Page 55 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

46 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

−1
The error transform with PPF can be expressed as s 2 = μ 2R (z 2 ).Then
2
substituting (3.27)into(3.26)yields
−1
˙ μ 1
˙ z 1 = r 1 μ 2R (z 2 ) +¯χ 1 − ϑ 2,1 − s 1 . (3.28)
2
μ 1
Hence, the virtual control ¯χ 1 to stabilize (3.28)isgiven by
r 1z 1 μ 2 ˙ μ 1
¯ 2
¯ χ 1 =−k 1z 1 − δ 1 + s 1 + ϑ 2,1 , (3.29)
|r 1z 1 μ 2 |+ 1 μ 1
where k 1 > 0, δ 1 > 0, and 1 > 0 are the design parameters.
¯
Step 2: In order to avoid the use of derivative of ¯χ 1,welet ¯χ 1 go through
aHGTDas

˙
ϑ 1,2 = ϑ 2,2 (3.30)
α
ϑ 2,2 = H 2 − ρ 1,2 [ϑ 1,2 −¯χ 1 ] − ρ 2,2 [ϑ 2,2 /H] β
˙
where ρ 1,2 and ρ 2,2 are also the design parameters. The derivative of z 2 is
given as
˙ μ 2 ˙ μ 2
˙
˙
˙ z 2 = r 2 ˙ s 2 − s 2 = r 2 ˆ x 2 − ¯χ 1 − s 2 (3.31)
μ 2 μ 2
where r 2 = (1/2μ 2 )[1/(ρ 2 + δ ) − 1/(ρ 2 − δ 2 )],and ρ 2 = s 2 /μ 2.
¯
2
Based on the system (3.3)withobserver(3.20), the derivative of (3.31)
is
˙ μ 2
˙ z 2 = r 2 k f (x 3 − x 1 ) − ϑ 2,2 − s 2 (3.32)
μ 2
By defining s 3 = x 3 −¯χ 2 as another intermediate error, one obtains
x 3 = s 3 +¯χ 2 . (3.33)

Substituting (3.33)into(3.32)yields

−1
˙ μ 2
˙ z 2 = r 2 k f (μ 3R (z 3 ) +¯χ 2 − x 1 ) − ϑ 2,2 − (3.34)
3 s 2
μ 2
Choose the virtual control ¯χ 2 as
r 2z 2 μ 2
1 ˙ μ 2 3
¯
¯ χ 2 = − k 2z 2 + ϑ 2,2 + s 2 +ˆ x 1 − δ 2 , (3.35)
k f μ 2 |r 2z 2 μ 3 |+ 2
where k 2 > 0, δ 2 > 0, and 2 > 0 are the design parameters.
¯