state feed back t he Er ro r equat io n Process mode Process dt dx d Desired response to command sig nals Desired response d /Amam+B Cortrol Contrd T He clased-loop system Error /(A-BD)+BMu/A(6)+B(6) Az+bu-A B Parametric ation A(6)=A Hence Bc(8)=Bm df=Ame+(a-Am-BL)X+(BM-Bm) Compatibility conditions =Ame+(a(0)-Am )x+(Bc()-Bm)uc A-Am=BL =Ane+y(0-6) B=BM t he lyapunoy func tion THe eror eq aton de adaptat io n of Fee d fo rw ard gain ry Error v(e,6)/;ePe+(0-0)y(0-0°) e/(|G(p)-|oG(P)ua/‖G(p)(6-6°)u Hence Introduce a reaization of G(s T Qe+r(0-0)tpE+(0-8or do eQe+(0-8 e=cx WHere Q positive defi nite and Candidate for lyapunoy function AmP+PAm/-Q Adaptation law v/2aPa+(-9 dg WHere A P+PA/-Q dv DIscUSS O KJ. As TToM aNd B. WIITENMaEkState Feedback Process model dx dt = Ax + Bu Desired response to command signals dxm dt = Amxm + Bmuc Control law u = M uc ￾ Lx The closed-loop system dx dt = (A ￾ BL)x + BMuc = Ac()x + Bc()uc Parametrization Ac( 0 ) = Am Bc( 0 ) = Bm Compatibility conditions A ￾ Am = BL Bm = BM The Error Equation Process dx dt = Ax + Bu Desired response dxm dt = Amxm + Bmuc Control law u = M uc ￾ Lx Error e = x ￾ xm de dt = dx dt ￾ dxm dt = Ax + Bu ￾ Amxm ￾ Bmuc Hence de dt = Ame + (A ￾ Am ￾ BL) x + (BM ￾ Bm) uc = Ame + (Ac() ￾ Am) x + (Bc () ￾ Bm) uc = Ame + ￾  ￾  0
The Lyapunov Function The error equation de dt = Ame + ￾  ￾  0
Try V (e; ) = 1 2 ￾ eT P e + ( ￾  0)T ( ￾  0)
Hence dV dt = ￾ 2 eT Qe + ( ￾  0 ) T P e + ( ￾  0 )T d dt = ￾ 2 eT Qe + ( ￾  0 )T  d dt + T P e where Q positive denite and ATmP + P Am = ￾Q Adaptation law d dt = ￾ T P e gives dV dt = ￾ 2 e TQe Discuss! Adaptation of Feedforward Gain Error e = (kG(p) ￾ k0G(p))uc = kG(p)( ￾  0)uc Introduce a realization of G(s) dx dt = Ax + B( ￾  0 )uc e = Cx Candidate for Lyapunov function V = 1 2 ￾ x T P x + ( ￾  0) 2
where AT P + P A = ￾Q c K. J. Åström and B. Wittenmark 7