MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 10.1155/2015/659521 659521 Research Article Adaptive Stabilization Control for a Class of Complex Nonlinear Systems Based on T-S Fuzzy Bilinear Model Xing Jinsheng Shi Naizheng Shanmugam Lakshmanan School of Mathematics & Computer Science Shanxi Normal University Linfen 041004 China snnu.edu.cn 2015 1312015 2015 24 07 2014 15 12 2014 1312015 2015 Copyright © 2015 Jinsheng Xing and Naizheng Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper proposes a stable adaptive fuzzy control scheme for a class of nonlinear systems with multiple inputs. The multiple inputs T-S fuzzy bilinear model is established to represent the unknown complex systems. A parallel distributed compensation (PDC) method is utilized to design the fuzzy controller without considering the error due to fuzzy modelling and the sufficient conditions of the closed-loop system stability with respect to decay rate α are derived by linear matrix inequalities (LMIs). Then the errors caused by fuzzy modelling are considered and the method of adaptive control is used to reduce the effect of the modelling errors, and dynamic performance of the closed-loop system is improved. By Lyapunov stability criterion, the resulting closed-loop system is proved to be asymptotically stable. The main contribution is to deal with the differences between the T-S fuzzy bilinear model and the real system; a global asymptotically stable adaptive control scheme is presented for real complex systems. Finally, illustrative examples are provided to demonstrate the effectiveness of the results proposed in this paper.

1. Introduction

It is known that bilinear models can describe many physical systems and dynamical processes in engineering fields [20, 21]. There are two main advantages of the bilinear system. One is that it provides a better approximation to a nonlinear system than a linear one, and another is that many real physical processes may be appropriately modelled as bilinear systems when the linear models are inadequate. Considering the advantages of bilinear systems and T-S fuzzy control, the fuzzy control based on the T-S fuzzy model with bilinear rule consequence attracted the interest of researchers . The T-S fuzzy bilinear model may be suitable for some classes of nonlinear plants. The robust stabilization for continuous-time fuzzy system with local bilinear model was studied in , and then the result was extended to the fuzzy system with time-delay only in the state . The problem of robust stabilization for discrete-time fuzzy system with local bilinear model was investigated in . Reference  focuses on the problem of nonfragile guaranteed cost control for a class of T-S discrete-time fuzzy bilinear systems. Based on the parallel distributed compensation approach, the sufficient conditions were derived such that the closed-loop system was asymptotically stable and the cost function value was no more than a certain upper bound in the presence of the additive controller gain perturbations. In , an observer-based fuzzy control design was given for discrete-time T-S fuzzy bilinear systems. In [27, 28], authors proposed robust stability conditions for stochastic fuzzy impulsive recurrent neural networks with time-varying delays and uncertain stochastic fuzzy recurrent neural networks with mixed time-varying delays.

However, when there are differences between T-S fuzzy bilinear model and reality systems, these results will not be applied.

Considering the differences of the fuzzy model and the reality systems, in the paper, a stable adaptive fuzzy control for complex nonlinear systems is presented based on multiple inputs T-S fuzzy bilinear system with parameters uncertainties. In consideration of the modelling error, an adaptive fuzzy control is proposed to compensate for the issues. At first, the concept of the so-called PDC and LMI approach is employed to design the state feedback controller without considering the error caused by fuzzy modelling. The sufficient conditions with respect to decay rate α are derived in the sense of Lyapunov asymptotic stability. Then the error caused by fuzzy modelling is considered; an adaptive compensation term is designed to reduce the effect of the modelling error. The contributions of this paper are as follows: (i) the differences between T-S fuzzy bilinear model and the real system are considered in the modelling and analysis; (ii) a global asymptotical stable adaptive control scheme is presented for real systems; (iii) a sufficient condition of the closed-loop systems is given. Finally theoretical analysis verifies that the state converges to zero and all signals of the closed-loop systems are bounded.

2. Problem Statement and Basic Assumptions

Consider the nonlinear system in the following form: (1) x ˙ i = x i + 1 , i = 1 , , n - 1 , x ˙ n = f x + g T x u , where x = ( x 1 , x 2 , , x n ) T R n and u R m are the vectors of state and control input, respectively. f ( x ) is the unknown continuous function; g ( x ) is the vector of unknown continuous control gain function which satisfies g j x g j min > 0 , j = 1,2 , , m .

Definition 1 (see [<xref ref-type="bibr" rid="B19">19</xref>]).

System (1) under the input being zero is globally asymptotically stable with decay rate α , if there exists a scalar α > 0 , such that (2) V ˙ ( x ( t ) ) - 2 α V ( x ( t ) ) , where V ( x ( t ) ) = x T ( t ) P x ( t ) is the Lyapunov function candidate and P > 0 .

Lemma 2 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Given two matrices A and B with appropriate dimensions, one has A T B + B T A A T A + B T B .

In this paper, our objective is to design an adaptive fuzzy controller so that the closed-loop systems are asymptotically stable; that is, the states of the closed-loop system converge to zero and all signals of the closed-loop systems are bounded.

System (1) can be expressed in terms of the T-S fuzzy model as follows: (3) Plant    rule    i IF    z 1 t    is    M 1 i , z 2 t    is M 2 i , , z g ( t )    is    M g i THEN    x ˙ ( t ) = A i x ( t ) + B i u ( t ) + C i u t x ( t ) + Q Δ f i ( x , u ) i = 1,2 , , r , where z ( t ) = z 1 t , z 2 t , , z g ( t ) R g are known premise variables that may be functions of the state variables, M j i is the fuzzy set, r is the number of the rules, Q = 0 , , 0,1 T , C i u t = j = 1 m C i , j u j , C i , j R n × n is constant matrices, and Δ f i ( x , u ) = ( Δ a i x + Δ b i u + { Δ C i u } x ) denotes the model error of the i th bilinear model in the i th fuzzy space (also called the i th fuzzy rule).

The modelling error terms are defined as follows: (4) Δ a i = Δ a i 1 , Δ a i 2 , , Δ a i n R 1 × n , Δ b i = Δ b i 1 , Δ b i 2 , , Δ b i m R 1 × m , Δ C i u t = j = 1 m Δ C i , j u j , Δ C i , j = Δ c i , j 1 , Δ c i , j 2 Δ c i , j n R 1 × n . A i R n × n ,   B i R n × m , and C i , j are constant matrices which have of the following forms: (5) A i = 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 a i 1 a i 2 a i 3 a i ( n - 1 ) a i n , B i = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b i 1 b i 2 b i 3 b i ( m - 1 ) b i m , C i , j = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c i , j 1 c i , j 2 c i , j 3 c i , j ( n - 1 ) c i , j n .

By using the fuzzy inference method with a singleton fuzzification, product inference, and centre average defuzzification, the overall fuzzy model is of the following form: (6) x ˙ t = i = 1 r h i ( z ( t ) ) A i x ( t ) + B i u ( t ) + { C i u ( t ) } x ( t ) + Q Δ f i x , u , where (7) h i z t = w i z t j = 1 r w j z t , w i z t = j = 1 g M j i z j t .

We assume w i ( z ( t ) ) 0 , i = 1 r w i ( z ( t ) ) > 0 , for all t . Therefore, we have (8) h i z t 0 , i = 1 r h i z t = 1 , i = 1,2 , , r .

By comparison with (1) and (6), it is easy to see that (9) f x = i = 1 r h i z t a i + Δ a i x t , g j x = i = 1 r h i z t b i , j + Δ b i , j + c i , j x t + Δ C i , j x t , where a i = a i 1 , a i 2 , , a i n R 1 × n ,    b i = b i 1 , b i 2 , , b i m R 1 × m , and   c i , j = ( c i , j 1 , c i , j 2 , , c i , j n ) .

Remark 3.

From now on, unless confusion arises, arguments such as z ( t ) in h i ( z ( t ) ) will be omitted just for notational convenience.

3. Control Design and Stability Analysis

System (6) can be represented by following the T-S fuzzy model without considering the modelling error; that is, Δ f i ( x , u ) 0 ,   i = 1,2 , , r . Consider (10) Plant    rule    i IF    z 1 t    is    M 1 i , z 2 ( t )    is    M 2 i , , z g ( t )    is    M g i THEN    x ˙ ( t ) = A i x ( t ) + B i u ( t ) + C i u t x ( t ) i = 1,2 , , r .

By using the fuzzy inference method with a singleton fuzzification, product inference, and centre average defuzzification, the overall fuzzy model is of the following form: (11) x ˙ t = i = 1 r h i A i x t + B i u t + C i u t x t .

Based on the idea of PDC, the j th state-feedback controller is designed as follows: (12) Plant    rule    i IF    z 1 t    is    M 1 i , z 2 ( t )    is    M 2 i , , z g ( t )    is    M g i THEN    u q , j = ρ j F i , j x ( t ) 1 + x T ( t ) F i , j T F i , j x ( t ) , i = 1,2 , , r , where F i , j R 1 × n is a vector to be determined and ρ j > 0 is a scalar to be assigned.

The overall fuzzy control law can be represented by (13) u q , j = i = 1 r h i ρ j F i , j x ( t ) 1 + x T ( t ) F i , j T F i , j x ( t ) = i = 1 r h i ρ j sin θ i , j = i = 1 r h i ρ j F i , j cos θ i , j x t , where (14) sin θ i , j = F i , j x ( t ) 1 + x T ( t ) F i , j T F i , j x ( t ) , cos θ i , j = 1 1 + x T ( t ) F i , j T F i , j x ( t ) , θ i , j - π 2 , π 2 , 1 i r , 1 j m .

Substituting (13) into (11), one can get the closed-loop system (15) x ˙ t = i = 1 r j = 1 r h i h j A i + k = 1 m B i , k ρ k F j , k cos θ j , k + k = 1 m C i , k ρ k sin θ j , k x t , where B i , k denotes the k th column of the B i .

Theorem 4.

Given positive scalars ρ k ( 1 k m ), if there exist a symmetric positive definite matrix U and some constant matrices W i , j , such that LMIs (16) and (17) hold, (16) A i U + U A i T + ρ - + 2 α U * * B - i W - i - I - C - i U - I - < 0 , 1 i r , (17) M i * * * * B - i W - j - I - B - j W - i - I - C - i U - I - C - j U - I - < 0 , 1 i < j r , where ρ - = k = 1 m ρ k 2 ,   M i = A i U + U A i T + A j U + U A j T + 2 ρ - + 2 α U , (18) B - i W - i = B i , 1 W i , 1 B i , m W i , m , C - i = C i , 1 C i , m , I - = I I . Then, the FBS (15) is globally asymptotically stable with decay rate α via the fuzzy feedback controller (13), and the gains can be determined by F i , j = W i , j U - 1 .

Proof.

Consider the Lyapunov function candidate as follows: (19) V 1 x t = x T t P x t , where P = U - 1 .

Applying Schur complement lemma, inequality (16) can be written as follows: (20) U A i T + A i U + k = 1 m ρ k 2 + k = 1 m B i , k W i , k T B i , k W i , k + k = 1 m C i , k U T C i , k U + 2 α U < 0 . Premultiplying and postmultiplying (20) by P , respectively, we have (21) A i T P + P A i + k = 1 m ρ k 2 P P + k = 1 m B i , k F i , k T B i , k F i , k + k = 1 m C i , k T C i , k + 2 α P < 0 . Applying a similar procedure to inequality (17), we can obtain (22) A i T P + P A i + A j T P + P A j + 2 k = 1 m ρ k 2 P P + k = 1 m B i , k F j , k T B i , k F j , k + k = 1 m C i , k T C i , k + k = 1 m B j , k F i , k T B j , k F i , k + k = 1 m C j , k T C j , k + 2 α P < 0 . The time derivative of V 1 is (23) V ˙ 1 x t = x ˙ T P x + x T P x ˙ . By substituting (15) into (23), we can get (24) V ˙ 1 x t = i = 1 r j = 1 r h i h j x T A i + k = 1 m B i , k ρ k F j , k cos θ j , k k = 1 m C i , k ρ k sin θ j , k T + k = 1 m C i , k ρ k sin θ j , k T P + P A i + k = 1 m B i , k ρ k × F j , k cos θ j , k + k = 1 m C i , k ρ k sin θ j , k T k = 1 m C i , k ρ k sin θ j , k x = i = 1 r h i 2 x T Λ i , i x + i < j r h i h j x T Λ i , j x , where (25) Λ i , i = A i T P + P A i + k = 1 m ρ k sin θ i , k C i , k T P + P C i , k + k = 1 m ρ k cos θ i , k B i , k F i , k T P + P B i , k F i , k , Λ i , j = A i T P + P A i + k = 1 m ρ k sin θ i , k C i , k T P + P C i , k + k = 1 m ρ k cos θ i , k B i , k F j , k T P + P B i , k F j , k + A j T P + P A j + k = 1 m ρ k sin θ j , k C j , k T P + P C j , k + k = 1 m ρ k cos θ j , k B j , k F i , k T P + P B j , k F i , k . First, by premultiplying and postmultiplying Λ i , i by U , we can obtain (26) U Λ i , i U = U A i T + A i U + k = 1 m ρ k sin θ i , k U C i , k T + C i , k U + k = 1 m ρ k cos θ i , k U B i , k F i , k T + B i , k F i , k U . According to Lemma 2, we can get the following: (27) ρ k B i , k F i , k U + ρ k U B i , k F i , k T cos θ i , k ρ k 2 cos 2 θ i , k + B i , k W i , k T B i , k W i , k , U C i , k T ρ k sin θ i , k + C i , k U ρ k sin θ i , k ρ k 2 si n 2 θ i , k + C i , k U T C i , k U . From (26) and (27), we can obtain (28) U Λ i , i U U A i T + A i U + k = 1 m ρ k 2 + k = 1 m C i , k U T C i , k U + k = 1 m B i , k W i , k T B i , k W i , k .

Applying similar procedures (26)–(28) to Λ i , j , we can obtain (29) U Λ i , j U U A i T + A i U + U A j T + A j U + 2 k = 1 m ρ k 2 + k = 1 m B i , k W j , k T B i , k W j , k + k = 1 m C i , k U T C i , k U + k = 1 m B j , k W i , k T B j , k W i , k + k = 1 m C j , k U T C j , k U . Substituting (28) and (29) into (24), we obtain (30) V ˙ 1 x t i = 1 r h i 2 x T P U A i T + A i U + k = 1 m ρ k 2 + k = 1 m B i , k W i , k T B i , k W i , k + k = 1 m C i , k U T C i , k U P x + i < j r h i h j x T P U A i T + A i U + U A j T k = 1 m ρ k 2 + A j U + 2 k = 1 m ρ k 2 + k = 1 m B i , k W j , k T B i , k W j , k + k = 1 m C i , k U T C i , k U + k = 1 m B j , k W i , k T B j , k W i , k + k = 1 m C j , k U T C j , k U P x . By substituting (21) and (22) into (30), we can obtain (31) V ˙ 1 x t - 2 α P x = - 2 α V 1 x t .

Then, by Definition 1, the closed-loop fuzzy system (15) is globally asymptotically stable with decay rate α . This completes the proof of Theorem 4.

Next, the modelling error in (6) is considered and an adaptive compensation term is adopted to reduce the effects of the modelling error.

Adopt the fuzzy controller in the following form: (32) u = u q , j + u s , j , j = 1,2 , , m , where compensator u s , j will be designed later.

Substituting (32) into (6) yields (33) x ˙ t = i = 1 r j = 1 r h i h j A i + k = 1 m B i , k ρ k F j , k cos θ j , k + k = 1 m C i , k ρ k sin θ j , k x ( t ) + Q Δ a i + k = 1 m Δ b i , k ρ k F j , k cos θ j , k + k = 1 m Δ C i , k ρ k sin θ j , k x ( t ) + k = 1 m B i , k + Q Δ b i , k + C i , k x ( t ) + Q Δ C i , k x t u s , j k = 1 m B i , k ρ k F j , k cos θ j , k . Suppose that there exists an unknown constant λ such that (34) λ i = 1 r j = 1 r Δ a i + k = 1 m Δ b i , k ρ k F j , k cos θ j , k + k = 1 m Δ C i , k ρ k sin θ j , k . Then, from 0 h i ( z ( t ) ) 1 and (35) i = 1 r j = 1 r h i h j Δ a i + k = 1 m Δ b i , k ρ k F j , k cos θ j , k + k = 1 m Δ C i , k ρ sin θ j , k i = 1 r j = 1 r Δ a i + k = 1 m Δ b i , k ρ k F j , k cos θ j , k + k = 1 m Δ C i , k ρ sin θ j , k we have (36) i = 1 r j = 1 r h i h j Δ a i + k = 1 m Δ b i , k ρ k F j , k cos θ j , k + k = 1 m Δ C i , k ρ sin θ j , k λ . It is easy to see that we can choose a function vector H λ such that (37) i = 1 r j = 1 r h i h j Δ a i + k = 1 m Δ b i , k ρ k F j , k cos θ j , k + k = 1 m Δ C i , k ρ k sin θ j , k = λ H λ and H λ 1 .

Remark 5.

Here assumption (34) is reasonable in many real systems due to its boundedness, such as chaotic system ; for example, Example 1 in the paper satisfies the assumption. On the other hand, the uncertain terms of the considered systems in the existing literature [2, 18, 2224] satisfy the condition of (34).

Denote (38) ω t = H λ x t . Substituting (37) and (38) into (33), we can obtain the following feedback system: (39) x ˙ t = i = 1 r j = 1 r h i h j A i + k = 1 m B i , k ρ k F j , k cos θ j , k + k = 1 m C i , k ρ k sin θ j , k x ( t ) + k = 1 m B i , k + Q Δ b i , k + C i , k x ( t ) + A i + k = 1 m B i , k ρ k F j , k cos θ j , k Q Δ C i , k x t u s , j + Q λ ω t . Choose the adaptive compensator as follows: (40) u s , j = - 1 2 m g j min γ 2 Q T P c ^ x t , j = 1,2 , , m , where c = λ 2 , c ^ is the parameter estimation of c , and γ > 0 is a gain constant.

Choose the adaptive law as follows: (41) c ^ ˙ = 1 2 η γ 2 x T t P Q Q T P x t , where η > 0 is a gain constant which determines the rate of adaptation.

Substituting (40) into (39) yields (42) x ˙ t = i = 1 r j = 1 r h i h j A i + k = 1 m B i , k ρ k F j , k cos θ j , k - Q T P c ^ x t 2 m g j min γ 2 + k = 1 m C i , k ρ k sin θ j , k x ( t ) + k = 1 m B i , k + Q Δ b i , k + C i , k x ( t ) + Q Δ C i , k x ( t ) × A i + k = 1 m B i , k ρ k F j , k cos θ j , k - Q T P c ^ x t 2 m g j min γ 2 + Q λ ω t .

Theorem 6.

Consider the uncertain nonlinear system (1) with control law defined by (32), (13), and (40) and the parameter updated by the adaptive law (41). If there exist a symmetric positive definite matrix P and some matrices F i , j ( 1 i , j r ) satisfying the LMIs (16) and (17) and the design parameter is chosen as (43) 0 < γ < 2 α λ min ( P ) , then the closed-loop system (42) is asymptotically stable and all signals of the closed-loop system (42) are bounded.

Proof.

Consider the Lyapunov function candidate (44) V = x T ( t ) P x ( t ) + η c ^ - c 2 , where η > 0 .

Let (45) G i , j = A i + k = 1 m B i , k ρ k F j , k cos θ j , k + k = 1 m C i , k ρ k sin θ j , k .

The time derivative of V is (46) V ˙ = x ˙ T P x + x T P x ˙ + 2 η c ^ - c c ^ ˙ .

Substituting (39) into (46), we can obtain (47) V ˙ = i = 1 r j = 1 r h i h j x T t k = 1 m Δ C i , k ρ k sin θ j , k × G i , j T P + P G i , j + 2 P Q k = 1 m Δ C i , k ρ k sin θ j , k × Δ a i + k = 1 m Δ b i , k ρ k F j , k cos θ j , k + k = 1 m Δ C i , k ρ k sin θ j , k x ( t ) + 2 x T ( t ) P k = 1 m B i , k + Q Δ b i , k + C i , k x ( t ) + k = 1 m Δ C i , k ρ k sin θ j , k Q Δ C i , k x ( t ) u s , k + 2 η ( c ^ - c ) c ^ ˙ = i = 1 r j = 1 r h i h j x T t G i , j T P + P G i , j x ( t ) + 2 x T ( t ) P Q λ ω ( t ) + 2 x T ( t ) P Q j = 1 m g j x u s , j + 2 η c ^ - c c ^ ˙ . From the proof of Theorem 4, we get (48) V ˙ - 2 α x T ( t ) P x ( t ) + 2 x T ( t ) P Q λ ω ( t ) + 2 x T t P Q j = 1 m g j x u s , j + 2 η c ^ - c c ^ ˙ . It is easy to see that (49) 2 x T P Q λ ω - γ 2 ω 2 + γ 2 ω 2 = - γ 2 ω - 1 γ 2 x T P Q λ 2 + 1 γ 2 x T P Q λ 2 Q T P x + γ 2 ω 2 λ 2 γ 2 x T P Q Q T P x + γ 2 ω 2 .

Substituting (49) into (48) yields (50) V ˙ - 2 α x T ( t ) P x ( t ) + λ 2 γ 2 x T P Q Q T P x + γ 2 ω 2 + 2 x T t P Q j = 1 m g j x u s , j + 2 η c ^ - c c ^ ˙ .

Substituting (40) and (41) into (50), we obtain (51) V ˙ - 2 α x T ( t ) P x ( t ) + ( - c ^ + c ) 1 γ 2 x T P Q Q T P x + 2 η ( c ^ - c ) c ^ ˙ + γ 2 ω 2 = - 2 α x T ( t ) P x ( t ) + γ 2 ω 2 - 2 α λ min ( P ) x t 2 + γ 2 ω 2 , where ω 2 = ω T ω = x T H λ T H λ x H λ T H λ · x 2 x 2 .

By choosing 0 < γ < 2 α λ min ( P ) , we can get V ˙ < 0 ; then, we have that the states x ( t ) 0 as t approaches infinity via LaSalle invariance principle and V ( t ) is bounded. From (44), we can obtain that states x and c ^ are bounded; therefore, the boundedness of u s , j is ensured from (40). Similarly, from (13), we can obtain that u q , j is bounded. Then it can be proved that ( 1 ) the closed-loop system (42) is asymptotically stable and ( 2 ) all signals of the closed-loop system (42) are bounded.

4. Simulations

In this section, we will give two examples to show the efficiency of the proposed approach. The first example is an unknown chaotic system, and the second example is a parameter uncertain T-S fuzzy bilinear system with multiple inputs.

Example 1.

Consider the following chaotic system with control input: (52) x ˙ 1 = x 2 , x ˙ 2 = - 0.1 x 2 - x 1 3 + 12 cos t + u . When u ( t ) = 0 and the initial states are chosen as x ( 0 ) = ( 2,2 ) T , the states phase portrait of system (52) is shown in Figure 1.

System (52) can be modelled as the following T-S fuzzy bilinear model: (53) Rule    1    IF    x 1 ( t )    is    about    0 THEN    x ˙ ( t ) = ( A 1 + Δ A 1 ) x ( t ) + ( B 1 + Δ B 1 ) u ( t ) + ( C 1 + Δ C 1 ) x ( t ) u ( t ) ; Rule    2    IF    x 1 ( t )    is    about ± 2 THEN    x ˙ ( t ) = ( A 2 + Δ A 2 ) x ( t ) + ( B 2 + Δ B 2 ) u ( t ) + ( C 2 + Δ C 2 ) x ( t ) u ( t ) ; Rule    3    IF    x 1 ( t )    is    about ± 4 THEN    x ˙ ( t ) = ( A 3 + Δ A 3 ) x ( t ) + ( B 3 + Δ B 3 ) u ( t ) + ( C 3 + Δ C 3 ) x ( t ) u ( t ) , where x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) T , A 1 = A 2 = A 3 = 0 1 - 1 - 1 , B 1 = B 2 = B 3 = 0 - 1 , and C 1 = C 2 = C 3 = 0 1 1 1 . Choose α = 0.3 , ρ = 0.09 , F 1 = 0 - 1 , F 2 = - 1 - 1 , and F 3 = - 1 - 1 . By solving LMIs (16)-(17), one can obtain (54) P = 9.3637 4.6993 4.6993 10.6246 , λ min ( P ) = 5.2528 . Utilize the controllers (32), (13), and (40) and the parameter updated law (41) to control system (52). The design parameters are chosen as η = 2 , γ = 2 < λ min ( P ) , the initial conditions are chosen as x ( 0 ) = ( 2 , - 2 ) T , c ^ ( 0 ) = 0 , and the relationship functions are selected as shown in Figure 2. The simulation results are shown in Figures 3, 4, 5, and 6. In Figures 36, the curves of states, control input, and adaptive updated parameter for the T-S fuzzy bilinear system are drawn by solid lines, respectively, while the curves of states, control input, and adaptive updated parameter for T-S fuzzy linear system are depicted by dotted lines, respectively. By comparison, the convergence rates of the states of two systems are almost the same, though the state and control amplitudes of T-S fuzzy bilinear system (FBS) are smaller than T-S fuzzy linear system (FLS). Thus, the proposed method has some advantages of performance over the existing approach .

The phase portrait of the chaotic system.

The relationship functions.

The state x 1 response curves.

The state x 2 response curves.

The control curves.

The curves of adaptive updated parameters.

Example 2.

Consider the following parametric uncertain multiple inputs bilinear fuzzy system: (55) Rule    1    IF    x 1 ( t )    is    L 1 THEN:   x ˙ ( t ) = A 1 x ( t ) + B 1 u ( t ) + C 1 u t x t + Q Δ a 1 x t + Δ b 1 u t + Δ C 1 u t x t ; Rule    2    IF    x 2 ( t )    is    L 2 THEN:   x ˙ ( t ) = A 2 x ( t ) + B 2 u ( t ) + C 2 u t x t + Q Δ a 2 x t + Δ b 2 u t + Δ C 2 u t x t , where x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) T , A 1 = A 2 = 0 1 - 1 - 1 , B 1,1 = 0 1 , B 1,2 = 0 - 1 , B 2,1 = B 2,2 = 0 1 , C 1,1 = 0 0 - 1 1 , C 1,2 = C 2,1 = C 2,2 = 0 0 - 1 - 1 , α = 0.3 , ρ 1 = 0.05 , ρ 2 = 0.04 , F 1,1 = - 1.2 - 1.8 , F 1,2 = - 0.8 - 0.9 , F 2,1 = - 0.5 - 1.2 , and F 2,2 = - 1 - 1 ; using LMI technique to solve (16)-(17), we can get a feasible solution as (56) P = 16.8099 8.5379 8.5379 18.0100 , λ min ( P ) = 8.8510 . Apply the controllers (32), (13), and (40) and the parameters updated law (41) to system (55). The design parameters are chosen as η = 2 , γ = 2 < 2 α λ min ( P ) . The initial conditions are x ( 0 ) = ( 2 , - 0.8 ) T , c ^ ( 0 ) = 0 . The simulation results are shown in Figures 7, 8, 9, 10, and 11.

Through the comparison between T-S fuzzy linear model and bilinear one, we can see that the settling time of the systems is almost the same under the same initial conditions, although responses of T-S fuzzy bilinear system (FBS) state amplitudes are smaller than T-S fuzzy linear system (FLS), and the demand of the control input of the system is low. Thus, the proposed method has better dynamic performances than the existing ones based on T-S fuzzy linear model.

Responses of system state x 1 (T-S FBS: solid line, T-S FLS: dotted line).

Responses of system state x 2 (T-S FBS: solid line, T-S FLS: dotted line).

Control input u 1 (T-S FBS: solid line, T-S FLS: dotted line).

Control input u 2 (T-S FBS: solid line, T-S FLS: dotted line).

Adaptive parameter c (T-S FBS: solid line, T-S FLS: dotted line).

5. Conclusion

This paper proposes a new modelling method based on the multiple inputs T-S fuzzy bilinear model which is used to approximate nonlinear system; the parallel distributed compensation (PDC) method is utilized to design the fuzzy controller without considering the error caused by fuzzy modelling. The sufficient conditions with respect to decay rate α are derived by linear matrix inequalities (LMIs). The error caused by fuzzy modelling is considered and the method of adaptive control is used to reduce the effect of the modelling error. By Lyapunov stability criterion, the resulting closed-loop system is proved to be asymptotically stable. Finally, two illustrative examples are provided to show that the approach based T-S fuzzy bilinear systems have some advantages of performance over the existing methods based on T-S fuzzy linear system. The future research work is to extend the approach to general system, such as discrete-time systems, stochastic systems, and time-delay systems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This project was supported by the Soft Science Foundation of Shanxi Province (2011041033-3).