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1 King s Research Portal DOI:.49/iet-cta.7.88 Document Version Peer reviewed version Link to publication record in King's Research Portal Citation for published version (APA): Song, G., Lam, H. K., & Yang, X. (7). Membership-function-dependent stability analysis of interval type- polynomial fuzzy- model-base control systems. Iet Control Theory And Applications, (7), Citing this paper Please note that where the full-text provided on King's Research Portal is the Author Accepted Manuscript or Post-Print version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version for pagination, volume/issue, and date of publication details. And where the final published version is provided on the Research Portal, if citing you are again advised to check the publisher's website for any subsequent corrections. General rights Copyright and moral rights for the publications made accessible in the Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognize and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the Research Portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the Research Portal Take down policy If you believe that this document breaches copyright please contact librarypure@kcl.ac.uk providing details, and we will remove access to the work immediately and investigate your claim. Download date:. Mar. 9

2 IET Research Journals Regular Paper Membership-Function-Dependent Stability Analysis of Interval Type- Polynomial Fuzzy-Model-Based Control Systems Ge Song H. K. Lam Xiaozhan Yang 3 3 Department of Informatics, King s College London, Strand, London WCR LS, UK ge.song@kcl.ac.uk ISSN doi: Abstract: In this paper, the stability analysis for interval type- (IT) polynomial fuzzy-model-based (PFMB) control system using the information of membership functions is investigated. In order to tackle uncertainties, IT membership functions are used in the IT polynomial fuzzy model and IT polynomial fuzzy controller. To improve the design flexibility and reduce the implementation costs, the IT polynomial fuzzy controller does not need to share the same premise membership functions nor the same number of rules with the IT polynomial fuzzy model. The stability of IT PFMB control system is investigated based on the Lyapunov stability theory and both sets of membership function independent (MFI) and membership function dependent (MFD) stability conditions are derived on the basis of the sum-of-squares (SOS) approach. To make the stability conditions membership function dependent, the boundary information of IT membership functions is used in the stability analysis. To extract richer information of IT membership functions, the operating domain is partitioned into sub-domains. In each sub-domain, the boundary information of IT membership functions and those of the upper and lower membership function are obtained. Furthermore, to further relax the conservativeness, a switching polynomial fuzzy controller, together with the informations obtained in each sub-domain, is employed in investigating the stability analysis. Numerical Examples and simulation results are given to demonstrate the validity of MFD and MFD switching methods. Introduction The Takagi-Sugeno (T-S) fuzzy model, which was firstly proposed in, ], is an effective method to represent the nonlinear systems. It represents the nonlinear plant by an average weighted summation of local linear systems. The weights of local linear systems, defined by membership functions 3], embed the system s nonlinearity. In order to control the nonlinear system representing by T-S fuzzy model, a fuzzy controller, which is the average weighted summation of local linear controllers, is employed. In general, there are three types of fuzzy controllers 4 7], e.g. parallel distributed compensation (PDC) fuzzy controller, in which the fuzzy controller shares the same premise membership functions as the fuzzy model; partially matched fuzzy controller, in which it has the same number of rules as but different premise membership functions from the fuzzy model; and imperfectly matched fuzzy controller, in which neither the number of rules nor the premise membership functions are the same as the fuzzy model. In terms of imperfectly matched fuzzy controller, since it does not need to share the same premise membership functions with the fuzzy model and the number of rules can be freely chosen, design flexibility and robustness property can be enhanced 5]. Connecting a fuzzy controller with the nonlinear plant (represented by the T-S fuzzy model) in a closed loop forms the T-S fuzzy-model-based (FMB) control system. The stability of T-S FMB control system can be analyzed and guaranteed by stability conditions in terms of linear-matrix inequalities (LMIs) obtained on the basis of quadratic Lyapunov function. The T-S FMB control system faces great success and has a wide range of applications on, to name a few, tracking control systems, time-delay control systems, chaotic control systems etc. 8 4]. However, issues related to stability analysis and control design still exist on using the T-S FMB control systems 5], such as stability conditions are non-convex, or the analysis results are too conservative. In the above discussion, since the membership functions contain no uncertainty information, the control system cannot deal with the system uncertainties directly. Once uncertainties appear, the grades of membership will become uncertain in value leading to conservative stability conditions when uncertainty is not considered in the stability analysis. Recently, the T-S fuzzy model is extended to a more general polynomial fuzzy model 6 9]. The nonlinear plant can be represented more effectively because polynomials are allowed in the consequent part. Due to the existence of polynomials in the consequent part, a better expressing capability compared with the T-S fuzzy model is demonstrated, and thus the number of rules in polynomial fuzzy model is generally fewer than that in the T-S fuzzy model. The polynomial fuzzy model together with the polynomial fuzzy controller connected in a closed loop forms the polynomial fuzzy-model-based (PFMB) control system. To facilitate the stability analysis of PFMB control system, a more general Lyapunov function, namely polynomial Lyapunov function, is used and it can provide more relaxed analysis and stability conditions than the results obtained in quadratic Lyapunov function. However, because of the existence of monomials, the LMI-based approach cannot be used to get the solution for the stability conditions. Then, a SOSbased approach ] is employed 6, ], where a feasible solution (if any) can be found by, for example, SOSTOOLS ]. In the literature such as the work mentioned above, most of the stability analysis are MFI that the information of membership functions is not taken into account and thus it potentially leads to conservative analysis results. To relax the conservativeness of stability analysis results through the use of membership function information, different types of membership functions such as piecewise-linear membership functions 8], polynomial membership functions 7] and mismatched premise membership functions 3] have been proposed for the MFD stability analysis. In terms of uncertainties which can be a form of parameter uncertainty, measurement uncertainty, saturation etc, a type- fuzzy set was proposed to represent uncertain grade of membership functions 4 6]. Type- fuzzy sets can be regarded as a collection of type- fuzzy sets that there are two membership functions, namely primary and secondary membership functions, in it and the additional information including system uncertainties can be captured by the c The Institution of Engineering and Technology 5

3 secondary membership functions. Hence, the type- fuzzy logic systems (FLSs) is useful when the nonlinear system faces uncertainties. For type- FLSs 7], it involves the operations of fuzzification, inference, and output processing. The crisp input is mapped into a fuzzy set, which is in general a type- fuzzy set in type- FLSs. Then, the input type- fuzzy sets are mapped to the output fuzzy sets by the inference engine combined with rules. The structure of If- Then rules in type- FLSs are similar to type- FLSs that the only difference is that some or all of the fuzzy sets involved are type- fuzzy sets. In the output process, the first step is to obtain a typereduced set from the output type- fuzzy set by the type reduction process and then a crisp output from the type- FLS can be obtained by defuzzifying the type-reduced set. However, because of the computational complexity of using a general type- fuzzy set, an interval type- (IT) fuzzy set whose secondary membership function is constant, which is expressed as a particular case of the general type- fuzzy set, has been used. In the context of FMB control system, the work in 8] considered an IT fuzzy model to represent a nonlinear plant subject to uncertainties, and the uncertainties can be effectively captured by the region between the lower and upper membership functions, which is defined as the footprint of uncertainties (FOU). However, because the information contained in the FOU eliminates the favorable property of type- FMB analysis approach to facilitate the stability analysis, conservative stability analysis results will usually be obtained. Thus to relax the stability analysis results, the information of lower and upper membership functions are utilized. So far, the concept of IT FMB control has been extended to various control stratety and systems 9 36] such as state and output feedback control 3, 34], control of nonlinear networked systems 3], filter design 3, 33, 35] of the IT fuzzy system, control of time-varying delay system 34, 36] etc. Yet, most of them focus on the control methodology and the stability analysis is approached by existing techniques. Furthermore, the T-S fuzzy model is the main stream on these work but the polynomial fuzzy model 9, 37] is rarely considered in the literature. Although 9, 37] investigated the stability analysis and tracking control of IT PFMB control systems, the method they used is on the basis of type- PFMB control system, i.e. approximating the IT membership functions by an embedded type- membership functions and considering the influences of the modeling errors between the embedded Type- membership functions and IT membership functions. In this paper, the stability analysis of IT PFMB control system is conducted and the stability conditions are obtained in terms of SOS. By applying IT fuzzy sets into the polynomial fuzzy model, system uncertainties can be captured by the lower and upper membership functions. In addition, the information of IT membership functions such as the polynomial function or constant approximated upper and lower membership functions and the boundary informations of the approximation are considered in the analysis. With such information, slack matrices are introduced to relax the conservativeness of the stability analysis results. Based on the Lyapunov stability theory, stability conditions in terms of SOS are derived to achieve a stable IT PFMB control system. Moreover, to further relax the stability conditions, an IT switching polynomial fuzzy controller is employed in conducting the stability analysis. The whole operating domain are divided into sub-domains. In each sub-domain, a local IT polynomial fuzzy controller is designed with the local information of membership functions introduced by some slacks matrices to facilitate the stability analysis. Since the feedback gains can be designed separately in each sub-domains, the feedback compensation capability can be enhanced and the stability conditions are more relaxed. The rest of the paper is organized as follows. In Section, an IT polynomial fuzzy model, an IT polynomial fuzzy controller, an IT switching polynomial fuzzy controller, an IT PFMB control system and an IT switching PFMB control system are presented. In Section 3, the stability issue is discussed based on the Lyapunov stability theory and the SOS-based stability conditions are derived. In Section 4, numerical examples and simulation results are given to illustrate the merits of proposed approach. Conclusion is given in Section 5. Notations and Preliminaries In this section, notations and preliminaries of the IT polynomial fuzzy model, the polynomial fuzzy controller including the IT polynomial fuzzy controller and the IT switching polynomial fuzzy controller, the IT PFMB control system and the IT switching PFMB control system are presented.. Notations Throughout the paper, the following notations are applied 38]. The monomial in x(t) = x (t), x (t),..., x n(t)] T is defined as x λ (t), xλ (t),..., xλn n (t), where λ l, l=,,..., n, are nonnegative integers. The degree of a monomial is defined as λ = n l= λ l. A polynomial p(x(t)) is defined as a finite linear combination of monomials with real coefficients. A polynomial p(x(t)) is an SOS if it can be rewritten as p(x(t)) = j= q j(x(t)), where q j (x(t)) is a polynomial. Hence, it can be seen that p(x(t)) if it is an SOS. If polynomial p(x(t)) is an SOS, then it can be represented in the form of ˆx(t) T Qˆx(t), where Q is a positive semidefinite matrix. The problem of finding a Q can be formulated as a semi-definite programme. The notations of M >, M, M < and M denote a positive, semi-positive, negative, semi-negative definite matrix M, respectively.. IT Polynomial Fuzzy Model The dynamics of the nonlinear plant with uncertainties can be represented by an IT polynomial fuzzy model with p rules of which the antecedents are of type- fuzzy sets, and the consequent part is a local polynomial dynamic system. The i-th rule is of the following format 39]: Rule i : if f (x(t)) is M i and,..., and f Ψ (x(t)) is M i Ψ then ẋ(t) = A i (x(t)) ˆx(x(t)) + B i (x(t)) u(t), () i where M α is an IT fuzzy set of rule i corresponding to the function f α(x(t)), α=,,..., Ψ; i=,,..., p; Ψ is a positive integer; A i (x(t)) R n N and B i (x(t)) R n m, which are polynomial matrices in x(t), are the known polynomial system and input matrices, respectively; x(t) R n is the system state vector and ˆx(x(t)) R N is a vector of monomials in x(t); u(t) R m is the input vector. It is assumed that ˆx(x(t)) = iff x(t) =. The firing strength of the i-th rule is in the following interval sets, where w i (x(t)) w i (x(t)), w i (x(t))], i =,,..., p, () w i (x(t)) = Ψ µ M i (f α(x(t))), (3) α α= Ψ w i (x(t)) = µ M i (f α α(x(t))), (4) α= in which µ M i α (f α(x(t))), ] and µ M i α (f α(x(t))), ] denote the lower and upper grades of membership governed by their lower and upper membership functions, respectively. The definition of IT membership functions makes the property µ M i α (f α(x(t))) µ M i α (f α(x(t))) hold, which leads to w i (x(t)) w i (x(t)) for all i. Then the IT polynomial fuzzy model is defined as, ẋ(t) = w i (x(t))v i (x(t))(a i (x(t))ˆx(x(t)) + B i (x(t))u(t)) i= + w i (x(t))v i (x(t))(a i (x(t))ˆx(x(t)) + B i (x(t))u(t)) i= c The Institution of Engineering and Technology 5

4 where = w i (x(t))(a i (x(t))ˆx(x(t)) + B i (x(t))u(t)), (5) i= w i (x(t)) = w i (x(t))v i (x(t)) + w i (x(t))v i (x(t)), (6) w i (x(t)) =, (7) i= in which v i (x(t)), ] and v i (x(t)), ] are nonlinear functions, which are not necessary to be known but exist to represent the uncertainty, that have the property v i (x(t)) + v i (x(t)) = for all i..3 IT Polynomial Fuzzy Controller An IT polynomial fuzzy controller with c rules is employed to stabilize the nonlinear plant represented by an IT polynomial fuzzy model (5). The j-th rule is of the following format 9]: Rule j : if g (x(t)) is Ñ j and,..., and g Ω(x(t)) is Ñ j Ω then u(t) = G j (x(t))ˆx(x(t)), (8) where Ñ j β is the IT fuzzy term of rule j corresponding to the function g β (x(t)), β=,,..., Ω; j=,,..., c; Ω is a positive integer; and G j (x(t)) R m N, j=,,..., c, are polynomial feedback gains need to be determined. The firing strength of the j-th rule is in the following interval sets, where m j (x(t)) m j (x(t)), m j (x(t))], j =,,..., c, (9) in which νñ j β m j (x(t)) = m j (x(t)) = Ω β= Ω β= νñ j (g β (x(t))), () β νñ j (g β (x(t))), () β (g β (x(t))), ] and νñ j (g β (x(t))), ] β denote the lower and upper grades of membership governed by the their lower and upper membership functions, respectively. The definition of IT membership functions makes the property ν N j β (g β (x(t))) νñ j β (f β (x(t))) hold, which leads to m j (x(t)) m j (x(t)) for all j. Then the IT polynomial fuzzy controller is defined as follows, where m j (x(t)) = u(t) = m j (x(t))g j (x(t))ˆx(x(t)), () j= m j (x(t))λ j (x(t)) + m j (x(t))λ j (x(t)) c k= (m k (x(t))λ k (x(t)) + m k(x(t))λ k (x(t))), (3) m j (x(t)) =, (4) j= in which λ j (x(t)), ] and λ j (x(t)), ] are nonlinear functions that have the property λ j (x(t)) + λ j (x(t)) = for all j, and (3) is the type reduction..4 IT Switching Polynomial Fuzzy Controller To enhance the IT polynomial fuzzy controller s capability of feedback compensation, the whole operation domain Φ is divided into D connected sub-domains Φ d, i.e., Φ = D d= Φ d, where d=,,..., D. As an extension of the controller proposed in 4] which uses constant feedback gains, polynomial feedback gains are used. The j-th rule of IT switching polynomial fuzzy controller with c rules is of the following format: Rule j : if g (x(t)) is Ñ j and,..., and g Ω(x(t)) is Ñ j Ω then u(t) = G jd (x(t))ˆx(x(t)), (5) where Ñ j β is the IT fuzzy term of rule j corresponding to the function g β (x(t)), β=,,..., Ω; j=,,..., c; Ω is a positive integer and G jd (x(t)) R m N, j=,,..., c, d=,,..., D, are polynomial feedback gains need to be determined. Then, the overall IT switching polynomial fuzzy controller is defined as follows: u(t) = m j (x(t))g jd (x(t))ˆx(x(t)), x Φ d, d (6) j= where m j (x(t)) is defined by (3) and satisfies the property that c j= m j(x(t)) =..5 IT PFMB Control System The IT polynomial fuzzy controller () is considered to close the feedback loop of IT polynomial fuzzy model (5). Then the IT PFMB control system can be represented as follows: ẋ(t) = w i (x(t))(a i (x(t))ˆx(x(t)) + B i (x(t)) i= m j (x(t)) j= G j (x(t))ˆx(x(t))) = w i (x(t)) m j (x(t))(a i (x(t)) + B i (x(t))g j (x(t))) ˆx(x(t)). (7).6 IT switching PFMB control system When an IT switching polynomial fuzzy controller (6) is used to close the feedback loop of IT polynomial fuzzy model (5), an IT switching PFMB control system, which is of the following format, can be obtained. ẋ(t) = D d= ξ d (x(t)) w i (x(t)) m j (x(t))(a i (x(t))+ B i (x(t))g jd (x(t)))ˆx(x(t)), (8) where ξ d (x(t)) = for x Φ d, otherwise ξ d (x(t)) =, d=,,..., D. 3 Stability Analysis The control objective is to determine the polynomial feedback gains G j (x(t)) (G jd (x(t)) when IT switching polynomial fuzzy controller (6) is considered) to make the IT PFMB control system (7) (IT switching PFMB control system (8)) asymptotically stable, i.e., x(t) as time t. In this section, MFI and MFD stability conditions are developed. In the following analysis, for brevity, w i (x(t)) is denoted as w i (x) and m j (x(t)) is denoted as m j (x). The time t associated c The Institution of Engineering and Technology 5 3

5 with the variables is dropped for the situation without ambiguity, e.g. x(t) and ˆx(x(t)) are denote as x and ˆx(x), respectively. From the IT PFMB control system (7), the relation between ˆx(x) and ẋ can be obtained as follows: ˆx(x) = ˆx(x) dx x dt = T(x)ẋ = w i (x) m j (x)(ãi(x) + B i (x)g j (x))ˆx(x), (9) where Ãi(x) = T(x)A i (x), Bi (x) = T(x)B i (x) and T(x) R N n with its (i, j)th entry defined as follows, T ij (x) = ˆx i(x) x j, i =,,..., N; j =,,..., n. () To facilitate the stability analysis, let A k i (x) denote the k-th row of A i (x), K={k, k,..., k m} denote the row indices of B i (x) whose corresponding row is equal to zero, and define x=(x k, x k,..., x km ) 5]. The following polynomial Lyapunov function is considered to investigate the system stability of (7). V (t) = ˆx(x) T X( x) ˆx(x), () where < X( x) = X( x) T R N N is a positive definite symmetric polynomial matrix. According to the Lyapunov stability theory, if X( x) can be found that satisfies V (t) > for x and V (t) < for x, the IT PFMB control system (7) is asymptomatically stable. 3. MFI Stability Analysis The stability analysis of the IT PFMB control system (7) is conducted in this section. The MFI stability conditions in the form of SOS are derived using the Lyapunov-based approach. Considering the polynomial Lyapunov function (), the time derivative of V (t) is obtained as follows: V (t) = ˆx(x) T X( x) ˆx(x) + ˆx(x) T dx( x) ˆx(x) dt + ˆx(x) T X( x) ˆx(x) = ˆx(x) T X( x) ˆx(x) + ˆx(x) T ( n k= X( x) ẋ )ˆx(x) k + ˆx(x) T X( x) ˆx(x). () Since B k i (x) = for k K, we can obtain Otherwise, ẋ k = w i (x)a k i (x)ˆx(x), k K. (3) i= X( x) =, k / K. (4) By substituting (9), (3) and (4) into (), it can be obtained that V (t) = w i (x) m j (x)(ãi(x) + B i (x)n j (x)x( x) ) ˆx(x)] T X( x) ˆx(x) + ˆx(x) T ( n k= X( x) w i (x) m j (x)a k i (x)ˆx(x))ˆx(x) + ˆx(x) T X( x) w i (x) m j (x)(ãi(x) + B i (x)n j (x)x( x) )ˆx(x) = h ij (x)ˆx(x) T X( x) X( x)ãi(x) T + N j (x) T Bi (x) T + Ãi(x)X( x) + B i (x)n j (x) n X( x) + X( x) A k i (x)ˆx(x)x( x)]x( x) ˆx(x) x k= k = h ij (x)z(x) T Q ij (x) + X( x) n k= X( x) A k i (x)ˆx(x)x( x)]z(x), (5) where h ij (x) w i (x) m j (x), z(x) = X( x) ˆx(x), Q ij (x) = X( x)ãi(x) T + N j (x) T Bi (x) T + Ãi(x)X( x) + B i (x)n j (x), N j (x) R m N is a polynomial matrix for all j, G j (x) = N j (x)x( x), and A k i (x)ˆx(x) is a scalar. Then, the last term in (5) can be rewritten by the following Lemma 5]. Lemma. For any invertible polynomial matrix X( x), the following holds, X( x) = X( x) X( x) X( x). (6) Proof: Since X( x) is invertible, we have X( x)x( x) = I. Differentiating both sides with respect to x k, we have X( x) X( x) + X( x) X( x) =. Hence, the following relation holds X( x) X( x) X( x) = X( x). By substituting (6) into (5), it can be obtained that V (t) = h ij (x)z(x) T Q ij (x) n X( x) A k i (x)ˆx(x)]z(x). (7) x k= k In order to keep V (t) (equality holds for x = ), Q ij (x) n X( x) k= A k i (x)ˆx(x) is a sufficient condition that must be held. The MFI stability conditions can be summarized in the following theorem. Theorem. The IT PFMB control system (7) consisting of IT polynomial fuzzy model (5) and IT polynomial fuzzy controller () connected in a closed loop, is asymptotically stable if there exist a symmetric polynomial matrix X( x) = X( x) T R N N 4 c The Institution of Engineering and Technology 5

6 and a polynomial matrix N j (x) R m N such that the following SOS-based conditions are satisfied. υ T (X( x) ε ( x)i)υ is SOS; (8) υ T (Q ij (x) k K X( x) A k i (x)ˆx(x) + ε (x)i)υ is SOS; i, j. (9) where ε ( x) > and ε (x) > are predefined polynomial scalars; υ R N is an arbitrary vector independent of x; Q ij (x) = Ã i (x)x( x) + X( x)ãi(x) T + B i (x)n j (x) + N j (x) T Bi (x) T for i=,,..., p, j=,,..., c; Ãi(x) = T(x)A i (x); B i (x) = T(x)B i (x); T(x) R N n is a polynomial matrix with its (i, j)th entry defined in (); A k i (x) RN and B k i (x) Rm, i=,,..., p, k =,,..., n, denote the k-th row of A i (x) and B i (x), respectively; and the polynomial feedback gains are defined as G j (x) = N j (x)x( x), j=,,..., c. Remark. Since no information about membership functions is used in the above analysis, it is valid for all kinds of membership functions, therefore, this stability condition is conservative. In order to reduce conservativeness, the information of membership functions can be introduced into the stability analysis process to obtain more relaxed stability conditions, which is discussed in the following sections. 3. MFD Stability Analysis Two types of stability analysis, namely MFD stability analysis and MFD switching stability analysis, are given in this section. In the MFD stability analysis, some slack matrices are used to introduce the information of IT membership functions and an IT polynomial fuzzy controller () can be obtained to control the nonlinear system represented by the IT polynomial fuzzy model (5). On the basis of the MFD stability analysis, the MFD switching stability analysis is developed to generate an IT switching polynomial fuzzy controller (6), which demonstrates stronger stabilizability due to the switching control strategy. 3.. MFD Stability Analysis: In this section, some slack matrices are introduced on the basis of Theorem to bring the information of IT membership functions into the stability conditions, which could effectively relaxed the conservativeness of the stability analysis results. Recalling that, the whole operation domain Φ is divided into D sub-domains Φ d, i.e., Φ = D d= Φ d, where d =,,..., D. In each sub-domain, the lower and upper bounds h ijd (x) and h ijd (x) of the IT membership functions h ij (x), the minimum and maximum value σ ijd and σ ijd of the upper bound h ijd (x), and the minimum and maximum value γ ijd and γ ijd of the lower bound h ijd (x), can be obtained. h ijd (x) and h ijd (x) are functions of x, while σ ijd, σ ijd, γ ijd and γ ijd are constants. According to the property of the IT membership functions, the following properties hold. h ijd (x) h ij (x) h ijd (x) (3) γ ijd h ijd (x) γ ijd (3) σ ijd h ijd (x) σ ijd (3) Then, we have h ij (x) h ijd (x) ; h ijd (x) h ij (x) ; h ijd (x) γ ijd ; γ ijd h ijd (x) ; h ijd (x) σ ijd ; σ ijd h ijd (x) for all d. From (7), we have V (t) = h ij (x)z(x) T Q ij (x) n k= X( x) A k i (x)ˆx(x)] z(x) = h ij (x)z(x) T Θ ij (x)z(x), (33) where Θ ij (x) = Q ij (x) X( x) k K A k i (x)ˆx(x) for all i =,,..., p, j =,,..., c. Combining the slack matrices R ijd (x) = R ijd (x) T R N N, R ijd (x) = R ijd (x) T R N N, S ijd (x) = S ijd (x) T R N N, S ijd (x) = S ijd (x) T R N N, U ijd (x) = U ijd (x) T R N N, U ijd (x) = U ijd (x) T R N N with the inequalities obtained above, when x Φ d, d =,,..., D, the following inequality can be obtained by extending the terms in the right-hand side of (33), V (t) h ij (x)z(x) T Θ ij (x)z(x) + h ijd (x))z(x) T R ijd (x)z(x) + ( h ij (x) (h ijd (x) h ij (x))z(x) T R ijd (x)z(x) + (h ijd (x) γ ijd )z(x) T S ijd (x)z(x) + (γ ijd h ijd (x))z(x) T S ijd (x)z(x) + (h ijd (x) σ ijd )z(x) T U ijd (x)z(x) + (σ ijd h ijd (x))z(x) T U ijd (x)z(x) = h ij (x)z(x) T {Θ ij (x) + R ijd (x) R ijd (x) + h rsd (x)(s rsd (x) R rsd (x) S rsd (x)) r= s= + h rsd (x)(r rsd (x) + U rsd (x) U rsd (x)) + γ rsd S rsd (x) γ rsd S rsd (x) + σ rsd U rsd (x) σ rsd U rsd (x)]}z(x), x Φ d ; d =,,..., D. (34) According to Lyapunov stability theory, the nonlinear system is asymptomatically stable if V (t) > and V (t) < are achieved for all x. The stability analysis results are summarized in the following theorem. Theorem. The IT PFMB control system (7) consisting of IT polynomial fuzzy model (5) and IT polynomial fuzzy controller () connected in a closed loop, is asymptotically stable if there exist symmetric polynomial matrices X( x) R N N, R ijd (x) R N N, R ijd (x) R N N, S ijd (x) R N N, S ijd (x) R N N, U ijd (x) R N N, U ijd (x) R N N and a polynomial matrix N j (x) R m N such that the following SOS-based conditions are satisfied. υ T (X( x) ε ( x)i)υ is SOS; (35) υ T {Θ ij (x) + R ijd (x) R ijd (x) + h rsd (x) r= s= c The Institution of Engineering and Technology 5 5

7 (S rsd (x) R rsd (x) S rsd (x)) + h rsd (x)(r rsd (x) + U rsd (x) U rsd (x)) + γ rsd S rsd (x) γ rsd S rsd (x) + σ rsd U rsd (x) σ rsd U rsd (x)] + ε (x)i}υ is SOS, υ T R ijd (x)υ is SOS, i, j, d; υ T R ijd (x)υ is SOS, i, j, d; υ T S ijd (x)υ is SOS, i, j, d; υ T S ijd (x)υ is SOS, i, j, d; υ T U ijd (x)υ is SOS, i, j, d; υ T U ijd (x)υ is SOS, i, j, d; i, j, d; (36) where ε ( x) > and ε (x) > are predefined polynomial scalars; υ R N is an arbitrary vector independent of x; Θ ij (x) = Ã i (x)x( x) + X( x)ãi(x) T + B i (x)n j (x) + N j (x) T Bi (x) T k K X( x) A k i (x)ˆx(x) for i=,,..., p, j=,,..., c ; Ã i (x) = T(x)A i (x); B i (x) = T(x)B i (x); T(x) R N n is a polynomial matrix with its (i, j)th entry defined in (); A k i (x) R N and B k i (x) Rm, i=,,..., p, k =,,..., n, denote the k-th row of A i (x) and B i (x), respectively; h ijd (x) and h ijd (x) are the lower and upper bounds of the IT membership functions h ij (x), respectively, determined in prior and have property that h ijd (x) h ij (x) h ijd (x), for all i, j, d; similarly, γ ijd and γ ijd are the minimum and maximum value of the lower bound h ijd (x) determined in prior and have the property that γ ijd h ijd (x) γ ijd for all i, j, d; σ ijd and σ ijd are the minimum and maximum values of the upper bound h ijd (x) determined in prior and have the property that σ ijd h ijd (x) σ ijd for all i, j, d; and the polynomial feedback gains are defined as G j (x) = N j (x)x( x), j =,,..., c. Remark. Theorem considers the information of membership functions in the stability conditions. The more information of membership functions are considered, i.e. the more sub-domains are used, the more relaxed stability conditions can be obtained. However, the computational complexity will be increased for solution solving when the number of sub-domains is increasing. Hence, it is suggested that a small number of sub-domains can be used in the beginning. If no feasible solution is found, try again by increasing the number of sub-domains gradually. 3.. MFD Switching Stability Analysis: In this section, an IT switching polynomial fuzzy controller (6) is employed. Considering the polynomial Lyapunov function candidate (), the time derivative of V (t) is given by V (t) = ˆx(x) T X( x) ˆx(x) + ˆx(x) T ( = + ˆx(x) T X( x) ˆx(x) D d= n k= X( x) ẋ )ˆx(x) k ξ d (x) h ij (x)ˆx(x) T X( x) X( x)ãi(x) T + N jd (x) T Bi (x) T + Ãi(x)X( x) + B i (x)n jd (x) n X( x) + X( x) A k i (x)ˆx(x)x( x)]x( x) ˆx(x) x k= k D = ξ d (x) h ij (x)z(x) T Q ijd (x) d= n k= X( x) A k i (x)ˆx(x)]z(x), (37) where Q ijd (x) = X( x)ãi(x) T + N jd (x) T Bi (x) T + Ã i (x)x( x) + B i (x)n jd (x). G jd (x) = N jd (x)x( x), where N jd (x) R m N, j=,,..., c, d=,,..., D, are polynomial matrices to be determined. From (37), it can be obtained that V (t) = D d= ξ d (x) h ij (x)z(x) T Θ ijd (x)z(x), (38) where Θ ijd (x) = Q ijd (x) n X( x) k= A k i (x)ˆx(x). Then introducing the information of IT membership functions by some slack matrices to relax conservative. We use the same boundary information (3) (3) and (3) in Section 3... By combining these informations with the slack matrices R ijd (x) = R ijd (x) T R N N, R ijd (x) = R ijd (x) T R N N, S ijd (x) = S ijd (x) T R N N, S ijd (x) = S ijd (x) T R N N, U ijd (x) = U ijd (x) T R N N, U ijd (x) = U ijd (x) T R N N, the following inequality can be obtained. V (t) D d= R ijd (x) + ξ d (x) h ij (x)z(x) T {Θ ijd (x) + R ijd (x) r= s= h rsd (x)(s rsd (x) R rsd (x) S rsd (x)) + h rsd (x)(r rsd (x) + U rsd (x) U rsd (x)) + γ rsd S rsd (x) γ rsd S rsd (x) + σ rsd U rsd (x) σ rsd U rsd (x)]}z(x) i, j, d. (39) According to Lyapunov stability theory, the nonlinear system is asymptomatically stable if V (t) > and V (t) < for all x are achieved. The stability analysis results are summarized in the following theorem. Theorem 3. The IT switching PFMB control system (7) consisting of IT polynomial fuzzy model(5) and IT switching polynomial controller (6) connected in a closed loop, is asymptotically stable if there exist symmetric polynomial matrices X( x) R N N, R ijd (x) R N N, R ijd (x) R N N, S ijd (x) R N N, S ijd (x) R N N, U ijd (x) R N N, U ijd (x) R N N and a polynomial matrix N jd (x) R m N such that the following SOS-based conditions are satisfied. υ T (X( x) ε ( x)i)υ is SOS; (4) υ T {Θ ijd (x) + R ijd (x) R ijd (x) + h rsd (x) r= s= (S rsd (x) R rsd (x) S rsd (x)) + h rsd (x)(r rsd (x) + U rsd (x) U rsd (x)) + γ rsd S rsd (x) γ rsd S rsd (x) + σ rsd U rsd (x) σ rsd U rsd (x)] + ε (x)i}υ is SOS, υ T R ijd (x)υ is SOS, i, j, d; υ T R ijd (x)υ is SOS, i, j, d; υ T S ijd (x)υ is SOS, i, j, d; υ T S ijd (x)υ is SOS, i, j, d; i, j, d; (4) 6 c The Institution of Engineering and Technology 5

8 υ T U ijd (x)υ is SOS, i, j, d; υ T U ijd (x)υ is SOS, i, j, d; where ε ( x) > and ε (x) > are predefined polynomial scalars; υ R N is an arbitrary vector independent of x; Θ ijd (x) = Ã i (x)x( x) + X( x)ãi(x) T + B i (x)n jd (x) + N jd (x) T Bi (x) T X( x) k K A k i (x)ˆx(x) for i=,,..., p, j=,,..., c and d=,,..., D; Ãi(x) = T(x)A i (x); B i (x) = T(x)B i (x); T(x) R N n is a polynomial matrix with its (i, j)th entry defined in (); A k i (x) RN and B k i (x) Rm, i=,,..., p, k=,,..., n, denote the k-th row of A i (x) and B i (x), respectively; h ijd (x) and h ijd (x) are the lower and upper bounds of IT membership functions h ij (x), respectively, determined in prior and have property that h ijd (x) h ij (x) h ijd (x), for all i, j, d; similarly, γ ijd and γ ijd are the minimum and maximum values of the lower bound h ijd (x), respectively, determined in prior and have the property that γ ijd h ijd (x) γ ijd for all i, j, d; σ ijd and σ ijd are the minimum and maximum values of the upper bound h ijd (x), respectively, determined in prior and have the property that σ ijd h ijd (x) σ ijd for all i, j, d; and the polynomial feedback gains are defined as G jd (x) = N jd (x)x( x), j =,,..., c and d =,,..., D. Remark 3. Since the switching control scheme is employed in Theorem 3, the number of decision variables such as N jd (x) are increased. Thus the computational demand for finding a feasible solution to stability conditions are increased accordingly. Hence, compared with Theorem under the same number of sub-domains being used, solving a feasible solution to the stability conditions in Theorem 3 will be further increased. 4 Simulation Example In this section, two examples are provided to verify the stability analysis results by applying the stability conditions in Theorems to 3. The first example is a numerical example which demonstrates that the information of the lower and upper membership functions, and the switching control scheme can offer more relaxed stability analysis results comparatively when they are included in the stability analysis. In the second example, it is to demonstrate that the developed stability conditions can be applied to a nonlinear plant, inverted pendulum, with physical meaning. 4. Numerical Example To demonstrate the effectiveness of the proposed approaches, a nonlinear system represented by an IT polynomial fuzzy model is given. Considering the three-rule IT polynomial fuzzy model with ˆx(x) = x = x x ] T,.59.x A (x ) = 7.9.5x....63x A (x ) = x.35. a.x A 3 (x ) = +.3x ] B (x ) =, ] 8 B (x ) =, ] b + 6 B 3 (x ) =. ], ], ], where a and b are constant parameters to be determined. The lower and upper membership functions for the IT polynomial fuzzy model are chosen as w (x ) = /( + e ( (x+3.5)) ), w 3 (x ) = /( + e ( (x 3.5)) ), w (x ) = /( + e ( (x+.5)) ), w 3 (x ) = /( + e ( (x.5)) ), w (x ) = w (x ) w 3 (x ) and w (x ) = w (x ) w 3 (x ). Meanwhile, the lower and upper membership functions for the IT polynomial fuzzy controller (IT switching polynomial fuzzy controller), which are different from those of the IT polynomial fuzzy model, are chosen as m (x ) = m (x ) = for x < 5. x +4.8 for 5. x 4.8 for x > 4.8 for x < 4.8 x +5. for 4.8 x 5. for x > 5. m (x ) = m (x ) and m (x ) = m (x ). b a Fig. : Feasible region indicated by for Theorem. By considering different values of a and b in the IT polynomial fuzzy model, the influence to feasible regions given by the proposed stability conditions are studied and compared. A larger size of feasible region implies less conservative stability conditions. b a Fig. : Feasible region for Theorem with sub-domain, 4 subdomains and sub-domains. represents the feasible region for sub-domain; represents the feasible region for 4 sub-domains while represents the feasible region for sub-domains.,, c The Institution of Engineering and Technology 5 7

9 We choose the region 5 < a < and 5 < b < 9 with both at the interval of, and X( x) as a constant matrix, and N j (x) being a function of monomials (, x, x ), and ε ( x) = ε (x) =., the feasible region for Theorem is shown in Figure. Then the whole operating domain is divided into several sub-domains to compare the feasible regions of Theorem and Theorem 3 with that of Theorem. We consider three cases that D =, 4 and. When D =, it means that the only sub-domain is the whole operating domain, i.e., Φ = Φ (, + ). When the operation domain is divided into 4 sub-domains, the sub-domain Φ d is characterized as Φ ( 5], Φ ( 5 ], Φ 3 ( 5] and Φ 4 (5 ), and when it is divided into sub-domains, the sub-domain Φ d is characterized as Φ ( 8], Φ ( 8 6], Φ 3 ( 6 4], Φ 4 ( 4 ], Φ 5 ( ], Φ 6 ( ], Φ 7 ( 4], Φ 8 (4 6], Φ 9 (6 8] and Φ (8 ). The boundary informations of the IT membership functions in each sub-domain are given in Tables to 4. In this case, setting h ijd (x) = γ ijd and h ijd (x) = σ ijd, choosing R ijd (x), R ijd (x), S ijd (x), S ijd (x), U ijd (x), U ijd (x), and X( x) as constant symmetric matrices, N jd (x) and N j (x) being the function of monomials (, x, x ), and ε ( x) = ε (x) =., the feasible region of Theorem is shown in Figure while that of Theorem 3 is shown in Figure 3 using the same settings. Remark 4. The membership functions of the IT polynomial fuzzy controller are different from those of the IT polynomial fuzzy model in this example. Since there are no entries of the row of the input matrices B i (x) are zero for all i, x is chosen to be constant matrix. The predefined scalars γ ijd, γ ijd, σ ijd and σ ijd, can be computed numerically as the extrema of the upper and lower membership functions in the corresponding sub-domain. To verify the analysis results, the phase plots of system states are shown in Figures 4 to 7. For simulation purposes, the IT membership functions of the IT polynomial fuzzy model are chosen as follows: w (x ) = sin(5x ) + w (x ) = cos(5x ) + w (x ) + ( sin(5x ) + )w (x ), (4) w (x ) + ( cos(5x ) + )w (x ), (43) w 3 (x ) = w (x ) w (x ). (44) When system state x changes, the membership functions of the IT polynomial fuzzy model change, which represent the occurrence of uncertainties. x(t) x(t) Fig. 4: Phase plot of x (t) and x (t) for Theorem of 4 subdomains with a = and b = where the initial conditions are indicated by. 9 b a Fig. 3: Feasible region for Theorem 3 with sub-domain, 4 subdomains and sub-domains. represents the feasible region for sub-domain; represents the feasible region for 4 sub-domains while represents the feasible region for sub-domains. x(t) For Theorem, there are 8 feasible points shown in Figure, which means that the nonlinear system represented by the IT polynomial fuzzy model can be stabilized at the particular a and b accordingly. As for Theorem, referring to Figure, 8 feasible points for sub-domain, 9 feasible points for 4 sub-domains and 4 feasible points for sub-domains are obtained. For Theorem 3, referring to Figure 3, 8 feasible points are obtained for subdomain, 3 feasible points for 4 sub-domains and 8 feasible points for sub-domains. Hence, it can be concluded that both Theorem and Theorem 3 could effectively relax conservativeness for the IT PFMB control systems. Comparing with Theorem, Theorem 3 gets more relaxed results, i.e., the feasible regions for both 4 and subdomains are larger than that of Theorem when the same boundary informations of the IT membership functions are applied x(t) Fig. 5: Phase plot of x (t) and x (t) for Theorem of subdomains with a = and b = where the initial conditions are indicated by. The membership functions of the IT polynomial fuzzy controller (IT switching polynomial fuzzy controller) are chosen as follows: m (x ) =.5m (x ) +.5m (x ), (45) m (x ) = m (x ). (46) 8 c The Institution of Engineering and Technology 5

10 For Theorem with 4 sub-domains, considering a = and b =, the phase plot is shown in Figure 4. The polynomial feedback gains obtained by the SOSTOOLS are as follows: ] X( x) = , G (x ) =.6858 x x x.44 x ], G (x ) =.578 x +.77 x x.95 x +.668]. For Theorem with sub-domains, considering a = and b =, the phase plot is shown in Figure 5. The polynomial feedback gains obtained by the SOSTOOLS are as follows: ] X( x) = , G (x ) =.534 x x x x ], G (x ) = x x x 7.43 x +.956]. For Theorem 3 with 4 sub-domains, considering a = and b =, the phase plot is shown in Figure 6. The polynomial feedback gains obtained are as follows: ] X( x) = 3.44, G (x ) =.76 3 x x x x 5.3], G (x ) = x x x.434 x +.66], G 3 (x ) =.68 3 x x x x ], G 4 (x ) = x x x x ], G (x ) =.64 3 x.73 3 x x x ], G (x ) =.89 3 x +.6 x x.8 x ], G 3 (x ) = x 3.54 x x x ], G 4 (x ) = x x x x ]. For Theorem 3 with sub-domains, considering a = and b = 8, the phase plot is shown in Figure 7. The polynomial feedback gains obtained are as follows: x(t) x(t) Fig. 6: Phase plot of x (t) and x (t) for Theorem 3 of 4 subdomains with a = and b = where the initial conditions are indicated by ] X( x) = , G (x ) =.48 3 x x x x ], G (x ) = x x x x ], G 3 (x ) = x x x x 5.55], G 4 (x ) = x x x x 6.633], G 5 (x ) =.7 5 x x x.96 x ], G 6 (x ) =5.6 4 x x x x +.5 ], G 7 (x ) = x.5 3 x x x ], G 8 (x ) =.4 3 x x x x ], G 9 (x ) = x x x x c The Institution of Engineering and Technology 5 9

11 ], +. ]. G, (x ) = x x x.36 3 x ], G (x ) = x x x.39 3 x ], G (x ) = x x 6.46 x(t) x x ], G 3 (x ) = x x x x.33], G 4 (x ) = x x x x ], G 5 (x ) = x x x x ], G 6 (x ) = x x x x +.74 ], G 7 (x ) = x.64 3 x x.6 3 x ], G 8 (x ) =.39 3 x x x.47 3 x ], G 9 (x ) =.93 3 x x x x +.55 ], G, (x ) = x x x x x(t) Fig. 7: Phase plot of x (t) and x (t) for Theorem 3 of subdomains with a = and b = 8 where the initial conditions are indicated by. It can be seen from Figures 4 to 7 that the obtained IT polynomial fuzzy controllers are able to stabilize the nonlinear plant subject to uncertainty by driving its system states to the origin. 4. Inverted Pendulum In this section, the stability of an inverted pendulum is investigated to verify the application of the proposed approaches. Consider an inverted pendulum subject to parameter uncertainty as the nonlinear plant. The control task is to apply Theorem and Theorem 3 to find the proper feedback gains to stabilize the inverted pendulum. The dynamic equation for the inverted pendulum 8] is given by θ = gsin(θ(t)) amps θ(t) sin(θ(t))/ acos(θ(t))u(t) 4S/3 am pscos, (θ(t)) (47) where θ(t) is the angular displacement of the inverted pendulum, g = 9.8 m/s, m p m pmin m pmax ] = 3] kg is the mass of the pendulum, M c M cmin M cmax ] = 8 6] kg is the mass of the cart, a = m p+m c, S = m is the length of the pendulum, and u(t) is the force applied on the cart. In the investigation, m p and M c are treated as the parameter uncertainties. The following 4-rule polynomial fuzzy model is adopted to describe the inverted pendulum: Rule i :Iff (x) is M i and f (x) is M i Then ẋ(t) = A i (x)ˆx(x) + B i (x)u(t), i =,, 3, 4. (48) After combining all the fuzzy rules, we have: where ẋ = 4 ( w i Ai (x)ˆx(x) + B i (x)u(t) ), (49) i= ˆx = x = x x ] T = θ(t) θ(t)] T, x = 5π 5π ], x = 5 5], c The Institution of Engineering and Technology 5

12 Table Boundary informations of IT membership functions of numerical example for sub-domain. γ ijd = h ijd (x) γ ijd σ ijd h ijd (x) = σ ijd γ ijd = h ijd (x) γ ijd σ ijd h ijd (x) = σ ijd γ =. γ =. σ =. σ =. γ =. γ =.4576 σ =. σ =.5885 γ =. γ =.68 σ =. σ =.379 γ =. 3 γ 3 =.68 σ 3 =. σ 3 =.379 γ =. γ =.4576 σ =. σ =.5885 γ =. 3 γ 3 =. σ 3 =. σ 3 =. Table Boundary informations of IT membership functions of numerical example for 4 sub-domains. γ ijd = h ijd (x) γ ijd σ ijd h ijd (x) = σ ijd γ ijd = h ijd (x) γ ijd σ ijd h ijd (x) = σ ijd γ =.8 γ =. σ =.94 σ =. γ =. 3 γ 3 =.4 σ 3 =. σ 3 =.394 γ =. γ =. σ =. σ =.85 γ =. 3 γ 3 =.4 σ 3 =.6 σ 3 =.394 γ =. γ =.738 σ =. σ =.8 γ =. 3 γ 3 =.47 σ 3 =.36 σ 3 =.4895 γ =. γ =. σ =. σ =.36 γ = γ 3 =.4576 σ 3 =.8 σ 3 =.5885 γ =. 3 γ 3 =. σ 3 =. σ 3 =.6 γ =. 33 γ 33 =.68 σ 33 =.85 σ 33 =.379 γ =. 3 γ 3 =. σ 3 =. σ 3 =. γ =.4 33 γ 33 =.8 σ 33 =.394 σ 33 =.94 γ =.4 γ =.8 σ =.394 σ =.94 γ =. 4 γ 4 =. σ 4 =. σ 4 =. γ =. γ =.68 σ =.85 σ =.379 γ =. 4 γ 4 =. σ 4 =. σ 4 =.6 γ =.738 γ =.4576 σ =.8 σ =.5885 γ =. 4 γ 4 =. σ 4 =. σ 4 =.36 γ =. γ =.47 σ =.36 σ =.4895 γ =. 4 γ 4 =.738 σ 4 =. σ 4 =.8 γ =. 3 γ 3 =.4 σ 3 =.6 σ 3 =.394 γ =. 34 γ 34 =. σ 34 =. σ 34 =.85 γ =. 3 γ 3 =.4 σ 3 =. σ 3 =.394 γ =.8 34 γ 34 =. σ 34 =.94 σ 34 =. Table 3 Boundary informations of IT membership functions of numerical example for sub-domains with d = to 6. γ ijd = h ijd (x) γ ijd σ ijd h ijd (x) = σ ijd γ ijd = h ijd (x) γ ijd σ ijd h ijd (x) = σ ijd γ =.989 γ =. σ =.9959 σ =. γ =.4 4 γ 4 =.5478 σ 4 =.78 σ 4 =.75 γ =. γ =. σ =. σ =. γ = γ 4 =.68 σ 4 =.98 σ 4 =.379 γ =. γ =.4 σ =. σ =. γ =.59 4 γ 4 =.458 σ 4 =.3468 σ 4 =.5857 γ =. γ =. σ =. σ =. γ =.45 4 γ 4 =.7 σ 4 =.45 σ 4 =.63 γ =. 3 γ 3 =. σ 3 =. σ 3 =. γ =.5 34 γ 34 =.8 σ 34 =.4 σ 34 =.79 γ =. 3 γ 3 =. σ 3 =. σ 3 =. γ =. 34 γ 34 =. σ 34 =. σ 34 =.35 γ =.94 γ =.989 σ =.977 σ =.9959 γ =.4 5 γ 5 =.4 σ 5 =.394 σ 5 =.78 γ =. γ =. σ =. σ =. γ =.4 5 γ 5 =.5 σ 5 =.394 σ 5 =.8 γ =.4 γ =.9 σ =. σ =.758 γ =.47 5 γ 5 =.4576 σ 5 =.4895 σ 5 =.5885 γ =. γ =. σ =. σ =. γ =.7 5 γ 5 =.47 σ 5 =.63 σ 5 =.4895 γ =. 3 γ 3 =. σ 3 =. σ 3 =. γ =.8 35 γ 35 =.4 σ 35 =.79 σ 35 =.394 γ =. 3 γ 3 =. σ 3 =. σ 3 =. γ =. 35 γ 35 =.4 σ 35 =.35 σ 35 =.394 γ = γ 3 =.94 σ 3 =.75 σ 3 =.977 γ =. 6 γ 6 =.4 σ 6 =.35 σ 6 =.394 γ =. 3 γ 3 =.498 σ 3 =. σ 3 =.98 γ =.8 6 γ 6 =.4 σ 6 =.79 σ 6 =.394 γ =.9 3 γ 3 =.59 σ 3 =.758 σ 3 =.3468 γ =.7 6 γ 6 =.47 σ 6 =.63 σ 6 =.4895 γ =. 3 γ 3 =.45 σ 3 =. σ 3 =.45 γ =.47 6 γ 6 =.4576 σ 6 =.4895 σ 6 =.5885 γ =. 33 γ 33 =.5 σ 33 =. σ 33 =.4 γ =.4 36 γ 36 =.5 σ 36 =.394 σ 36 =.8 γ =. 33 γ 33 =. σ 33 =. σ 33 =. γ =.4 36 γ 36 =.4 σ 36 =.394 σ 36 =.78 Table 4 Boundary informations of IT membership functions of numerical example for sub-domains with d = 7 to. γ ijd = h ijd (x) γ ijd σ ijd h ijd (x) = σ ijd γ ijd = h ijd (x) γ ijd σ ijd h ijd (x) = σ ijd γ =. 7 γ 7 =. σ 7 =. σ 7 =.35 γ =. 9 γ 9 =. σ 9 =. σ 9 =. γ =.5 7 γ 7 =.8 σ 7 =.4 σ 7 =.79 γ =. 9 γ 9 =. σ 9 =. σ 9 =. γ =.45 7 γ 7 =.7 σ 7 =.45 σ 7 =.63 γ =. 9 γ 9 =. σ 9 =. σ 9 =. γ =.59 7 γ 7 =.458 σ 7 =.3468 σ 7 =.5857 γ =.4 9 γ 9 =.9 σ 9 =. σ 9 =.758 γ = γ 37 =.68 σ 37 =.98 σ 37 =.379 γ =. 39 γ 39 =. σ 39 =. σ 39 =. γ =.4 37 γ 37 =.5478 σ 37 =.78 σ 37 =.75 γ = γ 39 =.989 σ 39 =.977 σ 39 =.9959 γ =. 8 γ 8 =. σ 8 =. σ 8 =. γ =. γ =. σ =. σ =. γ =. 8 γ 8 =.5 σ 8 =. σ 8 =.4 γ =. γ =. σ =. σ =. γ =. 8 γ 8 =.45 σ 8 =. σ 8 =.45 γ =. γ =. σ =. σ =. γ =.9 8 γ 8 =.59 σ 8 =.758 σ 8 =.3468 γ =. γ =. σ =. σ =. γ =. 38 γ 38 =.498 σ 38 =. σ 38 =.98 γ =. 3 γ 3 =. σ 3 =. σ 3 =. γ = γ 38 =.94 σ 38 =.75 σ 38 =.977 γ = γ 3 =. σ 3 =.9959 σ 3 =. A = A = f min B = B 3 = ], A 3 = A 4 = ], B f = B 4 = min f max ]. f max The IT membership functions are defined as shown in Table 5. ], In Table 5, we have f (x) = g ampsx cos(x ) ( sin(x ) ), 4S/3 am pscos (x ) x f (x) = acos(x ) 4S/3 am pscos (x ), c The Institution of Engineering and Technology 5

13 Table 5 Lower and Upper Membership Functions for the Interval Type- Fuzzy Model of the Inverted Pendulum. Lower and upper membership functions µ M (f (x)) = µ M (f (x)) µ M (f (x)) = µ M 3 (f (x)) fmax f(x) = f max f min ; fmax f(x) = f max f min ; µ M 3(f (x)) = µ M 4(f (x)) µ M (f (x)) = µ M 4(f (x)) = f(x) f min f max f min ; = f(x) f min f max f min with x (t) =, m p = m pmax with m p = m pmax = 3kg and M c = M cmin = 8kg = 3kg and M c = M cmax = 6kg µ M (f (x)) = µ M (f (x)) µ M (f (x)) = µ M 3(f (x)) fmax f(x) fmax f(x) = f max f min ; = f max f min µ M 3(f (x)) = µ M 4(f (x)) µ M (f (x)) = µ M 4(f (x)) = f(x) f min f max f min ; = f(x) f min f max f min ; with x = x max, m p = m pmax with m p = m pmin = kg = 3kg and M c = M cmin = 8kg and M c = M cmin = 8kg Through a Taylor series based approach 4], the minimum and maximum values of f (x) and f (x) are as follows: f min =.893x +.53, f max = x , f min =.388x x.765, f max =.97x x.895. The lower and upper grades of membership are respectively defined as: w L i (x) = µ M i (x) µ M i (x), w U i (x) = µ M i (x) µ M i (x) for all i. According to the IT PFMB fuzzy model, a two-rule IT polynomial fuzzy controller is adopted to stabilize the inverted pendulum. The following two-rule IT polynomial fuzzy controller is adopted to describe the inverted pendulum: Rule j : If x is Ñ j Then u(t) = G j x, j =,. (5) After combining of all the fuzzy rules, we have u(t) = m (x )G x + m (x )G x. (5) where m (x ) and m (x ) are the IT membership functions of the polynomial fuzzy controller. The upper and lower bounds of the membership functions of the fuzzy controller are as follows: m (x ) = m (x ) = for x < 5π x +5π/ for 5π 5π/ x 5π/ x for x 5π/ 5π for x > 5π for x < 5π.9(x +5π/) for 5π 5π/ x.9(5π/ x ) for x 5π/ 5π for x > 5π m (x ) = m (x ), m (x ) = m (x ) and m (x ) + m (x ) =. We choose λ j (x ) = λ j (x ) =.5, j =,. x(t) x(t) Time(sec) Time(sec) Fig. 8: The top figure is the time responses of x (t) and the bottom one is the time responses of x (t) for Theorem. x(t) x(t) Time(sec) Time(sec) Fig. 9: The top figure is the time responses of x (t) and the bottom one is the time responses of x (t) for Theorem 3. During the simulation, for demonstration purposes, we set m p =.5kg and m c = kg. Based on Theorem, we set ε ( x) = ε ( x) =., X( x) and N j (x) as a constant matrix. The operating domain of x is divided into 3 sub-domains, which are characterized as Φ = 5π 5π 36 ), Φ = 5π 36 5π 36 ) and Φ 3 = 5π 36 5π ]. The boundary information of the IT membership functions are given in Tables 6. The feedback gains are achieved as G = ], G = ], and X = ] The state response for Theorem is shown in Figure 8, where the solid lines are under the initial condition x() = 5π ], the dashed lines are under the initial condition x() = π 6 ], the dotted lines are under the initial condition x() = π 6 ] and the dash-dot lines are under the initial condition x() = 5π ]. Based on Theorem 3, we set ε ( x) = ε ( x) =., X( x) and N j (x) as a constant matrix. The operation domain of x is divided into sub-domains, which are characterized as Φ = 5π ), Φ = 5π ]. The boundary information of the IT membership functions are given in Table 7. The feedback gains are achieved as G = ], G = ], G = ], G = ], and X = ] The state response for Theorem 3 is shown in Figure 9, where the solid lines are under the initial condition x() = 5π ], the dashed lines are under the initial condition x() = π 6 ], the dotted lines are under the initial condition x() = π 6 ] and the dash-dot lines are under the initial condition x() = 5π ]. It can be seen from Figs. 8 and 9 that both IT polynomial fuzzy controller and IT switching polynomial fuzzy controller designed c The Institution of Engineering and Technology 5