the tracking error equation
Z
e
pQ 1
1 =
¯
u
R
p
pQm
in the following state-space form
In+ q− 1
˙ e = −θ∗
1
− − −−
e + θ∗ 2¯ up, e ∈ Rn+ q
(7.7.22)
0
e 1 = C e
where C
= [1 , 0 , . . . , 0] ∈ Rn+ q and θ∗ 1 , θ∗ 2 are the coefficient vectors of RpQm −
sn+ q, ZpQ 1, respectively.
Let eo = e − ˆ e be the state observation error. Then from the equation for ˆ e in
Table 7.5 and (7.7.22), we have
˙ˆ e = Ac( t)ˆ e + ˆ
KoC eo
˙ eo = Aoeo + ˜
θ 1 e 1 − ˜
θ 2¯ up
(7.7.23)
where
In+ q− 1
A
o =
−a∗
− − −−
0
520
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
is a stable matrix; Ac( t) = ˆ
A − ˆ
B ˆ
Kc and ˜
θ 1 = θ 1 − θ∗ 1 , ˜ θ 2 = θ 2 − θ∗ 2. The plant input
and output satisfy
yp = C eo + C ˆ e + ym
R
Q
u
pX Q 1
1 Y Rp
p
=
¯
u
y
A∗
p +
A∗
p
(7.7.24)
where X( s) , Y ( s) are polynomials of degree n + q − 1 that satisfy (7.3.25) and A∗( s) is a Hurwitz polynomial of degree 2( n+ q) − 1. Equation (7.7.24) for up is established
in the proof of Theorem 7.3.1 and is used here without proof.
Step 2. Establish e.s. for the homogeneous part of (7.7.23). Let us first
examine the stability properties of Ac( t) in (7.7.23). For each frozen time t, we
have
det( sI − Ac) = det( sI − ˆ
A + ˆ
B ˆ
Kc) = A∗c( s)
(7.7.25)
i.e., Ac( t) is stable at each frozen time t. If ˆ
Zp( s, t) Q 1( s) , ˆ
Rp( s, t) Qm( s) are strongly
coprime, i.e., ( ˆ
A, ˆ
B) is strongly controllable at each time t, then the controller gains
ˆ
Kc may be calculated at each time t using Ackermann’s formula [95], i.e.,
ˆ
Kc = [0 , 0 , . . . , 0 , 1] G− 1
c
A∗c( ˆ
A)
where
Gc = [ ˆ
B, ˆ
A ˆ
B, . . . , ˆ
An+ q− 1 ˆ
B]
is the controllability matrix of the pair ( ˆ
A, ˆ
B). Because ( ˆ
A, ˆ
B) is assumed to be
strongly controllable and ˆ
A, ˆ
B ∈ L∞ due to θp ∈ L∞, we have ˆ
Kc ∈ L∞. Now,
˙ˆ
d
K
˙
c = [0 , 0 , . . . , 0 , 1]
−G− 1
c
GcG− 1
c
A∗c( ˆ
A) + G− 1
c
A∗
dt c( ˆ
A)
Because θp ∈ L∞ and ˙ θp ∈ L 2, it follows that ˙ˆ
Kc( t) ∈ L 2, which, in turn, implies
that
˙
Ac( t) ∈ L 2. From Ac being pointwise stable and
˙
Ac( t) ∈ L 2, we have that
Ac( t) is a u.a.s matrix by applying Theorem 3.4.11. If ˆ
Zp( s, t) Q 1( s) , ˆ
Rp( s, t) Qm( s)
are not strongly coprime but ˆ
Zp( s, t) , ˆ
Rp( s, t) Qm( s) are, the boundedness of ˆ
Kc
and
˙ˆ
Kc( t) ∈ L 2 can still be established by decomposing ( ˆ
A, ˆ
B) into the strongly
controllable and the stable uncontrollable or weakly controllable parts and using
the results in [95] to obtain an expression for ˆ
Kc. Because Ao is a stable matrix the
homogeneous part of (7.7.23) is e.s.
Step 3. Use the properties of the L 2 δ norm and the B-G Lemma to establish
boundedness. As in Example 7.4.2., we apply Lemmas 3.3.3, 3.3.2 to (7.7.23) and
(7.7.24), respectively, to obtain
ˆ
e
≤ c C eo
yp
≤ c C eo + c ˆ e + c ≤ c C eo + c
(7.7.26)
up
≤ c ˆ
e + c yp ≤ c C eo + c
7.7. STABILITY PROOFS
521
where
·
denotes the L 2 δ norm for some δ > 0.
We relate the term C eo with the estimation error by using (7.7.23) to express
C eo as
C eo = C ( sI − Ao) − 1(˜
θ 1 e 1 − ˜
θ 2 ¯ up)
(7.7.27)
Noting that ( C, Ao) is in the observer canonical form, i.e., C ( sI − Ao) − 1 =
αn+ q− 1( s) , we have
A∗( s)
o
¯
n
s¯ n−i
C eo =
(˜
θ
A∗
1 ie 1 − ˜
θ 2 i ¯ up) ,
¯
n = n + q − 1
i=0
o
where ˜
θi = [˜
θi 1 , ˜
θi 2 , . . . , ˜
θi¯ n] , i = 1 , 2. Applying Swapping Lemma A.1 to each
term under the summation, we have
s¯ n−i ˜
Λ
s¯ n−i
θ
p( s) Q 1( s)
˜
θ
e
A∗
1 ie 1 =
1 i
1 + Wci( s) ( Wbi( s) e 1) ˙˜
θ 1 i
o( s)
A∗o( s)
Λ p( s) Q 1( s)
and
s¯ n−i ˜
Λ
s¯ n−i
θ
p( s) Q 1( s)
˜
θ
¯
u
A∗
2 i ¯
up =
2 i
p + Wci( s) ( Wbi( s)¯
up) ˙˜
θ 2 i
o( s)
A∗o( s)
Λ p( s) Q 1( s)
where Wci, Wbi, i = 0 , . . . , n + q − 1 are transfer matrices defined in Lemma A.1
with W ( s) =
s¯
n−i
. Therefore, C e
Λ
o can be expressed as
p( s) Q 1( s)
Λ
¯
n
s¯ n−i
s¯ n−i
C e
p( s) Q 1( s)
˜
o
=
θ
e
¯
u
A∗
1 i
1 − ˜
θ 2 i
p
+ r 1
o( s)
Λ
Λ
i=0
p( s) Q 1( s)
p( s) Q 1( s)
Λ
α
α
=
p( s) Q 1( s)
˜
θ
n+ q− 1( s) e
n+ q− 1( s) ¯ u
A∗
1
1 − ˜
θ 2
p
+ r 1 (7.7.28)
o( s)
Λ p( s) Q 1( s)
Λ p( s) Q 1( s)
where
Λ
¯
n
r
p( s) Q 1( s)
1 =
W
A∗
ci( s)[( Wbi( s) e 1) ˙˜
θ 1 i − ( Wbi( s)¯ up) ˙˜
θ 2 i]
o( s)
i=0
From the definition of ˜
θ 1, we have
˜
θ 1 αn+ q− 1( s) = θ 1 αn+ q− 1( s) − θ∗ 1 αn+ q− 1( s)
=
ˆ
Rp( s, t) Qm( s) − sn+ q − Rp( s) Qm( s) + sn+ q
= ( ˆ
Rp( s, t) − Rp( s)) Qm( s) = ˜
θa αn− 1( s) Qm( s)
(7.7.29)
where ˜
θa = θa − θ∗a is the parameter error. Similarly,
˜
θ 2 αn+ q− 1( s) = ˜ θb αn− 1( s) Q 1( s)
(7.7.30)
522
CHAPTER 7. ADAPTIVE POLE PLACEMENT CONTROL
where ˜
θb = θb − θ∗. Using (7.7.29) and (7.7.30) in (7.7.28), we obtain
b
Λ
Q
1
1
C e
p( s) Q 1( s)
˜
m( s)
o
=
θ
e
¯
u
A∗
a αn− 1( s)
1 − ˜
θb αn− 1( s)
p
+ r 1
o( s)
Q 1( s) Λ p( s)
Λ p( s)
Λ
Q
Q
=
p( s) Q 1( s)
˜
θ
m( s)
y
m( s)
u
A∗
a αn− 1( s)
p − ˜
θb αn− 1( s)
p
o( s)
Q 1( s)Λ p( s)
Λ p( s) Q 1( s)
+ r 1
(7.7.31)
where the second equality is obtained using
Qm( s) e 1 = Qm( s) yp,
Q 1( s)¯ up = Qm( s) up
Noting that
1
1
αn− 1( s)
u
y
Λ
p = φ 1 ,
αn− 1( s)
p = −φ 2
p( s)
Λ p( s)
we use Swapping Lemma A.1 to obtain the following equalities:
Qm( s) ˜
Q
θ
m( s)
y
˙˜ θ
Q
a φ 2 = − ˜
θa αn− 1( s)
p + Wcq Wbq ( s) φ 2
a
1( s)
Q 1( s)Λ p( s)
Qm( s) ˜
Q
θ
m( s)
u
˙˜ θ
Q
b φ 1 = ˜
θb αn− 1( s)
p + Wcq Wbq ( s) φ 1
b
1( s)
Q 1( s)Λ p( s)
where Wcq, Wbq are as defined in Swapping Lemma A.1 with W ( s) = Qm( s) . Using
Q 1( s)
the above equalities in (7.7.31) we obtain
Λ
C e
p( s) Qm( s) ˜
o = −
θ
A∗
p φ + r 2
(7.7.32)
o( s)
where
Λ
r
p( s) Q 1( s)
˙˜
˙˜
2 = r 1 +
W
θ
θ
A∗
cq ( s) Wbq ( s) φ 1
b + Wcq ( s) Wbq ( s) φ 2
a
o( s)
From Table 7.5, the normalized estimation error satisfies the equation
m 2 = −˜
θp φ
which can be used in (7.7.32) to yield
Λ
C e
p( s) Qm( s)