Attractors and Partial Stability of Nonlinear Dynamical Systems

Z* = {x EX:

~(t)

Suppose that the function cp satisfies to the following condition of regularity.

rank

=

1/J(x),


(5)

=

~(t,xo)

= cp(x(t,xo»,

defined for t E [0,00). Definition 3. The point ~ = 0 is called a partial equilibrium of the system (1),(5) when, for all Xo E Z* and t:::: 0, the solutions ~(t) = ~(t,xo» satisfy the identity ~(t,

xo) = O.

(6)

Definition 4. The system (1),(5) at the equilibrium point ~ = 0 is called partially (uniformly) asymptotically stabile when there exists a neighborhood £(Z*) such that, for all Xo E £(Z*), it holds (7) lim ~(t, xo) = 0

(3)

where 1/J is a smooth mapping from Z* to the open simply connected set Z. A specific behavior of the system when its solutions satisfy equation (2) and the relevant trajectories belong to the hypersurface (submanifold) is associated with the following notion.

=

and consider the solutions of the system (1), (5)


Assumption 1. For all x E Z*, it holds

(4)

t-oo

uniformly with respect to Xo E £(Z*).

Remark 1. In the general case, the notions introduced do not imply the existence of state equilibriums and hence the system's asymptotic stability with respect to some state points.

826

Sufficient conditions of the partial stability are given by the following proposition (Miroshnik et al ., 2000) . Let us introduce a smooth Lyapunovlike function V(x) and the positive definite functions Vl(~)' V2(~)' w(~) . Theorem 1. If, in the set £(Z·) C X, there exists a function V(x) such that

to demonstrate attractivity of simplest geometric objects (planes) and propose a simplified treatment of Theorem 1. Suppose that x = (~, z) and equation (5) takes the simple form ~ = [J O]x , where ~ E :=: eRn-v , z E Zo c RV , :=: X Zo = X . Rewrite (1) in the form

(8)

fz(~ ,

then (i) for all Xo E Z· it holds ~(t,

xo) = 0,

iF.(~ ,

t ;::: 0,

(9)

(10)

z), z),

(11)

where if. and fz are smooth vector function. We study the solutions ~(t) = ~(t ,

i.e. the point ~ = 0 is a partial equilibrium of the system (1) , (5), (ii) the system (1) , (5) at the point ~ = 0 is partially asymptotically stable.

xo) ,

t ;::: O.

of the system (10)-(11) with respect to the point ~ = 0 and the set (peace of the plane) in Rn :

Zo = {x E X

Remark 2. The concept of partial equilibrium is associate with the system 's invariance. In fact, the set Z· is represented by the points for which ~ = 0, and therefore the trajectory of the system lying on the set corresponds to identity (6) . Hence ~ = 0 is a partial equilibrium iff for arbitrary points Xo E Z· the solutions x(t, xo) of the system (1) belong to the invariant submanifold Z· . However, in the general case, the invariant submanifold Z· of a partially stable system fails to be an attracting set, because (7) is not insure that the relevant trajectories approach Z· according to (4), and conversely.

c

Rn : ~

= 0, z

E Zo}.

First, we represent the following notions (Miroshnik et al., 2000) . Definition 5. The point ~ = 0 is called a partial equilibrium of the system (10)-(11) when , for all Xo E Zo and t;::: 0, the solutions ~(t) = ~(t , xo» satisfy the identity ~(t , xo) =

O.

Definition 6. The system (10)-(11) at the partial equilibrium point ~ = 0 is called asymptotically stable with respect to part of variables when there exists a neighborhood £(Zo) such that, for all Xo E £(Zo), it holds

The brief review represented above demonstrates the necessity of studding relations of the concepts and finding unified local conditions, under which partial stability of the system and attractivity of the relevant sets can be established. Taking into account that Theorem 1 does not give a constructive way for design of Lyapunov functions, the problem of simplification of sufficient condition for the both cases also arises. That is why we start the consideration with a special problem of stability with respect to part of variables which will allows one to make use techniques of partial approximation and to design simplified quadratic Lyapunov functions .

lim ~ (t,xo) = 0

(12)

t - oo

uniformly with respect to Xo E £(Zo) ' Let us emphasize that, in opposite to the notions of partial stability given in (Rumyamtsev, 1957; Vorotnikov, 1998), Definition 5 does not require that the system has usual equilibrium points in Rn (see Remark 1) but the set Zo is obligatory invariant. That give an opportunity to generalize the relevant concepts for unstable (in Rn) systems arid simultaneously exclude the systems having on Zo solely isolated stable equilibrium point . This is a basic peculiarity of the following result , being a modification of the known Rumyantsev' theorem .

3. STABILITY WITH RESPECT TO PART OF VARIABLES

A problem of stability with respect to part of variables (see Rumyamtsev, 1957; Vorotnikov, 1998) can be considered as a special case of the partial stability problem. This case gives an opportunity

Theorem 2. If in the set £(Zo) a function V(x) such that

827

C X

there exists

where Vl(~)' V2(~)' w(~) are positive definite functions, then (i) for all Xo E ZO' it holds ~(t,

t ~ 0,

xo) = 0,

(ii) system (10)-(11) at the point ~ totically stable with respect to ~.

(14)

for all z E Zo . Then, for Xo E £(ZO'), the behavior of the system (1) is described by ZI differential equations of the system

= fzo(z),

(16)

= fz(O, z) .

where fzo(z)

On the other hand, if, in addition, the system (10)-(11) at the point ~ = 0 is asymptotically stable with respect to ~, then the plane ZO' is an attractor, and conversely. In fact, the distance from the set can be here found as

dist(x, ZO') =

IH

and therefore expressions (5) and (12) are equivalent. Thus, one can represent the following statement.

Theorem 3. Suppose that the hypothesis of Theorem 2 hold. Then the plane ZO' is an attractor of the system (10)-(11) . The results mentioned do not point out a constructive way for finding appropriate Lyapunov functions. The relevant local solutions and the system's properties in the neighborhood £(ZO') can be obtained by using the First Lyapunov method. Let us accomplish partial linearization of the system and rewrite (10)-(11) in the form A€(z)~,

i

fzo(z)

Re >'i {A(z)} > -00,

+ Bz(z)~,

(17) (18)

where A€ = afda~ I€=o, Bz = afz/a~ I€=o, and suppose that the higher order terms odz, ~), oz(z,~) of appropriate Taylor series obey the standard conditions (Fradkov et aI., 1999) . Now the system is represented by the linear nonstationary part (17), which parameters are generated by the nonlinear model (18) .

(19)

and there exist a function >.(z) such that sup Re >'i {A(zo)}

= 0 is asymp-

Identity (14) shows that the point ~ = 0 is a partial equilibrium of the system. Therefore, under the conditions mentioned, x(t, xo) E ZO' for all t ~ 0, and therefore the set ZO' is obviously an invariant set of the system (see Definition 1). The both properties can be verified by using the condition (15)

z

Let for all z E Zo

< >.(z).

(20)

zEZo

Then we can choose a Lyapunov-like function V(x) as

V(x) = ~T P(z)~,

(21)

where the matrix P = pT is found as a solution of Lyapunov-like equation

A(zf P(z)

+ P(z)A(z)

= -Q + 2>.(z)P(z) (22)

and Q = QT > o. By using Lyapunov Lemma and Theorems 2-3, one can prove the following result.

Theorem 4. Suppose that, in the neighborhood £(ZO') c X, the system (10)-(11) satisfies the hypothesis (15), (19)-(20). If, additionally, there exists >'0 > 0 obeying, for all z E Zo, the inequality

?(z) + 2(>.(z) - >'o)P(z) < o.

(23)

then (i) the system (10)-(11) at the point ~ = 0 is asymptotically stable with respect to ~, (ii) the set ZO' is an invariant set and an attractor of the system (10)-(11). The theorem proposes the sufficient conditions of asymptotic stability with respect to part of the variables and those of attractivity of ZO' . On the other hand, under the same conditions, there exists and can be simply constructed appropriate Lyapunov functions (21) satisfying the hypothesis of Theorems 2-3.

4. ATTRACTIVITY AND PARTIAL

STABILITY CONDITIONS

In order to obtain conditions of attractivity of an arbitrary submanifold Z' and to analyze the partial stability of the general system (1), (5), we transform the system and reduce the problem to that of stability with respect to part of variable studded in Section 3. Such a transformation is accomplished by using the coordinate change (3),(5), i.e.

I

828

~z 1 =

1

cp(x) 1j;(x)

I·

(24)

In the vicinity of the hypersurface the vector € characterizes a deviation from Z* and, as it was shown in (Fradkov et al., 1999; Miroshnik et al., 2000), can be directly connected with the function dist(x, Z*) (see below).

obtain the task-oriented model of the system in the form (10)-(11), where

h

t»,

ff. (z,O) = o.

I

Note that, for all Xo E Z* and t ~ 0, the system behavior is completely described by l/ equations of the model (16). In view of the equivalence of the initial and transformed systems for the case considered the properties of partial stability are also preserved. Theorem 7. Suppose that Assumption 2 holds. Then there exists a neighborhood £(Z*) such that, for all Xo E £(Z*) and (Eo, zo) E £(Zo), the following are equivalent: (i) the system (1), (5) at the point E = 0 is partially asymptotically stable; (ii) the system (10)-(11) at the point E = 0 is asymptotically stable with respect to E.

I

and without loss of the generality suppose the following. Assumption 2. There exists a neighborhood £ (Z*) such that (i) for all x E £(Z*) the Jucobian matrix of the

I: I #

Under the hypothesis of Theorem 7, the results of Section 3 including Theorems 2 and 3 can be used. As, in general, the properties of attractivity of Z* and partial stability of the system are not equivalent (see Remark 2), for the further analysis it is necessary to introduce additional metric restrictions. Define the metric matrix

0;

(ii) the mapping (24) is a local diffeomorphism from £ (Z*) onto £ (Zo) with the smooth inverse

x = r(z,€).

(28)

Zo o

(25)

8cp 8X / 8ljJ/&x '

mapping (24) is invertible: det

r(s, e). (27)

On the other hand, in the new coordinate chart (€, z), the nonlinear hypersurface is represented by a sheet of the central plane (subspace) Under the condition (28) (identical to (15)) this plane is an invariant set and the point E = 0 is a partial equilibrium of the transformed system (10)-(11).

Taking into account that the set Z* obeys Assumption 1 and it is an embedded submanifold, whereas vector z is introduced as a vector of local coordinates, one can conclude that the mapping (24) is a local diffeomorphism. Let us introduce the Jacobian matrix of the mapping (24): 1

0

When Z* is an invariant submanifold and the point E= 0 is a partial equilibrium of the system (1), according to Theorem 5 and in view of the definition (27) of the function h, one can write

(ii) the smooth submanifold Z* is invariant submanifold of the system (1); (iii) the point € = 0 is a partial equilibrium of the system (1),(5).

I=

r(s, e), fz = (IJI f)

Theorem 6. Suppose that Assumption 2 holds. Then there exists a neighborhood £(Z*) such that for all Xo E £(Z*) and t E [0, T), T > 0 the system (1) is equivalent to the model (10)-(11) .

Theorem 5. Suppose that Assumption 2 holds. Then the following are equivalent: (i) for all x E Z*

cl>(x) lJI(x)

0

Here equation (11) describes longitudinal dynamics of the system (1) z = z(t, zo, Eo) E Z, and equation (10) corresponds to the transversal dynamics (or deviation) E = E(t, zo, Eo) E S.

First, we note that the identity €(t) == 0 corresponds to the system's trajectory x(t, xo) belonging to the submanifold Z* (where the behavior of the system is uniquely defined by the vector z( i.e. it is a criterion of the set invariance (see Definition 1). On the other hand, the same expression gives a condition under which the point €(t) = 0 is a partial equilibrium of the system (1),(5) (Definition 3). Then diff~rentiating (5) with respect to time and substituting (1), one obtains, that for all x E Z* it holds (8cp/8x)f = O. This is the basis of the known criterions (Fradkov et al., 1999; Miroshnik et al., 2000).

8cp 8x f (x) = 0;

= (cl> f)

(26)

The assumption establishes a principal possibility to study nonlinear systems in the neighborhood of Z* and to analyze their stability by using equivalent models similar to those of considered in Section 3. In fact, differentiating equation (24) with respect to time and substituting (1) and (26), we

which characterizes a distortion of the Euclidian metrics when passing to coordinate chart (E, z) .

829

Then (see Fradkov et al., 1999; Miroshnik et al., 2000) , for small enough ~ one can find that

Recall that the Lyapunov function V(x) can be chosen in the form (21).

dist(x , Zo) = I~IQ~'

The theorem represents sufficient conditions of attractivity which differ from the hypothesis of Theorem 1 and 4 owing to the additional assumption of metric regularaty (30) . The same conditions establish the identity of the notions of submanifold attractivity and partial stability of the system.

e

where I~I~~ = Qef The latter shows that coincidence of the expressions (4) and (7), as well as that of the corresponding attractivity properties of the invariant submanifolds Z* and Zo, takes place under the following condition of metric regularity. Assumption 3. For all x E £(Z*) it holds q~ I :S Q(x) :S q~I,

where q2 ;:::: ql >

(30)

o.

Assumption 3 is more restrictive than Assumption 2 and ensures that the coordinate change (24) does not arise unlimited distortion of space metrics at least in the vicinity of Z* . Then , under the supposition that the system (1) is equivalent to the model (10)-(11) for all t ;:::: 0, the attractivity of Z* can be associated with partial stability of the system (10)-(11).

Theorem 10. Suppose that Assumption 3 holds. Then there exists a neighborhood £(Z*) such that , for all Xo E £(Z*), the following are equivalent: (i) the set Z* is an invariant submanifold and attractor of the system (1), (ii) the system (1), (5) at the point ~ = 0 is partially asymptotically stable. It should be noted that Assumption 3, playing the principal role for establishing attractivity of the submanifold Z* , can be omitted when partial stability of the system is solely investigated.

REFERENCES

Theorem 8. Suppose that Assumption 3 holds. Then there exists a neighborhood £(Z*) such that, for all Xo E £(Z*) and (~o , zo) E £(Zo) , the following are equivalent: (i) the set Zo is invariant submanifold and attractor of the system (1), (ii) the system (10), (11) at the point ~ = 0 is partially asymptotically stable with respect to ~ .

Burdakov, S.F., I.V.Miroshnik and E.R. Stelmakov (2001) . Trajectory Motion Control of Mobile Robots. Nauka, SaintPetersburg (in Russian) . Canudas de Wit, C., B. Siciliano and G. Bastin (1996) . Theory of Robot Control. Springer-Verlag, London.

Thus, the verification of attractivity of the submanifold Z* is reduced to checking the stability properties of the transversal dynamics model (10) which is disturbed by the model of longitudinal dynamics (ll). Those kind of partial stability conditions, as well as approaches to partial linearization of the system and design of the Lyapunov functions, were represented in Section 3 (see Theorems 2-4). They enable one to give the following statement, generalizing Theorems 1 and 4 for the problems of attractivity of an arbitrary smooth submanifold Z* .

Elkin, V.I. (1998) Reduction of Nonlinear Controlled Systems. A Diferentially Geometric Approach. Kluwer Acad. Pub., Dordrecht. Fradkov, A.L. , LV. Miroshnik and V .O. Nonlinear and Nikiforov (1999). Adaptive Control of Complex Systems. Kluwer Acad. Pub., Dordrecht. Isidori , A. (1995) Nonlinear Control Systems. 3nd edition. Springer-Verlag, Berlin. Miroshnik, LV., A.L. Fradkov and V.O. Nikiforov (2000). Nonlinear and Adaptive Control of Complex Dynamical Systems. Nauka, Saint-Petersburg (in Russian). Rumyantsev, V.V. (1957) On stability of motions with respect to part of variables. Vestnik of Moscow University, ser. Mat . and Mech., no. 4, pp. 9-16.

Theorem 9. Suppose that Assumption 3 holds . Then the set Z* is an invariant submanifold and an attractor of the system (1), if one of the following conditions holds: (i) there exists a Lyapunov function V(x) such that Vl(~) :S V(x):S V2(~) ' V(x) :S -w(~), (31) (ii) the system obeys the conditions (28) , (19)(20) . and there exists AO > 0 which, for all z E Z , satisfies inequality (23) .

Vorotnikov, V.L (1998) Partial Stability and Control. Birkhauser, New York.

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