Stability Analysis and Controller Design for Discrete-Time Fuzzy Systems With Time-Varying Delay

Stability Analysis and Controller Design for Discrete-Time Fuzzy Systems With Time-Varying Delay ⋆ Dimitri C. Viana ∗ Valter J. S. Leite ∗∗ M´arcio F. Miranda ∗∗∗ ∗ CEFET–MG,

campus II, Belo Horizonte, MG, 30480-000, BRAZIL (e-mail: [email protected]) ∗∗ CEFET–MG, campus V, Divin´opolis, MG, 35502-036, BRAZIL (e-mail: [email protected]) ∗∗∗ UFMG, COLTEC, Belo Horizonte, MG, 31270-010, BRAZIL (e-mail: [email protected])

Abstract: Convex conditions for stability analysis and control synthesis for discrete-time fuzzy systems with time-varying delay are proposed in this paper. The conditions depend on the variation rate of the time-delay and are obtained by considering a parameter dependent Lyapunov-Krasovskii functional. To reduce the conservatism, besides the standard techniques to include extra matrix variables, some extra equations related to the control signal are also added. It is shown that this approach leads to a less expensive convex formulation that the standard way. In all cases, the conditions are formulated as an LMI (linear matrix inequality) feasibility test, that can be efficiently solved in polynomial time by specialized numerical solvers. Numerical examples are presented to compare and illustrate the efficiency of the proposed conditions. Keywords: time-delay systems, time-varying delay, fuzzy systems, controller synthesis, stability analysis, linear matrix inequality (LMI). 1. INTRODUCTION A great advantage of Takagi-Sugeno (T-S) fuzzy models is that they allow to represent nonlinear systems by means of a convex combination of linear systems, called submodes. Since 1990, various fuzzy control approaches have been proposed and those based on T-S models have been used with success in real world applications, mainly due the representation of nonlinear systems by means of weighted linear models Tanaka and Wang [2001]. Among the issues concerned with these systems, there are unsolved questions related to time-delay, specially with stability analysis and synthesis. See Feng [2006] and references therein. Efforts to reduce both the conservatism of stability analysis and the computational burden have been done in several directions, including the use of multiple Lyapunov function Mozelli et al. [2009]. It is well known that delays can cause performance degeneration or even instability. Delays can be found due to digital control implementations, transport lag or as intrinsic properties of the systems. They can also appear as deliberate delays introduced in control actions to achieve a stable closed-loop system [Gu et al., 2003, pp. 3], as in the case of stabilization of vibratory systems as done by Du and Zhang [2008]. In this paper, static state feedback control law is pursued for stabilizing T-S fuzzy discrete-time systems with delayed states. The proposed control law employs delayed states which may enlarge the set of stabilizable systems. This kind of feedback can be straightforward applied in discrete-time models, once only a finite number of past values of states needs to be handle, but cannot be applied in continuous-time systems without more elaborated assumptions. Problems of stability analysis and stabilization ⋆ This work was supported by Brazilian agencies FAPEMIG and CNPQ.

of discrete-time fuzzy systems with time-varying delay remain open, even being relevant problems, many frequently related to applications. Different from what is usual in the literature, the conditions presented here use parameter dependent LyapunovKrasovskii (L-K) functionals. Conditions based on constant and parameter independent L-K functional have been presented in Lam and Leung [2007], where the system is continuous on time and the fuzzy controller is discrete on time. Also, consider Tian et al. [2009] and Li and Liu [2009] for continuous-time fuzzy systems with norm bounded uncertainty. In Wu [2008] and Zhang et al. [2007], observed-based control and filtering design are considered for discrete-time fuzzy system subjected to multiple constant delays under QS assumptions. A fuzzy LK functional is introduced in Wu [2006] for fuzzy discretetime systems with constant delay in the state. More recently, an investigation on the stability of discrete-time fuzzy systems with constant delay and design of a fuzzy state feedback gain for T-S models has been proposed in Lam and Leung [2008]. Constant L-K candidate matrices are used conjointly with a convex formulation that includes some extra matrices. All these papers proposals are concern with convex formulations for optimization problems that can be efficiently solved by means of available LMI solvers, see Sturm [1999]. The main contribution of this note is to present four convex conditions, two for asymptotically stability analysis and two for parallel-distributed-compensation (PDC) controller design for discrete-time fuzzy systems with time varying delay. All the conditions depend on variation rate of time-delay. It is worth to stress that the novellty is not the L-K functional used, but the manipulations presented: in the stability analysis new conditions are obtained by using distinct sets of equations associated to the dynamics of the fuzzy system. This strategy yields a

less numerically expensive formulation with complementary results w.r.t. standard LMI machinery used in robust control. In the PDC controller design case, besides the distinct sets of dynamics equations, an instrumental variable is employed to get an new convex stabilization condition. It is verified that the achieved condition may allow to obtain a wider stabilizable region, at less expensive numerical computations. This note is organized as follows: some definitions and the problem statement are presented in the next section. Then, the main results are given in Section 3: the stability analysis conditions are stated followed by the synthesis conditions. Two different approaches are employed leading to complementary results. In both cases, quadratic stability (QS) based approach can be recovered as special cases. This section ends with a numerical complexity analysis of the feasibility convex problems proposed. Some numerical examples are given in Section 4, where all the conditions are compared. Finally, the conclusions are stated in Section 5. Notation: The notation used is quite standard: xt is the state at time t ∈ N. R is the set of real numbers and N (N∗ ) stands for the set of the natural numbers (excluded the 0). I and 0 are the identity and the null matrices of appropriate dimensions, respectively. M = block-diag{M1 , M2 } stands for the blockdiagonal matrix M made up by the matrices M1 and M2 at the main diagonal. M > 0 (M < 0) means that M is positive (negative) definite. M T stands for the transpose of M. ⋆ is used to indicate diagonally symmetric blocks in the LMIs. w t (dt ) means w (t)(d(t)), the normalized grade of membership of each local system at time t and w i (t) is its i-th entry. 2. PROBLEM STATEMENT Consider a discrete-time delayed fuzzy system modeled in the T-S framework with the i-th rule given by (see Tanaka and Wang [2001] for details): Rule-i (i = 1, . . . , N) IF z1 (t) is Mi1 AND . . . AND zr (t) is Mir , THEN xt+1 = Ai xt + Adi xt−dt + Bi ut , t = 0, 1, . . .

(1)

where, Mi j is the fuzzy set, z j (t) is the j-th premise variable, which are supposed to be independent from ut , at the t-th sample. N is the number of fuzzy rules, r is the number of premise variables. xt = x(t) ∈ Rn is the state vector at the t-th sample, ut = u(t) ∈ R p is the input control vector with p entries, 0 < d ≤ dt ≤ d¯ < ¥ is the time-varying state delay which is limited by |dt+1 − dt | ≤ d ; d ∈ N : d = d¯ − d (2) n×n n×n n×p Ai ∈ R , Adi ∈ R and Bi ∈ R are known and constant matrices related to the dynamics of the time-delay system and to the control input signal. Each i-th consequent is called subsystem. Thus, for a given state xt the dynamic of the system is inferred as follows: xt+1 = A(w t )xt + Ad (w t )xt−dt + B(w t )ut (3) where w by

t

is the vector [w 1 (t) . . . w r

w i (t) =

j=1 N

N

(t)]T

with each entry given

and µMi j (z j (t)) represents the membership grade of z j (t) w.r.t. the fuzzy term Mi j , i = 1, . . . , N, j = 1, . . . , r, at the instant t. Therefore, Ni=1 w i (t) = 1, w i (t) ∈ [0, 1] ∀ i, yielding N

i=1 j=1

In this paper, the following PDC control law is considered ut = K(w t )xt + Kd (w t )xt−dt N

w i (t)Ki xt +

= i=1

N i=1

w i (t)Kdi xt−dt

(6)

where [K|Kd ]i ∈ R p×2n are the fuzzy state feedback gains that assures the asymptotic stability of (3)-(5). Therefore the delayed fuzzy closed-loop system (3)-(5) with (6) is given by ˆ t )xt + Aˆ d (w t )xt−dt xt+1 = A(w (7) with ˆ t ) ≡ A(w t ) + B(w t )K(w t ) A(w ˆ Ad (w t ) ≡ Ad (w t ) + B(w t )Kd (w t )



(8)

It is worth to mention that, if delay dt is not known, then it is enough to make Kdi = 0, i = 1, . . . , N, in (6), recovering the usual control rule for state feedback systems: ut = K(w t )xt . If dt is available, then the possibility of using K(w t ) and Kd (w t ) may improve the performance of the closed-loop system (7) or, as shown in this note, enlarge the stabilizable set of fuzzy systems. The objective here is to propose convex conditions solving the following problems: Problem 1. (Stability analysis). Given d subject to (2), determine if the delayed fuzzy system (7), with known [K|Kd ]i , i = 1, . . . , N, is asymptotically stable. Problem 2. (Controller design). Given d subject to (2) and the discrete-time fuzzy system with time-varying delay (3)-(5), determine, if possible, [K|Kd ]i ∈ R p×2n , i = 1, . . . , N, such that the PDC control law (6) leads to an asymptotically stable closed-loop systems (7). The following lemma is employed in this paper to obtain the proposed conditions. Lemma 1. (Finsler’s Lemma). Let x ∈ Rn , Q (w t ) = Q (w t )T ∈ Rn×n and B (w t ) ∈ Rm×n such that rank(B (w t )) < n. The following statements are equivalent: i) x T Q (w t )x < 0, ∀ x : B (w t )x = 0, x 6= 0 ii) ∃ X ∈ Rn×m : Q (w t ) + X B (w t ) + B (w t )T X T < 0 Proof. The proof follows similar steps given in de Oliveira and Skelton [2001], replacing the precisely known matrices by parameter dependent matrices. 3. MAIN RESULTS The following Lyapunov-Krasovskii candidate functional is considered in this paper V (t) =

(4) µMi j (z j (t))

(5)

i=1

µMi j (z j (t))

r

w i (t)[A|Ad |B]i

[A|Ad |B](w t ) =

3 v=1

where

Vv (w t ,t)

(9)

V1 (w t ,t) = xtT P(w t )xt , V2 (w t ,t) =

t−1 j=t−dt 1−d

V3 (w t ,t) =

(10)

xTj Q(w j )x j , t−1

ℓ=2−d¯ j=t+ℓ−1



M˜ (w t ) = 

(11)

xTj Q(w j )x j ,

P(w t ) =

w i (t)Pi ,

N

Q(w t ) =

i=1

(12)

w i (t)Qi

(13)

i=1

N

N

w ℓ (t + 1)w

=

m (t − dt )

ℓ=1 m=1 N

N

+2 i=1 j=i+1 N

N

N

N

"

N i=1

˜ (i, i, ℓ, m) w 2i (t)M

# ˜ w i (t)w j (t)M (i, j, ℓ, m) =

(2 − 0 j−i)w ℓ (t + 1)w

ℓ=1 m=1 i=1 j=i

3.1 Convex Stability Analysis

m (t − dt )

˜ (i, j, ℓ, m) × w i (t)w j (t)M

Theorem 1. If there exist symmetric and positive definite matrices Pi ∈ Rn×n , Qi ∈ Rn×n , i = 1, . . . , N, matrices X1 , X2 , X3 of adequate dimensions and a scalar b = d + 1 such that the following LMIs are verified  ˜  M11 M˜ 12 M˜ 13 M˜ (i, j, ℓ, m) =  ⋆ M˜ 22 M˜ 23  < 0, ˜ 33 ⋆ ⋆ M

i, ℓ, m = 1, . . . , N, j = i, . . . , N (14) ˜ = XT − ˜ T ˜ = X T − X Aˆ , M where M11 = Pℓ + X1 + X1 , M 12 3 2
1 i j T 13 ˜ X1 Aˆ di j , M22 = 0.5 b (Qi + Q j ) − (Pi + Pj ) − Aˆ i j X2T − X2 Aˆ i j , M˜ 23 = −(Aˆ Tij X3T + X2Aˆ di j ), M˜ 33 = −(Qm + Aˆ Tdi j X3T + X3 Aˆ di j ), Aˆ i j = 0.5(Ai + Bi K j + A j + B j Ki ) and Aˆ di j = 0.5(Adi + Bi Kd j + Ad j + B j Kdi ), then the delayed fuzzy system with known matrices [A|Ad |B|K|Kd ]i given by (7)-(8) is asymptotically stable. Proof. Clearly the requirement Pi > 0, Qi > 0, i = 1, . . . , N, assures the positivity of (9) with the choice (10)-(12). To be a Lyapunov-Krasovskii functional and thus assure the asymptotically stability of (7)-(8) it is necessary that D V (w t ,t) < 0 (15) Therefore, evaluating (9) as done by Leite and Miranda [2008], the following upper bound for (15) can be obtained T D V (w t ,t) ≤ xt+1 P(w

x T block-diag{P(w

(19)

M˜ (w t ) =

In the sequel, four convex conditions are proposed, two for stability analysis and two for PDC controller design. In the analysis case, the difference between the proposed conditions relies on the use of distinct sets of equations associated to the dynamics of the fuzzy system. In the PDC controller design case, besides the distinct sets of dynamics equations, an instrumental variable is employed to get a new stabilization condition.

Considering an  T T T , (16) can be rewritten as xt+1 xtT xt−d t

+ X1

⋆ ⋆

Therefore (19) can be recovered from (14) by noting that 1

with Pi > 0 and Qi > 0, i = 1, . . . , N.

T t+1 )xt+1 + xt [b Q(w t ) − P(w t )]xt T − xt−d Q(w t−dt )xt−dt < 0 t augmented space given by x ∈ R3n

T t+1 ) + X1

ˆ t) X2T − X1 A(w ˆ t )T X2T − X2 A(w ˆ t) b Q(w t ) − P(w t ) − A(w ⋆  X3T − X1 Aˆ d (w t )
ˆ t )T X3T + X2 Aˆ d (w t )  − A(w
<0 T T ˆ ˆ − Q(w t−dt ) + Ad (w t ) X3 + X3 Ad (w t )

and matrices P(w t ) and Q(w t ) can assume different values at each instant t, being defined as N

P(w

(16) :x =

t+1 ),

b Q(w t ) − P(w t ), −Q(w t−dt )}x < 0 (17) and it is possible to assemble (7) as   ˆ t ) −Aˆ d (w t ) x = 0 (18) I −A(w Thus, by means of Lemma 1 with (17), (18) and X =  T T T T ∈ R3n×n , it is possible to get X1 X2 X3

which completes the proof. Note that, if only the closed-loop matrices (8) are available, then conditions presented in Leite and Miranda [2008] are more suitable to evaluate the stability of the system, once no crossed terms due the product B(w t )K(w t ) or B(w t )Kd (w t ) need to be handle. Another analysis condition is presented in the following theorem. Although using the same L-K candidate functional, the resulting conditions may lead to a different set of stable systems identified by this new condition. This different set occurs thanks to the enlarged set of equations used in their derivation. Also, the number of LMI rows is significantly reduced w.r.t. Theorem 1 while the number of free variables is increased. Theorem 2. If there exist symmetric and positive definite matrices Pi ∈ Rn×n , Qi ∈ Rn×n , i = 1, . . . , N, matrices X11 , X12 , X21 , X22 , X31 , X32 , X41 , X42 of adequate dimensions and a scalar b = d + 1 such that the following LMIs are verified  

M11 M12 M13 M14 M22 M23 M24  < 0, ⋆ M33 M34  ⋆ ⋆ M44

⋆ M (i, j, ℓ) =   ⋆ ⋆

i, j, ℓ = 1, . . . , N

(20)

T + AT X T + X K + X A , where M11 = b Qi − Pi + KiT X11 11 i 12 i i 12 T T T T T + M12 = Ki X21 + Ai X22 + X11 Kdi + X12 Adi , M13 = KiT X31 T T T T T T Ai X32 − X11 + X12 Bi , M14 = Ki X41 + Ai X42 − X12 , M22 = T T T T T −Qℓ + Kdi X21 + ATdi X22 + X21 Kdi + X22 Adi , M23 = Kdi X31 + T −X +X B, M T X T + AT X T − X , M ATdi X32 = K 21 22 i 24 22 33 = di 41 di 42 T T T T T T −X31 + Bi X32 − X31 + X32 Bi , M34 = −X41 + Bi X42 − X32 and T − X , then the delayed fuzzy system with M44 = Pj − X42 42 known matrices [A|Ad |B|K T |KdT ]i given by (7)-(8) is asymptot-

ically stable.

Proof. Clearly the requirement Pi > 0, Qi > 0, i = 1, . . . , N, assures the positivity of (9) with the choice (10)-(12). To be a 1

Define 00 = 1 and 0k = 0, k ∈ N∗ .

Lyapunov-Krasovskii functional and thus assure the asymptotically stability of (7)-(8) it is necessary to verify (16). Then, considering an augmented space given by x ∈ R3n+p : x =  T T T T , (16) can be rewritten as xt xt−dt utT xt+1 x T block-diag{b Q(w t ) − P(w t ), − Q(w t−dt ), 0, P(w t+1 )}x < 0 (21) and it is possible to assemble (3) and (6) as   K(w t ) Kd (w t ) −I 0 x =0 (22) A(w t ) Ad (w t ) B(w t ) −I

Thus, by means of Lemma 1 with (21), (22) and X mounted with matrix blocks given by Xi j , i = 1, . . . , 4, j = 1, . . . , 2 of adequate dimensions, it is possible to get   M11 (w t ) M12 (w t ) M13 (w t ) M14 (w t ) ⋆ M22 (w t ) M23 (w t ) M24 (w t )  M (w t ) =   ⋆ ⋆ M33 (w t ) M34 (w t )  < 0 (23) ⋆ ⋆ ⋆ M44 (w t )

T + A(w )T X T + where M11 (w t ) = b Q(w t )− P(w t )+ K(w t )T X11 t 12 T T TX X11 K(w t ) + X12 A(w t ), M12 (w t ) = K(w t ) X21 + A(w t )T X22 11 T T Kd (w t ) + X12 Ad (w t ), M13 (w t ) = K(w t )T X31 + A(w t )T X32 − T + A(w )T X T − X , X11 + X12 B(w t ), M14 (w t ) = K(w t )T X41 t 12 42 T T M22 (w t ) = −Q(w t−dt )+Kd (w t )T X21 +Ad (w t )T X22 +X21Kd (w t ) T + A (w )T X T − X + +X22 Ad (w t ), M23 (w t ) = Kd (w t )T X31 t 21 d 32 T T T −X , M (w ) X22 B(w t ), M24 (w t ) = Kd (w t ) X41 +Ad (w t )T X42 22 33 t T + B(w )T X T − X + X B(w ), M (w ) = −X T + = −X31 t 31 32 t 34 t 32 41 T T − X and M (w ) = P(w T B(w t ) X42 32 44 t t+1 ) − X42 − X42 . Therefore (23) can be recovered from (20) by noting that N

N

N

M (w t ) =

w ℓ (t − dt ) w j (t + 1) w i (t) M (i, j, ℓ)

ℓ=1 j=1 i=1

which completes the proof. Remark 1. Theorems 1 and 2 are not equivalent. In the tests performed by authors Theorem 2 leads to a wider domain of stability. See Section 4.

As done in the analysis case, two different conditions with the same functional are derived in the sequel. The first follows quite standard steps to achieve a convex controller design (see, for example, Leite and Miranda [2008]), while the second takes an instrumental variable to obtain an augmented space of search leading, in general, to less conservative results. Theorem 3. If there exist symmetric and positive definite matrices Pi ∈ Rn×n , Qi ∈ Rn×n , i = 1, . . . , N, matrices X1 , Wi , Wdi , i = 1, . . . , N of adequate dimensions and a scalar b = d + 1 such that the following LMIs are verified  ¯  M11 M¯ 12 M¯ 13 M¯ (i, j, ℓ, m) =  ⋆ M¯ 22 0  < 0, ¯ 33 ⋆ ⋆ M

i, ℓ, m = 1, . . . , N, j = i, . . . , N (24) ¯ ˜ ¯ where M11 = M11 , M12 = −0.5 X1 (ATi + ATj ) + W j BTi +
¯
¯ T T T T Wi BTj M 13 = −0.5 X1 (Adi + Ad j ) + Wd j Bi + Wdi B j , M22 =
¯ = −Q , then, the discrete 0.5 b (Qi + Q j ) − (Pi + Pj ) and M 33 m time fuzzy system with time-varying delay (3)-(5) is asymptotically stable under control law (6) with and

By using an extra equation with an instrumental variable, the following condition can be stated. Theorem 4. If there exist symmetric and positive definite matrices Pi ∈ Rn×n , Qi ∈ Rn×n , i = 1, . . . , N, matrices X11 , X12 , X21 , X22 , X31 , X32 , X41 , X42 , X51 , X52 , X53 , Wi , Wdi , i = 1, . . . , N of adequate dimensions and a scalar b = d + 1 such that the following LMIs are verified   Mˆ 11 Mˆ 12 Mˆ 13 Mˆ 14 Mˆ 15  ⋆ M ˆ M ˆ M ˆ M ˆ   22 23 24 25   Mˆ (i, j, ℓ) =  ⋆ ⋆ Mˆ 33 Mˆ 34 Mˆ 35   < 0,  ˆ 44 M ˆ   ⋆ ⋆ ⋆ M 45  ˆ ⋆ ⋆ ⋆ ⋆ M 55

i, j, ℓ = 1, . . . , N (26) T T ˆ ˆ 12 = Bi X T + X11 BT , where M11 = b Qi − Pi + Bi X12 + X12 Bi , M i 22 T −X , M ˆ = B XT − X , M ˆ = B XT + A XT , Mˆ 13 = Bi X32 12 14 i 42 11 i 52 i 53 15 T + X BT , M ˆ 23 = Bi X T − X22 , M ˆ 24 = Mˆ 22 = −Qℓ + Bi X21 21 i 31 T −X , M ˆ = Bi X T + A X T , M ˆ 33 = −X T − X32, M ˆ 34 = Bi X41 21 25 di 53 51 32 T T T ˆ ˆ ˆ −X42 − X31 , M35 = −X52 + Wi , M44 = −X41 − X41 , M45 = T + W and M ˆ = P − X T − X , then, the discrete time −X51 j di 55 53 53 fuzzy system with time-varying delay (3)-(5) is asymptotically stable under control law (6) with −T −T Ki = Wi X53 and Kdi = Wdi X53 (27) ˆ t ), Aˆ d (w t )) is asymptotically Proof. Note that if the pair (A(w T T ˆ ˆ stable, then the pair (A (w t ), Ad (w t )) is also stable. In this case, it is possible to use an instrumental variable n t = B(w t )T xt to rewrite (3) as xt+1 = A(w t )T xt + K(w t )T n t

3.2 Convex Feedback Gain Design

Ki = WiT X1−T

Proof. The proof can be obtained by replacing in (14) Aˆ i j and Aˆ di j by 0.5(Ai + Bi K j + A j + B j Ki )T and 0.5(Adi + Bi Kd j + Ad j + B j Kdi )T , respectively, choosing X2 = 0 and X3 = 0, T and making the changing of variables Wi = X1 KiT , Wdi = X1 Kdi ¯ 11 < 0. noting the regularity of X1 thanks to M

Kdi = WdiT X1−T

(25)

By choosing

+ Ad (w t )T xt−dt + Kd (w t )T n t−dt

(28)

 T T T T x = xtT xt−d ∈ R3n+2p , (29) n tT n t−d xt+1 t t (16) can be similarly rewritten as in (21) and (28) yields   0 B(w t )T 0 −I 0  B(w t )T (30) 0 −I 0 0 w = 0 T T T T A(w t ) Ad (w t ) K(w t ) Kd(w t ) −I Hereafter, it sufficient to follow similar steps of the proof of Theorem 2: (26) can be obtained by setting X13 = X23 = X33 = X43 = 0 in Xi j , i = 1, . . . , 5, j = 1, . . . , 3, making the changing T , W = K X T and noting the regularity of variables Wi = Ki X53 di di 53 ˆ of X53 is due to M55 < 0. Remark 2. In Theorem 4 is not employed the usual replacement ˆ t ) by (A(w t ) + B(w t )K(w t ))T pursued in the literature. of A(w Remark 3. If Kd = 0, conditions stated in Theorem 4 can be T  T T simplified by considering x ∈ R3n+p : x = xtT xt−d n tT xt+1 t instead of (29). In this case the number of scalar variables and LMI rows are reduced. Note that, when delay value is not known, i.e., xt−dt is not available for feedback, then conditions (24) and (26) can be used with Wdi = 0, i = 1, . . . , N.

It is worth to note that all theorems presented in this paper encompass the case where the delay is constant, i.e., for d = 0. Also, the quadratic stability approach, frequently used in delayfree and fuzzy systems, is recovered from (14), (20), (24) and (26) if the functional matrices in (9)-(13) do not depend on w t . In these particular cases, the conditions are computationally less expensive than those presented here, but at the cost of being, in general, more conservative.

2.5 2 1.5 b 1

3.3 Numerical complexity The numerical complexity of the conditions presented in this paper can be determined by the number of scalar variables, K , and the number of LMI rows, L , involved in the optimization problems. For example, using the program SeDuMi, see Sturm [1999], the number of floating point operations performed to solve
these problems has an order given by O K 2 L 5/2 + L 7/2 . The values of K and L of the feasibility tests proposed in this paper are presented in Table 1. With

1.9 1.65

0.5 00

1.4 1

1

2 a

1.2

3

1.4

1.6

4

Fig. 1. Stability regions identified by Theorem 1 (solid, blue) and Theorem 2 (dashed, red).

Table 1. Number of scalar variables (K ) and LMI rows (L ) for each proposed conditions

detect stability in different regions of the parameter space a × b. Although complementary, the areas of the regions determined by the conditions stated in Theorem 1 and Theorem 2 are 1.14 and 3.54, respectively. Thus, the region identified by Theorem 2 is more than 210% larger than that determined by Theorem 1. Moreover, as discussed in Subsection 3.3, the less conservative SeDuMi, when n (or N) increases, then the relative numerical region achieved by Theorem 2 is identified with a smaller complexity OTh.2 /OTh.1 and OTh.4 /OTh.3 become smaller than 1. computational burden than with Theorem 1. For example, with p = 1, OTh.2 /OTh.1 and OTh.4 /OTh.3 are less Example 2. (Controller design). Consider the fuzzy system with than 1 for N > 2. Thus, it is clear that, in general, conditions in delayed states investigated in Example 1, with a = 1.2, b = theorems 2 and 4 are less expensive than their correspondent in 1.6 and c = 1. Suppose that an uncertain parameter r ∈ R theorems 1 and 3. affects all systems matrices in a multiplicative way, that is, the three subsystems become r Vi = r [A|Ad |B]i , i = 1, 2, 3 and 2 4. NUMERICAL EXAMPLES the fuzzy rules for this system are µM11 (x1 (t)) = e−(x1 (t)+5) /8 , −x (t)2 /4.5 and µ −(x1 (t)−5)2 /8 . The M31 (x1 (t)) = e Example 1 concerns with the stability analysis of a fuzzy µM21 (x1 (t)) = e 1 system within a parameter space. In the second example, the objective here is to determine feasible design regions in d × r plane, with d given in (2). The achieved regions below stabilization problem by control law (6) is illustrated. each curve provided by Theorem 3 (solid line) and TheoExample 1. (Stability analysis). Consider the discrete-time fuzzy rem 4 (dotted line) are shown in Fig. 2. In both cases, the system with time varying delay defined in (1)-(5) with N = 3, conditions were used twice, considering that the delay dt is n = 2, r = 1 and subsystem matrices given by Vi = [A|Ad |B]i , available (circle marks) and considering that it is not (square i = 1, 2, 3:   marks). Thus, in the latter, conditions (24) and (26) were c 0.5 0.1 0 0 V1 = 0.51 −0.1 −0.25 0 1 (31) used with Wdi = 0 yielding Kdi = 0, i = 1, 2, 3. It is clear that, if dt is available on line, the region stabilizable by con  ditions of Theorem 3 is improved by Theorem 4. On the 0.1 0.25 0.25 0 1 V2 = 0.15 −0.5 0.15 0 −0.5 (32) other hand, when dt is not available a lager feasible design   region is obtained by applying Theorem 3. With r = 1 and a 0.75 0.19 0 b V3 = 0.25 −0.8 0.06 0 0.5 (33) d = 11 only conditions from Theorem 4 were feasible. In this case, a controller has been designed by Theorem 4 and This system has been investigated by Lam and Leung [2008] some time-simulations has been performed for the closedfor constant delays, c = 0.1 and intervals 0 ≤ a ≤ 1.6 and loop system. In this case, the PDC controller is given by (6) 0 ≤ b ≤ 1.8. The inferred fuzzy controller is given by (6), with with K1 = [ −2.0771 −0.8002 ], K2 = [ −0.4160 −0.2041 ], Kdi = 0 and Ki computed to place the eigenvalues of Ai + Bi Ki , K3 = [ −0.6532 −0.2549 ], Kd1 = [ −0.0759 −0.0141 ], Kd2 = i = 1, 2, 3, all at −0.5. That is, for each pair (a, b), a new set of [ −0.2776 −0.0021 ], Kd3 = [ −0.1291 −0.0030 ]. In the Fig. 3 Ki , i = 1, 2, 3, is evaluate by using Ackerman’s algorithm with it is shown the state behaviors of the closed-loop delayed fuzzy a double pole in −0.5, see Chen [1999]. system with d¯ = 12 and d = 1. In the top of Fig. 4 it is shown the The objective here is to investigate, on the parameter space control signal and, in the bottom, the time-varying delay ema × b, a region such that this system is asymptotically stable ployed in the time simulations. The following initial conditions T for a time varying delay d = 1. In Fig. 1, the border of the were assumed in this time-simulation: x(t) = [ 2.0 −1.5 ] , regions identified by conditions proposed in Theorem 1 (solid t = −12, . . . , 0. Thus, it has been shown that conditions from line) and Theorem 2 (dotted line) are shown. As it can be Theorem 4 can lead to less conservative results at a smaller noted, the analysis conditions are not equivalent, once they can computational burden. Th. 1 2 3 4

K

L

(n2 +n)N+3n2 (n2 +n)N+3n2 +p2 +4np (n2 +2np+n)N+n2 (n2 +2pn+n)N+n2 +4p2 +6np

1.5nN 4 +1.5nN 3 +2nN (3n+p)N 3 +2nN 1.5nN 4 +1.5nN 3 +2nN (3n+2p)N 3 +2nN

varying delay. It has been shown that some of the proposed conditions require less computational burden, due the use of some extra equations associated to the control signal. In all cases, it has been considered a parameter dependent LyapunovKrasovskii functional and extra matrix variables, leading to less conservative conditions. Numerical examples are presented to compare and illustrate the efficiency of the proposed conditions. It is expected that the approach presented here can be used with more involved functionals and, also, extended to cope with some performance index such as H¥ guaranteed cost.

1.16 1.14 1.12 1.1

r

1.08 1.06 1.04 1.02 1

2

4

6

d

8

10

12

Fig. 2. Relation between the uncertain parameter r and the variation band size of dt . 2 1

x1 (t) 0 −1

0

5

10

15

20

25

30

0

5

10

15

20

25

30

2 1

x2 (t) 0 −1 −2

t

Fig. 3. The behaviors of the states x1 (t) and x2 (t) 0.5 0

u(t)−0.5 −1 −1.5

0

5

10

15

20

25

30

0

5

10

15

20

25

30

15 10

d(t) 5 0

t

Fig. 4. Control signal, u(t), and time-varying delay, d(t) 5. CONCLUSION Convex conditions depending on the maximum variation rate of the delay have been presented for stability analysis and control synthesis for discrete-time fuzzy systems with time-

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