Simulation, bifurcation, and stability analysis of a SEPIC converter controlled with ZAD

Received Jun 11, 2019 Revised Sep 30, 2019 Accepted Oct 7, 2019 This article presents some results of SEPIC converter dynamics when controlled by a center pulse width modulator controller (CPWM). The duty cycle is calculated using the ZAD (Zero Average Dynamics) technique. Results obtained using this technique show a great variety of non-linear phenomena such as bifurcations and chaos, as parameters associated with the switching surface. These phenomena have been studied in the present paper in numerical form. Simulations were done in MATLAB.


INTRODUCTION
Research on dynamic systems has been applied to different fields such as biology, power converters, impact oscillators, mechanical systems, etc. where a large number of phenomena [1] of a non-linear nature [2,3] are presented. Dynamic systems defined in pieces are very important topics of study in theoretical and experimental matters, being investigated in depth in recent years. An example of systems defined in pieces, are DC-DC voltage converters, which allow the control of output voltage from a given voltage source; that is, they act as bridges for energy transfers between sources and loads, both of direct current [4]. This leads naturally to the question of how to transfer energy from the source to the load with amplitude, which needs a 1 voltage, with the minimum loss of power. Multiple applications are presented by these converters including power sources in computers, distributed power systems, power systems in electric vehicles, etc [5,6]. Therefore, this study has been a source of research in the fields of dynamic systems. Power converters introduce a series of non-linearities in the switching process, which is why they have been studied as variable structure systems. In [7] controllers were designed in sliding mode to work with this type of converter. Later, Carpita [8] designed a controller based on a sliding surface given by a linear proportion of the error and the derivative of the error. These two results allow working with a robust, stable, and efficient controller. However, by generating a discontinuous action of this controller, a "chattering" phenomenon arises in the system [9], which implies an increase in the ripple and distortion at the exit. With the purpose of eliminating the "chattering" phenomenon, several techniques have been proposed to find a control scheme that guarantees a fixed switching frequency. For example, in [7], it is proposed to synthesize a controller that guarantees a zero average of voltage error through a technique known as Zero Average Current Error (ZACE). Fossas and his colleagues proposed a new control technique for power converters in which an auxiliary output is set and a digital control action is defined that guarantees zero average in the auxiliary output in each iteration, maintaining a fixed frequency of commutation, robustness, and stability. This technique is known as ZAD (Zero Average Dynamics) and consists of the definition of a switching surface on which the system is evolved on average. In [10] , it has been implemented making use of the switching surface (x( )) = ( 1 ( ) − 1 ) + (̇1( ) −̇1 ), where good results are shown in terms of robustness and low output error. In [11,12] it is also applied to analyze the dynamics present in the boost converter to study present non-linear phenomena, driven by a center aligned pulse width modulation converter (CPWM).
In the present article, the ZAD technique has been implemented to control a SEPIC converter, which has been used to control boost and buck converters in previous Works [11,13,14]. A linear combination of the error in voltage and current has also been taken as the switching surface ( ) and from this, the calculation of duty cycle has been done with which the system is evolved in a period of time T [15]. Finally, bifurcations that arise in the evolution of 1-periodic orbits have been characterized and the presence of chaos has been determined from certain values of constants associated with the commutation surface, which were taken as bifurcation parameters [16,17].

RESEARCH METHOD
An SEPIC converter [18] is a DC to DC converter belonging to the family of fourth-order converters [19]. This device can supply more or less voltage than the input voltage. The basic scheme of a SEPIC converter is shown in Figure 1, where is the input voltage, 1 is the current in 1 inductor, is the switch, is the diode, 1 is the voltage in 1 capacitor, is load resistance, 2 s the current in 2 inductor, 2 is the voltage in 2 capacitor. The basic principle of the SEPIC converter consists of two different states, depending on the state of the switch . When switch is closed, the status is and the input source connects to 1 coil at the same time as the diode is polarized inversely. As a consequence, the intensity that circulates through 1 inductance grows linearly, storing energy. In this situation, the 1 capacitor feeds the 2 inductor and the tension of 2 is delivered to the load. When switch is open, the status andthe energy previously stored in 1 coil together with the input is transferred to the 1 input capacitor and the energy stored in the 2 inductor is transferred to 2 and to the load.
Two modes of operation are distinguished in the SEPIC converter, depending on the currents by the inductors canceled during the operation period : Continuous Driving Mode ( ) and Discontinuous Driving Mode ( ). In this article, we will study the dynamics of the SEPIC converter in . The dynamics of the SEPIC converter are governed by the solution of this system of differential equations: (1) In this system, 1 , 1 , 2 , and 2 are status variables 1 , 2 , 1 , 2 , and ;, are the parameters and ∈ {0,1} is the control variable. In system (1), if = 1, then SEPIC is (topology 1) and when = 0, SEPIC is (topology 2). Making the change of variables: are defined, then a dimensional system is obtained for the dynamics of the SEPIC converter: where the new parameters of the system are , , and . Each topology of the system can be expressed in compact form by where ∈ {1,2} and The solution of each topology can be expressed as where

Pulse width modulation
The control scheme [20] that will be used in this study corresponds to center-aligned modulation by pulse width in such a way that a time interval [nT, (n + 1)T] is divided into three subintervals, where the first and the last have the same length, as shown in Figure 2. Commutations are made according to the scheme {1,0,1} and, therefore, the system will operate as follows: where is the period, is the duty cycle and the time during which the system is operated in status . After choosing the control scheme, we must decide how to calculate the time that the system must remain in conduction; that is, we must choose a criterion that allows us to calculate (period to period) the duty cycle . In this paper, we will calculate it using ZAD (Zero Average Dynamics) control technique [2] and based on the fact that being a variable structure system, the principles of control in sliding modes can be applied in such a way that the error dynamic is zero on average in each iteration [21,22].

CONTROL STRATEGY 3.1. ZAD control strategy
This technique consists of defining a switching surface ( ( )) = 0 in which the system will evolve on average [12]. In this paper, the switching surface given by the equation ( ( )) = ( 1 ( ) − 1 ) + (̇1( ) −̇1 ), using this switching surface and the technical control ZAD, we obtain the next expression for the duty cycle: where is a real number between 0 and T, if < 0 or > 0. Expression (7) is redefined, saying that the system is saturated. For this situation, the following selection is made in each period:

Flip bifurcation
The simulation described below is presented when we consider the SEPIC converter as a reducer; in addition, we take as reference values the vector (0.0544 1 0.1237 0.44) . Figure 3 shows a configuration of the state variables for which there is a 1T-periodic orbit that goes from stable to unstable, which indicates a bifurcation point. The point of interest is when 3 ∈ [40, 60]. The diagrams of these figures were obtained by varying 3 in the specified range with 1 = 25, 2 = −15, 4 = −10, = 0.18, = 0.2683, = 0.7021, and = 3.5583. Reviewing the eigenvalues of the Jacobian matrix associated with the Poincaré application in Table 1 and observing Figure 3, we find the resulting bifurcation is of the flip type [23] because one of these values goes from being stable to unstable, crossing through −1 for a value of the parameter 3 = 51.96.
This type of bifurcation is characterized by a period doubling; that is, the system goes from having a 1T-periodic orbit to having 2T-periodic orbits. Table 1 presents the values of the Poincaré map. An analysis of these allows confirmation that this is a flip-type bifurcation [24,25] because the proper value that goes from stable to unstable does it crossing by −1 in the interval 3 ∈ (51.40, 52.30) as shown in Figure 4. This type of bifurcation occurs because the orbit 1 -periodic becomes unstable and an orbit 2 -periodic is born, i.e., a period doubling occurs [12].   Figure 4. Graphic variation of eigenvalues by varying 3

Neimar-saker bifurcation
When 1 is varied in the interval [−10, −2] and we take as fixed values 2 = 1, 3 = −6, 4 = 2.5, = 0.18, = 0.2683, = 0.7021, = 3.5583, and, as a condition, initial (1.1241 1 0.5621 2) , we also find a bifurcation in the dynamics of the SEPIC converter as seen in Figure 5. In the bifurcation diagrams of Figure 5 a change in the dynamics of the system is observed, which is characteristic of a Neimar-Saker type bifurcation. To characterize this bifurcation, we analyze the evolution of the eigenvalues of the Jacobian matrix of the Poincaré map near the point of bifurcation 1 = −2.42. We observe in Table 2 and Figure 6 that the complex and conjugated eigenvalues of the Poincaré application near the bifurcation point, its module approaches 1, which characterizes the Neimar-Saker-type bifurcation.   Figure 6. Graphic variation of eigenvalues by varying 1

EXISTENCE AND CHAOS CONTROL
To guarantee the presence of chaos in the system, we use the exponents of Lyapunov, which determine the proximity or divergence of two orbits that were initially close. The i-th Lyapunov exponent is given by the expression: where is the i-th value of Poincaré map, its Jacobian, and is the i-th eigenvalue of . The presence of a positive Lyapunov exponent in a system whose trajectories evolve within a finite zone of the state space guarantees chaotic behavior [26]. On the other hand, the sum of all Lyapunov exponents in a chaotic attractor must be negative [27].
To determine the presence of chaos [16,28] and its respective control, we will use the values with which the flip bifurcation was obtained. Figure 7 shows the existence of positive Lyapunov exponents for values of 3 greater than 51.96 . Additionally, Figure 3 shows that for values of 3 ∈ (51.40,52.30), the system evolves in a bounded region of the state space. Therefore, we can see that the system presents chaotic behavior for values of 3 greater than 51.96.

Chaos control with FPIC technique
Because our system is non-autonomous, it is excited with an external signal . Any method of chaos control must stabilize the unstable orbits, and for this it must necessarily assure that proper values of the Jacobian matrix of the Poincaré map are within the unit circle (stability border). In this sense, several control strategies have been designed. To control the chaos presented by the SEPIC converter with ZAD, we use the FPIC (Fixed Point Induced Control) technique [29], which was designed by [14] and has been used numerically in [13,30]. This is based on the continuity of proper values theorem and helps stabilize period one or more orbits in unstable and/or chaotic systems and does not require the measurement of state variables. It forces the system to evolve to the fixed point; therefore, it is necessary to have prior knowledge of the control signal equilibrium point. The equilibrium point obtained for the system is given by the transposed vector: * = ( The duty cycle is to apply ZAD and FPIC control techniques is given by: where is determined by (7) and * by (13).
Next we apply the FPIC technique to the dynamics of the SEPIC converter. Figures 8 (a) shows that by choosing = 0.003, the area in which the system exhibits chaotic behavior decreases. Figure 8 (b) shows that for the FPIC constant = 0.005 the area in which the system exhibits chaotic behavior continues to decrease. Finally in Figure 8 (c) when the FPIC constant = 0.006 the area in which the system exhibits chaotic behavior has almost completely disappeared.

CONCLUSION
By analyzing the dynamics of the SEPIC converter numerically using the ZAD technique with CPWM, we can observe the presence of non-linear phenomena such as quasi-periodicity and chaos. The presence of bifurcations was detected by varying the parameters 1 and 3 associated to the switching surface, being the flip and Neimar-Saker bifurcations, respectively. Chaotic behavior can be controlled by introducing the FPIC technique. However, it is important to see how the FPIC technique influences other behaviors of the system, such as regulation.