Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P3 docx

Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach Kazuo Tanaka, Hua O Wang Copyright ᮊ 2001 John Wiley & Sons, Inc ISBNs: 0-471-32324-1 ŽHardback.; 0-471-22459-6 ŽElectronic CHAPTER LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS The preceding chapter introduced the concept and basic procedure of parallel distributed compensation and LMI-based designs The goal of this chapter is to present the details of analysis and design via LMIs This chapter forms a basic and important component of this book To this end, it will be shown that various kinds of control performance specifications can be represented in terms of LMIs The control performance specifications include stability conditions, relaxed stability conditions, decay rate conditions, constrains on control input and output, and disturbance rejection for both continuous and discrete fuzzy control systems w1᎐3x Other more advanced control performance considerations utilizing LMI conditions will be presented in later chapters 3.1 STABILITY CONDITIONS In the 1990’s, the issue of stability of fuzzy control systems has been investigated extensively in the framework of nonlinear system stability w1᎐18x Today, there exist a large number of papers on stability analysis of fuzzy control in the literature This section discusses some basic results on the stability of fuzzy control systems In the following, Theorems and deal with stability conditions for the open-loop systems Theorem can be readily obtained via Lyapunov stability theory The proof of Theorem is given in w4, 7x 49 50 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS THEOREM wCFSx The equilibrium of the continuous fuzzy system Ž2.3 with uŽ t s is globally asymptotically stable if there exists a common positi®e definite matrix P such that AT P q PA i - 0, i i s 1, 2, , r , Ž 3.1 that is, a common P has to exist for all subsystems THEOREM wDFSx The equilibrium of the discrete fuzzy system Ž2.5 with uŽ t s is globally asymptotically stable if there exists a common positi®e definite matrix P such that AT PA i y P - 0, i i s 1, 2, , r , Ž 3.2 that is, a common P has to exist for all subsystems Next, let us consider the stability of the closed-loop system By substituting Ž2.23 into Ž2.3 and Ž2.5., we obtain Ž3.3 and Ž3.4., respectively CFS r r x ˙Ž t s Ý Ý h i Ž z Ž t h j Ž z Ž t Ä A i y Bi Fj x Ž t Ž 3.3 is1 js1 DFS r x Ž t q s r Ý Ý h i Ž z Ž t h j Ž z Ž t Ä A i y Bi Fj x Ž t Ž 3.4 is1 js1 Denote Gi j s A i y Bi Fj Equations Ž3.3 and Ž3.4 can be rewritten as Ž3.5 and Ž3.6., respectively CFS r x ˙Ž t s Ý h i Ž z Ž t h i Ž z Ž t Gii x Ž t is1 r q2 Ý Ý hi Ž z Ž t h j Ž z Ž t is1 i-j ½ Gi j q Gji Ž x t Ž 3.5 DFS r x Ž t q s Ý h i Ž z Ž t h i Ž z Ž t Gii x Ž t is1 r q2 Ý Ý hi Ž z Ž t h j Ž z Ž t is1 i-j ½ Gi j q Gji Ž x t Ž 3.6 STABILITY CONDITIONS 51 By applying the stability conditions for the open-loop system ŽTheorems and to Ž3.5 and Ž3.6., we can derive stability conditions for the CFS and the DFS, respectively THEOREM wCFSx The equilibrium of the continuous fuzzy control system described by Ž3.5 is globally asymptotically stable if there exists a common positi®e definite matrix P such that T Gii P q PGii - 0, ž Gi j q Gji Ž 3.7 T / PqP ž Gi j q Gji / F 0, i - j s.t h i l h j / ␾ Ž 3.8 Proof It follows directly from Theorem For the explanation of the notation i - j s.t h i l h j / ␾ , refer to Chapter THEOREM wDFSx The equilibrium of the discrete fuzzy control system described by Ž3.6 is globally asymptotically stable if there exists a common positi®e definite matrix P such that T Gii PGii y P - 0, ž Gi j q Gji T / ž P Ž 3.9 Gi j q Gji / y P F 0, i - j s.t h i l h j / ␾ Ž 3.10 Proof It follows directly from Theorem The fuzzy control design problem is to determine Fj ’s Ž j s 1, 2, , r which satisfy the conditions of Theorem or with a common positive definite matrix P Consider the common B matrix case, that is, B1 s B2 s иии s Br In this case, the stability conditions of Theorems and can be simplified as follows COROLLARY Assume that B1 s B2 s иии s Br The equilibrium of the fuzzy control system Ž3.5 is globally asymptotically stable if there exists a common positi®e definite matrix P satisfying Ž3.7 COROLLARY Assume that B1 s B s иии s Br The equilibrium of the fuzzy control system Ž3.6 is globally asymptotically stable if there exists a common positi®e definite matrix P satisfying Ž3.9 52 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS T In other words, the corollaries state that in the common B case, Gii P q PGii - implies ž Gi j q Gji T / PqP ž Gi j q Gji / F0 T and Gii PGii y P - implies ž Gi j q Gji T / ž P Gi j q Gji / yPF0 To check stability of the fuzzy control system, it has long been considered difficult to find a common positive definite matrix P satisfying the conditions of Theorems 5᎐8 A trial-and-error type of procedure was first used w4, 7, 9x In w19x, a procedure to construct a common P is given for second-order fuzzy systems, that is, the dimension of the state is It was first stated in w11, 12, 17x that the common P problem for fuzzy controller design can be solved numerically, that is, the stability conditions of Theorems 5᎐8 can be expressed in LMIs For example, to check the stability conditions of Theorem 7, we need to find P satisfying the LMIs P ) 0, ž Gi j q Gji T Gii P q PGii - 0, T / PqP ž Gi j q Gji / F 0, i - j s.t h i l h j / ␾ , or determine that no such P exists This is a convex feasibility problem As shown in Chapter 2, this feasibility problem can be numerically solved very efficiently by means of the most powerful tools available to date in the mathematical programming literature 3.2 RELAXED STABILITY CONDITIONS We have shown that the stability analysis of the fuzzy control system is reduced to a problem of finding a common P If r, that is the number of IF-THEN rules, is large, it might be difficult to find a common P satisfying the conditions of Theorem Žor Theorem This section presents new stability conditions by relaxing the conditions of Theorems and Theorems and 10 provide relaxed stability conditions w1᎐3x First, we need the following corollaries to prove Theorems and 10 COROLLARY r r Ý h2 Ž z Ž t y r y Ý Ý h i Ž z Ž t h j Ž z Ž t G 0, i is1 is1 i-j RELAXED STABILITY CONDITIONS 53 where r Ý h i Ž z Ž t s 1, hi Ž z Ž t G is1 for all i Proof It holds since r r Ý h2 Ž z Ž t y r y Ý Ý h i Ž z Ž t h j Ž z Ž t i is1 is1 i-j s ry1 r Ý Ý Ä hi Ž z Ž t y h j Ž z Ž t G Q.E.D is1 i-j COROLLARY If the number of rules that fire for all t is less than or equal to s, where - s F r, then r r Ý h2 Ž z Ž t y s y Ý Ý h i Ž z Ž t h j Ž z Ž t G 0, i is1 is1 i-j where r Ý h i Ž z Ž t s 1, hi Ž z Ž t G is1 for all i Proof It follows directly from Corollary THEOREM wCFSx Assume that the number of rules that fire for all t is less than or equal to s, where - s F r The equilibrium of the continuous fuzzy control system described by Ž3.5 is globally asymptotically stable if there exist a common positi®e definite matrix P and a common positi®e semidefinite matrix Q such that T Gii P q PGii q Ž s y Q - ž Gi j q Gji T / PqP ž Gi j q Gji Ž 3.11 / y Q F 0, i - j s.t h i l h j / ␾ where s ) Ž 3.12 54 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS Proof Consider a candidate of Lyapunov function V Ž x Ž t s x T Ž t Px Ž t , where P ) Then, ˙ VŽ xŽ t s r r Ý Ý hi Ž z Ž t h j Ž z Ž t xT Ž t is1 js1 = Ž A i y Bi Fj T P q P Ž A i y Bi Fj x Ž t r s Ý h2 Ž z Ž t x T Ž t i T Gii P q PGii x Ž t is1 r qÝ Ý hi Ž z Ž t h j Ž z Ž t xT Ž t is1 i-j = ž Gi j q Gji where T / PqP ž Gi j q Gji / xŽ t , Gi j s A i y Bi Fj From condition Ž3.12 and Corollary 4, we have ˙ VŽ xŽ t F r Ý h2 Ž z Ž t x T Ž t i T Gii P q PGii x Ž t is1 r qÝ Ý h i Ž z Ž t h j Ž z Ž t x T Ž t Qx Ž t is1 i-j r F Ý h2 Ž z Ž t x T Ž t i T Gii P q PGii x Ž t is1 r q Ž s y Ý h2 Ž z Ž t x T Ž t Qx Ž t i is1 r s Ý h2 Ž z Ž t x T Ž t i T Gii P q PGii q Ž s y Q x Ž t is1 ˙ If condition Ž3.11 holds, V Ž x Ž t - at x Ž t / Q.E.D THEOREM 10 wDFSx Assume that the number of rules that fire for all t is less than or equal to s, where - s F r The equilibrium of the discrete fuzzy control system described by Ž3.6 is globally asymptotically stable if there exist a common positi®e definite matrix P and a common positi®e semidefinite matrix Q such that T Gii PGii y P q Ž s y Q - 0, ž where s ) Gi j q Gji T / ž P Gi j q Gji / Ž 3.13 y P y Q F 0, i - j s.t h i l h j / ␾ , Ž 3.14 RELAXED STABILITY CONDITIONS 55 Proof Consider a candidate of Lyapunov function V Ž x Ž t s x T Ž t Px Ž t , where P ) Then, ⌬V Ž x Ž t s V Ž x Ž t q y V Ž x Ž t r s r r r Ý Ý Ý Ý hi Ž z Ž t h j Ž z Ž t hk Ž z Ž t hl Ž z Ž t is1 js1 ks1 ls1 = x T Ž t GiTj PG k l y P x Ž t s r r r Ý Ý Ý Ý hi Ž z Ž t h j Ž z Ž t hk Ž z Ž t hl Ž z Ž t is1 js1 ks1 ls1 =x T Ž t F r Ž Gi j q Gji T P Ž Gk l q Gl k y P x Ž t r Ý Ý hi Ž z Ž t h j Ž z Ž t xT Ž t HiT PHi j y P x Ž t j is1 js1 r s r r T Ý Ý hi Ž z Ž t h j Ž z Ž t x Ž t HiT j is1 js1 P Hi j y P xŽ t r s Ý h2 Ž z Ž t x T Ž t i T Gii PGi i y P x Ž t is1 r q2 Ý T Ý hi Ž z Ž t h j Ž z Ž t x Ž t is1 i-j HiT j P Hi j y P xŽ t , where Hi j s Gi j q Gji From condition Ž3.14 and Corollary 4, the right side of the above inequality becomes r F Ý h2 Ž z Ž t x T Ž t i T Gii PGi i y P x Ž t is1 r q2 Ý Ý h i Ž z Ž t h j Ž z Ž t x T Ž t Qx Ž t is1 i-j r F Ý h2 Ž z Ž t x T Ž t i T Gii PGii y P x Ž t is1 r q Ž s y Ý h2 Ž z Ž t x T Ž t Qx Ž t i is1 r s Ý h2 Ž z Ž t x T Ž t i T Gii PGii y P q Ž s y Q x Ž t is1 If condition Ž3.13 holds, ⌬V Ž x Ž t - at x Ž t / Q.E.D 56 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS Corollary is used in the proofs of Theorems and 10 The use of Corollary would lead to conservative results because s F r Remark 11 It is assumed in the derivations of Theorems 7᎐10 that the weight h i Ž z Ž t of each rule in the fuzzy controller is equal to that of each rule in the fuzzy model for all t Note that Theorems 7᎐10 cannot be used if the assumption does not hold This fact will show up again in a case Žcase B of fuzzy observer design given in Chapter If the assumption does not hold, the following stability conditions should be used instead of Theorems 7᎐10: GiTj P q PGji - in the CFS case and GiTj PGji y P - in the DFS case These conditions imply those of Theorems 7᎐10 These conditions may be regarded as robust stability conditions for premise part uncertainty w18x Fig 3.1 Feasible area for the stability conditions of Theorem RELAXED STABILITY CONDITIONS Fig 3.2 57 Feasible area for the stability conditions of Theorem The conditions of Theorems and 10 reduce to those of Theorems and 8, respectively, when Q s Example This example demonstrates the utility of the relaxed conditions in the CFS case Consider the CFS, where r s s s 2, A1 s y10 , B1 s , A2 s a y10 , B2 s b The local feedback gains F1 and F2 are determined by selecting wy2 y2x as the eigenvalues of the subsystems in the PDC Figures 3.1 and 3.2 show the feasible areas satisfying the conditions of Theorems and for the variables a and b, respectively In these figures, the feasible areas are plotted for a ) and b ) 20 A common P Žand a common Q satisfying the conditions of Theorem ŽFigure 3.1 and Theorem ŽFigure 3.2 exists if and only if the system parameters a and b are located in the feasible areas under a ) and b ) 20 It is found in these figures that the conditions of Theorem lead to conservative results 58 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS 3.3 STABLE CONTROLLER DESIGN This section presents stable fuzzy controller designs for CFS and DFS We first present a stable fuzzy controller design problem which is to determine the feedback gains Fi for the CFS using the stability conditions of Theorem The conditions Ž3.7 and Ž3.8 are not jointly convex in Fi and P Now multiplying the inequality on the left and right by Py1 and defining a new variable X s Py1 , we rewrite the conditions as yXAT yA i X q XFiT BiT q Bi Fi X ) 0, i yXAT yA i X y XAT y A j X i j qXFjT BiT q Bi Fj X q XFiT B jT q B j Fi X G Define Mi s Fi X so that for X ) we have Fi s Mi Xy1 Substituting into the above inequalities yields yXAT yA i X q MiT BiT q Bi Mi ) 0, i yXAT yA i X y XAT y A j X i j qM jT B iT q Bi M j q MiT B jT q B j Mi G Using these LMI conditions, we define a stable fuzzy controller design problem Stable Fuzzy Controller Design: CFS Find X ) and Mi Ž i s 1, , r satisfying yXAT yA i X q MiT B iT q Bi Mi ) 0, i Ž3.15 yXAT yA i X y XAT y A j X i j qM jT BiT q Bi M j q MiT B jT q B j Mi G 0, i - j s.t h i l h j / ␾ Ž 3.16 where X s Py1 , Mi s Fi X Ž 3.17 The above conditions are LMIs with respect to variables X and Mi We can find a positive definite matrix X and Mi satisfying the LMIs or determination that no such X and Mi exist The feedback gains Fi and a common P can be obtained as P s Xy1 , from the solutions X and Mi Fi s Mi Xy1 Ž 3.18 68 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS Multiplying both side of the above inequality by block-diag w X I x gives r Ý hi Ž z Ž t is1 X MiT Mi ␮2 I G Hence we arrive at the condition Ž3.47 This derivation is more direct and compact Q.E.D The LMIs are available for both CFSs and DFSs A design problem of stable fuzzy controllers satisfying the input constraint can be defined as follows: Find X ) 0, Y G 0, and Mi Ž i s 1, , r satisfying Ž3.23 and Ž3.24 wor Ž3.27 and Ž3.28.x and Ž3.46 and Ž3.47 3.5.2 Constraint on the Output THEOREM 12 Assume that the initial condition x Ž0 is known The constraint y Ž t F ␭ is enforced at all times t G if the LMIs x Ž x Ž X T X XC iT Ci X ␭2 I G 0, Ž 3.53 G0 Ž 3.54 hold, where X s Py1 Proof The proof can be completed in the same procedure as in Theorem 11 The LMIs are available for both CFSs and DFSs A design problem of stable fuzzy controllers satisfying the output constraint can be defined as follows: Find X ) 0, Y G 0, and Mi Ž i s 1, , r satisfying Ž3.23 and Ž3.24 wor Ž3.27 and Ž3.28.x and Ž3.53 and Ž3.54 3.6 INITIAL STATE INDEPENDENT CONDITION The above LMI design conditions for input and output constraints depend on the initial states of the system This means that the feedback gains Fi must be again determined using the above LMIs if the initial states x Ž0 change This is a disadvantage of using the LMIs on the control input and output We modify the LMI constraints on the control input and output, where x Ž0 is unknown but the upper bound ␾ of x Ž0.5 is known, that is, x Ž0.5 F ␾ To encompass a large set of initial states, we can set ␾ to be a large quantity even if x Ž0 is unknown Of course, a large ␾ could lead to conservative designs The modified LMI is accomplished by the following results DISTURBANCE REJECTION 69 THEOREM 13 Assume that x Ž0.5 F ␾ , where x Ž0 is unknown but the upper bound ␾ is known Then, x T Ž Xy1 x Ž F Ž 3.55 ␾2I F X , Ž 3.56 if where X s Py1 Proof From Ž3.56., Xy1 F ␾2 I Therefore, x T Ž Xy1 x Ž F ␾2 x T Ž x Ž F Q.E.D Note that Ž3.55 is equivalent to Ž3.46 and Ž3.53 The condition Ž3.56 can be used instead of Ž3.55 A design example using the initial state independent condition will be presented in Chapter 3.7 DISTURBANCE REJECTION This section presents a disturbance rejection fuzzy controller design for the Takagi-Sugeno fuzzy models Consider the following CFS with disturbance w1x: r x ˙Ž t s Ý h i Ž z Ž t Ä A i x Ž t q B i u Ž t q Ei © Ž t , Ž 3.57 is1 r yŽ t s Ý h i Ž z Ž t Ci x Ž t , Ž 3.58 is1 where © Ž t is the disturbance The disturbance rejection can be realized by minimizing ␥ subject to sup © Ž t 2/0 yŽ t 52 ©Ž t 52 F ␥ Ž 3.59 THEOREM 14 wCFSx The feedback gains Fi that stabilize the fuzzy model and minimize ␥ in Ž3.59 can be obtained by sol®ing the following minimization problem based on LMIs 70 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS minimize ␥ x, M , , M r subject to X ) 0, ž y Ä XAT y M jT B iT q A i X y B i M j i qXAT y MiT B jT q A j X y B j Mi j y Ž Ei q E j 2 / y Ž Ei q E j T X Ž Ci q C j G 0, ␥ 2I 0 Ž Ci q C j X T I i F j s.t h i Ž z Ž t l h j Ž z Ž t / ␾ , Ž 3.61 where Mi s Fi X Proof Suppose there exists a quadratic function V Ž x Ž t s x T Ž t Px Ž t , P ) 0, and ␥ G such that, for all t, ˙ V Ž x Ž t q yT Ž t y Ž t y ␥ ©T Ž t © Ž t F Ž 3.62 for Ž3.57 and Ž3.58 By integrating Ž3.62 from to T, we obtain T ˙ H0 Ž V Ž x Ž t q y T Ž t y Ž t y ␥ © T Ž t © Ž t dt F By assuming that initial condition x Ž0 s 0, we have VŽ xŽT q T H0 Ž y T Ž t y Ž t y ␥ © T Ž t © Ž t dt F Since V Ž x ŽT G 0, this implies yŽ t 52 ©Ž t 52 F␥ Ž 3.63 71 DISTURBANCE REJECTION Therefore the L2 gain of the fuzzy model is less than ␥ if Ž3.62 holds We derive an LMI condition from Ž3.62 From Ž3.62., x ˙T Ž t Px Ž t q x T Ž t Px Ž t ˙ r r Ý h i Ž z Ž t h j Ž z Ž t x T CiT C j x Ž t y ␥ ©T Ž t © Ž t qÝ is1 js1 r s r Ý Ý h i Ž z Ž t h j Ž z Ž t x T Ž t Ž A i y Bi Fj T Px Ž t is1 js1 r r qÝ Ý h i Ž z Ž t h j Ž z Ž t x T Ž t P Ž A i y Bi Fj x Ž t is1 js1 r r qÝ Ý h i Ž z Ž t h j Ž z Ž t x T CiT C j x Ž t y ␥ ©T Ž t © Ž t is1 js1 r r q Ý h i Ž z Ž t © T Ž t EiT Px Ž t q is1 r s Ý h i Ž z Ž t x T Ž t PEi © Ž t is1 r Ý Ý hi Ž z Ž t h j Ž z Ž t x T Ž t ©T Ž t is1 js1 Ž A i y Bi Fj T P qP Ž A i y Bi Fj = qC iT C j EiT P PEi xŽ t F ©Ž t Ž 3.64 y␥ I From Ž3.64., we have the following conditions: r yÝ r Ý h i Ž z Ž t h j Ž z Ž t Ä Ž A i y Bi Fj T P is1 js1 qP Ž A i y Bi Fj q C iT C j r yP Ý h i Ž z Ž t Ei is1 G r y Ý h i Ž z Ž t EiT P ␥ 2I is1 Ž 3.65 72 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS The left-hand side of Ž3.65 can be decomposed as follows: r y r Ý Ý h i Ž z Ž t h j Ž z Ž t is1 js1 T = Ä Ž A i y B i Fj P q P Ž A i y B i Fj r yP Ý h i Ž z Ž t Ei is1 r y ␥ 2I Ý h i Ž z Ž t EiT P is1 r y r Ý Ý h i Ž z Ž t h j Ž z Ž t CiTC j is1 js1 r yÝ r Ý h i Ž z Ž t h j Ž z Ž t is1 js1 T = Ä Ž A i y B i Fj P q P Ž A i y B i Fj s r yP Ý h i Ž z Ž t Ei is1 r y ␥ 2I Ý h i Ž z Ž t EiT P is1 r y Ý h i Ž z Ž t CiT is1 r Ž 3.66 G Ý h i Ž z Ž t Ci is1 Inequality Ž3.67 is equivalent to r y r Ý Ý h i Ž z Ž t h j Ž z Ž t is1 js1 T = Ä Ž A i y Bi Fj P qP Ž A i y B i Fj r y r yP r Ý h i Ž z Ž t Ei Ý h i Ž z Ž t CiT is1 is1 G Ž 3.67 ␥ I 0 Ý h i Ž z Ž t EiT P I / y P Ž Ei q E j is1 r Ý h i Ž z Ž t Ci is1 Inequality Ž3.67 can be rewritten as r r Ý Ý hi Ž z Ž t h j Ž z Ž t is1 js1 ž T y Ä Ž A i y B i Fj P q P Ž A i y B i Fj T q Ž A j y B j Fi P q P Ž A j y B j Fi T y Ž Ei q E j P Ž Ci q C j 1 Ž Ci q C j T G ␥ 2I 0 I 73 DISTURBANCE REJECTION Therefore, we have ž T y Ä Ž A i y Bi Fj P q P Ž A i y B i Fj T q Ž A j y B j Fi P q P Ž A j y B j Fi 1 / Ž Ci q C j T G ␥ 2I Ž Ci q C j 0 T y Ž Ei q E j P 2 y P Ž Ei q E j I Ž 3.68 By multiplying both side of Ž3.68 by block-diag Ä X I I , Ž3.61 is obtained, where X s Py1 Q.E.D Next, consider the following DFS with disturbance w21x: r x Ž t q s Ý h i Ž z Ž t Ä A i x Ž t q B i u Ž t q Ei © Ž t , Ž 3.69 is1 r yŽ t s Ý h i Ž z Ž t Ci x Ž t , Ž 3.70 is1 where © Ž t is the disturbance The disturbance rejection can be realized by minimizing ␥ subject to sup ® Ž t 2/0 yŽ t 52 ©Ž t 52 F ␥ Ž 3.71 THEOREM 15 wDFSx The feedback gains Fi that stabilize the fuzzy model and minimize ␥ in Ž3.71 can be obtained by sol®ing the following LMIs: minimize ␥ X , M1, , Mr subject to X ) 0, X ž ␥ 2I Ž A i X y Bi M j qA j X y B j Mi / Ž Ci q C j X Ž Ei q E j ž Ž A i X y Bi M j qA j X y B j Mi Ž Ei q E j T T / X Ž Ci q C j X I i F j s.t h i l h j / ␾ , where X ) and Mi s Fi X G 0, 0 T Ž 3.72 74 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS Proof Suppose there exists a quadratic function V Ž x Ž t s x T Ž t Px Ž t , P ) 0, and ␥ G such that, for all t, ⌬V Ž x Ž t q y T Ž t y Ž t y ␥ © T Ž t © Ž t F Ž 3.73 for Ž3.69 and Ž3.70 From Ž3.73., we obtain T Ý Ä ⌬V Ž x Ž t q yT Ž t y Ž t y ␥ ©T Ž t © Ž t F ts0 By assuming that initial condition x Ž0 s 0, we obtain T Ý Ž yT Ž t y Ž t y ␥ ©T Ž t © Ž t F VŽ xŽT q Ž 3.74 ts0 Since V Ž x ŽT G 0, this implies yŽ t 52 ©Ž t 52 F␥ Therefore the L2 gain of the fuzzy model is less than ␥ if Ž3.73 holds We derive an LMI condition from Ž3.73.: ␥ © T Ž t © Ž t y y T Ž t y Ž t y ⌬V Ž x Ž t T r s ␥ © Ž t ©Ž t y x Ž t T T r y ž Ý h i Ž z Ž t Ci is1 /ž r / Ý h i Ž z Ž t Ci x Ž t is1 r ½Ý Ý ½Ý Ý is1 js1 r Ý h i Ž z Ž t Ei © Ž t is1 r =P T r h i Ž z Ž t h j Ž z Ž t Ž A i y Bi Fj x Ž t q r h i Ž z Ž t h j Ž z Ž t Ž A i y Bi Fj x Ž t q is1 js1 Ý h i Ž z Ž t Ei © Ž t is1 q x T Ž t Px Ž t s xT Ž t ©T Ž t T T P © Ž t xŽ t ©Ž t ␥ 2I r y x Ž t T r ½Ý Ý h i Ž z Ž t h j Ž z Ž t A i y Bi Fj is1 js1 r =P r ½Ý Ý is1 js1 h i Ž z Ž t h j Ž z Ž t A i y Bi Fj Ei xŽ t ©Ž t Ei 5 75 DISTURBANCE REJECTION T r yx T Ž t žÝ h i Ž z Ž t Ci is1 s xT Ž t / žÝ / h i Ž z Ž t Ci x Ž t is1 ©T Ž t T r žÝ Py = r h i Ž z Ž t Ci is1 r / žÝ h i Ž z Ž t Ci is1 / ␥ 2I r T y x Ž t © Ž t T r ẵí í T x t â t h i Ž z Ž t h j Ž z Ž t A i y B i Fj Ei is1 js1 r =P r ½Ý Ý h i Ž z Ž t h j Ž z Ž t A i y Bi Fj xŽ t G ©Ž t Ei is1 js1 From the Schur complement, we obtain the LMI condition: T r Py žÝ žÝ is1 r = / / h i Ž z Ž t C i h i Ž z Ž t C i is1 r r Ý Ý h i Ž z Ž t h j Ž z Ž t is1 js1 = Ä A i y B i Fj r ␥ 2I T Ý h i Ž z Ž t EiT is1 r r Ý Ý h i Ž z Ž t h j Ž z Ž t is1 js1 = Ä A i y B i Fj r Py1 Ý h i Ž z Ž t Ei is1 r P r Ý Ý h i Ž z Ž t h j Ž z Ž t is1 js1 = Ä A i y Bi Fj r ␥ 2I s T Ý h i Ž z Ž t EiT is1 r r Ý Ý h i Ž z Ž t h j Ž z Ž t is1 js1 = Ä A i y Bi Fj ž Py1 is1 T r y r Ý h i Ž z Ž t Ei Ý h i Ž z Ž t Ci is1 0 / r Ý h i Ž z Ž t Ci is1 0 G Ž 3.75 76 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS Inequality Ž3.75 is equivalent to r P r Ý Ý h Ž z Ž t h Ž z Ž t i j is1 js1 = Ä A i y B i Fj 0ž i is1 i / r ␥2 I T T r Ý h Ž z Ž t C Ý h Ž z Ž t E i T i is1 r ž r Ý Ý h Ž z Ž t h Ž z Ž t i j is1 js1 = Ä A i y B i Fj / r i Py1 i 0 Ý h Ž z Ž t E I is1 r Ý h Ž z Ž t C i i is1 r r sÝ Ý hi Ž z Ž t h j Ž z Ž t is1 js1 P ␥ 2I T Ž Ei q E j P y1 Ž A i y B i F j q A j y B j Fi T ŽCi q C j Ž A i y B i F j q A j y B j Fi T Ž Ei q E j 0 = 2 Ž Ci q C j 0 I G Ž 3.76 Therefore, P ␥ 2I Ž A i y Bi Fj q A j y B j Fi Ž Ci q C j Ž Ei q E j Ž A i y Bi Fj q A j y B j Fi T 2 Ž Ci q C j T Ž Ei q E j T y1 0 I P i F j s.t h i l h j / ␾ G 0, Ž 3.77 By multiplying both sides of Ž3.77 by block-diagw X I I I x, Ž3.72 is obtained, where X s Py1 Q.E.D A design example for disturbance rejection will be discussed in Chapter 3.8 DESIGN EXAMPLE: A SIMPLE MECHANICAL SYSTEM Let us consider an example of dc motor controlling an inverted pendulum via a gear train w22x Fuzzy modeling for the nonlinear system was done in w3x, DESIGN EXAMPLE: A SIMPLE MECHANICAL SYSTEM 77 w23x and w24x The fuzzy model is as follows: Plant Rule IF x 1Ž t is M1 , THEN ½ x ˙Ž t s A1 x Ž t q B1 u Ž t , y Ž t s C1 x Ž t Ž 3.78 Plant Rule IF x 1Ž t is M2 , THEN ½ x ˙Ž t s A x Ž t q B2 u Ž t , y Ž t s C2 x Ž t Ž 3.79 Here, x Ž t s x1Ž t x Ž t x Ž t A s 9.8 y10 T , , y10 B1 s 0 , 10 C1 s w 0 x A2 s 0 y10 , y10 B2 s 0 , 10 C2 s w1 0x The angle of the pendulum is x 1Ž t , x Ž t s ˙1Ž t , and x Ž t is current of the x motor The M1 and M2 are fuzzy sets defined as °sin x Ž t , M Ž x Ž t s~ x Ž t ¢ 1, 1 x Ž t / 0, x Ž t s 0, M Ž x Ž t s y M1 Ž x Ž t This fuzzy model exactly represents the dynamics of the nonlinear mechanical system under y␲ F x 1Ž t F ␲ Note that the fuzzy model has a 78 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS common B matrix, that is, B1 s B The fuzzy controller design of the common B matrix cases is simple in general To show the effect of the LMI-based designs, we consider a more difficult case, that is, we change B as follows: B2 s 0 20 3.8.1 Design Case 1: Decay Rate We first design a stable fuzzy controller by considering the decay rate The design problem of the CFS is defined as follows: maximize X , Y , M1, , Mr ␣ subject to X ) 0, Y G 0, Ž3.39 and Ž3.40 Fig 3.3 Design examples and DESIGN EXAMPLE: A SIMPLE MECHANICAL SYSTEM 79 We obtain ␣ s 5.0, F1 s w 282.3129 62.4176 3.2238 x , F2 s w 110.4644 24.9381 1.2716 x , 105.108 P s Xy1 s 20.4393 1.05294 20.4393 4.29985 0.23680 1432.034 Q s Xy1 YXy1 s 299.8039 16.26773 1.05294 0.23680 ) 0, 0.01567 299.8039 63.19188 3.449801 16.26773 3.449801 G 0.190786 The dotted line in Figure 3.3 shows the responses of y Ž t ws x 1Ž t x and uŽ t 3.8.2 Design Case 2: Decay Rate H Constraint on the Control Input It can be seen in the design example that max t uŽ t s 624 In practical design, there is a limitation of control input It is important to consider not only the decay rate but also the constraint on the control input The design problem that considers the decay rate and the constraint on the control input is defined as follows, where ␮ s 100 and x Ž0 s w0 10 0xT : maximize X , Y , M1, , Mr ␣ subject to X ) 0, Y G Ž3.39., Ž3.40., Ž3.46., and Ž3.47 The solution is obtained as ␣ s 4.23, F1 s w 38.3637 9.9338 0.7203 x , F2 s w 18.2429 6.4771 0.5118 x , P s 0.1578 0.03847 0.002738 0.03847 0.009995 0.000742 0.001250 Q s 0.000281 4.275 = 10y5 0.002738 0.000742 ) 0, y5 5.831 = 10 0.000281 0.0001215 6.332 = 10y6 4.275 = 10y5 6.332 = 10y6 G 1.976 = 10y6 80 LMI CONTROL PERFORMANCE CONDITIONS AND DESIGNS The real line in Figure 3.3 shows the responses of y Ž t Žs x 1Ž t and uŽ t The designed controller realizes the input constraint max t uŽ t s 99.3 - ␮ 3.8.3 Design Case 3: Stability H Constraint on the Control Input It is also possible to design a stable fuzzy controller satisfying the constraint on the control input, where ␮ s 100 Find X ) 0, Y G 0, and Mi Ž i s 1, , r satisfying Ž3.23., Ž3.24., Ž3.53., and Ž3.54 The solution is obtained as F1 s w 13.0065 3.6948 0.1786 x , F2 s w 7.7309 2.7900 0.1163 x , 0.0335 P s 0.0106 0.0015 0.0106 0.0036 0.0005 0.0015 0.0005 ) 0, 0.0001 0.0522 Q s 0.0203 0.0040 0.0203 0.0082 0.0016 0.0040 0.0016 G 0.0003 The dotted line in Figure 3.4 shows the responses of y Ž t ws x 1Ž t x and uŽ t It can be found that max t uŽ t s 38.1 - ␮ Fig 3.4 Design examples and REFERENCES 81 3.8.4 Design Case 4: Stability H Constraint on the Control Input H Constraint on the Output The response of the control system in the design example has a large output error Žmax t y Ž t s 2.16 since the constraint on the output is not considered in the fuzzy controller design To improve the response, we can design a fuzzy controller by adding the constraint on the output Find X ) 0, Y G 0, and Mi Ž i s 1, , r satisfying Ž3.23., Ž3.24., Ž3.46., Ž3.47., and Ž3.54 where ␮ s 100 and ␭ s The solution is obtained as F1 s w59.2819 9.3038 0.5580x, F2 s w33.7254 7.4115 0.4122x, 0.5478 P s 0.0519 0.0034 0.0519 0.0098 0.0006 0.0034 0.0006 ) 0, 0.0001 0.9936 Q s 0.0334 0.0075 0.0334 0.0118 0.0008 0.0075 0.0008 G 0.0001 The real line in Figure 3.4 shows the responses of y Ž t ws x 1Ž t x and uŽ t The response of the control system satisfies the constraints max t uŽ t s 93 - ␮ and max t y Ž t s 1.25 - ␭ REFERENCES K Tanaka, T Taniguchi, and H O Wang, ‘‘Model-Based Fuzzy Control of TORA System: Fuzzy Regulator and Fuzzy Observer Design via LMIs that Represent Decay Rate, Disturbance Rejection, Robustness, Optimality,’’ Seventh IEEE International Conference on Fuzzy Systems, Alaska, 1998, pp 313᎐318 K Tanaka, T Ikeda, and H O Wang, ‘‘Design of Fuzzy Control Systems Based on Relaxed LMI Stability Conditions,’’ 35th IEEE 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T Gii P q PGii - 0, ž Gi j q Gji Ž 3.7 T / PqP ž Gi j q Gji / F 0, i - j s.t h i l h j / ␾ Ž 3.8 Proof It follows directly from Theorem For the explanation of the notation i - j s.t h i l h... the conditions of Theorems 5᎐8 A trial-and-error type of procedure was first used w4, 7, 9x In w19x, a procedure to construct a common P is given for second-order fuzzy systems, that is, the dimension... Q.E.D is1 i-j COROLLARY If the number of rules that fire for all t is less than or equal to s, where - s F r, then r r Ý h2 Ž z Ž t y s y Ý Ý h i Ž z Ž t h j Ž z Ž t G 0, i is1 is1 i-j where