Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach Kazuo Tanaka, Hua O. Wang Copyright ᮊ 2001 John Wiley & Sons, Inc. Ž. Ž . ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic CHAPTER 12 NEW STABILITY CONDITIONS AND DYNAMIC FEEDBACK DESIGNS This chapter presents a unified systematic framework of control synthesis wx 1᎐5 for dynamic systems described by the Takagi-Sugeno fuzzy model. In comparison with preceding chapters, this chapter provides two significant extensions. First we provide a new sufficient condition for the existence of a quadratically stabilizing state feedback PDC controller which is more general and relaxed than the existing conditions. Second, we introduce the notion of Ž. dynamic parallel distributed compensation DPDC and we provide a set of sufficient LMI conditions for the existence of quadratically stabilizing dy- namic compensators. In this chapter, the notation M ) 0 stands for a positive definite symmet- Ž. TT n=n ric matrix M; L L A, P s AP q PA is defined as a mapping from ᑬ n=nn=n Ž TT . TT = ᑬ to ᑬ . The same holds for L L A , Q s AQ q QA. The term yT Ž y1 . T P is the same as P . From this chapter onward, we will use italic symbols such as A and B instead of A and B.Inaddition, to lighten the Ž. Ž. Ž. Ž. notation, we will use x, y, z, p, and u instead of x t , y t , z t , p t , and Ž. u t , respectively. Another notable point regarding the notation is that we Ž. Ž. will use pt or p instead of zt as premise variables. This is because z is used as performance variables in Chapters 13 and 15 which are based on the setting presented in this chapter. The symbol x X denotes the transposed vector of x.Weoften drop the p and just write h , but it should be kept in i mind that the h ’s are functions of the variable p. i The summation process associated with the center of gravity defuzzifica- Ž. Ž. tion in system 2.3 and 2.4 can also be viewed as an interpolation between the vectors Axq Bubased on the value of the parameter p. The parameter ii p can be given several different interpretations. First, we can assume that the 229 NEW STABILITY CONDITIONS AND DYNAMIC FEEDBACK DESIGN 230 parameter p is a measurable external disturbance signal which does not Ž. Ž. depend on the state or control input of the system 2.3 and 2.4 . Using this Ž. Ž. interpretation, equations 2.3 and 2.4 describe a time-varying linear system. Second, we can assume that the parameter p is a function of the state, Ž. Ž. Ž. p s fx. Using this interpretation, equations 2.3 and 2.4 describe a nonlinear system. As a slight modification to this interpretation, we can assume that the parameter p is a function of the measurable outputs of the Ž. system, p s fy. Finally, we can assume that p is an unknown constant Ž. Ž. value, in which case equations 2.3 and 2.4 describe a linear differential Ž. inclusion LDI . In most cases, we can only derive a benefit from the fuzzy rule base description if we know the values of the parameters, so we will not usually consider this last interpretation. It is also possible to interpret p using a combination of these approaches. 12.1 QUADRATIC STABILIZABILITY USING STATE FEEDBACK PDC In this section, we consider the special form of parameter-dependent state feedback which mirrors the structure of the T-S model, that is, parallel Ž.wx distributed compensation PDC 19, 20 The PDC controller structure consists of fuzzy rules: Control Rule i Ž. Ž. IF ptis M иии and ptis M , 11ilil Ž. Ž. THEN ut s Kxt, i where i s 1,2, .,r. The output of the PDC controller is r u s hKx.12.1 Ž. Ý ii i s1 Remark 37 Note that the notation for PDC here is in slightly different form from earlier chapters where the PDC controller is of the following form r u sy hFx.12.2 Ž. Ý ii i s1 Ž. X Let us consider the Lyapunov function candidate Vxs xPx, where P ) 0. Taking the time derivative of this function along the flow of QUADRATIC STABILIZABILITY USING STATE FEEDBACK PDC 231 the system, rr d X TT Vxs hhx L L A , P q KBPq PB K x 12.3 Ž. Ž . Ž . Ž. ÝÝ ij i j i i j dt i s1 js1 rr 1 X TT s hhx L L A q A , P q KBP Ž. Ž ÝÝ ij i j j i 2 i s1 js1 qPB K q K T B T P q PB K x.12.4 Ž. . ij ij ji If for each 1 F i F j F r there exists a symmetric n = n matrix T s T T ij ij such that L L A q A , P q K T B T P q PB K q K T B T P q PB K - T , ᭙ , ᭙ Ž. ij ji ijij jiijij 12.5 Ž. and T иии T 11 1 r . . T s - 0, 12.6 Ž. . T иии T 1 rrr then rr d X Vx- hhxT x Ž. ÝÝ ij ij dt i s1 js1 T XXXX wxwx s hx . hx T hx . hx 1 r 1 r - 0. In order to express these inequalities as LMI conditions, we need to use a y1 ˆ transformation. Define Q s P , T s QT Q, and M s KQ. Pre- and ij ij i i Ž. postmultiplying equation 12.5 by Q produces the expression TT TT TT ˆ L L A q A , Q q MBq BMq MBq BM-T , Ž. ij jiijijjiij i F j s.t. h l h / or.12.7 Ž. ij We also know that T - 0 if and only if ˆˆ T иии T 11 1 r . . ˆ T s - 0. 12.8 Ž. . ˆˆ T иии T 1 rrr The resulting LMI conditions are summarized in the following theorem: NEW STABILITY CONDITIONS AND DYNAMIC FEEDBACK DESIGN 232 Ž. THEOREM 45 The T-S model 2.3 is quadratically stabilizable in the large Ž. ®ia a state feedback PDC controller 12.1 if there exist Q ) 0, M , i s 1,2, .,r, i ˆ Ž. Ž. and T such that the LMI conditions 12.7 and 12.8 ha®e feasible solutions. ij The ith gain of the PDC controller is gi®en by K s MQ y1 12.9 Ž. ii and the Lyapuno® function is gi®en by V s x T Q y1 x. 12.10 Ž. Remark 38 The above theorem is a generalization of the stability condition wx wx wx given in 17 and 20 . It is also weaker than the LMI condition given in 25 , in which case T becomes tI. The above theorem can be further relaxed if ij ij we know the structure of the fuzzy membership function: ⅷ Sometimes there is no overlap between two rules, that is, the product of the h and the h may be identically zero. In this case, the above ij Ž. theorem can be relaxed by dropping the condition 12.7 corresponding Ž. to the i and j in 12.7 . ⅷ If only s - r rules can fire at the same time, then the conditions of this theorem can be further relaxed to only require that all the diagonal s = s principal submatrices of T are negative definite. 12.2 DYNAMIC FEEDBACK CONTROLLERS In this section we introduce the concept of a DPDC, and we derive a set of LMI conditions which can be used to design a stabilizing DPDC. In order to derive the LMI design conditions, it is useful to begin with a parameter-dependent linear model described by the equations xts Apxtq Bput, Ž. Ž . Ž. Ž . Ž. ˙ yts Cpxt, 12.11 Ž. Ž . Ž. Ž . Ž. Ž. Ž. where xt, yt, and ut denote the state, measurement, and input vectors, Ž. respectively. The variable pt is a vector of measurable parameters. In general, these parameters may be functions of the system states, external disturbances, and time. Note that the T-S model is in this form. A parameter-dependent dynamic compensator is a parameter-dependent linear system of the form xts Apxtq Bpyt, Ž. Ž . Ž. Ž . Ž. ˙ cccc ut s Cpxtq Dpyt. 12.12 Ž. Ž . Ž. Ž . Ž. Ž . cc c DYNAMIC FEEDBACK CONTROLLERS 233 Defining the augmented system matrix Apq BpD pCp BpC p Ž. Ž. Ž.Ž. Ž. Ž. cc Aps Ž. cl BpCp Ap Ž.Ž. Ž. cc and the augmented state vector T TT xts xt xt , Ž. Ž. Ž. cl c the resulting closed-loop dynamic equations are described by the equation xts Apxt. 12.13 Ž. Ž . Ž. Ž . ˙ cl cl cl Ž. The system 12.11 is said to be quadratically stabilizable via an s-dimen- sional parameter-dependent linear compensator if and only if there exists an s-dimensional parameter-dependent controller and a positive definite matrix P ) 0 such that cl PA pq A T pP- 0. 12.14 Ž. Ž. Ž . cl cl cl cl Ž. Remark 39 If we fix the value of p, equation 12.14 represents a sufficient condition for the existence of a set of linear, time-invariant controller Ž. Ž. Ž. Ž. Ž . matrices Ap, Bp, Cp, and Dpwhich will stabilize the system 2.3 ccc c Ž. and 2.4 at the fixed value of p. The unknown controller does not enter Ž. linearly into equation 12.14 , so this equation does not represent an LMI wx condition. However, the authors of the paper 14 present a transformation procedure which results in a modified set of inequalities which are linear in the unknown data. In what follows, we perform this transformation pointwise with respect to p. We will first partition the constant matrices P and P y1 into components: PP 11 12 P s cl T PP 12 22 and QQ 11 12 y1 P s , cl T QQ 12 22 and we will also define the matrices QI 11 ⌸ s 1 T Q 0 12 and IP 11 ⌸ s P ⌸ s . 2 cl 1 T 0 P 12 NEW STABILITY CONDITIONS AND DYNAMIC FEEDBACK DESIGN 234 Ž. Equation 12.14 will hold if and only if ⌸ T PA p⌸ q ⌸ T A T pP⌸ - 0. Ž. Ž. 1 cl cl 11cl cl 1 This equation can also be rewritten as ⌸ T Ap⌸ q ⌸ T A T p ⌸ - 0. Ž. Ž. 2 cl 11cl 2 Writing out the first term on the left-hand side of this equation, we have I 0 Apq BpD pCp BpC p Q I Ž. Ž. Ž.Ž. Ž. Ž. Ž. cc11 T PP BpCp Ap Q0 Ž.Ž. Ž. 11 12 cc12 s Ep. Ž. If we define the new variables A A p s PApq BpD pCp Q q PB pCpQ Ž. Ž. Ž. Ž.Ž. Ž.Ž. Ž. 11 c 11 12 c 11 q PBpC pQ T q PApQ T , Ž. Ž. Ž. 11 c 12 12 c 12 B B p s PBpD pq PB p, Ž. Ž. Ž. Ž. 11 c 12 c C C p s DpCpQq CpQ T , Ž. Ž.Ž. Ž. c 11 c 12 D D p s Dp, Ž. Ž. c Ž. then the matrix Epcan be rewritten as ApQ q Bp C C pApq Bp D D pC p Ž. Ž.Ž. Ž. Ž. Ž.Ž. 11 Eps , Ž. A A pPApq B B pCp Ž. Ž. Ž. Ž. 11 and the closed-loop stability condition can be expressed as Epq E T p - 0 Ž. Ž. or T Ž. Ap Ž. L L Ap, Q Ž. 11 T T ž/ T ž/ Ž. Ž.Ž. Ž. qBp D D pC p q A A p Ž. Ž. Ž. Ž. qBp C C p q C C pB p - 0 T Ž. L L Ap, P Ž. Ž. Ž. A A p q Ap 11 T T TT T ž/ ž/ Ž. Ž. Ž. qCp D D pB p Ž.Ž. Ž. Ž. q B B pC p q Cp B B p together with the constraint that P ) 0. cl DYNAMIC FEEDBACK CONTROLLERS 235 This last condition holds if and only if ⌸ T P ⌸ ) 0, 1 cl 1 or I 0 QI 11 T ⌸⌸s 12.15 Ž. 21 T PP Q0 11 12 12 QI 11 s ) 0. 12.16 Ž. IP 11 We also have the constraint that PQ q PQ T s I. 12.17 Ž. 11 11 12 12 We will now assume that the parameter-dependent plant can be described by a fuzzy T-S model using r model rules. In this case, the parameter-depen- dent plant can be described by the equation r Ap Bp A B Ž. Ž. ii s hp , Ž. Ý i Cp 0 C 0 Ž. i i s1 Ž. where hpsatisfies the normalization condition, r hpG 0 and hps 1. Ž. Ž. Ý ii i s1 Ž. The matrix Epcan be written as Ep Ep Ž. Ž. 11 12 E s , 12.18 Ž. Ep Ep Ž. Ž. 21 22 where rr Eps hphp Aq BD p C Q Ž. Ž. Ž. Ž. Ž. ÝÝ Ž 11 ij iicj11 i s1 js1 qBC p Q T , 12.19 Ž. Ž . . ic 12 rr Eps hphp Aq BD p C , 12.20 Ž. Ž. Ž. Ž. Ž . Ž. ÝÝ 12 ij iicj i s1 js1 rr Eps hphpP Aq BD p C Q Ž. Ž. Ž. Ž. Ž. ÝÝ Ž 21 ij 11 iic j11 i s1 js1 qPB pCQq PBC pQ T Ž. Ž. 12 ci11 11 ic 12 qPA pQ T , 12.21 Ž. Ž . . 12 c 12 NEW STABILITY CONDITIONS AND DYNAMIC FEEDBACK DESIGN 236 rr Eps hphp P Aq BD p C Ž. Ž. Ž. Ž. Ž. ÝÝ Ž 22 ij 11 iic j i s1 js1 qPB pC. 12.22 Ž. Ž . . 12 ci We are now ready to introduce dynamic parallel distributed compensators for this system. In general, a DPDC can have cubic, quadratic, or linear parameterization. For a given T-S model, the choice of a particular DPDC parameterization will be influenced by the structure of the T-S subsystems. In the following subsections, we discuss each of these three parameterizations. 12.2.1 Cubic Parameterization Controller Synthesis. In this section, we will assume that the controller has the form rr r ijk xts hphph pAxt Ž. Ž. Ž. Ž. Ž. ˙ ÝÝÝ cijkcc i s1 js1 ks1 rr ij q hphpByt, 12.23 Ž. Ž. Ž. Ž . ÝÝ ijc i s1 js1 rr r ij i ut s hphpCxtq hpDyt, 12.24 Ž. Ž. Ž. Ž. Ž. Ž. Ž . ÝÝ Ý ijcc i c i s1 js1 is1 or equivalently that rrr ijk Aps hphph pA, 12.25 Ž. Ž. Ž. Ž. Ž . ÝÝÝ cijkc i s1 js1 is1 rr ij Bps hphpB, 12.26 Ž. Ž. Ž. Ž . ÝÝ cijc i s1 js1 rr ij Cps hphpC, 12.27 Ž. Ž. Ž. Ž . ÝÝ cijc i s1 js1 r i Dps hpD. 12.28 Ž. Ž. Ž . Ý cic i s1 Using this controller form, we can rewrite the equations for the matrix DYNAMIC FEEDBACK CONTROLLERS 237 Ž. Epas rr r Eps hphph p Ž. Ž. Ž. Ž. ÝÝÝ 11 ijk i s1 js1 ks1 = A q BD j CQq BC jk Q T , 12.29 Ž. Ž. Ž. iick11 ic 12 rr r j Eps hphph p Aq BDC , 12.30 Ž. Ž. Ž. Ž. Ž . Ž. ÝÝÝ 12 ijk iick i s1 js1 ks1 rr r j Eps hphph pP Aq BDC Q Ž. Ž. Ž. Ž. Ž. Ž ÝÝÝ 21 ijk 11 iick11 i s1 js1 ks1 qPB ij CQ q PBC jk Q T q PA ijk Q T , 12.31 Ž. . 12 ck11 11 ic 12 12 c 12 rr r Eps hphph p Ž. Ž. Ž. Ž. ÝÝÝ 22 ijk i s1 js1 ks1 = PAq BD j C q PB ij C , 12.32 Ž. Ž. Ž. 11 iick 12 ck and we have that rr r A A p s hphph p A A Ž. Ž. Ž. Ž. ÝÝÝ ijk ijk i s1 js1 ks1 rr r j J hphph pP Aq BDC Q Ž. Ž. Ž. Ž. Ž ÝÝÝ ijk 11 iick11 i s1 js1 ks1 qPB ij CQ q PBC jk Q T q PA ijk Q T , . 12 ck11 11 ic 12 12 c 12 rr B B p s hphp B B Ž. Ž. Ž. ÝÝ ij ij i s1 js1 rr jij J hphpPBDq PB , Ž. Ž. Ž. ÝÝ ij 11 ic 12 c i s1 js1 rr C C p s hphp C C Ž. Ž. Ž. ÝÝ ij ij i s1 js1 rr iijT J hphpDCQq CQ , Ž. Ž. Ž. ÝÝ ij cj11 c 12 i s1 js1 r D D p s hp D D Ž. Ž. Ý i i i s1 r i J hpD. Ž. Ý ic i s1 NEW STABILITY CONDITIONS AND DYNAMIC FEEDBACK DESIGN 238 Ž. The matrix Epthen becomes rr r Eps hphph pE Ž. Ž. Ž. Ž. ÝÝÝ ijk ijk i s1 js1 ks1 rr r s hphph p Ž. Ž. Ž. ÝÝÝ ijk i s1 js1 ks1 = AQ q B C C A q B D D C i 11 iiik jk j . 12.33 Ž. A A PAq B B C 11 ik ijk ij The closed-loop stability condition then becomes rr r hphph p Ž. Ž. Ž. ÝÝÝ ijk i s1 js1 ks1 TT TT L L A , Q q B C C q C C BAq B D D C q A A Ž. i 11 iiiik jk jk j i jk =-0. 12.34 Ž. T T T Ž. A A q A q B D D C L L A , P q B B C q C B B Ž. ii k i 11 kk ijk j ij ij So the system will be stable if the following LMI holds. TT TT L L A , Q q B C C q C C BAq B D D C q A A Ž. i 11 iiiik jk jk j i jk - 0, T T T Ž. A A q A q B D D C L L A , P q B B C q C B B Ž. ii k i 11 kk ijk j ij ij ᭙i, j, k. 12.35 Ž. Ž. Ž. THEOREM 46 The T-S model 2.3 and 2.4 is globally quadratically stabi- Ž.Ž. Ž. lizable ®ia a DPDC controller 12.25 ᎐ 12.28 if the LMI conditions 12.16 and Ž. 12.35 are feasible with LMI ®ariables Q , P , A A , B B , C C , and D D . The 11 11 ijk ij jk j controller is gi®en by A ijk s P y1 A A y PB c CQ y PBC c Q T ž c 12 12 ij k 11 11 ijk 12 ijk yPAq BD c CQ Q y1 , 12.36 Ž. Ž. / 11 iijk11 12 B ij s P y1 B B y PBD c , 12.37 Ž. ž/ c 12 11 ij ij C ij s C C y D c CQ Q yT , 12.38 Ž. ž/ cij11 12 ij D i s D D , 12.39 Ž. c i where P , P , Q , and Q satisfy the constraint P Q q PQ T s I. 11 12 11 12 11 11 12 12 [...]... gi®en by y1 Aic s P12 Ž T Ai y P12 Bci CQ11 y P11 BCci Q12 yP11 Ž A i q BDci C Q11 Qy1 , 12 Ž 12.126 Bi y P11 BDci , Ž 12.127 i Cc s Ž Ci y Dci CQ11 QyT , 12 Ž 12.128 Dci Ž 12.129 y1 Bci s P12 Ž s Di , T where P11 , P12 , Q11 , and Q12 satisfy the constraint P11 Q11 q P12 Q12 s I EXAMPLE 253 Fig 12.1 The ball and beam system 12.3 EXAMPLE In this section, we consider a ball-and-beam system which... iii h3 Ž p A c q i is1 Ý Ý 3h2 Ž p h j Ž p A cii j i is1 j-i r qÝ Ý 3h2j Ž p h i Ž p A ci j j is1 j-i r qÝ Ý Ý 6 h i Ž p h j Ž p h k Ž p A ci jk , Ž 12.44 is1 j-i k-j r Bc Ž p s Ý r h2 Ž p Bcii q i is1 r Cc Ž p s Ý Ý 2 h i Ž p h j Ž p Bci j , r Ý h2 Ž p Ccii q Ý Ý 2 h i Ž p h j Ž p Cci j , i is1 Ž 12.45 is1 j-i Ž 12.46 is1 j-i r Dc Ž p s Ý h i Ž p Dci Ž 12.47 is1 Consequently, the... The closed-loop system will be r x ˙cl s Ý h i Ž p Aicl x cl , Ž 12.101 is1 where Aicl s A i q BDc Ci i BCc Bc Ci Aic Ž 12.102 The closed-loop system Ž12.101 will be stable with quadratic Lyapunov function if there exists a symmetric positive matrix P such that L Ž Aicl , P - 0, ᭙ i Ž 12.103 Define T T Ai s P12 Aic Q12 q P12 Bc Ci Q11 q P11 BCi Q12 q P11 Ž A i q BDc Ci Q11 , B s P12 Bc q P11... Aicji q A cjii i c is1 j-i r qÝ 1 Ý 3h2j Ž p h i Ž p 3 Ž Aicj j q A cji j q A cj ji is1 j-i r qÝ Ý Ý 6 hi Ž p h j Ž p hk Ž p is1 j-i k-j = 1 6 Ž Aicjk q Aick j q A cji k q A cjk i q Ak i j q Ak ji , c c r Bc Ž p s Ý h2 Ž p Bcii i is1 r qÝ 1 Ý 2 h i Ž p h j Ž p 2 Ž Bci j q Bcji , is1 j-i r Cc Ž p s Ý h2 Ž p Ccii i is1 r qÝ 1 Ý 2 h i Ž p h j Ž p 2 Ž Cci j q Ccji , is1 j-i r Dc Ž p s Ý h i... q CiT B T q Ai 0 - 0, ᭙ i Ž 12.105 The controller is gi®en by y1 Aic s P12 Ž Ai y P12 Bc Ci Q11 i T yP11 BCc Q12 yP11 Ž A i q BDci Ci Q11 Qy1 , 12 y1 Bc s P12 Ž Ž 12.106 B y P11 BDc , Ž 12.107 i Cc s Ž Ci y Dc Ci Q11 QyT , 12 Ž 12.108 Dc s D , Ž 12.109 T where P11 , P12 , Q11 , and Q12 satisfy the constraint P11 Q11 q P12 Q12 s I Linear Parameterization: Common C The case corresponding to... Ž P11 Ž A i q Bi Dc C j Q11 q P12 Bci C j Q11 is1 js1 T T qP11 Bi Ccj Q12 q P12 Aicj Q12 , r E22 Ž p s Ž 12.68 r Ý Ý h i Ž p h j Ž p Ž P11 Ž A i q Bi Dc C j q P12 Bci C j , Ž 12.69 is1 js1 and we have that r r A Ž p s Ý Ý h i Ž p h j Ž p Ai j is1 js1 r J r Ý Ý h i Ž p h j Ž p Ž P11 Ž A i q Bi Dc C j Q11 is1 js1 T T qP12 Bci C j Q11 q P11 Bi Ccj Q12 q P12 Aicj Q12 , r B Ž p s Ý h i... theorem: THEOREM 47 The fuzzy control system of the T-S model Ž2.3 and Ž2.4 is globally quadratically stabilizable ®ia a DPDC controller Ž12.73 ᎐ Ž12.76 if the LMI conditions Ž12.16., Ž12.85.,and Ž12.86 are feasible with LMI ®ariables Q11 , P11 , Ti j , Ai j , Bi , Ci , and D The controller is gi®en by y1 i A ci j s 1 P12 2 Ai j y P12 B c C j Q11 y P12 2 ž T BcjCi Q11 y P11 Bi C jc Q12 T yP11 Bj Cic... COROLLARY 6 The fuzzy control system of the T-S model Ž2.3 and Ž2.4 is globally quadratically stabilizable ®ia a DPDC controller Ž12.44 ᎐ Ž12.47 if the following LMIs are feasible with LMI ®ariables Q11 , P11 , Ai , Ai j , Ai jk , Bi , Bi j , Ci , Ci j , and Di : DYNAMIC FEEDBACK CONTROLLERS Q11 I I ) 0, P11 243 Ž 12.55 Wi - 0, Ž 12.56 Wi j - 0, Ž 12.57 Wi jk - 0 Ž 12.58 The controller is gi®en in a... considerably In terms of E Ž p our stability condition can be written as EŽ p q ET Ž p - 0 or r r r Ý Ý Ý h i Ž p h j Ž p h k Ž p Ž Ei jk q EiTjk - 0 is1 js1 ks1 This can be rewritten as r T Ý h3 Ž p Ž Eiii q Eiii i is1 r qÝ Ý 3h2 Ž p h j Ž p i is1 j-i 1 Ž Eii j q Ei ji q Ejii 3 1 q r qÝ Ý 3h2j Ž p h i Ž p is1 j-i 1 3 3 Ž Eii j q Ei ji q Ejii T Ž Ej ji q Eji j q Ei j j 1 q Ž Ej ji q Eji j q... C T BiT 0 - 0, ᭙ i Ž 12.115 The controller is gi®en by: y1 Aic s P12 Ž T Ai y P12 Bci CQ11 y P11 Bi Cc Q12 yP11 Ž A i q Bi Dc C Q11 Qy1 , 12 y1 Bci s P12 Ž Ž 12.116 Bi y P11 Bi Dc , Ž 12.117 Cc s Ž C y Dc CQ11 QyT , 12 Ž 12.118 Dc s D , Ž 12.119 T where P11 , P12 , Q11 , and Q12 satisfy the constraint P11 Q11 q P12 Q12 s I Linear Parameterization: Common B and Common C Consider the case of . Copyright ᮊ 2001 John Wiley & Sons, Inc. Ž. Ž . ISBNs: 0-4 7 1-3 232 4-1 Hardback ; 0-4 7 1-2 245 9-6 Electronic CHAPTER 12 NEW STABILITY CONDITIONS AND DYNAMIC. stabilizable via an s-dimen- sional parameter-dependent linear compensator if and only if there exists an s-dimensional parameter-dependent controller