ˆ
x =
1
[( a
s+ a
m − ˆ
a) x + ˆ bu]
(SP)
m
by considering the parameterization of the plant given by (4.2.15). Equa-
tion (SP) is widely used for parameter estimation and is known as the series-
parallel model [123]. The estimation method based on (SP) is called the equa-
tion error method [123, 172]. Various other models that are a combination of
(P) and (SP) are generated [123] by considering different parameterizations
for the plant (4.2.13).
The estimation error 1 = x − ˆ x satisfies the differential equation
˙1 = −a 1 + ˜ aˆ x − ˜ bu
(P1)
for model (P) and
˙1 = −am 1 + ˜ ax − ˜ bu
(SP1)
for model (SP) where
˜ a = ˆ a − a,
˜ b = ˆ b − b
are the parameter errors. Equations (P1) and (SP1) indicate how the param-
eter error affects the estimation error 1. Because a, am > 0, zero parameter
error, i.e., ˜ a = ˜ b = 0, implies that 1 converges to zero exponentially. Be-
cause ˜ a, ˜ b are unknown, 1 is the only measured signal that we can monitor
in practice to check the success of estimation. We should emphasize, how-
ever, that 1 → 0 does not imply that ˜ a, ˜ b → 0 unless some PE properties
are satisfied by ˆ
x, x, u as we will demonstrate later on in this section. We
should also note that 1 cannot be generated from (P1) and (SP1) because
˜ a and ˜ b are unknown. Equations (P1) and (SP1) are, therefore, only used
for the purpose of analysis.
Let us now use the error equation (SP1) to derive the adaptive laws for
estimating a and b. We assume that the adaptive laws are of the form
˙
˙
ˆ a = f
ˆ
1( 1 , ˆ
x, x, u) ,
b = f 2( 1 , ˆ x, x, u)
(4.2.16)
where f 1 and f 2 are functions of measured signals, and are to be chosen so
that the equilibrium state
ˆ ae = a, ˆ be = b,
1 e = 0
(4.2.17)
4.2. SIMPLE EXAMPLES
153
of the third-order differential equation described by (SP1) (where x ∈ L∞
is treated as an independent function of time) and (4.2.16) is u.s., or, if
possible, u.a.s., or, even better, e.s.
We choose f 1 , f 2 so that a certain function V ( 1 , ˜ a, ˜ b) and its time derivative ˙
V along the solution of (SP1), (4.2.16) are such that V qualifies as a Lya-
punov function that satisfies some of the conditions given by Theorems 3.4.1
to 3.4.4 in Chapter 3. We start by considering the quadratic function
1
V ( 1 , ˜ a, ˜ b) = ( 2
2 1 + ˜ a 2 + ˜ b 2)
(4.2.18)
which is positive definite, decrescent, and radially unbounded in R 3. The
time derivative of V along any trajectory of (SP1), (4.2.16) is given by
˙
V = −a
2
m 1 + ˜
ax 1 − ˜ bu 1 + ˜ af 1 + ˜ bf 2
(4.2.19)
˙
and is evaluated by using the identities ˙ˆ a = ˙˜ a, ˆ b = ˙˜ b, which hold because a
and b are assumed to be constant.
If we choose f 1 = − 1 x, f 2 = 1 u, we have
˙
V = −a
2
m 1 ≤ 0
(4.2.20)
and (4.2.16) becomes
˙
˙
ˆ a = −
ˆ
1 x,
b = 1 u
(4.2.21)
where 1 = x − ˆ x and ˆ x is generated by (SP).
Applying Theorem 3.4.1 to (4.2.18) and (4.2.20), we conclude that V
is a Lyapunov function for the system (SP1), (4.2.16) where x and u are
treated as independent bounded functions of time and the equilibrium given
by (4.2.17) is u.s. Furthermore, the trajectory 1( t) , ˆ a( t) , ˆ b( t) is bounded for all t ≥ 0. Because 1 = x − ˆ x and x ∈ L∞ we also have that ˆ x ∈ L∞;
therefore, all signals in (SP1) and (4.2.21) are uniformly bounded. As in the
example given in Section 4.2.1, (4.2.18) and (4.2.20) imply that
lim V ( 1( t) , ˜ a( t) , ˜ b( t)) = V∞ < ∞
t→∞
and, therefore,
∞
∞
2
1
˙
1
1( τ ) dτ = −
V dτ =
( V 0 − V∞)
0
am 0
am
154
CHAPTER 4. ON-LINE PARAMETER ESTIMATION
where V 0 = V ( 1(0) , ˜ a(0) , ˜ b(0)), i.e., 1 ∈ L 2. Because u, ˜ a, ˜ b, x, 1 ∈ L∞, it follows from (SP1) that ˙1 ∈ L∞, which, together with 1 ∈ L 2, implies that
˙ˆ
1( t) → 0 as t → ∞, which, in turn, implies that ˙
ˆ a( t) , b( t) → 0 as t → ∞.
˙
It is worth noting that
ˆ
1( t) , ˙
ˆ a( t) , b( t) → 0 as t → ∞ do not imply that
˜ a and ˜ b converge to any constant let alone to zero. As in the example of
Section 4.2.1, we can use (4.2.18) and (4.2.20) and establish that
lim (˜ a 2( t) + ˜ b 2( t)) = 2 V∞
t→∞
which again does not imply that ˜ a and ˜ b have a limit, e.g., take
√
√
˜ a( t) =
2 V∞ sin 1 + t, ˜ b( t) =
2 V∞ cos 1 + t
The failure to establish parameter convergence may motivate the reader
to question the choice of the Lyapunov function given by (4.2.18) and of the
functions f 1 , f 2 in (4.2.19). The reader may argue that perhaps for some
other choices of V and f 1 , f 2, u.a.s could be established for the equilibrium
(4.2.17) that will automatically imply that ˜ a, ˜ b → 0 as t → ∞. Since given
a differential equation, there is no procedure for finding the appropriate
Lyapunov function to establish stability in general, this argument appears
to be quite valid. We can counteract this argument, however, by applying
simple intuition to the plant equation (4.2.13). In our analysis, we put no
restriction on the input signal u, apart from u ∈ L∞, and no assumption is
made about the initial state x 0. For u = 0, an allowable input in our analysis,
and x 0 = 0, no information can be extracted about the unknown parameters
a, b from the measurements of x( t) = 0 , u( t) = 0, ∀t ≥ 0. Therefore, no
matter how intelligent an adaptive law is, parameter error convergence to
zero cannot be achieved when u = 0 ∀t ≥ 0. This simplistic explanation
demonstrates that additional conditions have to be imposed on the input
signal u to establish parameter error convergence to zero. Therefore, no
matter what V and f 1 , f 2 we choose, we can not establish u.a.s. without
imposing conditions on the input u. These conditions are similar to those
imposed on the input u in Section 4.2.1, and will be discussed and analyzed
in Chapter 5.
In the adaptive law (4.2.21), the adaptive gains are set equal to 1. A
similar adaptive law with arbitrary adaptive gains γ 1 , γ 2 > 0 is derived by
4.2. SIMPLE EXAMPLES
155
considering
1
˜ a 2
˜ b 2
V (
2
1 , ˜
a, ˜ b) =
+
2
1 + γ 1
γ 2
instead of (4.2.18). Following the same procedure as before we obtain
˙
˙
ˆ a = −γ
ˆ
1 1 x,
b = γ 2 1 u
where γ 1 , γ 2 > 0 are chosen appropriately to slow down or speed up adapta-
tion.
Using (4.2.18) with model (P1) and following the same analysis as with
model (SP1), we obtain
˙
˙
ˆ a = −
ˆ
1 ˆ
x, b = 1 u
(4.2.22)
and
˙
V = −a 21 ≤ 0
Hence, the same conclusions as with (4.2.21) are drawn for (4.2.22).
We should note that ˙
V for (P1) depends on the unknown a, whereas for
(SP1) it depends on the known design scalar am. Another crucial difference
between model (P) and (SP) is their performance in the presence of noise,
which becomes clear after rewriting the adaptive law for ˆ a in (4.2.21), (4.2.22)
as
˙ˆ a = −( x − ˆ x)ˆ x = ˆ x 2 − xˆ x
(P)
˙ˆ a = −( x − ˆ x) x = −x 2 + xˆ x
(SP)
If the measured plant state x is corrupted by some noise signal v, i.e., x is
replaced by x + v in the adaptive law, it is clear that for the model (SP),
˙ˆ a will depend on v 2 and v, whereas for model (P) only on v. The effect of
noise ( v 2) may result in biased estimates in the case of model (SP), whereas
the quality of estimation will be less affected in the case of model (P). The
difference between the two models led some researchers to the development
of more complicated models that combine the good noise properties of the
parallel model (P) with the design flexibility of the series-parallel model (SP)
[47, 123].
156
CHAPTER 4. ON-LINE PARAMETER ESTIMATION
Simulations
We simulate the parallel and series-parallel estimators and examine the ef-
fects of the input signal u, the adaptive gain and noise disturbance on their
performance. For simplicity, we consider a first-order example y =
b u
s+ a
with two unknown parameters a and b. Two adaptive estimators
˙
˙
ˆ a = −
ˆ
1 ˆ
x, b = 1 u
˙ˆ x = −ˆ aˆ x + ˆ bu, 1 = x − ˆ x
and
˙
˙
ˆ a = −
ˆ
1 x,
b = 1 u
˙ˆ x = −amˆ x + ( am − ˆ a) x + ˆ bu, 1 = x − ˆ x
based on the parallel and series-parallel model, respectively, are simulated
with a = 2 and b = 1. The results are given in Figures 4.2 and Figure 4.3,
respectively. Plots (a) and (b) in Figure 4.2 and 4.3 give the time response
of the estimated parameters when the input u = sin 5 t, and the adaptive
gain γ = 1 for (a) and γ = 5 for (b). Plots (c) in both figures give the results
of estimation for a step input, where persistent excitation and, therefore,
parameter convergence are not guaranteed. Plots (d) show the performance
of the estimator when the measurement x( t) is corrupted by d( t) = 0 . 1 n( t), where n( t) is a normally distributed white noise.
It is clear from Figures 4.2 (a,b) and Figure 4.3 (a,b) that the use of a
larger value of the adaptive gain γ led to a faster convergence of ˆ a and ˆ b
to their true values. The lack of parameter convergence to the true values
in Figure 4.2 (c), 4.3 (c) is due to the use of a non-PE input signal. As
expected, the parameter estimates are more biased in the case of the series-
parallel estimator shown in Figure 4.3 (d) than those of the parallel one
shown in Figure 4.2 (d).
4.2.3
Vector Case
Let us extend the example of Section 4.2.2 to the higher-order case where
the plant is described by the vector differential equation
˙ x = Apx + Bpu
(4.2.23)
4.2. SIMPLE EXAMPLES
157
3
3
aˆ
aˆ
2
2
bˆ
bˆ
1
1
0
0
0
50
100
150
200
0
50
100
150
200
sec
sec
(a)
(b)
3
3
aˆ
aˆ
2
2
bˆ
bˆ
1
1
0
0
0
10
20
30
0
50
100
150
200
sec
sec
(c)
(d)
Figure 4.2 Simulation results of the parallel estimator. (a) u = sin 5 t, γ = 1, no
measurement noise; (b) u = sin 5 t, γ = 5, no measurement noise; (c) u =unit
step function, γ = 1, no measurement noise; (d) u = sin 5 t, γ = 1, output x is
corrupted by d( t) = 0 . 1 n( t), where n( t) is a normally distributed white noise.
where the state x ∈ Rn and input u ∈ Rr are available for measurement,
Ap ∈ Rn×n, Bp ∈ Rn×r are unknown, Ap is stable, and u ∈ L∞. As in the
scalar case, we form the parallel model
˙ˆ x = ˆ
Apˆ x + ˆ
Bpu, ˆ x ∈ Rn
(P)
where ˆ
Ap( t) , ˆ
Bp( t) are the estimates of Ap, Bp at time t to be generated by
158
CHAPTER 4. ON-LINE PARAMETER ESTIMATION
3
3
aˆ
aˆ
2
2
bˆ
bˆ
1
1
0
0
0
100
200
300
0
100
200
300
sec
sec
(a)
(b)
3
3
aˆ
2
2
aˆ
bˆ
1
1
bˆ
0
0
0
10
20
30
0
100
200
300
sec
sec
(c)
(d)
Figure 4.3 Simulation results of the series-parallel estimator. (a) u = sin 5 t,
γ = 1, no measurement noise; (b) u = sin 5 t, γ = 5, no measurement noise;
(c) u =unit step function, γ = 1, no measurement noise; (d) u = sin 5 t, γ = 1,
output x is corrupted by d( t) = 0 . 1 n( t), where n( t) is the normally distributed white noise.
an adaptive law, and ˆ
x( t) is the estimate of the vector x( t). Similarly, by
considering the plant parameterization
˙ x = Amx + ( Ap − Am) x + Bpu
where Am is an arbitrary stable matrix, we define the series-parallel model
as
˙ˆ x = Amˆ x + ( ˆ
Ap − Am) x + ˆ
Bpu
(SP)
4.2. SIMPLE EXAMPLES
159
The estimation error vector 1 defined as
1 = x − ˆ
x
satisfies
˙1 = Ap 1 − ˜
Apˆ x − ˜
Bpu
(P1)
for model (P) and
˙1 = Am 1 − ˜
Apx − ˜
Bpu
(SP1)
for model (SP), where ˜
Ap = ˆ
Ap − Ap, ˜
Bp = ˆ
Bp − Bp.
Let us consider the parallel model design and use (P1) to derive the
adaptive law for estimating the elements of Ap, Bp. We assume that the
adaptive law has the general structure
˙ˆ
Ap = F 1( 1 , x, ˆ x, u) , ˙ˆ
Bp = F 2( 1 , x, ˆ x, u)
(4.2.24)
where F 1 and F 2 are functions of known signals that are to be chosen so that
the equilibrium
ˆ
Ape = Ap, ˆ
Bpe = Bp,