Fuzzy rule based combination of linear a

Fuzzy Sets and Systems 156 (2005) 153 – 184 www.elsevier.com/locate/fss Fuzzy rule-based combination of linear and switching state-feedback controllers H.K Lam∗ , F.H.F Leung Centre of Multimedia Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received 20 February 2004; received in revised form May 2005; accepted 24 May 2005 Available online 20 June 2005 Abstract This paper presents a fuzzy rule-base combined controller, which is a fuzzy rule-based combination of linear and switching state-feedback controllers, for nonlinear systems subject to parameter uncertainties The switching state-feedback controller is employed to drive the system states toward the origin When the system state approaches the origin, the linear state-feedback controller will gradually replace the switching state-feedback controller The smooth transition between the linear and switching state-feedback controllers is governed by the fuzzy rules By using the fuzzy rule-based combination technique, the proposed fuzzy rule-base combined controller integrates the advantages of both the linear and switching state-feedback controllers but eliminates their disadvantages As a result, the proposed fuzzy controller provides good performance during the transient period and the chattering effect is removed when the system state approaches the origin Stability conditions will be derived to guarantee the system stability Furthermore, a saturation function is employed to replace the switching component to alleviate the chattering during the transient period By using the proposed fuzzy rule-based combination technique, the steady state error introduced by the saturation function can be eliminated Application examples will be given to show the merits of the proposed approach © 2005 Elsevier B.V All rights reserved Keywords: Fuzzy model based control; Fuzzy controller; Stability ∗ Corresponding author E-mail address: hak_keung_lam@yahoo.com.hk (H.K Lam) 0165-0114/$ - see front matter © 2005 Elsevier B.V All rights reserved doi:10.1016/j.fss.2005.05.021 154 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 Introduction Control of nonlinear systems is a challenging task because no systematic mathematical tools exist to help find necessary and sufficient conditions to guarantee the stability and performance The problem becomes more complex if some of the system’s parameters are uncertain Fuzzy control is good at handling ill-defined and complex systems However, effective stability analysis and systematic controller design methodologies are still lacking Recently, stability analysis of fuzzy-model-based control systems, which is made of a Takagi–Sugeno– Kang (TSK) fuzzy plant model [15,17] and a fuzzy controller, has become a hot research topic Different stability conditions have been published In [21], it has been proven that the fuzzy control system is guaranteed to be stable if there exists a common solution to a set of Lyapunov equations Relaxed stability conditions have been derived in [5,6,18,20,24,27] under the assumption that the fuzzy controllers shares the same premises as those of the fuzzy plant models This assumption implies that the nonlinear plant is completely known Other stability analysis methods of fuzzy-model-based control systems were also proposed In [28], a linear controller was designed based on the fuzzy plant model In this approach, the fuzzy plant model is separated into two parts, nominal and varying parts The nominal part is handled by the linear controller The varying part is treated as an uncertainty Hence, when the magnitude of the varying part is too large, the linear controller will fail to control the plant In [10,11], stability conditions were derived based on the matrix measures [23] of the sub-system matrices In [9], a nonlinear statefeedback controller designed based on the fuzzy plant model was also proposed In [3], the stability conditions of a number of Riccati equations were derived Constructive algorithms were proposed to help find the solution of the derived stability conditions However, in most works, the parameter uncertainties of the system which commonly appear are practically not considered In some published work, fuzzy logic has been combined with the traditional sliding-mode controller to combine their advantages together In [1], a fuzzy sliding-mode controller using the sliding-surface function as the input of the fuzzy system has been reported Since only one variable is taken as the input of the fuzzy system, the number of fuzzy rules can be greatly reduced In [4,22], a fuzzy system was employed to estimate the values of the control gains of the sliding-mode controller Adaptive laws have been derived to update the rules of the fuzzy systems As the switching function of the sliding-mode controller is approximated by a continuous function, the chattering effect can be alleviated In [16], an adaptive fuzzy controller has been proposed to generate the control signals by estimating the values of the unknown system parameters of the systems Based on these estimated parameter values, tracking control can be achieved by the sliding-mode control However, in these approaches, the way to determine the fuzzy rules is still an open question Furthermore, the approximation error of the fuzzy systems will introduce steady-state error to the system states or even cause the system unstable In [25], to compensate the approximation error of the fuzzy system, switching elements were still needed in the controller Other techniques have also been proposed to alleviate the chattering effect A saturation function was proposed to replace the switching element or include the system states in the switching element [13] In [26,29], a two-phase variable structure controller was proposed The distance of the system states from the sliding surface was considered during the controller design For these approaches, the chattering effect may disappear only when the system state reaches the equilibrium, otherwise, the steady-state error may appear in the system states Switching control [2,7,8] is good at handling nonlinear systems subject to parameter uncertainties With this method a good system performance and global system stability can be guaranteed, however, an H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 155 undesired chattering effect will occur Linear state-feedback controller designed based on a linear model of the nonlinear system offers a simple and systematic design methodology However, this control approach guarantees the system stability locally, i.e the system state is within a small operating domain Throughout this paper, the fuzzy controller which is a fuzzy rule-based combination of the linear state-feedback and the switching controllers, is proposed The fuzzy rule-based combination technique combines their advantages of both approaches and eliminates their disadvantages Consequently, a fuzzy rule-base combined controller, which provides good system performance, guarantees global system stability, and has no chattering effect when the system state approaches the origin, can be obtained Furthermore, a saturation function is employed to replace the switching component to alleviate the chattering effect during the transient period By using the proposed fuzzy rule-based combination technique, the steady state error introduced by the saturation function can be eliminated This paper is organized as follows In Section 2, the fuzzy plant model and the proposed fuzzy rule-base combined controller will be presented In Section 3, the design of the fuzzy rule-base combined controller and the stability analysis will be presented In Section 4, application examples will be given A conclusion will be drawn in Section Fuzzy plant model and fuzzy rule-base combined controller We consider a multivariable fuzzy-model-based control system comprising a TSK fuzzy plant model and a fuzzy rule-base combined controller connected in closed loop 2.1 TSK fuzzy plant model Let p be the number of fuzzy rules describing the nonlinear plant subject to bounded parameter uncertainties The ith rule is of the following format: Rule i: IF f1 (x(t)) is M1i and and f (x(t)) is M i THEN x˙ (t) = Ai x(t) + Bi u(t), (1) where M i is a fuzzy term of rule i corresponding to the function f (x(t)), = 1, 2, , , i = 1, 2, , p, is a positive integer; Ai ∈ n×n and Bi ∈ n×m are known constant system and input matrices, respectively; x(t) ∈ n×1 is the system state vector and u(t) ∈ m×1 is the input vector The system dynamics are described by p x˙ (t) = wi (x(t))(Ai x(t) + Bi u(t)), (2) i=1 where p wi (x(t)) = 1, wi (x(t)) ∈ [0 1] for all i (3) i=1 wi (x(t)) = M1i (f1 (x(t))) × M2i (f2 (x(t))) × · · · × M i p k=1 (f (x(t))) M1k (f1 (x(t))) × M2k (f2 (x(t))) × · · · × M k (f (x(t))) (4) 156 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 and M i (f (x(t))), = 1, 2, , n, which denotes the grade of membership corresponding to the fuzzy term M i , is a nonlinear function of the system states and the parameter uncertainties 2.2 Fuzzy rule-base combined controller The proposed fuzzy rule-base combined controller is a fuzzy rule-based combination of linear and switching state-feedback controllers It has two rules in the following format: Rule : IF q(x(t)) is ZE THEN u(t) = Gx(t), Rule : IF q(x(t)) is NZ THEN u(t) = (5) p nj (x(t))Gj x(t), (6) j =1 where rule is for the linear state-feedback controller and rule is for the switching state-feedback controller; ZE (zero) and NZ (non-zero) are the fuzzy terms; G and Gj ∈ m×n , j = 1, 2, , p, are the feedback gains; The value of nj (x(t)) governed by the switching scheme will be discussed later; q(x(t)) is a function of system state to be designed The membership function corresponding to ZE will cover the region with q(x(t)) = while that of NZ will cover the region with q(x(t)) > The inferred output of the fuzzy rule-base combined controller is given by p u(t) = m1 (x(t))Gx(t) + m2 (x(t)) nj (x(t))Gj x(t), (7) j =1 where mk (x(t)) = 1; mk (x(t)) ∈ [0 1], k = 1, (8) k=1 m1 (x(t)) = ZE (q(x(t))) and NZ, respectively ZE (q(x(t))) ZE (q(x(t))) + NZ (q(x(t))) NZ (q(x(t))) and m2 (x(t)) = NZ (q(x(t))) ZE (q(x(t))) + NZ (q(x(t))) ; are the grades of membership corresponding to the fuzzy terms ZE and Stability analysis and design of fuzzy rule-base combined controller In this section, the system stability analysis and the design of the feedback gains will be presented From (2) and (7), writing wi (x(t)), nj (x(t)) and mk (x(t)) as wi , nj and mk respectively, and with the H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 property of p i=1 p wi =  k=1 p i=1 mk =  k=1 wi mk = 1, the closed-loop system are as follows:  p wi Ai x(t) + Bi m1 Gx(t) + m2 x˙ (t) = i=1 j =1 p p = m1 nj Gj x(t)   i=1  p wi Ai x(t) + Bi  wi (Ai x(t) + Bi Gx(t)) + m2 i=1 157 nj Gj x(t) (9) j =1 There are two cases to investigate the stability of (9) and will be detailed as follows 3.1 Uncertain input matrix The fuzzy plant model of (2) is assumed to have the following property: p wi Bi = (x(t))Bm , B(x(t)) = (10) i=1 where Bm ∈ n×m is a constant matrix; (x(t)) is an unknown non-zero scalar (because wi (x(t)) is unknown) but with single sign and known bounds, i.e | (x(t))| ∈ [ max ] It should be noted that because (x(t)) = is required, B(x(t)) = is assumed From (9) and (10) and writing (x(t)) as , we have p p x˙ (t) = m1 wi Ai x(t) + i=1  +m2  wi Bi Gx(t) i=1 p  p wi B i  wi Ai x(t) + i=1 i=1  p nj Gj x(t) j =1 p p = m1 Ao x(t) + Bo Gx(t) +  +m2  wi (Ai − Ao )x(t) + i=1 i=1 p wi Bm Gi x(t) + Bm i=1  p = m1 (Hx(t) + Hx(t)) + m2  j =1 p wi Hi x(t) + i=1 p p p wi (Ai + Bm Gi )x(t) − i=1 wi (Bi − Bo )Gx(t) p  nj Gj x(t)  ( nj − wj )Bm Gj x(t) , (11) j =1 where Ao = i=1 wi (0)Ai and Bo = i=1 wi (0)Bi are the constant system and input matrices with the uncertain parameters inside wi taking the nominal values; H = Ao + Bo G and Hi = Ai + Bm Gi p p are constant matrices; H = i=1 wi (Ai − Ao ) + i=1 wi (Bi − Bo )G is a matrix which is a function 158 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 of wi To investigate the stability of (11), the following Lyapunov function candidate is considered: V = x(t)T Px(t), where P ∈ n×n (12) is a constant symmetric positive definite matrix From (12) V˙ = x˙ (t)T Px(t) + x(t)T Px˙ (t) From (11) and (13)  (13)  V˙ = m1 (Hx(t) + Hx(t)) + m2  p nj − wj Bm Gj x(t) Px(t) wi Hi x(t) + i=1  T p  +x(t)T P m1 (Hx(t) + Hx(t)) + m2  j =1 p p wi Hi x(t) + i=1  nj − wj Bm Gj x(t) j =1 p = m1 x(t)T HT P + PH + HT P + P H x(t) + m2 i=1 p wi x(t)T HiT P + PHi x(t) ( nj − wj )x(t)T PBm Gj x(t) +2m2 j =1 p = m1 x(t) (−Q + H P + P H)x(t) − m2 T wi x(t)T Qi x(t) T i=1 p ( nj − wj )x(t)T PBm Gj x(t), +2m2 j =1 where Q = −(HT P + PH) and Qi = −(HiT P + PHi ) are constant symmetric positive definite matrices; T >0 (·) denotes the minimum eigenvalue of the input argument Let (Q − ( H P + P H)) V˙ p −m1 (Q − (  +2m2  H P + HP)) x(t) T − m2 wi x(t)T Qi x(t) i=1 p p nj x(t)T PBm Gj x(t) + j =1  wj |x(t)T PBm Gj x(t)| j =1 p −m1 x(t)  +2m2  − m2 wi x(t)T Qi x(t) i=1 p p nj x(t)T PBm Gj x(t) + j =1 j =1  wj |x(t)T PBm Gj x(t)| , (14) H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 159 where is a constant scalar and · denotes the l2 vector or induced matrix norm [23] Let the switching law of nj be defined as, nj = − Ksgn( )sgn(x(t)T PBm Gj x(t)) , j = 1, 2, , p, (15) where K > is a constant scalar to be determined and sgn(·) denotes the sign function From (14) and (15) V˙ p − m1 x(t) − m2 p i=1 As K > 1, wj ∈ [0 1] and (| |/ seen that p − 2m2 j =1 K| | − wi x(t) Qi x(t) + 2m2 T K| | j =1 ) 1, we have −(K| |/ ) + wj |x(t)T PBm Gj x(t)| + wj for all j Hence, it can be + wj |x(t)T PBm Gj x(t)| Thus, we have, V˙ p − m1 x(t) − m2 wi x(t)T Qi x(t) (16) i=1 Furthermore, ZE (q(x(t))) is designed such that (Q − ( HT P + P H)) > for the value of (q(x(t))) > (its normalized grade of membership m = 0) Q , i = 1, 2, , p is deigned to be i ZE symmetric positive definite Hence, it can be concluded that V˙ (equality holds when x(t) = 0) This implies that x(t) → as t → ∞ The analysis result is summarized by the following theorem p Theorem The closed-loop system of (9), with B(x(t)) = i=1 wi (x(t))Bi = (x(t))Bm and Bm is a constant matrix, is guaranteed to be asymptotically stable if there exists a constant symmetric matrix P such that the following linear matrix inequalities hold: P > 0, Q = −(HT P + PH) > 0, Qi = −(HiT P + PHi ) > 0, i = 1, 2, , p, p where H = Ao + Bo G; Hi = Ai + Bm Gi , Ao = i=1 wi (0)Ai and Bo = matrices The switching law of the switching controller is designed as ni = − Ksgn( )sgn(x(t)T PBm Gi x(t)) , p i=1 wi (0)Bi are constant i = 1, 2, , p with a constant K > and | (x(t))| ∈ [ max ] The membership function ZE (q(x(t))) is designed p such that (Q−( HT P+P H)) > for the value of ZE (q(x(t))) > where H = i=1 wi (Ai − p Ao ) + i=1 wi (Bi − Bo )G 160 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 3.2 Constant input matrix The case that B(x(t)) = p p i=1 wi Bi = B which is a constant input matrix is considered From (9)   wi Ai x(t) + BGx(t) + m2  x˙ (t) = m1 i=1 p p wi Ai x(t) + B i=1 nj Gj x(t) j =1 p = m1 Ao x(t) + BGx(t) + wi (Ai − Ao )x(t) i=1  p +m2  p wi (Ai + BGi )x(t) − i=1 j =1 p = m1 (Hx(t) + Ax(t)) + m2  nj Gj x(t) wi BGi x(t) + B i=1   p p (nj − wj )BGj x(t) , wi Hi x(t) + i=1  (17) j =1 p where A = i=1 wi (Ai − Ao ) is a matrix which is a function of wi and Hi = Ai + BGi is a constant matrix To investigate the stability of (17), the Lyapunov function candidate of (12) is considered From (13) and (17), similar to the proof of the previous case, we have   p V˙ = m1 (Hx(t) + Ax(t)) + m2  p wi Hi x(t) + i=1  T (nj − wj )BGj x(t) Px(t) j =1  p +x(t)T P m1 (Hx(t) + Ax(t)) + m2  p wi Hi x(t) + i=1  (nj − wj )BGj x(t) j =1 p = m1 x(t)T (HT P + PH + AT P + P A)x(t) + m2 i=1 p wi x(t)T (HiT P + PHi )x(t) (nj − wj )x(t)T PBGj x(t) +2m2 j =1 p = m1 x(t)T (−Q + AT P + P A)x(t) − m2 wi x(t)T Qi x(t) i=1 p (nj − wj )x(t)T PBGj x(t) +2m2 j =1 Let (Q − ( AT P + P A)) > and nj = −Ksgn(x(t)T PBGj x(t)), j = 1, 2, , p (18) H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 161 From (17) and (18), we have, V˙ p −m1 (Q − (  +2m2  A P + P A)) x(t) T − m2 wi x(t)T Qi x(t) i=1 p p nj x(t)T PBGj x(t) + j =1 wj |x(t)T PBGj x(t)| j =1 p −m1 x(t)  − m2 p wi x(t) Qi x(t) + 2m2 (−K + wj ) |x(t)T PBGj x(t)| T i=1 (19) j =1 As K > and wj ∈ [0 1], we have −K + wj < for all j Hence, V˙ (equality holds when x(t) = 0) This implies that x(t) → as t → ∞ The analysis result is summarized by the following theorem p Theorem The closed-loop system of (9), with B(x(t)) = i=1 wi (x(t))Bi = B and B is a constant matrix, is guaranteed to be asymptotically stable if there exists a constant symmetric matrix P such that the following linear matrix inequalities hold: P > 0, Q = −(HT P + PH) > 0, Qi = −(HiT P + PHi ) > 0, i = 1, 2, , p, where H = Ao + BG; Hi = Ai + BGi , Ao = of the switching controller is designed as ni = −Ksgn(x(t)T PBGi x(t)), p i=1 wi (0)Ai are constant matrices The switching law i = 1, 2, , p with a constant K > The membership function ZE (q(x(t))) is designed such that p P A)) > for the value of ZE (q(x(t))) > where A = i=1 wi (Ai − Ao ) (Q − ( AT P + It should be noted that the membership functions of the fuzzy rule-base combined controller are designed such that m1 = and m2 = in the operating domain around the origin and m1 = and m2 = in the operating domain far away from the origin Hence, in the operating domain that m1 = and m2 = 0, the switching state-feedback controller will be employed to drive the system state to the origin When the system state inside the operating domain that both m1 and m2 = 0, both the linear and switching state-feedback controllers will be employed The ratio of contribution of each controller is determined by the values of m1 and m2 The linear state-feedback controller will become dominant and the contribution of the switching state-feedback controller will vanish when the system state is inside the operating domain around the origin (i.e m1 = and m2 = 0) Inside this domain, the chattering effect introduced by the switching state-feedback controller totally disappears as only the linear state-feedback controller takes place to handle the nonlinear plant To alleviate the chattering effect during the transient period, a saturation function will be employed to replace the sign function The saturation function is 162 defined as follows:      sat(z) = −1     z T H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 for z Tz 1, − 1, T otherwise, for (20) where T is a non-zero positive scalar to be designed The saturation function may introduce steadystate error to the system state The magnitude of the steady-state error is related to the value of T The value of T should be designed such that the system state is able to be driven into the region with the value of ZE (q(x(t))) > ( (Q − ( HT P + P H)) > for the uncertain input matrix case or T P + P A)) (Q − ( A > for the constant input matrix case inside this region) Once the system states are inside this region, the linear state-feedback controller will gradually replace the switching state-feedback controller As a result, the chattering effect and the steady-state error will be eliminated eventually when the linear state-feedback controller completely dominates the control process The feedback gains of the linear and switching state-feedback controllers can be embedded in the LMI stability conditions of Theorems and [6,20] Hence, the feedback gains can be obtained by solving the LMIs The details can be found in [6,20] and will not be reiterated in this paper In this paper, a framework of the fuzzy rule-base combined controller, which combines the linear and switching state-feedback controllers, is proposed for nonlinear systems subject to parameter uncertainties In general, any controller can be combined by using the fuzzy rule-based combination techniques However, the main concern is to consider the characteristics of the controllers to be combined to benefit the control process Taking the fuzzy rule-based combination of linear and switching controllers as an example, the linear state-feedback controller provides good local stabilization ability without chatter effect and weaker robustness property while the switching state-feedback controller provides good global stabilization ability with chattering effect and stronger robustness property It can be seen that the characteristics of these two controllers can complement the deficiencies of each other Hence, the fuzzy rule-based combination technique can be employed to combine their advantages which are good to the control process By employing the proposed fuzzy rule-based combination technique to combine the conventional controllers, the good characteristics of the conventional controllers can be combined to form a better solution for controlling the nonlinear systems However, it can be seen that the LMI stability conditions are derived under the proposed framework of the fuzzy rule-base combined controller Hence, the LMI stability conditions are needed to be derived for combining different kinds of controllers The proposed fuzzy rule-base combined controller is subject to the following limitations: One, the nonlinear system should be represented by a fuzzy plant model in form of (2) Two, the input matrix p should exhibit the property that B(x(t)) = i=1 wi (x(t))Bi = (x(t))Bm where Bm is a constant matrix p (Theorem 1) or B(x(t)) = i=1 wi (x(t))Bi = B where B is a constant matrix (Theorem 2) If these conditions are satisfied, Theorems or can be applied to design a fuzzy rule-base combined controller for the nonlinear systems Application examples Two application examples will be given in this section Their details are given as follows 170 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 2.5 1.5 1.5 0.5 x2(t) (rad/s) x1(t) (rad) -0.5 0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2.5 -2 (a) Time (sec) 10 Time (sec) Time (sec) (b) 10 8000 6000 4000 u(t) (N) 2000 -2000 -4000 -6000 -8000 (c) 10 Fig System responses and control signals of the inverted pendulum with the proposed fuzzy rule-base combined controller using saturation function under m = kg and M = kg: (a) x1 (t); (b) x2 (t); (c) u(t)   f1  A2 =    k − Jm and   f1  max A4 =    k − Jm 0 0 0 f2max k − Jm gr2 f2max k − Jm gr2 0 Bm − Jm 0 Bm − Jm     ,    f1  max A3 =    k − Jm    ;      B=  Jm    ;  0 0 f2min k − Jm gr2 0 Bm − Jm       H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 171 2.5 1.5 1.5 x2(t) (rad/s) x1(t) (rad) 0.5 -0.5 0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -2.5 (a) Time (sec) 10 Time (sec) (b) 10 8000 6000 4000 u(t) (N) 2000 -2000 -4000 -6000 -8000 (c) 10 Time (sec) Fig 10 System responses and control signals of the inverted pendulum with the proposed fuzzy rule-base combined controller using saturation function under m = kg and M = 20 kg: (a) x1 (t); (b) x2 (t); (c) u(t) g l(t) m Gear Ratio: 1:gr m(t) u(t) k Fig 11 Single-link flexible joint l 172 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 1.5 0.8 0.6 0.4 x2(t) (rad/s) x1(t) (rad) 0.5 -0.5 0.2 -0.2 -0.4 -1 -0.6 -1.5 -0.8 -2 -1 (a) 10 x 104 0.5 x4(t) (rad/s) -0.5 -1.5 -6 -2 (c) -8 10 Time (sec) 10 10 x 104 -2 -4 -1 Time (sec) 1.5 (b) Time (sec) x3(t) (rad) (d) Time (sec) Fig 12 System responses of the single-link flexible joint with the proposed fuzzy rule-base combined controller under m = 10 kg: (a) x1 (t); (b) x2 (t); (c) x3 (t); (d) x4 (t) f1min = −4.9033 and f1max = −3.1217, f2min = −3.3333 × 104 and f2max = −2.2222 × 104 of which f1min f1 (x(t)) f1max and f2min f2 (x(t)) f2max that can be obtained analytically; wi = M1i (f1 (x(t))) × M2i (f2 (x(t))) k=1 ( M1k (f1 (x(t))) × M2k (f2 (x(t)))) ; M1 (f1 (x(t))) = −f1 (x(t)) + f1max f1max − f1min for = 1, 2; M (f1 (x(t))) = − M1 (f1 (x(t))) for = 3, 4; M2 (f2 (x(t))) = (−f2 (x(t)) + f2max )/ 1 (f2max − f2min ) for = 1, and M (f2 (x(t))) = − M1 (f2 (x(t))) for = 2, The fuzzy plant models 2 and its parameters are obtained by the algorithm which is detailed in [19] H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 173 x 105 u(t) (Nm) -1 -2 -3 Time (sec) 10 Fig 13 Control signal of the fuzzy rule-base combined controller for the single-link flexible joint under x(0) = 0 m = 10 kg T and The linear state-feedback controller of the proposed fuzzy rule-base combined controller is designed based on the following system and input matrices:   0  f1 f2o  o    0  Ao = wi (0)Ai =    k Bm  k i=1 − − Jm Jm gr2 Jm 3k 3k and B where f1o = 3g 2l − mo l , f2o (x(t)) = − mo l gr and mo = 10.75 kg The linear state-feedback gain is designed as G = [−98369.0376 −15770.0000 −7.1378 −3.0500] such that the eigenvalues of H = Ao + BG are located at −1, −1.5, −2.5 and −3 which is arbitrarily chosen The feedback gains of the switching controller are designed as G1 = [−91601.2829 − 14686.6835 − 7.1377 − 3.0500], G2 = [−137401.9743 −22030.0252 −7.1377 −3.0500], G3 = [−60087.5725 2434.5081 −7.8503 −3.0500] and G4 = [−90131.4087 3651.7622 − 7.8503 − 3.0500] such that the eigenvalues of Hi = Ai + BGi , i = 1, 2, 3, are located at −1, −1.5, −2.5 and −3 With the MATLAB LMI toolbox, it can be found that   83311.2191 −31987.8179 6.6626 0.7922  −31987.8179 40773.7767 −4.4047 −0.4979   P= −4  6.6626 −4.4047 7.6616 × 10 8.6689 × 10−5  0.7922 −0.4979 8.6689 × 10−5 1.5555 × 10−5 such that the stability conditions in Theorem are satisfied With the help of genetic algorithm [12], it = 3.6450 × 10−6 > for |x1 (t)| 0.2 According to can be found that (Q − ( AT P + AP)) 174 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 1.5 0.8 0.6 0.4 x2(t) (rad/s) x1(t) (rad) 0.5 -0.5 0.2 -0.2 -0.4 -1 -0.6 -1.5 -2 -0.8 (a) -1 10 x 104 1.5 0.5 -0.5 Time (sec) 10 10 x 104 -2 -1 -4 -1.5 -6 -2 (b) x4(t) (rad/s) x3(t) (rad) Time (sec) -8 (c) 10 (d) Time (sec) Time (sec) Fig 14 System responses of the single-link flexible joint with the proposed fuzzy rule-base combined controller under m = 11.5 kg: (a) x1 (t); (b) x2 (t); (c) x3 (t); (d) x4 (t) Theorem 2, the switching law is defined as ni = K sgn(x(t)T PHi x(t)), i = 1, 2, 3, with an arbitrary (t)| chosen K = 1.01 ZE (q(x(t))) and NZ (q(x(t))) are designed as shown in Fig with q(x(t)) = |x0.2 Fig 12 shows the system responses of the single-link flexible joint with the proposed fuzzy rule-base T T combined controller under the initial state conditions of x(0) = 0 , x(0) = 0 , x(0) = T − 0 , x(0) = − 0 T and m = 10 kg Fig 13 shows the control signal of the fuzzy rule-base T combined controller under x(0) = 0 and m = 10 kg Figs 14 and 15 show the system responses and the control signal of the fuzzy rule-base combined controller under different initial state conditions with m = 11.5 kg It can be seen that the proposed fuzzy rule-base combined controller can stabilize the single-link flexible joint under different initial state conditions and parameter values Furthermore, the chattering effect is eliminated when the system states enter the region of the linear controller H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 175 x 105 u(t) (Nm) -1 -2 -3 Time (sec) 10 Fig 15 Control signal of the fuzzy rule-base combined controller for the single-link flexible joint under x(0) = 0 m = 11.5 kg T and For comparison purpose, a sliding-mode [2] and a published fuzzy controller [24] will be employed to handle the single-link flexible joint The controller designs are given in Appendices A and B, respectively Fig 16 shows the system responses of the sliding-mode, the published fuzzy and the proposed fuzzy rulebase combined controllers Referring to this figure, it can be seen that the proposed fuzzy controller provides the best performance in terms of faster transient responses and shorter settling time Fig 17 shows the control signals of the sliding-mode and the published fuzzy controllers In this figure, it can be seen that the maximum magnitude of the control signal of the proposed fuzzy rule-base combined controller is larger than that of the sliding-mode controller However, the chattering effect of the proposed fuzzy rule-base combined approach is eliminated when the system state approaches the origin Although the published fuzzy controller provides a smooth control signal, the stability of the fuzzy controller is in fact not guaranteed as the solution to the stability conditions [24] cannot be found To alleviate the chattering effect during the transient period, a saturation function with T = 105 is employed to replace the sign function in the proposed fuzzy rule-base combined controller It can be seen from Figs 18 and 19 that the chattering effect in the system states and control signals are significantly alleviated, and there is no steady-state error appearing in the system states Conclusion A fuzzy rule-base combined controller, which is a fuzzy rule-based combination of linear and switching state-feedback controllers, has been proposed to handle nonlinear systems subject to parameter uncertainties The fuzzy rule-base combined controller and the switching law have been designed under the consideration of the system stability By using the fuzzy rule-based combination technique, the fuzzy rule-base combined controller has integrated the advantages of both linear and switching state-feedback controllers but eliminated their disadvantages As a result, the proposed fuzzy controller provides good 176 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 0.5 1.5 x2(t) (rad/s) x1(t) (rad) 0.5 -0.5 -1 -1.5 -0.5 -1 -2 15 Time (sec) (a) x 104 0 -0.5 -1 -2 -8 (c) 10 Time (sec) -10 15 (d) 15 10 15 x 104 -4 -6 10 -2 -1.5 Time (sec) 0.5 -2.5 (b) x4(t) (rad/s) x3(t) (rad) 10 Time (sec) Fig 16 System responses of the single-link flexible joint with the sliding-mode (solid lines), the published fuzzy (dotted lines) and the proposed fuzzy rule-base combined (dash-dotted lines) controllers: (a) x1 (t); (b) x2 (t); (c) x3 (t); (d) x4 (t) performance during the transient period and the chattering effect has been removed when the system state approaches the origin A saturation function has been employed to replace the switching component to alleviate the chattering during the transient period The proposed fuzzy rule-based combination technique is also able to eliminate the steady state error introduced by the saturation function Application examples have been given to illustrate the merits of the proposed approach Acknowledgements The work described in this paper was fully supported by a Grant from the The Hong Kong Polytechnic University (Project No G-YX31) H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 177 x 105 1.5 0.5 u(t) (Nm) -0.5 -1 -1.5 -2 -2.5 -3 10 (a) 15 Time (sec) x 104 u(t) (Nm) -2 -4 -6 -8 -10 -12 (b) Time (sec) 10 Fig 17 Control signals of the sliding-model and the published fuzzy controllers under m = 11.5 kg: (a) u(t) of the sliding-mode controller; (b) u(t) of the published fuzzy controller Appendix A The sliding-mode controller [2] for the inverted pendulum and the single-link flexible joint will be derived Considering the following Lyapunov function candidate: V = 21 s(x(t))T s(x(t)), where s(x(t)) = Fx(t) is the switching plane and F ∈ (A.1) m×n V˙ = s(x(t))T s˙ (x(t)) = s(x(t))T F(A(x(t))x(t) + B(x(t))u(t)), where A(x(t)) = i=1 wi (x(t))Ai and B(x(t)) = (A.2) i=1 wi B i 178 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 1.5 0.8 0.6 0.4 0.5 x2(t) (rad) x1(t) (rad) -0.5 0.2 -0.2 -0.4 -1 -0.6 -1.5 -0.8 -2 -1 (a) Time (sec) 10 1.5 0.5 -0.5 (c) Time (sec) 10 Time (sec) 10 -2 -4 -1.5 -6 -1 x 104 x4(t) (rad) x3(t) (rad) x 104 -2 (b) Time (sec) -8 10 (d) x 105 u(t) (Nm) -1 -2 -3 (e) Time (sec) 10 Fig 18 System responses and control signals of the single-link flexible joint with the proposed fuzzy rule-base combined controller using saturation function under m = 10 kg: (a) x1 (t); (b) x2 (t); (c) x3 (t); (d) x4 (t); (e) u(t) H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 1.5 0.8 179 0.6 x2(t) (rad/s) x1(t) (rad) 0.4 0.5 -0.5 0.2 -0.2 -0.4 -1 -0.6 -1.5 -2 -0.8 (a) x 104 0.5 -1.5 -6 5 Time (sec) 10 10 -2 -4 -1 x 104 1.5 -2 (b) -0.5 (c) -1 10 x4(t) (rad/s) x3(t) (rad) Time (sec) -8 10 (d) Time (sec) Time (sec) x 105 u(t) (Nm) -1 -2 -3 (e) Time (sec) 10 Fig 19 System responses and control signals of the single-link flexible joint with the proposed fuzzy rule-base combined controller using saturation function under m = 11.5 kg: (a) x1 (t); (b) x2 (t); (c) x3 (t); (d) x4 (t); (e) u(t) 180 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 A.1 Inverted pendulum f1 (x(t)) and B(x(t)) = (x(t))Bm where Bm = 0 ¯ 0 and A2 (x(t)) = f1 (x(t)) Hence, from (A.2), For the inverted pendulum, we have A(x(t)) = ¯1+A ¯ (x(t)) where A ¯1 = Let A(x(t)) = A we have ¯1 +A ¯ (x(t)))x(t) + (x(t))Bm u(t)) V˙ = s(x(t))T F((A T ¯ ¯ (x(t))x(t) + s(x(t))T (x(t))FBm u(t) = s(x(t)) FA1 x(t) + s(x(t))T FA (A.3) Let the control signal be u(t) =− ¯ x(t)| + sgn(s(x(t)))|F||A ¯ 2max (x(t))| |x(t)| + s(x(t))) K(FBm )−1 sgn( (x(t)))(sgn(s(x(t)))|FA , (A.4) ¯ (x(t)), K > and max(·) denotes the maximum ¯ 2max (x(t))| = A where |A value of the arguments From (A.3) and (A.4), we have ¯ x(t)| + |s(x(t))T | |F| |A ¯ (x(t))| |x(t)| − K| (x(t))| |s(x(t))T | |FA ¯ x(t)| V˙ |s(x(t))T | |FA max(|f1min |,|f1max |) 0 − K| (x(t))| 1− − ¯ 2max (x(t))| |x(t)| − |s(x(t))T | |F| |A K| (x(t))| K| (x(t))| K| (x(t))| min s(x(t))T s(x(t)) ¯ x(t)| + |s(x(t))T | |F| |A ¯ (x(t))| |x(t)| |s(x(t))T | |FA ¯ 2max (x(t))| |x(t)| − K| (x(t))| s(x(t))T s(x(t)) |s(x(t))T | |F| |A With the facts that K| (x(t))| K| (x(t))| ⇒ |s(x(t))T | |s(x(t))T | min ¯ (x(t))| |A ¯ 2max (x(t))| ⇒ |F| |A ¯ (x(t))| |x(t)| |F| |A ¯ 2max (x(t))| |x(t)|, it can be concluded that and |A ¯ (x(t))| |x(t)| − K| (x(t))| |s(x(t))|T |F| |A ¯ 2max (x(t))| |x(t)| |s(x(t))T | |F| |A (A.5) From (A.5) and the fact that − V˙ − K| (x(t))| K| (x(t))| s(x(t))T s(x(t)) 0, it can be seen that (A.6) Consequently, V˙ (equality holds when s(x(t)) = 0) which implies s(x(t)) → as t → ∞ When s(x(t)) = Fx(t) = (⇒ s˙ (x(t)) = F˙x(t) = 0) occurs where F = [F1 F2 ], the system dynamics of the inverted pendulum are reduced to F2 x(t) ˙ = −F1 x(t) K is chosen to be 11 by trial-and-error for best performance We select F = [2 1] such that the eigenvalue of the reduced system is located at −2 which is similar to the design criterion of the fuzzy rule-base combined controller H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 181 A.2 Single-link flexible joint For the single-link flexible joint, we have  0  f1 (x(t)) f2 (x(t))  0 A(x(t)) =    k Bm k − − − Jm Jm Jm gr2 Let       and   B(x(t)) = B =   Jm  ¯1 +A ¯ (x(t)) A(x(t)) = A and    f1 (x(t)) ¯ (x(t)) =  A  0 0 0   ¯ (x(t)) =  where A   k − Jm f2 (x(t)) 0 0 0 0 k − Jm gr2 0 Bm − Jm             0  0 Hence, from (A.2), we have, ¯1 +A ¯ (x(t)))x(t) + Bu(t)) V˙ = s(x(t))T F((A The control signal is chosen as ¯ x(t)| + sgn(s(x(t)))|F| |A ¯ 2max (x(t)| |x(t)| + s(x(t))), u(t) = −K(FB)−1 (sgn(s(x(t))) |FA (A.7) where  0  max(|f1 |, |f1 |) max ¯ 2max (x(t))| =  |A  0 0  max(|f2min |, |f2max |)   0 0 ¯ (x(t)) A With the sliding-mode controller of (A.7) and following similar derivation procedure, it can be shown that V˙ (equality holds when s(x(t)) = 0) which implies s(x(t)) → as t → ∞ Similarly, when s(x(t)) = Fx(t) = occurs where F = [F1 F2 F3 F4 ], we have F4 x4 (t) = −F1 x1 (t)−F2 x2 (t)−F3 x3 (t) Hence, the system dynamics of the single-link flexible joint are reduced to      x˙1 (t) x1 (t) f2 (x(t))   f (x(t))  x˙2 (t)  =   (A.8)  F F2 F3  x2 (t) x˙3 (t) x3 (t) − − − F4 F4 F4 182 H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 K is chosen to be by trail-and-error for best performance Under the similar design criterion of the fuzzy rule-base combined controller for the single-link flexible joint, the eigenvalues of the reduced f1 +f1 f2 +f2 system of (8), with f1 (x(t)) and f2 (x(t)) taking the values of max and max , are located at −1, −2 and −3, respectively Consequently, we have F = [6.5135×104 −2.5180×104 6.0000 1.0000] Appendix B The fuzzy controllers [24] for the inverted pendulum and the single-link flexible joint will be given in this appendix The rules of the fuzzy controller are of the following format: j j Rule j : IF f¯1 (x(t)) is N1 AND f¯2 (x(t)) is N2 Then u(t) = Gj x(t), j = 1, 2, 3, (B.1) The output of the fuzzy controller is defined as mj (x(t))Gj x(t) u(t) = (B.2) j =1 The membership functions of the fuzzy controller of (B.2) are designed as the nominal membership functions (the uncertain parameters take the nominal values) of the fuzzy plant model of the nonlinear plant: j N1 mj = (f¯1 (x(t))) × j N2 (f¯2 (x(t))) ¯ ¯ l=1 ( N1l (f1 (x(t))) × N2l (f2 (x(t)))) ¯ ¯1 (x(t))) = −f1 (x(t)) + f1max ( f N1 f1max − f1min ¯ N1 (f1 (x(t))) =1− ¯ N11 (f1 (x(t))) −f¯2 (x(t)) + f2max f2max − f2min ¯ (f¯2 (x(t))) = − N (f2 (x(t))) N ¯ N2 (f2 (x(t))) = 2 for ; = 1, 2; for = 3, 4; for = 1, and for = 2, The fuzzy control system is guaranteed to be stable [24] if there exists a symmetry positive-definite matrix T P + PH are negative definite where H = A + B G P such that all Hij ij ij i i j B.1 Inverted pendulum For the inverted pendulum, the inputs of the fuzzy controller are defined as g − ao mo lx2 (t)2 cos(x1 (t)) sin(x1 (t)) f¯1 (x(t)) = and x1 (t) 4l/3 − ao mo l cos2 (x1 (t)) ao cos(x1 (t)) f¯2 (x(t)) = − 4l/3 − ao mo l cos2 (x1 (t)) H.K Lam, F.H.F Leung / Fuzzy Sets and Systems 156 (2005) 153 – 184 183 The feedback gains are designed based on the parallel distributed compensation (PDC) design approach [24] such that the eigenvalues of Hii , i = 1, 2, 3, 4, are the same as those of the fuzzy rule-base combined controller Consequently, we have G1 = [67.9887 22.6629], G2 = [256.4103 85.4701], G3 = [118.9802 22.6629] and G4 = [448.7179 85.4701] B.2 Single-link flexible joint For the single-link flexible joint, the inputs of the fuzzy controller are defined as 3g f¯1 (x(t)) = 2l sin(x1 (t)) x1 (t) − 3k mo l and f¯2 (x(t)) = − 3k mo l gr The feedback gains are designed based on the PDC design approach [24] such that the eigenvalues of Hii , i = 1, 2, 3, 4, are the same as those of the fuzzy rule-base combined controller Consequently, we have G1 = [−91601.2829 − 14686.6835 − 7.1377 − 3.0500], G2 = [−137401.9743 − 22030.0252 − 7.1377 − 3.0500], G3 = [−60087.5725 2434.5081 − 7.8503 − 3.0500] and G4 = 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