**5. Bifurcations in Buck converter with delayed ZAD-FPIC**

14 Nonlinearity, Bifurcation and Chaos - Theory and Applications

define *F*.*P*. as (*ref* , *γref*). Its eigenvalues (or characteristic multipliers) determine stability properties of the fixed point. The 1-periodic orbit is asymptotically stable if all characteristic multipliers have magnitude less than one (|*mi*| < 1); it is unstable if at least one eigenvalue

One-delay control law implies that the duty cycle *dk* depends on state variables in the instant (*k* − 1)*T*, i.e., *dk* = *c*1*x*1((*k* − 1)*T*) + *c*2*x*2((*k* − 1)*T*) + *c*3. Two additional state variables can be defined *x*3(*kT*) = *x*1((*k* − 1)*T*) and *x*4(*kT*) = *x*2((*k* − 1)*T*). Therefore, *dk* = *c*1*x*3(*kT*) +

> *f*1(*x*1(*kT*), *x*2(*kT*), *x*3(*kT*), *x*4(*kT*)) *f*2(*x*1(*kT*), *x*2(*kT*), *x*3(*kT*), *x*4(*kT*)) *x*1(*kT*) *x*2(*kT*)

> > *∂ f*<sup>1</sup> *∂x*3(*kT*)

> > *∂ f*<sup>2</sup> *∂x*3(*kT*)

*<sup>∂</sup>x*4(*kT*) *<sup>∂</sup> <sup>f</sup>*<sup>2</sup>

1000 0100

*∂ f*<sup>1</sup>

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

> � *<sup>∂</sup>***<sup>f</sup>** *∂***x***<sup>i</sup>* � *F*.*P*.

*∂ f*<sup>2</sup> *∂x*4(*kT*)

*Anj* (*x*(*k*))

�� � � �

� *<sup>∂</sup>***<sup>f</sup>** *∂***x***<sup>i</sup>* � *F*.*P*.

). In this case, we

(33)

(34)

(35)

). In this case, we

The matrix **A***n*<sup>0</sup> should be evaluated in the fixed point (**A***n*<sup>0</sup> =

In this case, Poincaré map (30) can be written as equation (33).

= = = =

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

Jacobian matrix of the system with *τ* = 1 can be computed with equation (34).

*∂ f*<sup>1</sup> *∂x*1(*kT*)

*∂x*1(*kT*)

*∂ f*<sup>1</sup> *∂x*2(*kT*)

*∂ f*<sup>2</sup> *∂x*2(*kT*)

Four characteristic multipliers are computed. The 1-periodic orbit is asymptotically stable if

Characteristic multipliers for delayed PWM control law with *τ* > 1 can be computed following the same procedure. However, The order of Jacobian matrix increases as the delay

Now, we compute Lyapunov exponents using a numeric routine. This algorithm is based on the definition of Lyapunov exponents. Equation (35) synthesizes this procedure. Poincaré map is used to compute the values of state variables. Jacobian matrix should be known to

Floquet exponents, Lyapunov exponents and characteristic multipliers are interconnected to each other. Mathematical expressions to relate each approach are synthesized in figure 5. For

*x*1((*k* + 1)*T*) *x*2((*k* + 1)*T*) *x*3((*k* + 1)*T*) *x*4((*k* + 1)*T*)

**A***n*<sup>1</sup> =

define *F*.*P*. as (*ref* , *γref* ,*ref* , *γref*).

compute the eigenvalues *qi* in each iteration *k*.

**4.2. Equivalence between stability methods**

number grows.

The matrix **A***n*<sup>1</sup> should be evaluated in the fixed point (**A***n*<sup>1</sup> =

the four characteristic multipliers have magnitude less than one.

*λ<sup>i</sup>* = *Lim M*→∞ � 1 *M*

*M* ∑ *k*=0 log � � �*qi* �

has magnitude greater than one (|*mi*| > 1).

*c*2*x*4(*kT*) + *c*3.

In this section, we analyze types of bifurcations in the buck converter controlled with Delayed ZAD-FPIC scheme using the procedure based on Floquet exponents described in previous sections. We transform Floquet exponents in characteristic multipliers using the equivalences shown in figure 5.

If at least one characteristic multiplier is outside of the unit circle then the system has an unstable fixed point and nonlinear phenomena as quasi-periodicity and chaos could be present. In the boundary, the smooth bifurcations (flip, fold and Neimark-Sacker) are present. The presence of the three smooth bifurcations in the same converter is not common and this fact has not been reported widely in Digital-PWM switched converters [14].

Control parameters *ks* and *N* can be varied in **R** with the exception of *ks* = 0 and *N* + 1 = 0 (because the control law is not defined there). Parameter *τ* can be varied in **Z**. The 3D-parameter space (*ks*, *N*, *τ*) is discontinuous due to the discrete delays (*τ* = 0, 1, 2, 3, ) and the undefined planes (*ks* = 0 and *N* + 1 = 0). Figure 8 (*a*) shows a representation of the control parameter space.

The two-dimensional plane (*ks*, *N*) can be divided into four regions: region I: *ks* > 0 and *N* > −1; region II: *ks* < 0 and *N* > −1; region III: *ks* < 0 and *N* < −1 ; and region IV: *ks* > 0 and *N* < −1. Fold zones, flip zones and Neimark-Sacker zones can be identified in the control space. The fold bifurcation is an alarm for duty cycle saturation in *d* = 0% or *d* = 100%; the flip bifurcation signals a doubling period and the Neimark-Sacker bifurcation is related to 2D-torus birth.

Computer simulations are given for the purpose of illustration and verification. Next, we present the three bifurcations types in the 3D-parameter space.
