On the stability of a class of nonlinear hybrid systems

ELSEVIER

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

IfAC

--

PUBLICATIONS

www.elsevier.comllocate/ifac

ON THE STABILITY OF A CLASS OF NONLINEAR HYBRlD SYSTEMS L.Burlion .'•• , T.Ahrned-Ali ••• and F .Larnnabhi-Lagarrigue·

• LSS, SUPELEC-C.N.R.S, Universite Paris-XI Sud, 3 Rue Joliot Curie, 91192 Gi/-sur- Yvette, France •• LRBA, BP 914, 27207 Vernon Cedex , France ••• ENSIETA, 2 Rue Franl;ois Verny, 29806 Brest Cedex 9, France

Abstract: In this paper, we propose new conditions to guarantee the stability of nonlinear sampled data systems. In order to do this, we extend lyapunov theory to this class of systems. Copyright @2004 IFAC Keywords: Sampled-data systems, Nonlinear systems, Stabilizing controllers, Hybrid systems, Lyapunov methods

sampling time is sufficiently small. We here generalize this approach to the case of dynamical control laws : this time, the discretization is applied on the dynamical law so that the closed loop system has both a continuous and a discrete dynamic. That's why we have to deal with the stability of nonlinear hybrid sampled-data systems. The analysis of a nonlinear sampled-data system is of great interest but has always been deduced by the analysis of its corresponding linearized sampled-data system (Ye, Michel and Hou, 1998). We here propose to directly consider the nonlinear system in order to study global stability property. As we shall see in the next section, the class of sampled-data systems is a subclass of the class of time-dependent impulsive dynamical systems: the stability of nonlinear impulsive systems have been studied by many researchers ; to the best of our knowledge, the most general extension of Lyapunov's concepts to this class of systems was made in (Ye, Michel and Hou, 1998) : that's why we used its main result to solve our problem. The paper is organized as follows : in the section 2, we show that the class of sampled-data systems is a subclass of the class of time-dependent impulsive dynamical systems and we deduce a stability

1. INTRODUCTION

This paper deals with the analysis of the stability of nonlinear sampled-data feedback control systems. Nonlinear sampled-data feedback control systems consist of the interconnection of a nonlinear plant (described by a system of first order ordinary differential equations) and a digital controller ( described by a system of first order nonlinear difference equations ) along with the interface elements (AID and DI A converters). Thus, such a class of systems contents the class of nonlinear systems which are stabilized by a discretized dynamical feedback. Our main contribution is to prove the heuristic result which claims that the discretization of continuous feedback control laws required by their practical implementation doesn't affect the stability of the system provided the sampling time is small enough: moreover, we obtain two inequalities (5) from which one can deduce a lower bound for the sampling period In the litterature, it has been shown in (Hermann, Spurgeon and Edwards, 1999) that the discretization of stabilizing nonlinear control law with a Lipschitz property preserves the stability of the initial non linear system provided the (constant)

429

theorem for nonlinear sampled-data systems. In the section 3, we prove our main result. We finally conclude by numerical results and comments.

the solutions are right-continuous having left-side limits at each Tic and are differentiable almost everywhere. We also remark that 0 is an equilibrium because le(O) = 0 and Id(O) = 0 : to prove asymptotic stability, one can use the following theorem:

2. DESCRIPTION OF THE CLASS OF SYSTEMS

In this paper, we say that a function !/i : ]R+

Theorem I (Ye, Michel and Hou, 1998)(adapted to our notations and restricted to autonomous systems)

---+

]R+ is of class }C if it is continuous, strictly increasing and !/i(0) = O. It is of class }Coo if in addition !/i(x) ---+ 00 as x ---+ 00

Consider the class of system with impulse effects described by equations above. Assume there exist h E C[lR+, lR+j such that h(O) = 0, !/i,!/i1,!/i2 E}C, and V : ]Rn -> ]R+ such that \;/x E ]Rn, \;/k EN:

We first describe the class of hybrid systems used in this paper. Throughout this paper, the state x belongs to the space (X = ]Rn, 11.11) and the control u belongs to U = ]Rm. We consider the class of nonlinear sampled-data control systems described by :

!/i1(llxID :'S V(x) :'S !/i2(llxll), V(x(t)) :'S h(V(X(TIc))),

x(t) = I(x(t)) + g(X(t))Uk ,t E h, Tk+d { Uk+1=h(uk,X(tk)) ,t=Tk, kEN

\;/t E

\;/t

E]R+

[Tk,Tlc+d

and that in addition:

where:

where:

• {Tk, kEN j TO = 0 < T1 < ... < Tk < Tk+ 1 < ... } is an unbounded closed (Tn converges

DV(X(TIc)))

to +00 without finite accumulation) discrete subset of]R+ such that T = SUp(TIc+1 - Tic) IcEN exists. • I: X ---+ X is Lipschitz on X, and 1(0) = 0 • 9 : X ---+ XU is bounded • h: U x X ---+ U is Lipschitz with respect to x and h(.,O) = 0

~ TIc+1 1-

Tic

[V(X(TIc+d) - V(X(TIc))]

then the equilibrium x = 0 is uniformly asymptotically stable.D This theorem is easily extended to prove global asymptotic stability by adding the hypothesis limllxll--->+oo !/ii(X) = +00, i E {1,2} (see for instance (Khalil, 1992)). Note that it's only required that V at times Tic when jump occurs, be non increasing and that between Tic and TIc+1, V only needs to be bounded by a continuous function of V(X(TIc)) (For instance, h can be increasing i).

In this case, the solution x exists, is unique for a given initial condition and is differentiable almost everywhere. We also notice that the function U : t ---+ u(t) = UIc, t E [Tic, TIc+d, kEN is piecewise constant and then Lebesgue measurable. Let's now consider the class of impulsive differential system with impulses at fixed times described by

x(t) = le(x(t)) ,t i Tic { t..x(t) = x(t) - x(t-) = Id(X(t-)) ,t = Tic, k E N* where:

• {TIc,k E N;TO = 0 < T1 < ... < Tic < TIc+1 < ... } is an unbounded closed (Tn converges

Remark: The class of nonlinear sampled-data systems is included in the class of nonlinear (time-dependent) impulsive dynamical systems, once we add the control U to the state. In this case, the state is (x,u) where u is defined by :

U(t) = UIc u(t) = 0

to +00 without finite accumulation) discrete subset of ]R+ such that T = SUp(TIc+1 - Tic) IcEN exists. • le : X ---+ X is Lipschitz on X, and 1(0) = 0 • Id: X ---+ X, and Id(O) = 0 • x(t-) ~ lim x(s)

,tE[TIc-1,TIc[, kEN,t

i

Tic

This remark enables us to write a similar theorem for nonlinear sampled-data systems:

Theorem 11

s---+t-

Since le is Lipschitz, the solution x exists and is unique for a given initial condition j moreover,

Consider the class of nonlinear sampled-data systems described by equations above.

430

where {3, Cl, C2, C3, C4 are strictly positive constants. In this case, x tends to 0 asymptotically (according to La Salle's principle).

Assume there exist h E C[R+, R+j such that h(O) = 0, 4> E /C, 4>1,4>2 E /Coo, and V : Rn X R m -> R+ such that Vx ERn, Vk E N, VUk E Rm :

4>1(llxll,lIukll)

~ V(X,Uk) ~

4>2(lIxll,llukll)

V(X(t),Uk) ~ h(V(X(Tk),Uk)),

Let us consider a time-discretized control :

Vt E h,Tk+l[

Vk E N, Vt E [Tk' Tk+d, u(t) = U~(Tk) = UC(Tk)

and that in addition:

where: Tk

~ Tk+1 1-

Tk

T>0

Theorem III (Hermann, Spurgeon and Edwards, 1999) There exits a sufficiently small T such that for any T, T < T the discretized control law for the above system is (asymptotically) stable.

where: DV(X(Tk),Uk))

= kT,

[V(X(Tk+d,Uk+d

-V(X(Tk),U k )] 3.2 Discretized dynamical Feedback

then the equilibrium (X,Uk) = 0 is (globally) uniformly asymptotically stable.

The purpose of this article is to extend the above approach to dynamical feed backs ; we also add the hypothesis that the sampling period is timevarying but upper bounded. As we shall see, this boils down to considering the class of nonlinear sampled data systems.

Proof: By using the remark and the first Theorem, the proof is straightforward. 0

Let us consider the following class of nonlinear systems

3. STABILIZATION OF NON LINEAR SYSTEMS BY MEANS OF DISCRETIZED FEEDBACKS

i {

3.1 Discretized static Feedback

(1)

where x E Rn, and U E Rm. We suppose that the following assumptions hold:

Let us consider the following class of nonlinear systems i

= j(x) + g(x)u

u =a(x,u)

BI: There exists a function V : Rn which satisfies

= j(x) + g(x)u

X

Rm

---+

R

where x E Rn, and U E R m. We suppose that the following assumptions hold:

AI: There exist a function V: Rn X R m ---+ R+ and a control U = uC(x), Kc-Lipschitz which satisfy Vx E X:

B2 : f is ft-Lipschitz and j(O)

=0

B3 : There exists gl > 0 such that IIgll

:s gl·

B4 : Vx,y,u,v:

1I:(X,U)-:(y,V)II:s cs(llx-yll+llu-vll) 8V

8u (0,0) = 0

A2 : f is jl-Lipschitz A3 : the open-loop system has no finite escape time for any bounded input u

A4 : There exists gl

> 0 such that Ilgll

B5 : The function a(x, u) acts as a dynamical feedback such that V(x,u) E X x U, 8V 8V 8x [j(x)+g(x)ul+ 8u [a(x, u)]

~ gl.

:s -{3(ll xIl 2 +ll uIl 2 )

Moreover: Vx, y, u, v

431

Ila(x, u)-a(y, v)1I :s aM(llx-YII+llu-vll)

a(O,O) = 0

W exists almost everywhere and Vt E]Tk, Tk+1 [,

where l3,c1,c2,c3,c4,cs,aM are strictly positive constants. In this case, (x, u) tends to 0 asymptotically (according to La Salle's principle).

. 8V 2 2 W(t) :S -,B(llxI1 + Il u kll ) - 8u IX,Uk [a(x, Uk)] 8V + 8u IX(Tk+,),Q(X(Tk),Uk)(t-Tk)+Uk a(x(Tk), Uk) 2 2 8V :S -13(llxll + Ilukll ) - 8u x,u.[a(x,uk) - a(x(Tk 1

Let us consider, what we will call its associated nonlinear sampled-data system:

x(t) = f(x(t)) + g(X(t))Uk , t E [Tk, Tk+d { Uk+1 = Uk + (Tk+1 - Tk)a(uk,x(tk)) ,t = Tk, kEN where :{Tk,k E NiTo = 0 < T1 < ... < Tk < Tk+1 < ...} is an unbounded closed (Tn converges to +00 without finite accumulation) discrete subset of lR+ such that T = SUp(Tk+1 - Tk) exists.

+

8u IX(Tk+,),Q(X(Tk),Uk)(t-Tk)+Uk (w

W

8u IXoUk

)

a(x(Tk),Uk) 2 2 :S -13(llxII + Il u kl1 ) + cs(llukll + IlxlDllx - X(Tk)1 +Tcsalt(llx(Tk)11 + Ilukll)2 + csaMllx(Tk+d -:z (1lxh)ll + IluklD

kEN

Both systems (1),(8) have the same initial condition. This also means that if T is small enough, we approximate the control of the first system u(.) by a piecewise affine function Uk (.).

X(t) - X(Tk) = =

Theorem IV There exists a sufficiently small TM such that for any T < TM, the nonlinear closed-loop sampleddata system considered above is globally asymptotically stable.

it Tk

+ g(X(S))Ukds

f(x(s))

1~ f(x(s) -

X(Tk))

+ [f(x(s))

- f(x(s) -

+g(X(S))Ukds Hence,

Proof:

Ilx(t) - x(Tk)lI:s (Tk+1 - Tk)(hllx(Tk)11 + g111 u kll)

We have: . 8V V(x, Uk) = 8x [f(x)+g(X)Uk] +0

,Vt

+ 1~ hllx(s) -

Eh, Tk+d

X(Tk)lI ds

By Gronwall's inequality:

Thanks to the Hypothesis B5, \It E]Tk, Tk+1 [

Ilx(t) - X(Tk)11 :S (TH1 - Tk)(fIllx(Tk) 11

+ gIllukIDeh(t-

:S T(f11Ix( Tk)11 + g111 u k IDeM We then have: (1Ix(t) - X(Tk) 11 ~ Ilx(t)II-lIx(TkID We consider a right-continuous function W defined as follows: Vt E [Tk, Tk+d ,

hT Ilx(t)ll:S Ilx(Tk)11 +T(hllx(Tk)1I +g11I u kll)e :S (1 +The M ) Ilx(Tk)11 + (Tg1 ehT ) lIukll

W(t) = V(x(t), Uk) + t - Tk 8V a(x(Tk), Uk) 0 8u IX(Tk+,),Q(X(Tk).Uk)S+Uk ds

(3)

1

We also have,

Remark:

IIx(t) - X(Tk+1)11

Vk EN,

+ X(Tk)

- X(Tk)11

:S 2T(hllx(Tk)11 + g11lukll)ehT

2 W(t):S -13(llxII + Ilukl12) + cs(llukll + IlxlDllx - X(Tk)1

+Tcsnlt(llx(Tk) II

---I

=

V(X(Tk),Uk-d

=

V(X(Tk),Uk) =

+ V(X(Tk),Uk)

(4)

Finally, by using (2),(3) and (4), we obtain:

W(Tk-) = V(X(Tk), uk-d + a(x(Tk-d, uk-d x Tk-Tk-1 8V o 8u X(Tk),Q(X(Tk_,),Uk_,)S+Uk_1 ds

= W(T:)

X(Tk+d

:S Ilx(t) - X(Tk)11 + Ilx(Tk+1) - X(Tk)11

Although V is only right-continuous at each Tk, we built W such that W is continuous everywhere. Indeed,

1

= Ilx(t) -

- V(X(Tk),Uk-l)

+csaMllx(Tk+l) - xii (1Ixh)11 + Ilukll) 2 2 :S A1(T)llx(Tk)11 + A2(T)lI u kI1 + A3(T)lIx(Tk)lllluk :S (A1(T)

W(Tk)

432

+ II u kl1)2

+ A3t))llx(Tk)1I2+(A2(T) + A3t)) 111

where:

where:

+ T/1 ef, -r? + TC5 ((1 + Thef, -r)/l ef,

>'1 (T) = -,8(1

T

=

+ T2gie 2f, T) + TC5 ((1 + 20M )gl ef,-r

-,8(1

+ o~ + Tgie2f,-r) >'3(T)

+ . {TCl,C2 } C5 mln

0 M

(~+ 20 MT)V 2

(20M + 1)(/1 + 9l) e h -r

3.3 Linear Case Let's consider the following linear sampled-data system (the sampling time is constant),

= -2,8T(1 + T/Ief,-r)9lefl-r + TC5 (20~

+

=V

We then conclude by using the theorem 11.

+ o~ +20Mhefl-r) >'2 (T)

h(V)

x(t) = Ax(t) + BUk ,t E [Tk, Tk+d { Uk+l = Uk + T((C - Im)uk + DX(tk))

+ 2T /Igle 2f,-r)

, t = Tk = kT, (8)

By integrating the first equality, we have: T Vk E N, x(Tk+d = eA-r X(Tk)+(l eA(T-')BdS)Uk

Then if: (5)

Thus, (remark : these two inequalities prove the existence of TM and give a lower bound of TM in practise) there exists a real number, such that 0 < ,

-max(>'l(T)

+ A3t) ,

>'2(T)

+ A3J-r»)

where:

~

and: Then, the system is uniformly asymptotically stable if!' the matrix H (T) is Schur stable. This means that T has to be small enough so that the eigenvalues of H(T) remain in the open unit circle. We remark that the well known lemma 5.1 of (Ye, Michel and Hou, 1998) is a particular case (T = 1) of this result. In the Non Linear case, we notice that DV(X(Tk)) = ~ J;'.+1 W plays the role of H(T).

Thus, by integration and by using the fact that W(Tk) = V+(Tk) = V(Tk) we deduce that

DV(X(Tk))

1

= -[V(X(Tk+l)) - V(X(Tk))] T

~ -,T(llx T .1I 2 + Ilukll2)

(7)

Moreover, since: Vt EJTk, Tk+d,

4. NUMERICAL SIMULATION RESULTS

I

· () W' ( ) (() ) av In this subsection, we consider the system: V t = t + 0 X Tk , Uk8u x(-r.+,),a(x(-r.),u.)(t--r.)+u. . 2 ~ 0 + c5 0 MII U kll(llx(Tk)1I

+ Ilukll)

+ c5o~(llx(Tk)11 + Il u kl1)2(t - Tk) ~ c5 0 Mll u kll(lIx(Tk)1I + Ilukll) + c5o~T(lIx(Tk)11 + lI u kl1)2 ~ C50M[ (1 + OMT)lIuk112 + (1 + 2oMT)lIukllllx(Tk)11 + 0MT Ilx(Tk)11 2 ]

~ C5 0 M

G+

20 MT) (1Ixh)11

2

+ Il u kl1 2)

~ mln. / Cl, C2 }C50M(-23 +20MT)V(X(Tk),Uk)

(9)

where K l > O. We consider the function V :

V(Xl,X2,U)

1 2 1 2 2 = "2X2+"2(Xl+u) +x l

. 2 2 2 V = -K l x 2 - (Xl + u) + 2XlU + 2X1X2 = -(Kl - 2xdx~ - xi - u 2 (10) We use the norm: Ilxll = IXll + IX21. The example verifies the different hypothesis' of our theorem when:

Hence, by integration : Vt E h,Tk+l[,

Xl = U +X2 . -- Xl - K lX2 + U X2 { U = -2u - Xl - X2 - X~

V(X(t),Uk) ~ h(V(X(Tk),Uk))

433

• we interest us to the set {XI (0), X2(0), U(O)jVt, 12xI(t)1 < K I +{31 and IX2(t)1 < K 2 } where K 2 , (31 > O. In this case :

v ~ -(3IX~ -

xi - u

~ - min{l,(3d(x~

o

+ xi)

- u

u2

...,

o

o

..

11

o

1;c5=1

o

For K I = 10, according to the simulation, one can choose: 0 < (31 ~ 9.6, K 2 = 0,4. In this case, the following inequalities (5):

{ >'2(7)

>'3t) < 0

+ -2- < 0

o

I

2

S

..

S

•

1

•

t

10

o

1

2

3

..

S

•

1

•

•

10

12

,

S

•

1

•

1

2

J

,

...

10

I 'I 10

, . .

T

10

= 0.35

1

2

:I

..

S

•

1

•

•

10

1

2

,

..

6

•

1

•

•

10

1

2

,

..

,

5

•

T

7

•

•

to

= 0.65

H. Ye, A.N. Michel and L. Hou (1998). " Stability Theory for Hybrid Dynamical Systems ", IEEE Transactions on Automatic Control, vo1.43, no.4, pp 461-474. H. Ye, A.N. Michel and L. Hou (1998). " Stability Analysis of Systems with Impulse Effects", IEEE Transactions on Automatic Control, vo1.43, pp 1719-1723. G. Hermann, S.K. Spurgeon and C. Edwards (1999). "Discretization of Sliding Mode based Control Schemes" , in Proc. of the IEEE Conference on Decision and Control, pp 42574262. H.K. Khalil (1992). " Nonlinear Systems", Prentice Hall. R. Sepulchre, M. Jankovic and P. Kokotovic (1997). " Constructive Nonlinear Control", Springer-Verlag, London. D. Nesic, A.R. Teel and P.V. Kokotovic (1999). " Sufficient conditions for stabilization of sampled-data systems via discrete-time approximations" , Systems and Control Letters, vo1.38, pp 259-270

•:;;;:: ! ..

•

REFERENCES

>'3\7)

'~~::"I -l: ' ;;I -1 0

;:

1

1 / ; ,

Fig. 3. Discretized Feedback.

are satisfied when 7 ~ 0.01. Our theorem provides a lower limit for 7M but, of course, doesn't give the " highest limit" .

,~p

•

,~~

Numerically, for K l = 10, the nonlinear closedloop sampled-data system is stable for 7 ~ 0.6.

+

S

'l~;";:1

X,

>'1(7)

..

~~ o

IIg(x)1I =11 (~) 11 ~ 2 • we set : Cl = 4 ; C2 = 4 ; C3 = 2 ; C4 = and for all

:t

Fig. 2. Discretized Feedback.

_xtX2) ~ (K2+ K l )l x 21 + IX11

(Xl

2

'~E"

Hence: (3 = min{l, ~(3d We also have: aM = max{2, K 2 + I} because: (for alllx21,IY21 < K 2 and for all XI,YI,U,V, la(x,u) - a(y,v)1 ~ 21u - vi + IXl - Yl! + (K 2 + 1)jx 2 - Y21 ) h =max{1,K l +K2 } ; gl =2 Indeed, for alllx21 < K 2 , for all Xl, IIf(x)11 = 11

1

-t

2

~ -~ min{l, (3d(lX11 + IX21)2 -

•

;:k" .... :I

2

•

•

Fig. 1. Continuous Feedback

434