θy + η ,
u ≤ c ˜
θy + ηu
and, therefore, the fictitious normalizing signal mf satisfies
m 2
2
f = 1 + u 2 + y 2 ≤ 1 + c( ˜
θy 2 + η 2 + ηu )
(9.3.50)
Step 2. Use the swapping lemmas and properties of the L 2 δ norm to upper
bound ˜
θy with terms that are guaranteed by the adaptive law to have small in
m.s.s. gains. We use the Swapping Lemma A.2 given in Appendix A to write the
identity
˜
α
α
1
α
θy = (1 −
0
)˜
θy +
0
˜
θy =
( ˙˜
θy + ˜
θ ˙ y) +
0
˜
θy
(9.3.51)
s + α 0
s + α 0
s + α 0
s + α 0
where α 0 > 0 is an arbitrary constant. Now from the equation for y in (9.3.49) we
obtain
˜
θy = ( s + am)( η − y)
9.3. ROBUST MRAC
671
which we substitute in the last term in (9.3.51) to obtain
˜
1
( s + a
θy =
( ˙˜
θy + ˜
θ ˙ y) + α
m) ( η − y)
s + α
0
0
s + α 0
Therefore, by choosing δ, α 0 to satisfy α 0 > am > δ > 0, we obtain
2
˜
1
s + a
θy ≤
( ˙˜
θy + ˜
θ ˙ y ) + α
m
( η + y )
s + α
0
0
s + α
∞δ
0
∞δ
Hence,
˜
2
θy ≤
( ˙˜
θy + ˜
θ ˙ y ) + α
α
0 c( η
+ y )
0
where c = s+ am
s+ α
∞δ . Using (9.3.46), it follows that
0
y ≤
m 2 + c ˙˜
θφ
therefore,
˜
2
θy ≤
( ˙˜
θy + ˜
θ ˙ y ) + α
α
0 c(
m 2 + ˙˜
θφ + η )
(9.3.52)
0
The gain of the first term in the right-hand side of (9.3.52) can be made small by
choosing large α 0. The m.s.s. gain of the second and third terms is guaranteed by
the adaptive law to be of the order of the modeling error denoted by the bound
∆2, i.e., m, ˙˜
θ ∈ S(∆22). The last term has also a gain which is of the order of the
modeling error. This implies that the gain of ˜
θy is small provided ∆2 is small
and α 0 is chosen to be large.
Step 3. Use the B-G Lemma to establish boundedness. The normalizing prop-
erties of mf and θ ∈ L∞ guarantee that y/mf , φ/mf , m/mf ∈ L∞. Because
˙ y ≤ |a| y + ∆ m( s) ∞δ u + u
it follows that ˙ y /mf ∈ L∞. Due to the fact that ∆ m( s) is proper and analytic in
Re[ s] ≥ −δ 0 / 2, ∆ m( s) ∞δ is a finite number provided 0 < δ ≤ δ 0. Furthermore, η /mf ≤ ∆ ∞ where
∆
∆
m( s)
∞ =
(9.3.53)
s + am ∞δ
and, therefore, (9.3.52) may be written in the form
˜
c
θy ≤
( ˙˜
θm
α
f
+ mf ) + α 0 c( mmf + ˙˜
θmf + ∆ ∞mf )
(9.3.54)
0
Using (9.3.54) and η /mf ≤ c∆ ∞, ηu /mf ≤ c∆ ∞ in (9.3.50), we obtain
1
m 2
2
f ≤ 1 + c
+ α 2
α 2
0∆2
∞
m 2 f + c ˜ gmf
0
672
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
where ˜
g 2 = |˙˜ θ| 2 + α 2
+ α 2
α 2
0 | m| 2 + α 2
0 | ˙˜
θ| 2 and α 0 ≥ 1. For c 1
α 2
0∆2
∞
< 1, we have
0
0
t
m 2
2
f ≤ c + c ˜
gmf
= c + c
e−δ( t−τ)˜
g 2( τ ) m 2 f( τ) dτ
0
Applying the B-G Lemma III, we obtain
t
t
t
m 2
˜
g 2( τ ) dτ
˜
g 2( τ ) dτ
f ≤ ce−δtec 0
+ cδ
e−δ( t−s) ec s
ds
0
Because ˙˜
θ, m ∈ S(∆22), it follows that
t
1
c
˜
g 2( τ ) dτ ≤ c∆22
+ α 20 ( t − s) + c
s
α 20
Hence, for
1
c∆22
+ α 2
α 2
0
< δ
(9.3.55)
0
we have
t
e−δtec
˜
g 2( τ ) dτ
0
≤ e−¯ αt
where ¯
α = δ − c∆2
1
2
+ α 2
α 2
0
which implies that mf is bounded. The boundedness
0
of mf implies that all the other signals are bounded too. The constant δ in (9.3.55)
may be replaced by δ 0 since no restriction on δ is imposed except that δ ∈ (0 , δ 0].
The constant c > 0 may be determined by following the calculations in each of the
steps and is left as an exercise for the reader.
Step 4. Obtain a bound for the regulation error y. The regulation error, i.e.,
y, is expressed in terms of signals that are guaranteed by the adaptive law to be of
the order of the modeling error in m.s.s. This is achieved by using the Swapping
Lemma A.1 for the error equation (9.3.45) and the equation m 2 = −˜
θφ+ η to obtain
(9.3.46), which as shown before implies that y ∈ S(∆22). That is, the regulation
error is of the order of the modeling error in m.s.s.
The conditions that ∆ m( s) has to satisfy for robust stability are summarized
as follows:
1
1
c
+ α 2
+ α 2
α 2
0∆2
∞
< 1 ,
c∆22
0
< δ 0
0
α 20
where
∆
∆
∆
m( s)
m( s)
∞ =
,
∆
s + a
2 =
m
s + a
∞δ
m
0
2 δ 0
The constant δ 0 > 0 is such that ∆ m( s) is analytic in Re[ s] ≥ −δ 0 / 2 and c denotes finite constants that can be calculated. The constant α 0 > max { 1 , δ 0 / 2 } is arbitrary and can be chosen to satisfy the above inequalities for small ∆2 , ∆ ∞.
9.3. ROBUST MRAC
673
Let us now simulate the above robust MRAC scheme summarized by the equa-
tions
u = −θy
˙
z − θφ
θ = γ φ − σsγθ,
=
m 2
1
1
φ =
y, z = y −
u
s + am
s + am
m 2
= 1 + ms, ˙
ms = −δ 0 ms + u 2 + y 2 , ms(0) = 0
where σs is the switching σ, and applied to the plant
1
y =
(1 + ∆
s − a
m( s)) u
where for simulation purposes we assume that a = 1 and ∆ m( s) = − 2 µs with
1+ µs
µ ≥ 0. It is clear that for µ > 0 the plant is nonminimum phase. Figure 9.2
shows the response of y( t) for different values of µ that characterize the size of
the perturbation ∆ m( s). For small µ, we have boundedness and good regulation
performance. As µ increases, stability deteriorates and for µ = 0 . 35 the plant
becomes unstable.
General Case
Let us now consider the SISO plant given by
yp = G 0( s)(1 + ∆ m( s))( up + du)
(9.3.56)
where
Z
G
p( s)
0( s) = kp
(9.3.57)
Rp( s)
is the transfer function of the modeled part of the plant. The high frequency
gain kp and the polynomials Zp( s) , Rp( s) satisfy assumptions P1 to P4 given
in Section 6.3 and the overall transfer function of the plant is strictly proper.
The multiplicative uncertainty ∆ m( s) satisfies the following assumptions:
S1. ∆ m( s) is analytic in Re[ s] ≥ −δ 0 / 2 for some known δ 0 > 0.
S2. There exists a strictly proper transfer function W ( s) analytic in Re[ s] ≥
−δ 0 / 2 and such that W ( s)∆ m( s) is strictly proper.
674
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
2
µ = 0.23
1.5
µ = 0.2
µ = 0.1
µ = 0.05
1
µ = 0.0
0.5
tracking error
0
-0.5
-10
5
10
15
20
25
30
35
40
time (sec)
Figure 9.2 Simulation results of the MRAC scheme of Example 9.3.2 for
different µ.
Assumptions S1 and S2 imply that ∆ ∞, ∆2 defined as
∆ ∞ = W ( s)∆ m( s)
, ∆
∞δ
2 =
W ( s)∆ m( s)
0
2 δ 0
are finite constants. We should note that the strict properness of the overall
plant transfer function and of G 0( s) imply that G 0( s)∆ m( s) is a strictly
proper transfer function.
The control objective is to choose up and specify the bounds for ∆ ∞, ∆2
so that all signals in the closed-loop plant are bounded and the output yp
tracks, as close as possible, the output of the reference model ym given by
Z
y
m( s)
m = Wm( s) r = km
r
Rm( s)
for any bounded reference signal r( t). The transfer function Wm( s) of the
reference model satisfies assumptions M1 and M2 given in Section 6.3.
The design of the control input up is based on the plant model with
∆ m( s) ≡ 0 and du ≡ 0. The control objective, however, has to be achieved
for the plant with ∆ m( s) = 0 and du = 0.
9.3. ROBUST MRAC
675
We start with the control law developed in Section 6.5.3 for the plant
model with ∆ m( s) ≡ 0, du ≡ 0, i.e.,
up = θ ω
(9.3.58)
where θ = [ θ 1 , θ 2 , θ 3 , c 0] , ω = [ ω 1 , ω 2 , yp, r] . The parameter vector θ is to be generated on-line by an adaptive law. The signal vectors ω 1 , ω 2 are
generated, as in Section 6.5.3, by filtering the plant input up and output yp.
The control law (9.3.58) will be robust with respect to the plant uncertainties
∆ m( s) , du if we use robust adaptive laws from Chapter 8, instead of the
adaptive laws used in Section 6.5.3, to update the controller parameters.
The derivation of the robust adaptive laws is achieved by first developing
the appropriate parametric models for the desired controller parameter vec-
tor θ∗ and then choosing the appropriate robust adaptive law by employing
the results of Chapter 8 as follows:
We write the plant in the form
Rpyp = kpZp(1 + ∆ m)( up + du)
(9.3.59)
and then use the matching equation
(Λ − θ∗ 1 α) Rp − kp( θ∗ 2 α + Λ θ∗ 3) Zp = ZpΛ0 Rm
(9.3.60)
where α = αn− 2( s) = [ sn− 2 , · · · , s, 1] , (developed in Section 6.3 and given
by Equation (6.3.12)) satisfied by the desired parameter vector
θ∗ = [ θ∗ 1 , θ∗ 2 , θ∗ 3 , c∗ 0]
to eliminate the unknown polynomials Rp( s), Zp( s) from the plant equation
(9.3.59). From (9.3.59) we have
(Λ − θ∗ 1 α) Rpyp = (Λ − θ∗ 1 α) kpZp(1 + ∆ m)( up + du)
which together with (9.3.60) imply that
Zp( kp( θ∗ 2 α + Λ θ∗ 3) + Λ0 Rm) yp = (Λ − θ∗ 1 α) kpZp(1 + ∆ m)( up + du) Filtering each side with the stable filter 1 and rearranging the terms, we
Λ Zp
obtain
α
R
α
k
m
p
θ∗ 2
+ θ∗
y
u
Λ
3
yp + Z p = kpup − kpθ∗ 1
p
m
Λ
Λ − θ∗
+ k
1 α
p
(∆
Λ
m( up + du) + du)
676
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
or
α
α
k
Λ − θ∗
θ∗
m
1 α
1