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

, ns ∈ L∞, θk ∈ ∞

618

CHAPTER 8. ROBUST ADAPTIVE LAWS

(ii) , m ∈ S( η 2 ) , ∆ θ

) where ∆ θ

m 2

k ∈ D( η 2

m 2

k = θk+1 − θk and







k 0+ N

t



k + N T

0

s

D( y) =

{x

x

y( τ ) dτ + c



k}

k xk ≤ c 0

1 

k= k

tk

0

0

for some c 0 , c 1 ∈ R+ and any k 0 , N ∈ N + .

(iii) If ns, φ ∈ L∞ and φ is PE with a level of excitation α 0 > 0 that is

independent of η, then

(a) ˜

θk = θk − θ∗ converges exponentially to the residual set

D 0 = ˜

θ |˜

θ| ≤ c( σ 0 + ¯ η)

where c ≥ 0 , ¯

η = sup t | η |.

m

(b) There exists a constant ¯

η∗ > 0 such that if ¯

η < ¯

η∗, then ˜

θk

converges exponentially fast to the residual set

Dθ = ˜

θ |˜

θ| ≤ c¯

η

Proof Consider the function

V ( k) = ˜

θk Γ − 1 ˜ θk

(8.5.109)

Using ˜

θk+1 = ˜

θk + ∆ θk in (8.5.109), where ∆ θk = Γ tk+1( ( τ ) φ( τ) − w( τ) θ

t

k) dτ , we

k

can write

V ( k + 1) = V ( k) + 2˜

θk Γ − 1∆ θk + ∆ θk Γ − 1∆ θk

tk+1

= V ( k) + 2˜

θk

( ( τ ) φ( τ ) − w( τ ) θk) dτ

(8.5.110)

tk

tk+1

tk+1

+

( ( τ ) φ( τ ) −w( τ ) θk) dτΓ

( ( τ ) φ( τ ) −w( τ ) θk) dτ

tk

tk

Because ˜

θ φ = − m 2 + η, we have

k

tk+1

tk+1

tk+1 2

tk+1

˜

m 2

η 2

θk

( τ ) φ( τ ) dτ =

( − 2 m 2 + η) dτ ≤ −

dτ +

dτ

t

2

2 m 2

k

tk

tk

tk

(8.5.111)

8.5. ROBUST ADAPTIVE LAWS

619

where the last inequality is obtained by using the inequality −a 2 + ab ≤ − a 2 + b 2 .

2

2

Now consider the last term in (8.5.110), since

tk+1

tk+1

( ( τ ) φ( τ ) − w( τ ) θk) dτΓ

( ( τ ) φ( τ ) − w( τ ) θk) dτ

tk

tk

t

2

k+1

≤ λm

( ( τ ) φ( τ ) − w( τ ) θk) dτ

tk

where λm = λmax(Γ − 1), it follows from the inequality ( a + b)2 ≤ 2 a 2 + 2 b 2 that tk+1

tk+1

( ( τ ) φ( τ ) − w( τ ) θk) dτΓ

( ( τ ) φ( τ ) − w( τ ) θk) dτ

tk

tk

t

2

2

k+1

φ( τ )

tk+1

≤ 2 λm

( τ ) m( τ )

dτ

+ 2 λm

w( τ ) θkdτ

t

m( τ )

k

tk

tk+1

tk+1 |φ| 2

≤ 2 λ

2

m

( τ ) m 2( τ ) dτ

dτ + 2 λmσ 2 sT 2 s|θk| 2

t

m 2

k

tk

tk+1

≤ 2 λ

2

mTs

( τ ) m 2( τ ) dτ + 2 λmσ 2 sT 2 s|θk| 2

(8.5.112)

tk

In obtaining (8.5.112), we have used the Schwartz inequality and the assumption

|φ| ≤ 1. Using (8.5.111), (8.5.112) in (8.5.110), we have

m

tk+1

tk+1 |η| 2

V ( k + 1) ≤ V ( k) − (1 − 2 λ

2

mTs)

( τ ) m 2( τ ) dτ +

dτ

t

m 2

k

tk

− 2 σ

˜

sTsθk θk + 2 λmσ 2

s T 2

s |θk| 2

tk+1

≤ V ( k) − (1 − 2 λ

2

mTs)

( τ ) m 2( τ ) dτ + ¯

η 2 Ts

tk

− 2 σsTs(˜

θk θk − λmσ 0 Ts|θk| 2)

tk+1

≤ V ( k) − (1 − 2 λ

2

mTs)

( τ ) m 2( τ ) dτ + ¯

η 2 Ts

tk

1

|θ∗| 2

− 2 σsTs ( − λ

(8.5.113)

2

mσ 0 Ts) |θk| 2 −

2

where the last inequality is obtained by using ˜

θ θ

− |θ∗| 2 . Therefore, for

k

k ≥ |θk| 2

2

2

1 − 2 σ 0 λmTs > 0 and 1 − 2 λmTs > 0, it follows from (8.5.113) that V ( k + 1) ≤ V ( k) whenever

¯

η 2 + σ

|θ

0 |θ∗| 2

k| 2 ≥ max

M 20 , σ 0(1 − 2 λmσ 0 Ts)

Thus, we can conclude that V ( k) and θk ∈ ∞. The boundedness of , m follows

immediately from the definition of

and the normalizing properties of m, i.e.,

φ , η ∈ L

m m

∞.

620

CHAPTER 8. ROBUST ADAPTIVE LAWS

To establish (ii), we use the inequality

σ

σ

σ ˜

s

s

sθk θk

=

θ

θ

2 k θk + ( 2 k θk − σsθk θ∗)

σ

σ

≥

s |θ

s |θ

2

k| 2 + 2 k|( |θk| − 2 |θ∗|)

Because σs = 0 for |θk| ≤ M 0 and σs > 0 for |θk| ≥ M 0 ≥ 2 |θ∗|, we have σs|θk|( |θk|−

2 |θ∗|) ≥ 0 ∀k ≥ 0, therefore,

σ

σ ˜

s

sθk θk ≥

|θ

2

k| 2

which together with 2 σ 0 λmTs < 1 imply that

σs(˜

θk θk − λmσ 0 Ts|θk| 2) ≥ cσσs|θk| 2

(8.5.114)

where cσ = 1 − λ

2

mσ 0 Ts. From (8.5.114) and the second inequality of (8.5.113), we

have

tk+1

tk+1 η 2

(1 − 2 λ

2

mTs)

( τ ) m 2( τ ) dτ + cσσs|θk| 2 ≤ V ( k) − V ( k + 1) +

dτ

t

m 2

k

tk

(8.5.115)

√

which implies that m ∈ S( η 2 ) and

σ

). Because | | ≤ | m| (because

m 2

sθk ∈ D( η 2

m 2

m ≥ 1 , ∀t ≥ 0 ), we have ∈ S( η 2 ).

m 2

Note from (8.5.112) that ∆ θk satisfies

k 0+ N

k 0+ N

t

k

k+1

0+ N

(∆ θ

2

k) ∆ θk

≤ 2 λmTs

( τ ) m 2( τ ) dτ + 2 λmT 2 s

σ 2 s|θk| 2

k= k

tk

0

k= k 0

k= k 0

t

k

k

0+ N

0+ N

≤ 2 λ

2

mTs

( τ ) m 2( τ ) dτ +2 λmT 2 s

σ 2 s|θk| 2 (8.5.116)

tk 0

k= k 0

Using the properties that m ∈ S( η 2 ) , σ

), we have

m 2

sθk ∈ D( η 2

m 2

k 0+ N

tk 0+ N η 2

(∆ θk) ∆ θk ≤ c 1 + c 2

dτ

m 2

k= k

tk

0

0

for some constant c 1 , c 2 > 0. Thus, we conclude that ∆ θk ∈ D( η 2 ).

m 2

Following the same arguments we used in Sections 8.5.2 to 8.5.6 to prove pa-

rameter convergence, we can establish (iii) as follows: We have

tk+1

tk+1

˜

φ( τ ) φ ( τ )

φ( τ ) η( τ )

θk+1 = ˜

θk − Γ

dτ ˜

θk + Γ

dτ − Γ σsθkTs

t

m 2( τ )

m 2( τ )

k

tk

8.5. ROBUST ADAPTIVE LAWS

621

which we express in the form

˜

θk+1 = A( k)˜

θk + Bν( k)

(8.5.117)

where A( k) = I − Γ tk+1 φ( τ) φ ( τ) dτ , B = Γ, ν( k) =

tk+1 φ( τ) η( τ ) dτ − σ

t

sθkTs.

k

m 2( τ )

tk

m 2( τ )

We can establish parameter error convergence using the e.s. property of the homo-

geneous part of (8.5.117) when φ is PE. In Chapter 4, we have shown that φ being

PE implies that the equilibrium ˜

θe = 0 of ˜

θk+1 = A( k)˜

θk is e.s., i.e., the solution ˜

θk

of (8.5.117) satisfies

k

|˜

θk| ≤ β 0 γk + β 1

γk−i|νi|

(8.5.118)

i=0

for some 0 < γ < 1 and β 0 , β 1 > 0. Since |νi| ≤ c(¯

η + σ 0) for some constant c ≥ 0,

the proof of (iii) (a) follows from (8.5.118). From the definition of νk, we have

|νk| ≤ c 0 ¯

η + c 1 |σsθk| ≤ c 0 ¯

η + c 1 |σsθk| |˜ θk|

(8.5.119)

for some constants c 0 , c 1 , c 1, where the second inequality is obtained by using

˜

σ

θ θ

k

k

s|θk| ≤ σs

≤

1

|σ

M

sθk| | ˜

θk|. Using (8.5.119) in (8.5.118), we have

0 −|θ∗|

M 0 −|θ∗|

k− 1

|˜

θk| ≤ β 0 γk + β 1¯ η + β 2

γk−i| σsθi||˜

θi|

(8.5.120)

i=0

where β 1 = c 0 β 1 , β

1 −γ

2 = β 1 c 1.

To proceed with the parameter convergence analysis, we need the following

discrete version of the B-G Lemma.

Lemma 8.5.1 Let x( k) , f ( k) , g( k) be real valued sequences defined for k =

0 , 1 , 2 , . . . ,, and f ( k) , g( k) , x( k) ≥ 0 ∀k. If x( k) satisfies k− 1

x( k) ≤ f ( k) +

g( i) x( i) ,

k = 0 , 1 , 2 , . . .