where A =
|b|
√
, ϕ = (sgn( b) − 1)90 ◦ − tan − 1 1 , which can be shown to
1+ a 2
a
satisfy (5.2.6). Therefore, for u = sin t the signal vector φ carries sufficient
information about a and b, φ is P E and θ( t) → θ∗ exponentially fast.
We say that u = sin t is sufficiently rich for identifying the plant (5.2.3),
i.e., it contains a sufficient number of frequencies to excite all the modes of
the plant. Because u = c 0 = 0 can excite only the zero frequency gain of the
plant, it is not sufficiently rich for the plant (5.2.3).
5.2. PARAMETER IDENTIFIERS
255
Let us consider the second order plant
b
y =
1 s + b 0
u = G( s) u
(5.2.7)
s 2 + a 1 s + a 0
where a 1 , a 0 > 0 and G( s) has no zero-pole cancellations. We can show
that for u = sin ω 0 t, y( t) at steady state does not carry sufficient infor-
mation to be able to uniquely determine a 1 , a 0 , b 1 , b 0. On the other hand,
u( t) = sin ω 0 t + sin ω 1 t where ω 0 = ω 1 leads to the steady-state response y( t) = A 0 sin( ω 0 t + ϕ 0) + A 1 sin( ω 1 t + ϕ 1) where A 0 = | G( jω 0) |, ϕ 0 = G( jω 0) , A 1 = | G( jω 1) |, ϕ 1 = G( jω 1). By measuring A 0 , A 1 , ϕ 0 , ϕ 1 we can determine uniquely a 1 , a 0 , b 1 , b 0 by solving four algebraic equations.
Because each frequency in u contributes two equations, we can argue that
the number of frequencies that u should contain, in general, is proportional
to the number of unknown plant parameters to be estimated.
We are now in a position to give the following definition of sufficiently
rich signals.
Definition 5.2.1 A signal u : R+ → R is called sufficiently rich of order
n if it consists of at least n distinct frequencies.
2
For example, the input
m
u =
Ai sin ωit
(5.2.8)
i=1
where m ≥ n/ 2, Ai = 0 are constants and ωi = ωk for i = k is sufficiently
rich of order n.
A more general definition of sufficient richness that includes signals that
are not necessarily equal to a sum of sinusoids is presented in [201] and is
given below.
Definition 5.2.2 A signal u : R+ → Rn is said to be stationary if the
following limit exists uniformly in t 0
1
t 0+ T
Ru( t) = lim
u( τ ) u ( t + τ ) dτ
T →∞ T
t 0
256
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
The matrix Ru( t) ∈ Rn×n is called the autocovariance of u. Ru( t) is a positive semidefinite matrix and its Fourier transform given by
∞
Su( ω) =
e−jωτ Ru( τ) dτ
−∞
is referred to as the spectral measure of u. If u has a sinusoidal component at
frequency ω 0 then u is said to have a spectral line at frequency ω 0 and Su( ω) has a point mass (a delta function) at ω 0 and −ω 0. Given Su( ω), Ru( t) can be calculated using the inverse Fourier transform, i.e.,
1
∞
Ru( t) =
ejωtS
2 π
u( ω) dω
−∞
Furthermore, we have
∞ Su( ω) dω = 2 πRu(0)
−∞
For further details about the properties of Ru( t) , Su( ω), the reader is referred
to [186, 201].
Definition 5.2.3 A stationary signal u : R+ → R is called sufficiently
rich of order n, if the support of the spectral measure Su( ω) of u contains
at least n points.
Definition 5.2.3 covers a wider class of signals that includes those specified
by Definition 5.2.1. For example, the input (5.2.8) has a spectral measure
with 2 m points of support, i.e., at ωi, −ωi for i = 1 , 2 , . . . m, where m ≥ n/ 2, and is, therefore, sufficiently rich of order n.
Let us now consider the equation
φ = H( s) u
(5.2.9)
where H( s) is a proper transfer matrix with stable poles and φ ∈ Rn. The
P E property of φ is related to the sufficient richness of u by the following
theorem given in [201].
Theorem 5.2.1 Let u : R+ → R be stationary and assume that H( jω 1) ,
. . ., H( jωn) are linearly independent on Cn for all ω 1 , ω 2 , . . . , ωn ∈ R, where
ωi = ωk for i = k. Then φ is P E if, and only if, u is sufficiently rich of
order n.
5.2. PARAMETER IDENTIFIERS
257
The proof of Theorem 5.2.1 is given in Section 5.6.
The notion of persistence of excitation of the vector φ and richness of the
input u attracted the interest of several researchers in the 1960s and 1970s
who gave various interpretations to the properties of P E and sufficiently
rich signals. The reader is referred to [1, 171, 201, 209, 242] for further
information on the subject.
Roughly speaking, if u has at least one distinct frequency component for
each two unknown parameters, then it is sufficiently rich. For example, if
the number of unknown parameters is n, then m ≥ n distinct frequencies
2
in u are sufficient for u to qualify as being sufficiently rich of order n. Of
course, these statements are valid provided H( jω 1) , . . . , H( jωn) with ωi = ωk
are linearly independent on Cn for all ωi ∈ R, i = 1 , 2 , . . . , n. The vectors
H( jωi) , i = 1 , 2 , . . . , n may become linearly dependent at some frequencies
in R under certain conditions such as the one illustrated by the following
example where zeros of the plant are part of the internal model of u.
Example 5.2.1 Let us consider the following plant:
b
y = 0( s 2 + 1) u = G( s) u
( s + 2)3
where b 0 is the only unknown parameter. Following the procedure of Chapter 2, we
rewrite the plant in the form of the linear parametric model
y = θ∗φ
where θ∗ = b 0 is the unknown parameter and
s 2 + 1
φ = H( s) u,
H( s) = ( s + 2)3
According to Theorem 5.2.1, we first need to check the linear independence of
H( jω 1) , . . . , H( jωn). For n = 1 this condition becomes H( jω) = 0 , ∀ω ∈ R. It is clear that for ω = 1 , H( j) = 0, and, therefore, φ may not be PE if we simply choose
u to be sufficiently rich of order 1. That is, for u = sin t, the steady-state values of
φ, y are equal to zero and, therefore, carry no information about the unknown b 0.
We should note, however, that for u = sin ωt and any ω = 1 , 0, φ is PE.
Remark 5.2.1 The above example demonstrates that the condition for the
linear independence of the vectors H( jωi) , i = 1 , 2 , · · · , n on Cn is suf-
ficient to guarantee that φ is P E when u is sufficiently rich of order n.
258
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
It also demonstrates that when the plant is partially known, the input
u does not have to be sufficiently rich of order n where n is the order of
the plant. In this case, the condition on u can be relaxed, depending
on the number of the unknown parameters. For further details on the
problem of prior information and persistent of excitation, the reader
is referred to [35].
In the following sections we use Theorem 5.2.1 to design the input signal
u for a wide class of parameter estimators developed in Chapter 4.
5.2.2
Parameter Identifiers with Full-State Measurements
Let us consider the plant
˙ x = Ax + Bu, x(0) = x 0
(5.2.10)
where C = I , i.e., the state x ∈ Rn is available for measurement and A, B
are constant matrices with unknown elements that we like to identify. We
assume that A is stable and u ∈ L∞.
As shown in Chapter 4, the following two types of parameter estimators
may be used to estimate A, B from the measurements of x, u.
Series-Parallel
˙ˆ x = Amˆ x + ( ˆ
A − Am) x + ˆ
Bu
(5.2.11)
˙ˆ
A = γ 1 1 x ,
˙ˆ
B = γ 2 1 u
where Am is a stable matrix chosen by the designer, 1 = x − ˆ x, γ 1 , γ 2 > 0
are the scalar adaptive gains.
Parallel
˙ˆ x = ˆ
Aˆ
x + ˆ
Bu
(5.2.12)
˙ˆ
A = γ 1 1ˆ x ,
˙ˆ
B = γ 2 1 u
where 1 = x − ˆ x and γ 1 , γ 2 > 0 are the scalar adaptive gains.
As shown in Chapter 4, if A is a stable matrix and u ∈ L∞ then
ˆ
x, ˆ
A, ˆ
B ∈ L∞;
˙ˆ
A( t) , ˙ˆ
B( t) , 1 ∈ L 2 ∩ L∞ and 1( t) and the elements
of ˙ˆ
A( t) , ˙ˆ
B( t) converge to zero as t → ∞.
5.2. PARAMETER IDENTIFIERS
259
For the estimators (5.2.11) and (5.2.12) to become parameter identifiers,
the input signal u has to be chosen so that ˆ
A( t) , ˆ
B( t) converge to the un-
known plant parameters A, B, respectively, as t → ∞.
For simplicity let us first consider the case where u is a scalar input, i.e.,
B ∈ Rn× 1.
Theorem 5.2.2 Let ( A, B) be a controllable pair. If the input u ∈ R 1 is suf-
ficiently rich of order n + 1 , then the estimates ˆ
A, ˆ
B generated by (5.2.11) or
(5.2.12) converge exponentially fast to the unknown plant parameters A, B,
respectively.
The proof of Theorem 5.2.2 is quite long and is presented in Section 5.6.
An example of a sufficiently rich input u for the estimators (5.2.11),
(5.2.12) is the input
m
u =
Ai sin ωit
i=1
for some constants Ai = 0 and ωi = ωk for i = k and for some integer
m ≥ n+1 .
2
Example 5.2.2 Consider the second-order plant
˙ x = Ax + Bu
where x = [ x 1 , x 2] , and the matrices A, B are unknown, and A is a stable matrix.
− 1
0
Using (5.2.11) with Am =
, the series-parallel parameter identifier is
0
− 1
given by
ˆ
˙
− 1
0
ˆ a
b
ˆ
x =
(ˆ
x − x) +
11( t)
ˆ a 12( t)
x +
1( t)
u
0
− 1
ˆ a
ˆ
21( t)
ˆ a 22( t)
b 2( t)
where ˆ
x = [ˆ
x 1 , ˆ x 2] and
˙ˆ aik = ( xi − ˆ xi) xk,
i = 1 , 2; k = 1 , 2
˙ˆ bi = ( xi − ˆ xi) u, i = 1 , 2
The input u is selected as
u = 5 sin 2 . 5 t + 6 sin 6 . 1 t
260
CHAPTER 5. IDENTIFIERS AND ADAPTIVE OBSERVERS
which has more frequencies than needed since it is sufficiently rich of order 4. An
input with the least number of frequencies that is sufficiently rich for the plant
considered is
u = c 0 + sin ω 0 t
for some c 0 = 0 and ω 0 = 0.
If u is a vector, i.e., u ∈ Rq, q > 1 then the following theorem may be
used to select u.
Theorem 5.2.3 Let ( A, B) be a controllable pair. If each element ui, i =
1 , 2 , . . . q of u is sufficiently rich of order n + 1 and uncorrelated, i.e., each ui
contains different frequencies, then ˆ
A( t) , ˆ
B( t) converge to A, B, respectively,
exponentially fast.
The proof of Theorem 5.2.3 is similar to that for Theorem 5.2.2, and is
given in Section 5.6.
The controllability of the pair ( A, B) is critical for the results of The-
orems 5.2.2 and 5.2.3 to hold. If ( A, B) is not a controllable pair, then
the elements of ( A, B) that correspond to the uncontrollable part cannot be
learned from the output response because the uncontrollable parts decay to
zero exponentially fast and are not affected by the input u.
The complexity of the parameter identifier may be reduced if some of
the elements of the matrices ( A, B) are known. In this case, the order of
the adaptive law can be reduced to be equal to the number of the unknown
parameters. In addition, the input u may not have to be sufficiently rich of
order n + 1. The details of the design and analysis of such schemes are left
as exercises for the reader and are included in the problem section.
In the following section we extend the results of Theorem 5.2.2 to the case
where only the output of the plant, rather than the full state, is available
for measurement.
5.2.3
Parameter Identifiers with Partial-State Measurements
In this section we concentrate on the SISO plant
˙ x = Ax + Bu,
x(0) = x 0
(5.2.13)
y = C x
5.2. PARAMETER IDENTIFIERS
261
where A is a stable matrix, and y, u ∈ R 1 are the only signals available for
measurement.
Equation (5.2.13) may be also written as
y = C ( sI − A) − 1 Bu + C ( sI − A) − 1 x 0
(5.2.14)
where, because of the stability of A, t = L− 1 {C ( sI − A) − 1 }x 0 is an exponentially decaying to zero term. We would like to design an on-line parameter
identifier to estimate the parameters A, B, C. The triple ( A, B, C) contains
n 2 + 2 n unknown parameters to be estimated using only input/output data.
The I/O properties of the plant (5.2.14) at steady state (where t = 0), how-
ever, are uniquely determined by at most 2 n parameters. These parameters
correspond to the coefficients of the transfer function
y( s)
b
= C ( sI − A) − 1 B = msm + bm− 1 sm− 1 + . . . + b 0
(5.2.15)
u( s)
sn + an− 1 sn− 1 + . . . + a 0
where m ≤ n − 1.
Because there is an infinite number of triples ( A, B, C) that give the same
transfer function (5.2.15), the triple ( A, B, C) associated with the specific
or physical state space representation in (5.3.13) cannot be determined, in
general, from input-output