Affine parallel distributed compensator design for affine fuzzy systems via fuzzy Lyapunov function

Affine parallel distributed compensator design for affine fuzzy systems via fuzzy Lyapunov function

Engineering Applications of Artificial Intelligence 37 (2015) 407–416 Contents lists available at ScienceDirect Engineering Applications of Artificial...

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Engineering Applications of Artificial Intelligence 37 (2015) 407–416

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Affine parallel distributed compensator design for affine fuzzy systems via fuzzy Lyapunov function Mokhtar Sha Sadeghi n, Mostafa Rezaei, Mohammad Mardaneh Control and Power Engineering Departments, Electrical and Electronics Engineering Faculty, Shiraz University of Technology, Shiraz, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 31 December 2013 Received in revised form 7 July 2014 Accepted 29 September 2014 Available online 5 November 2014

This paper develops a novel stability analysis and robust controller design method for affine fuzzy systems. The emphasis of the paper is to present more relaxed stability conditions based on nonquadratic fuzzy Lyapunov function and affine parallel distributed compensation. At first, diffeomorphic transformations are used to treat more general class of nonlinear systems in a unified manner. Then, by introducing slack matrices, the Lyapunov matrices are decoupled from the feedback gain matrices and controller affine terms which lead to eliminate the structural constraints of Lyapunov matrices and consequently reduces the conservativeness of the proposed approach. Because of the bias terms, the stabilization conditions are obtained in terms of bilinear matrix inequalities. A nonsingular state transformation together with using the S-procedure and also slack variables lead to derive the stabilization conditions in the formulation of linear matrix inequalities which can be solved by convex optimization techniques. Moreover, H 1 controller is used to reject the disturbances. Finally, the merit and applicability of the proposed approach are demonstrated via comparative numerical and industrial case studies. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Affine fuzzy systems Stability analysis Fuzzy Lyapunov function Linear matrix inequality Disturbance rejection

1. Introduction Practical and industrial systems are inherently nonlinear, so that the concentration of many research works in control engineering belongs to stability analysis and controller design synthesis of nonlinear complex systems. In this area, fuzzy control is one of the very active research fields to tackle the problem. The main reasons of increasingly interesting to use fuzzy systems are its simplicity and effectiveness (Kim et al., 2005; Sha Sadeghi et al., 2014a). Originally, the fuzzy control had been introduced as a model free controller design approach which was based on heuristic and expert human knowledge (Feng, 2006; Wang and Yang, 2012). Lack of systematic design procedures to guarantee the stability and performance of the control systems and its shortcomings cause to turn the researchers attentions to develop model-based fuzzy control (Kim and Kim, 2001). One of the most popular research field in the model-based fuzzy control is Takagi– Sugeno (TS) fuzzy model-based control (Takagi and Sugeno, 1985). In TS fuzzy model, the overall nonlinear dynamics is subdivided to several local dynamics. The local dynamics are appeared in the consequent parts of the fuzzy rules which their antecedents show

n

Corresponding author. Tel.: þ 98 9122764681. E-mail addresses: [email protected] (M.S. Sadeghi), [email protected] (M. Rezaei), [email protected] (M. Mardaneh). http://dx.doi.org/10.1016/j.engappai.2014.10.008 0952-1976/& 2014 Elsevier Ltd. All rights reserved.

the local operating regions. Then, the overall fuzzy model of the original nonlinear system is obtained by fuzzy blending of these local models (Kim et al., 2005, 2013). Generally, the TS fuzzy systems are classified into two categories as linear and affine fuzzy models (Kim and Kim, 2001, 2002). The linear TS system means that the TS fuzzy system which its consequent part is linear and does not have a constant term. Moreover, the affine fuzzy system (AFS) has affine consequent part and has a constant bias term. More easily stability analysis and controller design of linear TS fuzzy system persuade the researchers that their studies are more concentrated on the linear TS fuzzy system than the affine one (Kim and Kim, 2001; Kim et al., 2005, 2013). Nonetheless, the need to obtain more exact modeling for some nonlinear plants, representing the equivalent TS fuzzy model with less fuzzy rules for complicated nonlinear systems, more natural and appealing to human than the linear one (Chang et al., 2011), and the more capability of the affine TS fuzzy system to model complex nonlinear system cause to be devoted an increasing amount of research work on stability analysis and controller design based on the affine TS fuzzy system and lead the researchers to further focus on the this area (Cao et al., 1997; Chang and Shing, 2005; Chang and Yeh, 2007; Chang et al., 2011, 2012; Fantuzzi et al., 1996). A conventional method for stability analysis and controller design for T–S fuzzy systems is based on Lyapunov stability theorem (Slotine and Li, 1991). A stability analysis and controller synthesis methodology

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M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416

is proposed in Kim and Kim (2002) based on quadratic stability for a continuous AFS. Stability conditions are derived in terms of linear matrix inequalities (LMIs), but in synthesis part, the stabilization conditions are converted to bilinear matrix inequalities (BMIs) and solved numerically in an iterative manner. The fuzzy local controllers have the affine form in Kim and Kim (2002). Another analysis and design method was proposed in Kim et al. (2005) for both continuous and discrete-time fuzzy systems. Two diffeomorphic state and input transformations were introduced such that the input matrices of all local dynamics of TS fuzzy system are the same and then by considering diagonal form for Lyapunov matrix, the stabilization conditions are stated in terms of LMIs, but in many cases, a Lyapunov matrix in diagonal form might not be found. In Chang et al. (2009), by applying the passivity theory and Lyapunov theory, stability conditions are derived to guarantee stability and passivity of closed-loop affine fuzzy systems. At first, the conditions are derived in terms of BMIs which cannot be solved by the optimal convex programming algorithm. For obtaining stability conditions, new auxiliary variables are introduced and an iterative LMI is proposed to determine and update the auxiliary variables to find the solution with convex programming algorithm. This method suffers from the initialization of variables and tolerates some conservativeness in spite of introducing more relaxation in stability conditions than previous works. In Wang and Yang (2013), a PDC state feedback controller is designed for continuous-time affine fuzzy model. The authors introduced extra slack variables to decouple the Lyapunov matrix and the system matrix such that the controller parameterization is independent of the Lyapunov matrix. By using diffeomorphic state and input transformations, the conditions are stated in the form of LMIs. Also, the method does not require any structural constraint on the Lyapunov matrix. Afterwards, H 1 controller is designed to overcome the disturbance. Finally, in Kim et al. (2013), a fuzzy Lyapunov function is used for reducing conservativeness in stability conditions of AFS and the conditions are described in the form of LMIs. Anyway, the controller design is not discussed in Kim et al. (2013). One of the most important research fields in stability analysis and controller design in the form of LMIs is to reduce the conservativeness of the LMIs conditions. Several ideas and method have been developed in the literature (Kim et al., 2005, 2013; Sha Sadeghi et al., 2014b; Wang and Yang, 2012, 2013). The methods are divided mainly into two categories as quadratic and nonquadratic stabilities depending on the Lyapunov function definitions. As the literature survey shows, the most stability analysis conditions of AFS are based on quadratic Lyapunov functions. The only references which utilizes a fuzzy Lyapunov function is Kim et al. (2013) which does not cover the controller synthesis. In this paper, a systematic approach is proposed for the stabilization of closed loop AFSs. This approach is based on the fuzzy Lyapunov function and affine parallel distributed compensation (APDC) controller. At first, the stabilization conditions of AFSs with constant input matrix, by applying lyapunov theorem and S-procedure are derived in the form of BMIs which cannot be solved by convex optimization algorithms. Therefore, some slack matrices are utilized to decouple the Lyapunov matrices from the system matrices and convert the proposed conditions in the form of LMIs. In other words, no conservative iterative LMIs are required to solve the control problem. These considerations provide the most relaxation in the stability and controller design conditions up to existing and newly published researches which the comparative simulation examples illustrate the effectiveness of the method. The remainder of the paper is organized as follows: In Section 2, affine TS fuzzy model is presented for two cases where the input matrices are the same and where the input matrices are functions of the states. Controller design and stabilizations conditions are derived in terms of LMI in Section 3. The H 1 synthesis for disturbance rejection is discussed in Section 4. Simulation and

comparison results are given in Section 5 and finally the paper is concluded in Section 6.

2. Affine fuzzy modeling How to obtain the affine fuzzy model of a nonlinear system is investigated in this section. Two cases are examined: (i) the case where the input matrices are the same for all local dynamics in the consequent parts of the fuzzy rules and (ii) the case where the input matrices are functions of the states. 2.1. Affine fuzzy modeling with a constant input matrix The nonlinear system with constant input matrix is considered as following: x_ ¼ F ðx; uÞ ¼ f ðxÞ þ Gu

ð1Þ

where xðtÞ A ℝ is the state vector, u A ℝ is the control input, f ðxÞ A ℝn , and G A ℝnm is a constant matrix with full column rank. Because of constant input matrix, the system representation in the form of (1) may seem some restrictive. Nevertheless, many practical plants such as ball-and-beam (Wang, 1997) or permanent magnet synchronous motor (PMSM) (Kim and Youn, 2002) can represented in the form of (1). By Taylor expansion of (1) around operating points, the affine fuzzy model is derived as follows: n

m

Plant Rule i : If x1 ðt Þ is M i1 and:::and xn ðt Þ is M in ; then x_ ðt Þ ¼ Ai xðt Þ þ Buðt Þ þ μi ;

i ¼ 1; 2; …; r

ð2Þ

where M i1 ; M i2 ; …; M in are fuzzy sets; r is the number of fuzzy rules, Ai A ℝnn is the system matrix; B A ℝnm is the input matrix, and μi A ℝn is the constant term. By applying singleton fuzzifier, product inference engine and center-average defuzzifier, the overall affine fuzzy model can be obtained as following: x_ ðt Þ ¼

  r   ∑ri ¼ 1 ωi ðxðt ÞÞ Ai xðt Þ þBuðt Þ þ μi ¼ ∑ hi ðxðt ÞÞ Ai xðt Þ þ Buðt Þ þ μi r ∑i ¼ 1 ωi ðxðt ÞÞ i¼1

ð3Þ

where hi ðxðt ÞÞ ¼

ωi ðxðt ÞÞ ; ∑ri ¼ 1 ωi ðxðt ÞÞ

0 rhi ðxðt ÞÞ r 1;

n

ωi ðxðt ÞÞ ¼ ∏ M ij ðxj ðtÞÞ j¼1

r

∑ hi ðxðt ÞÞ ¼ 1

ð4Þ

i¼1

hi ðxðt ÞÞ, is the normalized membership function and M ij ðxj ðtÞÞ is the grade of xj ðtÞ in M ij . In the following, xðt Þ and uðt Þ are denoting by x and u for simplicity. 2.2. Affine fuzzy modeling with a function input matrix Consider a nonlinear system with a function input matrix of the following form: x_ ¼ Fðx; uÞ ¼ f ðxÞ þ GðxÞu

ð5Þ

where x A ℝ is the state vector, u A ℝ is the control input, and f ðxÞ A ℝn and GðxÞ A ℝnm have full column rank. If the column of GðxÞ be involutive, then there exists a diffeomorphic state transform (Kim et al., 2005) n

x ¼ ΦðxÞ

m

ð6Þ

and an input diffeomorphic transform u ¼ MðxÞu

ð7Þ

such that the nonlinear system (5) is represented as the following system with a constant input matrix G. x_ ¼ f ðxÞ þ Gu

ð8Þ

M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416

The diffeomorphic transforms (6) and (7) help to treat the system (1) and (5) in a unified manner (Wang and Yang, 2013). The following lemma is introduced for shorting the proof process of the main theorems in the next section: Lemma 1. Let ϕ be a symmetric matrix and P be a positive-definite matrix. The following statements are equivalent (Wang and Yang, 2012): (a) ϕ þ PA þ AT P o0: (b) For a large enough constant α 4 0, there exists a matrix F such that ! ϕ 2αP P þ ðA þ αIÞT F o 0:  F  FT P þ F T ðA þ αIÞ

409

In the following Theorem, sufficient stability analysis conditions under which the closed-loop system (11) is stable, are presented. Fuzzy Lyapunov function is used for stability analysis and S-procedure is utilized to achieve feasible conditions in terms of bilinear matrix inequalities (BMIs). Theorem 1. The affine fuzzy system (11) is asymptotically stable if there exist positive definite matrices P i ¼ P Ti ði ¼ 1; 2; …; rÞ, symmetric matrix M ¼ M T which satisfies P i þ M 4 0, and scalars τiq Z0 such that P ϕ þ ðAi þ BK i ÞT P i þ P i ðAi þ BK i Þ o 0 for the fuzzy rule i ði A I ξ Þ and 0

n

T B P ϕ þ ðAi þ BK i Þ P i þ P i ðAi þ BK i Þ  ∑ τiq T iq q¼1 B B B @ n

ð13Þ 1 n   P i μi þ Bσ i  ∑ τiq uiq C q¼1 C C o0 n C A  ∑ τiq viq q¼1

ð14Þ

3. Stabilization conditions for affine fuzzy system In this section, the new stability conditions for the AFS are derived based on nonquadratic fuzzy Lyapunov function and parallel distributed compensator (PDC) with affine structure.

Affine parallel distributed compensation (APDC) is described as following: Controller Rule i : If x1 is M i1 and:::and xn is M in ; i ¼ 1; 2; …; r

ð10Þ

where hi ðxÞ and ωi ðxÞ is defined as (4). The closed-loop system is obtained by substituting (10) in (3): r   x_ ðt Þ ¼ ∑ hi ðxÞ ðAi þBK i Þx þ ðμi þ Bσ i Þ

ði ¼ 1; 2; …; n; q ¼ 1; 2; …; nÞ

ð11Þ

i¼1

r

r

r

i¼1 r

ρ¼1

i¼1

V_ ðxÞ ¼ ∑ hi ðxÞx_ T P i x þ ∑ h_ ρ ðxÞxT P ρ x þ ∑ hi ðxÞxT P i x_ ¼ ∑ h_ ρ ðxÞxT P ρ x ρ¼1

n r þ ∑ hi ðxÞ xT ðAi þ BK i ÞT P i x þ xT P i ðAi þ BK i Þx i¼1

 T  o þ μi þ Bσ i P i x þxT P i μi þ Bσ i From (4), it is concluded that r

r

ρ¼1

ρ¼1

∑ hρ ðxÞ ¼ 1 - ∑ h_ ρ ðxÞ ¼ 0

ð17Þ

So, the following matrix statement with slack matrix M is proposed: r

∑ h_ ρ ðxÞxT Mx ¼ 0

ð18Þ

By defining r

Fuzzy Lyapunov function is a fuzzy blending of quadratic Lyapunov functions as following: r

ð16Þ

ρ¼1

3.2. Fuzzy Lyapunov function candidate

V ðxÞ ¼ ∑ hi ðxÞxT P i x

ð15Þ

Proof. Consider fuzzy Lyapunov function (12)

ð9Þ

The APDC controller shares the same antecedent with AFS (2). Each fuzzy controller rule has a constant term in the consequent part. By applying singleton fuzzifier, product inference engine and center-average defuzzifier, the overall APDC controller is obtained as follows: r ∑r ωi ðxÞðK i x þ σ i Þ uðt Þ ¼ i ¼ 1 r ¼ ∑ hi ðxÞðK i x þ σ i Þ ∑i ¼ 1 ωi ðxÞ i¼1

F iq ðxÞ  xT T iq x þ 2uTiq x þ viq r 0

and T iq , uiq

for all x which activates rule i ðhi ðxÞ a 0Þ (Kim and Kim, 2001, 2002).

3.1. Affine parallel distributed compensator

then u ¼ K i x þ σ i ;

for the other fuzzy rules, where P ϕ ¼ and viq are defined such that

∑ri ¼ 1 ∅i ðP i þMÞ

ð12Þ

i¼1

In contrast to conventional quadratic Lyapunov function, fuzzy Lyapunov function provides more relaxed stability conditions (Abdelmalek et al., 2007; Kim et al., 2013; Mozelli et al., 2009; Tanaka et al., 2003). The following assumptions are required for the stability analysis and also controller design formulations development: Assumption 1. Let I ξ be the set of indices for the fuzzy rules that contains the origin ðx ¼ 0Þ, then for i A I ξ , the constant term μi in (1) is assumed to be 0. This assumption assures that the origin is the equilibrium point of affine fuzzy system (Kim and Kim, 2001, 2002). Assumption 2. Assume that the derivatives of the membership functions are bounded as jh_ ρ ðxÞj r ϕρ , where ϕρ 4 0 ðρ ¼ 1; 2; …; rÞ (Abdelmalek et al., 2007; Kim et al., 2013; Tanaka et al., 2003).

P ϕ : ¼ ∑ ϕρ ðP ρ þ MÞ ρ¼1

and consider (18) and Assumptions 1 and 2 one has r

V_ ðxÞ ¼ ∑ h_ ρ ðxÞxT ðP ρ þ MÞx ρ¼1

n r þ ∑ hi ðxÞ xT ðAi þ BK i ÞT P i x þ xT P i ðAi þ BK i Þx i¼1

 T  o þ μi þ Bσ i P i x þxT P i μi þ Bσ i r

r ∑ ϕρ xT ðP ρ þ MÞx ρ¼1

n r þ ∑ hi ðxÞ xT ðAi þ BK i ÞT P i x þ xT P i ðAi þ BK i Þx i¼1

 T  o þ μi þ Bσ i P i x þxT P i μi þ Bσ i r  T  ¼ ∑ hi ðxÞ xT P ϕ x þ xT ðAi þ BK i ÞT P i x þ xT P i ðAi þ BK i Þx þ μi þ Bσ i P i x i¼1

  þ xT P i μi þ Bσ i n   o ¼ ∑ hi ðxÞ xT P ϕ þ ðAi þ BK i ÞT P i þP i ðAi þ BK i Þ x i A Iξ

ð19Þ

410

M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416

n o  T   þ ∑ xT P ϕ þ ðAi þ BK i ÞT P i þ P i ðAi þ BK i Þ x þ μi þ Bσ i P i x þ xT P i μi þ Bσ i

AFS (3) will be written as

i2 = Iξ

ð20Þ From (20), it is obtained that V_ ðxÞ o 0, if

ð28Þ

i¼1

  xT P ϕ þ ðAi þ BK i ÞT P i þP i ðAi þ BK i Þ x o0

where ð21Þ A^ i ¼ ψAi ψ  1 ;

for the fuzzy rule iðiA I ξ Þ and  T   xT fP ϕ þ ðAi þ BK i ÞT P i þ P i ðAi þ BK i Þgx þ μi þ Bσ i P i x þ xT P i μi þ Bσ i o 0

ð22Þ

for other fuzzy rules. By applying congruence Lemma (Boyd, 1994), inequality (21) obtains (13). Also, based on the region of x corresponding to each rule, the following inequality is held: F iq ðxÞ  xT T iq x þ 2uTiq x þ viq r 0

  r ^ þ μ^ x_~ ¼ ∑ hi ðxÞ A^ i x~ þ Bu i

ði ¼ 1; 2; …; n;

I mm 0ðn  mÞm

! ;

μ^ i ¼ ψμi

Similarly, the APDC controller, the closed-loop system and the Lyapunov function are respectively rewritten as   r ð29Þ u ¼ ∑ hi ðxÞ K^ i x~ þσ i i¼1 r

x_~ ¼ ∑ hi ðxÞ i¼1

q ¼ 1; 2; …; rÞ ð23Þ

B^ ¼ ψB ¼

n

  o ^ i A^ i þ B^ K^ i x~ þ μ^ i þ Bσ

ð30Þ

and r

Now by considering inequality (23) and using S-procedure (Boyd, 1994), inequality (22) holds, if there exist τiq 4 0, such that n  T   xT fP ϕ þ ðAi þ BK i ÞT P i þ P i ðAi þ BK i Þgx þ μi þ Bσ i P i x þ xT P i μi þ Bσ i  ∑ τiq F iq ðxÞ o 0 q¼1

ð24Þ for all x which activates rule i; ðhi ðxÞ 4 0Þ. Inequality (24) can be rewritten as follows: 0 n T  T B P ϕ þ ðAi þ BK i Þ P i þ P i ðAi þ BK i Þ  ∑ τiq T iq q¼1 x B B 1 B @ n  x  o0 1

1 n   P i μi þ Bσ i  ∑ τiq uiq C q¼1 C C n C A  ∑ τiq viq q¼1

x~ ¼ ψx

ð26Þ

where ψ is chosen such that the inner product of ψ by B equals to

0ðn  mÞm

ð31Þ

i¼1

where K^ i ¼ K i ψ  1 ;

 T P^ i ¼ ψ  1 P i ψ  1

Therefore, (13) and (14) convert to (32) and (33), respectively, as follows:  T   ð32Þ P^ ϕ þ A^ i þ B^ K^ i P^ i þ P^ i A^ i þ B^ K^ i o 0 0

 T   n ^ ^ ^ ^ ^^ ^ ^ ^^ B P ϕ þ Ai þ BK i P i þ P i Ai þ BK i  ∑ τiq T iq q¼1 B B B @ n

1   n ^ i  ∑ τiq u^ iq P^ i μ^ i þ Bσ C q¼1 C Co0 n C A  ∑ τiq viq q¼1

ð33Þ

ð25Þ

Finally, by applying congruence Lemma, (25) obtains (14). The proof is completed. ▢ Multiplying the Lyapunov matrices and the controller gains leads to obtain the conditions of Theorem 1 in terms of BMIs which must be solved by iterative LMI methods (Kim and Kim, 2001, 2002) or nonconvex programming which results local optimal solutions. Therefore, if iterative LMI is utilized to solve the inequalities of Theorem 1, it requires an initial feasible solution which is very conservative. In the following, the attempt is made to derive conditions of Theorem 1 in terms of LMIs. To do this, a nonsingular transformation is needed as follows (Kim et al., 2005):

I mm

V ðxÞ ¼ ∑ hi ðxÞx~ T P^ i x~

!

According to Theorem 1 and the nonsingular transformation (26), new nonquadratic stabilization conditions for the closed-loop system (29) are obtained in Theorem 2. The BMIs stability conditions are converted to LMIs ones using Lemma 1 by separating the controllers matrices from Lyapunov ones, and instead, the constraints are imposed on slack matrices. Theorem 2. The affine fuzzy system (30) is asymptotically stable if T there exist positive definite matrices P^ i ¼ P^ i ði ¼ 1; 2; …; rÞ, matrices W, F 1 , F 2 , F 3 with appropriate dimensions where ! ! W1 0 0 F 11 W¼ ; F1 ¼ W2 W3 F 21 F 31

and M ¼ M T which satisfies P^ i þ M 4 0 and scalars τiq Z 0 such that for large enough α; k 40 0 1

T Yi T ^ ^ ^ ^ P P  2α P þ A W þ þkW B ϕ C i i i ð34Þ 0 @ Ao0

:

W WT

n

for the fuzzy rule i ði A I ξ Þ and 0

A common approach for choosing ψ is of the following form: 0

1 1 BT A BT B @ ψ¼ Ξ

ð27Þ

where Ξ is chosen such that its rows are independent from each other and its inner product of Ξ by B is equal to zero. Therefore, the

B P^ ϕ ∑nq ¼ 1 τiq T^ iq  2αP^ i B B B B Bn B B B B n @ n

 ∑nq ¼ 1 τiq u^ iq  ∑nq ¼ 1 τiq viq 2α

T Yi T P^ i þ A^ i F 1 þ þαF 1 0

T Vi þ μ^ Ti F 1 þ αF 2 0

1 0 1 þαF 3

n

F 1  F T1

 F T2

n

n

 F 3 F T3

C C C C C Co0 C C C C A

ð35Þ for other fuzzy rules, where P^ ϕ ¼ ∑ri ¼ 1 ϕi ðP^ i þ MÞ. And T^ iq , u^ iq and

M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416

viq are defined such that T F^ iq ðxÞ  x~ T T^ iq x~ þ 2u^ iq x~ þviq r0

ði ¼ 1; 2; …; n; q ¼ 1; 2; …; rÞ

ð36Þ

for all x which active rule i ðhi ðxÞ a 0Þ. And the fuzzy controller matrices are obtained from K^ i ¼ W T 1 Yi 0 B B B B B B B B B @

for the fuzzy rules Ri ði A I ξ Þ

n

q¼1

n

T

1 þ αF 3

n

n

F 1  F T1

 F T2

n

n

n

 F 3 F T3

q¼1

and T K^ i ¼ F 11 Yi;

T σ i ¼ F 11 V i for the other fuzzy rules

8 > > > K^ T B^ T F 1 ¼ K^ Ti n½ I mm > > < i

h 0mðn  mÞ F 1 ¼ K^ T i

h > > > > σ T B^ T F 1 ¼ σ Ti n I mm > : i

i h 0mðn  mÞ F 1 ¼ σ Ti

0

" i F 11 "

0

ð38Þ

0 F 22

F 21

i F 11

0

# #

F 22

F 21

Proof. From (32), one has  T   T P^ ϕ þ A^ i þ B^ K^ i P^ i þ P^ i A^ i þ B^ K^ i ¼ ϕ þ A P þ PA o0 0

 T   n ^ ^ ^ ^ ^^ ^ ^ ^^ B P ϕ þ Ai þ BK i P i þ P i Ai þ BK i  ∑ τiq T iq q¼1 B B B @ n n

i h T 0 ¼ Yi

h ¼ σ Ti F 11

i h 0 ¼ V Ti

^ ^ B P ϕ  ∑ τiq T iq q¼1 B B ¼B n @  ∑ τiq u^ T iq q¼1

 ∑ τiq u^ iq C q¼1 C Cþ n C  ∑ τiq viq A

P^ i 0

T ¼ Yi

!

A^ i þ B^ K^ i 0

^ i μ^ i þ Bσ 0

!

q¼1

A^ i þ B^ K^ i

^ i μ^ i þ Bσ

0

0

!T

P^ i

0

0

1

!

i



0 ¼

0ðn  mÞm Yi

" i W1 W2

0 W3

#

h T ¼ K^ i W 1

i 0

T

0

ð47Þ

4. H1 controller design T

¼ ϕ þ P A þA P o 0

ð40Þ

By applying Lemma 1, it is concluded that for a large enough constants k; α 4 0 there exist slack matrices W and F such that ! ϕ  2kP P þ ðA þkIÞT W o0 ð41Þ P þ W T ðA þ kIÞ W WT for the fuzzy rule i ði A I ξ Þ and 0 1 ϕ  2αP P þ ðA þ αIÞT F @ Ao0 P þ F T ðA þ αIÞ  F  FT

ð46Þ

for the other fuzzy rules. Substituting (46) and (47) in (44) and (45), respectively, obtains (34) and (35). The proof is completed. ▢

q¼1

0 1

h T T K^ i BT W ¼ K^ i n I mm h

1   n ^ i  ∑ τiq u^ iq P^ i μ^ i þ Bσ C q¼1 C C n C A  ∑ τiq viq

ð45Þ

i Y T i 0 ¼ 0 i V T i 0 ¼ 0

for the fuzzy rule i ði A I ξ Þ and ð39Þ

1

n

C C C C C Co0 C C C A

for the other fuzzy rules. Inequality (44) is of the form BMI, due to T the multiplication of K^ i by W. Also, (45) is BMI, according to the T T T T^ ^ ^ terms K i B F 1 and σ i B F 1 . In the following, the attempt is made to ^ F 1 and W, derive (44) and (45) in terms of LMI. By substituting B, one has

h T ¼ K^ i F 11

and from (33), one concludes that

þ

1 0

 ∑ τiq viq  2ασ Ti B^ F 1 þ μ^ Ti F 1 þ αF 2

n

0

for the fuzzy rule i ði A I ξ Þ and

T T P^ i þ A^ i F 1 þ K^ i B^ T F 1 þ αF 1

 ∑ τiq u^ iq

q¼1

and substituting them into (41) and (42), one concludes that for large enough α; k 4 0 ! T T P^ ϕ 2kP^ i P^ i þ A^ i W þ K^ i B^ T W þ kW o0 ð44Þ n W W T

ð37Þ

n

P^ ϕ  ∑ τiq T^ iq  2αP^ i

411

In this section, the object is to decrease the effect of disturbance input on the system output. To do this, H1 synthesis is used such that L2 -norms of disturbance wðtÞ to the output zðtÞ be less than γ, under zero initial conditions i.e.  ‖zðtÞ2 ‖ oγ : ‖wðtÞ2 ‖ Consider the following AFS:   r x_~ ðt Þ ¼ ∑ hi ðxðt ÞÞ A^ i x~ ðt Þ þ B^ 1 uðt Þ þ B^ 2i wðtÞ þ μ^ i

ð42Þ

for the other fuzzy rules. By defining W and F as following: ! ! ! W1 F 11 0 0 F1 0 W¼ ; F¼ ; F1 ¼ ð43Þ W2 W3 F2 F3 F 21 F 31

i¼1 r

~ zðt Þ ¼ ∑ hi ðxðt ÞÞL^ xðtÞ

ð48Þ

i¼1

where xðtÞ A ℝn is the state vector, uðtÞ A ℝm is the control input, wðtÞ A ℝp is the disturbance input vector, and zðtÞ A ℝq is the output. Following the symbols and notations defined previously in the

412

M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416 r

paper, the system and input matrices are as following: A^ i ¼ ψAi ψ  1 ;

I mm 0ðn  mÞm

B^ 1 ¼ ψ B1 ¼

¼ ∑ h_ ρ ðxÞx~ T ðP^ ρ þ MÞx~

! ;

μ^ i ¼ ψμ;

B^ 2i ¼ ψ B2i ;

ρ¼1

L^ ¼ Lψ  1

 T    T r T T þ ∑ hi ðxÞ x~ A^ i þ B^ 1 K^ i P^ i x~ þ x~ P^ i A^ i þ B^ 1 K^ i x~ þ μ^ i þ B^ 1 σ i P^ i x~

ð49Þ

i¼1

  T þ x~ T P i μ^ i þ B^ 1 σ i þwT B^ 2i P^ i x~ þ x~ T P^ i B^ 2i w ) r T ^T ^ 2 T ~ ~ þ ∑ hi ðxÞx L Lx γ w w

and using the APDC controller (29), the closed-loop system is obtained as    n o r x_~ ðt Þ ¼ ∑ hi ðxðt ÞÞ A^ i þ B^ 1 K^ i x~ ðt Þ þ B^ 1 σ i þ μ^ i þ B^ 2i wðt Þ ð50Þ

i¼1

i¼1

The following theorem presents new sufficient conditions in terms of LMIs to guarantee the H 1 performance level of the disturbance attenuation. To do this, Lemma 1 and the nonsingular transformation (26) are used.

T B P^ ϕ þ L^ L^  2kP^ i B B B B n B B B n @

o   T þ x~ T P^ i μ^ i þ B^ 1 σ i þwT B^ 2i P^ i x~ þ x~ T P^ i B^ 2i w  γ 2 wT w   T   T T ¼ ∑ hi ðxÞ x~ T P^ ϕ þ A^ i þ B^ 1 K^ i P^ i þ P^ i A^ i þ B^ 1 K^ i þ L^ L^ x~ þ wT B^ 2i P^ i x~

0

0

 γ 2 I  2kI

T B^ 2i W 1 þ kW 2

n

 W 1  W T1

 W T2

n

n

 W 3  W T3

n

I þ kW 3

C C C C Co0 C C C A

ð51Þ for the fuzzy rule i ði A I ξ Þ and 0

 ∑ τiq u^ iq

0

q¼1

n

Λ14

0

0

αF 22

0

αF 32

αF 33  F T31

n

 ∑ τiq viq  2α

T B^ 2i F 11 þ αF 21

T Vi μ^ Ti F 11 þ þαF 31

n

n

 F 11 F T11

F T21

 F T32 F 33  F T33

 γ 2 I 2αI

0 n

q¼1

0

n

n

n

 F 22 F T22

n

n

n

n

o0

C C C C C C C C C C C C C C C C A

( þ ∑ hi ðxÞ

r

  o T þ x~ T P^ i μ^ i þ B^ 1 σ i þwT B^ 2i P^ i x~ þ x~ T P^ i B^ 2i w  γ 2 wT w

n

Λ11 ¼ P^ ϕ þ L^ L^  ∑ τiq T^ iq  2αP^ i q¼1

T



W1

0

W2

W3

F 11 ¼



Yi 0

!

  T   T T x~ T P^ ϕ þ A^ i þ B^ 1 K^ i P^ i þ P^ i A^ i þ B^ 1 K^ i þ L^ L^ x~ þ wT B^ 2i P^ i x~ þ x~ T P^ i B^ 2i w  γ 2 wT w  ¼

;

W1 ¼

F 111

0

F 112

F 113

þ αF 11 0

W 21

W 22

!

! ;

0

F 11 B F ¼ @ F 21 F 31

0 F 22 F 32

0

1

T F 111 Y i;

T

  T   x~ T T 2 o0 P^ ϕ þ A^ i þ B^ 1 K^ i P^ i þ P^ i A^ i þ B^ 1 K^ i þ L^ L^ P^ i B^ 2i B^ 2i P^ i  γ I w

ð55Þ for the fuzzy rules i ði A I ξ Þ and



 T    T   T x~ T P^ ϕ þ A^ i þ B^ 1 K^ i P^ i þ P^ i A^ i þ B^ 1 K^ i þ L^ L^ x~ þ μ^ i þ B^ 1 σ i P^ i x~ þ x~ T P^ i μ^ i þ B^ 1 σ i T

þ wT B^ 2i P^ i x~ þ x~ T P^ i B^ 2i w  γ 2 wT w o 0

for the fuzzy rules Ri ðiA I ξ Þ T σ i ¼ F 111 V i for the other fuzzy rules

Proof. The following inequality guarantees the stability of AFS (50) and H 1 performance level γ (Wang and Yang, 2012). V_ ðxÞ þ zT ðt Þzðt Þ  γ 2 wT ðt Þwðt Þ o0 ð53Þ One has V_ ðxÞ þ zT ðt Þzðt Þ  γ 2 wðt ÞT wðtÞ

ð56Þ

ði ¼ 1; 2; …; n; q ¼ 1; 2; …; rÞ

ð57Þ



 T    T   T x~ T P^ ϕ þ A^ i þ B^ 1 K^ i P^ i þ P^ i A^ i þ B^ 1 K^ i þ L^ L^ x~ þ μ^ i þ B^ 1 σ i P^ i x~ þ x~ T P^ i μ^ i þ B^ 1 σ i

0 C A; F 33

The matrices of APDC controller are obtained from T Yi K^ i ¼ W 11

x~ w

Now by considering above inequality (57) and using S-procedure (Boyd, 1994), inequality (56) holds if there exist τiq Z 0, such that:

T

W 11

ð54Þ

Inequality (53) holds, if

T F^ iq ðxÞ  x~ T T^ iq x~ þ 2u^ iq x~ þ viq r 0

i¼1

Λ14 ¼ P^ i þ A^ i F 11 þ



 T    T T x~ T P^ ϕ þ A^ i þ B^ 1 K^ i P^ i þ P^ i A^ i þ B^ 1 K^ i þ L^ L^ x~ þ μ^ i þ B^ 1 σ i P^ i x~

for the other fuzzy rules hold. Also, based on the region of x corresponding to each rule and transformation (26), the following inequality is held:

P^ ϕ ¼ ∑ ∅i ðP^ i þ MÞ T

o

i2 = Iξ

ð52Þ

for the other fuzzy rules, where

K^ i ¼

þ x~ T P^ i B^ 2i w  γ 2 wT w

1

n

B Λ11 B B B B n B B B B n B B B B B n B B n @

i A Iξ

1

T Yi T P^ i þ A^ i W 1 þ þ kW 1 0



 T    T T x~ T P^ ϕ þ A^ i þ B^ 1 K^ i P^ i þ P^ i A^ i þ B^ 1 K^ i þ L^ L^ x~ þ μ^ i þ B^ 1 σ i P^ i x~

i¼1

Theorem 3. The affine fuzzy system (47) with APDC controller (29) is asymptotically stable and L2 -norm of the disturbance wðtÞ to the output zðtÞ be less than γ, if there exist positive definite matrices P i ¼ P Ti ði ¼ 1; 2; …; rÞ, matrices W, W 1 , F and F 11 , symmetric matrix M which satisfies P^ i þ M 4 0, and scalars τiq Z 0 such that for large enough α; k 4 0 0

(

r

r ∑ hi ðxÞ

n

T þ wT B^ 2i P^ i x~ þ x~ T P^ i B^ 2i wðt Þ  γ 2 wT w  ∑ F^ iq ðxÞτiq r0

ð58Þ

q¼1

for all x which active rule iðhi ðxÞ a 0Þ. Inequality (58) can be rewritten as following: 9 08 < P^ ϕ þ ðA^ i þ B^ 1 K^ i ÞT P^ i þ P^ i ðA^ i þ B^ 1 K^ i Þ = 1T B T B: n ^ ^ ^ ; þ L L  ∑q ¼ 1 τiq T iq B C B @wA B B B n 1 @ 0

x~

n

P^ i B^ 2i  γ2 I n

1 0 1 P^ i ð^μi þ B^ 1 σ i Þ  ∑nq ¼ 1 τiq u^ iq C C x~ CB C [email protected] w Ao 0 C C 0 A 1  ∑nq ¼ 1 τ iq viq

ð59Þ By applying congruence Lemma, from (55) and (59), one concludes the following conditions: ! T P^ ϕ þ ðA^ i þ B^ 1 K^ i ÞT P^ i þ P^ i ðA^ i þ B^ 1 K^ i Þ þ L^ L^ P^ i B^ 2i o0 ð60Þ n  γ2 I

M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416

for the fuzzy rule i ði A I ξ Þ and 0 B B B @

T P^ ϕ þ ðA^ i þ B^ 1 K^ i ÞT P^ i þ P^ i ðA^ i þ B^ 1 K^ i Þ þ L^ L^ ∑nq ¼ 1 τiq T^ iq

P^ i B^ 2i

P^ i ð^μi þ B^ 1 σ i Þ  ∑nq ¼ 1 τiq u^ iq

n

 γ2 I

0

n

n

 ∑nq ¼ 1 τiq viq

nonlinear synchronous generator system and comparison results and advantages of the proposed approach are presented.

1 C C Co0 A

ð61Þ for the other fuzzy rules. By rewriting (60) and (61) as T

T

ϕ þ PA þ A P o 0 and ϕ þ PA þA P o 0, respectively, and applying Lemma 1, one concludes that for large enough constants k; α 4 0 there exist slack matrices W and F such that ! ϕ  2kP P þ ðA þkIÞT W o0 ð62Þ P þ W T ðA þ kIÞ W WT and for the fuzzy rule i ði A I ξ Þ and ! ϕ  2αP P þ ðA þ αIÞT F o0 P þ F T ðAþ αIÞ  F  FT for the other fuzzy rules, where ! T P^ i 0 P^ ϕ þ L^ L^ ϕi ¼ ; P¼ 2 0 n γ I

ð63Þ

0

!

1

;



A^ i þ B^ 1 K^ i

B^ 2i

0

0

!

ð64Þ

B B B ϕ ¼B i B B @

n

0

n

 γ2 I

n

n

q¼1

0 A

n

P^ ϕ þ L^ T L^  ∑ τiq T^ iq

B ¼@

 ∑ τiq u^ iq

1

C C C C; 0 C C n  ∑ τiq viq A q¼1

A^ i þ B^ 1 K^ i

B^ 2i

u^ i þ B^ 1 σ i

0

0

0

0

0

0

5.1. Numerical example 1 Consider the following AFS (Kim et al., 2005; Wang and Yang, 2013): Plant Rule i : If x3 ðt Þ is M i then x_ ðt Þ ¼ Ai xðt Þ þ Buðt Þ þ μi ;

0

P^ i B P¼@ 0 0

0 1 0

0

1

C 0 A; 1

i ¼ 1; 2; 3 ð66Þ

where 0

1þb

0 B  A1 ¼ A3 ¼ B @ 1 0

1 1

4  3π

0

1

0 B  A2 ¼ B @ 1 0

C  1 þ 1π C A; 1 1π

0

and 0

413

0 o ao 10;

1 1

1

0 1 0 B C B¼@0A 1

C  1  1π C A; 1 1þπ

1

3 B C μ1 ¼  μ3 ¼ @ 1 A; 1

0 ob o 10;

4 þ 3π

1þ a

0 1 0 B C μ0 ¼ @ 0 A 0

ð67Þ

The membership functions M i ; ði ¼ 1; 2; 3), are shown in Fig. 1 and the parameters a in A2 and b in A1 and A3 will take different values in order to check the feasibility and conservativeness of the stabilization analysis conditions. The matrices of S-procedure corresponding to each operation region are calculated as following: For the fuzzy rule R1 which  ð3π=2Þ r x3 r  ðπ=2Þ 0

0 B T 13 ¼ @ 0 0

0 0 0

1

0

0

C 0 A; 1

B u13 ¼ @

0

0

 1

1 C  A; 2π

 v13 ¼

 2  3π 2  5



3π 2

 

2π 5



ð68Þ

q¼1

1 C A

ð65Þ

Assume W and F as ! ! W1 W 11 0 0 W¼ ; W1 ¼ ; W2 W3 W 21 W 22 0 1 ! 0 0 F 11 F 111 0 B 0 C F ¼ @ F 21 F 22 A; F 11 ¼ F 112 F 113 F 31 F 32 F 33 Finally, substituting (64) and (65) into (58) and (59), leads to the conditions (51) and (52), respectively. The proof is completed. ▢

For the interval of the premise variable which contains the origin (the fuzzy rule R2 ), no matricesare necessary  for S-procedure. For the fuzzy rule R3 which 2π=5 r x3 r π=2 : 0 1 0 1 0 0 0 0   2π 3π B C B C 0 ; v T 33 ¼ @ 0 0 0 A; u33 ¼ @ ¼  A 33   5 2 1 2π 3π 2 5 þ 2 0 0 1 ð69Þ The parameters ϕρ ðρ ¼ 1; 2; 3Þ, defined in Assumption 2, are assumed to be as ϕ1 ¼ ϕ2 ¼ ϕ3 ¼ 1. Fig. 2 shows the stability region for Theorem 9 in Wang and Yang (2013) and Theorem 2 in this paper with α ¼ k ¼ 1. In Fig. 2, the  mark shows feasible stability region based on the method of Wang and Yang (2013) and o mark shows feasible stability region 10

5. Simulation examples

8 7 6

b

In this section, two examples are presented. The first one is presented to show the reduction of stability analysis conditions conservativeness compared to newly published works. In the second example, the nonlinear dynamics of a practical nonlinear synchronous generator system and the equivalent TS fuzzy are presented. Then, PDC controller is designed based on the TS fuzzy matrices. Finally, the designed controller is applied to the original

9

5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

9

10

a Fig. 1. The membership functions of the open-loop affine fuzzy system.

Fig. 2. Stability regions (Theorem 9 in Wang and Yang, 2013)—“  ”, the proposed approach—“o”).

M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416

based on the proposed method in this paper. Fig. 2 reveals that the proposed approach provides wider stability region compared to Wang and Yang (2013). More relaxed stability analysis conditions have been achieved because of using fuzzy Lyapunov function and the slack matrices that decoupled Lyapunov matrix from the system matrices which provide more degrees of freedom for LMI conditions. The proposed approach requires no structural restrictions on the form of Lyapunov matrices. Therefore, the new proposed approach provides more relaxed stabilization conditions compared to Wang and Yang (2013).

4 x1 x2

3

x3

2

States

414

1

0

5.2. Numerical example 2 This example illustrates the comparison of the H 1 performance of the proposed approach compared to Wang and Yang (2013). Consider the following continuous-time affine fuzzy system (Wang and yang, 2013): Plant Rule i : If x1 ðt Þ is M i ; ( x_ ðt Þ ¼ Ai xðt Þ þ B1 uðt Þ þB2i wðtÞ þ μi then ; zðt Þ ¼ Lxðt Þ

-1

-2

0

1

0

4 3π

1

B A1 ¼ A3 ¼ B @ 1 0

C  1 þ 1π C A;

1 1

0

L1 ¼ ½ 0

1

0

B A2 ¼ B @ 1 0

0 1 0 B C B C μ1 ¼  μ3 ¼ @ 1 A; μ2 ¼ @ 0 A; 1 0 0 1 0 1 0:4 0:3 B C B C B22 ¼ @ 0:5 A; B23 ¼ @ 0:6 A 0 1 3

1

1 1π

0

1

i ¼ 1; 2; 3

4 þ 3π

1

1

C  1 1π C A;

1

1 þ 1π

1

0

ð70Þ

0:3189

0:6

1

B C B21 ¼ @ 0 A; 0:1

0

ð71Þ

0 B P3 ¼ @

n

0:1680 2:2734

n

n

1 0:1427  1:6848 C A; 2:6198

0:3146

0:1635

n

2:2691

n

n

0:1393

0 B P2 ¼ @

1

5

6

7

8

9

10

Table 1 Parameters values of the generator.

0 1 0 B C B1 ¼ @ 0 A 1

0:3106 n

0:1662 2:2724

n

n

1 0:1256  1:6466 C A 2:5806

 1:6809 C A

ð72Þ

2:6148

Parameters

Values of parameter

f0 D H ω Pm Vs x0dΣ xdΣ E0q0

50 Hz 0:8 8s 1 p:u: 0:79 p:u: 1 p:u: 1:1108 p:u: 2:3108 p:u: 1:2723 p:u:

δ0 xd xd T d0 kA Rf u0

The disturbance is considered as w ¼ 15 e sin ð20πtÞ. The disturbance attenuation level is obtained as γ ¼ 0:9094 in Wang and Yang (2013). But, using the proposed approach, the optimal disturbance level is obtained as γ opt ¼ 0:1712, which shows the priority to the conditions derived based on the fuzzy Lyapunov function. In this case, Lyapunov matrices are obtained as B P1 ¼ @

4

Fig. 3. Responses of the states.

 0:1t

0

3

Time(sec)

where 0

2

X ad

x_ 3 ðt Þ ¼  þ

60 3 1:5 p:u: 0:3 p:u: 3s 10 0:0045 p:u: 7:2942  10  4 p:u: 1:3 p:u:

 x  x0 xdΣ  d d x3 ðt Þ þ E0q0 þ V s cos ðx1 ðt Þ þ δ0 Þ 0 T D0 xdΣ T D0 x0dΣ

kA xad ðu0 þ uðtÞÞ T D0 Rf

ð73Þ

where the star (n) in the symmetric Lyapunov matrices denotes the transpose of its symmetric element. Fig. 3 shows the states evolution. The proposed approach effectively can deal with disturbance input. The parameters ϕρ ðρ ¼ 1; 2; 3Þ, defined in Assumption 2, are assumed to be as ϕ1 ¼ ϕ2 ¼ ϕ3 ¼ 1.

The state variables are defined as

5.3. Application to control a nonlinear synchronous generator system

where δ is the angular position of the rotor of generator with respect to a synchronously reference which is selected here to be the infinite bus, δ0 is the value of δ under steady operating condition, ω is the angular velocity of the rotor; E0q is the electromotive force (EMF) in q-axis of generator, E0q is transient EMF in the q-axis of generator, u is the control input. The parameters of the dynamic equations are presented in Table 1. The Jacobian linearization is employed to find the affine subsystems with the chosen operation points (Kim et al., 2005). After blending the subsystems and the membership functions, the AFS is provided. Hence, the affine fuzzy modeling of synchronous

The dynamic of nonlinear synchronous generator system is proposed as following to demonstrate the effectiveness of the proposed design method (Chang et al., 2009): x_ 1 ðt Þ ¼ 2πf 0 x2 ðt Þ þ wðtÞ 0

  1 P m  xV0 s x3 ðt Þ þE0q0 sin ðx1 ðt Þ þ δ0 Þ D ω0 B dΣ C x_ 2 ðt Þ ¼  x2 ðt Þ þ @ ðx  x0 ÞV 2 A H H þ d d0 s sin ðx ðt Þ þδ Þ cos ðx ðt Þ þ δ Þ 1 0 1 0 xdΣ x dΣ

x1 ¼ δ δ0 x2 ¼ ω  ω0 x3 ¼ Eq'  Eq0'

M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416

generator system is represented as following:

415

30

Plant Rule i : If x1 ðt Þ is M i ; then x_ ðt Þ ¼ Ai xðt Þ þ B1 uðt Þ þ B2i wðtÞ þ μi 3

2

6 7  0:1 0:0563 5; A1 ¼ 4  0:0948  0:1802 0 0:6937 2 3 2 3 0 1 6 7 6 7 B21 ¼ 4 0 5; μ1 ¼ 4 0:0028 5 0:0376 0

6 B1 ¼ 4

0

314:1593

2

0 6 A2 ¼ 4  0:1009  0:3121 2 0 6 A3 ¼ 4  0:0585  0:3604

L¼½1

0

3 0  0:0975 7 5;  0:6937 3 0 7  0:1126 5;  0:6937

314:1593  0:1 0 314:1593  0:1 0

3

0

20

7 5;

0

x1(degree)

where 2

962:963

2 3 1 6 7 B22 ¼ 4 0 5; 0 2 3 1 6 7 B23 ¼ 4 0 5; 0

2 3 0 6 7 μ2 ¼ 4 0 5 0 2

0

1 B T 11 ¼ @ 0 0

0 0 0

6 7 μ3 ¼ 4  0:0139 5 0:0085

-5

0

1 0 C 0 A; 0

B u11 ¼ @

1 π  12ð  30  10Þ180 C 0 A;

 v11 ¼

0

 30π 180



 10π 180

1 0 C 0 A; 0

0

B u31 ¼ @

1 π  12ð10 þ 30Þ180 C 0 A; 0



 10π 30π  v31 ¼ 180 180

The parameters ϕρ ðρ ¼ 1; 2; 3Þ, defined in Assumption 2, are selected to be as ϕ1 ¼ ϕ2 ¼ ϕ3 ¼ 3. The Lyapunov matrices P i ði ¼ 1; 2; 3Þ are obtained as following: B P1 ¼ @

7:7891

0 B P3 ¼ @

1.5

2

n

68:6740 121650

n

n

1

1:7285  965:8728 C A;

0 B P2 ¼ @

11:6183

7:7361

82:2046

n

121670

n

n

n

 38:2006 132870

n

n

8:6641

1

1:6632  962:4531 C A:

2.5

3

3.5

4

4.5

3.5

4

4.5

Fig. 5. Response of x1 .

0.5

x 10

-3

0

ð77Þ

0

1

Time(sec)

1

-0.5

x 2(degree/sec)

0 0 0

0.5



For the interval of the premise variable which contains the origin (the fuzzy rule R2 ), no matrices are necessary for S-procedure. For the fuzzy rule R3 which 10 3 rx1 r 30 3 : 1 B T 31 ¼ @ 0 0

0

ð75Þ

ð76Þ

0

10

0

3

and the membership functions are shown in Fig. 4. The matrices of S-procedure corresponding to each operation region are calculated as following: For the fuzzy rule R1 which  30 3 rx1 r  10 3 : 0

15

5

0

0

25

ð74Þ

-1 -1.5 -2 -2.5 -3 -3.5

0

0.5

11:6083

By applying the proposed approach, a robust PDC controller is designed such that the nonlinear system be asymptotically stable with H 1 performance γ. The disturbance wðtÞ is chosen as a white noise with zero-mean and unit variance. A generalized eigenvalue problem was solved to obtain an optimal value for the disturbance attenuation level. The optimal H 1 performance γ optimal is equal to

1.5

2

2.5

3

Fig. 6. Response of x2 .

14:2681

ð78Þ

1

Time(sec)

3:3495  113100 C A

0:2216. The scaling values α and k are calculated as α ¼ 55 and k ¼ 0:5 and in this case, the controller gains are obtained as following: K 1 ¼ ð  0:0093 5:4458  0:0616Þ; σ 1 ¼ 2:2618  10  4 K 2 ¼ ð  0:0099 0:0806  0:0100Þ; σ 2 ¼ 0 K 3 ¼ ð  0:0055 5:4350  0:0670Þ; σ 3 ¼  0:0014 The states evolution and controller effort are presented in Figs. 5– 7. The proposed approach effectively can deal with disturbance input. Also the controller designed by the proposed approach can stabilize the nonlinear faster than the one in Chang et al. (2009). For example, in Chang et al. (2009), x1 converges to its equilibrium point in about 2 s but the convergence time of state x1 is about 1 s. Moreover, there is no over shoot in the system response of the closed loop original system with the designed controller.

6. Conclusions

Fig. 4. Membership functions of x1 ðtÞ.

Stability analysis of continuous-time AFS was investigated in this paper. More relaxed stabilization conditions were achieved by using nonquadratic fuzzy Lyapunov function and affine PDC

416

M.S. Sadeghi et al. / Engineering Applications of Artificial Intelligence 37 (2015) 407–416

0.05 0 -0.05

x 3(p.u)

-0.1 -0.15 -0.2 -0.25 -0.3 -0.35

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time(sec) Fig. 7. Response of x3 .

controller. Also, some ideas were utilized to formulate the stabilization conditions in terms of LMIs which can be solved by convex optimization algorithm. Diffeomorphic state transformation, introducing slack matrices, nonsingular state transformation, Sprocedure and using some lemmas help us to obtain the objective of less conservativeness stabilization condition. Afterwards, robust controller was designed based on the stability conditions to guarantee an H 1 performance level to attenuate disturbance. Finally, some comparative numerical and industrial examples were presented to demonstrate the validity and effectiveness of the proposed method. Simulation results showed that wider stability region is achieved by the proposed approach. References Abdelmalek, I., Goléa, N., Hadjili, M., 2007. A new fuzzy Lyapunov approach to nonquadratic stabilization of Takagi–Sugeno fuzzy models. Int. J. Appl. Math. Comput. Sci. 17 (1), 39–51. Boyd, S.P., 1994. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, PA. Cao, S.G., Rees, N.W., Feng, G., 1997. Analysis and design for a class of complex control systems Part I: fuzzy modelling and identification. Automatica 33 (6), 1017–1028.

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