- Email: [email protected]
Control of the saddle-node and transcritical bifurcations
Copyright © IFAC Nonlinear Control Systems. Stuttgart, Germany, 2004
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocate/ifac
CONTROL OF THE SADDLE- ODE AND TRANSCRITICAL BIFURCATIONS Femando Verduzco t
Universidad de Sonora, Mexico
Abstract: In this paper, the control of the saddle-node and transcritical bifurcations in nonlinear systems is treated. A new approach is presented to find sufficient conditions in terms of the original vector fields. The analysis of the system dynamics is reduced to dimension one through the center manifold theorem. Copyright © 2004 IFAC Keywords: saddle-node bifurcation, transcritical bifurcation, center manifold, bifurcation control.
1. INTRODUCTION
2. STATEMENT OF THE PROBLEM Consider the nonJinear system
It is known that parameterized family of dynamical systems can exhibit complex dynamics around bifurcation points. Bifurcation control, an emerging field of research, refers to the design of controllers that permit modify the bifurcating properties of the parameterized family, see (Chen et al., 20(0) for an overview of this field.
~ == F(~)
+ G(~)u,
(1)
where ~ E lRn is the state and u E lR is the control input. The vector fields F and G are assumed to be sufficiently smooth, with F(O) O. Assume that
DF(O) ==
There exist some approximations to the bifurcation control problem, it can be mentioned linear and nonlinear state feedback control, harmonic balance, quadratic invariants in normal forms and time-delayed feedback. See (Abed and Fu, 1987; Chang et aI., 2000; Kang, 1988a; Kang, 1988b; Hamzi and Kang, 2003; Kang, 1996; Kang, 2000; Verduzco, 2004) for methods that use state feedback to control codimension one stationary bifurcations. Among those methods, it can be distinguished three different methodologies, depending of the tool employed. The projection method is used in (Abed and Fu, 1987), while in (Chang et al., 2(00) the authors use the Brunovsky form. In (Verduzco, 2004) is employed a new approach, that uses just the Jordan form. In this paper are complemented the results obtained in the last one.
(0o 0) Js
with Js E lR(n-l)x(n-l) a
HUlwitz matrix. Suppose that
F(~)
=
G 1 (~) ) ( Z ) . G(~) == ( G2(~) ,and ~ == w ' WIth z E Il w E IRn - 1 , Ft, G I : lR x lRn - 1 -+ Il and F 2 , G 2 : IR x lRn - 1 -+ lRn - 1 . Then, expanding system (1) around ~ == 0 yields
z == F 21 (z, w) + F 31 (z, w) + ... +(b 1 + mlz + M 2w + .. ·)u tU == J SW
b
!\J[4W
+ .. ·)u,
= (~). DG(O)
t m 1 M?) 'thb ( M 3 M~ w i l E Il b2 E IRn- ,and
This work was partially supported by CONACyT
681
(2)
+ F 22 (z, w) + ...
+(b2 + M 3 z + where G(O) =
1
(~i~l),
=
M
where
1 01 0) 0 (o0-{31L o
P ==
,p-l ==
(10 01 0) 0,
I 0 (3IL o I A o E IR(n-1)x1,L 1 E IR1x (n-1), T A 2 == (A 21 ,···, A 2,n_d , AI, L 2, A 2j E IR(n-1)x(n-1). L o == lS1b2, and I E IR(n-l)x(n-1) is the identity matrix, put the linear part in Jordan form. Then we We wish to design a control law u == u(z, J..L), with J..L a whereao E
a
have the system (5) into standard form
real parameter, such that the original system (1) undergoes the saddle-node and transcritical bifurcations at ~ == 0 and J..L == 0, and that we could control them, i.e., that we could decide the direction of the bifurcation.
~) == (~{3~b1 ~)
( iJ
Consider the control law
+
(3)
where /31,132 E IR. Now, using the control law (3) in system (2) we obtain the closed-loop system
z == {3lb l J..L +:F1(z, w, J..L) W == (3lb 2J..L
(:) y
ls
0
(6)
!(X,J..L,Y)) (
0
,
g(x,J..L,Y)
where
f(x, J..L, y) ==:F1(x, J..L, Y - (31J..LL o) (7) == {31 m l X J1. + {31J1.M2 (y - (31J1.L o) + (ao + {32bd x 2 + xL 1 (y - f31J-LL o) +21 (y - (31J1.L o)T A 2 (y - (31J..LL o) + ... ,
(4)
+ lsw + :F2(z, w, J..L),
where
:F1(z, w, J..L) == F 21 (z, w) + {32b1Z2 +(31J..L(mlz + ]\;[2W) + ... ,
0
g(x,J-L,Y) ==:F2(x,J..L,Y - (31J..LL o). and
We seek a center manifold
:F2(z, W, J..L) == F 22 (z, w) + {32b2Z2 +(3IJ..L(M3 z + ]\;[4 W) + ... such that h(O,O) == 0, Dh(O,O) == 0 and B k E IR(n-l) x 1 for k == 1,2,3. Substituting (8) into (6) and using the chain rule, we obtain
Then, our goal is to find {31 and (32 such that system (4) undergoes the saddle-node and transcritical bifurcations, and can be controllable. For this, we use the center manifold theory.
lsh(x, J..L) + g(x, J..L, h(x, J..L)) 8h(x, J-L) 8x [f(x, J..L, h(x, J..L))] == O.
(9)
3. CENTER MANIFOLD We don't need to calculate the center manifold h(x, J1.) because the dynamics on the center manifold is determined until the quadratics terms, as we will see at the theorem 1.
3.1 Quadratic terms Equation (4) represents a J..L-parameterized family of systems, which we can write as an extended system
3.2 Dynamics on the center manifold (5)
On the center manifold the dynamics is given by
± == (31 blJ..L + f(x, J..L, h(x, J..L)),
(10)
where, from (7), In this form, the system has a bi-dimensional center manifold through the origin. To find this manifold, we should change coordinates to put the linear part in diagonal form, but is not possible. The next change of coordinates
f (x, J..L, h (x, J..L)) == 131 (m 1
-
L 1 L o ) J..LX
(11 )
+ (ao + 132 b1) x 2 + ... Then, substituting (11) in (10) we obtain
(~) =p (~),
± == {31 b1J-L + {31 (ml - LILo) J-Lx + (ao + 132 b1) x 2 + ....
682
(12)
4. BIFURCATION CONTROL
where
~ E
!Rn is the state and u E !R is the control
input. Suppose that F(O)
In this section we will find sufficient conditions to ensure that system (12) undergoes the saddle-node and transcritical bifurcations.
with Js
G(O)
E
= 0, DF(O) = (~Js)
jR(n-1) x(n-1)
a Hurwitz matrix, and
= (:~).
1. If b1 =I- 0 and JC == 0, then there exist such tha~ with the control law
4.1 Bifurcation Theorem
/31
and
/32
See (Wiggins, 1990) for the following theorem of the classical bifurcation theory. the system undergoes the saddle-node bifurcation. Furthermore, it is possible to select /31 and /32 in a such way that we can determine the direction of the bifurcation. 2. If b1 == 0 and JC =I- 0, then there exists /31 such that, with the constant control law
Theorem 1. Consider the sufficiently smooth system
x ==
f(x,p),
x E lR,
pER
(13)
Assume that (0,0) is a non hyperbolic equilibrium, Le., 1(0,0) == 0 and ~ (0, 0) == O.
u ==
a) The system (13) undergoes a saddle-node bifurcation at (0,0) if
8f
8p (0, 0) =I- 0,
/31P
the system undergoes the transcritical bifurcation. Furthermore, it is possible to select /31 in a such way that we can determine the direction of the bifurcation.
and
82 f
8x 2 (0,0) =I- 0
The case b1 == 0 deserves a di fferent treatment. With a different control law is possible to control the transcritical and pitchfork bifurcations. This case was reported in (Verduzco, 2004).
b) The system (13) undergoes a transcritical bifurcation at (0,0) if
5. CONCLUSIONS
and
In this paper we have established sufficient conditions to ensure the control of the saddle-node and transcritical bifurcations in nonlinear control systems. The linear part of the driven vector field is assumed in Jordan normal form, and the dynamical analysis is reduced to one dimension through the center manifold theorem. This document complement the results reported in (Verduzco, 2004).
4.2 Control law design In this section we state the main results in this paper.
Theorem 2. Consider the system REFERENCES where JC
== m1
1
- L 1 J S b2 .
Abed, E.H. and J.H. Fu (1987). Local feedback stabilization and bifurcation control, ii. stationary bifurcation. Systems and Control Letters 8, 467473. Chang, D.E., W. Kang and A.J. Krener (2000). Normal forms and bifurcation of control systems. Proc. 39th IEEE CDC. Chen, G., J.L. Moiola and H.O. Wang (2000). Bifurcation control: theories, methods, and applications. International Journal of Bifurcation and Chaos 10, 511-548. Harnzi, B. and W. Kang (2003). Resonant terms and bifurcations of nonlinear control systems with one uncontrollable mode. Systems and Control Letters 49, 267-278.
1. If b1 =I- 0 and JC == 0, then there exist /31 and /32 such that the system undergoes the saddle-node bifurcation. Moreover, we can select /31 and /32 in a such way that we can determine the direction of the bifurcation. 2. If b1 == 0 and JC =I- 0, then there exists /31 such that the system undergoes the transcritical bifurcation. Moreover, it is possible to select /31 in a such way that we can determi ne the direction of the bifurcation.
Corollary 3. Consider the system
~ == F(~)
+ G(~)u,
683
Kang, W. (I 988a). Bifurcation and normal form of nonlinear control systems, part i. SIAM J. ControL Optimization 36, 193-212. Kang, W. (I 988b). Bifurcation and normal form of nonlinear control systems, part ii. SIAM J. ControL Optimization 36, 213-232. Kang, W. (1996). Invariants and stability of control systems with transcritical and saddle-node bifurcations. Proc. of the 36th IEEE CDC. Kang, W. (2000). Bifurcation control via state feedback for systems with a single uncontrollable mode. SIAM J.Control Optimization 38, 14281452. Verduzco, F. (2004). Controlling from the center manifold codimension one bifurcations. Asian ControL Conference. Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag. New York.
684
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocate/ifac
CONTROL OF THE SADDLE- ODE AND TRANSCRITICAL BIFURCATIONS Femando Verduzco t
Universidad de Sonora, Mexico
Abstract: In this paper, the control of the saddle-node and transcritical bifurcations in nonlinear systems is treated. A new approach is presented to find sufficient conditions in terms of the original vector fields. The analysis of the system dynamics is reduced to dimension one through the center manifold theorem. Copyright © 2004 IFAC Keywords: saddle-node bifurcation, transcritical bifurcation, center manifold, bifurcation control.
1. INTRODUCTION
2. STATEMENT OF THE PROBLEM Consider the nonJinear system
It is known that parameterized family of dynamical systems can exhibit complex dynamics around bifurcation points. Bifurcation control, an emerging field of research, refers to the design of controllers that permit modify the bifurcating properties of the parameterized family, see (Chen et al., 20(0) for an overview of this field.
~ == F(~)
+ G(~)u,
(1)
where ~ E lRn is the state and u E lR is the control input. The vector fields F and G are assumed to be sufficiently smooth, with F(O) O. Assume that
DF(O) ==
There exist some approximations to the bifurcation control problem, it can be mentioned linear and nonlinear state feedback control, harmonic balance, quadratic invariants in normal forms and time-delayed feedback. See (Abed and Fu, 1987; Chang et aI., 2000; Kang, 1988a; Kang, 1988b; Hamzi and Kang, 2003; Kang, 1996; Kang, 2000; Verduzco, 2004) for methods that use state feedback to control codimension one stationary bifurcations. Among those methods, it can be distinguished three different methodologies, depending of the tool employed. The projection method is used in (Abed and Fu, 1987), while in (Chang et al., 2(00) the authors use the Brunovsky form. In (Verduzco, 2004) is employed a new approach, that uses just the Jordan form. In this paper are complemented the results obtained in the last one.
(0o 0) Js
with Js E lR(n-l)x(n-l) a
HUlwitz matrix. Suppose that
F(~)
=
G 1 (~) ) ( Z ) . G(~) == ( G2(~) ,and ~ == w ' WIth z E Il w E IRn - 1 , Ft, G I : lR x lRn - 1 -+ Il and F 2 , G 2 : IR x lRn - 1 -+ lRn - 1 . Then, expanding system (1) around ~ == 0 yields
z == F 21 (z, w) + F 31 (z, w) + ... +(b 1 + mlz + M 2w + .. ·)u tU == J SW
b
!\J[4W
+ .. ·)u,
= (~). DG(O)
t m 1 M?) 'thb ( M 3 M~ w i l E Il b2 E IRn- ,and
This work was partially supported by CONACyT
681
(2)
+ F 22 (z, w) + ...
+(b2 + M 3 z + where G(O) =
1
(~i~l),
=
M
where
1 01 0) 0 (o0-{31L o
P ==
,p-l ==
(10 01 0) 0,
I 0 (3IL o I A o E IR(n-1)x1,L 1 E IR1x (n-1), T A 2 == (A 21 ,···, A 2,n_d , AI, L 2, A 2j E IR(n-1)x(n-1). L o == lS1b2, and I E IR(n-l)x(n-1) is the identity matrix, put the linear part in Jordan form. Then we We wish to design a control law u == u(z, J..L), with J..L a whereao E
a
have the system (5) into standard form
real parameter, such that the original system (1) undergoes the saddle-node and transcritical bifurcations at ~ == 0 and J..L == 0, and that we could control them, i.e., that we could decide the direction of the bifurcation.
~) == (~{3~b1 ~)
( iJ
Consider the control law
+
(3)
where /31,132 E IR. Now, using the control law (3) in system (2) we obtain the closed-loop system
z == {3lb l J..L +:F1(z, w, J..L) W == (3lb 2J..L
(:) y
ls
0
(6)
!(X,J..L,Y)) (
0
,
g(x,J..L,Y)
where
f(x, J..L, y) ==:F1(x, J..L, Y - (31J..LL o) (7) == {31 m l X J1. + {31J1.M2 (y - (31J1.L o) + (ao + {32bd x 2 + xL 1 (y - f31J-LL o) +21 (y - (31J1.L o)T A 2 (y - (31J..LL o) + ... ,
(4)
+ lsw + :F2(z, w, J..L),
where
:F1(z, w, J..L) == F 21 (z, w) + {32b1Z2 +(31J..L(mlz + ]\;[2W) + ... ,
0
g(x,J-L,Y) ==:F2(x,J..L,Y - (31J..LL o). and
We seek a center manifold
:F2(z, W, J..L) == F 22 (z, w) + {32b2Z2 +(3IJ..L(M3 z + ]\;[4 W) + ... such that h(O,O) == 0, Dh(O,O) == 0 and B k E IR(n-l) x 1 for k == 1,2,3. Substituting (8) into (6) and using the chain rule, we obtain
Then, our goal is to find {31 and (32 such that system (4) undergoes the saddle-node and transcritical bifurcations, and can be controllable. For this, we use the center manifold theory.
lsh(x, J..L) + g(x, J..L, h(x, J..L)) 8h(x, J-L) 8x [f(x, J..L, h(x, J..L))] == O.
(9)
3. CENTER MANIFOLD We don't need to calculate the center manifold h(x, J1.) because the dynamics on the center manifold is determined until the quadratics terms, as we will see at the theorem 1.
3.1 Quadratic terms Equation (4) represents a J..L-parameterized family of systems, which we can write as an extended system
3.2 Dynamics on the center manifold (5)
On the center manifold the dynamics is given by
± == (31 blJ..L + f(x, J..L, h(x, J..L)),
(10)
where, from (7), In this form, the system has a bi-dimensional center manifold through the origin. To find this manifold, we should change coordinates to put the linear part in diagonal form, but is not possible. The next change of coordinates
f (x, J..L, h (x, J..L)) == 131 (m 1
-
L 1 L o ) J..LX
(11 )
+ (ao + 132 b1) x 2 + ... Then, substituting (11) in (10) we obtain
(~) =p (~),
± == {31 b1J-L + {31 (ml - LILo) J-Lx + (ao + 132 b1) x 2 + ....
682
(12)
4. BIFURCATION CONTROL
where
~ E
!Rn is the state and u E !R is the control
input. Suppose that F(O)
In this section we will find sufficient conditions to ensure that system (12) undergoes the saddle-node and transcritical bifurcations.
with Js
G(O)
E
= 0, DF(O) = (~Js)
jR(n-1) x(n-1)
a Hurwitz matrix, and
= (:~).
1. If b1 =I- 0 and JC == 0, then there exist such tha~ with the control law
4.1 Bifurcation Theorem
/31
and
/32
See (Wiggins, 1990) for the following theorem of the classical bifurcation theory. the system undergoes the saddle-node bifurcation. Furthermore, it is possible to select /31 and /32 in a such way that we can determine the direction of the bifurcation. 2. If b1 == 0 and JC =I- 0, then there exists /31 such that, with the constant control law
Theorem 1. Consider the sufficiently smooth system
x ==
f(x,p),
x E lR,
pER
(13)
Assume that (0,0) is a non hyperbolic equilibrium, Le., 1(0,0) == 0 and ~ (0, 0) == O.
u ==
a) The system (13) undergoes a saddle-node bifurcation at (0,0) if
8f
8p (0, 0) =I- 0,
/31P
the system undergoes the transcritical bifurcation. Furthermore, it is possible to select /31 in a such way that we can determine the direction of the bifurcation.
and
82 f
8x 2 (0,0) =I- 0
The case b1 == 0 deserves a di fferent treatment. With a different control law is possible to control the transcritical and pitchfork bifurcations. This case was reported in (Verduzco, 2004).
b) The system (13) undergoes a transcritical bifurcation at (0,0) if
5. CONCLUSIONS
and
In this paper we have established sufficient conditions to ensure the control of the saddle-node and transcritical bifurcations in nonlinear control systems. The linear part of the driven vector field is assumed in Jordan normal form, and the dynamical analysis is reduced to one dimension through the center manifold theorem. This document complement the results reported in (Verduzco, 2004).
4.2 Control law design In this section we state the main results in this paper.
Theorem 2. Consider the system REFERENCES where JC
== m1
1
- L 1 J S b2 .
Abed, E.H. and J.H. Fu (1987). Local feedback stabilization and bifurcation control, ii. stationary bifurcation. Systems and Control Letters 8, 467473. Chang, D.E., W. Kang and A.J. Krener (2000). Normal forms and bifurcation of control systems. Proc. 39th IEEE CDC. Chen, G., J.L. Moiola and H.O. Wang (2000). Bifurcation control: theories, methods, and applications. International Journal of Bifurcation and Chaos 10, 511-548. Harnzi, B. and W. Kang (2003). Resonant terms and bifurcations of nonlinear control systems with one uncontrollable mode. Systems and Control Letters 49, 267-278.
1. If b1 =I- 0 and JC == 0, then there exist /31 and /32 such that the system undergoes the saddle-node bifurcation. Moreover, we can select /31 and /32 in a such way that we can determine the direction of the bifurcation. 2. If b1 == 0 and JC =I- 0, then there exists /31 such that the system undergoes the transcritical bifurcation. Moreover, it is possible to select /31 in a such way that we can determi ne the direction of the bifurcation.
Corollary 3. Consider the system
~ == F(~)
+ G(~)u,
683
Kang, W. (I 988a). Bifurcation and normal form of nonlinear control systems, part i. SIAM J. ControL Optimization 36, 193-212. Kang, W. (I 988b). Bifurcation and normal form of nonlinear control systems, part ii. SIAM J. ControL Optimization 36, 213-232. Kang, W. (1996). Invariants and stability of control systems with transcritical and saddle-node bifurcations. Proc. of the 36th IEEE CDC. Kang, W. (2000). Bifurcation control via state feedback for systems with a single uncontrollable mode. SIAM J.Control Optimization 38, 14281452. Verduzco, F. (2004). Controlling from the center manifold codimension one bifurcations. Asian ControL Conference. Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag. New York.
684