Switching Stabilization Control for a Set of Nonlinear Time-Varying Systems

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SWITCHING ST ABILIZA TION CONTROL FOR A SET OF NONLINEAR TIME-VARYING SYSTEMS Liu Yong and Gao Weibing Th l' Sl'l 'l'I l lh R l'.Il'Ilrch Di" i.,iv lI . Bfiji llg IlIsl ill/ll' vf A l'I'o l/ aUlics all d Asl ra l/a l/ lin . Beijillg . PRC

liIotivated by Byrnes and Wi lle.s (1984), the set of all pos s ible plant s i s divided into several subsets in ter.s of Lyapunov controllable regions by the finite covering theore • . Switching control law is then utilized for the purpose of stabi lization, which, different fro. general variable s tructure control strategy, posse sses a finite nuaber switche s before asy.ptotic stability of the syste. i s achieved . To this end,we apply the idea of .ulti-step swit ches working in Fu and Bar.ish (1986) to the plants in a set of nonlinear syste.s . The controller generated in such a way can therefore be considered as a universal controller in so.e sense. The adaptation of the controller to the uncertainties of plants plays a key role in the controller, whi ch,once with its structure deter.ined, can work well for any plant in this set of syste.s . It will be also show that only a fini te nuaber of swi tches occurs and then the controller re.ains fixed with respect to a corre sponding Lyapunov controllable pair . Lyapunov direct .ethod will be used for des i gning of both control law and switching index. Throughout the paper, we as suae that the current state of the sys te.s is available through direct .easure.ent s for the purpose of state feedback . The .a i ~ result of this paper could be considered as the extension of adaptive s tabilization .ethod for linear syste.s proposed by Fu and Bar.ish (1986) to adaptive control for nonlinear syste • .

ABSTRACT, In this paper, a stabi lizing controller is designed for a finite or infinite set of nonlinear ti.e-varying syste. s wi th unlutown parueters under certain hypotheses on the set of the plant s, nuely, co.pactness and Lyapunov controllabi li ty assuaptions . A swit ching control law is ut i lized for the purpose of stabi lization, which, different fro. general variable s tructure control (YSC) s trategy, possesses a finite nuaber of swi tches before asy.ptoti c stabi li ty of the syste.s is achieved. Lyapunov direct .ethod will be used for designing both the control law and the switching index. The .ain result of thi s paper could be considered as the extension of adaptive stabilization .ethod for linear syste.s proposed by Fu and Bar.ish (1986) to adaptive control for nonlinear syste • . I.

Introduction

Fu and Bar.ish (1986) proposed a new .ethod for adaptive stabi lization wi thout a .ini.ua phase assuaption and without knowledge of the s ign of the high frequency gain. Instead, they include a co.pactne ss require.ent on the set of possible plant s and assuae that the upper bound on the order of the plant i s known. Note that a co.pactnes s assuaption can be considered as a bound assu.ption on the parueters of the plants, the possible plant s sati sfying this bound assuaption for. a subset of the whole possible unlutown syste.s for which an adaptive universal controller i s required. In their work, Fu and Bar.ish in fa c t divided the set into several subset s in ter. s of feedback gain .atrices by the finite covering theore • . Vnder the additional hypotheses, a piecewise linear ti.e-invariant swi tching control law, swi tching uong the subsets one by one, is generated which can lead to a guarantee of Lyapunov stability and an exponential rate of convergence for the state . It has been shown through the cons truction of a practical adaptive controller that a nonlinear control law (adaptive), when discretized by a co.bination of several linear ones, can perfor. as well under certain restrictions on the plants. The algori th. can also be considered in principle as a .ixture of adaptive control and variable structure control theories . In this paper, we consider the proble. of parueter adaptive control for nonlinear ti.e - varying syste.s . A stabilizing controller is designed under the assu.ption on a fini te or infini te set of the plants, nuely, ca.pactness and Lyapunov controllabi li ty assu.ptions ( Kappos and Sas try 1986; Gershwin and Jacobson 1971>.

2. Set of the plants In thi s paper, the proble. of adaptively s tabi lizing a set of nonlinear ti.e varying syste.s are considered with Lyapunov direct .ethod . First,suppose a finite upper bound on state di.ension N+g(x, t)u (1) where state x E R- for so.e n
25 1

252

ler

Liu Yong and Gao We ibing

could

be

founded.

controllability concepts Kappos

is

proposed and

Lyapunov switching

(1986)

Based

dire c t

on

aethod

adaptive

problea, we throughout

give the

this

paper, a For

this

Lyapunov

g(x, t), we can easily find an

purpose, the

around systea a such that for each a ' E E( a) (wi th

the and

by Gershwin and Jocob s on

Sastry

requireaent .

In

introduced . are

lhis

iaproved

a

and

sue diaension of a)

aeet

our

S.(x, t), there al s o ex ist s the inequality

to

concept,

is eaployed for generating

si aple

neighbourhood

(1971)

controllabi li ty

controller. Before

open

the

notation which will be used

paper. For any differentiable sc alar or

wi th

the

sue

V. ( x, t)

L. ' =D, V. (x, t )+D, V. (x, t) [f' (x, t)+g' (x, t)S. (x, t) J

the

considering

and

E(a)

<-y, . • (

11

Cons equent ly, an taking

xii )

open

covering of '1'. is

the union of all the

range s over '1' • .

neigborhood s

generated

by

E( a)

a

as

Recalling the coapactness assuaption on

vector function V(x" . . . ,x.), denote D.VC) a s the partial

each

derivative or the Jocobi aatri x with re spect to

ith

controllable pairs (V• . "S• . ,), (V•. "S• . ,), ... , (V• .•• • "

if

S• . •••• ) and associated open subsets E• . "E• . " .. . , E• . •••• such that for each a E E•. '"'' there ex ists a Lyapunov controllable pair (V' ...... ,S' . .... ,).

the

variable. Defini tion Systea (I) is s aid to

be

Lyapunov

there exist a scalar fun c tion

V(x, I)

controllable and

a

control

11

x

a

fini te

set

'I' wi th the orders less than N, we can

)
11

ex tract

11

xII )

L=D, V(x, t)+D. V(x, t) [ f( x, I) +g(x, I)S(x, t)J < - y ,(

11

xII)

where y. C), i =1, 2, 3. are K- functions . A K- function is

decoupled

s tates of

appropriate

therefore

aake

plant

the

z =Az

(VC),SC»,siaply denoted by

V(x, z, t )=V(x, t )+z' Pz

nued

Lyapunov

controllable pa i r .

u(x,

Reaark 1

always

di.ension

possess e s

an

Z,

of Lyapunov contrOllability for

s ufficient both

local

conditions and

add to

in soae

it

and

N--
t) =S(x, t)

where V(x, t) i s the Lyapunov function

Kappos and Sastry (1986) gave

Lyapunov

Lyapunov controllable pair in such a way

posi tive and non - de c rease fun c tion wi th y (0) =0. The pair (V,S), i s

of

Since the upper bound on the orders of the plants

u=S(x, t) E U satisfying y ,(

'1'. , we can

global

for

the

original

sys tea, P>O, s a t i s f ies A' P+PA=- W, W>O. Indeed, the condi t ional subsystea

and

the

addi tional

tera

in

the

Lyapunov

cases. However, the condi tion dV(X( x»;t O for global

case

fun c tion iapose no effect on the dynuics of the original

seeas to be incorre cl si nce there is alway s dV =O at

x=O,

sys tea .

provided V is a Lyapunov func tion . We use

Using the fact given above, we can then

notation ~ and 'I' to denote the se t

the

all the possible plants and s et

of

all

the

of

possible

adding a subsy s tea

Lyapunov controllable plant s ~ =(

to

the

orig i nal one

quadrati c fora tera to

the

original

( 2a)

re spec tively. Therefore a set of the

'I' =(a =( f,g) E ~.a is Lyapunov controllable}

(2b)

controllable pair i s obtained which

'1'. =( a =( f, g) E '1'. a has the diaension of n}

( 2c)

a=
We aay fir s t consider the cas e where

the

nuaber

of

always

aake

each V•.• ,., be coae an N- diaensional Lyapunov function by and

adding

a

Lyapunov

function

augaented

Lyapunov

is

fitted

for

all

possible plants. Such being the case, we siaply take

the

s et s of Lyapunov controllable pairs (V,',S,'), (V,',S,'),

eleaents in 'I' is finite.But this case can be included in

... , (V.', S.')

the following case.Ac tually, the fin i tene ss as s uaption can

Lyaunov c ontrollable

be

be

range s froa 1 to N. Now, for any fixed i E ( 1, 2, .. . ,v), the

of

se t s

relaxed. Instead, a

iaposed on

the set

'I'

coapactness c ontaining

eleaents which can lead to the

assuaption inf i ni te

siai lar

aay

nuaber

result s

as

the

as the union of the sets of the augaented pairs

'1','=( a =«(, g) E '1'.

finiteness assuaption doe s. For the purpos e of s tating our aain results, the a ss uaptions aentioned above are

foraaly

a

which

are

generated

posse sse s

the

Lyapunov

i E (1, 2, ... v) can be defined whi ch s atisfy the requireaents

As s uap t i on 1

le . . a.

of

the

eleaent s

in '1'. is finite . Ass uap t i on

n

controllable pair (V,',S,')}

given as follows . Each '1'. is a finite s et , i.e. the nuaber of

as

3. Construction of the switching controller

2

Each '1'. i s a coapact set . In thi s section, Le . . a 1

the

foraal

construction

swit ching controller whi ch achieves the desired

Let '1'. n=1,2, . .. ,N. be the set

given

by

(2c)

and

satisfy Assuaption 2, then we can find a fini te nuaber

of

sets '1',' , '1',', .. . , '1'.' , such that

ideas behind the construction are

(3)

froa their work, we replac e the their s by

ass uaed

that

there

exi s ts a Lyapunov controtlable pair (V.,S.),such that 11

)
11 x

11 x

parueter

II )

of

aonitoring

Lyapunov function

controllable pai~ since

in

in

nonlinear

function each

case

of

Lyapunov aonitoring not

be

constructed. For each i E (I, 2, ... , vI and each

11 )

Since functional L. depends continuousl y

use

fun c tion like those of Fu and Baraish (1986) can

L. =D, V. ( x, I)+D, V. (x, t) [f(x, I) +g( x, I) S. (x, t) J

< -y , . • (

froa

especially froa Fu and Baraish (1986), However, different

exists a fi x ed Lyapunov controllable pair (V. ' ,S, ') .

y , . • ( 11 x

taken

adaptive control and variable structure control theories,

U· .. , '1'. ' ='1'

is

a

s tability for the c losed - loop systea is provided . The basic

and for each i E(1, 2, ... ,v } and any a=
of

Lyapunov

s wit ching instant which is defined on

f(x, t)

and

pai r to

a =
decide

when

control law switches froa ith s tep to (i+1)th s tep .

the

253

Switching Stabilization Control for a Set of Nonlinear Time-varying Systems

t, =inf{t,t>t,_" V,·(x(I),I) - V,·(x(t,_.),t, _,) > - J .,~ y... ( 11 x( s) 11 )ds} h(t) =i where h(t) is


the

switching

index

which

need

to

be

designed for each interval of tiae [t, _" t, ).Subsequentty, the control law is recursively generated using the foraular u( I)=S•• " • (x, I) (6) The key step of the procedure is to decide when the swi tches occur . I t is well known that the swi tching instants should be deterained by the estiaates of output / state regulating errors or by soae functions of thea. In this paper,we utilize Lyapunov functions to sever the purpose which vary with respect to the output / state regulating errors. We consider the

switching

control

law

with

its

swi tching index and swi tching instant given above . First, we c laia that for soae i here is t, =00, i. e. the generation of t, is terainated and the control reuins u=S,"(x, t) for tE [t, _"oo) . To this end,let a =(f,g) E IjI with arbitrary initial condition x(O)=x. and note that, in accordance wi th Leua I, a E 1jI," for soae i
.eans


(4)

)< -1'

Ob) Y • . , ( 11 x(s) 11 )ds

tl =00

We now ci te a leua for convenience, the proof of which is oai ted Leua 2 If i) f(t) is continues with bounded derivative for t>t •. ii) G( x) is continuous and positive, G(x)=O if and only if x=O . iii)J..G[f(l)]dt
state

and

prove

the

aain

Therefore 11 x( t) 11

< [y •. , " ] -,

Hence 11x( t, ) 11

< [y • . , . ] . •

y • . ,. ( 11

x(

y • . ,' ( 11

t, _. ) 11 )

x(

t, _. ) 11 )


(12)

Define the func tion

.'r·

$,=[Y • . y • . ,. <13 ) It can be easily proved that $" jE{1,2, ... ,v}, are non-decrease Cunction s provided tha t Y • . : C), Y • • : C ) E K and y •. ,·(r)b, c=a-' p(a)
contradiction. Note the fact above, then 11 x( t) 11 < $, $, _. . .. $ • ( 11 X. 11 ) Note that $, are continuous and bounded x( t) is bounded. Froa t, =00, we have

(14) t >t, _. Cunctions, then

J ,~,y •.• ( 11 x(s) 11 )dst. _. So J:;.1' " have

( 11

x( s) 11 )ds

exists . Then, froa leua 2,

U!.x( 1)=0. As yapototic stable property then guaran ted .

oC

the closed-loop sy s tea

we

is

Reaark 3 Since Lyapunov asyaptotic stability is ass ured Cor each plant, we aay aodify our switching control law to deal with uncertain systeas with their noainal systeas contained in 1jI, provided the uncertainties are bounded but possibly tiae - dependent . The technique developed for control oC uncertain systeas (Corless and Leitaann 1981; Baraish et at. 1983) can be eaployed in this case . This problea will be a topic oC Curther study.

result of this paper.

4 . Quadratic stabllizatlon and an example Theorea 1 Consider the set of the possible nonlinear systeas IjI with the corresponding set of Lyapunov controllable pairs, and switching control law (6). For any a=(f, g)E 1jI, when controlled by (6), the closed - loop systea has a solution for any ini tial condi tion x(O)=x., which is asyaptotically stable. Proof, The existence property of the s olution for the closed-loop systea is easily built by the assU8ption that the right - hand side of systea (I) is always Caratheodory. We have claiaed that the switching index h(t) converges to soae i . So le t is the fi rs t tiae when t, =00, that aeans t"j E{ 1,2, i - l} is finite and the switch stops for the reaainder of the tiae. The aain effort here devoted to the proof is to bound the state x with respect to a given ini tial condi tion . Indeed, for the given plant o=(f, g) E IjI and at the step j
In thi s section, we restrict our interests oC stabilizing the set of possible nonlinear plants to counterpart oC quadratic stabilization,which can lead to exponential stabi lization oC any plant contained in the se t . DeCini tion 2 Systea (I) is s aid to be quadratically stabilizable iCC there exist a positive definite syuetric aatrix P, a posi tive constant a and a continuous feedback control S(x, t) satisfying the Collowing condi tions L=2x' P [f(x, I)+g(x, I)S( x, I)] < - a 11 xII' Define a new set Q ={o =(C,g),o is quadratically stabilizable}

siailar but better propertie s will be derived Cor the systeas wi thin the quadratically stabi lizable set Q. To

problea in the tiae intervat

thi s

T, =[ t, _"t,) Making use of Lyapunov control lab le pai r of 1jI,.

results are given as follows . t. =inf{t, t>t • .• ,x·( t)P. ·x(t)- x·(t. _. )P. "x( t. _.)

t E T" we have y • . , • ( 11 x( t) 11 ) < V, • (x( t ), I)

for


In the context of quadratic s tabi lization, a posi tive deCinite quadrati c Cora is used as Lyapunov function . A

end, soae

new

deCini tions

and

>- J 't~, a. 11 x 11 • d t } h(t ) =i

the corresponding

254

Liu Yong and Gao Weibing

u=S...

C ')

Reaark 4

-(XI t)

<1>,=« A ••• [P,J )/ A.,. [P,J }'" For any ini tial condi lion x(O)=x., there is x( t, )

11

11

< <1>, <1>, _, _ . . <1>, (

11

X.

Consider the case of first

initial

condition. It is

shown that the first switching instant is t, =O, i.e. (16)

11 )

De fine

the checking function at the beginning of the

•••

=aax( <1>" j E [1, 2, ... , v] }

In practice, this saall - tiae delay.

then 11 x( t, _, ) 11 < [<1> ••• J ' -, 11 X. 11 ' lIi thin the reaainder of this section, we

provide

a

the

state of the closed-loop systea burns up with respect kind

of

switches

to

evolution.

should

have

a

5.Conclusions

counterexaaple to illustrate our theoretical resul ts. Exaaple I This is

Adaptive control aethod and variable structure control an

exaaple

wi th

two classes of nonlinear

plants which, when coabined together,fora the set I/I.Note that

in

this

exople

the division of 1/1 into 1/1,'

is

natural and therefore trivial. y=2yu, -x' u,

wi th

their

paroeters

to of

unknown . The in this

set

stable in the sense of Lyapunov, however, only after a finite nuaber of swi tches

occurs . Actually, the

switching

procedure

could

be

considered as a procedure autoaatically searching probable control law oong several possible control laws

x=cy-bx' u, y=-2cx' - y' u,

for the unknown systea . With this switching controller,we

bE [0.500,2.000]' c E [-1. 000,1. OOOJ 1/1,' ={ 0, =( f, g), bE [0.500,2 . 000],

can deal with a wider class of stabilization problems for

c E [-1. 000, 1. OOO]} The Lyapunov controllable pairs for 1/1,' and 1/1,' are respec t i ve ly

Consider a systea

do

A key restriction of this scheae

wi th rises

the general due

to

the

Lyapunov controllable . Therefore in a practical point of view, the

0

obtained switching control law is of a bit difficulty

E 1/1,' with b=1.000 and

lie want to show after a fini te tiae systea is controlled by the first swi tch

uncertain systeas than we can variable structure control aethod .

assuaption that each plant in the set shOuld be

{V, '=x' +y', S,' = [-I, y]' } (V,' =x' +y', S,' = [2y, I]' )

pair, a

systeas

asyaptotically this is done

a E [2.000,4.000] 1/1, '=( o,=(f,g), aE [2.000,4 . 000J} 0,

nonlinear

conlrol law we construct can aeke any plant

x=axu, +x' yu,

0,

aethod are co.bined in this paper in a particular way cope with adaptive stabilization probleas for a set

occurs

for

checking Lyapunov

the

c=O . OOO. whi le

the

controllable

to

be used. However, if unifora ulti.ate boundedness properly rather

than

asyaptotic

concerned, several

stability

sufficient

for

the

conditions

plants

for

is

Lyapunov

control

law

which is

controllability

one, that

is

actually

arbitrarily saall neighborhood of

the origin

what the systea require. In fact , the systea is now reduced

obtained,e.g. the condition given

by

to

which can find its use in the scheae proposed here.On the other hand, it would be worthwhi le searching other ways to

adaptively changed to the second

x=-x·u.. yo_yO u,

of

nonlinear

systeas

outside

an

have

been

Liu et at. (1988),

study the probleas without Lyapunov controllability assuaption . On this way, it .ay be possible to use a set of

lie now prepare to calculate the first swi tching instant when the systea is initially controlled by the first

positive definite function as a aeasure.ent

Lyapunov controllable pair . The closed-loop systea becoaes

state

regulating

errors

within

each

for

output /

particular state

x=-x'y

region and the controller would not stop switching

y=y'

the region unti 1 soae stabi li ty properties are garanteed.

Since 1/1,' is in fact quadratically stabilizable with its

The

Lyapunov controllable pair, we have

future research .

idea

like

this

will

probably

oong

be eaployed in the

L, '=-2(ax'+2y' )<-4(x'+y') and there fore we de fine wi th t. =0 and x( t. )=x. y(

and

ACKNOWLEDGEMENT

t. )=Y. t,=inf( t, t>O,

[x'(t)+y'(l)J-[x·'+y·'J>-4J" [x'(s) +y' (s) J dsl

On the other hand,we can get the explicit solution of the closed-loop systea by direct calculus.

This work is supported by the National Science Fund of China REFERENCE Barai sh, B. R. , Corless, N., and Le i taann, G., 1983, SIAM J. Control and Optiaization, 21, 264 . Byrnes,C.I,and Isidori,A., 1984, Proc . 23rd IEEE Conf on Decision and Control,PP.1569-1573;l986 Byrnes, C. I., and Willeas,I.C . , 1984,Proc. IEEE Conf. on Decision and Control, pp.15H - 1577.

The direct calculation shows that with two typical initial conditions X. =-1. 000,

y. =1. 000,

t, =0 . 000

X. =-3.000,

y. =0.800,

t,=0.504

Corless,N.I., and Leit.ann, G. , 1981, IEEE Trans.autoa. Control, 26, 1139. Fu, N.Y . ,and Baraish,B.R. , 1986 , IEEE Trans . autoa. Control, 31, 1097.

Switching Stabilization Control for a Set of Nonlinear Time-varying Systems

Gershwin,S.B. ,and Jocobson,D.H., 1971, IEEE Trans.auto•. Conlrol, 16,37 . Hahn, W., 1967, St.bi li Iy of Mali on (New York, Springer-Ver lag). Kappos, E., and Sas try, S., 1986, Proe. IEEE. Conf. on Deci s ion and Conlrol, pp. 973 - 974 . Khal i I, H. K., and Saber i, A., 1987, IEEE. Trans. auloa. Control, 32,1031. Liu, Y., Cheng, M., and Gao, W.B., 1988,1. Op I i.i. Theo. and Appt. 50. Morse, A. S., 1980, IEEE. Trans. au to• . Con t ro I, 25, 433 ;198 2, Proc . CNRS Colloqui. on Develop.ent and Utilizalion of Mathe .. tical Models in AulOllalic Conlrol, pp . 733-740. Narendra, K. , Lin, Y. H., and Valavani, L. S., 1980, IEEE. Trans. auto •. Control, Z5, 440. Nussbaua,R.D., 1983,Syste.s and Control Lellers, 3,243. Ulkin, V. l., 1977, l. E. E. E. Trans. aulo... Con trol, 22, 212. Wi lle.s, 1. C. , and Byrnes, C. I., 1984,Proc. INRIA Conf. Analysis and Opti.ization of Sysle.s,pp.49-57.

255