a d aptatio n of Fee d fo rw ard gain Dervative of v Process d V Px+xP. =A+B(-6) 2Ax+Bu(-)2 Adaptation law 2P(Ax+Bu(a)+(e-)正 B IP dt ub Pa Can wefi nd p such tHat Ada a BP=C d dt -yuc b Pa T He adaptation law tHen becomes Q DIScUSS Kalm an-Yakubovic hlem m a definition 5 m m ary A rationa transfer furction g witH real coefficierits is positive real(PR)if Lyapunov Stability T Hed Stability concep G(s)≥0KRs≥0 Lyapunov tHeorem a transfer function G is strictly positive reai How to use rt? (SPR)if G(s-E)is positive rea for some rea Adaptive laws witH Guaranteed stati Simpl e aeslgn b roceau re Find control law LEMMA 1 T He transfer fuction Denve Error Equation G(s=c(sI-A)-lB Find Lyapunov function CHoose adjustment law so tHat is strictly positive rea if and orly if tHere exist positive dfi nite matrices P and Q sucH tHat ●Rear AP+PA=-Q Strong similarities witH Mh rule No norma iz ation O K. AsTEoM aNd B. WitleNMalkAdaptation of Feedforward Gain Derivative of V dV dt = 2  dxT dt P x + xT P dx dt  + ( ￾  0 ) d dt = 2 ￾Ax + Buc( ￾  0 )
T P x + 2 xT P ￾Ax + Buc ( ￾  0 )
+ ( ￾  0 ) d dt = ￾ 2 xT Qx + ( ￾  0 )  d dt + uc BT P x Adaptation law d dt = ￾ uc BT P x gives dV dt = ￾ 2 x TQx Discuss Output Feedback Process dx dt = Ax + B( ￾  0 )uc e = Cx Adaptation law d dt = ￾ uc BT P x Can we nd P such that BT P = C The adaptation law then becomes d dt = ￾ uce Kalman-Yakubovich Lemma Definition 5 A rational transfer function G with real coecients is positive real (PR) if Re G(s)  0 for Re s  0 A transfer function G is strictly positive real (SPR) if G(s ￾ ") is positive real for some real " > 0. Lemma 1 The transfer function G(s) = C(sI ￾ A)￾1B is strictly positive real if and only if there exist positive denite matrices P and Q such that AT P + P A = ￾Q and BT P = C Summary  Lyapunov Stability Theory { Stability concept { Lyapunovs theorem { How to use it?  Adaptive laws with Guaranteed Stability  Simple design procedure { Find control law { Derive Error Equation { Find Lyapunov Function { Choose adjustment law so that dV =dt  0  Remark { Strong similarities with MIT rule { Actually simpler { No normalization c K. J. Åström and B. Wittenmark 8