Derivation of Adaptive Law A First Order System The error Process y-yI dy -ay bu be d +a+b6 b dt=-am ym+bme p+a+b82 Controller b261 +a+b02)2“c=p+a+b62 u(t)=61u2(t)-62y(t) Approximate ideal parameters p+a+b62≈p+am 81=00=m 62= dt p+am Block Diagram Simulation G(s) I Input and output l s+a Parameters p+am 62 dt Example a= 1, 6=0.5, am=bm=2 C K.J. Astrom and B WittenmarkA First Order System Process dy dt
−ay + bu Model dym dt
−am ym + bmuc Controller u(t)
θ 1uc(t) −θ 2 y(t) ideal parameters θ 1
θ0 1
bm b θ 2
θ0 2
am − a b Derivation of Adaptive Law The error e
y − ym y
bθ 1 p + a + bθ 2 uc e θ 1
b p + a + bθ 2 uc e θ 2
− b2 θ 1 (p + a + bθ 2)2 uc
− b p + a + bθ 2 y Approximate p + a + bθ 2  p + am Hence dθ 1 dt
−γ  am p + am uc  e dθ 2 dt
γ  am p + am y  e Block Diagram − Σ Π + e u y Σ Π Π Π − + uc Gm (s) G(s) θ 1 θ 2 γ s − γ s am s + am am s + am dθ 1 dt
−γ  am p + am uc  e dθ 2 dt
γ  am p + am y  e Example a
1, b
0.5, am
bm
2. Simulation Input and output 0 20 40 60 80 100 −1 1 0 20 40 60 80 100 −5 0 5 Time Time ym y u Parameters 0 20 40 60 80 100 0 2 4 0 20 40 60 80 100 0 2 Time Time θ 1 θ 2 γ
5 γ
1 γ
0.2 γ
5 γ
1 γ
0.2 c K. J. Åström and B. Wittenmark 3