Adaptive tuning of bifurcation for time-varying nonlinear systems

where 'I' E RP and ~ E R I are estimates of y and z. Recall that the essentially bounded function B( t) is called PE if there exist positive constants L and S, such, that for any t

~

0

t+L

J B( s ) B( s ) T ds ~ Sip,

Ass u m pt ion 1. For any Lebesgue measurable and essentially bounded inputs J.l( t ), d( t) and any E

•

Let us discuss the above assumptions. The Assumption 1 ensures existence of original system solutions for all t ~ 0, see also paper (Angeli, and Sontag, 1999) for necessary and sufficient conditions of forward completeness. Assumption 2 claims that the system (2), (3) has bounded solution in the bifurcation point J.l = J.lo for pointed class of inputs from

is directly measured, that is a realistic assumption for some examples (e.g. for some biological systems). To consider the systems which are on the border of stability let us introduce the set 1) of all input func-

initial conditions x( 0)

•

Ass u m pt ion 4. For given input dE 1) signal B( y( t)) is persistently excited (PE) (Fomin, et af.,

The posed problem differs from standard adaptive observer design problem due to presence of the feedback via J.l in equations (2), (3), i.e. it can be classified as adaptive observer based controller design. An additional difficulty is in that the solutions of the system (2), (3) may not be assumed bounded for any values of J.l since it is not the case near bifur-

= J.lo.

0, where I I is

identity matrix with dimension I x I .

t~oo

cation point J.l

~

t

where I p is identity matrix with dimension p x p .

Rn system (2), (3) has well

defined solution x( t) for all t pleteness property).

~

0 (forward com-

To solve the posed problem it is suggested to adjust the estimates J.l of unknown parameters J.lo by the

•

speed gradient algorithm: Ass u m pt ion 2. For given signal dE 1) solutions ofthe system y = Al Y + A 2Z +
~ = -y p] B( Y) T (y - 'I' ), y > 0 .

Therefore, the proposed adaptive observer based controller is described by equations (4)-(6). The main result of the paper is as follows.

Z= A 3 z+
are bounded for any initial conditions x(O)

E

The 0 rem 1. Let for the system (2), (3) Assumptions 1-4 hold. Then solution of system (2)-(6) is bounded. Furthermore, let one of the following conditions hold: 1) k 2 > 0;

bounded input 8( t) = col( 8f (t) 8~ (t)) provided that there exists a function X E K£ with property 1 8( t ) I

~ X( 11 8 11, t ), t ~ 0 .

(6)

•

2) A 2 =0;

x: R~o x R~o ---» R~o

3) system S= A 3 s has an invariant solution s( t),

is from class K£ if it is positive definite and nondecreasing in the first argument for any fiXed second one, and strictly decreasing to zero in the second argument for any fixed first one.

such, that A 2 set) = b for a vector bE RI and

Note that a continuous function

B( y( t)) is non constant function of time t ~ 0 for

given signal d E 1) ; 4) system S= A 3 s has an invariant solution s( t), such, that A 2 s( t) =t:- const and B( y( t) ) = B . Then the following limit relation holds:

Ass u m pt ion 3. There exist some matrices K},

K 2 with dimensions (p x p), (I x p) and positive

686

.

(7)

s= A 3 s

has an invariant

lim Jl( t ) = Jl 0

IAI=L:laijl·

t~oo

Additionally, if the system

i,j

Then auxiliary input 8 satisfy the constraint

solution s( t) such, that A 2 s( t ) ~ consl, then the

18( I ) I~ [ IA l + K I I+ IK 2 I+] p( IX( 0 ) I, kIt ),

relation (7) holds for almost all inputs d E 'D . Proof. Let us consider differential equations described dynamics of observer estimation errors e=y-'I', £=z-~, which can be written as follows: e=-K 1e+A 2£+B(y(t»)(Jl-Jlo); (8)

+LJ,y+L~

and variables 'I' and ~ are bounded. Now the boundedness of the overall system solution follows from the boundedness of e and £. Finally, let us prove the convergence of Jl( t) to its

t = A 3 £- K 2 e.

(9) For nonautonomous dynamical system (6), (8)-(9) (with unbounded regressor function B(Y(/»)) it is possible to introduce a Lyapunov function

desired value Jlo. First of all note, that according to

e

W( e,£, w) = eT PIe + £T P2£+ y-l (Jl- JlO)T (Jl- Jlo). Its time derivative with Assumption 3 takes the fonn:

W~ -k1 1e 1

2

-k 2 1£1

2

also goes to zero with I .

X=co/(yT zT 'I'T~T JlT) is well defined for all t ~ O.

and the system is

asymptotically stable with respect to part of variables e. The presence of multiplicative coefficient k1 in the above inequality can be explained in the following way. Let

Oziraner, 1987) one can obtain, that the system is asymptotically stable with respect to variable £. In this case equality (10) can be reduced to B( y( t) )( Jlo - Jloo ) = 0 , and the relation (7) follows from Assumption 4 and result from (Fomin, et aI., 1981). The same arguments are valid for the case of the second condition of the Theorem, then A 2 = 0 . Left hand side of constraint (10) is a solution of subsystem (9), which on co -limit trajectories has autonomous form t=A 3 £, (11)

~ -I e(t) 1 ~ Ie(/) I ~ p(1 X(O) 1,1 ), 2

then introducing time reparameterisation 1= k 1 't one can rewrite the last property as follows:

W(t)~-kde(t)12 ~ le(t)l~p(IX(O)I,klt), that was needed to prove. Further, it is possible to rewrite observer equations as follows: \jI = A 1'I' + A 2~ +
+(
and possibilities of fulfilling (10) depend on properties of system (11) solutions. If third condition is unsatisfied for B( y( I) ) = const and there exists an

~=A3~+
+ (
Iyet) I

E

holds for all t

~

invariant solution £( t) of (11) with A 2 £( I)

N

instant of time and N c RP is some compact set (Angeli, and Sontag, 1999). It follows from continuity assumption imposed on functions
I

=b

for

constant vector b E Ri , then equality (10) can be true for non zero vector Jlo - Jloo' However, such situa-

T , where T > 0 is arbitrary finite

I

~

where Jloo E Rq is an asymptotic value of Jl( t) . if Equality (10) can be satisfied A 2 £(/) = Jlo - Jloo = 0, I ~ 0 and in this case the Theorem is proven. Let us prove the property (7) in other cases. Let the fITst condition be true and k2 > O. Using arguments from (Rumyantsev, and

I,k

W(/)

Additionally, from

0, I ~ + 00 follows. Hence, from (8) the following relation is satisfied along the u> -limit trajectories of the system: A 2 £( I) = B( y( t) )( Jlo - Jloo ), I ~ 0, (10)

)le t)

Further, using results from Rumyantsev, and Oziraner (1987) we claim that the estimate Ie(t) 1~ p(1 X(O) l t )

p E KL

~ +00.

boundedness of y and attractiveness of e property

Since W~ 0 the variables e, £, Jl are bounded. Hence from Assumption 1 the whole system (2)-(6) is forward complete and solution of the system

holds for some function

e

continuity property of function B variables and are bounded functions of time. Combining this fact with attractiveness property of e we conclude (similarly to (Fomin, et al., 1981; Ljung, 1978)) that e(/)

tion is excluded by the third condition of the Theorem and if B( y(t) ) ~ consl , then equality (10) can be true only for b = 0 . Indeed, left side is a constant while right hand side is a non zero function of time, that is a contradiction. If the fourth condition holds, left hand side of constrain (10) is a non constant function of time, while the right hand side is a non zero constant. This contradiction shows that both sides are equal to zero and system (11) have to converge to its trivial solution A 2 £(t) = O.

I

where i = 1,2, L~ is Lipschitz constant for function

181 (t) I~ IAl + KIll e( t ) I+ LJ,y Ie( t ) I' 1 8 2 ( t ) I ~ I K 2 11 e( I ) I+ L7v Ie( I ) I'

Now let us analyze the general case (fifth condition). Suppose that system (11) admits non constant time invariant solution £( I) with time-varying

where norm of matrix is calculated as the sum of all its elements:

687

A 2 t( t ) -:;; 0, t

~

0, such, that equality (10) is true.

B( y( t) ) -:;; const

Then the function

sions such that P(A+KC)+(A+KC)T P < 0,

too (if it were

constant the fourth condition of the Theorem could be applied). Therefore, (I 0) is true for all t ~ 0 if and only if time derivatives of right and left hand sides of ( 10) are also equal to each other for all t ~ 0 : dB(y(t) )[ A 2 A 3 t(t)= A]y(t)+A 2 z(t)+ d y( t) (12)

PB(y)=CTG(y). Then systems (4) and (6) take fonn V= A 'I'+
Jl = -y G( y) T (y -

C \11)

and all the statements of Theorem 1 hold. In other words, the relaxed assumption is valid if the linear part of the system (l a) is passifiable which is equivalent to its hyper-rninimum phaseness (see • Fradkov, et al., 1999).

+
In order to illustrate verification of the conditions of the Theorem and Assumptions below some examples are presented.

on structure properties of matrix B derivative). Therefore, in general case equality (10) can hold with A 2 t(t) -:;; 0, t ~ 0 for any given X(O) only for restricted subclass of d in set V. Indeed, t( t) is a

I I

bounded solution of linear autonomous system (I 1) and all such invariant solutions are oscillation functions of time with some frequencies. If for some frequency of signal d 1 equality (12) is true, then for any

I I I

I

I

I

I I I

I I

I I

I I

I

I

I

I

I

I

I

-------l-------L------L-~----J-~--------~------Ir------r------,---

40

I I I I I

20 ----

-- --

I I I

I

- ---

I I I.

I I I

I I

I I

-- ---

-----

i------~------~------i-----I I I I I I

a.

I I

I I

I I

--o.. . . . .'----_-.. . .-_-__ . -_-_-...,&....L._-_-_-_--_-....L_-_-__-_-_-.......l-_-_-_-_----I o 40 80 --~------~------r------~-----I I I I I I

I I

I I

I I

I

I

I

I

~

0.8 b. -2

10

5

:

J

I J

:

I I I

j! ! o ,---t-----r---------r--------Vi

:

-0.5

-0.4

I

I I I

5

10

-0.8

c.

2

I. Note, that in the case then A 2 = 0

I

I

I

I

I

I

I

I

~---i------~------~------i-------

1:

:

:

:

~

----J-------L------L------J-------

-----~------~------~------~------I

I

I

I

I

I

I

I

I

I

I

I

_IL

LI

~

40

I JI

_

80

--Vl I

(for example, this is the situation with y = x , then all

-

components of state space vector are directly measured) it is possible to use in proof of Theorem 1 another Lyapunov function -1

I

~-------~------}------~-------

o

t

Fig. 1. Trajectories for neural integrator.

T

I

-----~------~------~------~------I I I I

:

I I I

I

r~~j,-------r------!------!-------

e2

I I I

' I

I

0.4 -

o -----JC--l---------l---------

b· •5

------t-------~------t------1-------

__

/~-'-r

-.J../

-

I

------~------~------~------~---l-

60

------~

Remark

I

------~------~------~------~-----I I I I

a.

other choice of the frequency of d 1 this equality would be violated. This fact finishes the proof. _

~

I

-

I I

I I I I I I

T

I I I

I

I

I I I

I I I

I

I

I

I

I

I

I

I

I

----.------~------~------~-----I I I I I

W(e,w)=e P]e+y (J.!O -J.!) (J.!O -J.!) instead of W . And that is more, in this case there is no need to estimate variable z in adaptive observer, that in such situation takes form (6) and (4): V = Al y( t) +
I

----~------._------r------~------

I

I

I

: : : : J}-----l------J-------l------l-----I

I

I

I

I

,

I

I

I

I

I

I

I

I

I

I

0'"--------'---.....1.---"-----'----' o 40 80 Fig. 2. Resonance regime tuning for the pendulum trajectories.

Rem ark 2. Assumption 3 can be relaxed. It is sufficient to assume that in the general form of Lurie system (la) there exist matrices K, P = pT > 0 and matrix-function G( y) of appropriate dimen-

3. APPLICATIONS

1. Neural integrator. In work (Moreau, et aI., 2003)

688

the following model of neural integrator was used x= (Jl - Jlo ) x + d to design a dynamical feedback Jl provided bifurca-

which in bifurcation point Jl = Jlo is described as series connection of nonlinear pendulum subsystem (Y, Zl ) and linear oscillation second order subsystem

tion tuning for case d = O. For this system with Y = x adaptive observer equations (4}-(6) take form:

(zl' z2) forced by external input d. For bounded inputs Jl( t) and d( t) this system has right hand side

~ = k, (y - 'V ) + d(t) ;

bounded by the norm of space vector, the last fact means, that the system admits Assumption 1. Using mentioned decomposition of the system on series connection of subsystems it is possible to show that

Jl = -Y Y ( Y -

'V ), Y > 0 , For this system Assumption I is satisfied as for linear autonomous system, Assumption 2 holds for d( I) = sine rol ), 00 > 0 which form set of admissible

for d(t) = asin( oot) with a ER, 00 E R~o/{J2) this system possesses conditions of Assumptions 2 and 4. Assumption 3 also holds for adaptive observer:

inputs 1) during simulation. Assumption 4 also follows for such class of inputs. Conditions of Theorem I are valid with A 2 = O. The examples of system trajectories are shown in Fig. 1 for Jlo

= 1,

00

~=SI+kl(Y-'V);

= I,

SI = -

Y = 1 and k l = 1 .

;

S2 = -2s 1 +d(t);

2. Resonance tuning of a pendulum. Let us consider the pendulum equations xl =x2;

i

sine Y ) + ( Y - 'V ) + S2

Jl = -y Y ( Y -

'V ) .

2 2 =(Jl-00 )sin(xl )+d(t),

where 00 > 0 is unknown natural frequency of the pendulum; Jl E R is the adjusted parameter, as before, that is introduced to tune pendulum frequency to the desired value; d( t) = sine ood t) is a sinusoidal signal with known exciting frequency ood > 0 . The problem is to tune the resonance regime of the pen" . h th ' e tunmg error Jl = 00 2 - 00 d2 du Ium to dImlnlS when natural frequency of the pendulum and external input d coincide and pendulum solution exhibits oscillations with infinite amplitude. Such a problem is important for practice, since resonance regime with growing amplitude arises only for coinciding frequencies of the pendulum and the input d . It can be generated by designer for the case of known frequency ood' while the natural frequency of the pendulum 00 may depend on uncertain and unpredictable external factors and its exact value is unmeasured. Let y = col( Xl xl) and pendulum equations can be rewritten in form (2), (3) as follows Xl = x2; x2

= -oo~ sine Xl ) + (Jl- Jlo ) sine Xl ) + d( I),

where Jlo =

00

2

-

oo~. In this case also A 2

e

= 0 and,

-1

due to Remark 1 adaptive observer takes form

W2 =-oo~sin(YI )-k l (Y2 Jl = -y sine Yl ) (Y2 - 'V 2 ).

2

It is possible to show, that conditions of the Theorem hold in this example, especially PE condition is satisfied for sinusoidal function of runaway argument. Trajectories of this system simulation are shown in

Fig. 2 for

00

=

J3 '

ood

=I ,

_·· __··-t-·-··-T

c.

-\V2 )+d(t);

. _.

~

:

-0.5

··---:-"-·· i . _ -!

-"-l--"-"-~-"-"--:

-+--

-_..

~

_.~

._-

i----+---1--

~

Y= k l = I .

o _.- ·--··--f-··_·_·_··_+···_··__·· _· _. ·t·.._··_·_··_··_·~ . _._-- _.. ;

! ~

; !

! !

-l~:+=~=====~== o

3. Oscillalor with changing zero dynamics. Let us consider a dynamical system of the form (2), (3): -'v = zl + (Jl- Jlo Lv; 2- 1 = -sin(y) + z2;

40 80 Fig. 3. Trajectories for complex oscillator.

22 = -2z l + d(t),

689

Here A 2

*0

Fomin, V.N., A.L. Fradkov and V.A. Yakubovich

and system (11) has fonn: Et

(1981). Adaptive control of dynamical plants.

= E2;

Moscow: Science, p. 448. (in Russian) Fradkov, A.L. (1995). Adaptive synchronisation of hyper-minimum-phase systems with nonlinearities. Proc. of jrd IEEE Mediterranean Symp. on New Directions in Control. Limassol, 1, pp. 272 -277. Fradkov, A.L., LV. Miroshnik and V.O. Nikiforov

E2 = - 2E t, while solution of y -subsystem for pointed class of inputs d performs nonlinear oscillations with different frequency. Thus, fifth condition of Theorem 1 is satisfied. An example of system trajectories is presented in Fig. 3 for Jlo = 1, k 1 = 0.5, 'Y = 5, a = 1 and

0)

(1999). Nonlinear and adaptive control of complex systems. Kluwer Academic Publishers, 528

= 2.

p. Fradkov, A.L., H. Nijmeijer and A. Markov (2000). Adaptive observer-based synchronisation for communications. Intern. J. of Bifurcation and Chaos, 10, 12, pp. 2807 - 2814. Fradkov, A.L., V.O. Nikiforov and B.R. Andrievsky (2002). Adaptive observers for nonlinear nonpassifiable systems with application to signal transmission. Proc. 41 th IEEE Con! Decision and Control, Las Vegas, 10 -13 Dec. Krstic, M., L Kanellakopoulos and P.V. Kokotovic

4. CONCLUSION In this work a new adaptive control problem is posed, which is oriented on adaptive tuning of bifurcation regimes of nonlinear systems. This problem formulation differs from previously proposed in (Moreau, and Sontag, 2003; Moreau, et al., 2003) due to external exciting input presence is taken into account, a general form nonlinear system is considered and in common case only part of variables assumed to be available for measurements. An adaptive output feedback controller is proposed which tunes a nonlinear uncertain dynamical system to its bifurcation point under some mild conditions. This solution is based on theory of adaptive observers design proposed in (Fradkov, 1995; Fradkov, et ai., 1999). Possible interpretations of proposed solution in biological applications can help in investigation of those systems. Rather restrictive assumption on relative degree of the system can be relaxed using construction of adaptive observers from (Efimov, and Fradkov, 2003; Fradkov, et aI., 2002) and it is a way for further investigations. Series of examples with computer simulations demonstrate good potentialities of the proposed solution.

(1995). Nonlinear and adaptive control design. New York Wiley. Leung, A.Y.T., J.J. Chen and G. Chen (2004). Resonance control for a forced single-degree-offreedom nonlinear system. Intern. J. of Bifurcation and Chaos, to appear. Ljung, L. (1978). Convergence analysis of parameteric identification methods. IEEE Trans. Aut. Confr., AC-21, pp. 779 - 881. Loria, A., Panteley, E., D. Popovic and A.R. Teel (2002). 8-Persistency of excitation: a necessary and sufficient condition for unifonn attractivity. Proc. 41 th IEEE Conf. Decision and Control, Las Vegas, 10 -13 Dec., pp. 3506 - 3511. Moreau, L. and E.D. Sontag (2003). Balancing at the border of instability. Physical Review, 68, pp. 1-4. Moreau, L., E.D. Sontag and M. Arcak (2003). Feedback tuning of bifurcations. Systems and Control Letters, 50, pp. 229 - 239. Rumyantsev, V.V. and A.S. Oziraner (1987).

5. ACKNOWLEDGEMENT This work is partly supported by grant 02-01-00765 of Russian Foundation for Basic Research, by Russian Science Support Foundation and by Program of Presidium of Russian Academy of Science N~ 19.

Stability and stabilization of motion with respect to part ofvariables. Moscow: Science, 263 p. (in Russian)

REFERENCES Angeli, D. and E.D. Sontag (1999). Forward completeness, unboundedness observability, and their Lyapunov characterizations. Systems and Control Letters, 38, pp. 209 - 217.

Bifurcation control: theory and applications. Lecture Notes in Control and Information Sciences (2003) (Chen, G., D.J. Hill and X. Yu (Eds.)). Berlin: Springer. Chen, G., J.L. Moiola and H.O. Wang (2000). Bifurcation control: theories, methods, and applications. Intern. J. ofBifurcation and Chaos, 10, pp. 511 - 548. Efimov, D.V. and A.L. Fradkov (2003). Adaptive Nonlinear Partial Observers With Application To Time-Varying Chaotic Systems. IEEE Conf. Control Applications. Istanbul, June 23-25.

690