|θ( t) |
s, σs =
σ
− 1
if M
(8.4.18)
0
0 ≤ |θ( t) | ≤ 2 M 0
M 0
σ 0
if |θ( t) | > 2 M 0
shown in Figure 8.4 where the design constants M 0 , σ 0 are as defined in
the discontinuous switching σ-modification. In this case the adaptive law is
given by
˙ θ = γ 1 u − γσsθ, 1 = y − θu
and in terms of the parameter error
˙˜ θ = γ 1 u − γσsθ, 1 = −˜ θu + η
(8.4.19)
The time derivative of V (˜
θ) = ˜ θ 2 along the solution of (8.4.18), (8.4.19) is
2 γ
given by
2
˙
d 2
V = − 2
˜
1
˜
0
1 + 1 η − σsθθ ≤ −
− σ θθ +
(8.4.20)
2
s
2
Now
σ ˜
sθθ
= σs( θ − θ∗) θ ≥ σs|θ| 2 − σs|θ||θ∗|
≥ σs|θ|( |θ| − M 0 + M 0 − |θ∗|)
8.4. MODIFICATIONS FOR ROBUSTNESS: SIMPLE EXAMPLES 561
i.e.,
σ ˜
sθθ ≥ σs|θ|( |θ| − M 0) + σs|θ|( M 0 − |θ∗|) ≥ 0
(8.4.21)
where the last inequality follows from the fact that σs ≥ 0 , σs( |θ| − M 0) ≥ 0
and M
˜
0 > |θ∗|. Hence, −σsθθ ≤ 0. Therefore, in contrast to the fixed σ-
modification, the switching σ can only make ˙
V more negative. We can also
verify that
−σ ˜
˜
sθθ ≤ −σ 0 θθ + 2 σ 0 M 2
0
(8.4.22)
which we can use in (8.4.20) to obtain
2
˙
d 2
V ≤ − 1 − σ ˜
θθ + 2 σ
0
(8.4.23)
2
0
0 M 2
0 + 2
The boundedness of V and, therefore, of θ follows by using the same pro-
cedure as in the fixed σ case to manipulate the term σ ˜
0 θθ in (8.4.23) and
express ˙
V as
˙
d 2
|θ∗| 2
V ≤ −αV + 2 σ
0
0 M 2
0 +
+ σ
2
0
2
where 0 ≤ α ≤ σ 0 γ which implies that ˜
θ converges exponentially to the
residual set
γ
Ds = ˜
θ |˜
θ| 2 ≤
( d 2
α 0 + σ 0 |θ∗| 2 + 4 σ 0 M 20)
The size of Ds is larger than that of Dσ in the fixed σ-modification case
because of the additional term 4 σ 0 M 20. The boundedness of ˜ θ implies that
θ, ˙ θ, 1 ∈ L∞ and, therefore, property (i) of the unmodified adaptive law in
the ideal case is preserved by the switching σ-modification when η = 0 but
η ∈ L∞.
As in the fixed σ-modification case, the switching σ cannot guarantee
the L 2 property of 1 , ˙ θ in general when η = 0. The bound for 1 in m.s.s.
follows by integrating (8.4.20) to obtain
t+ T
t+ T
2
σ ˜
2
sθθdτ +
1 dτ ≤ d 2
0 T + c 1
t
t
where c
˜
1 = 2 sup t≥ 0[ V ( t) − V ( t + T )] , ∀t ≥ 0 , T ≥ 0. Because σsθθ ≥ 0 it follows that
σ ˜
˜
sθθ, 1 ∈ S( d 2
0). From (8.4.21) we have σsθθ ≥ σs|θ|( M 0 −
|θ∗|) and, therefore,
( σ ˜
θθ)2
σ 2
s
˜
s |θ| 2 ≤
≤ cσ θθ
( M
s
0 − |θ∗|)2
562
CHAPTER 8. ROBUST ADAPTIVE LAWS
for some constant c that depends on the bound for σ 0 |θ|, which implies that
σs|θ| ∈ S( d 20). Because
| ˙ θ| 2 ≤ 2 γ 2 21 u 2 + 2 γ 2 σ 2 s|θ| 2
it follows that | ˙ θ| ∈ S( d 20). Hence, the adaptive law with the switching- σ
modification given by (8.4.19) guarantees that
(ii)
1 , ˙
θ ∈ S( d 20)
In contrast to the fixed σ-modification, the switching σ preserves the ideal
properties of the adaptive law, i.e., when η disappears ( η = d 0 = 0), equation
(8.4.20) implies (because −σ ˜
˜
sθθ ≤ 0) that 1 ∈ L 2 and
σsθθ ∈ L 2, which,
in turn, imply that ˙ θ ∈ L 2. In this case if ˙ u ∈ L∞, we can also establish
as in the the ideal case that 1( t) , ˙ θ( t) → 0 as t → ∞, i.e., the switching
σ does not destroy any of the ideal properties of the unmodified adaptive
law. The only drawback of the switching σ when compared with the fixed
σ is that it requires the knowledge of an upper bound M 0 for |θ∗|. If M 0 in
(8.4.18) happens not to be an upper bound for |θ∗|, then the adaptive law
(8.4.18) has the same properties and drawbacks as the fixed σ-modification
(see Problem 8.7).
(c) 1 -modification [172]. Another attempt to eliminate the main drawback
of the fixed σ-modification led to the following modification:
w( t) = | 1 |ν 0
where ν 0 > 0 is a design constant. The adaptive law becomes
˙ θ = γ 1 u − γ| 1 |ν 0 θ, 1 = y − θu
and in terms of the parameter error
˙˜ θ = γ 1 u − γ| 1 |ν 0 θ, 1 = −˜ θu + η
(8.4.24)
The logic behind this choice of w is that because in the ideal case 1 is
guaranteed to converge to zero (when ˙ u ∈ L∞), then the leakage term w( t) θ
will go to zero with 1 when η = 0; therefore, the ideal properties of the
adaptive law (8.4.24) when η = 0 will not be affected by the leakage.
8.4. MODIFICATIONS FOR ROBUSTNESS: SIMPLE EXAMPLES 563
The time derivative of V (˜
θ) = ˜ θ 2 along the solution of (8.4.24) is given
2 γ
by
˜
˙
θ 2
|θ∗| 2
V = − 2
˜
1 + 1 η − | 1 |ν 0 θθ ≤ −| 1 |
| 1 | + ν 0
− ν
− d
2
0
2
0
(8.4.25)
where the inequality is obtained by using ν ˜
˜
θ 2
|θ∗| 2
0 θθ ≤ −ν 0
+ ν
. It is
2
0 2
clear that for |
˜
θ 2
|θ∗| 2
|θ∗| 2
1 | + ν 0
≥ ν
+ d
( ν
+ d
2
0 2
0, i.e., for V ≥ V 0 =
1
γν
0
0)
0
2
we have ˙
V ≤ 0, which implies that V and, therefore, ˜
θ, θ ∈ L∞. Because
1 = − ˜
θu + η and u ∈ L∞ we also have that 1 ∈ L∞, which, in turn, implies
that ˙ θ ∈ L∞. Hence, property (i) is also guaranteed by the 1-modification
despite the presence of η = 0.
Let us now examine the L 2 properties of 1 , ˙ θ guaranteed by the unmod-
ified adaptive law ( w( t) ≡ 0) when η = 0. We rewrite (8.4.25) as
2
2
˜
˙
d 2
|
θ 2
|θ∗| 2
d 2
V ≤ − 1 + 0 − |
˜
θ 2 − |
1 − 1 |ν 0
+ |
+ 0
2
2
1 |ν 0
1 |ν 0 θ∗ ˜
θ ≤ − 2
2
1 |ν 0
2
2
by using the inequality −a 2 ± ab ≤ − a 2 + b 2 . If we repeat the use of the
2
2
same inequality we obtain
2
˙
d 2
|θ∗| 4
V ≤ − 1 + 0 + ν 2
(8.4.26)
4
2
0
4
Integrating on both sides of (8.4.26), we establish that 1 ∈ S( d 20 + ν 20).
Because u, θ ∈ L∞, it follows that | ˙ θ| ≤ c| 1 | for some constant c ≥ 0 and
therefore ˙ θ ∈ S( d 20 + ν 20). Hence, the adaptive law with the 1-modification
guarantees that
(ii)
1 , ˙
θ ∈ S( d 20 + ν 20)
which implies that 1 , ˙ θ are of order of d 0 , ν 0 in m.s.s.
It is clear from the above analysis that in the absence of the disturbance
i.e., η = 0, ˙
V cannot be shown to be negative definite or semidefinite unless
ν
˜
0 = 0. The term | 1 |ν 0 θθ in (8.4.25) may make ˙
V positive even when η = 0
and therefore the ideal properties of the unmodified adaptive law cannot
be guaranteed by the adaptive law with the 1-modification when η = 0
unless ν 0 = 0. This indicates that the initial rationale for developing the
1-modification is not valid. It is shown in [172], however, that if u is PE,
564
CHAPTER 8. ROBUST ADAPTIVE LAWS
then 1( t) and therefore w( t) = ν 0 | 1( t) | do converge to zero as t → ∞ when η( t) ≡ 0 , ∀t ≥ 0. Therefore the ideal properties of the adaptive law can be
recovered with the 1-modification provided u is PE.
Remark 8.4.1
(i) Comparing the three choices for the leakage term w( t), it is clear that
the fixed σ- and 1-modification require no a priori information about
the plant, whereas the switching- σ requires the design constant M 0 to
be larger than the unknown |θ∗|. In contrast to the fixed σ- and 1-
modifications, however, the switching σ achieves robustness without
having to destroy the ideal properties of the adaptive scheme. Such
ideal properties are also possible for the 1-modification under a PE
condition[172].
(ii) The leakage −wθ with w( t) ≥ 0 introduces a term in the adaptive law
that has the tendency to drive θ towards θ = 0 when the other term
(i.e., γ 1 u in the case of (8.4.9)) is small. If θ∗ = 0, the leakage term
may drive θ towards zero and possibly further away from the desired
θ∗. If an a priori estimate ˆ
θ∗ of θ∗ is available the leakage term −wθ
may be replaced with the shifted leakage −w( θ − ˆ
θ∗), which shifts the
tendency of θ from zero to ˆ
θ∗, a point that may be closer to θ∗. The
analysis of the adaptive laws with the shifted leakage is very similar to
that of −wθ and is left as an exercise for the reader.
(iii) One of the main drawbacks of the leakage modifications is that the es-
timation error 1 and ˙ θ are only guaranteed to be of the order of the
disturbance and, with the exception of the switching σ-modification, of
the order of the size of the leakage design parameter, in m.s.s. This
means that at steady state, we cannot guarantee that 1 is of the order
of the modeling error. The m.s.s. bound of 1 may allow 1 to exhibit
“bursting,” i.e., 1 may assume values higher than the order of the mod-
eling error for some finite intervals of time. One way to avoid bursting
is to use PE signals or a dead-zone modification as it will be explained
later on in this chapter
8.4. MODIFICATIONS FOR ROBUSTNESS: SIMPLE EXAMPLES 565
(iv) The leakage modification may be also derived by modifying the cost
2
function J( θ) = 1 = ( y−θu)2 , used in the ideal case, to
2
2
( y − θu)2
θ 2
J( θ) =
+ w
(8.4.27)
2
2
Using the gradient method, we now obtain
˙ θ = −γ∇J = γ 1 u − γwθ
which is the same as (8.4.9). The modified cost now penalizes θ in
addition to 1 which explains why for certain choices of w( t) the drifting
of θ to infinity due to the presence of modeling errors is counteracted.
8.4.2
Parameter Projection
An effective method for eliminating parameter drift and keeping the param-
eter estimates within some a priori defined bounds is to use the gradient pro-
jection method to constrain the parameter estimates to lie inside a bounded
convex set in the parameter space. Let us illustrate the use of projection for
the adaptive law
˙ θ = γ 1 u, 1 = y − θu
We like to constrain θ to lie inside the convex bounded set
g( θ) = θ θ 2 ≤ M 20
where M 0 ≥ |θ∗|. Applying the gradient projection method, we obtain
γ
˙
1 u
if |θ| < M 0
˜
θ =
˙ θ =
or if |θ| = M
(8.4.28)
0 and θ 1 u ≤ 0
0
if |θ| = M 0 and θ 1 u > 0
1
= y − θu = −˜
θu + η
which for |θ(0) | ≤ M 0 guarantees that |θ( t) | ≤ M 0 , ∀t ≥ 0. Let us now
analyze the above adaptive law by considering the Lyapunov function
˜
θ 2
V = 2 γ
566
CHAPTER 8. ROBUST ADAPTIVE LAWS
whose time derivative ˙
V along (8.4.28) is given by
− 21 + 1 η
if |θ| < M 0
˙
V =
or if |θ| = M
(8.4.29)
0 and θ 1 u ≤ 0
0
if |θ| = M 0 and θ 1 u > 0
Let us consider the case when ˙
V = 0, |θ| = M 0 and θ 1 u > 0. Using the
expression 1 = −˜
θu + η, we write ˙
V = 0 = − 21 + 1 η − ˜ θ 1 u. The last term
in the expression of ˙
V can be written as
˜
θ 1 u = ( θ − θ∗) 1 u = M 0sgn( θ) 1 u − θ∗ 1 u
Therefore, for θ 1 u > 0 and |θ| = M 0 we have
˜
θ 1 u = M 0 | 1 u| − θ∗ 1 u ≥ M 0 | 1 u| − |θ∗|| 1 u| ≥ 0
where the last inequality is obtained by using the assumption that