Page 236 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Introduce a new state variable β 2 and let ¯z 2 pass through a first-order
filter with time constant τ 2 > 0, and we have
˙
τ 2 β 2 + β 2 =¯z 2 ,β 2 (0) =¯ z 2 (0). (15.25)

Define the filter error as
y 2 = β 2 −¯z 2 . (15.26)

Substituting (15.26)into(15.25), we can obtain
¯ z 2 − β 2 y 2
˙
β 2 = =− . (15.27)
τ 2 τ 2
Step 2: Consider the definition

(15.28)
˙ z 2 = z 3
and denote the intermediate error as

(15.29)
s 2 = z 2 − β 2
Then, we can calculate the derivative of s 2 as

s ˙ 2 = z 3 − β 2 (15.30)
˙
Choose a virtual control ¯z 3 as

˙
¯ z 3 =−k 2s 2 − s 1 + β 2 (15.31)
where k 2 > 0 is a positive constant.
Again, introducing a new state variable β 3 and let ¯z 3 pass through a
first-order filter with time constant τ 3 > 0, we have

˙
τ 3 β 3 + β 3 =¯z 3 ,β 3 (0) =¯ z 3 (0). (15.32)
Define the filter error as

y 3 = β 3 −¯z 3 . (15.33)

Substituting (15.33)into(15.32), we can obtain

¯ z 3 − β 3 y 3
˙
β 3 = =− . (15.34)
τ 3 τ 3