W
(9.8.24)
Λ( s)
m( s)∆ m( s)
2 δ 0
it follows that in (9.8.23) ˜
g ∈ S( f 0 + ∆22 + d 20 ).
m 2
Robust Adaptive Law of Table 9.4 We have m 2 = z − ˆ
z = −˜
θ φp − η, i.e.,
˜
θ φp = −η − m 2. We need to relate ˜
θ φp with ˜
θ ω. Consider the identities
˜
θ φp = ˜
θ 0 φ 0 + ˜ c 0 yp, ˜ θ ω = ˜ θ 0 ω 0 + ˜ c 0 r
where ˜
θ 0 = [˜
θ 1 , ˜ θ 2 , ˜ θ 3] , ω 0 = [ ω 1 , ω 2 , yp] and φ 0 = Wm( s) ω 0. Using the above equations, we obtain
˜
θ 0 φ 0 = ˜ θ φp − ˜ c 0 yp = − m 2 − ˜ c 0 yp − η
(9.8.25)
9.8. STABILITY PROOFS OF ROBUST MRAC SCHEMES
753
Let us now use the Swapping Lemma A.1 to write
Wm( s)˜
θ ω = ˜
θ 0 Wm( s) ω 0 + ˜ c 0 ym + Wc( Wbω )˙˜ θ
Because ˜
θ 0 φ 0 = ˜ θ 0 Wm( s) ω 0, it follows from above and (9.8.25) that
Wm( s)˜
θ ω = − m 2 − ˜ c 0 yp − η + ˜ c 0 ym + Wc( Wbω )˙˜
θ
(9.8.26)
Substituting for
1
yp = ym +
W
c∗
m( s) ˜
θ ω + ηy
0
in (9.8.26) and using η = c∗ 0 ηy, where
Λ( s) − θ∗
η
1 α( s)
y =
W
c∗
m( s)[∆ m( s)( up + du) + du]
0Λ( s)
we obtain
˜ c
W
0
m( s) ˜
θ ω = − m 2 −
W
c∗
m( s) ˜
θ ω + Wc( Wbω )˙˜
θ − c 0 ηy
0
Because 1 + ˜ c 0 = c 0 and 1 ∈ L
c∗
c∗
c
∞, we have
0
0
0
c∗
W
0
m( s) ˜
θ ω =
− m 2 + W
c
c( Wbω ) ˙˜
θ − c 0 ηy
(9.8.27)
0
Rewriting (9.8.13) as
˜
θ ω = F
˜
1( ˙˜
θ ω + ˜
θ ˙ ω) + F W − 1
m ( Wmθ ω)
and substituting for Wm( s)˜
θ ω from (9.8.27), we obtain
˜
θ ω = F 1(˙˜
θ ω + ˜
θ ˙ ω)
c∗
+ F W − 1 0
m
− m 2 + W
c
c( Wbω ) ˙˜
θ − c 0 ηy
(9.8.28)
0
Following the same approach as with the adaptive law of Table 9.3, we obtain
˜
c
θ ω ≤ c ˜
gmf + c
+ αk
α
0 ∆ ∞
mf + cd 0
(9.8.29)
0
where ˜
g ∈ S( f 0 + η 2 ) or ˜ g ∈ S( f
) and ˜
g 2 = |˙˜ θ| 2 + α 2 k
m 2
0 + ∆2
2 + d 20
m 2
α 2
0 ( | n 2
s| + | ˙˜
θ| 2 + 2).
0
Step 3. Use the B-G Lemma to establish boundedness. The bound for ˜
θ ω
in (9.8.21), (9.8.23), and (9.8.29) has exactly the same form for all three adaptive
754
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
laws given in Tables 9.2 to 9.4. Substituting for the bound of ˜
θ ω in (9.8.12) we
obtain
1
m 2
2
f ≤ c + c ˜
gmf
+ c
+ α 2 k
α 2
0 ∆2
∞
m 2 f + cd 20
0
For
1
c
+ α 2 k
α 2
0 ∆2
∞
< 1
(9.8.30)
0
we have
m 2
2
f ≤ c 0 + c ˜
gmf
where c 0 depends on d 20, which may be rewritten as
t
m 2 f ≤ c 0 + c
e−δ( t−τ)˜
g 2( τ ) m 2 f( τ) dτ
0
Applying the B-G Lemma III, we obtain
t
t
t
m 2
˜
g 2( τ ) dτ
˜
g 2( τ ) dτ
f ≤ ce−δtec 0
+ c 0 δ
e−δ( t−s) ec s
ds
0
Because ˜
g ∈ S( f 0+∆2 i+ d 20 ) where ∆
m 2
i = ∆02 for the SPR-Lyapunov based adaptive
law and ∆ i = ∆2 for the adaptive laws of Tables 9.2 and 9.4, we obtain
t
d 2
t
d 2
0
t
0
m 2
t
dτ
dτ
m 2( τ )
( t−s)
m 2( τ )
f ≤ ce− δ 2 ec 0
+ c 0 δ
e− δ 2
ec s
ds
(9.8.31)
0
provided
δ
c( f 0 + ∆2 i) ≤
(9.8.32)
2
where c in (9.8.32) is proportional to 1 + α 2 k
α 2
0
and can be calculated by keeping
0
track of all the constants in each step. The constant c also depends on the H∞δ and
H 2 δ norms of the transfer functions involved and the upper bound for the estimated
parameters.
To establish the boundedness of mf , we have to show that cd 20 < δ for all
m 2( t)
2
t ≥ 0 or for most of the time. The boundedness of mf will follow directly if we
modify the normalizing signal as m 2 = 1+ n 2 s, n 2 s = β 0 + ms and choose the constant β 0 large enough so that
cd 20
cd 2
δ
≤
0 ≤
m 2( t)
β 0
2
∀t ≥ 0. This means that m is always larger than the level of the disturbance. Such a
large m will slow down the speed of adaptation and may in fact improve robustness.
A slow adaptation, however, may have an adverse effect on performance.
The boundedness of signals can be established, however, without having to
modify the normalizing signal by using the properties of the L 2 δ norm defined over
9.8. STABILITY PROOFS OF ROBUST MRAC SCHEMES
755
an arbitrary interval [ t 1 , t] given by Lemma 9.8.2, 9.8.3 and by repeating steps 1 to
3 as follows:
We apply Lemma 9.8.2 to (9.8.9) to obtain
u
2
2
2
2
p
, y
≤ ce−δ( t−t 1) m 2( t
+ c∆2
+ cd 2
t,t
p
1) + c ˜
θ ωt,t
1
t,t 1
1
∞ upt,t 1
0
where we use the fact that ˜
θ ωt 2 δ , y
, u
≤ cm( t). Therefore, the
0
pt 2 δ 0
pt 2 δ 0
fictitious signal m 2 = e−δ( t−t 1) m 2( t
2 + y
2 satisfies
f
1) + up
p
1
t,t 1
t,t 1
m 2
2
f 1 ≤ ce−δ( t−t 1) m 2( t 1) + c ˜
θ ωt,t
+ c∆2
+ cd 2
1
∞m 2
f 1
0 ∀t ≥ t 1 ≥ 0
Following the same procedure as in step 2 and using Lemma 9.8.2, 9.8.3 we obtain
˜
1
θ ω 2 ≤ ce−δ( t−t 1) m 2( t
2
1) + c (˜
gmf )
+ c
+ α 2 k
+ cd 2
1 t,t 1
α 2
0
∆2 ∞m 2 f 1
0
0
where ˜
g is as defined before. Therefore,
1
m 2
2
f ≤ ce−δ( t−t 1) m 2( t 1) + c (˜
gmf ) t,t
+ c
+ α 2 k
+ cd 2
1
1
1
α 2
0
∆2 ∞m 2 f 1
0
0
Using (9.8.30), we obtain
m 2
2
f ≤ ce−δ( t−t 1) m 2( t 1) + c (˜
gmf ) t,t
+ cd 2
1
1
1
0 ,
∀t ≥ t 1
or
t
m 2 f ( t) ≤ c + ce−δ( t−t 1) m 2( t
e−δ( t−τ)˜
g 2( τ ) m 2 ( τ ) dτ
1
1) + c
f 1
t 1
Applying the B-G Lemma III, we obtain
t
t
c
˜
g 2( τ ) dτ
t
m 2
˜
g 2( τ ) dτ
t
f ( t) ≤ ce−δ( t−t 1)(1 + m 2( t 1)) e
1
+ cδ
e−δ( t−s) ec s
ds
1
t 1
∀t ≥ t 1 ≥ 0. Because ˜ g ∈ S( f 0 + ∆2 i + d 20 ), it follows as before that for cf m 2
0 + c∆2
i ≤
δ/ 2, we have
t
c
d 2 /m 2 dτ
m 2( t) ≤ m 2
t
0
f ( t) ≤ ce−δ/ 2( t−t 1)(1 + m 2( t
1
1
1)) e
t
t
+ cδ
e−δ/ 2( t−s) ec
d 2 /m 2( τ ) dτ
s
0
ds
(9.8.33)
t 1
∀t ≥ t 1, where the inequality m 2( t) ≤ m 2 follows from the definition of m . If we
f
f
1
1
establish that m ∈ L∞, then it will follow from Lemma 9.8.1 that all signals are
bounded. The boundedness of m is established by contradiction as follows: Let us
assume that m 2( t) grows unbounded. Because θ ∈ L∞, it follows that
m 2( t) ≤ ek 1( t−t 0) m 2( t 0)
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CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
for some k 1 > 0, i.e., m 2( t) cannot grow faster than an exponential. As m 2( t) grows unbounded, we can find a t 0 > ¯
α > 0 and a t 2 > t 0 with ¯
α > t 2 − t 0 such that
m 2( t 2) > ¯
αek 1 ¯ α for some large constant ¯
α > 0. We have
¯
αek 1 ¯ α < m 2( t 2) ≤ ek 1( t 2 −t 0) m 2( t 0)
which implies that
ln m 2( t 0) > ln ¯
α + k 1[¯
α − ( t 2 − t 0)]
Because ¯
α > t 2 − t 0, it follows that ln m 2( t 0) > ln ¯
α, i.e., m 2( t 0) > ¯
α for t 0 ∈
( t 2 − ¯
α, t 2).
Let t 1 = sup τ≤t {arg( m 2( τ) = ¯ α) }. Since m 2( t
2
0) > ¯
α for all t 0 ∈ ( t 2 − ¯
α, t 2),
it follows that t 1 ≤ t 2 − ¯
α and m 2( t) ≥ ¯
α ∀t ∈ [ t 1 , t 2) and t 2 − t 1 ≥ ¯
α. We now
consider (9.8.33) with t 1 as defined above and t = t 2. We have
t 2
m 2( t 2) ≤ c(1 + ¯
α) e−β( t 2 −t 1) + cδ
e−β( t−s) ds
t 1
where β = δ − cd 20 . For large ¯
α, we have