−1 (Sij l − 2Ql ) + πll (Ql − Rl ) − 2 ϒ22 = μ−1 (Sij l − 2Ql ) −





πls Rl −

s



πlp Rl

p

πls (Ql + Rl )

s k , ζ k , Q1 and Q2 , are shown at the Theorem 2, then the system (11) is robustly stochastically stable. Moreover, ζ1K 2K K K the state-feedback fuzzy controller gains from (9) are given by

Kj,l = Yj,l Z −1 , j ∈ IR , l ∈ N

(50)

Proof. If the conditions (45)–(49) hold, then we have r r   i=1 j =1 r r  

1 θi (t)θj (t)ϒij,lsp < 0,

(51)

2 θi (t)θj (t)ϒij,ls < 0.

(52)

i=1 j =1

Similarly to the proof of Theorem 2, by using Lemmas 1 and 4, it is easy to show that if the conditions (45) and (46) are satisfied, then the following inequalities hold: ⎡ −Z − Z T ∗ ∗ ∗ ∗ ∗ ∗ ⎤ 1 ⎢ ⎢ 0 ⎢ ⎢ E Z ⎢ ci,l ⎢ Z ⎢ ⎣ 0 0

2 l DlT 0 0 k )T (ζ1K 0

∗ −l I 0 0 0 k )T (ζ2K

∗ ∗ −l I 0 0 0

∗ ∗ ∗ −μSij l 0 0

∗ ∗ ∗ ∗ −Q1K 0

∗ ⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥≤0 ∗ ⎥ ⎥ ∗ ⎦ −Q2K

(53)

  where 1 = Aci,l Z + Bi,l Yj,l + Ql , 2 = μ−1 (Sij l − 2Ql ) + πll (Ql − Rl ) − s πls Rl − p πlp Rl , Eci,l = E1i,l + E2i,l Kj,l . By applying a similar approach to that in Theorem 2 to (53), we can obtain   3 ∗ ∗ (53) ⇔ Di,l Ql −l I ≺0 (54) ∗ 0 −l I l Eci,l       where 3 = Ql ATci,l + Aci,l Ql + s∈TK πls Ql Q−1 πls Ql Q−1 s Ql − Rl + πll (Pl − Wl ) + s Ql − Rl + s∈T UK  −1 s∈Tl πls (Ql Qs Ql − Rl ). K

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Pre- and post-multiplying left side of (54) by diag{Pl , σl I, σl I } and its transpose, we obtain (37) for l ∈ Tl . Denote Yj,l = Kj,l Z, σl = l−1 and apply the Schur complement and Lemma 3 to the above inequality (54), we obtain that  (54) ⇔ ATci,l (z)Pl + Pl Aci,l + πls (Ps − Wl ) + πll (Pl − Wl ) +



s∈TK

πls (Ps − Wl ) +

s∈TUK



T T πls (Ps − Wl ) + σl E1i,l E1i,l + σl−1 Pl Di,l Di,l Pl ≺ 0.

(55)

s∈TlK

Likewise, it can be verified that LMIs (47)–(49) are sufficiency for (52). Therefore, if LMIs (45)–(49) hold, the uncertain closed-loop system (11) is stochastically stable according to Theorem 3. Then, uncertain MJFS (1) can be stabilized with the state feedback controller (9), and the desired fuzzy controller gains are given by (50). The proof is completed. 2 4. An illustrative example In this section, we present a numerical example to show the effectiveness of the proposed methods. Consider a single-link robot arm in [8]. Its dynamics is described by D(t) 1 MgL sin(θ (t)) − θ˙ (t) + u(t) (56) θ¨ (t) = − J J J where θ (t) the angular position of the arm; u(t) the control input; M the payload mass; J the moment of inertia; g the acceleration of gravity; L the arm length; and D(t) the coefficient of viscous friction. Their parameter setting (g, L) = (9.81, 0.5). As assumed by [8] and [4], the parameter D(t) = D = 2 is time invariant, and the parameters M and J have four different modes as shown in Table 1. Let the state (x1 (t), x2 (t)) = (θ (t), θ˙ (t)). Under the condition −179.4270 < θ(t) < 179.4270, the nonlinear term sin(θ (t)) in (56) can be represented as [15] sin(x1 (t)) = h1 (x1 (t)) · x1 (t) + h2 (t)(x1 (t)) · β · x1 (t)

(57)

with β = 10−2 /π , where h1 (x1 (t)), h2 (x1 (t)) ∈ [0, 1], and h1 (x1 (t)) + h2 (x1 (t)) = 1. By solving Eq. (57), the membership functions h1 (x1 (t)) and h2 (x1 (t)) can be represented as follows: $ sin(x1 (t))−β·x1 (t) , x1 (t) = 0 x1 (t)·(1−β) (58) h1 (x1 (t)) = 1, x1 (t) = 0. $ x1 (t)−sin(x1 (t)) x1 (t)·(1−β) , x1 (t) = 0 h2 (x1 (t)) = (59) 0, x1 (t) = 0. It can be seen from (58)–(59) that h1 (x1 (t)) = 1, h2 (x1 (t)) = 0 when x1 (t) is about 0 rad, and h1 (x1 (t)) = 0, h2 (x1 (t)) = 1 when x1 (t) is about π or −π rad. This builds the two-rule T–S fuzzy model of the single link robot arm parameterized by [8]. Plant Rule 1 : IF x1 (t) is “about 0 rad,”     THEN x(t) ˙ = A1,l + A1,l x(t) + B1,l + B1,l u(t), Plant Rule 2 : IF x1 (t) is “about π rad or − π rad,”     THEN x(t) ˙ = A2,l + A2,l x(t) + B2,l + B2,l u(t) where

" ! " 1 0 1 , A1,2 = , −D −gL −0.8D ! " ! " 0 1 0 1 , A1,4 = , A1,3 = −gL −0.5D −gL −0.4D

A1,1 =

!

0 −gl

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Table 1 Modes of the parameters M, and J . Mode l

M

J

1 2 3 4

1 1.25 2 2.5

1 1.25 2 2.5

Table 2 Different transition probabilities matrices. Case I

1

2

3

4

1 2 3 4

−0.5 ? 0.2 ?

? −0.6 0.3 ?

0.3 ? ? 0.3

? 0.3 ? ?

Case II 1 2 3 4

!

−0.7 ? 0.5 ?

? −0.4 0.3 ?

0.1 0.1 ? 0.4

0.1 ? ? ?

" ! " 1 0 1 , A2,2 = , −D −βgL −0.8D " " ! ! 0 1 0 1 , A2,4 = , A2,3 = −βgL −0.5D −βgL −0.4D ! " ! " 0 0 B1,1 = B2,1 = , B1,2 = B2,2 = , 1 0.8 ! " ! " 0 0 B1,3 = B2,3 = , B1,4 = B2,4 = . 0.5 0.4 A2,1 =

0 −βgL

The uncertain parameters A1,l (t), A2,l (t), B1,l (t) and B2,l (t), l ∈ [1, 2, 3, 4] satisfy (2) and (3) with A1,l (t) = Dl F1,l (t)E11,l , A2,l (t) = Dl F2,l (t)E12,l , B1,l (t) = Dl F1,l (t)E21,l , B2,l (t) = Dl F2,l (t)E22,l , ! " 0 Dl = , E11,l = E12,l = [ 0 0.1D ] 1 E21,l = E22,l = 0. The two cases for the partly known transition rates matrix are considered in Table 2. We now attempt to design a mode-dependent state feedback fuzzy controller in the form of (9) to stochastically stabilize the single-link robot arm. To demonstrate the effectiveness of proposed method, Theorem 2 of Sheng [8], Shen [4] and Theorem 4 are applied to the design of a mode-dependent fuzzy controller such that the closed-loop nominal system is stable with partly unknown transition probabilities. Sheng and Shen only considered nominal fuzzy Markovian systems as partially unknown and there is no feasible solution to the stabilization problem. However, the method proposed here still works well and the controller gain matrices are obtained by solving the conditions given in Theorem 4. With the choice of μ = 0.2, we apply Theorem 4 to the design of a mode-dependent fuzzy controller such that the closed-loop system is stochastically stable. This demonstrates the proposed stabilization conditions based on free-weighting matrix method is less conservative than previous results. By solving the LMIs given in Theorem 4, the fuzzy controller gains are obtained as follows:

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14

Fig. 1. The trajectories of states x1 and x2 : Case I.

Case I ! K1,1 = !

0.7874 −4.0854

"T

! , K1,2 =

3.9425 −6.8974

!

"T

"T −3.7942 −10.6485 ! "T ! "T −5.4878 1.1025 K2,1 = , K2,2 = −10.0789 −7.8456 "T "T ! ! −5.9745 −8.5496 K2,3 = , K2,4 = −6.8456 −10.5414 K1,3 =

"T

1.1025 −5.6682

, K1,4 =

Case II ! K1,1 = !

0.8774 −4.1824

"T

! , K1,2 =

3.8745 −6.5444

!

"T

"T −3.8742 −10.7485 ! "T ! "T −5.7418 1.2025 K2,1 = , K2,2 = −10.1519 −7.3441 ! ! "T "T −5.4416 −8.1514 K2,3 = , K2,4 = −6.7431 −10.4478 K1,3 =

"T

1.1925 −5.7482

, K1,4 =

Now, we apply the designed fuzzy static controller in the form of (9) with the above gain matrices to the nonlinear systems in (56). The initial condition can be chosen as x0 = [0.5π 2]T and η0 = 1. Figs. 1–2 show the simulation results of the proposed method. As one can immediately witness all trajectories are well guided to the origin. These results indicate the designed fuzzy static feedback controller can effectively stabilize the single robot arm. Despite the partially unknown transition probabilities, the designed robust fuzzy controllers are feasible and effective ensuring the resulting closed-loop system systems are stable.

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Fig. 2. The trajectories of states x1 and x2 : Case II.

5. Conclusion This paper has proposed a new methodology of robust state feedback fuzzy controller for a class of uncertain Markovian jump nonlinear systems with partial transition probability. Our primary contributions lie in the use of the free weighting matrix method to derived a less conservative stochastic stability criterion of the uncertain MJFSs in terms of LMIs. A new improved LMI formulation is used to alleviate the interrelation between the stochastic Lyapunov matrix and system matrices containing controller variables in the derivation process. Moreover, a sufficient condition for the mode-dependent state feedback fuzzy controller design has been derived for the Markovian jump fuzzy systems for all admissible parameter uncertainties. Finally, a simulation result has successfully verified the effectiveness of the proposed methodology. Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2012R1A2A2A01014088) and the Human Resources Development program (No. 20124030200040/2) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy. References [1] Y. Ji, H.J. Chizeck, Jump linear quadratic Gaussian control in continuous time, IEEE Trans. Autom. Control 37 (12) (1992) 1884–1892. [2] C. Souza, Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems, IEEE Trans. Autom. Control 51 (5) (2006) 836–841. [3] H. Wu, K. Cai, Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control, IEEE Trans. Syst. Man Cybern., Part B, Cybern. 36 (3) (2006) 509–516. [4] M. Shen, D. Ye, Improved fuzzy control design for nonlinear Markovian-jump systems with incomplete transition descriptions, Fuzzy Sets Syst. 217 (2) (2013) 80–95. [5] J. Dong, G. Yang, Fuzzy controller design for Markovian jump nonlinear systems, Int. J. Control. Autom. Syst. 5 (6) (2007) 712–717. [6] N. Arrifano, V. Oliveria, Robust H∞ fuzzy control approach for a class of Markovian jump nonlinear systems, IEEE Trans. Fuzzy Syst. 14 (6) (2006) 738–754. [7] S. Nguang, W. Assawinchaichote, P. Shi, Robust H∞ control design for fuzzy singularly perturbed systems with Markovian jumps: an LMI approach, Inf. Sci. 177 (1) (2007) 893–908. [8] L. Sheng, M. Gao, Stabilization for Markovian jump nonlinear systems with partly unknown transition probabilities via fuzzy control, Fuzzy Sets Syst. 161 (21) (2010) 2780–2792.

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