Robust Adaptive Control by Petros A. Ioannou, Jing Sun - HTML preview

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∞.

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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

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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