Piecewise Controller Design for Affine Fuzzy Systems

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This paper studies the problem of state feedback controller design for a class of nonlinear systems, which are described by continuous-time affine fuzzy models. A convex piecewise affine controller design method is proposed based on a new dilated linear matrix inequality (LMI) characterization, where the system matrix is separated from Lyapunov matrix such that the controller parametrization is independent of the Lyapunovmatrix. In contrast to the existing work, the derived stabilizability condition leads to less conservative LMI characterizations and much wider scope of the applicability. Furthermore, the results are extended to H∞ state feedback synthesis. Finally, two numerical examples illustrate the superiority and effectiveness of the new results.

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Piecewise Controller Design for Affine Fuzzy Systems

  1. 1. Piecewise controller design for affine fuzzy systems via dilated linear matrix inequality characterizations Huimin Wang a , Guang-Hong Yang a,b,n a College of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China b State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, PR China a r t i c l e i n f o Article history: Received 4 November 2011 Received in revised form 5 April 2012 Accepted 28 June 2012 Available online 21 July 2012 Keywords: Affine fuzzy systems Stabilizability Slack variable Linear matrix inequalities (LMIs) State feedback a b s t r a c t This paper studies the problem of state feedback controller design for a class of nonlinear systems, which are described by continuous-time affine fuzzy models. A convex piecewise affine controller design method is proposed based on a new dilated linear matrix inequality (LMI) characterization, where the system matrix is separated from Lyapunov matrix such that the controller parametrization is independent of the Lyapunov matrix. In contrast to the existing work, the derived stabilizability condition leads to less conservative LMI characterizations and much wider scope of the applicability. Furthermore, the results are extended to H1 state feedback synthesis. Finally, two numerical examples illustrate the superiority and effectiveness of the new results. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction In the nonlinear control area, an important approach to nonlinear control system design is to model the considered nonlinear systems as Takagi and Sugeno (T–S) fuzzy systems [1]. By incorporating linguistic information from human experts and ‘‘blending’’ some locally linear systems, the T–S fuzzy model has been proved to be a well universal approximator [2]. As a result, the conventional linear system theory can be applied to the analysis and synthesis of the class of nonlinear control systems (see [3–11] and the references therein). Usually, T–S fuzzy systems can be classified into linear fuzzy models and affine fuzzy models [12]. The main difference between them is that the latter considers a constant bias term in each fuzzy rule, which makes the function approximation capabilities of T–S fuzzy systems be improved substantially [6,13]. Moreover, the stability theory of affine fuzzy models can be extended to the linear ones [14]. As such, the research on affine fuzzy systems is expected to be interesting and significant. Recently, an increasing amount of work has been devoted to analysis of affine fuzzy systems [12,14,15]. These results considered the structural information in the rule base and introduced S-proce- dure to decrease the conservatism of the stability analysis. However, due to the constant bias term and the introduced S-procedure, most existing results on control synthesis for affine fuzzy systems are obtained in the form of bilinear matrix inequalities (BMIs) [12,16–18], and it is impossible to convert these BMIs to LMIs by simply using the inverse of the Lyapunov matrix P. As we know, to deal with such non-convex problems, we need to design an iterative LMI algorithm and obtain an initial feasible solution, which is usually conservative and even impracticable. Although a lot of efforts have been spent on improving these weaknesses, the efficient and effective method has not yet to be developed. In [19], Kim et al. obtained a convex state feedback controller design condition through limiting the Lyapunov matrix P to a diagonal structure. Although the result is more solvable, this constraint may lead to difficulty since such a Lyapunov matrix might not exist in many cases, especially for highly nonlinear complex systems. In addition, it is also noted that the results are obtained based on the assumption that the input matric B is a common one, i.e., _xðtÞ ¼ Pr i ¼ 1 hiðxðtÞÞðAixðtÞþ BuðtÞ þmiÞ. For the case Bi aB, the convexifying techniques are not efficient. This is a much simpler system, and leads to much narrower scope of the applicability. In [20], the authors proposed two approaches to robust H1 output feedback controller design for affine fuzzy systems. It is shown that the synthesis conditions can be formulated in the terms of LMIs that can be efficiently solved by interior-point methods. To the authors’ best knowledge, few efficient attempts have been made on convex state feedback controller design for general continuous-time affine fuzzy systems, which motivates us for this study. In past years, the technique of decoupling the Lyapunov matrix and the system matrix by adding slack variables has been developed, and it can effectively decrease the conservatism in multi-objective control problems [21–23]. In [24], by introducing slack variables, the authors derived a convex stabilizability condition, and there is no need to impose any structural constraint on the Lyapunov matrix. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.06.014 n Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China. Tel.: þ86 24 83681939. E-mail addresses: wanghuimin702@yahoo.cn (H. Wang), yangguanghong@ise.neu.edu.cn, yang_guanghong@163.com (G.-H. Yang). ISA Transactions 51 (2012) 771–777
  2. 2. However, the obtained method can only suit for the system with a common input matrix. In this paper, a convex piecewise affine controller design method is proposed based on a new dilated linear matrix inequality (LMI) characterization, where the system matrix is separated from Lyapunov matrix such that the controller parame- trization is independent of the Lyapunov matrix. It is noted that the controller design method is obtained in the formulation of LMIs in conjunction with a search of scaling parameters, which can provide more relaxed conditions and deal with more general systems than the existing results. Furthermore, an extended H1 performance analysis of a class of affine fuzzy systems is presented, and the H1 controller synthesis condition is derived. The structure of this paper is as follows: following the introduc- tion, the system description and the problem under consideration are given in Section 2. In Section 3, lemmas which are used throughout are proposed. A quadratic stabilizability condition in the form of LMIs is presented in Section 4. Section 5 proposes an extended H1 controller design of affine fuzzy systems. Numerical examples are given in Section 6 to show the superiority and effectiveness of proposed method. Finally, conclusions are drawn in Section 7. Notation: For a symmetric matrix M, M40 (Mo0) means that it is positive definite (negative definite). MT denotes the transpose of matrix M. The symbol n is used in some matrix expressions to denote the transposed elements in the symmetric positions of a matrix. 2. System description and problem statement 2.1. System description The following continuous-time affine T–S system can be used to represent a complex nonlinear system with both fuzzy infer- ence rules and local analytic models as follows Ri : If x1ðtÞ is Mi1 and . . . and xnðtÞ is Min; Then _xðtÞ ¼ AixðtÞþBiuðtÞþmi ð1Þ where xT ðtÞ ¼ ðx1ðtÞ x2ðtÞ Á Á Á xnðtÞÞ; Mi1,Mi2, . . . ,Min are fuzzy vari- ables, Ri ði ¼ 1; 2, . . . ,rÞ denotes the ith fuzzy rule and r is the number of rules. Ai, Bi, mi are constant matrices with appropriate dimensions. In (1), it can be seen that the local subsystem includes a constant affine term mi, which can approximate the original non- linear system more accurately [12]. By using the fuzzy inference method with a singleton fuzzifier, product inference, and center average defuzzifiers, the overall affine T–S model is represented as _xðtÞ ¼ Pr i ¼ 1 wiðxðtÞÞðAixðtÞþBiuðtÞþmiÞ Pr i ¼ 1 wiðxðtÞÞ ¼ Xr i ¼ 1 hiðxðtÞÞðAixðtÞþBiuðtÞþmiÞ, ð2Þ where wiðxðtÞÞ ¼ Yn j ¼ 1 MijðxjðtÞÞ: Denote hiðxðtÞÞ ¼ wiðxðtÞÞ Pr i ¼ 1 wiðxðtÞÞ then 0rhiðxðtÞÞr1, Xr i ¼ 1 hiðxðtÞÞ ¼ 1: MijðxjðtÞÞ is the grade of membership of xj(t) in Mij, and hiðxðtÞÞ is said to be the normalized membership function. In this paper, the membership functions of the fuzzy proposi- tions are trapezoidal, and we will partition the state-space into operating regions and interpolation regions [25]. In each operat- ing region, there exists some l such that hlðxðtÞÞ ¼ 1, and all other membership functions evaluate to zero and the dynamic of the system is given by _xðtÞ ¼ AlxðtÞþBluðtÞþml. In between operating regions, there are interpolation regions where 0ohlðxðtÞÞo1, and the system dynamics are given by a convex combination of several affine systems. This decomposition of the state-space into operating and interpolation regions will be central in our analysis. Here, fSigi AF DRs denotes a polyhedral partition of the state, and F denotes the set of cell indexes. For each cell Si, the set K(i) contains the indexes for the system matrices used in the interpolation within that cell. For operating regions, K(i) contains a single element. Furthermore, we let F0 DF be the set of indexes for cells that contain origin and F1 DF be the set of indexes of the cells that do not contain the origin. Hence, in each cell, the global system in (2) can be expressed by a blending of mAKðiÞ subsystems: _xðtÞ ¼ ~AixðtÞþ ~BiuðtÞþ ~mi, xðtÞASi, iAF, ð3Þ where ~Ai ¼ X mAKðiÞ hmðxðtÞÞAm, ~mi ¼ X mA KðiÞ hmðxðtÞÞmm, ~Bi ¼ X mA KðiÞ hmðxðtÞÞBm, hmðxðtÞÞ40, X mA KðiÞ hmðxðtÞÞ ¼ 1: 2.2. Problem statement In this paper, we assume that the states x(t) of system (3) can be measured and used to design controllers. The piecewise state feedback controller is designed as follows: uðtÞ ¼ KixðtÞþsi, xðtÞASi, iAF: ð4Þ The value of si in (4) is assumed to be 0 while the cells contain origin. Applying the state feedback controller (4) to system (3), the resulting closed-loop system is given by _xðtÞ ¼ ð ~Ai þ ~BiKiÞxðtÞþð ~Bisi þ ~miÞ xðtÞASi, iAF: ð5Þ Then, the purpose of this paper is to design a piecewise state feedback controller described by (4) such that the resulting closed-loop system, which is given in (5), is quadratically stable. 3. Preliminaries To facilitate control system design, the following lemmas are presented and will be used in the later developments. Lemma 1. Let F be a symmetric matrix, P be a positive-definite matrix. The following statements are equivalent: (a) FþPAþAT Po0. (b) For a large enough constant a40, there exist matrices F, P such that FÀ2aP PþðAþaIÞT F PþFT ðAþaIÞ ÀFÀFT " # o0: ð6Þ Proof. ðaÞ ) ðbÞ: If the statement (a) is satisfied, there exists a sufficiently large positive scalar a1 such that 8a4a1 fþPAþAT Pþ 1 2a AT PAo0: H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777772
  3. 3. Rewrite the above inequality as FÀ2aPþ a 2 Pþ 1 a ðAþaIÞT P PÀ1 Pþ 1 a PðAþaIÞ o0 by applying Schur complement, we get FÀ2aP Pþ 1 a ðAþaIÞT P Pþ 1 a PðAþaIÞ À 2 a P # o0: By taking F ¼ ð1=aÞP, then the statement (b) is obtained. ðbÞ ) ðaÞ: Multiplying (6) by ½I ðAþaIÞT Š from the left and by its transpose from the right leads to the statement (a). Then the proof is completed. Remark 1. (1) Note that the statement (b) remains sufficient for the statement (a) by using the property of congruence with the matrix ½I ðAþaIÞT Š while some constraints are imposed on F. (2) This lemma provides a new dilated LMI characterization, and while Fo0, through setting a ¼ 0, it can be reduced to the result in [23]. Lemma 2. The continuous affine fuzzy system (3) is quadratically stable with u 0 in the large if there exist a common positive definite matrix P ¼ PT and tiq Z0 ðiAF1; q ¼ 1; 2, . . . ,nÞ such that AT mPþPAm o0 ð7Þ for iAF0, mAKðiÞ and AT mPþPAmÀ Pn q ¼ 1 tiqTiq PmmÀ Pn q ¼ 1 tiquiq mT mPÀ Pn q ¼ 1 tiquT iq À Pn q ¼ 1 tiqviq 2 4 3 5o0 ð8Þ for iAF1, mAKðiÞ, where Tiq, uiq, viq are defined in Appendix such that FiqðxðtÞÞ xT ðtÞTiqxðtÞþ2uT iqxðtÞþviq o0 while xðtÞASi, iAF. Proof. The proof procedure is similar to Theorem 1 in [12], and it is omitted here. By Lemma 1, the following novel quadratic stability analysis result is derived, which is equivalent to Lemma 2. Lemma 3. The conditions of Lemma2 are feasible if and only if there exist a common positive definite matrix P ¼ PT , matrices Wi, Fi (iAF) with appropriate dimensions, and scalars tij Z0 ðiAF1; j ¼ 1; 2, . . . ,nÞ such that, for large enough positive constants a,k À2kP PþðAm þkIÞT Wi PþWT i ðAm þkIÞ ÀWiÀWT i # o0 ð9Þ for iAF0, mAKðiÞ and FiÀ2aP P þðbAm þaIÞT Fi P þFT i ðbAm þaIÞ ÀFiÀFT i 2 4 3 5o0 ð10Þ for iAF1, mAKðiÞ, where Fi ¼ À Pn j ¼ 1 tijTij À Pn j ¼ 1 tijuij À Pn j ¼ 1 tijuT ij À Pn j ¼ 1 tijvij 2 4 3 5, P ¼ P 0 0 1 , bAm ¼ Am mm 0 0 : ð11Þ Proof. Rewriting (8) as Fi þP bAm þ bA T mP o0, where Fi,bAm,P are defined as (11), and applying Lemma 1, we can obtain this conclusion immediately. 4. Quadratic stabilizing controller design From Lemma 3, the novel sufficient condition under which the closed-loop fuzzy system (5) is quadratically stable is given in the following forms. Lemma 4. The continuous affine fuzzy system (5) is quadratically stable in the large if there exist P40, matrices Fi, Wi ðiAFÞ with appropriate dimensions, and scalars tij Z0 ðiAF1; j ¼ 1; 2, . . . ,nÞ such that, for large enough positive constants a,k À2kP PþðAm þBmKi þkIÞT Wi PþW T i ðAm þBmKi þkIÞ ÀWiÀW T i 2 4 3 5o0 ð12Þ for iAF0, mAKðiÞ and FiÀ2aP P þðAim þaIÞT Fi P þFT i ðAim þaIÞ ÀFiÀFT i # o0 ð13Þ for iAF1, mAKðiÞ, where Aim ¼ Am þBmKi Bmsi þmm 0 0 ð14Þ and Fi,P are defined as (11). It can be seen that the conditions of Lemma 4 are with non- convex forms because of controller gains on one side and free matrices Fi (or Wi) on the other side. Due to the S-procedure, i.e., Fi, it is difficult to convert the non-convex conditions such as the inequality (13) to convex ones by the equivalent transformation. To overcome this difficulty, the following assumption is consid- ered here: Assumption 1. Assume that Bl,l ¼ 1; 2, . . . r, are of full column rank, and let invertible matrices Tl,l ¼ 1; 2, . . . r, such that TlBl ¼ ImÂm 0 for l ¼ 1; 2, . . . r: ð15Þ Remark 2. For each Bl, the corresponding Tl generally is not unique. A special Tl can be obtained by Tl ¼ ðBT l BlÞÀ1 BT l P # and P is a matrix composed of the rows which are mutually independent and perpendiculars to the columns of Bl. Based on Lemma 4 and Assumption 1, a new quadratic stabilizability condition for the affine fuzzy system (3) is obtained as follows. Theorem 1. The continuous affine fuzzy system (3) is quadratically stabilizable in the large by the piecewise controller (4) if there exist P ¼ PT 40, matrices Wi, Fi1, Fi2, Fi3 with appropriate dimensions, where Wi ¼ Wi1 0 Wi2 Wi3 # , Fi1 ¼ Fi11 0 Fi21 Fi22 # H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777 773
  4. 4. and Yi ARmÂn ,Vi ARmÂ1 , scalars tij Z0 ðiAF; j ¼ 1; 2, . . . ,nÞ such that, for large enough positive constants a,k À2kP PþAT mTT mWi þkT T mWi þ Yi 0 T n ÀTT mWiÀWT i Tm 2 6 6 4 3 7 7 5o0 ð16Þ for iAF0, mAKðiÞ, and À Xn j ¼ 1 tijTijÀ2aP À Xn j ¼ 1 tijuij PþAT mTT mFi1 þ Yi 0 T þaTT mFi1 0 n À Xn j ¼ 1 tijvijÀ2a Vi 0 T þmT mTT mFi1 þaFi2 1þaFi3 n n ÀTT mFi1ÀFT i1Tm ÀFT i2 n n n ÀFi3ÀFT i3 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 o0 ð17Þ for iAF1, mAKðiÞ. The local gains of the fuzzy controller are given by Ki ¼ WÀT i1 Yi, iAF0; Ki ¼ FÀT i11Yi, si ¼ FÀT i11Vi, iAF1: Proof. Assuming Wi ¼ X lA KðiÞ flTT l Wi, Fi ¼ P lA KðiÞflTT l Fi1 0 Fi2 Fi3 # with appropriate dimensions, where Wi ¼ Wi1 0 Wi2 Wi3 # , Fi1 ¼ Fi11 0 Fi21 Fi22 # , and the function fl is defined as fl ¼ 1, l ¼ m, 0, else: ( Then we have TmBm ¼ ImÂm 0ðnÀmÞÂm # : Substituting them to (12) and (13), the conditions (16) and (17) are obtained. Remark 3. It is noted that in [19], the Lyapunov matrix P is limited to a diagonal structure. In this paper, by applying the proposed decoupling technique, the derived result does not need to impose any constraint on the Lyapunov matrix P. In addition, the results in [19] are obtained based on the assumption that the input matric B is a common one, this is a much simpler system, and leads to much narrower scope of the applicability. By introducing extra slack variables Fi, Wi, the proposed method in this paper can deal with more general systems whose input matrices are uncommon, and the controller design conditions are obtained in the formulation of linear matrix inequalities. 5. H1 controller design In this section, we will investigate the H1 control synthesis problem for a class of affine fuzzy systems. The objective of this section is to design a suitable controller based on the approach proposed in the above section, such that the induced L2-norm of the operator from disturbance w(t) to the controlled output z(t) is less than under zero initial conditions JzðtÞJ2 ogJwðtÞJ2 for all nonzero wðtÞAL2. Consider the following affine fuzzy system: _xðtÞ ¼ Xr l ¼ 1 hlðxðtÞÞðAlxðtÞþml þB1luðtÞþB2lwðtÞÞ, zðtÞ ¼ LxðtÞ, ð18Þ where xðtÞARn is the state vector, uðtÞARp is the control input, wðtÞARm is the disturbance input which is assumed to belong to L2½0,1Þ, and zðtÞARq is the controlled output. ½AlŠnÂn, ½mlŠnÂ1, ½B1lŠnÂp, ½B2lŠnÂm, ½LŠqÂn ðl ¼ 1; 2, . . . ,rÞ are constant matrices, and B1ls are of full column rank. Similar to the partition in Section 2, the global system in (18) can be expressed by a blending of mAKðiÞ subsystems _xðtÞ ¼ ~AixðtÞþ ~mi þ ~B1iuðtÞþ ~B2iwðtÞ, zðtÞ ¼ LxðtÞ, xðtÞASi, iAF, ð19Þ where ~Ai ¼ X mAKðiÞ hmðxðtÞÞAm, ~mi ¼ X mA KðiÞ hmðxðtÞÞmm, ~B1i ¼ X mAKðiÞ hmðxðtÞÞB1m ~B2i ¼ X mAKðiÞ hmðxðtÞÞB2m, hmðxðtÞÞ40, X mAKðiÞ hmðxðtÞÞ ¼ 1: Lemma 5. The affine fuzzy system (19) with u 0 is quadratically stable with a guaranteed H1 disturbance attenuation level g if there exist a symmetric matrix P40 and scalars tiq Z0 ðiAF; q ¼ 1; 2, . . . ,nÞ such that AT mPþPAm þLT L PB2m BT 2mP Àg2 I # o0 ð20Þ for iAF0, mAKðiÞ, and AT mPþPAm þLT LÀ Pn q ¼ 1 tiqTiq PB2m PmmÀ Pn q ¼ 1 tiquiq n Àg2 I 0 n n À Pn q ¼ 1 tiqviq 2 6 6 4 3 7 7 5o0 ð21Þ for iAF1, mAKðiÞ; where Tiq, uiq, viq are defined as in Appendix. Proof. Consider the following Lyapunov function: VðxðtÞÞ ¼ xT ðtÞPxðtÞ: It is well known that it suffices to show the following inequality: _V ðxðtÞÞþzT ðtÞzðtÞÀg2 wT ðtÞwðtÞo0 to prove that the affine fuzzy system (19) is asymptotically stable with a given H1 performance g under zero initial conditions. Based on the above Lyapunov function, we get _V ðxðtÞÞþzT ðtÞzðtÞÀg2 wT ðtÞwðtÞ ¼ xðtÞ wðtÞ 1 2 6 4 3 7 5 T ~A T i PþP ~Ai þLT L P ~B2i P ~mi n Àg2 I 0 n n 0 2 6 6 4 3 7 7 5 xðtÞ wðtÞ 1 2 6 4 3 7 5: Then, by taking into consideration the partition information with iAF1, we have _V ðxðtÞÞþzT ðtÞzðtÞÀg2 wT ðtÞwðtÞr xðtÞ wðtÞ 1 2 6 4 3 7 5 T H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777774
  5. 5. ~A T i PþP ~Ai þLT LÀ Pn q ¼ 1 tiqTiq P ~B2i P ~miÀ Pn q ¼ 1 tiquiq n Àg2 I 0 n n À Pn q ¼ 1 tiqviq 2 6 6 6 4 3 7 7 7 5 xðtÞ wðtÞ 1 2 6 4 3 7 5: By expanding the fuzzy-basis functions, then the condition (21) is obtained. Similarly, for the case iAF0, the condition (20) can be easily obtained where ~mi ¼ 0 and the S-procedure is not involved. Theorem 2. The affine fuzzy system (19) with the state feedback control law (4) is quadratically stabilizable with a guaranteed H1 disturbance attenuation level g if there exist P40, matrices Wi, Fi11, Fi21, Fi22, Fi31, Fi32, Fi33 with appropriate dimensions, where Fi11 ¼ Fi1 0 Fi2 Fi3 # , Wi ¼ Wi1 0 Wi2 Wi3 # , Yi ARmÂn , Vi ARmÂ1 and scalars tiq Z0 ðiAF; q ¼ 1; 2, . . . ,nÞ such that, for large enough positive constants a,k xi 11 n n n n n 0 Àg2 IÀ2aI n n n n xi 31 0 xi 33 n n n xim 41 xim 42 xim 43 ÀTT mFi11ÀFT i11Tm n n 0 IþaFT i22 aFT i32 ÀFi21 ÀFi22ÀFT i22 n 0 0 1þaFT i33 ÀFi31 ÀFi32 ÀFi33ÀFT i33 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 o0 ð22Þ for iAF1, mAKðiÞ, and LT LÀ2kP n n n 0 Àg2 IÀ2kI n n Lim 31 WT i1TmB2m þkW T i2 ÀTT mWi1ÀWT i1Tm n 0 IþkW T i3 ÀWi2 ÀWi3ÀWT i3 2 6 6 6 6 6 4 3 7 7 7 7 7 5 o0 ð23Þ for iAF0, mAKðiÞ, where xi 11 ¼ LT LÀ Xn q ¼ 1 tiqTiqÀ2aP, xi 31 ¼ À Xn q ¼ 1 tiquT iq, xi 33 ¼ À Xn q ¼ 1 tiqviqÀ2a, xim 41 ¼ PþaFT i11Tm þFT i11TmAm þ Yi 0 , xim 43 ¼ FT i11Tmmm þ Vi 0 þaFT i31, xim 42 ¼ FT i11TmB2m þaFT i21, Lim 31 ¼ PþkW T i1Tm þWT i1TmAm þ Yi 0 : The controller gains are given by Ki ¼ WÀT i1 Yi, iAF0; Ki ¼ FÀT i1 Yi, si ¼ FÀT i1 Vi, iAF1. Proof. The proving procedure is similar to Theorem 1, thus its details are omitted here. Remark 4. When k and a in (16), (17) and (22), (23) are set to be fixed parameters, the problems become convex and can be solved by employing the LMI Toolbox. To find the optimal values of corresponding parameters, in this paper, we will first solve the feasibility problem of LMIs (16), (17) and (22), (23) by using LMI Toolbox under a set of initial scaling parameters. Then, for the optimization problem, applying a numerical optimization algo- rithm, such as the program fminsearch, and then a locally convergent solution to the problem is obtained. Remark 5. The results given in this paper are obtained based on the assumption that the input matric Bls are of full column rank, which leads to much wider scope of the applicability than the common input matrix. Nevertheless, if there exists invertible matrices Tl,l ¼ 1; 2, . . . r, such that TlBl ¼ IncÂnc 0 0 0 for l ¼ 1; 2, . . . ,r, where nc om, which means that the input matrices are not full column rank, and then a novel piecewise affine state feedback controller is desirable with the form uðtÞ ¼ ~KixðtÞþ ~si, xðtÞASi, iAF, ð24Þ where ~Ki ¼ Knc i 0 # , ~si ¼ snc i 0 # : The corresponding state feedback control synthesis method is proposed as follows. Corollary 1. The continuous affine fuzzy system (3) is quadratically stabilizable in the large by the piecewise controller (24) if there exist P ¼ PT 40, matrices Wi, Fi1, Fi2, Fi3 with appropriate dimensions, where Wi ¼ Wi1 0 Wi2 Wi3 # , Fi1 ¼ Fi11 0 Fi21 Fi22 # and Yi ARncÂn , Vi ARncÂ1 , scalars tij Z0 ðiAF; j ¼ 1; 2, . . . ,nÞ such that, for large enough positive constants a,k, the linear matrix inequalities in (16) and (17) are satisfied, and the local gains of the fuzzy controller are given by Knc i ¼ WÀT i1 Yi, iAF0; Knc i ¼ FÀT i11Yi, snc i ¼ FÀT i11Vi, iAF1: 6. Examples 6.1. Example 1 The following example illustrates the merits of the new stability condition (Theorem 1) compared to the approach in [19]. Consider a continuous affine fuzzy plant composed of the following three rules: Ri: If x3ðtÞ is Mi, then _xðtÞ ¼ AixðtÞþBuðtÞþmi; i ¼ 1; 2,3 where A1 ¼ A3 ¼ 0 1þb 4À 2 p ðEþ1Þ À1 À1 À1þ 2 p E 0 1 1À 2 p E 2 6 6 4 3 7 7 5, m1 ¼ Àm3 ¼ À2ðEþ1Þ 2E À2E 2 6 4 3 7 5, B ¼ 0 0 1 2 6 4 3 7 5, A2 ¼ 0 1þa 4þ 2 p ðEþ1Þ À1 À1 À1À 2 p E 0 1 1þ 2 p E 2 6 6 4 3 7 7 5, m2 ¼ 0 0 0 2 6 4 3 7 5, xðtÞ ¼ ðx1ðtÞ x2ðtÞ x3ðtÞÞT , and E is set to 0.5. The membership functions M1,M2 and M3 are depicted in Fig. 1. The parameters a in A2, and b in A1,A3, will take values in a prescribed grid, in order H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777 775
  6. 6. to check the feasibility of the associated fuzzy control synthesis problem under two different approaches. With the space partition defined in this paper, we can have the cells S1 ¼ xðtÞ9À 3p 2 rxðtÞrÀ p 2 , S2 ¼ xðtÞ9À p 2 rxðtÞrÀ 2p 5 , S3 ¼ xðtÞ9À 2p 5 rxðtÞr 2p 5 , S4 ¼ xðtÞ9 2p 5 rxðtÞr p 2 , S5 ¼ xðtÞ9 p 2 rxðtÞr 3p 2 : As stated earlier, the piecewise controller is designed as follows: uðtÞ ¼ KixðtÞþsi ði ¼ 1; 2,3; 4,5Þ and s3 ¼ 0 because of F0 ¼ f3g in this example. Fig. 2 shows the parameter regions where the stability of the fuzzy control system is ensured by respectively using the controller synthesis method in [19] and the method proposed in this paper with a ¼ k ¼ 1. In this figure, the  mark indicates the existence of feasible stabilizing regulators proved by [19] (and, of course, also by Theorem 1 in this paper); the J mark indicates parameter values for which stabilizability is proved in this paper, but not in [19]. It is apparently seen that our stabilization region is larger than the results in [19]. 6.2. Example 2 In this section, the following affine fuzzy system with uncom- mon input matrices is considered. It is noted that, the approaches given in [19,24] cannot be applied for this class of systems. Ri: If x1ðtÞ is Mi, Then _xðtÞ ¼ AixðtÞþmi þB1iuðtÞþB2iwðtÞ, zðtÞ ¼ LxðtÞ, i ¼ 1; 2,3, where A1 ¼ A3 ¼ 0:9817 1 0 À0:5317 À1 À1 3:0451 1:7320 0 2 6 4 3 7 5, m1 ¼ Àm3 ¼ À0:9831 1:0347 À2:5836 2 6 4 3 7 5, A2 ¼ 1:3183 1 0 À1:2883 À1 À1 4:9549 1 0 2 6 4 3 7 5, m2 ¼ 0 0 0 2 6 4 3 7 5, B11 ¼ B13 ¼ 1 0 0 2 6 4 3 7 5, B12 ¼ 1 0:5 0 2 6 4 3 7 5, L ¼ ½0 1 0Š, B21 ¼ 0:2 0 0:1 2 6 4 3 7 5, B22 ¼ 0:1 0:2 0 2 6 4 3 7 5, B23 ¼ 0:3 0:1 1 2 6 4 3 7 5: xðtÞ ¼ ðx1ðtÞ x2ðtÞ x3ðtÞÞT . The membership functions M1,M2 and M3 are depicted in Fig. 3. The objective is to design a piecewise state feedback controller in the form of (4) such that the resulting closed-loop system is asymptotically stable with H1 performance g under wðtÞ ¼ 15eÀ0:1t sinð20ptÞ and zero initial conditions. By applying Theorem 2 and the fminsearch, we can get a piecewise state feedback controller with the controller gains are given as follows. Moreover, the optimal scaling Fig. 1. Membership functions. -2 -1 0 1 2 3 4 0 1 2 3 4 5 6 a b Fig. 2. Stabilization region based on Theorem 3 in [19] (  ) and Theorem 1 in this paper with a ¼ k ¼ 1 (  ,J). Fig. 3. Membership functions. 0 5 10 15 20 25 30 35 40 45 50 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 x(1) x(2) x(3) Fig. 4. State responses of the closed-loop system. H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777776
  7. 7. parameters are obtained as a ¼ 1:3710 and k¼0.8275 with the initial values a ¼ k ¼ 0:5, and the optimal H1 performance g ¼ 0:8449. K1 ¼ ½À7:8209À1:3782À0:1890Š, s1 ¼ 0:9831, K2 ¼ ½À21:7245À2:3324À0:8507Š, s2 ¼ 0:6985, K3 ¼ ½À7:9316À1:3810À0:1591Š, s3 ¼ 0, K4 ¼ ½À21:8990À2:3496À0:8616Š, s4 ¼ À0:7012, K5 ¼ ½À7:9193À1:3906À0:2011Š, s5 ¼ À0:9831: To show the effectiveness of the obtained results, simulations have been carried out. Fig. 4 shows the state responses of the corresponding closed-loop system under initial conditions xð0Þ ¼ ½À1:5 2 3ŠT . 7. Conclusion In this paper, the problem of control synthesis for a class of continuous-time affine fuzzy systems has been discussed. First, a lemma is provided to decouple the system matrix and the Lyapunov matrix, and then a new stability analysis condition based on the lemma is presented and the corresponding controller design method is obtained in the formulation of LMIs in conjunction with a search of scaling parameters, which can provide more relaxed conditions and deal with more general systems than the existing results. Furthermore, an extended H1 performance analysis of a class of affine fuzzy systems is presented, and the H1 controller synthesis condition is derived. Finally, two examples are used to illustrate the superiority and effectiveness of the proposed method. Although this paper addressed the convex quadratic stability and stabilizability of the affine fuzzy system, the common Lyapunov function could lead to very conservative results. To reduce the conservatism, the results on dynamic output feedback controller design for discrete-time affine fuzzy systems based on piecewise Lyapunov function have been obtained in [26], and the works on analogous design and synthesis method based on piecewise Lyapunov function for continuous-time affine fuzzy system are under investigation. Acknowledgement This work was supported in part by the Funds for Creative Research Groups of China (no. 60821063), National 973 Program of China (Grant no. 2009CB320604), the Funds of National Science of China (Grant no. 60974043), the Funds of Doctoral Program of Ministry of Education, China (20100042110027). Appendix The range of x(t) is represented by the following system of at most n linear inequalities while xðtÞASi: for x1ðtÞ, x1ðtÞrai1 or x1ðtÞZbi1 or ai1 rx1ðtÞrbi1 for x2ðtÞ, x2ðtÞrai2 or x2ðtÞZbi2 or ai2 rx2ðtÞrbi2 ^ for xn(t), xnðtÞrain or xnðtÞZbin or ain rxnðtÞrbin Thus, the S-procedure is obtained with FiqðxðtÞÞ xT ðtÞTiqxðtÞþ 2uT iqxðtÞþviq r0, and the structure of matrices Tiq, uiq, viq is similar to [12]. References [1] Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics 1985;15(1):116–32. [2] Tanaka T, Wang HO. Fuzzy control systems design and analysis: a linear matrix inequality approach. New York: Wiley; 2001. [3] Wang HO, Tanaka K, Griffin M. Parallel distributed compensation of nonlinear systems by Takagi and Sugeno fuzzy model. In: Proceedings of the FUZZ- IEEE’95, Yokohama, Japan; 1995. p. 531–8. [4] Sala A, Arin˜o C. 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