Switching structure state and parameter estimator for MIMO non-linear robust control

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Switching structure state and parameterestimator for MIMO non-linear robust controlJoel Correa MartÍnez a & Alex S. Poznyak aa Depto de control Automa´tico, CINVESTAV-IPN, Av.IPN 2508, esq.Calz.Ticomań, A.P. 14-740, Me´xico

Version of record first published: 08 Nov 2010.

To cite this article: Joel Correa MartÍnez & Alex S. Poznyak (2001): Switching structure state and parameterestimator for MIMO non-linear robust control, International Journal of Control, 74:2, 175-189

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Switching structure state and parameter estimator for MIMO non-linear robust control

JOEL CORREA MARTIÂ NEZ{ and ALEX S. POZNYAK{*

In this paper the problem of simultaneous robust state and parameter estimation for a class of MIMO non-linear systemsunder mixed uncertainties (unmodelled dynamics as well as observation noises) it tackled.

A switching gain robust òbserver± identi® er’ is introduced to obtain the corresponding estimates. This is achieved byapplying an observer to the so-called nominal extended system, obtained from the original system without any uncer-tainties and considering the parameters as additional constant states. As it is shown, in general the extended system canlose the global observability property, supposed by valid for the original non-extended system, and a special procedure isneeded to provide a good estimation process in this situation. The suggested adaptive observer has the Luenberger typeobserver structure with switching matrix gain that guarantees a good enough upper bound for the identi® cation errorperformance index. The Van der Monde generalized transformation is introduced to derive this bound which turns out tobe tight’ (it is equal to zero in the absence of both noises and unmodelled dynamics). The example dealing with aninverted pendulum illustrates the high eŒectiveness of the suggested approach.

1. Introduction

Modern identi® cation theory (Ljung 1979, EykhoŒ

and Parks 1990, Ljung and Gunnarsson 1990) basically

deals with the problem of the e� cient extraction of sig-

nal and system dynamic properties based on availabledata measurements. The non-linear system identi® cation

is traditionally concerned with two main directions:

. estimation of parameters based on direct and com-

plete space state measurements; and

. state space estimation (® ltering) of completely

known non-linear dynamics.

The ® rst direction, dealing with the parameter iden-ti® cation technique and applied to diŒerent classes of

non-linear systems, has been extensively studied in

many aspects during the last three decades. Basically,

the class of linear and non-linear systems whose

dynamics depend linearly on the unknown parameterswas considered, where external noises of the stochastic

nature were assumed (see, for example, Ljung 1979,

Poznyak 1980, Bai 1990, Han-Fu Chen and Lei Guo

1991, Sheikholeslam 1995, Song 1997). The general spe-

ci® c feature of these publications is that the exact state

space vector measurements are assumed to be available.

The second systematic contribution, concerning theobserver construction problem for non-linear systems in

the presence of complete information about non-linear

dynamics, was done in Williamson (1977), Krener and

Isidori (1983), Krener and Respondek (1985) and Xiaand Gao (1989) . Most of these results deal with the

situation when there are no observation noises. It is

possible to obtain a set of rather restrictive conditions

and the dynamics of the observation errors is assumed

to be linear. In Walcott and Zak (1987) and Walcott

et al. (1987), a class of observers for non-linear systems

subjected to bounded non-linearities or uncertainties

was suggested. A canonical form and a necessary and

su� cient observability condition for a class of non-

linear systems which are linear with respect to inputs

were stated in Gauthier and Bornard (1981) . The

extended Luenberger observer for a class of SISO non-

linear systems was designed in Zeitz (1987). These

results were extended in Birk and Zeitz (1988) for a

class of MIMO non-linear systems. The exponentially

convergent observer was derived in Gauthier et al.

(1992) for non-linear systems observable for any input

signal. The more advanced results were obtained in

Ciccarella et al. (1993) which were based on the simple

assumptions on regularity. The global asymptotic con-

vergence of the estimated states towards the true states

was shown.

A much more di� cult situation arises in the case

where we have to construct state and parameter esti-

mates simultaneously in the presence of both internal

(unknown parameters and unmodelled dynamics) and

external (observation noises) uncertainties.

Traditionally, the approach dealing with such problems

is called adaptive ® ltering (TornambeÁ 1989, Haykin

1991, Ljung 1979), when the state observer uses current

estimates of parameters, or adaptive identi ® cation

(Grewal and Glover 1976, Siferd and Maybeck 1982,

Bernsten and Balchen 1983, Tunali and Tarn 1987,

BortoŒ and Spong 1990, Krause and Khargonekar

1990, Unbehauen and Rao 1990), when the identi® er is

constructed based on current state estimates. To solve

this di� cult problem, the high-gain type observers were

International Journal of Control ISSN 0020± 7179 print/ISSN 1366± 5820 online # 2001 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

INT. J. CONTROL, 2001, VOL. 74, NO. 2, 175 ± 189

Received 1 April 1999. Revised 1 April 2000.Communicated by Professor V. Utkin.

* Author for correspondence. e-mail: [email protected]

{ CINVESTAV-IPN, Depto de control AutomaÂ tico,Av.IPN 2508, esq. Calz.TicomaÂ n, A.P. 14-740, MeÂ xico.

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suggested by TornambeÁ (1989) . Ljung (1979) studied the

asymptotic behaviour of the extended Kalman ® lter to

be applied for the identi® cation of linear stochastic dis-

crete time systems but, unfortunately, the conditions of

the convergence in mean square sense turned out to bevery complex for checking. In Haykin (1991) a variety of

recursive estimating algorithms which converge to the

optimum Wiener solution (in some statistical sense)

was considered and numerous engineering applications

of adaptive ® lters were discussed. The identi® ability con-

ception was constructively discussed in Grewal andGlover (1976) , Siferd and Maybeck (1982) and Tunali

and Tarn (1987) where necessary and su� cient con-

ditions for system identi® ability were investigated in

view of the relations between non-linear observability,

functional expansion and the uniqueness theorem onnon-linear realization theory. The approach based on

H1-theory results and applied to parameter identi® -

cation problems in the presence of non-parametric

dynamic uncertainty was suggested in Krause and

Khargonekar (1990). The augmented system was intro-duced in BortoŒand Spong (1990) where the identi® er

based on an extended Luenberger observer was con-

structed and the su� cient conditions for global conver-

gence of its parameter estimates were discussed. The

most advanced techniques for determining the observa-

bility and identi® ability properties are based on the

mathematical branch of diŒerential algebra; thisapproach was extensively developed by Diop and

Fliess (1991) and, particularly, the concept of identi® a-

bility employing the notion of characteristic sets was

suggested by Ljung and Glad (1994). A comprehensivesurvey concerning continuous-time approaches to

system identi® cation, studied until 1990, can be found

in Unbehauen and Rao (1990).

In this paper, based on the traditions of BortoŒand

Spong (1990), Ciccarella et al. (1993), and our previouspaper (see Poznyak and Correa 2000) which deals with

SISO non-linear systems, a switching structure robust

state and parameter estimator is designed for a class of

MIMO non-linear systems and an upper bound for the

corresponding estimate error functional, which turns

out to be a linear combination of external and internal

uncertainties levels, is derived. In the absence of anyuncertainties and noise perturbations , the global asymp-

totic error stability directly follows from the main the-

orem ( tightness’ property). The simulation example (an

inverted pendulum with an unknown friction eŒect) isconsidered to illustrate the eŒectiveness of the suggested

approach.

2. Uncertain systems and problem setting

Consider the class S of non-stationary non-linear

systems (NLS) with multi inputs and multi outputs

(MIMO) containing mixed uncertainties which may

include both an unmodelled dynamics part and noises

of deterministic nature, that is

S :_zt ˆ f …t; zt; ut; c† ‡ ¯1…t; zt; ut; c†; ztˆ0 ˆ z0

yt ˆ h…t; zt; ut; c† ‡ ¯2…t; zt; ut; c†

(

where t 2 <‡ :ˆ ft : t ¶ 0g, zt 2 <n is a state vector at

time t, c 2 C ³ <q is the constant vector of unknown

parameters de® ned within a connected set C, yt 2 <p is

an output vector and ut 2 U ³ <m is a control action

vector, both at time t. The functions

¯1…¢† 2 D1 ³ <n; ¯2…¢† 2 D2 ³ <p

characterize mixed uncertainties. The class S of NLS is

assumed to be consistent, that is, for any ® xed pair

…c; z0† 2 C £ <n and for any input sequence futg there

exists a strong solution fz…t; z0; ut; c†g of the correspond-

ing Cauchy problem. A control strategy futg is said tobe admissible if it is smooth enough and provides the

consistency condition for S.

For the consistent class S de® ne the class S0 of, so-

called nominal non-extended systems, given by

S0 :_·zt ˆ f …t; ·zt; ut; c†; ·ztˆ0 ˆ z0

·yt ˆ h…t; ·zt; ut; c†

(

For each ® xed c 2 C, the corresponding nominal non-

extended system denotes a given NLS without any

uncertainties.

Assumption 1: For any admissible control strategyfutg and for any NLS from S there exists ¶ < 0 such

that for i ˆ 1; 2

Wi :ˆ…1

sˆ0

e¶…t¡s† sup¯i…¢†2Di

k¯i…s; zs; us; c†k ds < 1

Remark 1: Any uniformly bounded uncertainties evi-

dently satis® es Assumption 1.

The objective of this study is to design a system(named òbserver± identi® er’ ) which can generate simul-

taneously `good’ estimates of the states and the

unknown parameters of NLS from S, and to derive a

tight upper bound for the performance index character-

izing the corresponding estimation process.In this paper, to solve this problem, the approach

suggested in Poznyak and Correa (2000) for SISO

non-linear systems is extended for the class of MIMO

uncertain NLS. The employed strategy is to construct an

observer for the extended system, obtained by consider-

ing the parameter vector as additional constant states.Nevertheless, this approach conveys some interesting

problems that justify the introduction of the of O"-

observability concept and the implementation of the

estimator with a switching structure.

176 J. C. Martõ Ânez and A. S. Poznyak

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3. O"-Observability concept for MIMO non-linear

systems

For N :ˆ n ‡ q, de® ne the extended state vector

wt :ˆ ‰zTt cTŠT 2 <N …1†

and rewrite S in the extended form as the uncertain

system

_wt ˆ Fw…t; wt; ut† ‡ ¢w;1…t; wt; ut†; wtˆ0 ˆ w0

yt ˆ hw…t; wt; ut† ‡ ¯w;2…t; wt; ut†

9=

;

…2†

where

Fw…¢† :ˆfw…¢†

0

" #

; ¢w;1…¢† :ˆ¯w;1…¢†

0

" #

; w0 :ˆz0

c

" #

…3†

and

fw…t; wt; ut† :ˆ f …t; zt; ut; c†jw

¯w;i…t; wt; ut† :ˆ ¯i…t; zt; ut; c†jw; i ˆ 1; 2

hw…t; wt; ut† :ˆ h…t; zt; ut; c†jw

9>>>=

>>>;…4†

Moreover, de® ne the input vector derivatives

Ukt :ˆ ‰u1;t _u1;t ¢ ¢ ¢ u

…k†1;t ¢ ¢ ¢ um;t _um;t ¢ ¢ ¢ u

…k†m;tŠT

The same way as for S0, for (2) de® ne the nominal

extended system relative to S as

_·wt ˆ Fw…t; ·wt; ut†; ·wtˆ0 ˆ ·w0

·yt ˆ hw…t; ·wt; ut†

9=

; …5†

The following de® nitions are based on concepts and

results introduced in Gauthier and Bornard (1981),

Tunali and Tarn (1987), Birk and Zeitz (1988) and

BortoŒand Spong (1990). They will be fundamental

throughout this paper.

De® nition 1: The consistent class S0 is said to be

completely uniformly locally observable on a set

M± » <n if there exists a set of p non-negative integer

numbers l± :ˆ fl±;1; l±;2; . . . ; l±;pg with l±;1 ‡ l±;2 ‡ ¢ ¢ ¢ ‡l±;p ˆ n and l¤± :ˆ maxi l±;i such that for any t 2 <‡,any U¤

± ¡ 1 2 Uml¤± , any c 2 C and any ·zt 2 M± , the co-

ordinate transformation given by

·±t :ˆ F±…t; ·zt; U l¤± ¡1; c† …6†

is a diŒeomorphism between sets M± and

F±…t; M± ; U l¤± ¡1t ; c†

where

F±…¢† :ˆ ‰h1…¢† LF±h1…¢† ¢ ¢ ¢ L

l±;1¡1

F±h1…¢† ¢ ¢ ¢

hp…¢† LF±hp…¢† ¢ ¢ ¢ L

l±;p¡1

F±hp…¢†ŠT

and LkF±

…hi†…i ˆ 1; . . . ; p† denotes the Lie derivative of

the output function hi along the vector ® eld

F±…t; ·zt; Uml¤±t ; c† :ˆ ‰FT…t; ·z; ut; c† 1 _u1;t ¢ ¢ ¢

u…l¤± †1;t ¢ ¢ ¢ _um;t ¢ ¢ ¢ u

…l¤± †m;t ŠT

As it is known, (6) is a diŒeomorphism if and only if

the òbservability matrix’ de® ned by

Q±;t :ˆ Q±…t; ·zt; U l¤± ¡1t ; c† :ˆ

@F±…t; ·zt; U l¤± ¡1t ; c†

@·z…7†

is non-singular for any

…t; ·zt; Ul¤±

¡1

t ; c† 2 <‡ £ M± £ Uml¤± £ C

De® nition 2: The extended nominal system given by

(5) is completely uniformly locally observable in

Mx ³ <N if there exists a set of p non-negative integernumbers lx :ˆ flx;1; lx;2; . . . ; lx;pg with lx;1 ‡ lx;2 ‡ ¢ ¢ ¢ ‡lx;p ˆ N and l¤x :ˆ maxi lx;i such that for every

…t; U l¤x¡1t † 2 <‡ £ Uml¤x the òbservability matrix’

Qt :ˆ Qx…t; ·wt; U l¤x¡1

t † :ˆ @Fx…t; ·wt; U l¤x¡1†

@ ·w…8†

is non-singular 8 ·wt 2 Mx where the coordinate transfor-

mation

·xt :ˆ Fx…t; ·wt; U l¤x¡1t † …9†

is de® ned as

Fx…¢† :ˆ ‰h1w…¢† LFxh1w…¢† ¢ ¢ ¢ L

lx;1¡1Fx

h1w…¢†

¢ ¢ ¢ hpw…¢† LFxhpw…¢† ¢ ¢ ¢ L

lx;p¡1

Fxhpw…¢†ŠT

with

Fx…t; ·wt; U l¤xt † :ˆ ‰Fw…t; ·wt; ut† 1 _u1;t ¢ ¢ ¢ u

…l¤x†1;t ¢ ¢ ¢

_um;t ¢ ¢ ¢ u…l¤

x†m;t ŠT

Based on the de® nitions given above, we can easily

conclude that the simultaneous observability and iden-

ti® ability properties for the class S0 is implied by the

complete uniform observability of the correspondingextended system (5).

The main test to check the complete uniform

observability property for the nominal extended system

consists in the veri® cation of the inequality

det …Qx…t; ·wt; U l¤x¡1t †† 6ˆ 0 …10†

MIMO non-linear robust control 177

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condition that can serve as a tool to determine the set of

singular times and inputs (times and inputs which make

the system unobservable).

Remark 2.1: By the approach similar to one given in

Poznyak and Correa (2000) and derived for SISOsystems, a standard high-gain non-linear observer for the

extended system will be implemented within the special

trajectory subset.

Assumption 2: Within the class S we will consider the

subclass for which the nominal systems are completelyuniformly observable in global sense and for which

det …Qt† is dependent only on ·w.

In the extended space (1) de® ne the function C…w† and

the non-linear submanifold O » <N as

C…w† :ˆ j det Qtj …11†

O :ˆ fw 2 <N : det Qt ˆ 0g …12†

where Qt is the observability matrix given by (8).

De® nition 3: The nominal extended system (5) is said

to be O"-observable within the non-empty subset

L"…O† :ˆ f ·wt 2 <N : C > "g » <N ; " > 0

if the observability matrix (8) is "-non-singular inside

this set, that is

det Qt > " …13†

Based on this de® nition the extended space <N can

be split into the two disjoint subsets L"…O† and ·L"…O†where

·L"…O† :ˆ f ·wt 2 <N : C… ·wt† µ "g

is the O"-non-observable (`practically observable’ ) sub-

space and L"…O† is the `practically observable’ subset.

4. Switching structure state and parameter estimator

For any estimate trajectory wt 2 <N de® ne the

sequence of stopping times’ f½ng …n ˆ 0; 1; 2; . . .†, fromwhich the subsequences f½2kg and f½2k‡1g…k ˆ 0; 1; 2; . . .† correspond to the instants when the pro-

cess fwtg leaves the O"-observable subset and returns

back, respectively (see ® gure 1)

½0 :ˆ 0; ½1 :ˆ inf ftjwt 2 L"…O†g

½2k :ˆ min ftjws 2 ·L"…O† 8s > ½2k¡1g …14†

½2k‡1 :ˆ inf ftjws 2 L"…O† 8s ¶ ½2k …15†

De® ne also the characteristic function

Àt :ˆ0; ½2k µ t µ ½2k‡1

1; ½2k¡1 < t < ½2k

(

…16†

and two gain matrices Kobt , Knob

t 2 <N£p

Kobt :ˆ QtKx; Knob

t :ˆQ±;tK±

0

" #…17†

where Qt is the observability matrix (8) of the extended

system

Kx :ˆ

Kx;1 0 ¢ ¢ ¢ 0

0 Kx;2 ¢ ¢ ¢ 0

..

. ... . .

. ...

0 0 ¢ ¢ ¢ Kx;3

2

66666664

3

77777775

2 <N£p

with

Kx;i :ˆ

kx;i;1

..

.

kx;i;lx;i

2

66664

3

777752 <lx;i£1

to be chosen, and Q±;t 2 <n£n is the observability matrix

(7) for the class S0. The gain matrix K± 2 <n£p isassumed to be of the same structure as Kx (with the

appropriate substitutions Kx;i by K±;i and lx;i by l± ; i)

and should be chosen as well.

The main idea of the method suggested in this paper

is as follows: if the current trajectory of the extendedstate is in the `good’ (O"-observable) subset, then a stan-

dard estimation approach is suggested to be applied

using a high gain observer for the extended state; but

if the trajectory of the extended state estimate turns out

to be within the `bad’ (O"-unobservable) subspace, then

the parameter estimates are suggested to be kept frozen’and only the states of the nominal non-extended class S0

are estimated using the same high gain observer

approach, a priori assuming that this nominal non-

extended class is completely uniformly observable. The


Figure 1. Stopping times.

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following variable (switching) structure state and par-

ameter estimator realizes the procedure

_wt ˆ Fw…t; wt; ut† ‡ ‰Knobt ‡ Àt…Kob

t ¡ Knobt †Š

£ …yt ¡ hw…t; wt; ut†† …18†

with ® xed initial conditions

ztˆ0

ctˆ0

" #ˆ

z0

c0 2 C

" #

5. Upper estimate of error bound

In this section the suggested estimator (18) will be

shown to possess the robustness property’ within somesubclass of S ful® lling the following additional assump-

tions:

Assumption 3: Any nominal non-extended system from

the class S0 is O"-observable, that is, there exists " > 0

such that

L"…O† 6ˆ 1

As it is shown in Krener and Isidori (1983), if the

dynamics of S0 are completely uniformly observable,

then the new coordinates ±t :ˆ F±…t; zt; U l¤± ¡1

t ; c†, given by

(6) satisfy

S¤0 :

_±t ˆ A±±t ‡ B±H±…t; ±t; U l¤±t ; c† ‡ ¯±;1…t; ±t; U l¤± ¡1

t ; c†

±tˆ0 ˆ F±…0; z0; U l¤± ¡1; c†

yt ˆ C±±t ‡ ¯±;2…t; ±t; U l¤± ¡1t ; c†

8>>>><

>>>>:

where

¯±;1…t; ±t; U l¤± ¡1t ; c† :ˆ Q±;t¯1…t; zt; ut; c†j

zˆF¡1± …t;±t;U

l¤±

¡1

t ;c†

¯±;2…t; ±t; U l¤± ¡1t ; c† :ˆ ¯2…t; zt; ut; c†j

zˆF¡1± …t;±t;U

l¤±

¡1

t ;c†

and the vector function H±…¢† is de® ned as

H±…t; ±t; U l¤±

t ; c† :ˆ

Ll1;±

F±h1…t; F¡1

± …t; ±t ; U l¤± ¡1

t ; c†; U l¤±

t ; c†

Ll2;±

F±h2…t; F¡1

± …t; ±t ; U l¤± ¡1

t ; c†; U l¤±

t ; c†

..

.

Llp;±

F±hp…t; F¡1

± …t; ±t ; U l¤± ¡1

t ; c†; U l¤±

t ; c†

2

66666666664

3

77777777775

2 <p

…19†

For the same reasons, if system (5) is O"-observable

inside L"…O†, the coordinates xt :ˆ Fx…t; wt; U l¤x¡1t †, given

by (9), satisfy

_xt ˆ Axxt ‡ BxHx…t; xt; U l¤xt † ‡ ¢x;1…t; xt; U l¤x¡1

t †

xtˆ0 ˆ x0 :ˆ Fx…0; w0; U l¤x¡1t †

yt ˆ Cxxt ‡ ¢x;2…t; xt; U l¤x¡1t †

9>>>>=

>>>>;

…20†

where, by (3) and (4), the uncertain terms can be

expressed as

¢x;1…t; xt; U l¤x¡1t † :ˆ Qt¢w;1…t; w; ut†jwˆF¡1

x …t;xt;Ul¤x¡1

t †

¢x;2…t; xt; U l¤x¡1t † :ˆ ¯w;2…t; w; ut†jwˆF¡1

x …t;xt;Ul¤x¡1

t †

and the vector function Hx…¢† is de® ned as

Hx…t; xt; U l¤xt ; c† :ˆ

Ll1;x

fxh1…t; F¡1

x …t; xt; U l¤x¡1t ; c†; U l¤x

t †

Ll2;x

Fxh2…t; F¡1


t †

..

.

Llp;x

Fxhp…t; F¡1


t †

2

6666666664

3

7777777775

2 <p£1

…21†

Here the nominal parts of S¤0 and (20) are expressed in,

so-called, generalized Brunovsky canonical form (or

generalized observer form where the pairs …A± ; B±† and

…Ax; Bx† are controllable. Meanwhile the pairs …A± ; C±†and …Ax; Cx† are observable.

Within the O"-observable subset L"…O† there exists the

inverse transformation of (9) is given by

wt ˆzt

c

" #ˆ F¡1

x …t; xt; U l¤x¡1t † ˆ

F¡1z …t; xt; U l¤x¡1

t †

F¡1c …t; xt; U l¤x¡1

t †

2

4

3

5

…22†

but inside of the O"-non-observable subset ·L"…O†

kF¡1x …t; xt; Dl¤x¡1

t †k ¶ O…"¡1† …23†

that may lead to a singularity eVect.

Assumption 4: There exists a non-empty set L"…O†where the functions Hx…t; x; U l¤x

t † and F¡1c …t; x; U l¤x¡1

t † are

completely quasi-Lipschitz in x, that is, for any

…t; U l¤x¡1t † 2 <‡ £ Uml¤x , for any trajectory xt satisfying

(20) and for any x 0 2 L"…O† there exist positive con-

stants LHx…"† and LF¡1

xc …"† such that

kHx…t; xt; U l¤xt † ¡ Hx…t; x 0; U l¤x

t †k µ LHx…"†kxt ¡ x 0†k

kF¡1xc …t; xt; U l¤x¡1

t † ¡ F¡1xc …t; x 0; U l¤x¡1

t †k µ LF¡1xc

…"†kxt ¡ x 0k

Assumption 5: The functions


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H±…t; ±; U l¤±t ; c† Fx…t; w; U l

¤x¡1

t †

F¡1xz …t; x; U l¤x¡1

t †; F¡1± …t; ±; U l¤± ¡1

t ; c†

F±…t; z; U l¤± ¡1

t ; c†

are completely globally Lipschitz, that is, for any

…t; U l¤x¡1

t † 2 <‡ £ Uml¤x it follows that

kH±…t; ±; U l¤±

t ; c† ¡ H±…t; ± 0; U l¤±

t ; c 0†k µ LH±…k± ¡ ± 0k ‡ kc ¡ c 0k†

kFx…t; w; U l¤x¡1t † ¡ Fx…t; w 0; U l¤x¡1

t k µ LFxkw ¡ w 0k

kF¡1xz …t; x; U l¤x¡1

t † ¡ F¡1xz …t; x

0; U l¤x¡1

t †k µ LF¡1xz

kx ¡ x0k

kF¡1± …t; ±; U l¤

±¡1

t ; c† ¡ F¡1± …t; ± 0; U l¤± ¡1

t ; c†k µ LF¡1±

k± ¡ ± 0k

kF±…t; U l¤± ¡1

t ; c† ¡ F±…t; z0; U l¤

±¡1

t ; c†k µ LF±kz ¡ z

0k

for all …±; c†, …± 0; c 0† 2 <n £ C, all w, w 0 2 <N; x,

x 0 2 <N , all ±, ± 0 2 <n and all z, z 0 2 <n with some

LH±, LFx

, LF¡1xz

, LF¡1±

, Lz, LF±2 …0; 1†.

Remark 3: From the de® nitions given above, it fol-

lows that the Lipschitz constants LHxand LF¡1

xcin-

crease if the value of " decreases, since, in this case,

the region of the observable part L"…Ot† has the ten-

dency to increase too.

The main contribution of this study is given in the

next theorem which establishes a bound for the estima-

tion error provided by the suggested variable structure

state and parameter estimator (18).

Select the gain matrices K±;i and Kx;i…i ˆ 1; . . . ; p†,participating in (18), in such a way that the matrices

…A± ¡ K±C±† and …Ax ¡ KxCx† would have diŒerent

real negative eigenvalues, that is

¶±;n < ¶±;n¡1 < ¢ ¢ ¢ < ¶±;1 < 0

¶x;N < ¶x;N¡1 < ¢ ¢ ¢ < ¶x;1 < 0

De® ne the generalized Van der Monde matrix associated

to the block companion matrix M having the set of

p-blocks

VM :ˆ

VM;1 0 ¢ ¢ ¢ 0

0 VM;2 ¢ ¢ ¢ 0

..

. ... . .

. ...

0 0 ¢ ¢ ¢ VM;p

2

66666664

3

77777775

…24†

with

VM;i :ˆ

¶li ¡1i;1 ¶

li¡2i;1 ¢ ¢ ¢ 1

¶li ¡1i;2 ¶

li¡2i;2 ¢ ¢ ¢ 1

..

. ... . .

. ...

¶li ¡1i;li

¶li¡2i;li

¢ ¢ ¢ 1

2

66666666664

3

77777777775

2 <li£li

where f¶i;1; . . . ; ¶i;lig is the set of eigenvalues corre-

sponding to the ith block. This matrix serves to diago-

nalize the exponential matrix derived from M

eMt ˆ V¡1M eL

Mt VM

where

LM :ˆ

LM;1 0 ¢ ¢ ¢ 0

0 LM;2 ¢ ¢ ¢ 0

..

. ... . .

. ...

0 0 ¢ ¢ ¢ LM;p

2

6666666664

3

7777777775

with

LM;i :ˆ diag‰¶i;1; . . . ; ¶i;liŠ

De® ne the constants

To :ˆ infr

f…½2r ¡ ½2r¡1†g

Tno :ˆ infr

f‰½2r¡1 ¡ ½2…r¡1†Šg

9>=

>;…25†

a :ˆ exp …¡¬xTo†‰2c21zc3c ‡ c2

1z ‡ c21c ‡ c2

1zc23c

‡ 2c21cc3c ‡ c2

1cc23c ‡ …c2

1z ‡ c21c†c2

3z exp …¡2¬±Tno†Š1=2

…26†

b :ˆ exp …¡¬xTo†c4z

c1z

c1c

2

4

3

5 ‡c2z

c2c

2

4

3

5

®®®®®®

®®®®®®…27†

with

0 < ¬± :ˆ maxi

f¶±;ig ‡��l¤±

qLH±

kV¡1A± ¡K± C±

k

0 < ¬x :ˆ maxi

f¶x;ig ‡��l¤x

pLHx

…"†kV¡1Ax¡KxCx

k

9>>=

>>;…28†

and


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c1z :ˆ LFxLF¡1

xzkVAx¡KxCx

k kV¡1Ax¡KxCx

k

c1c :ˆ LFxLF¡1

xc…"†kVAx¡KxCx

k kV¡1Ax¡KxCx

k

c2z :ˆ LF¡1xz

kVAx¡KxCxk kV¡1

Ax¡KxCxk…kKxkW2 ‡ W1†

c2c :ˆ LF¡1xc

…"†kVAx¡KxCxk kV¡1

Ax¡KxCxk…kKxkW2 ‡ W1†

c3z :ˆ LF±LF¡1

±kVA± ¡K± C±

k kV¡1A± ¡K± C±

k

c3c :ˆ

��l¤±

qLH±

j¶z±;1j kV¡1

A± ¡K± C±k

c4z :ˆ LF¡1±

kVA± ¡K± C±k kV¡1

A± ¡K± C±k…kK±kW2 ‡ W1†

9>>>>>>>>>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>>>>>>>>>;

…29†

Theorem 1: Under Assumptions 1± 5, for any admissi-ble control, any uncertain system from S and for any

realized trajectory fztg satisfying the persistent excita-

tion condition

…1

tˆ0

Àt dt ˆ 1 …30†

the variable structure state and parameter estimator (18)provides the following upper bound for the corresponding

estimation error

lim supt!1

zt ¡ zt

ct ¡ c

" #®®®®®

®®®®®

µ…c2

2z ‡ c22c†1=2 if 9T : Àt ˆ 1 8t > T

b=…1 ¡ a† …if a < 1† in another case…31†

8<

:

The condition a < 1 establishes the relation betweenthe measure of the uncertainties, the selected value " and

the chosen gain matrices in (18).

If the persistent excitation condition’ (30) is not ver-

i® ed, the estimation error remains bounded when Àt ˆ 0for all

t > ·T :ˆ supt

ft : Àt ˆ 1g

The proof of this theorem is given in the Appendix.

6. Test simulations for inverted pendulum with

unknown friction

The dynamical behaviour of the inverted pendulum,

represented in ® gure 2, is governed by the equations

M ‡ m mL cos ³t

mL cos ³t mL2

2

4

3

5�st

�³t

2

4

3

5 ¡mL… _³t†2 sin ³t

0

2

4

3

5

¡0

mgL sin ³t

2

4

3

5 ‡·1…t; st; _st; ³t; _³t†

·2…t; st; _st; ³t; _³t†

2

4

3

5 û

¡k _³t

2

4

3

5

_stˆ0

_³tˆ0

2

4

3

5 ˆ_s0

_³0

2

4

3

5;stˆ0

³tˆ0

2

4

3

5 ˆs0

³0

2

4

3

5

where the terms ·1…¢† and ·

2…¢† include possible noises

and unmodelled dynamics. The term k _³t represents the

friction eŒect. The displacement st of the car and the

angle position ³t of the arm are assumed to be available

at any time t. These measurements may be contaminated

by unknown bounded disturbances ¯3;t and ¯4;t, respect-

ively. De® ning the state vector as

zt ˆ ‰z1;t; z2;t; z3;t; z4;tŠT ˆ ‰st; _st; ³t; _³tŠT

the dynamics and the output of the system can be repre-

sented in the form

_zt ˆ

z2;t

f1…zt; k†

z4;t

f2…zt; k†

2

6666664

3

7777775‡

0

¯1…t; zt†

0

¯2…t; zt†

2

6666664

3

7777775

yt ˆz1;t

z3;t

" #‡

¯3;t

¯4;t

" #

; ztˆ0 ˆ z0

9>>>>>>>>>>>>>=

>>>>>>>>>>>>>;

…32†

where


Figure 2. Inverted pendulum.

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f1…zt; k† :ˆ …mL2z24;t ¡ gL cos z3;t† sin z3;t ‡ kz4;t cos z3;t

‡ LutŠ‰L…M ‡ m sin2 z3;t†Š¡1

f2…zt; k† :ˆ ¡‰m2…L2z24;t cos z3;t ¡ gL† sin z3;t

‡ …M ‡ m†kz4;t ‡ mL…ut cos z3;t ¡ Mg sin z3;t†Š

£ ‰mL2…M ‡ m sin2 z3;t†Š¡1

The damping factor k will be considered as the unknown

parameter in the system. From (32), one can see that the

nominal system results in

_·zt ˆ

·z2;t

f1…·zt; k†

·z4;t

f2…·zt; k†

2

6666664

3

7777775

·yt ˆ·z1;t

·z3;t

" #

; ·ztˆ0 ˆ z0

9>>>>>>>>>>>>>=

>>>>>>>>>>>>>;

…33†

To check the observability property of this system,

de® ne the vector ® eld as

·F…·zt; k; ut† :ˆ

·z2;t

f1…·zt; k†

·z4;t

f2…·zt; k†

2

66666664

3

77777775

The direct calculation of the ® rst derivatives of the

nominal output functions along ·F leads to

·x1;t :ˆ ·z1;t

·x2;t :ˆ _·x1;t ˆ L ·F…·z1;t† ˆ ·z2;t

·x3;t :ˆ ·z3;t

·x4;t :ˆ _·x3;t ˆ L ·F…·z3;t† ˆ ·z4;t

The Jacobian matrix of this transformation with respect

to the state ·zt (which corresponds to the observability

matrix for (33) as well as for (32) is given by the identity

matrix of order 4. As a result, this system is concluded to

be completely uniformly observable. De® ne the state

vector of the extended system as

w :ˆ ‰z1 z2 z3 z4 k†T

that implies

_wt ˆ

w2;t

f1…wt†

w4;t

f2…wt†

0

2

66666666664

3

77777777775

‡

0

¯1…t; wt†

0

¯2…t; wt†

0

2

66666666664

3

77777777775

yt ˆw1;t

w3;t

2

4

3

5 ‡¯3;t

¯4;t

2

4

3

5; wtˆ0 ˆ w0 :ˆz0

k

" #

9>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>;

…34†

with nominal dynamics given by

_·wt ˆ

·w2;t

f1… ·wt†

·w4;t

f2… ·wt†

0

2

66666666664

3

77777777775

; ·wtˆ0 ˆ w0

·yt ˆ·w1;t

·w3;t

2

4

3

5

9>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>;

…35†

To construct the coordinate transformation for (35)

de® ne the vector ® eld

F… ·wt; ut† :ˆ

·w2;t

f1… ·wt†

·w4;t

f2… ·wt†

0

2

66666666664

3

77777777775

Deriving twice the ® rst nominal output function ·y1;t and

once the second one ·y2;t along the direction of the

de® ned vector ® eld F , it follows that

·xt :ˆ Fx… ·wt; ut† :ˆ

·w1;t

·w2;t

g… ·w; ut†

·w3;t

·w4;t

2

66666666664

3

77777777775

where


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g… ·w; ut† :ˆ ‰mL ·w24;t sin ·w3;t ‡ ut

¡ …mg sin ·w3;t ¡ ·w4;t ·w5;tL¡1† cos ·w3;tŠ

£ …M ‡ m sin2 ·w3;t†¡1

So, the observability matrix for the nominal extended

system …@Fx=@ ·wt† is given by

Qx… ·wt; ut† :ˆ

1 0 0 0 0

0 1 0 0 0

0 0 q33 q34 q35

0 0 1 0 0

0 0 0 1 0

2

66666666664

3

77777777775

with

q33 :ˆ ‰mL ·w24;t cos ·w3;t ‡ …mg sin ·w3;t ¡ ·w4;t ·w5;tL

¡1†

£ sin ·w3;t ¡ mg cos2 ·w3;tŠ

£ …M ‡ m sin2 ·w3;t†¡1 ¡ 2m sin ·w3;t cos ·w3;t

£ ‰mL·w24;t sin ·w3;t ‡ ut ¡ …mg sin ·w3;t ¡ ·w4;t ·w5;tL

¡1†

£ cos ·w3;tŠ…M ‡ m sin2 ·w3;t†¡2

q34 :ˆ ‰2mL ·w4;t sin ·w3;t ‡ ·w5;tL¡1 cos ·w3;tŠ…M ‡ m sin2 ·w3;t†¡1

q35 :ˆ ·w4;tL¡1 cos ·w3;t…M ‡ m sin2 ·w3;t†¡1

It is easy to see that this observability matrix loses itsrank over the manifold

·w4;t cos ·w3;t ˆ 0

that is, over this manifold the system (35) (as well as(34)) is non-uniformly observable. Note that the non-

observable manifold does not depend on either the

parameter or the input. The inverse observability

matrix is

Q¡1x … ·wt; ut† ˆ

1 0 0 0 0

0 1 0 0 0

0 0 0 1 0

0 0 0 0 1

0 0 qi53 qi54 qi55

2

66666666664

3

77777777775

where

qi53 :ˆ L…M ‡ m sin2 ·w3;t†… ·w4;t cos ·w3;t†¡1

qi54 :ˆ ‰2m2L2 ·w24;t cos ·w3;t sin2 ·w3;t ‡ 2mLut cos ·w3;t sin ·w3;t

¡m…M ‡ m†L2 ·w24;t cos ·w3;t ‡ m2L2 ·w2

4;t cos 3 ·w3;t

‡ mMgL…cos2 ·w3;t ¡ sin2 ·w3;t† ¡ m2gL sin2 ·w3;t…1 ‡ cos2 ·w3;t†

‡ …M ‡ m…1 ‡ cos2 ·w3;t†† ·w4;t ·w5;t sin ·w3;t

‡ m2gL…cos2 ·w3;t ¡ cos4 ·w3;t†Š‰…M ‡ m sin2 ·w3;t†·w4;t cos ·w3;tŠ¡1

qi55 :ˆ ¡…2mL2 ·w4;t sin ·w3;t ‡ ·w5;t cos ·w3;t†…·w4;t cos ·w3;t†¡1

Applying now the main result of this paper, we can

suggest the robust estimator for the pendulum system

as the switching scheme …wtˆ0 ˆ ·w0†

_wt ˆ

w2;t

f1…wt†

w4;t

f2…wt†

0

2

66666666664

3

77777777775

‡ ‰ÀwQ¡1x …wt; ut†K1

‡ …1 ¡ Àw†K2Š yt ¡w1;t

w3;t

" #Á !

where the characteristic function Àw is de® ned as

Àw :ˆ0 if jw4;t cos w3;tj µ "

1 if jw4;t cos w3;tj > "

8<

:

with

K1 :ˆ

k11 0

k12 0

k13 0

0 k14

0 k15

2

666666664

3

777777775

; K2 :ˆ

k21 0

k22 0

0 k23

0 k24

0 0

2

666666664

3

777777775

The gain vectors K11 :ˆ ‰k11; k12; k13ŠT, K12 :ˆ ‰k14; k15ŠTand K21 :ˆ ‰k21; k22ŠT, K22 :ˆ ‰k23; k24ŠT are selected, re-

spectively, in such a way that the following polynomials

become stable

…¶ ¡ ¶11†…¶ ¡ ¶12†…¶ ¡ ¶13† ˆ ¶3 ‡ k11¶2 ‡ k12¶ ‡ k13

…¶ ¡ ¶14†…¶ ¡ ¶15† ˆ ¶2 ‡ k14¶ ‡ k15

…¶ ¡ ¶21†…¶ ¡ ¶22† ˆ ¶2 ‡ k21¶ ‡ k22

…¶ ¡ ¶23†…¶ ¡ ¶24† ˆ ¶2 ‡ k23¶ ‡ k24

The selected roots ¶ij < 0 determine the convergence

rate within the corresponding estimate region.


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To carry out the numerical simulation, the following

values and functions have been chosen

k ˆ 0:5; ut ˆ sin …t†

¯1…t† ˆ 0:005square …100t†

¯2…t† ˆ 0:005sawtooth …100t†

The noises ¯3…t† and ¯4…t† are as shown in ® gures 3 and

4; the vectors were chosen as

K11 ˆ ‰30; 300; 1000ŠT and K12 ˆ K21 ˆ K22 ˆ ‰20; 100; 0ŠT

that correspond to the multiple eigenvalue ¶ ˆ ¡10.

Finally, we selected " ˆ 0:3.

The simulation results are shown in ® gures 5± 9

where the observers have had null initial conditions

while the system has z0 ˆ ‰1; 0:5; 0; ¡ 0:8ŠT.


Figure 3. Noise over y1;t.

Figure 4. Noise over y2;t.

Figure 5. The position of M, z1;t.

Figure 6. The velocity of M, z2;t.

Figure 7. The position of m, z3;t.

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Note that in this example the obtained approxima-tion is quite sensitive to the uncertainties (noise) level.

7. Conclusions

Attempting to estimate simultaneously state andparameters in dynamical systems brings a serious prob-

lem (singularity eŒects) in the implementation of the

standard approaches based on ® xed structure observers.

Additionally, the incorporation of uncertainties and

general bounded noises in the modelling equations

introduces some di� culties in the estimation process insuch a way that it is necessary to consider global proper-

ties and one has to accept not a convergent to zero error

but only a bounded one. In this paper these details are

clari® ed and the new algorithm of time-varying (switch-

ing) structure providing a good estimate for state and

parameter vectors within a class of uncertain MIMO

non-linear systems is suggested. The proposed technique

permits us to obtain the idea about the estimation qual-

ity, deriving the expression for the error upper boundand providing the possibility to carry out the estimation

of the state and parameters simultaneously avoiding any

singularities by the implementation of the O"-observa-

bility concept. The suggested algorithm realizes the

observer for the extended system, obtained by the con-

sideration of the parameters as constant states. Due tothe singularities induced by this approach, the switching

gain structure for the estimator is required. It is shown

that this strategy provides (under general enough

assumptions on the class of non-linear systems) a good

upper bound for the estimation error. It turns out to betight’ , that is, in the absence of any unmodelled

dynamics and any external measurement noises, this

bound is equal to zero which leads to the global asymp-

totic stability of the estimation error.

Appendix

Proof of Theorem 1: To simplify the proof presenta-tion, separate it into three basic parts: the analysis of

the suggested estimator over the O" observable subset

L"…O†, the analysis over the O"-non-observable subset·L"…O† and, ® nally, the proof of main result.

Part 1: Estimation over L"…O†. Consider a time in-terval …½2k¡1; ½2k† …k ˆ 1; 2; . . .† where the states wt of

the suggested time-varying structure observer (18) be-

long to L"…O†. In this case the following lemma holds.

Lemma 1:

kzt ¡ ztk µ LF¡1xz

Gt;½2k¡1…36†

and

kct ¡ ck µ LF¡1xc

…"†Gt;½2k¡1…37†

where

Gx;t;½2k¡1:ˆ kVAx¡KxCx

k kV¡1Ax¡KxCx

k

£ …e¡¬x…t¡½2k¡1†L©xkw½2k¡1

¡ w½2k¡1k

‡ kKxkW2 ‡ W1† …38†

and

0 < ¬x :ˆ maxi

f¶x;ig ‡��l¤x

pLHx

…"†kV1¡Ax¡KxCx

k

Proof: Multiplying both sides of (18) by Qt (8) and

using the transformation xt ˆ Fx…t; wt; U l¤x¡1t †, based on

(9), for the suggested time-varying structure observer,

the following canonical form can be obtained


Figure 8. The velocity of m, z4;t.

Figure 9. The parameter k.

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_xt ˆ Axxt ‡ BxHx…t; xt; U l¤xt † ‡ Kx…yt ¡ Cxt†

x½2k¡1ˆ Fx…½2k¡1; w½2k¡1

; U l¤x¡1½2k¡1

†

9=

; …39†

De® ne the estimation error as et :ˆ xt ¡ xt. In view of

(20) and (39), the error dynamics can be written as

_et ˆ A0xet ‡ Bx‰Hx…t; xt ‡ et; U l¤xt † ¡ Hx…t; xt; U l¤x

t †

‡ Kx¢x;2…t; xt; U l¤x¡1t † ¡ ¢x;1…t; xt; U l¤x¡1

t † …40†

with the initial condition e2k¡1 :ˆ x2k¡1 ¡ x2k¡1. Here

A0x is a stable matrix de® ned as A0x :ˆ Ax ¡ KxCx. By

the relation

eA0xt ˆ V¡1A0x

eLx t VA0x

from (40) it follows that

VA0xet ˆ eLx…t¡½2k¡1†VA0x

e½2k¡1

‡…t

½2k¡1

eLx…t¡s† VA0xBx‰Hx…s; xs ‡ es; U l¤x

s †

¡ Hx…s; xs; U l¤xs †Š ds

‡…t

½2k¡1

eLx…t¡s† VA0x‰Kx¢x;2…s; xs; U l¤x¡1

s †

¡ ¢x;1…s; xs; U l¤x¡1s †Š ds

Taking into account kVA0xBxk ˆ

��l¤x

pand Assumption 4

it follows that

kVA0xetk µ kVA0x

e½2k¡1k e¶x;1…t¡½2k¡1†

‡��l¤x

pLHx

…"†kV¡1A0x

k…t

½2k¡1

e¶x;1…t¡s†kVA0xesk ds

‡ kVA0xk

…t

½2k¡1

e¶x;1…t¡s†

£ …kKxk k¢x;2…s; xs; U l¤x¡1s †k

‡ k¢x;1…s; xsU l¤x¡1s †k† ds

The application of Remark 1 to Lemma 2 in Ciccarella

et al. (1993) and Assumption 1 implies

kVA0xetk µ kVA0x

e½2k¡1k e…¶x;1‡

��l¤x

pLHx

…"†kV¡1A0x

k†…t¡½2k¡1†

‡ kVAoxk…kKxkW2 ‡ W1†

In view of Lemma 1 in Ciccarella et al. (1993) there exist

¬x > 0 and ¶x;1 < 0 such that

kVA0xetk µ e¡¬x…t¡½2k¡1†kVA0x

e½2k¡1k

‡ kVA0xk…kKxkW2 ‡ W1† < 1

or, taking into account the transformation (9) and thede® nition (38), if follows that

ketk µ Gx;t;½2k¡1…41†

The use of Assumption 4 in (41) leads to the desired

result

kzt ¡ ztk ˆ kF¡1xz …xt† ¡ F¡1

xz …xt†k µ LF¡1xz

ketk

kct ¡ ck ˆ kF¡1xc …xt† ¡ F¡1

xc …xt†k µ LF¡1xc

…"†ketk

So, the lemma is proved. &

Part 2: Estimation over ·L"…O†. Consider the time

interval ‰½2k; ½2k‡1Š …k ˆ 0; 1; 2; . . .† where the states wt

of the observer (18) belong to ·L"…O†. The next lemma

states the observation error for the state zt, when the

parameter estimates remain ® xed: ct ˆ c½2k.

Lemma 2:

kzt ¡ ztk µ LF¡1±

G±;t;½2k…42†

where

G±;t;½2k:ˆ kVA± ¡K± C±

k kV¡1A± ¡K± C±

k

£ ‰e¡¬± …t¡½2k†LF±kz½2k

¡ z½2kk ‡ kK±kW2 ‡ W1Š

‡

��l¤±

qLH±

j¶±;1j kV¡1A± ¡K± C±

k kc½2k¡ ck …43†

with 0 < ¬± :ˆ maxif¶±;ig ‡��l¤±

qLH±

kV¡1A± ¡K± C±

k.

Proof: The proof is essentially as before. Applying

the transformation ±t :ˆ F±…t; zt; U l¤±¡1

t ; c† to the obser-

ver (18), we obtain the Brunowsky form

_±t ˆ A± ±t ‡ B±H±…t; ±; U l¤±

t ; c½2k† ‡ K±…yt ¡ C± ±t†

±tˆ½2kˆ F±…½2k; z½2k

; U l¤± ¡1½2k

; c½2k†

9>=

>;

…44†

De® ne the estimation error as "t :ˆ ±t ¡ ±t. Considering

S¤0 and (44), the error dynamic equation can be rewritten

as

_"t ˆ A0±"t ‡ B±‰H±…t; ±t ‡ "t; U l¤±t ; c½2k

† ¡ H±…t; ±t; U l¤±t ; c½2k

†Š

‡ K±¯±;2…t; ±t; U l¤± ¡1

t ; c½2k† ¡ ¯±;1…t; ±t; U l¤± ¡1

t ; c½2k†

"tˆ½2kˆ "½2k

:ˆ ±½2k¡ ±½2k

9>>>>>>=

>>>>>>;

…45†

where A0± :ˆ A± ¡ K±C± . Applying to (45) the same

diagonalization process as in Lemma 1, it follows that


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VA± ¡K± C±"t ˆ eL± …t¡½2k†VA± ¡K± C±

"½2k‡

…t

½2k

eL± …t¡s†VA± ¡K± C±B±

£ ‰H±…s; ±s ‡ "s; U l¤±s ; c½2k

†

¡ H±…s; ±s; U l¤± ¡1

s ; c†Š ds

‡…t

½2k

eL± …t¡s†VA± ¡K± C±‰K±¯±;2…s; ±s; U l¤± ¡1

s ; c†

¡ ¯±;1…s; ±s; U l¤± ¡1s ; c†Š ds

Taking into account that kVa± ¡K± C±B±k ˆ

��l¤±

qand in

view of Assumption 4, it follows that

kVA± ¡K± C±"tk µ kVA± ¡K± C±

"½2kk e¶±;1…t¡½2k†

‡…t

½2k

e¶±;1…t¡s†��l¤±

qLH±

kV¡1A± ¡K± C±

k kVA± ¡K± C±"sk ds

‡…t

½2k

e¶±;1…t¡s†��l¤±

qLH±

kc½2k¡ ck ds

‡…t

½2k

e¶±;1…t¡s†kVA± ¡K± C±k…kK±k

£ k¯±;2…s; ±s; U l¤± ¡1s ; c†k

‡ k¯±;1…s; ±s; U l¤±¡1

s ; c†k ds

Applying Assumption 1 to this inequality, we obtain

kVA± ¡K± C±"tk µ kVA± ¡K± C±

"½2kk

£ e…¶±;1‡

��l¤±

pLH±

kV¡1A± ¡K± C±

k†…t¡½2k†

‡

��l¤±

qLH±

j¶±;1j kc½2k¡ ck

‡ kVA± ¡K± C±k…kK±kW2 ‡ W1†

By the analogous manner as in Lemma 1, one can con-

clude that there exist constants ¬± > 0 and ¶±;1 < 0 such

that

kVA± ¡K± C±"tk µ e¡¬± …t¡½2k†kVA± ¡K± C±

"½2kk

‡

��l¤±

qLH±

j¶±;1jkc½2k

¡ ck

‡ kVA± ¡K± C±k…kK±kW2 ‡ W1† < 1

or, in view of the de® nition (43) and Assumption 4

k"tk µ G±;t;½2k

from which the estimate (42) follows. The lemma is

proved. &

Part 3: Main results. De® ne the vector estimation

errors of state and parameters as

ez;t :ˆ kzt ¡ ztk; ec;t :ˆ kct ¡ ctk …46†

Employing de® nitions (46) and (29) and under the

assumption that w0 2 ·¤"…O† (if this is not the case, it

is possible to take ½0 ˆ ½1), based on (36), (37) and(42), for the kth cycle corresponding to the time interval

‰½2…k¡1†; ½2k†, k ˆ 1; 2; . . . the following inequalities are

satis® ed:

if Àt ˆ 1

ez;t µ c1z exp ‰¡¬x…t ¡ ½2k¡1†Škw½2k¡1¡ w½2k¡1

k ‡ c2z

µ c1z exp ‰¡¬x…t ¡ ½2k¡1†Š…ez;½2k¡1‡ ec;½2k¡1

† ‡ c2z

ec;t µ c1c exp ‰¡¬x…t ¡ ½2k¡1†Škw½2k¡1¡ w½2k¡1

k ‡ c2c

µ c1c exp ‰¡¬x…t ¡ ½2k¡1†Š…ez;½2k¡1‡ ec;½2k¡1

† ‡ c2c

9>>>>>>>=

>>>>>>>;

…47†

if Àt ˆ 0

ez;t µ c3z exp ‰¡¬±…t ¡ ½2…k¡1††Šez;½2…k¡1†‡ C4z

‡ c3cec;½2…k¡1†; ec;t µ ec;½2…k¡1†

…48†

Denote the errors (46) at the end of the kth cycle by

e¤;k :ˆ e¤;½2k; ¤ 2 fz; cg

and

ek :ˆ ‰ez;k ec;kŠT …k ¶ 1†

Note that the errors in the beginning of the estimation

process can be represented as

e0 :ˆ ‰ez;0 ec;0ŠT

De® ne also the positive cone <2‡ :ˆ f…¹1; ¹2† : ¹1; ¹2 ¶ 0gwith the following ordering: we say that x1 µ x2 if there

exists an element x3 2 <2‡ such that x1 ‡ x3 ˆ x2. So, ifx1 µ x2…x1; x2 2 <2‡† then kx1k µ kx2k. Since ek 2 <2‡

8k ¶ 0 and applying (47) and (48), it follows that the

errors at the end of the kth cycle satisfy the vector

inequality

ek µ Akek¡1 ‡ k; k ¶ 1 …49†

with


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Ak :ˆ exp ‰¡¬x…½2k ¡ ½2k¡1†Š

c1zc3z exp ‰¡¬±…½2k¡1 ¡ ½2…k¡1††Š c1z…1 ‡ c3c†

c1cc3z exp ‰¡¬±…½2k¡1 ¡ ½2…k¡1††Š c1c…1 ‡ c3c†

2

4

3

5

k :ˆ exp ‰¡¬x…½2k ¡ ½2k¡1†Šc4z

c1z

c1c

2

4

3

5 ‡c2z

c2c

2

4

3

5

The back iteration, applied to (49), leads to

ek µYk

rˆ1

Ar

Á !

e0 ‡Xk

sˆ1

Yk

rˆs‡1

Ar

Á !

s

Yk

rˆk‡1

Ar :ˆ I

that implies

kekk µ ke0kYk

rˆ1

kArk ‡Xk

sˆ1

k skYk

rˆs‡1

kArk …50†

with

kArk ˆ exp ‰¡¬x…½2r ¡ ½2r¡1†Šf2c21zc3c ‡ c2

1z ‡ c21c ‡ c2

1zc23c

‡ 2c21cc3c ‡ c2

1cc23c ‡ …c2

1z ‡ c21c†c2

3z

£ exp ‰¡2¬±…½2r¡1 ¡ ½2…r¡1††Šg1=2

Taking into account the persistency excitation con-

dition (30) if follows that:

(1) if t ! 1, k ! 1. In view of the de® nitions (26),(27) and (25) the right-hand side of (50) can be

estimated as

ke½2kk ˆ kekk µ ke0kak ‡ b

Xk

sˆ1

as¡1

then if a < 1 it is easy to conclude that

lim supk!1

ke½2kk µ b

1 ¡ a…51†

This inequality serves only for the time sub-

sequence corresponding to the ® nal times ½2k of

each cycle k. To prove the same bound for alltimes t 2 ‰0; 1†, de® ne the vector sequence con-

taining as its components the maximum errors

values

Emaxt :ˆ

Emaxz;t

Emaxc;t

" #

:ˆsup½¶t

ez;½

sup½¶t

ec;½

2

64

3

75 …52†

Based on this de® nition, from expressions (47)

and (48), for the kth cycle, it follows that

et :êz;t

ec;t

" #µ Ek :ˆ

Ez;k

Ec;k

" #

:ˆc3zez;½2…k¡1†

‡ c3cec;½2…k¡1†‡ c4z

ec;½2…k¡1†

2

4

3

5

that, for (52) leads to

Emaxt µ Ek 8 2 ‰½2…k¡1†; ½2k¡1†

from which, in view of (49), it follows that

Ek ˆ ek¡1 ˆ e½2…k¡1† ; k > 1

Combining this result with (51) we ® nally derivethat

lim supt!1

ketk µ b=…1 ¡ a†

(2) if t ! 1, k < 1. In this case there exists at time

instant T such that Àt ˆ 1 8t > T and then it isveri® ed that kAkk ! 0 and k kk ! k‰c2z; c2cŠTkas t ! 1, so

lim supt!1

ketk µ …c22z ‡ c2

2c†1=2

The theorem is proved. &

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