MRAS with MIT and Lyapunov rule hat do we know? The robustness issue · What happens when G≠G0? . What is Ro bust ness? How sens it ive is the result to the 叫G(s) assumpt io ns? hat are the critical assum pt io ns? What about the assum pt io ns in the Model stability proof Ot her fields Adapt at io n of Feedfo rward gain Lyapunov versus MIT once more · A first order system yme (MIT) dt (SPR) Analysis (SPR) Analysis e=3-gn=M6(p)(0(()-4Cm(n()+6g(a)=4gcn dt +r(ho gm e)kG(Buc)=yko Gmue +y0kavgluc Guc)]=rboavg(gmu)] The equilibrium para meters are dtt mucho G(Bue)=rko Gme avg(G Averaged equations BMITF avgi(Gmue)(Guc)H G+y0kkoavgi(Gmue)(Guc))=yk3avg(Gmus)2) ko avgluc(much d+70 kavgfuel(Gue))=rk avgfuc(Gm ue)y navg(gmc)(Gu]>0 (MIT) The equilibrium para ravguc(Gu)>0(SPR G ggmu(G espr- ko avgfuc(gm ue)) CK.J. Astrom and BWittenmarkThe Robustness Issue  What is Robustness? { How sensitive is the result to the assumptions? { What are the critical assumptions? { What about the assumptions in the stability proof { Other elds  Adaptation of Feedforward Gain { Lyapunov versus MIT once more  A rst order system MRAS with MIT and Lyapunov Rule  What do we know?  What happens when G 6= G0? θ Σ – Model Process + Σ – Model Process + y y e e θ uc uc kG(s) kG(s) k0G(s) k0G(s) y m y m − γ s − γ s Π Π Π Π (a) (b) d^  dt = ￾ yme (MIT) d^  dt = ￾ uce (SPR) Analysis d^ dt = ￾ yme (MIT) d^  dt = ￾ uce (SPR) e = y ￾ym = kG(p)  ^ (t)uc(t)￾k0Gm(p)uc(t) Hence d^  dt + (k0Gmuc)kG(^ uc) = k0Gmuc d^  dt + uckk0G(^ uc) = k0Gmuc Averaged equations d  dt +  kk0avgf(Gmuc)(Guc)g = k2 0 avgf(Gmuc)2g d  dt +  kavgfuc(Guc)g = k0avgfuc(Gmuc)g The equilibrium parameters are  MIT = k0 k avgf(Gmuc)2g avgf(Gmuc)(Guc)g  SPR = k0 k avgfuc(Gmuc)g avgfuc(Guc)g Analysis d  dt +  kk0avgf(Gmuc)(Guc)g = k2 0 avgf(Gmuc)2g d  dt +  kavgfuc(Guc)g = k0avgfuc(Gmuc)g The equilibrium parameters are  MIT = k0 k avgf(Gmuc)2g avgf(Gmuc)(Guc)g  SPR = k0 k avgfuc(Gmuc)g avgfuc(Guc)g avgf(Gmuc)(Guc)g > 0 (MIT) avgfuc(Guc)g > 0 (SPR) c K. J. Åström and B. Wittenmark 3