An Example Example daptation of feedforward gain Process y= kG(p)u G(s) a18+ e =y==ym Approximate CE dt s3+a1s2+a2s+M=0 Parameter equation u=yymuok Stability condition dt +yym(kg(p)Ouc)=yy yym uck <a Why approximate? A thought experiment ym de d t t rymuc(G(p)a)=r(ym)2 Characteristic equation s+rymuchG(s Key parameter u=youch Example G(s)=九 Summary Modified Algorithm . The ide MIT rule Model following dt=rpe The mit rule Normalized adaptation law · The error equation e(t)=(G(p, 0)-Gm (p))uc(t) dt a+pp · a gradient procedure · Approximations normalization de dt C K.J. Astrom and B WittenmarkAn Example daptation of feedforward gain y
kG(p)u ym
k0G(p)uc u
θuc e
y − ym dθ dt
−γ yme Parameter equation dθ dt + γ ym (kG(p)θuc)
γ y2 m Why approximate? A thought experiment dθ dt + γ yo muo c (kG(p)θ)
γ (yo m)2 Characteristic equation s + γ yo muo ckG(s)
0 Key parameter µ
γ yo muo ck Example G(s)
1 s+1 Example Process G(s)
1 s2 + a1s + a2 Approximate CE s3 + a1s2 + a2s + µ
0 µ
γ yo muo ck. Stability condition γ yo muo ck < a1a2 0 20 40 60 80 100 −0.1 0.1 0 20 40 60 80 100 −1 1 0 20 40 60 80 100 −10 10 Time Time Time (a) ym y (b) ym y (c) y ym Modified Algorithm MIT rule dθ dt
γ ϕ e Normalized adaptation law dθ dt
γ ϕ e α + ϕTϕ 0 20 40 60 80 100 −0.1 0.1 0 20 40 60 80 100 −1 1 0 20 40 60 80 100 −10 10 Time Time Time (a) ym y (b) ym y (c) ym y Summary • The idea – Model following – The MIT rule • The error equation e(t)
(G(p,θ) − Gm(p))uc(t) • A gradient procedure dθ dt
γ ϕ e ϕ
G(p,θ ) θ uc • Approximations • Normalization dθ dt
γ ϕ e α + ϕTϕ c K. J. Åström and B. Wittenmark 5