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