CONTINUOUS SYSTEM mras MRAS for first-order system with Bad parameters good control? INPUT y uc OUTPUT u The closed loop transfer function is STAtE ym th1 th2 x1 x2 deR dym dth1 dth2 dx1 dx2 Gc(s) 61G(s 61b +62G(s) +2b u=th1*uc-th2*y 62 dym=-am*ym+bm*uc dx1=-am*x1+ dth1=-gamma*e* dth2=-gamma*e*x2 am: 2 model para gamma: 2 adaptation gain END Error and Parameter Convergence Consider adaptation of feedforward gain (k-和0ue=k(6-60)u de Determination of Adaptation gain y22(0-) ult problem Solution Approximations give insight 6(t)=6°+(6(0)-6")e Leads to modified algorithms ()dr Exponential convergence with persistant excitation C K.J. Astrom and B WittenmarkCONTINUOUS SYSTEM mras "MRAS for first-order system with " Gm=bm/(s+am) INPUT y uc OUTPUT u STATE ym th1 th2 x1 x2 DER dym dth1 dth2 dx1 dx2 u=th1*uc-th2*y dym=-am*ym+bm*uc dx1=-am*x1+am*uc dx2=-am*x2-am*y e=y-ym dth1=-gamma*e*x1 dth2=-gamma*e*x2 am:2 "model parameter bm:2 "model parameter gamma:2 "adaptation gain END Bad Parameters Good Control? The closed loop transfer function is Gcl(s)
θ 1G(s) 1 + θ 2G(s)
θ 1b s + a + θ 2b 01234 −1 0 1 2 θ 2 θ 1 Error and Parameter Convergence Consider adaptation of feedforward gain e
(kθ − k0)uc
k(θ − θ0 )uc with θ0
k0/k dθ dt
−γ k2u2 c (θ − θ0) Solution θ (t)
θ0 + (θ (0) −θ0)e−γ k2It where It
Z t 0 u2 c (τ ) dτ Exponential convergence with persistant excitation Determination of Adaptation Gain • A difficult problem • Approximations give insight • Leads to modified algorithms c K. J. Åström and B. Wittenmark 4