Chapter 2 The Analytical Analysis of the System
2.2 Stability Analysis by Lyapunov Direct Method
dynamics of the system of rotational machine with fly-ball governor is described by a three-dimensional autonomous system.
2.2 Stability Analysis by Lyapunov Direct Method
Find the equilibrium points of the system and determine the stability of them. These equilibrium points can be found from equation (2.4) as
)
Substituting equation (2.5) into equation (2.4), and expanding to Taylor series, it becomes
y
q
and the terms higher than one degree have not been written down. Let
, then , , and .
>0
> F
q A>0 B>0 C>0 D>0
First, asymptotical stability of the origin of equation (2.6) can be studied by using Lyapunov direct method. Construct the quadratic Lyapunov function candidate in the form
.
Let be the principal minor determinants of the characteristic matrix of the quadratic form, then
pi (i=1,2,3)
By Sylvester’s criterion [14], is positive definite if and if that all are positive: are positive definite. Let
A11 A22 A33 A12 A13 V&
−
CD
is negative definite. By Sylvester’s theorem, the sufficient condition for be positive definite is founded:
V
From the Lyapunov asymptotic stability theorem, we conclude that the origin is asymptotically stable. Furthermore, the result is the same as analysis by linearized system.
For the given range for disturbance ε, we can find the allowable range for the initial disturbances Ω by the Lyapunov function. Let
2 2 2
2 x y z
R = + +
Using the method of Lagrange’s multiplier, we form the Lagrange’s function . From
3
differential equations for disturbances are
In order to determine the instability of the origin of equation(2.8), the quadratic Lyapunov function candidate is assumed in the form
.
The derivative of V with respect to τ along the trajectories of system is given by
which is positive definite. There exists the region in the neighborhood of the origin of equation(2.2.8). Its boundaries in the y-z plane are
So, by the Lyapunov instability theorem, the origin is unstable.
Chapter 4
Conclusions
The dynamic system of the rotational machine with a hexagonal centrifugal governor exhibits a rich variety of nonlinear behaviors as certain parameters are varied. Due to the effect of nonlinearity, regular or chaotic motions may occur. In this thesis, both analytical and computational methods have been employed to study the dynamical behaviors of the nonlinear system.
The conditions of stability and instability of fixed points have been determined by using the Lyapunov direct method in Chapter 2.
The computational analyses have been studied in Chapter 3. For autonomous system, the periodic and chaotic motions of the autonomous system have been obtained by the numerical methods such as phase trajectory, power spectrum, Poincaré map and Lyapunov exponents. The changes of parameters play a major role for the nonlinear system. The chaotic motion has been detected by using Lyapunov exponents and Lyapunov dimensions. In spite of that these methods are different, the results obtained matches each other. In order to improve the performance of a dynamical system or avoid the chaotic phenomena, four methods: delayed feedback control, linear feedback control, optimal control and adaptive control algorithm are used to control the chaotic motion to any assigned periodic motion effectively. Especially, for our system, the delayed feedback control is the best method compared with the others, for which the control time is minimum and the delay time and the gain can be
easily selected. Synchronization of two chaotic oscillators is studied in this thesis. For coupled systems, increase of coupling strength leads to the occurrence of phase synchronization. We show that one of the zero Lyapunov exponents becoming negative cannot be used as a sufficient criterion for the occurrence of phase-locking synchronization. However, one of the negative Lyapunov exponents approaching zero gives a clue of the appearance of phase synchronization of a coupled system.
For anticontrol of chaos of autonomous system, two different procedures to design the controller have been presented. The periodic motion of the system is disappeared and replaced by chaotic motion effectively by adding a linear and a nonlinear feedback term, respectively. Chaos synchronization of the autonomous governor system has been presented by adding linear feedback term, adding sinusoidal term and adaptive feedback methods. Chaos synchronization is also attained by a recursive procedure, backstepping design that combines the choice of a Lyapunov function for selecting a proper controller. Besides, the parameter of chaotic system is estimated from time sequences for chaos synchronization has been studied.
In section 3.3, the periodic and chaotic motions of the autonomous system with time-delay have been obtained by the numerical methods such as phase trajectory, time history and power spectrum. In order to improve the performance of a dynamical system or avoid the chaotic phenomena, two methods, adaptive control algorithm and linear feedback control, are used to control the chaotic motion to periodic motion effectively. Synchronization of two chaotic oscillators is studied. For two chaotic systems, increase of coupling strength leads to the occurrence of completely synchronization and phase synchronization. Finally, anticontrol of chaos is also studied in this section.
The chaotic dynamics of the nonautonomous system have been obtained by the numerical methods such as power spectrum, Poincaré map and Lyapunov exponents in section 3.4. All these phenomena have been displayed in bifurcation diagrams. More information of the behaviors of the periodic and the chaotic motion can be found in parametric diagrams. Chaotic motion is the motion which has a sensitive dependence on initial condition in deterministic physical systems. The chaotic motion has been detected by using Lyapunov exponents and Lyapunov dimensions. The presence of chaotic behavior is generic for suitable nonlinearities, ranges of parameters and external force, where one wish to avoid or control so as to improve the performance of a dynamical system. Eight methods are used to control chaos effectively.
Especially, we can control the chaotic motion to any assigned periodic motion by addition of period force, periodic impulse control, the delayed feedback control, external force feedback control, optimal control and adaptive control algorithm. For our system, the delayed feedback control is the best method compared with the others.
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Fig. 2.1. Physical model of the system
Fig. 3.1. (a) Phase portrait and Poincaré map for q=2.3 (b) q=3 (c) phase portrait for q=5.5 (d) Poincaré map for q=5.5
Fig. 3.2 (a) Power spectrum for q=2.3 (b) q=5.5
Fig. 3.3. The Lyapunov exponents for q between 2.3 and 5.5
Fig. 3.4. Phase portrait of controlled system (a) τ =5, (b) τ =4
via adaptive feedback control Fig. 3.5. Phase portrait of controlled system
Fig. 3.6. Phase portrait of controlled system via linear feedback control
Fig. 3.7. Phase portrait of controlled system via optimal control
Fig. 3.8 Chaos synchronization via a linear feedback approach
Fig. 3.9. Chaos synchronization via adaptive feedback with k1=10 and k2=1
The projection of the attractor in the x-y plane(a) and x-z plane(b) with the Fig. 3.10.
coupling strength K =0.1.
ig. 3.11. (a) shows the maximum absolute difference of two phases. (b) shows the F
maximum absolute difference of two trajectories. (c) The cross-correlation function )
0
ξ( with τ =0
Fig. 3.12. The Lyapunov exponents of coupled system.
Fig. 3.13. Phase portrait of uncontrolled system (a1 =a2 =a3 = 0).
Fig. 3.14 Phase portrait of controlled system with a1 =0.2,a2 =−0.1,a3 =−0.1.
Fig. 3.15. parameter diagram of (a) a2 versus a3 for a1 =0.2, (b) versus or
a2 a1
f a3 =−0.1.
Fig. 3.16 Phase portrait of controlled system with ε =0.01.
Fig. 3.17. Chaos synchronization via a linear feedback with k1 =0.2.
Fig. 3.18. Chaos synchronization via a nonlinear feedback with k2 =0.2.
Fig. 3.19. Chaos synchronization via adaptive feedback with k3=0.5 and k4=0.2.
Fig. 3.20. Chaos synchronization via backstepping design.
Fig. 3.21. The minimum value of U with respect to the coupling constant k1
Fig. 3.22. The difference U versus the parameter q for k1 =0.2.
Fig. 3.23. Time evolution of the parameter q by the random optimization process.
m
P1 P2
k
Fig. 3.24. A mass-spring system.
Fig. 3.25. (a) Phase portrait for q=3 (b) q=5.5
Fig. 3.26. (a) Time history ,(b) power spectrum for q=5.5
Fig. 3.27. Controlled system via adaptive feedback.
Fig. 3.28. (a)Phase portrait of controlled system via linear feedback control for K2 = 0.5, (b) K2 = 5
Fig. 3.29. Chaos synchronization via a unidirectional linear feedback approach for K=3.
Fig. 3.30. Chaos synchronization via a mutual linear feedback approach for K=1.5.
Fig. 3.31. Chaos synchronization via a mutual linear feedback approach for K=3.
Fig. 3.32. Chaos synchronization via a mutual nonlinear feedback approach for K=1.5.
Fig. 3.33. The maximum absolute difference of mean frequency between two chaotic subsystems.
Fig. 3.34. (a) A sawtooth function, (b) Phase portrait of controlled system.
(a) (b)
(c) (d)
Fig. 3.35 (a) Phase portrait and Poincaré map for q=2.07 (b) q=2.14 (c) phase portrait for q=2.21 (d) Poincaré map for q=2.21
(a)
(b)
Fig. 3.36 (a) Power spectrum for q=2.07 (b) q=2.21
Fig. 3.37 Bifurcation diagram of q versus x
(a)
(b)
Fig. 3.38 (a) Parametric diagram of q versus b (b) q versus ω
Fig. 3.39. The largest Lyapunov exponent for q between 2.07 and 2.21
Fig. 3.40. Bifurcation diagram of T versus x
Fig. 3.41. Bifurcation diagram of v versus x
Fig. 3.42. Bifurcation diagram of ρ versus x
Fig. 3.43. Bifurcation diagram of K versus x
Fig. 3.44. (a) Parameter converge to q=2.07 from chaotic motion q=2.21 (b) Phase portrait and Poincaré map of controlled system
Fig. 3.45 (a) Parameter converge to q=2.14 from chaotic motion q=2.21 (b) Phase portrait and Poincaré map of controlled system
ig. 3.46 (a) Phase portrait and Poincaré map of controlled system (b) the error signal F
Fig. 3.47. Bifurcation diagram of versus x
K1
K1
Fig. 3.48. Bifurcation diagram of versus x
K2
K2
Fig. 3.49. Chaos synchronization via linear feedback with k =0.2.