Stabilization of Invariant Manifolds for Nonlinear Nonaffine Systems

Copyright © IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998

STABILIZATION OF INVARIANT MANIFOLDS FOR NONLINEAR NONAFFINE SYSTEMS Anton S. Shiriaev . , .. Alexander L. Fradkov·

• Institute jor Problems of Mechanical Engineering, 61, Bolshoy V.O., St.Petersburg, 199178 RUSSIA . e-mail: [email protected] •• Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7034 Trondheim, NORWAY. e-mail: [email protected] (current address)

Abstract: This paper is devoted to the solution of the problem of control of the nonlinear systems oscillations. The approach is based on the detailed investigations of nonlinear control systems with feedback control law formed by speed-gradient method. The main results of the paper are new sufficient conditions which guarantee that the solutions of the closed loop system achieve the prescribed set of the phase space. Copyright © 1998 IFAC

Keywords: Nonlinear control, energy control, speed-gradient method.

1. INTRODUCTION

u

Control of nonlinear oscillatory systems is an important problem with variety of application in mechanics, physics, vibrational technology, control education, etc. (Chernousko et al., 1980; Astrom, and Furuta, 1996; Wen and Kreutz-Delgado, 1991; Bloch et al., 1992; Vorotnikov, 1997; Fradkov, and Pogromsky, 1998). Control of oscillatory systems requires achieving non-classical control goals (Swinging, synchronization) as well as describing and analyzing complex motions of the closed loop system. The new approach to stabilization of the desired level of oscillations for Hamiltonian systems based on speed-gradient method (Fradkov, 1979) and energy objective functions was proposed in (Fradkov, 1994; Fradkov et al., 1995; Fradkov, 1996; Fradkov et al., 1997) It was shown that the new control algorithms allow to achieve the desired energy hypersurface with arbitrary small control intensity.

R"' and with smooth right hand sides f :

It is assumed that the nonnegative objective function Q(x), Q : xn -> Rl of form Q(x) = Ih(xW,

(2)

where h(x) is a smooth vector function, h : xn -> Rk is given ( 1·1 stands for the Euclidean norm). The following control problem is considered: to define the causal regulator and to find the conditions guaranteeing that for any solution x(t) = x(t,xo) of the closed loop system with initial conditions Xo from some set V E X n , the limit relation

lim Q(x(t)) = 0

t ..... oo

(3)

is valid. Below the set V is chosen as either V = xn (global achievement of control goal (3)) or 1) = {x E xn : Q(x) ~ c} with some positive constant c. It is convenient to introduce the auxiliary output

In this work we consider the nonaffine controlled system x(t) = f(x(t), u(t))

E

xn -> T X n, where T xn is a bundle of xn.

(1)

with a state vector x E X n, where xn is a ndimensional smooth manifold, an input function

y(t)

213

= h(x(t)).

(4)

Then the problem can be reformulated in the equivalent form: to define regulator and to determine conditions guaranteeing that for any solution x = x(t, xo) of the closed loop system with the initial conditions Xo E V the limit relation lim y(t)

t-CXl

functions whenever it does not lead to confusion. List now the main assumptions which we will use in the paper:

(AI) for any point x of the phase space xn, x E xn, the inequality L/o(x)Q(x) $ 0 is valid;

=0

(A2) for any point x of the set {x E xn LgoQ(x) = O} the identity LJoh(x) == 0 is valid;

holds.

(A3) the functions Rj(x, u), j = 1, . .. , m from (5) satisfy the inequality

The goal (1) is related to the asymptotic partial stabilization of the system (1) with respect to the vector y(t) which has a long history (see, e.g. (Rumyantsev, 1970; Vorotnikov, 1993; Vorotnikov, 1997» . However the problem of nonlocal partial stabilization of manifolds was not previously considered for nonaffine nonlinear systems evolving on manifolds. Most of existing results are local and do not cover the case of unbounded goal sets which is important for control of the oscillatory or rotatory motion.

max IRj (x , u)1 $ p(x) ,

lul9

= n,y =

(6)

j = 1, ... , m, Vx, where p(x) is a smooth scalar function.

Remark 1. The asSumption (A3) is valid if, for example, the functioA ,/(x , u) has polynomial rate of growth with respect 'of the input function u or the condition 11V'~fll $ C holds for all x E xn.

Note that the well studied problem of stabilization of the single equilibrium point is just a special case of the above problem. In a number of papers, see, e.g. (Byrnes et al., 1991; Lin, 1996) the efficient passivity-based algorithms for stabilization of equilibrium were suggested. We will show that the results of (Byrnes et al., 1991; Lin, 1996) are just special cases of our results for the case k

:

Definition 2. The function Q(x) is said to be proper if for any nonnegative number c the set {x E xn : Q(x) $ c} is compact. To solve the posed problem we will use the control algorithm suggested by the speed-gradient method (Fradkov, 1979; Fradkov, 1990)

x.

u(x(t) = -'/'V'uQ(x(t)

This paper continues the previous investigations of the authors (see (Fradkov et al. , 1997; Shiriaev, 1997; Shiriaev, 1998; Fradkov, and Pogromsky, 1998» and extends the previous results to the controlled systems of the general form (1).

or, more generally,

u(x(t» = -1l'(V'uQ(x(t))) .

(7)

Here Q(x) is the full derivative along the solutions of (1) and 1l'(z) is a vector-function forming sharp angle with z, i. e. w(zV z > 0 for z =f: O. To be more precise, we calculate the speed-gradient of Q with respect to the reduced or " affinized" system

2. PRELIMINARlES AND ASSUMPTIONS The system under consideration always can be expressed in the affine-like form

X = fo(x)

+ go(x)u.

Introduce the distribution

m

X = fo(x)

+ go(x)u +

L uj(Rj(x, u)u), (5)

S(x) = span {Lgoh(x) , LJoLgoh(x) , LJoLgoh(x) , .. . },

j=1

where span{v}, V2 , ... } is a linear space generated by columns of matrices v}, V2, • ... It is easy to see that for any point x the dimension of linear space S(x) cannot be greater than k. Consider the set P of points x, x E xn, where the linear spaces S (x) have not full dimension, i. e.

where

fo(x)

= f(x,O) ,

(8)

of

go(x) = au (x, 0),

Rj(x, u) are m x m-matrices and Uj , 1 $ j < m are the components of the vector u. Define operator Lb(x)a(x) where a(x), b(x) are smooth functions (possibly vector functions of appropriate dimensions) in a standard way

P = {x E and denote

xn : dimS(x) < k},

n c P the set

n= {xo E xn:

aa(x) Lb(x)a(x) = a;-b(x).

x

=

the whole trajectory (9) = 0 lies in P}.

x(t,xo) of (1) with u

T

We will use notation V' xa(x) = 8a~:) for the gradient vector and V';a(x) for the hessian matrix of function f (x). We will omit the argument of the

In the next section we will see that the introduced distribution S(x) and the sets P, are very useful in the problem of interest.

n

214

°

3. MAIN RESULTS

for some c > the set Qc is compact. Let the set satisfy the inclusion

Theorem 3. Consider the controlled system (5) . Suppose that the assumptions (AI), (A2) , (A3) are valid, the nonne~ative function Q(x) is proper on xn and the set n satisfies the inclusion

nc c Qo = {x E xn : Q(x) = O} .

ne Qo = {x

E

xn : Q(X)

=

a}.

nc

Then for the solution of closed loop system (5) with the regulator of the form

(10)

Then for any solutions of the closed loop system (5) with the regulator of the form

u(x) = -a(x)w([LgoQ(x))T),

(16)

and an arbitrary initial conditions Xo from the set {xo E xn : Q(x) S c} the control goal (3) is achieved. Here the functions w(z) and a(x) are from theorem 3.

(11)

the control goal is achieved, i. e. lim Q(x(t)) = 0.

See the proof of theorem 5 in the appendix.

t-+oo

Here w(z) is an arbitrary smooth vector function w(z) : Rm --+ R m such that

ZTW(Z) ~ "p(z)lIw(z)11 2 > 0, IIw(z)1I SI,

Remark 6. In the examples the inclusion (16) can be replaced by the more rough dimensional condition

'Vz =I 0, (12)

dimS(x) = k,

"p (z) is a smooth scalar function, "p (z) ~ "po > 0, 'Vz; and a(x) is a smooth scalar function satisfying the inequality 0< a(x) S &(x) , where a(x) = m(l

nc

+ UI8Q(x)j8xllp(x)j2) '

~ is a positive number,

°<

~

Remark 7. It can be proved that among regulators (11) described in theorem 3 there exists the regulator providing arbitrarily small value of control, i. e. 'Vc > there exist (a(x), w(z) , "p(z) , 1/Jo , ~) such that regulator (11) defined by these parameters satisfies the inequality lIull Se.

< min {"po, m} and

the function p(x) is from the assumption (A3).

°

See the proof of theorem 3 in the appendix.

Remark 4. Theorem 3 remains also true for the case when instead of the relations (12) the function W satisfies the strict pseudo-gradient condition (see (Fradkov, 1990))

ZTW(Z) ~ "p(z)lIw(z)11 6 > 0, Ilw(z)11 S 1,

The assumptions on the function Q(x) to be proper in theorem 3 or the compactness of {x E xn : Q(x) S c} for some c > in theorem 5 have mainly the technical meaning. Now we extend the Theorem 3 and theorem 5 to the situations where these assumptions are violated. To this end we impose additional condition guaranteeing, first, continuability of all solutions of the closed loop system (1) , (11) to the interval [0, +00) and, second, the existence of a nonempty w-limit set for any trajectory of the closed loop system (1) , (11).

°

'Vz =I 0, (13)

where "p(z) is a smooth scalar function, 1/J(z) ~ "po > 0, 'Vz, and 2 ~ 6 > -00. Indeed, in this case

IIw(z)1I 6 2: IIw(z)1I 2 ,

'V z

and the inequality (12) is valid. Now we investigate in detail the case where the set Qc = {x E xn : Q(x) S c} is compact for some positive constant c> 0. Denote

Pc

= {x

E

xn : dimS(x) < k} n Qc ,

nc = {xo E xn : the whole trajectory x = x(t,xo) of (1) with u =

One possible formulation of such a condition appears in the following theorem.

(14)

°lies in Pc}.

(17)

Indeed, the dimensional condition (17) implies that the set P, P = {x E xn : dimS(x) < k} , lies in the set Qo . Hence, c P c {x E xn : Q(x) = O} and the inclusion (16) is fulfilled.

2·~

A

'Vx E Qc \ Qo .

(15)

Theorem 5. Consider the controlled system (5) . Suppose that the assumptions (AI), (A2), (A3) are valid, the function Q(x) is nonnegative and

Theorem 8. Consider the controlled system (1). Suppose that the assumptions (AI), (A2), (A3) are valid, the function Q(x) is nonnegative and that for some c > the functions I(x , u) , LgoQ(x), LfLgoQ(x) are bounded on the set A = Qc x {u : lul SI} . Take the regulator of the form

°

u(x) = -a(x)w([LgoQ(x)f),

215

(18)

where w(z) is an arbitrary smooth vector function w(z) : Rm -4 Rm such that

where

ZTW(Z) 2: 1/J!(z)lw(z)1 2 > 0, Iw(z)l::; 1, 19 1/72(z)lzl2: Iw(z)1 > 0, Vz f= O. () Here 1/71 (z), 1/72 (z) are any smooth scalar functions, 1/J!(Z) 2: 1/7~ > 0, 1/7~ 2: 1/72(Z) > 0, Vz ; a(x) is any smooth scalar function such that 8a(x)/8x is bounded on A and that the inequality 0 < c&(x) ::; a(x) ::; &(x) is valid, where 0 < c < 1, • a(x) = m(1

8Q(x) R 1 (X ,U ) ~ LR(x,u)Q(x) =

[ 8Q(x) D( ) 8x oL'-In x,u and the matrices Rj(x, u), j = 1, .. . , m are defined in (5). Using the estimation (6) we derive the next one

IILR(x,u)Q(x)II ::; mp{x)

2· f3 + [II8Q(x)/8xllp(x)]2) '

Q{x(t» ::; LfoQ(x)-a(x)Lgo V(x)w{[LgoQ(x)f)+ mp(x)

118~~x) '11 ~\~(~)211W([LgoQ(x)f)II2.

It is obviously that 0 « a(x) « 1 and 0 « IIu(x)II ::; 1, Vx. By assumption the function W satisfies the inequalities (12) then

is compact. Let the inclusion (16) be valid. Then for the solution of closed loop system (5), (18) with an arbitrary initial condition Xo E Qc the control goal (3) is achieved.

Q{x(t» ::; a(x)IIw([Lgo Q(x)f)II 2 x x ( mp{x)

See the proof of theorem 8 in the appendix.

'f/;(z)

=1+

Iz12, 1/70

= 1,

118~~X) 1 a(x) -1/7([Lgo Q(x)f»)

::;

::; a(x)IIw([LgoQ(xW)II2[f3 -1/7oJ ::; O.

Theorem 3, theorem 5 and theorem 8 remain true if k = n and the goal function Q(x) is positive definite: Q(x) > 0 for x f= 0, Q(O) = O. For this special case the theorem 5 recovers the result of (Lin, 1996) where the sufficient conditions of zerostate stabilizability by the regulator of the form (11) with the coefficients

= 1 +ZlzI 2 '

11 ~~ 11·

Continue (22) taking account of last estimation and the form of control law (11)

and f3 is positive number, 0 < f3 < min{1/7o, m}. Suppose that there exists b > 0 such that each connected component of the set

lJ!(z)

:

I

Thus the value of the function Q does not increase along x(t), Q{x{t» ::; Q(xo) for all t E [0, t.) , and x{t) belongs to the compact invariant set Qc. Hence x(t) is well defined on [0, +00) and has nonempty w-limit set ,0 with LfoQ(x) = 0, LgoQ(x) = 0, Vx E ,0, and Q(/O) = d, where d is some nonnegative number. If d equals to zero then theorem 5 1) is proved. Otherwise, take any point xo E ,0 and consider the solution x(t) = x{t, xo) of the closed loop system (1), (11).

(21)

were obtained. Our results, in contrast to (Lin, 1996) are not the restricted by the dimensional relation dim u = dim y . Moreover, our theorem 3, theorem 5 and theorem 8 describe the wider family of the stabilizing regulators.

In accordance with our arguments proved above along x(t), the value of the control u is zero, moreover, LfoQ{x{t» = 0, LgoQ(x(t» = and Q{x{t» = d for all t 2: O. Hence

°

LgoQ(x(t» = h{x{t)V Lgoh{x(t» == 0, Vt . (23) 4. APPENDIX Differentiating the last identity along the trajectory x(t) we have

Let us first prove the theorem 5.

LfoLgoQ{x{t» = [Lfoh{x(t»JT Lgoh(x{t»+ ( ) +h{X{t»T LfoLgoh(x{t» = 0, Vt. 24

Proof of theorem 5. Take any point Xo E Qc and consider the solution x(t) = x(t,xo) of the closed loop system (1), (11) . By smoothness of !(x, u), h(x, u), a(x), w(x) and Q(x), the trajectory x(t) is well defined on the interval [0, t.). Calculating the time derivative of V along x(t) we have

By the assumption (A2) the first term of (24) is equal to zero, hence

Consecutively differentiating the identity (23) in such way (see (24» we obtain that for an arbitrary

216

vector S of the linear space S(x(to)) with the fixed time to E [0,00), sE S(x(to)), the relation

and as we have shown in the proof of theorem 5 the solution x(t) belongs to A whenever it is well defined. Hence we conclude that Ix(t)1 grows not faster than a linear function, and, therefore, x(t) is well defined on [0, +00). In particular, we also conclude that the inequality (27) is valid for 'if t 2: 0, and that the value of the integral

(26)

holds and this is true for any to. Moreover, we know that the scalar function Ih(x(t))1 is constant for all t, hence h(x(to)) =f. O. Otherwise h(x(t)) == o and the control goal is achieved.

+00

J

a(x(r))lz(x(r))1 2dr

Thus we conclude that the relation h(x(t)) =f. 0, 'ift can be valid only for whole trajectory of unforced (1) which for all t lies in the set PnQe' We denoted this set previously as By theorem assumption (16) the set lies in zero level of the function Q(x), and theorem 5 is proved. 0

ne.

ne

is finite. Consider the derivative of the function a(x)lz(x)12 along the the trajectory x(t). We have

1i(a(x)lz(x)1 2)lx=x(t) = 80' (29) T 8z Izl2 8x f (x, u(x)) + 2az 8x f (x, u(x)).

Proof of theorem 3. The validity of theorem 3 follows from the proof of theorem 5. Indeed, by assumption the function Q is proper on X n , then taking advantage of the theorem 5 proof arguments we obtain that any solution x(t) = x(t,xo) of the closed loop system (1), (11) is well defined on [0, +00), bounded and Q(x(t)) ::; Q(xo). Therefore the trajectory x(t) possesses non-empty w-limit set "(0 and along any trajectory x(t) E "(0 of the closed loop system (1), (11) the relations Q(x(t)) = d, LfoQ(x(t)) = 0, LgoQ(x(t)) = 0 hold for all t 2: O.

By virtue of the assumptions the right side of (29) is uniformly bounded for t E [0, +00) . Therefore by the finiteness of the integral (28) we derive that lim a(x(t))lz(x(t)W = O.

t-+oo

The last limit relation implies that lim t-++oo

Following again the arguments of the theorem 5 proof we derive that x(t) E where c = Q(xo). It is clear that

ne,

nc

(28)

o

111Jf([LgoQ(x(t))]T)112

- 0

1 + [11 &Q~~(t» IIp(x(t))J2 -

.

and by virtue of the assumption the trajectory x( t) belongs to to the one of the compact connected component of the set D6, see (20). Thus the trajectory x(t) possesses non-empty compact wlimit set "(0. Following the arguments the theorem 3 proof we derive the validity of theorem 8. 0

ne c

By assumption Qo, hence Qo for any c 2: 0 and x(t) E Qo for all t 2: O. Theorem 3 is proved. 0

5. CONCLUSION

Proof of theorem 8. Take any point Xo E Qe and consider the solution x(t) = x(t,xo) of the closed loop system (1), (11) . Following the arguments of the theorem 5 proof we obtain that the trajectory x(t) is well defined on [0, t.) and along this trajectory the following relation

In this paper we consider the problem of the global (local) stabilization invariant manifold of the general nonlinear controlled systems. It is assumed that this manifold can be described by zero value set of some smooth nonnegative scalar function. The main contribution of the paper are the new sufficient conditions and the description of the set of state feedback regulators providing the asymptotic stability of the goal manifold with the specified region of attraction.

t

Q(x(t)) - Q(xo) ::; -C

J

a(x(r))lz(x(r)Wdr(27)

o holds for t E [0, t.). Here z(x) C = (VJo - {3) > O.

= IJf ([LgoQ(X)]T),

Even in a particular case when the desired attractor of the controlled system consists of one point of equilibrium (zero-state stabilization problem) we suggest the new sufficient condition guaranteeing global (local) asymptotic stabilizability. The proof of the obtained results are based on the detailed analysis of the w-limit sets of the closed loop system with the regulators derived by the speedgradient method.

Let us show that x(t) is well defined on [0, +00). Indeed, x(t) satisfies the equation

x = f(x, u(x)) and from the proof of theorem 5 we know that lu(x)1 ::; 1. By assumption the function f(x, u) is bounded on the set A = Qe x {u ; lul ::; 1}

217

The work was supported in part by RFBR (grant 96-01-01151), INTAS (project 94-965) and the Russian Federal Programme "Integration" (project 2.1-589) .

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