Parametric resonance in the Rayleigh–Duffing oscillator with time-delayed feedback

Commun Nonlinear Sci Numer Simulat 17 (2012) 4485–4493

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Parametric resonance in the Rayleigh–Duffing oscillator with time-delayed feedback M. Siewe Siewe a,b,⇑, C. Tchawoua a, S. Rajasekar c a

Laboratoire de Mécanique, Département de Physique, Faculté des Sciences, University of Yaounde I, Box 812, Yaounde, Cameroon Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria 0002, South Africa c School of Physics, Bharathidasan University, Tiruchirapalli 620 024, Tamilnadu, India b

a r t i c l e

i n f o

Article history: Received 13 September 2011 Received in revised form 27 November 2011 Accepted 23 February 2012 Available online 3 March 2012 Keywords: Resonance Stability Delay feedback Parametric excitation

a b s t r a c t We investigate the principal parametric resonance of a Rayleigh–Duffing oscillator with time-delayed feedback position and linear velocity terms. Using the asymptotic perturbation method, we obtain two slow flow equations on the amplitude and phase of the oscillator. We study the effects of the frequency detuning, the deterministic amplitude, and the time-delay on the dynamical behaviors, such as stability and bifurcation associated with the principal parametric resonance. Moreover, the appropriate choice of the feedback gain and the time-delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time-delay can broaden the stable region of the non-trivial steady-state solutions and enhance the control performance. Theoretical stability analysis is verified through a numerical simulation. Crown Copyright Ó 2012 Published by Elsevier B.V. All rights reserved.

1. Introduction Since the work of Verhulst (1804–1849) in population dynamics, delay differential equations (DDEs), are used to model dynamical systems in many scientific and engineering domains, e.g., population dynamics, neural networks, chemistry, climatology, economy, and so on [1]. In particular, many biological systems and controlled mechanical systems call for a description by means of DDEs. In practice, unavoidable time-delays frequently appear in the controlled mechanical or structural systems, especially in hydraulic actuators used in active suspensions of ground vehicles and active tendons of tall buildings. In recent years great attention has been paid to the study of dynamics of DDEs [2,3]. It has been found that time-delay not only makes the systems retarded, but often it is a source for birth of a limit cycle, lose of stability and occurrence of bifurcations and chaos [4]. There is a growing need to deepen our understanding on the complexity induced, particularly, by multiple time-delays. Shayer and Campbell [5] and Liao and Chen [6] studied the Hopf bifurcation and double-Hopf bifurcation respectively in DDEs with multiple time-delay feedback. Xu and Yu [7] investigated the delay-induced bifurcation in a non-autonomous system with time-delayed velocity feedback. The effect of time-delay on the collective dynamics of coupled limit cycle oscillators at Hopf bifurcation was studied [8]. Over the last years considerable interest has been paid to the study of combined effect of parametric excitation and time-delayed control on dynamical systems. Parametric excitation occurs in a wide variety of engineering applications. For example, parametric excitations are found in MEMS/NEMS devices when illuminated within an interference field of a

⇑ Corresponding author at: Laboratoire de Mécanique, Département de Physique, Faculté des Sciences, University of Yaounde I, Box 812, Yaounde, Cameroon E-mail addresses: [email protected] (M.S. Siewe), [email protected] (C. Tchawoua), [email protected] (S. Rajasekar). 1007-5704/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.02.030

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continuous wave laser [9,10], in high-speed milling applications [11,12], in a model of a towed mass underwater for application to submarine dynamics [13], in the pumping of a swing [14], in a nanowire system driven by an oscillating electric field [15] and in mass-loaded string systems [16] such as elevators, cranes and cable-stayed bridges. There are several studies in parametrically driven systems with time-delayed feedback. It has been shown that in a parametrically excited Duffing–van der Pol oscillator under linear-plus-nonlinear state feedback control with a time-delay, nontrivial steady state responses may lose their stability by a saddle-node or Hopf bifurcation as parameters vary [17]. In the parametrically driven van der Pol oscillator, the effects of time-delay on the existence and characteristics of limit cycles were analysed [18]. The interaction effect of fast vertical parametric excitation and time-delay on self-oscillation in the van der Pol oscillator was reported [19]. It was shown that vertical parametric excitation, in the presence of time-delay, could suppress self-excited vibrations. The main objective of the present paper is to investigate the dynamics of a parametrically excited Rayleigh–Duffing oscillator with time-delayed feedback position and linear velocity terms. This system is well known for its relevance to many applications, for example, vehicle dynamics, flutter of plates and shells, oil-film journal bearings, and so on. The lowest-order approximate solution of the model oscillator is constructed using the asymptotic perturbation (AP) [20] which is based on the method of averaging. We observe that the amplitude peak of the parametric resonance can be reduced by means of a correct choice of the time-delay and the feedback gain. We discuss how the existence region of steady-state solutions is modified by the feedback control and show the existence regions of the nontrivial solution in the plane of the parametric excitation amplitude and the detuning parameter for an uncontrolled system, a controlled system without time-delay, and with time-delays. We present a bifurcation analysis and parametric excitation-response and frequency–response curves. Then, we perform a stability analysis of the bifurcation of the model and obtain the sufficient conditions for stable nontrivial solutions. We verify the theoretical stability analysis through a numerical simulation.

2. Controlling parametric resonance in a system with position and velocity time-delay We consider the dissipative parametric Rayleigh–Duffing oscillator with two kinds of time-delayed coupling given by

  €x  el 1  x_ 2 x_ þ ðd þ ea cos tÞx þ ekx3  ebxðt  sÞ  ecxðt _  sÞ ¼ 0;

ð1Þ

where e is a positive parameter, l; d, a; k; b, and c are constants and s is the time-delay parameter. It has been shown that [21–23] (and references therein) in the absence of time-delay and parametric force, Eq. (1) can approximate the dynamics of the real quarter-car vehicle under external forcing. Parametric excitation appears as a consequence of periodic variations of one or more parameters in the vibration system, such as the stiffness or the damping. Among these parameters, the stiffness of the system is closely related to the natural frequency of the system and therefore has a great effect on the self-excited vibration [24]. When concerned with the reduction of amplitude of vibration in mechanical systems, it is well known that all controllers exhibit a certain time-delay during operation. In fact, most of the controlled mechanical systems are found to have a time-delay [25]. In the system (1) we consider time-delay in both linear position and velocity terms. We detune-off of 2:1 resonance by setting d ¼ 1=4 þ ed1 . According to the method of averaging [20,26], in the absence of the parameter e, Eq. (1) reduces to € x þ x=4 ¼ 0, with the solution:

x ¼ q cos

  1 tþu ; 2

x_ ¼ 

q 2

sin

  1 tþu : 2

ð2Þ

For small positive values of the parameter e, we look for oscillatory solutions in the form (2) which are not necessarily periodic solutions. Treating q and u as unknown functions of time t, variation of parameters gives the following equations

 1 t þ u G; 2   1 qu_ ¼ e cos t þ u G; 2

q_ ¼ e sin



ð3aÞ ð3bÞ

where

G¼

          t q t lq3 3t kq3 3t þ u  4d1 þ 3kq2 cos þu  þ 3u  þ 3u  1 sin sin cos 2 2 2 2 2 32 4 4 4           aq 3t t 1 c 1  þ u þ cos  u þ q b cos ðt  sÞ þ u  sin ðt  sÞ þ u : cos 2 2 2 2 2 2 

lq 3q2

ð4Þ

Using the method of averaging, replacing the right-hand sides of Eq. (3) by their averages over one period of the system with e ¼ 0, we obtain

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Z





e 2p 1 e dt; tþu G sin 2 2p 0   Z 2p e 1 e dt tþu G qu_ ¼  cos 2 2p 0 q_ ¼ 

ð5aÞ ð5bÞ

with

         2 e ¼  1 lq 1  q sin2 1 t þ u sin 1 t þ u  ðd1 þ a cos t Þq cos 1 t þ u  kq3 cos3 1 t þ u G 2 2 2 2 2 4     1 1 1 ~ cos ~  cq ~ ; ~ sin þ bq ðt  sÞ þ u ðt  sÞ þ u 2 2 2

ð6Þ

~ ¼ qðt  sÞ and u ~ ¼ uðt  sÞ. Evaluating the integrals in Eq. (5) we find where q

  3q2 ~ cosðu  u ~ sinðu  u ~ þ s=2Þ þ cq ~ þ s=2Þ ; þ aq sin 2u  2bq 4 16   e 3kq3 ~ cosðu  u ~ þ s=2Þ þ cq ~ þ s=2Þ : ~ sinðu  u qu_ ¼ 2d1 q þ þ aq cos 2u  2bq 4 2

q_ ¼

e





lq 1 

ð7aÞ ð7bÞ

_ are OðeÞ. Equations (7) show that q_ and u ~ and u ~ as For small s we expand q

q~ ¼ qðt  sÞ ¼ qðtÞ  sq_ ðtÞ þ s2 q€ ðtÞ þ . . . ; _ ðtÞ þ s2 u € ðtÞ þ . . . : ~ ¼ uðt  sÞ ¼ uðtÞ  su u

ð8aÞ ð8bÞ 2

~ and u ~ by q and u in Eqs. (7) since q_ ; u _ and q €; u € are OðeÞ and Oðe Þ respectively. This Eqs. (8) indicate that we can replace q reduces an infinite-dimensional problem in functional analysis to a finite-dimensional problem by assuming the product es as small. Introducing the new time scale t ¼ ~t=e we have the following averaged equations:

    3q2 l 1 þ a sin 2u  2b sinðs=2Þ þ c cosðs=2Þ ; 4 16   q 3kq2 qu_ ¼ 2d1 þ þ a cos 2u  2b cosðs=2Þ þ c sinðs=2Þ : 4 2

q_ ¼

q

ð9aÞ ð9bÞ

The equilibrium points of system (9), obtained from a set of nonlinear algebraic equations by letting the right-hand sides of Eq. (9) be zero, correspond to the steady state responses of the system Eq. (1). A trivial solution is q ¼ 0. When q – 0, we obtain

q40  2Pq20 þ Q ¼ 0

ð10Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 ¼ P  P 2  Q ;

ð11Þ

and

where

    2 lc

bl l2

; P¼ s =2Þ  c k þ s =2Þ þ cosð sinð  2kd 2bk þ 1 2 8 4 8 3 k2 þ l64

ð12aÞ

4

½K  4bc sin s þ 4ðcd1  blÞ sinðs=2Þ þ 2ðcl  4bd1 Þ cosðs=2Þ; Q¼ 2 9 k2 þ l64

ð12bÞ

K ¼ l2 þ 4d21 þ 4b2 þ c2  a2 :

ð12cÞ

We restrict our study to the case of positive sign in Eq. (11). For real q0 we require (i) P; Q > 0, P2 > Q or (ii) P-arbitrary, Q < 0. We choose the parameters in such a way that any one of these two conditions is satisfied so that q0 is given by Eq. (11) with positive sign. Using these conditions, from Eq. (11), the boundary separating the domain where the control is efficient (reduction in the amplitude of oscillation) to the domain where it is inefficient is given by

2 3

 2  Q 21 2 2 2 4 a ¼ l þ 4d1 ðPc þ Pu Þ þ Q 1 P4 þ Pu Pc þ 2 l2 5; ðPc þ Pu Þ2 9 k þ 64 2 c

1

where Pu and Pc are respectively the function corresponding the uncontrolled and controlled system. Q 1 is given by

ð13Þ

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4

α

c

3

β>γ

2

1

0

β<γ

0

1

2

3

τ

4

Fig. 1. Boundary criterion for the effectiveness of the control of the oscillation amplitude for b ¼ 0:35 (red color curve) and b ¼ 0:1 (blue color curve). The values of the other parameters are l ¼ 0:002; k ¼ 1; d1 ¼ 1; c ¼ 0:25. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

1.5

1.5

β>γ

β>γ

1

1

α=2

ρ

ρ

0

0

α=2

0.5

α=1 0.5

α=1 0

−0.5

0

δ1

0.5

0

0

1

2

3

τ

Fig. 2. q0 versus (a) d1 with s ¼ 0:005 and (b) s with d1 ¼ 0:2 for two fixed values of a. The values of the other parameters are l ¼ 0:002; k ¼ 1; c ¼ 0:25 and b ¼ 0:35.

 4

4b2 þ c2  4bc sin s þ 4ðcd1  blÞ sinðs=2Þ þ 2ðcl  4bd1 Þ cosðs=2Þ : Q1 ¼ l2 2 9 k þ 64

ð14Þ

Fig. 1 shows the variation of ac with the time-delay s for three values of b; 0:3; 0:5 and 0.8, with l ¼ 0:1, d1 ¼ 1; k ¼ 1 and c ¼ 0:25. ac for c > b is lower than that of c < b. In Fig. 1, in the region below the curve the control is efficient in reducing the amplitude of the oscillations. This result can be applied to reduce the amplitude of vibration in the quater-car model. In Fig. 2 we have plotted the frequency- and force-response curves for two values of a. When the amplitude of the parametric excitation a is below the boundary domain in Fig. 1 the steady-state response shown in Fig. 2 is lower. Next, Fig. 3 demonstrates the effect of s and d1 on q0 for two different values of control gain parameter b. The maximum peak amplitude of the steady-state solutions in Fig. 3(a) corresponding to b ¼ 0:1 are relatively lower than those in Fig. 3(b) where b ¼ 0:3. Fig. 3 confirms the technical control in reducing the amplitude of vibration. Further, in Fig. 3, for a certain range of values of timedelay s and detuning parameters d1 the steady-state solution becomes zero. That is, the system exhibits the amplitude death phenomenon and is due to the stabilization of fixed points by appropriate time-delayed feedback and detuning of the parameter d. In the next section, we study separately the effects of gain parameters alone, parametric excitation alone and both on the response amplitude of the system (1). 3. Effects of gain parameters and parametric excitation in nonlinear resonance 3.1. Effects of parametric excitation alone We consider the system (1) with s ¼ 0, but b; c – 0. In Fig. 4 we have plotted the frequency- and force-responses of the nontrivial solution. q0 is multivalued for a range of values of d1 for both a ¼ 0:85 and 5. However, for both the values of a the

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Fig. 3. Three-dimensional plot of q0 versus s and d1 for (a) b ¼ 0:1 and (b) b ¼ 0:5 where c ¼ 0:25 and a ¼ 2.

(a) 4.5

(b) 3.5

4

3

α=5.0

2.5

2.5 2

δ =0.04

0

α=0.85

1

ρ

0

3

ρ

δ =2.5 1

3.5

2

1.5

1.5 1

1 0.5 0 −4

0.5 −2

0

2

4

δ

6

8

0

10

0

2

4

α

1

6

Fig. 4. q0 versus (a) d1 and (b) a. The values of the parameters in the system (1) are l ¼ 0:002; k ¼ 1; c ¼ 0:25; b ¼ 0:1; upper and lower curves correspond to a ¼ 5. The value of a for the two inner curves is 0.85.

(a) 4

3

β=1.0

2.5

0

2.5 2

ρ

0

s ¼ 0. In the subplot (a) the

3.5

β=0.4

3

ρ

10

(b) 4

3.5

1.5

1

1

0.5

0.5 0

2

4

δ

1

6

8

10

γ=1.0

2

1.5

0 −2

8

0

γ=0.4

0

2

4

6

δ

8

10

1

Fig. 5. Variation of q0 with the detuning parameter d1 for (a) c ¼ 0 and (b) b ¼ 0. The values of the other parameters are l ¼ 0:8; k ¼ 1; a ¼ 0 and s ¼ 0:008. In the subplot (a) the upper and lower curves correspond to b ¼ 1. The value of b for the two inner curves is 0.4. In the subplot (b) the upper and lower curves correspond to c ¼ 1. The value of cfor the two inner curves is 0.4.

blue color curves trace the nontrivial stable solution while the red color curves mark the nontrivial unstable steady-state solution. The reduction effect due to parametric amplitude is quite important and is found to be visible only in a small region of the detuning parameter surrounding the origin. The instability region increases with increase in the value of parametric

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(a) 4

(b) 4

3.5

3.5

α=5.0

3

3

0

2

2.5

α=0.85

ρ

ρ

0

2.5

1.5

δ1=0.1

1.5

1

1

0.5 0 −4

2

δ =1.5

0.5 −2

0

2

4

δ

6

8

1

0

10

0

5

10

15

α

1

Fig. 6. Response amplitude q0 versus (a) d1 and (b) a for l ¼ 0:8; k ¼ 1; c ¼ 0:25; b ¼ 0:1 and s ¼ 0:008. The blue color curves are stable branches while the red color curves are unstable. In the subplot (a) the upper and lower curves correspond to a ¼ 5. The value of a for the two inner curves is 0.85. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

amplitude. The influence of a on q0 is presented in Fig. 4(b) for two values of the detuning parameter d1 . Hopf bifurcation with multi-solution occurs as the detuning parameter d1 increases. The saddle-node bifurcation disappears for lower values of the detuning parameter.

(a)

10

Unstable domain

α

8

6

4

Stable domain 2 −6

−4

−2

δ1

0

2

4

(b) 10 8

Unstable domain

α

6

4

2

0 −6

Stable domain −4

−2

δ1

0

2

4

Fig. 7. Regions in ða; d1 Þ parameter space where the nontrivial solution of the system (1) given by Eq. (15) are stable and unstable. The values of the parameters are l ¼ 0:1; k ¼ 1; c ¼ 0:25; s ¼ 0:005 and (a) b ¼ 0:1 and (b) b ¼ 0:5.

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3.2. Effects of control gain parameters in the absence of parametric excitation Next, we consider the system (1) with a ¼ 0 and s – 0. Fig. 5(a) and (b) show the response amplitude q0 versus d1 for c ¼ 0; b – 0 and c – 0; b ¼ 0 respectively. In both the figures red color curves are unstable branches. We notice that the two control gain parameters increase the region of the detuning parameter surrounding the origin where nontrivial values of q0 are possible and is more visible when the time-delayed position is alone considered than the case where the timedelayed velocity is alone considered. As the detuning parameter increases to a large value, the peak of amplitude is greater in the case of time-delayed position and the domain of instability also increase. 3.3. Effects of combined control gain parameters and parametric excitation Next, we assume that all the parameters in the system (1) are nonzero. Fig. 6(a) and (b) show the plot of q0 as a function of d1 and a respectively. Fig. 6(a) and (b) are similar to the Fig. 4(a) and (b) respectively. Consequently, statements made on Fig. 4 can be extended to Fig. 6 also. Referring to Figs. 4–6 it is clear that an appropriate choice of the control gain parameter can reduce the response amplitude. 4. Stability of the nontrivial steady-state solution In this section we perform the stability analysis of the steady state solutions given by Eq. (11). We superpose small perturbations in the steady-state solution and then linearize the resulting equations. Subsequently, we consider the eigenvalues of the corresponding system of first-order differential equations with constant coefficients (the Jacobian matrix). A positive real root (eigenvalue) indicates an unstable solution, whereas if the real parts of all the eigenvalues are negative then the steady-state solution is stable. When the real part of an eigenvalue is zero, a bifurcation occurs. A change from complex conjugate eigenvalues with negative real parts to complex conjugate eigenvalues with positive real parts is an indication of the occurrence of a supercritical or subcritical Hopf bifurcation. The question of which possibility actually occurs depends on the nonlinear terms. The eigenvalues of the Jacobian matrix satisfy the equation

10

4

5

2

dx/dt

(b) 6

x

(a) 15

0

0

−5

−2

−10

−4

−15

0

200

400

600

−6 −15

800

−10

−5

t

10

4

5

2

dx/dt

(d) 6

x

(c) 15

0

−2

−10

−4

0

200

400

t

5

10

15

5

10

15

0

−5

−15

0

x

600

800

−6 −15

−10

−5

0

x

Fig. 8. x versus t and x versus x_ for a ¼ 2 (subplots (a) and (b)) and a ¼ 10 (subplots (c) and (d)) with d1 ¼ 0; parameters are as used in Fig. 7.

e ¼ 0 and b ¼ 0:001. The values of the other

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(b) 6

10

4

5

2

dx/dt

(a) 15

x

0

0

−5

−2

−10

−4

−15

0

200

400

600

800

−6 −15

−10

−5

t

(d) 6

10

4

5

2

dx/dt

(c) 15

x

0

−2

−10

−4

0

200

400

5

10

15

5

10

15

0

−5

−15

0

x

600

800

−6 −15

−10

−5

t

0

x

Fig. 9. Same as Fig. 8 but for a ¼ 3 (subplots (a) and (b)) and a ¼ 10 (subplots (c) and (d)) with d1 ¼ 2;

n2 þ pn þ q ¼ 0;

e ¼ 0:001 and b ¼ 0:5.

ð15Þ

where

1  3lq20  16l þ 32b sinðs=2Þ  16c cosðs=2Þ ; 32 q ¼ q40 þ A1 q20 þ A0 ;  1

32kd1 þ l2 þ 2ð12k þ lÞK 1  4kK 2 ; A1 ¼ 2 l 2 6 64  3k

ð16bÞ

   4

a2  l2  4d21 þ 2K 1 ðl  d1 Þ  2K 2 d1  c2 þ 4b2  4cb sin s ; A0 ¼ 2 l 2 9 64  3k

ð16dÞ

K 1 ¼ c cosðs=2Þ  2b sinðs=2Þ;

ð16eÞ

p¼

K 2 ¼ c sinðs=2Þ  2b cosðs=2Þ:

ð16aÞ

ð16cÞ

From the Routh–Hurwitz criterion [20], the steady-state response is asymptotically stable if and only if p > 0 and q > 0 which keep the real parts of the eigenvalues negative. The stability domain of the non-trivial solutions q0 given by Eq. (11), when parametric excitation and time-delayed position (resp. time-delayed velocity) are combined, in the plane ðd1 ; aÞ is shown in Fig. 7. Fig. 7(a) (resp. Fig. 7(b)) corresponds to the case b < c (resp. b > c). The ðd1 ; aÞ plane is divided into three regions where two of them are stable and one is unstable. The stable domain increases when the time-delay control gain position is greater than the time-delay control gain velocity (see Fig. 7(a) and (b)). In order to verify the validity of the theoretical analysis presented in the present work we numerically integrated the equation of motion (1). Specifically, we tested the stability diagram (7) for two fixed values of d1 . Referring to Fig. 7(a) where b < c, for d1 ¼ 0, according to the theoretical treatment employed, the nontrivial value of q0 is stable for a < ac ¼ 3:03 and unstable for a > ac . That is, a stable periodic solution with amplitude q0 can occur for a < 3:03 while for a > 3:03 there is no stable definite periodic solution. In Fig. 8(a) and (b) the trajectory plot and phase portrait are shown for a ¼ 2ð< 3:03Þ. The solution shown is periodic. For a ¼ 10ð> 3:03Þ the motion is quasiperiodic and is shown in Figs. 8(c) and (d). As an another example for numerical verification, we consider Fig. 7(b). For d1 ¼ 2, theoretically one can expect a definite periodic solution for a < ac ¼ 3:24 and nonperiodic solution for a > ac . Fig. 9 shows the numerical solution for a ¼ 3 and 10. For a ¼ 3 and

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10 the long time motions are periodic and quasiperiodic respectively. The theoretical predictions are in good agreement with the numerical simulation. 5. Conclusion In this paper, we studied the parametric resonance in Eq. (1) that involves the interaction of parametric excitation with time-delayed position and velocity. Our analytical results are based on slow flow system Eqs. (9) that were obtained using first-order averaging. We studied the condition of reducing amplitude of vibration as well as the nonlinear response and the stability of the steady-state solutions and the bifurcations that accompanied response and stability changes: saddle-nodes, Hopf bifurcations. By the stability analysis of the parametric Rayleigh–Duffing system, we have shown that consideration of either time-delayed velocity alone or time-delay position and velocity could be better to enhance the stability domain. We point out that the control gains parameters as well as parametric amplitude play an important role as a relevant factor in onset properties of stability. This observation indicates that the time-delay is a sensitive factor for the bifurcation of the system and instability. One can suppress the instability oscillations by properly choosing a time-delay and control gains parameters. Acknowledgments The first author M. Siewe Siewe indebted to the University of Pretoria for its financial support to do research work as a senior visiting Fellow and also indebted to Professor Xiaohua Xia, Head of the Department of Electrical, Electronic and Computer Engineering for hosting him to undertake part of this work. We are thankful to the referees for their comments and suggestions which improved the quality and presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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