because ˙˜ ai, ˙˜ bi and u, y are available for measurement.
✷
We can now use (5.4.3) instead of (5.4.2) to develop an adaptive law for
generating θ = ˆ bp , ˆ ap
. Using the results of Chapter 4, it follows that the
adaptive law is given by
˙ θ = Γ φ,
= ˜
y = y − ˆ
y
(5.4.7)
where Γ = Γ > 0.
We summarize the main equations of the adaptive observer developed
above in Table 5.3. The structure of the adaptive observer is shown in
Figure 5.2.
Theorem 5.4.1 The adaptive observer presented in Table 5.3 guarantees
that for any bounded input signal u,
(i)
all signals are u.b.
(ii)
˜
y( t) = y − ˆ
y → 0 as t → ∞.
˙
(iii) ˙ˆ a
ˆ
p( t) , bp( t) ∈ L 2 ∩ L∞ and converge to zero as t → ∞.
In addition, if u is sufficiently rich of order 2 n, then
(iv) ˜
x( t) = xα( t) − ˆ x( t) , ˜ ap( t) = ˆ ap( t) − ap, ˜ bp( t) = ˆ bp( t) − bp converge to zero exponentially fast.
5.4. ADAPTIVE OBSERVER WITH AUXILIARY INPUT
283
Table 5.3 Adaptive observer with auxiliary input
In− 1
Plant
˙ x
α = −ap
· · · xα + bpu,
y = [1 0 . . . 0] xα
0
ap = [ an− 1 , . . . , a 0] , bp = [ bn− 1 , . . . , b 0]
In− 1
˙ˆ x =
−ˆ
a
· · · ˆ
x+ˆ b
Observer
p( t)
p( t) u+( a∗−ˆ
ap( t))( y−ˆ y)+ v
0
ˆ
y = [1 0 . . . 0]ˆ
x
˙ˆ bp = Γ φ( y − ˆ y) , Γ = Γ > 0
˙
Adaptive law
ˆ ap
α
( s)
α
( s)
φ =
n− 1
u, − n− 1
y
Λ( s)
Λ( s)
0
˙
v =
, v =
n
ˆ b
v
i=1 − [ Wi( s) y] ˙
ˆ an−i+[ Wi( s) u] n−i
Wi( s) = ( sI − Λ0) − 1 di,
d 1 = −λ;
di = [0 ... 0 , 1 , 0 ... 0] , i = 2 , . . . , n
Auxiliary input
↑
( i − 1)
In− 2
Λ
0 = −λ
· · · , λ = [ λn− 2 , . . . , λ 0]
0
det( sI − Λ0) = Λ( s) = sn− 1 + λ αn− 2( s)
In− 1
(i) a∗ is chosen such that A∗=
−a∗
· · · is stable
0
Design variables
(ii) The vector λ is chosen such that Λ0 is stable
and [1 0 . . . 0] ( sI − A∗) − 1
1
is SPR.
λ
284
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
u
✲
y
Plant
ν
Auxiliary
✛
+ ❄
♥ ✲˜
y
✛
Σ
Signal Generator
− ✻
✲
Observer
✲
ˆ
x ✲
✁
C
✲
ˆ
y
(Equation (5.4.1))
✻
✲
˜
y
Parameter
✲
✲
Estimator
ˆ ap, ˆ bp
Figure 5.2 Structure of the adaptive observer with auxiliary input signal.
Proof The main equations that describe the stability properties of the adaptive
observer are the error equation (5.4.3) that relates the parameter error with the
state observation error and the adaptive law (5.4.7). Equations (5.4.3) and (5.4.7)
are analyzed in Chapter 4 where it has been shown that e, ˜
y ∈ L 2 ∩ L∞ and ˜
θ ∈ L∞
for any signal vector φ with piecewise continuous elements. Because u, y ∈ L∞, it
follows that φ ∈ L∞ and from (5.4.3) and (5.4.7) that ˙ θ, ˙ e, ˙˜ y ∈ L∞ and ˙ θ ∈ L 2.
Using e, ˜
y ∈ L 2
L∞ together with ˙ e, ˙˜ y ∈ L∞, we have ˜ y( t) → 0, e( t) → 0 as t → ∞, which implies that ˙ θ( t) → 0 as t → ∞. From ˙ θ ∈ L 2 ∩ L∞, it follows that v ∈ L 2 ∩ L∞ and v( t) → 0 as t → ∞. Hence, all inputs to the state equation (5.4.2) are in L∞, which implies that ˜ x ∈ L∞, i.e., ˆ x ∈ L∞.
In Chapter 4 we established that if φ, ˙ φ ∈ L∞ and φ is P E, then the error
equations (5.4.3) and (5.4.7) guarantee that ˜
θ( t) → 0 as t → ∞ exponentially fast.
In our case φ, ˙ φ ∈ L∞ and therefore if u is chosen so that φ is P E then ˜
θ( t) → 0 as
t → ∞ exponentially fast, which implies that e( t) , ˜
y( t) , ˙ θ( t) , v( t) , ˜ ap( t) y, ˜ bp( t) u and, therefore, ˜
x( t) converge to zero exponentially fast. To explore the PE property of
φ, we note that φ is related to u through the equation
1
α
φ =
n− 1( s)
u
Λ( s)
−αn− 1( s) C ( sI − A) − 1 bp
Because u is sufficiently rich of order 2 n, the P E property of φ can be established
by following exactly the same steps as in the proof of Theorem 5.2.4.
✷
The auxiliary signal vector ¯
v can be generated directly from the signals
5.4. ADAPTIVE OBSERVER WITH AUXILIARY INPUT
285
φ and ˙˜
θ, i.e., the filters Wi for y, u do not have to be implemented. This
simplification reduces the number of integrators required to generate φ, ¯
v
considerably and it follows from the relationship
α
( sI − Λ
n− 1( s)
0) − 1 di = Qi
, i = 1 , 2 , . . . , n
Λ( s)
where Qi ∈ R( n− 1) ×( n− 1) are constant matrices whose elements depend on
the coefficients of the numerator polynomials of ( sI − Λ0) − 1 di (see Problem
2.12).
As with the adaptive Luenberger observer, the adaptive observer with
auxiliary input shown in Table 5.3 requires the input u to be sufficiently
rich of order 2 n in order to guarantee exact plant state observation. The
only difference between the two observers is that the adaptive Luenberger
observer may employ any one of the adaptive laws given in Tables 4.2, 4.3,
and 4.5 whereas the one with the auxiliary input given in Table 5.3 relies
on the SPR-Lyapunov design approach only. It can be shown, however, (see
Problem 5.16) by modifying the proof of Lemma 5.4.1 that the observation
error ˜
y may be expressed in the form
˜
y = −˜
θ φ
(5.4.8)
by properly selecting the auxiliary input v and φ. Equation (5.4.8) is in the
form of the error equation that appears in the case of the linear parametric
model y = θ∗ φ and allows the use of any one of the adaptive laws of
Table 4.2, 4.3, and 4.5 leading to a wide class of adaptive observers with
auxiliary input.
The following example illustrates the design of an adaptive observer with
auxiliary input v and the generation of v from the signals φ, ˙ θ.
Example 5.4.1 Let us consider the second order plant
−a
b
˙ x
1
1
1
α
=
x
u
−a
α +
0
0
b 0
y = [1 0] xα
where a 1 > 0 , a 0 > 0 , b 1 , b 0 are unknown constants and y, u are the only available signals for measurement.
286
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
We use Table 5.3 to develop the adaptive observer for estimating xα and the un-
known plant parameters. We start with the design variables. We choose a∗ = [4 , 4] ,
λ = λ 0 , Λ( s) = s + λ 0. Setting λ 0 = 3 we have
s + 3
[1 0][ sI − A∗] − 1
1
=
3
( s + 2)2
which is SPR.
The signal vector φ =
s u, 1 u, − s y, − 1 y
is realized as follows
s+3
s+3
s+3
s+3
φ = [ φ 1 , φ 2 , φ 3 , φ 4]
where
φ 1 = u − 3 ¯
φ 1 ,
˙¯
φ 1 = − 3 ¯
φ 1 + u
φ 2 = ¯
φ 1
φ 3 = −y + 3 ¯
φ 3 ,
˙¯
φ 3 = − 3 ¯
φ 3 + y
φ 4 = − ¯
φ 3
with ¯
φ 1(0) = 0 , ¯
φ 3(0) = 0.
For simplicity we choose the adaptive gain Γ = diag { 10 , 10 , 10 , 10 }. The adap-
tive law is given by
˙ˆ
˙
b
ˆ
1 = 10 φ 1( y − ˆ
y) ,
b 0 = 10 φ 2( y − ˆ y)
˙ˆ a 1 = 10 φ 3( y − ˆ y) , ˙ˆ a 0 = 10 φ 4( y − ˆ y)
0
and the signal vector v =
by
¯
v
3
3
˙
1
1
˙
¯
v =
y
˙ˆ a
u ˆ b
y
˙ˆ a
u ˆ b
s + 3
1 −
s + 3
1 −
s + 3
0 +
s + 3
0
˙
˙
= 3 ˙ˆ a ¯
ˆ ¯
¯
ˆ ¯
1 φ 3 − 3 b 1 φ 1 − ˙
ˆ a 0 φ 3 + b 0 φ 1
= 10( y − ˆ
y)( − 3 φ 3 φ 4 − 3 φ 1 φ 2 + φ 24 + φ 22)
The observer equation becomes
ˆ
˙
−ˆ a
b
4 − ˆ a
ˆ
x =
1
1
ˆ
x +
1
u +
1
( y − ˆ
y)
−ˆ a
ˆ
0
0
b 0
4 − ˆ a 0
0
+
( y − ˆ
y) − 3 φ
10
3 φ 4 − 3 φ 1 φ 2 + φ 2
4 + φ 2
2
The input signal is chosen as u = A 1 sin ω 1 t + A 2 sin ω 2 t for some A 1 , A 2 = 0 and ω 1 = ω 2.
5.5. ADAPTIVE OBSERVERS FOR NONMINIMAL PLANT MODELS 287
5.5
Adaptive Observers for Nonminimal Plant
Models
The adaptive observers presented in Sections 5.3 and 5.4 are suitable for
estimating the states of a minimal state space realization of the plant that
is expressed in the observer form. Simpler (in terms of the number of inte-
grators required for implementation) adaptive observers may be constructed
if the objective is to estimate the states of certain nonminimal state-space
representations of the plant. Several such adaptive observers have been pre-
sented in the literature over the years [103, 108, 120, 123, 130, 172], in this
section we present only those that are based on the two nonminimal plant
representations developed in Chapter 2 and shown in Figures 2.2 and 2.3.
5.5.1
Adaptive Observer Based on Realization 1
Following the plant parameterization shown in Figure 2.2, the plant (5.3.1)
is represented in the state space form
˙ φ 1 = Λ cφ 1 + lu, φ 1(0) = 0
˙ φ 2 = Λ cφ 2 − ly, φ 2(0) = 0
˙ ω = Λ cω,
ω(0) = ω 0 = B 0 x 0
(5.5.1)
η 0 = C 0 ω
z = y + λ φ 2 = θ∗ φ + η 0
y = θ∗ φ − λ φ 2 + η 0
where ω ∈ Rn, φ = [ φ 1 , φ 2 ] , φi ∈ Rn, i = 1 , 2; Λ c ∈ Rn×n is a known stable matrix in the controller form; l = [1 , 0 , . . . , 0] ∈ Rn is a known vector such
that ( sI − Λ c) − 1 l = αn− 1( s) and Λ( s) = det( sI − Λ
Λ( s)
c) = sn + λ αn− 1( s),
λ = [ λn− 1 , . . . , λ 0] ; θ∗ = [ bn− 1 , bn− 2 , . . . , b 0 , an− 1 , an− 2 , . . . , a 0] ∈ R 2 n are the unknown parameters to be estimated; and B 0 ∈ Rn×n is a constant
matrix defined in Section 2.4.
The plant parameterization (5.5.1) is of order 3 n and has 2 n unknown
parameters. The state ω and signal η 0 decay to zero exponentially fast with
a rate that depends on Λ c. Because Λ c is arbitrary, it can be chosen so that
η 0 , ω go to zero faster than a certain given rate. Because φ 1(0) = φ 2(0) = 0,
288
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
the states φ 1 , φ 2 can be reproduced by the observer
˙ˆ φ
ˆ
1
= Λ cφ 1 + lu,
ˆ
φ 1(0) = 0
˙ˆ