0
can use the boundedness of the signals to show that ˙ e 1 ∈ L∞ which together with
e 1 ∈ L 2 imply that e 1( t) → 0 as t → ∞.
6.8. STABILITY PROOFS OF MRAC SCHEMES
423
Step 5. Establish parameter convergence. First, we show that φ, φp are PE if
r is sufficiently rich of order 2 n. From the definition of φ, we can write
( sI − F ) − 1 gup
( sI − F ) − 1 gy
φ = H( s)
p
y
p
r
where H( s) = Wm( s) if the adaptive law of Table 6.5(B) is used and H( s) = L− 1( s) if that of Table 6.5(A) is used. Using up = G− 1
p ( s) yp and yp = Wm( s) r + e 1, we
have
φ = φm + ¯
φ
where
( sI − F ) − 1 gG− 1
p ( s) Wm( s)
( sI − F ) − 1 gG− 1
p ( s)
( sI − F ) − 1 gW
( sI − F ) − 1 g
φ
m( s)
m = H ( s)
W
r, ¯
φ = H( s)
e 1
m( s)
1
1
0
Because e 1 ∈ L 2, it follows from the properties of the PE signals that φ is PE if
and only if φm is PE. In the proof of Theorem 6.4.1 and 6.4.2, we have proved that
( sI − F ) − 1 gG− 1
p ( s) Wm( s)
( sI − F ) − 1 gW
φ
m( s)
0 =
W
r
m( s)
1
is PE provided r is sufficiently rich of order 2 n. Because H( s) is stable and minimum
phase and ˙ φ 0 ∈ L∞ owing to ˙ r ∈ L∞, it follows from Lemma 4.8.3 (iv) that
φm = H( s) φ 0 is PE.
From the definition of φp, we have
( sI −F ) − 1 gG− 1
p ( s) Wm( s)
Wm( s)( sI −F ) − 1 gG− 1
p ( s)
( sI − F ) − 1 gW
W
φ
m( s)
m( s)( sI − F ) − 1 g
p = Wm( s)
W
r +
e 1
m( s)
Wm( s)
1
1
Because φp has the same form as φ, the PE property of φp follows by using the
same arguments.
We establish the convergence of the parameter error and tracking error to zero
as follows: First, let us consider the adaptive laws of Table 6.5. For the adaptive
law based on the SPR-Lyapunov approach (Table 6.5(A)), we have
= WmL( ρ∗ ˜
θ φ − ˜
ρξ − n 2 s)
˙˜ θ = −Γ φ sgn( kp/km)
(6.8.20)
424
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
where Cc ( sI − Ac) − 1 Bc = Wm( s) L( s) is SPR. The stability properties of (6.8.20) are established by Theorem 4.5.1 in Chapter 4. According to Theorem 4.5.1 (iii),
˜
θ( t) → 0 as t → ∞ provided φ, ˙ φ ∈ L∞, φ is PE and ξ ∈ L 2. Because ξ =
Wc( Wbω ) ˙ θ and ω ∈ L∞, ˙ θ ∈ L 2, we have ξ ∈ L 2. From up, yp ∈ L∞ and the expression for φ, we can establish that φ, ˙ φ ∈ L∞. Because φ is shown to be PE,
the convergence of ˜
θ( t) to zero follows. From ˜
θ( t) → 0 as t → ∞ and ˜
ρξ, n 2 s ∈ L 2
and the stability of Ac, we have that e( t) → 0 as t → ∞. In fact, the convergence
of e( t) to zero follows from the properties of the Lyapunov-like function used to
analyze (6.8.20) in Theorem 4.5.1 without requiring φ to be PE.
The proof for the adaptive law of Table 6.5(B) follows from the above arguments
by replacing L− 1( s) with Wm( s) and using the results of Theorem 4.5.2 (iii).
For the adaptive laws of Table 6.6, we have established in Chapter 4 that without
projection, ˜
θ( t) → 0 as t → ∞ exponentially fast provided φp is PE. Since projection
can only make the derivative ˙
V of the Lyapunov-like function V , used to analyze the
stability properties of the adaptive law, more negative the exponential convergence
of ˜
θ to zero can be established by following exactly the same steps as in the case of
no projection.
6.8.3
Proof of Theorem 6.6.2: Indirect MRAC
We follow the same steps as in proving stability for the direct MRAC scheme:
Step 1. Express yp, up in terms of ˜
θ ω. Because the control law for the indirect
MRAC is the same as the direct MRAC scheme, equations (6.8.2), (6.8.3) still hold,
i.e., we have
1
1
y
˜
˜
p = Wm( s)
r +
θ ω , u
θ ω
c∗
p = G− 1
p ( s) Wm( s)
r +
0
c∗ 0
As in the direct case, we define the fictitious normalizing signal
m 2
2
2
f = 1 + up
+ yp
where
·
denotes the L 2 δ-norm for some δ > 0. Using the same arguments as in
the proof of Theorem 6.5.1, we have
m 2 f ≤ c + c ˜ θ ω 2
Step 2. We upper bound ˜
θ ω with terms that are guaranteed by the adaptive
laws to have L 2 gains. From (6.6.25), we have
1
θ
ˆ¯
1 αn− 2( s) = Λ( s) −
Z
ˆ
p( s, t) · ˆ
Q( s, t)
(6.8.21)
kp
6.8. STABILITY PROOFS OF MRAC SCHEMES
425
1
θ 2 αn− 2( s) + θ 3Λ( s) =
[ ˆ
Q( s, t) · ˆ
R
ˆ
p( s, t) − Λ0( s) Rm( s)]
(6.8.22)
kp
Consider the above polynomial equations as operator equations. We apply (6.8.21)
to the signal Wm( s) u
y
Λ( s)
p, and (6.8.22) to Wm( s)
Λ( s)
p to obtain
1
W
θ
ˆ¯
m( s)
1 Wm( s) ω 1
= Wm( s) up −
Z
u
ˆ
p( s, t) · ˆ
Q( s, t)
p
k
Λ( s)
p
1
W
θ
m( s)
2 Wm( s) ω 2 + θ 3 Wm( s) yp
=
[ ˆ
Q( s, t) · ˆ
R
y
ˆ
p( s, t) − Λ0( s) Rm( s)]
p
k
Λ( s)
p
Combining these two equations, we have
1
W
θ
ˆ¯
m( s)
0 Wm( s) ω 0
= Wm( s) up −
Z
u
ˆ
p( s, t) · ˆ
Q( s, t)
p
k
Λ( s)
p
1
W
+
[ ˆ
Q( s, t) · ˆ
R
m( s) y
ˆ
p( s, t) − Λ0( s) Rm( s)]
p
(6.8.23)
k
Λ( s)
p
where θ 0 = [ θ 1 , θ 2 , θ 3] , ω 0 = [ ω 1 , ω 2 , yp] . Repeating the same algebraic manipulation, but replacing θ, θp by θ∗, θ∗p in the polynomial equations, we have
¯
Z
W
θ∗
p( s) Q( s)
m( s)
0 Wm( s) ω 0
= Wm( s) up −
u
k
p
p
Λ( s)
Q( s) R
W
+
p( s) − Λ0( s) Rm( s)
m( s) y
k
p
(6.8.24)
p
Λ( s)
where Q( s) is the quotient of Λ0( s) Rm( s) /Rp( s) whose order is n∗ − 1 and n∗ is the relative degree of Gp( s). Subtracting (6.8.24) from (6.8.23), we have
˜
1
W
1
W
θ
ˆ¯
m( s)
ˆ
m( s)
0 Wm( s) ω 0
= −
Z
u
Q( s, t) · ˆ
R
y
ˆ
p( s, t) · ˆ
Q( s, t)
p +
p( s, t)
p
k
Λ( s)
ˆ
Λ( s)
p
kp
1 Λ
1 Λ
−
0( s) Rm( s) W
0( s) Rm( s) W
ˆ
m( s) yp −
m( s) yp
k
Λ( s)
k
Λ( s)
p
p
¯
Z
R
+
p( s) Q( s) W
p( s) Q( s) W
k
m( s) up −
m( s) yp
pΛ( s)
kpΛ( s)
= ef − e 1 f + e 2 f
(6.8.25)
where ˜
θ 0 = θ 0 − θ∗ 0 and
1
W
1
W
e
ˆ¯
m( s)
ˆ
m( s)
f = −
Z
u
Q( s, t) · ˆ
R
y
ˆ
p( s, t) · ˆ
Q( s, t)
p +
p( s, t)
p
k
Λ( s)
ˆ
Λ( s)
p
kp
426
CHAPTER 6. MODEL REFERENCE ADAPTIVE CONTROL
e 1 f , e 2 f are defined as the terms in the second and third brackets of (6.8.25), re-
spectively. Because c 0 = km , c∗
, Λ( s) = Λ
ˆ
k
0 = km
k
0( s) Zm( s) and ¯
Zp( s) up = Rp( s) yp,
p
p
we have
e 1 f = ( c 0 − c∗ 0) yp, e 2 f = 0
(6.8.26)
Using (6.8.26) in (6.8.25) and defining ωp = [ ω 0 , W − 1
m ( s) yp]
we can write
˜
θ Wm( s) ωp = ef
(6.8.27)
Because ˜
θ ω = c∗ 0 ˜
θ ω
c
p, proved in Section 6.8.2 (see equation (6.8.14) ), we use
0
Swapping Lemma A.2 to write
˜
c∗
θ ω = 0 F
c
1( s, α 0)( ˙˜
θ ωp + ˜
θ ˙ ωp) + F ( s, α 0)˜
θ ωp
(6.8.28)
0
where F ( s, α 0) and F 1( s, α 0) are as defined in Section 6.8.2 and satisfy
c
F 1( s, α 0) ∞δ ≤
,
F ( s, α
α
0) W − 1
m ( s) ∞δ ≤ cαn∗
0
0
for any α 0 > δ > 0. Applying Swapping Lemma A.1 to Wm( s)˜
θ ωp and using
(6.8.27), we obtain
˜
θ ωp = W − 1
m (