- Email: [email protected]
Chaos Control of Lorenz Model via Direct Gradient Descent Control
IFAC
Copyright <0 IFAC Nonlinear Control Systems, SI. Petersburg, Russia, 200 I
[: 0
C>
Publications www.elsevier.comllocatelifac
CHAOS CONTROL OF LORENZ MODEL VIA DIRECT GRADIENT DESCENT CONTROL Takeshi Yamaguchi' Kiyotaka Shimizu'
• Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, JAPAN
Abstract: This paper is concerned with controlling chaos. Lorenz model, a well known chaotic system, is picked up as a controlled plant in which manipulated variables are additive control inputs. For a controlling technique, we apply a method called the direct gradient descent control(DGDC), which is a dynamic state feedback controller for general nonlinear systems. DGDC manipulates control inputs directly so as to decrease a performance function like the squared error, based on the gradients of the performance function and the steepest descent method. Several simulation results demonstrate the usefulness of the proposed method for the Lorenz chaotic system. Copyright fl) 20011FAC Keywords: Controling Chaos, Nonlinear Control, Dynamic state Feedback, Steepest descent, Stabilization, Regulator control
1. INTRODUCTION
Affter 1990, the studies of analysis and control about chaos have been extensively made. The work of pioneer, Ott, Grebogi and York, is called O-G-Y method(Ott, et al.,1990) . They proposed a method to stabilize an unstable periodic orbit by making some small time-dependent perturbation on one of adjustable parameters. After that, various techniques have been proposed in the field called chaos control. By the way, driving a strange attractor to a periodic orbit was the main work of chaos control, but driving to a desired steady state is also important. So, the purpose of this paper is asymptotical stabilization to the desired state. Studies on nonlinear feedback control have been extensively made in recent years . The direct gradient descent control (DGDC) was proposed by Shimizu et al. (1998), which directly manipulates control inputs so as to decrease a performance function such as the squared error from the desired equilibrium state, based on the gradient of performance function with respect to control inputs. Stability of DGDC can be proved by the comparison function method(Shimizu, et al.,1999).
ling Lorenz Chaos model in which control inputs are additive to the state equation or multiplicative to the state. In (Yu, 1996), by using VSC, asymptotical convergence to the unstable equilibrium was achived. But, the control input is multiplied to the nonlinear term. In (Tian, 1997), nonlinear state feedback is used for the additive control input. Moreover, in (Chen, 1994), by Lyapunov method, asymptotical convergence to the unstable equilibrium is achieved . In (Fradkov and Pogromsky, 1998) it is illustrated that the speed gradient algorithm can control Lorenz model. In this paper, we add the control input to the Lorenz model, and aim to make the chaotic motion converge asymptotically to the unstable equilibrium. We corroborate that the direct gradient descent cotrol, which is effective to nonlinear general system, is also effective to the chaotic system. 2. DIRECT GRADIENT DESCENT CONTROL Let us consider a general non linear plant. The aim of control is to manipulate a control input u(t) so that a performance function F(x(t)) decreases : (la)
decrease F(x(t)), U(t)
In this paper, we adopt Lorenz model for a controlled object , and apply the direct gradient descent control. There are many studies on control-
subj.to
789
x(t)
= f(x(t),u(t)),
x(O)
= Xo, (lb)
where x(t) E Rn is a state vector and u(t) ERr is a control vector. Under appropriate assumptions plant (lb) has a unique continuous solution x : x(t), t ~ 0 for any continuous u : u(t), t ~ O. We denote the state trajectory x associated with a given u by x(u), whose value at t will be denoted by x(t;u). Problem (1) is such a problem that we directly manipulates the control input u(t) so as to decrease the performance function F(x(t)). As a mean of on-line control for problem (1) let us execute the gradient descent method (the steepest descent method) with respect to u(t). As a class of admissible control for the fixed interval [0, tj, we consider the space U[O ,t] consisting of r-dimensional-vector-valued continuous functions, and define the inner product :
J t
(u, v)U 1o •t1 ~
u(Tf VeT) dT
(2)
o Let us define a functional by t::.
[uj = F(x(t;u))
where .c ~ diag( 001, ' . . , Oor), 00; > 0, is a proportional canstant. Substituting (5) into (6) yields
it(t) = -.cfu(x(t;u),u(t)fFx(x(t;u)f . (7) However, this control law has only the work of decreasing F(x(t)). Since stability of control input u(t) is not taken in account, it is considered that the control law (7) is not enough for the stability of closed loop system. Accordingly, to make sure of stability we consider to add a penalty term on u( t) and let us introduce a penalty function P(u(t)). We consider a control law which decreases both the performance function F(x(t)) and penalty function P(u(t)) simultaneously. We introduce the quadratic penalty function as 1 P(u(t)) = 2(u d - u(t)f R(Ud - u(t)), (8) where Ud is the corresponding control and R is (normally diagonal) (r x r) positive definite matrix. Thus gradient descent control algorithm with the penalty on u(t) becomes as follows:
(3)
of time interval [0, t], and the derivative of [uj with respect to u as bellow.
it
it(t)
[Assumption 1) f is continuously differentiable [Assumption 2) f u' Fx are Lipschitz continu-
=
-.c{/u(x(t;u),u(t)fFx(x(t;u)f -R(Ud-U(t))} , u(O) = Uo . (10)
Indeed, this will be executed by simultaneous differential equations
ous in (x,u).
x(t) it(t)
Then we have the following Lemma.
= f(x(t), u(t)), x(O) = xo, = -.c {/u(x(t), u(t))T Fx(x(t))T
[Lemma 1) Under Assumptions 1 and 2, the oper-
ator x(t; .) : U[O,tl ~ R is Gateax differentiable, and its lacobian is given, at right-end time t, as follows: V'x(t; u)(t)
= fu(x(t; u), u(t)f
(9)
Substituting (5) and (8) into (9), we obtain
We make the following assumptions.
in (x,u).
= -.c {V' 4>[u)(t) + V' P( u(t))}
-R(Ud - u(t))} , u(O)
(11) = uo.(12)
We call the control law (10) the direct gradient descent control(DGDC) . 3. STABILITY OF DGDC
(4)
(Proof) See (Shimizu, et al., 1998). 0 From Lemma 1, we obtain the derivative of (3):
In this section, we investigate a sufficient condition for asymptotical stability of DGDC (11),(12).
[Theorem 1) Under Assumptions 1 and 2, the functional : U[O ,t] ~ R, deffined by (3), is Gateax differentiable, and the gradient V' [u] E U[o.t] is given, at right-end time t, by
For simplicity put
V'[u](t) = fu(x(t;u), u(t))TFx(x(t; u)f· (5) (Proof) See (Shimizu, et al., 1998). These expression of derivative function can be obtained by caluculating Gateax differential and using the Riesz-F'retchet Theorem. 0 By using the so defined gradient function, we apply the steepest descent method at each time t E [0,00) as on-line control law for problem (1). Namely, u(t) is modified by gradient descent control algorithm
it
=
-.cV'[u](t) , u(O)
=
Uo.
G(x,u) ~ fu(x,uf Fx(xf
+ Pu(x)T.(13)
Now we make the Taylor expansion of the system (11), (12) in the neighborhood of (Xd, Ud) = (0,0) and express it with a linear term plus more than 2 degree one as follows.
X(t)] [ u(t)
=
[f(X(t), u(t)) ] -.cG(x(t), u(t)) (0,0)
+ D [x(t)] + u(t)
[91g2(X(t), (x(t), U(t))] u(t))
(14)
where
D ~ [fx(x,u)
fu(X,U)] -.cGx(x , u) -.cGu(x, u)
(6)
790
(15) (0 ,0)
For (21) and (22), D is concretely obtained as and 9i(x(t), u(t)), i = 1,2 , are the sum of more than 2 degree terms. Since the first term in (14) is zero at the equilibium (0,0), (14) becomes X(t)] [ u(t)
=D
[X(t)] u(t)
+
D=[
Jx(O,O)
Ju(O , O)] -£R .
-£Ju(O,ofQ
(23)
And assume: [Assumption 5]By choosing proper Q, Rand £, matrix D given by (23) can be made asymptotically stable, that is, there exist w > 0 and a > 0 such that lIeDtl1 :S ae- wt .
[91(X(t),U(t))] (16) 92(x(t), u(t)) ,
and its solution satisfies the following integral equation. X(t)] =eDt [X(O)] [ u(t) u(O)
Applying Theorem 2, we obtain: [Corollary l]Consider DGDC (21),(22). In regard to the expanded system (16) with D given by (23) corresponding to the DGDC (21),(22), assume that Assumption 5 and 4 holds. Then
t
+eDtJe- DT [91(X(r) , u(r))] dr (17) 92(x(r), u(r))
°
We assume the following assumptions.
the norm of
[:~g]
is bounded from the above
[Assumption 3JBy choosing proper F(x), P(u) and £, matrix D given by (15) is asymptotically stable, that is, there exist w > 0 and a > 0 such that lIeDtll :S ae- wt .
and evaluated with (18). Furthermore if [ :~~~ ]
[Assumption 4] There exist constants 8 > 0 and > 0 such that
stable, and we have
satisfies (19), (16) is exponentially asymptotically
(J
11
[:~~:m: :~m ] :S (J :~~n !rH 11
:m]
is bounded from
above as shown in the following expression :
D
(18)
11
[:~~n
11
<
(al~O(J) t :~~~]
-t
°as t
[_£:TQ
Y=
- t 00.
-~R] '
(24)
ex,
(26)
where x E Rn,u E Rr,y E Rr are state, input and output vector respectively. [Definition 1] (Relative degree(Isidori,1995)) The linear system (25),(26) has a relative degree { ql , ... , qr} if
o Problem with quadratic performance function is important from an application point of view. Consider the problem 1 decrease F(x(t)) = -x(t)T Qx(t), (20a) U(ij 2 subj.to x(t) = J(x(t), u(t)), x(O) = Xo. (20b)
u(t)
=
In preparation let us consider a linear system whose input and output have the same dimension x = Ax + Bu, (25)
(19)
(Proof) See (Shimizu, et al., 1999). Based on the comparsion function method, it can be proved by using generalized Gronwall-Bellman inequality.
Then DGDC system (11),(12) becomes x(t) = J(x(t), u(t)) , x(O) = Xo,
- t 00.
We have to choose Q, Rand £ to satisfy Assumption 5.
then DGDC (11), (12) is exponentially asymptotically stable, and we have [
as t
Let us consider problem (20). Putting A ~ Jx(O,O) and B = Ju(O,O), D given by (21) becomes
11[:mlll s [('11 [:~~m~ -~fp'-'-dTl ~~-, Further, if it holds that
-t
Corollary 1 assures that if the linearized system of DGDC (11),(12) is asymptotically stable (i.e, D is Hurwitz), there exists asymptotically stable region of DGDC in the neighborhood of the equiriblium and its evaluation can be made. But it is not so easy to choose Q, Rand £ satisfing Assumption 5. In the following, let us study a method for it.
11 [
[Theorem 2]In regard to the expanded system (14) i.e. (16), assume that Assumption 3 and 4 hold. Then the norm of [
[:m] °
(i) For all k
< qi, {) (k)
!i
= 1,2, . .. ,r = 0 , for all 1 :S j i
uu· ..) th J. (n e r x r matnx -{)-.uJ gular.
[8y/q,)]
:S
r ,
. nonstn.
ts
i:Si, j:Sr
If the linear system (25),(26) has relative degree {I, ... , I} , then there exists the nonsingular coordinate transformation
(21)
= -£ {J u(x(t), u(t)f Qx(t) + Ru(t)} , u(O) = Uo . (22)
791
which transforms the linear system (25) ,(26) into a normal form
= QllZ + Q12e , {= Q21Z + Q22e + CBu , y = e. z
Because of the assumption of theroem, (32) is Hurwitz, (37) is asymptotically stable, and thus system (33), (34) is minimum phase. Futher matrix CB for Proposition 2 corresponds to
(27)
(28)
(29) Here z = QllZ is called zero dynamics, and when zero dynamics is asymptotically stable, the linear system (25) is said to be minimum phase.
[BTQ ,
v
[Proposition 1](Isidori, 1995) Suppose that the system (25) has the relative degree {I, . . . , I}, and matrix CB is positive definite. And assume that the system (25), (26) is minimum phase. Then there exists the output feedback control
is asymptotically stable.
which asymptotically stabilize the system (25).
(31)
In general, the Riccati algebraic equation is given;
AP+AT p-PBR- 1B T p+Q'=0. (41) (32)
We make the following assumptions.
is Hurwitz, then there exists a matrix .c diag{a1' . .. , ar}, a; > 0 such that (31) is asymptotically stable.
[Assumption 6]{A, B} is stabilizable. [Assumption 7]{ A, C} is detectable, where C is
(Proof) (25) can be rewritten as
any matrix such that
= BTQx + Ru.
We can conclude that if P is used as Q of squared error performance fucntion (20a) , then (32) is Hurwitz, namely the condition of Theroem 3 is satisfied. Based on the previous discussion, we propose the following design procedure.
(34)
At first , we show that the linear system (33) ,(34) is minimum phase.
oil
ov
= ~ (Rv) = R
ov
er C ~ Q' .
Under Assumption 6, the Riccati algebraic equation (41) has a unique solution P ;:- o. If Assumption 7 is verified, then x(t) = (A-BR- 1BT P)x(t) is asymptically stable(Anderson, 1990).
where v ERr , v = U is a new control input, x and u are state. And set an output y E Rr as y
0
It is difficult to choose matrices Q, R such that (32) is Hurwitz for high order plant. So we propose the decision method of Q and R , using the Riccati algebraic equation.
[Theorem 3) To check asymptotical stability of D given by (24), consider a dynamic system
Then if R is positive and A _BR- 1B T Q
, a; > 0 (39)
Theorem 3 implies that, if we choose Q and R such that (32) is Hurwitz, then there exists L that satisfies Assumption 5.
Then, we consider about the way of choosing Q, Rand .c to satisfy Assumption 5. Applying Proposition 1, we obtain the following theorem.
-~R] [:~~n ·
= -.cy, .c ~ diag{a1, .. . , ar}
such that
u=-Ky,K~diag{l\:l ' .. . , I\:r} , 1\:;>0 , (30)
= [ _.c~T Q
(38)
which is positive definite . Applying Proposition 1 to the system (33) , (34) , we conclude that there exists output feedback
Using the obvious properties of the system (25) , we can obtain the following proposition.
[~~~n
R][~]=R>O,
(35)
[Algorithm A]
implies that relative degree of (33) , (34) is { 1, .. . , I} as R > 0 . Then
I:::.
I:::.
[1J Calculate A = /x(O,O) , B = /u(O , O) at the equilibrium x = 0, U = of plant (20b). [2] For Q' satisfing Assumption 7 and R > 0 get a solution P of the Riccati algebraic equation (41). [3J Set quadratic performance funciton (20a) as 1 F(x) = "2x(tf Px , (42)
transposes system (33) ,(34) into a normal form
z=(A-BR- 1BT Q) z+BR- 1e , {=BTQ (A - BR- 1B T Q) z+B T QBR- 1e+Rv,
°
and calculate the DGDC (21),(22).
y={ .
Under the above design method , DGDC's parameter .c exists such that locally asymptoticar stability of non linear plant is obtained.
So the zero dynamics of (33) , (34) is given by
792
4. CONTROL OF LORENZ CHAOS (Simulation Results)
D=
The Lorenz model was modelled and studied in (Lorenz, 1960) as follows : Xl =--aXI(t)+UX2(t), (43a)
X2 =-XI(t)X3(t)+1'XI(t)-X2(t),
(43b)
where Q is a symmetric matrix . To confirm asymptotical stability of D based on Theorem 3, we checked the stability condition from the characteristic equation of (A - br-1bT Q) by RouthHurwitz method, that follows 1 41 1 r - qll > - - and r - qll > 0 3 722 r - 1(11 -"3qll - 28ql2 ) <-"""3
~X3(t).
(44b)
(44d)
From (11) and (12), we obtain the DGDC as u(t) = -et{ ql1XI (t)+QI2X2(t)+qI3X3(t) +ru(t)} (45) We set u(O) obtained as
= o.
where A
= Ax(t) + bu(t) ,
=
(46)
b= P
And the eigenvalues of A will be s = -22.8277, 11.8277, -2.6667, so the system is unstable. Next , to examine the controllability of the system, we check the controllability matrix as
M = [b Ab A 2bJ =
0
=
23.712 18.491 0.000 18.491 14.454 0.000 [ 0.000 0.000 0.188
1>
0
(48)
For the performance function (20a) , we set Q = P . Changing et of DGDC (45) as 25,30,40, we obtained the results for x(O) = (1 , 2,3) as shown in Fig. 1 . We could confirm that any et makes D asymptotically stable. If we set et larger, then the oscillation will decrease quickly and the speed of converge will become faster. These results indicate the following matter. "Calculate Q of (A - br-IbTQ) obtained by zero dynamics, using the Riccati algebraic equation , and adjust et freely, then the DGDC can asymptotically stabilize chaotic system such as Lorenz model."
[~ ~~O !~g8l o
> 0
Now, since the system is stabilizable, we use Algorithm A to calculate Q from the Riccati algebraic equation (41) . First we choose Q' = I , then C = I. Then the {A, C} is observable, so Assumption 7 is satisfied. A solution P of the Riccati equation (41) for linearization {A, b} is obtained as follows .
The linearization of (44b-d) is
x(t)
2~4 q12 - ~qll)
(47)
If we choose Q and r which satisfy (47) , then (A - br-1bTQ) will be Hurwitz. From Theorem 3, under such Q and r, there exists et > 0, so we can choose suitable et . Namely if there exist such Q, r and et , then the DGDC (45) will be asymptotically stable near the origin. Solving the inequality (47), we see that if we choose Qll and r which satisfies r- I Ql1 > 270, we can stabilize (A _br-IbT Q) . Under such Ql1 and r, there exists et > 0, and so choosing et suitably, we obtained an asymptotically stable trajectory. Using the following method, however, we got a better control result, so we will omit the previous way because of page limitation.
X2(t) = -Xl (t)X3(t) +28xI (t) -X2(t) ,(44c) X3(t) = Xl (t)X2(t) -
1
+ 720 + r- 1 ( -
(44a)
subj.to xI(t)=-lOxI(t)+lOx2(t)+U(t),
I
8 224 3qll+"""3qI21: >720 2 1 (r - 1qll + ~1 ){ _ 73 + r-I (28q12 + 13 qll) }
r-
Taking the desired state Xd as (Xld, X2d, X3d) = (0,0,0) , we simulated the DGDC for the Lorenz model. First of all, we consider controlling Xl. That is, controlling a flow of direction x(horizontal direction that the roll lines) and z (virtical direction) .
,,(t)
0
0 0 8 3
-etql2 -etQ13
(43c) Here Xl is proportional to the intensity of convective motion, X2 is proportional to the temperature difference between ascending and descending currents, and X3 is proportional temperature profile. Constants U > 0 and l' are called Prandtl number and the Rayleigh number respectively. The constant b is related to the given space. If we put U = 10, l' = 28 and b = ~, the unstable equilibium of Lorenz model will be caluculated as (Xl , X2, X3) = (0 , 0, 0), (6y2, 6y2, 27) , (6y2, 6y2, 27) .
The problem can be formulated as 1 decrease F(x(t)) = -2 x(t)TQx(t),
10 -1
0
but rank of M is not 3, so the system isn't controllable. Then, to examine the stabilizability, by the similar transformation of {A , b} we seperate it to the controllable part and uncontrollable part. However, {A , b} is already seperated, and the uncontrollable part is - ~ , so we can know it is asymptotically stable. Therefore the linear 1ized system (46) is stabilizable. Meanwhile, the D given by (24) becomes as follows .
It is very difficult to decide Q by trial and error. But if we have Q = P such as satisfying the Riccati equation, then there exists et which stabi-
793
'"~
:;; '"
:s:...
3 2.5 2 1.5 1 0.5 0 -0.5 ·1
"
XI(l)
\
X2(1 ) x](I) "------.
...
"
\
'" S
:;;
'\
w
'--'--"
0
.;
=25
0.5
1.5
2
2.5
3 2.5 2 1.5 1 0.5 0 -0.5
x,(I) -
x2( r ) .l')(I) ._-----
'"~
5 ~
.;
a=3O
0.5
3
~
0
~
10
11(,)-
10
2.5
i
"
V 0.5
2.5
'. .......
--. 1.5
a=4O
2
2.5
3
11(,)
-
·10
·20
i
-30
-20 -30 -40
·so
=30
-60
1.5
\\
0
-SO
a-25 0.5
I\ '"
10
-40
-40
--
time
-10
-10 -20 -30
0
x,(I)
.,(,) .__._._.-
i\
0
11(,)-
J'\..
0
-SO -60
XI(I)
\,
bme
time
20
1.5
3 2.5 2 1.5 1 0.5 0 -0.5 -1
<>=40
-60
0
0.5
1.5
2.5
0
3
time
time
0.5
1.5
2.5
time
Fig.! In case of control input added to Xl ,
lizes the system, and so we can execute easily the DGDC which guarantees asymptotical stability.
And we can get almost desirable convergence speed of the closed-loop system by adjusting .c.
Next , we considered controlling X2 , adding the in· put to X2 . That is, controlling the proportional of the temperature difference between ascending and decending flows , The problem is then expressed as
6. REFERENCES
decreaseF(x(t))
1
= -x(tfQx(t) ,
2 subj.to XI(t)= -lOxI(t)+lOx2(t), u(t)
Anderson, B.D.O . and J.B .Moore (1990) . Optimal Control -Linear Quadratic Methods-, Prentice Hall, Inc, New Jersey. Chen, Y.H. and M.Y.Chou (1994). Continuous feedback approach for controlling chaos. Phys. Rev. E, 50-3, pp.2331-2334. Fradkov, A.L. and A.Y .Pogromsky (1998) . Introduction to Control of Oscillations and Chaos, World Scientific, Singapore. Isidori , A. (1995) . Nonlinear Control Systems, 3rd ed .. Springer-Verlag. London. Lorenz, E.N. (1960) . Deterministic Nonperiodic Flow. Journal Of The Atmospheric Sciences, 20, pp.130-141. Ott, E., C.Grebogi and J .A.Yorke (1990) . Controlling chaos. Phys. Rev. Lett. ,64, pp.15001501. Shimizu, K., S.lto and S.Suzuki (1998) . Tracking Control of General Nonlinear System by Direct Gradient Descent Method. IFAC Symp. on Nonlinear Control System Design 98, 1 of 3 , pp.185-190. Shimizu, K., K.Otsuka and J.Naiborhu (1999) . Improved Direct Gradient Descent Control of General Nonlinear Systems. European Control Conference ECC'99 Proceedings(CDROM), pdf file no.F0676. Tian , Y. P. (1997) . Controlling Chaos Via Continuous Nonlinear State Feedback. Proceedings of the 36th IEEE Conference on Decision and Control , pp.1502-1507. Yu, X. (1996). Controlling Lorenz chaos. International Journal of System Science, 27-4 , pp.355-359.
(49a) (49b)
X2(t) = -Xl (t)X3(t) +28xI (t) -X2(t)+u(t) , (49c) (49d) From the linearized system, the system was known to be stabilizable at the origin. As the same in the previous case, we could calculate the condition for zero dynamics (A - br-IbT Q) to be Hurwitz through Algorithm A. Then changing a freely, we experienced obtaining desirable performances without harming its stability. From various simulation results it was confirmed that the DGDC achieved excellent performances by use of Algorithm A. 5. CONCLUSION In the paper, we applied DGDC to stabilize a chaotic system, Lorenz model, and obtained satifactory performances. Our results were good enough, compared to the methods proposed in (Chen 1994; Tian, 1997; Yu, 1996). It is interesting that DGDC can carry out stable control even for the chaotic system with conplex behavior. Moreover, Algorithm A offers an easy way how to choose parameters Q, Rand .c such that asymptotical stability of DGDC is assured. The most significant thing is that choosing Q, R and determining .c can be made independently.
794
Copyright <0 IFAC Nonlinear Control Systems, SI. Petersburg, Russia, 200 I
[: 0
C>
Publications www.elsevier.comllocatelifac
CHAOS CONTROL OF LORENZ MODEL VIA DIRECT GRADIENT DESCENT CONTROL Takeshi Yamaguchi' Kiyotaka Shimizu'
• Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, JAPAN
Abstract: This paper is concerned with controlling chaos. Lorenz model, a well known chaotic system, is picked up as a controlled plant in which manipulated variables are additive control inputs. For a controlling technique, we apply a method called the direct gradient descent control(DGDC), which is a dynamic state feedback controller for general nonlinear systems. DGDC manipulates control inputs directly so as to decrease a performance function like the squared error, based on the gradients of the performance function and the steepest descent method. Several simulation results demonstrate the usefulness of the proposed method for the Lorenz chaotic system. Copyright fl) 20011FAC Keywords: Controling Chaos, Nonlinear Control, Dynamic state Feedback, Steepest descent, Stabilization, Regulator control
1. INTRODUCTION
Affter 1990, the studies of analysis and control about chaos have been extensively made. The work of pioneer, Ott, Grebogi and York, is called O-G-Y method(Ott, et al.,1990) . They proposed a method to stabilize an unstable periodic orbit by making some small time-dependent perturbation on one of adjustable parameters. After that, various techniques have been proposed in the field called chaos control. By the way, driving a strange attractor to a periodic orbit was the main work of chaos control, but driving to a desired steady state is also important. So, the purpose of this paper is asymptotical stabilization to the desired state. Studies on nonlinear feedback control have been extensively made in recent years . The direct gradient descent control (DGDC) was proposed by Shimizu et al. (1998), which directly manipulates control inputs so as to decrease a performance function such as the squared error from the desired equilibrium state, based on the gradient of performance function with respect to control inputs. Stability of DGDC can be proved by the comparison function method(Shimizu, et al.,1999).
ling Lorenz Chaos model in which control inputs are additive to the state equation or multiplicative to the state. In (Yu, 1996), by using VSC, asymptotical convergence to the unstable equilibrium was achived. But, the control input is multiplied to the nonlinear term. In (Tian, 1997), nonlinear state feedback is used for the additive control input. Moreover, in (Chen, 1994), by Lyapunov method, asymptotical convergence to the unstable equilibrium is achieved . In (Fradkov and Pogromsky, 1998) it is illustrated that the speed gradient algorithm can control Lorenz model. In this paper, we add the control input to the Lorenz model, and aim to make the chaotic motion converge asymptotically to the unstable equilibrium. We corroborate that the direct gradient descent cotrol, which is effective to nonlinear general system, is also effective to the chaotic system. 2. DIRECT GRADIENT DESCENT CONTROL Let us consider a general non linear plant. The aim of control is to manipulate a control input u(t) so that a performance function F(x(t)) decreases : (la)
decrease F(x(t)), U(t)
In this paper, we adopt Lorenz model for a controlled object , and apply the direct gradient descent control. There are many studies on control-
subj.to
789
x(t)
= f(x(t),u(t)),
x(O)
= Xo, (lb)
where x(t) E Rn is a state vector and u(t) ERr is a control vector. Under appropriate assumptions plant (lb) has a unique continuous solution x : x(t), t ~ 0 for any continuous u : u(t), t ~ O. We denote the state trajectory x associated with a given u by x(u), whose value at t will be denoted by x(t;u). Problem (1) is such a problem that we directly manipulates the control input u(t) so as to decrease the performance function F(x(t)). As a mean of on-line control for problem (1) let us execute the gradient descent method (the steepest descent method) with respect to u(t). As a class of admissible control for the fixed interval [0, tj, we consider the space U[O ,t] consisting of r-dimensional-vector-valued continuous functions, and define the inner product :
J t
(u, v)U 1o •t1 ~
u(Tf VeT) dT
(2)
o Let us define a functional by t::.
[uj = F(x(t;u))
where .c ~ diag( 001, ' . . , Oor), 00; > 0, is a proportional canstant. Substituting (5) into (6) yields
it(t) = -.cfu(x(t;u),u(t)fFx(x(t;u)f . (7) However, this control law has only the work of decreasing F(x(t)). Since stability of control input u(t) is not taken in account, it is considered that the control law (7) is not enough for the stability of closed loop system. Accordingly, to make sure of stability we consider to add a penalty term on u( t) and let us introduce a penalty function P(u(t)). We consider a control law which decreases both the performance function F(x(t)) and penalty function P(u(t)) simultaneously. We introduce the quadratic penalty function as 1 P(u(t)) = 2(u d - u(t)f R(Ud - u(t)), (8) where Ud is the corresponding control and R is (normally diagonal) (r x r) positive definite matrix. Thus gradient descent control algorithm with the penalty on u(t) becomes as follows:
(3)
of time interval [0, t], and the derivative of [uj with respect to u as bellow.
it
it(t)
[Assumption 1) f is continuously differentiable [Assumption 2) f u' Fx are Lipschitz continu-
=
-.c{/u(x(t;u),u(t)fFx(x(t;u)f -R(Ud-U(t))} , u(O) = Uo . (10)
Indeed, this will be executed by simultaneous differential equations
ous in (x,u).
x(t) it(t)
Then we have the following Lemma.
= f(x(t), u(t)), x(O) = xo, = -.c {/u(x(t), u(t))T Fx(x(t))T
[Lemma 1) Under Assumptions 1 and 2, the oper-
ator x(t; .) : U[O,tl ~ R is Gateax differentiable, and its lacobian is given, at right-end time t, as follows: V'x(t; u)(t)
= fu(x(t; u), u(t)f
(9)
Substituting (5) and (8) into (9), we obtain
We make the following assumptions.
in (x,u).
= -.c {V' 4>[u)(t) + V' P( u(t))}
-R(Ud - u(t))} , u(O)
(11) = uo.(12)
We call the control law (10) the direct gradient descent control(DGDC) . 3. STABILITY OF DGDC
(4)
(Proof) See (Shimizu, et al., 1998). 0 From Lemma 1, we obtain the derivative of (3):
In this section, we investigate a sufficient condition for asymptotical stability of DGDC (11),(12).
[Theorem 1) Under Assumptions 1 and 2, the functional : U[O ,t] ~ R, deffined by (3), is Gateax differentiable, and the gradient V' [u] E U[o.t] is given, at right-end time t, by
For simplicity put
V'[u](t) = fu(x(t;u), u(t))TFx(x(t; u)f· (5) (Proof) See (Shimizu, et al., 1998). These expression of derivative function can be obtained by caluculating Gateax differential and using the Riesz-F'retchet Theorem. 0 By using the so defined gradient function, we apply the steepest descent method at each time t E [0,00) as on-line control law for problem (1). Namely, u(t) is modified by gradient descent control algorithm
it
=
-.cV'[u](t) , u(O)
=
Uo.
G(x,u) ~ fu(x,uf Fx(xf
+ Pu(x)T.(13)
Now we make the Taylor expansion of the system (11), (12) in the neighborhood of (Xd, Ud) = (0,0) and express it with a linear term plus more than 2 degree one as follows.
X(t)] [ u(t)
=
[f(X(t), u(t)) ] -.cG(x(t), u(t)) (0,0)
+ D [x(t)] + u(t)
[91g2(X(t), (x(t), U(t))] u(t))
(14)
where
D ~ [fx(x,u)
fu(X,U)] -.cGx(x , u) -.cGu(x, u)
(6)
790
(15) (0 ,0)
For (21) and (22), D is concretely obtained as and 9i(x(t), u(t)), i = 1,2 , are the sum of more than 2 degree terms. Since the first term in (14) is zero at the equilibium (0,0), (14) becomes X(t)] [ u(t)
=D
[X(t)] u(t)
+
D=[
Jx(O,O)
Ju(O , O)] -£R .
-£Ju(O,ofQ
(23)
And assume: [Assumption 5]By choosing proper Q, Rand £, matrix D given by (23) can be made asymptotically stable, that is, there exist w > 0 and a > 0 such that lIeDtl1 :S ae- wt .
[91(X(t),U(t))] (16) 92(x(t), u(t)) ,
and its solution satisfies the following integral equation. X(t)] =eDt [X(O)] [ u(t) u(O)
Applying Theorem 2, we obtain: [Corollary l]Consider DGDC (21),(22). In regard to the expanded system (16) with D given by (23) corresponding to the DGDC (21),(22), assume that Assumption 5 and 4 holds. Then
t
+eDtJe- DT [91(X(r) , u(r))] dr (17) 92(x(r), u(r))
°
We assume the following assumptions.
the norm of
[:~g]
is bounded from the above
[Assumption 3JBy choosing proper F(x), P(u) and £, matrix D given by (15) is asymptotically stable, that is, there exist w > 0 and a > 0 such that lIeDtll :S ae- wt .
and evaluated with (18). Furthermore if [ :~~~ ]
[Assumption 4] There exist constants 8 > 0 and > 0 such that
stable, and we have
satisfies (19), (16) is exponentially asymptotically
(J
11
[:~~:m: :~m ] :S (J :~~n !rH 11
:m]
is bounded from
above as shown in the following expression :
D
(18)
11
[:~~n
11
<
(al~O(J) t :~~~]
-t
°as t
[_£:TQ
Y=
- t 00.
-~R] '
(24)
ex,
(26)
where x E Rn,u E Rr,y E Rr are state, input and output vector respectively. [Definition 1] (Relative degree(Isidori,1995)) The linear system (25),(26) has a relative degree { ql , ... , qr} if
o Problem with quadratic performance function is important from an application point of view. Consider the problem 1 decrease F(x(t)) = -x(t)T Qx(t), (20a) U(ij 2 subj.to x(t) = J(x(t), u(t)), x(O) = Xo. (20b)
u(t)
=
In preparation let us consider a linear system whose input and output have the same dimension x = Ax + Bu, (25)
(19)
(Proof) See (Shimizu, et al., 1999). Based on the comparsion function method, it can be proved by using generalized Gronwall-Bellman inequality.
Then DGDC system (11),(12) becomes x(t) = J(x(t), u(t)) , x(O) = Xo,
- t 00.
We have to choose Q, Rand £ to satisfy Assumption 5.
then DGDC (11), (12) is exponentially asymptotically stable, and we have [
as t
Let us consider problem (20). Putting A ~ Jx(O,O) and B = Ju(O,O), D given by (21) becomes
11[:mlll s [('11 [:~~m~ -~fp'-'-dTl ~~-, Further, if it holds that
-t
Corollary 1 assures that if the linearized system of DGDC (11),(12) is asymptotically stable (i.e, D is Hurwitz), there exists asymptotically stable region of DGDC in the neighborhood of the equiriblium and its evaluation can be made. But it is not so easy to choose Q, Rand £ satisfing Assumption 5. In the following, let us study a method for it.
11 [
[Theorem 2]In regard to the expanded system (14) i.e. (16), assume that Assumption 3 and 4 hold. Then the norm of [
[:m] °
(i) For all k
< qi, {) (k)
!i
= 1,2, . .. ,r = 0 , for all 1 :S j i
uu· ..) th J. (n e r x r matnx -{)-.uJ gular.
[8y/q,)]
:S
r ,
. nonstn.
ts
i:Si, j:Sr
If the linear system (25),(26) has relative degree {I, ... , I} , then there exists the nonsingular coordinate transformation
(21)
= -£ {J u(x(t), u(t)f Qx(t) + Ru(t)} , u(O) = Uo . (22)
791
which transforms the linear system (25) ,(26) into a normal form
= QllZ + Q12e , {= Q21Z + Q22e + CBu , y = e. z
Because of the assumption of theroem, (32) is Hurwitz, (37) is asymptotically stable, and thus system (33), (34) is minimum phase. Futher matrix CB for Proposition 2 corresponds to
(27)
(28)
(29) Here z = QllZ is called zero dynamics, and when zero dynamics is asymptotically stable, the linear system (25) is said to be minimum phase.
[BTQ ,
v
[Proposition 1](Isidori, 1995) Suppose that the system (25) has the relative degree {I, . . . , I}, and matrix CB is positive definite. And assume that the system (25), (26) is minimum phase. Then there exists the output feedback control
is asymptotically stable.
which asymptotically stabilize the system (25).
(31)
In general, the Riccati algebraic equation is given;
AP+AT p-PBR- 1B T p+Q'=0. (41) (32)
We make the following assumptions.
is Hurwitz, then there exists a matrix .c diag{a1' . .. , ar}, a; > 0 such that (31) is asymptotically stable.
[Assumption 6]{A, B} is stabilizable. [Assumption 7]{ A, C} is detectable, where C is
(Proof) (25) can be rewritten as
any matrix such that
= BTQx + Ru.
We can conclude that if P is used as Q of squared error performance fucntion (20a) , then (32) is Hurwitz, namely the condition of Theroem 3 is satisfied. Based on the previous discussion, we propose the following design procedure.
(34)
At first , we show that the linear system (33) ,(34) is minimum phase.
oil
ov
= ~ (Rv) = R
ov
er C ~ Q' .
Under Assumption 6, the Riccati algebraic equation (41) has a unique solution P ;:- o. If Assumption 7 is verified, then x(t) = (A-BR- 1BT P)x(t) is asymptically stable(Anderson, 1990).
where v ERr , v = U is a new control input, x and u are state. And set an output y E Rr as y
0
It is difficult to choose matrices Q, R such that (32) is Hurwitz for high order plant. So we propose the decision method of Q and R , using the Riccati algebraic equation.
[Theorem 3) To check asymptotical stability of D given by (24), consider a dynamic system
Then if R is positive and A _BR- 1B T Q
, a; > 0 (39)
Theorem 3 implies that, if we choose Q and R such that (32) is Hurwitz, then there exists L that satisfies Assumption 5.
Then, we consider about the way of choosing Q, Rand .c to satisfy Assumption 5. Applying Proposition 1, we obtain the following theorem.
-~R] [:~~n ·
= -.cy, .c ~ diag{a1, .. . , ar}
such that
u=-Ky,K~diag{l\:l ' .. . , I\:r} , 1\:;>0 , (30)
= [ _.c~T Q
(38)
which is positive definite . Applying Proposition 1 to the system (33) , (34) , we conclude that there exists output feedback
Using the obvious properties of the system (25) , we can obtain the following proposition.
[~~~n
R][~]=R>O,
(35)
[Algorithm A]
implies that relative degree of (33) , (34) is { 1, .. . , I} as R > 0 . Then
I:::.
I:::.
[1J Calculate A = /x(O,O) , B = /u(O , O) at the equilibrium x = 0, U = of plant (20b). [2] For Q' satisfing Assumption 7 and R > 0 get a solution P of the Riccati algebraic equation (41). [3J Set quadratic performance funciton (20a) as 1 F(x) = "2x(tf Px , (42)
transposes system (33) ,(34) into a normal form
z=(A-BR- 1BT Q) z+BR- 1e , {=BTQ (A - BR- 1B T Q) z+B T QBR- 1e+Rv,
°
and calculate the DGDC (21),(22).
y={ .
Under the above design method , DGDC's parameter .c exists such that locally asymptoticar stability of non linear plant is obtained.
So the zero dynamics of (33) , (34) is given by
792
4. CONTROL OF LORENZ CHAOS (Simulation Results)
D=
The Lorenz model was modelled and studied in (Lorenz, 1960) as follows : Xl =--aXI(t)+UX2(t), (43a)
X2 =-XI(t)X3(t)+1'XI(t)-X2(t),
(43b)
where Q is a symmetric matrix . To confirm asymptotical stability of D based on Theorem 3, we checked the stability condition from the characteristic equation of (A - br-1bT Q) by RouthHurwitz method, that follows 1 41 1 r - qll > - - and r - qll > 0 3 722 r - 1(11 -"3qll - 28ql2 ) <-"""3
~X3(t).
(44b)
(44d)
From (11) and (12), we obtain the DGDC as u(t) = -et{ ql1XI (t)+QI2X2(t)+qI3X3(t) +ru(t)} (45) We set u(O) obtained as
= o.
where A
= Ax(t) + bu(t) ,
=
(46)
b= P
And the eigenvalues of A will be s = -22.8277, 11.8277, -2.6667, so the system is unstable. Next , to examine the controllability of the system, we check the controllability matrix as
M = [b Ab A 2bJ =
0
=
23.712 18.491 0.000 18.491 14.454 0.000 [ 0.000 0.000 0.188
1>
0
(48)
For the performance function (20a) , we set Q = P . Changing et of DGDC (45) as 25,30,40, we obtained the results for x(O) = (1 , 2,3) as shown in Fig. 1 . We could confirm that any et makes D asymptotically stable. If we set et larger, then the oscillation will decrease quickly and the speed of converge will become faster. These results indicate the following matter. "Calculate Q of (A - br-IbTQ) obtained by zero dynamics, using the Riccati algebraic equation , and adjust et freely, then the DGDC can asymptotically stabilize chaotic system such as Lorenz model."
[~ ~~O !~g8l o
> 0
Now, since the system is stabilizable, we use Algorithm A to calculate Q from the Riccati algebraic equation (41) . First we choose Q' = I , then C = I. Then the {A, C} is observable, so Assumption 7 is satisfied. A solution P of the Riccati equation (41) for linearization {A, b} is obtained as follows .
The linearization of (44b-d) is
x(t)
2~4 q12 - ~qll)
(47)
If we choose Q and r which satisfy (47) , then (A - br-1bTQ) will be Hurwitz. From Theorem 3, under such Q and r, there exists et > 0, so we can choose suitable et . Namely if there exist such Q, r and et , then the DGDC (45) will be asymptotically stable near the origin. Solving the inequality (47), we see that if we choose Qll and r which satisfies r- I Ql1 > 270, we can stabilize (A _br-IbT Q) . Under such Ql1 and r, there exists et > 0, and so choosing et suitably, we obtained an asymptotically stable trajectory. Using the following method, however, we got a better control result, so we will omit the previous way because of page limitation.
X2(t) = -Xl (t)X3(t) +28xI (t) -X2(t) ,(44c) X3(t) = Xl (t)X2(t) -
1
+ 720 + r- 1 ( -
(44a)
subj.to xI(t)=-lOxI(t)+lOx2(t)+U(t),
I
8 224 3qll+"""3qI21: >720 2 1 (r - 1qll + ~1 ){ _ 73 + r-I (28q12 + 13 qll) }
r-
Taking the desired state Xd as (Xld, X2d, X3d) = (0,0,0) , we simulated the DGDC for the Lorenz model. First of all, we consider controlling Xl. That is, controlling a flow of direction x(horizontal direction that the roll lines) and z (virtical direction) .
,,(t)
0
0 0 8 3
-etql2 -etQ13
(43c) Here Xl is proportional to the intensity of convective motion, X2 is proportional to the temperature difference between ascending and descending currents, and X3 is proportional temperature profile. Constants U > 0 and l' are called Prandtl number and the Rayleigh number respectively. The constant b is related to the given space. If we put U = 10, l' = 28 and b = ~, the unstable equilibium of Lorenz model will be caluculated as (Xl , X2, X3) = (0 , 0, 0), (6y2, 6y2, 27) , (6y2, 6y2, 27) .
The problem can be formulated as 1 decrease F(x(t)) = -2 x(t)TQx(t),
10 -1
0
but rank of M is not 3, so the system isn't controllable. Then, to examine the stabilizability, by the similar transformation of {A , b} we seperate it to the controllable part and uncontrollable part. However, {A , b} is already seperated, and the uncontrollable part is - ~ , so we can know it is asymptotically stable. Therefore the linear 1ized system (46) is stabilizable. Meanwhile, the D given by (24) becomes as follows .
It is very difficult to decide Q by trial and error. But if we have Q = P such as satisfying the Riccati equation, then there exists et which stabi-
793
'"~
:;; '"
:s:...
3 2.5 2 1.5 1 0.5 0 -0.5 ·1
"
XI(l)
\
X2(1 ) x](I) "------.
...
"
\
'" S
:;;
'\
w
'--'--"
0
.;
=25
0.5
1.5
2
2.5
3 2.5 2 1.5 1 0.5 0 -0.5
x,(I) -
x2( r ) .l')(I) ._-----
'"~
5 ~
.;
a=3O
0.5
3
~
0
~
10
11(,)-
10
2.5
i
"
V 0.5
2.5
'. .......
--. 1.5
a=4O
2
2.5
3
11(,)
-
·10
·20
i
-30
-20 -30 -40
·so
=30
-60
1.5
\\
0
-SO
a-25 0.5
I\ '"
10
-40
-40
--
time
-10
-10 -20 -30
0
x,(I)
.,(,) .__._._.-
i\
0
11(,)-
J'\..
0
-SO -60
XI(I)
\,
bme
time
20
1.5
3 2.5 2 1.5 1 0.5 0 -0.5 -1
<>=40
-60
0
0.5
1.5
2.5
0
3
time
time
0.5
1.5
2.5
time
Fig.! In case of control input added to Xl ,
lizes the system, and so we can execute easily the DGDC which guarantees asymptotical stability.
And we can get almost desirable convergence speed of the closed-loop system by adjusting .c.
Next , we considered controlling X2 , adding the in· put to X2 . That is, controlling the proportional of the temperature difference between ascending and decending flows , The problem is then expressed as
6. REFERENCES
decreaseF(x(t))
1
= -x(tfQx(t) ,
2 subj.to XI(t)= -lOxI(t)+lOx2(t), u(t)
Anderson, B.D.O . and J.B .Moore (1990) . Optimal Control -Linear Quadratic Methods-, Prentice Hall, Inc, New Jersey. Chen, Y.H. and M.Y.Chou (1994). Continuous feedback approach for controlling chaos. Phys. Rev. E, 50-3, pp.2331-2334. Fradkov, A.L. and A.Y .Pogromsky (1998) . Introduction to Control of Oscillations and Chaos, World Scientific, Singapore. Isidori , A. (1995) . Nonlinear Control Systems, 3rd ed .. Springer-Verlag. London. Lorenz, E.N. (1960) . Deterministic Nonperiodic Flow. Journal Of The Atmospheric Sciences, 20, pp.130-141. Ott, E., C.Grebogi and J .A.Yorke (1990) . Controlling chaos. Phys. Rev. Lett. ,64, pp.15001501. Shimizu, K., S.lto and S.Suzuki (1998) . Tracking Control of General Nonlinear System by Direct Gradient Descent Method. IFAC Symp. on Nonlinear Control System Design 98, 1 of 3 , pp.185-190. Shimizu, K., K.Otsuka and J.Naiborhu (1999) . Improved Direct Gradient Descent Control of General Nonlinear Systems. European Control Conference ECC'99 Proceedings(CDROM), pdf file no.F0676. Tian , Y. P. (1997) . Controlling Chaos Via Continuous Nonlinear State Feedback. Proceedings of the 36th IEEE Conference on Decision and Control , pp.1502-1507. Yu, X. (1996). Controlling Lorenz chaos. International Journal of System Science, 27-4 , pp.355-359.
(49a) (49b)
X2(t) = -Xl (t)X3(t) +28xI (t) -X2(t)+u(t) , (49c) (49d) From the linearized system, the system was known to be stabilizable at the origin. As the same in the previous case, we could calculate the condition for zero dynamics (A - br-IbT Q) to be Hurwitz through Algorithm A. Then changing a freely, we experienced obtaining desirable performances without harming its stability. From various simulation results it was confirmed that the DGDC achieved excellent performances by use of Algorithm A. 5. CONCLUSION In the paper, we applied DGDC to stabilize a chaotic system, Lorenz model, and obtained satifactory performances. Our results were good enough, compared to the methods proposed in (Chen 1994; Tian, 1997; Yu, 1996). It is interesting that DGDC can carry out stable control even for the chaotic system with conplex behavior. Moreover, Algorithm A offers an easy way how to choose parameters Q, Rand .c such that asymptotical stability of DGDC is assured. The most significant thing is that choosing Q, R and determining .c can be made independently.
794