θ + (˜
θ R˜
θ) 12 (
)
− w ˜
θ θ
m t 2 β
Using the inequality −a 2 + ab ≤ − a 2 + b 2 , we obtain
2
2
˜
2
˙
θ R˜
θ
1
η
V ≤ −
+
(
)
− w ˜
θ θ
(8.5.59)
2
2
m t 2 β
Let us now consider the following two choices for w:
(A) Fixed σ For w = σ, we have −σ ˜
θ θ ≤ − σ |˜
θ| 2 + σ |θ∗| 2 and therefore
2
2
˜
2
˙
θ R˜
θ
σ
σ
1
η
V ≤ −
−
|˜
θ| 2 +
|θ∗| 2 +
(
)
(8.5.60)
2
2
2
2
m t 2 β
Because R = R
≥ 0, σ > 0 and ( η )
≤ c
m t 2 β
m for some constant cm ≥ 0, it
follows that ˙
V ≤ 0 whenever V (˜
θ) ≥ V 0 for some V 0 ≥ 0 that depends on ¯
η, σ and
|θ∗| 2. Hence, V, ˜
θ ∈ L∞ which from (8.5.44) implies that , m, ns ∈ L∞. Because
R, Q ∈ L∞ and θ ∈ L∞, we also have that ˙ θ ∈ L∞.
598
CHAPTER 8. ROBUST ADAPTIVE LAWS
Integrating on both sides of (8.5.60), we obtain
1
t
1
t
(˜
θ R˜
θ + σ|˜
θ| 2) dτ
≤ V ( t
σ|θ∗| 2 dτ
2
0) − V ( t) +
t
2
0
t 0
1
t
τ
+
e−β( τ−s) η 2( s) dsdτ
2 t
m 2( s)
0
0
By interchanging the order of the double integration, i.e., using the identity
t
τ
t
t
t 0
t
f ( τ ) g( s) dsdτ =
f ( τ ) g( s) dτ ds +
f ( τ ) g( s) dτ ds
t 0
0
t 0
s
0
t 0
we establish that
t
τ
1
t η 2
c
e−β( τ−s) η 2( s) dsdτ ≤
dτ +
(8.5.61)
t
m 2( s)
β
m 2
β 2
0
0
t 0
where c = sup η 2( t)
t
, which together with V ∈ L
m 2( t)
∞ imply that
√
R 12 ˜
θ, σ|˜
θ| ∈ S( σ + η 2 /m 2)
√
It can also be shown using the inequality −σ ˜
θ θ ≤ − σ |θ| 2 + σ |θ∗| 2 that
σ|θ| ∈
2
2
S( σ + η 2 /m 2). To examine the properties of , ns we proceed as follows: We have
d ˜ θ R˜ θ = 2˜ θ R˙˜ θ+ ˜ θ ˙ R˜ θ
dt
t
= − 2˜
θ RΓ R˜
θ + 2˜
θ RΓ
e−β( t−τ) φ( τ ) η( τ ) dτ
0
m 2( τ )
(˜
θ φ)2
− 2 σ ˜
θ RΓ θ − β ˜
θ R˜
θ +
m 2
where the second equality is obtained by using (8.5.57) with w = σ and the expres-
sion for ˙
R given by (8.5.47). Because = −˜ θ φ+ η , it follows that
m 2
d
t
˜
θ R˜
θ = − 2˜
θ RΓ R˜
θ + 2˜
θ RΓ
e−β( t−τ) φ( τ ) η( τ ) dτ
dt
0
m 2( τ )
η
− 2 σ ˜
θ RΓ θ − β ˜
θ R˜
θ + ( m −
)2
m
2
Using ( m − η )2 ≥
m 2 − η 2 , we obtain the inequality
m
2
m 2
2 m 2
d
t
≤
˜
θ R˜
θ + 2˜
θ RΓ R˜
θ − 2˜
θ RΓ
e−β( t−τ) φ( τ ) η( τ ) dτ
2
dt
0
m 2( τ )
η 2
+ 2 σ ˜
θ RΓ θ + β ˜
θ R˜
θ +
(8.5.62)
m 2
8.5. ROBUST ADAPTIVE LAWS
599
Noting that
t
2 ˜
θ RΓ
e−β( t−τ) φ( τ ) η( τ ) dτ
0
m 2( τ )
t
t
≤ |˜
θ RΓ | 2 +
e−β( t−τ) |φ( τ ) | 2 dτ
e−β( t−τ) η 2( τ ) dτ
0
m 2( τ )
0
m 2( τ )
where the last inequality is obtained by using 2 ab ≤ a 2 + b 2 and the Schwartz
inequality, it follows from (8.5.62) that
2 m 2
d
≤
˜
θ R˜
θ + 2˜
θ RΓ R˜
θ + 2 σ ˜
θ RΓ θ + β ˜
θ R˜
θ
2
dt
t
t
η 2
+ |˜
θ RΓ) | 2 +
e−β( t−τ) |φ( τ ) | 2 dτ
e−β( t−τ) η 2( τ ) dτ +
0
m 2( τ )
0
m 2( τ )
m 2
√
Because R 1 ˜
2 θ,
σ|θ| ∈ S( σ + η 2 /m 2) and φ , θ, R ∈ L
m
∞, it follows from the above
inequality that , ns, m ∈ S( σ + η 2 /m 2). Now from (8.5.57) with w = σ we have
| ˙ θ| ≤
Γ R 12 |R 12 ˜
θ| + σ Γ |θ|
1
1
t
2
t
2
+ Γ
e−β( t−τ) |φ( τ ) | 2 dτ
e−β( t−τ) η 2( τ ) dτ
0
m 2( τ )
0
m 2( τ )
which can be used to show that ˙ θ ∈ S( σ + η 2 /m 2) by performing similar manipu-
√
lations as in (8.5.61) and using
σ|θ| ∈ S( σ + η 2 /m 2).
To establish the parameter error convergence properties, we write (8.5.57) with
w = σ as
˙
t
˜
θ = −Γ R( t)˜
θ + Γ
e−β( t−τ) φη dτ − Γ σθ
(8.5.63)
0
m 2
In Section 4.8.4, we have shown that the homogeneous part of (8.5.63), i.e., ˙˜
θ =
−Γ R( t)˜
θ is e.s. provided that φ is PE. Noting that
t e−β( t−τ) φη dτ ≤ c¯ η for some
0
m 2
constant c ≥ 0, the rest of the proof is completed by following the same procedure
as in the proof of Theorem 8.5.1 (iii).
(B) Switching σ For w( t) = σs we have, as shown in the proof of Theorem 8.5.2,
that
˜
θ θ
σ ˜
sθ θ ≥ σs|θ|( M 0 − |θ∗|) ≥ 0 ,
i.e., σs|θ| ≤ σs
(8.5.64)
M 0 − |θ∗|
and for |θ| = |˜
θ + θ∗| > 2 M 0 we have
σ
σ
−σ ˜
0
0
sθ θ ≤ −
|˜
θ| 2 +
|θ∗| 2
2
2
600
CHAPTER 8. ROBUST ADAPTIVE LAWS
Furthermore, the inequality (8.5.59) with w( t) = σs can be written as
˜
2
˙
θ R˜
θ
1
η
V ≤ −
+
(
)
− σ ˜
θ θ
(8.5.65)
2
2
m t
s
2 β
Following the same approach as the one used in the proof of part (A), we can
show that ˙
V < 0 for V > V 0 and for some V 0 > 0, i.e., V ∈ L∞ which implies that
, θ, n
˜
˜
s ∈ L∞. Integrating on both sides of (8.5.65), we obtain that R 12 θ,
σsθ θ ∈
S( η 2 /m 2). Proceeding as in the proof of part (A) and making use of (8.5.64), we
show that , ns ∈ S( η 2 /m 2). Because
σ 2
˜
s |θ| 2 ≤ c 1 σsθ θ
where c 1 ∈ R+ depends on the bound for σ 0 |θ|, we can follow the same procedure
as in part (A) to show that ˙ θ ∈ S( η 2 /m 2). Hence, the proof for (i), (ii) of part (B)
is complete.
The proof of part (B) (iii) follows directly by setting η = 0 and repeating the
same steps.
The proof of part (B) (iv) is completed by using similar arguments as in the
case of part (A) (iii) as follows: From (8.5.57), we have
˙
t
˜
θ = −Γ R( t)˜
θ + Γ
e−β( t−τ) φη dτ − Γ wθ, w = σs
(8.5.66)
0
m 2
In Section 4.8.4, we have proved that the homogeneous part of (8.5.66) is e.s. if
φ ∈ L∞ and φ is PE. Therefore, we can treat (8.5.66) as an exponentially stable
linear time-varying system with inputs σsθ, t e−β( t−τ) φη dτ. Because |η| ≤ ¯
η and
0
m 2
m
φ ∈ L
m
∞, we have
t
e−β( t−τ) φη dτ ≤ c¯
η
0
m 2
for some constant c ≥ 0. Therefore, the rest of the parameter convergence proof
can be completed by following exactly the same steps as in the proof of part (A)
(iii).
✷
-Modification
Another choice for w( t) in the adaptive law (8.5.46) is
w( t) = | m|ν 0
where ν 0 > 0 is a design constant. For the adaptive law (8.5.47), however,
the -modification takes a different form and is given by
w( t) = ν 0 ( t, ·) m( ·) t 2 β
8.5. ROBUST ADAPTIVE LAWS
601
where
z( τ ) − θ ( t) φ( τ )
( t, τ ) =
, t ≥ τ
m 2( τ )
and
t
( t, ·) m( ·) 22 β =
e−β( t−τ) 2( t, τ ) m 2( τ ) dτ
0
We can verify that this choice of w( t) may be implemented as
1
w( t) = ( r 0 + 2 θ Q + θ Rθ)2 ν 0
z 2
˙ r 0 = −βr 0 +
, r
m 2
0(0) = 0
(8.5.67)
The stability properties of the adaptive laws with the -modification are
given by the following theorem.
Theorem 8.5.5 Consider the adaptive law (8.5.46) with w = | m|ν 0 and
the adaptive law (8.5.47) with w( t) = ν 0 ( t, ·) m( ·) 2 β. Both adaptive laws guarantee that
(i)
, ns, θ, ˙ θ ∈ L∞.
(ii)
, ns, ˙ θ ∈ S( ν 0 + η 2 /m 2) .
(iii) If ns, φ ∈ L∞ and φ is PE with level α 0 > 0 that is independent of η,
then ˜
θ converges exponentially fast to the residual set
D = ˜
θ |˜
θ| ≤ c( ν 0 + ¯ η)
where c ≥ 0 and ¯
η = sup |η|
t
.
m
Proof The proof of (i) and (ii) for the adaptive law (8.5.46) with w( t) = ν 0 | m|
follows directly from that of Theorem 8.5.3 by using the Lyapunov-like function
V = ˜ θ Γ − 1