H filtering for continuous-time T-S fuzzy

Transcription

1 H filtering for continuous-time T-S fuzzy 1 systems with partly immeasurable premise variables Jiuxiang Dong and Guang-Hong Yang Abstract This paper is concerned with H filtering problem for T-S fuzzy systems with partly immeasurable premise variables. By using measurable premise variables of fuzzy models as the premise variables of fuzzy filters, a new fuzzy filter scheme is constructed. Further based on a class of line integral fuzzy Lyapunov functions, a convex condition for designing H filters is proposed. In contrast to the existing approaches, the new condition can take full use of measurable premise variables for less conservative design. A numerical example is given to illustrate the effectiveness of the proposed method. Index Terms LMIs. T-S fuzzy control systems, immeasurable premise variables, H filter, linear matrix inequalities Jiuxiang Dong is with the College of Information Science and Engineering, Northeastern University, Shenyang, , P. R. China. s: dongjiuxiang@ise.neu.edu.cn Guang-Hong Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang, , P. R. China. s: yangguanghong@ise.neu.edu.cn

2 2 I. INTRODUCTION In the past several decades, there have been considerable interest in the problem of control synthesis and filter design for nonlinear systems based on Takagi-Sugeno T-S fuzzy models, and many significant advances have been achieved, where quadratic Lyapunov function approaches [1]-[4] are widely employed. Since a common Lyapunov matrix is used for all local models of fuzzy systems, the quadratic Lyapunov function approach often leads to conservative results. Then parameter dependent Lyapunov functions or called fuzzy Lyapunov functions [5]-[8], piecewise Lyapunov functions [9]-[11], k-sample variation method for various Lyapunov functions [12] are respectively proposed for reducing the conservatism introduced by using quadratic Lyapunov functions. On the other hand, by sharing the same fuzzy rules with the fuzzy models, parallel distributed compensation PDC scheme [13] is widely used for designing fuzzy controllers and filters. In addition, a number of alternative schemes are also proposed for less conservative design, such as the switching PDC scheme [14], the non-pdc scheme [5], [15], local nonlinear feedback scheme [16], [17] and so on. It is well known that the state variables in control systems aren t always measurable on line, then the state estimation and filter problems for nonlinear systems are always one of the hot research areas and many excellent results has been achieved. These results can be roughly divided into two categories by virtue of the accessibility of priori statistical information. If the priori statistical information on the external disturbances is available, then the statistical information can be used for designing filters, for example, the celebrated Kalman filtering has been widely applied in practical engineering systems. When the statistical information of the external disturbances is unavailable, H filters can be used only with the assumption that the external noise is energy bounded. It provides a guaranteed noise attenuation level by minimising

3 3 the H norm from external noise to the estimation error [18]. In recent years, the problem of filter design of T-S fuzzy systems has attracted a great interest of the researchers. By using an augmentation technique, a markovian jump system approach is proposed for designing filters for nonuniformly sampled T-S fuzzy systems in [19]. For a class of discrete-time T-S fuzzy systems in sensor networks with additive filter gain uncertainties, a non-fragile distributed H filter design condition is given by using robust control approaches in [20]. By constructing a Lyapunov-Krasovskii function with a factor α, a sufficient condition for designing an exponential H filter for switched delay systems is presented in [21]. By constructing general sensor failure model for T-S fuzzy model-based networked control systems, a condition for designing reliable H filters is proposed in [22]. By considering sensor faults as an auxiliary state variable, the robust fault detection observer design is formulated as an H /H problem for T-S fuzzy systems in [23]. Note that the premise variables of the fuzzy filters are the same as ones of the fuzzy models in the above mentioned results, which implies that all premise variables of fuzzy models should be accessible on line. However, the premise variables of many T-S fuzzy models aren t always accessible on line, for example, the premise variables of some T-S fuzzy models might be dependent on the immeasurable states. In general, there are two classes of approaches for constructing T-S fuzzy models, i.e., identification from input-output data and direct derivation from given nonlinear system equations [13]. The premise variables of the T-S fuzzy models constructed from the second approach are often dependent on system states, then those premise variables dependent on immeasurable states cannot be accessed and these unaccessible premise variables are also called as immeasurable premise variables with a bit of confusion concept in this paper. For T-S fuzzy systems with immeasurable premise variables, the above-mentioned methods

4 4 aren t applicable, then various techniques are proposed for designing filters and controllers for the class of T-S fuzzy systems in the past decade, see [24]-[38]. By merging unaccessible fuzzy rule weights into uncertainties, the filter and control designs are converted into H filtering and control problems of uncertain fuzzy systems in [39], [24], [26]. By using the estimation of the unaccessible fuzzy rule weights, the conditions for designing fuzzy controllers and filters are proposed in [28]-[32]. For a class of T-S fuzzy systems with fuzzy rule weight functions satisfying Lipschitz-like conditions, the relations between the estimation errors of fuzzy rule weights and the states are used for less conservative control synthesis condition in [33]-[35]. Note that all premise variables of fuzzy controllers or filters need to be estimated or merged into uncertainties in these results. If there exist partly measurable premise variables in fuzzy models, then the partly measurable premise variables cannot be directly used for designing fuzzy controllers and filters in the above-mentioned approaches, which might lead to conservative results. In fact, fuzzy models of many engineering systems often are simultaneously with immeasurable and measurable premise variables, for example, wastewater treatment plants [37], tank systems [38], two-wheeled vehicles [40], bioreactor models [41] and so on. In order to overcome the shortcoming of the existing approaches, a new H filtering scheme with the partly measurable premise variables and a new line-integral fuzzy Lyapunov function are proposed for continuoustime T-S fuzzy systems with immeasurable premise variables, and by using the new scheme and the new Lyapunov function, a convex condition for designing H filters is presented. The new method makes the best of the available premise variable information, then it can give less conservative results. An example is given to illustrate the effectiveness of the new approach. The paper is organized as follows. Section II presents system description and some notations. A new filter scheme for T-S fuzzy systems with partly immeasurable premise variables is proposed

5 5 and the corresponding design condition is given in Section III. A numerical example is given to illustrate the effectiveness of the new proposed method in Section IV. Concluding remarks are given in Section V. Notation: For a symmetric block matrix, is used for the blocks induced by symmetry, for example G 11 G 11 G T 21 G T 31 G 21 G 22 = G 21 G 22 G T 32 G 31 G 32 G 33 G 31 G 32 G 33 The superscript T stands for matrix transposition and the notation G T denotes the transpose of the inverse matrix of G. HeG = G + G T. II. SYSTEM DESCRIPTION Consider the following fuzzy system model: Plant Rule i 1 i 2 i p : IF v 1 t is M 1i1 and v 2 t is M 2i2,, v p t is M pip THEN ẋt = A i1 i 2 i p xt + B wi1 i 2 i p wt yt = C yi1 i 2 i p xt zt = C zi1 i 2 i p xt + D zi1 i 2 i p wt 1 xt R nx is the state, wt R nw is the disturbance input, yt R ny is the measurable output, zt R nz is the signal to be estimated, v l t, l = 1,, p are the premise variables, M lil, l = 1,, p, i l = 1,, r l denotes a v l t-based fuzzy set and they are linguistic terms characterized by fuzzy membership functions M lil v l t, where r l is the number of v l t-based fuzzy sets. It is easily to be obtained that the fuzzy rule base consists of r = p r i IF-THEN i=1

6 6 rules. The membership functions M lil v l t are generally non-linear functions of the premise variables v l t, l = 1,, p, which depend on the external or internal variables to the system. The internal variables might be measurable such as {ut, yt} or immeasurable states of the system. If some premise variables are dependent on the immeasurable states, then they are unaccessible on line. With a bit of confusion concept in this paper, these unaccessible premise variables are called as immeasurable premise variables. If the T-S fuzzy model 3 is constructed from a given nonlinear system equation, then it often contains immeasurable premise variables. For example, the techniques of sector nonlinearity in fuzzy model construction and local approximation in fuzzy partition spaces are widely used for obtaining T-S models [13]. If these techniques are used for constructing T-S fuzzy models, then the premise variables of the fuzzy models may be dependent on the system states, which implies that it is impossible to have access to the values of the premise variables dependent on immeasurable states. Without loss of generality, assume that the premise variables v i t, i = 1,, p 0 are measurable and the premise variables v i t, i = p 0 + 1,, p are immeasurable and the following new full-order filter scheme is proposed, where the premise variables of the filter consist of the measurable premise variables of the fuzzy model 3. Filter Rule i 1 i 2 i p0 : IF v 1 t is M 1i1,, v p0 t is M p0 i p0 THEN ˆxt = Âi 1 i 2 p 0 ˆxt + ˆB i1 i 2 p 0 yt ẑt = Ĉi 1 i 2 p 0 ˆxt + ˆD i1 i 2 p 0 yt 2 where ˆxt R nx is the state of the filter, ẑt R nz is the estimation of zt, and the parameters Âi 1 i 2 p 0, ˆBi1 i 2 p 0, Ĉ i1 i 2 p 0, ˆDi1 i 2 p 0 in the filter are to be determined, such that the

7 7 resulted closed-loop system is asymptotically stable and satisfies an H performance constraint. By using the fuzzy inference method with a singleton fuzzifier, product inference, and center average defuzzifiers, the final T-S fuzzy model 1 and the filter 2 are obtained as: r 1 r 2 r p p M lil v l t A i1 i 2 i p xt + B wi1 i 2 i p wt i 1 =1 i 2 =1 i p=1 l=1 ẋt = r 1 r 2 r p p M lil v l t and Let yt = zt = ˆxt = ẑt = r 1 r 2 i 1 =1 i 2 =1 r 1 i 1 =1 i 2 =1 r 1 i 1 =1 i 2 =1 r 1 r p i p=1 r 1 r 2 i 1 =1 i 2 =1 r 2 r p r 2 r 2 i 1 =1 i 2 =1 i p=1 r p0 i p0 =1 r p0 i p0 =1 i 1 =1 i 2 =1 p i p=1 l=1 M lil v l t C yi1 i 2 i p xt l=1 r p p M lil v l t p i p=1 l=1 M lil v l t C zi1 i 2 i p xt + D zi1 i 2 i p wt l=1 p r 1 r 2 i 1 =1 i 2 =1 p0 l=1 r 1 i 1 =1 i 2 =1 p0 l=1 r 1 r p i p=1 l=1 M lil v l t M lil v l t Âi1i2 ip0 ˆxt + ˆB yt i1i2 ip0 r 2 r p0 p 0 i p0 =1 l=1 M lil v l t M lil v l t Ĉi1i2 ip0 ˆxt + ˆD yt i1i2 ip0 r 2 i 1 =1 i 2 =1 r p0 p 0 i p0 =1 l=1 M lil v l t 3 4 then µ lil v l t = M li l v l t r l, for 1 l p, 1 i l r l 5 M ljl v l t j l =1 r l i l =1 µ lil v l t = 1, for 1 l p 6 and from 5 and 6, the fuzzy system 3 and the filter 4 can be rewritten as follows: r 1 r 2 r p p ẋt = µ lil v l t A i1 i 2 i p xt + B wi1 i 2 i p wt i 1 =1 i 2 =1 i p=1 l=1

8 and yt = zt = ˆxt = ẑt = r 1 r 2 i 1 =1 i 2 =1 r 1 r 2 i 1 =1 i 2 =1 r 1 r 2 i 1 =1 i 2 =1 r 1 r 2 i 1 =1 i 2 =1 r p i p=1 r p i p=1 r p0 p µ lil v l t C yi1 i 2 i p xt l=1 p l=1 p0 i p0 =1 l=1 r p0 p0 i p0 =1 µ lil v l t C zi1 i 2 i p xt + D zi1 i 2 i p wt µ lil v l t Âi1i2 ip0 ˆxt + ˆB yt i1i2 ip0 l=1 µ lil v l t Ĉi1i2 ip0 ˆxt + ˆD i1i2 ip0 yt In this paper, a set-theoretic description for the T-S fuzzy system 7 and the filter 8 is proposed, and the equivalence class theory in set theory will be applied for designing fuzzy filters with the help of the set-theoretic description. In order to give the new description of the fuzzy system 7, some sets are defined as follows: S ρ = {1, 2,, r ρ }, 1 ρ p p S = S 1 p = S ρ = S 1 S p = {i 1,, i p : i 1 S 1 i p S p } ρ=1 p 0 S 1 p0 = S ρ = S 1 S p0 = {i 1,, i p0 : i 1 S 1 i p0 S p0 } S p0 +1 p = ρ=1 p S ρ = S p0 +1 S p = {i p0 +1,, i p : i p0 +1 S p0 +1 i p S p } ρ=p where i 1, i 2,, i p is an ordered p-tuple, represents a classic logical operator conjunction. The ordered p-tuple i 1, i 2,, i p can be viewed as a permutation of p elements, and is also denoted as i 1 i 2 i p in this paper. For a simple expression, the follow notations are used in this paper, i = i 1 p = i 1 i 2 i p i 1 p0 = i 1 i 2 i p0

9 9 i p0 +1 p = i p0 +1i p0 +2 i p p µ i = µ i 1 p = µ i1 i 2 i p = µ lil v l t l=1 p 0 µ i 1 p0 = µ i1 i 2 i p0 = µ lil v l t l=1 µ i = µ p0 +1 p i p0 +1i p0 +2 i p = then we can easily obtain p µ lil v l t 10 l=p 0 +1 i 1 p0, i p0 +1 p = i 1 p = i µ i 1 p0 µ i p0 +1 p = µ i 1 p = µ i By using the above notations, the fuzzy model rule and filter rule can be rewritten as follows: Plant Rule i: IF v 1 t is M 1i1 and v 2 t is M 2i2,, v p t is M pip THEN ẋt = A i xt + B w i wt yt = C y i xt zt = C z i xt + D z i wt Filter Rule i 1 p0 : IF v 1 t is M 1i1 and v 2 t is M 2i2,, v p0 t is M p0 i p0 THEN ˆxt = Â i 1 p0 ˆxt + ˆB i 1 p0 yt ẑt = Ĉ i 1 p0 ˆxt + ˆD i 1 p0 yt and the simple set-theoretic expressions for the T-S fuzzy system 7 and the fuzzy filter 8 are

10 10 given as follows: and Furthermore, let ˆxt = ẑt = ẋt = i S 1 p µ i yt = zt = i S 1 p µ i C y i xt i S 1 p µ i i 1 p0 S 1 p0 µ i 1 p0 i 1 p0 S 1 p0 µ i 1 p0 A i xt + B w i wt Cz i xt + D z i wt 11  i 1 p0 ˆxt + ˆB i 1 p0 yt Ĉ i 1 p0 ˆxt + ˆD i 1 p0 yt 12 Aµ = µ i A i, B wµ = i S 1 p C z µ = µ i C z i, D zµ = i S 1 p i S 1 p µ i B w i, C yµ = i S 1 p µ i D z i i S 1 p µ i C y i µ = µ i 1 p0  i 1 p0, ˆBµ = µ i ˆB 1 p0 i 1 p0 i 1 p0 S 1 p0 i 1 p0 S 1 p0 Ĉµ = µ i 1 p0 Ĉ i 1 p0, ˆDµ = µ i ˆD 1 p0 i 1 p0 i 1 p0 S 1 p0 i 1 p0 S 1 p0 then 11 and 12 can respectively be rewritten as: ẋt =Aµxt + B w µwt yt =C y µxt zt =C z µxt + D z µwt 13 and ˆxt =µˆxt + ˆBµyt

11 11 ẑt =Ĉµˆxt + ˆDµyt 14 With the plant 13 and the filter 14, we get: ẋ cl t = Ax cl t + Bwt z e t = Cx cl t + Dwt 15 where x cl t = xt, z et = zt ẑt and ˆxt A B = C D Aµ 0 B w µ ˆBµC y µ µ 0 C z µ ˆDµC y µ Ĉµ D zµ In this paper, our main purpose is to design the filter 4, i.e., determine the filter parameters  1 p0, ˆB1 p0, Ĉ 1 p0, ˆD1 p0, such that the closed-loop system 15 under zero initial condition satisfies the following H performance, z T e tz e tdt = t=0 t=0 t=0 16 zt ẑt T zt ẑtdt γ 2 w T twtdt 17 III. MAIN RESULT In this section, a new class of line-integral fuzzy Lyapunov functions are firstly given. Subsequently, a convex condition for designing H filters is proposed. A. Line-integral fuzzy Lyapunov function candidate In spire of the line-integral Lyapunov function [42], the following line-integral function is considered, κx = 2 < Ψψ, dψ > 18 Γ0,x

12 12 where Γ0, x is a path from the origin 0 to the current state x, ψ R nx is a dummy vector for the integral, <, > denotes an inner product, dψ is an infinitesimal displacement vector and Ψx is a state vector function with Ψx = Φ 0 + Φµx 19 where Φµ = diag[d 11 µ d 22 µ d nxnx µ], φ 11 φ 12 φ 1nx φ 12 φ 22 φ 2nx Φ 0 = a 20b d hh µ = l Ω h φ 1nx φ 2nx φ nxn x r l i l =1 µ lil d hhlil, for h = 1, 2,, n x 20c Ω h = {l : 1 l p and v l t is only dependent on x h t} for h = 1, 2,, n x 20d From the above definitions, Φµ can be rewritten as follows: with Φ i = Φ i 1 i p = diag Φµ = µ i Φ i 21 i S 1 p [ l Ω 1 d 11lil d 22lil l Ω 2 ] l Ω nx d nxn xli l 22 From 19, we have that Ψ ix x j = Ψ jx x i = φ ij, for 1 i j n x, which implies that the line-integral function 18 is path-independent. Then, the time-derivative of κx becomes κx =x T Φ 0 + Φµ ẋ + ẋ T Φ 0 + Φµ x

13 13 =x T Φ 0 + Φµ Aµxt + B w µwt + Aµxt + B w µwt T Φ0 + Φµ x 23 In the following lemma, a new class of line-integral fuzzy Lyapunov candidate functions are proposed. Lemma 1: If there exist matrices X = X T, N, Φ 0 = Φ T 0, Φ i = ΦT i, i S 1 p satisfying Φ 0 + Φ i N T N > 0, for i S 1 p 24 X where S 1 p, Φ i are the same as in 9 and 22, respectively, then the following function V x clt is a Lyapunov function candidate. V x cl t =κxt + 2x T tn ˆxt + ˆx T tx ˆxt =2 < Φ 0 + Φµψ, dψ > +2x T tn ˆxt + ˆx T tx ˆxt 25 Γ0,x where κxt and Φµ are respectively the same as in 18 and 21. Proof: It follows from 24 that there exist positive definite matrices P = P = P 11 N T N, satisfying X P 11 N T N, X P 11 < Φ 0 + Φ i < P 11 0 < P < Φ 0 + Φ i N T N < P X 26a 26b Note that Ψx = Φ 0 + Φµx is path-independent, then substitute νx for ψ in 25 with v is a scalar, we have V x cl =2 < Φ 0 + Φµνx, xdν > +2x T N ˆx + ˆx T X ˆx Γ0,1

14 14 =2 =2 Γ0,1 Γ0,1 Combiniting it and 26, it yields that 2 2 Γ0,1 Γ0,1 νx T Φ 0 + Φµ T xdν + 2x T N ˆx + ˆx T X ˆx x T Φ 0 + Φµxνdν + 2x T N ˆx + ˆx T X ˆx x T P 11 xνdν + 2x T N ˆx + ˆx T X ˆx V x cl x T P11 xνdν + 2x T N ˆx + ˆx T X ˆx i.e., 2x T P 11 x 2x T P11 x Γ0,1 Γ0,1 νdν + 2x T N ˆx + ˆx T P 22ˆx V x cl νdν + 2x T N ˆx + ˆx T P22ˆx which can be rewritten as follows: x T P 11 x + 2x T N ˆx + ˆx T X ˆx = x T clp x cl V x cl x T P11 x + 2x T N ˆx + ˆx T X ˆx = x T cl P x cl which implies that V 0 = 0, V x cl > 0 for x cl 0, and V x cl as x cl. Therefore, the scalar function V x cl t is a Lyapunov function candidate. Before the condition for designing H filters is proposed, some mapping, equivalent relations and useful lemmas are given. Let r = p p r ρ, it is follows from S ρ = r ρ that S 1 p = S ρ = r, i.e., the set S 1 p ρ=1 consists of r elements. Therefore, there exists a 1 1 mapping p q : S ρ {1, 2,, r} ρ=1 Based on a lexicographic order, a particular mapping q can be chosen as follows: ρ=1 qi 1 i 2 i p

15 15 p 1 =i p + i p 1 1r p + i p 2 1r p r p 1 + i p 3 1r p r p 1 r p i 1 1 p 1 ρ 1 =i p + i p ρ 1 r p l 27 ρ=1 l=0 Define a mapping st from Z 2 + to Z 2 + Z + denotes the positive integer set, l=0 r p l stab = ā b, ā b 28 which is an arrangement of the permutation ab. By using the mapping st, the following binary relations on Z 2 + can be obtained, R 1 = {i 1 j 1 : sti 1 j 1 = stj 1 i 1 } 29a. 29b R p = {i p j p : sti p j p = stj p i p } 29c It is easily shown that the relation R i, i = 1,, p are reflexive, symmetric, and transitive, which implies they are equivalence relations. The following lemmas are useful. Lemma 2: [43] pp. 12 If M is an equivalence relation on X, then the set X/M = { x M : x X} is a partition of X. Conversely, for each partition of X, there exists an equivalence relation M o on X, such that X/M o = { x Mo : x X} is the partition. Lemma 3: If scalars 0 µ lil 1, r l i l =1 ζ, ϱ = 1, 2,, r with r = p r l satisfy i 1 j 1 S 1 l=1 i pj p S p T i1 i 2 i pj 1 j 2 j p µ lil = 1, and matrices T i j, i, j, S 1 p, H ζϱ = H T ϱζ, i 1 j 1 S 1 i pj p S p H qi1 i pqj 1 j p, for S 1 S 2 1/R 1,, S p S 2 p/r p 30

16 H 11 H 12 H 1r H 21 H 22 H 2r < H r1 H r2 H rr where q is the same as in 27, then i.e., r 1 r 2 r p r 1 r 2 r p i 1 =1 i 2 =1 i p=1 j 1 =1 j 2 =1 j p=1 µ 1i1 µ 2i2 µ pip µ 1j1 µ 2j2 µ pjp T i1 i pj 1 j p < 0 16 Proof: See Appendix. i S 1 p j S 1 p µ i µ j T i j < 0 Remark 1: In Lemma 3, R 1,, R p are the same as in 29, they are equivalence relations. The set S 2 l /R l is a partition of S 2 l. S l is a element of the partition S 2 l /R l, which implies that S l is an equivalence class of the set S 2 l with the relation R l. If S 1 = {1, 2}, S 2 = {1, 2}, then S 2 1 = {11, 12, 21, 22}, S 2 2 = {11, 12, 21, 22}, S 2 1/R 1 = {{11}, {12, 21}, {22}}, S 2 2/R 2 = {{11}, {12, 21}, {22}} B. H filter design In this subsection, based on the line-integral Lyapunov function 25 and the fuzzy filter scheme 2, a convex condition for designing H filters is proposed. The following Lemma is useful. Lemma 4: Given matrices Φ 0 = Φ T 0, Y = Y T, Φ i = ΦT i, i S 1 p and an invertible matrix N, satisfying Φ 0 Y > 0 32

17 17 Φ i + Y > 0, i S 1 p 33 then it is possible to determine a double invertible matrices P µ and P 1 µ, such that P i = Φ 0 + Φ i N T P µ = Φ 0 + Φµ N T N > 0 34 X N > 0 35 X Y + Φµ 1 P 1 µ = N 1 Φ 0 + ΦµY + Φµ 1 + N 1 Y + Φµ 1 Φ 0 + ΦµN T + N T N Φ ΦµY + Φµ 1 Φ 0 + Φµ Φ 0 + Φµ where Φ i, Φµ are the same as in 21 and X = N T Φ 0 Y 1 N. Proof: It is easily to show from that N T > 0 36 Φ 0 + Φ i > 0, i S 1 p and Φ 0 + Φ i Φ 0 Y > 0, i S 1 p Applying Schur complement to the above inequality and 32, then we have that Φ 0 + Φ i I Left- and right-multiplying the above inequality by I > 0, i S 1 p Φ 0 Y 1 I 0 and its transpose, then it yields 0 N T

18 18 that P i = Φ 0 + Φ i N T N > 0, i S 1 p N T Φ 0 Y 1 N i.e., 34 holds and then it is easily obtained from the above inequalities that i.e, 35 holds. Φ 0 + Φµ N T N > 0 N T Φ 0 Y 1 N On the other hand, from 33 and the invertible matrix N, we can construct the following matrix Y + Φµ 1 N 1 Φ 0 + ΦµY + Φµ 1 + N 1 Y + Φµ 1 Φ 0 + ΦµN T + N T N Φ ΦµY + Φµ 1 Φ 0 + Φµ Φ 0 + Φµ N T and can obtain Φ 0 + Φµ N T N Y + Φµ 1 X N 1 Φ 0 + ΦµY + Φµ 1 + N 1 Y + Φµ 1 Φ 0 + ΦµN T + N T N Φ ΦµY + Φµ 1 Φ 0 + Φµ Φ 0 + Φµ = I 0 0 I where the infer process of the blocks 2, 1 and 2, 2 of the above matrix are given as follows: N T Y + Φµ 1 + N T Φ 0 Y 1 Φ 0 + ΦµY + Φµ 1 + I =N T I + Φ 0 Y 1 Φ 0 + Φµ + Y + Φµ Y + Φµ 1 N T N T

19 19 =N T I + Φ 0 Y 1 Φ 0 + Y Y + Φµ 1 =0 I N T Y + Φµ 1 Φ 0 + ΦµN T + N T Φ 0 Y 1 Φ 0 + ΦµY + Φµ 1 Φ 0 + Φµ Φ 0 + Φµ N T =I + N T I + Φ 0 Y Φ Φµ Y + Φµ Y + Φµ 1 Φ 0 + ΦµN T =I then 36 holds. Thus, the proof is complete. Remark 2: A set of positive definite matrices P µ are given in 35, the blocks 1, 1 of these matrices are dependent on the membership functions. Although only one block is dependent on µ in P µ, all blocks in P 1 µ are dependent on µ. The special structure is helpful for less conservative results. Based on Lemmas 1 and 4, the following theorem can be obtained. Theorem 1: If there exist matrices Φ 0 = Φ T 0, Y = Y T, Φ i = ΦT i, i S 1 p, µ, ˆBµ, Ĉµ, ˆDµ and an invertible matrix N satisfying 32, 33 and W 11 W 12 W 13 W 14 W 22 W 23 W 24 < 0 37 γ 2 I W 34 I where S 1 p and Φµ are respectively the same as in 9 and 21, W 11 =He Y + ΦµAµ W 12 =Y + ΦµAµ Φ 0 Y N T  T µn T + A T µφ 0 + Φµ + C T y µ ˆB T µn T W 13 =Y + ΦµB w µ

20 20 W 14 = C z µ ˆDµC T y µ + ĈµN 1 Φ 0 Y W 22 =He Φ 0 + ΦµAµ + N ˆBµC y µ W 23 =Φ 0 + ΦµB w µ W 24 = C z µ ˆDµC T y µ W 34 =D z µ then the system 15 is asymptotically stable, and the H -norm of the system 15 from exogenous input to the filter error is less than or equal to γ. Proof: Let P i = Φ 0 + Φ i N N T N T Φ 0 Y 1 N Φ 0 + µ i Φ i N P µ = i S 1 p = Φ 0 + Φµ N T N T Φ 0 Y 1 N N T N N T Φ 0 Y 1 N Applying Lemma 4 to 32 and 33, one can obtain P i > 0, P µ > 0 Combining it and Lemma 1, it follows that V t =κxt + 2x T tn ˆxt + ˆx T tx ˆxt =2 < Φ 0 + Φµψ, dψ > +2x T tn ˆxt + ˆx T tx ˆxt 38 Γ0,x is a Lyapunov candidate with X = N T Φ 0 Y 1 N. Note that V t = κxt + 2ẋ T tn ˆxt + 2x T tn ˆxt + 2ˆx T tx ˆxt

21 21 Combining it and 23, it yields that then we have that V t =2x T Φ 0 + Φµ ẋ + 2ẋ T tn ˆxt + 2x T tn ˆxt + 2ˆx T tx ˆxt =2x T clt Φ 0 + Φµ N T N ẋclt X V t + zt ẑt T zt ẑt γ 2 w T twt T =2x T clt Φ 0 + Φµ N ẋclt + x cl t C T [ C N T X wt D T x cl t wt T 0 0 x cl t 0 γ 2 I wt =2x T cltp µẋ cl t + x cl t wt T T C T D T [ C x cl t 0 0 x cl t wt 0 γ 2 I wt T = x cl t He P µa P µb + wt B T P µ γ 2 I where A, B, C, D are the same as in 16. On the other hand, let then D C T D T ] x cl t wt [ C D ] x cl t wt I I Γ 1 =, Γ 2 = Y + Φµ Φ 0 + Φµ N 1 Φ 0 Y 0 0 N T ] x cl t D 39 wt Γ T 1 P µ = Γ T 2 40a

22 22 Γ T 1 P µaγ 1 = Γ T 2 AΓ 1 Y + ΦµAµ = NµN 1 Φ 0 Y + Φ 0 + ΦµAµ + N ˆBµC y µ Y + ΦµAµ Φ 0 + ΦµAµ + N ˆBµC y µ Y + ΦµB w µ Γ T 1 P µb = Γ T 2 B = Φ 0 + ΦµB w µ [ ] CΓ 1 = C z µ ˆDµC y µ + ĈµN 1 Φ 0 Y C z µ ˆDµC y µ 40b 40c 40d By using 40, then 37 can be rewritten as follows: Γ HeP µa P µb C T Γ I 0 B T P µ γ 2 I D T 0 I 0 < I C D I 0 0 I Since Γ 1 is invertible, 41 is equivalent to HeP µa P µb C T B T P µ γ 2 I D T < 0 C D I Applying Schur complement lemma to the above inequality, then it yields that HeP µa P µb + C T [ ] C D < 0 42 B T P µ γ 2 I D T

23 23 Left- and right-multiplying the above inequality by x cl t wt T x cl t HeP µa P µb + wt B T P µ γ 2 I for x cl t 0 wt C T D T T [ C and its transpose, it follows that ] x cl t D < 0, wt Combining it and 39, we have that V t + zt ẑt T zt ẑt γ 2 w T twt < 0 43 Integrating both sides of this inequality yields 0 V t + =V V 0 + <0 0 zt ẑt T zt ẑt γ 2 w T twt zt ẑt T zt ẑt γ 2 w T twt Using the fact that x0 = 0 and V 0, we obtain zt ẑt T zt ẑtdt γ 2 w T twtdt 0 0 Hence, 17 holds and the H performance is fulfilled. If the disturbance wt = 0, then from 43, we have V t < 0. Hence, the system 15 is asymptotically stable. Thus, the proof is complete. Remark 3: Since the product of µ lil vt, l = 1,, p, i l = 1,, r l appears in the closed-loop system, Theorem 1 may be nonlinear in µ lil vt. Hence, it need to be checked for all values of µ lil vt. Applying Lemma 3 to the condition of Theorem 1, then Theorem 1

24 24 can be turned into a finite number of matrix inequalities. Further by using variable substitution technique, we can obtain the following LMI-based condition. Theorem 2: For a given positive scalar γ, if there exist matrices Y = Y T, Φ 0 = Φ T 0, Φ i = ΦT i, H q iq j = HT, A q jq i i 1 p0, B i 1 p0, C i 1 p0, D i 1 p0, i, j S 1 p, i 1 p0 S 1 p0 satisfying the linear matrix inequalities 32, 33 and T i j H q iq j, for S 1 S 2 1/R 1,, S p S 2 p/r p i 1 j 1 S 1 i pj p S p i 1 j 1 S 1 i pj p S p 44 H 11 H 12 H 1r H 21 H 22 H 2r < H r1 H r2 H rr where S 1 p, S 1 p0, S l and R l, l = 1,, p are the same as in 9 and 29, respectively; Φ i, i S 1 p are the same as in 22, He Y + Φ j A i Y + Φ j A i + AT i Φ 0 + Φ j + CT y i BT j 1 p0 T i j = HeA i T Φ 0 + Φ j + B j 1 p0 C y i Y + Φ j B w i C T z i CT y i DT j 1 p0 + C j T 1 p0 Φ 0 + Φ j B w i C T z i CT y i DT j 1 p0 γ 2 I D T z i I then the filter with local gains A j 1 p0 46 Â j 1 p0 = N 1 A j 1 p0 Φ 0 Y 1 N, ˆB j 1 p0 = N 1 B j 1 p0

25 25 w k m y Fig. 1: Mass-Spring system. Ĉ j 1 p0 = C j 1 p0 Φ 0 Y 1 N, ˆD j 1 p0 = D j 1 p0 for j 1 p0 S 1 p0 47 guarantee that the system 15 is asymptotically stable, and the H -norm from exogenous input to the filter error is less than or equal to γ. Proof: The proof is easily obtained from Lemma 3 and Theorem 1, therefore it is omitted. IV. EXAMPLE In this section, we consider a mass m sliding on a horizontal surface and attached to a vertical surface through a spring [44], which is shown in Fig. 1. The simple nonlinear mass-spring system is given as follows: mÿ + ky kβ 2 y 3 + c 1 ẏ + c 2 ẏ ẏ w = 0 where y is the displacement from a reference position and measurable, k is the spring constant, w is the external disturbance. The parameters of the system are m = 5, k = 50, c 1 = 2, c 2 = 0.5, β = Let x 1 = y, x 2 = ẏ, then the state space equation of the nonlinear system is ẋ 1 ẋ 2 = 0 1 k1 β2 x 2 1 c 1+c 2 x 2 m m x 1 x w 1

26 [ ] y = 1 0 x 1 x 2 26 We assume that y [ a, a], ẏ [ b, b], a > 0, b > 0 and by using sector nonlinearity approach [13], then the nonlinear system can be exactly represented by the following T-S fuzzy model: Plant Rule 11: IF x 2 1t is M 11 and x 2 t is M 21 THEN ẋt = A 11 xt + B w11 wt Plant Rule 12: yt = C y11 xt IF x 2 1t is M 11 and x 2 t is M 22 THEN ẋt = A 12 xt + B w12 wt Plant Rule 21: yt = C y12 xt IF x 2 1t is M 12 and x 2 t is M 21 THEN ẋt = A 21 xt + B w21 wt Plant Rule 22: yt = C y21 xt IF x 2 1t is M 12 and x 2 t is M 22 THEN ẋt = A 22 xt + B w22 wt yt = C y22 xt

27 27 where the membership functions M 11 = a2 x 2 1 t, M a 2 12 = x2 1 t, M a 2 21 = b x 2, M b 22 = x 2 b 0 1 A 11 = A 21 = k m c 1 m, A 12 = 0 1 k1 β2 a 2 m c 1 m 0 1 k m, A 22 = B w11 = B w12 = B w21 = B w22 = 1 [ ] C y11 = C y12 = C y21 = C y22 = 1 0 c 1+c 2 b m 0 1 k1 β2 a 2 m c 1+c 2 b m, and From 5, we have µ 11 x 1 t = a2 x 2 1 t a 2 x 2 t b. Let The global fuzzy model is ẋt = yt = 2 2 i 1 =1 i 2 =1 2 i 1 =1 i 2 =1, µ 12 x 1 t = x2 1 t, µ a 2 21 x 2 t = b x 2t, µ b 22 x 2 t = µ 1i1 x 1 tµ 2i2 x 2 t A i1 i 2 xt + B wi1 i 2 wt 2 µ 1i1 x 1 tµ 2i2 x 2 tc yi1 i 2 xt α qi1 i 2 t = µ 1i1 x 1 tµ 2i2 x 2 t, A qi1 i 2 = A i1 i 2, B wqi1 i 2 = B wi1 i 2, C yqi1 i 2 = C yi1 i 2, for i 1 = 1, 2, i 2 = 1, 2 48 where q is the same as in 27. Then the corresponding conventional description of the above fuzzy model can be obtained as follows: ẋt = yt = 4 ς=1 α ς t A ς xt + B wς wt 4 α ς tc yς xt ς=1

28 28 Note that x 2 t is immeasurable, then the premise variable x 2 t cannot be used in fuzzy filters, which implies that the conventional fuzzy filter, ˆxt = ẑt = 4 ς=1 4 ς=1 α ς tâς ˆxt + ˆB ς yt α ς tĉς ˆxt + ˆD ς yt isn t applicable due to the unaccessible α q i t = µ 1i 1 x 1 tµ 2i2 x 2 t. For the class of T-S fuzzy systems with immeasurable premise variables, the H filter design methods in [39] and [27] can be used as far as we know. The methods in [39] can directly be used for the conventional model, but for applying the methods in [27], the nonlinear system needs to be constructed by the technique of [27] as the following form: with l 1 t = a2 x 2 1 t a 2 ẋt = yt = 4 l ς t + h ς t A ς xt + B wς wt ς=1 4 l ς t + h ς tc yς xt ς=1, l 2 t = 0, l 3 t = x2 1 t, l a 2 4 t = 0, h 1 t = a2 x 2 1 t x a 2 b 2 t, h 2 t = a 2 x 2 1 t x a 2 b 2 t, h 3 t = x2 1 t x a 2 b 2t, h 4 t = x2 1 t x a 2 b 2t, h 1 = 0, h 1 = 1, h 2 = 1, h 2 = 0, h 3 = 0, h 3 = 1, h 4 = 1, h 4 = 0, A ς, B wς, C yς are the same as in 48. Assume that the estimated signal zt = x 2 t, and we apply Theorem 2, the methods in [39] and [27] to design H filters, respectively. The methods in [39] and [27] are infeasible, but a feasible solution is obtained by Theorem 2. The fact illustrates that the new method is more effective than the existing ones. In order to avoiding too large filter gains, some constraint conditions need to be added in Theorem 2 as follows: Y + Φ i > 0.5I, for i S

29 50I A i 1 p0 50I B i 1 p0 50I C i 1 p0 50I D i 1 p0 A T i 1 p0 < 0, for i 1 p0 S 1 p0 50I B T i 1 p0 < 0, for i 1 p0 S 1 p0 50I C T i 1 p0 < 0, for i 1 p0 S 1 p0 50I D T i 1 p0 < 0, for i 1 p0 S 1 p0 50I 29 By using Theorem 2 with the above constraint conditions and N = I, it can be obtained that the optimal H performance index γ = and the filter gains  1 =,  = ˆB 1 =, ˆB2 = [ ] [ ] Ĉ 1 = , Ĉ 2 = ˆD 1 = , ˆD2 = [ T The fuzzy filter 12 is used to do simulation with the initial condition x0 = 0 1], and the exogenous disturbance input 1, 4 t 5 wt = 0, other The responses of zt, ẑt and zt ẑt are given in Figs From Figs. 2-3, it can be

30 zt hatz z,hatz Time sec Fig. 2: Responses of zt and ẑt seen that the fuzzy filter can obtain good estimation and robust performance, which shows that the effectiveness of the new method. V. CONCLUSION For T-S fuzzy systems with partly immeasurable premise variables, a new filter scheme with partly measurable premise variables are proposed. Based on the new filter scheme and a class of line integral fuzzy Lyapunov functions, a convex condition for designing H filters has been given. Because the partly measurable variables are directly used in the proposed filter scheme, the new condition can make better use of the partly measurable premise variables for less conservative results than the existing ones. A numerical example has been given to illustrate

31 z e Time sec Fig. 3: Response of the estimation error z e t = zt ẑt the effectiveness of the new approach. VI. APPENDIX The proof of Lemma 3 Let β q i = µ 1i 1 µ 2i2 µ pip where q is the same as in 27. [ ] Left and right-multiplying 31 by β 1 I β 2 I β r I and its transpose, then it follows that r r β ζ β ϱ H ζϱ < 0 ζ=1 ϱ=1

32 32 Because q is a 1-1 mapping from S to {1, 2,, r}, the above inequality is equivalent to β q i β q j H q iq j < 0 i S j S Combining it and S = p S g, we have that g=1 β q i β q j H q iq j < 0 i 1 j 1 S 2 1 i pj p S 2 p Since R g is the equivalent relation over the set S 2 g, g = 1,, p, then apply Lemma 2 to the above inequality, we have that β q i β q j H q iq j i 1 j 1 S 2 i 1 pj p S 2 p = S 1 S 2 1 /R 1 S p S 2 p /Rp β q i β q j H q iq j < 0 49 i 1 j 1 S 1 i pj p S p If i 1 j 1, ī 1 j 1 S 1,, i p j p, ī p j p S p, then sti 1 j 1 = stī 1 j 1,, sti p j p = stī p j p, which implies that µ 1i1 µ 1j1 = µ 1ī1 µ 1 j 1,, µ pip µ pjp = µ pīp µ p j p. Therefore, we have β q i β q j =µ 1i 1 µ 2i2 µ pip µ 1j1 µ 2j2 µ pjp =µ 1i1 µ 1j1 µ 2i2 µ 2j2 µ pip µ pjp =µ 1ī1 µ 1 j 1 µ 2ī2 µ 2 j 2 µ pīp µ p j p =µ 1ī1 µ 2ī2 µ pīp µ 1 j 1 µ 2 j 2 µ p j p =β qī1 ī pβ q j 1 j p =β q ī β q j From the above statements, it can be seen that the value of β q i β q j is independent on the choice of the element in these given sets S 1, S 2,, S p. Then for all i 1 j 1 S 1,, i p j p S p, the value of β q i β q j is unique, so that we can denote the unique value of β q i β q j for all i 1j 1 S 1,

33 33, i p j p S p as ϖs 1, S 2,, S p. Further, we have that β q i β q j H q iq j i 1 j 1 S 1 i pj p S p = ϖs 1, S 2,, S p H q iq j i 1 j 1 S 1 i pj p S p =ϖs 1, S 2,, S p i 1 j 1 S 1 i pj p S p H q iq j Combining it and 49, it yields that β q i β q j H q iq j S 1 S 2 1 /R 1 S p S 2 p /Rp i 1 j 1 S 1 i pj p S p = ϖs 1, S 2,, S p i 1 j 1 S 1 <0 S 1 S 2 1 /R 1 S p S 2 p /Rp i pj p S p H q iq j Consider it, 30 and ϖs 1, S 2,, S p 0, we can obtain that ϖs 1, S 2,, S p i 1 j 1 S 1 S 1 S 2 1 /R 1 which implies that S 1 S 2 1 /R 1 S p S 2 p /Rp = S 1 S 2 1 /R 1 S p S 2 p /Rp = S 1 S 2 1 /R 1 S p S 2 p /Rp = S 1 S 2 1 /R 1 S p S 2 p/r p S p S 2 p/r p i pj p S p T i j < 0 ϖs 1, S 2,, S p T i j i 1 j 1 S 1 i pj p S p ϖs 1, S 2,, S p T i j i 1 j 1 S 1 i pj p S p β q i β q j T i j i 1 j 1 S 1 i pj p S p µ 1i1 µ 2i2 µ pip µ 1j1 µ 2j2 µ pjp T i j i 1 j 1 S 1 i pj p S p = µ 1i1 µ 2i2 µ pip µ 1j1 µ 2j2 µ pjp T i j i 1 j 1 S 2 1 i pj p S 2 p

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