Tracking problem on output values under effect of external disturbances

Victor A. Utkin, Svetlana A. Krasnova, Anna G. Akhobadze

Abstract—The paper deals with tracking problem in linearmultidimensional dynamic systems for predeterminedtrajectory of output variables under unmeasurable externaldisturbances. The step-by-step transformation procedure ofinitial linear system to joint controllability and observabilityblock form with respect to output values subject to externaldisturbances was presented. Tracking problem synthesis offeedback and observer was performed with respect to the sametransformed coordinates without inverse transform.

I. INTRODUCTION

A Tracking Problem was considered without externaldisturbances in [1]. Subject to external distributions oneshould solve synthesis problems of invariant with respect tooutput variables simultaneously with observation problemfor at least some components of the state vector.

The presented approach is based on the block concept [2,3] and synthesis methods of systems with discontinuouscontrols in the feedback and observer of state acted insliding mode [1, 4]. Using of discontinuous controlsprovides invariability of closed systems to external andparametrical disturbances and, moreover, decomposition ofprocedure synthesis of high-dimensional systems onindependent elementary subproblems of lower dimensions[4].

This paper is structured as follows. Part 3 provides theconstructive analysis procedure of formulated problemsolvability on the basis of the plant model reduction to jointblock controllability and observability forms with respect tooutput variables subject to external disturbances (BCFO).Part 4 describes BCFO-based decomposition synthesisprocedures of basic algorithm of controller and conditionobserver with respect to the same modified coordinates. Theresults of this paper can be useful in automation of widerange of technological processes, including robotic systems.

II. SYSTEM DEFINITION AND PROBLEM STATEMENT

Consider a linear multivariate system,, 1 DxyQBuAxx =++= h& (1)

where nRXx ⊂∈ is the state, 0

1mRy ∈ is the output,

pRu ∈ is the control, DQBA ,,, are the known constantmatrix with corresponding dimension. Without loss ofgenerality, it’s assumed that nmyD <== 01dimrank ,

npuB <== dimrank , 0mp ≥ . qRt ∈)(h is the vector ofexternal disturbances, which components are assumed to beunknown non-smooth limited functions of time.

V.A. Utkin, S.A. Krasnova, A.G Akhobadze are with Institute of ControlSciences, Russian Academy of Sciences, 65 Profsoyuznaya Str., 117997Moscow, Russia; vicutkin@ipu.rssi.ru

The goal of this paper is a feedback synthesis for system(1) that provides convergence of output variables 1y to

program trajectory 0)( mRtg ∈ with predeterminedconvergence rates on the assumption that only outputvariables are accessible for measurements.

III. PAPER SPECIFICATIONS

The proposed below procedure consists of nonsingulartransformations of system (1) to BCFO. Initial systemstructure properties (observability, controllability/invariancewith respect to output variables) that depend on matrixes

QDBA ,,, and define possibility and architectonics ofdecomposed feedback synthesis of tracing problem for giventrajectories are being opened in the course of procedure.

Step 1.1) Splitting: Regroup components of the state vector

),(col 11 xxx()= , 0

1mRx ∈)

, 0

1mnRx −∈(

so that 0det 11 ≠D is

met in the linear decomposition 1121111 xDxDDxy() +==

what allows one to set up a one-to-one correspondencebetween output variables and some part of the state vectorcoordinates:

=

1

1

1

111 x

y

x

xHx ((

)

a ,

=

− 0

121111

mnIO

DDH , 0det 11 ≠H ,

where I is the singular matrix and O is a zero matrix withcorresponding dimension. The executable nonsingulartransformation allows one to represent system (1) in theform of two subsystems, namely,

.

,

11111111

11111111

h

h

RuSxGyCx

QuBxDyAy(((((

&(

(&

+++=

+++= (2)

Further, the complex analysis of matrix 111 ,, QBD isperformed in the first equation of system (2) (the first BCFOblock).

2) Check of invariance and observability conditions.Invariance condition: If <1rankB )rank( 11 QB , then in thefirst equation of the system (2) conditions of matching arenot met ( 11 ImIm BQ ∉ ) and the procedure is finished (posedproblem has no decision), because a invariance of outputvariables 1y to the external disturbances in the asymptoticsense can not be ensured [5]. Otherwise

)rank(rank 111 QBB = . (3)In case (3), the condition of invariability is performed. Let

11rank pB = , if 01 mp = , then the procedure is finished

because BCFO is obtained on the first step. If 01 mp < , thenone should check next condition.

978-1-4244-2200-5/08/$25.00 ©2008 IEEE 238

Observability conditions: If ),rank(rank 111 QDQ = thenvariables of the second equation of system (2) are notobservable with respect to output variables and theprocedure is finished (posed problem has no decision),because under 11 ImIm QD ∈ it is impossible to organize

virtual vector of output variables on the basis of 11xD(

whichis invariant to external disturbances for the secondsubsystem (2) [1]. Otherwise

)rank(rank 111 QDQ < (4)such a possibility is obtained, the observability conditionunder external disturbances is met and we can begin nextstep if 011 ,0 mpp <≠ .

3) Reduction of the first block to the regular formconcerning the control: By permutation of strings the vectorof output variables is introduced as )~,(col 111 yyy

)= and thefirst equation of system (2) is split as follows:

,~~~~~

,

11111111

11111111

h

h

QuBxDyAy

QuBxDyAy

+++=

+++=(&

))())&)

.~dim~

rank

,dim

111

1101

pyB

mpmy

==

=−=)

Let us eliminate the control vector components from thefirst part of the first equation by nonsingular transformation

11111~yLyy −=

), where += 1111

~BBL

), +

1

~B is a pseudo inverse

matrix of matrix 1

~B :

−+++= h11111111 QuBxDyAy))())

&

=+++− + )~~~~

(~

111111111 hQuBxDyABB()

,111111 hQxDyA ++ (

,~~

11111111 ABBAA +−=))

,~~

11111 DBBDD +−=))

11111

~~QBBQQ +−=

)).

Let us prove that OQ =1 what follows from theinvariability condition (3) of the first equation of the system(2) and from permanency of matrix range withinnonsingular transformations. Genuinely:

=

1

1

11

1

1

111 ~~~~~)(

Q

Q

B

O

Q

Q

B

BQB

))

,

⇒== 1111 rank)(rank pBQB

11

1

11

1

1

1~~rank~~rank pQ

Q

B

O

Q

Q

B

B=

=

⇒

))

,

.0rank~

rank 1111 OQQpB =⇒=⇒=Sequence of these transformations to the regular form is

marked as follows0det),~,(col 1211112 ≠= HyyyH . (5)

In the issue of the transformation (5) the system (2)assumes the following form

,111111 xDyAy(& +=

,~~~~~

11111111 hQuBxDyAy +++= (& (6)

.11111111 hRuSxGyCx(((((&( +++=

4) Nonsingular transform of control and disturbancesvector components: In the second sub-block of system (6)

1p coordinates of the control vector, conformable to the

basis columns of the matrix 1

~B , are fixed as control for 1p

components of vector 1~y and eliminated from the second

block of system (6). At that, corresponding part of theexternal vector disturbances h will be also eliminated fromthe last subsystem because of the (3). Let us performfollowing nonsingular coordinate replacements of controland disturbance vectors:

,

,

1

1

1

114

1

1

1

113

=

=

hhh

hf

H

u

v

u

uHu

(

a

)

a

,~~

,~~

1

1

121114

121113

=

=

−

−

qq

pp

IO

QQH

IO

BBH

(7)

where ),(col 11 uuu)= , ,,, 11

111ppp RuRvu −∈∈)

0~

det 11 ≠B ,

),~~

(~

12111 BBB = )~

(~

11211

111 uBvBu −= −); == 11 dimdim fh(

,~

rank 111 pqQ ≤== ),,(col 11 hhh (= 1

1qqR −∈h ,

,~~

~~~

))1()11(()1)11((

))1(1()11(

1413

1211

1

=

−×−×−

−××

qqqpqqp

qqqqq

QQ

QQQ

).~

(~

,~~~

11211

111121

111314 hh QfQOQQQQ −==− −− (

Let us set

=

−11113

1 ~~ 1

QQ

IZ

q, subject to (7) two last equations

of the system (6) will be represented as follows:

).~

(

,~~~~

11111111111111

111111111

vfZPRuSxGyCx

fZvxDyAy

+++++=

+++=((((((

&(

(&

hLet us eliminate 11,vf from the right part of the last

equation of given system by nonsingular replacement

11211~yLxx −= (

, where 112 PL(

= :

0det,,~~

1512

151

1

1

115

0

1 ≠

−=

=

−

HIL

OIH

x

y

x

yH

mn

p( .

In the issue the last equation of system (6) will berepresented as follows:

1111111111 hRuSxGyCx +++=& (8)

where .dimdim 011 mnxx −== (

5) Check of tracing problem solvability: If in the first

equation of the system (6) 111 dimrank myD =< , then thetracing problem with respect to all output vectorcomponents has no decision and the procedure is finished.

Otherwise ( 11rank mD = ), the tracing problem has a

decision, and the virtual controls 112 xDy = , 1

2mRy ∈ is

assigned to output variables 1y . Let us proceed to thesecond step of the procedure, where the similartransformations are applied to the system (8), where the

vector 112 xDy = is interpreted as virtual output.The first block of BCFO is obtained in the form of two

sub blocks

,~~~~, 1111111121111 fZvxDyAyyyAy +++=+= && (9)

239

when the previous notation of changed matrix 1111

~, AA are

leaved unaltered for simplicity.Let us describe the i -th step of the procedure where

similar transformations are applied to the system

,, 11

1

111111,11 −−

−

=−−−−−−− =+++= ∑ iii

i

jiiiiiijjii xDyRuSxGyCx h& (10)

where 0...dim 2101 ≠−−−−= −− ii mmmnx , −=− pui 1dim

0... 11 ≠−−− −ipp , 0...dim 111 ≠−−−= −− ii qqqh , =iydim

0rankdim 111 ≠=== −−− iii mDy , 1dimdim −< ii xy . It issupposed that mentioned conditions of the finish procedurewere not met on the previous steps.

Step i1) Splitting: Regroup components of state vector so, that

in linear ),(col1 iii xxx()=− , 1−∈ im

i Rx)

, decomposition

iiiii xDxDy()

21 += condition 0det 1 ≠iD is met, what allowsone to set up a one-to-one correspondence between outputvariables and some part of the state vector coordinates:

.,110 ...

21111

=

=

−−−−−−

immmn

iii

i

i

i

iii IO

DDH

x

y

x

xHx ((

)

a

Subject to this replacement the system (10) is representedas follows:

.

,

111

111

−−=

−−=

+++=

+++=

∑

∑

iiiiii

i

jjiji

iiiiii

i

jjiji

uSRxGyCx

uBQxDyAy

(((((&(

(&

h

h (11)

2) Check of invariance and observability conditions.Invariance condition: If )rank(rank iii QBB < , then theprocedure is finished because it is impossible to obtain fullinvariance of variables iy to external disturbances in

asymptotic sense. Genuinely, iy variables are treated as

virtual control for variables 1−iy of previous ( 1−i )-thblock, which again act as virtual control of the part ofcomponents ( 2−i )-th block, and so on till 1y . Thus, part ofdecisions will not be invariant to external disturbances inasymptotic sense. Otherwise )rank(rank iii QBB = and

invariance condition is met. Let us set ii pB =rank . If

1−< ii mp , then we can check next condition.

Observability conditions: If ),rank(rank iii QDQ = thenvariables of the second equation of system (11) are notobservable with respect to output variables and theprocedure is finished (posed problem has no decision).Otherwise ( )rank(rank i ii QDQ < ), so the observabilitycondition is met and we can begin next step if

1,0 −<≠ iii mpp .

3) Reduction of the i -th block to the regular formconcerning the control: Perform nonsingular transformationsimilar to 12H

,0det),~,(col 22 ≠= iiiii HyyyHallows one to represent system (11) as follows:

,

,~~~~~

,

111

111

1

−−=

−−=

=

+++=

+++=

+=

∑

∑

∑

iiiiii

i

jjiji

iiiiii

i

jjiji

ii

i

jjiji

RuSxGyCx

QuBxDyAy

xDyAy

h

h

(((((&(

(&

(&

(12)

where .dim,~

rank~dim 1 iiiiiii mpmypBy =−=== −

4) Nonsingular replacement of control and disturbancesvector components: Let us perform following nonsingularcoordinate replacements of controllability and disturbancesvectors:

,

,

41

31

=

=

−

−

i

i

i

iii

i

i

i

iii

fH

u

v

u

uHu

hhh

h(

a

)

a

,0det,~~

,0det,~~

4...

214

321

3

1

≠

=

≠

=

−−−

−

iqqq

iii

ipp

iii

HIO

QQH

HIO

BBH

i

i

where ),,(col1 iii uuu)=− ,...21 ipppp

i Ru −−−∈ ipii Rvu ∈,

),

)~~

(~

21 iii BBB = , 0~

det 1 ≠iB , )~

(~

21

1 iiiii uBvBu −= −);

),,(col1 iii hhh (=− iqqq

i R −−−∈ ...1h , == ii fdimdimh(

iii pqQ ≤==~

rank , ),~

(~

21

1 iiiii QfQ hh −= −(

,~~

~~~

))...21()(())((

))...21(()(

43

21

=

−−−−×−×−

−−−−××

iqqqqiqipiqiqip

iqqqqiqiqiq

ii

ii

iQQ

QQQ .

~~~2

1134 OQQQQ iiii =− −

Let us set

=

−113

~~

ii

q

iQQ

IZ i , then two last equations of

system (12) will be represented as follows:

).~

(

,~~~~

1

1

iiiiiiiiii

i

jjiji

iiiii

i

jjiji

fZvPRuSxGyCx

fZvxDyAy

+++++=

+++=

∑

∑

=

=

((((((&(

(&

h

Let us eliminate ii vf , from the right part of the lastequation of given system by nonsingular transform

iiii yLxx ~2−= (

, where ii PL(

=2 :

.0det,,~~

52

55

1...10

≠

−=

=

−−−−

ii

p

ii

i

i

ii H

IL

OIH

x

y

x

yH

immmn

i

(

In the issue the last equation of system (12) will berepresented as follows:

iiiiii

i

jjiji RuSxGyCx h+++= ∑

=1

& , (13)

where 110 ...dimdim −−−−−== iii mmmnxx(

.5) Check of tracing problem solvability: If in the first

equation of system (12) iii myD =< dimrank , then thetracing problem with respect to all output vectorcomponents has no decision and the procedure is finished.

Otherwise ( ii mD =rank ), the tracing problem has decision,

and iy variables assigned to the virtual control iii xDy =+1 ,

240

imi Ry ∈+1 . With additional condition 0dim ≠iu we go to

the next ( )1+i -th step of the procedure, which includessimilar transformations applied to system (13), where thevector 1+iy is treated as virtual exit and so on.

The i -th block of BCFO is obtained in the form of twosub-blocks

.~~~~,

11iiiii

i

jjijiii

i

jjiji fZvxDyAyxDyAy +++=+= ∑∑

==

&& (14)

The described procedure is finished by the finite numberof steps, because in the issue of each step the finitedimension of unreformed state vector ),(col1 iii xyx a− is

reduced by the value equal to 0dim 1 ≠= −ii my .

Step m (the last)

At the m -th step the following system is considered:

,

,

11

111111

1

1,11

−−

−−−−−−

−

=−−

=

+++= ∑

mmm

mmmmmm

m

mm h

xDy

RuSxGyCxj

jj&(15)

where 0...dim 2101 ≠−−−−= −− mm mmmnx , −=− pu 1dim m

0... 11 ≠−−− −mpp , 0rankdim 11 ≠== −− mmm mDy .

The fulfillment of the condition 1dimdim −= mm xy in

system (15) means, that all space of state nRXx ⊂∈ will

be one-to-one reflected to the space nRYy ⊂∈ , where

),...,(col 1 myyy = , 1dim −= ii my , m,1=i and nmi

i =∑−

=

1

0

m.

From system (15) by shortcut transformation 11 −= mmm xHy ,

11 −= mm DH the following equation is obtained:

.111

−−=

++= ∑ ij

jj QuByAy hmmm

m

mm& (16)

This equation is the last block of full BCFO, if

mm yB dimrank = (complex check of the fulfillment of the

invariance and solvability conditions of tracing problem).Otherwise, posed problem has no decision.

If at the m -th step conditions 1dimdim −< mm xy ,

mm yB dimrank = are met, then system (1) after

transformations 1mH , 3mH – 5mH will be transformed into

non-full BCFO in the following manner

,11

+=

+= ∑ i

i

jjiji yyAy& (17)

;1,1,~~~~

1

−=+++= ∑=

mm

mifZvxDyAy iiii

jjiji

&

,1

mmmm

m

mm ZfvxDyAyj

jj +++= ∑=

,1

mmmm

m

mm hRSuxGyAxj

jj +++= ∑=

&

where )~,(col iii yyy = , ,dimdim 1 iii myy == + ∑−

=+=

1

0

m

iimn

mxdim+ , ∑=

=+m

m1

dimi

i pup , iii pvy == dim~dim .

In general, any sub-blocks with respect to variables iy~ on

ground of 0=ip can be absent in system (17). With the aimof the system work capacity, unreformed variables of thelast block mx should meet requirements of decision

limitation. The block form (17) is called BCFO due todouble function that is performed by vectors 1+iy in the firstsub-blocks:

– on the one hand, within the observation problem solvingthey are to be estimated at the i -th step and interpreted as avirtual output with full dimension for the )1( +i -th block;

– on the other hand, within the tracing problem solvingthey are interpreted as virtual control with full dimension,purposefully chosen for support of required dynamic of iy

variables. Then, at the )1( +i -th step (according to the blockapproach [2, 3]) the problem of disparity stabilizationbetween real and needed virtual control is being solved.

Thus, synthesis of the both problems is performed on thebasis of the BCFO (17) with respect to the same transformedvariables and shared for two consequently solvingelementary subproblems of lower dimensions than the initialsystem.

The required dynamic of the vector iy~ components is

performed directly by choosing of true controls of iv . Thefact that at the BCFO (17) the coordination conditions aremet (external disturbances if belong to control space iv ),allows one to use to the full extent known approaches ofcompensation and noise cleaning.

In order to point relations between blocks, let us performadditional reduction of the state vector of system (17)

),,...,(col 1,1, mm iiiii yyyy −+= , jij py =dim , 2,1 −= mi ,

where by 0=jp some components can be absent. This

reduction allows one to represent system (17) in thefollowing manner

,,

,...,,

;...;~~~~

,~;~~~

1

1,111

2

1,2,2

322,211,22211,11

22221

2

211,12121111

11

m

m

mmm

m

mmmm

mmmmmmm

m

m

m

yyAyyyAy

yyAyAyyyAy

fZvxDyAy

yyAyfZvyAy

jjj

jjj

jjij

jjj

+=+=

++=+=

+++=

+=++=

∑∑

∑

∑

−

=−−−

−

=−−

=

=

&&

&&

&

&&

(18)

∑=

+++=m

mmmmmmm1

.~~~

jjj fZvxDyAy&

It is seen, equations of system (18) with respect to eachcomponent group of the output vector

),...,,,~(col 1131211 myyyyy = are served as first blocks of

block-controlled and simultaneously block-observedsubsystems. Dimension of state variables of subsystems of

system (18) ip , m,1=i , can be interpreted as joint

241

controllability and observability indexes of system (1) withrespect to output variables subject to external disturbances.

Consequently, within the presented procedure thesubsystems of fewer dimensions than the initial system arebeing investigated. Solvability analysis of the posedproblem seems to be natural and transparent enough. Theresults formulation is significantly simplified in comparisonwith formulation in the initial system terms (1).

The presented procedure constructability consists incomplex investigation of invariability and observabilityconditions and tracing problem solvability. At that, theobtained form (18) is revealed as basic for thedecomposition synthesis of observability and tracingproblems with respect to the same transformed coordinates.

IV. DECOMPOSITION SYNTHESIS

Let us assume that as a result of the presented procedureof analysis for the initial system (1), the full BCFO (17) wasobtained, which is used for the basis of decompositionsynthesis of feedback that includes solving of observabilityproblem and, properly, synthesis of control. Without loss ofgenerality let us consider block-controlled and block-

observable u -subsystem ( mu ,2= ) of system (18), which

subject to simplified symbols ( uii yy =: , 1,1 −= ui ,

uu yy ~:= ; 1: −∈=∗ imii Ryy , m,1=i , uvv =: and i.e.) can be

represented in the following manner

∑∑=

∗+

=

∗ ++=−=+=m

uuu1

11

,;1,1,j

jji

i

jjiji fvyAyiyyAy && (19)

where upvyi == dimdim , vector uumu fZxDf~~

+= is

treated as external limited disturbances. Feedback synthesisprocedures on the basis of system (19) that were developedin this part are similar for the rest subsystems (18).

A. Basic control algorithmThe tracing problem for the program trajectories

upRtg ∈)(1 ( ( ) ugggggH == :,~,col 112 , ),...,(col 2 mggg = )

with respect to 1y coordinates of system (18) is posed. Onthe assumption that upsetting control are described bysmooth functions of time, it is required to provideasymptotic convergence of output variables to giventrajectories with prescribed rate of convergence:

)()(lim 11 tgtyt

=∞→

. Given problem is reduced to the system

stabilization with respect to mismatches:

,0111 →−= gye upRe ∈1 . (20)This problem is solved procedurally on the basis of block

approach [2, 3] and reduced to consequent solving of uelementary problems with dimension up .

Step1: Let us construct differential equation with respectto the mismatches (20):

12111111 gyyAgye &&&& −+=−= ∗ . (21)

It system (21) vector 2y is supposed as virtual control,

which is chosen as follows of −−= )( 112 eUy 1111 gyA &+∗

where ))(),...,((col)( 11111111 uu pp eUeUeU = is a stabilize

vector-function that allows one to compensate crosscoupling and obtain asymptotic convergence to giventrajectories, i.e. decision of the system )( 11 eUe −=& satisfies(20) with prescribed rates of convergence.

In the solving problem, the transformed state variables areconsidered as a virtual control, at choosing of view ofstabilize function )( 11 ii eU , it is required to take into accountlimitations that are imposed in system (1) on state variablessubject to realized transformations. Basic requirements aresmoothness and limitation of state variables signals.

According to the block approach, the stabilizationproblem of the mismatches between actual and chosenvirtual controls is solved at the second step:

0)( 11111122 →−++= ∗ gyAeUye & . (22)In view of the (22) the closed subsystem (21) is presented

in the form of 2111 )( eeUe +−=& .Step2: Let us construct differential equation with respect

to the mismatches (22):

),)(( 2111

1132221212 eeU

eU

gyyAyAe +−∂∂

+−++= ∗∗∗∗ &&& (23)

where vector 3y is supposed to be a virtual control, which ischosen in the following manner

))(()( 2111

11222121223 eeU

eU

gyAyAeUy +−∂∂

−+−−−= ∗∗∗∗ && .

The closed system (23) became )( 22 eUe −=& .According to the block approach, the stabilization

problem of the mismatches between actual and chosenvirtual controls is solved at the third step:

0),( 122212121233 →−+++= ∗∗∗∗∗ gyAyAeeUye && , (24)

where upRe ∈3 , )(),( 2211

1212 eUe

e

UeeU +

∂∂

=∗ & . Subject to the

(24) the closed subsystem (23) is presented in the followingmanner 3222 )( eeUe +−=& , and so on.

In the issue of the mentioned procedure at the last u -th stepequations of the closed system (19) will be represented inthe following manner

,(.);1,1,)( 1 nju u +=−=+−= + eieeUe iiii && (25)

where own motions of variables )( iii eUe −=& areasymptotically stable.

In the u -th subsystem (25) all previous stepsconstructions are provided by actual control withcompensate component in the form of:

.)( juu −−= eUv (26)That leads the last equation (25) to the following manner

)( uuu eUe −=& , 0lim =∞→ ue

t. (27)

Behavior of variables of the closed diagonalized system(25), (27) conforms to the following logic chain:

242

,00

...00

1112

1

gyee

ee

→⇒→⇒→⇒

⇒⇒→⇒→ −uu (28)

that solves the posed tracing problem.One should notice that if it is possible to form actual

control in the control system in the form of discontinuousfunction

ueMv sign−= (29)

where 0const >=M , then if the condition (.)j>M is

met, the sliding mode [4] will occur by finite time in the lastsubsystem (25) on variety 0=ue , after whichcorrespondences of the (28) will be obtained in sequence.As it is seen, only information about current values ofvariable ue and additionally current values of (.)j arerequired for realization of the basic control algorithm (29).

B. Solving of observability problemIn this part, the problem of information support for basic

control algorithms (26), (29) is solved by state observer onsliding modes [1]. The fact that the considered subsystem(19) serves simultaneously as block-observable and block-controlled in the aggregate with using of the block approachin the feedback forming allows one to solve theobservability problem with respect to variables of the closedsubsystem (25). Let us construct the state observer onsliding modes for system (25) in the following manner

,;1,1,)( 1 uuu wvziwzzUz iiiii +=−=++−= + && (30)

where õp

i Rz ∈ is the state vector, õp

i Rw ∈ is a correctiveobserver actions, that are sequentially constructed in theform of discrete functions in order to solve system

stabilization problem with respect to iii ze −=e , ue pi R∈

which owing to (25), (30) is represented in the form of:

,(.);1,1,)( 1 uu jeueee wiwU iiiii −=−=−+∆−= + && (31)where the Lipshitz's conditions are met for the vector-function )()()( iiiiiii zUå+zU=åU −∆ , namely

iiii LU ee ≤∆ )( , 1,1 −= ui . (32)

Let us formulate decomposition synthesis procedure ofobserver corrective actions (30) which splits into twoindependent solvable elementary subtasks.

At the first step let us construct discontinuous correctiveactions 111 signeMw = in the first observer block (30) on

measured variables 111 )( zgy=å −− , where 01 >M isamplitude of discontinuous correction, which value ischosen on the basis of sufficient conditions for slidingmodes existing, namely

.0)(

)sign)((0

2112111

11211111

eeee

eeeeee

>⇒<−+−≤

≤−+∆−⇒<

MML

MUT

TT & (33)

If the condition (33) is met, then the sliding mode isperformed on variety 1111 }0{ ezS =⇒== e by the finite

time 01 >t . At 1tt > from the static equations we can get

estimations 2eq1eq121 0 eee =⇒=−= ww& , which values are

obtained from first-order liner filters with small timeconstant:

.0,,lim, 11eq110

eq11111

>∈=+−=→

kRwwk p

k

utttt& (34)

The obtained current estimations (34) are used at thesecond step for constructing of discontinuous correction inthe second observer block (30) 122 signtMw = 22signeM= ,and so on [1]. In the issue, feed to the last of observer block(30) of discontinuous corrective actions == 1-sign uuu tMw

0const,sign >== uuu e MM , where −⇒< (.)(0 jeee uuuTT &

)sign uu eM− ,0)( jje uuu >⇒<−≤ MMT will lead to

initiation of sliding mode by the theoretically finite time

1−> uu tt on variety uuuuu e ezSS =⇒== − }0{ 1 I , whatsolves the stabilization problem and consequentlyobservability problem. From the static equation under utt >we have estimations jje uuu =⇒=−= eqeq 0 ww& , which

values one can obtain from outputs of linear filters similar to(34). Subject to these constructions, the basic controlalgorithms are realized by observer estimations similar to(26), (29).

V. CONCLUSIONS

The tracing problem with respect to output variables inlinear multidimensional dynamic systems under externaldisturbances is solved under the assumption that only outputvariables can be estimated and distribution vectorcomponents are interpreted as uncontrolled limited functionsof time. The procedure of reducing of general lineardynamic systems to the controllability and observabilityblock form with respect to output variables subject toexternal disturbances is presented. The procedure allows oneto reveal joint observability and controllability properties ofthe initial system with respect to output variables andconstructively formalize the solvability conditions of tracingproblem. On the basis of the given form, decompositionfeedback synthesis procedures were presented, includingdecision of problem observability and controllabilityproblem with respect to the same transformed blocks ofvariables.

REFERENCES

[1] S. A. Krasnova, V. A. Utkin, “Cascade synthesis of dynamicsystems state observer”, M.: Sciense, 2006 (in Russian).

[2] Drakunov, S at al., “Block control principle I” // Automationand Remote Control, 1990, Vol. 51, No. 5, pp. 601–609.

[3] Utkin Victor A., “Invariaance and autonomy in systems withseparable motion” // Automation and Remote Control, 2001,Vol. 62, No. 11, pp. 1825-1843.

[4] Utkin V., “Sliding modes in Control and Optimization”,Berlin: Springer-Verlag, 1992.

[5] B. Drazenovic, “The invariance conditions in variablestructure systems” // Automatica. 1969. V.5. N.3. pp. 287–295.

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