Uniting of global and local controllers under acting disturbances

Copyright © IFAC Nonlinear Control Systems, Stuttgart, Germany, 2004

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UNITING OF GLOBAL AND LOCAL CONTROLLERS UNDER ACTING DISTURBANCES EfimovD.V. Control ofComplex Systems Laboratory Institute ofProblem ofMechanical Engineering, Bolshoi av., 61, v.o.. St-Petersburg, 199/78 Russia [email protected] Abstract: A switching control is presented, which unites global stabilising control law and local optimal one. In the presence of disturbances this switching control ensures boundedness of all trajectories of the system and additionally provides output attractiveness property, if disturbance is absent. Copyright © 2004 IFAC Keywords: uniting control, hybrid control, switching, disturbances. bounded external disturbances. In Section 2 all definitions and notations can be found, in Section 3 main result formulation and proofs are summarized. Conclusion in Section 4 finishes the paper.

I. INTRODUCTION The problem of uniting of global and local controllers is rather not new. Frequently in practical applications a situation is appeared, then for some plant it is possible to design a global stabilizing control law, which ensures global boundedness of the system solution in the presence of disturbances without desired performance, and local control law, that guarantees to solution optimality in some sense for absent disturbances. It is necessary to design unite controller, which inherits properties of both controllers. Really, the most of well known techniques, such as backstepping, forwarding, feedback linearization and passivation (Fomin, et aI., 1981; Isdori, 2000; Krstic, et aI., 1995; Sepulchre, et aI., 1997), which solve task of robust system stabilization, usually do not address the problem of system performance even in some small neighborhood of the origin. In these approaches it is supposed, that the main goal consists in disturbance rejecting or compensating. But in real applications the quality of transient processes is very important and a compromise should be found. Uniting controller can be considered as a such compromise, if it equals to the optimal controller near the origin and provide boundedness of system solution, then disturbances presence is detected.

2. DEFINITIONS AND STATEMENTS Let the model of dynamical system under consideration can be presented as follows: = r(x,u,d), = h(x), (I)

x

x E Rn is state space vector of the system;

where yE R

y

m

is output vector; function

schitz continuous,

r( 0, 0, 0 ) =

tinuous and h( 0)= 0;

U

r

is locally Lip-

0 ; function h is con-

E RP and dE Rq are con-

trol and disturbance inputs respectively, it is supposed that they are Lebesgue measurable and essentially bounded functions of time:

1 U III 0,1) = esssup nu( .)I}< +00, TE[ 0,1)

lid 11[0,1)

= esssup{ld( ')I}< +00, TE[O,I)

classes of such functions for u and d we denote as

Mp and M q respectively (here 1·1 denotes usual Euclidean norm, "u lilo, +(0)

The problem of uniting control was solved in (Teel, and Kapoor, 1997), where a dynamic time invariant controller was proposed, which converges to local optimal controller near the origin only under some special conditions. In paper (Morin, et aI., 1998) a static time invariant controller was presented under hard verified condition of existing continuous path between global and local controllers. In works (Prieur, 200 I; Prieur, and Praly, 1999) an example was found, which does not allow any continuous or even discontinuous time invariant controllers. Additionally, in these works several solutions of uniting control problem were proposed (continuous, discontinuous, hybrid and time-varying) for case of external disturbances absence. A kind of uniting controls for chained nonholonomic systems was developed in (Prieur, and Astolfi, 2002), where robust properties of such controller with respect to some sufficiently small disturbances were analyzed. Here we present a hybrid uniting control for systems with not measured

= I1 U 11)·

Thus, for any

initial conditions Xo E Rn and any u E Mp and dE M q , solution of (1) x(t,xo,u,d) is well defined at the least locally on some time interval [0, T) (further we will omit dependence of Xo and u,d ifit is clear from the context and will simply write x( t) or for systems without controls x( t, xo, d ); denote output solution as y(t,xo,u,d) or for system without T

controls

= +00

y( t, xo, d )

correspondingly).

If

for all initial conditions and any U E Mp

and d E :Mq , then system (1) is called forward complete. As usually, it is said, that function p: R~o ~ R~o belongs to ctass K, if It-!S strictly incr~~ing and

405

p( 0) = 0; P E 1(." if p E K and p( s) ~ co for s ~co

i=fg(x,d)=flx,Ug(x1d), y=h(x),

(radially unbounded). Function ~: R~o x R~ ~ R?o

possesses the following properties for any to

is from class 1(£, if it belongs to class K for the first argument (for any fixed second) and strictly decreases to zero for the second (for any fixed first ar-

initial conditions

gument). The distance from given point ~ E Rn to any non empty subset .J:\. from Rn is denoted as ~

Iqj\ =dist[ ~,.J:\.] = 'lE0\ inf [~, Tt], Ug

or equivalently

control-

VE>OVK>O 3T=T(E,K,d)~to : Ix{to)I~K,

I

sup x( t, x( to 1d u/ : Rn ~

admits the following properties: A. Forward completeness for any d

E

system solution (property A of Assumption 2) and attractiveness (property B of Assumption 2) of some neighborhood of the origin

(2)

:Mq , i.e. there

X g ={xER

+x{~x4(1d( ')1 j
(ISS) property with respect to compact set X g , if it

~

is possible to reformulate Assumption 2 as GS and AG properties for set X g :

pEK;

Ix{ t,x{ to 1d )Ix ~ {I x( to )Ix

and locally output asymptotically stable, i.e. for all

to~O

and X(tO)EX p ,

there exists some function

g

Xp={xERn:lxl~p(K)}

lim I x{t,x{t01d )Ix

I~
13 E 1(£, such, that

Necessary and sufficient conditions for forward completeness property can be found in (Angeli, and Sontag, 1999). Local stability property ensures, that solutions of system (2) x( t, x( to 10) starting with

+

0-(11 d 11),

~ y(lld 11)·

but GS property does not follow from property A in Assumption 2. The main difference consists in dependence from initial conditions, while in Assumption 2 a boundedness by state space vector is supposed, GS property needs boundedness with respect to distance to the set. Note also, that if AG property with respect to set X g introduced in Assumption 2

x( to) at set X/ ~ X p stay in set X p for all t ~ to. Let us stress, that Assumption I says nothing about optimality of system (2) solution for case of disturbance d absence. In property B of Assumption 2 function 13 reflects character of output of system (2)

holds uniformly with respect to initial distances to the set and input disturbances: V&>OVK>O 3T=T{&,K)~tO: Ix{to)l x ~K,

decreasing to zero with initial conditions lying in set X p ' It is supposed, that such decreasing is optimal in some sense, but for our proofs such supposition is not important one and it is not introduced explicitly. For example, control u/ can be obtained as a solution of optimal control task for linearized model of system (1) with d = 0 (Fradkov, eta!., 1999). ~

g

) g

In paper (Sontag, and Wang, 1996) equivalence of properties GS and AG to ISS property with respect to non empty closed compact set was proven. It is easy to see, that property B from Assumption 2 exactly coincides with AG property with respect to set X g ,

1y( t,x( t010 )1 ~ 13(1 x( to )I,t - to) for all t ~ to· •

Ass u m p t ion 2. There exist a constant k

:Ixl~k}

Rem ark I. From another point of view, control U g provides for system (3) input-to-state stability

B. If d( t) == 0 for all t ~ 0, then solution of system (2) is locally stable:

~ Ix(t,x(to),o)l~p(lx(to)l),

n

uniformly with respect to initial conditions. Term "practical" used above in Assumption 2 formulation means that both properties A and B hold with some static error k .

)+ X2(I x( to ) I )+

x( to) E X/, X/ = {x E Rn : 1x 1~ K }

•

Thus, global control u g provides boundedness of

exist some functions Xl' X2, X3, X4 E Kx, and positive constant X, such, that (Angeli, and Sontag, 1999)

1x( t, x( to 1 d ) I~ XI ( t - to

)1 ~ y(ll d 11)+ k + E, d E :Mq .

I?T

RP, such,

i=f/(x,d)=f(x,u/(x1d), y=h(x),

:Mq :

Ix{t,x{t01d)l~y(lldll)+k,

lim I~+
I. There exist a constant K > 0

and a continuous local control that system

E

1

lers for system (I). Ass u m p t ion

and d

0,

A. Practical global stability (GS), i.e. there exists some function 0' E K , such, that I x( t, x( to d )1 ~ 0'(1 x( to)1 )+ 0'(11 d 11 )+ k . B. Practical asymptotic gain (AG) with some function yE Kx,:

in this way 11;\0 = 11;\ . Now we are ready to introduce main properties of local u / and global

x( to) E Rn

(3) ~

g

~y(lldll)+E, Vde:Mq ,

sUPlx{t,x{t01d)lx I?T

g

then such property is called uniform asymptotic gain and it is equivalent to ISS property with respect to set X g (Sontag, and Wang, 1996).

0

and a continuous global control u g : Rn ~ RP ,

Above discussion deals with necessary link from Assumption 2 to ISS property with respect to some compact set, but it is clear that ; :'ftc;ent relation

such, that system

406

hoJ.::-; ..without cny additional requirements. In ~ther words, if some control easures for system (I) ISS property with respect to set X g , then it can be con-

V£>OVK>O 3T=T(£,K,0)~/0 :

Ix( to)1 ~ K, sup Ix( t, x( 10 ~ 0 )1 ~ k + £ . t~T

If k < K , then there exists some £ > 0, such, that k+£=K and VK>O 31) =Tl(K,K,O)~/O :

sidered as a global control Ug . Necessary and sufficient conditions for ISS Control Lyapunov function existence with respect to the origin for affine in controls nonlinear systems were presented in paper (Liberzon, et aI., 2001), where also continuous control law provided such property was developed. Sufficient conditions for input-to-output stabilization and controls were obtained in (Efimov, 2002), that result also can be used to calculate control u g . •

Ix(to)I~K, suplx(/,x(to~O)I~K. Property (4) means that if disturbance all

n

to, then there exists a finite time instant

can be utilized for indirect disturbance d presence detecting. Of course it is possible a situation, then trajectory reaches for set XI, but disturbance d is

and dE:Mq :

acting on the system and switching to control ul causes instability of system solution. According to Assumption I control ul does not guarantee any boundedness of system solution for system (I) with disturbance, only forward completeness can be obtained. Such critical situation will be analyzed later, during proof of our main result. Before it we should make some remarks about solutions existence in switching systems with disturbances.

The first two goals fix requirements to stability properties of closed loop system, while the last goal ensures optimality property for system solution near the origin.

3. MAIN RESULTS

If we can directly measure external disturbance d or simply detect its presence in the system, the following simple switching or so-called supervision algorithm solves proposed uniting task for k < K: if disturbance presence is detected or x( t) \i!: X I' then u(x)=ug(x), and U(X)=U/(X) otherwise. Indeed control Ug is connected to system (I) always, except to case of disturbances absence and then system trajectory enters a small neighborhood of the origin. In such situation property A of Assumption 2 provides GS property for the system and property B with k < K ensures, that there exists a finite time instant, such, that solution reaches for set XI if disturbance

Frequently, in switching systems under acting of disturbances a strange behaviour can arise, which is called chattering regime. Such chattering regime originates from fast switching that can take place in the system due to disturbances presence. Classical definitions of differential equations or differential inclusions solution do not suit well for dynamical system in chattering regime, hence, some special methods are used in logic-based switching control theory to prevent such regime arising. In this work we will firstly use so-called dwell time technique (Morse, 1995), which is traditionally used only in linear switching systems due to finite time escape phenomena in nonlinear systems. So, supervision algorithm can be described as follows: t j ) if t < t D ;

i(

i(t)=

is absent. Local control ul is linked with plant only

t ~ 0,

for

switching to control ul is desirable. Such property

- practical GS; - output global asymptotic stability (oGAS) for case d( t) == 0, I ~ 0 ; - there exists a neighborhood 3 of the origin, such, that uniting controller equals to control ul on this set.

on set XI with d( t) == 0,

I ~

d( t) == 0

Tt ~ to ~ 0 uniformly with respect to initial conditions on compact sets, such, that trajectory of system (3) is entering into set XI and stays there. Further

The basic desired properties of uniting controller, that should be synthesized for system (I) starting from Asswnptions 1 and 2, can be formulated as follows for any 10 ~O, x(to)ER

(4)

t~T

when it provides

1g,/'

if X(t)E XI;

ifx(/)\i!:Xp;ift~tD;

i( t j}

convergence of output y to zero. Combining this oLAS property with GS property and global attractiveness of set XI received by U g control, we ob-

1: = I,

(5)

otherwise;

t( t J = 0,

tain oGAS for case d( t) == 0, t ~ 0 . It is clear, that in

where auxiliary variable t represents internal supervisor timer dynamics, t D > 0 is dwell time constant

this situation 3 = XI .

and t j' j = 0, I, 2,... are moments of switching (moments then signal

But due to problem formulation, disturbance d is not measured, and such algorithm needs a modification, that deals with task of indirect disturbances detecting. Control Ug , due to property B of Assumption 2, for

i( t)

changes its value), j is

number of the last switching. The operating of algorithm (5) can be explaining in the following way: after each switching internal timer t is initialised to zero, while t < t D signal i( t) can not change its value. Dwell time presence in algorithm (5) help us to prevent fast switching arising in the system. Then

case of disturbance d absence provides attractiveness of set X g :

407

dwell time i~over, signd i( t) can be set up t'.l I, if a trajectory reaches for set X,; signal i( t) would be set up to g, if variable x leaves set X p ' From As-

Proof. If t E .0.\ then initial conditions always belong to set X, and according to Assumption I system is forward complete. By defmition of switching algorithm (5), t E .0.\ while solution stays in set X p '

sumption I solutions of system (2) stay in set X p

d( t ) == 0,

starting in X, for

If a trajectory leaves set X p ' then after time interval with maximum length t D supervisor (5) switches on global controller. Hence proposed upper estimate should hold for the system in this case.

t ~ to, thus, if control

u, can not provide forward invariant property of set X p ' it means that disturbances are acting on the system and control u g have to be switched on. There are two situations with

x( t ) EN,

where N = X p

i( t) = I

or

i( t) = g

Claim 2.Let tEn 2 .n 2 ={tE[to,T):i(t)=g}.

and

IX, , then switching is

Then all trajectories are bounded

1x( t )Ix ~ cr( R( x( to 1d))+ cr{11 d 11). •

not needed and signal i( t) saves its current value. Indeed, situation with i( t) = I and x( t ) E X p is ad-

Proof. According to property A of Assumption 2 solution in this case is bounded in the following manner:

missible for local controller for case of disturbance absence. Then i( t) = g and x( t ) E N we should wait to switch on local controller until trajectory crosses set N due to workability property of local controller was claimed in Assumption I only for trajectories with initial conditions belonged to set X,. The main properties of such uniting controller are substantiated in the following theorem.

Ix( t ) I ~ 0{ Ix( t j ) I)+ cr( 1 d 1 )+ k , where t j' j = 0, I, 2,... are instants of time, then global controller was plugged to the plant (I). For j > 0, x( t j) by continuity property admits estimate from Claim I, thus the last estimate can be rewritten as follows:

Ix( t )15 cr( R( x( to 1d))+ cr(11 d11)+ k ,

The 0 rem I. Let Assumptions I and 2 be troe and k < K . Then system x=f(x,u;(x1d), y=b(x), iE{!,g} (6) with supervision algorithm (5) possesses the following properties for all

x( to) E Rn

from that claimed inequality can be obtained. It is obvious, that [to, T) = .0.\ u .0. 2 , but solution of

switching system is bounded for all nz, z = 1,2 , thus, T = + ex> and the following inequality is satisfied for all t ~ 0 :

and to ~ 0:

I. Global stability with respect to set X g for any dE :Mq :

Ix(t )Ix ~cr(R(x(t01d))+cr{lldll)·

•

Ix(t )Ix ~cr(R(x(to),d))+cr(lldll).

Indeed, if estimate from Claim 2 holds, then estimate from Claim I is also satisfied, due to property

•

{ XlcD)+X2(P(K))+

R( (1d)x to

+X3(tDx4(ll d ll))+x,

-max

}

cr(s)~s.

x(to) .

2. Global output attractiveness for d( t ) == 0, t ~ to :

"iE>O ::3Ty =Ty (x(to)): ly(t)I~Eforallt~Ty"

Proof. Firstly, from definition of supervision algorithm (5), on any time interval [Ts , Te ) with Ts

~

to

~

0, Te > Ts

'

such, that d( t) == 0 for all t ~ Td . Then two situations are possible. First i( Td ) = g , then according to property (4) there exists a switching time instant Ti,

a finite number of switches

N[ T, ,T,) is possible and the following upper estimate

such, that x( T,) E X, and signal i is switched to position I. Then basing on property B from Assumption lone can conclude, that system possesses output attractiveness property. Second i( Td ) = I .

holds: N[T 3"

Te -Ts T)~I+--. e 'CD

Between switches uniting controller is continuous and equals to local u, or global u g controllers,

Then either

x( Td

)E

X p and trajectory leaves set

X p ' in such case there exists a switching time in-

which are continuous functions for all x E Rn . Thus resulting control is piecewise continuous function of time and solution of system (5), (6) is continuous and defmed at the least locally [to,T). Secondly, let us investigate stability properties of local and global controllers separately.

stant T" such, that x( Ti ) E X, and signal i is switched to position I; or trajectory stays in set X p for all t ~ Td and according to property B of Assumption I local attractiveness of output y is obtained. All statements of the theorem are proved. _

Claim 1. Let tEn) • .0. 1 ={tE[to,T):i(t)=/}.

Proposed in Theorem I solution does not handle third property listed in control goals in Section I. It is hard to specify some neighborhood of the origin, where uniting controller equals to local one. It is a ccnse-

Then all trajectories are bounded: 1x(t)1

The last thing, that we should base is output attractiveness for system (5), (6) for case of disturbance d absence. Suppose, that there exists some Td ~ to ,

~ X\(tD)+X2(P(K ))+X3( tD x4(lI d ll ))+X.

408

quence of dwell time technique, indeed, starting out::side of set X p a trajectory of system (3) with globai

a!gorAlm (7)possessesthe-following properties for

controller can in general reaches the origin for time less than 't D .

I. Global stability with respect to set X g for any

It is worth to note, that forward completeness property stated in Assumption I and dwell time were not really needed in proof of Theorem I. Forward completeness property was imposed to guarantee existence of system (2) solutions for 't S 't D' In fact, forward completeness requirement can be weakened to merely existence of system (2) solution for all initial conditions from set X p on fInite time interval

I x( t )Ix so( R( x( to)}}+ a(11 d11),

dE M q :

•

R(x(to))= max{p(K ~ x(to)}· 2. Global output attractiveness for d( t );:: 0, t ~ to :

technique using. Let us say that d E M; , if d E M q and additionally 11 d 11 S r, r E [ 0, + 00 ). Then it is

1i( t J otherwise,

where t j' j

= 1,2,3,...

possible to introduce an upper bound for right hand sides of systems (2) and (3) on set N as follows: sup (

Number

context means that signal

= max

F( r

1

.

} always exists and finite due to conti-

of sets N

and M;, it can infinitely increase with

r ----) + 00, but it always stays finite for finite r.

> 0 that systems (2) or

Hence, the smallest time

't r

(3) need to cross set N lowing manner:

can be estimated in the fol-

Term "otherwise" in such

i( t )

{f g ( x, d )}

sup

nuity property of functions f l , f g and compactness

(7)

is number of the last switching. As in algorithm (5), the system is equivalent to (2) on set XI and to sys.

{fl(X,d}},]

xeN,de:M; xeN,de:M;

are moments of switching, j

tem (3) on set Rn /X p

)

F r

ifx(t}EXI; ifx(t)l1OX p ;

E

Proof. The main steps of proof of the present theorem coincide with the same from proof of Theorem I. Differences are originated by implicit dwell time

norm of input d and can be considered as a dwell time analogue for the system. Such technique was used in (Prieur, 200 I; Prieur, and Praly, 1999) for construction of uniting controllers for case of external disturbances d absence and it is called hysteresis switching in logic-based control theory (Morse, 1995):

i(t)= g,

I I

> 0 :J Ty = Ty ( x( to)} : y( t} S E for all t ~ Ty . 3. For all t~to: X{t}EX I ~ i(t}=I. V

with length is bigger than 't D' It is obvious, that proof of Theorem I can be used in such way without any modifIcations. Compact set N plays a role of buffer, where trajectories of system (2) spend time while crossing this set, such time depends on Loo

/,

all X(tO}E Rn and to ~ 0:

p(K }-K 't r

is not changing on set

N . Set N prevents fast switching arising and due to its compactness helps to avoid problem of unboundedness of system (2) solutions for case of disturbance d presence. Hence, it is possible to modify Assumption 1 as follows.

=

F(r}

Further one can repeat all steps of Theorem I proof with substitution 't D = 't r ' On any time interval

[Ts ' Te ) with Ts ~ to ~ 0, Te > Ts , a finite number of switches N[ T, ,r,) is possible and the following upper estimate holds: Te -Ts N[r r) S I + - - .

Ass u m p t ion 3. There exist a constant K > 0

6'

and a continuous local control uI : Rn ----) RP, such, that system (2) with d( t) ;:: 0 for all t ~ 0 admits the following properties: A. Solution ofsystem (2) is locally stable: X(tO)E XI ~ X(t)E X p .

C

't

r

Between switches uniting controller is continuous and equals to local u, or global u g controllers, which are continuous for all x E Rn . Thus resulting control is piecewise continuous function of time and solution of system (6), (7) is continuous and defined at the least locally [ to, T).

8. Solution ofsystem (2) is locally output asymptotically stable, i.e. for all to ~ 0 and x( to) E X p ,

13 E 7([., such, that Iy( t,x( to ~O )1 S 13(1 x( to~,t -to) for all t ~ to· •

there exists somejunction

Claim 3.Let tED!. Then X(t}EX p

.

Proof. The statement simply follows from defmition of algorithm (7).

In the last assumption all requirements to system (2) deal with its unperturbed by disturbance version. Such modification allows to simplify problem of local controller construction for system (1).

C I a i m 4. Let t E O 2 , Then all trajectories are

bounded

Ix(t}lx so(R(x(to))}+a(lldll)·

•

The 0 rem 2. Let Assumptions 2 and 3 be true and k < K, p( s} > s. Then system (6) with supervision

Proof. According to property A of Assumption 2 solution in this case is bvunded:

409

Ix( t )1 ~ cr(1 x(t j)I)+ cr{1I d 11)+ k , where t j' j

= 0,1,2,...

5.

are instants of time, then

~CKNOWLEDGEMENT

The author would like to thank Dr. I. Polushin for helpful comments and discussion.

global controller was plugged to the plant (1). For j > 0, x(t j ) by continuity property admits estimate

Ix(t )1~cr(R(x(to)))+cr(lIdl\)+k.

The paper is supported by Russian Science Support Foundation and by Program of Russian Academy of Science Presidium NQ 19.

It is clear, that [to,T)=OI u02 and according to

REFERENCES

from Claim 3 and

Claims 3 and 4 solution of system (6), (7) is bounded, thus, T = + 00 and the following inequality is satisfied for all t ~ 0 :

Ix( t)lx

Angeli, D., and E.D. Sontag (1999). Forward completeness, unboundedness observability, and their Lyapunov characterizations. Systems and Control Letters, 38, pp. 209 - 217. Efimov, D. (2002). Universal formula for output asymptotic stabilization. Proc. of 15th IFA C World Congress, T-We-M 07 4, Barselona, Spain. Efunov, D. (2003). Switching adaptive control of affine nonlinear system. Proc. of European Control Conference, Cambridge, UK. Fomin, V.N., A.L. Fradkov and V.A. Yakubovich (1981). An adaptive control of dynamical plants. Moscow: Science, 448 p. (in Russian) Fradkov, A.L., LV. Miroshnik and V.O. Nikiforov (1999). Nonlinear and adaptive control of complex systems. Kluwer Academic Publishers, 528

~ cr{ R( x( to)))+ cr(/I d 11)·

g

Output attractiveness for system (6), (7) for case of disturbance d absence can be obtained by the same line of consideration as in proof of Theorem 1 taking into account Assumption 3. The third point of the theorem is a corollary of supervisory algorithm (7) properties. _ Due to results of Theorem 2 system (6), (7) possesses all requirements posed in control goals. In practice applications d E:.M; with r ~ R < + 00 and with proper choice of function p (which defines the size of set :N') it is possible to obtain desired minimal value of time value of time

't r 't r

for any d

E

p.

:.M: . Such minimal

Isidori, A. (2000). Nonlinear control systems. Berlin: Springer-Verlag. Krstic, M., I. Kanellakopoulos and P.V. Kokotovic (1995). Nonlinear and adaptive control design. New York, Wiley. Liberzon, D., E.D. Sontag and Y. Wang (2001). Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation. Systems and Control Letters. Morse, A.S. (1995). Control using logic-based switching. In: Trends in control (A. Isidory (Ed.)), Springer-Verlag, pp. 69 - 113. Morin, P., R.M. Murray and L. Praly (1998). Nonlinear rescaling of control laws with application to stabilization in the presence of magnitude saturation. Proc. ofNOLCOS'98. Prieur, C. (2001). Uniting local and global controllers with robustness to vanishing noise. Math. Control Signals Systems, 14, pp. 143-172. Prieur, C. and A. Astolfi (2002). Robust stabilisation of chained systems via hybrid control. Proc. of CDC'2002, pp. 522 - 527. Prieur, C. and L. Praly (1999). Uniting local and global controllers. Proc. ofCDC'99, pp. 12141219. Sepulchre, R., M. Jankovic and P.V. Kokotovic (1997). Constructive nonlinear control. New York, Springer. Sontag, E.D. and Y. Wang (1996). New characterization of the input to state stability property. IEEE Trans. Aut. Contr., 41, pp. 1283 -1294. Teel, A.R. and N. Kapoor (1997). Uniting local and global controllers. Proc. ofECC'97.

is needed in practice to realize su-

pervisory algorithm (7) due to it includes a part working in discrete time (supervision algorithm (7)) and its implementation can be performed only with some minimum time step. Rem ark 2. Let global controller provide practical stability and attractiveness properties, stated in Assumption 2, only on some finite time interval, i.e. there is a time instant tl > to, such, that inequalities

Ix( t, x( to ~d )1 ~ cr(1 x( to)I)+ cr(11 d11)+ k ; Ix(t\>x(to~d)l~k hold for

to ~ 0 , initial conditions

dE:.M q , tE[to,tl],

crEK

x( to) E

Rn ,

and k~O. In other

words, global controller ensures, that from any initial conditions solution stays bounded on some finite time interval [to, tl ] and in the end of this time interval trajectory enters into set XI; in common case for t > t1 system can produce unstable solution. Then it is enough to prove a version of Theorem 2 with such requirements imposed on global controller, due to into set XI system always should be switched to local variant. _

4. CONCLUSION The paper develops a solution of uniting controller synthesis problem for case of essentially bounded disturbances presence.

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