An LMI Approach to Robust Controller Designs of akagi-sugeno fuzzy Systems with Parametric Uncertainties Li Qi and Jun-You Yang School of Electrical Engineering Shenyang University of echnolog Shenyang, PRChina qili1973@omcom Abstract his paper proposes a new robust stability condition for akagi-sugeno (-S) fuzzy control system with parametric uncertainties he condition is represented in the form of linear matrix inequalities (LMIs) and is less conservative than some relaxed quadratic stabilization conditions published recently in the literature by collecting the interrelations of fuzzy subsystems into a set of matrices, not just into a single matrix Based on the LMI-based conditions derived, one can easily synthesize controllers for state feedback robust stabilization for -S fuzzy control system with parametric uncertainties Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable Finally, the validity of the proposed approach is successfully demonstrated in the the control of simulation table Index erms -S fuzzy model, parametric uncertainties, robust stability, linear matrix inequalities I INRODUCION here have been several recent studies concerning the stability and the synthesis of controllers and observers for nonlinear systems described by akagi-sugeno (-S) fuzzy models Based on quadratic Lyapunov function some papers have discussed the feedback control and the state estimation for fuzzy systems 1-6,12-14 For example in 2, stability conditions for fuzzy control systems are reported to relax the conservatism of the basic conditions by considering the interrelations of every two fuzzy subsystems o less of conservatism other methods to relaxed quadratic stability conditions are also proposed in 6 and 7, collecting the interrelations of fuzzy subsystems into a single matrix, More recently, in 8 new stability conditions are obtained by relaxing the stability conditions of the previous works, which collect the interrelations into a set of matrices However, in the above paper, the state feedback controllers with parametric uncertainties are not considered Consequently the robustness of the closedloop fuzzy model is not guaranteed In 3,9-1 an approach to design an robust fuzzy control of uncertain fuzzy models was proposed Unfortunately, the synthesis conditions are very conservative In 11 another approach was proposed with less conservativeness, which based on 7 In this paper news derived stability conditions for robust state feedback stabilization relaxing the conservatism of the conditions obtained by the previous works (such as 11 and references therein) are given he proposed results are based on the work of 8 developed in the case of state feedback control Motivated by the aforementioned works, sufficient conditions are derived for robust asymptotic state feedback robust stabilization using quadratic Lyapunov function hese stability conditions are solved using optimization techniques A -S model of simulation table is used to show the effectiveness of the derived conditions he following notations are used in this paper P>means P is a real symmetric positive definite matrix A denotes transpose of matrix A R nm denotes the set of real matrices of dimensions n m he symbol denotes the transpose elements in the symmetric positions II PRELIMINARIES In order to consider parametric uncertainties in the -S fuzzy systems, we propose the continuous-time -S fuzzy system in which the ith rule is formulated in the following form: Continuous-time -S fuzzy model: Plant Rule i =1, 2,,q : IF x 1 (t) is Γ i 1 and and x n (t) is Γ i n HEN ẋ(t) =(A i + A i )x(t)+(b i + B i )u(t) (1) where Γ i j is a fuzzy set, x(t) Rn is the state vector, u(t) R m is the control input vector, A i R nn and B i R nm are system matrix and input matrix, respectively, A i and B i are time-varying matrices with appropriate dimensions, which represent parametric uncertainties in the plant model, and q is the number of rules of this -S fuzzy model he defuzzified output of the -S fuzzy system (1) is represented as follows: ẋ(t) = h i (x(t))((a i + A i )x(t)+(b i + B i )u(t)) (2) 978 1 4244 1674 5/8/$25 c 28 IEEE CIS 28
where h i (x(t)) = ω i(x(t)) ω i (x(t)) ω i (x(t)) = n Γ i j (x j(t)) in which Γ i j (x(t)) is the grade of membership of x j(t) in Γ i j Some basic propertied of ω i (x(t)) are ω i (x(t)) ω i (x(t)) >,, 2,,q It is clear that h i (x(t)) h i (x(t)) = 1,, 2,,q Next, a fuzzy model of a state-feedback controller for the continuous-time -S fuzzy model is formulated as follows: Controller Rule i= 1, 2,,q : IF x 1 (t) is Γ 1 j and and x n(t) is Γ n j HEN (3) u(t) = K i x(t), (4) where K i R mn are constant control gains to be determined Since the plant rules (1) have time-varying uncertain matrices, it is not easy to design the controller gain matrices In order to find these gain matrices K i, the uncertain matrices should be removed under some reasonable assumptions Hereforth we assume, as usual, that the uncertain matrices A i and B i are admissibly norm-bounded and structured Assumption 1: he parameter uncertainties considered here are norm-bounded, in the form Ai B i = Di F i (t) E 1i E 2i where D i, E Ai and E Bi are known real constant matrices of appropriate dimensions, and F i (t) is an unknown matrix function with Lebesgue-measurable elements and satisfies F i (t) F i (t) I, in which I is the identity matrix of appropriate dimension III ROBUS SABILIZAION OF HE -S FUZZY MODEL Lemma 1: Given constant matrices D and E and a symmetric constant matrix S of appropriate dimensions, the following inequality holds: S + DFE + E F D < where F satisfies F F R, if and only if for some ɛ> S + ɛ 1 E ɛd R ɛ 1 E I ɛd < Consider a continuous-time -S fuzzy model (2), the objective is to design a -S fuzzy-model-based state-feedback controller for robust stabilization of the system (2) in the form ω i (x(t))k i (t)x(t) u(t) = = h i (x(t))k i x(t) ω i (x(t)) (5) he closed-loop system of (2) and (5) is found in (6) ẋ(t) = q h i (x(t))h j (z(t))(a i + A i (B i + B i )K j )x(t) =( q h i (x(t)) q h i (x(t))h j (x(t)) (A i + A i (B i + B i )K j )x(t) he main result on the global asymptotic stability of the continuous-time -S fuzzy model with parametric uncertainties is summarized in the following theorem heorem 1: he continuous-time -S fuzzy system (2) is asymptotically stabilizable via the -S fuzzy model-based state-feedback controller (5),if there exist matrices Q > ; N i, i = 1, 2,,q; Υ iii, i = 1, 2,,q; Υ jii = Υ iij and Υ iji, i = 1, 2,,q, j i, j = 1, 2,,q and Υ ijl =Υ lji, Υ ilj =Υ jli, Υ jil =Υ lij, i =1, 2,,q 2, j = i +1,,q 1, l = j +1,,q and some scalars ɛ ijl >,i,j,l =1,,q such that the following LMIs are satisfied: Υ 1i1 Υ 1i2 Υ 1iq Υ 2i1 Υ 2i2 Υ 2iq >, (7) Υ qi1 Υ qi2 Υ qiq i =1, 2,,q Φ iii E 1i Q E 2i N i ɛ iii I ɛ iii Di ɛ iii I i =1, 2,,q (6) < (8) Because of Scarcity of space, LMI (9) and LMI (1) are listed in the top of next page Proof: Consider the Lyapunov function candidate V (x(t)) = x(t) Px(t) (11) where P is a time-invariant, symmetric and positive definite matrix V (x(t)) is positive definite and radially unbounded he time derivative of V (x(t)) is V (x(t)) = ẋ(t) Px(t)+x(t) P ẋ(t) (12) he equilibrium of (2) is quadratically stable if V (x(t)) = x(t) { q h i h j (A i B i K j) P that is +P q h i h j (A i B i K j)}x(t) <, x(t) h i h j {(A i Bi K j ) P +P (A i Bi K j )} < (13) where A i = A i + A i,b i = B i + B i
Ψ iij E 1i Q E 2i N i ɛ iij I E 1i Q E 2i N j ɛ iji I E 1j Q E 2j N i ɛ jii I (ɛ iij + ɛ iji )Di (ɛ iij + ɛ iji )I ɛ jii Dj ɛ jii I i =1, 2,,q,j =1, 2,,q,j i ijl E 1i Q E 2i N j ɛ ijl I E 1i Q E 2i N l ɛ ilj I E 1j Q E 2j N i ɛ jil I E 1j Q E 2j N l ɛ jli I E 1l Q E 2l N i ɛ lij I E 1l Q E 2l N j ɛ lji I (ɛ ijl + ɛ ilj )D (ɛ i ijl +ɛ ilj )I (ɛ jil + ɛ jli )D (ɛ j jil +ɛ jli )I (ɛ lij + ɛ lji )Dl (ɛ lij +ɛ lji )I where i =1, 2,,q 2,j = i +1,,q 1,l = j +1,,q < (9) < (1) Φ iii = QA i + A i Q N i B i B i N i +Υ iii Ψ iij = 2QA i + QA j +2A iq + A j Q (N i + N j ) B i N i B j B i(n i + N j ) B j N i +Υ iij +Υ iji +Υ iij ijl = 2Q(A i + A j + A l ) (N j + N l ) B i (N i + N l ) B j (N i + N j ) B l +2(A i + A j + A l )Q B i (N j + N l ) B j (N i + N l ) B l (N i + N j ) +Υ ijl +Υ ilj +Υ jil +Υ ijl +Υ ilj +Υ jil with the fuzzy local state feedback gains are K i = N i Q 1, i =1, 2,q Let Q = P 1, N i = K i P 1 Premultiply and postmultiply (13) by Q, we get Λ c = h 3 i Λ 1 + j i q 2 h i h j Λ 2 + q 1 j=i+1 l=j+1 h i h j h l Λ 3 If (14)-(16)are feasible, that is QA i + A i Q N i B i B i N i + Q A i + A i Q N i B i B i N i < Υ iii (14) Λ 1 = (QA i + A i Q N i B i B i N i ) +(Q A i + A i Q N i B i B i N i ) Λ 2 = (2QA i + QA j +2A iq + A j Q (N i + N j ) Bi Ni B j B i(n i + N j ) B j N i ) +(2Q A i + Q A j +2 A iq + A j Q (N i + N j ) Bi Ni B j B i(n i + N j ) B j N i ) Λ 3 = (2Q(A i + A j + A l ) (N j + N l ) Bi (N i + N l ) Bj (N i + N j ) Bl +2(A i + A j + A l )Q B i (N j + N l ) B j (N i + N l ) B l (N i + N j )) (2Q( A i + A j + A l ) (N j + N l ) Bi (N i + N l ) Bj (N i + N j ) Bl +2( A i + A j + A l )Q B i (N j + N l ) B j (N i + N l ) B l (N i + N j )) 2QA i + QA j +2A iq + A j Q (N i + N j ) Bi Ni B j B i(n i + N j ) B j N i +2Q A i +Q A j +2 A iq + A j Q (N i + N j ) Bi Ni B j B i(n i + N j ) B j N i < Υ iij Υ iji Υ iij 2Q(A i + A j + A l ) (N j + N l ) Bi (N i + N l ) Bj (N i + N j ) Bl ) +2(A i + A j + A l )Q B i (N j + N l ) B j (N i + N l ) B l (N i + N j ) +2Q( A i + A j + A l ) (N j + N l ) Bi (N i + N l ) Bj (N i + N j ) Bl +2( A i + A j + A l )Q B i (N j + N l ) B j (N i + N l ) B l (N i + N j ) < Υ ijl Υ ilj Υ jil Υ ijl Υ ilj Υ jil (15) (16)
so (17) hold Λ c < q h 3 i Υ iii q q 2 q 1 j=i+1 l=j+1 h 2 i h j(υ iij +Υ iji +Υ iij ) j i h i h j h l (Υ ijl +Υ ilj +Υ jil +Υ ijl +Υ ilj +Υ jil ) Υ 1i1 Υ 1i2 Υ 1iq = Z r Υ 2i1 Υ 2i2 Υ 2iq h i Z Υ qi1 Υ qi2 Υ qiq (17) where Z = h 1 I h 2 I h q I hus if (7) holds, then Λ c < hat is system (2) is asymptotically stable about its zero equilibrium via the fuzzy controller (5) hen, using Assumption 1, (14) can be represented as where Φ iii + D i F i (t)(e 1i Q E 2i N i ) +(E 1i Q E 2i N i ) F i (t) D i < Φ iii = QA i + A i Q N i B i B i N i +Υ iii (18) Using Lemma 1 repeatedly, the matrix inequality (18) holds for all F i (t) satisfying F i (t) F i (t) I if and only if there exists a constant ɛ 1/2 iii > such that Φ iii + ɛ 1/2 iii (E 1i Q E 2i N i ) ε 1/2 iii D i ε 1/2 iii (E 1i Q E 2i N i ) ε 1/2 iii D i = Φ iii + (E 1i Q E 2i N i ) D i ε 1 iii I (E1i Q E 2i N i ) ε iii I < D i (19) Applying Schur complement to (19), we get Φ iii E 1i Q E 2i N i ε iii I < Di ε 1 iii I which is equivalent with inequality (8) In the similar procedures, we can obtain he LMIs (9) and (1) from (15) and (16), respectively his completes the proof of heorem 1 IV EXAMPLE OF SIMULAION ABLE Consider a -S fuzzy system of simulation table defined by the following rules: Plant Rule 1: If x 3 (t) is Positive, hen ẋ(t) =(A 1 + A 1 )x(t)+(b 1 + B 1 )u(t) Plant Rule 2: If x 3 (t) is Zero, hen ẋ(t) =(A 2 + A 2 )x(t)+(b 2 + B 2 )u(t) Plant Rule 3: If x 3 (t) is Negative, hen ẋ(t) =(A 3 + A 3 )x(t)+(b 3 + B 3 )u(t), where x 3 (t) represent the angular velocity of rotor, which adopt triangular degree of membership According to the Brush-less Dc Motor selected,we get the parameters as follows: A 1 = A 2 = A 3 = 13 83 36274 83 13 19 1 13 36274 13 19 1 13 83 36274 83 13 19 1 B 1 = B 2 = B 3 = 544 he norms of parameter uncertainties are chosen as follows: 2 D 1 = D 2 = D 3 = 2 2 2 E A1 = E A2 = E A3 = E B1 = E B2 = E B3 = 1 1 1 1 By heorem 1, we solve LMIs (7)-(1) and give the main solution matrices as follows: 3368 1792 11 Q = 38249 1792 1499 2562 11 2562 4995 N 1 = 59291 69394 5149 167879 N 2 = 5165 5988 16791 N 3 = 59291 69394 5149 167879 So the feedback gains are K 1 = 15487 18142 9473 8241 K 2 = 15521 9852 84416
K 3 = 15487 18142 9473 8241 We can then determine the fuzzy controller (5) according to these feedback gains and h 1,h 2,h 3 V CONCLUSION his paper presents new stability conditions for robust asymptotic stabilization of Nonlinear Systems with Parametric Uncertainties he derived results are based on the use of the quadratic Lyapunov function and solved using LMI optimization techniques he proposed stability conditions are less conservative than previous results and include previous ones as special cases he approach used in this paper can be also applied to solve the fuzzy robust static output feedback satbilization problems REFERENCES 1 K Kiriadis Fuzzy model-based control of complex plants, IEEEran on Fuzzy systems, 1998, 6(4), 517-529 2 K anaka, Ikeda, O Wang Fuzzy regulator and fuzzy observer: relaxed stability conditions and LMI based design, IEEE rans on Fuzzy Systems, 1998, 6(2), 25-256 3 H J Lee, J B Park, G Chen Robust fuzzy control of nonlinear systems with parametric uncertainties, IEEE rans on Fuzzy Systems, 21, 9(2), 369-379 4 SCCao,NWRees,GFengHinf control of uncertain fuzzy continuoustime systems, Fuzzy sets and systems, 2, 115, 171-19 5 A Hajjaji, A Ciocan, D Maquin and J Ragot State estimation of uncertain multiple model with unknown inputs, IEEE 43th Conference on Decision and Control, Bahamas, 24, Dec, 14-17 6 E Kim, H Lee New approaches to relaxed quadratic stability condition of fuzzy control systems, IEEE rans on Fuzzy systems, 2, 8(5), 523-534 7 Liu X D, Zhang Q L New approach to controller designs based on fuzzy observer for -S fuzzy systems via LMI, Automatica, 23, 39(5), 1571-1582 8 Fang C H, Liu Y S, Kau S W, Hong L, Lee C H A new LMI-based approach to relaxed quadratic stabilization of -S fuzzy control systems, IEEE ransaction on Fuzzy Systems, 26, 14(3), 386-397 9 Shaocheng ong, Han-Hiong Li Observer-based robust fuzzy control of nonlinear systems with parametric uncertainties, Fuzzy Sets and Systems, 22, 131, 165-184 1 Mohammed CHADLI, Ahmed E1 HAJJAJI Comment on Observerbased robust fuzzy control of nonlinear systems with parametric uncertainties, Fuzzy Sets and Systems, 26, 157, 1276-1281 11 Mohammed CHADLI, Ahmed E1 HAJJAJI Output robust stabilization of uncertain akagi-sugeno model, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Confenrence 25 Seville, Spain, 25, December 12-15 12 HOWang, K, anaka, and MFGriffin An approach to fuzzy control of nonlinear systems: Stability and design issues, IEEE rans on Fuzzy Systems, 1996, Feb 4(1), 14-23 13 J Joh, Y H Chen,and R Langari On the stability issues of linear akagi-sugeno fuzzy models, IEEE rans Fuzzy Systems, 1998, Aug 6(3), 42-41 14 W J Wang and C S Sun Relaxed stability condition for -S fuzzy systems, Proc 9th IFSA World congress, 21, 221-226