.
.
. In− 1
A∗ =
.
.
(5.3.9)
−a∗ .
. . .
... 0
is a stable matrix and ˆ ap( t) and ˆ bp( t) are the estimates of the vectors ap and
bp, respectively, at time t.
A wide class of adaptive laws may be used to generate ˆ ap( t) and ˆ bp( t)
on-line. As an example, we can start with (5.3.7) to obtain as in Section
2.4.1 the parametric model
z = θ∗ φ
(5.3.10)
5.3. ADAPTIVE OBSERVERS
271
where
α
α
φ =
n− 1( s) u, − n− 1( s) y
= φ
Λ( s)
Λ( s)
1 , φ 2
sn
z =
y = y + λ φ
Λ( s)
2
Λ( s) = sn + λ αn− 1( s)
and
θ∗ = [ bn− 1 , bn− 2 , . . . , an− 1 , an− 2 , . . . , a 0]
is the parameter vector to be estimated and Λ( s) is a Hurwitz polynomial
of degree n chosen by the designer. A state-space representation for φ and z
may be obtained as in (5.2.18) by using the identity ( sI − Λ c) − 1 l = αn− 1( s)
Λ( s)
where (Λ c, l) is in the controller canonical form and det( sI − Λ c) = Λ( s).
In view of (5.3.10), we can choose any adaptive law from Tables 4.2,
4.3 and 4.5 of Chapter 4 to estimate θ∗ and, therefore, ap, bp on-line. We
can form a wide class of adaptive observers by combining (5.3.8) with any
adaptive law from Tables 4.2, 4.3 and 4.5 of Chapter 4 that is based on the
parametric plant model (5.3.10).
We illustrate the design of such adaptive observer by using the gradient
algorithm of Table 4.2 (A) in Chapter 4 as the adaptive law. The main
equations of the observer are summarized in Table 5.1.
The stability properties of the class of adaptive observers formed by
combining the observer equation (5.3.8) with an adaptive law from Tables
4.2 and 4.3 of Chapter 4 are given by the following theorem.
Theorem 5.3.1 An adaptive observer for the plant (5.3.6) formed by com-
bining the observer equation (5.3.8) and any adaptive law based on the plant
parametric model (5.3.10) obtained from Tables 4.2 and 4.3 of Chapter 4
guarantees that
(i)
All signals are u.b.
(ii) The output observation error ˜
y = y − ˆ
y converges to zero as t → ∞.
(iii) If u is sufficiently rich of order 2 n, then the state observation error
˜
x = xα − ˆ x and parameter error ˜
θ = θ − θ∗ converge to zero. The rate
of convergence is exponential for all adaptive laws except for the pure
least-squares where the convergence is asymptotic.
272
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
Table 5.1 Adaptive observer with gradient algorithm
.
.
. In− 1
˙ x
.
.
x
Plant
α = −ap .
. . . α + bpu, xα ∈ Rn
... 0
y = [1 , 0 . . . 0] xα
.
.
. In− 1
˙ˆ x=
.
.
ˆ
x+ˆ b
Observer
−ˆ
ap( t) . . . .
p( t) u+( a∗ −ˆ
ap( t))( y− ˆ y)
... 0
ˆ
y = [1 0 . . . 0]ˆ
x
˙ θ = Γ φ
θ = ˆ bp ( t) , ˆ ap ( t)
,
= z−ˆ z,
m 2
Adaptive law
ˆ
z = θ φ, Γ = Γ > 0
α
( s)
α
( s)
φ =
n− 1
u, − n− 1
y
Λ( s)
Λ( s)
z = sn y
Λ( s)
Design variables
a∗ is chosen so that A∗ in (5.3.9) is stable; m 2 =
1 or m 2 = 1 + φ φ; Λ( s) is a monic Hurwitz poly-
nomial of degree n
Proof (i) The adaptive laws of Tables 4.2 and 4.3 of Chapter 4 guarantee that
, m, ˙ θ ∈ L 2 ∩L∞ and θ ∈ L∞ independent of the boundedness of u, y. Because u ∈
L∞ and the plant is stable, we have xα, y, φ, m ∈ L∞. Because of the boundedness
of y, φ we can also establish that , m, ˙ θ → 0 as t → ∞ by showing that ˙ ∈ L∞
(which follows from ˙ θ, ˙ φ ∈ L∞), which, together with ∈ L 2, implies that → 0 as
t → ∞ (see Lemma 3.2.5). Because m, φ ∈ L∞, the convergence of m, ˙ θ to zero
follows.
The proof of (i) is complete if we establish that ˆ
x ∈ L∞. We rewrite the observer
5.3. ADAPTIVE OBSERVERS
273
equation (5.3.8) in the form
˙ˆ x = A∗ˆ x + ˆ bp( t) u + ( ˆ
A( t) − A∗) xα
(5.3.11)
Because θ = ˆ bp ( t) , ˆ ap ( t)
, u, xα ∈ L∞ and A∗ is a stable matrix, it follows that
ˆ
x ∈ L∞. Hence, the proof of (i) is complete.
(ii) Let ˜
x = xα − ˆ x be the state observation error. It follows from (5.3.11),
(5.3.6) that
˙˜ x = A∗˜ x − ˜ bpu + ˜ apy, ˜ x(0) = xα(0) − ˆ x(0)
(5.3.12)
where ˜ bp = ˆ bp − bp, ˜ ap = ˆ ap − ap. From (5.3.12), we obtain
˜
y = C ˜
x( s) = C ( sI − A∗) − 1( −˜ bpu + ˜ apy) + t
where t = L− 1 C ( sI − A∗) − 1 ˜ x(0) is an exponentially decaying to zero term.
Because ( C, A) is in the observer canonical form, we have
α
[ sn− 1 , sn− 2 , . . . s, 1]
C ( sI − A∗) − 1 =
n− 1( s)
=
det( sI − A∗)
det( sI − A∗)
Letting Λ ∗( s) = det( sI − A∗), we have
1
n
˜
y( s) =
sn−i −˜ b
Λ ∗( s)
n−iu + ˜
an−iy + t
i=1
where ˜ bi, ˜ ai is the ith element of ˜ bp and ˜ ap, respectively, which may be written as
Λ( s) n sn−i
˜
y( s) =
−˜ b
Λ ∗( s)
Λ( s)
n−iu + ˜
an−iy + t
(5.3.13)
i=1
where Λ( s) is the Hurwitz polynomial of degree n defined in (5.3.10). We now apply
Lemma A.1 (see Appendix A) for each term under the summation in (5.3.13) to
obtain
Λ( s) n
sn−i
sn−i
˜
y =
−˜ b
u + ˜ a
y
Λ ∗( s)
n−i Λ( s)
n−i Λ( s)
i=1
−Wci( s) ( Wbi( s) u) ˙˜ bn−i + Wci( s) ( Wbi( s) y) ˙˜ an−i + t (5.3.14)
where the elements of Wci( s) , Wbi( s) are strictly proper transfer functions with the
same poles as Λ( s). Using the definition of φ and parameter error ˜
θ = θ − θ∗, we
rewrite (5.3.14) as
Λ( s)
n
˜
y =
−˜
θ φ +
W
Λ ∗( s)
ci( s)
− ( Wbi( s) u) ˙˜ bn−i + ( Wbi( s) y) ˙˜ an−i
+ t
i=1
274
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
✲
Luenberger
u
✲
y
ˆ
x
Plant
✲
✲
Observer
(Equation 5.3.8)
✲ Parameter ✛
✻
Estimation
(ˆ ap, ˆ bp)
Figure 5.1 General structure of the adaptive Luenberger observer.
Because ˙˜ bn−i, ˙˜ an−i ∈ L 2 ∩ L∞ converge to zero as t → ∞, u, y ∈ L∞ and the
elements of Wci( s) , Wbi( s) are strictly proper stable transfer functions, it follows
from Corollary 3.3.1 that all the terms under the summation are in L 2 ∩ L∞ and
converge to zero as t → ∞. Furthermore, from m 2 = z − ˆ
z = −˜
θ φ, m ∈ L∞, ∈
L 2 ∩ L∞ and ( t) → 0 as t → 0, we have that ˜
θ φ ∈ L 2 ∩ L∞ converges to zero as
t → ∞. Hence, ˜
y is the sum of an output of a proper stable transfer function whose
input is in L 2 and converges to zero as t → ∞ and the exponentially decaying to
zero term t. Therefore, ˜ y( t) → 0 as t → ∞.
(iii) If φ is P E, then we can establish, using the results of Chapter 4, that
˜ ap( t) , ˜ bp( t) converge to zero. Hence, the input −˜ bpu + ˜ apy converges to zero, which, together with the stability of A∗, implies that ˜
x( t) → 0. With the exception of the
pure least-squares, all the other adaptive laws guarantee that the convergence of
˜ bp, ˜ ap to zero is exponential, which implies that ˜ x also goes to zero exponentially
fast .
The PE property of φ is established by using exactly the same steps as in the
proof of Theorem 5.2.4.
✷
The general structure of the adaptive observer is shown in Figure 5.1.
The only a priori knowledge assumed about the plant (5.3.1) is that it is
completely observable and completely controllable and its order n is known.
The knowledge of n is used to choose the order of the observer, whereas
the observability of ( C, A) is used to guarantee the existence of the state
space representation of the plant in the observer form that in turn enables
us to design a stable adaptive observer. The controllability of ( A, B) is not
needed for stability, but it is used together with the observability of ( C, A)
to establish that φ is P E from the properties of the input u.
5.3. ADAPTIVE OBSERVERS
275
Theorem 5.3.1 shows that for the state xα of the plant to be estimated
exactly, the input has to be sufficiently rich of order 2 n, which implies that
the adaptive law has to be a parameter identifier. Even with the knowledge of
the parameters ap, bp and of the state xα, however, it is not in general possible
to calculate the original state of the plant x because of the usual nonunique
mapping from the coefficients of the transfer function to the parameters of
the state space representation.
Example 5.3.2 Let us consider the second order plant
−a
b
˙ x =
1
1
x +
1
u
−a 0 0
b 0
y = [1 , 0] x
where a 1 , a 0 , b 1 , b 0 are the unknown parameters and u, y are the only signals available for measurement.
Using Table 5.1, the adaptive observer for estimating x, and the unknown pa-
rameters are described as follows: The observer equation is given by
ˆ
˙
−ˆ a
b
9 − ˆ a
ˆ
x =
1( t)
1
ˆ
x +
1( t)
u +
1( t)
( y − ˆ
y)
−ˆ a
ˆ
0( t)
0
b 0( t)
20 − ˆ a 0( t)
ˆ
y = [1 , 0] ˆ
x
−a∗
where the constants a∗
1
1
1 = 9 , a∗
0 = 20 are selected so that A∗ =
has
−a∗ 0 0
eigenvalues at λ 1 = − 5 , λ 2 = − 4.
The adaptive law is designed by first selecting
Λ( s) = ( s + 2)( s + 3) = s 2 + 5 s + 6
and generating the information vector φ = [ φ 1 , φ 2 ]
˙
− 5 − 6
1
φ 1 =
φ
u
1
0
1 +
0
˙
− 5 − 6
− 1
φ 2 =
φ
y
1
0
2 +
0
and the signals
z = y + [5 , 6] φ 2
ˆ
z =
ˆ b 1 , ˆ b 0 φ 1 + [ˆ a 1 , ˆ a 0] φ 2
= z − ˆ
z
276
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
The adaptive law is then given by
˙ˆ b
˙
1
ˆ a 1
˙
= γ
= γ
ˆ
1 φ 1 ,
˙
2 φ 2
b
ˆ a
2
2
The adaptive gains γ 1 , γ 2 > 0 are usually chosen by trial and error using simulations
in order to achieve a good rate of convergence. Small γ 1 , γ 2 may result in slow
convergent rate whereas large γ 1 , γ 2 may make the differential equations “stiff” and
difficult to solve numerically on a digital computer.
In order for the parameters to converge to their true values, the plant input u
is chosen to be sufficiently rich of order 4. One possible choice for u is
u = A 1 sin ω 1 t + A 2 sin ω 2 t
for some constants A 1 , A 2 = 0 and ω 1 = ω 2.
5.3.3
Hybrid Adaptive Luenberger Observer
The adaptive law for the adaptive observer presented in Table 5.1 can be
replaced with a hybrid one without changing the stability properties of the
observer in any significant way. The hybrid adaptive law updates the param-
eter estimates only at specific instants of time tk, where tk is an unbounded
monotonic sequence, and tk+1 − tk = Ts where Ts may be considered as the