the boundedness of x, u. We consider the estimation error equation
˙1 = −am 1 + ˜ ax
(4.3.10)
where ˜ a = ˆ a − a and propose the same function
2
˜ a 2
V = 1 +
(4.3.11)
2
2
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CHAPTER 4. ON-LINE PARAMETER ESTIMATION
as in Section 4.2.2. The time derivative of V along (4.3.9) and (4.3.10) is
given by
˙
V = −a
2
m 1 ≤ 0
(4.3.12)
Because x is not necessarily bounded, it cannot be treated as an independent
bounded function of time in (4.3.10) and, therefore, (4.3.10) cannot be de-
coupled from (4.3.8). Consequently, (4.3.8) to (4.3.10) have to be considered
and analyzed together in R 3, the space of 1 , ˜ a, ˆ x. The chosen function V
in (4.3.11) is only positive semidefinite in R 3, which implies that V is not a
Lyapunov function; therefore, Theorems 3.4.1 to 3.4.4 cannot be applied. V
is, therefore, a Lyapunov-like function, and the properties of V , ˙
V allow us
to draw some conclusions about the behavior of the solution 1( t) , ˜ a( t) with-
out having to apply the Lyapunov Theorems 3.4.1 to 3.4.4. From V ≥ 0 and
˙
V = −a
2
m 1 ≤ 0 we conclude that V ∈ L∞, which implies that 1 , ˜
a ∈ L∞,
and 1 ∈ L 2. Without assuming x ∈ L∞, however, we cannot establish any
bound for ˙˜ a in an Lp sense.
As in Section 4.3.1, let us attempt to use normalization and modify
(4.3.9) to achieve bounded speed of adaptation in some sense. The use of
normalization is not straightforward in this case because of the dynamics
introduced by the transfer function
1
, i.e., dividing each side of (4.3.7)
s+ am
by m may not help because
x
1
x
u
=
( a
+
m
s + a
m − a)
m
m
m
For this case, we propose the error signal
1
1
= x − ˆ
x −
n 2
(˜ ax − n 2
s + a
s =
s)
(4.3.13)
m
s + am
i.e.,
˙ = −am + ˜ ax − n 2 s
where ns is a normalizing signal to be designed.
Let us now use the error equation (4.3.13) to develop an adaptive law for
ˆ a. We consider the Lyapunov-like function
2
˜ a 2
V =
+
(4.3.14)
2
2
4.3. ADAPTIVE LAWS WITH NORMALIZATION
167
whose time derivative along the solution of (4.3.13) is given by
˙
V = −a
2
m
− 2 n 2 s + ˜ a x + ˜ a ˙˜ a
Choosing
˙˜ a = ˙ˆ a = − x
(4.3.15)
we have
˙
V = −a
2
m
− 2 n 2 s ≤ 0
which together with (4.3.14) imply V, , ˜ a ∈ L∞ and , ns ∈ L 2. If we now
write (4.3.15) as
˙
x
˜ a = − m m
where m 2 = 1 + n 2 s and choose ns so that x ∈ L
m
∞, then m ∈ L 2 (because
, ns ∈ L 2) implies that ˙˜ a ∈ L 2. A straightforward choice for ns is ns = x,
i.e., m 2 = 1 + x 2.
The effect of ns can be roughly seen by rewriting (4.3.13) as
˙ = −am − n 2 s + ˜ ax
(4.3.16)
and solving for the “quasi” steady-state response
˜ ax
s =
(4.3.17)
am + n 2 s
obtained by setting ˙ ≈ 0 in (4.3.16) and solving for . Obviously, for
n 2 s = x 2, large s implies large ˜ a independent of the boundedness of x, which, in turn, implies that large s carries information about the parameter error
˜ a even when x ∈ L∞. This indicates that ns may be used to normalize the
effect of the possible unbounded signal x and is, therefore, referred to as
the normalizing signal. Because of the similarity of s with the normalized
estimation error defined in (4.3.5), we refer to
in (4.3.13), (4.3.16) as the
normalized estimation error too.
Remark 4.3.1 The normalizing term n 2 s in (4.3.16) is similar to the non-
linear “damping” term used in the control of nonlinear systems [99].
It makes ˙
V more negative by introducing the negative term − 2 n 2 s in
the expression for ˙
V and helps establish that ns ∈ L 2. Because ˙ˆ a is
168
CHAPTER 4. ON-LINE PARAMETER ESTIMATION
4
3.5
3
2.5
a
˜
2
1.5
α=10
1
α=2
0.5
α=0.5
00
5
10
15
20
25
30
35
40sec
Figure 4.4 Effect of normalization on the convergence and performance of
the adaptive law (4.3.15).
bounded from above by
n 2 s + 1 = m and m ∈ L 2, we can con-
clude that ˙ˆ a ∈ L 2, which is a desired property of the adaptive law.
Note, however, that ˙ˆ a ∈ L 2 does not imply that ˙ˆ a ∈ L∞. In contrast
to the example in Section 4.3.1, we have not been able to establish
that ˙ˆ a ∈ L∞. As we will show in Chapter 6 and 7, the L 2 property
of the derivative of the estimated parameters is sufficient to establish
stability in the adaptive control case.
Simulations
Let us simulate the effect of normalization on the convergence and perfor-
mance of the adaptive law (4.3.15) when a = 0 is unknown, u = sin t, and
am = 2. We use n 2 s = αx 2 and consider different values of α ≥ 0. The
simulation results are shown in Figure 4.4. It is clear that large values of α
lead to a large normalizing signal that slows down the speed of convergence.
4.3. ADAPTIVE LAWS WITH NORMALIZATION
169
4.3.3
General Plant
Let us now consider the SISO plant
˙ x = Ax + Bu, x(0) = x 0
(4.3.18)
y = C x
where x ∈ Rn and only y, u are available for measurement. Equation (4.3.18)
may also be written as
y = C ( sI − A) − 1 Bu + C ( sI − A) − 1 x 0
or as
Z( s)
C adj( sI − A) x
y =
u +
0
(4.3.19)
R( s)
R( s)
where Z( s) , R( s) are in the form
Z( s) = bn− 1 sn− 1 + bn− 2 sn− 2 + · · · + b 1 s + b 0
R( s) = sn + an− 1 sn− 1 + · · · + a 1 s + a 0
The constants ai, bi for i = 0 , 1 , . . . , n − 1 are the plant parameters. A
convenient parameterization of the plant that allows us to extend the results
of the previous sections to this general case is the one where the unknown
parameters are separated from signals and expressed in the form of a linear
equation. Several such parameterizations have already been explored and
presented in Chapter 2. We summarize them here and refer to Chapter 2
for the details of their derivation.
Let
θ∗ = [ bn− 1 , bn− 2 , . . . , b 1 , b 0 , an− 1 , an− 2 , . . . , a 1 , a 0]
be the vector with the unknown plant parameters. The vector θ∗ is of di-
mension 2 n. If some of the coefficients of Z( s) are zero and known, i.e.,
Z( s) is of degree m < n − 1 where m is known, the dimension of θ∗ may be
reduced. Following the results of Chapter 2, the plant (4.3.19) may take any
one of the following parameterizations:
z = θ∗ φ + η 0
(4.3.20)
y = θ∗λ φ + η 0
(4.3.21)
170
CHAPTER 4. ON-LINE PARAMETER ESTIMATION
y = W ( s) θ∗λ ψ + η 0
(4.3.22)
where
u
u
z = W 1( s) y,
φ = H( s)
,
ψ = H
,
η
y
1( s)
y
0 = c 0 eΛ ctB 0 x 0
θ∗λ = θ∗ − bλ
W 1( s) , H( s) , H 1( s) are some known proper transfer function matrices with
stable poles, bλ = [0 , λ ] is a known vector, and Λ c is a stable matrix which
makes η 0 to be an exponentially decaying to zero term that is due to non-
zero initial conditions. The transfer function W ( s) is a known strictly proper
transfer function with relative degree 1, stable poles, and stable zeros.
Instead of dealing with each parametric model separately, we consider
the general model
z = W ( s) θ∗ ψ + η 0
(4.3.23)
where W ( s) is a proper transfer function with stable poles, z ∈ R 1 , ψ ∈ R 2 n
are signal vectors available for measurement and η 0 = c 0 eΛ ctB 0 x 0. Initially
we will assume that η 0 = 0, i.e.,
z = W ( s) θ∗ ψ
(4.3.24)
and use (4.3.24) to develop adaptive laws for estimating θ∗ on-line. The effect
of η 0 and, therefore, of the initial conditions will be treated in Section 4.3.7.
Because θ∗ is a constant vector, going from form (4.3.23) to form (4.3.20)
is trivial, i.e., rewrite (4.3.23) as z = θ∗ W ( s) ψ + η 0 and define φ = W ( s) ψ.
As illustrated in Chapter 2, the parametric model (4.3.24) may also be a
parameterization of plants other than the LTI one given by (4.3.18). What
is crucial about (4.3.24) is that the unknown vector θ∗ appears linearly in
an equation where all other signals and parameters are known exactly. For
this reason we will refer to (4.3.24) as the linear parametric model. In the
literature, (3.4.24) has also been referred to as the linear regression model.
In the following section we use different techniques to develop adaptive
laws for estimating θ∗ on-line by assuming that W ( s) is a known, proper
transfer function with stable poles, and z, ψ are available for measurement.
4.3. ADAPTIVE LAWS WITH NORMALIZATION
171
4.3.4
SPR-Lyapunov Design Approach
This approach dominated the literature of continuous adaptive schemes [48,
149, 150, 153, 172, 178, 187]. It involves the development of a differential
equation that relates the estimation or normalized estimation error with
the parameter error through an SPR transfer function. Once in this form
the KYP or the MKY Lemma is used to choose an appropriate Lyapunov
function V whose time derivative ˙
V is made nonpositive, i.e., ˙
V ≤ 0 by
properly choosing the differential equation of the adaptive law.
The development of such an error SPR equation had been a challeng-
ing problem in the early days of adaptive control [48, 150, 153, 178]. The
efforts in those days were concentrated on finding the appropriate transfor-
mation or generating the appropriate signals that allow the expression of the
estimation/parameter error equation in the desired form.
In this section we use the SPR-Lyapunov design approach to design adap-
tive laws for estimating θ∗ in the parametric model (4.3.24). The connection
of the parametric model (4.3.24) with the adaptive control problem is dis-
cussed in later chapters. By treating parameter estimation independently
of the control design, we manage to separate the complexity of the estima-
tion part from that of the control part. We believe this approach simplifies
the design and analysis of adaptive control schemes, to be discussed in later
chapters, and helps clarify some of the earlier approaches that appear tricky
and complicated to the nonspecialist.
Let us start with the linear parametric model
z = W ( s) θ∗ ψ
(4.3.25)
Because θ∗ is a constant vector, we can rewrite (4.3.25) in the form
z = W ( s) L( s) θ∗ φ
(4.3.26)
where
φ = L− 1( s) ψ
and L( s) is chosen so that L− 1( s) is a proper stable transfer function and
W ( s) L( s) is a proper SPR transfer function.
Remark 4.3.2 For some W ( s) it is possible that no L( s) exists such that
W ( s) L( s) is proper and SPR. In such cases, (4.3.25) could be prop-
erly manipulated and put in the form of (4.3.26). For example, when
172
CHAPTER 4. ON-LINE PARAMETER ESTIMATION
W ( s) = s− 1 , no L( s) can be found to make W ( s) L( s) SPR. In this s+2
case, we write (4.3.25) as ¯
z =
s+1
θ∗ φ where φ = s− 1 ψ and
( s+2)( s+3)
s+1
¯
z = 1 z. The new W ( s) in this case is W ( s) =
s+1
and a wide
s+3
( s+2)( s+3)
class of L( s) can be found so that W L is SPR.
The significance of the SPR property of W ( s) L( s) is explained as we
proceed with the design of the adaptive law.
Let θ( t) be the estimate of θ∗ at time t. Then the estimate ˆ
z of z at time
t is constructed as
ˆ
z = W ( s) L( s) θ φ
(4.3.27)
As with the examples in the previous section, the estimation error 1 is
generated as
1 = z − ˆ
z
and the normalized estimation error as
= z − ˆ
z − W ( s) L( s) n 2 s = 1 − W ( s) L( s) n 2 s (4.3.28)
where ns is the normalizing signal which we design to satisfy
φ ∈ L
m
∞,
m 2 = 1 + n 2 s
(A1)
Typical choices for ns that satisfy (A1) are n 2 s = φ φ, n 2 s = φ P φ for
any P = P
> 0, etc. When φ ∈ L∞, (A1) is satisfied with m = 1, i.e.,
ns = 0 in which case = 1.
We examine the properties of
by expressing (4.3.28) in terms of the
parameter error ˜
θ = θ − θ∗, i.e., substituting for z, ˆ
z in (4.3.28) we obtain
= W L( −˜
θ φ − n 2 s)
(4.3.29)
For simplicity, let us assume that L( s) is chosen so that W L is strictly proper
and consider the following state space representation of (4.3.29):
˙ e = Ace + Bc( −˜
θ φ − n 2 s)
(4.3.30)
= Cc e
where Ac, Bc, and Cc are the matrices associated with a state space repre-
sentation that has a transfer function W ( s) L( s) = Cc ( sI − Ac) − 1 Bc.
4.3. ADAPTIVE LAWS WITH NORMALIZATION
173
The error equation (4.3.30) relates
with the parameter error ˜
θ and is
used to construct an appropriate Lyapunov type function for designing the
adaptive law of θ. Before we proceed with such a design, let us examine
(4.3.30) more closely by introducing the following remark.