m[∆ m( up + du) + du]
α
α
model
z = Wmup, φp = [ Wm
u
y
Λ
p, Wm Λ p, Wmyp, yp]
= z−ˆ z, ˆ
z = θ φ
Normalized
m 2
p, θ = [ θ 1 , θ 2 , θ 3 , c 0]
m 2 = 1 + m
estimation
s, ˙
ms = −δ 0 ms + u 2 p + y 2 p, ms(0) = 0
error
Constraint
B( θ) = c 0 − c 0 sgn( kp ) ≤ 0 for some c
km
0 > 0
satisfying 0 < c 0 ≤ |c∗ 0 |
f
if |c 0 | > c 0
Projection
Pr[ f ]=
or if |c 0 | = c 0 and f ∇B ≤ 0
operator
f − Γ ∇B( ∇B) f otherwise
( ∇B) Γ ∇B
∇B = −[0 , . . . , 0 , 1] sgn( kp/km)
Γ = Γ > 0
Robust
˙ θ = Pr[ f] , |c 0(0) | ≥ c 0
adaptive law
(a) Leakage
f = Γ φp − wΓ θ, w as given in Table 8.1
(b) Dead zone
f = Γ φp( + g) , g as given in Table 8.4
Pr[Γ φp]
if |θ| < M 0
(c) Projection
˙ θ =
or |θ| = M
0 and (Pr[Γ φp]) θ ≤ 0
( I − Γ θθ )Pr[Γ φ
θ Γ θ
p]
otherwise
where M 0 ≥ |θ∗|, |θ(0) | ≤ M 0 and Pr[ ·] is the projec-
tion operator defined above
Properties
(i) , ns, θ, ˙ θ ∈ L∞; (ii) , ns, ˙ θ ∈ S( f 0 + η 2 ), where
m 2
f 0 is as defined in Table 9.2
9.3. ROBUST MRAC
683
˜
θ ω
η 1 = ∆ m( up + du) + du
r
❄ u
❄
✲ c∗
✲ ❧
p
y
✲
+ +
+
✲
p
✲
0
Σ
❧
Σ
Gp( s)
+ +
✒✂✍ ✻
+
❄
❄
✂
✂
( sI − F ) − 1 g
( sI − F ) − 1 g
θ∗ ✛ ω 1
1
ω 2
θ∗ ✛
2
θ∗ ✛
3
Figure 9.3 The closed-loop MRAC scheme in the presence of unmodeled
dynamics and bounded disturbances.
Step 2. Use the swapping lemmas and properties of the L 2 δ norm to
bound ˜
θ ω from above with terms that are guaranteed by the robust adap-
tive laws to have small in m.s.s. gains. In this step we use the Swapping
Lemma A.1 and A.2 and the properties of the L 2 δ-norm to obtain the ex-
pression
˜
1
θ ω ≤ c gmf + c(
+ αk
α
0 ∆ ∞) mf + cd 0
(9.3.67)
0
where α 0 > max { 1 , δ 0 / 2 } is arbitrary and g ∈ S( f 0 + ∆2 i + d 20 ) with ∆
m 2
i =
∆02 , k = n∗ + 1 in the case of the adaptive law of Table 9.2, ∆ i = ∆2 , k = n∗
in the case of the adaptive laws of Tables 9.3 and 9.4, and d 0 is an upper
bound for |du|.
Step 3. Use the B-G Lemma to establish boundedness. Using (9.3.67)
in (9.3.66) we obtain
1
m 2
2
f ≤ c + c gmf
+ c
+ α 2 k
α 2
0 ∆2
∞
m 2 f + cd 20
0
We choose α 0 large enough so that for small ∆ ∞
1
c
+ α 2 k
α 2
0 ∆2
∞
< 1
0
684
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
We then have m 2
2
f ≤ c + c gmf
for some constants c ≥ 0, which implies
that
t
m 2 f ≤ 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
t
t d 2
g 2( τ ) dτ ≤ c( f
0
0 + ∆2
i )( t − s) + c
dτ + c
s
s m 2
∀t ≥ s ≥ 0, it follows that for c( f 0 + ∆2 i) < δ/ 2 we have
t
t
t
m 2
t
( d 2 /m 2) dτ
( d 2 /m 2) dτ
0
( t−s)
0
f ≤ ce− δ 2 ec 0
+ cδ
e− δ 2
ec s
ds
0
The boundedness of mf follows directly if we establish that c d 20 < δ/ 2,
m 2( t)
∀t ≥ 0. This condition is satisfied if we design the signal n 2 s as n 2 s = β 0 + ms where β 0 is a constant chosen large enough to guarantee c d 20
< c d 20 <
m 2( t)
β 0
δ/ 2 , ∀t ≥ 0. This approach guarantees that the normalizing signal m 2 =
1 + n 2 s is much larger than the level of the disturbance all the time. Such a
large normalizing signal may slow down the speed of adaptation and, in fact,
improve robustness. It may, however, have an adverse effect on transient
performance.
The boundedness of all signals can be established, without having to
modify n 2 s, by using the properties of the L 2 δ norm over an arbitrary interval
of time. Considering the arbitrary interval [ t 1 , t) for any t 1 ≥ 0 we can
establish by following a similar procedure as in Steps 1 and 2 the inequality
t
d 2
c
0
m 2( t) ≤ c(1 + m 2( t
( t−t 1)
1)) e− δ 2
e t 1 m 2 dτ
t
t
d 2
c
0
+ cδ
e− δ ( t−s)
2
e s m( τ)2 dτ ds, ∀t ≥ t 1 ≥ 0
(9.3.68)
t 1
We assume that m 2( t) goes unbounded. Then for any given large number
¯
α > 0 there exists constants t 2 > t 1 > 0 such that m 2( t 1) = ¯
α, m 2( t 2) >
f 1(¯
α), where f 1(¯
α) is any static function satisfying f 1(¯
α) > ¯
α. Using the fact
that m 2 cannot grow or decay faster than an exponential, we can choose f 1
9.3. ROBUST MRAC
685
properly so that m 2( t) ≥ ¯
α ∀t ∈ [ t 1 , t 2] for some t 1 ≥ ¯
α where t 2 − t 1 > ¯
α.
Choosing ¯
α large enough so that d 20 /¯ α < δ/ 2, it follows from (9.3.68) that
m 2( t
¯
α
2) ≤ c(1 + ¯
α) e− δ 2 ecd 20 + c
We can now choose ¯
α large enough so that m 2( t 2) < ¯
α which contradicts
with the hypothesis that m 2( t 2) > ¯
α and therefore m ∈ L∞. Because m
bounds up, yp, ω from above, we conclude that all signals are bounded.
Step 4. Establish bounds for the tracking error e 1 . Bounds for e 1 in
m.s.s. are established by relating e 1 with the signals that are guaranteed by
the adaptive law to be of the order of the modeling error in m.s.s.
Step 5. Establish convergence of estimated parameter and tracking error
to residual sets. Parameter convergence is established by expressing the pa-
rameter and tracking error equations as a linear system whose homogeneous
part is e.s. and whose input is bounded.
The details of the algebra and calculations involved in Steps 1 to 5 are
presented in Section 9.8.
Remark 9.3.2 Effects of initial conditions. The results of Theorem 9.3.2
are established using a transfer function representation for the plant.
Because the transfer function is defined for zero initial conditions the
results of Theorem 9.3.2 are valid provided the initial conditions of
the state space plant representation are equal to zero. For nonzero
initial conditions the same steps as in the proof of Theorem 9.3.2 can
be followed to establish that
t
m 2 f( t) ≤ c + cp 0 + cp 0 e−δt + c
e−δ( t−τ) g 2( τ ) m 2 f( τ) dτ
0
where p 0 ≥ 0 depends on the initial conditions. Applying the B-G
Lemma III, we obtain
t
t
t
m 2
g 2( s) ds
g 2( s) ds
f ( t) ≤ ( c + cp 0) e−δtec 0
+ δ( c + cp 0)
e−δ( t−τ) ec τ
dτ
0
where g ∈ S( f 0 + ∆2 i + d 20 /m 2) and f 0 , ∆ i are as defined in Theorem 9.3.2. Therefore the robustness bounds, obtained for zero initial
conditions, will not be affected by the non-zero initial conditions. The
686
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
bounds for mf and tracking error e 1, however, will be affected by the
size of the initial conditions.
Remark 9.3.3 Robustness without dynamic normalization. The results of
Theorem 9.3.2 are based on the use of a dynamic normalizing signal
ms = n 2 s so that m =
1 + n 2 s bounds both the signal vector φ and
modeling error term η from above. The question is whether the signal
ms is necessary for the results of Theorem 9.3.2 to hold. In [168, 240],
it was shown that if m is chosen as m 2 = 1 + φ φ, i.e., the same
normalization used in the ideal case, then the projection modification
alone is sufficient to obtain the same qualitative results as those of
Theorem 9.3.2. The proof of these results is based on arguments over
intervals of time, an approach that was also used in some of the original
results on robustness with respect to bounded disturbances [48]. The
extension of these results to modifications other than projection is not
yet clear.
Remark 9.3.4 Calculation of robustness bounds. The calculation of the
constants c, δ, ∆ i, ∆ ∞ is tedious but possible as shown in [221, 227].
These constants depend on the properties of various transfer func-
tions, namely their H∞δ, H 2 δ bounds and stability margins, and on
the bounds for the estimated parameters. The bounds for the esti-
mated parameters can be calculated from the Lyapunov-like functions
which are used to analyze the adaptive laws. In the case of projection,
the bounds for the estimated parameters are known a priori. Because
the constants c, ∆ i, ∆ ∞, δ depend on unknown transfer functions and
parameters such as G 0( s) , G− 1
0 ( s) , θ∗, the conditions for robust stabil-
ity are quite difficult to check for a given plant. The importance of the
robustness bounds is therefore more qualitative than quantitative, and
this is one of the reasons we did not explicitly specify every constant
in the expression of the bounds.
Remark 9.3.5 Existence and uniqueness of solutions. Equations (9.3.65)
together with the adaptive laws for generating θ = ˜
θ + θ∗ used to
establish the results of Theorem 9.3.2 are nonlinear time varying equa-
tions. The proof of Theorem 9.3.2 is based on the implicit assumption
that these equations have a unique solution ∀t ∈ [0 , ∞). Without
9.3. ROBUST MRAC
687
this assumption, most of the stability arguments used in the proof of
Theorem 9.3.2 are not valid. The problem of existence and unique-
ness of solutions for a class of nonlinear equations, including those of
Theorem 9.3.2 has been addressed in [191]. It is shown that the sta-
bility properties of a wide class of adaptive schemes do possess unique
solutions provided the adaptive law contains no discontinuous modifi-
cations, such as switching σ and dead zone with discontinuities. An
exception is the projection which makes the adaptive law discontinuous
but does not affect the existence and uniqueness of solutions.
The condition for robust stability given by (9.3.64) also indicates that the
design parameter f 0 has to satisfy certain bounds. In the case of switching σ
and projection, f 0 = 0, and therefore (9.3.64) doesnot impose any restriction
on the design parameters of these modifications. For modifications, such as
the -modification, fixed σ, and dead zone, the design parameters have to
be chosen small enough to satisfy (9.3.64). Because (9.3.64) depends on
unknown constants, the design of f 0 can only be achieved by trial and error.
9.3.4
Robust Indirect MRAC
The indirect MRAC schemes developed and analyzed in Chapter 6 suffer
from the same nonrobust problems the direct schemes do. Their robustifica-
tion is achieved by using, as in the case of direct MRAC, the robust adaptive
laws developed in Chapter 8 for on-line parameter estimation.
In the case of indirect MRAC with unnormalized adaptive laws, robusti-
fication leads to semiglobal stability in the presence of unmodeled high fre-
quency dynamics. The analysis is the same as in the case of direct MRAC
with unnormalized adaptive laws and is left as an exercise for the reader.
The failure to establish global results in the case of MRAC with robust but
unnormalized adaptive laws is due to the lack of an appropriate normaliz-
ing signal that could be used to bound from above the effect of dynamic
uncertainties.
In the case of indirect MRAC with normalized adaptive laws, global
stability is possible in the presence of a wide class of unmodeled dynamics
by using robust adaptive laws with dynamic normalization as has been done
in the case of direct MRAC in Section 9.3.3.
688
CHAPTER 9. ROBUST ADAPTIVE CONTROL SCHEMES
We illustrate the robustification of an indirect MRAC with normalized
adaptive laws using the following example:
Example 9.3.3
Consider the MRAC problem for the following plant:
b
yp =
(1 + ∆
s − a
m( s)) up
(9.3.69)
where ∆ m is a proper transfer function and analytic in Re[ s] ≥ −δ 0 / 2 for some
known δ 0 > 0, and a and b are unknown constants. The reference model is g