Adaptive Lecture04 2005
Transcript of Adaptive Lecture04 2005
523 M1380: Adaptive Control Systems
Lecture 4: On-Line Parameter Estimation Spring 2005
In many applications, plant (model) structure may be known, but its parameters
may be unknown and time-varying due to change in operation conditions, aging of
equipment, etc. Thus, the off-line parameter estimation is inefficient.
On-line estimation schemes refer to those estimation schemes that provide fre-
quent estimates of plant parameters by properly processing the plant I/O data on-line.
The essential idea behind is the comparison of the observed system response y(t) with
the output of a parameterized model ),(ˆ ty θ , whose structure is the same as that of
plant model. Then, θ(t) is adjusted continuous so that ),(ˆ ty θ approaches y(t) as t in-
creases. (Under certain input conditions, being close to y implies that θ(t) is close
to the unknown θ
y*.)
The on-line estimations procedure, therefore, involves 3 steps:
Step 1: Select an appropriate plant parameterization.
Step 2: Select an adaptive law for generating or updating θ(t).
Step 3: Design the plant input so that θ(t) approaches θ* as t → ∞.
Remark 4.0.1: In adaptive control, where the convergence of θ(t) to θ* is usually not
one of the objectives, the first two steps are the most important ones.
4.1 SCALAR EXAMPLE: ONE UNKNOWN PARAMETER
In this section, a scalar example is used to illustrate the importance of the nor-
malized in identification.
4.1.1 Estimation without Normalization
Consider the plant
),()( tuty ∗θ= (4.1.1)
where θ* is unknown, and y(t) and u(t) are measurable. If u and y are measured in a
noise-free manner, then
)()()(
tutyt =θ , 0)( ≠tu .
1
However, disadvantages lie in:
(i) numerical problem when u(t) ≈ 0, and
(ii) noise effect of measurements of y(t) and u(t) will cause wrong estimations.
Remedy is possible to use a recursive (on-line), division-free scheme.
Alternatively, let be the estimation value of y(t) in the form of
. Define
)(ˆ ty
)()()(ˆ tutty θ=
, ~ ˆ1 uuuyy θ−=θ−θ=−=ε ∗
where , ~ ∗θ−θ=θ and define
.) (21
21)ˆ( 22
1 uyJ θ−=ε=θ
Then, adjustment of θ is trying to minimize J(θ), which naturally leads to the “gradi-
ent method” as:
uuuyJ 1] [)( γε=θ−γ=θ∇γ−=θ& , 0)0( θ=θ ,
where γ > 0 is a scaling constant. For stability analysis of the estimator, construct the
following Lyapunov function candidate:
2~21)~( θγ
=θV ,
subject to ,~1uγε=θ=θ && then which implies that (or ).
Since
,0)~( 21 ≤ε−=θV& ∞∈θ L~
∞∈θ L
∞<ττε=−=ττ− ∫∫∞
∞
∞
0
210
0 )()( dVVdV& ,
we have . Now, assuming that u ∈ L∞∩∈ε LL21 ∞, we have Ad-
ditionally, if we assume that
.21 ∞∩∈γε=θ LLu&
∞∈Lu& , then it follows that ∞∈θ−θ−=ε Luu &&& ~ ~1 . By
Barbalat’s Lemma, ε1(t) → 0 as t → ∞ and hence 0)(~→θ t& as t → ∞.
On the other hand, note that
).0(~)(~ )(
0 2
θ∫=θττγ−
tdu
et
Therefore, converges to zero if and only if )(~ tθ
00
20 )( TduTt
tα≥ττ∫
+, 0≥∀t ,
for some α0, T0 > 0, which is referred to as persistence excitation (PE).
2
4.2.1 Estimation with Normalization
Consider the plant (4.1.1) again and assume u and y are piecewise continuous but not
necessarily bounded, and θ∗ is to be estimated. Let uy ˆ θ≡ , uyyy ˆ1 θ−=−≡ε ,
where θ(t) is estimation θ∗ at t.
⇒ Minimization problem: 221 ) (min)(min uyJ θ−=θ θθ is ill posed because
. ∞∉Lyu,
⇒ An alternative is uy ∗θ≡ , myy = , m
uu = where and n22 1 snm += s is
chosen so that ∞∈ Lmu . A straightforward choice of ns is ns ≡ u such that
and thus, 22 1 um += ∞∈Lyu , . Thus, , ˆ uy θ≡ and uyyy ˆ
1 θ−=−≡ε .
⇒ Minimization problem: 2
212
21 ) (min) (min)(min 2 uyuyJ
mθ−=θ−=θ θθθ
is well-posed.
Using the gradient method, we obtain
u1εγ=θ& , ,0>γ
or
um
u 21 γε=ε
γ=θ& ,
with 21
mε≡ε , where ε is called the normalized estimation error. Note that
mu
mu
m ~ ~
221 θ
−=θ
−=ε
=ε ,
where ∗θ−θ≡θ~ , which implies that θγ−=θ=θ
~ ~ 2u&& . Construct
γθ
=θ2
~)~(
2
V .
Then,
0~ 2222 ≤ε−=θ−= muV&
such that , εm ∈ L∞∈θθ L~ , 2. Because ∞∈θ Lu ~ , , if follows that mu ~ θ−=ε and εm ∈
L∞ . Hence we have 2 LLumr ∩∈ε=θ ∞& . Since
uumdtd && ~ ~ )( θ−θ−=ε ,
if we assume that ∞∈ Lu& , then ∞∈ε Lmdtd )( , which together with εm ∈ L2 implies
3
that εm(t) → 0 as t → ∞ by Barbalat’s Lemma. Then, as t → ∞. 0)(~→θ t&
Remark 4.2.1: Despite y, u may be unbounded, adaptive law can still be designed
such that θ ∈ L∞ and . 2LL ∩∈θ ∞&
4.2 NORMALIZED ADPTIIVE LAWS BASED ON SPR-LYAPUNOV DESIGN
APPRAOCH
Consider an SISO plant as follows:
.,
T xCyBuAxx
=
+=&
Two kinds of parameterizations: (i) , or (ii) , where
, , ,
0T η+φθ= ∗λy 0
T)( η+ψθ= ∗λsWy
T01210121 ] , , , , , , , , ,[ aaaabbbb nnnn KK −−−−
∗ =θ λ∗∗
λ −θ=θ b TT ] ,0[ λ=λb
⎥⎦
⎤⎢⎣
⎡=φ
yu
sH )( , , ⎥⎦
⎤⎢⎣
⎡=ψ
yu
sH )(1
and W(s) is one strictly proper transfer function with stable poles, stable zero (i.e.,
minimum phase) and relative-degree one. A general model is the form of
, which is linear parameter model or linear regression model. ψθ= ∗T)(sWz
Choose L(s) so that L−1(s) is a proper stable transfer function and W(s)L(s) is a
proper SPR transfer function. Then
,)()( Tφθ= ∗sLsWz , ψ=φ − )(1 sL
with its estimate as so that the estimation error is and
normalized estimation error is
φθ= Tˆ)()(ˆ sLsWz zz ˆ1 −≡ε
21 )()( snsLsW ε−ε=ε ,
where ns is the normalizing signal such that
∞∈φ Lm
, . 22 1 snm +=
(Note that the example of ns is , or , where P > 0.) Thus, φφ= T2 sn φφ= PnsT2
) ~)(()( 2TsnsLsW ε−φθ−=ε ,
where ∗θ−θ≡θ~ . Now, let (Ac, Bc, Cc) be the state space representation of W(s)L(s),
then
4
,
), ~(T
2T
eC
nBeAe
c
scc
=ε
ε−φθ−+=&
and
.)()()( 1Tccc BAsICsLsW −−=
Since W(s)L(s) is SPR, there is a matrix Pc > 0 such that
,,TT
ccc
ccccc
CBPLqqPAAP
=ν−−=+
for some vector q, matrix Lc > 0, and a small constant ν > 0, by either KYL Lemma (if
(Ac, Bc, Cc) is minimal) or MKY Lemma (if (Ac, Bc, Cc) is nonminimal). To design the
adaptive law, construct a Lyapunov like function
2
~~
2),~(
1TT θΓθ+=θ
−ePeeV c ,
where such that 0>Γ
),~(~22
1
~~) ~(22
1
1T22TTT
1T2TTTT
εφΓ−θΓθ+ε−ν
−−=
θΓθ+ε−φθ−+ν
−−=
−
−
&
&&
sc
sT
ccc
neLeeqqe
nBPeeLeeqqeV
and hence, the adaptive law is selected as follows:
εφΓ=θ& . (4.2.1)
i.e., .~εφΓ=θ& Obviously, ∞∈θθε Le ~ , , , , and 2 , Lns ∈εε , so that and 2Lm∈ε
( ) 2Lm
m ∈⎟⎠⎞
⎜⎝⎛ φεΓ=θ& ,
which is independent of the boundedness of φ . We summarize the property of this
design by the following theorem:
Theorem 4.2.1: The SPR-Lyapunov adaptive law guarantees that
(i) , ∞∈θε L ,
(ii) , 2 , , Lns ∈θεε &
independent of the boundedness properties of φ.
Remark 4.2.1: For stable plants, ∞∈φ L . Then, ∞∈ε L& and hence ε → 0, ε1 → 0 as t
→ ∞ by Barbalat’s Lemma provided , or . φφ= T2 sn φφ= PnsT2
5
One important property of the adaptive law is the convergence of θ to the un-
known vector θ∗. Such a property is achieved for a special class vector signals φ de-
scribed by the following definition.
Definition 4.2.1 (Persistence of Excitation (PE)): A piecewise continuous signal
vector is PE in RnRR: →φ + n with a level of excitation α0 > 0, if there are con-
stants α1, T0 > 0 such that
IdT
ITt
t 0
T
01
0 )()(1α≥ττφτφ≥α ∫
+, 0≥∀t .
Remark 4.2.2: φ is PE if and only if 0
2T11
0
0)]([ α≥ττφ≥α ∫
+Tt
tT dq , , where q
is any constant vector in R
0≥∀t
n with 1=q .
Before we guarantee the convergence of θ to θ∗ by the PE condition, useful
lemmas are introduced as follows.
Lemma 4.2.1 (Uniformly Complete Observability (UCO) with Output Injection):
Assume that there exits constants ν > 0, kv ≥ 0, such that for all t0 ≥ 0, K(t) ∈ Rn × l sat-
isfies
v
vt
tkdK ≥ττ∫
+0
0
2)(
Then (C, A + KCT), where C ∈ Rn × l, A ∈ Rn × n, is a UCO pair if and only if (C, A +
KCT) is a UCO pair.
Lemma 4.2.2: If w: [0, ∞) → Rn is PE, ,∞∈Lw& and H(s) is a stable, minimum phase,
proper rational transfer function, then w1 = H(s)w is PE.
Lemma 4.2.3: Consdier
1T
0
2
2T
11
0YCy
YYBYAY
c
cc
==
φ−=&
&
where Ac is a stable matrix, (Cc, Ac) is observable, and φ ∈ L∞. If φf defined as
φ−≡φ −cccf BAsIC 1T )(
satisfies
6
0 ,)()(11
T
02
0 ≥∀α≥ττφτφ≥α ∫+
tIdT
ITt
t ff
for some constant α1, α2, T0 > 0, then the system above is UCO.
Now the convergence of θ to θ∗ by the PE condition is proved in the following.
Corollary 4.2.1: If , and φ is PE, then the former adaptive law guar-
antees θ(t) → θ
∞∈φφ Lns& , ,
∗ exponentially fast.
Proof: Consider
⎪⎭
⎪⎬
⎫
=εφεΓ=θ
ε−θ−+=
eC
nBeAe
c
scc
T
2T
~)~(
&&
(4.2.2)
that describe the stability properties of the adaptive law. In proving the Theorem 4.1.1,
we have also shown that the time derivative of
,)( εφΓ=θ∇Γ−=θ J&
where and satisfies 0T >Γ=Γ 0T >= cc PP
2εν′−≤V&
for some constant ν′ > 0. Defining
⎥⎦
⎤⎢⎣
⎡Γ
==⎥⎦
⎤⎢⎣
⎡
φΓφ−−
= −1TT
T
T2T
00
21 ,]0[ ,
0)( c
cc
csccc PPCC
CBnCBA
tA
we rewire (4.2.2) as
xCxtAx T ,)( =ε=&
and express the above Lyapunov-like function V and its derivative as
2TTTT
T
)( εν′−=ν′−≤++=
=
xCCxxPPAPAxVPxxV
&&
where . This implies that 0=P&
.0)()()()( TT ≤ν′++ tCtCPtAtPA
Using Theorem 3.3.4 (See Lecture 3), we can establish the equilibrium xe = 0 (i.e., ee
= 0 and eθ~= 0) is u.a.s., equivalent e.s. provided (C, A) is a UCO pair.
Since the (C, A) and (C, A + KCT) have the same UCO property, where
,2
⎥⎦
⎤⎢⎣
⎡
φΓ−≡ scnB
K
7
is bounded (see Lemma 4.1.1). We can therefore establish that (4.2.1) is UCO by
showing that is a UCO pair. The system corresponding to (C, A + KCT) is as
1T
0
2
2T
11
0YCy
YYBYAY
c
cc
==
φ−=&
&
Because φ is PE and is stable and minimum-phase and , it
follows that
ccc BAsIC 1T )( −− ∞∈φ L&
∫τ σ−τ σσφ≡τφ
)(T )()(t c
Acf dBeC c
is also a SPR (refer to Lemma 4.1.2); therefore, there exists constants α1, α2, T0 > 0
such that
0 ,)()(11
T
02
0 ≥∀α≥ττφτφ≥α ∫+
tIdT
ITt
t ff
We can conclude that (C, A + KCT) is UCO (see Lemma 4.1.3) which implies (C, A) is
UCO. Hence, we conclude that the equilibrium xe = 0 (i.e., ee = 0 and eθ~= 0) is e.s. in
the large.
If W(s) is minimum phase, one may choose L(s) = W −1(s) leading to W(s)L(s) = 1.
Then,
2
T
21
~
mmφθ
−=ε
=ε .
Consider
2
~~)~(
1T θΓθ=θ
−
V
so that provided the adaptive law is chosen 22mV ε−=&
εφΓ=θ& .
4.3 NORMALIZED GRADIETN ALGOIRHTM
4.3.1 The Gradient Algorithm Based on Instantaneous Cost
Consider the following cost functions:
2
2T22
2)(
2)(
mzmJ φθ−
=ε
=θ (Quadratic cost function),
where
8
2
T
mz φθ−
=ε
is the normalized estimation error based on the estimate θ of θ∗. We have
,)( εφΓ=θ∇Γ−=θ J& (Instantaneous adaptive Law)
i.e., θ(t) is chosen at each time t to minimize the square of the error. The performance
of the instantaneous adaptive law is summarized as follows.
Theorem 4.3.1: The instantaneous adaptive law guaranteed that
(i) ∞∈θθεε Lns& , , , ,
(ii) , 2 , , Lns ∈θεε &
independent of the boundedness of the signal vector φ and
(iii) if ns, φ ∈ L∞ and φ is PE, then θ → θ∗ exponentially fast.
Proof: From the adaptive law, we have
.~εφΓ=θ& (4.3.1)
We choose the Lyapunov-like function
.2
~~ 1T θΓθ=
−
V (4.3.2)
Then along the trajectory of the adaptive law, we have
.0~ 22T ≤ε−=φεθ= mV& (4.3.3)
Hence, , which implies that ∞∈θ LV ~ , ∞∈εε Lm , . In addition, we establish from the
properties of V, that , which implies that V& 2Lm∈ε 2 , Lns ∈εε . Now, from the
adaptive law, we have
mm
φεΓ≤θ=θ &&~
which together with ∞φ ∈ Lm and ∞∈ε LLm I2 implies that and the
proof of (i) and (ii) is complete.
∞∈θ LL I&2
The proof for (iii) is given now. The parameter error equation may be written as
⎪⎭
⎪⎬⎫
θ=θ=θ
~)(
~)(~
T0 tCy
tA& (4.3.4)
where
9
mym
tCm
tA ε=φ
−=φφ
Γ−= 0
TT
2
T
,)( ,)(
This system is analyzed using the Lyapunov-like function (4.3.2) that led to (4.3.3)
along the trajectory of this adaptive law. We need to establish that the equilibrium
0~=θe of (4.3.4) is e.s. We achieve that by using Theorem 3.3.4 (See Lecture 3) as
follows. Let P =Γ−1, then
2
~~T θθ=
PV
and
θθ−=θ++θ=~~~)(~
21 TTTT CCPPAPAV &&
where . This implies that 0=P&
0)()(2)()( TT ≤++ tCtCPtAtPA
According to Theorem 3.3.4, 0~=θe is e.s. provided (C, A) is UCO. Using Lemma
4.2.1, we have that (C, A) is UCO if (C, A + KCT) is UCO for some that satisfies the
condition of Lemma 4.2.1. We choose
mK φ
Γ−=
leading to A + KCT = 0. We consider the following system that corresponds to the pair
(C, A + KCT), i.e.,
⎭⎬⎫
−===
φ YYtCyY
m
T
)(0
T0
& (4.3.5)
The observability grammian of (4.3.5) is given by
∫+
τττφτφ
=+Tt
td
mTttN
2 )()()() ,(
Because φ is PE and m ≥ 1 is bounded, it follows that immediately that the grammian
matrix N(t, t + T) is positive definite for some T > 0 and for all t ≥ 0, which implies
that (4.3.5) is UCO which in turn implies that (C, A) is UCO; thus, the proof is com-
plete.
4.2.1 The Gradient Algorithm Based on Integral Cost
Consider another cost functions:
10
∫ τττε=θ τ−β−t t dmteJ
0
22)( )(),(21)( (Integral cost function),
where
ε=ετ
τφθ−τ=τε ),( ,
)()()()(),( 2
T
ttm
tzt
is the normalized estimation error at time τ based on the estimate θ(t) of at time
. We have
∗θ
τ≥t
∫ ττφτ
τφθ−τΓ=θ∇Γ−=θ τ−β−t t d
mtzeJ
0 2
T)( )(
)()()()()(& , (Integral adaptive law)
i.e. θ(t) is chosen at each time t to minimize the integral square of the errors on all
past data that are discounted exponentially. That is,
n
nn
QQmzQQ
RRm
RR
tQtR
R ,0)0( ,
R ,0)0( ,
)]()([
2
2
T
∈=φ
−β−=
∈=φφ
+β−=
+θΓ−=θ
×
&
&
&
The performance of the integral adaptive law is summarized as follows.
Theorem 4.3.2: The integral adaptive law guarantees that
(i) ∞∈θθεε Lns& , , , ,
(ii) , 2 , , Lns ∈θεε &
(iii) 0)(lim =θ∞→
tt
& , and
(iv) if ns, φ ∈ L∞, and φ is PE, then θ converges exponentially to θ*.
Proof: Because ∞φ ∈ Lm , it follows that R, Q ∈ L∞ and, hence, θ behaves as a LTV
system with bounded input. Substituting for in the differential equation for
Q, we verify that
φθ= ∗Tz
∗∗τ−β− θ−=θττφτφ
−= ∫ )( )()(
0 2
T)( tRd
meQ
t t ,
and hence,
.~)(~θΓ−=θ=θ tR&& (4.3.6)
Consider the Lyapunov-like function
2
~~)~(
1T θΓθ=θ
−
V (4.3.7)
11
such that
0~)(~T ≤θθ−= tRV& . (4.3.8)
Since R(t) = RT(t) ≥ 0, ∀t ≥ 0, it follows that V, , ∞∈θ L~
=θθ 21
)~~( TR 2~
21
LR ∈θ .
From 2
T~
mφθ−=ε and ∞
φ ∈θ Lm ,~ , we conclude that ε and εm, therefore, εns ∈ L∞.
From (4.3.6), we have
∞∩∈θΓ≤θ LLRR 2~ 2
121&
which together with R ∈ L∞ and ∞∩∈θ LLR 2~
21
imply that . Since ∞∩∈θ LL2&
∞∈θ LR&& ,~ , it follows from (4.3.6) that , which together with ∞∈θ L&&2
~ L∈θ& , implies
that
0)(~)(lim)(lim =θΓ=θ∞→∞→
ttRttt
& .
To show that εm ∈ L2, we proceed as follows. We have
θθβ−θΓθ−ε=θθ~~~~2~)(~ TT22T RRRmtR
dtd .
so that
∫∫∫ τθθβ+τθΓθ+θθ=τεttt
dRdRRRdm
0
T
0
TT
0
22 ~~~~2~~ .
Because as , and 0)(~)()(~T →θθ ttRt 0→t 2~
21
LR ∈θ , it follows that
. ∞<τε=τε ∫∫∞
∞→
0
22
0
22lim dmdmt
t
i.e. . 2Lm∈ε
The proof for (iv) is given now. In proving (i) to (iii), we have shown (4.3.8)
from (4.3.7). From the differential equation on R, we have
∫ τττφτφ
= τ−β−t t dm
etR
0 2
T)(
)()()()(
Because φ is PE and is bounded, we have
12
Ie
de
dm
edm
etR
T
t
Tt
T
Tt tt
Tt
t
0
0
0
0
0
1
T0
0 2
T)(
2
T)(
)()(
)()()(
)()()()(
β−
−
β−
− τ−β−
−
τ−β−
β≥
ττφτφα′≥
τττφτφ
+τττφτφ
=
∫
∫∫
for any t ≥ T0, where , 0001 Tα′α=β)(
10 2sup
tmt=α′ and α0, T0 > 0 are constants given
by the definition of PE. Hence,
VeetRV TT 00 )(2~~~)(~min1
T1
T β−β− Γλβ−≤θθβ−≤θθ−=&
for any t ≥ T0, which implies that
00)( ),()( 0 TtTVetV Tt ≥≤ −α−
where . Using 0)(2 min1Te β−Γλβ−=α
VV )(2)(2 maxmin Γλ≤θ≤Γλ
we have that
)(0
min
max)(0max
0202 )(~)()()()(2)(~ TtTt eTetVt −−−− αα
θΓλΓλ
≤Γλ≤θ
Thus, θ(t) converges exponentially to θ* with a rate of 2α .
Remark 4.3.1: 0 as t → ∞ without any additional condition on φ
and m.
)( →θ∇Γ−=θ J&
Remark 4.3.2: θ(t) converges to a trajectory that minimizes the integral asymptoti-
cally with time. Furthermore, if ∞∈φ Lns , , and φ is PE, then
)(2
0min
max 0)(~)()()(~ tt
ett−
α−
θΓλΓλ
≤θ , 0Tt ≥∀ ,
where , )(2 min
10 Γλβ=α β− Te 0001 α′α=β T ,
)(1
0 2suptmt=α′ and α0, T0 are the constants
defining PE of φ. Hence, larger α0 and larger )(min Γλ will guarantee faster conver-
gence of )(tθ to zero.
4.4 NORMALIZED LEAST SQUARES
Consider a simple plant:
,nduy +θ= ∗
where dn is a noise disturbance. Consider the following two approaches:
13
Bad approach:
)()(
)()()(
ττ
+θ=ττ
=θ ∗
ud
uyt n
for some τ < t for which u(τ) ≠ 0.
Better approach:
,))()()((21)(
0
2∫ ττθ−τ=θt
dutyJ
⇒ ,0)()()()()(
0
2
0 =ττθ+τττ−=θ∇ ∫∫
ttdutduyJ
⇒ (Least-squares estimate). ∫∫ τττ⎟⎠⎞⎜
⎝⎛ ττ=θ
− ttduydut
0
1
0
2 )()()()(
Example 4.4.1: u(t) ≡ 1, ∀t ≥ 0 and dn has a zero average value,
∗
→∞
∗
→∞→∞θ=ττ+θ=τττ=θ ∫∫
t
nt
t
ttdd
tduy
tt
0
0 )(1lim)()(1lim)(lim .
For general linear model, , the estimate of z and the normalized es-
timation error are generated as: , and
φθ= ∗Tz z
φθ=z T
2
T
2
ˆm
zm
zz φθ−=
−=ε , ,1 22
snm +=
)()(21
)())()()((
21)( 00
T0
0 2
2T)( θ−θθ−θ+τ
ττφθ−τ
=θ β−τ−β−∫ Qedm
tzeJ tt t ,
where Q0 > 0, β ≥ 0, θ0 = θ(0). Because ∞φ ∈ Lmm
z , , J(θ) is a convex function of θ
over Rn at each time t. Hence, any local minimum is also global and satisfies
0))(( =θ∇ tJ , ,0≥∀t
i.e.
0)()(
)()()(21))(()(
0 2
T)(
00 =ττφ
ττφθ−τ
−θ−θ=θ∇ ∫ τ−β−β− t tt dm
tzetQeJ
which yields the so-called non-recursive least-squares algorithm:
⎥⎦
⎤⎢⎣
⎡τ
ττφτ
+θ=θ ∫ τ−β−β− t tt dm
zeQetPt
0 2)(
00
)()()()()( ,
where 1
0 2
T)(
0
)()()()(
−
τ−β−β−⎥⎦
⎤⎢⎣
⎡τ
ττφτφ
+= ∫t tt d
meQetP .
14
Because , and φ(τ)φ0T00 >= QQ T(τ) is positive semi-definite, P(t) exists at each time
t. Then
( ) 0111 =⎟⎠⎞
⎜⎝⎛+= −−− P
dtdPPPPP
dtd &
We can show that
Pm
PPP 2
T
φφ−β=& , , 1
00)0( −== QPP
⇒ (Continuous-time recursive least-squares algorithm). φε=θ P&
Supplementary: Why for Least-squares? φε=θ P&
Since
⎥⎦
⎤⎢⎣
⎡τ
ττφτ
+θ=θ ∫ τ−β−β− t tt dm
zeQetPt
0 2)(
00
)()()()()( ,
it follows that
∗τ−β−β−− θτττφτφ
+θ=θ ∫t tt d
meQettP
0 2
T)(
00 1
)()()()()( ,
Hence,
∗∗τ−β−β−−− θφφ
+θτττφτφ
β−θβ−=θ+θ ∫ )()()(
)()()()()()()( 2
T
0 2
T)(
00 11
tmttd
meQettPttP
t tt&& .
Recall that
)()()()()( 2
T11
tmtttPtP φφ
+β−= −−& .
Therefore,
),()()()()(
)()()ˆ(
)()()())()(()(
)()()()(
)()()())()(()()()()(
21
2
2
T1
2
T1
2
T111
ttttmt
tmtzz
tmttttPt
mtP
tmttttPttPttP
φε=φε
=φ−
=
θφφ
+θβ−θ⎟⎟⎠
⎞⎜⎜⎝
⎛ττφτφ
+β−−=
θφφ
+θβ−θ−=θ
∗−−
∗−−− &&
and hence, the least square is given by
).()()()( tttPt φε=θ&
4.4.1 Pure Least-Squares Algorithm
Set β = 0, we have
15
2mPPP
Tφφ−=& , , φε=θ P&
where P is usually called the covariance matrix. In terms of the P−1, we have
,2
T1
mP
dtd φφ
=−
which implies that P−1 may grow without bound. In the matrix case, this means P may
become arbitrarily small and slow down adaptation in some directions. This is the
so-called covariance wind-up problem that constitutes one of the main drawbacks of
the pure least-squares algorithm.
Theorem 4.4.1: The pure least-squares algorithm guarantees that
(i) , ∞∈θθεε LPns , , , , &
(ii) , 2 , , Lns ∈θεε &
(iii) θ=θ∞→ )(lim tt , where θ is a constant vector, and
(iv) if ns, φ ∈ L∞, and φ is PE, then θ(t) converges to θ* as t → ∞.
Proof: We have that , i.e. P(t) ≤ P0≤P& 0. Because P(t) is nonincreasing and bounded
from below (i.e., P(t) = PT(t) ≥ 0, ∀t ≥ 0), it has a limit, i.e.,
PtPt =∞→ )(lim .
where 0T ≥= PP is a constant matrix. Let us now consider
,0~~~)~( 2
T1111 =εφ+
θφφ=θ+θ−=θ −−−−
mPPPPP
dtd &&
Hence, ),0(~)(~)( 10
1 θ=θ −− PttP and therefore, ),0(~)()(~ 10 θ=θ −PtPt and
),0(~)(~lim 10 θ=θ −
∞→PPt
t
which implies that
.)0(~)(lim 10 θ≡θ+θ=θ −∗
∞→PPt
t
Because P(t) ≤ P0 and )0(~)()(~ 10 θ=θ −PtPt , we have ∞∈θθ L~ , , which, together with
∞φ ∈ Lm , implies that ∞
φθ ∈−=ε Lm m
T~ and hence ∞∈εε Lns , . Let us now consider the
function. Consider
2
~~ 1θθ=
−PVT
,
The time derivative of along the trajectory of this adaptive law is given by
16
0222
~~~ 222222
2
TTT ≤
ε−=
ε+ε−=
θφφθ+φθε=
mmmm
V&
which implies that V ∈ L∞, εm ∈ L2; therefore, ε, εns ∈ L2, we have. From the adap-
tive law, we have
mm
P εφ
≤θ&
Because P, mφ , εm ∈ L∞ and εm ∈ L2, we have , which completes the
proof (i), (ii), and (iii).
∞∩∈θ LL2&
The proof of (iv) is given now. In proving (i) to (iii), we have shown that satisfies
the following equation
),0(~)()(~ 10 θ=θ −PtPt
We now show that P(t) → 0 as t → ∞ when φ satisfies the PE condition. Because P −1
satisfies
2
T1
mP
dtd φφ
=−
using the condition that φ is PE, i.e.,
ITdTt
t 00
T )()( α≥ττφτφ∫+
for some constant α0, T0 > 0, it follows that
ImT
TtI
mTnd
mPtP
t00
0
000
0 2
T11 1
)()()()0()( α
⎟⎟⎠
⎞⎜⎜⎝
⎛−≥
α≥τ
ττφτφ
=− ∫−−
Therefore
000
0
00
0
11 ,11)0()( TtImT
TtI
mT
TtPtP ≥∀
α⎟⎟⎠
⎞⎜⎜⎝
⎛−≥
α⎟⎟⎠
⎞⎜⎜⎝
⎛−+≥ −−
which implies that
0
1
000
1
00
0
,11)( TtImTTtI
mT
TttP ≥∀⎥
⎦
⎤⎢⎣
⎡α⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎥
⎦
⎤⎢⎣
⎡ α⎟⎟⎠
⎞⎜⎜⎝
⎛−≤
−−
Since P(t) ≥ 0 for all t ≥ 0 and the right-hand side of the above inequality goes to zero
asymptotically, we can conclude that P(t) → 0 as t → ∞ as. Hence the proof of (iv) is
complete.
Remark 4.4.1: Convergence rate of θ(t) to is not guaranteed to be exponential
even when φ is PE. In fact,
∗θ
17
ITtmtP
00 )()(
α−≤ , )0(~
)()(~
00
10 θ
α−≤θ
−
TtmPt , 0Tt ≥∀ ,
where )(sup 2 tmm t= . i.e. )(~ tθ is guaranteed to converge to zero with a speed of
1/t.
4.4.2 Pure Least-Squares with Covariance Resetting
To avoid covariance wind-up problem, we modify least-squares by incorporating co-
variance resetting mechanism:
,)( ,
,
002
T
IPtPm
PPP
P
r ρ==φφ
−=
εφ=θ
+&
&
where tr is the time for which 1min ))(( ρ≤λ tP , and ρ0 > ρ1 > 0 are some design scalars
such that ItP 1)( ρ≥ , . 0≥∀t
Theorem 4.4.2: The pure least-squares algorithm with covariance resetting has the
following properties:
(i) , ∞∈θθεε Lns& , , ,
(ii) , 2 , , Lns ∈θεε &
(iii) if ns, φ ∈ L∞, and φ is PE, then θ(t) converges to θ* exponentially fast.
Proof: The covariance matrix P(t) has elements that are discontinuous functions of
time due to the resetting. At the discontinuity or resetting point tr,
therefore, Between discontinuities,
,)( 00 IPtP r ρ==+
.)( 10
1 ItP r−+− ρ= 0)(1 ≥− tPdt
d , i.e., P−1(t2) −
P−1(t1) ≥ 0, ∀ t2 ≥ t1 ≥ 0 such that tr ∉ [t1, t2], which implies that
On the other hand, because of resetting, .0 ,)( 10
1 ≥∀ρ≥ −− tItP .0 ,)( 1 ≥∀ρ≥ tItP .
Hence, we have .0 ,)( ,)( 10
11110 ≥∀ρ≥≥ρρ≥≥ρ −−− tItPIItPI
Let us consider the function
2
~~ 1θθ=
−PVT
Since is a bounded positive definite symmetric matrix, it follows that V is decrescent
and radially unbounded in the space of . Along the trajectory of this adaptive law,
we have
θ~
18
θθ+ε−=θθ+θθ=−
−− ~)(~
21~~~)(~
21 1
T221T1
T
dtPdmP
dtPdV &&
Between the resetting points, it follows
02
)(21 22
2
2T22 ≤
ε−=
φθ+ε−=
mm
mV&
∀ t ∈ [t1, t2], where [t1, t2] is any interval in [0, ∞) for which tr ∉ [t1, t2]. On the other
hand, at the discontinuity of P, we have
.~)]()([~21)()( 11T θ−θ=− −+−+
rrrr tPtPtVtV
Because it follows that ,)( ,)( 10
110
1 ItPItP rr−−−+− ρ≥ρ=
0)()( ≤−+rr tVtV
which implies that V is a nonincreasing function of time for all t ≥ 0. Hence, V ∈ L∞
and limt → ∞ V(t) = V∞. Since the points of discontinuities form a set of measure zero, it
follows that 2 , Lm ∈εε . From V ∈ L∞, and we have ,10
111 IPI −−− ρ≥≥ρ ,~
∞∈θ L
which implies that . Using ∞∈εε Lm , 2LLm I∞∈ε and ,)( 10 ItPI ρ≥≥ρ we have
and the proof of (i) and (ii) is, therefore, complete. 2LL I&∞∈θ
The proof of (iii) is similar to the proof of Theorem 4.4.1 (iii) and is omitted.
4.4.3. Modified Least-Squares with Forgetting Factor
When β > 0, the problem of P(t) becoming arbitrarily small in some directions no
longer exist. But P(t) may grow without bound since for . Thus,
modification is the following:
0>P& 0>βP
⎪⎪⎩
⎪⎪⎨
⎧≤
φφ−β
=
εφ=θ
otherwise ,0
)( if , 02
T
PtPm
PPPP
P
&
&
where . 0)0( 0 >= PP
4.5 NORMALIZED ADAPTIVE LAWS WITH PROJECTION
Consider the linear parametric model:
φθ= ∗T)(sWz , . nR∈θ∗
Sometimes it could be advisable to design adaptive laws that are constrained to search
19
for estimates of θ* in the set where θ* is located. The advantages lie in:
(i) to speed up convergence,
(ii) to reduce large transients when θ(t) is far away from θ*, and
(iii) to constrain θ(t) such that it always satisfies certain properties (Con-
strained parameter estimation).
t∀
4.5.1 Gradient Algorithm with Projection
Let us start with the gradient method as follows:
min )(θJ
subject to S∈θ
where S is a convex with smooth boundary almost everywhere. Let S be given by
},0)(|R{ ≤θ∈θ= gS n
where g : Rn → R a smooth function. The solution of the constrained minimization
problem follows from the gradient projection method is given by:
⎪⎩
⎪⎨⎧
∇Γ∇Γ∇
∇∇Γ+∇Γ−
≤∇∇Γ−∂∈θ∈θ∇Γ−≡∇Γ−=θ otherwise ,
0)( and )( ifor )(Int if ,)Pr(
T
T
T
Jgg
ggJ
gJSSJJ&
or
⎪⎩
⎪⎨⎧
εφΓ∇Γ∇
∇∇Γ−εφΓ
≤∇εφΓ∂∈θ∈θεφΓ=εφΓ+=θ otherwise ,
0)( and )( ifor )(Int if ,)Pr( T
T
gggg
gSS
T
&
where θ(0) ∈ S..
Theorem 4.5.1: The gradient projection adaptive law retain all the properties that are
established in the absence of projection, and in addition guarantees that ,
provided θ(0) ∈ S and θ
S∈θ
0≥∀t * ∈ S.
Brief Proof: Whenever , , which implies that points either in-
side or along the tangent plane of
)(S∂∈θ 0≤∇θ g& θ&
)(S∂ at point θ. Because S∈θ )0( , it follows that
θ(t) will never leave S, . 0≥∀t
Next, the difference of the adaptive law, when projection is applied, lies in the
additional term:
Jgg
ggQ ∇Γ∇Γ∇
∇∇Γ= T
T
.
20
To show that this additional term will not try to make more positive, we see that V&
⎪⎩
⎪⎨⎧
>∇∇Γ−∂∈θ∇Γ∇Γ∇
∇∇θ=Γθ −
otherwise , 0
0)( and )( if ,~~ TT
TT
1T gJSJgg
ggQ
and that 0)(~ TT ≥∇θ−θ=∇θ ∗ gg when )(S∂∈θ because S is convex. Therefore,
⎪⎩
⎪⎨⎧
>∇∇Γ−∂∈θ≤∇Γ∇
∇∇Γ∇θ=Γθ −
otherwise , 0
0)( and )( if 0,))(~(~ TT
T
1 gJSgg
gJgQ
T
T .
In other words, the additional term in V introduced by projection can only
make more negative.
Q1T~ −Γθ &
V&
4.5.2 Least-Squares with Projection
Consider the adaptive law as follows:
⎪⎩
⎪⎨⎧
εφ∇Γ∇
∇∇−εφ
≤∇εφ∂∈θ∈θεφ=εφ=θ otherwise ,
0)( and )(or )(Int if ,)(
T
T
T
Pgg
ggPP
gPSSPPPr
&
where S∈θ )0( , { }0)(| ≤θ∈θ= gRS n , and
⎪⎩
⎪⎨⎧
≤∇εφ∂∈θ∈θφφ
−β=otherwise ,0
0)( and )(or )(Int if , T2
T
gPSSm
PPPP&
where . 0)0( 0 >= PP
4.6 BILINEAR PARAMETRIC MODEL
As shown in Lecture 2, a certain class of plants can be parameterized in terms of their
desired controller parameters that are related to the plant parameters via a Diophantine
equation. Such parameterizations and their related estimation problem arise in direct
MRAC. Recall that the bilinear parametric model means
)]()[( 0T zsWz +ψθρ= ∗∗
where ρ* is an unknown constant; z, ψ, z0 are signals that can be measured and is a
known proper transfer function with stable poles.
For simplifying, we assume sgn(ρ*) is given here.
4.6.1 SPR-Lyapunov Design
We rewrite the bilinear parametric model in the form
21
)]()[()( 1T zsLsWz +φθρ= ∗∗
where and L(s) is chosen so that L is proper and stable
and WL is proper and SPR. The estimate of z and the normalized estimation error ε
are generated as
ψ=φ= −− )( ,)( 10
11 sLzsLz
z
)]()[()(ˆ 1T zsLsWz +φθρ=
2)()(ˆ snsLsWzz ε−−=ε
where ns is designed to satisfy
221 1 , , snmLmz
m+=∈
φ∞
and ρ(t), θ(t) are the estimates of at time t, respectively. Letting ∗∗ θ−θ=θρ−ρ=ρ
~ ,~
it follows that
]~)[()( 2T1
TsnzsLsW ε−φρθ−ρ−φθρ=ε ∗∗
Now φθρ−φθρ−=φρθ−φθρ ∗∗∗ TTTT ~~ which implies that
1T2T ],~~)[()( znsLsW s +φθ=ξε−ξρ−φθρ−=ε ∗
A minimal state representation is given by
⎭⎬⎫
=εε−φρθ−ρ−φθρ−+= ∗∗
eCnzBeAe
c
sccT
2T1
T )~(& (4.6.1)
where is SPR. The adaptive law is now developed by
considering the Lyapunov-like function
)()()( 1T sLsWBAsIC ccc =− −
γρ
+θΓθ
ρ+=θ−
∗
2
~
2
~~
2),~(
21TT ePeeV c
where satisfies the algebraic equations implied by the KYL Lemma, and
, γ > 0. Along the trajectory of the adaptive law, we have
0T >= cc PP
0T >Γ=Γ
γρρ
+θΓθρ+ε−ξρε−φθερ−ν
−−= −∗∗&&&~~~~~~
221T22TT
TT
sc neLeeqqeV
where ν > 0, Since .0T >= cc LL )sgn( ∗∗∗ ρρ=ρ , it follows that by choosing
⎪⎭
⎪⎬⎫
γεξ=ρ=ρρεφΓ=θ=θ ∗
&&
&&
~)sgn(~
(4.6.2)
we have
22
022
22TTT
≤ε−ν
−−= sc neLeeqqeV&
The performance of this adaptive law is summarized in the following.
Theorem 4.6.1: This SPR-Lyapunov adaptive law guarantees that:
(i) , ∞∈ρθε L , ,
(ii) , 2 , , , Lns ∈ρθεε &&
(iii) If φ, ∈ Lφ& ∞, φ is PE and ξ ∈ L2, then θ(t) converges to θ* as t → ∞.
(iv) If ξ ∈ L2, ρ converges to a constant ρ independent of the properties of φ.
Brief Proof: The proof of (i) and (ii) follows directly form the properties of by fol-
lowing the same procedure as the linear parametric model case and is left as an exer-
cised for the students. The proof of (iii) is establish by using the results of Corollary
4.2.1 to show the homogeneous part of (4.6.1) with ξρ~ treated as an external input
together with the equation of (4.6.2) form an e.s. system. Since 2~ L∈ξρ and Ac is
stable, it follows that 0)(~),( →θ tte as t →∞. The proof of (iv) follows from ε, ξ ∈
L2 and the inequality
∞<⎟⎠⎞⎜
⎝⎛ ττξ⎟
⎠⎞⎜
⎝⎛ ττεγ≤ττξτε≤ττρ ∫∫∫∫
∞∞ 21
21
0
2
0
2
0
0 )()()()()( dddd
tt&
which implies 1L∈ρ& . Hence, we conclude that has ρ(t) a limit ρ .
Remark 4.6.1: The lack of convergence of ρ to ρ∗ is due to ξ ∈ L2. If, however, are
such that is PE, then we can establish by following the same approach as in the proof
in Corollary 4.2.1 that converge to zero exponentially fast. For ξ ∈ L2, the vector [φ T,
ξ] T can not be PE even when φ is PE.
4.6.2 Gradient Algorithm
For the gradient method, we rewritten the model as
)( 1T zz +φθρ= ∗∗
where . The estimate of z and the normalized estimation
error ε are generated as
ψ=φ= )( ,)( 01 sWzsWz z
)(ˆ 1T zz +φθρ=
21
T
2)(ˆ
mzz
mzz +φθρ−=
−=ε
23
where ns is designed to satisfy
221 1 , , snmLmz
m+=∈
φ∞
We consider the cost function
2
21
T22
2)(
2 mzzmJ
∗∗∗ ρ−ξρ+ρξ−φθρ−=
ε=
Using the gradient method, we obtain
⎭⎬⎫
γεξ=ρεφρΓ=θ ∗
&
&1
which indicates that a implementable form
⎭⎬⎫
γεξ=ρρεφΓ=θ ∗
&
& )sgn(1
for
)sgn(1∗∗
∗∗ ρΓ=ρ
ρΓ
=ρΓ
Remark 4.6.2: Strictly speaking, J is not a convex function of ρ, θ over Rn + 1 because
of the dependence of ξ on θ. Let us, however, ignore this dependence and treat ξ as an
independent function of time.
The performance of this adaptive law is summarized in the following.
Theorem 4.6.2: This instantaneous adaptive law guarantees that:
(i) , ∞∈ρθρθεε Lns && ,, , ,
(ii) , 2 , , , Lns ∈ρθεε &&
(iii) If ns, φ ∈ L∞, φ is PE and ξ ∈ L2, then θ(t) converges to θ* as t → ∞.
(iv) If ξ ∈ L2, ρ converges to a constant ρ independent of the properties of φ.
The proof from that of the linear parametric model and Theorem 4.6.1 and is left as an
exercise for the students.
The extension of the integral adaptive law and least-squares algorithms to the
bilinear parametric model is more complicated and difficult to the implement due to
the appearance of the unknown ρ∗ in the adaptive laws. This problem is avoided the
knowledge of a lower bound for in addition to sgn(ρ∗). (The detailed introduction is
24
given in [Ioannou & Sun, 1996].)
4.7 HYBRID ADAPTIVE LAWS
For gradient algorithm, we have
φεΓ=θ & , 2
ˆm
zz −=ε ,
where is the output of the linear parametric model and . Then, φθ= T*z φθ= Tz
,)()(1
1 ∫+ ττφτεΓ+θ=θ +
k
k
t
tkk d )0(0 θ=θ , K,2,1 ),( =θ=θ ktkk
Theorem 4.7.1: Let m, Ts = (tk+1 − tk), and Γ be chosen such that
12
T
≤φφ
m, , 1≥m
and
rT ms ≥λ−2 for , where 0>r )(max Γλ=λm .
Then the hybrid adaptive law guarantees that
(i) , ∞∈θ lk
(ii) , , where 2lk ∈θ∆ 2 , LLm ∩∈εε ∞ kkk θ−θ=θ∆ +1 ,
(iii) If and φ is PE, then as ∞∈φ Lm, ∗θ→θk ∞→k exponentially fast.
Proof: Let kkkV θΓθ= − ~~)( 1T where ∗θ−θ=θ kk~ . We have
kkkkV θ∆Γθ∆+θ=∆ − ~)~~2()( 1T
where , which implies that )()1()( kVkVkV −+=∆
⎟⎠⎞⎜
⎝⎛ ττφτεΓ⎟
⎠⎞⎜
⎝⎛ ττφτε+ττφτεθ=∆ ∫∫∫
+++ 111
T
T )()()()()()(~2)( k
k
k
k
k
k
t
t
t
t
t
tk dddkV
Because )(~ 2 tm Tk φθ−=ε and ( ) Γ≥Γλ=λ )(maxm , we have
2
22 11
)()(
)()()()(2)( ⎟⎟⎠
⎞⎜⎜⎝
⎛τ
ττφ
ττελ+τττε−≤∆ ∫∫++ k
k
k
k
t
tm
t
td
mmdmkV .
Using Schwartz inequality, we can establish that
∫
∫∫∫+
+++
τττε≤
τ⎟⎟⎠
⎞⎜⎜⎝
⎛ττφ
τττε≤⎟⎟⎠
⎞⎜⎜⎝
⎛τ
ττφ
ττε
1
111
22
2
222
)()(
)()(
)()()()(
)()(
k
k
k
k
k
k
k
k
t
ts
t
t
t
t
t
t
dmT
dm
dmdm
m
such that
25
∫+ τττελ−−=∆ 1
22 )()()2()( k
k
t
tms dmTkV
So, if , then 02 >>λ− rT ms 0)( ≤∆ kV , which implies that is a nonincreasing
function and thus the boundedness of ,
)(kV
)(kV kθ~ and kθ follows. Hence, we have
)2()1()0()()(1
0
22
ms
t
TkVVdmk
λ−+−
≤τττε∫+
which yields that
)1(lim +∞→
kVk
exists, and 22 LLLLm ∩∈ε⇒∩∈ε ∞∞ since 1≥m
Similarly, we can obtain
∫+ τττελ≤θ∆θ∆ 1
222T )()(k
k
t
tmskk dmT
such that
∞<τττελ≤θ∆θ∆ ∫∑∞∞
=
0
222
1
T )()( dmT msk
kk
which implies , and thus completes the proof of (i) and (ii). 2lk ∈θ∆
The proof of (iii) is given now. From the proof of (1) and (ii), we have
∫+ τττελ−−=−+=∆ 1
22 )()()2()()1()( k
k
t
tms dmTkVkVkV
which implies
∑∫−
=
++
+
τττελ−−=−+1
1
221 )()()2()()(n
i
t
tmsik
ik
dmTkVnkV (4.7.1)
for any integer n. We now write
∫
∫∫++
+
++
+
++
+
ττ
τφθ−θ+τφθ=
τττφθ
=τττε
+
+
1
11
2
2T1
T
2
2T1
22
)()]()~~()(~[
)()](~[)()(
ik
ik
ik
ik
ik
ik
t
tkkk
t
tkt
t
dm
dm
dm
Using 22212)( yxyx −≥+ , we write
∫∫∫++
+
++
+
++
+
ττ
τφθ−θ−τ
ττφθ
≥τττε ++ 111
2
2T1
2
2T1
22
)()]()~~[(
)()](~[
21)()( ik
ik
ik
ik
ik
ik
t
tkk
t
tk
t
td
md
mdm
Since )()(ττφ
m is bounded, we have
2
2
2T1 ~~
)()]()~~[(1
kiks
t
tkk cTd
mik
ik
θ−θ≤ττ
τφθ−θ+
+∫++
+
(4.7.2)
where
26
)()(
sup 2
2
ττφ
=m
c
From the hybrid adaptive algorithm, we have
nidik
k
t
tkik , ,2 ,1 ,)()(~~
K=ττφτε=θ−θ ∫
+
+
Using the Schwartz inequality and the boundedness of )()(ττφ
m ,
∫+ τττε≤θ−θ +
nk
k
t
tskik dmciT
222,)()(~~ (4.7.3)
Using (4.7.2) and (4.7.3), we have
∫∫∫+
+
++
+
++
+
τττε−τττφθ
≥τττε + nk
nk
ik
ik
ik
ik
t
ts
t
tkt
tdmiTcd
mdm
2222
2
2T1
22 )()()()](~[
21)()( 11
which leads to
∑ ∫∫
∑∫∫−
=
+
−
=
⎥⎦
⎤⎢⎣
⎡τττε−τ
ττφθ
≥
τττε=τττε
+
+
++
+
++
+
+
1
0
2222
2
2T1
1
0
22
22
)()()()](~[
21
)()()()(
1
1
n
i
t
ts
t
tk
n
i
t
t
t
t
nk
nk
ik
ik
ik
ik
nk
k
dmiTcdm
dmdm
and hence it follows that
k
t
tk
ks
t
t
ik
ik
nk
k
dmTcnn
dm θτττφθ
θ−+
≥τττε ∫∫++
+
+ + ~)()](~[~
]2/)1(1[21)()( 1
2
2T1T
22
22 (4.7.4)
Since φ is PE and 1 ≤ m < ∞, there exist constants 0 , , 012 >α′α′ T such that
Idm
ITt
tk
1
2
2T1
20
)()](~[
α′≥τττφθ
≥α′ ∫+
+
for any t. Hence, for any integer k, n where n satisfies nTs ≥ T0 we have
1T
2
2
2T1T )(~~~
)()](~[~ 1 α′
λ≥θθα′≥θτ
ττφθ
θ ∫++
+
+
mkkk
t
tk
kkVd
mik
ik
(4.7.5)
Using (4.7.4) and (4.7.5) in (4.7.1), we obtain the following inequality:
)(]2/)1(1[2
)2()()( 221 kV
TcnnTkVnkV
s
ms
−+α′λ−
−=−+
hold for any n with nTs ≥ T0. Now it follows that
)()( kVnkV γ≤+
with
1]2/)1(1[2
)2(1 221 <
−+α′λ−
−=γs
ms
TcnnT
27
Therefore,
)0())2(())1(())1(()( 2 VnkVnkVnnkVknV kγ≤≤−γ≤−γ≤+−= L
or
k
mmkn
VknV )()0()(~λ
λ≤
λ≤θ
yields that 0~→θkn exponentially, which together with the property of the hybrid
adaptive algorithm (i.e., kk θ≤θ +~~
1 ), implies θk converges to θ∗ exponentially. The
proof of (iii) is complete.
ASSIGNING READING
[1] Chapter 4 in [Ioannou & Sun, 1996].
[2] Chapter 2 in [Sastry & Bodson, 1989].
28